Yann Mambrini

Particles in the
Dark Universe
A Student’s Guide to Particle Physics
and Cosmology
Particles in the Dark Universe
Yann Mambrini

Particles in the Dark

Universe
A Student’s Guide to Particle Physics
and Cosmology
Yann Mambrini
Laboratory of the Physics of the Two
Infinities Irène Joliot-Curie (IJCLab)
CNRS/University Paris-Saclay
Orsay, France

ISBN 978-3-030-78138-5 ISBN 978-3-030-78139-2 (eBook)

https://doi.org/10.1007/978-3-030-78139-2

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‘If you are doing everything well, you are not
doing enough’
Howard Georgi.
To my parents, Jacques and Annick
Foreword

Like all successful areas in physics, cosmology is based on an interplay between

theory and experiment. In cosmology, experiment typically means observation, and
what we observe is light. We assume that the spectral properties of light are literally
universal, and as a result, the observed shifted spectra, allow us to determine the
dynamics of the galaxies and clusters of galaxies hosting the light-producing gas
and stars. These dynamics indicated the presence of significantly more matter than
could be accounted for by the observed luminosity. What began as the problem
of missing mass (or more aptly phrased as the problem of missing light—as the
presence of mass without accompanying light) became to be known as the problem
of dark matter.
Modern cosmology or big bang cosmology originated in the late 1940s from
work principally by Alpher, Gamov, and Herman in their attempts to understand
the pattern of element abundances observed in the Universe. Recognizing that
nuclear reactions needed sufficient temperatures and densities to be effective, they
envisioned the currently expanding Universe at early times as a primeval nuclear
reactor. While this theory of big bang nucleosynthesis was only successful in
accounting for the abundance of elements through Li, as a by-product it led to
the prediction by Alpher and Herman of the cosmic microwave background with
a temperature of 1–5 K. This was of course famously discovered in 1964 by Penzias
and Wilson, with the interpretation by Dicke, Peebles, Roll, and Wilkinson in 1965
as a direct remnant of the big bang. The work on element abundances however
shifted from cosmological to stellar, as most elements are produced in stars or as
a result of stellar explosions. Nuclear astrophysics began its boom in the 1950s,
highlighted by the work of Burbidge, Burbidge, Fowler, and Hoyle.
The 1950s also saw a boom in particle physics, with the discovery of new
particles. By the late 1960s, there emerged a new Standard Model for particle
physics, one that has held up remarkably well culminating in the discovery of the
Higgs boson at the LHC at CERN in 2012. The role of particle physics in cosmology
also began in the 1960s, although the role of neutrinos in cosmology was known
in the early work on nucleosynthesis. At the time, it was not known if neutrinos
possessed mass or if they were truly massless particles. Indeed, the Standard Model
was first described in terms of massless neutrino states, and the determination of
neutrino mass, in some sense, became the first evidence of physics beyond the
Standard Model.
ix
x Foreword

Even a tiny neutrino mass, of order a few eV, would have an enormous impact on
cosmology. Cosmological limits on neutrino masses began in the 1960s and excelled
in the 1970s with limits on particle masses and lifetimes. Thus was born the field of
particle astrophysics or as it is more commonly called: astro-particle physics.
As it was becoming clear that some form of dark matter was a necessity to explain
several very distinct observations, its nature was and remains unknown. Could it be
normal matter in a non-luminous state (very difficult as normal matter couples to
light and easily shines in one wavelength or another), or could it represent some
change in Einstein gravity? Another possibility is that dark matter resides in particle
form. As just noted, neutrinos with eV masses could in fact supply all of the dark
matter in the Universe. There was one catch though, it is the wrong kind of dark
matter. In the early 1980s, work on the formation of structure in the Universe led
to the realization that all forms of dark matter were not equal. Structure in the
Universe begins to form when the Universe becomes dominated by matter, that is,
dark matter, which can be classified as either hot or cold depending on whether or
not it is relativistic at the time structure formation begins. Neutrinos, as very light
dark matter candidates, would be hot, and as a result, they would erase most all
small-scale structures, leaving the Universe very different from the one we observe.
In contrast, cold dark matter (CDM) preserves structures on small scales and is
in effect in excellent agreement with observations. Hence, a Standard Model of
cosmology, known as CDM, where , refers to a dark energy component which
may simply be Einstein’s cosmological constant.
Is dark matter then a new particle? Particle physics has a long history of solving
problems by introducing a new particle, necessitated by either theory or experiment.
Examples are plentiful. The positron was needed to complete Dirac’s theory of
relativistic quantum mechanics. The neutrino was needed to explain the missing
energy in Tritium decay. In later examples , the charm quark is needed to explain
the suppression of flavor-changing neutral currents. Indeed, one can argue that the
entire third generation of quarks and leptons was needed to account for CP violation
in weak interactions. The Higgs boson was proposed to explain the breaking of the
electroweak gauge symmetry. Thus, to a particle physicist, there is nothing unusual
about proposing the existence of a new particle which solves a problem.
Two examples of proposed dark matter candidates are worth mentioning. Both
were proposed to solve other problems in particle physics, but yet could play an
important role as dark matter. The smallness of the neutron electric dipole moment
is an indication that CP violation in strong interactions is very small. A priori, there
is no reason for it to be small, and a symmetry was proposed as an explanation.
With this symmetry, there is a new very light scalar particle called the axion which
could account for the dark matter. Though the axion is light, unlike the neutrino, it
is in fact cold due to the mechanism leading to presence in the Universe. A second
example comes from an extension of the Poincare algebra known as supersymmetry.
Supersymmetric transformations lead to (roughly) a doubling of the number of
particle types, and the lightest of these is expected to be stable. Often assumed to
be the partner of either the photon, Z, or Higgs bosons, the lightest supersymmetric
particle is another well-studied dark matter candidate.
Foreword xi

This book is about particle dark matter. It is also not for the faint hearted. It is
a serious exposé of particle dark matter, and in particular the mechanisms which
lead to its production in the Universe. After a brief introduction to the history and
motivation for dark matter, Yann Mambrini jumps straight to a description of the
Friedmann-Robertson-Walker Universe and Inflation. Apart from the resolution of
classic cosmological problems, inflation is key to producing density fluctuations,
which lead to structure formation, and reheating, producing a thermal bath from
which the radiation-dominated era in standard cosmology is born. Yann Mambrini
spends considerable effort on the latter and begins with a somewhat non-traditional
approach by examining dark matter production during the period of reheating after
inflation.
Whether produced immediately after inflation or through equilibrium production
and subsequent thermal freeze-out, the thermodynamics of the early Universe is a
necessary component. The book covers the radiation-dominated era in great detail,
including a detailed discussion of thermalization and decoupling. When applied to
dark matter, Yann Mambrini provides a complete derivation for the computation of
the relic density for cold dark matter. All this is done generically with a few simple
examples.
The second part of the book examines the state of dark matter today. Split into
two chapters, the reader will find all that’s necessary for computations of direct
detection rates as well as the complications involved in indirect detection.
The book could be titled “Everything you need to know about particle dark
matter.” However, Yann Mambrini shies away from the particular dark matter
candidate. There is no lengthy discussion of supersymmetry or axion dark matter,
the discussion throughout is very generic and can be applied to nearly any dark
matter model, though many examples are supplied.
As an added bonus, the Appendices in the book are mini-text books in them-
selves. All of the basic relativity, particle physics, neutrino physics, and statistics
are placed in separate appendices with a final one on useful values in cosmology
and particle physics.

Minneapolis, MN, USA Keith Olive

New York, NY, USA P. J. E. Peebles
Paris, France Joseph Silk
January 2021
Preface

I wrote this book out of frustration. Several excellent works and reviews are on the
market. I cite some of them at the beginning of the appendix with their particularity.
However, a vast majority of them has been written by astrophysicists, and as a
student, I always needed the equivalent of the Particle Data Booklet to compute
cross sections in specific processes, especially fighting with the “2π-like” factors.
On the other hand, particle physics textbooks were very light in the treatment of
thermodynamics of the Universe or radiative effects like the loss of energy of a
charged particle in astrophysical framework. In other words, I was always juggling
between the Kolb and Turner book [1] to solve subtleties in Boltzmann equation, the
Jackson [2] textbook to clarify the radiative effects of a model at the Galactic scale,
the Jungman, Kaminowski and Griest [3] review for some amplitudes computations
and Shifmann et al. papers for details in the nucleus composition. This work is a
humble attempt of a particle physicist to propose a textbook which can be useful
for any phenomenologist who is interested in astroparticle physics, especially in its
dark matter aspects.
I also tried to unify units. In such a vast field as dark matter research, we have
to deal with cosmological scales (when solving Boltzmann equation or computing
reheating temperatures), astrophysical scales (when looking at propagation of
cosmic rays or indirect detection of dark matter) and of course microscopic scale
when computing interaction cross section. It is always possible to convert Jansky to
Joule, cm s−1 to GeV−2 . . . However, as a particle physicist by formation, I decided
2

to work in a unified GeV-framework. From this point of view, comparison between

energy loss, expansion rates or annihilation cross section is straightforward. In any
case, all conversion factors are given in the Appendix and I also give the results in
the more “natural” units for people who are used to their own scale-related units.
I want to prevent reading of the manuscript from A to Z, but want the reader
to look at the table of contents, or even better, the index, picking a word they
would like to understand better and read the corresponding section. Indeed, I
wanted a chronological structure, starting from the Big Bang, inflation, reheating,
thermal Universe, CMB, and Current Universe, presenting at each step the possible
mechanisms of dark matter production. This is also the structure I use for the courses
I give at the University. At first sight, this structure seems elegant and logical, but it
may be that some elements necessary to a step are at the next step (need to know a
xiii
xiv Preface

thermal distribution even if the Universe is not yet thermalised. for example). The
reader should therefore not hesitate to skip from chapter to chapter. The advantage
of this presentation is that one can begin reading the book at any stage. No need
to read the first chapters to understand the mechanisms involved in the direct or
indirect detection of dark matter, for example. I thought this book as an efficient
tool above all.
This book should stir up curiosity in reader more than teaching them a complete
history of the Universe. I also advice the reader to keep this book in a place easily
accessible: anytime they read an article related to dark matter, or physics beyond the
standard model, they should find the answer or at least a hint of their answer in this
book. If not, they should not hesitate to send me an email concerning a subject not
treated or too lightly treated here. This textbook is far from complete and will evolve
thanks to the comments of all the readers. To appreciate at its best the reading of the
manuscript, some symbol ∗ and ∗∗ are added at the beginning of each section of
chapter depending on its technical difficulty. So we advice the readers to skip them
at a first lecture.

Orsay, France Yann Mambrini

February 2021
Acknowledgements

I want to especially thank all the people who contributed directly or indirectly to the
production of this book. Especially Genevieve Belanger, Soo-Min Choi, Emilian
Dudas, Marcos Garcia, Andreas Goudelis, Lucien Heurtier, Kunio Kaneta, Keith
Olive and Mathias Pierre.
Above all, I must thank those who, through my discussions, my meetings and by
reading their writings, have always been able to fuel my flame. First, Keith Olive
who guided me throughout my career, for whom physics is an art of living. Keith
taught me to feel the phenomena behind the equations and managed to transmit
to me a part of his intuition. His background in particle physics, astrophysics or
cosmology is just incredible. Thank you, Keith.
I would also like to thank Jim Peebles. He doesn’t know it, but for years I closed
my lectures or seminars insisting that, from my point of view, I couldn’t understand
how the Nobel Committee hadn’t yet awarded him the ultimate prize in our field.
This was done in 2019, and I was lucky enough to be in Stockholm at the time of the
nomination. I read and reread his seminal articles, and their clarity of presentation
and calculation make them works of art in scientific literature.
Another physicist that was often present, indirectly, during my career was Joe
Silk. Joe was always accessible “next door” at the Institute of Astrophysics of Paris
every time I had a question, a doubt, or needed an explanation about an observation
or theoretical subtleties. His kindness is matched only by his incredible scientific
talent.
I would also like to warmly thank Lisa Scalone, from Springer Publishing, for
her incredible patience, her boundless kindness and her constant support in this
adventure that I did not imagine to be so trying at the beginning of the project. This
book would definitely not be in your hands without her.
Finally, I would like to dedicate this book to my mentor, Pierre Binetruy, without
whom none of this would have been possible. He introduced me to the fabulous
world of theoretical physics, from my university years to the heart of my scientific
career. He is missed by all of us, and I hope that wherever you are, you enjoy the
journey as you read this book.

xv
Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
1.1 The First Dark Matter Paper . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
1.2 Local Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6
1.3 Anomalies in Rotation Curves of Galaxies . . . . . . .. . . . . . . . . . . . . . . . . . . . 7
1.4 Cluster Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9
1.5 Gravitational Lensing.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12
1.6 Bullet Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14
1.7 Comparison of Three Matter Abundance .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 16
1.8 Cosmic Microwave Background (CMB) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19
1.9 Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 21

Part I The Primordial Universe

2 Inflation and Reheating [MP → TRH ] . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25

2.1 The Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25
2.1.1 The Hubble Law .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26
2.1.2 The Friedmann Equations in a Dust Universe . . . . . . . . . . . . . 29
2.1.3 The Friedmann Equations in a Radiative Universe . . . . . . . . 32
2.1.4 The Friedmann–Lemaitre–Robertson–Walker
(FLRW) Metric . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 33
2.1.5 Friedmann’s Equation in General Relativity . . . . . . . . . . . . . . . 39
2.1.6 Another Look on the Hubble Expansion . . . . . . . . . . . . . . . . . . . 46
2.1.7 The Comoving Distance or Codistance . . . . . . . . . . . . . . . . . . . . 50
2.2 Inflation [10−43 − 10−37 s]. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 53
2.2.1 The Horizon Problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 53
2.2.2 The Flatness Problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 56
2.2.3 The Inflaton .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 59
2.2.4 The Equation of Motion.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 61
2.2.5 The Equation of Motion (Generalization) .. . . . . . . . . . . . . . . . . 62
2.2.6 The Slow-Roll Regime .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 63
2.2.7 The Coherent Oscillation Regime . . . . . .. . . . . . . . . . . . . . . . . . . . 67
2.2.8 The General Case, V (φ) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 68

xvii
xviii Contents

2.2.9 Constraint from Perturbations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71

2.2.10 Preheating and Dark Matter.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 76
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] .. . . . . . . . . . . . . . . . . . 83
2.3.1 The Context .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 84
2.3.2 The (Non-thermal) Distribution Function .. . . . . . . . . . . . . . . . . 90
2.3.3 End of the Thermalization Process: Transition
Toward a Thermal Bath . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97
2.3.4 Dark Matter Production During the Non-thermal
Phase of the Reheating . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] . . . .. . . . . . . . . . . . . . . . . . . . 106
2.4.1 Understanding the Reheating . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106
2.4.2 Non-instantaneous Reheating .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 110
2.4.3 Producing Dark Matter During the Reheating Phase . . . . . . 118
2.5 The Thermal Era [10−28 − mχ ] . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125
2.5.1 Instantaneous Reheating and Instantaneous
Thermalization.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130
3 A Thermal Universe [TRH → TCMB ] . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131
3.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131
3.1.1 A Brief Thermal History of the Universe in Some
Dates and Numbers.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131
3.1.2 Statistics of Gas, Pressure, and Radiation: The
Classic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135
3.1.3 Statistics of Gas, Pressure, and Radiation: The
Quantum Case . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 138
3.1.4 In the Primordial Plasma . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 140
3.1.5 Degrees of Freedom .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 144
3.1.6 Time and Temperature . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 148
3.1.7 The Entropy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 154
3.1.8 The Meaning of Decoupling . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 157
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? .. . . . . . . . . . . . 160
3.2.1 The Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 160
3.2.2 Approximate Solution .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 160
3.2.3 What Is Happening After the Decoupling? . . . . . . . . . . . . . . . . 161
3.2.4 Transfer of Energy and Thermalization . . . . . . . . . . . . . . . . . . . . 164
3.3 The Case of Light Species. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 172
3.3.1 The Neutrino Decoupling . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 172
3.3.2 The Tremaine-Gunn Bound .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 176
3.3.3 Dark Radiation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 178
3.3.4 The Recombination: Decoupling of the Photons . . . . . . . . . . 181
3.3.5 The Dark Ages, or Re-Ionization .. . . . . .. . . . . . . . . . . . . . . . . . . . 184
3.4 The Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185
3.4.1 The Context .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185
3.4.2 Overview.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 188
Contents xix

3.4.3 The Deuterium Formation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 191

3.4.4 The Lithium Problem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 192
3.5 Producing Dark Matter in Thermal Equilibrium .. . . . . . . . . . . . . . . . . . . . 194
3.5.1 The Boltzmann Equation.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195
3.5.2 Overview.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 196
3.5.3 Solving the Equation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 199
3.5.4 The Lee-Weinberg Bound .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 208
3.5.5 Two Exceptions to the Boltzmann Equation . . . . . . . . . . . . . . . 210
3.6 Non-thermal Production of Dark Matter . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 213
3.6.1 The Idea .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 213
3.6.2 Axion as a Dark Matter Candidate . . . . .. . . . . . . . . . . . . . . . . . . . 217
3.6.3 The Special Case of the Gravitino .. . . . .. . . . . . . . . . . . . . . . . . . . 222
3.6.4 Non-thermal Production Through Decays . . . . . . . . . . . . . . . . . 225
3.7 Extracting Information from the CMB Spectrum . . . . . . . . . . . . . . . . . . . . 231
3.7.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 231
3.7.2 To Find the Components of the Universe . . . . . . . . . . . . . . . . . . 232
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237

Part II Modern Times [TCMB → T0 ]

4 Direct Detection [T0 ] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 241

4.1 Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 241
4.2 Velocity Distribution of Dark Matter: f (v) . . . . . .. . . . . . . . . . . . . . . . . . . . 243
dσ
4.3 Measuring a Differential Rate: d|q| 2 . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 246
4.3.1 Kinematics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 246
4.3.2 Differential Rate . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 248
4.4 Structure Function of the Nucleus: F (q) .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 248
4.5 Computing a Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 251
4.6 Being More Realistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 254
4.6.1 Taking into Account the Earth Velocity . . . . . . . . . . . . . . . . . . . . 254
4.6.2 Annual Modulation of the Signal . . . . . . .. . . . . . . . . . . . . . . . . . . . 256
4.7 Influence of the Structure of the Nucleons . . . . . . .. . . . . . . . . . . . . . . . . . . . 257
4.8 Spinorial Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 261
4.9 More About the Effective Approach .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 264
4.9.1 Validity of the Approach . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 264
4.9.2 Effective Operators .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 264
4.9.3 Gluons and Heavy Quarks Contributions . . . . . . . . . . . . . . . . . . 272
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 274
5 In the Galaxies [T0 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 275
5.1 The Anatomy of the Milky Way . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 276
5.1.1 Internal Characteristics .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 276
5.1.2 The Color of the Sky: The Diffuse Gamma Ray
Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 277
5.1.3 Galactic Coordinates, Velocity of the Sun
and of the Earth .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 278
xx Contents

5.2 Computation of a Flux.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 281

5.3 Example of the Isothermal Profile . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 284
5.4 Radiative Processes in Astrophysics Part I: The
Non-Relativistic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 287
5.4.1 Maxwell Equations .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 288
5.4.2 Loss of Energy of a Moving Charged Particle . . . . . . . . . . . . . 288
5.4.3 Coulomb and Ionization Losses . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293
5.4.4 Thomson Scattering . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 296
5.4.5 Cyclotron Radiation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 298
5.4.6 Bremsstrahlung Radiation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299
5.5 Notions of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 300
5.5.1 Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 300
5.5.2 Lorentz Transformations .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 300
5.5.3 Relativistic Larmor’s Formula . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 305
5.5.4 Doppler Effect .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 305
5.5.5 Transformations on the Energies . . . . . . .. . . . . . . . . . . . . . . . . . . . 306
5.5.6 Fizeau Experiment . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307
5.6 Radiative Processes in Astrophysics Part II:
The Relativistic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 308
5.6.1 Relativistic Coulomb Scattering or Ionization Losses . . . . . 308
5.6.2 Inverse Compton Scattering.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 310
5.6.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 311
5.6.4 Relativistic Bremsstrahlung .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 312
5.6.5 Energy Losses: Summary . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 313
5.7 Ultra-High Energetic (UHE) Processes . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 314
5.7.1 Cosmic Rays Case . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 315
5.7.2 Photons and Neutrinos Cases . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 316
5.8 Indirect Detection of Gamma Ray . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 317
5.8.1 The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 317
5.8.2 Galactic Halo . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 319
5.8.3 Adiabatic Compression Mechanism .. . .. . . . . . . . . . . . . . . . . . . . 319
5.9 The Tricky Case of the Galactic Center . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 321
5.9.1 The Idea .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 321
5.9.2 Dark Matter Density Profiles . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 322
5.9.3 Gamma-Ray Flux from Dark Matter Annihilation . . . . . . . . 325
5.10 Dark Matter and Synchrotron Radiation. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 330
5.10.1 Neglecting Diffusion . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 330
5.10.2 Synchrotron Loss of Energy . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 331
5.10.3 Taking into Account Spatial Diffusion .. . . . . . . . . . . . . . . . . . . . 335
5.10.4 General Astrophysical Setup∗ . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 338
5.11 Sommerfeld Enhancement . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 344
5.11.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 344
5.11.2 Solving the Schrodinger’s Equation . . . .. . . . . . . . . . . . . . . . . . . . 345
5.11.3 The Coulomb Potential. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 348
5.11.4 The Yukawa Interaction .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 348
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5.12 Structure Formation Constraints . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 350

5.12.1 Free Streaming . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 351
5.12.2 Jeans Radius and Mass . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 354
5.12.3 The Influence of Dark Matter .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 357
5.12.4 Correlation Function . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 360
5.12.5 Power Spectrum P (k) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 361
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 362

A Cosmology and Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 363

A.1 Useful Cosmology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 363
A.1.1 Lorentz Transformation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 363
A.1.2 Friedmann Equation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 366
A.1.3 The Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 367
A.2 Basics of General Relativity . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 369
A.2.1 The Context .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 369
A.2.2 Measuring a Length, a Surface, or a Volume.. . . . . . . . . . . . . . 370
A.2.3 The Einstein–Hilbert Action (I) . . . . . . . .. . . . . . . . . . . . . . . . . . . . 371
A.2.4 Tooling with the Metric . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 373
A.2.5 A Geometrical Approach . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 375
A.2.6 The Riemann Tensor . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 378
A.2.7 The Einstein Equation of Fields in Vacuum.. . . . . . . . . . . . . . . 379
A.2.8 Adding Matter Fields. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 381
A.2.9 The Perfect Fluid Stress–Energy–Momentum Tensor .. . . . 381
A.2.10 Deflection Angle . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 384
A.3 Matter/Radiation Domination .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 386
A.4 Thermodynamical Fundamental Relations . . . . . . .. . . . . . . . . . . . . . . . . . . . 387
A.5 Classical Thermodynamic: The Laplace’s Law . .. . . . . . . . . . . . . . . . . . . . 387
A.6 Tooling with Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 389
A.6.1 Function (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 389
A.6.2 The Riemann Zeta Function ζ(z) . . . . . . .. . . . . . . . . . . . . . . . . . . . 389
A.6.3 Modified Bessel Function of the 2nd Kind Kn (z) . . . . . . . . . 390
A.6.4 Useful Integrals .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 391
A.6.5 Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 393
B Particle Physics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 395
B.1 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 395
B.1.1 Decay Rates and Cross Sections . . . . . . . .. . . . . . . . . . . . . . . . . . . . 395
B.2 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 397
B.2.1 General Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 397
B.2.2 Majorana Rules . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 399
B.2.3 Standard Model Couplings.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 402
B.3 Diracology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 404
B.3.1 Matrices .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 404
B.3.2 Dirac Equation.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 406
B.3.3 The Spin Matrix . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 407
xxii Contents

B.3.4 Proca Equation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 408

B.3.5 Rarita–Schwinger Equation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 409
B.3.6 Parity Operator . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 410
B.3.7 The Charge Conjugate Operator .. . . . . . .. . . . . . . . . . . . . . . . . . . . 412
B.3.8 The Majorana Case . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 414
B.3.9 Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 417
B.3.10 Mandelstam Variables .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 418
B.3.11 The Generators Tia . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 419
B.4 Lorentz Invariant Scattering Cross Section and Phase Space.. . . . . . . 420
B.4.1 The FERMI’s Golden Rule . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 420
B.4.2 Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 425
B.4.3 Computing the Phase Space . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 426
B.4.4 Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 429
B.4.5 Spin-Independent Diffusion, Elastic Scattering .. . . . . . . . . . . 441
B.4.6 Decaying Particles .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 442
B.4.7 Higgs Lifetime.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 446
B.4.8 Majorana Case . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 449
B.4.9 Vector Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 449
B.5 s-Wave, p-Wave, Helicity Suppression and All That .. . . . . . . . . . . . . . . 451
B.5.1 Velocity Suppression . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 451
B.5.2 Spin Selection . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 451
B.5.3 Application to Specific Models .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 452
B.5.4 Helicity Suppression . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 453
B.5.5 Summary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 454
B.6 Schrodinger Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 455
B.6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 455
B.6.2 Solutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 457
B.7 The Strong-CP Problem .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 458
B.7.1 QCD Lagrangian . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 458
B.7.2 The Axionic Peccei–Quinn Solution . . .. . . . . . . . . . . . . . . . . . . . 461
B.8 Useful Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 463
B.8.1 Gamma Spectrum . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 463
B.8.2 Positron Spectrum .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 464
B.8.3 Antiproton Spectrum . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 466
B.9 Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 466
B.9.1 Standard Model .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 466
B.9.2 Singlet Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 468
B.9.3 Extra U(1) and Kinetic Mixing . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 469
C Neutrino Physics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 471
C.1 Astrophysical and Cosmological Sources of Neutrino .. . . . . . . . . . . . . . 471
C.1.1 Solar Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 471
C.1.2 Atmospheric Neutrinos.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 473
C.2 Ultra-High Energetic Neutrinos .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 473
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C.3 Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 473

C.3.1 Dirac Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 473
C.3.2 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 474
C.4 The See-Saw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 476
C.4.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 476
C.4.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 477
C.4.3 The Specific Case mL M = 0 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 478
C.4.4 An Application: Coupling to a Scalar Field (Majoron) .. . . 479
D Useful Statistics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481
D.1 5σ and p-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481
D.1.1 5σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481
D.1.2 p-Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 483
D.2 Systematics vs Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 483
D.3 Look-Elsewhere Effect (LEE) . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 485
D.3.1 Generality .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 485
D.3.2 Applying the LEE Effect to the Higgs Discovery.. . . . . . . . . 487
D.3.3 Applying the LEE Effect to the Dark Matter Searches . . . . 488
D.4 Bayesian vs Frequentist Approach .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 488
E Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 491
E.1 Useful Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 491
E.1.1 Cosmology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 491
E.1.2 Particle Physics .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 492
E.2 Tables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 493
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 497
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 499
About the Author

Yann Mambrini is research director at the French

CNRS (Centre National de la Recherche Scientifique)
in the Irene Joliot Curie Laboratory, University Paris-
Saclay. His research interests concern the field of par-
ticle physics and fundamental interactions. He works
actively on extensions of the standard model, from
supergravity to grand unified theories, and more specif-
ically on their cosmological consequences, especially
involving dark matter aspects. He was awarded with the
Prix d’Excellence Scientifique of the National Research
Council in 2010, 2014 and 2018. Currently, he is lectur-
ing at Ecole Normal Supérieure. One of his passions is
the concept of time, from its measurement to its nature,
and the history of ideas in physics. He wrote several
popular books on the subject: Histoires de Temps and
Le Siècle des révolutions Scientifiques, Ed. Ellipses
and Newton à la Plage, Ed. Dunod. He shares this
passion with magic, with which he presents shows for
the general public mixing illusions, time and mysteries
of the Universe.

xxv
Introduction
1

Abstract

The twentieth century studies evidenced the existence of a new form of matter
which have inspired interest in modern physics scenario. It has been named “Dark
Matter” (DM), exotic name but with a clear meaning: a component of matter that
does not emit luminous radiation. Beginning from a study presented by Zwicky
in 1933 [4] who analyzed the motion of individual galaxies in the Coma cluster,
subsequently other observations have indicated the presence of dark matter from
the kinematics of gravitationally bound systems and rotating spiral galaxies,
the effects of gravitational lensing of background objects, various evidences
among which the observation of the Bullet Cluster, until recent results from the
PLANCK satellite. Furthermore, the dark matter appears to have an important
role in the formation of the structures, in the evolution of galaxies and also
has effects on non-uniformity observed cosmological microwave of background
radiation. Before going into detailed analyses of each step structuring the dark
matter presence and interaction in our Universe, we will first introduce in this
chapter the most important evidences, explaining where the dark matter may
intervene to resolve the oddities observed before listing the general features of
dark matter particles.

1.1 The First Dark Matter Paper

There are many books that retrace in a more or less faithful and more or less
exhaustive way the history of dark matter research and its interpretations. I advise
the reader to refer to three very accessible references. First of all, the summary by
G. Bertone and D. Hooper [5] is an excellent introduction to the field. They retrace
the steps that led to the hypothesis of a dark matter in the form of a particle. R.H.
Sanders in [6] summarizes very well, on the astrophysical side, the disappointments
and difficulties that had to be overcome before admitting the presence of missing

© Springer Nature Switzerland AG 2021 1

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2_1
2 1 Introduction

mass in our Universe. In [7, 8] you will find a complete summary of the dark matter
candidates, their properties, and the status of the detection prospect. Finally, the
excellent review by J. Peebles [9] reminds us of the steps that had to be taken in
order to have an image of an expanding cosmos dominated by predominantly black
components. Concerning inflationary models, there is no more complete review than
the one proposed by Keith Olive [10] and the seminal articles by A.H. Guth [11] and
A. Linde [12].
Each of my colleagues working in the so-called dark matter field will have their
own interpretation of facts and history. I remember a question I was asked at a
seminar in Moscow: “What is, for you, the first paper dealing with dark matter?”
This question, which seemed trivial, turned out to be much more difficult to answer.
And I turned it into a seminar that I still manage to give for Master’s students.
First of all, we have to agree on the term “Dark Matter.” What we consider in this
textbook is a dark matter in the form of a particle or a field. For a very long time the
presence of missing mass has been debated. Then, once its presence was irrefutable,
the problem was not to explain its presence, but to understand why it was not seen
(light path deflected, obscured by interstellar gas? Stars too old to be visible?). It
took several years, and the work of Peebles and Ostriker on the stability of galaxies,
to realize that ‘something more’ was needed. The neutrino was obviously the first
candidate, quickly discarded because of its too light mass and relativistic character.
It was not until the 1970s that the first paper mentioning the need for a new particle
appeared, and its authors calculated the expected relic density, based on data from
the cosmological diffuse background and astronomical observations of the time.

Prehistory

The first mention of the word “Dark Matter” appeared in a work by Poincaré in
1908. In “The Milky Way and the Theory of Gases,” Poincaré follows Lord Kelvin’s
1904 hypothesis and proposes to treat stars in galaxies as atoms in a gas. He then
deduced, by comparing the movement of the Earth around the Sun with that of the
Sun around the nearest star Proxima, an approximation of the mass of the Milky
Way. Assuming an approximately constant star density, he deduced that our galaxy
should be made up of a billion stars. This value was quite close to the number of
stars then visible in our galaxy (we know today that it contains 200 to 400 times
more stars). So he concluded

There is not dark matter, or at least not so much as there are of shining matter.

Of course, his constant density hypothesis finally made his calculation wrong,
most of the dark matter being present around the galactic center which is a strong
gravitational well. But here, we must appreciate the revolutionary idea of treating a
galaxy as a gas, and applying a kinetic theory to it, to calculate its mass. The second
appearance of the word «dark matter» in the literature is in a paper of the physicist
1.1 The First Dark Matter Paper 3

Jan Oort from Netherland in 1932. While he was analyzing the radial velocities, he
noticed a discrepancy with Newton’s law. He computed that only one-third of the
dynamically inferred mass was present in bright visible stars. It is clear from the
context that, as characterizing the remainder as «dark» («Dunkle Materie»), Oort
was describing all matter which is not in the form of visible stars with luminosity
comparable or larger than that of the Sun. Gas and dusts between the stars were the
main constituents of his «invisible mass» that should be found (for him) soon. The
main reason evoked at this time was the presence of low luminosity objects (dead
stars) or large absorbing gas. Imagining a new dark component took a very long
time to physicists, who even preferred to modify the law of gravity at large scale
before invoking a new particle. In this sense, the first real work underlining that the
missing mass could be problematic is Fritz Zwicky in 1933.
It is by observing the Coma cluster that Zwicky realizes a problem in the
movement of the galaxies that make it up. The Coma cluster is a highly regular
gravitationally bound system of thousands of galaxies at a distance of about
100 Mpc. The astrophysicist noticed that each of its galaxies was following a
movement that seemed fast compared to the movements of closer galaxies such
as Andromeda. He then applied the virial theorem to this cluster of 800 galaxies
of ∼109 solar masses and deduced that the average speed of each of them must be
of the order of 80 km/s. What was his surprise when he measured the individual
velocities of the galaxies in this cluster, and that they were close to 1000 km/s. The
only possible explanation was the presence of invisible matter which increased the
gravity potential. Hence his conclusion at the end of his article was

In order to obtain the observed value of an average Doppler effect of 1000 km/s or more,
the average density in the Coma system would have to be at least 400 times larger than
that derived on the grounds of observations of luminous matter. If this would be confirmed
we would get the surprising result that dark matter is present in much greater amount than
luminous matter.

This result was then completely forgotten and no one took Zwicky’s comment
seriously. Indeed, large scale astrophysics had only just emerged following the
discovery of Hubble, and many physicists believed that the problem of missing mass
will be solved when they better understand the mechanisms of light absorption in
the interstellar/internebulae medium. In fact, the missing mass problem has long
been viewed as a missing luminosity problem: why do not we see the astrophysical
bodies that should be responsible for Newtonian dynamics. On the other hand,
several scientists tried rather to modify (already in the 30s) the law of attraction
in r12 . It is not surprising. The theory of general relativity was then only in its
observational beginnings and was much less untouchable than today. Introducing
a new component of matter into the Universe was much more esoteric. It was then
that the analysis of galaxies began.
4 1 Introduction

The Galactic Scale

The history of measurements of the rotation curves of galaxies dates back to 1914
when Vesto Slipher at the Lowell Laboratory observed the Andromeda galaxy. It is
the closest galaxy to us, at a distance of ∼800 kpc, but estimated at 210 kpc at this
time due to the approximate determination of the Hubble constant. Slipher noticed
that the speeds of the stars measured to the left of the galaxy bulge were approaching
us at higher speeds (∼320 km/s) than those on the right side of the central bulge
(∼280 km/s). This observation corresponds to that of a disc spinning in front of
us. In 1918, Pease at the Mount Wilson Observatory measured the speed of rotation
over a radius of 600 pc (the central part of Andromeda). Its result could be expressed
by the formula

Vc = −0.48r − 316, (1.1)

where Vc is the measured circular speed (in km/s) at a distance r from the central
Andromeda bulge. This central part seems to rotate at a constant angular speed. It
is Horace Babcock who, in his PhD thesis in 1939, extended the study to larger
scale, up to 24 kpc from the center. Measuring still a constant angular velocity, he
concluded

..constant angular velocity discovered for the outer spiral arms is hardly to be anticipated
from current theories of galactic rotations.

From the computation of the density, he deduced a total mass for Andromeda
of 1011 solar mass, equivalent to a mass-to-luminosity ratio M/L = 50. He then
concluded

This last coefficient is much greater than that for the same relation in the vicinity of the
Sun.

As we can see, for Babcock at the scale of galaxies, just like for Zwicky at the
scale of galaxy clusters, a problem remains. But for the moment, no explanation is
really satisfactory. When Karl Jansky observes radio waves of synchrotron radiation
from the galactic center in 1932, and Oort and Van de Hulst study the sky by
observing the spectral line 21cm from the hydrogen atom in 1957, astrophysicists
realize that the presence of an invisible matter, responsible for the rotation of the
closest galaxies, extends well beyond the visible domain. Van de Hulst does not
insist too much in his article on the flatness of the rotation curve. But, by calculating
the mass of M31 (Andromeda), he deduces that it is much heavier than the Milky
Way. The “dark matter” hypothesis does not (yet) hit the galactic scale.
1.1 The First Dark Matter Paper 5

Stabilization of the Structures

In the 70s, Moore’s law of exponential development describing the evolution of

computing power over time affected astrophysical studies: the computing power
doubling every two years, it was possible at the end of the 60s to develop electronic
machines for the digital resolution of complex problems (technically, it was the
replacement of vacuum tubes by transistors that gave a big leap in the field). Franck
Hohl carried out in 1971 one of the first “N-body” simulations (100,000 stars!!)
to test the stability of galactic structures, with a disk of particles supported in
equilibrium almost entirely by rotation. He noticed that an elongated spiral shape
forms after 2 revolutions, but quickly the kinetic energy diffuses the particles and the
final state is a pressure dominated gas with large elongated axisymmetric ellipses.
To keep the observed spiral shape, Miller, Pendergast, and Quirk tried to stabilize
the model by artificially adding energy loss. Nevertheless, the heating of the gas
destroyed the spiral structure after a few revolutions. It was at this point that the
proposition of the presence of a dark halo came to the rescue and is mentioned for
the first time in an article.
In 1973, P.J.E. Peebles and J.P. Ostriker [13] noticed that the individual random
velocities of stars in our galaxies (around 30–40 km/s) are much smaller than the
average circular motion (around 200 km/s). So not only is the system unstable as
noted by Hohl et al., but it also shows that the galaxies seem to be dominated by a
cold gravitational system and not by kinetic pressure. Peebles and Ostriker noticed
that if at least 28% of the kinetic energy is stored in the rotation, the system is
unstable and destabilized very quickly. However, in our Milky Way, the rotational
speed is around 200 km/s, while the individual velocities approach 40 km/s, which
gives a ratio of 50% if one takes into account only the visible stars, well over the
limit of stability. The ingenious idea of Peebles and Ostriker is then to add an
additional component to the galaxy, a dark halo Udark , which contributes at least
50% of the mass around the position of the Sun:

U → U + Udark , (1.2)

where U represents the total gravitational potential of the galaxy. This spheroidal
system increases the gravitational potential energy but adds nothing to the rotational
energy. The ratio (kinetic energy)/(gravitational energy) would then be reduced and
perhaps stability restored. In their 1973 seminal paper, they conclude

Adding an extended component corresponding to the “halo” [. . . ] apparently will stabilize

the system if the halo mass is equal to or somewhat greater than the disk mass.

Vera C. Rubin and W. Kent Ford [14] also analyzed the rotation curve of the
Andromeda galaxy, up to 120 minutes of arc, and proposed also the presence of a
halo surrounding all the nebulae, even if their conclusion states

.. extrapolation beyond that distance is a matter of taste.

6 1 Introduction

A study by D.H. Rogstad and G.S. Shostak in 1972 showed that the presence
of a dark halo is also necessary in several galaxies surrounding the Milky Way. In
1985, Dekel and Silk went further, showing in a work ahead of its time [16] how
dark matter could explain the formation of dwarf galaxies, finding that the relation
between their mass and radius can fit in a cosmological model which includes a
right component of cold dark matter.
After several independent works by Dolgov, Zeldovich, Cowsik, Mc Clelland,
Hut, Lee, and Weinberg, which give limits on the dark matter mass from obser-
vations, one has to wait for the paper by Steigman et al. in 1978 [17] to see a
first particle candidate, beyond the Standard Model, including the computation of
its relic abundance and a complete analysis of the equilibrium issue. Preceding
important works proposed neutrinos as dark matter, showing they cannot fill the
entire missing mass, or even massive neutrino–like particles, but [17] is the first to
generalize to a yet unknown electroweak particle. I am sure that a lot of colleagues
will dispute my choice. However, I consider it as the first dark matter paper in
the sense that they study, for the first time, the cosmological consequences of the
existence of any stable, massive, neutral lepton, particle that is now more known
under the acronym “WIMP” for Weakly Interacting Massive Particle. Since then,
several models have been proposed to fit a candidate in a more general framework.
The more solid proposition (until now) stays the supersymmetric version of the
Standard Model, which includes two serious candidates: the neutralino (the lightest
partner of the weak gauge bosons) and the gravitino (supersymmetric partner of
the graviton). The first complete study on the subject was made by Ellis, Hagelin,
Olive, Nanopoulos, and Srednicki in their seminal work [18], where they proposed
scenarios with Higgsino (partner of the Higgs), photino (partner of the photon), and
gravitino dark matter, computing their relics from the Big Bang. After this brief
historical introduction, one needs to see in more detail how the local abundance has
finally been measured, and how we can extrapolate its distribution all over the large
scale structures of the Universe.

1.2 Local Dark Matter

The dynamical density of matter in the Solar vicinity can be estimated using vertical
oscillations of stars around the galactic plane. The orbital motions of stars around
the galactic center play a much smaller role in determining the local density.
Oort in 1932 [15] indicated, in his analysis, as members of a “star atmosphere,”
a statistical ensemble in which the density of stars and their velocity dispersion
define a “temperature” from which one obtains the gravitational potential. The
result contradicted grossly the expectations: the potential provided by the known
stars was not sufficient to keep the stars bound to the galactic disk because the
density of visible stellar populations by a factor of up to 2, and so the galaxy should
rapidly be losing stars [19]. Since the galaxy appeared to be stable, there had to
be some missing matter near the galactic plane, Oort thought, exerting gravitational
attraction. This limit is often called the Oort limit. This used to be counted as the
1.3 Anomalies in Rotation Curves of Galaxies 7

first indication for the possible presence of dark matter in our galaxy: the amount of
invisible matter in the Solar vicinity should be approximately equal to the amount
of visible matter. A modern calculation of the local relic abundance using several
observables can be found in the work of R. Catena and P. Ullio in [20]. Their result
gives a local amount of dark matter of ρ0  0.39 GeV/cm3 , which corresponds
mainly to 100,000 particles passing through us per cm2 per second.
It is important to notice that such a value for a local amount of dark matter
(0.39 GeV/cm3 ) is very small. To collect one gram of dark matter, we would have to
put together all the particles contained in a volume comparable to that occupied by
the entire Earth. Indeed, the dark matter is much more present near the galactic
center. That also explains why we can safely use the Newton laws in the solar
system. Of course, when one looks at larger scale, the situation is completely
different: between the Sun and the next star, for instance, the void is huge. A quick
calculation shows that the density of the planets in the solar system in a sphere that
extends halfway to the next star, Proxima Centauri, is about the same as the density
of dark matter.

1.3 Anomalies in Rotation Curves of Galaxies

Radiowave radiation from interstellar gas, in particular that of neutral hydrogen, is

not strongly absorbed or scattered by interstellar dust [21]. It can therefore be used to
map and to study the motion of neutral hydrogen clouds concentrated in spiral arms.
We can therefore determine the angular velocity of the gas at different distances
from the galactic center r, and plot the corresponding rotation curve v = v(r).
The most convincing and direct evidence for dark matter on galactic scales comes
mainly from the observations of the rotation curves of galaxies. Rotation curves are
usually obtained by combining observations of the 21 cm line with optical surface
photometry: if θ is the angle between the velocity of the star and the line of sight, the
velocity components can be written as vr = v cos θ and vt = v sin θ . The tangential
velocity vt results in the proper motion, which can be measured by taking photos at
intervals of several years or decades. The radial velocity vr can be measured from
Doppler shift of the stellar spectrum, in which the spectral lines are often displaced
toward the blue or red. The blueshift means that the star is approaching, while the
redshift indicates that it is receding. From 1940 (Oort’s) numerous observations in
spiral galaxies showed, in outer regions of galaxies, an anomaly in rotation velocity
that can be translated in a high M/L, mass-to-luminosity ratio. Observed rotation
curves usually exhibit that the central part of the galaxy rotates like a rigid body,
i.e. v ∝ r, and then the velocity reaches a maximum value, or plateau. At this point
we would expect a decreasing velocity outward, as the third Kepler law suggests,
instead there is a characteristic flat behavior until edges of galaxies where few light
is emitted. Let us consider for simplicity a spherical distribution of matter in the
galaxy. In Newtonian dynamics, the virial theorem determines the circular velocity,
8 1 Introduction

which is expected to be

v2 GM(r) GM(r)
= ⇒ v= , (1.3)
r r2 r
r
where M(r) = 4π 0 ρ(a)a 2da is the mass of the matter of density ρ(a) √ contained
in a sphere of radius r. From Eq. (1.3), v(r) should fall following 1/ r outside the
optical disk where M should be constant without the presence of dark matter. The
fact that the observation gave v(r)  constant implies the existence of a halo with
M(r) ∝ r and then ρ(r) ∝ 1/r 2 confirming that a large part of the mass is present
in the outer part of the galaxy and not in the visible disk. This is well illustrated
in Fig. 1.1. The mass distribution is obtained from rotational curves, determined
from light distribution of luminous components in the galaxy. By photometry the
estimated mass in our galaxy between the galactic center (GC, Sagitarius A*) and
the Sun (at a distance r = 8 kiloparsec (kpc) from the GC) is M = 9 × 1010M ,
while for the outer edge of galaxy, where the luminosity decreases exponentially,
the component of luminous is negligible. At this distance from the galactic center,
the rotational velocities verify approximately the Keplerian law as the visible matter
dominates the matter distribution at such distances from the GC.

Fig. 1.1 Rotation curve of a galaxy with the different components extracted from observations:
bulge, disk, and dark halo. OpenStax University Physics, extracted from the book University
Physics Volume 1 under Creative Commons Attribution 4.0 International license
1.4 Cluster Dark Matter 9

Below r some discrepancies in the rotational velocity can be explained by the

presence of an invisible mass halo, called dark matter halo or dark halo, around our
galaxy. This dark component of matter would be spherically distributed in a halo
extended until 230 kpc from galactic center and having a density profile following
ρ0
ρ(r) =     , (1.4)
r 2
r
a 1+ a

a being a typical scale of the halo depending on the galaxy. With this profile the
galaxy behaves like 1/r at the center (r a) and 1/r 3 in the edges r a. With
this calculation the mass of halo of dark matter must be 5.4 × 1011M within 50
kpc and 2.5 × 1012M at 230 kpc [22]. The profile proposed in Eq. (1.4) is called
NFW profile and was proposed by Navarro Frenk and White in [23], whereas the
first measurement of the rotation curve was made by Ford and Rubin in [14].

1.4 Cluster Dark Matter

Another mass discrepancy was found by Zwicky in 1933 [24]. He measured

redshifts of galaxies in the Coma cluster and found that the velocities of individual
galaxies with respect to the cluster mean velocity are much larger than those
expected from the estimated total mass of the cluster, calculated from masses
of individual galaxies. His article was in fact more a review on the status of
observational cosmology in 1933. It is just at the end of its article that he concludes
on the remark concerning the probable presence of dark matter in the Coma cluster.
Stars move in galaxies and galaxies in clusters along their orbits, and those are
virialy bound systems: the orbital velocities are balanced by the total gravity of
the system, similar to the orbital velocities of planets moving around the Sun in
its gravitational field. In the simplest dynamical framework, one treats clusters of
galaxies as statistically steady, spherical, self-gravitating systems of N objects of
average mass m, and average orbital velocity v. The total kinetic energy E of such
a system is then

1
E= Nmv 2 .
2
If the average separation is r, the potential energy of N(N − 1)/2 pairings is

1 Gm2
U = − N(N − 1) .
2 r
10 1 Introduction

The virial theorem states that for such a system,

U
E=− .
2
The total dynamic mass M can then be estimated from v and r from the cluster
volume

2rv 2
M = Nm = .
G
Zwicky was the first to use the virial theorem to infer the existence of unseen matter.
He found that the orbital velocities are almost a factor of ten larger than expected
from the summed mass of all galaxies belonging to the clusters, and this implies that
the average mass of galaxies within the cluster has a value about 400 times greater
than expected from their luminosity. The gravity of the visible galaxies in the cluster
would be far too small for such fast orbits, so something extra was required. This is
known as the “missing mass problem,” and he proposed that most part of the missing
matter was a dark, non-visible form of matter, which would provide enough of the
mass and gravity to hold the cluster together.
There exists another method to determine the mass of cluster: the temperature of
the hot intracluster gas, like the galaxy motion, traces the cluster mass. Indeed, hot
gas inside the clusters emits X-ray radiation through bremsstrahlung process (see
Sect. 5.6.4 for more details). Observations show that the gas is in hydrodynamic
GMr ρ
equilibrium (dFgrav = dFpress = dP dr = − r 2 with Mr the inner total mass
inside the radius r), and it moves in the gravitational field of cluster in orbits with
velocities depending on the mass of the cluster. Through spectroscopic analysis of
hot gas, we can obtain density and temperature of gas as a function of galactic
distance r. With these parameters, we can get mass distribution of cluster. For
example, gas mass of the Coma cluster is Mgas = 1.05 × 1014M , which is
larger than the visible mass Mvis = 1.5 × 1013M , but not sufficiently to explain
the value extracted from the virial theorem, that is, Mvir = 3.3 × 1015M . We
illustrate in Fig. 1.1 the different components of the galactic structure and their
relative abundance as a function of their distance from galactic center.
N-body simulations also give information on the possible distribution of dark
matter in our galaxy. This is a daunting task especially when one considers that
even the three-body problem—the problem of describing the orbits of three celestial
bodies under their reciprocal gravitational attraction—is extraordinarily difficult
and can only be solved in some simplified cases. How can we therefore hope to
solve the problem of computing the reciprocal interactions of all particles in the
Universe? The problem is that, in principle, one has to calculate for each particle
the attraction of every other particle in the universe. Eric Holmberg, an ingenious
Swedish scientist, found an original solution to the problem in 1941. He decided
to simulate the intersection of two galaxies using 74 light bulbs, together with
photocells and galvanometers, using the fact that light follows the same inverse
1.4 Cluster Dark Matter 11

square law as the gravitational force. He then calculated the amount of light received
by each cell and manually moved the light bulbs in the direction that received the
most intense light.
The first application of computer calculations to gravitational systems was
probably by John Pasta and Stanislaw Ulam in 1953. Their numerical experiments
were performed on the Los Alamos computer, which by then had already been
applied to a variety of other problems, including early attempts to decode DNA
sequences and the first chess-playing program. Two young astrophysicists, the
Toomre brothers, had access in the early 1970s to one of the NASA’s two IBM 360-
95 computers, completed with high-resolution graphics workstations and auxiliary
graphics–rendering machines (computing facilities far in advances of any other
astrophysics laboratory). They set up a series of simulations of galaxy grazings and
collisions using a simple code that described the galaxies as two massive points
surrounded by a disk of test particles. The outcome of the analysis was a very
influential paper, published in 1972, that contained a detailed discussion of the role
of collisions in the formation of galaxies. Together with their paper, the Toomre
brothers also created a beautiful 16 mm microfilm movie “Galactic Bridges and
Tails” that you can find at http://kinotonik.net/mindcine/toomre. A photo of the
simulation is represented in Fig. 1.2.
Numerical simulations have improved immensely since these pioneering
attempts, thanks to a dramatic increase in computing power. Modern supercom-
puters allow us to simulate entire universes by approximating their constituents
with up to ten billion particles, as in the case of the Millennium simulation shown
in Fig. 1.2, which was run at the Max Planck Society’s Supercomputing Center
in Garching, Germany. This has a computing power ten million times larger than
the old IBM 360-95 used by the Toomre brothers. We usually parameterize our
“ignorance” of the exact distribution obtained by the N-body simulation systems
with an empirical formula for the dark matter halo in the cluster or in our Milky
Way:
ρ0
ρDM (r) =  α  3−α ,
r
rs 1+ r
rs

where rs is a typical scale (usually the distance from the Sun to the galactic
center), and ρ0 is the normalization constant, determined from the observation of
the local density of dark matter in the vicinity of the Sun. The values of α are
determined by observations or after running simulations and are approximately
between 0 ≤ α ≤ 3/2. For a more detailed analysis on the profile structure and how
one obtains them, have a look at Sect. 5.8.2. It is however important to notice that
N-body simulations deal with “particle” mass of the order of several solar masses
and that one should extrapolate their results down to electroweak masses, which can
infer some errors. Moreover, the smaller dimension scale that these simulations can
reach is of the order of the kiloparsec, so all spatial extrapolations around the parsec
galactic center can be dubious, especially, when we do not know exactly which kind
of ingredient and initial conditions are taken by the simulations. That explains the
12 1 Introduction

Fig. 1.2 N-body simulation in 1972 with 120 particles (top) and in 2014 with ten billions of
particles (bottom), from the Illustris project, https://www.illustris-project.org/, licensed under the
Creative Commons Attribution-Share Alike 4.0 International license

orders of magnitude existing between prediction in indirect detection of dark matter

when most of the annihilation processes occur around the galactic center, where the
precisions of the simulations are the worst, without real visible observables to test
them.

1.5 Gravitational Lensing

The theory of general relativity teaches us that gravitational field curves the space-
time metric, whereas the particles or photons travel in geodetic trajectory. An
observable consequence of this effect is the gravitational lensing: a photon in a
1.5 Gravitational Lensing 13

gravitational field moves as if it possessed mass, and light rays therefore bend
around gravitating masses. Thus celestial bodies can serve as gravitational lenses
probing the gravitational field, whether baryonic or dark without distinction, and
thus can probe the dark massive component of any celestial object. If we consider
that a trajectory of light ray in a gravitational filed with a spherical symmetry
(r, θ, φ) is represented as

d2 1 1 GM
+ =3 2 .
dφ 2 r r r

The solution of this equation can be thought as a perturbation of special relativity

(without gravitational field), and the deflection angle is derived in the Appendix,
Eq. (A.106):

GM
δ=4 ,
r0 c2

r0 being the closest distance from the light ray to the massive body causing the
deflection. The deflexion provided for a light ray that enters in gravitational field of
the Sun is δ  1.75" and has been measured by Eddington in 1919. Since photons
are neither emitted nor absorbed in the process of gravitational light deflection, the
surface brightness of lensed sources remains unchanged. Changing the size of the
cross section of a light bundle only changes the flux observed from a source and
magnifies it at a fixed surface brightness level. We can categorize three classes of
gravitational lensing as follows:

• Strong lensing, the photons move along geodesics in a strong gravitational

potential, which distorts space as well as time, causing larger deflection angles
and requiring the full theory of general relativity. The images in the observer
plane can then become quite complicated because there may be more than one
null geodesic connecting source and observer. Strong lensing is a tool for testing
the distribution of mass in the lens rather than purely a tool for testing general
relativity. The masses of clusters of galaxies determined using this method
confirm the results obtained by the virial theorem and the X-ray data.
• Weak lensing, which refers to deflection through a small angle when the
light ray can be treated as a straight line, and the deflection as if it occurred
discontinuously at the point of closest approach (the thin-lens approximation in
optics). One then only invokes special theory of relativity, which accounts for the
distortion of clock rates. This kind of lensing allows to determine the distribution
of dark matter in clusters as well as in superclusters: the lensing mass estimate is
almost twice as high as that determined from X-ray data.
14 1 Introduction

• Microlensing, if the mass of the lensing object is very small, one merely observes
a magnification of the brightness of the lensed object. Microlensing of distant
quasars by compact lensing objects (stars, planets) has also been observed and
used for estimating the mass distribution of the lens-quasar systems. A fraction
of the invisible baryonic matter can lie in small compact objects. To find the
fraction of these objects in the cosmic balance of matter, special studies have
been initiated, based on the microlensing effect. This process is used to find
Massive Compact Halo Objects (MACHOs), small baryonic objects as planets,
dead stars, or brown dwarfs, which emit so little radiation that they are invisible
most of the time. A MACHO may be detected when it passes in front of a star
and the MACHO’s gravity bends the light, causing the star to appear brighter.
Some authors claimed that up to 20% of the dark matter in our galaxy can be in
low-mass stars (Fig. 1.3).

1.6 Bullet Cluster

The Bullet Cluster (1E 0657-558) consists of two colliding clusters of galaxies. It
is at a co-moving radial distance of 1.141 Gpc (3.721 Gly). Gravitational lensing
studies of the Bullet Cluster are claimed to provide the best evidence to date for
the existence of dark matter [25]. At a statistical significance of 8σ , it was found
that the spatial offset of the center of the total mass from the center of the baryonic
mass peaks cannot be explained with an alteration of the gravitational force law. In
Fig. 1.4, we show the Chandra X-ray Observatory image of this cluster taken in 2004
[26]. This cluster was formed after the collision of two large clusters of galaxies.
Hot gas detected by Chandra in X-rays is seen as two pink clumps in the image and
contains most of the “normal” or baryonic matter in the two clusters. The bullet-
shaped clump on the right is the hot gas from one cluster, which passed through
the hot gas from the other larger cluster during the collision. An optical image from
Magellan and the Hubble Space Telescope shows the galaxies in orange and white.
The blue areas in this image show where astronomers find most of the mass in the
clusters. The concentration of mass is determined using the effect of gravitational
lensing, where light from the distant objects is distorted by intervening matter. Most
of the ordinary visible matter in the clusters is clearly separate from the matter
responsible of the gravitational lensing, giving direct evidence that nearly all of the
matter in the clusters is dark: the hot gas in each cluster was slowed down by a force
like air resistance, whereas the dark matter was not slowed by the impact because
it does not directly interact with the gas itself or if not through gravity. Therefore,
during the collision, the lumps of dark matter from the two clusters moved ahead of
the hot gas, producing the separation of dark matter and baryonic matter. If hot gas
was the most massive component in the clusters, as proposed by alternative theories
of gravity, this effect would not be seen: this result shows that dark matter is required
at least in the Bullet Cluster.
1.6 Bullet Cluster 15

Fig. 1.3 Top: effect of gravitational lensing of a galaxy acting on the light emitted from a galaxy
on its way to the Earth, ALMA (ESO/NRAO/NAOJ), L. Calçada (ESO), Y. Hezaveh et al. Bottom:
distorted images due to the gravitational lensing system called SDSS J0928+2031 observed by
the Hubble telescope. ESA/Hubble image released by the ESA under the Creative Commons
Attribution 4.0 Unported license
16 1 Introduction

Fig. 1.4 Bullet Cluster photo in X-ray (red) superimposed with the gravitational lensing (blue),
exposition time of about 140 h and megaparsec scale [26] ( NASA/CXC/M. Weiss)

1.7 Comparison of Three Matter Abundance

From all the measurements described above, one can estimate the relative mass
ρg
contribution of the galaxies g = ρcrit , ρg being the density of mass in the galaxies
and ρcrit the critical density of the Universe (see Sect. 2.1.6 for more details) to be
g  0.03 − 0.07. This estimation comes from combining the mean luminosity per
unit volume produced by galaxies Lg and the mean mass to light ratio M/L

M
ρg = Lg ×   6 × 10−31 h2 g cm3 ,
L

which gives1 g  0.03 − 0.07. At the level of the cluster of galaxies, estimations
of the gravitational lens produced by Abell 2218 or Abell 1689 give a similar result,
with a mean M/L around 10 times the one in galaxies (almost constant for scales
above the Mpc [27]), and coherent with other observations, which gives clust er 
0.2−0.4. The discrepancy between the three values of i can then also be attributed
to the presence of non-luminous dark matter, which may play an important role in

1ρ
c = 1.78 × 10−29 h2 , see Sect. 2.1.6.
1.7 Comparison of Three Matter Abundance 17

structure formation. For a mass scale R > 1.5 h−1 Mpc (typical galaxy radius), the
mass-to-light ratio of superclusters of galaxies confirms that there does not exist an
additional quantity of dark matter at higher scale, R = 6 h−1 Mpc.
We can also estimate the contribution from baryonic material by comparing the
observed abundances of light elements (deuterium, 3 He, 4 He, and 7 Li) with the
predictions of primordial nucleosynthesis computations, which gives us b  0.04.
This value is obtained from the standard Big Bang nucleosynthesis and (except from
the lithium “problem”) corresponds to recent observations. For the curious people,
they can go to have a look at Sect. 3.4 for more details. In few words, according to
the Big Bang model, the Universe began in an extremely hot and dense state. For
the first few seconds, it was so hot that atomic nuclei could not form, and space was
filled with a hot soup of protons, neutrons, electrons, photons, and other short-lived
particles. Occasionally a proton and a neutron collided and sticked together to form
a nucleus of deuterium (a heavy isotope of hydrogen), but at such high temperatures
they were broken immediately by high-energy photons. When the Universe cooled
down, these high-energy photons became rare enough that it became possible for
deuterium to survive, during a short period before the expansion freezes out the
production process. This narrow time window is called the deuterium bottleneck.
These deuterium nuclei could keep sticking to more protons and neutrons, forming
nuclei of 3 He, 4 He, lithium, and beryllium. This process of element formation is
called nucleosynthesis. The denser proton and neutron gas is at this time, the more
of these light elements will be formed. As the Universe expands, however, the
density of protons and neutrons decreases and the process slows down. Neutrons are
unstable (with a lifetime of about 15 min) unless they are bound up inside a nucleus.
After a few minutes, therefore, the free neutrons are gone and nucleosynthesis
stops. There is only a small window of time in which nucleosynthesis can take
place, and the relation between the expansion rate of the Universe (related to the
total matter density) and the density of protons and neutrons (the baryonic matter
density) determines how much of each of these light elements are formed in the early
Universe. Figure 1.5 shows the computed abundance of deuterium D (2 H), 3 He, 4 He,
and 7 Li (compared with (H) hydrogen). The abundances are all shown as a function
of ηb , the baryon-to-photon ratio, which is related to b by b = 0.004 h−2 10η−10 b
.
The estimates of the primordial values of the relative abundances of these elements
obtained by WMAP [28] and PLANCK [29] appear2 to be in agreement with
nucleosynthesis predictions, but only if the density parameter in baryonic material
is b h2 = 0.02.

2η
b = 6.19 × 10−10 as measured by WMAP.
18 1 Introduction

Fig. 1.5 Relative abundance of light elements (relative to hydrogen) as function of the ordinary
matter relative to photon (ηb ). Licensed under the Creative Commons Attribution-Share Alike 4.0
International license
1.9 Alternatives 19

1.8 Cosmic Microwave Background (CMB)

The relic abundance of dark matter can also be extracted from the analysis of the
Cosmic Microwave Background (CMB), radiation originating from the propagation
of photons in the early Universe once they decoupled from matter. This is the
recombination era (see Sect. 3.3.4 for more details). In 1965, this radiation was
detected by Penzias and Wilson, and this discovery was a powerful confirmation
of the Big Bang theory. WMAP [28] and more recently the European PLANCK
satellite [29] gave us the more precise photo of the Universe and its ingredients.
After many decades of experimental efforts, the CMB is known to follow with
extraordinary precision a black-body spectrum corresponding to a temperature T =
2.726 K. It is also quasi-isotropic withbtemperature fluctuations (called anisotropy)
−3 − 10−5 . The variations of temperature in the CMB can be
T  10
of the order δT
expressed as a sum of spherical harmonics Ylm

∞ l
δT
(θ, φ) = alm Ylm (θ, φ), (1.5)
T
l=2 m=−l

l
where alm gives us the variance Cl = |alm |2  = 2l+1 1
m=−l |alm | . If the
2

temperature fluctuations are assumed to be Gaussian, as it appears to be the case,

all of the information contained in CMB maps can be encoded into the power
spectrum, essentially giving the behavior of l(l + 1)Cl /2π as a function of l.
WMAP and PLANCK data could map universal fluctuations after removing the
dipole anisotropy (l = 1) and galactic and extragalactic contaminations. To extract
information from CMB, we must consider a cosmological model with fixed number
of parameters. Comparing each position of the peak and its height, one can deduce
the deep composition of the Universe. We represent the result of PLANCK in
Fig. 1.6. This spectrum was fitted by a model that considers a Universe with a
cosmological constant  and a cold dark component of matter (CDM ). The
position of the first peak determines m h2 . Combining the measurements of the
temperature power spectrum with a determination of the Hubble constant h, the
PLANCK team found the total mass density parameter m = 0.3175. The ratio
of amplitudes of the second-to-first Doppler peaks determines the baryonic density
parameter b = 0.048; the dark matter component is then DM = 0.2695.

1.9 Alternatives

To be complete, we should say that there exist some alternatives to the dark matter
scenario. Indeed, all the evidences of dark matter that we have discussed above
relied on the strong assumption that we know the law of gravity at all scales.
Is it possible that by changing the laws of gravity we can avoid this mysterious
component of the universe? In 1983, Mordehai Milgrom proposed to get rid of
dark matter altogether and to replace the known laws of gravitation with the so-
20 1 Introduction

Fig. 1.6 The sky observed by WMAP satellite (top) and composition of the Universe deduces by
PLANCK (bottom). NASA/WMAP Science Team, ESA

called MoND paradigm, short for “modified Newtonian dynamics.” The price to
be paid was to abandon general relativity, a theory that is particularly appealing to
many physicists because of its elegance and formal beauty. But there is no dogma
in physics, and history has taught us that our theories can always be refined and
improved. Milgrom proposed that the law of gravity is modified below a certain
acceleration, that is, when the gravitational force becomes very weak.
This proposal is very clever, because it bypasses one of the main difficulties of
theories of modified gravity: the easiest way to construct them is to introduce a new
distance scale, above which the gravitational force is modified from its characteristic
inverse square law. But observations tell us that modifications of gravity (or,
alternatively, the presence of dark matter) are observed on different scales in
different systems. MoND is surprisingly accurate on the scale of galaxies, and it
even addresses some mysterious correlations between properties of galaxies that
find no explanation in the standard dark matter paradigm. A few years later, Jacob
Bekenstein even embedded Milgrom’s proposal into a more relativistic theory called
References 21

TeVeS, for “tensor-vector-scalar” theory, promoting it from a phenomenological

model to a more fundamental theory. It is without doubt an interesting proposal
which attracted and is still attracting substantial interest.
There is a problem with these theories, however. Indeed, as soon as we move
from the scale of galaxies to the scale of galaxy clusters, they fail to reproduce the
observational data. Perhaps the biggest challenge to MoND-like theories, and one of
the most direct proofs of the existence of dark matter, is provided by systems like the
so-called Bullet Cluster discussed previously in Fig. 1.4, the system of two clusters
of galaxies that have recently collided, with one of the two passing through the
bigger one like a bullet (hence the name). To explain this observation with MoND-
like theories, one has to postulate the existence of additional matter, in the form
of massive neutrinos, for instance. But there is an ensign between the properties
required of the neutrinos and current data, and, in general, it is not very appealing
to require at the same time a modification of gravity and the presence of some form
of dark matter.
It is however still perfectly possible that it is through a modification of the laws of
gravity that we will be able to explain the motion and the shape of cosmic structures,
and it is important that part of the research effort of the scientific community should
focus on this possibility. Fortunately, as Bekenstein says,

The increasing sophistication of the measurements in [gravitational lensing and cosmology]

should eventually clearly distinguish between the various modified gravity theories, and
between each other and General Relativity.

The dark matter paradigm will remain a conjecture until we finally put our hands
on the particles by measuring their properties in our laboratories. Before describing
the techniques that have been devised so far to detect dark matter however, we need
to understand a little bit our thermal history and the behavior of the dark matter in
the primordial Universe.

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 Part I
The Primordial Universe

Et fiat lux. . . Not exactly. The Universe was not first dominated by radiation but by
matter in the form of an inflation, 13.8 billion years ago. This field is at the same
time responsible of the very fast inflationary phase (in the first 10−37 s), followed by
a reheating phase (until ∼ 10−20 s) and then a thermal phase the following 380,000
years, before matter dominates during 6 billion years. Then the expansion of the
universe is globally (but not locally) driven by the cosmological constant. In the
first chapter, we propose to review in detail the three phases preceding a thermal
world (inflation, thermalization, and reheating) and look at how dark matter can be
produced in each of these epochs. Then we will study the evolution of the primordial
plasma and how particles can be decoupled from it, neutrino as well as dark matter.
Inflation and Reheating [MP → TRH ]
2

Abstract

The inflation and reheating phases of the Universe concern a period where the
Universe changes very quickly from a vacuum/constant density domination to an
oscillation/matter domination and then a radiation domination. These transitions
are not only fast but also violent. We will analyze in detail each of these phases,
insisting on the possibility of producing dark matter before reaching the thermal
equilibrium. But before, one needs to understand the equation that will lead our
expanding Universe: the Hubble law.

2.1 The Context

When dealing with the physics of the primordial Universe, one needs to study
high energy processes in a specific space-time metric and evolution, which is
determined by its content (matter or radiation) through the Hubble scaling. The
interplay between radiation and matter also plays a role in the inflaton decay and
the reheating. These entanglements can seem at first extremely complex on the
formal side (Special and General Relativity, Quantum Field Theories, Unification,
etc.) as well as on the physics side (inflation, thermodynamics in an expanding
Universe, finite temperature effects, etc.). Nevertheless, paradoxically, the whole
framework can be summarize by a set of “Mann’s” equations: Boltzmann’s and
Friedmann’s equations are summarized in Eqs. (2.206–2.208) and can be directly
used by the researcher who already knows the context. They both can be derived
from General Relativity principles, as we show in Appendix A.2. However, a pure
Newtonian approach can also lead to the same system of equations (forgetting the
cosmological constant, which is a consequence of a non-flat space-time). This can be
seen as an intermediate step. The reader who is not directly interested in the General
Relativity approach can directly jump to Sects. 2.1.1 and 2.1.2 to have a Newtonian

© Springer Nature Switzerland AG 2021 25

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2_2
26 2 Inflation and Reheating [MP → TRH ]

approach of the Hubble law and Friedmann equations, respectively, whereas a

classical description of the Boltzmann equation can be found in Sect. 2.3.1.

2.1.1 The Hubble Law

According to cosmological principle, the Universe is homogeneous and isotropic

over distances of the order Gpc but becomes highly inhomogeneous for scale below
100 Mpc. It is also an observed fact that the Universe expands (and cools down)
following the Hubble law since around 13.8 billions of years. The Hubble law that
governs the expansion of the Universe is fundamental in many aspects. Indeed, it
governs the physics in the primordial Universe and the decoupling of particles from
the thermal bath, acts on the structure formation of galaxies and clusters of galaxies,
and determines the mean free path of relativistic particles in our present Universe.
Understanding this law is then a critical step to understand all the processes involved
in the evolution of the Universe. In this section, we will show how the basic
Newtonian concepts give a relatively clear picture of the phenomena.
In an expanding, homogeneous, and isotropic Universe, the relative velocities of
observers obey the Hubble law: the recession velocity of observer B with respect to
A, vAB is

vAB = H (t)rAB , (2.1)

where rAB is the relative position of B with respect to A.

Homogeneous and isotropic

Homogeneity and isotropy are fundamental because it means that our physics
and the results we obtain from our computation are valid in the whole
Universe. In other words, cosmology as a science can exist.1 We can then
distinguish a Homogeneous Universe, which is the same wherever you are,
and an Isotropic Universe, which looks the same on every angle you look
at (see Fig. 2.1). Notice that there exists a kind of anthropic link between
the two notions. Indeed, if a Universe is isotropic but not homogeneous, the
only possibility is that we are at the center of the Universe (Fig. 2.1, right).
This is an Aristotelian conception of the world. In other words, combining
the homogeneous and isotropic constraints tells us that we are not in a
unique position in the Universe. A space that is isotropic about every point
is necessary homogeneous.

(continued)

1 Hubble never had the Nobel prize because cosmology was not recognized as a scientific discipline
until his death in 1953.
2.1 The Context 27

It is possible to show that the unique law that respects a Homogeneous

and Isotropic Universe is in fact the Hubble law. Indeed suppose three
observers A, B, and C (Fig. 2.2) define the velocity function of an observer
i, v i = f i (r, t). The observer A sees the point C moving away following
vCA = f A (rAC , t), whereas the observer B sees C receding at a velocity
vCB = f B (rBC , t). The observer A then believes that B sees C moving
following its own law. Then for A, the velocity of C observed by B is the
velocity of C observed by himself (A) minus the velocity of B relative to
himself: vCB = f A (rAC , t) − f A (rAB , t). If the Universe is homogeneous
and isotropic, A and B should observe the same law in any direction. One
can then write f B = f A , and so, equalizing the velocity for both observers,
vCB = f (rBC , t) = f (rAC , t) − f (rAB , t). For C= A, on obtains f (rBA , t) =
−f (rAB , t). Noting rBC = rBA − rCA , one deduces f (rBA − rCA , t) =
f (rBA , t) − f (rCA , t). Developing f (x) = i Hi (t)x i , it is straightforward
to see that only the linear term survives this condition. We have just proven
that the homogeneity and isotropy of the Universe impose f to be a linear
function of r, f (r, t) = v(r, t) = H (t)r. The Hubble law is indeed the unique
law which respects a homogeneous and isotropic Universe.

This law means that for an object B situated at a distance twice as large as the
object C from the observer A, the velocity of B is twice larger than the one of C. If
we go back in time and consider constant velocity, it will take the same time for C

Homogeneous non-isotropic Universe : Isotropic non-homogeneous Universe :

uniform mass distribution, but preferred direction density gradient, isotropic for the dot observer,
not the star

Fig. 2.1 Left: a homogeneous but not isotropic Universe: the dot and the star see the same
Universe, but not isotropic ones because one direction is preferred. Right: An isotropic but non-
homogeneous Universe: The dot sees the same Universe whatever direction he looks at, but the
star does not see the same Universe as the dot
28 2 Inflation and Reheating [MP → TRH ]

to join A as B because even if being further away, its velocity compensates its larger
distance. Measuring the Hubble parameter nowadays (H0  67 km/s/Mpc), we can
compute the time t0 needed for B and C to join A, t0  1/H0  14.2 billions years,
which as a good approximation corresponds to the age of the Universe (13.8 billion
years). One can also understand why the Hubble law is valid only at scales above
100 Mpc. Indeed, at such distances, the velocity due to the expansion is of the order
of 6700 km/s, which is larger than the peculiar velocities of galaxies (of the order
of hundreds of km/s). For instance, the Andromeda galaxy, which is at a distance of
0.7 Mpc from the Milky Way (see Sect. 5.1), possesses a negative recession velocity:
Andromeda falls toward us at a velocity of around 300 km/s: at such low distance,
the Hubble expansion has little influence on the Newton law of attraction.
The computation is in fact a little bit more subtile as the Hubble parameter is not
constant and depends on the Universe content (matter, radiation, or cosmological
constant). This law is easy to interpret if one considers points on a radius of an
expanding sphere of radius a(t) as illustrated in Fig. 2.2. One can define rAB (t) =
χAB a(t), and then vAB = χAB ȧ(t). χAB is called the comoving coordinate. This
corresponds to a system of coordinates in which matter is at rest all the time, which

Fig. 2.2 Illustration of the A

Hubble law (see text for
details). Up: Universe with B
expansion parameter a(t) and VAB
comoving coordinate χ.
Bottom: velocity of a point C
viewed by observers A and B
a(t) C
BC

VAC

VBC

VAC
C

VAB
A B
2.1 The Context 29

does not evolve with time either and can be considered as an initial condition. Then
ȧ(t)
vAB = H (t)rAB ⇒ H (t) = . (2.2)
a(t)

2.1.2 The Friedmann Equations in a Dust Universe

2.1.2.1 The Hubble Parameter

It becomes interesting to solve the equation of motion of an observer B at a distant
rAB of A, in a homogeneous and isotropic space in expansion dominated by matter
(dust dominated2), i.e. respecting the Hubble law. If one defines ρ the mass density
that we assume homogeneous between A and B, one can write
 3
4π 3 a0
M= r ρ(t), ⇒ ρ(t) = ρ0 , (2.3)
3 a(t)

where r = rAB = χa(t), and M is the mass inside the radius r. On the other hand,
the Newtonian equation of motion for the observer B of mass mB is

GmB M 4πG 4πG a03

mB r̈ = mB χ ä(t) = − , ⇒ ä(t) = − ρ(t)a(t) = − ρ0 .
χ 2 a 2 (t) 3 3 a 2 (t)
(2.4)

Multiplying by ȧ on both sides of Eq. (2.4) and integrating with constant of

integration k, one obtains

1 ˙2 4πG a3
a = ρ0 0 + k, (2.5)
2 3 a(t)

ȧ(t) 2 8πG k 8π k
⇒ = H 2 (t) = ρ(t) + 2 = ρ(t) + 2 ,
a(t) 3 a (t) 3MP2 l a (t)

or

ρ(t) k
H 2 (t) = 2
+ 2 , (2.6)
3MP a (t)

2A matter dominated Universe is called a “dust” Universe. It is a Universe composed of matter

whose pressure p is negligible compared to its energy density ρ.
30 2 Inflation and Reheating [MP → TRH ]

MP l
where we used H (t) = ȧ(t)/a(t) and G = 1
MP2 l
= 1
8πMP2
, MP = √ 
8π
2.4 × 1018 GeV being the reduced Planck mass. This is the Friedmann equation
that we obtained in a pure Newtonian formalism and which determines entirely the
physics of an expanding mass dominated (“dust dominated”) Universe (up to the
cosmological constant). We will see that the equation is still valid in the General
Relativity framework, ρ being the density of energy.
In fact, Eq. (2.6) can also be seen as the conservation of energy equation, k being
the initial energy. Indeed, if we rewrite Eq. (2.6)

1 2 4πG ρ0 a03 1
ȧ − = ȧ 2 + V (a) = k, (2.7)
2 3a 2
we recognize a familiar expression we used to tool with in high school when dealing
with the classical rocket problem and escape velocity: for a given gravitational
potential (V ), what is the escape velocity (ȧ) needed to escape the attraction (k ≥ 0).
This velocity is also called the critical velocity. In a sense, a positive k means
that at the beginning of the expansion, the kinetic energy was larger than the
potential attraction: the Universe evolves in a continuous expansion with time. On
the opposite side, a negative k drives the Universe, dominated by the gravitational
attraction, to a future collapse on itself. In a relativistic approach, we will see
that k corresponds to the curvature of the Universe in a Friedmann–Robertson–
Walker metric. We can go further in the “rocket” analogy. Indeed, historically
speaking, astronomers gathered much more data concerning the expansion of the
Universe (especially from Hubble’s observations) before knowing about its mass
composition. Even in the 1960s, the value of the mass density in the Universe
was mainly given by approximations of the star component in the galaxies, and
the common accepted value was ρnow  10−31g cm−3 . The measurements of star
velocities were best known before having access to the Cosmological Microwave
Background. The problem was then the opposite: knowing the escape velocity
ȧ from the Hubble constant, what is the critical density ρc that allows such an
expansion. From Eq. (2.7), we obtain

3H 2
k ≥ 0 ⇒ ρ ≤ ρc ≡ , (2.8)
8πG
 a0 3
where we used ρ = ρ0 a . We can then rewrite Eq. (2.6)

4πG ρc a 2 ρ 4πG ρc a 2
k= 1− = [1 − (t)] , (2.9)
3 ρc 3
ρ
where (t) = ρc .
2.1 The Context 31

It is interesting to compute another useful cosmological parameter called the

deceleration parameter q(t)

1 ä(t)
q(t) = − . (2.10)
H 2(t) a(t)

Historically speaking, this was the main source of information on the matter
content of the Universe before the precise measurement of the CMB and its
anisotropies. From Eq. (2.4), one obtains

4πG 1 ρ(t) 1
q(t) = ρ(t) = = (t). (2.11)
3H 2(t) 2 ρc (t) 2

In the early 60s until the 80s, the matter content of the present Universe was
limited by the measurement of q0 = q(t0 )  2.5 from the observation of
the deceleration of nearby clusters of galaxies. This limit gives (t0 )  5, far
above the present limit given by the measurement of the anisotropies of the CMB
( CMB (t0 )  0.3).

2.1.2.2 The Continuity Equation

To compute any physical processes in the early Universe, we need to have the
expression of the radius, or scale factor a as a function of time, which means solving
Eq. (2.6). For that, we first need an expression of ρ as a function of a and t. This
is obtained through the continuity equation, tightly related to the conservation of
energy. In the case of a Universe dominated by dust, which means a non-relativistic
matter of density ρm , the density is given by

M  3
0 a0
ρm ∝ , or ρm = ρm , (2.12)
a3 a
which gives
 a 3 ȧ
0
ρ̇m = −3ρm
0
= −3Hρm , (2.13)
a a
implying

ρ̇m + 3Hρm = 0. (2.14)

Notice that the right-hand side of the previous equation is null because we
considered an isolated system. If energy is injected or even lost (by the decay of
a particle, for instance), “0” should be replaced by the source term.
32 2 Inflation and Reheating [MP → TRH ]

2.1.3 The Friedmann Equations in a Radiative Universe

The treatment described in the previous section is elegant but, being Newtonian,
cannot deal with a Universe dominated by relativistic species (as it is the case during
the reheating phase, dominated by a gas with pressure = 1/3 of the energy density,
see Eq. (2.204) for the precise calculation) or by a cosmological constant (as it is
the case nowadays, corresponding to an effective negative pressure). It also does not
take into account the curvature of space due to the gravitational deformation of its
metric. Moreover, the notion of absolute time is absent, whereas lengths depend on
the referential. We can, however, also describe a Universe dominated by radiative
(relativistic) species from a classical perspective.
A system of relativistic particles is fundamentally different from a system of
static (massive) particles due to its pressure. Indeed, if we consider a system of
dust made of particles “i” and compress it, its total energy is not modified, as we
do not create or extract particles from the system. Their individual energies (only
given by their mass because they do not move) are constant. Thus, the total energy,
Em , is given by Em = ρm × V = i mi , in a volume V . This is not the case
anymore for relativistic particles because they exert a pressure p in the system. In
the same way that a bicycle pump heats when we compress its piston because of
the increasing pressure of the air inside the pump, we understand quite well that
the same phenomenon appears in an expanding Universe: the pressure does work,
modifying the internal energy of the system:

dE = −pdV . (2.15)

The negative sign reflects the fact that the internal energy decreases for an
increasing volume and vice versa (the same way than a bicycle pump.3) To imagine
physically the origin of such energy, we need to understand that in a closed system,
which is the case of the whole Universe as no particle can escape from it, diluting
the space will decrease the energy of each relativistic particle and thus of the entire
system (by redshift,4 for instance). The pressure is just defined as the proportionality
coefficient between the energy lost and the volume gained. In other words, “how
much energy I lose per unit of volume I gain.” It is then common and natural to
express it as a function of the total energy density

p = wρ, (2.16)

w being a constant depending on the nature of the content of Universe (0 for dust,
1/3 for a relativistic gas, −1 for the cosmological constant, etc., see Sect. 2.3.1.2).

3A more complete statement can be found in Sect. 3.1.2.

4 Theredshift is the propensity for a relativistic particle to have its wavelength changed in an
expanding Universe or if it is emitted by a moving source. See Sect. 2.1.4.3 for more details.
2.1 The Context 33

V being proportional to a 3 , we can rewrite Eq. (2.15) for a gas with density of energy
ρR

dρR = −3(ρR + p)d ln a (2.17)

or

ρ̇R + 3H (ρR + p) = 0. (2.18)

Notice that this equation is not only valid for a relativistic gas of particles but
also for any kind of energy density content of the Universe, as long as one knows the
relation between the pressure p and the density ρ. We also considered a completely
isolated system. If, by any means, the Universe can be populated by another source
of energy, the “0” on the right side should be replaced by its density rate of injection.
Equation (2.18) should then be replaced by

ρ̇i + 3H (ρi + pi ) = 0. (2.19)

i

At the end, it seems that we have a perfect set of equations, and people can ask
“why the need to go further toward a complex General Relativity approach?” First
of all, it is true that the Newton theory being a limit of Einstein theory in a flat space-
time, away from extreme deformations (like a black hole or the very early Universe)
the classical approach is valid. However, for a curved space-time, the integration
constant k has a deep meaning and can be measured. There were two periods in our
history where the curvature was dominating the Universe: nowadays and during the
violent expansion phase of inflation. In both cases, a General Relativity treatment
is not at all a refinement but an obligation, as we will see when dealing with the
inflationary sector. Indeed, for matter or radiation, the energy–momentum is well
defined and we can treat them classically. For a less conventional potential, in a
curved space-time, the metric and energy content should be analyzed with care as
we detailed in Appendix A.2. We will present in the following section the basics one
needs to understand to deal with the Friedmann equations in a General Relativity
framework. Let us begin first with the difference between the metric in a flat space
and in a curved space.

2.1.4 The Friedmann–Lemaitre–Robertson–Walker (FLRW) Metric

2.1.4.1 Generalities
The construction of a general space-time metric of the Universe is based on
the hypothesis that the Universe is homogenous and isotropic. This is called the
34 2 Inflation and Reheating [MP → TRH ]

cosmological principle and is empirically justified on scales larger than 100 Mpc.
So one should first build a 3D spatial metric plunged in a 4D space. Using Cartesian
coordinates (x, y, z, w) but replacing (x, y, z) by spherical coordinates (r, θ, φ), we
have for the infinitesimal space interval dl

dl 2 = dρ 2 + ρ 2 d 2
+ dw2 ,

where d 2 = dθ 2 + sin2 θ dφ 2 is shorthand for the solid angle. For a positive

curvature, one can also write

x 2 + y 2 + z2 + w 2 = ρ 2 + w 2 = R 2 ,

where R is the curvature radius, independent of the position (x, y, z) by the

hypothesis of homogeneity. We then have

ρdρ + wdw = 0.

Therefore,

ρ2 2 ρ2
dw2 = dρ = 2 dρ 2
w 2 R − ρ2

and so

ρ 2 dρ 2
dl 2 = dρ 2 + + ρ2d 2
R2 − ρ 2

giving

dρ 2
dl 2 = + ρ2d 2
.
1 − (ρ/R)2

This is a homogeneous, isotropic 3D space of (positive) curvature 1/R 2 . Notice

that if R → ∞, we recover the Euclidean 3D metric dl 2 = dρ 2 + ρ 2 d 2 . Setting
r = Rρ , we can express

dr 2
dl 2 = R 2 + r 2d 2
1 − r2

with 0 ≤ r ≤ 1. This is the expression of the metric in a 3D sphere, corresponding

to a positive curvature. The coordinates (r, θ, φ) will play an important role as
“comoving” coordinate we discussed in the previous section. This corresponds to
the metric in a unit 3D-sphere or, in other words, in sphere of radius R = 1 constant
with time. Negative and zero curvatures are also possible. For the negative case, the
2.1 The Context 35

radius condition should be written as

dl 2 = dρ 2 + ρ 2 d 2
− dw2 and ρ 2 − w2 = −R 2 .

This is the equivalent of the hyperboloid surfaces in 2D. We then obtain


dr 2
dl 2 = R 2 + r 2d 2
.
1 + r2

We can then combine all the possible curvatures with the generic expression

dr 2
dl 2 = R 2 + r 2d 2
1 − kr 2

with k = +1 for a positive curvature, k = 0 for a flat one, and k = −1 for a negative
curvature. In general we must allow for R to be an arbitrary function of time R(t)
(not position since that would destroy homogeneity). The coordinates (r, θ , φ) are
called comoving coordinates, in the sense that, in this system or coordinates, even
if the Universe expands, the distance between two fixed observers does not change.
In this referential, only proper movements have dynamic. If we apply the definition
of the invariant metric of special relativity defined in Eq. (A.11)

ds 2 = c2 dt 2 − dl 2 ,

we obtain, in a curved space,


dr 2
ds = c dt − R (t)
2 2 2 2
+ r 2 dθ 2 + r 2 sin2 θ dφ 2 . (2.20)
1 − kr 2

This is the Friedmann–Robertson–Walker metric. It was first derived by Fried-

mann in 1922 and then more generally by Robertson and Walker in 1935. It applies
to any metric theory of gravity, not just General Relativity.

2.1.4.2 Geometry of the Universe

We can analyze in more detail the properties of this metric. We have just seen that
we have three cases:

• k = 1: positive curvature, closed Universe,

• k = 0: zero curvature, flat Universe (flat space, not flat space-time), and
• k = −1: negative curvature, open Universe.
36 2 Inflation and Reheating [MP → TRH ]

An alternative form of the metric is often useful. For k = 1, setting r = sin χ,

the interval becomes

ds 2 = c2 dt 2 − R 2 (t)(dχ 2 + sin2 χd 2
),

which can be written by generalization

ds 2 = c2 dt 2 − R 2 (t)(dχ 2 + Sk2 (χ)d 2

),

where

S1 (χ) = sin χ, S0 (χ) = χ, S−1 (χ) = sinh χ. (2.21)

This convention has the great advantage of describing the metric in a “flat” way
for d = 0 (in a straight geodesic line, as the line of sight of a photon) as we will
see when computing the redshift or expressing the Hubble law.

2.1.4.3 Redshift
This convention of variables (t, χ, ) is less obvious to understand the physics lying
beyond the equations, but much more practical to deal with formal calculations.
This becomes clear when one needs to compute processes like the redshift, for
example. Indeed, the wavelength of light from astronomical sources is a crucial,
easily measured observable. Consider two pulses of light emitted at times t = te
and t = te + δte by an object at χ toward an observer at the origin who picks them
up at t = t0 and t = t0 + δt0 . The light is emitted in a solid angle d =0 and follows
a geodesic (ds = 0).
For photons traveling toward the origin, since ds = 0

cdt = R(t)dχ.

Therefore, because χ represents a comoving coordinate, which is fixed and

independent of time,
 t0  t0 +δt0
cdt cdt
χ= = .
te R(t) te +δte R(t)

Subtracting the first integral from the second,

 t0 +δt0  te +δte
cdt cdt
− = 0.
t0 R(t) te R(t)
2.1 The Context 37

For small intervals, R(t) is almost constant,5 so

δt0 δte
= .
R(t0 ) R(te )

Therefore the redshift z, which is defined as the relative difference between the
observed wavelength and the emitted one (z = λ0λ−λe
e
), is given by

λ0 νe δt0 R(t0 )
1+z = = = = .
λe ν0 δte R(te )

2.1.4.4 The Hubble Law

We can now try to find the Hubble law expression in a general space-time metric.
The universal “fluid” (=galaxies) is at rest in comoving coordinates χ, θ , and
φ. Expansion of the Universe is encoded in the scale factor R(t). Consider the
instantaneous physical distance (or proper distance) to a galaxy at radius χ
 χ
dP = R(t)dχ = R(t)χ.
0

Since χ is fixed, the rate of recession of the galaxy is

d Ṙ
v= (dP ) = Ṙχ = dP .
dt R
Identifying

Ṙ ȧ
H (t) = = ,
R a

where we have defined a dimensionless scale factor R(t) = R0 a(t), R0 being the
present Universe radius, we then have

v = H (t)dP ,

which is the Hubble law, while H (t) is the Hubble “constant” = H (t0 ) = H0 today.
Hubble’s law is thus a direct outcome of homogeneity and isotropy and has the same
expression in a curved space than in a flat space.

5 Whereas te and t0 are usually very spaced: R(te ) = R(t0 ).

38 2 Inflation and Reheating [MP → TRH ]

2.1.4.5 Measuring the Size of the Universe

This is a little exercise I use to give to master students for oral exam:
Knowing that the Universe began to be dominated by a cosmological constant at
t  10 Gyrs, after being dominated by matter, what is the size of the Universe?
Naively speaking, students want to answer “13.8 billion light-years,” forgetting
that, since the emission of the CMB radiation, which indeed occurred 13.8 billion
years ago6 at a time t = tCMB , the Universe has expanded. The point source of
the radiowave we receive now (at time t = t0 ) is presently at a distance further
than ctCMB . To compute the distance of this point source, one needs to compute the
distance traveled by the light, while the Universe is expanding continuously. The
exercise is quite similar to the one with an ant on an inflating balloon, who tried to
reach the north pole while following a meridian, like the light follows a geodesic on
its way from the CMB to us. But the curvature is not the point here. The important
fact is that the structure of the space dilates with time.

Exercise Considering an ant walking on the meridian of an inflating balloon,

walking at v=1 cm/s toward the north pole of the balloon which is 2 cm away. The
balloon inflate at a rate of H = 0.4 cm/s per centimeters. Show that the time to reach
the north pole is given by

ln(5)
t=  4 s. (2.22)
0.4
Hint: during a time dt, the distance that the ant still needs to travel is

d =  × H × dt − vdt. (2.23)

Solving the equation,  = 0 gives you the time of the travel. Compute also the
distance of the point of origin when the ant arrived to the north pole.

We can then do the same exercise for our Universe. Depending on its composi-
tion, the scaling factor a evolves differently with time. For instance, in the case of a
matter or radiation dominated Universe,
 α
t
a(t) ∝ t α = a0 (2.24)
t0

6 Or 13.8 Gyrs−380,000 years to be more precise because the CMB took place 380,000 years after
the Big Bang as we will discuss in Sect. 3.3.4.
2.1 The Context 39

with α = 23 for a matter domination and α = 12 for a radiation domination. We can

then compute the distance λ0i traveled by the light between ti and t0 :
 0 ct0
R dχ = R0 a(t) dχ = cdt ⇒ R0 χi0 = R0 dχ = (2.25)
i a0 (1 − α)
ct0
⇒ λ0i = R0 a0 χi0  , (2.26)
1−α

where we have supposed, in the last equation, t0 ti . We can then write the distance
λ traveled by the light from ti t0 to t0

λ = 3ct0 [matter domination] ; λ = 2ct0 [radiation domination]. (2.27)

Exercise Show that in the case of a cosmological constant  dominated Universe

(in other words, with a constant Hubble parameter H0 ), the distance traveled by the
light, from ti to t0 is

r −1 a0
λ = c(t0 − ti ) , with r = [ domination]. (2.28)
ln(r) ai

As a first approximation, if we consider that the Universe was dominated by the

radiation during almost 380,000 years, then 13.8 billion years of matter domination,
we can write
 
λt ot = c × 2 × 380,000 + 3 × 13.8 × 109 = 41.4 × 109 light − years (2.29)

for the radius of the observable Universe or a diameter of ∼83 billion light-years.
Fortunately, this naive approach is quite ok and gives a result not so far from the
reality, which is 92 billion light-years. For the exact calculation, one needs to take
into account 2 important points: the Universe has been dominated by a cosmological
constant at a redshift z = 0.326 and the evolution between the two dominations is
smooth. As it is almost always the case, it is easiest to integrate the free path with
respect to the scale factor a and not the time t. We compute the exact solution in
Sect. 2.1.7.2.

2.1.5 Friedmann’s Equation in General Relativity

After having studied in detail the subtleties of a curved space-time metric, we

can generalize it to a metric with local deformations. This is the aim of General
Relativity, where the gravity is the source of local deformations of space-time.
40 2 Inflation and Reheating [MP → TRH ]

The reader will easily understand that it is impossible to give a complete and fair
treatment of such a complex subject in a book devoted on the dark Universe. A lot
of fantastic textbooks exist on the subject. I personally love the D’Inverno one [1]
for its way to avoid complex machinery, keeping concentrated on the essential, and
the Hartle book [2] for some refinements. The subject is developed in greater detail
in the book of Weinberg [3]. In this section, we will give the necessary tools needed
to understand where and how do the fundamental Friedmann equations are affected
by a locally curved metric. To find the Friedmann solution to Einstein equations, we
will just need (obviously) the Einstein equations (A.88) and the Robertson–Walker
metric (2.20):

1
Gμν = R μν − Rg μν = 8πG T μν + g μν (2.30)
2

dr 2
ds 2 = c2 dt 2 − R 2 (t) + r 2 dθ 2 + r 2 sin2 θ dφ 2 , (2.31)
1 − kr 2

where G = 1/MP2 l is the gravitational coupling (MP l is the Planck mass = 1.22 ×
1019 GeV) and  the cosmological constant. I can understand the need to clarify
the origin of Eq. (2.30), that is, the reason why we devoted a complete appendix on
the subject for any student who feels the need to recover the Einstein equations of
General Relativity. Even if most readers are surely familiar with these equations,
Appendix A.2 is available for the untrained reader. However, for the study of a
primordial Universe with an (almost) flat geometry, I think the reader can accept,
at a first step, this set of equations that will lead to the fundamental Friedmann
relations.

2.1.5.1 The Friedmann Equations

Concretely speaking, the idea is to write the set of equations (2.30) for all possible
values of (μ, ν) parameters, given a definite metric like the one given by (2.31).
Naively speaking one could think that 16 equations should be taken into account,
but in fact, because of the homogeneous and isotropic conditions, only two relations
are of interest for us: the G00 and Gii ones.7 Using the Robertson–Walker metric
normalized to c = 1,

R 2 (t)
g00 = gt t = 1 ; grr = − ; gθθ = −R 2 (t)r 2 ; gφφ = −R 2 (t)r 2 sin2 θ,
1 − kr 2
(2.32)

7 Infact, all the Gii conditions are the same due to the isotropic principle: none of the 3 directions
can be distinguished from the others.
2.1 The Context 41

μ 
we obtain the following Christoffel symbols:8 νρ = 1 μα
2g ∂ν gαρ + ∂ρ gαν

− ∂α gνρ

R Ṙ
t
rr = ; θθ
t
= r 2 R Ṙ; φφ
t
= r 2 sin2 θ R Ṙ;
1 − kr 2
kr Ṙ
r
rr = ; θθ
r
= −r(1 − kr 2 )φφ
r
= −r(1 − kr 2 ) sin2 θ ; trr = ;
1 − kr 2 R
1 Ṙ
θ
rθ = ; φφ
θ
= − cos θ sin θ ; tθθ =
r R
φ 1 φ φ Ṙ
rφ = ; θφ = cot θ ; t φ = , (2.33)
r R
all the other components vanishing.

Exercise Derive the above expression, and compare the Christoffel symbols for a
Cartesian metric.

α
We can then deduce the Riemann and the Ricci tensors Rμβν and Rμν = Rμαν
α :

α
Rμβν = σβ
α σ
μν − σα ν μβ
σ
+ ∂β μν
α
− ∂ν μβ
α
(2.34)

R̈ R̈ (Ṙ)2 k
Rμν = Rμαν
α
⇒ R00 = Rt t = −3 ; R = Rμμ = −6 − 6 2 − 6 2 .
R R R R

Exercise With the help of the results (2.33), prove that Rtt t t = 0, Rtrrt = − R̈
R,
φ
Rtθθ t = − R̈
R , Rt φt = − R , and then deduce Eq. (2.34). Do the same with the
R̈

Ṙ) +2k 2
other components of the Ricci tensor to show that Rrr = R R̈+2( , Rθθ =
   1−kr 2
r R R̈ + 2(Ṙ) + 2k and Rφφ = r sin θ R R̈ + 2(Ṙ) + 2k , giving the Ricci
2 2 2 2 2

scalar (2.34).

Combining Eqs. (2.30), (2.32), and (2.34), we obtain

 2
1 Ṙ k
G00 = Gt t = Rt t − Rgt t = 3 +3 = 8πGTt t + gt t = 8πGρ + 
2 R R2
(2.35)

8 See Appendix A.2.4 for details.

42 2 Inflation and Reheating [MP → TRH ]

or

 2
Ṙ 8πG k 
= H2 = ρ− 2 + , (2.36)
R 3 R 3

where we have considered the stress-energy tensor of a perfect fluid of density of

energy ρ (see Eq. A.96), as it should be the case in the early Universe.9 This first
time-like component of the Einstein equations is in fact the general expression of the
Hubble law we obtained by a Newtonian approach in Eq. (2.6), where the “k” term
is directly linked to the metric (curvature) of the space-time, and a new term, the
cosmological constant, is present. Even if not necessary, this term was at first, not
included by Einstein himself in his solutions. But observations of certain types of
supernovae in 1995 confirmed the presence of the cosmological constant , which
in fact dominates the expansion of the Universe since almost 4 billion years.

2.1.5.2 The Deceleration Equation

To be complete, we have now to compute the Gii component of the Einstein
equations.
If one considers the case of a perfect fluid of energy density ρ and pressure P ,
we can use the expression (A.97) for T μν

dx μ dx ν
T μν = (ρ + P ) − g μν P , (2.37)
dτ dτ
and noticing that (we let the reader prove it)
  2

R̈ Ṙ k
Gii = −2 − − 2 gii ,
R R R

we obtain
  2

R̈ Ṙ k dxi dxi
−2 − − 2 gii = 8πG (ρ + P ) − gii P + gii .
R R R dτ dτ

9 During the phase of inflation, the energy–momentum tensor should depend on the dynamics of
the scalar inflaton as we will see in Sect. 2.2.
2.1 The Context 43

If we place ourselves in the rest frame (comoving frame) of the perfect fluid
dxφ
dxr
dτ = dx
dτ = dτ = 0, we then can write
θ 10

 2
R̈ Ṙ k
2 + + = −8πGP + . (2.38)
R R R2

It is not easy to understand the meaning of Eq. (2.38) by itself. However,

eliminating the Hubble rate Ṙ
R given by Eq. (2.36), we obtain an equation for the
acceleration.

R̈ 4πG 
=− (ρ + 3P ) + , (2.39)
R 3 3
also called the the Raychaudhuri equation.

This equation is in fact a deceleration equation, as R̈ is always negative if one

neglects the cosmological constant. It is interesting to go back a century ago, when
Einstein tries to find a static solution of his General Relativity equations (2.36) and
(2.39). Asking for R̈ = 0, we obtain  = 4πG(ρ + 3P ) = 4πGρ for a dust matter,
and Ṙ = 0 implies then 4πGρ = Rk2 . In other words, the conditions

k
 = 4πGρ = (2.40)
R2
are the conditions for a static Universe, which should not be flat. Eddington
already noticed that this equality represents a curious Universe where a dynamical
variable (ρ) should be exactly equal to a constant of Nature () to ensure a static
Universe. Moreover, one should also remark that modifying slightly locally the
density of matter will render all the system unstable. On the other hand, supposing
a homogeneous density of matter (as proposed by Friedmann and Lemaitre) to
solve the cosmological principle of Einstein was not a priori so obvious, especially
when we observe such an inhomogeneous sky every nights. The main result of the
Hubble discovery was that the Universe was in fact homogeneous and isotropic
at largest scale, and this was far to be obvious in the 20s. The deceleration
was the parameter measured by astrophysicists, giving constraints to the density
of the Universe nowadays, before the discovery of the Cosmological Microwave
Background (CMB). Indeed, the beauty of this equation is the absence of the
curvature k, rendering it easily testable. To be more precise, physicists were using

10 There are subtleties in this argument. One way to see it is to think that in the rest frame of a
perfect fluid, a particle with a velocity dx θ
dτ , for instance, will have a counterpart of another particle
dxθ
of velocity − dτ , canceling the velocity part of the stress–energy tensor Tii (2.37).
44 2 Inflation and Reheating [MP → TRH ]

the measurement of the deceleration parameter


R̈R R̈ 1
q=− =− , (2.41)
(Ṙ)2 R H2

the minus sign having been added in the definition to render q positive because
historically physicists believed the Universe should contract under the gravitational
forces. This was of course much before the discovery of the cosmological constant.
Another combination of (2.38) and (2.36) by elimination of the term R̈R gives

ρ̇ = −3H (ρ + P ), (2.42)

which is a generalization of the energy conservation equations (2.14) and (2.18)

Exercise Recover (2.42) from the energy conservation equation Dμ T μν = 0.

In summary, the set of Friedmann equations that we will need to study the
evolution of the early Universe can be expressed as


H2 = 8πG
3 ρ − k
R2
+ 
3 (2.43)
ρ̇ = −3H (ρ + P ) = −3H (1 + w)ρ with P = wρ.

Remark To obtain this set of equations, we have supposed a “stable” source of

energy density ρ. If there exist decay processes with a width  (which is the case for
the inflaton), one needs to add a − × ρ term on the right-hand side of Eq. (2.42). In
the same manner, if a source of energy is injected in the volume under consideration,
a term of the form + × ρsource should be included. We also want to point out that
the relation between pressure and energy density, P = wρ, also called equation of
state, has a classical physical interpretation called the Laplace law as we show in
Appendix A.5 and more specifically in Eq. (A.115).

Teaching Friedmann equations

When I give lectures at a bachelor level, or when I know that students do
not have sufficient training in General Relativity, I present things differently,
the goal always being to arrive at Friedmann–Lemaitre’s equations. First of
all, the virial theorem gives, by equalizing, the mean kinetic energy and the
mean potential energy in a (non-relativistic) gas system of density ρ, made of

(continued)
2.1 The Context 45

identical particles of masses m:

1 Gmρ × 4π
3 R 
3
m v2  = , (2.44)
2 R

or, getting rid of the mean,

 2
Ṙ 8πρ 
H2 = =G + , (2.45)
R 3 3

where we added a cosmological term , as Einstein did to counterbalance

the gravitational attraction and keep a static Universe. The internal energy
U = ρV of the gas at a pressure P respects

dR
dU = dρV + 3ρ V = −P dV ⇒ ρ̇ + 3H (ρ + P ) = 0. (2.46)
R
Deriving the expression for H (2.45), using (2.46), one obtains

R̈ 4π(ρ + 3P ) 
= −G + . (2.47)
R 3 3
In the classical convention we use in this textbook, writing R = a × R0
(R0 being the present radius of the Universe) and G = 1 2 , MP = 2.4 ×
8πMP
1018 GeV, we finally have

 2
ȧ ρ 
= H2 = 2
+ ;
a 3MP 3
ρ̇ + 3H (ρ + P ) = 0;
ä ρ + 3P 
=− 2
+ . (2.48)
a 6MP 3

An interesting point, noted of course by Einstein, is that, if you want to

force a static Universe, the condition ä = 0 requires
ρ
= . (2.49)
2MP2

(continued)
46 2 Inflation and Reheating [MP → TRH ]

Notice also that we recover in Eq. (2.48) the fact that the attractive force is
proportional to 1/a 2 and the repulsive one, driven by , to a: an object twice
as far as an observer will be 4 times less attracted by the matter, but twice
more repulsed by the cosmological constant term . This can give a hint to
the student why at small distances, the attractive gravity dominates and we can
apply Newton laws, whereas at large scales, one needs to look at the forces
induced by the cosmological term.

Exercise Supposing that in the Universe today, matter dominates over radiation,
compute the value of  needed to have a static Universe (ä = 0). Compute then the
Hubble constant and compare with its initial value after the inflation. What can you
conclude?

2.1.5.3 The Cosmological Constant Case

The other possibility to consider is a Universe dominated by the cosmological
constant . Far from being artificial or an exercise, this is in fact the present situation
we observe since almost 4 billion years. If we look at Eq. (2.30), we see that it
corresponds to an energy–momentum tensor Tμν  = g , or in other words, in a
μν
flat metric (k = 0) from (A.96):
⎛ ⎞ ⎛ ⎞
 0 0 0 ρ 0 0 0
⎜0 − 0 0 ⎟ ⎜0 P 0 0⎟

8πG Tμν = gμν  = ⎜
⎝0
⎟ = 8πG ⎜ ⎟, (2.50)
0 − 0 ⎠ ⎝0 0 P 0⎠
0 0 0 − 0 0 0 P

which corresponds to an equation of state P = wρ with w = −1, where ρ is


constant and can be identified to 8πG , and ä > 0: we are indeed in the presence of a
Universe in an accelerating expansion. Notice that Lemaitre in 1934 remarked that
this vacuum energy density is not changed by a velocity transformation. Indeed, Tμν 

is proportional to the Minkowski metric tensor ημν and therefore is unchanged by

a Lorentz transformation. In other words, ρ does not define a preferred frame of
motion, and it is not from the same nature that the ether in special relativity.

2.1.6 Another Look on the Hubble Expansion

We can take a more detailed look to the Hubble expansion rate predicted by the
Friedmann equations
 2
ȧ 8πG k 
H =2
= ρ− 2
+ (2.51)
a 3 2
a R0 3
2.1 The Context 47

with k the curvature factor (0, ±1). Compared to the formulae of the preceding
section, a is a dimensionless parameter defined as R(t) = R0 × a(t), R0 being
the present time radius of the Universe (and then a ≤ 1), and ρ = ρr + ρm
is the energy density of radiation and matter in the Universe. The presence of
the cosmological constant , which appeared in the 90s through observational
measurements, changed completely the fate of the Universe. Indeed, without it, the
Universe had 3 options:

• If k = −1, the expansion slows down all the time, but without stopping.
• If k = 0, the expansion slows down and stops at t = ∞.
• If k = +1, the expansion slows down, stops, and then turns over to contraction.

All these different destinies are related by a slowing down Universe. Indeed, in
all the cases, the ρ (∝ 1/a 3 if matter dominated and 1/a 4 if radiation dominated, see
Sect. 3.1.6) term or curvature terms were decreasing as a is increasing. The presence
of a positive  term inverts all the process, giving nowadays an accelerating
expanding Universe.
If the Universe is matter dominated, ρ = ρ0 /a 3 , ρ0 being the density of matter
today. We can also define

3H 2 ρM ρR
ρc = , M = , R = ,
8πG ρc ρc
k 
k =− and  = , (2.52)
R 2 H02 3H02

where ρc is the critical density. It corresponds to the density we would expect if the
Universe is flat (k = 0). Any deviation on ρc can be interpreted as a measurement of
the space curvature. Nowadays, for a value of H0 = 100h km s−1 Mpc−1 = 2.13 ×
10−42 h GeV (corresponding to the speed of a galaxy 1 Mpc away, h being measured
to be in 2019, h  0.74, whereas 2018 Planck measurement gave h  0.68), one
obtains, reminding that G = 1/MP2 l ,

3H02
ρc0 = = 1.05 × 10−5 h2 GeV cm−3 = 1.88h2 × 10−29 g cm−3 . (2.53)
8πG
The CMB measurements as well as the type Ia supernovae observations seem
to favor a flat Universe with a matter component composing a fraction M =
ρM /ρc0 = 0.3 of the critical density. Notice that we can define densities ρk and
ρ to have a uniform definition of H

8πG
H2 = (ρR + ρM + ρk + ρ )
3
3k 
⇒ ρk = − , ρ = . (2.54)
8πG R 2 8πG
48 2 Inflation and Reheating [MP → TRH ]

One can then compute which fraction of this matter is composed of baryonic
(1 GeV as it is mainly proton/hydrogenic clouds) matter from the data given by
WMAP or PLANCK of the ratio of baryon to photon number density [4]
nB
η= = 6.12 × 10−10. (2.55)
nγ

Knowing the temperature of photons nowadays (2.725 K), one can deduce the
number density of photon Eq. (3.42): 411 cm−3 , and so the number density of
baryon is nb = η×nγ = 2.4×10−7cm−3 . Considering 1 GeV baryon, and h = 0.71,
from the value of ρc0 computed in (2.53), we deduce

1 GeV × nb
b =  0.044. (2.56)
ρc0

As we notice, this value is far to be sufficient to account for the matter component
of the Universe ( M = 0.3), which implies the need for a dark component.
Substituting this into the Friedmann equation, Eq. (2.51), and replacing a with
a = 1/(1 + z), where z is the redshift defined in Sect. 2.1.4.3, gives
 
H 2 (z) = H02 0
R (1 + z)
4
+ 0
M (1 + z)
3
+ 0
k (1 + z)2 + 0
 , (2.57)

or as function of a,


0 −4 0 −3 0 −2
H (a) = H0 Ra + Ma + ka + 0
 (2.58)

Exercise Considering a Universe dominated by matter and a cosmological constant,

show that the age of the Universe can be written as
 1 √
a
tU =  da. (2.59)
0 H0 0
M + a
3

Deduce the age of the Universe for the Einstein–de Sitter model ( 0M = 1, 0
 =
0) and the Universe corresponding to the observed content ( 0M = 0.311, 0
 =
0.689).
2.1 The Context 49

As we already discussed in Sect. 2.1.5.2, before having access to the CMB

measurement, the deceleration parameter was the best mean to compute the relic
abundance of matter. Indeed, from (2.41), we can deduce in a flat space,

1
q0 = 0
M − 0
. (2.60)
2

It was then common to extract 0M = 2q0 , q0 being measured by astrophysicists

(in the hypothesis of null cosmological constant).
In 1932, Einstein and de Sitter proposed a model with M = 1 and R =
k =  = 0. The simple measurement of the sign of q0 could distinguish their
proposition from models dominated by  . This became known as the Einstein–
de Sitter cosmological model. It is interesting to see how Einstein, following the
experimental discovery of a non-static Universe, totally rejected the idea of the
cosmological constant that he himself had introduced. One can feel his bitterness
in a letter written in 1947 to Georges Lemaitre:
“ Since I have introduced this term, I had always a bad conscience. . . I cannot help
to feel it strongly and I am unable to believe that such an ugly thing should be
realized in nature.”
It is also interesting to see that in 1931, Georges Lemaitre proposed a model
where a part of the energy density is taken by the cosmological constant. The
main advantage is that, at this time, the age of the Earth was measured to be
tEarth  1.6 × 109 years (from radioactive decay sources), whereas the expansion
timescale was believed to be H0−1 = 1.8 × 109 years. Lemaitre concluded that
the Einstein–de Sitter model predicting an age of a matter dominated Universe
t = 23 H0−1 would conflict with the radioactive decay age. This is the reason why he
proposed to introduce the cosmological constant. In a letter to Einstein dated July
30, 1947, he wrote
“that the cosmological constant is necessary to get a time-scale of evolution which
would definitely clear out from the dangerous limit imposed by the known duration
of geodesic ages,”
letter to which Einstein replied that “it offers a possibility, it may even be the right
one.” Einstein was finally even thinking to reintroduce the cosmological constant
to address the issue of the age of the Earth. The proposition of Lemaitre is easy
to understand by a quick look to Eq. (2.39). We clearly see that the role of the
matter is to decelerate the Universe. The higher is the recession velocity, the further
we are from the asymptotic limit, the younger is the Universe, whereas for a
cosmological constant Universe, the higher is the recession velocity, the older the
Universe is. Both phenomena counterbalance, which explains why adding a little
bit of cosmological constant in ages the Universe. Paradoxically, more precise
measurements of H0 in the 1950s will once again bring the age of the Earth within
the limits of the age of the Universe, once again rendering the cosmological constant
obsolete, until the revolution of 1998, which will bring  back into the limelight.
50 2 Inflation and Reheating [MP → TRH ]

Exercise From Eq. (2.57), and taking M = 0.3,  = 0.7, R = k = 0, show

that z , the redshift when the cosmological constant density began to dominate the
evolution of the Universe is z = 0.32 (∼9 Gyrs). Show that the condition R̈ = 0
from Eq. (2.39) gives z = 0.67, corresponding age of the Universe is 6.4 Gyrs.
This phase of the Universe was called by Lemaitre, the “hoovering Universe”,
when, he believed, the slowdown of the expansion was favorable to the formation
of galaxies. Comment. Now, suppose that the Universe is composed of M = 0.1
and  = 0.9, whereas R = 10−4 , show that at the redshift z = 1010, where
(z=1010 ) −36 . Same question
the first light elements are produced, the ratio (z=10 10 ) = 10
R
if the Universe is composed today of 10% matter and 90% curvature. Show in this
10 )
case that k (z=10 −16 . What you can conclude from these numbers? Do you
(z=10 10 ) = 10
R
understand the Weinberg proposition of multiverse?

2.1.7 The Comoving Distance or Codistance

2.1.7.1 Generalities
Sometimes, when computing a physical process, one needs to know the phenomena
characteristic (spectra, lifetime, temperature, etc.) at a given distance and a given
time t (or equivalently, redshift z). At that redshift z, the Universe was quite different
than today, and its metric was also reduced by a factor 1/(1 + z). One should then
be careful when computing the signal observed now, from a source producing it
at a high redshift. For instance, let us think about the spectrum of a source, which
emits a monochromatic signal continuously in the Universe. One has to take into
account that the energy emitted at the distance d(z) has been redshifted along its
way from the source to the Earth. As a consequence, the observed signal on the
Earth is not monochromatic anymore, but a sum of redshifted signal, on a distance
which evolves also with time (as z evolves with time). Let us put it in numbers.
The first step is to compute the distance of the source from the Earth at a redshift z,
which we call the comoving distance χ. In other words, the distance x that a photon
has crossed if it has been produced at a redshift z. For a massless particle like the
photon, we can write

dt
dt 2 − a 2 (t)dx 2 = 0 ⇒ dx = . (2.61)
a(t)
da(t )/dt
In the meantime, we know that by definition, H (t) = a(t ) , which implies

da(t) a0 dz
dx = , a= ⇒ dx = − ,
a 2 (t)H (t) (1 + z) a0 H (t)
2.1 The Context 51

where we kept a0 (= 1) for a better understanding.11 The codistance dχ from the

Earth can then be written dχ = −dx or

dχ 1
= .
dz a0 H (z)

We now need to find the expression of H (z), which obliviously depends on

the redshift as, if the cosmological constant stays the same, the density of matter
evolves with time due to the expansion of the metric. Using Eq. (2.57), one obtains

H (z) = H0 (1 + z)4 0
R + (1 + z)3 0
M + (1 + z)2 0
k + 0
 ⇒ (2.62)
dχ 1
=  .
dz 3
a0 H0 (1 + z) 2 (1 + z) + + 0 /(1 + z) +
 /(1 + z)
0 0 3
R M k

Depending on the problem, one then can integrate from z0 = 0 to any redshift z
to compute the consequences of an event on the Earth, which is occurring regularly
since the redshift z, like a decaying dark matter, for instance.

2.1.7.2 The Size of the Universe (bis)

Using the comoving distance is also the easiest way to compute the exact size (and
age) of the Universe. Indeed, in Sect. (2.1.4.5), we gave a solution, under the form
of an exercise for students. The real computation of the size of the Universe is not
so much complicated but should be done with the scale factor (or redshift) as the
variable to integrate on. It is indeed the easiest dynamical variable, as the time is
more difficult to define because it depends itself already on the composition of the
Universe. As in (2.25), we need to compute the distance λ traveled by the light
within an expanding Universe:

dλ = R0 a(t) dχ = c dt
  1
dt da
⇒ R0 χ = c =c 2
, (2.63)
a(t) 0 a H (a)

where we used dt = da
aH . Rewriting Eq. (2.62) as function of a,

H (a) = H0 0 a −4 + 0 a −3 + 0 a −2 + 0 (2.64)
R M k 

11 Weremind the reader that by convention, the radius of the Universe at a time t is written R(t) =
a(t)R0 , where R0 is the present radius of the Universe, of the order of 46 Gpc.
52 2 Inflation and Reheating [MP → TRH ]

Fig. 2.3 Evolution of the Universe from the CMB emission till now

giving

 1
c da
λ = R0 χ =   44.7×109 lyrs.
H0 0 a2 0 a −4 + 0 a −3 + 0 a −2 + 0
R M k 
(2.65)

Using the values  = 0.69, 0M = 0.31, R = 8 × 10−5, H0−1 = 13.8 Gyrs,

and a flat Universe12 (k = 0). This result of 89.4 × 109 light-years is very near
from the one computed with our naive assumptions (2.29). This is coming from the
fact that light has (almost) not traveled before the CMB, and the -domination time
happened in a very late time (z  0.66), so the Universe can be considered as a
mater dominated Universe in its whole history from the CMB. To compute the age
of the Universe T , one needs to integrate
  1 da
T = dt =  13.2 Gyrs, (2.66)
0 aH (a)

not so far from the measured value T  13.8 Gyrs. The difference comes from the
naive integration we do. We illustrate in Fig. 2.3 the evolution of the Universe from
the time the light was emitted from the CMB till now, with the respective size of the
Universe in both cases (46 × 109 years now and 1100 46
= 42 × 106 light-years at the
CMB epoch.

12 A slightly more precise computation commonly used in the literature gives λ = 46.3 × 109 lyrs.
2.2 Inflation [10−43 − 10−37 s] 53

2.2 Inflation [10−43 − 10−37 s]

The cosmological standard model, from the reheating process to the galaxies
formation is consistent and can explain the presence of dark matter, nucleosynthesis,
the relative abundance of hydrogen, lithium and helium, structures formations,
and even the cosmological constant and the expansion rate of the Universe can
be included with minimum changes. We have to admit that few models have so
many predictions with so few hypothesis and parameters: almost all the physics
is included in Eq. (2.36). However, when putting some numbers, it seems that the
vanilla scenario exhibits some issues in the very early stage of the Universe, around
the initial singularity of the theory.

2.2.1 The Horizon Problem

The first problem that was noticed is usually labeled “horizon” problem, but also
“homogeneity” or “isotropy” issue. The idea is very simple. Indeed, the observable
Universe today (t0 = 13.8 Gyr) has a horizon, defined by13
0
dH = dH (t0 ) = c × t0  1.3 × 1026 m,

whereas this horizon was, at the very initial time, the Planck time ti = tP lanck =
10−43 s
ai
i
dH = dH (ti ) = dH (t0 ) × , (2.67)
a0

where ai = a(ti ) and a0 = a(t0 ) are the scale factors of the Universe at the time
ti and t0 , respectively. At ti , with a good approximation, the light has covered a
distance cti = 3 × 10−35 m, or in other words,

dHi
 5 × 1028. (2.68)
c × ti

That means that it would exist almost (5 × 1028 )3  1085 causally disconnected
“bubbles” in the very first instant of the Universe,14 rendering impossible to observe
−4 in the CMB spectrum. Another way
such a homogeneity δE E  T  10
δT

to see it is to say that our present horizon would be composed of 1085 initially
disconnected patches. If we apply the same reasoning to compute the number of

13 To be more precise, one should take into account the physical horizon distance, i.e. the actual

distance traveled by the light, which is given by Eq. (2.65), but this approximation is quite valid
for the argument.
14 The same result can be obtained by computing the entropy of the Universe in the very early time.
54 2 Inflation and Reheating [MP → TRH ]

causally disconnected bubbles at the CMB time, we obtain (we let the reader to
prove it)

CMB
dH
 30 (2.69)
c × tCMB

corresponding to more than 1000 regions that could not be connected.

To understand better the phenomena, we can rewrite Eq. (2.67) as

i
dH t0 ai ȧi
=  1, (2.70)
c × ti ti a0 ȧ0

where we have supposed in the last equation that a(t) ∝ t α , α being a constant,
which is effectively the case in every type of density domination we will encounter
(except during the inflation phase of course as we will demonstrate). But, as ȧ
decreases with time, the horizon distance will always be larger than the causally
connected region “c × t” when going back to time. That is, completely unavoidable
as we show in Fig. 2.4. However, if we suppose an expending phase, where ȧ
increases in a very short period of time before the radiative Universe, we can make
the horizon “re-entering” in the causally connected bubbles. How is it possible?
If we do not consider the presence of a cosmological constant, the evolution of
√
ȧ is given by the Hubble parameter H ∝ ρ, Eq. (2.43), where we supposed a flat
Universe (we will discuss the flatness problem in the following section). That means
that neither a radiative domination (ρ ∝ a −4 ⇒ ȧ ∝ a1 ) nor a matter dominated
Universe (ρ ∝ a −3 ⇒ ȧ ∝ √1a ) can induce a phase of increasing ȧ. However, if we
suppose the presence of a field whose density ρ is constant just after the Planck
time ti , we have
√ √
ρ ρ
H (ti ) = √ ⇒ ȧ = a √ > 0. (2.71)
3MP 3MP

To be more precise, if the Universe is dominated by ρ between the Planck time

i =
ti and a time tf , and we suppose that at ti all the horizon is causally connected (dH
f
cti ), we can compute how stretched is the horizon dH at the time tf supposing the
Universe dominated by ρ . Making use of (2.63), we can write
 f cda f c af
R0 a(t)dχ = cdt ⇒ R0 χf = ⇒ dH = R0 af χf = ,
i H a 2 H ai

f
where we supposed ai af . Asking for dH to fit the observable contracted horizon
from now to tf :

f af
dH = c × t0 , (2.72)
a0
2.2 Inflation [10−43 − 10−37 s] 55

Fig. 2.4 Top: an illustration of the evolution of the causally connected regions (ct in blue) with the
horizon (dH in red) in the case of a pure radiative Universe (without a phase of inflation). Bottom:
same but with the hypothesis of an inflationary phase prior to the radiative Universe

we obtain
c af af
= c × t0 , (2.73)
H ai a0

which gives

af H af a0
 = , (2.74)
ai H0 a0 af
56 2 Inflation and Reheating [MP → TRH ]

where we used in the last relation, H ∝ a12 in a radiation dominated Universe,15

H0  t10 , and H = H (af ) because H is constant during the inflation phase. Notice
that we transformed an equality into a  relation because the relation holds as long
as the horizon distance after the inflation phase is greater than the one computed
from now to tf . We summarize all the process in Fig. 2.4.

Exercise Integrating the expression (2.61), recover the preceding result by com-
puting the distance traveled by the light from ti ∼ 0 to t = tCMB and then
the same horizon distance from t = 0 to t0 . Show that the distance of the CMB
horizon today dHCMB can be expressed as a function of the present horizon d 0 by
√
0 = 1 + z × d CMB , which gives (2.69) for z = z
H
dH H CMB = 1000.

2.2.2 The Flatness Problem

Even if we do not consider the horizon problem, another issue arises in the vanilla
thermal Universe model, and it concerns its curvature. From the CMB measurement,
we know that k (2.52) is  0.02 (and is probably much less) at present time t0 ,
while the radiation density is R = ρρRc  10−4 . However, as we can see from
Eq. (2.51), the curvature density scales as a −2 whereas the radiation density scales
as a −4 . That means that the hierarchy between the radiation and curvature densities
increases drastically as we go back in time. If we assume by simplicity that both
densities are at most equal today (ρk ≤ ρR ), we obtain at the Planck time ti
 2 
ρk0 aa0i   2
ρk (ti ) ai 2
ρR0 10−13
=  4 ≤ =   10−64, (2.75)
ρR (ti ) a0 ρRi 1019
ρR0 aa0i

where we considered ρRi = ρR (ti )  MP4 l GeV4 and took the measured value
ρR0 = ρR (t0 )  (10−4 eV)4 . That means that the curvature should be very near
zero at the very early stage of the Universe, whereas it would be natural to consider
a homogeneous Universe with an equal amount of density of each kind in the very
beginning (that is, ρk (ti ) = ρR (ti )), especially if one considers a Universe emerging
from a quantum gravity phase. It corresponds to justify a fine tuning of more than
64 orders of magnitude.
The main problem comes from the fact that the ratio ρρRk increases with time,
 2
proportionally to aai , which very quickly reaches huge values. We can then play
the same game we did when we tried to find a solution to the horizon problem.
Indeed, if one supposes at the beginning of time, a reasonable density of curvature, a

15 Thatis the phase during which the Universe evolves the more, compared to the matter or dark
energy period.
2.2 Inflation [10−43 − 10−37 s] 57

phase of rapid expansion will dilute it sufficiently such that nowadays, the curvature
takes reasonable values again. Concretely speaking, if between the Planck time ti
and the end of this inflationary phase tf the Universe is dominated by a constant
density ρ , without yet any density of radiation, supposing the natural initial
condition ρki = ρ , we obtain
 2  2  2
f ai ai ai
ρk = ρki = ρ ⇒ ρk0 = ρ . (2.76)
af af a0

Then, if all the energy contained in ρ is converted into radiation at tf , one has
 4
f af
ρR = ρ ⇒ ρR0 = ρ . (2.77)
a0

Combining the two previous equations, we obtain the condition to respect the
observation ρk0 ≤ ρR0 :

ai af
≤ , (2.78)
af a0

which is surprisingly the same relation we obtained to solve the horizon problem,
Eq. (2.74). What tells Eqs. (2.74) or (2.78) is that the volume of the Universe should
have the same expansion rate between ti and tf that between tf and t0 . A convenient
measure of expansion is the so-called e-fold number defined as

N ≡ ln a. (2.79)

If we suppose (as we will see later on) that the energy scale of the radiation at
tf is ρR  (1016 GeV)4 , and knowing the present value ρR0 = (10−4 eV)4 , we can
f

deduce
f
a0 1 ρ
Nf 0 = N0 − Nf = ln = ln R0  67. (2.80)
af 4 ρR

We can then conclude that if inflation takes place at a scale of ∼1016 GeV, it
should last for a minimum of 67 e-folds. We then have a one-to-one correspondence
between the energy scale of the inflation and the number of e-folds necessary to
achieve it. The larger is the scale, the more e-folds are needed to dilute sufficiently
and counterbalance the evolution of the curvature density between tf and t0 .

Exercise Compute the number of e-folds necessary if the inflation scale is of the
order of TeV (solution: 37).
58 2 Inflation and Reheating [MP → TRH ]

Computing the necessary number of e-folds

For the students, there is a shorter formulation that gives, at first order, a
good approximation of the number of e-folds necessary to solve the horizon
problem. The first step is to compute the size of the horizon, from the initial
scale factor ai (which we will take at ti = tplanck  10−43 s) to the final stage
of the inflation af (at t = tf ). With the same method we used to compute
the size of the Universe, the distance traveled by the light from ti to tf can be
written as
 tf  af
cdt cda af c
dfH = af R0 χf = af = af 2
 , (2.81)
ti a ai H a ai H

where
√
we applied Eq. (2.63) and considered a constant Hubble rate H =
ρ
√  . We also supposed af ai . To solve the horizon problem, we want
3MP
that, at tf , the size of the horizon corresponds to the size of the Universe. In
other words, we want that at tf , all the points inside the volume (R0 af )3 are
causally connected:
 
af H
dfH = R0 af χf ⇒ ln  ln + ln af , (2.82)
ai H0

where we used R0  ct0  Hc0 . Taking a radiation dominated Universe

(which is the period where the Universe evolved the more), we can write

 
af tf 10−43
af = =  = 10−30, (2.83)
a0 t0 1017

which gives at the end, if one considers a potential ρ = V (φ) 

(1016 )4 GeV4 at unification scale,

af
Ne = ln  ln 1026  60, (2.84)
ai

where we took H0  10−42 GeV.

2.2 Inflation [10−43 − 10−37 s] 59

2.2.3 The Inflaton

The issues we discussed above can be solved by the introduction of a scalar field,
called inflaton.16 It will be responsible of the rapid expansion phase between
ti and tf but will also reheat the thermal bath through its decay into Standard
Model particles from tf during all its lifetime. We talk about reheating and not
heating because all the initial radiation that could have been produced before the
inflationary phase has been largely diluted. However, before going in detail into the
thermalization process, one needs to understand in greater detail the evolution of
the inflaton field φ with time. To find its equation of motion, we should minimize its
action. As we will see, we can minimize it with respect to the metric or with respect
to the field (and its derivative) itself. If we consider the inflaton as a classical field
with a Lagrangian density

1 μν
Lφ = g ∂μ φ∂ν φ − V (φ),
2
its action can be written as

√
Sφ = d 4 x −gLφ (2.85)

asking the invariance of Sφ under a metric transformation gives


 √ √ 
δSφ = d 4 x δ( −g)Lφ + −gδ(Lφ ) . (2.86)

Noticing that δg = gg μν δgμν (A.77) and g μν δgμν = −gμν δg μν (A.83) implying

√
√ −g
δ( −g) = − gμν δg μν , (2.87)
2
we can rewrite Eq. (2.86) as
 
1 4 √   1 √
δSφ = d x −g −gμν Lφ + ∂μ φ∂ν φ δg μν = d 4 x −g Tμν
φ
δg μν ,
2 2

which gives
φ
Tμν = ∂μ φ∂ν φ − gμν Lφ . (2.88)

16 Even if the original model of Starobinsky had no scalar, it was shown to be equivalent to a scalar
theory.
60 2 Inflation and Reheating [MP → TRH ]

Exercise From
1 μν  
LS = g ∂μ S ∗ ∂ν S + ∂ν S ∗ ∂μ S − V (S)
2
i  
Lψ = g μν ψ̄γμ ∂ν ψ + ψ̄γν ∂μ ψ − ∂μ ψ̄γν ψ − ∂ν ψ̄γμ ψ − mψ̄ψ
4
1
LV = − g μα g νβ Fαβ Fμν ,
4
where S is a complex scalar, ψ a Dirac fermion, and Vμ a vector with Fμν = ∂μ Vν −
∂ν Vμ , show that in the approximation of flat metric,

S
Tμν = (∂μ S ∗ ∂ν S + ∂ν S ∗ ∂μ S) − gμν LS (2.89)
i  
ψ
Tμν = ψ̄γμ ∂ν ψ + ψ̄γν ∂μ ψ − ∂μ ψ̄γν ψ − ∂ν ψ̄γμ ψ − gμν Lψ (2.90)
4
1
V
Tμν = −g αβ Fαμ Fβν + gμν Fαβ F αβ . (2.91)
4

Developing gμν = ημν + M1P hμν , where hμν is the graviton field, show that the
coupling of the matter to the graviton can be written as

1 μν i
Lhi = h Tμν . (2.92)
2MP

Notice that in General Relativity, the derivatives ∂μ should be thought as

covariant derivatives Dμ defined in Eq. (A.51). However, for a scalar, Dμ φ = ∂μ φ,
and for a vector, Dμ Vν − Dν Vμ = ∂μ Vν − ∂ν Vμ .
The fermionic case is much more tricky due to the fact that one should use
the geometric γ μ matrices (that you can think as being also bent by the metric
μ
g μν ) and defined them as γ μ = ea γ a , γ a being the classical dirac matrices in the
μ
Minkowski flat space and ea the vierbein, see Eq. (A.43) and Eq. (A.59). A detailed
and pedagogical computation of the stress–energy tensor of a free dirac field can be
found in [5]. The connection term present in Eq. (A.59) disappears in the flat space
approximation. To compute δLψ , do not forget to compute δγμ :

1
δγμ = δ(eμa γa ) = δ(eμa )γa = − gμα δg αν (eνa γa ), (2.93)
2

where we used from gμν = eμa eνb ηab : (prove it)

1 1
δeμa = δgμν eνa , δeμa = − δg μν eνa . (2.94)
2 2
2.2 Inflation [10−43 − 10−37 s] 61

The density of energy of the stress–energy tensor is given by the {00} component
φ
of Tμν ,

φ 1 2
ρφ = T00 = (∂t φ)2 − g00 Lφ = φ̇ + V (φ), (2.95)
2
where we have supposed a flat and homogeneous metric for the space-time, whereas
the pressure is the spatial {ii} component of the tensor17

φ 1 2
Pφ = Tii = (∂i φ)2 − gii Lφ = φ̇ − V (φ), (2.96)
2
φ
our matrix Tμν can then be written (A.96)
⎛ ⎞ ⎛1 ⎞
2 φ̇ + V (φ)
ρφ 0 0 0 2 0 0 0
⎜ ⎟ ⎜ ⎟
2 φ̇ − V (φ)
1 2
⎜0 Pφ 0 0⎟ ⎜ 0 0 0 ⎟
φ
Tμν =⎜ ⎟=⎜ ⎟.
⎝0 0 Pφ 0⎠ ⎝ 0 0 1 2
2 φ̇ − V (φ) 0 ⎠
0 0 0 Pφ 0 0 0 1 2
2 φ̇ − V (φ)

Notice that for a non-homogenous field φ, the energy density ρφ is

1 
ρφ = (∂t φ)2 + (∂i φ)2 + V (φ). (2.97)
2
This situation appears when one needs to deal with constraints from inhomo-
geneities present in the CMB spectrum. This remark is also valid at ti = tP lanck ,
where ∂i φ need not be zero but just “sufficiently” small to get washed out by
expansion.

2.2.4 The Equation of Motion

Implementing ρφ and Pφ in Eq. (2.42), we obtain the equation of motion for φ:

ρ̇φ + 3H (ρφ + Pφ ) = 0
⇒ φ̈ + 3H φ̇ + V  (φ) = 0, (2.98)

where V  (φ) stands for ∂φ V (φ).

17 We remind the reader that in our convention of the flat metric, g00 = g 00 = +1 and gii = g ii =
−1.
62 2 Inflation and Reheating [MP → TRH ]

Exercise Noticing that gμν is independent of φ, recover Eq. (2.98) applying the
√
Euler Lagrange equation on δS = −gLφ with respect to (φ, ∂μ φ). Hints: you
√ √
will need the relation (prove it) ∂t −g = 3H −g, and Eq. (2.166) can help you.

A quick look at Eq. (2.98) shows that φ behaves like a rolling-down ball, slowed
down by a friction term represented by H , the expansion of the Universe. That
is indeed the classical exercise given at high school to compute the limit velocity
reached by a falling body submitted by a friction force Ff = −kv = −k ẋ, H
playing the friction role of k3 . The equation of motion in a potential V (x) is then
ma = mẍ = −k ẋ − dx d
V (x), which is exactly the expression we obtained for φ
(2.98).
Notice also that this equation assumes a stable field φ. If, as we will see when
discussing the reheating phase, the φ field decays into lighter particles, to give rise
to the Standard Model bath, one needs to add to Eq. (2.98) a term of the form ˜ φ φ̇,
˜ φ being the width of the inflaton. The evolution of φ then becomes

φ̈ + 3H φ̇ + V  (φ) = −˜ φ φ̇. (2.99)

Each term of the equation above has a clear physical meaning. The evolution of
the density of the kinetic energy, frictional energy, and potential energy is converted
into the lost of energy by decay. It is always possible to solve numerically Eq. (2.99),
but we can find quite accurate analytical solution in two regimes of interests: the
slow roll regime and the coherent oscillating regime.

2.2.5 The Equation of Motion (Generalization)

We can generalize Eq. (2.98) to fields that interact with the inflaton. This is useful
when one needs to compute production of dark matter in the preheating phase. Let
us consider first the case of a scalar field S of mass mS interacting with the inflaton
φ through a dimensionless coupling proportional to λ:
 
√ 1 1 1 √
SS = d 4 x −g g μν ∂μ S∂ν S − m2s S 2 − λφ 2 S 2 = d 4 s −gLφS .
2 2 2
(2.100)

The Euler–Lagrange equation can be written as

√ ∂LφS √ ∂LφS
∂μ −g − −g = 0, (2.101)
∂μ S ∂S
2.2 Inflation [10−43 − 10−37 s] 63

which gives, for a flat metric,

ds 2 = c2 dt 2 − dx 2 − dy 2 − dz2 = c2 dt 2 − a(t)2 dχ 2 , (2.102)

√
and noticing that −g ∝ a(t)3 ,

∂V (S) |k|2
S̈ + 3H Ṡ + + 2 S = 0, (2.103)
∂S a
√
where we used ∂t −g = 3 ȧa = 3H and V (S) = 12 m2S S 2 + 12 λφ 2 S 2 and implicitly
|k|2
worked in Fourier space. The a2
term can be understood as a redshifted kinetic
energy and comes from

∂ |k|2
∂ i ∂i S = − S = − S. (2.104)
a 2∂χ 2 a2

2.2.6 The Slow-Roll Regime

2.2.6.1 The Context

To understand the evolution of the inflaton field, one needs to solve Eq. (2.98) or
(2.99) if one takes into account the possibility of decay. Of course, it is always
possible to do it numerically, but it is good to find analytical solutions to feel
the behavior of φ. The first stage is called the slow-roll regime, which name is
quite explicit. At the very beginning, we can first neglect the acceleration part φ̈.
Physically we are in the presence of a field “falling” in a potential V (φ) with a
friction term 3H . The only point, which differs from the classical falling body
Newtonian analogy we took, is that the friction parameter is not constant but
depends strongly on “the height” (φ).
Remember that, to realize inflation, we need a phase of constant density ρ = ρφ
during a very short period of time, between the Planck times ti and tf . A look
at Eq. (2.43) or (2.50) shows that we need the equation of state Pφ = wρφ with
w = −1. From Eqs. (2.95) and (2.96),

Pφ 1 2
2 φ̇ − V (φ)
w= = , (2.105)
ρφ 1 2
2 φ̇ + V (φ)

we deduce that the scalar field φ has the desired equation of state w = −1 only
if φ̇ 2 V (φ). In other words, during a short period of time, the kinetic energy
of the inflaton should be subdominant to realize the inflation phase. Writing Pφ =
−ρφ + φ̇ 2 helps us to understand that the inflation phase should last as long as the
64 2 Inflation and Reheating [MP → TRH ]

kinetic energy φ̇ 2 is kept sufficiently small compared to the potential energy V (φ),
where “as long as” should be understood as, as long as the 67ish e-folds of Eq. (2.80)
have not yet been reached. This condition is possible if V (φ) evolves very slowly
with φ between ti and tf , this is called the slow-roll regime and tf can be considered
as the time when inflation ends. From now on, we will then call it tend .

Exercise Combining Eqs. (2.39), (2.95) and (2.96), and neglecting Λ show that the
inflation condition R̈ 0 corresponds to V (φ) φ̇ 2 .

2.2.6.2 The V = 12 m2 φ 2 Case

We will first concentrate in the simplest quadratic example for V (φ). Indeed, even if
this model is in tension with data, this will help us to understand the dynamics of the
inflation. Concretely speaking, the idea is quite straightforward. We need to solve
the set of equations Eq. (2.43) for φ(t) and then extract H (t), which in turn will
give us the solution a(t). To begin with, one should first rewrite the set of equations
(2.43) as a function18 of φ:

φ̈ + 3H φ̇ + V  (φ) = 0 (2.106)
1 2
2 φ̇ + V (φ)
H2 = (2.107)
3 MP2
1 2 2
V (φ) = m φ . (2.108)
2
As we discuss in the previous section, at the beginning of the “falling” of the
inflaton, its acceleration φ̈ and its velocity φ̇ can be neglected. Equation (2.107)
then gives

mφ
H  √ , (2.109)
6MP

which we can replace in (2.106) to write

 
2 2
φ̇ = − m MP ⇒ φ(t)  φi − m MP (t − ti ), (2.110)
3 3

and as a consequence

mφ mφi m2
H = H (t) = √  √ − (t − ti ), (2.111)
6MP 6MP 3

18 We will not consider a decaying φ at this stage. This is justified because if ˜ φ is comparable
to the Hubble rate, then the slow-roll regime is not valid anymore: the inflaton decays too fast to
obtain the needed 67 e-folds.
2.2 Inflation [10−43 − 10−37 s] 65

where ti is the Planck time. Defining the end of inflation tend by m φ(tend ) <
|φ̇|(tend ), or in other words, by the time when the kinetic energy begins to reach
values similar to the potential energy, Eq. (2.110) gives

 
3 φi 2
tend − ti  , |φend | = MP . (2.112)
2 m MP 3

Another way to understand the condition for the inflation to end is to see that if
φ(tend ) = 0, the inflation effectively ends because V (φ) vanished for φ = 0, and the
kinetic part φ̇ dominates on V (φ), giving an end to the slow-rolling inflation regime.
Indeed, the negative solution for φ̇ shows that during inflation, φ will decrease,
inducing automatically a decrease in V (φ), whereas 12 φ̇ 2  m2 MP2 stays almost
constant. It is then unavoidable that there will exist a moment when φ̇ 2 < V (φ)
We have now all the tools in hand to compute a(t) and apply the condition (2.80)
to obtain a condition on φi in order to obtain sufficient e-folds for the inflation to
occur. Integrating H (t) in Eq. (2.111) from ti to tend , and supposing φi MP , we
have
φi2
2
a(tend )  ai e 4MP
, (2.113)

which means we need

√
φi = 4 × 67MP  16 MP (2.114)

to obtain the necessary 67 e-folds. It is worth noticing that having a field above,
the Planck mass is not problematic (in the sense, one does not need to look for a
quantum theory of gravity) as long as the energy density stays below the Planck
scale. For instance, for an inflaton mass of m = 1013 GeV and φi = 106 MP , the
potential V (φ) = 12 m2 φi2 = 12 (1019)4 GeV4 is still sub-Planckian at the Planck time
ti . We can also see that

2
Pφ = −ρφ + m2 MP2 , (2.115)
3
66 2 Inflation and Reheating [MP → TRH ]

which means that when φ has decreased to a value ∼ MP , the kinetic part begins to
be important in the equation of state: the inflaton will enter in an oscillatory regime
and will not behave anymore like a vacuum energy, but like a dust.19

Exercise Noticing that the total number of e-folds Ne can be generalized by

 tP l  tP l
H (t  )dt  a(te )
a(te ) = a(tP l ) × e te ⇒ Ne = ln = H (t  )dt  , (2.116)
a(tP l ) te

where tP l and te represent, respectively, the Planck time and the time when inflation
ends and compute the number of e-folds for a potential V (φ) = λφ 4 and V (φ)
generic.

Conditions for the slow-roll regime

It is interesting to write the general condition for the slow-roll regime, which
corresponds to neglecting the term φ̈ in Eq. (2.106). In other words,

3H φ̇  −V  (φ) (2.117)

φ̈ −3H φ̇  V (φ). (2.118)

Remarking that, during the inflation era, φ is almost constant (φ̇ 2

V (φ)),

ρ 1 2
2 φ̇ + V (φ) V (φ)
H2 = =  , (2.119)
3MP2 3MP2 3MP2

Equation (2.117) can be written as


V  (φ)2 MP2 M2 V  (φ) 2
φ̇  2
V (φ) ⇒  ≡ P 1. (2.120)
3V (φ) 2 V (φ)
dφ d
In the same way, writing d
dt = dt dφ , we can express condition (2.118)

d d V  (φ) V  (φ)V  (φ) MP2 V  (φ)

φ̈ = φ̇ = −φ̇ = V  (φ) ⇒ η ≡ 1,
dt dφ 3H 9H 2 V (φ)
(2.121)

where we supposed H  cst.

(continued)

19 Tobe more precise, as we will see in the next section, the inflaton will behave like a dust for a
quadratic potential V (φ), whereas it will behave like a radiation for a quartic potential.
2.2 Inflation [10−43 − 10−37 s] 67

Exercise Show that Eq. (2.121) can be recovered as a consequence of (2.120)

and (2.117).

2.2.7 The Coherent Oscillation Regime

In this regime, the field φ has “rolled down” the potential and oscillates. Another
way is to notice that V (φ) has decreased down to the value ∼ φ̇sr 2 and we cannot use

the approximation φ̇ mφ anymore. Indeed, while φ̇ was negative, it will become

positive (and φsr negative, see Eq. (2.110)), pushing V (φ) toward larger values, then
back until the moment where V (φ) will reach again φ̇ 2 . . . We clearly recognize an
oscillating body, transferring its energy between kinetic (φ̇ 2 ) and potential (V (φ)).
Rewriting the equation for the Hubble rate

φ̇ 2 + m2 φ 2
H2 = , (2.122)
6MP2

we can define a new variable θ by

√ √
φ̇ = 6MP H sin θ, mφ = 6MP H cos θ (2.123)

and implementing it in Eq. (2.106) to obtain

Ḣ = −3H 2 sin2 θ
3
θ̇ = −m − H sin 2θ. (2.124)
2
Exercise Recover the previous set of equations.

Noticing that for φ MP (after the end of inflation tend ), H  m MφP m, we

can solve the set of equations (2.124):
 −1
2 sin 2mt
H  1− (2.125)
3t 2mt
θ  −mt (2.126)
68 2 Inflation and Reheating [MP → TRH ]

and

√
2 2 MP cos mt
φ= √   . (2.127)
3 1 − sin 2mt mt
2mt

We show in Fig. 2.5 the evolution of φ as a function of the normalized time mt,
where Eq. (2.106) is solved numerically. We see clearly thetwo different regimes,
where φ evolves first linearly with t, following a slope − 23 MP as predicted in
(2.110), in the slow-roll regime (for low values of mt). Then, entering in the coherent
oscillation regime, φ oscillates with a frequency  m, in accordance with (2.124)
while still losing amplitude at a rate  3H . The inflation ends when t  φ−1 ,
or mt  mφ = 8π y2
if one considers an effective coupling to the standard model
fermions of the form L = yφ f¯f .

Exercise Show that the envelope of the oscillation follows a law in a −3/2, or
equivalently, 1t . Conclude that ρφ follows a dust equation of state.

2.2.8 The General Case, V (φ)

We can apply the same scheme in the case of a generic potential V (φ). In this case,
we need to solve the following set of equations:

φ̈ + 3H φ̇ + V  (φ) = 0 (2.128)
1 2
2 φ̇ + V (φ)
H2 = . (2.129)
3MP2

If, as before, we suppose that at the beginning the inflaton is “falling” without
large acceleration φ̈ nor large velocity φ̇, we can write
√
V (φ)
φ̇ 2 V (φ) ⇒ H  √ (2.130)
3MP
MP V  (φ)
φ̈ V  (φ) ⇒ φ̇  − √ √ (2.131)
3 V (φ)
2.2 Inflation [10−43 − 10−37 s] 69

)(0)=10 MP
2.0 × 10 19

1.5 × 10 19
)(GeV)

1.0 × 10 19

5.0 × 10 18

0 10 20 30 40 50
m t

2×1017

1 × 1017
)(GeV)

–1×1017

–2×1017

0 20 40 60 80 100
m t

Fig. 2.5 Evolution of the inflaton field φ as a function of the time (normalized to mφ ) in the case
of a quadratic potential V (φ) = 12 m2φ φ 2 , with φ(ti ) = 10 MP

and find the solution for a:

√
V (φ) d ln a d ln a dφ MP V  (φ) d ln a
H = √ = = =−√ √ (2.132)
3MP dt dφ dt 3 V (φ) dφ
 V (φ)
− 2 V  (φ) dφ
⇒ a = ai e MP
. (2.133)
70 2 Inflation and Reheating [MP → TRH ]

We then see that if one considers a potential of the form V (φ) ∝ φ k , we obtain

φi2 −φ 2
2k
a = ai e 2MP
. (2.134)

Asking for Ne = 67 e-folds, we obtain

√ 
φi  134 kMP = 2Ne kMP , (2.135)

which is of the same order as the value of φi we found in Eq. (2.114), and we recover
it for k = 2.
After the inflationary phase, we enter in the coherent oscillation phase. In this
case, one cannot neglect anymore φ̈ in Eq. (2.128). Multiplying the equation by φ
and taking the mean (noticing that on a period, φ φ̇ = 0), we obtain

− φ̇ 2  + 3 H φ̇φ + φV  (φ) = 0. (2.136)

Remarking that at the end of the inflation,

φ̇ φ
H  √ ⇒ H φ̇φ = √ φ̇ 2  φ̇ 2 , (2.137)
3MP 3MP

we can neglect the friction term H in Eq. (2.136), which gives

φ̇ 2   φV  (φ). (2.138)

We can then express the equation of state by the relation

Pφ 1
φ̇ 2  − V (φ) φV  (φ) − 2 V (φ) k−2
w= = 2
= = , (2.139)
ρφ 1
2 φ̇ 2  + V (φ) φV  (φ) + 2 V (φ) k+2

where we supposed V (φ) ∝ φ k in the last equation. Implementing the previous

relation between Pφ and ρφ in Eq. (2.98), we obtain the generic equation for ρφ

6k
ρ˙φ + Hρφ = 0. (2.140)
k+2

In a Universe, dominated by φ, Eq. (2.140) can be solved analytically and we

have
 6k
− k+2
a
ρφ (a) = ρφ (ai ) . (2.141)
ai
2.2 Inflation [10−43 − 10−37 s] 71

This last equation is very interesting because we recover that for k = 2,

ρφ ∝ a −3 , which means that ρφ behaves like a matter field, whereas for k = 4,
ρφ ∝ a −4 and behaves like a radiation field. Naively, it is surprising that we recover
the behavior of a massive field for an oscillating field. In fact, it is easy to understand
if one interprets Eq. (2.124) as the equation of a homogeneous field oscillating at
a frequency m. φ can then be considered as a coherent wave constituted of “φ-
particles,” or oscillators, with zero momentum (k = 0) and a frequency m. The
density of energy for φ can then be written as a sum of density of energies of
oscillators of number density nφ , in other words,

ρφ = nφ m, (2.142)

which is effectively the behavior of a massive field of density nφ . If one considers a

density of energy V (φ) = 12 m2 φ 2 , one obtains

1
nφ = mφ 2  1092 cm−3 (2.143)
2

for m = 1013 GeV and φ = MP , which can be considered as a rough value of the
density of entropy in the Universe after the inflation phase.

2.2.9 Constraint from Perturbations∗

2.2.9.1 Generalities
The inflationary models we discussed were homogeneous, i.e. we never took into
consideration any spatial distribution of the inflaton field. However, there is no doubt
that the CMB sky, measured by PLANCK satellite, shows density perturbations at
the level of
δρ
 5 × 10−5 . (2.144)
ρ

The subject of primordial perturbations is extremely complex and far beyond

the scope of this book.20 However, we would like to give some hints, and order of
approximations for the reader, to help him/her understanding the philosophy and
mechanics of perturbation theory.
Let us first have a rough estimate of the density perturbation we expect from a
non-homogeneous field δφ. In the absence of expansion, at the classical level, if one
considers a massless scalar field φ, the density of energy can be written using (2.88)
as
1 
ρ = Tφ00 = (∂t φ)2 + (∂i φ)2 . (2.145)
2

20 For a detailed study on the subject, see [6].

72 2 Inflation and Reheating [MP → TRH ]

At a given time,

1 1 1
δρ ∼ δ(∂i φ)2 = δ(k12 + k22 + k32 )φ 2 = k 2 δφ 2 . (2.146)
2 2 2

Remarking that in the volume δV = λ3 , with λ ∼ q1 the wavelength of the k

mode of δφ, we should have one quanta of energy k, one deduces

δρ 1 k 2 δφ 2
δρ × λ3 ∼ = = k ⇒ δφ  k. (2.147)
k3 2 k3
This means that a perturbation δφ  k generates an energy density perturbation
δρ ∼ k. In other words, if one asks for ρ to vary of a quantity of order k, one needs
also φ to vary by the same quantity, δφ ∼ k.

Exercise For a quantum treatment of the preceding analysis, consider a field (B.77)

d 3k
φ(x, t) = √ (eiEk t −ikx ak† + e−iEk t +ikx ak ), (2.148)
(2π)3/2 2Ek

where the creation and annihilation operators obey

[ak†, ak  ] = δ(k − k ). (2.149)

Show that

dk
φ 2 (x) = 0|φ(x, t)φ(x, t)|0 = Pφ (k) (2.150)
k

with

k2
Pφ (k) = . (2.151)
(2π)2

From Eq. (2.150), we can understand the power spectrum Pφ as the increase of
the perturbation δφ 2 = φ 2 (x) per decimal interval of momenta k. If we define
the amplitude of the quantum oscillation δφk as the variance of the field with
momentum k,

δφk ≡ φ = Pφ (k), (2.152)
2.2 Inflation [10−43 − 10−37 s] 73

we obtain

k
δφk = , (2.153)
2π

which is similar to the classical solution (2.147).

2.2.9.2 In an Expanding Universe

The main difference, in an expanding Universe, is that one needs to take into account
the redshift of the momenta k with the scale factor a(t):

k
q= . (2.154)
a(t)

If we write the inflaton field

(x, t) = c (t) + φ(x, t), (2.155)

c (t) being the (homogeneous) classical inflaton field, the equation of motion for
the perturbation φ is

k2
φ̈ + 3H φ̇ + φ + V  (c )φ = 0. (2.156)
a2
Exercise Using

1 √
Sφ = d 4 x −g[g μν ∂μ φ∂ν φ − V  (c )φ 2 ]
2

1
= dtd 3 xa 3 [φ̇ 2 − a −2 (∂i φ)2 − V  (c )φ 2 ], (2.157)
2

recover the expression (2.156) from the Euler–Lagrange equation (2.101).

Neglecting V  (φ) (slow-roll condition 2.117), we can briefly describe the

different regimes for Eq. (2.156). At first, when a is extremely small, the first and
third terms dominate. The equation is then a classical harmonic oscillator equation,
with a redshifting frequency (∼ ak ) decreasing with time. When the wavelength
(∼ ak ) reaches the horizon size21 H −1 , then the field φ freezes in, and the second
term of (2.117) dominates; φ̇ = 0 ⇒ φ is constant. In the first stage, φ is in a
subhorizon mode, transforming into a superhorizon mode in the second stage. Once

21 That is almost constant during the whole process.

74 2 Inflation and Reheating [MP → TRH ]

Fig. 2.6 Evolution of a fundamental mode, from the subhorizon to the superhorizon phase, when
it froze in

in the superhorizon phase, the vacuum fluctuations φ are frozen in, even through the
wavelength grows. At this frozen stage, one deduces from (2.153) (Fig. 2.6).

H
δφk  . (2.158)
2π

To have a feeling about the mechanics of the formation of overdense regions, we

notice that some regions will possess a density c − |δφ|, in a slow-roll regime,
where V (c − δφ) < V (c ) = V ( ). We illustrate it in Fig. 2.7. These regions,
with negative fluctuations, will exit the inflation phase before the regions with
positive fluctuations. These regions will lose energy density, whereas, during a small
period δt, the region with positive δφ will still have the constant inflationary density.
The time it will take for c to reach c − |δφ| is given by

˙ c δt.
δφ =  (2.159)

From the evolution of the density given in Eq. (2.43),

δρ H H2
ρ̇ + 3H (1 + w)ρ = 0 ⇒ ∼ −H δt = − δφ = − , (2.160)
ρ ˙c
 ˙c
2π 

where we used (2.158). This condition should be evaluated at the time the k-mode
exits the horizon. From the slow-roll condition (2.117), we deduce
 V 3/2(φ)
˙ c = − V (φ) ⇒ δρ ∼
 √ 3 . (2.161)
3H ρ 2π 3MP V  (φ)
2.2 Inflation [10−43 − 10−37 s] 75

V( )

Region with fluctuations

c+
exiting the inflation
mode after c

Region with fluctuations

c-
exiting the inflation mode
before c

Fig. 2.7 Illustration of inflation exit for regions with different fluctuations

If one considers the chaotic inflation type of potential V () = λn , one obtains
√
λ  2 +1
n
δρ
∼ √  5 × 10−5 , (2.162)
ρ n 2π 3MP3

where the last equality holds

√ from PLANCK measurement of the CMB. It is then
easy to extract limit on λ as a function of the number of e-folds Ne using
Eq. (2.135):
√ 4
mφ = 2λ  × 10−4 MP  1.8 × 1013 GeV (2.163)
Ne

in the case of a quadratic potential for Ne  50, and

93
λ × 10−10  7.4 × 10−14 (2.164)
Ne3

for a quartic potential.

76 2 Inflation and Reheating [MP → TRH ]

2.2.10 Preheating and Dark Matter∗

2.2.10.1 Parametric Resonance

Before treating the perturbative reheating process in the next section, we want to say
few words about a preheating phenomenon called the narrow resonance, which is
in fact a classical parametric resonance in the context of the preheating. The idea
is that, if one couples the inflaton field  to a scalar S, there is the possibility
of producing it with a large rate, increasing exponentially its occupation number,
through a mechanism of resonance. For illustration, let us consider the action
  
√ 1 μν √
S = d 4 x −g g ∂μ S∂ν S − V (S, ) = d 4 x −gL. (2.165)
2

The Euler–Lagrange equation is

∂ √ ∂ √
−gL = ∂μ −gL (2.166)
∂S ∂(∂μ S)
√ √ √
⇒ − −gV  (S) = ∂t [ −gg t t Ṡ] + ∂i [ −gg ii ∂i S]
√ ġ √ √
⇒ − −gV  (S) = − √ g t t Ṡ + −gg t t S̈ + ∂i [ −gg ii ∂i S].
2 −g

Using

√ 1 ġ ȧ
−g ∝ a 3 ⇒ ġ ∝ 6a 5ȧ ⇒ − = +3 = 3H, (2.167)
2 −g a

we obtain
∂V
S̈ + 3H Ṡ + S + = 0, (2.168)
∂S
ki
or, for a plane wave22 S ∝ eik.χ , implying ∂i S = 1 ∂
a ∂χi S = a S,

k2 ∂V
S̈ + 3H Ṡ + S+ =0 (2.169)
a2 ∂S

If one takes the potential V (S, ) = μS 2 , and suppose, at a first approximation
that a ∼ constant during this short preheating phase, for an inflaton field of the from

22 Keeping in mind that we can always decompose S in its fundamental modes, S = k sk e

ik.x , all
contributions add up to the number density n or energy density ρ.
2.2 Inflation [10−43 − 10−37 s] 77

1 × 1015 (0)=0.1 MP; m=108 GeV; k/a =mF/2

5 × 1014
S (GeV)

–5 × 1014

–1 × 1015

0 100 200 300 400 500

m t
Fig. 2.8 Illustration of the parametric (also called narrow) resonance in the context of preheating.
We can see clearly the exponential envelop of the periodic solution

 = 0 cos(m t), the equation one needs to solve is

k2
S̈ + 3H Ṡ + + μ0 cos(m t) S = 0. (2.170)
a2

Supposing a  constant, we can neglect H . This equation is one form of the

Mathieu equation, which is the equation for an oscillator with a time dependant
2
frequency ω2 (t) = ak 2 +μ0 cos(m t), and is present in a lot of classic phenomena
involving periodical force. It can be shown that for

m μ0 k m μ0
− < < + , (2.171)
2 2m a 2 2m

we enter in a regime where the solution grows exponentially with time.23 We can
understand it easily, from the shape of the Mathieu equation, where, periodically,
the coefficient cos(m t) becomes negative and drives the evolution of S toward an
exponential solution, periodically. The evolution of S is shown in Fig. 2.8. A more

23 This situation corresponds to a narrow resonance if one considers μ 0 . The regime μ  0

is a broad resonance regime but exhibits similar features [7].
78 2 Inflation and Reheating [MP → TRH ]

refined treatment necessitate to compute the Bogoliubov coefficient to extract the

occupation number [7], but we give in the following section a more intuitive view
of the phenomena, solving the equation for the density of the φ decay products. For
the analytical solution of the Mathieu equation (2.170), the reader is directed to [6],
which is without doubt the best textbook treating it, and [7], which is (paradoxically)
the seminal research paper on the subject and one of the clearer and more detailed
works in the literature.

Exercise Redo the previous analysis for a potential V (S, ) = 12 2 S 2 .

The reason why we will not observe this highly non-perturbative effect in a
classical (perturbative) vision, which we will use later, is quite delicate. It comes
mainly from the fact that in perturbation theory, we consider a mother particle at rest,
decaying into two daughter particles satisfying the mass shell relation k 2 = m2S . This
is not exactly the case when we look at the inflaton as a background field. Indeed,
even if we show in Eq. (2.141) that the inflaton field behaves like a set of matter
fields at rest with respect to the evolution of the scale factor a, we should rather
treat these fields as a set of coherently oscillating φ-fields and not as a physical
set of independent φ-fields. One should in fact take into account the effects of the
decay products on the width of the φ-fields themselves and the back reaction effects.
Another way to look at it is to look at this physics from the statistical point of view
of the Bose enhancement.

2.2.10.2 Narrow Resonance Interpreted as Bose Condensates

The result discussed above can be obtained numerically (as we have done) or ana-
lytically, via the Floquet method. However, there is a more intuitive interpretation in
terms of Bose enhancement. Indeed, the narrow resonance effect can be calculated
by considering the decay of the inflaton field into two bosons and by calculating
the number of particles occupancy in the final state. In the rest frame of the φ-
particle, the two particles produced have the same momentum, but with opposite
directions. Being bosonic by nature, if the final state is already occupied, there will
be an enhancement effect by a Bose factor, which we must therefore calculate.
Let us suppose an inflaton coupling to a real scalar S:

1 μν 1 m2
L= g ∂μ φ∂ν φ − V (φ) + g μν ∂μ S∂ν S − S S 2 − μφS 2 .
2 2 2
The production rate of S should be computed combining the decay process φ →
S S and the inverse decay S S → φ. The former is proportional to

φ→SS ∝ | nφ − 1, nk + 1, n−k + 1|aφ ak† a−k

†
|nφ , nk , n−k |2
= nφ (nk + 1)(n−k + 1),
2.2 Inflation [10−43 − 10−37 s] 79

where we classically defined the creation/annihilation operators ai† and ai (see

Sect. 4.9.2.4 and Eq. (4.72) in particular). The rate of the inverse decay is

SS→φ ∝ | nφ + 1, nk − 1, n−k − 1|aφ† ak a−k |nφ , nk , n−k |2 = (nφ + 1)nk n−k .

In the following, considering a homogeneous background, the density occupation

number depends only on the amplitude of the momentum and not its direction. We
can then fairly define nk = nk = n−k . Adding the two processes and noting that
ρ
nφ ≡ mφφ 1, we obtain

μ2
eff  φ (1 + 2nk ), with φ = (2.172)
32πm

where we used Eqs. (B.165) and (B.179). To compute the occupation number nk ,
we use the definition24 (B.85)

d 3k
ns = nk , (2.173)
(2π)3

the volume d 3 k being limited by the relation


1
Ek = k 2 + m2s + 2μφ(t) = mφ , (2.174)
2
or
2μ|dφ|
|dk| = . (2.175)
k

For a quadratic potential, V (φ) = 1

2 mφ , one can use the solution we obtained
(2.127)

8 MP
φ(t) = cos mφ t ≡  cos mφ t, (2.176)
3 mφ t

approximation already valid after only one oscillation. Noticing then that dφ ∼ 2
m
and k ∼ 2φ , we obtain

8μ π2
|dk|  ⇒ nk = ns , (2.177)
mφ μmφ

24 Becareful in the following notations; ns represents the density number of particle S (per unit of
space volume), whereas nk represents the occupation number (no units).
80 2 Inflation and Reheating [MP → TRH ]

and


2π 2
eff  φ 1 + ns . (2.178)
μmφ

2.2.10.3 Production of Dark Matter in the Preheating Era

It is then straightforward to compute the production of a dark matter species (or any
type of particles) by solving the Boltzmann equation

dns ρφ 2π 2 ρφ
= 2eff = 2φ 1 + ns , (2.179)
dt mφ μmφ mφ

the factor “2” taking into account the fact that 2 particles are produced by decay. A
quick look at the above equation shows where the explosive, exponential behavior
of particle production comes from. This is an effect directly related to the fact that
bosons will tend to accumulate in the phase space that already contains the largest
number of bosons, this term being proportional to ns . Solving Eq. (2.179), we obtain
for ns

μmφ  2π 2 μ φ t  μm  π μ t
φ
ns (t) = e − 1 = e 16 mφ − 1 . (2.180)
2π 2 2π 2

Being at a very early time (S t 1), we recover the narrow width condition
(μ ) for the explosive production to be efficient.
The natural question we are then entitled to ask is when does this exponential
production end? There are several cases, depending on the value of the width of
φ. When the perturbative decay dominates, we are in the perturbative regime, the
oscillations ends, and we can jump to the next chapter. But how long should we
wait to see the perturbative decay dominates, and dominates over what?

• The first (and most naive) condition is that the non-perturbative decay rate must
be less than the perturbative part, i.e. from (2.180),

π μ μ2  2
 φ = ⇒  2, (2.181)
16 mφ 32πmφ μ π

or, in other words, when the narrow width condition is not satisfied anymore.
However, we are in a period of time where φ H , which means, another
condition should be stronger.
2.2 Inflation [10−43 − 10−37 s] 81

• Indeed, the friction term appearing in the equation of motion is of the form

3H + φ (2.182)

(see Eq. (2.206)), for instance). A stronger condition to stop the preheating
production is then

π μ m μMP 16 3
 3H + φ ∼ 3H = 3 √ ⇒ 2
 , (2.183)
16 mφ 6MP mφ π 2

corresponding to the fact that the expansion rate dominates the production rate,
and the process is frozen.
• Last but not the least, an even stronger constraint is directly related to the
narrowness of the resonance. Indeed when a particle is produced with momentum
k, if by redshift this momentum exits from the phase space volume defined by
|dk| of Eq. (2.175) at a rate larger than the produced ones, the phase space is not
populated anymore. k being redshifted from time t1 to tP H (time of the end of
preheating phase), we can write
√
a(t1 ) ȧ(tP H ) |dk| −1 8 6μMP
|dk| = k dt  k H (tP H )dt ⇒ dt = H = ,
a(tP H ) a(tP H ) k m3φ

where we used a(t1 ) ∼ a(tP H ), the evolution being slow during the whole
m
process, k = 2φ , and |dk| = 4 μ mφ (2.175). The inflaton will then not populate
efficiently the region as soon as

 
π μ 3 πμ2 MP μ2 MP 21
dt =  1 ⇒  . (2.184)
16 mφ 2 m4φ m4φ 3π

This last constraint, which is the strongest of all, gives us the value of  at which
explosive production ceases to be effective. Remembering that (2.176)

8 MP
(t) = , (2.185)
3 mφ t
82 2 Inflation and Reheating [MP → TRH ]

and implementing it in Eq. (2.184), we can then calculate the time tP H from which
the reheating phase starts and the preheating phase ends:

μ2 MP2
tP H = 2π . (2.186)
m5φ

To give an idea, for a typical value of μ = 105 GeV, one obtains tP H 

10−19 GeV−1  7 × 10−44 s, where we took mφ = 2 × 1013 GeV from Eq. (2.163).
This time is very close to Planck’s time (5.4 × 1044 s). We can now compute
the relic abundance of a bosonic particle produced from tP l to tRH , the reheating
temperature, and compare it to the perturbative production.
Combining Eqs. (2.180), (2.184), and (2.186), we deduce the density of S at tP H ,
nPs H

m2 π μMP π μMP
μmφ 161 μMφ mtP H μmφ 4√6 m2φ μMP mφ 4√6 m2
nPs H = 2
e P = e  e φ , (2.187)
2π 2π 2 2π 2

where we have set  = MP in the last approximation.25 This density will be diluted
by a factor

  2
tRH 2 φ−1 64m12
φ
= = (2.188)
tP H tP H μ8 MP4

between its production and the end of reheating, which gives us

π μMP
√
μ9 MP5 4 6 m2
nPs H (tRH ) = e φ . (2.189)
128π 2m11
φ

We can then compare this production to the one obtained by the direct pertur-
bative decay of the inflaton at time tRH = φ−1 , solving Eq. (2.179) without taking
into account the enhancement effect but taking into account the expansion rate:

μ4 MP4
s (tRH ) =
nRH , (2.190)
96π 2 m3φ

25 Toobtain the exact result, one should in fact solve numerically the combined set of Eqs. (2.276)
and (2.179). Taking   MP in the overall factor of ns is a valid approximation.
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 83

40 nsRH (t RH)
Log[10,ns (GeV3)]

30

20

10 nsPH (t RH)

0
2 4 6 8 10 12
Log[10, m (GeV)]
Fig. 2.9 Number density of a scalar as a function of μ produced in the preheating phase (nPs H )
compared to the one produced perturbatively at reheating time (nRH
s )

where we used the result obtained in the reheating phase, see Eq. (2.248), with
t = φ−1 . This is justified because it is the dominant phase in this case. We show
in Fig. 2.9 the two densities as a function of μ. For μ  1010, the exponential
production dominates the perturbative one. The reader can find a nice detailed
treatment of the bosonic condensation effect we just discussed in [8].

Exercise Redo the previous analysis for a potential V (S, φ) = λφ 4 .

Now that we understood how evolves the inflaton field and density from ti to tend ,
it is time to couple it to matter, and allow it to decay in order to reheat the Universe
by the production of a thermal bath of Standard Model particles. This is the aim of
the next chapter.

2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s]

Thermalization of the hot and dense primordial plasma produced by the decay of
the inflaton φ is far from being a simple subject. And, to be honest, it is far to be
an understood subject. The reasons of such difficulties are multiple. We can say
that computing processes in a matter dominated Universe, driven by Friedmann
expansion mixed with a non-thermal statistics, renders the subject sensitive to a
lot of theoretical uncertainties and/or questionable hypothesis. Paradoxically, once
84 2 Inflation and Reheating [MP → TRH ]

the Universe has thermalized, the story is much easier to describe. A thermal bath
of relativistic particles is, in some sense, quite Universal. Things then become more
complicated every time particles decouple, or during the Big Bang Nucleosynthesis,
in other words, every time the Universe becomes colder and goes through a phase of
mixed matter–radiation domination. In this chapter, we will review in detail how a
set of particles, generated by the decay of the inflaton, begins to interact sufficiently
strongly to overcome the expansion rate.

2.3.1 The Context

Before digging into the thermalization process, let us emphasize that before the
Universe reaches its thermal bath phase, temperature has, by definition, no meaning.
One should then solve any set of equations as a function of the time t, which is
the dynamical parameter we have then in hands. The problem is then to compute
the evolution of the content of the Universe, which is composed of a mix of non-
relativistic species (behaving as a dust, like the inflaton φ) and relativistic species
(radiation like the Standard Model particles, or any kind of relativistic matter or
gauge fields that is produced by the decay of the inflaton). As we will see, dark
matter can be one of them.26 Of course, these laws can be justified by the Einstein
equations we studied in the preceding chapter, however, if we neglect the space-
time curvature (the parameter k), we can recover the Boltzmann–Friedmann set of
equations just asking for the conservation of energy.

2.3.1.1 The Boltzmann Equation for the Dust (Inflaton

or Non-relativistic Fields)
We call dust or matter field, generically, a field that obeys the relation27 ρM ∝ a −3 .
To follow its evolution with time in an expanding Universe, let us first write the law
of conservation of energy for a decaying matter field φM of mass mM , width M
and density nM :

d(nM a 3 )
= −M (nM a 3 ), (2.191)
dt
and multiplying by mM on both sides, we obtain

d(ρM a 3 )
= −M (ρM a 3 ), (2.192)
dt

26 Dueto the relative large mass of the inflaton (mφ  1013 GeV), Standard Model particles
produced by its decay are obviously ultra-relativistic and can be considered as a form of radiation.
27 ρ ∝ a −3(1+w) with w = 0 and p = wρ.
M
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 85

where ρM = nM mM is the energy density of the matter (dust) field φM . Developing

the derivative, the preceding equation gives

ρ̇M + 3HρM = −M ρM . (2.193)

Notice that we recover exactly Eq. (2.140), which describes the evolution of the
inflaton field φ for the case k = 2, but where we have added in the equation a term
M ρM as we allowed the possibility for the matter field φM to decay. As we have
seen in the previous section, this is not surprising, as the inflaton, in a quadratic
potential, behaves, during its coherent oscillation phase, as a pressureless field (k =
2 ⇒ w = 0). That will be our first ingredient to understand the thermalization:

ρ̇φ + 3Hρφ = −φ ρφ . (2.194)

Notice that this equation is valid for any type of matter field which can, for
instance, dominate the Universe even after the reheating has occurred.

2.3.1.2 The Boltzmann Equation for the Radiation (Relativistic Fields)

To find the evolution of the density of radiation, one needs to use the first law of
thermodynamics in a system of internal energy U = ρR a 3 in contact with another
system transmitting heat dQ = −d(ρφ a 3 ) (represented by the decaying inflaton φ)
in a volume V = a 3 . The amount of internal energy dU = dQ − pR dV (pR being
the pressure of the relativistic gas) received by the radiation sector is28
ρR 3
d(ρR a 3 ) = −pR da 3 − d(ρφ a 3 ) = − da − d(ρφ a 3 ) (2.195)
3
where we used the usual
ρR
pR = (2.196)
3

relation of a relativistic gas (w = 1

3 ), as explained in the frame below. We then
obtain the equation

ρ̇R + 4HρR = φ ρφ , (2.197)

where we used Eq. (2.192):

d(ρφ a 3 )
= −φ (ρφ a 3 ). (2.198)
dt

28 We make an important hypothesis of a constant number of degrees of freedom in the relativistic

bath during all the process. That will not be the case once the Universe cools down as some particles
may decouple as we will discuss in the next chapter.
86 2 Inflation and Reheating [MP → TRH ]

Pressure of a gas: pR = ρ3R

Let us recall where the relation pR = ρ3R in (2.196) comes from. For that,
one needs to compute the pressure π of a homogeneous and isotropic gas of
relativistic particles. Consider a surface S hit by particles with a distribution
of momentum f (p) during a time t. The pressure π is defined as the force
applied by the collisions of the particles per unit of surface, which can be
written
|F| p
π= , with F = , (2.199)
S t
p being the transferred momentum.

p
t
v

One particle arriving on the surface with an angle θ , will bounce with an
angle −θ and will transfer to the surface a momentum |p| = 2p cos θ .
During the time t, this corresponds to dn particles of velocity v with
distribution f (p) in a volume (S × cos θ )vt:

|p|t ot = 2p cos θ S cos θ vtdn, (2.200)

(continued)
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 87

which gives (considering a relativistic gas, v = c):

d 3p
|p|t ot = 2pc St cos θ 2 f (p) . (2.201)
(2π)3

The pressure, given by the momentum transfer per unit of surface and unit
of time, is

1 |p|t ot p2
dπ = =  cos2 θf (p) dpdφd cos θ, (2.202)
2 St (2π)3

where  = pc is the energy of the relativistic particle of momentum p, and

we have defined the solid angle by d 3 p = p2 dpd = p2 dpdφd cos θ .
The 12 factor in the definition of dπ comes from the fact that only half
of the distribution should be considered, the one embedding the particles
with momentum directing toward the surface (see figure the above). After
integrating on cos θ between −1 and +1, we obtain
  
p2
π = dπ = cos2 θ d cos θ f (p) dpdφ
(2π)3

2 p2
= dpdφ
f (p)
3 (2π)3
 
1 +1 p2 ρ
= d cos θ f (p) 3
dpdφ = , (2.203)
3 −1 (2π) 3
 d 3p
ρ = f (p) (2π) 3 being the energy density, which can be written for a
radiation π = pR , ρ = ρR :
ρR
pR = , (2.204)
3
which is precisely Eq. (2.196).

The factor “4” in front of H in Eq. (2.197) compared to the factor “3” in
Eq. (2.194) can also be understood as a redshift effect. Indeed, it is natural to write
an energy density for a radiation ρR ∼ nR ER ∝ a −3 × a −1 , the first term coming
from the dilution by expansion, and the second term (a −1 ) being the redshift of
the energy ER due to the expansion. In this case, dρ dt ∝ −4HρR . Quantitatively
R

speaking, it is easy to understand that the pressure in Eq. (2.195) acts negatively,
“adding” in a sense some dispersion and thus rendering then dilution faster than in
the matter case.
88 2 Inflation and Reheating [MP → TRH ]

A last way to recover Eq. (2.197) is to apply directly the General Relativity
equation (2.42) adding the decay rate in the right-hand side, and replacing P = ρ3R
or, in other words, to use the stress–energy tensor (A.96)
⎛ ⎞
ρ 0 0 0
⎜0 ρ 0 0 ⎟
R
Tμν =⎜ 3 ⎟
⎝0 0 ρ 0 ⎠ . (2.205)
3
0 0 0 ρ3

Combining results above, we obtain the set of equations to solve

dρφ
+ 3Hρφ = −φ ρφ (2.206)
dt
dρR
+ 4HρR = φ ρφ (2.207)
dt
ρφ ρR
H2 = + (2.208)
3MP2 3MP2

MP l
with MP = √ ≈ 2.4 × 1018 GeV (as usual in all the book) the reduced Planck
8π
mass29 and H = H (t) = 3t2 . The last expression assumes a Universe purely matter
dominated (see the box below for details). A radiation dominated Universe follows
the law H = 2t1 . This approximation can be justified by the fact that we are dealing,
for time t φ−1 , with a Universe where the inflaton dominates still largely the
dynamic.

2.3.1.3 The Influence of the Nature of the Inflaton

It can also be interesting to see how the set of Eqs. (2.206) and (2.207) changes for
the decay of an inflaton with a generic equation of state Pφ = wρφ . Indeed, even if
intuitive, the previous equations describe the evolution of the density of energy ρφ
while the width appearing on the right-hand side in the width of the field φ, φ , and
not “ρφ .” To find the conservation of energy equation, one needs to rederive it from
the original equation for φ, (2.99):

φ̈ + 3H φ̇ + V  (φ) = −φ φ̇. (2.209)

29 Notice that for φ = 0, we recover the classical evolutions (ρφ ∝ a −3 , ρR ∝ a −4 ) for

independent systems with entropy conservation, the a −4 corresponding to the redshifted energy
E (a )
ER (a) = R a f , and af being the radius of the Universe just after the inflation.
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 89

Multiplying both sides by φ̇ and taking the mean, we obtain

φ̇ φ̈ + 3H φ̇ 2  + φ̇V  (φ) = −φ φ̇ 2 . (2.210)

Noticing that (2.95)

φ̇ 2  = ρφ + Pφ , (2.211)

and

ρ̇φ = φ̇ φ̈ + φ̇V  (φ), (2.212)

we can write

ρ̇φ + 3H (ρφ + Pφ ) = −φ (ρφ + Pφ ), (2.213)

or, if we include the contribution to the radiation,

dρφ
+ 3Hρφ = −φ (1 + w)ρφ . (2.214)
dt
dρR
+ 4HρR = φ (1 + w)ρφ . (2.215)
dt

The Hubble parameter and time

We can have a simple expression of the Hubble parameter H (t) for the two
cases of Universe matter dominated and radiation dominated. Indeed, neglect-
ing the curvature and cosmological constant (approximation completely valid
in the early Universe), the Friedmann equation (2.6) can be written as
 2
ȧ 8πG ρ
H =
2
= ρ= , (2.216)
a 3 3MP2

with ρ = ρM = n × M ∝ a −3 M for a Universe dominated by a massive

field of mass M and ρ = ρR = n × ER ∝ a −3 a −1 for a radiation dominated
Universe (where the energy is redshifted by a factor a −1 due to the expansion).
In this context, solving the Friedmann equation (2.216) for a(t) gives H (t) =
3t for a matter dominated Universe and H (t) = 2t for a radiation dominated
2 1

Universe, justifying the common approximation H  1t

90 2 Inflation and Reheating [MP → TRH ]

2.3.2 The (Non-thermal) Distribution Function

2.3.2.1 Time Evolution of the Densities

When the reheating phase of the Universe begins, the inflaton staying in a coherent
oscillation regime, we cannot yet evoke a “temperature.” Whereas distribution
functions will be the classical thermal Boltzmann or Bose–Einstein distributions
when the Universe reaches thermal equilibrium, the situation is more complex in the
early phase of reheating. The time t is then the more justified dynamical parameter
to describe the evolution of the Universe.30 The first set of equations describes the
evolution of the matter field (the inflaton). During all this phase, the inflaton will
dominate the Universe. It is then justified to use the relation H (t) = 3t2 that we
extracted from Eq. (2.216). We are in a period much before the lifetime of φ, which
means t φ−1 , or H φ . The problem is then clearly set: one has a system
of 3 equations governing 3 parameters (ρφ , ρR , a) as a function of a dynamical
parameter, the time t. We can then first find an analytical solution of Eq. (2.206).

2.3.2.2 The Matter

Let us begin by the evolution of the inflaton density in time. Defining Xφ = t 2 ρφ
and using H = 3t2 , one can integrate Eq. (2.206) between31 tend and t

dXφ Xφ
= −t 2 φ 2 ⇒ Xφ (t) = Xφ (tend )e(t −tend )φ
dt t

tend 2 −(t −tend )φ
⇒ ρφ = ρφ (tend ) e . (2.217)
t
√
ρφ (tend )
Notice that, at the end of the inflation, H = 2
3tend = √ . We can then write
3MP

4 2
2
ρφ (tend )tend = M , (2.218)
3 P
which implies

4 MP2 −(t −tend )φ

ρφ (t)  e . (2.219)
3 t2

30 Itis possible to find in the scientific literature the parameter v = t × φ to keep the dynamical
parameter dimensionless.
31 We will consider in all this section the time t
end as being the time just at the end of the inflation,
at the beginning of the coherent oscillation.
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 91

2.3.2.3 The Radiation

We can make the same exercise for the radiation density of energy ρR . Replacing H
by 3t2 in Eq. (2.207) and defining XR = t 8/3 ρR , one needs to solve
 t
dXR
= t 8/3 φ ρφ ⇒ XR = φ ρφ (tend )tend
2
eφ tend z2/3e−zφ dz,
dt tend
(2.220)

or, using Eqs. (A.123) and (A.124),

−2/3 5 5
XR = φ 2
ρφ (tend )tend eφ tend γ ( , φ t) − γ ( , φ tend ) , (2.221)
3 3

where γ (α, x) is the incomplete gamma function. Then,

eφ tend 5 5
ρR = φ ρφ (tend )tend
2
8/3
γ ( , φ t) − γ ( , φ tend ) . (2.222)
t 3 3

In the regime φ t 1, we can use the approximation Eq. (A.124), to write

     
5/3 5/3
3 2 e
φ tend tend 4 φ MP2 tend
ρR  φ ρφ (tend )tend 1−  1− ,
5 t t 5 t t
(2.223)

2 = 4 M 2 , Eq. (2.218).
where we used ρφ (tend )tend 3 P

2.3.2.4 The Scale Factor

To compute the evolution of the scale factor as a function of time, one needs to recall
the definition of the Hubble parameter, Eq. (2.216),
 2/3
2 ȧ t
H (t) = = ⇒ a(t) = a(tend ) . (2.224)
3t a tend

2.3.2.5 Summary
Finally, the evolution of the energy densities and the scale factor in the reheating
phase of Universe can be written as follows:

4 MP2 −(t −tend )φ 4 MP2

ρφ (t)  e  (2.225)
3 t2 3 t2
  
4 φ MP2 tend 5/3
ρR (t)  1− (2.226)
5 t t
92 2 Inflation and Reheating [MP → TRH ]

 2/3
t
a(t) = a(tend ) , (2.227)
tend

where we have considered the regime tend t φ−1 , in other words, well before
the end of reheating, when the Universe is still largely dominated by the energy of
the inflaton coherent oscillations.

Exercise Find the solutions for ρφ and ρR considering a generic equation of state
for ρφ : pφ = wρφ .

We plot the evolution of ρφ (t), ρR (t), and a(t) in Fig. 2.10, where we fixed
tend = 10−37 s (see below). The width of the inflaton is fixed by its effective Yukawa
coupling yφ . Indeed, it is common in the literature to write the inflaton coupling to
the bath under the form (see Eq. B.182)

yφ2
Lφ,bat h = yφ φ f¯f ⇒ φ = mφ , (2.228)
8π

where f represents Standard Model fermions.32 The coupling in Eq. (2.228) is

obviously not gauge invariant and should “include” in some way all the degrees
of freedom of the Standard Model (and beyond the Standard Model) particles
joining the thermal bath. This type of term can be written in a gauge invariant
)(H ∗ f )
way through higher dimensional operators like φ (f H  2 , for instance.33 In any
case, naively, an invariant term φ|H |2 should effectively dominate the production
rate. The expression (2.228) is then convenient because the width can be expressed
as a function of a dimensionless parameter yφ but should only be considered as a
practical approach without real meaning, physics-wise.
We can make some comments on the Fig. 2.10. First of all, we notice that ρR
reaches a maximum at around t = tmax  5 × 10−36 s. The value of tmax is
in fact highly dependent on the value of tend , which is itself √very dependent on
ρ
the inflationary model being considered. Given that Hend = √ end with ρend =
3MP
m2φ φ2  m2φ MP2 , tend  m1φ  10−37 s for mφ = 3 × 1013 GeV. The behavior
of the radiation energy density is quite easy to understand. Indeed, at the beginning,
the density of radiation is extremely small due to inflation. Then it increases due
to the inflaton decay. However, as time passes, the dilution and redshift effect
tend to diminish the radiation density. It is then a fight between the injection of

32 Suppose a coupling to bosons is also possible.

33 Alternatively,one can also draw a loop diagram with two Higgs and a fermion which will give
the effective decay φ → f f .
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 93

2.5 × 1068

2.0 × 1068
rR (g.cm–3)

1.5 × 1068

1.0 × 1068

5.0 × 1067

0
–37.0 –36.5 –36.0 –35.5 –35.0

Log10 [t] (seconds)

100 mF=3.1013GeV
yF= 3.10–5
Log10[ r] (GeV.cm–3)

rF
95

90
rR
85

80

75

70
–36 –34 –32 –30 –28 –26
Log10 [t] (seconds)

Fig. 2.10 Evolution of the density of radiation energy (in g.cm−3 , top) and the density of inflaton
(in GeV.cm−3 , bottom) for mφ = 3 × 1013 GeV, yφ = 10−5 , and tend = 10−37 s (see the text for
details)

energy through the inflaton decay (proportional to t) and its dilution (proportional
to a −4 ∝ t −8/3 , see Eq. 2.227). This eventually results in an overall decrease of ρR
with time for t > tmax , albeit slower than a −4 (ρR ∝ 1t for tmax < t < treh ). Notice
also the value of the density, compared to the density of water (1 g.cm−3, density
94 2 Inflation and Reheating [MP → TRH ]

of the Universe at the CMB time) or the matter density of the Universe nowadays
(3 × 10−30 g.cm−3 ).
Observing the comparison between the inflaton density and the radiation density,
we notice that radiation reaches the inflaton density for a time t = treh  φ−1 
10−26 s for yφ = 10−5 (corresponding to a width φ = 119 GeV). This is the
beginning of the radiation domination era.34 This is also logical, as the inflaton
dilution is governed by the expansion rate a −3 (t), whereas the radiation still receives
energy from its decay, compensating partly the dilution factor.

Exercise Find the expression for tmax , the time when the radiation density reaches
its maximum.

2.3.2.6 The Distribution Function

In the preceding section, we computed the total density of radiation ρR . But the key
ingredient when one needs to compute any process is the energy distribution. More
precisely the distribution function is defined by

d 3p p2
dn = dn(p, p + dp) ≡ f (p) = f (p) dpd , (2.229)
(2π)3 (2π)3

where dn is the density of particles having momenta between p and p + dp and

d = d cos θ dφ the solid angle (in the momentum phase space). For the reader
who is not familiar with the concept of distribution function, we encourage him/her
to take a look at Sect. 3.1.3. The classical distributions f (p) = ep/kT1 ±1 one can
find in a lot of situations (Fermi–Dirac, Bose-Einstein, or even Maxwell) are only
valid when the particles are in a thermal bath, i.e. in equilibrium at a temperature T .
This is clearly not the case in the beginning of the reheating phase, where the rate
of interaction radiation-radiation (decay product f on decay product f ) is too weak
to counterbalance the expansion (n σ vf ↔f < H ). It is only when the number of
relativistic particles reaches a certain threshold that thermalization is achieved, at a
time tt h . We will discuss in great detail the thermal case in the following section.
To illustrate the process, let us imagine the inflaton decaying into two relativistic
particles with a rate φ per second. That means that every second, 2 φ particles
m
are created with an energy 2φ . After some time t, this energy will be redshifted
by the scale factor a(t), while the number density of particles produced at the time
t will also be diluted by a 3 (t). We illustrate the phenomena in the drawing below
(Fig. 2.11).

34 It is important to point out that the radiation has thermalized well before treh .
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 95

n n

tend t1

mφ mφ
(tt ( mφ
2/3
E end E
2 2 1 2

t2

mφ
( tt ( (t (
mφ 2/3 mφ t1 2/3
end E
2 2 2 2 2

Fig. 2.11 An illustration of the evolution of a spectrum for a decaying particle of mass mφ into
m
two particles. They are produced with energy 2φ at tend and then are redshifted at t1 while being
diluted by the expansion, and so on. The spectrum, which has initially a Dirac-delta form, becomes
flatter and flatter with time

More concretely, to compute the spectrum, one needs to know what is the density
of particles having their energies between E and E + dE at a time t. These particles
have been produced at a time ti , which is defined by35
 2/3  2/3
a(ti ) ti mφ ti
E = E(t) = E(ti ) × = E(ti ) = , (2.230)
a(t) t 2 t

which implies
  √
2E 3/2
3 2E 1/2
3 2 √
ti = t ⇒ dti = dE t = 3/2 t EdE. (2.231)
mφ mφ mφ mφ

In the meantime, the number density of particles produced at ti , with the energy
E(ti ) = 2φ , has been diluted by a scaling factor a −3 :
m

mφ a 3 (ti )
dn(E, t) = dn(E, E + dE)t = dn( , ti ) 3 . (2.232)
2 a (t)

35 By simplicity, we considered a 2-body decay in this section.

96 2 Inflation and Reheating [MP → TRH ]

Then, if φ × nφ (ti ) × a 3 (ti ) particles are produced per second at time ti , during
time dti given by Eq. (2.231), dti × φ × nφ × a 3 (ti ) particles are produced between
ti and ti + dti . This means
√ 
a 3 (ti ) 3 2√ ti 2
dn(E, t) = dti φ nφ (ti ) 3 = 3/2 EdEtφ nφ (ti ) , (2.233)
a (t) mφ t

where we used a ∝ t 2/3 (Eq. 2.227). Using ρφ = nφ × mφ , we can write

√ √
3 2 Eφ
dn(E, t) = 5/2
ρφ (ti )ti2 dE. (2.234)
mφ t

With the help of Eq. (2.225), valid for φ t 1,

4 2
ρ(t) = 3MP2 H 2 (t) = M ,
3t 2 P
one obtains

√ √  2/3
dn 4 2 E φ MP2 tend mφ mφ
= 5/2
, with <E< , (2.235)
dE mφ t t 2 2

d p 3
We can then extract the distribution function defined by dn = f (p) (2π) 3

√
8 2π 2 φ MP2
f (p) = 5/2
. (2.236)
p3/2 mφ t

dn
We show in Fig. 2.12 the spectrum dE as a function of the energy at different
− 37
epochs, from t = tend = 2 × 10 s ( m1φ ) to t = 10−36 s. We fixed the inflaton
width through its effective Yukawa coupling yφ as we discussed in Eq. (2.228).
A quick look at Fig. 2.12 teaches us interesting features of the beginning of the
reheating process. First of all, if one considers the end of inflation at t = tend =
10−37 s, the spectrum becomes flat quickly. At 10−36 s already there is no trace of
the monochromatic injection. Whereas the dependence on the width φ will just be
an uplift of the whole spectrum, the lowest energy particles that began to populate
the Universe will be the first to thermalize, much before the decay process of φ
finishes.
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 97

1 × 10 25

2.10–37s
8 × 10 24 yΦ=10–5
dn/dE (GeV2)

6 × 10 24

4 × 10 24
5.10–37s

2 × 10 24
10–36s

12.5 12.6 12.7 12.8 12.9 13.0 13.1

Log10[E] (GeV)

Fig. 2.12 Evolution of the spectrum of the products of inflaton decay at different times for yφ =
10−5

Exercise Compute the distribution function in the case of the decay of the inflaton
into n particles.

2.3.3 End of the Thermalization Process: Transition Toward

a Thermal Bath

2.3.3.1 Understanding the Process

We show in Fig. 2.13 the evolution of the distribution function in the plasma as
a function of time, from the end of the inflation (t  m1φ  10−37 s) to the
beginning of the thermal era (t  10−30 s). We see clearly how the shape evolves
from a delta function to a flat function (once the energies of the products begin
to redshift), to the appearance of a thermal “bump” for the less energetic particles
to a complete thermal distribution that will redshift also with time. We clearly see
how the spectrum passes from a delta function distribution, when all the particles
m
produced are given a momentum p = 2φ to a redshifted flat distribution as we study
in Fig. 2.12 to a Boltzmann (or Fermi–Dirac/Bose–Einstein) classical distribution,
as we will discuss in the following chapter devoted to the thermal bath. The complete
treatment of the thermalization is very complex and deserves a book (or even several
books) in itself. However, we can try to give an idea of how and why the transition
physically occurs.
To understand this phenomenon, one needs to remember that the processes that
are responsible for the thermalization are long range interaction, where the particles
98 2 Inflation and Reheating [MP → TRH ]

Fig. 2.13 Illustration of the evolution of the distribution function in the plasma with time, from
the beginning of the oscillation (t  m1φ  10−37 s) to the beginning of the thermalization era
(t  10−30 s)

exchange massless bosons or scalars between themselves (the electroweak phase

transition has yet to occur, and all the particles in the plasma are massless at this
energy scale). In this case, the interaction cross section between particles in the
plasma is proportional to 1s , s being the classical center of mass energy squared. It
means that the particles with the lowest energy will interact more efficiently between
themselves to form a thermal bath, competitive with the expansion of the Universe,
H , which has a tendency to make the Universe colder. In other words, at a time when
m
the particles (all produced at an energy E = 2φ ) having low redshifted energies are
sufficiently present, their interaction rate n σ v > H , and the Universe enters in a
thermal phase.
The transition phase between the thermalization and the radiation dominated
Universe is then a phase where the bath is thermal, in the sense that any particle
m
f produced with an energy 2φ is “caught” by particles in thermal bath because
their density is sufficient to ensure nbat h σ vbat h−f > H . The Universe is matter
dominated because the inflaton is still the source of the energy injected in the bath,
but its non-inflaton content is radiation. This is the time when the energy injected in
the primordial plasma is maximal: the production rate is large enough to compensate
the dilution due to the expansion rate. This feature is easy to see in Eq. (2.226),
where a maximum is reached for ρR before a decrease proportional to 1t . The
transition phase is terminated once the decay rate of the inflaton cannot compete
anymore with the inflation scale (φ < H ). The Universe then enters in a radiation
dominated era, until the last scattering time (the decoupling time, see Sect. 3.3.4).
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 99

2.3.3.2 Computing the End of Thermalization Process

The process of thermalization ends when any particle produced by the inflaton
m
decay, with an energy 2φ , is trapped by particles already in equilibrium with
themselves, in a time shorter than the Hubble timescale H −1 . The physics behind
the thermalization mechanism is quite complex and effects as LPM (Landau–
Pomeranchuk–Migdal) effect, splitting processes, and loss of energies in inelastic
scatterings have to be taken into account. The physics needed deserves a complete
book by itself, and we will then just keep the result of interest in our case: the energy
lost by inelastic scattering in a plasma of temperature T , for a particle of energy E,
with a gauge coupling g with the particles in the plasma is given by
  
dE E 4 2 E
∼g T 4 2
= xg T , (2.237)
dt inelastic T T

where we have embedded the uncertainties in the complex treatment of the physics
in x, which is of the order of unity. If we denote by tt h the time when the loss of
m m
energy E is equal to 2φ (i.e. when a produced particle with energy 2φ will lose
all its energy by the inelastic scattering with particles from the plasma), we obtain

E E mφ /2
xg T 4 2
= = −1 (2.238)
T t H

with
  ρ 1/4
1 ρφ R
H = and T = , (2.239)
MP 3 α

where we have define the temperature by analogy with the thermal bath (see
Eq. 3.35).

gρ π 2 4 gρ π 2
ρR = T = αT 4 with α = , (2.240)
30 30

where gρ is the sum of degrees of freedom in the thermal plasma.36 Replacing E by

mφ
2 in Eq. (2.238), we obtain

 3/8 
mφ ρR mφ ρφ
xg 4
3/8
= . (2.241)
2 α 2MP 3

36 When calculating g
ρ , one should be careful between the fermionic and bosonic states, as one can
see in Eq. (3.26): gρ = 106.75 (α  35) for the Standard Model, and gρ = 213.5 (α  70) in
supersymmetry.
100 2 Inflation and Reheating [MP → TRH ]

Using Eqs. (2.225) and (2.226) at time tt h , we obtain

 4/5  3/5  8/5 4/5

2 gρ π 2 1 mφ
tt h = , (2.242)
9 24 xg 4 6/5 3/5
MP φ

or

1.5  g 3/5  0.1 32/5 

mφ 1/5 
10−5
5/6
ρ
tt h = 8/5 × 10 30
s,
x 106.75 g 3 × 1013GeV yφ
(2.243)

where we used Eq. (2.228). We show in Fig. 2.14 the evolution of the inflaton density
ρφ and radiation density ρR as a function of time for different values of yφ . We see
that the thermalization time is reached before the domination of the Universe by the
radiation. This means that during a period of time, the evolution of the Universe is
dominated by the inflaton energy density, while the distribution function of the bath
is non-thermal (from tend to tt herm ). It then enters in a phase, still dominated by
the inflaton density, but with a thermal bath (from tbat h to trh , the reheating time),
before being ruled by the radiation density until the decoupling time (from trh to
tcmb ). Viewing Fig. 2.14, we understand easily that increasing yφ has two straight

100 m)=3.1013GeV 100 m)=3.1013GeV

Log10[r] (GeV.cm-3)

Log10[r] (GeV.cm-3)

ρΦ ρΦ
95 y)=10 –5
95 y)=10–3
90 90
ρR ρR
tthern=5.8 10–33
tthern=1.5 10–30

85 85

80 80

75 75

70 70
–36 –34 –32 –30 –28 –26 –36 –34 –32 –30 –28 –26
Log10[t] (seconds) Log10[t] (seconds)

100 m)=3.1013GeV
ρΦ
Log10[r] (GeV.cm-3)

95 y)=10–1

90
ρR
tthern=2.5 10–35

85

80

75

70
–36 –34 –32 –30 –28 –26
Log10[t] (seconds)

Fig. 2.14 Evolution of the inflaton density ρφ and radiation density ρR as a function of time for
different values of the inflaton coupling to the Standard Model yφ , compared to the thermalization
time for the same value of yφ
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 101

effects: the radiation density being larger and faster, the reheating period can be
reached earlier, and the thermalization time, for the same reason, tends also to shift
toward the end of inflation time and the reheating time. The interesting point is then
to understand how the production of dark matter can be computed in this phase of
the thermalization era.

2.3.4 Dark Matter Production During the Non-thermal Phase

of the Reheating

2.3.4.1 The Context

We have now all the tools in hand to compute the amount of dark matter
produced during the process of thermalization, from the end of inflation tend to the
thermalization time tt h . There are two possible sources: the inflaton decay and the
scattering of Standard Model particles already produced. In any case, the exercise
consists in solving the Boltzmann equation

dnχ
+ 3H nχ = R(t), (2.244)
dt
where R(t) is the production rate of dark matter, from the end of inflation tend to the
thermalization time tt h . Defining the variable Xχ = t 2 nχ , and using H = 3t2 from
Eq. (2.216), the solution of Eq. (2.244) can be written as
 2  t
tend 1
nχ (t) = nχ (tend ) + 2 t 2 R(t  )dt  . (2.245)
t t tend

Let us apply the calculation to the production of dark matter from decaying
inflaton and scattering from the (not yet thermal) particles in the plasma.

2.3.4.2 Direct production by inflaton decay

This source of production is always present, and very difficult to avoid, except by
invoking specific symmetries that would seclude completely the dark sector from
the inflationary sector. As we will see later, even if the dark matter is completely
secluded from the inflaton sector, loop processes are able to produce dark matter
in a sufficiently large amount to fill the Universe. To compute the density of a dark
matter candidate χ produced by direct decay of the inflaton is straightforward. One
needs to solve the Boltzmann equation for χ:

dnχ ρφ
+ 3H nχ = lBrχ φ , (2.246)
dt mφ
102 2 Inflation and Reheating [MP → TRH ]

where Brχ is the branching fraction of decay of φ into l dark matter particles.
Defining Xχ = t 2 nχ and using Eq. (2.225), we can write

dXχ 4 φ 2
= l Brχ M (2.247)
dt 3 mφ P

from that one can extract

 
4 φ MP2 tend Brχ 2 MP2 tend
nχ (t) = l Brχ 1− =l y 1− ,
3 mφ t t 6π φ t t
(2.248)

where we used Eq. (2.228) in the last expression. With the help of Eq. (2.242), one
can then compute the amount of dark matter in the Universe at t = tt h , tt h tend :

16/5 16/5
l Brχ 2 MP2 l Brχ yφ MP
χ (tt h ) =
ndec yφ = (2.249)
6π tt h (6π)(8π)3/5δ 1/5
mφ

with
 8/5  4/5  
7 × 106  gρ 3/5 0.1
3/5 32/5
1 2 gρ π 2
δ= 
xg 4 9 24 x 8/5 106.75 g
(2.250)

from Eq. (2.242), or

  
3/5  16/5 
g 32/5
1/5
l Brχ x 8/5 106.75 yφ 3 × 1013
nχ (tt h ) = .
7 × 10−32 gρ 0.1 10−
5
mφ
(2.251)

We remind the reader that the coefficient x represents the uncertainties and the
unknown thermalization exact process and is of the order of 1.

2.3.4.3 Production by Scattering

The second source of dark matter production is the thermal bath. The scattering of
Standard Model particles can lead to a large amount of dark matter, especially if
the annihilation cross section is highly energy dependent. To compute the density
of particles produced this way, one needs to solve the Boltzmann equation for
the dark matter field, Eq. (2.244). The rate R(t) can be computed directly from
the expressions (B.87) and (B.95). Indeed, the interaction rate (the number of
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 103

interactions per unit of time) dwf i is given by (B.86)

nf
! V
dwf i = V × (2π)4 δ (4) (pf − pi )|Tf i |2 d 3 pf .
(2π)3
f =1

To understand each term of the equation above, notice that Tf i has been defined
|S |2
from the S matrix element Sf i = 1 + (2π)4 δ (4) (pf − pi )Tf i , and wf i = Tf i ,
"n V d 3 p
which gives, in a phase space of volume37 V × d 3 p, dwf i = wf i × ff=1 (2π)3f
(and using the relation [(2π)4 δ (4) (pf − pi )]2 = V × T × (2π)4 δ (4) (pf − pi )).
However, the precedent reasoning was made in a non-covariant form (quantum
mechanics). As explained in Sect. B.4.1.4, one needs to normalize the wave
function by ψi → √2E 1
ψi to render the theory Lorentz invariant (normalization
iV
corresponding to 2Ei particles i in volume V ). We then define the matrix element
Mf i by (see Eq. B.87)
n ⎛ ⎞
nf
!i
1 ! 1
Tf i = √ ⎝  ⎠ Mf i .
i=1
2Ei V f =1
2E f V

The rate R (interaction per unit of time for a given initial state) is then given by
Eq. (B.88)
nf
V 1−ni !ni
1 ! d 3 pf
4 (4) 2
R = dwf i = (2π) δ (pf − pi )|M fi | . (2.252)
(2π)3nf i=1
2Ei
f =1
2Ef

However, in the situation where the momentum of initial state is not definite, but
follow a statistical law (as it is the case in the early Universe) one should multiply
this rate by the number of quantum states of the initial particles (1 and 2) weighted
by their distribution functions f1 and f2 , respectively:38

!
ni
V
d R̃ = fi d 3 pi × R, (2.253)
(2π)3
i=1

37 To see it easily, one can use uncertainty principle of Heisenberg, for instance, where the number
3 3p d3p
of states Nf is d xd h3
, and h̄ = 1 ⇒ h = 2π, which implies Nf = V(2π) 3 .
38 Noticethat the dark matter particle is never in thermal equilibrium with the plasma during the
whole production process.
104 2 Inflation and Reheating [MP → TRH ]

which gives, combining with Eq. (2.252),

 !
ni nf
!
d 3 pi d 3 pf
R̃ = V fi 3
(2π) 4 (4)
δ (pf − pi )|Mfi | 2
(2.254)
(2π) 2Ei (2π)3 2Ef
i=1 f =1

with R̃ the rate of interaction, per unit of time, in the volume V . To obtain the rate
R(t) for the Boltzmann equation (2.244), one just needs to divide R̃ by V , which
gives

 !
ni nf
!
R̃ d 3 pi d 3 pf
R(t) = = fi (2π) 4 (4)
δ (pf − pi )|M f i | 2
.
V (2π)3 2Ei (2π)3 2Ef
i=1 f =1
(2.255)

A little remark is here: we notice that Tf i has a dimension of (Energy)4 , and

then Mf i has dimension (Energy)4−ni −nf , which is obviously a dimensional for
2 → 2 scattering.
In our case of interest (2 → 2 scattering, 1 + 2 → 3 + 4), using the formula for
the 2-body phase space final state Eq. (B.100) gives

d 3 p3 d 3 p4 d 13
3 3
δ (4) (pf − pi ) = (2.256)
(2π) 2E3 (2π) 2E4 512π 6

with d 13 being the solid angle between particles 1 and 3 in the center of mass
frame. In the relativistic case for the initial particles,39 pi = Ei . We can then write

d 3 p1 d 3 p2 4π × 2π d cos θ12 E1 dE1 E2 dE2

= , (2.257)
2E1 2E2 4

where cos θ12 is the angle between particles 1 and 2 in laboratory frame. Combining
with Eq. (2.255), one obtains

 
E1 E2 dE1 dE2 d cos θ12
R(t) = f1 f2 |Mf i |2 d 13 , (2.258)
1024π 6

with f1 and f2 the distribution functions given by Eq. (2.236).

39 That is the case as the photons and/or Standard Model particles are the 1 and 2 particles, much
lighter than the energy at the reheating time, and therefore massless.
2.3 Reheating: Non-thermal Phase [10−37 − 10−30 s] 105

n+2 n
We can then suppose |M|2 =  n+2 (corresponding to σ ∼ n+2 ). Setting
s 2 s2

s = (P1 + P2 )2 , Pi being the quadrivector of incoming particle i, and considering

massless particles40 (m1 = m2 = 0), one can easily integrate Eq. (2.258), which
gives
  n+3 2   n+3 2
4φ2 MP4 mn−2
φ tend 3  tend 3
R(t) = 1− = 2 1−
(n + 3)2 (n + 4)πn+2 t 2 t t t
(2.259)

with

4φ2 MP4 mn−2

φ
= . (2.260)
(n + 3)2 (n + 4)πn+2
n+2
Exercise Compute the production rate in the t and u case (|M|2 ∝ t 2 and |M|2 ∝
n+2
u 2 ).

We can then use the solution Eq. (2.245) to compute nχ (tt h ), neglecting tend
tt h

13 26
n− 14
4 φ5 MP5 mφ 5

nscat
χ (tt h ) = , (2.261)
(n + 3)2 (n + 4)πδ n+2

with δ given by Eq. (2.250). We can then combine Eq. (2.249) with Eq. (2.261)
to obtain the ratio R between the production of dark matter generated by the
inflaton decay and the production due to the scattering of particles from the plasma.

40 There are different reasons to consider massless particles: we are in ultra-high energy regimes,

the particles are thus ultra-relativistic, and in the meantime, the electroweak phase transition giving
masses to the Standard Model sector has not yet occurred.
106 2 Inflation and Reheating [MP → TRH ]

We obtain

ndec
χ (tt h ) l Brχ 8π 2 (n + 3)2 (n + 4) n+2
R(tt h ) = =
nscat
χ (tt h ) 3 mnφ MP2
 2 n+2  n
l Brχ (n + 3)2 (n + 4) 10−5  1013
= ,
1.5 × 10−2n−5 yφ 1015 mφ
(2.262)

2.4 Reheating: Thermal Phase [10−30 − 10−28 s]

First of all, the first time a graduate student hears about “reheating” he/she should
naturally think that the Universe was cold, then hot, and then cold again before
having to “re”-heat. That is not exactly the case in fact. This misuse of language is
very similar to the one we find when discussing about “re-ionization,” “dark age,”
or “recombination” epochs that we will discuss later on. One reference that the
reader can read and is quite accessible to any kind of public is the mini-review
made by Lev Kofman [9] for the 60th birthday celebration of Igor Novikov. Lev
Kofman collaborated a lot with Andrei Linde and Alexei Starobinsky, two fathers
of inflationary models. The term “reheating” is an anachronism as he was used at
the time models of inflations supposed a hot and dense phase before an inflationary
phase which refresh the Universe, heated again by the inflaton decay: thus the name
“re-heating.” Contrarily to the previous section, we will consider from now on that
the standard model plasma is thermalized, which means that the standard model
particles are in thermal equilibrium with themselves. In this section we will then
neglect all the effects of thermalization in the preheating phase studied before.
It has the advantage of ignoring the uncertainties encoded in the x parameter of
Eq. (2.237) that was telling us when the bath enters in thermal equilibrium. In a
sense, in this chapter we will consider that the Universe story began at t = tt h . This
is a generic statement that we can make. The later we consider the initial condition,
the less dependent we are from any cosmological or early Universe construction.
The extreme Weakly Interacting Massive Particle (WIMP) case, as we will see, is
even completely independent from the reheating process as we suppose in this case,
a thermal equilibrium state between the dark matter and the Standard Model bath.

2.4.1 Understanding the Reheating

We will discuss in detail in Sect. 3.2.4 the general process of thermalization of a

dark bath heated by the standard model plasma. In this section, however, we will
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 107

concentrate on the processes happening before this epoch: the thermalization of

the standard model bath in itself. In other words, we will describe how a bath of
standard model particles produced by the decay of an inflaton reaches a thermal
equilibrium. The spectrum of the particles in the plasma at the production can
be flat or monochromatic, but the equilibrium is then established in two phases:
kinetic equilibrium first where the scattering (elastic or inelastic) or annihilation
2 → 2 processes redistribute the temperature to reach a kinetic equilibrium.
However, elastic scattering alone cannot bring the decay products of the inflaton into
thermal (chemical) equilibrium because the number densities in general will not be
correct. Annihilation, which does not change the number of particles, will neither
be of any help. Chemical equilibrium is reached after kinetic equilibrium when
2 → 3 collisions become operative (see Sect. 3.2 for the difference between kinetic
and chemical decoupling and Sect. 3.2.4 for some insights on the thermalization
process).
Supposing that the thermal bath is populated by standard model particles f and
f¯ produced by the inflaton decay, one needs first to approximate at which time
(and thus at which temperature) the decay occurs. A first approximation (called
“instantaneous reheating”) is that the decay of the inflaton φ begins to be effective
at a time td corresponding to a temperature Td respecting φ = H (Td ), φ being the
width of the inflaton.41 Indeed, one can interpret the inverse of the Hubble constant
H −1 (t) as the “doubling time” of the Universe: during the time t = 1/H (t), the
radius of the Universe has doubled its size by doubling the scaling factor42 a. With
this interpretation, the condition φ = H (Td ) corresponds to compute the time td
at which, for every decay φ → f f¯ the Universe has doubled its size, and the
(physical) density of its products f and f¯ is not diluted anymore by the expansion:
the Universe, after being matter dominated by the massive presence of the inflaton,
becomes radiatively dominated by (effectively) massless particles f, f¯. This occurs
much before the electroweak phase transition and justifies the massless hypothesis.

The Hubble constant in the thermal epoch

Instead of taking the time as the dynamical parameter, once the thermal
equilibrium is reached, it is easier (and more physical) to work with the
temperature as the dynamical parameter. We let the readers jump to Sect. 3.1.4
to recover the characteristics of number densities and energy densities of
a thermal Universe. If one considers a radiation dominated Universe, the
evolution of the Hubble parameter as a function of the temperature is easily

(continued)

41 Or more precisely, the branching ratio of the width into Standard Model particles. Decays into

dark or hidden sectors are allowed as we will see in the following section.
42 H (t) = ȧ ⇒ da = a H dt, so during dt =
H (t) , da = a: the radius of, the Universe has
1
a
doubled in size.
108 2 Inflation and Reheating [MP → TRH ]

derived through the Friedmann equation (2.6),

ρR (T )
H2 = , (2.263)
3MP2

2
with ρR (T ) = gρ π30 T 4 (Eq. 3.35) the radiative energy density of the
Universe, gρ being the effective degrees of freedom of the plasma (Eq. 3.37).
Replacing the expression of ρR (T ) in Eq. (2.263), we obtain
  g 1/2 T 2
gρ T 2 T2 ρ
H (T ) = π  0.33gρ1/2 = 3.42 . (2.264)
90 MP MP 106.75 MP

Before that time, particles produced in the bath were diluted by the expansion
(see Fig. 2.15 for an illustration of the process). Solving φ = H (Td ) with the
expression (2.264) and approximating at a first step Td  TRH , we obtain
 1/4 
90 
φ = H (Td ) ⇒ Td  TRH = MP φ  MP φ . (2.265)
gρ π 2

H( Td ) = *d

Inflaton
SM particles (f) density n(f) is constant

H( Td ) > *d

density n(f) decreases

H( Td ) < *d

density n(f) increases

Fig. 2.15 Illustration of the principle of dilution of the decay products of the inflaton fields. We
can easily notice that it is when the Universe reaches the temperature obeying d  H (Td ) that
the mechanism of population becomes efficient, and the decay products are not diluted anymore
by the expansion rate
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 109

In the case of Yukawa-type coupling yφ of the inflation to the particles of the

yφ2
bath: Lφ = yφ f¯f , we have φ = 8π mφ (Eq. 2.228), which gives finally
 1/4 
yφ 45
TRH = MP mφ . (2.266)
π 32gρ

The mass of the inflaton being extracted from WMAP/PLANCK data for
different kinds of inflationary models (see Sect. 2.2 for more details), a typical
mass of inflaton of 3 × 1013 GeV implies a reheating temperature of TRH =
5 yφ × 1014 GeV for gρ = 106.75, which is the Standard Model effective number
of degrees of freedom. This is the typical scale one can expect for a reheating
temperature from inflationary model constructions.

Instantaneous reheating and instantaneous thermalization: computing TRH

In the early Universe, the energy density is never totally dominated by one
or other species (inflation, radiation, matter, etc.). The processes are smooth,
and one usually needs to solve numerically the equations. That is why it
is possible to find different definitions of the reheating temperature TRH in
the literature. Once the inflaton is settled at its minimum, it decays with a
decay rate φ . One usually can consider that the thermalization process in
the early Universe is instantaneous. It means that during a doubling time
of the Universe, the new “photons” (by photons we mean all relativistic
particles in the primordial plasma) produced by the decay of the inflaton enter
automatically in equilibrium with the existing plasma [nγ σ vscat t ering (T ) >
H (T )]. This is called instantaneous thermalization. However, one can treat
the inflaton as a particle decaying instantaneously (instantaneous reheating)
or with a time delay. In the instantaneous reheating, we consider that the
inflaton decays when H = φ as we did in this section. The interpretation
being that the Universe is dominated by the radiation once the doubling time
of the Universe dominates the decay rate, see Fig. 2.15. The other possibility
is to consider that the time of reheating tRH can be approximated by φ−1 ,
the inflaton lifetime. This is justified by the fact that the radiation dominates
the energy budget of the Universe if the inflaton decays instantaneously at
the time φ−1 . A last possibility is to consider reheating as the time when
the density of radiation reaches the density of the inflaton (ρR = ρφ ). The
three definitions of the reheating time differ only by numerical factors of the
order one. We can see it by computing the reheating temperature in the second
case, where tRH = φ−1 . The Friedmann equation (2.6) should be written in
a matter dominated Universe, as the inflaton is the only source of energy of
density ρφ = nφ × mφ , nφ being the inflaton number density, which, as a

(continued)
110 2 Inflation and Reheating [MP → TRH ]

matter field, is proportional to a −3 :

   2/3
2
ȧ ρφ nφ mφ 3 nφ mφ
H =
2
= = ⇒ a(t) = t 2/3
a 3MP2 3MP2 2 3MP2

ȧ 21 2 2
⇒ H = = ⇒ H (tRH ) = = φ . (2.267)
a 3t 3tRH 3

To compute TRH , one supposes that just after tRH , the Universe is
π2 4
dominated by a radiation, of energy density ρR = gRH 30 TRH , with gRH the
relativistic degrees of freedom of the thermal bath at tRH . We then obtain

4 2 ρR (TRH ) gRH π 2 4
H 2 (TRH ) = φ = 2
= TRH ⇒ TRH
9 3MP 90MP2
 1/4 
40
= φ MP . (2.268)
gRH π 2

As we claimed above, another definition of the reheating temperature could

be when the expansion rate dominates on the decay rate, i.e. H (TRH ) = φ .
In this case all the results have to be “normalized” to a factor 2/3 every time
the Hubble parameter appears in the expression.

2.4.2 Non-instantaneous Reheating

The reheating temperature discussed above was calculated assuming an instanta-

neous conversion of the energy density of the inflaton field into radiation when the
decay width, φ , is equal to H . The reheating temperature is best regarded as the
temperature below which the Universe expands as a radiation dominated Universe,
−1/3
with the scale factor decreasing as a ∝ gS T −1 , which can be understood from an
entropy conservation argument. In this regard it has a limited meaning. For instance,
TRH should not be used as the maximum temperature of the Universe during the
reheating process. As we will see, the maximum temperature is in fact much larger
than TRH . One implication of this is that it is incorrect to assume that the maximum
abundance of a massive particle species of mass M, produced after inflation, is
suppressed by a factor e−M/TRH . We will, show in this section, how to calculate
the evolution of the temperature in a hot Universe, in presence of massive states.
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 111

Fig. 2.16 Difference

between an instantaneous R
treatment of the reheating
(top) and a treatment taking
into account the finite width
of the inflaton φ (bottom)
Instantaneous
reheating

1/TRH 1/T

Non-instantaneous
reheating

1/TRH 1/T

Let us consider first a model Universe with two components: the inflaton field
with an energy density ρφ and the radiation with the energy density ρR . We will
assume that the decay rate of the inflaton field energy density into light degrees
of freedom, from now on referred to as radiation, is φ H . This condition
ensures that we place ourselves in a regime where the decay of the inflaton is not yet
effective. This regime corresponds to the phase of domination of the inflaton, while
it begins to generate the thermal bath. During this first phase, the inflaton can be
considered as almost stable because its decay rate is below the expansion rate. We
will also assume that the light degrees of freedom are in local thermal equilibrium
(instantaneous thermalization.43) We show in Fig. 2.16 the difference between the
non-instantaneous decay of the inflaton and the instantaneous reheating treatment.
With the above assumption, one needs to describe the evolution of the density
of the different components of the Universe. We have 3 equations to deal with:
the Boltzmann equations for the inflaton field density and radiation density and the
Friedmann equation. We also have 3 parameters: the scale a, the time t (appearing
in the definition of Hubble parameter), and the temperature T . As the temperature
T is the only measurable quantity of the three, it seems natural to express t and a as

43 See Sect. 2.3 for the case of non-instantaneous thermalization.

112 2 Inflation and Reheating [MP → TRH ]

a function of T . This is done by combining the three equations cited above. Let us
begin by the inflaton density evolution.

2.4.2.1 Evolution of the Temperature During Reheating

To obtain the evolution of the density of energy and thus of the temperature, we
2
need to solve the set of Eqs. (2.206–2.208) with ρR = gρ π30 T 4 (Eq. 3.35). With the
hypothesis φ H in the first phase of reheating, the Universe is still dominated by
the inflaton, and Nφ = ρφ a 3 is thus almost constant during all the reheating process.
To solve the set of equations, one needs to choose first the dynamical parameter. It
was naturally the time t when we were analyzing processes in the preheating stage
of the Universe, before the formation of a thermal state. The more natural dynamical
parameter during the reheating phase, with the instantaneous reheating hypothesis,
is the temperature T . The first step is then to eliminate the two other dynamical
parameters t and a from the set of equations. Eliminating t is quite straightforward.
Indeed, noticing that

d da d da dT d dT d dT d
= = =aH =H ,
dt dt da dt da dT da dT d ln a dT
Equation (2.207) then becomes

dρR ρR ρφ
+ 4 dT = φ , (2.269)
dT a da H a dT
da

which gives, once we eliminate H through Eq. (2.208),


dρR ρR 3ρφ MP
+ 4 dT = φ . (2.270)
dT a da a dT
da

We see appearing little by little the solution. Indeed, considering ρφ a 3  constant

 3
⇒ ρφ = ρφi aai (ρφi and ai being the initial value of the density and scale factor,
2
just after the inflationary stage), and using ρR = gρ π30 T 4 = αT 4 , one can solve
Eq. (2.270) and obtain

1T4 ρφi ai3 MP
dX4 √
3 dT
4T +4 = 4 = φ 3 , (2.271)
da a a da a 5/2α
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 113

where we have defined X = a × T . Solving Eq. (2.271) for X, and expressing the
solution for T , one obtains

 a 4 √   a 3/2  a 5/2
4 i 12 3 i i
T = Ti4 + 2
φ MP ρφ
i
1 − . (2.272)
a gρ π a a

Notice that we recover the classical redshift evolution for the temperature if φ =
0. Indeed, this situation corresponds to no injection of entropy in the thermal bath.
Only adiabatic processes occur, and the thermal system is entropy conserved, T ×
a = constant.

Exercise Compute T as a function of a for a generic equation of state, ρφ ∝

a −3(1+w).

We show in Fig. 2.17 the evolution of the temperature as a function of the scaling
factor (normalized to its initial value, aai ). We notice the three distinct phases in the
evolution of T . After a sharp increase in the temperature, it decreases following a
a −3/8 law when inflaton density still dominates the Universe and finally a classical
entropy conserving a −1 law when T reaches TRH . The maximum temperature can
be easily extracted from Eq. (2.272), which occurs for values44
  2/5
√  3 3/5
a 8 4 15 3
=  1.5 and Tmax =  i
φ P ρφ
M .
ai max 3 2gρ π 2 8
(2.273)

The difference of slopes (− 38 compared to −1) between an inflaton dominated

Universe and a radiation dominated Universe can be understood from the fact that
the inflaton injects some energy continuously in the thermal bath, counterbalancing
(but not totally) the redshift due to the expansion. Once the inflaton has completely
disappeared (after TRH ), the adiabatic relation T × a holds as there are no any
external sources of energy to heat the bath anymore. This effect can also be seen
T2
from the expression of the Hubble rate. Whereas it is proportional to M P
in a
ρφ
radiative Universe, Eq. (2.264), from H = 2
2 , one can extract
3MP

gρ π 2 T 4
H = , (2.274)
36 φ MP2

44 We considered Ti = 0, in other words, no other thermal sources outside from the inflaton decay.
114 2 Inflation and Reheating [MP → TRH ]

4.0 × 1012

3.5 × 1012

3.0 × 1012
yΦ=10–5
12
T (GeV)

2.5 × 10

2.0 × 1012

1.5 × 1012

1.0 × 1012
yΦ=10–6

5.0 × 1011

0 2 4 6 8 10
a/ai

12 yΦ=10–6
Log10[T] (GeV)

11

10
yΦ=10–5

7
0 2 4 6 8 10
Log10[a/ai]

Fig. 2.17 Evolution of the temperature of the Universe as a function of the scaling parameter
a/ai for different values of yφ . One can clearly see the 3 stages of evolution: first, the temperature
increases up to a maximum value and then decreases following a a −3/8 law when the Universe is
still dominated by the inflaton energy density and finally a −1 when it becomes purely radiative

which means, naively speaking that “H decreases faster (as a function of T ) in

a matter dominated Universe than in a radiation dominated Universe.” In fact, it
is simply coming from the fact that the temperature decreases less in the inflaton
dominated Universe because it is a non-adiabatic process from the thermal bath
point of view: the latter receives energy injected by the inflaton decay.
Notice that in all the expressions above, we “condensed” by simplicity the initial
condition uncertainties into the parameter ρφi . In the literature, it is also common to
express the temperature as a function of Hi , the Hubble scale at the end of inflation.
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 115

The advantage of using ρφi comes from the possibility of expressing it as a function
of the inflaton potential. For instance, if we suppose a potential,

V (φ) = m2φ φ 2 ⇒ ρφi  m2φ φ2  m2φ MP2 , (2.275)

for oscillations with Planck scale amplitudes, expression that can be replaced easily
in Eq. (2.272). The fact that the maximum temperature of the Universe can be
much larger than the reheating temperature can have dramatic consequences on
the production of dark matter during the reheating phase as we will discuss in the
following section.

Exercise Recover the expression for Tmax and amax , and recompute it with a
generic equation of state for ρφ ∝ a −3(1+w) .

2.4.2.2 A Closer Look on the Hubble Constant∗

Before studying the production of dark matter in this early phase of the Universe,
we want to detail slightly the dependence of H on the temperature in the reheating
phase (Eq. 2.274), i.e. when the inflaton still dominates massively the Universe. To
be more general, we will do it in the case of an equation of evolution for ρφ
 a m
= ηa −m ,
i
ρφ = ρφi (2.276)
a
which is the simplified form of Eq. (2.141). Notice that the problem is in fact
more complex because the power m in (2.276) depends on the power of φ in the
inflationary potential V (φ), which in his turn can change the Boltzmann equation
for φ and then the relation (2.276). But we will not enter in such details here. We
can then generalize (2.270) to
√
3ηa − 2 MP
m
dρR ρR
+ 4 dT = φ , (2.277)
dT a da a dT
da

and using ρR = αT 4 , we obtain

√
3ηa − 2 MP
m
dT T4 1 dX4
4T3 +4 = 4 = φ , (2.278)
da a a da αa
gρ π 2
where as usual we have defined X = a × T and α = 30 . Solving (2.278) for X,
we can have the expression of T = f (a):
√  a − m +4  a 4
2φ 3ηMP − m i 2 i
T =
4
a 2 1− + Ti4 . (2.279)
α(8 − m) a a
116 2 Inflation and Reheating [MP → TRH ]

We can then deduce H (T ) in the limit a ai (around TR H , for instance):

√  −m
ρφ ηa 2 8 − m αT 4
H (T ) = √ = = (2.280)
3MP 3 MP 6 φ MP2

√  −m
ρφ ηa 2 8 − m αT 4
H (T ) = √ = = (2.281)
3MP 3 MP 6 φ MP2

gρ π 2
with α = 30 . We recover then Eq. (2.274)

5 αT 4
H (T ) = (2.282)
6 φ MP2

for a dust-like inflaton (m = 3). We then need to be careful when defining the
reheating temperature. Indeed, when we used in Eq. (2.267) H (tRH ) = 3tRH
2
=
√
2
3 φ and replaced H by H (TRH ) with ρ√R (TRH ) , we used in fact two different
3MP
definitions of the reheating time without really noticing it. The former equation tells
us that the reheating time is given by the inverse width of the inflaton field (first
definition), whereas the later one tells us that at TRH the Universe is dominated by
the radiation. However, it is not exactly true. To be exact, one should use the exact
form of H (T ) (2.282), evaluate it at TRH , and impose ρφ (TRH ) = ρR (TRH ) =
4 . We then obtain45
αTRH


4 4
αTRH
5 αTRH 2 6
2
= √ ⇒ TRH = √ φ MP , (2.283)
6 φ MP 3MP 5 3α


ρφ =αT 4
45 The approximation we need to apply here is that we considered H (TRH ) = √ RH , where
3MP
we neglected the radiation contribution in ρ because we supposed a Universe still dominated by
√ √ √
ρφ . Using H (TRH ) = √ρφ+ρR = √ 2ρR will also lead to a misleading factor of 2 because the
3MP 3MP
formulae (2.282) could not be applied if the Universe is not anymore dominated by the inflaton
field. The exact numerical solution lies between these two approximations.
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 117

which corresponds, when plugging it again in H (T ) (2.282), to the relation

2
H (TRH ) = φ (2.284)
3c

with c = 53 , which represents the (tiny) difference between the two definitions of
reheating temperature. With this definition of reheating temperature, one can write
 8
T gρ π 2 T 8
ρφ = ρφ (TRH ) = 4
. (2.285)
TRH 30 TRH

How to teach non-instantaneous reheating

I present in this little box a shortcut to derive the evolution of thermal energy
density as a function of the scale factor (the way I do it for my students).
Instead of writing Equation (2.269) as a function of T , one can directly write
it as a function of a:
dρR ρR φ ρφ
+4 = . (2.286)
da a H a
This equation has two advantages. First of all, it shows that it is the ratio
φ
H that governs the process and will determine when the reheating will stop,
justifying the approximation that at reheating time H (TRH )  φ . At this
temperature, a naive (but interesting) look at Eq. (2.286) gives ρR  ρφ
for φ ∼ H . But, more interesting is the fact that solving√
this equation is
ρφ
extremely easy if one considers ρφ ∝ a −3(1+w) and H  √ . Supposing a
3MP
constant φ , and using the variable X = ρR × a4, we let the reader check that

1 φ  a 4+δ
i
ρR = ρφ 1 − , (2.287)
4+δ H a

where δ = − 32 (1 + w), and we supposed ρR (ai ) = 0. This equation can even

be generalized to situations where φ depends also on a (where you have
inflaton scattering, for instance). In this case, δ = − 32 (1 + w) + δ , with
φ ∝ a δ . This equation is nice to show to students. It explicits the fact that

ρR is proportional to ρφ , with a coefficient of proportionality ∼ Hφ . We can
also write it in different ways, depending what are the initial conditions we

(continued)
118 2 Inflation and Reheating [MP → TRH ]

are interested in,

√  a 4+δ
3 √ i
ρR = φ MP ρφ 1 −
4+δ a
1 φ i  ai  32 (1+w)  a 4+δ
i
= ρφ 1− ...
4 + δ Hi a a

We then recover the condition (2.284), defining the reheating temperature

as ρφ = ρR , H = 4+δ
1
φ = 25 φ for a dust-like inflaton.

Exercise Compute the factor c of Eq. (2.284), noticing that at TRH ,

√
ρR (TRH )
ρφ (TRH ) = ρR (TRH ) ⇒ H (TRH ) = √ .
3MP

Show that taking the definition H (TRH ) = φ as the reheating temperature leads
to (for a dust-like inflaton)

αT 4 2 2 φ MP
H (T ) = 2
, TRH = √ √ (2.288)
2φ MP 5 α

instead of

5 αT 4 6 φ MP
H (T ) = 2
, TRH = √ √ (2.289)
6 φ MP4 5 3 α

when supposing ρφ (TRH ) = αTRH

4 .

In summary, one can always use Eq. (2.284) for the definition of the reheating
temperature, taking c = 1 if one considers tRH = φ−1 , c = 23 for H (TRH ) = φ ,
and c = 53 corresponding to the definition ρφ (TRH ) = αTRH 4 . The formulae are

easy to switch by a simple “redefinition” of φ .

2.4.3 Producing Dark Matter During the Reheating Phase

2.4.3.1 The Context

We just learned in the previous section how the reheating evolution should be treated
with care once we do not suppose instantaneous reheating. It is then natural to ask
how the physics of reheating can affect the production of dark matter. In fact, the
influence of reheating is especially large for production modes that are heavily
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 119

dependent on the energy (and thus the temperature). Every model that includes
higher dimensional operators or derivative couplings is concerned. It runs from
axion-like particles with couplings of the form aF μν Fμν to longitudinal modes of
U (1) gauge groups of the form ∂μ a (Stueckelberg notation) or goldstino in the case
of the gravitino if SUSY breaking scale is much above the reheating temperature.
Indeed, every time the scale of a broken symmetry lies above the mass of the
corresponding gauge boson, the only surviving mode is the longitudinal one. That
is, the Goldstone boson (goldstino in the case of the gravitino), which is reduced to
the derivative of the Higgs phase (or the imaginary part of the Higgs). This degree
of freedom, which is “eaten” in the standard Higgs mechanism, stays as a physical
degree of freedom in the low energy theory and in fact as the only polarization mode
of the massive gauge boson, appearing under the form of the derivative of the phase
of the Higgs and thus highly temperature dependent.
Those were just some examples where one should be cautious when looking at
the production of dark matter in the early Universe. Two effects should then be
treated separately:

• The effect of a non-instantaneous reheating. In this case, the inflaton does

not decay at a given time, but the bath produced while it still dominates the
energy budget of the Universe can already begin to produce dark matter (through
annihilation) before the Universe is dominated by the radiation.
• The effect of a non-instantaneous thermalization. This happens if the standard
model particles do not have time to thermalize before the production of dark
matter from the annihilation of standard particles just produced by the inflaton
decay (of energy mφ /2, see Sect. 2.3).

Indeed, the Universe in the presence of the inflaton is particular as we already

discussed in Sect. 2.4.2. To be complete, one should solve the set of three equations,
the combined ones of the radiation, the inflaton (conservation of energy), and the
dark matter (Boltzmann equation). We can then rewrite the set of Eqs. (2.206–2.208)
adding the possibility for dark matter production

dρφ
+ 3Hρφ = −φ ρφ
dt
dρR
+ 4HρR = φ ρφ
dt
ρφ ρR
H2 = 2
+
3MP 3MP2
dnχ
+ 3H nχ = R(T ), (2.290)
dt
120 2 Inflation and Reheating [MP → TRH ]

where nχ is the dark matter density. Several remarks should be done at this point.
First of all, the Boltzmann equation (the last one) is linked to the three other ones
through the dependence of t on T . Indeed, if you remember what we told in the
previous section, to express all the equations as a function of one variable (we
have chosen T because it corresponds to what is measured nowadays by CMB
experiments), one needs to eliminate the scale factor a and the time t by subtle
combinations of the Friedmann equation and the conservation of energy equations.
We will not obtain the simple relation T ∝ a −1 as we have in a pure radiation
domination. That can be understood as in those cases, one can treat the thermal
bath as an isolated system whose entropy is conserved. In the case of a massive
particle (inflaton of density ρφ ), which decays into radiation of density ρR , we are
in the presence of two open thermodynamical systems in contact. The energy of the
inflaton is diffused to the thermal bath. The total entropy is conserved, but not inside
the thermal bath that receives entropy from an external system, proportional to the
decay width of the inflaton as it is clear from Eq. (2.290).
In this section, we will suppose a non-instantaneous reheating, which means we
suppose that the decay of the inflaton “takes its times” in a sense. In other words,
φ H (T ). This can be considered as a “long-lived” inflaton. That means that,
during a time, the Universe will be dominated by the inflaton, while the thermal
bath will begin to form. Dark matter can then be produced during this period of
time, where the particles are very energetic because the temperature is very high,
around the inflaton mass. In the first step, we will consider instant thermalization.
That means that two photons (I call indifferently photon any relativistic particle of
the thermal bath) have time to scatter to a photon newly produced by the inflaton
to bring it to the thermal bath before the expansion forbids it. From Eq. (2.272) one
can write for a ai
 √ 1/4
 a 3/8 12 3 
i
T =β , with β = i
φ MP ρφ . (2.291)
a gρ π 2

d d
Once we know the dependence of T on a, one can define dt as a function of dT :

d d da d d dT 3 d
= = aH = aH = − H T , (2.292)
dt da dt da dT da 8 dT
from where we can express the Boltzmann equation of Eq. (2.290) as a function of
the temperature

dnχ nχ 8R(T )
−8 =− . (2.293)
dT T 3H T
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 121

We can then define a “covariant” Yield Y = Tn8 , permitting to “absorb” the

Hubble expansion in the Boltzmann equation, giving

dY 8 R(T )
=− . (2.294)
dT 3H T9

It is always possible to define a covariant Yield Y . Y = Tn3 in entropy conserved

system, Tn8 in mixed matter–radiation dominated Universe. . . The power of T in
the definition of Y depends only on the dependence of T on a and thus on how
the temperature evolves with the scale of the Universe. The fact of having T ∝
a −3/8 in the inflaton-type model means that the temperature decreases slower than
in typical radiation dominated Universe where T ∝ a −1 (which is logical as matter
as tendency to slow down expansion). We then need higher power of temperatures
to keep covariant Yields.

2.4.3.2 Production from Inflaton Decay

We can then easily compute the relic abundance of a dark matter χ obtained by
the inflaton decay if we allow a fraction BR of the inflaton decay into dark matter.
One then needs to solve the Boltzmann equation (2.294) between Tmax and TRH ,
defining the production rate (the number of dark matter particles produced per unit
of time and volume) as

ρφ (T ) ρφi  ai 3
Rdecay (T ) = BR φ = BR φ
mφ mφ a

ρφi T 8 gρ2 π 4 T8
= BR φ = BR × . (2.295)
mφ β 432 φ mφ MP2

We can then implement Eq. (2.274) and Eq. (2.295) into Eq. (2.294) to write

dYχdec 2 gρ π 2
= − BR × , (2.296)
dT 9 mφ T 5

which gives, after integration between Tmax and TRH ,

1 gρ π 2
Yχdec (TRH )  BR × 4
18 mφ TRH
gρ π 2 4
⇒ ndec
χ (TRH ) = BR × T (2.297)
18 mφ RH
122 2 Inflation and Reheating [MP → TRH ]

n (T )
with Yχ (T ) = χT 8 . Notice that, if we would not have taken into account the non-
instantaneous effect, Eq. (2.297) would have been written simply as

ρφ gρ π 2 4
χ (TRH ) = BR × nφ = BR ×
ndec = T , (2.298)
mφ 30 mφ RH

i.e. 35 times less.

The density of a dark matter of mass mχ measured today can be written using
Eq. (E.1)

ρ nχ (T0 ) × mχ
= = −5 2 (2.299)
ρc 10 h GeV.cm−3
 0  dec 
3
2 5 gs nχ (TRH ) T0 mχ 
⇒ h |decay = 10 RH −3
gs cm TRH 1 GeV
 0 
ndec  m 
3 gs χ (TRH ) χ
 × 108 RH 3
,
2 gs TRH 1 GeV

where we used46 T0 = 3.3 × 10−4 eV, and 1 cm−3  8 × 10−42 GeV3 . We can then
implement the expression (2.297) into Eq. (2.299) to obtain
 
gρ π 2 BR TRH  mχ 
h |decay
2
=
12 10−8 mφ 1 GeV
  13   m 
BR 10 GeV TRH χ
 0.1 . (2.300)
10−8 mφ 1010GeV 1 GeV

2.4.3.3 Production by Scattering

If one defines the scattering rate of production of dark matter from the thermal bath
T n+6
as47 R(T ) =  n+2 . As we did for the inflaton decay, we can combine Eq. (2.294)
and Eq. (2.274) to write

dYχscat 96 T n−7
=− φ MP2 , (2.301)
dT gρ π 2 n+2

46 And approximated 1.7 by 32 to have simplified expressions.

47 The power of T has been chosen so that n corresponds to the dependence of the cross section in
the temperatures: R(T ) ∝ n2SM σ v ∝ T 6 σ v.
2.4 Reheating: Thermal Phase [10−30 − 10−28 s] 123

which gives after integration between Tmax and TRH depending on the value of n:

96 TRH n+2
n < 6 : n(TRH ) = φ MP2
gρ π (6 − n)
2 
 8 
96 TRH Tmax
n = 6 : n(TRH ) = 2
 M
φ P
2
ln
gρ π  TRH
8 n−6
96 2 TRH Tmax
n > 6 : n(TRH ) = φ MP , (2.302)
gρ π 2 (n − 6) n+2

which gives, for the relic abundance, using Eq. (2.299),

 
48 c × 109 MP TRH n+2  mχ 
n<6: h |scat
2
= 0.1 × 
π 10gρ (6 − n) TRH  1 GeV
  
48 c × 109 MP TRH 8  mχ  Tmax
n=6: h2 |scat = 0.1 ×  ln
π 10gρ T RH  1 GeV TRH

48 c × 109 7
TRH n−6 M
Tmax  m 
P χ
n>6: h2 |scat = 0.1 ×  ,
π 10gρ (n − 6) n+2 1 GeV

where we used

2
gρ π 2 TRH
φ = cH (TRH ) = c (2.303)
90 MP

with c of the order one represents the freedom in defining the reheating temperature,
see Eq. (2.268). It then becomes interesting to combine the production of dark
matter by the thermal bath, Eq. (2.303) with the direct product of the inflaton decay
Eq. (2.300),

h2 |tot = h2 |scat + h2 |decay ,

we obtain

   n+2
48 c × 109 MP TRH
2
h |n<6 = 0.1 × 
π 10gρ (6 − n) TRH 
   m 
gρ π 2 BR TRH χ
+
12 10−9 mφ 1 GeV
124 2 Inflation and Reheating [MP → TRH ]

   
48 c × 109 MP TRH 8 Tmax
2
h |n<=6 = 0.1 ×  ln
π 10gρ TRH  TRH
   m 
gρ π 2 BR TRH χ
+ −9
12 10 mφ 1 GeV
 
48 c × 109 7 T n−6 M
TRH max P
h2 |n>6 = 0.1 × 
π 10gρ (n − 6) n+2
   m 
gρ π 2 BR TRH χ
+ . (2.304)
12 10−9 mφ 1 GeV

We show in Fig. 2.18 the result on the scan of the parameter space in the case
n = 4, 6, and 8, respecting the relic density constraint h2 = 0.11 for a large range
of dark matter mass for TRH = 1011 GeV and Tmax = 1013 GeV. We notice that

0 0

Log10[m] Log10[m]
–5 –5

5 5
Log10[BR]

Log10[BR]

–10 –10
0 0

–15 –5 –15 –5

TRH 11 13
TRH=1011GeV R =10 GeV, Tmax=10 GeV
–20 –20
13 14 15 16 17 18 13 14 15 16 17 18
Log10[Λ] Log10[Λ]
0

Log10[m]
–5

5
Log10[BR]

–10
0

–15 –5

TRH=1011GeV, Tmax=1013GeV
–20
13 14 15 16 17 18
Log10[Λ]

Fig. 2.18 Parameter space in the plane (, BR ) of points respecting PLANCK constraints, with
the corresponding dark matter mass m in the case n = 4, 8, and 8, respectively, see Eq. (2.304)
2.5 The Thermal Era [10−28 − mχ ] 125

the range of dark matter mass allowed is very large (from keV to Eev scale), that is,
a direct consequence of the strong power dependence of the relic abundance on the
scale . We also remark that a branching ratio of  1 is possible to obtain for dark
matter mass of keV, whereas PeV scale dark matter requires very tiny branching
fraction of the order of BR  10−18 to avoid an overclosure of the Universe.
We can also imagine the limit, in the plane (, m) where the production of dark
matter from the inflaton decay dominates on the production from annihilation. From
Eq. (2.304), we obtain the conditions
√ n M m
576 10 TRH P φ
BRmin |n<6 =
(6 − n)(gρ π 2 )3/2 n+2
√ 6 
576 10 TRH MP mφ Tmax
BRmin |n<6 = 2 3/2 8
ln
(gρ π )  TRH
√ 6 T n−6 M m
576 10 TRH max P φ
BRmin |n>6 = .
(n − 6)(gρ π )
2 3/2 n+2

2.5 The Thermal Era [10−28 − mχ ]

In this last section, we will consider the instantaneous reheating and instantaneous
thermalization phases. For dark matter production processes that do not exhibit
strong dependence on the energy and thus on the temperature, this treatment is
much more simpler and usually gives reasonably good results compared to the one
obtained numerically. We will in a sense repeat the previous exercise, considering
that the inflation decayed instantaneously, which means, the history of the Universe
begins at TRH . The advantage is that we do not have to make any assumptions on the
inflationary models, the thermalization process, or the inflaton width. In this case,
TRH is the only initial condition one needs.

2.5.1 Instantaneous Reheating and Instantaneous Thermalization

The Boltzmann equation in a Universe where the dark matter is not in thermal
equilibrium in the bath, because of its small density nχ , forbids the back reaction
nχ nχ → nSM nSM and can be written (Eq. 2.244):

dnχ
+ 3H nχ = R(T ) (2.305)
dt
with (see Eq. 2.258)
 
E1 E2 dE1 dE2 d cos θ12
R(T ) = f1 f2 |Mf i |2 d 13 , (2.306)
1024π 6
126 2 Inflation and Reheating [MP → TRH ]

for a process 1 + 2 → 3 + 4 with 1 and 2 the standard model particles of the bath and
3 and 4 the dark matter candidates, with f1 and f2 being the distribution function of
the incoming particles 1 and 2, and d 13 is the solid angle between the particles 1
and 3 in the center of mass frame.
To solve the Boltzmann equation, the strategy is always the same: eliminating
the time t to express it as a function of the temperature T (because rates and cross
sections depend on T ) using the (dimension) scale factor a as an intermediate state.
Indeed, the Hubble parameter H connects a and t through H = ȧa and also a and
T through the Friedmann equation H 2 (T ) = 8πρtot 2 = ρtot2 , MP being the reduced
3MP l 3MP
planck mass MP = √1 MP l = 2.4 × 1018 GeV. More concretely, one can write
8π

d d d dT d d dT d
=H ×a and = ⇒ =H . (2.307)
dt da da da dT dt d ln a dT
We see that, knowing H (T ) from the Friedmann equation, if one can obtain the
dependence, a = f (T ), the Boltzmann equation (2.305) only as a function of the
temperature T . Without loss of generalities, we can generalize the dependence of T
on a as T = a −m . In this case, (2.305) becomes

dn dn n R(T )
− HT m + 3H n = R(T ) ⇒ −3 =− . (2.308)
dT dT mT mH T
It can seem strange to “complexify” the Boltzmann equation introducing the m
dependence. In fact, it comes from the fact that in a lot of dark matter scenarios,
the physics occurs at temperatures well below the reheating temperature TRH , in a
regime where the Universe is totally dominated by radiation. In this case, m = 1
because the temperature is just redshifted by the expansion. That is also a (hidden)
consequence of the conservation of entropy: s × a 3 = cst ⇒ T ∝ a −1 . However,
in a Universe where entropy is injected in the thermal bath from the decay of the
inflaton, for instance, non-instantaneous reheating, the entropy is not conserved
(locally): the bath is continuously receiving energy from an outer system. In this
case, the relation s × a 3 does not hold anymore, and one needs to generalize it. It is
even more true when one considers non-instantaneous thermalization, where even
the concept of entropy and temperature does not make sense (or at least, not in the
usual understanding).
Let us now define Yields Y (T ) such that the Hubble term of Eq. (2.308),
−3 mTn
, cancels. That is equivalent, in the radiation dominated era (or any entropic
conservation framework), to the definition Y ∝ Tn3 ∝ n×a 3 , which permits to work
in a comoving frame,48 absorbing the effect of space dilatation in Y , which is in fact

48 Inthe literature, it is very common to define Y = ns and common to define Y = nnγ . In any case,
the dependence in both cases is of the form Y = cte × Tn3 . As we will deal with epochs where
(locally) the entropy will not be conserved (and thus the definition of s is more difficult), we prefer
to define Y = Tnn in this section, with n = 3 in a Universe with local entropy conservation.
2.5 The Thermal Era [10−28 − mχ ] 127

proportional to the total number of particles and not the density. If one generalizes
Y = Tnk , imposing m × k = 3, Eq. (2.308) becomes

dY −R(T )
= . (2.309)
mH T m +1
3
dT

Notice that we recover for m = 1, the classic Boltzmann equation in a radiation

dominated Universe dTdY
= − HR(T
TT3
)
, or if one defines Y = ns , dT
dY
= − R(T )
H T s . Let us
have a look at the solution of (2.309) in 2 different scenarios: radiation dominated
Universe and matter dominated Universe. The case of a Universe dominated by a
decaying matter (as an inflaton) has been treated in the previous section.

2.5.1.1 Radiation Dominated Universe

To obtain the dependence of the Hubble rate on the temperature H (T ), one needs to
solve the Friedmann equation (2.51) neglecting the curvature

8π ρR αT 4 α T2
H2 = GρR = 2
= ⇒ H = , (2.310)
3 3MP 3MP2 3 MP

MP l g π2
MP being the reduced Planck mass MP = √ and α = ρ30 [gρ = 106.75
8π
in the Standard Model, see Eq. (3.38)]. To know m, one needs to write the law of
conservation of energy

dρR dρR
+ 4HρR = 0 ⇒ H a + 4HρR = 0
dt da
d ln ρR
⇒ = −4 ⇒ ρR ∝ a −4 ⇒ T 4 ∝ a −4 , (2.311)
d ln a
which gives m = 1. We can then write


dY 3 R(T )
= −MP . (2.312)
dT α T6
128 2 Inflation and Reheating [MP → TRH ]

T n+6
If one defines R(T ) = n+2
,  representing the BSM scale, one obtains

  n+1
TRH 3 MP TRH
Y (T ) = dY ⇒ Y (T ) =
T α (n + 1)n+2
 n+1
3 MP TRH
⇒ n(T ) = T 3 × Y = T 3. (2.313)
α (n + 1)n+2

We integrated from T up to TRH because before TRH the Universe is not anymore
radiation dominated and one should then be careful in the treatment of the energy
conservation equations, which deals with matter + radiation components as we will
see below. From Eq. (2.313), one can deduce the relic abundance of a dark matter
candidates of mass mχ nowadays (T = T0 ):

n+1 
n(T0 ) × mχ g0 3 MP TRH mχ 
= ⇒ h = 1.6 × 10
2 8
,
ρc0 gRH αRH (n + 1) n+2 1 GeV
(2.314)

where g0 = 3.91 and gRH = 106.75 in the Standard Model, which gives
  n+1  n+2
−3 mχ TRH 1013 GeV
h  4 × 10
2
× 1011−3n .
(n + 1) GeV 1010 GeV 
(2.315)

6+n n
As we defined n from R(T ) =  T
n+2 , we can approximate σ v ∝ n+2 . In other
T

words, the exchange of a massive virtual field corresponds to n = 2, equivalent

to the exchange of a massless field with two (mass) reduced couplings. n = 4
corresponds to one reduced coupling and a massive exchanged fields, whereas n = 6
is the case of the exchange of a massive field and two reduced couplings, like in the
gravitino case in high-scale SUSY. Indeed, the couplings of the gravitino is Planck-
mass suppressed and the cross section is suppressed by the exchange of very massive
supersymmetric particles. For n = 6, for instance, we obtain
  7 8
mχ TRH 1013
h2 = 0.112 . (2.316)
7 × 107 GeV 1010 GeV 

It is interesting to note several points before developing further on the reheating

process. First of all, the dependence on the reheating temperature (to the power
7) is very important. It means that a small change in the temperature can affect
drastically the abundance. On the other way, it also means a weak dependence on
the dark matter mass, as it can be easily corrected by the reheating temperature.
For instance, a change of 107 in the dark matter mass corresponds “only” to one
order of magnitude in the reheating temperature. That is also an interesting remark:
2.5 The Thermal Era [10−28 − mχ ] 129

natural values of reheating temperatures of the order of 1010 lead to heavy dark
matter (around the PeV scale). That can also be understood easily by the fact that
producing dark matter in such circumstances is not so easy. The feeble amount of
matter is thus compensated by heavier candidates.

2.5.1.2 Matter Dominated Universe

When the Universe is matter dominated, by any massive field, the entropy is still
conserved (locally and globally). However, the dependence of H on T is different,
due to the Friedmann equation

ρ ρm N ×M
H2 = = , with ρm = . (2.317)
3MP2 3MP2 a3

In Eq. (2.317), ρm is the dominant density of the Universe, given by N particles

of mass M. That is of course an interpretation: replacing N × M by any mass scale
M does not change anything in the following. To obtain the dependence of ρm on a,
we just solved the equation of conservation of energy:

d d ȧ
ρm + 3Hρm = 0 ⇒ ȧ ρm + 3 ρm = 0 ⇒ ρm ∝ a −3 . (2.318)
dt da a
The dependence of a = f (T ) is the same as in the radiation dominated case.
Indeed, we can see the system massive particle + radiation as two independent
systems, which do not discuss together, isolated. The entropy is then conserved
individually in the two sectors (which was not the case in the previous section when
dealing with a Universe with an energy budget dominated by a decaying massive
inflaton). We can say that the total entropy S is the sum of a “matter” entropy Sm
and a “radiation” entropy SR , S = Sm + SR . The entropy of a massive particle is
zero, so S = SR = T 3 × a 3 = constant ⇒ a ∝ T −1 . An equivalent way is to
solve the energy conservation for the radiation

dρR dρR
+ 4HρR = 0 ⇒ H a + 4HρR = 0 ⇒ ρR ∝ a −4 ⇒ T ∝ a −1 ,
dt da
as we did in the radiation dominated Universe. The main difference between a matter
dominated and radiation dominated Universe is then only the liked between the time
t and the scale factor a through H in the Friedmann equation but not between the
temperature T and the scale factor a. We then obtain for the Yield Y

dY R(T ) R(T )MP a 3/2 MP R(T )

=− = − √ ∝ −√ . (2.319)
dT 3H T 4 3N × MT 4 3N × M T 11/2
130 2 Inflation and Reheating [MP → TRH ]

T n+6
Considering as above,49 R(T ) = n+2
, we can solve the preceding equation,
which gives

n+3/2
MP TRH
Y (T ) = √ ⇒ (2.320)
3N × M(n + 3/2) n+2

  n+3/2
5 × 1013 GeV mχ TRH
h2 = 0.1
N ×M (n + 3/2) GeV 10
10 GeV
 13 n+2
10 GeV
× ,


which gives, in the case n = 6,


 15/2 
5 × 1013 GeV  mχ 
8
TRH 1013 GeV
h2 = 0.1 .
N ×M 7.5 TeV 10
10 GeV 
(2.321)

However, in the case of the inflaton, the Universe is not simply matter dominated.
It is dominated by a particle (the inflaton), which decays into relativistic SM
particles (the radiation). One should then deal with coupled energy conservation
equations (2.207) to obtain the value of m and of H (T ).

References
1. R. d’Inverno, Introducing Einstein’s Relativity (Clarendon, Oxford, 1992), 383 p.
2. J.B. Hartle, Gravity: An Introduction to Einstein’s General Relativity (Pearson, London, 2002),
616 p.
3. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of
Relativity (John Wiley and Sons, Inc., Hoboken, 1972), 657 p.
4. E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011).
[arXiv:1001.4538 [astro-ph.CO]]
5. I.L. Shapiro, [arXiv:1611.02263 [gr-qc]]
6. D.S. Gorbunov, V.A. Rubakov, Introduction to the theory of the early universe: Cosmological
perturbations and inflationary theory. https://doi.org/.1142/7874
7. L. Kofman, A.D. Linde, A.A. Starobinsky, Phys. Rev. D 56, 3258–3295 (1997). https://doi.org/
10.1103/PhysRevD.56.3258 [arXiv:hep-ph/9704452 [hep-ph]]
8. V. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press).
9. L.A. Kofman, astro-ph/9605155

49 And integrating the coefficient of proportionality in the definition of  by simplicity.

A Thermal Universe [TRH → TCMB ]
3

Abstract

Once the Universe has terminated its inflationary phase, and produced a thermal
bath through the Inflaton decay, its evolution follows the evolution of a classical
plasma in an expanding Universe. The law undergoes (relativistic) statistical laws
and one can apply our knowledge of this field to the production and decoupling
of elements, from neutrino to dark matter. We propose in this chapter to review
in details the thermal evolution of the primordial plasma, and the possibility to
produce weakly interacting massive particles (WIMP) from it.

3.1 Thermodynamics

3.1.1 A Brief Thermal History of the Universe in Some Dates

and Numbers

The history of the Universe is rich and complex and depends strongly on its
temperature. In this section, we will give a quick snapshot of the thermal history of
the Universe that will be developed through the chapter. Each phase will be studied
in detail later on. Let us begin by recalling some temperatures, possibly realized
in the hot Universe, the related cosmic times, and the connection with microscopic
physics at the corresponding energies:

• T  0.1eV [t  1013 s  380,000 years]

Light nuclei and electrons form neutral atoms and the Universe becomes
transparent to photons. They decouple from the plasma (see Sect. 3.3.4) and are
observable today as cosmic microwave background (CMB).

© Springer Nature Switzerland AG 2021 131

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2_3
132 3 A Thermal Universe [TRH → TCMB ]

• T  0.1 − 10 MeV [t  102 − 10−2 s]

Light nuclei are formed from protons and neutrons (primordial nucleosynthesis,
BBN) and neutrinos decouple from the plasma (see Sect. 3.3.1).
• T  10 GeV [t  10−8 s]
Weakly interacting massive particles (WIMPs), the most popular dark matter
candidates, decouple from the plasma (see Sect. 3.5).
• T  100 GeV [t  10−10 s]
The Higgs vacuum expectation value forms, and all Standard Model particles
become massive. Baryon and lepton number changing “sphaleron process” are
no longer in thermal equilibrium.
• T  108 − 1011 GeV [t  10−22 − 10−28 s]
Baryogenesis via leptogenesis takes place and gravitino dark matter or particles
coupling via GUT interactions to the Standard Model (see Sect. 3.6) can be
thermally produced.
• T  1012 GeV [t  10−30 s]
This corresponds to the reheating process, where the thermal bath is created
as inflaton decay modes, perturbatively or through parametric resonance (see
Sect. 2.4).

This little thermal history is summarized in Figs. 3.1 and 3.2. Nowadays, the
known Universe is mainly composed by baryonic matter, photons, and neutrinos,
while the unknown components are dark matter and dark energy, which involve
physics beyond the Standard Model (BSM). The content of the Universe in particles

Fig. 3.1 Epochs of the hot early Universe with their cosmic times scale (The Astronomy Bum,
under the Creative Commons CC0 1.0 Universal Public Domain Dedication)
 Radiation dominated Matter dominated Radiation dominated Matter dominated

sections
Large increase of the Entropy due to Expansion rate H dominates decay Decoupling : the temperature is not
the decay of the rates of
and electrons: Radiation dominated era are in thermal equilibrium reheating temperature (minimum 2 formation of neutral atoms : the
with electromagnetically MeV up to GUT scale) are free to propagates to form the
charged species : CMB
n =3/4 nfermions

Dark Matter
3.1 Thermodynamics

Decay of

Temperature is below the binding

energy (entropy renormalized) of
Exponential suppression from the deuteron beginning of the
Boltzman factor Exp(-m/kT) Nucleosynthesis

10 -10^7 sec
1 MeV - 1 keV The neutron being heavier than the
proton, it freezes out with a more
3 10^9 K -3 10^6 K suppressed boltzmann factor,
1 yr= 3 10^7 s Exp(mp-mn/kT) =0.15-0.20
t(s)=10^13/E^2(eV)
T(K)=3 10^3 E(eV) Transition quarks/hadron
H(T)=5.5 T^2/Mpl

First galaxies formation

n He4 C12

200 MeV

Mdm/20

Big Time Treheating: 10^-5 s 1 second 1-3 minutes 300 000 yr 500 Gyr 15 Tyr
Energy =3H(Tr) 1 GeV 1 MeV 0.3 MeV 1 eV 0.02 eV 1 meV
Bang Temperature Tr=0.7 ^0.5 3 10^9 K 10^9 K 3000 K 60 K 2.7 K
3 10^12K
133

Fig. 3.2 The little thermal history of the Universe with each process described in the following
134 3 A Thermal Universe [TRH → TCMB ]

of the Standard Model is known thanks to a set of observables ranging from Cosmic
Microwave Background (CMB) measurements to interstellar gas data. The baryon
density, ρb , is obtained from a combination of CMB data and primordial deuterium
abundance extracted from the absorption spectrum of high-redshift quasars. The
mean value is
2 ρb 2
bh  0.022 = h (3.1)
ρc0
⇒ ρb  2.3 × 10−7 GeV/cm3 = 2.46 × 10−7 mp /cm3 .

mp being the proton mass and where we used ρc0 = 1.05 × 10−5 h2 GeV/cm3 , see
Eq. (2.56). This implies a baryonic density

nb = 2.46 × 10−7 cm−3 , (3.2)

which corresponds, roughly to 0.25 baryons per m3 , or 1011 solar mass per Mpc3 .
As we will see, whereas baryons dominate around the Earth, it is really subdominant
at larger scale. The most abundant species in interstellar space is the photon. As we
have seen, its density nγ is intimately related to the temperature of the thermal bath.1
This will be explicitly calculated later, see Eq. (3.27):

2ζ(3) 3
nγ = T = 411 cm−3 , (3.3)
π2 γ

where we took T = 2.725 K = 2.39 × 10−4 eV. Combining (3.2) and (3.3) we
obtain

nb
η= = 6 × 10−10, (3.4)
nγ

very near from the measured value of η given by PLANCK collaboration,

ηP LANCK = 6.12 × 10−10. (3.5)

1 Bydefault when one talks about temperature, T represents the photon temperature Tγ , or more
generically, the temperature of the thermal bath.
3.1 Thermodynamics 135

This means that for each baryon, we currently find on average 2 × 109 photons in
the Universe. The other Standard Model particle which coexists with the photons
and baryons in the interstellar medium is the neutrino. Its density follows the same
law than the photon, but with a different temperature (see Sect. 3.3.1):
 1/3 3
4 4
Tν = Tγ = 1.96 K (1.7 × 10−4 eV) ⇒ nν = 2
nγ  112 cm−3 ,
11 2 11

where the first ratio corresponds to the difference between the degrees of freedom2
of a Majorana (fermionic) neutrino ( 34 × 2) and a (bosonic) photon (2). This
corresponds roughly to 1/3 neutrino per photon. One could also multiply by 3 to
take into account the three neutrino flavors.
The last component is the dark matter, with a global density measured by Planck

−6 1 GeV
dm h  0.1 ⇒ nDM  10
2
cm−3 , (3.6)
Mdm

which is globally 5 times more than the baryonic one (3.2), and a local one measured
by astrophysical observations

1 GeV
nDM  0.3 cm−3 . (3.7)
Mdm

Let us go into the details to understand how the Standard Model of cosmology
combined with the Standard Model of particle physics to explain these numbers.

3.1.2 Statistics of Gas, Pressure, and Radiation: The Classic Case

Most of the matter in the Universe exists in gaseous form. As we just discussed, a
fraction (∼20%) of it is baryonic matter, like what we are familiar with in everyday
experience, and a larger fraction (∼80%) is non-baryonic dark matter, the nature of
which we try to identify in this chapter. We infer the presence of dark matter through
its gravitational interactions, all visible matter are therefore still baryonic. The
baryonic matter of the Universe consists mainly of Hydrogen (about three quarter
of the mass) and Helium (about a quarter). There is a small fraction of heavier
elements which, collectively, are referred to as metals. The abundance of metals in
the solar neighborhood is about 2% by mass, while the Hydrogen abundance is 71%.
Table 3.1 shows the mass fraction of several elements in the solar neighborhood.
As we saw in the previous chapter, a keypoint to understand the dynamic and the
evolution of a gas is to know the dependency of its essential parameters, i.e. pressure

2 The computation of the degrees of freedom will be detailed in Sect. 3.1.5.

136 3 A Thermal Universe [TRH → TCMB ]

Table 3.1 Mass fraction of H 0.71

different elements in the Solar
He 0.28
neighborhood
C 0.34 × 10−2
N 0.99 × 10−3
O 0.96 × 10−2
Ne 0.18 × 10−2
Na 0.35 × 10−4
Mg 0.66 × 10−3
Al 0.56 × 10−4
Si 0.70 × 10−3
S 0.3 × 10−3
Cl 0.47 × 10−5
Ar 0.11 × 10−3
Ca 0.65 × 10−4
Cr 0.18 × 10−4
Fe 0.13 × 10−2
Co 0.36 × 10−5
Ni 0.73 × 10−4

and distribution function of the energy density in the system. We will first see how
to calculate these quantities in a classical system.
The equation of state of a gas is the relation connecting its variables like pressure,
density, temperature, or internal energy. We will first concentrate on the case where
the energy of mutual interactions between particles are negligible. This type of gas is
composed by particles whose kinetic distribution follows random motion, a kinetic
energy Ek being associated with a momentum p(Ek ), the total energy being written
E = E0 +Ek with E0 = mc2 the rest energy. We remind the reader that the pressure
is defined as the rate of momentum transfer in a given direction through a unit of
area per unit time, and since the direction of the momentum is randomly distributed
in three dimensions,3 the pressure P is given by
 ∞
1
P = f (E)p(E)v(E)dE, (3.8)
3 0

where f (E)dE is the number of particles with kinetic energy between E and E +
dE in a gas unit volume, and v(E) is the velocity associated with the energy E. To
simplify, it will be assumed that the particles in the gas do not have internal degrees
of freedom (like spin or charges), and that the kinetic energy is entirely due to the

3 For 1
a more technical explanation of the factor 3 originated from an integration of cos2 θ (θ being
the scattering angle), see Eq. (2.203).
3.1 Thermodynamics 137

random translational motion.4 If the random motion in the gas is non-relativistic,

then we can write v = p/m and Ek = p2 /2m, resulting in
 ∞
2 2
P = f (E)Ek dE = u, (3.9)
3 0 3

where
 ∞
u= f (E)Ek dE (3.10)
0

is the internal energy density. This corresponds to the total kinetic energy of all
the particles per unit volume of the gas. If the particles are relativistic, v = c and
Ek = pc, giving
 ∞
1 1
P = f (E)Ek dE = u. (3.11)
3 0 3

Notice that the two relations P = 23 u and P = 13 u for the two limits are very
general and do not depend on the details of the distribution function f (E).

Exercise Using E 2 = (mc2 )2 +(pc)2 , recover the values for Ek in the two extreme
cases (relativistic and non-relativistic). Compare then the P = f (ρ) relations
obtained classically above with those calculated in the treatment of the inflaton
in Eq. (2.139). Which value of k corresponds to a relativistic inflaton? A non-
relativistic inflaton? Are the relations identical in both cases? If not, why?

The coefficient of proportionality between P and u has a physical meaning: it is

equal to γ − 1, where γ , called the “adiabatic index” (Eq. A.115), corresponding to
the ratio of specific heats for a thermal gas. Reversible adiabatic processes yield
P V γ = constant,5 (Eq. A.114) where V ∝ R 3 is the volume of the gas. As a
consequence one finds that for a specific adiabatic expansion, the total energy
U = uV of a gas being proportional to V1w drops as R −2 for a non-relativistic gas
and as R −1 for a relativistic gas, where V ∝ R 3 . Another way to obtain the same
result is to note that the energy of a relativistic species is Urel = pc and scales as
R −1 , whereas the energy of non-relativistic species is Unon−rel = 12 mv 2 and scales
as R −2 .

4 This assumption is obviously not valid in the case of the Standard Model, since all its particles
have internal degrees of freedom.
5 See Sect. A.5 for the demonstration of this law called Laplace’s law.
138 3 A Thermal Universe [TRH → TCMB ]

At this point, let us recall what we mean by a “thermal gas.” For a classical
gas, this means that all energy levels of the gas, both discrete and continuous, are
occupied according to the Boltzmann distribution:

N(E) ∝ g(E)e−E/ kT ,

where g(E) is the so-called density of states. In case of quantum statistics, the
corresponding distribution is

g(E)
N(E) ∝
e(E−μ)/ kT ±1

μ being the chemical potential and where the positive sign in the denominator
corresponds to a Fermi gas and the negative sign to a Bose gas. Quantum statistics
comes into play only when the number of particles per phase space cell of volume h3
is of order unity. For dilute gases, as encountered in most astrophysical situations,
classical description is quite adequate, whereas in the case of the dense primordial
plasma in the early Universe, a complete detailed quantum analysis is necessary.

3.1.3 Statistics of Gas, Pressure, and Radiation: The Quantum Case

3.1.3.1 Distribution Functions and Thermodynamics Quantities

From elementary quantum mechanics we can remember the “particle in the box”
interpretation of the phase space occupation number. Let suppose a cubic box
of length L (volume V = L3 ) with periodic boundary conditions. Solving the
Schrodinger equation to determine the energy and momentum eigenstates, we obtain
a discrete set of momentum eigenvalues

h
p= (nx i + ny j + nz k), ni = 0 ± 1, ±2, . . . , (3.12)
L

where h = 4.14 × 10−24 GeV is the Planck’s constant. The density of states in
momentum space p (i.e. the number of states per px py pz ) is thus6

L3 V
3
= 3,
h h

6 Another way to understand it is to imagine a square box made of sides corresponding to ni = 0

and ni = 1. Each side of the box has a size of h/L, so the box has a volume of h3 /L3 . However,
each of the eight corners of the box corresponds to a state, pertaining to 8 boxes surrounding it, so
counting as 1/8 of states par box. Then, a box of 8 corners contains 8 × 1/8 = 1 state in its h3 /L3
volume. In other words, px py pz /(h3 /L3 ) represents the number of states [number of boxes]
having momentum p in the range (px , py , pz ).
3.1 Thermodynamics 139

and the state density in phase space x, p is 1/ h3 . If the particle has g internal degrees
of freedom (e.g. spin), then the density of states becomes
g g
3
= ,
h (2π)3

where in the second equality we used natural units with h̄ = h/(2π) = 1. To

obtain the number density n of a gas of particles, we need to know how the
momentum eigenstates are distributed. This information is contained in the phase
space distribution function f (x, p, t). Because of homogeneity, the distribution
function should, in fact, be independent of the position x. Moreover, at such early
times isotropy requires that the momentum dependance is only in terms of the
magnitude of the momentum p = |p|. We will typically leave the time dependance
implicit: it will manifest itself in terms of the temperature dependance of the
distribution function. The particle density in phase space is then the density of states
multiplied by the distribution function:
g
× f (p). (3.13)
(2π)3

The number density of particles (in real space) is found by integrating (3.13) over
momentum

g
n= d 3 pf (p). (3.14)
(2π)3

To obtain the energy density of the gas of particles, we have to weight each
momentum eigenstates by its energy. To a good approximation, the particles in the
early Universe were weakly interacting. This allows us to ignore the interaction
energies between the particles and write the energy of a particle of mass m and
momentum p simply as

E(p) = m2 c 4 + p 2 c 2 . (3.15)

Integrating the product of (3.13) and (3.15) over momentum then gives the energy
density

g
ρ= d 3 pf (p)E(p). (3.16)
(2π)3

And, just like we did in Sect. 3.1.2, we can define the pressure in a quantum gas as

g p2
P = d 3 pf (p) , (3.17)
(2π)3 3E
140 3 A Thermal Universe [TRH → TCMB ]

where we rewrote Eq. (3.8) using the relation vi = pEi . Notice that we recover the
expressions (3.9) and (3.11) for E = p and E = p2 /2m, respectively.

3.1.4 In the Primordial Plasma

The original gas, denoted as “Ylem” by Gamow himself, exists under the form of
a complete ionized plasma of elementary particles. As far as their kinetic energy
is dominant on their potential energy, we can suppose that the primordial gas is
perfect.7 Moreover, the interaction rates between all these particles are often much
larger than the expansion rate of the Universe driven by the Hubble constant H .
The situation is then very different from the previous chapter, where at times around
the inflationary period, the expansion played an important role on the distribution
functions and dark matter production. After the reheating, the Universe entered
in a phase of thermodynamic equilibrium, thermal and chemical equilibrium. We
dedicate a complete Sect. 3.2 on the subject of the equilibriums coexisting in a hot
gas, but we want in this chapter to give the main ideas. From now on, to simplify
the presentation of the results, one will consider c = 1.
The thermodynamical evolution of a complex system, composed of multiple
particles, is expressed in terms of the chemical potentials μi for each of the particles
i. If one considers the reactions between particles Ai and Bi

A1 + A2 + . . . + An ↔ B1 + B1 + . . . + Bm , (3.18)

then the chemical equilibrium condition is written

μA1 + μA2 + . . . + μAn = μB1 + μB2 + . . . + μBm . (3.19)

This relation reflects the fact that the rate of production of particles “B” is the
same as the rate of destruction of these particles. Therefore, the concentration of
the species A and B does not vary over time. A system in thermal equilibrium is
a system where the temperature within the system is spatially uniform and kept
temporally constant thanks to efficient scatterings of particles interacting in the
plasma. Processes involved in thermal equilibrium do not change the nature or the
number of species. For instance:

• The plasma is in thermal equilibrium under collisional effect of the type:

e + γ → e + γ. (3.20)

7 It is important to note that this hypothesis will no longer be valid when it comes to the quark-

hadrons phase transition, where the strong interactions confine quarks inside protons, neutrons, or
pions.
3.1 Thermodynamics 141

• The chemical equilibrium is realized through reactions like

3γ ↔ e+ e− ↔ 2γ . (3.21)

From the last reaction, one can immediately deduce that the chemical potential of
photon is null (3μγ = 2μγ ) and that the chemical potential of positron and electrons
are opposite (μe− = −μe+ ). Moreover, from the very tiny ratio of the baryon to
photon ratio of today (nB /nγ  6 × 10−10), one deduces that μe− = μe+ . The two
previous hypotheses imply that μe− = μe+ = 0 and one can describe the primordial
plasma as a group of bosonic and fermionic population at temperature T with null
chemical potential. In this case, for a particle A, in a more general context its statistic
(homogeneous) distribution is given by
gA
fA (p) = (3.22)
e(E−μA )/ kT ± 1

with E 2 = m2 c4 +p2 c2 and ±1 correspond to fermionic (+) or bosonic (−) statistic.

gA is the internal (spin, helicity, polarization, color factors. . . ) degree of freedom of
the particle A (2 for a fermion, 3 for a massive vector, 2 for a photon. . . ). The
number density nA (T ) and energy density ρA (T ) of a species A is then given by
  ∞
gA gA (E 2 − m2A )1/2
nA (T ) = fA (p)d 3
p = EdE
(2π)3 2π 2 mA e(E−μA )/ kT ± 1
 ∞
gA p2 dp
=  (3.23)
2π 2 mA p 2 +m2A −μA / kT
e ±1
  ∞
gA gA (E 2 − m2A )1/2 2
ρA (T ) = E(p)f (p)d 3
p = E dE
(2π)3 2π 2 mA e(E−μA )/ kT ± 1

 ∞ p2 + m2A p2 dp
gA
=  . (3.24)
2π 2 mA p 2 +m2A −μA / kT
e ±1

In the relativistic limit (T m) and for negligible chemical potential (T μA )

one can extract an analytical expression of 3.24 using the relation
 ∞ 
xn
dx = (n + 1)ζ(n + 1)y(δ) (3.25)
0 ex − δ
142 3 A Thermal Universe [TRH → TCMB ]

 
with y(δ) = 1 if δ = 1 and 1 − 21n if δ = −1.  is the Euler function ((z + 1) =
z(z)) and ζ (s) is the zeta function defined by
∞
1
ζ (s) = , [ζ(0) = −1/2; ζ(1) = ∞; ζ(2) = π 2 /6; ζ(3)  1.2; ζ(4)
ns
n=1

= π 4 /90; ..].

We give a list of very useful integrals in the Appendix A.6.4.

Integrating Eq. (3.24) gives

π2
ρA (T )T mA ,μ = gA T 4 (bosons)
30
7 π2
ρA (T )T mA ,μ = gA T 4 (fermions) (3.26)
8 30
ζ(3)
nA (T )T mA ,μ = 2 gA T 3 (bosons)
π
3 ζ(3)
nA (T )T mA ,μ = gA T 3 (fermions). (3.27)
4 π2

For non-relativistic species (when the temperature of the plasma approaches

the mass of a particle A), the Boltzmann factor dominates the denominator in
Eq. (3.24), making bosonic and fermionic distribution
 identical. Developing (3.23)
 ∞ 2 −ax 2
2
for p /mA2 1 and using 0 x e = 4 a 3 we obtain
1 π


mA T 3/2 −(mA −μA )/T
nA (T )T mA = gA e (3.28)
2π

mA T 3/2 −(mA −μA )/T
ρA (T )T mA = gA mA e . (3.29)
2π

One can then compute the mean energy per particle for non-degenerate relativistic
species, E = ρn :

per part icle π4

ET mA = T  2.7 T (bosons)
30ζ(3)
per part icle 7π 4
ET mA = T  3.2 T (fermions). (3.30)
180ζ(3)
3.1 Thermodynamics 143

Taking concrete values, we can estimate the energy density of a gas composed
of relativistic particles at a temperature T . For that we need to reintroduce the
4
fundamental numbers c and h and multiply the expression (3.26) by k3 3 
h̄ c
1.15 ×10−21J.cm−3 .K −4 for the energy density and (k/h̄c)3 for the number density
(3.27). Remembering that gγ = 2, we obtain
 4
T
ργ (T )  9 × 1040 GeV.cm−3 ;
GeV
 3  3
T T
nγ (T )  20  3.3 × 1040 .
K GeV

When the particle is non-relativistic, but still

 having some kinetic energy, it is an
intermediate regime. If we develop E = m2 + p2  m + p2 /2min the non-
∞
relativistic limit, integrating on p and using 0 dxx 4e−ax = 233a 2 πa , we can
2

obtain the kinetic energy of a gas of non-relativistic particles:

 3/2
3gA 2mA T
Ec T mA = T e−mA /T
16 π
per particle Ec  3
⇒ Ec T mA = = T. (3.31)
nA (T )T mA 2

Gathering the mass and kinetic energy we finally obtain

per part icle 3

ET mA = m + T, (3.32)
2
which is obviously the same for a boson or a fermion because the exponential
suppression is the dominant contributor to the density and it washes up the spin-
statistics differences. In other words, one can say that, in average, a relativistic
particle has only kinetic energy (by definition) of 2.7 T if it is a boson, or 3.2 T
if it is a fermion. On the other hand, if the particle is non-relativistic, its average
kinetic energy is 1.5 T (fermion or boson).
Another possibility (especially if one needs analytical solutions) is to use the
Boltzmann distribution instead of the Fermi-Dirac or Bose-Einstein one. Indeed, in
E, μA ), fA (p)  gA e− T = fBoltzmann . In the Boltzmann
E
regimes where (T
E
approximation (e T 1), one can have an analytical solution of neq of a population
144 3 A Thermal Universe [TRH → TCMB ]

of particles with gA internal degrees of freedom

 
d 3p − E gA
|p|EdEe− T
E
neq ∼ gA e T = (3.33)
(2π)3 2π 2
   
gA m3A E E m
− mE TA
= (E/m A ) 2 − 1 d e A
2π 2 mA mA
 m 
gA m3A  2 −zx gA m2A T A
= z − 1 zdze = K2 (3.34)
2π 2 2π 2 T

K2 (x) being the modified Bessel function of second kind described in

Appendix A.6.3. In fact, we can show that this expression is still valid at the
order of 20 to 25% for T E. Indeed, comparing Eqs. (3.27) and (3.34), a
100 GeV Majorana particle (gA = 2) at a temperature of T = 108 GeV has a
density of 1.8 × 1023 GeV3 , whereas the Boltzmann approximation (3.34) gives
2. × 1023 GeV3 .

3.1.5 Degrees of Freedom

3.1.5.1 Computation of gρ (T )
So far we have considered plasma of specific species. The primordial plasma is a
mix between different kind of particles, some are relativistic, some are not. However,
since the energy density of relativistic species is much greater than that of no-
relativistic ones (Boltzmann suppressed), it suffices to include the relativistic species
only.8 The energy density can thus can be written
  ∞
π2 4 30 Ti 4
gi ( 2 − xi2 )1/2  2 d π2 4
ρ(T ) = T = gρ (T ) T
30 π2 T 2π 2 xi e−μi /T ± 1 30
i=all species
(3.35)

with xi = mi /T and
 
30 gi ∞ ( 2 − xi2 )1/2  2 d Ti 4
gρ (T ) = 2 (3.36)
π 2π 2 xi e−μi /T ± 1 T
i=all species

8 This is true in the early Universe, but not at a later time when eventually the rest masses of the

particles left over from annihilation begin to dominate and we enter a matter dominated era, see
Sect. 3.1.6.
3.1 Thermodynamics 145

which gives with a pretty good approximation, approximating the Boltzmann factor
as a step function:

 4  4
Tb 7 Tf
gρ (T )  gb + gf . (3.37)
T 8 T
b=bosons f =f ermions

The relative factor of 78 accounts for the difference in Fermi and Bose statistics.
The individual temperatures Tb and Tf are equal to the photon temperature of the
bath (Tγ = T ) as long as they are relativistic in equilibrium in the bath. One can
interpret gρ as being the effective internal degree of freedom of a boson which
composes the gas. The degrees of freedom of all the real relativistic species are
integrated in gρ .
As an example we can compute the total degrees of freedom of a gas in the
SU (3)c × SU (2)L × U (1)Y Standard Model. The contents are: 6 Dirac (4) quarks
colored (3) [6 × 4 × 3] plus 3 Dirac (4) leptons [3 × 4] plus 3 Majorana (2) neutrino
[3 × 2] plus 3 massive (3) vector fields Z 0 W ± [3 × 3], 9 non-massive vector fields
(2) (1 photon and 8 gluon) [9 × 2] and finally one real scalar (1) Higgs field [1]
which gives

7 427
gρSM = (3 × 3 + 9 × 2 + 1) + (6 × 4 × 3 + 3 × 4 + 3 × 2) = = 106.75.
8 4
(3.38)

Exercise Show that gρSU SY for the minimal supersymmetric Standard Model
(MSSM), where each boson has a fermionic partner and each fermion has a bosonic
partner,9 with the addition of a Higgs doublet and its fermionic part, is

7 915
gρSU SY = 122 + 122 × = = 227.75. (3.39)
8 4
Note that for the Higgs field, one can also do the calculation above the
electroweak breaking scale and count 4 degrees of freedom for the Higgs, and only
two for each gauge boson (which have not yet “eaten” the degrees of freedom of
the Higgs). If one adds a Dirac dark matter candidate, gρSM+DM = gρSM + 78 × 4 =
110.25. We represent in Fig. 3.3 the evolution of the degree of freedom gρ = geff
as function of the temperature. We observed that the effective number of relativistic
species decreases especially around 200 MeV where the quark/hadrons transition
absorbs the up-type, down-type, and gluonic degrees of freedom to form the first

9 Notice that the 2 degrees of freedom of the gravitino hμν and the 2 degrees of freedom of its
partner the gravitino are not counted here because they do not participate to the thermal bath.
146 3 A Thermal Universe [TRH → TCMB ]

geff
150

100
70
50

30

20
15

10

0.001 0.01 0.1 1 10 100 1000

T(GeV)
b decoupling
quark
Hadron tau top decoupling
phase decoupling
transition
Fig. 3.3 Effective degree of freedom of the primordial plasma as function of the temperature

nucleus (see the Big Bang Nucleosynthesis section for more details) going from
61.75 to 10.75. Below 511 keV, we enter in a region where the neutrino decoupling
plays a role.
We can compute the value of gρ nowadays, noticing that, a priori the 2
relativistic species still present in the bath are the photons and the neutrinos, even
if the neutrinos have a lower temperature Tν than the photons due to the entropy
transferred from the electrons to the photons when they decoupled (see Sect. 3.3.1),
Tν = (4/11)1/3Tγ = (4/11)1/3 × 2.725 ≈ 1.95 K (1.7 × 10−4 eV). What is left
is thus 1 family of neutrino (fermions) with 2 degrees of freedom (SU(2)) and a
temperature10 Tν = 1.95 K and a photon (boson) with 2 degrees of freedom and a
temperature Tγ = T0 = 2.725 K. We can then extract from Eq. (3.37),
 4/3
today 7 4
gρ =2+ ×2× = 2.45 (3.40)
8 11

10 At least two neutrino flavors being nowadays non-relativistic from the oscillations measurements,

see the text in the box below Eq. (3.90) for a discussion on the subject.
3.1 Thermodynamics 147

(gρ = 3.36 for 3 neutrino flavors) which implies

π2
ρrel = gρ T04 = 5.9 × 10−34 g cm−3 (3.41)
30
and
3
2ζ (3) 3 −3 2 4
nγ  336 cm−3 .
today
nγ = 2
T 0 = 411 cm ; ntoday
ν = 3 × (3.42)
π 2 11

3.1.5.2 QCD (Quark–Hadrons) Phase Transition

Before the strange quark had time to annihilate, something else happens: matter
undergoes the QCD phase transition (also called quark-hadron transition). This
takes place at T ∼ 150 MeV. While quarks are asymptotically free (i.e. weakly
interacting) at high energies, below 150 MeV, the strong interaction between the
quarks and the gluons becomes important. The quarks and gluons then form bound
three-quark systems, called baryons, and quark-antiquark pairs, called mesons, and
below this temperature we should treat these as our new degrees of freedom.11
The lightest baryons are the proton and the neutron. The lightest mesons are the
pions π ± , π 0 . Baryons are fermions mesons are bosons. There are many different
species of baryons and mesons, but all except the pions are non-relativistic below
the temperature of the QCD phase transition. Thus, the only particle species left
in large number are the pions, electrons, muons, neutrinos, and photons. The three
pions (spin 0) correspond to g = 3 internal degrees of freedom. We therefore get
gρ = 2 + 3 + 78 × (4 + 4 + 6) = 17.25. Soon after the QCD phase transition, the
pions and muons annihilate and gρ = 17.25 − 3 − 78 × 4 = 10.75. Next electrons
and positrons annihilate (see Sect. 3.3.1).

3.1.5.3 A Little History of gρ (T ): Summary

We show in the Table 3.2 the different values of gρ (T ) after the annihilation is over
assuming the next annihilation would not have begun yet. In reality they overlap
in many cases. The temperature on the left is the approximate mass of the particle
in question and indicates roughly when the annihilation begins. The temperature is
much smaller when the annihilation ends. Therefore top annihilation is placed after
the electroweak transition. The top quark receives its mass after the electroweak
transition, so annihilation only begins after the transition.

11 The assumption of a weakly interacting gas of particles still holds for the baryons and the mesons,

but not for the individual quarks and gluons.

148 3 A Thermal Universe [TRH → TCMB ]

Table 3.2 History of gρ (T )

T Particles gρ (T )
200 GeV All present 106.75
100 GeV Electroweak transition (No effect)
<170 GeV Top annihilation 96.25
<80 GeV W ± , Z0 , H 0 86.25
<4 GeV Bottom 75.75
<1 GeV charm, τ − 61.75
<150 MeV QCD transition 17.25
<100 MeV π ± , π 0 , μ− 10.75
<500 keV e− annihilation (7.25) 2 + 5.25(4/11)4/3 = 3.36

3.1.6 Time and Temperature

From the relations we have just obtained and thanks to the current data, we
can extract the evolution of temperature, number density, and energy density of
the Universe as a function of time. Indeed, from the measurement of the CMB
background temperature T0 we can deduce the present energy density of the photons
with Eq. (3.26),

π2
ργ0 = × 2 × T04 = 2.62 × 10−10 GeV/cm3 . (3.43)
30
Moreover, we saw that the neutrino temperature is 1.91 K in the absence of any kind
of “non-standard” phenomena that could increase the temperature or the effective
degrees of freedom of the neutrino (see Sect. 3.3.3 for some exceptions). This gives
us for the energy density of the neutrinos,

π2 7  4
ρν = × × 2 × 3 × T0ν = 1.78 × 10−10 GeV/cm3 , (3.44)
30 8
where we considered 3 types of neutrinos. This expression is therefore valid until
two types of neutrinos become non-relativistic. If we note by m1 , m2 , and m3 their
mass eigenstate, the measurements of the mass differences (|m3 −m1 | and |m2 −m1 |)
give

|m3 −m1 |  0.049 eV > |m2 −m1 |  0.008 eV > T0ν = 1.7×10−4 eV (3.45)

which would mean that at least two neutrinos are non-relativistic. So there are
two types of neutrinos, currently, that participate in the “hot” component of dark
matter. This will be discussed in more detail in Sect. 3.3.1. We also want to add
that the factor “3” we used in Eq. (3.44) considers an instantaneous decoupling
ν
of the neutrinos. The ratio TTγ is not exactly the same when we take into account
3.1 Thermodynamics 149

non-instantaneous processes and the neutrino oscillation effects, which effectively

corresponds to a neutrino number of Neff  3.045 where Neff is defined by

π2 7  4
ρνCMB = × × 2 × Neff × T0ν . (3.46)
30 8

Finally, the present day radiation is thus

ρR = ργ + ρν = 2.4 × 10−10 + 5.7 × 10−11  3 × 10−10 GeV/cm3 , (3.47)

where we took one family of neutrino.12 For the matter content, we measured around
1011 solar mass per megaparsec which implies a matter density

ρmatter = 1011 solar masses/Mpc3  4.1 × 10−6 GeV/cm3 (3.48)

in relatively good accordance with WMAP data which gives

ρbaryons+dark matter  1.6 × 10−6 GeV/cm3 ;

ρbaryons+dark matter+dark energy  ρc0 = 5.3 × 10−6 GeV/cm3 , (3.49)

from which we can deduce ρdark energy  3.7 × 10−6 GeV/cm3 . Using Eq. (2.53),
we can also write
ρmat t er 2 ρradiat ion 2
0
mat t er h
2
= h  0.15 0
radiat ionh
2
= h  2.9 × 10−5 .
ρc0 ρc0
(3.50)

Matter dominated the Universe during almost 9.8 billion years after the CMB.
Since then, the dark energy is the main contributor to the density. As we can see,
presently Universe density of matter is 5000 times greater than radiation density.13
However, it was not always the case. In the early Universe, all Standard Model
particles were relativistic and contributed to the radiation. The evolution over time
of a system dominated by “dust” (matter) is indeed very different from the evolution
of a Universe dominated by radiation.

12 Keeping in mind the possibility that all the three families of neutrino are massive and thus do not
contribute at all to the radiation density.
13 Be careful to not be confused with the ratio η = n /n  6 × 10−10 . Indeed, a photon energy
b b γ
is around 10−4 eV at present time, whereas a baryon mass is 1 GeV. ρb /ργ  1000 gives you
nb /nγ  10−10 . The number density of the photon is much larger than the number density of the
baryons nowadays, this is the baryogenesis concept.
150 3 A Thermal Universe [TRH → TCMB ]

The density of matter changes because the volume V of the Universe changes,
while no new matter is created: ρm ∼ 1/V ∝ 1/a 3 where a is the scaling factor
defined from the current radius of the Universe R0 : R(t) = R0 × a(t) (0 < a(t) <
1). For radiation, it is easier to think in terms of photons. Its number density evolves
as ∼ 1/V as well, however, each photon wavelength is redshifted, so energy of
individual photons also changes as Eγ ∼ 1/a. Therefore the energy density of
E
radiation evolves as Vγ ∝ 1/a 4. We can then deduce from Eq. (3.26) that the scale
factor evolves as a ∝ 1/T and thus the matter density evolves as ρm ∝ T 3 . From
the actual values obtained by WMAP we then have

3 4
T T
ρm = 1.6 × 10−6 GeV/cm3 ; ρR = 3 × 10−10 GeV/cm3 (3.51)
T0 T0

with T0 is the present temperature of the Universe T0 = 2.34 × 10−13 GeV. The
result is shown in Fig. 3.4 where we can see that the matter has begun to dominate at
a temperature of around 1 eV, corresponding to the decoupling time, see Sect. 3.3.4.
If we want to calculate the expression of energy densities as a function of
time rather than temperature, we must first calculate the Hubble constant H
which represents the relation between time and the scaling factor a. As a first
approximation, we can write H ≡ ȧ/a = 2t1 with ȧ = da dt (see discussions around
Eq. (2.216) for details). Combining Eqs. (2.263) and (3.26) we then deduce the time

U(GeV/cm3) T = 10−9 GeV T = 3* 10−13GeV

109
Dark energy domination
106

1000 Radiation Matter domination

domination
1 U
U M
R
0.001
U
DE

10−6
107 108 109 1010 1011 1012
1/T (GeV−1)
Fig. 3.4 Evolution of the different components of the Universe density as function of the
temperature of the photons computed from Eq. (3.51)
3.1 Thermodynamics 151

tR in the radiation dominated era:

 
ȧ 1 ρR π gρ T 2
H = = = 2
= (3.52)
a 2tR 3MP 3 10 MP
 2
−6 1 GeV
⇒ tR  2.4 × 10 √ s [Radiation dominated era : T > 1 eV].
gρ T

Later on, when the matter begins to dominate the Universe, the density is dominated
by ρm given in Eq. (3.51) and the time dependance now becomes tm :

2 ρm
H = = (3.53)
3tm 3MP2
 3/2
−2 1 GeV
⇒ tm  5.9 × 10 × s [Matter dominated era : T < 1 eV].
T

We can then inverse the previous relations to obtain the evolution of the temperature
of the Universe as function of time in the radiation era [TR (t)] and the matter era
[Tm (t)].
 1/2
1s
TR (t)  1.5 × 10−3 GeV
t
 2/3
1s
Tm (t)  0.15 × GeV. (3.54)
t

We represent the evolution of the temperature as function of the time in Fig. 3.5
where we can notice the “slowdown” of the decrease at t  100 s, corresponding to
a temperature T  0.2 MeV. This is the BBN epoch when the quarks/colors degrees
of freedom are converted to heat the photon of the plasma following the entropy
conservation. There is a second slowdown at around t  1013 s corresponding to
the recombination epoch (last scattering surface) when the Universe begins to be
matter dominated: the slop of T = f (t) is thus changing from −1/2 to −2/3 as it
is clear from Eq. (3.54).
We can compute zEQ , the redshift at the matter density energy begins to dominate
over the radiation density energy. For that, one needs first to compute the present
radiation density, which content neutrino and photons. We will suppose that the
neutrino was relativistic all the time, even if we know that at least two have
decoupled from the recombination time. That is a relatively good approximation.
 1/3
The neutrino temperature being Tν = 11 4
Tγ , with Tγ the photons temperature,
and having also 2 degrees of freedom (because only left-handed neutrino has been
152 3 A Thermal Universe [TRH → TCMB ]

T (GeV)
1
Radiation dominated

0.001

Matter dominated
10−6
BBN

10−9
Recombination

10−12
10−8 0.001 100 107 1012 1017
t (seconds)
Fig. 3.5 Evolution of the temperature as function of the time computed from Eq. (3.54)

detected so far), we can write using Eq. (3.26)

 4/3  4/3
7 4 π2 4 7 4
ρν = 3 × × ×2× Tγ = 3 × ργ = 0.68 ργ (3.55)
8 11 30 8 11

implying

ρR = ρν + ργ = 1.68 ργ , (3.56)

where ργ is the (measured) background radiation. Asking for γ (zEQ ) =

m (zEQ ), we obtain

0 (1 + z 4
R 1.68 × γ 1.68 γ EQ )
= = 0 (1 + z 3
=1
m m m EQ )
0.311
⇒ zEQ = = 3366  3400, (3.57)
1.68 × 5.5 × 10−5

corresponding to a time tEQ  40,000 years from the Big Bang.

Exercise Taking the dark energy density and matter density to be, respectively,
 = 0.689 and m = 0.311 show that the redshift z corresponding to the dark
energy domination,  (z ) = m (z ), is z = 0.304. Show that it happens ∼10
3.1 Thermodynamics 153

Gyrs after the Big Bang. Find then the temperature/time relation in the phase where
the Universe is dominated by dark energy, 10 Gyrs after the Big Bang.
Hints: use the relation in a − dominated Universe, 1 + z = eH0 t .

The cosmological constant problem

The introduction of the cosmological constant has proved to be a real
challenge in the history of the construction of theoretical models, as we
have seen in our discussion at the end of the Sect. 2.1.6. However, a model
dominated by a cosmological constant is the most natural if we want a
homogeneous and isotropic Universe as Albert Einstein imagined it. It is
indeed more difficult for the density of matter to be homogeneously and
isotropically distributed when it is dominated by unstable dynamic forces,
like the gravity forces. Moreover, this hypothesis, known as the cosmological
principle, was unnatural, especially in view of the night sky before Edwin
Hubble’s measurements. The discovery of Hubble’s law was above all the
confirmation of a homogeneous and isotropic Universe. The crucial point
is, if we assume the existence of this cosmological constant, what would
be its most natural value today? That is a reasonable question, leading to
what is commonly called “the cosmological constant problem,” a complete
review being accessible at [1]. The main idea is that combining Eq. (2.30)
with Eq. (2.89), for the Higgs Lagrangian (B.229), in the electroweak breaking
phase, we can write an “effective” electroweak cosmological constant EW

2
1 2 2 1 MH
V (H = v) = − MH v ⇒ EW = − v 2  −2 × 10−29 GeV2 ,
8 8 MP2

where we used Eq. (B.232) and v = 246 GeV. From Eq. (2.54), this value for
EW corresponds to a density ρ = MP2 EW = −1.2 × 108 GeV4 . This
number is huge and has to be compared with the values of the cosmological
0 = −47 GeV4 , 55 orders
 × ρc = 2.9 × 10
constant measured today, ρ 0

of magnitude less that what we should expect. We can always tune the
parameters, adding an uplifting term in the Higgs potential to counterbalance
exactly the V (H = v) term and obtain the cosmological constant measured
today. This is clearly not satisfactory and would imply that the vacuum energy
density was huge prior to the electroweak phase transition, not to mention the
fact that such fine tuning would not resist loop corrections. This is what is
called the cosmological constant problem.
154 3 A Thermal Universe [TRH → TCMB ]

3.1.7 The Entropy

To compute the entropy of the Universe, we need the second law of thermodynam-
ics. The first law was used to compute the relation between pressure and volume
in the Appendix A.5 in the case of adiabatic transformations. The second principle
of thermodynamics establishes the link between entropy, pressure, temperature, and
internal energy. It can be written as

T dS = dU + P dV = δQ, (3.58)

where dS is the variation of entropy at a temperature T under a pressure P ,

corresponding to an exchange of heat δQ. If one defines ρ as the energy density
(U = ρV ) and s the density of entropy (S = sV ), one obtains

dV
dρ − T ds = (T s − ρ − P ) . (3.59)
V
In equilibrium, the entropy density, energy density, and pressure are intensive
quantities that can be written as functions only of the temperature ρ(T ), s(T ),
P (T ), such that dρ − T ds ∝ dT . The coefficients in front of dT and dV are
then independent and must vanish separately because one is intensive (independent
of volume) and the other one is extensive (depends on the size of the system). This
relates the entropy density to the energy density and pressure

ρ +P
s= (3.60)
T
using (3.11) and (3.26) one has

2π 2
s= gs T 3 (3.61)
45

 3  3
Tb 7 Tf
gs = gb + gf . (3.62)
T 8 T
b=bosons f =f ermions

Except in the case where particles transfer their entropy to photons and not
to already decoupled particles (such as neutrinos), all particles have the same
temperature and we can reasonably approximate gs  gρ . One can also notice
that sγ  3.6 nγ , and constant entropy imposes s ∝ 1/R 3 . Then, working with
comoving frame densities (Yi = ni /s ∝ ni × R 3 ) is equivalent to working with
particle/photon ratio density Yi  ni /nγ . It is also important to notice that, like
3.1 Thermodynamics 155

n(T ), s(T ) evolves as T 3 . However, the degrees of freedom (especially the statistical
factors are 7/8 and not 3/4 as in n(T )) are the ones of a statistic in T 4 , as the entropy
is proportional to ρ(T )/T and not n(T ). Conservation of entropy is then directly
linked to the conservation of energy: ρ(Ti )/Ti = ρ(Tf )/Tf .
As in the case of degrees of freedom for energy density, we can calculate the
today
effective degrees of freedom for entropy density s today, gs , which is different
today
from gρ because the decoupling of the neutrino has created a difference between
the neutrino and photon temperature (see Sect. 3.3.1 for details). One obtains14

today 7 4
gs =2+ ×2×3× = 3.91. (3.63)
8 11

The fact that we considered 3 types of neutrinos (and not only one relativistic) is
subtle and explained in the box below Eq. (3.90). From that number one can deduce
the present entropy s0 , which will be very useful to compute relic abundance of
stable species

2π 2 today 3
s0 = gs T0 = 34.71 K3 = 2.2 × 10−38 GeV3 = 2909 cm−3 . (3.64)
45

Asking for constant entropy of the Universe, S = sR 3 , has several consequences.

First, from Eq. (3.61) the temperature of the photons and other relativistic particles
follows a law T ∝ R −1 . The other consequence is that, every time a particle
decouples from the thermal bath, this decoupling happens at constant entropy. It
means that this particle species “gives” its entropy to the relativistic particles to
which it is coupled and still present in the bath, before leaving it. This information
s
is in fact encoded in the degrees of freedom geff : after the decoupling of a particle
species i, the effective degrees of freedom in the bath decrease: geff s
→ gs − gi .
the entropy being constant (and so its density as the Universe do not have time
to evaluate during this process considered adiabatic) gs T 3 =cst implies that the
decoupling of a specie increases the temperature of the bath (i.e. of the photons)
following Tγafter = Tγbefore × (gsbefore /gsafter )1/3 . After this heating of the bath, the
temperature of the plasma follows the R −1 law.
On the other hand, the particle which has decoupled from the plasma follows
different laws if it is a massless or massive particle. Indeed, after decoupling the
energy of each massless particle is redshifted by the expansion of the Universe

14 The fact of taking into account three species of neutrino, even if it is known from the
measurement of matm and msol that some are non-relativistic nowadays, comes from the fact
that, once decoupled, the massive neutrinos still follow a classical relativistic distribution function
(see Sect. 3.2 for more details).
156 3 A Thermal Universe [TRH → TCMB ]

E(t) = E(tdec )R(tdec )/R(t) = Edec Rdec /R = p(t) in the relativistic case. As
n(t) decreases proportionally to R −3 because the density is “frozen” (no any way
of producing it thermally nor destroying it after decoupling time) the distribution
function f (p) = d 3 n/d 3 p is constant during the expansion. This implies eE/T =
eEdec /Tdec = eER/Rdec Tdec ⇒ T = Tdec Rdec /R ∝ R −1 . The relativistic species
thus follows the R −1 evolution after they decoupled from the plasma but are not
affected by the (gs )1/3 enhancement that affects the photons and relativistic species
still living in the bath.
The massive non-relativistic species (m Tdec ) follows another law after being
decoupled. Their kinetic momentum follows the classical redshift p = pdec Rdec /R
p2
from which it follows that the kinetic energy (∝ 2m ) of each particle red shifts
−2
as R : Ec = Ec Rdec /R . Conservation of the density function thus implies
dec 2 2
2 2 2 /R 2 ∝ R −2 . As a conclusion, after the
epdec /2mTdec = ep /2mT ⇒ T = Tdec Rdec
decoupling of a species “i”

Ti ∝ R −1 if relativistic Ti ∝ R −2 if non-relativistic. (3.65)

We have summarized this effect in Fig. 3.6

Exercise Taking the radius of the Universe found in Eq. (2.65) show that the total
entropy in the Universe is St ot = 9.3 × 1089 , the total number of photons Nγ =
1.3 × 1089, and the total number of baryons Nb = 7.9 × 1079. Compute the same
numbers for a radius of 46.3 × 109 lyrs.

R−1

T photon

dec
T massive
−2 dec g
s R−1
R T massless eff

R
Fig. 3.6 Evolution of the temperature of different species (relativistic/massless and non-
relativistic/massive) after their decoupling from the thermal bath. It can be, for instance, the dark
matter decoupling followed by the neutrino decoupling
3.1 Thermodynamics 157

<sig v>

Target

Rbefore Rafter
Fig. 3.7 Illustrative example of the decoupling epoch when the number of interactions is divided
by 2 during a time t due to the dilution of the target. The volume necessary to have 2 collisions
(Rbef ore ) is now just sufficient to give one collision (Raf ter )

3.1.8 The Meaning of Decoupling

In the previous paragraphs, we have often referred to the notion of decoupling

of primordial plasma, without really giving a definition of what is meant by
“decoupling.” Suppose a particle A interacting in the bath with a rate per particle
 = n σ v, n being the density of the target particle, and σ v the average cross
section times the relative velocity. t = 1/  represents the mean time between
two collisions. During this time t the Universe has expanded by a factor R
such as R/R = H t = H / . In another word, when H  , the size of
the Universe has doubled and the density n of the target has been divided by 8, as
the interaction rate  which is proportional to n. In another word, the time (or the
temperature) of the Universe when H   is the epoch where the particles decouple
from the bath and their interaction rates with the plasma decreases exponentially. We
illustrate it in Fig. 3.7. The exact way to treat the decoupling problem is to solve the
Boltzmann equation. However, the approximation H   to obtain the decoupling
time of particles is usually quite accurate. We will give two specific examples to
understand how the nature of the interaction can change drastically the temperature
of decoupling of species. We will consider two distinct cases:

• (i) interactions mediated by a massless gauge boson (like the photon)

• (ii) interactions mediated by a massive gauge boson (Z or Z  ).

The exchange of a massless gauge field between two particles S and S̃ can
be parameterized in a case of bosonic particles15 by a Lagrangian of the form
L = (Dμ S)(D μ S)† + (Dν S̃)(D ν S̃)† which includes the terms of interactions
ig pμ Aμ SS † + ig p̃ν Aν S̃ S̃ † , Aμ being the massless vectorial field (photon, for
instance) and g its coupling to the particles in the bath. One can then compute the
amplitude of the interaction (see Appendix C)

M = g 2 pμ ημν p̃ν /p2 = g 2 p.p̃/p2  E Ẽ(1 − cos θ )/E 2 ,

15 The analysis does not depend on the nature –fermionic/bosonic– of the particles we consider.
158 3 A Thermal Universe [TRH → TCMB ]

cos θ being the diffused angle between S and S̃ which implies

|M|2  g 4 E Ẽ(1 − cos θ )/E 2 .

Using Eq. (B.110), σ  |M|2 /64π 2 s, considering that the particles are relativistic
in the plasma and using Eq. (3.30) (mS,S̃ T ⇒ E  Ẽ  T ) one can deduce
σ  g /T implying when combined with Eq. (3.23)  = nS σ v = nS σ c 
4 2

g 4 T . The particle S will thus be decoupled from the primordial plasma when

 g 4 MP
1⇒  1 ⇒ T  g 4 Mp  1014GeV, (3.66)
H T
where we took g = 0.1 as illustration. It is important to notice that T in this case
is an upper bound, which means that as soon a T  1014 GeV the long-range force
mediated by the massless boson (as it is the case for electroweak interactions) will
always be sufficient to maintain charged relativistic particles in equilibrium in the
bath: the decoupling will appear only when the temperature will reach mS , where
the density will be exponentially suppressed by the Boltzmann factor (the term nS
in the expression of ).
In the case of the exchange of a massive gauge boson Z  (MZ   T ) the
amplitude of the reaction can be written M  g 2 E 2 /MZ2   g 2 T 2 /MZ2  ⇒  
g 4 T 5 /MZ4  . The decoupling temperature will then be given by the usual condition
/H  1:
 4/3  4/3
 g 4 MP T 3 MZ  −1/3 MZ 
1⇒ 1⇒T  Mp  0.1 MeV.
H MZ4  g 1 TeV
(3.67)

In this case we clearly see that the decoupling of particles charged only under SU (2)
(Z boson exchange) will decouple quite late in the history of the Universe (around
1 MeV) which is precisely the case of the neutrino, studied in detail in Sect. 3.3.1.
We represent in Fig. 3.8 the evolution of H (T ) and (T ) in different cases (taking
g = 0.1). The following relations help to understand the diagram.

log H  −2 log 1/T − 19 ;

log γ  − log 1/T − 4 ;
log MZ  −5 log 1/T − 4 log MZ  − 4.
3.1 Thermodynamics 159

Log

50

*MZ’=1016GeV *MZ’=1010GeV *MZ’=1 TeV

19
15

*J

−19
19 10
10 GeV 10 GeV 1 GeV 1/T

Fig. 3.8 Evolution of H (T ), γ (T ), and MZ (T ) for different masses of Z  and relativistic
species. The black dots show the temperature when the decoupling occurs (/H  1)
160 3 A Thermal Universe [TRH → TCMB ]

3.2 Chemical Decoupling or Kinetic/Thermal Decoupling?

3.2.1 The Main Idea

Because we will need to analyze the decoupling of a dark matter from the thermal
bath, it is important to point out that all the discussions above concerned mainly
the chemical decoupling, i.e. the temperature for which the production rate of a
species (Standard Model particles or dark matter) is too small to maintain it in
equilibrium with the thermal bath. In other words the thermal bath cannot change
anymore the number of dark species, so the name “chemical” decoupling in analogy
with chemical reactions. It is very important to realize that even though dark
matter is a chemically distinct particle species after this decoupling, it is still a
constituent of the local hot bath. Indeed, the relevant target density for elastic
scattering processes to maintain thermal equilibrium is provided by the number
density of relativistic Standard Model particles and thus decreases only at a rate
proportional to T 3 , and not exponentially as the “self” target, the dark matter itself.
Eventually, at a temperature Tk , the elastic scattering (and not annihilating) rate16
eq eq
scat = nSM σscat , with nSM the equilibrium density, cannot compete with that of
the expansion of the Universe, and the dark matter particles start to decouple from
kinetic equilibrium. Elastic scattering processes cease. In fact, to be precise, even
before the last scattering occurs, the temperature of the dark matter has no time to
be relaxed because of the Hubble expansion. It is thus the relaxation time which
will determine the kinetic decoupling of the dark matter, i.e. the time when the
temperature of the dark matter is not maintained anymore to the plasma one through
scattering and drops quickly with the scale factor a as a −2 instead of a −1 , as we
saw in Eq. (3.65). The temperatures of the dark matter particles χ and the radiation
background are then approximately related by Tχ  |p|2 /2mχ  ∼ T 2 /Tk ,
Tk  mχ being the temperature of the kinetic decoupling.

3.2.2 Approximate Solution∗

Let us estimate the relaxation time for a WIMP of mass mχ living in a thermal bath
of temperature T . Let suppose that after each shock, the Standard Model particles
transfer a momentum p (tri-vector), roughly given by its energy |p| ∼ T to the
dark matter corresponding to a velocity v. After the number of collisions Ncoll ,
the velocity of the dark matter is vNcoll = vNcoll −1 + v, implying

|vNcoll |2 = |vNcoll −1 |2 + 2|vNcoll −1 ||v| cos θ + |v|2 ,

that σscat vSM = σscat as the SM particles are still largely relativistic in the thermal bath
16 Notice

: vSM = c.
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 161

θ being the angle between the two colliding (dark matter and Standard Model)
particles. We can thus deduce vN 2
coll
 = v02  + Ncoll (v)2 . In other words, after
Ncoll collisions, the velocity (and thus momentum) gained by the dark matter (if
1/2
we suppose it at rest, for instance, at the beginning, v0 = 0) is pNcoll = Ncoll p
which should be equal to (2mχ Ec )1/2 ∼ (mχ T )1/2 for a dark matter particle at the
temperature T . Replacing p ∼ T we then can compute the number of collisions
needed to keep the dark matter in kinetic equilibrium:
mχ
Ncoll ∼ 1. (3.68)
T
The relaxation time (time needed to still keep the kinetic equilibrium, i.e. to obtain
Ncoll collisions) with a scattering scat is

Ncoll mχ 1 mχ 109 mχ
τr ∼ = eq ∼ 4 ∼  . (3.69)
scat T nSM σscat T σscat σscat
T 4 10 −9

Then, kinetic decoupling occurs approximately

 at 2 the temperature Tk , for which
−1 g T
τr (Tk ) = H (Tk ) which gives with H = 3 10ρ M
π
P

 1/2 
σscat mχ −1/2
Tk ∼ 10−4 gρ1/4 GeV (3.70)
10−9 GeV−2 100 GeV

corresponding roughly to the MeV scale for a 100 GeV dark matter and an
electroweak-like scattering cross section (σscat  σEW ∼ 10−9 GeV−2 ). When
the temperature of the plasma drops below Tk , the number of collisions needed to
still keep the dark matter in kinetic equilibrium is such that the time to reach it
becomes larger that the Hubble expansion time: the dark matter reaches the kinetic
decoupling. This effect can play an important role when the velocity appears in
processes like Sommerfeld enhancement, for instance. A more detailed analysis of
the thermalization in the dark bath is explained in Sect. 3.2.4.

3.2.3 What Is Happening After the Decoupling?

Once  (scattering or decay) falls below the expansion rate H , the particles decouple
from the plasma and propagate freely along geodesics of the space-time. The form
of the distribution function f (p) is conserved, while the momentum redshifts as
p(t) ∝ 1/a, implying p(t) = p(tdec )a(tdec )/a(t). The form of the distribution
function is indeed conserved as long as the particle is not in thermal bath with
another dark sector (which can also have decoupled from the primordial plasma
before) because of the conservation of energy/momentum of the decoupled system:
there is no source (thermal bath or collisions) that can modify the distribution in
162 3 A Thermal Universe [TRH → TCMB ]

energy of the decoupled gas. It follows that the distribution function for any t > tdec
is given by

a(t)
f (p, t > tdec ) = f ( p, tdec ). (3.71)
a(tdec )

We see that the distribution function of the decoupled particles is simply a rescaled
version of the distribution function at decoupling time tdec .

• Decoupling while relativistic: if a particle of mass m decouples when relativistic,

Tdec m, then the distribution function (3.71) takes the form

1 a(tdec )
f (p, t > tdec ) = , with T̃ (t) = Tdec . (3.72)
eE/T̃ (t ) ±1 a(t)

So the temperature of a decoupled relativistic species falls strictly as a −1 . If the

particle later becomes non-relativistic, we have E ∼ m but the distribution func-
tion keeps the form (3.72). This is an important point, especially when computing
the relativistic degrees of freedom for the present entropy, for instance. What is
important is that the particle was relativistic during the decoupling time. Once it
is decoupled, it is as if the thermal bath gave them a punch, and they continue
their way in space without any interaction at the same velocity, just redshifted by
the scale factor. The density number distribution is a relativistic one, respecting
n(T̃ ) ∝ T̃ 3 (3.27) even if T < m (no exponential suppression). In other words,
all particles that were relativistic during their decoupling time count as relativistic
degrees of freedom in the Universe nowadays. Notice that the conservation of the
entropy, in the radiation dominated era, imposes S = sR 3 = sa 3 R03 =cte, using
(3.61), one deduces T ∝ a −1 . This means that even after decoupling, a gas
of relativistic particle keep a “virtual” temperature T̃ following the one of the
thermal bath.
• Decoupling while non-relativistic: if, on the other hand, a particle is non-
relativistic at the time of decoupling, Tdec , then E  m + p2 /2m and the
distribution function is given by
 2
a(tdec )
f (p, t > tdec ) = e−m/Tdec e−p
2 /2mT̃
, T̃ (t) = Tdec . (3.73)
a(t)

So the temperature of decoupled non-relativistic species falls as a −2 , which

means much faster than the relativistic ones. Their number density thus decreases
not exponentially but as T 2 in a radiation dominated Universe.

It is important to emphasize that, despite the different scaling behavior of the

temperature for relativistic and non-relativistic species after decoupling, in both
cases the equilibrium distribution is maintained. The particles are not anymore in
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 163

h(p) T=100 GeV, m=10 GeV

a/adec=1
200
a/adec=1.2
150

100

50
a/adec=1.5

0
0 200 400 600 800 1000
p (GeV)
h(p) T=100 GeV, m=200 GeV
120

100 a/adec=1

80

60

40 a/adec=1.2

20
a/adec=1.5
0
0 200 400 600 800 1000
p (GeV)
p2
Fig. 3.9 Distribution function h(p) = 2π 2 × f (p) for T = 100 GeV for different values of the
scale factor a(t) in the relativistic fermion (top) and non-relativistic fermion (bottom)

thermal equilibrium but the shape of the distribution is maintained around a virtual
temperature T̃ . We illustrate the behavior of the two cases (relativistic and non-
relativistic) in the Fig. 3.9 for T = 100 GeV and two masses of particles: 10 and
200 GeV for the relativistic and non-relativistic case, respectively.
164 3 A Thermal Universe [TRH → TCMB ]

3.2.4 Transfer of Energy and Thermalization

3.2.4.1 Generalities
There exists a second Boltzmann equation which concerns the transfer of energy.
If a set of particles i is all produced by the same mechanism and the scattering is
more efficient than the expansion of the Universe, they naturally reach a common
thermal temperature (the one of the photons), T = Ti . However, in scenario where
the dark sector is not produced with the Standard Model ones or very feebly coupled
with the visible sector, the temperature of the dark sector increases slowly due to the
transfer of the energy from the thermal bath to the dark bath of energy density ρ 
and pression P  . If the density of dark particles is very small at the reheating time,
the Boltzmann equation for the transfer from the thermal bath energy density ρ to
ρ  from 1 + 2 → 3 + 4 reaction can be written

dρ 
+ 3H (ρ  + P  ) = n2EQ σ v(E1 + E2 ) (3.74)
dt
with nEQ being the density of particles in the bath. The phenomenon is illustrated in
Fig. 3.10. The right-hand side of Eq. (3.74) should also content terms proportional to
n2dark for the inverse transfer process ρ  (3 + 4) → ρ(1 + 2) but as we considered the
dark bath composed of particles of mass M with feeble interactions with the thermal
bath, this process can be neglected. It is a situation similar to FIMP (Freeze in
Massive Particle) or heavy mediators which will be discussed is Sect. 3.6. With the
hypothesis that the thermal bath is in equilibrium, combining Eqs. (3.74) and (3.23)
and the fact that in the relativistic case17 P  = ρ  /3 and dt
d
= −H T dT
d

 
d ρ  /ρ dρ  d(ρ  /ρ) n2EQ
ρ = − 4Hρ  ⇒ =− σ v(E1 + E2 ) (3.75)
dT dT dT ρH T

with

1 d 3 p1 d 3 p2 |M̄|2
σ v(E1 + E2 ) = f1 (E1 )f2 (E2 ) (E1 + E2 )d
n2EQ (2π)3 (2π)3 128π 2E1 E2
(3.76)

17 Notice that we made the supposition in this section that the dark matter bath is in thermal
equilibrium with itself (with a temperature T  ) or from the self-scattering of the dark matter on
itself or from the scattering of other particles of the dark sector on themselves, i.e. scatter (T ) >
H (T ). For a more detailed description of the mechanism of thermalization, see the next section.
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 165

Fig. 3.10 Illustration of the

Boltzmann equation
describing the transfer of Thermal bath
energy from the Standard (photons, temperature T)
Model bath to the dark bath density of energy U

E1 + E2

Dark bath
(dark particles, temperature T’)
density of energy U’

M̄ being the mean of the amplitude squared on the spin of the initial particles and fi
their statistical distribution. Following a classical procedure of integration that will
be more detailed when we will treat the WIMP case, see Eq. (3.180), we can write

n2EQ σ v(E1 + E2 )
 ∞   ∞  √
g1 g2 s
= s − 4M 2 |M̄|2 s dsd t t 2 − 1e−t T dt
32(2π)6 4M 2 1

implying
√ 
d(ρ  /ρ) g1g2 10 MP ∞ 
= −45 3/2
| M̄| 2 2
x x 2 T 2 − 4M 2 K2 (x) dxd
dT 512π 9T 3 gρ 2M
T
(3.77)

g1 and g2 being the internal degrees of freedom of particles 1 and 2. We can

then divide the integral (3.77) in two regimes. In the first one the particle can be
considered as relativistic (M T ), and the second one when the temperature drops
166 3 A Thermal Universe [TRH → TCMB ]

below the mass of the dark matter (T M) where the integral in Eq. (3.77) tends
to 0:
√ 
d(ρ  /ρ) g1 g2 10 MP ∞
= −45 3/2
|M̄|2 x 3 K2 (x) dxd [M T]
dT 512π 9T 2 gρ 0

d(ρ  /ρ)
=0 [T M] (3.78)
dT

3.2.4.2 A Specific Case: Exchanged of a Massless Gauge Boson*

To illustrate the general result obtained in Eq. (3.77) we will take a simple toy model,
based on mirror dark matter, where the dark matter, a Dirac fermion of mass Mχ ,
is charged under the electromagnetic charge with a coupling δ. This happens, for
instance, in the case of the presence of a kinetic mixing between an extra U(1)[dark
photon] and photon. This is also called “millicharged dark matter” as it is equivalent
to suppose that the dark matter is very feebly coupled to the electromagnetic fields
(coupling of the order δ 1). The amplitude can then be written
ημν
M = δ χ̄ γ μ χ qf ef¯γ ν f (3.79)
(p1 + p2 )2

pi being the quadrivector of the incoming particles, and qf the electromagnetic

charge of f . We than can compute

Mχ2
|M̄|2 = 16 qf2 e2 δ 2 (1 + cos2 θ ) + (1 − cos2 θ )
s

2 1 Mχ2
⇒ |M̄|2 d = 128π qf2 e2 δ 2 +
3 3 s

d(ρ  /ρ) 15 √ g1 g2 MP qf2 e2 δ 2  ∞ Mχ2
⇒ =− 10 3/2
2+
dT 4 π 8 T 3 gρ 2
Mχ
T
T2

× x 2 x 2 T 2 − 4Mχ2 K2 (x) dx.

Noticing that x 3 K2 (x)dx = 8, after integration on T we obtain, in the regime
T Mχ for Dirac annihilating particles (g1 = g2 = 4) and qf = 1 as an example,
 
d(ρ  /ρ) √ ααD MP ρ 45 ααD MP
= −15360 10 3/2
⇒ = 2560 (3.80)
dT π 6 T 2 gρ ρ π π 6 T gρ3/2

with α = e2 /4π and αD = δ 2 /4π. We made the hypothesis that ρ   0 at the

beginning of the transfers and integrated from the reheating temperature TRH until
T considering TRH T . Notice that we neglected in Eq. (3.76) a second term in
the right-hand side corresponding to the inverse transfer from the dark bath to the
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 167

Standard Model (photons) bath. This comes from the fact we supposed that the dark
system has not been populated by dark particles from the beginning of the thermal
history (no coupling to the inflaton, for instance).
Once the density of dark particles reaches (through the Boltzmann equation
describing the evolution of the thermal density of dark particles) an equilibrium
appears between ρ  and ρ similar to the one for the Yields Yi : d(ρ  /ρ)/dT 
0 ⇒ ρ  ∝ ρ. As the we supposed the dark bath is in thermal equilibrium

  4
(scatter (T ) > H (T )), we can use Eq. (3.23) and write ρρ = TT . Eq. (3.80)
then becomes
1/4 
MP T 3/4 T
T  = (ααD )1/4 (15360)1/4101/8 3/8
⇒
π 3/2 gρ 1 GeV

√ T 3/4
 3000 δ . (3.81)
1 GeV

The numerical result is shown in Fig. 3.11 where we plotted TT as a function of T for
Mχ = 100 GeV and different values of δ from 10−6 down to 10−9 . We clearly see
the two regimes T Mχ and T Mχ with the two behaviors given by Eqs. (3.78)
and (3.81), respectively. We also remark that the value δ  10−6 is the limit value
for which the dark bath does not enter in thermal equilibrium with the thermal bath.

T’/T
1.00
M  =100 GeV
0.50
= 10−6
= 10−7
0.20

0.10 = 10−8

0.05
= 10−9

0.02

2 3 4 5
10 10 10 10 10
T (GeV)

Fig. 3.11 Evolution of T  , temperature in the dark bath as function of T for different values of
the coupling δ and for Mχ = 100 GeV
168 3 A Thermal Universe [TRH → TCMB ]

F F

F F

J’
J’
F
F

F F

Fig. 3.12 Feynman diagrams of the χχ → χχ scattering which contributes to the thermalization
in the dark sector of temperature T 

Indeed, for any values of δ larger than 10−6 , the temperature T  (and thus the density
of energy ρ  ) will reach T (respectively, ρ) and then form a common thermal bath.

Exercise Do the same analysis with the Higgs-portal model.

3.2.4.3 Thermalization
In the previous sections, we always made the supposition (explicitly or implicitly)
that the system(s) we were studying were in thermal equilibrium. We can check
when this condition is realized and, if not realized, how the dark bath reaches
equilibrium through a (somewhat) complex thermalization process. The condition
for the dark bath to be in thermal equilibrium is scat t er/dark↔dark(T ) > H (T ).
This translates into, before the Universe had time to double in size, there was (at
least) one scattering. We show the different scattering processes in Fig. 3.12. The
scattering is present to keep the distribution function of the systems of particles
(Boltzmann, Fermi-Dirac, or Bose-Einstein) in equilibrium state.
These processes depicted are equivalent to the Bhabha scattering, and to a good
approximation, one can keep the s-channel exchange diagram and apply Eq. (3.181)

|M̄|2  gD
4
(1 + cos2 θ )
 √
T  gχ2 √ s 4 T 4 g 2
gD χ
⇒ σ vn2χ = sK1 |M̄|2 dsd = , (3.82)
32(2π)6 T 3(2π)6

where we made the hypothesis that the dark matter is in equilibrium at a temperature
T  . We want to know for which temperature T the condition of equilibrium
scat (T ) > H (T ). We then obtain at equilibrium using Eq. (3.27), nχ =
3 ζ(3) 3
4 π 2 gχ T :

gD4 T g
scat (T  ) = nχ σ vscat 
χ
. (3.83)
9ζ(3)(2π)4
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 169

Not thermalized

g = 0.001
1 D

0.001

g = 0.001
10−6 D

g = 0.1
10−9 D
g =1
D

10−12
104 106 108 1010 1012
T (GeV)
Fig. 3.13 Region of the parameter space (T , δ), T being the temperature of the Standard Model
bath, where we can consider that the dark bath of temperature T  is in equilibrium with itself
through scattering

Using the expression of H (T ) given by Eq. (2.264) and the relation between T  and
T we obtained18 in Eq. (3.81)
  g 4 √  1 GeV
scat (T  ) χ √ MP T
4g 5/4
2 gD D
= 10  2.6 × 1014 δ . (3.84)
H (T ) 3ζ(3)(2π)5 T T 0.1 T

We show in Fig. 3.13 the region of the parameter space where thermal equilib-
rium is satisfied. We notice that the temperature T of the Standard Model bath from
which the thermal equilibrium of the dark bath of temperature T  is attained is
smaller for small values of δ and of gD . The reasons are different for both cases.
Indeed, small δ implies a few energy transfer between the two systems, and thus
a low temperature T  , and as a consequence, a low density of dark particles: the
scattering is thus not sufficiently efficient to bring the new created dark matter from

18 We took Dirac fermions for the dark matter and particles 1 and 2: g1 = g2 = gχ = 4.
170 3 A Thermal Universe [TRH → TCMB ]

dN/dE
Dark matter at production time

1) Production of the dark species from the thermal bath Dark bath
Thermal bath

g F
SM G D

J’
SM F

<E> ~ T’ <E> ~ T E
3 3
dn = T’ dn = T

dN/dE
2) Thermalization of the dark species by the dark bath through scattering
F F
gD

J’

g
D
F F

dN/dE
3) Heating of the dark bath by the thermal bath from energy transfer

F
SM

J’
G g
D

SM F

<E> ~ T’ + H E
3
dn = (T’ + H )

Fig. 3.14 Summary of the different steps leading to thermalization of a dark bath

annihilation of the Standard Model in the thermal bath into a new dark equilibrium
state. A small value of gD renders the scattering less efficient also but due from the
too weak coupling of the self interaction of the dark matter on itself. We can also
remark that the condition of thermal equilibrium we supposed to obtain Fig. 3.11 is
valid.
We summarize in Fig. 3.14 the process of thermalization. The dark mater χ is
produced from the thermal (Standard Model) bath with a distribution of energy
dnχ /dEγ which follows the distribution of energy of the photons from the thermal
bath, by conservation of energy: Eγ = Eχ , so (Tχ )after production  Tγ = T .
However, the interaction between the dark matter and the photon is very weak
(proportional to δ) and thus, the scattering of the photons on the dark matter is
not sufficient to maintain the thermal equilibrium of χ. However, other χ particles
already produced before and in dark equilibrium at a temperature T  between
3.2 Chemical Decoupling or Kinetic/Thermal Decoupling? 171

themselves can scatter on the newly produced χ to bring it into the dark bath and
to follow the new distribution of energies dnχ /dEχ which defines the temperature
T  . But in the meantime, there was a small transfer of energy between the thermal
bath and the dark bath which has increased a little bit T  → T  + . Then the
process loops until T  reach T and then the dark matter enters in equilibrium with
the thermal bath, or until the temperature of the thermal bath T drops below the dark
matter mass Mχ : the photons are then not sufficiently energetic anymore to heat the
dark bath.

3.2.4.4 γ  Entering in the Dance

After some time the γ  will become sufficiently abundant to enter in thermal, and
chemical equilibrium with the dark matter χ. When it was only the mediator of the
self-scattering process, its density was not of real interest for the evolution of T  or
nχ . However, once it enters in the dark bath (χχ↔γ  γ  (T ) > H (T )), its effect on T 
will be equivalent to the inverse process of the electron decoupling (see Sect. 3.3.1).
Whereas electron gives its degrees of freedom to heat the bath of photons when it
decouples, in this case, from conservation of entropy, the inclusion of the γ  in the
thermal bath will decrease the temperature T  from entropy conservation in the dark
system, S  :
   
Sbefore ∝ gχ (Tbefore )3 ; Safter ∝ (gχ + gγ  )(Tafter )3
 1/3
  gχ
⇒ Tafter = Tbefore × . (3.85)
gχ + gγ 

This result can also be observed from the Boltzmann equation where a new term
expressing the effect of γ  on the abundance of nχ from its annihilation enters in the
game.

Exercise Recover this result by solving the Boltzmann equation.

We understood in this section how important is the process of thermalization,

and how complex can be the mechanism if it involved several species of particles.
Three kinds of equilibrium can co-exist in the primordial plasma: kinetic equi-
librium, where all the particles share the same distribution of energy; chemical
equilibrium, where the number density of particles are maintained constant from
annihilation/production processes; and thermal equilibrium which is the state of
kinetic + chemical equilibrium. We will now apply these techniques to the case
of the neutrino decoupling and the recombination time at the CMB epoch.
172 3 A Thermal Universe [TRH → TCMB ]

3.3 The Case of Light Species

3.3.1 The Neutrino Decoupling

Before the last scattering (CMB) period, the neutrino decoupled from the plasma.
Indeed, the neutrino is the only particle of the Standard Model which interacts only
weakly and not electromagnetically nor strongly. Because of that, the interaction
rate ν = nν σ v from the scattering reaction eν → eν or annihilation νν → ee
becomes smaller than the Hubble expansion rate H (T ) much earlier than for the
other particles of the Standard Model in the plasma. We can easily compute this rate
combining Eqs. (3.30) and (B.139)

G2F 3 ζ(3)
ν  nν 9.93 × T 2 , with GF = 1.1664 × 10−5 GeV−2 and nν = gν T 3 .
8π 4 π2
At such earlier time, the Universe is radiation dominated, which means H (T ) =
1/2
0.33gρ T 2 /MP , see Eq. (2.263). The condition ν (Tνdec )  H (Tνdec ) gives

3 ζ (3) 9.93  −5
2 2
1/2 Tνdec
2 1.16 × 10 T 5
νdec  0.33(10.75) ⇒ Tνdec  3.58 MeV.
4 π 2 8π MP
(3.86)

From 3.6 MeV, the neutrino density is frozen and will not increase even when
T  me = 511 keV because the process ee → νν is too weak compare to the
interaction with the only light particles left in the plasma at such low temperature,
the photons. The entropy in e± pair is transferred to the photon through the stronger
electromagnetic interaction (Thomson scattering), but not to the neutrino through a
too weak interaction.
When the temperature is below 3 MeV, the neutrinos are decoupled, which
means they are “invisible” in the thermal plasma where only electrons, positrons,
and photons survive. However, even if decoupled, their temperature followed the
classical law Tν3 R 3 = cte. This is due to the fact that the effective number of
neutrinos is fixed because they escaped from the plasma. The photons follow the
same law, which means than even after the neutrino decoupling, Tν = Tγ : there is no
“transfer” of the neutrino degrees of freedom to the temperature of the plasma from
entropy conservation, as the neutrino still participates to the entropy of the Universe.
They decoupled from the photon but still keep the same temperature and evolution.
The system can be seen as the composition of two independent boxes, hermetic
to each other. In one of the boxes the density of photons, electrons, and positrons
follows the thermal distribution law with a greater number of degrees of freedom
than in the “neutrino” box, whose neutrino energy follows a classical redshift law.
It is an abuse of language to talk about “temperature” in the neutrino system, since
there is no longer thermal equilibrium. However, as they are still relativistic, this
3.3 The Case of Light Species 173

3 MeV 511 keV

neutrino, electrons and photon electron and photon
in equilibrium plasma of photon
in equilibrium

Tv Tv= (4/11) 1/3 Tph

Tv = Te = Tph
Te = Tph Te = 0
Tph

Fig. 3.15 A brief history of the neutrino

abuse of language remains coherent. Each system keeps its own entropy, without
sharing it (see the box below for a more detailed explanation).
When the temperature drops below 511 keV, the electrons and positrons decouple
from the plasma. They transfer their part of entropy to the photons (and not to
the neutrinos which are already decoupled) and increase the photons temperature
relative to the neutrinos temperature by this way. The entropy in the system
(photons, electrons, positrons) being constant, we can write
e+e−γ γ
gs (Tγe+e−γ )3 R 3 = gs (Tγγ )3 R 3

e+e−γ γ γ
with gs = 2 + 4 × 78 = 11
2 and gs = 2 (see Eq. (3.62), Tγ being the photons
e+e−γ
temperature after the electrons decoupling and Tγ the plasma temperature
before the electrons decoupling19). We then obtain
 1  1
11 3 11 3
Tγγ = Tγe+e−γ = Tν . (3.87)
4 4

The present photons temperature being 2.725 K on deduce that the neutrinos
temperature today is
 1
4 3
Tν = × 2.725 = 1.95 K = 1.68 × 10−4 eV. (3.88)
11

The processes are summarized in Fig. 3.15.

As a final note we want to stress that there are finally 2 ways to obtain a freeze out
condition: ni (T ) × σ v < H (T ). Or the density decreases due to the exponential
suppression of the Boltzmann factor e−Ei /T (kinetic freeze out) when T  mi
(case of the quarks, gauge bosons or dark matter candidate as we will see later on),
or the particle is light and still relativistic, but the expansion rate H (T ) becomes

19 Two degrees of freedom for the photon, 2 fermionic states for the electron, plus 2 for the
positrons.
174 3 A Thermal Universe [TRH → TCMB ]

greater than the annihilation cross section σ v because of weak couplings (like in
the case of the neutrino, chemical freeze out). In the case of the recombination
temperature (decoupling of the photon) we will see that it occurred not at T =
13.6 eV which is the binding energy of the atom of hydrogen but a little bit later,
at 0.3 eV as we can see in Fig. 3.16, because what is important is not only that the
photons have a temperature below the binding energy BH but then nγ σ v is below
the rate to destroy and forbid the hydrogen atom formation. In other words, even
at T = Tγ = 1 eV, there are still sufficient photons present in the queue of the
distribution with T = 13.6 eV, able to destroy the structure of the hydrogen atom.
Another interesting point is the measurement of the relic abundance of neutrinos
nowadays. From Tν = 1.95 K ( 1.7 × 10−4 eV) one can compute from Eq. (3.27)
the neutrino density par family to be nν  112 cm−3 implying20 ρν0 = mν n0ν

ρν0 mν 112 cm−3 mν mν

ν = = −5 −3
≈ ⇒ νh
2
≈ .
ρc0 1.05 × 10 h GeV cm
2 94 h2 eV 94 eV
(3.89)

Asking for the neutrino abundance to be less than the CMB bound on M ∼ 0.3
and taking h  0.7 we can derive the limit on the mass of a stable neutrino:

mν  13.8 eV. (3.90)

ν

This cosmological bound to the mass of a stable, light neutrino species is often
referred to as the Cowsik-McClelland bound.

How many degrees of freedom are for the neutrinos nowadays?

When proposing lectures to master students, often appears the question “why
should we take 3 degrees of freedom for the calculation of the entropy
nowadays, whereas we know that at least 2 neutrino flavors have masses above
Tν and are then non-relativistic, out of equilibrium from their own bath.”
The question is very legitimate and arises when I present the computation
of the degrees of freedom for the entropy, Eq. (3.63) which takes into
today
account 3 generations of neutrino to obtain gs = 3.91. First of all,
even once a neutrino becomes non-relativistic (we remind that the relative

(continued)

20 Notice that the mass density of a massive neutrino does not follow the Boltzmann suppressed
evolution of classical massive particles as neutrino have already decoupled from the thermal bath,
and there density thus always follows the T 3 evolution by number of particle conservation (nν (t)×
a(t) ∝ n(t) × T −3 = cst), a(t) being the scale factor of the Universe.
3.3 The Case of Light Species 175

mass measurements extracted from the observations of the oscillation gives

m221  7.4×10−5 eV2 and m231  2.6×10−3 eV2 [3] implying at minima
(massless lightest neutrino), m2  8.6 × 10−3 eV and m3  5.1 × 10−2
eV, both masses being larger than Tν = 1.95 K = 1.7 × 10−4 eV), it does
not quit its thermal bath as does the dark matter, or even the electrons or
any Standard Model particles. Indeed, after their decoupling, neutrinos are
neither in thermal equilibrium with the photons nor with themselves. The
three flavors are out of equilibrium but with a thermal distribution: their
number density nν , for instance, is proportional to Tν3 , as it is the case for the
photons. However, Tν should not be considered as a “temperature” as there
is no thermal equilibrium between neutrinos. It should more be considered
as “the parameter in e−Eν /Tν in the distribution function.” That is usually
where the confusion is coming from. So, when computing the entropy, it
is indeed the three neutrino species that should be taken into account. By
conservation of the entropy S = s × a 3 nothing happens when Tν reaches
values below m2 or m3 because Tν has no thermal meaning. In the case of
the electrons, when Te− reaches me− , their degrees of freedom are transferred
to the photon through the thermal bath, conserving that way the entropy. In
the case of neutrinos when Tν  m3 , nothing happens as Tν has no thermal
meaning. To be more precise, if one separates S = Sγ + Sν into its radiation
and neutrino components, the neutrino part Sν = (sν1 + sν2 + sν3 ) × a 3 ∝
 1 3  2 3  3 3
Tν T T Ti
Tγ + Tνγ + Tνγ should be invariant, independently of the value mνi .
One should then really count the 3 flavors of neutrino in the computation of
the present entropy.
However, another subtlety appears when one needs to compute the pressure
of the neutrino gas today, and its contribution to the radiative energy density
ρrel (3.40). In this case, one should only consider the relativistic species, i.e.
the lightest neutrino and the photon. This explains why in the computation
today
of gρ we considered only one neutrino degrees of freedom and not
three. Remark that there is also the possibility that none of the neutrinos
is relativistic today, as we do not constraint their masses but their mass
differences. The same argument holds to compute the pressure of the neutrino
gas πν . The neutrino gas should be pressureless (equation of state with
wν = 0) as πρνν 1. This last relation can be computed exactly from
the distribution function of the massive neutrinos today but can also be
understood qualitatively from the kinetic energy of a non-relativistic gas with
pν2 πν Tν
a momentum distribution fν (pν ), Ecν = 2mν  ⇒ ρν ∝ mν 1.

(continued)
176 3 A Thermal Universe [TRH → TCMB ]

Exercises Recover the expression πρνν for a non-relativistic gas.

Show that a neutrino of 0.05 eV becomes non-relativistic at a redshift of
297, concluding that neutrinos are almost certainly relativistic at all epochs
where the radiation content in the Universe is significant.
Considering by simplification that 3 species of neutrino are relativistic,
show that the redshift where the radiation equal the matter is zEQ  3400.

3.3.2 The Tremaine-Gunn Bound

The Cowsik-McClelland limit discussed above is often counterbalanced by another

constraint imposed by Tremaine and Gunn in 1979. Supposing a massive neutrino
as the dark matter (which was one of the first valid candidate) they noticed that the
upper limit on the density in phase space of one spin state of one kind of neutrino is

1 1
fν = E

e T +1 2

where we defined

d 3p
dnν = fν (p) .
(2π)3

Notice that this upper limit on fν is only valid because ν is a neutrino. There does
not exist such a bound for bosons which tend to form a condensate by gathering into
a single state. We then obtain
 pmax 1 d 3p 1 |p|3max
nmax
ν  2
= .
0 2 (2π) 4π 2 3

Supposing a galaxy of mass M and radius R, conservation of energy gives us

 3
2m2ν GM m3ν 2GM 2
|p|2max = ⇒ nν  nmax
ν = , (3.91)
R 12π 2 R
or

4 8 M 3 G3 R 3 8
M= πR 3 nν mν  mν . (3.92)
3 81 π
3.3 The Case of Light Species 177

Noticing that (virial theorem) v 2  GM

R we can rewrite the limit

81 π
m4ν  (3.93)
8 R 2 vν G

which is called the Tremaine-Gunn bound. If we consider typical velocities vν 

300 km/s and R  20 kpc, we obtain

mν  15 eV, (3.94)

and even

mν  280 keV (3.95)

for a dwarf galaxy with vν  100 km/s and R  10 pc. Notice that the above
limits are incompatible with the Cowsik-McClelland bound (3.90). This is one of
the reasons why we usually exclude hot particles as possible candidates for dark
matter.
In general, light dark matter enters easily in conflict with the physics of structure
formations, destroying a large amount of substructure during the streaming time
between their decoupling from the bath and the time they become non-relativistic,
where they begin to be the web structuring the future galaxies and clusters of
galaxies. The machinery to compute the Jean mass generated during a phase where
the Universe is filled by a relativistic dark constituent is technically very complex
and necessitates months of simulations and statistical analysis for a trustworthy
result. However, the physics leading to the minimal halo mass obtained after the
smoothing passage of a dark component of mass21 mν can be understood with
simple orders of magnitude reasoning.22
Between its decoupling time and a time t, a neutrino streaming about with
a velocity v(t) would smooth primeval departures from a homogeneous neutrino
distribution on the length scale λs  v(t) × t, generating a smoothing mass
√
4π 3 π 3 MP3 3
Ms = ρ(t)λs = √ v , (3.96)
3 2 ρ
√
ρ
where we used t = 2H 1
and H = √ , see Eq. (3.52). Primeval mass density
3MP
fluctuations would be dissipated on scales smaller than Ms . Until the matter-
domination era, at tEQ , the neutrino is relativistic, and then v = c. A quick look

21 We keep in this paragraph the notation mν for the dark matter mass because it was the first
candidate of that sort proposed in the literature and still can be under the form of a sterile neutrino,
for instance.
22 For a more realistic and detailed study, the reader can have a look at Sect. 5.12.1 and Eq. (5.184).
178 3 A Thermal Universe [TRH → TCMB ]

at Eq. (3.96) shows that the mass Ms increases with time, proportionally to a 3/2, a
being the scale factor. On the other hand, once ν becomes non-relativistic, v ∝ a −1
and Ms ∝ a −3/2 . This decreasing effect illustrates the fact that the expansion freezes
the proto-structure, and no smoothing is sufficient enough to counterbalance the
expansion effect. There exists then a maximum mass Msmax below which the mass
distribution, at the start of the structure formation (that will grow Ms even further),
had been smoothed. This mass is attained at t = tν defined by Tν (tν ) = mν , just
before the time where the neutrino becomes relativistic. From (3.96) we obtain
√  2
3 5 MP3 10 eV
Msmax =  5 × 1014M . (3.97)
2 m2ν mν

This means that, at Tν = ( 11 ) Tγ  mν , all structures having masses below

4 1/3

∼1015 M would have been smoothed out, forbidding the presence of structures
(“seeds”) at the level of galaxies and dwarf galaxies, which is clearly not compatible
with the observations. To ensure the preservation of structures of the order of Ms 
5 × 108 M , one needs to add the constraint

mν  10 keV, (3.98)

which is a limit of the same order of magnitude than (3.95). These kinds of
constraints, resulting from structure formation or stability, tend to exclude hot dark
matter candidates, but still allowing a window for warm dark matter, i.e. above the
∼keV scale, see Fig. 5.24 for a more detailed analysis.

3.3.3 Dark Radiation

3.3.3.1 Generalities
A lot of work have been published recently focusing on increasing the effective
number of neutrino species in the Universe Neff defined in Eq. (3.46), trying to
solve the “issue” observed by spectrum of WMAP and certain telescopes. Indeed,
the relativistic degrees of freedom in have been measured Neff = 3.7+0.8 −0.7 (2σ )
4
from the inferred abundance of primordial He. The Atacama Cosmology Telescope
(ACT) and the South Pole Telescope have measured a damping tail of the CMB
spectrum and, in combination with data from WMAP, Baryon acoustic oscillation
(BAO), and the Hubble constant H0 also find a preference for a value of Neff greater
than three, obtaining Neff = 4.6±0.8 (1σ ) and Neff = 3.9±0.4 (1σ ), respectively.
The simplest solution is to introduce new particles, which decouple at a tem-
perature above the neutrino decoupling. The cosmic radiation content being usually
expressed in term of the effective number of thermally excited neutrino species,
Neff , the standard value being Neff = 3.045  3, the number of relativistic degrees
of freedom of a plasma containing the standard relativistic particles (photon γ and
3.3 The Case of Light Species 179

neutrino ν) with a new candidate ν̃ will then be written with the help of Eq. (3.26):

π2 7 7
ρR = ργ + ρν + ρν̃ = gγ Tγ4 + 3 × gν Tν4 + Nν̃ × gν̃ Tν̃4 (3.99)
30 8 8
  
π2 7 gν̃ Tν̃ 4 π2 7
= gγ Tγ4 + gν Tν4 3 + Nν̃ = gγ Tγ4 + gν Tν4 Neff
30 8 gν Tν 30 8

with gi the internal degree of freedom of particle i (2 for photons and neutrino) and
Nν̃ the number of types (generations) of particles ν̃ and
 4
gν̃ Tν̃
Neff = 3 + Nν̃ = 3 + Neff . (3.100)
gν Tν

Following the procedure described in the Sect. 3.3.1 for the decoupling of the
standard neutrino, we need to impose the conservation of entropy gsi Ti3 = constant:
 γν 1/3  γν 4/3
γ ν ν̃ γν Tν̃ gs gν̃ gs
gs Tν̃3 = gs Tν3 , ⇒ = γ ν ν̃
⇒ Neff = Nν̃ γ ν ν̃
Tν gs gν gs
(3.101)

γ ν ν̃
with the notation gs being the number of degrees of freedom while γ , ν, and ν̃ are
γν
still coupled while gs corresponds to the number of degrees of freedom when only
γ and ν are coupled. This is equivalent to the phenomena of the electron decoupling
transferring its degrees of freedom to the temperature of the photon. Indeed the ν̃
particle is massive and couples to the neutrino and photons (through the exchange
of a Z’, for instance) and, as it is massive, when it decouples, ν̃ disappears and thus
heats the plasma. As an example we can imagine right-handed neutrinos (Nν̃ = 3,
gν̃ = 2) decoupling at a temperature of 3 GeV. The number of degrees of freedom
γ ν ν̃
at this temperature is gs (3 GeV) = 81, where we used Tab.(3.2). Remembering
γ ν 43 34
that23 gs (3 MeV) = 43 4 we obtain Neff = 3 × ( 324 )  0.2.

3.3.3.2 One Example to Increase Neff *

Another possibility to increase Neff is to introduce a new particle χ which can play
for the neutrino the same role than the electron for the photon. In other words, if a
χ of mass mχ ∼ 10 MeV couples only with neutrino, it decouples from the plasma
after the neutrino decoupling temperature of ∼3 MeV (at Td  mχ /26 ∼ 0.4 MeV,
see Eq. 3.150) it will reheat only the neutrino temperature without modifying the
photon one, thus increasing Neff through the increase of Tν . The entropy can thus
be decomposed, at the temperature of the neutrino system decoupling Td into 2

23 Which corresponds to 3 active neutrinos, e± and photons.

180 3 A Thermal Universe [TRH → TCMB ]

different entropies S γ and S ν . Each entropy is thus constant after Td

γ
γ T3 S ν gs
S =S +S ; S =
γ ν γ
gs Tγ3 a 3 ; S =
ν
gsν Tν3 a 3 ⇒ ν3 = γ ν (3.102)
Tγ S gs

Sν
we can determine Sγ noticing that at T = Td , Tγ = Tν we then can write

γ  γ 1/3
Tν3 gsν |Td gs Tν gsν |Td gs
= γ ⇒ = γ . (3.103)
Tγ3 gs |Td gsν Tγ gs |Td gsν

We can notice indeed that if the neutrino is alone in its own plasma, gsν |Td = gsν
as neutrino of course cannot decouple from itself. We recover then the result
Tν 4 13
Tγ = ( 11 ) . Looking at Eq. (3.103) it is obvious that if neutrino is in equilibrium
in a system isolated from the thermal plasma, if one particle decouples from the
neutrino plasma it will transfer its entropy to the neutrino, increasing its temperature
Tν relative to Tγ . As a consequence, following Eq. (3.46) it is equivalent to increase
the effective degree of freedom Neff .

3.3.3.3 Another Example: The Case of the Mirror Dark Matter*

The mirror dark matter arises in models where you suppose a completely “mirrored”
world from the Standard Model like q  quarks or ν  neutrinos, even mirrored
hydrogen H  or oxygen O  . The interaction between the mirror world and our
visible Universe is only due to gravity (except if one allows for some kinetic mixing
between the photon γ and γ  ). The two worlds are supposed to be independently in
thermal equilibrium, but not in equilibrium with each other. One can then define two
different temperatures (and entropies) T and S for the visible thermal bath, and T  ,
S  for the mirror bath. Both entropy being constant, and the volume being obviously
the same, we can deduce that

S a 3s  s gs (T  )T 3 T gs (T ) 1/3
= 3 = = ≡ x 3
= constant ⇒ =x× .
S a s s gs (T )T 3 T gs (T  )
(3.104)

One can then deduce the total radiation density ρ̄R including the mirror bath

π2 π2 4 gρ (T  ) gs (T ) 4/3
ρ̄R = (gρ T 4 + gρ T 4 ) = T gρ 1 + x4
30 30 gρ (T ) gs (T  )
π
= gρ T 4 (1 + bx 4 ).
30
At the temperature we consider, above 1 MeV, b  1. The deviation from gρ can
be interpreted as a new neutrino effective degree of freedom. gρ with 3 neutrinos
(2 degrees of freedom each), 1 photon (2 degrees of freedom), and 1 electron (4
3.3 The Case of Light Species 181

degrees of freedom) gives

7 7 43
gρ = 3 × 2 × +2+4× = = 10.75.
8 8 4

The difference between ḡρ = gρ (1 + x 4) and gρ can be written as Neff × 2 × 78 =

1.75 Neff . Thus we have

Neff = 6.14 x 4 . (3.105)

Imposing the recent constraints, one obtains

Neff < 1 ⇒ x < 0.70. (3.106)

This means that the mirror matters should have an equilibrium temperature lower
than the observable one in our Universe. It is interesting to note that the mirror dark
matter is just a motivated framework. In any model where two different thermal
baths are present, the constraints from Neff on its temperature T  (3.106) are always
valid.

3.3.4 The Recombination: Decoupling of the Photons

This period of the Universe happens quite late (∼380,000 years after the Big
Bang) and is at the origin of the cosmic microwave background (CMB). Indeed,
at this time, 2 phenomena took place: the temperature of the plasma in equilibrium
decreased such that the photon living in the plasma did not have anymore sufficient
kinetic energy to destroy the forming atom of hydrogen. The would-be-free
electrons begin to form bound state (the binding energy of the hydrogen is 13.6 eV)
with the proton from the plasma to compose an electromagnetic neutral atom of
hydrogen. This is the recombination era. As a consequence of this process, the
population of free electrons drastically decreases opening the way to the photons
which do not scatter anymore on them and begin their very long journey toward
us to create the CMB, decoupling era as it is depicted in the illustration (2.3). This
corresponds to the last scattering surface. In this section, we will rederivate “a la
Gamow” the temperature of these two processes and show that they happened in a
very short period of time. As soon as the first atoms of hydrogen were formed, the
photons were liberated from the plasma.

3.3.4.1 The Recombination

To simplify the picture, we will suppose that the plasma is made of electrons (e),
protons (p), photons (γ ), and hydrogen atoms (H ) in thermal equilibrium:

p + e ↔ H + γ. (3.107)
182 3 A Thermal Universe [TRH → TCMB ]

The Universe being electrically neutral, the electrons and protons density are equal:
ne = np . The density of each species can thus be written
 3/2
mi T μi −mi
ni = gi e T (3.108)
2π

and, knowing from Eq. (3.21) that μγ = 0, the condition of chemical equilibrium
imposes μe + μp = μH implying
 3/2
gH 2π BH
nH = np ne e T (3.109)
gp ge me T

BH being the binding energy of the hydrogen, BH = me + mp − mH and were

we supposed mp  mH . For T BH , one has nH ∝ ne × np which can
be understood as the probability of forming a hydrogen is proportional to the
probability of presence of an electron multiplied by the probability of presence
of a proton. Once the temperature approaches the hydrogen atom binding energy
T ∼ BH , the exponential factor dominates the hydrogen density: the photons of
the backgrounds are not able to split the atom of hydrogen, and its formation is
exponential. If one defines the ionization fraction
np np ne
Xe = = = . (3.110)
nb np + nH nb

Using η = nγnb
= 6 × 10−10, Eq. (3.4), and nγ = 2 ζ(3)
π2
T 3 , Eq. (3.27), we can rewrite
Eq. (3.109)24

 3   3
2π 2 BH 2 T 2 BH
1 − Xe = η nγ Xe2 e T = Xe2 × 2.4 × 10−9 ζ(3) e T
me T π me
(3.111)

which is a second order equation called Saha equation that we can solve
straightforwardly. The result is shown in Fig. 3.16 where we plotted the ionization
fraction as function of the temperature. The exponential nature of the formation
of the atom of Hydrogen is clearly visible for T  0.3 eV. If one defines the
recombination temperature as the temperature where only 10% of the electrons are
still free. Solving Xe (Trec ) = 0.1 we obtained Trec = 0.297 eV.
3.3 The Case of Light Species 183

Xe
1.0
Electron proton plasma
0.8

0.6

0.4

0.2

Hydrogen Plasma
0.0
0.0 0.2 0.4 0.6 0.8 1.0
T (eV)
Fig. 3.16 Ionization rate Xe as function of the temperature (Eq. 3.111). The Hydrogen begins to
be formed for T  0.3 eV. The recombination time is defined when only 10% of the electrons are
still free

Photon

e−

Fig. 3.17 Thomson scattering of a photon off an electron

3.3.4.2 The Last Scattering Surface

After the formation of the atoms of hydrogen, electromagnetically neutral, the mean
free path of the photons increases dramatically as there are very few electrons left
in the plasma to interact. We compute in this section the temperature for which
the free path of the photons becomes larger than the horizon of the Universe
(Hubble distance), which is named the decoupling temperature. Photons interact
with electrons through the Thomson scattering process showed in Fig. 3.17. The

24 with ge = gp = 2 (spin 1/2) and gH = 4 (combination of two spins 1/2).

184 3 A Thermal Universe [TRH → TCMB ]

interaction rate can be written (see the Sect. 5.4.4 for more details)

γ = σT cne [s−1 ] (3.112)

 2
8π q2
with σT =  6.65 × 10−25 cm2 = 0.665 barn
3 4π0 me c2
 2500 [GeV−2 ]

γ−1 corresponds to the mean time between two collisions, thus cγ−1 is the
distance that a photon will go through before colliding an electron. A photon will
decouple at the temperature for which its mean path equals the Hubble horizon
H (T ), as we depicted in Fig. 3.7. In a matter dominated Universe (which is already
the state of the Universe 70,000 years after the Big Bang due to the presence of dark
matter) we have from Eq. (2.58)
  3/2
T
H (T ) = H0 0
M −4
2.35 × 10
  3/2
T
= 2.13 × 10−42 h 0 GeV, (3.113)
M
2.35 × 10−4

where T is expressed in eV. Asking for cγ−1 = H −1 (T ) one obtains H (T ) = ne σT

which can be written
  3/2
−42 Tdec
2500 ne (Tdec ) = 2.13 × 10 0 h2 . (3.114)
M
2.35 × 10−4 eV

Combining ne (T ) = Xe (T )nb with Eqs. (3.4) and (3.111) one can solve numerically
the condition (3.114). We obtained Tdec = 0.234 eV, which means that just after
the recombination, the photons become free. This results from the exponential
formation of the atoms of Hydrogen, implying that very quickly ne decreases and
no electron are available anymore for any Thomson scattering.25

3.3.5 The Dark Ages, or Re-Ionization

There is also another epoch called “dark ages” or “re-ionization time” which runs
from the decoupling/recombination time (380,000 years, see Sect. 3.3.4.1) till 100

25 The Thomson scattering is the low energy limit of the Compton scattering. Both effects are a

diffusion on the electromagnetic field by a charged particle. In the CMB, the electron being non-
relativistic, the Thompson limit is a valid one.
3.4 The Big Bang Nucleosynthesis 185

million or one billion years. At the recombination time, the rate of recombination
of electrons and protons to form neutral hydrogen was higher than the re-ionization
rate. Universe was opaque due to the large rate of scattering of photons on electrons,
and then became transparent once the first atom of hydrogen formed. The dark ages
of the Universe started at that point, because there was no light sources other than
the gradually redshifting cosmic background radiation. The second phase change
occurred once objects started to condense in the early Universe that were energetic
enough to re-ionize neutral hydrogen. As these objects formed and radiated energy,
the Universe reverted from being neutral, to once again being an ionized plasma.
This occurred between 150 million and one billion years after the Big Bang. At
that time, however, matter had been diffused by the expansion of the Universe, and
the scattering interactions of photons and electrons were much less frequent than
before electron-proton recombination. Thus, a Universe full of low density ionized
hydrogen will remain transparent, as is the case today.

3.4 The Big Bang Nucleosynthesis

3.4.1 The Context

The very early Universe was hot and dense, resulting in particle interactions
occurring much more frequently than today. For example, we saw that while photon
can today travel across the entire Universe without interacting with the interstellar
medium (deflection or capture), resulting in a mean-free path greater than 1028
cm (the horizon), the mean-free path of a photon when the Universe was 1 s old
was about the size of an atom. This resulted in a large number of interactions
which kept the interacting constituents of the Universe in equilibrium. As the
Universe expanded, the mean-free path of particles increased (thus decreasing the
rates of interactions) to the point where equilibrium conditions could no longer be
maintained. Different constituents of the Universe decoupled (fell out of equilibrium
with the rest of the Universe) at different times, which determined their abundance.
We have previously seen the history of the decoupling of the neutrino and the
photon. In this chapter, we will concentrate on the decoupling of the baryons. This
decoupling is a little bit more complicated (in a sense) than that of neutrinos. The
main reason is that even if it occurs around the same time (1 MeV), the neutrino is a
fundamental “free” particle, unable to form bound states with particles surrounding
it, which is not the case of the quarks and gluons. They are partons, the main
constituents of baryons, and form bound states beginning by protons and neutrons.
Then protons and neutrons begin to form the first nucleus (deuterium, helium, and
lithium), but why at T = 1 MeV?
Whereas T = 3 MeV corresponds to the decoupling of the neutrinos which
cannot “talk” anymore with the thermal bath due to the weakness of their interac-
tions, 511 keV is the decoupling scale for the electrons due to their underproduction
from a thermal bath with T  511 keV as shown in Fig. 3.15, 1 MeV corresponds
to the energy at which the photons are not able anymore to forbid nuclear bound
186 3 A Thermal Universe [TRH → TCMB ]

state through its scattering processes. The MeV scale is indeed the scale of nuclear
boundary energies. A nice summary of the mechanisms in play in these different
processes can be found in [2]. As we just described in the previous chapters, when
the temperature of the Universe reaches 1 MeV, the cosmic plasma consists of:

• Relativistic particles in equilibrium: photons, electrons, and positrons.

These interact among themselves via electromagnetic interaction e+ e− ↔ γ γ .
The abundances of these constituents are given by Fermi-Dirac and Bose-
Einstein statistics.
• Decoupled relativistic particles: neutrinos. They decoupled at around 3 MeV
due to their weak coupling with the photons and electrons (which couple together
quite “strongly” through the electromagnetic interaction). Even if decoupled
from the photon bath, they continue to be present in the Universe, contributing to
its entropy as we saw in Sect. 3.3.1.
• Non-relativistic parcels: the baryons. The baryon is present thanks to the
baryonic asymmetry in the Universe. Indeed, if there was as many baryons (b)
as antibaryon (b̄), after being frozen out they should annihilate b + b̄ → γ γ
(through a t-channel b exchange, for instance) which is clearly not the case as we
are made of baryons.

What happened is that there was an initial asymmetry between baryons and anti-
baryons

nb − nb̄
 10−10, (3.115)
s
throughout the early history of the Universe, until the anti-baryons were annihilated
away at about T  1 MeV. We can express the ratio baryon density to photon density
ηb at 1 MeV as function of the present ratio (n0b /n0γ ) which gives

ρb0 
1 GeV 1.05 × 10−5 h2 cm−3
ρc b
nb n0b
ηb ≡ (1 MeV) = 0 = m
= m
= b
nγ nγ nγ nγ m 411 cm−3
   2
1 GeV 1 GeV bh
= 2.4 × 10−8 b h2 = 5.8 × 10−10 ,
m m 0.02
(3.116)

where we used the observed values given in [3]. By a simple argument one
can understand why the ratio nb /nγ is constant with respect to the time after the
decoupling from the thermal plasma. Just from the Boltzmann equation dn dt =
b

−3H nb , with H = Ṙ
R. Imposing a constant entropy,

dS Ṙ 1 dT
S = R3 s ∝ R3 T 3 , =0 ⇒ =−
dt R T dt
3.4 The Big Bang Nucleosynthesis 187

implying

dnb 3 dT nb nb
=− nb ⇒ 3 ∝ = constant,
dt T dt T nγ

where we used the formulae (3.27) and (3.61) for nγ and s, respectively.
Therefore, there are orders of magnitude more relativistic particles than baryons
at about T  1 MeV (and nowadays of course). The keypoint of this observation
is that baryons were frozen out at 1 MeV (t ∼ 10−4 s), the number of photons and
baryons/antibaryon pairs must have been nearly equal, since they were constantly
exchanging (via pair production/annihilation). Indeed, the matter-anti-matter pairs
must also have been equal, i.e. the Universe was equal parts matter and anti-matter.
At freeze out, there must have been a slight excess of matter over anti-matter (1 part
in 109 ), so the present Universe is dominated by matter and not anti-matter.
The aim of this chapter is to understand how the baryons combine themselves
from 1 MeV town to keV scale to form the first light nucleus. This process
bears the name “Big Bang Nucleosynthesis” (or BBN) in contrast with the Solar
Nucleosynthesis developed by Hoyle and Fowler some years after. The first idea
of BBN genesis was the Gamow work in 1948 [4], see [5] for historical details.
If the baryons were in thermal equilibrium throughout the history of the Universe,
they will naturally produce lead and iron which are some of the more stable nuclear
element, with the higher binding energy. However, it is far to be the case as the
number of interactions between baryons decreases as the Universe evolve (as soon
as H (T )  σ vequilibrium ) letting baryon living with themselves but out of any
equilibrium. At that moment, the binding energy plays a role. Indeed, baryons will
tend naturally to form binding states, which lower their ground energetic state as
soon as the photons, scattering around, are not able to interfere in the binding
process by destroying newly formed bounded state.

Historical aspect of the BBN

When teaching the rudiments of the Big Bang Nucleosynthesis, I usually
begin my lecture by the method originally employed by Gamow and Alpher
in 1946–1948. I will not go into the historical details that can be found in the
excellent book [6] but give the main line of reasoning. Gamow was aware
that when the energy of the plasma dropped below the binding energy of
the deuterium (2.5 MeV) its dissociation by photons scattering stopped, and
the process of helium production through the deuterium interactions is open.
Even if not explicitly stated in his papers, we can suppose he knew the Saha
equation (3.109) and did not take the temperature to be Td  2 × 1010 K,
corresponding to the deuterium binding energy, but Td = 109 K  10−4 GeV,
which is the temperature obtained by solving the Saha equation. Physically,
this means that even at a temperature 10 times lower, there are still sufficient

(continued)
188 3 A Thermal Universe [TRH → TCMB ]

photons to dissociate deuterium. This temperature corresponds to a time

T2
(supposing a radiation dominated Universe and H ∼ MP
),

tBBN = 1027 GeV−1 = 667 s. ∼ 103 s. (3.117)

The baryonic density to initiate the nucleosynthesis can be found asking for
at least one production:

nb × σ v × tBBN  1 ⇒ nb  1017 cm−3 , (3.118)

where Gamow took for σ v the Bethe value, σ v  10−20 cm3 s−1 . In the 40’s,
the density of the Universe was evaluated by astrophysical observations to be
ρ(T0 ) = 10−30 kg/cm3  10−6 proton per cm3 , which gives for the present
temperature T0
 3
nb (Td ) Td
= ⇒ T0  20 K, (3.119)
nb (T0 ) T0

which is already quite an impressive prediction considering all the approxi-

mations made, and the very few data available at this time. Notice that in their
1948 paper, Alpher and Hermann took a more precise value Td = 0.6 × 109
K, which gives (good exercise) T0 = 5.5 K (!!).

3.4.2 Overview

To visualize the generation of the light nuclei, especially helium and lithium, one
needs to imagine the system of neutrons, protons, electrons, and neutrinos in thermal
equilibrium as long as the reaction

n + ν ↔ p + e− (3.120)

respects

p = nν σ v > H (T ). (3.121)

Notice that there are more reactions than (3.120) involved in the equilibrium pro-
cesses, like the neutron decay. The set of equations needs to be solved numerically,
but we want to give an idea of the primordial nucleosynthesis in this section. The
binding energy of deuterium or helium being around the MeV scale, the protons and
neutrons are highly non-relativistic, and their density number follows Eq. (3.28),
3.4 The Big Bang Nucleosynthesis 189

which means
nn Q
 e− T (3.122)
np

with Q = mn − mp =1.293 MeV. To determine the decoupling temperature given by

(3.121), one needs to compute the cross section of the scattering process n + ν →
p + e− , via the t-channel exchange of a W boson. Using the couplings given in
Appendix B.2.3 we can write an effective 4-fermions interaction “a la Fermi” for
T MW

g iημν g
Leff = √ ν̄γ μ (1 − γ 5 )e 2 √ n̄γ ν (cV − cA γ 5 )p. (3.123)
2 2 MW 2 2

Notice that the couplings to the protons and neutrons take a more complex form than
for the leptons, because they are bound states of quarks and cannot be determined
only on theoretical basis but should be measured. The most recent analysis gives
cV  1, cV  1.26. To be more precise, this correction accounts for the possibility
that gluons inside the nucleon split into quark-antiquark pairs. The amplitude
squared is then given by

g4
|M|2 = 2
2
(1 + 3cA )(Pn .Pν )(Pp .Pe )  16G2F (1 + 3cA
2
) mn E ν × mp E e ,
2MW
(3.124)

where the factor 3 comes from the spin structure on the nucleons, as we can see
from Eq. (4.46) and we supposed the neutron and protons almost at rest at the MeV
scale. Using the same method presented in the Appendix B.4.4.6 we obtain from
Eq. B.110

|M|2 G2 3
σ =  F (1 + 3cA
2
) (3.2T ) T , (3.125)
16πs π 2

where we approximate s = (Pp + Pe )2  m2p and make use of Eqs. (3.30) and
(3.31) for Eν and E , where we supposed Ee almost not relativistic. With the help
of Eq. (3.27) we can compute the decoupling temperature Td satisfying Eq. (3.121)
with the help of Eq. (2.264):

p = nν σ v  1.6 G2F T 5 ,

π 10.75 T 2
p  H (T ) = ⇒ T  0.86 MeV = Td , (3.126)
3 10 MP
190 3 A Thermal Universe [TRH → TCMB ]

where we supposed the radiation dominated Universe with gρ = 10.75 as we

computed in Table 3.2. Implementing (3.126) in (3.122), we obtain at the freeze
out temperature Td

nn 1
 . (3.127)
np FO 5

In fact the exact numerical value, solving the complete set of reactions like the
neutron decay, or electron-neutron scattering gives

 num.
nn 1
 . (3.128)
np FO 7

It is, however, surprising how a simple analytical analysis gives a result not so
far from the numerical one.
Once the protons and neutrons are frozen out, they will quickly bound to 4 He, the
most tightly bound of all the light nuclei. 1 neutron being present for 7 protons (2
neutron per 14 protons), the mass ratio of the Helium 4 He (composed of 2 neutrons
and 2 protons) on the total baryonic mass is then

n4 4 1
Yp = = = = 25%. (3.129)
baryons 2 + 14 4

This prediction was one of the more outstanding proofs of the Standard Model
of Cosmology, especially the superiority of the Big Bang Model above other
alternatives like the Steady State Universe of Gold, Bondi, and Hoyle. Notice
how lucky we (the Universe) are that the decoupling temperature Td of the couple
(proton, neutron) is very close to their mass difference. Indeed, if we take a closer
look at Eq. (3.122), it would be enough for the temperature to be just 10 times
smaller so that, at the moment of freeze out, there is not enough neutron in the
thermal bath to be able to form helium in large enough quantities.

Exercise Repeat the previous analysis, explicitly calculating the average interaction
rate:

1 d 3 pE d 3 pν d 3 pn (4)
p = δ (Pp + Pe − Pν − Pn )
(2π)5 2Ee 2Eν 2En
× fe (Ee )(1 − fν (Eν ))|M|2 (3.130)
3.4 The Big Bang Nucleosynthesis 191

with |M|2 given by the equation (3.124). Then show that

 3
G2F T Q
p = 1.636 (1 + 3cA
2
)m5e e− T [T Q, me ]
2π 3 me
7π
p = (1 + 3cA
2
)G2F T 5 [T Q, me ], (3.131)
60
and find the decoupling temperature of protons and neutrons.

3.4.3 The Deuterium Formation

Even if the method we developed in the previous paragraph gives sufficiently

accurate results to have an idea of the mechanisms in play during the Big Bang
Nucleosynthesis era, we show in this section how to be a little bit more precise.
Indeed, the formation of Helium is not instantaneous and passes through the
deuterium bottleneck phase. Indeed, the first reaction is the binding of a neutron
and a proton into a nucleus of deuterium (D):

n + p → D + γ. (3.132)

The binding energy of deuterium is BD = 2.2 MeV (see Appendix E.2), which
means that mD = mp + mn − BD = 1877.62 MeV. As soon as the temperature
of the Universe falls below 2.2 MeV, the photons from the thermal bath are not
sufficiently energetic to destroy a “molecule” of deuterium. So, naively, one would
expect that the formation process of the first deuterium appears at around 1–2 MeV.
It happens in fact much later due to the very weak concentration of baryons (protons
and neutrons) at this temperature, reduced by the measured baryon to photon ratio
(around 10−10) from the particle/antiparticle asymmetry (see above). Let us check
it quantitatively. At T  1 MeV, we are clearly in the regime mi T and can apply
the formulae (3.29) and compute, not forgetting that deuterium has gD = 3 because
of 3 spin states of D and gp = gn = 2:
 3/2
nD gD mD T
2π e−mD /T
=  3/2  3/2 (3.133)
nn np mp T
gn mn T
2π e−mn /T gp 2π e−mp /T
 −3/2  3/2
gD T mD
= e−(mD −mn −mp )/T
gn gp 2π mn mp
 3/2
3 2πmD
= eBD /T
4 mn mp T
192 3 A Thermal Universe [TRH → TCMB ]

because BD = mn + mp − mD . With reasonable approximations one can write

 3/2
nD 3 4π
 eBD /T . (3.134)
nn np 4 mp T

Noticing that at 1 MeV np  nn  nb /2 (the only baryons at 1 MeV are mainly

protons and neutron), we can rewrite Eq. (3.134)
 3/2  3/2
nD 3 4π 12ζ(3) T
 ηb nγ eBD /T = √ ηb eBD /T , (3.135)
nb 4 mp T π mp

where we have used nγ = 2ζ(3) π2

T 3 from Eq. (3.27). Having a look to this result,
we understand easily that as ηb 1 and mp 1 MeV, the temperature at which
the density of the deuterium is of the order of magnitude of the proton one is below
1 MeV. A simple solution of Eq. (3.135) gives
 
nD BD BD
log  −23 + ⇒  23. (3.136)
nb T T nD nb

We then expect the beginning of the BBN to occur at around 50 keV which is around
the minute scale if we look at the relation (3.52). Small baryon to photon ratio
thus inhibits nuclei production until the temperature drops well beneath the binding
energy (T BD ). This is why at temperatures T > 0.1 MeV virtually all baryons
are in the form of neutrons and protons. Around this temperature, the production of
deuterium and helium starts, but the reaction rates are too low to produce heavier
elements. The heavier elements are formed in stars (triple alpha process):
4
H e +4 H e +4 H e →12 C, (3.137)

but that is only much later. The early Universe is not sufficiently dense for this
reaction to take place, i.e. for three helium nuclei to find another on relevant
timescales. The three alpha processes have been found to take place in stars by
Burbidge, Burbidge, Fowler, and Hoyle in 1957 [7].

3.4.4 The Lithium Problem

Standard Big Bang Nucleosynthesis (SBBN) comprises the set of first-order

Boltzmann equation on the abundances of the different species

dYi dYi
= −H (T )T = (ij Yj + ikl Yk Yl + . . .). (3.138)
dt dT
3.4 The Big Bang Nucleosynthesis 193

Assuming thermal distribution and the expression of H (T ) given in Eq. (2.263)

one can solve the system of equations (3.138). This system is quite complex and
generally involves numerical solutions. It should be noted that this happens in a
Universe where the number of remaining baryons is very small (ηb = 6 × 10−10 )
and apart from the production of Helium 4 which can be seen in a fairly good
approximation as a mechanism for absorbing free neutrons before they decay,
the other mechanisms, as we have seen for deuterium formation, require nuclear
reactions whose rates are obviously proportional to ηb , as is explicit in Eq. (3.135).
Concerning the Lithium, it is formed through a reaction between Helium 4 and
Helium 3 following:
4
H e +3 H e →7 Be + γ →7 Li (3.139)

the last reaction resulting from the capture of an ambient electron by the beryllium
nucleus. This trace of Lithium, although extremely weak compared to helium-4 or
even deuterium, remains an important source of data for the study of physics beyond
the Standard Model. The most recent predictions give 7 Li/H ratio of the order of
5 × 10−10 , while observations are around 1.5 × 10−10 :
7 7
Li Li
 5 × 10−10;  1.5 × 10−10. (3.140)
H th H exp

It is not yet clear where this discrepancy comes from. It could come from the physics
of the formation of the first stars that could absorb some of the lithium produced
during the BBN phase. Research is still very active in this area. Another application
of the study of primordial nucleosynthesis is the study of metastable relic particles.
Indeed, if a particle disintegrates during the BBN cycle or a little later, this will
cause a change in the abundance ratios of the various primordial elements. One of
the most effective constraints is to require that the lifetime of metastable particles
respects τ  100 s. This is, for example, the case in models such as supersymmetry
with the partner of the graviton (the gravitino) as dark matter candidate. In this
case, the just slightly more massive supermetrics particle (usually the stau τ̃ , the
supersymmetric partner of the tau lepton) will have a relatively long lifetime due to
reduced gravitational coupling. The constraint ττ̃ < 100 s is very often applied in
this case. More generally, the constraints from BBN measurements applied also for
the searches of unstable relics. Indeed, very generally, the BBN bounds require that
unstable relics previously present in the Universe, decay with lifetime smaller than
100 s, unless the abundance of these particles is very small or only a tiny fraction
of these particles decay with energetic hadrons in the final state. Among the MSSM
particles, the latter condition is satisfied by the lightest neutrino or also the lightest
stau.
Before concluding we want to point out that several papers have studied the
Big Bang Nucleosynthesis, as well as from a model independent point of view
as for more specific models. A nice and clear introduction of the subject can be
194 3 A Thermal Universe [TRH → TCMB ]

t/sec
0.1 1 10 100 1000 104 105 106

H D b.n.
1
N Yp
10−2 SBBN f.o.

10−4 ν dec. n/p dec. D/H

e± ann. 3
He/H
10−6
T/H

10−8 7
Be/H
10−10 7
Li/H

10−12
6
Li/H
10−14
1000 100 10 1
T /keV

Fig. 3.18 Time and temperature evolution of all standard Big Bang nucleosynthesis relevant
nuclear abundances. The vertical arrow indicates the moment at T  8.5 × 108 K at which most
of the helium nuclei are synthetized. The gray vertical bands indicate main BBN stages. From left
to right: neutrino decoupling, electron-positron annihilation and n/p freeze out, D bottleneck, and
freeze out of all nuclear reactions. Proton (H) and neutrons (N) are given relative to nb , whereas
Yp denotes the 4 He mass fraction [2]

found in [2]. Obviously, historically speaking, the idea and first calculation of relic
abundance of nuclei were done by Alpher, Herman, and Gamow in [4]. Figure 3.18
is extracted from [2].

3.5 Producing Dark Matter in Thermal Equilibrium

Before BBN occurred, the dark matter had already decoupled from the thermal bath.
If it was relativistic during its decoupling the mechanism is exactly the same as in
the case of the neutrino described in the Sect. 3.3.1. However, if, as in the case
of a Weakly Interacting Massive Particles (WIMPs), its decoupling is done when it
becomes non-relativistic, the treatment is different. This is what we will see in detail
in this chapter.
3.5 Producing Dark Matter in Thermal Equilibrium 195

3.5.1 The Boltzmann Equation

As we already discussed, the Boltzmann equation formalizes the interactions of a

particle with the photons (and other relativistic species) in the thermal bath before
they decoupled from it. It can be written as:

Evolution of particle A per unit of volume = particle A created by annihilation of

particles in the bath B minus self-annihilation of A.

Putting this in equation, we obtain

 
1 d nA a 3
= σ vB→A n2B − σ vA→B n2A (3.141)
a3 dt
⇒ ṅA + 3H nA = σ vB→A n2B − σ vA→B n2A . (3.142)

Using H = a1 da dt and knowing that by definition, when particles A are in

equilibrium, its number per comoving frame is constant26 and writing the evolution
as function of the temperature from the relations T 3 × a 3 =cte (constant entropy)
⇒ da/a = −dT /T and H = da/dt a one can deduce dt d
= dTdt dT = −H T dT and
d d

write (forgetting from now on the index A)

dn n σ v  2 
−3 = n − n2eq . (3.143)
dT T HT

We can distinguish 3 regimes before giving an explicit solution of the equation.

(A) When the temperature of the plasma is large compared to the mass of the
particle A (T mA = m) and the Hubble expansion is negligible, the particle
A is relativistic and stays in thermal equilibrium with its other relativistic
companions, like the photons, and their number density is given by Eq. (3.27):
the Boltzmann equation (3.143) can be simplified as n  neq ∝ T 3 .
(B) When the temperature of the plasma reaches the value T  m, then the
value of neq ∝ e−m/T decreases very steeply, almost like a step function
Eq. (3.29) and the production rate of Eq. (3.143) proportional to n2eq is almost
completely suppressed. We can interpret it as if the photons in the plasma do
not possess sufficient kinetic energy (∝ T ) to produce particles A onshell.
We are left with a bath of particles A which does not “discuss” anymore
with the bath of the photons27 and can only annihilate. Their density then

26 d[nA ×a 3 ] = 0 when n = neq .

dt A A
27 More precisely, the particles A talk to the photons but cannot hear them.
196 3 A Thermal Universe [TRH → TCMB ]

decreases, but not as steeply as e−m/T as they have decoupled from the bath:
their evolution is dictated by their annihilation rate n2 σ v whose temperature
behavior depends strongly on the model and on σ v. Up to now, we are still not
in the “frozen” regime but in a semi-decoupled one (the Hubble parameter H is
still negligible compared to n σ v). The Eq. (3.143) can then be approximated
2
by dTdn
 n HσTv .
(C) After a while, the Hubble parameter H dominates on the annihilation rate
 = n σ v and we recover the decoupling limit (“freeze out”) we discussed
previously /H  1. In this regime, the number of particles A per comoving
frame is constant and frozen and the Boltzmann equation (3.143) can be simply
approximated by dn/dT  3n/T ⇒ n ∝ T 3 . The law has the same behavior
than a relativistic particle in thermal bath but for a different reason (expansion
rate in this case). In fact, fundamentally speaking, the reason is not so different.
The keypoint being the conservation of the entropy which, at the same time,
is responsible for the thermal equilibrium and for the evolution of the scale
factor as function of the temperature: asking for a constant number of particle
per comoving frame gives similar evolution for a relativistic particle in thermal
equilibrium with the photons than for a particle out of equilibrium which
density evolve solely due to the expansion. These 3 regimes (A), (B), and (C)
are summarized in Fig. 3.19.

3.5.2 Overview

Neglecting the evolution of gs (T ) we can write (3.143) in a more workable way.

Indeed, introducing Y = n/s, s being the entropy density, and supposing adiabatic
processes, we can deduce sa 3 = S =cste ⇒ ṡ/s = −3ȧ/a = −3H ⇒ Ẏ /Y =
ṅ/n − ṡ/s = ṅ/n − 3H . We then deduce

dY
ṅ + 3H n = s Ẏ = −H s T × , (3.144)
dT

where we have used as previously d

dt = dT d
dt dT = −H T d
dT . We can then write

dY σ vs  2 
= Yeq − Y 2
dx Hx

6.6 × 109 gs  m  σ v  
 −26 cm3 s−1
2
Yeq − Y2 (3.145)
x2 gρ
1/2 GeV 3 × 10
3.5 Producing Dark Matter in Thermal Equilibrium 197

n
equilibrium (chemical + kinetic)
n(A)
neq(A)
n( )

T −3
out of chemical equilibrium

freeze out
<  v>
e −m/T

T −3

100 1 0.01 m/T

A B C

Fig. 3.19 Schematically evolution of a massive particle A as a function of the temperature

following the Boltzmann equation (red-full) in comparison with the photon thermal bath density
(brown-dotted) and the equilibrium state (blue-dashed) i the regimes A, B, and C (see the text for
details). Kinetic equilibrium is equivalent to thermal equilibrium

with (see Sect. 3.1.3 for more details)

m 2π 2 m3
x= , s= gs 3 ,
T 45 x
neq 45 π 1/2 gA 3/2 −x

Yeq = = x e (3.146)
s 2π 4 8 gs
gA 3/2 −x
 0.14 x e (for x 3)
gs
gA 3
 0.278 (for x 3) if A is a bosons [fermion], (3.147)
gs 4

gA being the internal degrees of freedom of the particle A, and H is given by

Eq. (2.264):

π gρ m2
H (x) = . (3.148)
3 10 x 2 MP
198 3 A Thermal Universe [TRH → TCMB ]

To quantify the above argument concerning the transition between the thermal
equilibrium and the freeze out time (the region B). Defining the deviation from
the equilibrium  such as Y = (1 + )Yeq , after neglecting 2 , d/dx 
Eq. (3.145) becomes
  
H 3 gs (x − 3/2) x 1 GeV 3 × 10−26 cm3 s−1
 x−  5.5 × 1010 e .
2eq 2 g x 1/2 m σ v
(3.149)

We can easily understand from Eq. (3.149) that the yield Y will departure from
Yeq when eq = neq σ v begins to be small. For m = 100 GeV and σ v =
3 × 10−26 cm3 s−1 one obtains
m
  1 (decoupling condition) ⇒ x =  26, (3.150)
T

which means that the decoupling does not begin when T  m but much later, when
T  m/26. At this temperature, Y  2Yeq > Yeq .
Another way to understand the decoupling phenomenon is by writing Eq. (3.145)
as
dY eq 1  2 
= Yeq − Y 2 . (3.151)
dx H xYeq

It is then easy to understand that in early Universe, as long as eq H Yeq ,

the number density Y tries to track the equilibrium value Yeq . This is because Y
wants to change to match Yeq . However, eq is decreasing. Eventually eq  H at
some “time” xf . From that point on, dY/dx becomes small and Y does not want to
change anymore. We are left with Y (x)  Y (xf ), so that the number of particles
per comoving volume has frozen out. For neutrinos this occurs while the species are
still relativistic (see above), whereas for WIMPs, this occurs when the particles are
already non-relativistic (see below).
That is also a fundamental point to underline. We made the hypothesis, in the
neutrino case as in the WIMP case, that the dark matter, or the neutrino, is in
thermal equilibrium with the bath before decoupling from it. In one case (neutrino)
the decoupling comes from the fact that its coupling to the bath is too weak to
counterbalance the expansion, whereas in the dark matter case, it comes from
the density, reduced by the Boltzmann suppression factor e−m/T . Their exist a
third possibility, which is the production from the thermal bath while not being
in equilibrium with it. In this case, all the particles are relativistic, the initial ones
(bath) as the final ones (dark matter) and the treatment above is not valid anymore
as we will see in Sect. 3.6.
3.5 Producing Dark Matter in Thermal Equilibrium 199

Computing the freeze out temperature

I present here, a simplified way to recover quickly the relation (3.150)
between the decoupling temperature and the dark matter mass. As we
explained in Sect. 3.1.8, the freeze out occurs when the Hubble expansion rate
begins to dominate the annihilation rate of the process dm dm → SM SM.
In other words, when

T2
ndm σ vdm→sm  H (T )  . (3.152)
MP

Using the non-relativistic expression for the dark matter density ndm (3.29),
we can write for the decoupling temperature Td :

− Tm Td2
(mTd )3/2 e d σ v 
MP
m 3 5
⇒  ln MP + ln m − ln Td + 4 ln gdm  ln MP − ln m + 4 ln gdm
Td 2 2
m g 4 MP
⇒  ln dm ,
Td m

g4
where we supposed σ v  Tdm2 , and approximated Td  m on the right-hand
−1
T = 28.5 for m = 100 GeV and gdm = 10 ,
side of the equation. This gives m
which is quite in good agreement with (3.150). In fact, this ratio is not really
sensitive to the dark matter mass and varies from 33 to 19 for dark matter
masses between 1 GeV and 1 PeV.

3.5.3 Solving the Equation

Before solving the equation (3.145), let us write it in a convenient way


dY 2gs π 10 m MP   λ  2 
= σ v Y 2
eq − Y 2
= Y − Y 2
(3.153)
dx 15 gρ x 2 x 2 eq

with

2gs π 10
λ= mMP σ v.
15 gρ
200 3 A Thermal Universe [TRH → TCMB ]

We can also consider gρ = gs , which is the case at such temperature as non-massive

degrees of freedom like neutrino has not decoupled yet. Combining Eq. (B.112) and
(B.137), a typical electroweak cross section can be written for a fermionic dark
matter

|M|2 2G2F s 2 s
σEW v    10−9 GeV−2 , (3.154)
2 × 2 × 8πs 2 × 2 × 8πs (2 × 10)2

where we normalized to a 10 GeV dark matter. We then obtain λ  σEW vmMP 

1011 1 for a 100 GeV particle (m = 100 GeV). As a consequence, such a large
value of λ depletes very quickly the density of the particle (even if much slower than
the equilibrium density which is Boltzmann suppressed).

3.5.3.1 s-wave
The equation (3.153) is a type of Riccati equation with no analytic solution. Despite
the fact that it is not exactly solvable, we can still see through by invoking some
physics intuition. We know that all happen around x ∼ 1. In this region we can see
that the left-hand side of (3.153) is O(Y ) while the right-hand side is O(λY ). We
just understood that λ 1, so the right-hand side must have a cancelation in the
Y 2 − Yeq
2 term.

After freeze out, Yeq will continue to decrease according to the thermal sup-
pression e−m/T , so that Y Yeq . This happens at late times x 1 where the
Boltzmann equation reduces to

dY λ(x)
≈ − 2 Y2 (3.155)
dx x
s-wave annihilation is characterized by σ v = cst. In this case, the resolution of the
Eq. (3.153), integrating between the freeze out time xf and the present time x0 , one
obtains (noticing x0 xf and Y0 Yf )

1 1 1 1 xf
− =λ − ⇒ Y0  (3.156)
Y0 Yf xf x0 λ

with xf , time of freeze out obtained as a first approximation by solving

neq (xf ) σ v = H (xf ) (3.157)

which, for a weakly cross section, gives xf  20 (a more precise solution will be
given in the following section). From Y0 one can then deduce the relic abundance of
the particle A of mass m:

today
ρA mnA mY0 s0 m xf 2π 2 gs T03
A = 0
= 0 = 0
= (3.158)
ρc ρc ρc λ 45ρc0
3.5 Producing Dark Matter in Thermal Equilibrium 201

with ρc0 = 3H02/8πG = 1.88h2 g cm−3 = 1.05 × 10−5h2 GeV cm−3 as we

today
computed in Eq. (2.53) and the value of gs = 3.91 as we found in Eq. (3.63).
With these numbers, we obtain

xf 8.8 × 10−7
A = √ GeV cm−3 . (3.159)
σ v gρ ρc0
(10−9 GeV−2 )

Taking gρ  100 (see Fig. 3.3), xf ∼ 20 (Eq. (3.170)) and the value of ρc0 of
Eq. (2.53) we can write28

0.17
Ah
2
 σ v
. (3.160)
(1.2×10−26 cm3 s−1 )

This is often called “WIMP miracle.” Indeed, we see that for a typical elec-
troweak cross section the relic abundance A reaches 0.17/ h2  0.3 which is the
measured value of the matter content in the Universe. Some corrections have to be
taken into account: the velocity at decoupling time is not c, the value of xf should
be computed iteratively (see next section for a more complete calculation) and the
dependence on the effective degree of freedom or mass of dark matter should be
looked carefully. However, this approximation is surprisingly quite accurate in any
models with s-wave dominated annihilation process.

3.5.3.2 General Solution

Now that we understood how to compute the relic abundance in a specific case,
we can apply the same method in the generic case, developing σ v = a + bv 2 ,
v being the relative velocity between the two annihilating particles.29 Notice that
in the Boltzmann equation, it is not σ v which enters in the definition of λ in
Eq. (3.153) but the thermal averaged cross section σ v. At the temperature of
interest at freeze out (xf = m/Tf ≈ 20 as we will compute more in detail later
on) we can consider that the annihilating particles “1” and “2” are non-relativistic
and thus their distributions (3.22) can be approximated by a Boltzmann distribution
fi  e−Ei /T  e−(m+pi /2m)/T . One thus can write
2

  3
d p1 d 3 p2 (a + bv 2 )e−E1 /T e−E2 /T
σ v = 1 2   3 3 −E1 /T e −E2 /T
. (3.161)
1 2 d p1 d p2 e

28 σ v has been normalized to a typical electroweak cross section for a 100 GeV particle:

10−9 GeV−2 = 1.2 × 10−26 cm3 s−1 , Eq. (3.154). 

(pi .pj )2 −m2i m2j
29 Wedefine the relative velocity between two particles i and j by vij = Ei Ej , with pi
and Ei being four-momentum and energy of particle i.
202 3 A Thermal Universe [TRH → TCMB ]

We can then develop

a + bv 2  = a + b v 2 

with, Eq. (5.71)

|p2 − p1 | |p2 − p1 |
v=  .
γm m

We then have

v 2 = (|p1 |2 + |p2 |2 − 2p1 p2 cos θ )/m2 ,

θ being the angle between the two colliding particles. Noticing by symmetry that
cos θ  = 0 and |p1 |2  = |p2 |2 , using Eq. (A.122) we then can write30
 2 2 2 −p12 /2mT √
1 p1 dp1 (p1 /m )e 2 3/8 π(2mT )5/2 T
v2  = 2  2 = √ = 6 = 6x −1
−p12 /2mT m2 1/4 π(2mT )3/2 m
1 p1 dp1 e
(3.162)

giving

6b
σ v = a + . (3.163)
x
We can now solve the Boltzmann equation (3.153) in the regime x xf to
obtain Y0 = Y (x0 ) nowadays. Defining  = Y − Yeq , we can neglect Yeq and
dYeq /dx which are negligible due to the Boltzmann suppression at such late times.
Expressing

2gs π 10
λ = σ vλ0 , with λ0 = mMP . (3.164)
15 gρ

Equation (3.153) can be rewritten

 
d a+ 6b
x λ0
− 2 . (3.165)
dx x2

30 See the Sect. 3.5.5.1 for another way to lead the integration for the mean σ v.
3.5 Producing Dark Matter in Thermal Equilibrium 203

After integration from xf → x0 and noticing f = (xf ) (x0 ) = 0 due to

the larger value of λ0 , using (gρ (xf ) = gs (xf ))

2gs π 10 2π 2 0 3
λ0 = mMP , s0 = g T , and ρc0 = 3H02MP2 (3.166)
15 gρ 45 s 0

one obtains
xf 1 mY0 s0
Y0  0 =   ⇒ 0 = (3.167)
λ0 a + 3 b ρc0
xf
⎛ ⎞

π gρ T03 ⎝ xf ⎠ 4 × 10−2 xf
= ≈  .
9 10 H02 MP3 a + 3 xb h2 gρ (xf ) (a+3b/xf )
f 3×10−26 cm3 s−1

To complete the solution we need to compute a more precise value of xf . For that
we can “define” xf such that the temperature when the deviation of the density
from the equilibrium value is of the order of the equilibrium density itself. Putting
in equation, one needs to solve (xf ) = cYeq (xf ), c being of the order of unity and
can be calculated exactly numerically. We will keep it as a free parameter during
all our calculation. However, in this regime, we cannot neglect anymore Yeq and
dYeq /dx as around xf the evolutions of the densities are very large and supposing
the evolution of (x) negligible with respect to the evolution of Yeq (d/dx
dYeq /dx). We can then write (3.153)
 
d dYeq λ0 a + 6b
x  
=− −  2Yeq +  (3.168)
dx dx x2
with, Eq. (3.146)

45  π 1/2 gA 3/2 −x
Yeq = x e = αx 3/2 e−x
2π 4 8 gs

which gives (considering xf 1)


λ0 α 6b
e xf = √ a+ c(c + 2) (3.169)
xf xf
204 3 A Thermal Universe [TRH → TCMB ]

which gives using (3.166)


45 1 gA MP m(a + 6b/xf )
xf = ln c(c + 2) √ 
8 2π 3 xf gs (xf )
m gA (a + 6b/xf )
≈ ln 2c(c + 2) × 108  (3.170)
gs (xf ) 3 × 10−26 cm3 s−1

gA being the internal degree of freedom of the annihilating particle, and c an

order unity parameter which is normally determined numerically from the solution
of the Boltzmann equation. It is usually set equal to 0.5 for analytical approximation.
Equation (3.170) gives xf  23 for a 100 GeV mass particle which confirms our
approximation xf 1.

3.5.3.3 Hot Dark Matter

It can be interesting to see what is happening with a particle which decoupled while
still relativistic (a neutrino-like particle, but decoupling at a freeze out time xf ). In
this case,

neq 45 geff
Yeq (x) = = ζ(3) (3.171)
s 2π 4 gs (x)

with geff = gA ( 34 gA ) for a bosonic (fermionic) dark matter. Applying the same
reasoning than above, we can easily show that as dYeq /dx = 0 around the freeze out
time, contrarily to the cold/massive case where dYeq /dx ∝ Yeq . As a consequence,
(x) ≈ 0 and so Y follows Yeq and stays constant after the freeze out:

45 geff
Y0 = Yeq (xf ) = ζ(3)
2π 4 gs (xf )
geff gs0 8m(T0 )3 ζ(3) 9.6 × 10−2 geff  m 
⇒ 0 = ≈ .
gs (xf ) 3πH02 MP2 h2 gs (xf ) eV

From this result, we can obtain an upper limit for a hot dark matter (right-handed
neutrino like, for instance, with geff = 2×3/4 and gs (xf )  10) from the condition
h2  0.1 we obtain m  7 eV.
3.5 Producing Dark Matter in Thermal Equilibrium 205

WIMP in brief
When I teach the Boltzmann equation or the WIMP paradigm, I like to
recall the historical way it was done in 1977. It allows for a reasonably good
WIMP-relic abundance calculation, without the need of the heavy Boltzmann
equation machinery. Several great physicists (Weinberg, Lee, Hut, Zeldovich,
Dolgov. . . ) got the same idea at the same time. Using arguments similar to
the one used by Gamow to compute the present CMB temperature (3.117),
they noticed (see above) that for a “hot” species, the relic abundance can be
written (I keep “ν” for historical reason, a heavy neutrino being, at this time,
the more popular dark matter candidate):
mν
νh
2
 0.1 . (3.172)
10 eV

They remarked that a 3 GeV candidate would need a suppression of 3 × 108

in its density to still be compatible with the cosmological observations.
mν
Considering the Boltzmann suppression factor of e Tν , this corresponds to
Tν  m20ν , or, considering a radiation dominated Universe,
 2
1 GeV
tν  6.67 × 10−4 s. (3.173)
mν

Asking for the annihilation to stop at tν (freeze out time, time when you have
less than one annihilation)

nν × σ v × mν  1 ⇒ n  1030 cm−3 , (3.174)

where we used the electroweak cross section σ v = 3 × 10−27 cm3 s−1 ,

corresponding to a present density
 3  2
T0 n(T0 )mν 1 GeV
n(T0 ) = nν ⇒ h2 =  . (3.175)
Tν ρc / h2 mν

We recover in this simple manner a cosmological limit mν  3 GeV, called

the Lee-Weinberg bound.

3.5.3.4 Another Approach to Average the Annihilation Cross Section

We just saw that the thermal average of the annihilation cross section times velocity
can be done by expanding the cross section at low relative velocity, σ v = a + 6b x
from σ v = a +bv 2 , v being the relative velocity between the two colliding particles.
However, there are cases where the approximation is not valid anymore due to some
divergences. An integral formulation of the solution becomes then useful.
206 3 A Thermal Universe [TRH → TCMB ]

Let us consider the case of two dark matter particles with energy E1 and E2 and
momenta p1 and p2 colliding with an angle cos θ . It is important to note that in
this case, one should make the computation in the “gas rest frame” and not in the
mass frame of the two colliding particles, because one should take into account a
statistical population of colliding particles with a statistical distribution of momenta
and angles. First of all, one should compute the cross section of two particles “1”
and “2” of masses M1 and M2 into two particles “3” and “4” of masses m3 and
m4 representing the relativistic particles in the plasma. From Eq. (B.110), taking
into account that (P1 .P2 )2 − M12 M22 = E1 E2 v12 , v12 being the relative velocity
between the particle 1 colliding the particle 2 (from now denoted simply by v) we
deduce

dσ |M̄|2 s 2 − 2m23 s − 2m4 s + (m3 − m4 )2
= (3.176)
d 128π 2s E1 E2 v

|M̄|2 being the squared mean of the amplitude over the initial spin states. As a good
approximation, for the temperature of interest, we can relatively safely consider that
the particles living in the bath in thermal equilibrium with the massive particles 1
and 2 are massless (they are electrons, neutrinos, quarks, and gauge bosons mainly).
The massive ones decoupled from the bath (Boltzmann suppression) very quickly
and do not “discuss” anymore with the other particles in thermal equilibrium.
Moreover, we will concentrate on annihilation of dark matter (the generalization
to coannihilation is straightforward) and we will thus suppose M1 = M2 = M.
Note that during all the demonstration, we will consider annihilation of dark matter
1 + 2 → 3 + 4, the computation would be the same if one considers the production
process 3 + 4 → 1 + 2.
The average of the annihilations cross section per solid angle31 can be written
(considering m2 = m3 s):

 d 3 p1 d 3 p2 |M̄|2
f1 (E1 )f2 (E2 ) (2π) 3 (2π)3 128π 2 E E d A
v dσ  = 
1 2
= (3.177)
d 3 p1 d 3 p2
f1 (E1 )f2 (E2 ) (2π)3 (2π)3 n2eq

31 We must be careful that the solid angle d is the one between the outgoing particles m3 and m4
in the center of mass of the colliding particles, to not confuse with the solid angle of the colliding
particles cos θ in which we perform the statistical average.
3.5 Producing Dark Matter in Thermal Equilibrium 207

neq being the density of the colliding particles at equilibrium (the dark matter
candidate in our case). The phase space d 3 p1 d 3 p2 can be rewritten as

1
d 3 p1 d 3 p2 = 4π|p1 |E1 dE1 4π|p2 |E2 dE2 d cos θ
2
giving

|M̄|2
A= f1 f2 |p1 |dE1 |p2 |dE2 d cos θ d . (3.178)
(2π)4 64π 2

As M usually is expressed as function of the center of mass energy it would be

easier to make a benefit change of variables:

E+ = E1 + E2
E− = E2 − E1
s = (P1 + P2 )2 = 2M 2 + 2E1 E2 − 2|p1 ||p2 | cos θ. (3.179)

Using the Jacobian and the new limit on integration one obtains

  
 √
∞ ∞ E− =+ E+ 2 −s 1−4M 2 /s
f1 f2 |M̄|2
A= √  √ d dE+ dE− ds.
s=4M 2 E+ = s E− =− E+2 −s 1−4M 2 /s (2π)4 256π 2
(3.180)

For the regime where T  M (in other words when the dark matter candidate
is not relativistic at the freeze out temperature) one can approximate the Fermi-
Dirac or Bose-Einstein statistical distribution f1 and f2 by the Boltzmann one =
f  e−E1 /T ; f2  e−E2 /T . We let the reader have a look to the Sect. 3.6 to see how
this change in the case of non-thermal production, when the approximation T  M
is not valid anymore. In the present case, f1 f2 = eE+ /T and one can integrate over
E− and E+ directly using the definition of modified Bessel function K1 (z) of the
Appendix A.6.3:
√ 
 ∞  ∞ e−
s − 4M 2 E+
E+
2 −s
T
A= √ √ dE+ dsd
s=4M 2 E+ = s (2π)6 32 s
 ∞  √
T s
= s − 4M K1
2 |M̄|2 dsd = n2eq vdσ . (3.181)
32(2π)6 4M 2 T
208 3 A Thermal Universe [TRH → TCMB ]

This result is one of the more important of the chapter and will be very useful to
solve the Boltzmann equation analytically. One can also compute neq as it is done
in Eq. (3.34)

∞ √ √ 
T
32(2π)6 4M 2
s − 4M 2 |M̄|2 K1 Ts dsd
vdσ  =
m2 T
m 2

2π 2
K2 T
 ∞  √
1 2 s
= s − 4M |M̄| K1
2 dsd . (3.182)
256π T M 4 K22 (m/T )
2
4M 2 T

It is also interesting to notice that if we would have looked at the production

process 3 + 4 → 1 + 2 the result would have been the same. Indeed, setting m3 =
m4 = M in the Eq. (3.176) (the final√state being in this case the dark matter), one
would have obtain the same factor s − 4M 2 of the the 1 + 2 → 3 + 4 case,
generated this time from the integration on E− , Eq. (3.181). This equation is thus
valid for the annihilation as for the production process.

3.5.4 The Lee-Weinberg Bound

For any kind of cold dark matter, we can apply Eq. (3.160) and find the minimal
mass for a typical WIMP (heavy neutrino) still in accordance with WMAP limits.
Indeed, in the case of a heavy neutrino ν of mass mν , the Lagrangian needed to
compute the annihilation cross section is

g   g  
f¯γ μ cV − cA γ 5 f .
ν 5 f f
Lν = Zμ ν̄γ μ cVν − cA γ ν+
2 cos θW 2 cos θW
(3.183)

In the regime MZ mν we can write the Zμ propagator as

ημν + PμZ PνZ /MZ2 iημν

i ≈
PZ2 − MZ2 MZ2

which gives

g2    
f f
M=i 2
v̄(pν̄ )γ μ cVν − cA γ u(pν ) ū(pf )γμ cV − cA γ 5 v(pf¯ )
ν 5
4 cos2 θW MZ
(3.184)
3.5 Producing Dark Matter in Thermal Equilibrium 209

f f f f f
with cV = T3L − 2qf sin2 θW and cA = T3L , T3L being the isospin of the particle
f . We keep the ciν as free parameter for our calculation. Computing |M|2

g2
2  2  2
f f
|M| = 32
2
cf cV + cA (3.185)
4 cos2 θW MZ2 f
 2  ν 2   
× cVν + cA pf .pν pf¯ .pν̄ + pf .pν̄ pf¯ .pν
   ν 2 
2
+ 2m2ν cVν − cA pf .pf¯

with cf being the color factor of SU (3) charged particles. At the limit of zero
velocity, Eν  mν and Eq. (3.185) can be written as

8g 4 m4ν  ν 2  2  2
f f
|M|2 = cV cf cV + cA
cos4 θW MZ4 f

1 |M|24π g 4 m2ν  ν 2  2  2
f f
⇒ σv  = c cf cV + cA
2
4 32π s 16π cos θW MZ V
4 4
f

f f
f cf [(cV ) + (cA ) ]  6.2. If we impose h2  0.1,
For light masses, 2 2
−26 −1 −9 −2
Eq. (3.160) gives σ v  2 × 10 cm s = 1.7 × 10 GeV we then obtain
3

2
mν  4.1 GeV for νh  0.1. (3.186)

Although it is often called the Lee–Weinberg bound [8], it was discovered indepen-
dently by a number of people.32 A viable way of building light dark matter models
avoiding the Lee–Weinberg bound is by postulating new light bosons. This increases
the annihilation cross section and reduces the coupling of dark matter particles to
the Standard Model making them consistent with accelerator experiments.
This limit was used during a long time in several publication as it justified
relatively heavy (∼100 GeV, or electroweak scale) dark matter. However, since the
nineties, several lighter candidates appeared and when one looks in more details the
solution to the Boltzmann equation, we find several exceptions. The main point is
that the expression (3.184) is somewhat too simplistic as (1) we cannot decouple
the Z boson so easily every time (especially near its pole mass), and (2) maybe the
particle exchanged is not the Z boson, but a Z  or a supersymmetric particle. In
any case, it is important to keep in mind that the Lee-Weinberg bound is only valid
for a dark matter candidate coupled only to the Z boson, far from its pole and with

32 For instance, by Pete Hut [9].

210 3 A Thermal Universe [TRH → TCMB ]

electroweak like strength coupling. We will review some exceptions to this bound
in the following discussion.

3.5.5 Two Exceptions to the Boltzmann Equation∗

The two exceptions we will discuss in this section are the pole region and the
kinematic threshold. You can find a detailed analysis of such regime in [10].

3.5.5.1 The Pole Region

Computing the annihilation cross section of a particle χ through an intermediate
particle exchanged, with a mass Mex gives
αD αV s
σv = , (3.187)
(s − Mex
2 )2 + M 2  2
ex ex

where αD , αV ∼ 0.01 corresponds to the coupling of the exchanged particle to the

dark sector (D) and the visible one (V ). ex is the total width of the exchanged
4m2χ
particle and33 s = 1−v 2 /4
. If one defines

 2  2
2mχ ex
r= and  =
Mex Mex

one obtains
r
αD αV 1−v 2 /4
σv = 2 2
. (3.188)
Mex
1− r
1−v 2 /4
+

In the case of a Z exchange,  = (2.5/91.2)2  7.5 × 10−4 , and one can understand
easily why the velocity in Eq. (3.188) should be treated with care. At zero relative
velocity, the pole occurs at r = 1 while in general it occurs at r = 1 − v 2 /4. The
value of the cross section is then
αD αV
σ vpole = 2 
. (3.189)
Mex

However, as we saw in the previous section, in relic abundance calculation, the cross
section must be thermal averaged. The standard method was to Taylor expand it to
first order in v 2 and then substitutes v 2  = 6 mTχ = x6 (see Eq. (3.162) and below).

33 See
Eq. (B.111) and discussion below for details and the expression (B.152) for a concrete
example of a Z  exchanged.
3.5 Producing Dark Matter in Thermal Equilibrium 211

However, for small values of  the expansion in v 2 breaks down near the pole and
the standard method gives extremely poor results, even allowing the cross section to
become negative!
Two methods have usually been used to treat the pole region, with quantitatively
bad results. A common approach has been to factorize the “pole factor” P (v 2 ) =
 −1
2
1− r
1−v 2 /4
+ before making the Taylor expansion and then multiply the

result by the pole factor at zero velocity which would give

αD αV r 3 αD αV r 3 1
σ v1 = 1+ P (0) = 1+ . (3.190)
Mex2 2x Mex2 2x (1 − r)2 + 

Another possibility is to approximate σ v by just substituting v 2 → 6

x in σ v

 1
6 αD αV r 1−3/2x
σ v2 = σ v v 2 = = 2  2 . (3.191)
x Mex
1− r
1−3/2x +

Both approximations can be valid far from the pole, but a real numerical treatment
should be used when approaching it. The numerical thermalized cross section σ v
is found from the calculation of the mean of σ v in the center of mass frame (p1 =
−p2 )
  3
d p1 d 3 p2 δ 3 (p1 + p2 )(σ v)e−E1 /T e−E2 /T
σ v = 1  2  3 −E1 /T e −E2 /T
(3.192)
1 2 d p1 d p2 δ (p1 + p2 )e
3 3

in the non-relativistic limit, one can approximate

p2
E1  E2  m + with |p1 | = |p2 | = |p| = γ (v1 )mv1 = γ (v/2)mv/2,
2m
v being the relative velocity between particle1 and particle 2 (see discussions after
Eq. (B.111) for this point) and γ (v1 ) = 1/ 1 − v12 . We then obtain with x = m
T
and the help of the functions (A.122)
 
p2 (σ v)e−p /mT dp ∞
2
x 3/2
dvv 2 (σ v)e−xv
2 /4
σ v = σ vnum =  = √ .
p2 e−p /mT dp
2
2 π 0
(3.193)
212 3 A Thermal Universe [TRH → TCMB ]

Log[ V v] in GeV −2
−5

−6

−7
<V v >num

−8

−9 D2 r D2
2 2
M ex r M ex
−10
0 2 4 6 8 10
r

Log[ V v] in GeV −2
−4

−5 Vv!2
Vv!1

−6
Vv!num

−7

−8
0.6 0.7 0.8 0.9 1.0 1.1 1.2
r
2m
Fig. 3.20 Numerical thermal averaged cross section σ vnum near a pole as function of r = Mexχ
in comparison with the two asymptotic behaviors (top) and with the two approximations (called
“1” and “2” in the text, bottom)

The results are shown in Fig. 3.20 where we plotted σ v as function of r for x = 25
and Mex = 91 GeV, αD = αV = 0.01. As expected, in the first method, the pole
appears at r = 1, whereas in the second one appears at r = 1 − 2x 3
= 0.94.
α2r
We also show the behavior of the two asymptotic values σ v = 2
Mex
and σ v =
3.6 Non-thermal Production of Dark Matter 213

α2
2 corresponding to
rMex
the average cross section far from the pole where there is no
dependence on the velocity, σ v = σ v in this case.

3.5.5.2 Kinematic Threshold

Suppose a channel where 2mχ  m1 + m2 , m1 and m2 being the mass of the two
Standard Model-like particles. The annihilation channel χ χ → 1 2 should then be
forbidden, except in the early Universe where the temperature at the decoupling time
m
T  25χ can be sufficient for s to reach the threshold m1 + m2 . Since the χ particles
are Boltzmann suppressed, the annihilation takes place at a certain rate. If the masses
of the annihilation products are not too much greater, this kind of annihilation can
dominate the cross section. This possibility as well as the coannihilation processes
are very well described in [10].

3.6 Non-thermal Production of Dark Matter

3.6.1 The Idea

The previous discussion was based on a fundamental hypothesis: all the particles,
including the dark matter, are thermal, which means they are produced, at the
reheating temperature in equilibrium with the photons. There is another possibility
that we should take into account. If some particles, including the dark matter, are
not produced after the reheating time with the thermal bath, their evolution follows a
completely different way. In this case, one should neglect in Eq. (3.153) the second
term corresponding to the annihilation of the dark matter, because its number density
Y is too small, and only consider the production rate from the particles living in the
bath. The evolution of the number density Y is then given by
 
dY λ 2 2gs π 10 σ v 2 2gs π 10 σ v
 2 Yeq = mMP 2 Yeq = mMP 2 2 n2eq
dx x 15 gρ x 15 gρ s̃ x

s̃ being the density of entropy of the system (3.61) at the temperature T and x = m
T.
Using Eq. (3.181) (A = σ vn2eq ) we obtain
 
dY 2gs π 10 mMP T4 ∞ 
= z z2 − 4x 2K1 (z)|M|2 dzd (3.194)
dx 15 gρ x 2 16(2π)6s̃ 2 2m
T

√
s 2π 2
with z = T . Developing s̃ = 45 gs T
3 we obtain

√  ∞ 
dY 270 10 MP 1
= √ z z2 − 4x 2K1 (z)|M|2 dzd .
dx π m gs gρ 16(2π)8 2x
(3.195)
214 3 A Thermal Universe [TRH → TCMB ]

We can find solutions to this equation for canonical regimes. Before studying a
microscopic defined model, let us first suppose a simplified model with a mediator
H coupling to the SM and dark matter with a strength α  . The amplitude squared
can then be approximated (to some numerical factors of the order of unity) by

(α  )2 s 2 (g  )2
|M|2  with α  = . (3.196)
(s − MH 2 )2 4π

At the temperature of interest (around the reheating time) we can safely neglect
√ the
mass of the dark matter compared to the center of mass energy s: m s. We
then have√ to consider 2 cases: very heavy
√ mediators, with M H T RH and thus
MH s or light mediator, MH s. We will distinguish the two cases in the
following.

3.6.1.1 Case A: Heavy Mediator: MH  TRH

In this case, |M|2 can be approximated by |M|2 ≈ (α  )2 s 2 /MH
4 . At the temperature

of interest, we can consider that gρ = gs . The Eq. (3.195) then becomes

√ √
dY 48(α  )2 270 10 MP m3 4 270 10 MP
= 7 3 4
⇒ Y (T ) = 9 3  )4
3
TRH −T3
dx (2π ) x 4 M (2π ) (M H /g
2 2
gs π H gs π
(3.197)

TRH being the reheating temperature. From Eq. (3.197) we can see that all the
dark matter are produced around the reheating temperature. Very quickly Y reach
its maximum value Y∞ = Y (T → 0). Physically speaking, it means that only
the photon and relativistic particles which possess large energy (around TRH ) can
contribute to the production of the dark matter candidate. As soon as the temperature
decreases below TRH the production process is too weak compared to the expansion
rate. From Eq. (3.197) one can also deduce the relation between the heavy particle
mediator and the temperature of reheating to produce sufficiently dark matter to
respect WMAP bound, h2 = 0.1. We can indeed write
 
ρc0 100 GeV 135 × 10−3 H02 MP2
Y∞ = = GeV−1 ⇒ (3.198)
m s0 0.1 m 2π 2 gs0 T03
√  
3 T3 
 4 3 10 gs (T 0 ) 16 × 10 3 TRH 0 0.1  m 
MH = (g )
4
GeV
4π gs3/2 (TRH ) 3(2π)7 H02 MP 100 GeV
 1/4  1/4
5  m 0.1 3/4
⇒ MH  4.6 × 10 g TRH GeV1/4 . (3.199)
100 GeV h2
3.6 Non-thermal Production of Dark Matter 215

Fig. 3.21 Feynman diagram

representing the process of
f F
production of dark matter
from annihilation of the
Standard Model particles
q qF
living in the bath through the f
exchange of a mediator H
H

f F
< Vv >

The last expression34 gives us the minimum value required by MH for each
reheating temperature to avoid the overclosure of the Universe ( h2 < 0.1):
 m 1/4
= 4.6 × 105 g 
min 3/4
MH TRH GeV1/4 . (3.200)
100 GeV
The results are shown in Fig. 3.21.
It is important to notice several keypoints at this stage. First of all, the approxi-
mation concerning the distribution (Maxwell-Boltzmann approximation) we used to
obtain Eq. (3.195) is not valid anymore for large temperature. Strictly speaking we
should take the Fermi-Dirac expression of the distributions f1 and f2 . A complete
numerical analysis gave us a 44% of differences between the two treatments. The
reader can have a look at Sect. 2.4.3 where we treat this case in detail. Secondly, α  is
in fact the interaction strength multiplied by the charges of the interacting particles,
so factor of unities can also enter in the results: α  = qDM2 q 2 α
SM real . Finally our
condition is valid in the approximation MH > TRH (which was our hypothesis)
which corresponds to, from Eq. (3.199)
 1/4  1/4
100 GeV h2
g  > 2.1 × 10−6 TRH GeV−1/4 ,
1/4
(3.201)
m 0.1

lower bound which stay perturbative in any cases. There is, however, other scenario
possible if the mediator is light (let suppose massless for a good approximation).
We will study them in the next section.

3.6.1.2 Case B: Light Mediator: MH  s, Weak-Like coupling: α   αEW

When the particle exchanged between the dark sector and the visible world is very
light or even massless, the thermal history is quite different. Depending on the value
of the coupling α  , the dark matter can reach the thermal equilibrium before that
the temperature of the plasma drops below the mass of the dark matter candidate

  
h2
34 As an indication, Y∞  3.3 × 10−12 100 GeV
m 0.1 .
216 3 A Thermal Universe [TRH → TCMB ]

m, or later. The former case corresponds to relatively large value of α  (of the
order of electroweak coupling) and we recover the classical WIMP-like scenario,
even if the dark matter was not produced with the SM ones at the reheating time,
but progressively. In the former case, however, the coupling is so feeble that the
dark matter abundance reaches the critical
√ value before reaching the thermal bath
equilibrium. Approximating MH s, we can approximate |M|2 ≈ (α  )2 . The
equation (3.195) becomes
√
dY 270 10 (α  )2 MP
= √ 7
(3.202)
dx π gs gρ 4(2π) m
√ 
270 10 (α  )2 MP 4.2 × 1011 GeV
⇒ Y (T ) = √  (α  )2
π gs gρ 4(2π)7 T T

contrarily to the heavy mediator case, the evolution is very fast with decreasing
temperature. For a relativistic particle, one can compute the temperature for which
the dark matter number density reach its equilibrium number:

neq (T ) 45ζ(3)geff
Yeq = = ≈ 2.6 × 10−3 geff (3.203)
s(T ) 2π 4 gs (T )

giving Yeq  7.8 × 10−3 for a Dirac fermion. If we suppose a classical coupling
(α  ∼ 10−3 ) and using Eq. (3.202) one can see that the equilibrium is reached for
T  5 × 107 GeV so very quickly after the reheating. Once Y = Yeq , dY dx  0
from Eq. (3.153), and we recover the classical WIMP thermal history with freeze
out scenario. In fact, for weakly like coupling dark matter, there is no real difference
if the candidate is produced at the reheating time at the same time that the Standard
Model particles (through the decay of the inflaton, for instance) or if it is produced
by Standard Model particles annihilation because in the later case it reaches its
thermal value vary fast. The process is depicted in Fig. 3.22.

3.6.1.3 Case C: Light Mediator: MH  s, Feebly Like Coupling:

α   αEW
In this scenario, the coupling is so feeble that the dark matter particle is not
sufficiently produced by SM annihilation: when the temperature of the thermal bath
reaches T  m, the production is completely frozen “in.” The particles produced by
this mechanism are called FIMP as Freeze In (or Feebly Interacting) Massive Parti-
cle. The evolution follows the Eq. (3.202) but with very tiny values of α  . The value
of the coupling needed to respect WMAP can be obtained asking for the number
density to reach the relic abundance density just before the decoupling temperature
(Tf  23m
, Eq. 3.170) where the particles of the plasma do not have sufficient energy
to produce the dark matter species (exponential suppression). Strictly speaking,
we cannot really talk about “decoupling temperature” or “decoupling time” as the
particle was never coupled to the plasma. This happens when Y (Tf ≈ 23 m
) = Y∞
3.6 Non-thermal Production of Dark Matter 217

dY D (Y − Yeq ) = 0
dT rel −3
Yeq = 8*10

Thermal WIMP Y D −T
Y D 1/ T

dm −12
Y0 = 3.3*10
3 3
Y D ( T RH− T )
HMDM Gravitino
Y D (TRH − T) Y D 1/T
FIMP

TRH 10 8 m/20 T0
T (GeV)
Fig. 3.22 Three different ways to produce dark matterfrom theStandard
 2  Model bath and obtain
the correct critical relic abundance Y∞  3.3 × 10−12 100mGeV h
0.1 : Thermal WIMP, Heavy
Mediated Dark Matter (HMDM), and Feebly Interacting Dark Matter (FIMP)

which gives

m  2
 2 gs2 −1 4 MP H0
Y = Y∞ ⇒ (α ) = √ 3 × 10 (2π)  10−22
23 0.1 gρ T03
⇒ α  = αF I MP ≈ 10−11 (3.204)

this scenario is summarized in Fig. 3.22.

3.6.2 Axion as a Dark Matter Candidate∗

Several dark matter candidates are analyzed in this book. We want to detail in
this section the calculation of the relic abundance for one of the first that appears
in the literature: the axion dark matter proposed by Pierre Sikivie and Laurence
Abbott [11].

3.6.2.1 The Thermal Production

Axion particles have different origins. It was introduced in 1977 to solve the Strong-
CP problem (see Appendix B.7 for details). One property of an axion φa is its
218 3 A Thermal Universe [TRH → TCMB ]

coupling to the Standard Model:

φa
La = Fμν F̃ μν , (3.205)
fa

where Fμν is the field strength of a gauge boson Aμ , F̃μν = 12 μνρσ F ρσ and fa
a typical breaking scale of the theory. Historically, the coupling involved only the
gluonic fields, but it is now common to find in the literature this kind of candidate
under the name ALP for Axion Like Particle. One possibility of production is from
the thermal bath. As we will see, processes of the type

A + A → A → A + φa (3.206)

can produce φa in a sufficiently large amount.

Exercise Is this process possible for Abelian gauge group?

Suppose that at the earliest stages of the Universe the axions were in thermal
equilibrium. This is possible if the cross section of the reactions (3.206) respects35

T2
n σ vAA↔Aφa  H (T )  , (3.207)
MP

where we supposed a Universe dominated by the radiation (2.264),

√
with a number
density n. Approximating the amplitude of the process M ∼ g3 fas , where A is a
√
gluon field and s the center of mass energy, we deduce
 2
3 |M|
α3 T 3 2T2 fa 1014
n σ v ∼ T ∼ > ⇒ T > Td = GeV,
8πs 2 fa2 MP 1016 GeV α3
(3.208)

g2
with α3 = 4π3 and Td represents the decoupling temperature. At Td , we can write
na (Td ) = ζ(3)
π2 d
T 3 (Eq. 3.27) and implement it in Eq. (E.1) to write

2 8 g0 n(Td ) ma
a h = 1.6 × 10 , (3.209)
gd Td3 1 GeV

35 See the Sect. 3.1.8 for a deeper understanding of this decoupling condition.
3.6 Non-thermal Production of Dark Matter 219

where gd is the effective degree of freedom at the decoupling time that we will take
gd = 106.75 (3.38), whereas the present one is g0 = 3.91. We then obtain for the
thermal component of a

t h h2 ma
a
 , (3.210)
0.1 140 eV

which is in tension with constraints from free-streaming (see Sect. 5.12.1). One
also needs to add a condition on fa from the fact that its lifetime τa = a−1 should
exceed X−rays constraints at this mass range, which means a  10−52 GeV, or

m3a  m 
−52 a
a→A A  < 10 GeV ⇒ fa > 1.5 × 10 15
GeV.
4πfa2 140 eV
(3.211)

This lower bound on fa implemented in (3.208) means that Td  1012 GeV, which
is in tension with the maximum temperature of the thermal bath if one considers
that the primordial plasma is produced from the inflaton decay. We understand that
the thermal production of axion is not really convincing, but another mechanism
can populate the Universe of ultra-light axions, without being excluded by free-
streaming constraints or the temperature during the reheating. This is called the
misalignment mechanism.

3.6.2.2 The Misalignment Mechanism

Another possibility is indeed to consider the axion field as a background field, i.e.
an homogeneous field varying with time, in a very similar way the inflaton does.
Before applying it directly to the QCD-axion, let us look in detail the calculation of
the relic abundance for a generic background field φa (t). From its action
 
√ √ 1 m2
Sa = d 4 x −gLa = d 4 x −g gμν ∂ μ φa ∂ ν φa − a φa2 , (3.212)
2 2

the equation of motion

∂ √ ∂ √
−gLa − ∂ μ μ −gLa = 0 (3.213)
∂φa ∂∂ φa

for a field φa = φa (t) reduces to

φ̈a + 3H φ̇a + m2a φa = 0. (3.214)

Exercise Recover the previous equation.

220 3 A Thermal Universe [TRH → TCMB ]

A solution of the form φa (t) = f (t) cos ma t can be extracted from this equation,
with
3
f˙ = − f
3 3
⇒ f (t) ∝ t − 4 ∝ a − 2 , (3.215)
4t
where we supposed a radiation dominated Universe. We then obtain
a 3
i 2
φa (t) = φai cos ma t , (3.216)
a
where ai is the scale factor at the time where ma  H , in other words when
the oscillation modes begin to dominate the energy density. Before this time, for
scale factors below ai where H ma , the equation (3.214) can be reduced to
φa =constant≡ φai if one takes φ̇ai = 0.
The energy density ρa is then (2.95)

1 2 m2a 2 ma i 2  a i  3
ρa = φ̇a + φa = (φ ) . (3.217)
2 2 2 a a
Notice that ρa behaves like a dust, whatever is the value of ma . From the
perturbation point of view, and the structure formation constraints, we do not have
the free-streaming tensions we had for the neutrino. Indeed, the background field
φa (t) can be viewed as a sum of oscillators at rest around which structures can
begin to form in the early thermal stages, respecting conditions (5.200) well before
the recombination time. φa is indeed a cold dark matter candidate. To compute its
relic abundance, we use Eq. (E.1) with

ρa ma  i 2  ai 3
na (T ) = = φa (3.218)
ma 2 a

which gives

ma (φai )2 g0 ma
ah
2
∼ 0.8 × 108 , (3.219)
(ma MP )3/2 gi 1 GeV
or

mis h2  m 1  φai
2  m 1  θ f 2
a 2 a 2 i a
a
= =
0.1 0.4 eV 1012 GeV 0.4 eV 12
10 GeV
(3.220)

φi
where θi = faa is introduced because such and axionic field appears often as a
pseudo-Nambu-Goldstone boson of a broken chiral symmetry described by a field
3.6 Non-thermal Production of Dark Matter 221

i φa
 = A(x) e fa ≡ A(x) eiθ . The misalignment terminology comes from the fact
that, at the beginning of the domination of the oscillation modes, the angle θi = 0.

3.6.2.3 QCD-Axion Dark Matter

In the case of the QCD-axion, two additional features should be taken into account.
Firstly, the fact that the axion mass is generated through quantum corrections and
therefore is dependent on fa . Secondly, the mass being a radiative product, it also
depends on the temperature and the time, rendering the equation of motion (3.214)
a little bit more complex. Let first find the equation for ρa (t) in the case of a varying
mass ma (t). Multiplying Eq. (3.214) by φ̇a , we obtain

1 d 2 m2 d
φ̇a + 3H φ̇a2 + a φa2 = 0, (3.221)
2 dt 2 dt
or

d 1 2 m2a 2
φ̇ + φ − ma ṁa φa2 + 3H φ̇a2 = 0. (3.222)
dt 2 a 2 a

Taking the mean of the previous relation, and noticing that for ma H , φ̇a2  =
ma (t) φa , where the averages are over the oscillation period, we obtain
2

ṁa ma
ρ̇a − ρa + 3Hρa = 0 ⇒ ρa ∝ 3 . (3.223)
ma a

It is remarkable that even if the mass is time-dependent, the density still follows the
law of a dust-like component.
To find the temperature Ti for which the oscillation mode begins to dominate,
one needs to solve

10 3
ma (Ti )  H (Ti ) ⇒ Ti  ma MP QCD
6 4
(3.224)
gi π

where we supposed
 4  4
QCD QCD
ma (T QCD ) = ma (QCD ) = ma . (3.225)
T T

The evolution of the mass for the QCD-axion needs a highly non-trivial computa-
tion, the previous relation being a reasonable approximation. For T  QCD , ma ∼
constant. Applying once more Eq. (E.1) with

ma (Ti ) i 2 na (Ti ) (φai )2  g  7 π 7

i 12 6
na (Ti ) = (φa ) ⇒ 3
= 1/6 7/6 2/3
,
2 Ti 2ma MP QCD 10 3
(3.226)
222 3 A Thermal Universe [TRH → TCMB ]

mπ fπ
and taking φai = fa θi and fa = 2ma from Eq. (B.225), we obtain

  g  7  π  7 (m f )2
qcd 2 g0 i 12 6 π π 1
a h = 2 × 10 7
7/6 2/3
θi2 7/6
gi 10 3 MP QCD ma
 7
1.5 × 10−6 eV
qcd 2 6
a h
⇒  θi2 , (3.227)
0.1 ma

where we took QCD = 200 MeV.

3.6.3 The Special Case of the Gravitino∗∗

3.6.3.1 What Is a Gravitino

It is obviously very far from the objective of the book to describe the supersymmetry
theory of fields, let alone to discuss the supergravity foundations. However, we will
try in this section to give the main clues that should help the reader to appreciate
the motivations and the gravitino-SM coupling with a minimum of hypothesis.
The literature is full of very good textbook treating the supergravity models. For
a complete introduction, we suggest [12] and [13].
In 1972, Volkov and Akulov noticed that the Nature possesses a massless
boson, the photon, in which mass is protected by the gauge invariance of the
electromagnetic interaction. At this epoch, another particle seemed also massless,
the neutrino. Knowing that any broken symmetry generates a massless goldstone
boson, they proposed in [14] to extend the idea to spinors, introducing a “Goldstone
fermion,” what we presently call a “goldstino.” Enlarging the Poincaré group to
spinors transformation, we can consider an upgraded space-time (ψ, Xμ ), ψ being
a spinor. If we suppose that the neutrino, or any other spinor field ψ is massless, it
should respect the condition

iγ μ ∂μ ψ = 0. (3.228)

This equation is invariant under transformations of the Poincaré group and the chiral
transformations as well as under a new kind of translation of the type

ψ →ψ +ζ ; Xμ → xμ = Xμ , (3.229)
3.6 Non-thermal Production of Dark Matter 223

where ζ is a constant spinor, anticommuting with ψ. Replacing the transformation

(3.229) by

ψ →ψ +ζ ; ψ̄ → ψ̄ + ζ̄
i  μ 
Xμ → x μ = Xμ + 4
ζ̄ γ ψ − ψ̄γ μ ζ ,
2
we create a group structure with ten commuting and four anticommuting parameters.
This is typical from a supersymmetric transformation. Looking at an infinitesimal
transformation ζ = dψ

i  
dx μ = dXμ + 4
d ψ̄γ μ ψ − ψ̄γ μ dψ , (3.230)
2
we can define the vierbein

∂Xa i  a 
eμa = μ
= δμa + 4
ψ̄γ ∂μ ψ − ∂μ ψ̄γ a ψ (3.231)
∂x 2
such that from the definition of the metric

∂Xa ∂Xb μ ν
ds 2 = ηab dXa dXb = gμν dx μ dx ν = ηab dx dx
∂x μ ∂x ν
we deduce

∂Xa ∂Xb i  
gμν = ηab = ηab eμa eνb = ημν + 4 ψ̄γμ ∂ν ψ − ∂ν ψ̄γμ ψ
∂x μ ∂x ν 
= ημν + δgμν .

The coupling to the Standard Model fields can then be calculated from

1 μν i   μν
Lψ = δgμν TSM ⊃ 4
ψ̄γμ ∂ν ψ − ∂ν ψ̄γμ ψ TSM (3.232)
2 2
μν
TSM being the stress-energy tensor of the Standard Model. This coupling appears
when the scale of the supersymmetry, represented by36 , is much above the energy
we consider for our calculation (inflaton mass in the case of reheating processes). ψ
can be considered as the goldstino, or the longitudinal mode of the gravitino. In the
case of the Volkov-Akulov paper, they considered ψ to be the neutrino.
For energies above the supersymetry breaking scale, one needs to look at the
effect of other particles in the spectrum, especially the supersymmetric partners of


36 To be more precise,  ∼ MP m3/2 , m3/2 being the gravitino mass.
224 3 A Thermal Universe [TRH → TCMB ]

the particles present in the thermal bath. The more natural coupling one can write,
implies gravitino, gluon and its partner, the gluino:

1 μ  
L3/2 = ψ̄3/2 γμ γ α , γ β G̃ Gαβ , (3.233)
4MP

where Gaμν = ∂μ Aaν − ∂ν Aaμ + g3 fabc Abμ Acν , Aaμ being the gluon field. Considering
the longitudinal mode, reminding the classical longitudinal mode for the gauge
μ
boson (∂μ θ , θ being the parameter of the transformation), we can write ψ3/2 =
∂ μ ψ, implying

i ∂μ ψ̄ μ  α β 
L3/2 = − γ γ , γ G̃ Gαβ , (3.234)
4MP m3/2

which gives after integration by part and applying the Dirac equation on the gluino,

M3  
L3/2 = ψ̄ γ α , γ β G̃ Gαβ (3.235)
4MP m3/2

with M3 the (onshell) gluino mass.

3.6.3.2 MSUSY < TRH

In supersymmetric models, the gravitino is the partner of the graviton. If the
supersymmetric spectrum lies below the gravitino mass, the gravitino is produced
also from the annihilation of Standard Model particles from the bath, mainly the
ones charged under SU (3). The more common process is the annihilation of gluons
g into gluino37 g̃ of mass M3 and gravitino G̃ of mass m3/2 through the exchange
of a gluon: g + g → g → g̃ + G̃. The sum of all the scattering process has been
calculated and gives, with a good approximation:

2 4g32 T 6 M32
σ v3/2 neq  2 2 1 + (3.236)
π MP 3m23/2
√
dY 1 270 10 mMP
⇒ = √ σ v3/2 n2eq
dx 4gs gρ π x 2T 6
√ 
270 10 mg32 M32
 1 +
3
x 2 π 2 MP 3m23/2
gs2 π

37 Supersymmetric partner of the gluon.

3.6 Non-thermal Production of Dark Matter 225

which implies after integrating on x

√ 
270 10 g32 M32
Y (T ) = 1+ TRH − T . (3.237)
3
MP π 2 3m23/2
gs2 π

The evolution of the population of gravitino is thus proportional to the temperature

and its rate of production is softer than in the case of the heavy mediator we
discussed in the previous section. Moreover, the ratio of the gluino to gravitino mass
becomes important in this case. In specific scenarios where m3/2 M3 (anomaly
mediation models, for instance) one obtains, with g32 ∼ 0.5 ∼ (g3GU T )2 at such high
temperatures,

−12 TRH
Y∞  2 × 10 10
(3.238)
10 GeV

giving a reheating temperature of TRH  108 GeV for a 10 TeV gravitino.38 We

have represented the evolution of the gravitino production mechanism in Fig. 3.22.

3.6.4 Non-thermal Production Through Decays

3.6.4.1 Generalities
Another possibility is that heavy particles like scalars, Z  , or moduli fields can decay
into the dark matter candidate to populate the Universe. Let us consider a particle A
in thermal equilibrium in the bath, and decaying into 2 dark matter candidates χ of
momentum p1 and p2 : 2mχ  MA . Supposing that initially the dark matter is not
produced in the thermal bath: it can happen if the dark matter is not coupled with
the inflaton (through symmetries, for instance). In this case, the number of particles
produced per second from the decay of the parent A is dnχ /dt = A nA , with A
being the width of A. We can then rewrite Eqs. (3.142) and (3.143) neglecting the
annihilation rates (as the annihilation process is an order of coupling weaker than
the decay) (Fig. 3.23)

dnχ dYχ A  dYχ

+ 3H nχ = gA nA A , ⇒ = −gA YA ⇒
dt dT HT dx
A 
= gA YA (3.239)
Hx

  
h2
38 After using Y∞  3.3 × 10−12 100 GeV
m 0.1 from Eq. (3.198).
226 3 A Thermal Universe [TRH → TCMB ]

Fig. 3.23 Feynman diagram

of a particle A in thermal bath
F
decaying into 2 dark matter
candidates χ

with x = mχ /T , gA the number of degrees of freedom of A and Yi = ni /s,

s being the entropy density given by Eq. (3.61), s = 2π 2 /45gs T 3 . In fact, in this
specific case, each parent A decays into two dark matter candidates. To include
all the possibilities, like the decay into 3 or 4 particles, we have included this
multiplicity in the definition of gA . It is important to notice that A  is the averaged
width of the parent A and not the pure width. We need to stop some time on this
issue. Indeed, there is a little subtlety in the Boltzmann equation that is not obvious
at the first glance.39 The real Boltzmann equation (3.239) should in fact be written
as40
dnχ
+ 3H nχ
dt
  
d 3 pA d 3 p1 d 3 p2
= gA fA (2π)4 δ 4 (pA − p1 − p2 )|M|2
(2π)3 2EA (2π)3 2E1 (2π)3 2E2
 3
d pA MA dY dY dY
= gA fA A = gA nA A  = S = −H T s = H xs (3.240)
(2π)3 EA dt dT dx

fA being the statistical distribution function of A in the thermal bath and x =

mA /T . Comparing with the general formula Eq. (B.157) we note the presence of
d 3 pA
the Lorentz invariant phase space (2π) 3 2E in the right hand side of (3.240). This is
A
somewhat similar to the annihilation process, where the “2EA ” factor necessary to
satisfy Lorentz invariance in Eq. (3.240) is absorbed in the “2MA /2EA ” factor, the
“2E1 2E2 ” factor in annihilating case is absorbed in the “v” of σ v from the formula

4 (P1 .P2 )2 − M12 M22 = 2E1 2E2 × v (3.241)

39 This remark is also valid in the case where the dominant process is the annihilation one.
40 We have incorporated in gA the multiplicity of the decay rate as A decays into 2 dark matter
particles.
3.6 Non-thermal Production of Dark Matter 227

that one finds in Eq. (B.106). Another way to see it is that the factor M EA = γ
A 1

corresponds to the time contraction factor. Some particles A being in the thermal
bath with some high velocities are a smaller effective width (a longer lifetime). A
is the width of A at rest.
We can then solve step by step the evolution equation for Yχ supposing that A
is in thermal equilibrium and that all the physics is happening for temperatures of
the plasma T much above MA , the thermal distribution can be approximated by the
Maxwellian one from T = MA to the reheating temperature TRH , fA  e−EA /T
we then can write
 TRH
dY dEA
H xs = MA A e−EA /T pA 4π
dx MA (2π)3
 ∞ 
MA A
= e−EA /T 2 − M 2 dE
EA A A
2π 2 MA

T MA2 A MA dY MA3 A
= K1 ⇒ = K1 (x). (3.242)
2π 2 T dx H x 2s2π 2

K1 being the first modified Bessel function of the second kind that the reader can
find in Sect. A.6.3. Using the common relations (3.27) for s(x) and (3.148) for H (x)

2π 2 MA3 π gρ MA2
s(x) = gs 3
, H (x) =
45 x 3 10 x 2 MP

we finally obtain the equation

√ 
270 10 1 MP A gA ∞ 3
Y0 = Y (x → ∞) = √ x K1 (x)dx. (3.243)
π gs gρ 8π 4 MA2 TMA
RH

To have an idea of the order of magnitude of the yield of the dark matter produced
from the decay of a massive particle of mass MA , we can approximate A 
gD2 M /(16π), g being a gauge coupling in the case of a vectorial particle A, a
A D
Yukawa-type coupling in the case of a scalar one.41 Noticing that gS  gρ at the
energy of interest, we can write
√ 2g M  ∞
270 10 gD A P
Y0  x 3 K1 (x)dx. (3.244)
π 128π 5MA TMA
RH

We then need to distinguish

∞ two cases: 1) MA < TRH , A is in thermal equilibrium
∞
with the bath and MA x 3 K1 (x)dx  0 x 3 K1 (x)dx = 4.7 and 2) MA > TRH .
TRH

41 See Eq. (B.182) and (B.185) for the exact expressions.

228 3 A Thermal Universe [TRH → TCMB ]

In this case, using the equations (A.122) and the approximation of K1 (x) given in
Sect. A.6.3 we have
  
∞ √  ∞ π − TMA MA 5/2
2π  M t 6 e−t dt 
2
x 3 K1 (x)dx  e RH . (3.245)
X A
TRH
2 TRH

Applying then Eq. (3.167)

mY0 s0 g 0 2π 2 (T0 )3
0 = 0
= mχ Y0 s 2 ⇒ 0h
2
 2.8 × 108 mχ Y0 (3.246)
ρc 135MP (H0 )2

which gives
mχ
0h
2
 2 × 1022 gD
2
[MA < TRH ],
MA
 3/2
mχ − TMA MA
0h
2
 5 × 10 21 2
gD e RH [MA > TRH ]. (3.247)
MA TRH

To respect WMAP constraints one thus needs gD  10−11 if A and the dark matter
are at TeV scale in the plasma (first case) or MA  4 × 1010, for TRH = 109 GeV
and a 100 GeV weakly interacting particle (gD  10−1 ) in the second case.

3.6.4.2 An Example: Decay of the Gravitino to Populate Dark Matter∗∗

In several minimal supersymmetry extension of the Standard Model, the dark
matter candidate is a wino, fermionic partner of the W a gauge vector. The wino
being charged under SU(2), the coupling to the Standard Model bath (added to the
coannihilation process with its charged supersymmetric partner which have the same
mass at tree level) underproduced dark matter. Typical relic abundance for a wino
dark matter is around h2  10−3 There are different ways to overcome this issue.
One solution is to populate the Universe through the decay of the gravitino G̃ → χχ
of mass m3/2. As we discussed earlier on, the gravitino is super-weakly interacting
and decouples from the plasma at an early stage when the plasma has temperature
Td , around the reheating temperature TRH , Td  TRH . Its density is then fixed by
TRH by Eq. (3.238) YG̃  2 × 10−12(TRH /1010 GeV). The equilibrium condition
can be written G̃  H (T ).
To understand physically speaking what is happening in the plasma, it is useful
to express Eq. (3.242) as function of G̃ and H only:

dYχ 
∝ G̃ xK1 (x). (3.248)
dx H
As we did when we tried to understand the Boltzmann equation and the decoupling
effect for annihilating dark matter, we see that a similar phenomenon exists for
a decaying particle into dark matter. Indeed, when the temperature of the plasma
3.6 Non-thermal Production of Dark Matter 229

dY
reaches G̃  H (T = T3/2 ), dxχ  0. In other words, the number of dark matter
particle per co-volume (Yχ ) resulting from the decay of the gravitino is constant.
Physically speaking one can understand it as during the doubling Hubble time
t = 1/H , the decay of G̃ populates the Universe with 2 particles: the density
is thus constant. We need to impose that this decay occurs before the Big Bang
Nucleosynthesis (BBN) time to avoid any disturbance in the nucleosynthesis process
(TBBN  1 MeV). Indeed, at least one possible channel of decay must include either
a photon, a charged lepton or a meson, each of which would be energetic enough to
destroy a nucleus if it strikes one. One can show that enough such energetic particles
will be created in the decay as to destroy almost all the nuclei created in the era of
nucleosynthesis, in contrast with observations. In fact, in such a case the Universe
would have been made of hydrogen alone, and star formation would probably be
impossible.
From the expression of G̃ [15]


1 nm  m3/2
3 3/2
10 TeV
G̃ = nv + = 24 sec, (3.249)
4 12 MP2 l m3/2

where nv = 12 and nm = 49 √ are the number of vector and chiral matter multiplets,
respectively, and MP l = 8πMP . The condition H (T3/2) = G̃ can then be
written in the radiation dominated era
  1/4 
1/4  1/2
45 45 nv + n12m 3/2
T3/2 = MP l G̃ = m3/2
4gρ π 3 4gρ π 3 MP2 l 32π
 1/4 
10.75 m3/2 3/2
 0.24 MeV. (3.250)
gρ (T3/2 ) 10 TeV

We thus can observe that for a gravitino lighter than 10 TeV, its decay happens at
a temperature below TBBN . This is what is called as the gravitino problem. As the
result of the late time decays of the gravitino, the wino (or any supersymmetric
particle which can play the role of the dark matter) is non-thermally produced at
around T3/2 which is lower than the freeze out temperature of the wino. After the
decay of all the gravitino, the resultant relic density for χ is given by the gravitino
yield, Eq. (3.238) applied to the formula (3.247) which gives
 
mχ  TRH
decay 2
χ h  0.168 × 10
. (3.251)
300 GeV 10 GeV

Altogether, the relic abundance of a wino (or any non-thermally produced dark
matter from a decay) is the sum of the thermal production plus the non-thermal
one

χh
2
= t hermal 2
χ h + decay 2
χ h . (3.252)
230 3 A Thermal Universe [TRH → TCMB ]

As a conclusion, the thermal history of the Universe in the presence of a

metastable gravitino begins by the decoupling of the gravitino itself at a temperature
Td around TRH when H (Td )  σ vSM SM→G̃ G̃ . Then the dark matter candidate
χ decouples (freezes out) at a temperature around Tχ = mχ /20 which can give
χh
2 0.1, corresponding to H (Tχ ) = σ vSM SM→χ χ .
The population of dark matter is, however, still populated by the gravitino decay.
Finally the end of the decay (kind of freeze out too) of the gravitino G̃ → χχ at
T3/2 , stabilized to obtain χ h2  0.1 at a temperature T3/2 respecting H (T3/2) =
G̃→χχ . Several comments are in order. First, it should be noted that the entropy
produced by the decay of the gravitino is negligible since the energy density of the
gravitino at the decay time is subdominant. Second, it should be also noted that the
annihilation of the wino after the non-thermal production is negligible, since the
yield of the non-thermally produced wino is small enough. Indeed,
 
H (T3/2) 10−24 cm3 s−1 1 MeV
Yχdecay Yχannihilat ion   10−9 × .
s σ v σ v T3/2

The dark matter is thus largely diluted and not able to annihilate anymore (no return
to the thermal bath). A summary is shown in Fig. 3.24.

Y
: decaying gravitino

Wino rel −3
Yeq = 8*10

Gravitino
dm −12
Y0 = 3.3*10

Y wino thermal

TRH Td TFO T3/2 T BBN

T
Decoupling of Decoupling of Gravitino
gravitino Wino temperature

Fig. 3.24 Evolution of the yields Y of a wino dark matter and through decay of the gravitino
3.7 Extracting Information from the CMB Spectrum 231

3.7 Extracting Information from the CMB Spectrum

3.7.1 Generalities

The study of the anisotropies in the CMB spectrum is far beyond the scope of this
textbook. The literature is full of very complete and precise accounts on the subject.
The reader who is especially interested can, for instance, look at [16]. In this section,
we just want to give some intuitive insights about how physicists extract information
from the CMB measurement. Therefore, the reader should not expect here a rigorous
study of two-point correlation functions, but rather qualitative estimates. We suggest
to start by the Sect. 5.12.3 in order to approach our discussion with confidence.
The saga of the CMB prediction and measurement is a long story that is
admirably told in [17]. We already have shown the Gamow’s way to find the present
microwave temperature of T0  2.7 K from the BBN mechanism, see Eq. (3.119).
But what about the anisotropy of the CMB? This issue was already raised in the
late 60’s. Indeed, the first anisotropy that was predicted (and observed) is the one
generated by the motion of the Earth inside the homogenous radiation, mainly due
to the motion of the Earth in the solar system, the Sun in the Milky Way, the Milky
Way in the local group. . . We can estimate the velocity of the Earth with respect to
the CMB of v ∼ 300 km/s ∼ 10−3 c. This corresponds to a dipole anisotropy (also
called “aether drift”) between the temperature pointing to the direction of motion,
and the temperature pointing backward, of

v T
T ∼ 2 T0 ⇒  10−3 . (3.253)
c T0

Exercise Show that the received frequency ν of an emitted photon of original

frequency ν  , from an angle θ with respect to the direction of motion of the observer
with velocity v is

1 v2 ν
ν= = 1− × . (3.254)
t c 2 1 − c cos θ
v

Hint: have a look at Sect. 5.6.2. Deduce then that


2
1 − vc2  v 

T =T  T 1 + cos θ . (3.255)
1 − vc cos θ c

Once the spectrum has been renormalized by the aether drift, one is left with two
other possible sources of anisotropy: the one generated by the Sachs-Wolfe effect
described in Sect. 5.12.3 and more specifically by Eq. (5.208)

δTSW
 2 × 10−5 ⇒ δTSW  50 μK, (3.256)
T0
232 3 A Thermal Universe [TRH → TCMB ]

at large scale (2 degrees in the sky, reminding that the moon covers half a degree)
corresponding to the difference of temperature needed to form the actual patchwork
of large scale structures observed nowadays. This level of anisotropy is clearly
visible on the left part of the spectrum in Fig. 1.6.
The second effect is related to the horizon problem we discussed in Sect. 2.2.1.
Two photons which were separated by a distance smaller than the horizon size at
decoupling time could have been in contact and could have shared information, by
mutual scattering. We then expect a larger correlation between points below the
horizon size. We know the radius of our Universe is roughly 50 Glyrs, Eq. (2.29).
The CMB being emitted at a redshift zCMB ∼ 1000, we deduce that the radius of
the Universe at the decoupling time is roughly RCMB = 50 Mlyrs. The horizon
diameter being dH = 2 × ctCMB  760000 lyrs, we deduce that the angle under
which is seen the horizon is

dCMB 7.6 × 105

θCMB =   1.5 × 10−2 radian  1 degree. (3.257)
RCMB 50 × 106

We clearly see in Fig. 1.6 that the behavior above 1 degree is radically different that
the behavior below 1 degree. At small angle we are in presence of photons which
were part of the primordial plasma, in equilibrium with baryons, and probably dark
matter, before the decoupling time. It means that the spectrum we observe is the one
expected from oscillations present in any plasma, with wavelengths depending on
the constitution of the plasma.

3.7.2 To Find the Components of the Universe

The astrophysicists play with a set of parameters to extract the information about the
composition of the Universe. The best fit gives the value of H0 , the present Hubble
rate, and the i ’s we gave in appendix. Let us detail the influence of each of them
one by one.

3.7.2.1 Influence of the Matter, m

Suppose first a flat Universe with a null cosmological constant. It is easy to
understand that the distance traveled by the light, before the decoupling time,
depends on the rate of expansion H0 and on the matter content. We already saw
that the matter already dominated the Universe at decoupling time (Eq. 3.57), we
then can write following Eq. (2.25)
 χCMB  aCMB 3/2
cda 2 c aCMB
dCMB = aCMB × R0 dχ = aCMB 2
=  ,
0 0 a H (a) H0 0
m
(3.258)
3.7 Extracting Information from the CMB Spectrum 233

R0 , H0 , and 0m being the present radius of the Universe, the present Hubble
parameter, and the present matter relic density, whereas aCMB is the scale factor
at decoupling time. In the meantime, RCMB keeping the same value, RCMB =
aCMB × R0 , we obtain
√
dCMB 2 c aCMB 2c
θCMB = =  =  √ . (3.259)
m 1 + zCMB
RCMB R0 H0 0
m R0 H0 0

We understand then that a decrease in the value of 0m would shift all the
spectrum of Fig. 1.6 to the left (larger angle), whereas an increase would shift it
to the right (smaller angle). This can be physically understood by the fact that a
Universe with more matter has a tendency to grow “faster” due to a larger Hubble
rate and then needs a shorter time to reach the same size, meaning a shorter horizon
size (and vice versa). We illustrate our interpretation in Fig. 3.25.

3.7.2.2 Influence of the Curvature, k

The influence of the curvature of space-time on the spectrum of CMB anisotropies
is very similar to the case just discussed. Indeed, a modification of the curvature
(positive or negative) will have as a direct consequence a modification of the
observer’s viewing angle. Indeed, this strongly resembles a deformation effect that
we find in general relativity, reproducing magnifying effects on the trajectory of

Fig. 3.25 Illustration of the expected shift in the spectrum in a flat Universe, without cosmological
constant, for different values of 0m : 0.1 and 1
234 3 A Thermal Universe [TRH → TCMB ]

light rays. For a curvature k = −1; 0; 1, the angle becomes

dCMB
k
θCMB = , (3.260)
Sk RCMB

with Sk defined in Eq. (2.21), in other words

2c 1
k=−1
θCMB  √
R0 H0 0 sinh(1/(1 + z
m CMB )) 1 + zCMB
2c 1
k=0
θCMB =  √
R0 H0 0
m 1 + zCMB
2c 1
k=1
θCMB =  √ .
R0 H0 0 sin(1/(1 + z
m CMB )) 1 + zCMB
(3.261)

Exercise Recover the previous set of equations.

We illustrate our result in Fig. 3.26 where we plotted the expected spectrum for an
open (k = −1), flat (k = 0) and close (k = +1) Universe. In the case of a close
space (k = +1), we expect to see the illusion of a larger horizon, coming from the
convex curvature of the light ray, and vice versa for an open space.

Fig. 3.26 Illustration of the expected shift in the spectrum in a for different curvature of the
Universe: open (k = −1), flat (k = 0) and closed (k = +1)
3.7 Extracting Information from the CMB Spectrum 235

3.7.2.3 Influence of the Cosmological Constant, 

If one adds to the game the presence of a cosmological constant, the distance
traveled by the light to reach its horizon at tCMB becomes a little bit more complex:

aCMB × c aCMB da
dCMB =   . (3.262)
H0 0 0 0
m
a+ 0a 4
m

On the other hand, the radius of the Universe nowadays should also be modified,
compared to the one computed with the approximation of a matter dominated
Universe. We can write for the radius at CMB:

aCMB × c 1 da
RCMB =   (3.263)
H0 0 0 0
m
a+ 
0 a4
m

where we used Eq. (2.64). Between 0 and aCMB , the Universe is never dominated
0

by the cosmological constant, one can then safely neglect the term 0 , which is not
m
the case when computing the radius RCMB . We then obtain

3/2
dCMB 2a
θCMB = =  1 CMB . (3.264)
RCMB  da
0 0
a+  4
0 a
m

0
We illustrate in Fig. 3.27 the influence of 0 on the position of the spectrum. We
m
clearly see that increasing the cosmological constant component decrease the radius
of the Universe and increase the angle under which the horizon at decoupling time
is seen.

3.7.2.4 Influence of the Baryons, b

Finally, the presence of the baryons also affects the spectrum. Combining it with
the determination of 0m , we can deduce the cold dark matter density, 0cdm . The
effect of baryons is slightly more subtle than those described above. The presence
of baryon affects first the horizon size, the same way the curvature does. Indeed,
until now, we considered that the photons were traveling in vacuum. Whereas that
is true from the decoupling time to now, that is not the case before, because they
are in a medium governed by mutual interactions, and one should consider its sound
speed and not its vacuum celerity to compute the horizon size. The information
travels at this epoch at the sound speed cs , not at c. We detail the computation of the
sound speed around Eq. (5.190). If we suppose that the Universe is only composed
236 3 A Thermal Universe [TRH → TCMB ]

Fig. 3.27 Influence of the presence of the cosmological constant on the CMB spectrum for
different values of 

of baryons and photons, we should then use for dCMB

 aCMB cdt
dCMB = aCMB  , (3.265)
0 a 1 + 34 ρρRb

where ρb and ρR are, respectively, the baryon and photon energy density. Indeed,
we suppose that the neutrino has decoupled from the thermal bath at decoupling
time, thus not affecting the sound speed of the photons.
However, this effect is mild, as the proportion of baryons to photons is not
gigantic at this epoch. The more interesting effect induced by the presence of
baryons is on the amplitude of the first peak. Indeed, in a gravitational potential
generated (for instance) by the presence of dark matter (but could also be only
baryonic) , the conservation of energy imposes the pressure to adapt following

δP + (ρ + P ) = 0, (3.266)

with ρ = ρb + ρR , P = Pb + PR = PR and where we neglected the expansion of

the Universe and supposed adiabatic evolution. With δP = δPR = 13 δρR , we can
rewrite Eq. (3.266)

δρR
3 ρb
= −4 (3.267)
ρR (1 + 4 ρR )
References 237

Fig. 3.28 Influence of the baryonic component of the primordial plasma on the CMB spectrum

or

δT 1 δρR 3 ρb
= =  1+ . (3.268)
T 4 ρR 4 ρR

We then see that the relative abundance between baryons and photons will modify
the height of the peaks, as one can see in Fig. 3.28.

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annurev.nucl.012809.104521 [arXiv:1011.1054 [hep-ph]]
3. M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98(3), 030001 (2018). https://doi.org/
10.1103/PhysRevD.98.030001
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238 3 A Thermal Universe [TRH → TCMB ]

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Press, Cambridge, 2009)
 Part II
Modern Times [TCMB → T0 ]
Direct Detection [T0 ]
4

Abstract

Once we understood how dark matter can be produced in the early Universe,
from the inflationary phase to the thermal one, passing through the reheating
process, it is time to question the possibility to detect this dark component
largely present in our galaxy. The most natural way to detect it is through direct
interaction with a nucleus on Earth. Direct Detection (DD) was first proposed
by Goodman and Witten (Phys Rev D 31: 3059, 1985). The idea was to study
the scattering of dark matter particles from the Galactic halo with targets on
earth and extract informations about their masses and couplings to the Standard
Model. Because dark matter is weakly interacting with standard matter, such
experiments require large detectors (from 10 kilos for the first version several
tons for the future projects) and deeply buried underneath the surface of the Earth
to avoid contamination from cosmic rays. This is similar to neutrino detection
experiments in the principle, except that dark matter is supposed to be quite
massive. While neutrino scatterings generate a Cherenkov shower, a WIMP
bumps into a heavy nuclei and experimentalists try to measure the energy of
recoil produced by the interaction. A nice and pedagogical review can be found
in Arcadi et al. (Eur Phys J C 78(3): 203, 2018).

4.1 Generality

The general strategy of direct detection is to measure the recoil energy of a

target nucleus produced by its elastic collision with a dark matter candidate, to
distinguish it from the background, before comparing it with theoretical predictions
(parameterized by the dark matter mass and its coupling to matter). A nice and
pedagogical review can be found in [1]. Interactions with a nucleus act on three

© Springer Nature Switzerland AG 2021 241

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2_4
242 4 Direct Detection [T0 ]

steps: first the interaction with the partons (quarks and gluons) is given by the
Lagrangian and the calculation of Feynman amplitudes. The second step is to deduce
from the parton interaction and the nucleon interaction that depends strongly on the
structure of the nucleons. Results from lattice QCD or chiral perturbation theories
χP T are the fundamental tools to extract the component of the nucleons (and often
both approaches disagree). The third and last step is to pass from the nucleon level
to the nuclear level. At this stage, the distribution function of the nucleon inside
the nucleus is crucial. Different form factors (Woods–Saxon, for instance) enter in
the game. We will first concentrate on each step one by one. The physical quantity
connecting experiments to theoreticians’ community is the elastic nuclear recoil
spectrum dR/dER , where R is the recoil event rate (the number of events per unit
of time and per unit of mass of detector) and ER is the energy of the recoiling
nucleus. The majority of experiments are based on ionization, scintillation, phonon
techniques, or some combinations of these. They have in common the same basic
theoretical interpretation: the differential energy spectrum of such nuclear recoils is
expected to be featureless and smoothly decreasing, with (for the simplest case of a
detector stationary in the galactic frame) the typical form:

dR R0 −ER /E0 k
= e , (4.1)
dER E0 k

where ER is the recoil energy, E0 is the most probable incident kinetic energy of
a dark matter particle of mass mχ , k is a kinematic factor 4mχ MT /(mχ + MT )2
for a target nucleus of mass MT , R is the event rate, and R0 is the total event rate.1
We will reconstruct this expression step by step in the following sections (see 4.31
for the complete form). Since typical galactic dark matter velocities are of order
v  10−3 c, values of mχ in the 10–1000 GeVc−2 range would give typical recoil
energies in the range 5–500 keV (∼ 12 mχ v 2 ). All the experimental efforts lie on
discriminating in Eq. (4.1) the signal from the background composed for instance
of neutrons produced in muons cosmic rays, or to develop methods to distinguish
nuclear recoils from electron recoils that allow to reject gamma and beta decay
backgrounds. More generically, one can deduce the differential rate R per energy
of recoil ER , dR/dER for the interaction with an incoming particle of mass mχ
and velocity vχ with a distribution f (vχ ) on a nucleus of mass MN  Amp per
kilogram of target (mp being the proton mass):

N0 ρ0 dσ
dR = vχ d|q|2 f (vχ )d 3 vχ , (4.2)
A mχ d|q|2

1 See Eq. (4.15) with θ ∗ = π to understand the appearance of the factor k in (4.1).
4.2 Velocity Distribution of Dark Matter: f (v) 243

N0 being the Avogadro number2 (6.02 × 1026 kg−1 ), ρ0 the local mass density of
the dark matter, and q the momentum transferred from the WIMP to the nucleus.
We will compute each of the terms of this equation in the following sections.
In this section we will reconstruct step by step Eq. (4.1) and show its validity
limits. We will also correct its, adding the dependence on the Earth velocity,
escape velocity or form factor of the nucleus. We will then go into the details of
the experimental strategies settled nowadays to detect such events. The unit for
differential events or background rates is conventionally expressed as 1 event keV−1
kg−1 day−1 called ’differential rate unit’ (dru). When integrated over the recoil
energy, one can talk about “total rate unit” (tru) expressed in events kg−1 day−1 .
When integrated between two energies (threshold energy and escape velocity for
instance), we usually refer to “integrated rate unit” (iru) also expressed in events
kg−1 day−1 .

4.2 Velocity Distribution of Dark Matter: f (v)

The differential particle density, the number of particles with velocities on Earth
(hitting the detector) comprised between v and v ± dv, can be written as
n0
dn = f (v, vE )d 3 v, (4.3)
C
with vE = vEart h/H alo the Earth velocity relative to thehalo (see Sect. 5.1.3 for
v
details) and where C is a normalization constant such that 0 esc dn = n0 , vesc being
the local escape velocity; in other words,
 2π  +1  vesc
C= dφ d cos θ f (v, vE )v 2 dv. (4.4)
0 −1 0

n0 is the mean dark matter particle number density (ρ0 /mχ ), whereas v = vχ/Eart h
is the velocity of the dark matter in the reference frame of the experiment, the
velocity that is effectively measured by the collaboration (through the recoil of
the nucleus for instance). As a good approximation, one can represent the annual
modulation of the orbit of the Earth around the Sun by

t − 152.5 days
vE = 232 + 15 cos 2π km s−1 . (4.5)
365.25 days

2 As a reminder, the Avogadro number 6.02 × 1026 kg−1 corresponds by definition to the number
of atoms of Carbon 12 in 12 grams of Carbon, or in other words, the number of mass (nucleons)
per gram of material. N 0
A corresponds then to the number of nucleus of atomic mass A in 1 gram
of substance.
244 4 Direct Detection [T0 ]

Considering that the dark matter statistical distribution of its velocity in the halo
follows a Maxwellian distribution with a most probable velocity3 v0 , and observing
that vχ/H alo = vχ/Eart h + vEart h/H alo = v + vE , one obtains

− 12 mχ vχ/H
2 −vχ/H
2 2
= e−(v+vE )
2 /v 2
f (v, vE ) = e alo / kT ≡e alo /v0 0 . (4.6)

For vesc = ∞ and using Eq. (A.122) we can compute

 vesc =∞ −
2
vχ/H alo
1 v02 2 2 3/2
d cos θ dφe vχ/H alo dvχ/H alo = 1 ⇒ C = (πv0 ) ,
C 0

which gives4

ρ0 e−(v+vE ) /v0
2 2

dn = dφd cos θ v 2 dv. (4.7)

mχ (πv02 )3/2


As we could have expected, the number density of dark matter (n0 = dn) is
independent of the Earth velocity. The (normalized) distribution

v2
e−(v+vE ) /v0
2 2
v 2 f (v, vE ) = 2 3/2
(4.8)
(πv0 )

is maximum for v = v0 . Considering that the Earth is moving with a velocity

vE  v0 around the galactic center, it corresponds to an absolute dark matter
velocity vχ/H alo = v + vE  0: a static halo. We show in Fig. 4.1 some examples
of distributions for different values of v0 .
If one would have taken a particular value for vesc (and not ∞), the result one
would have obtained is

2 3/2 vesc 2 vesc −vesc
2 /v 2
Cvesc =∞ = (πv0 ) erf − 1/2 e 0 , (4.9)
v0 π v0

Exercise Calculate this value of Cvesc =∞ .

with erf the Gauss error function (see A.128 for details). If we take v0 = 230 km s−1
and vesc = 600 km s−1 , we obtain Cvesc =∞ /Cvesc =600  0.9965, which can justify
in the majority of cases the approximation vesc = ∞. For our calculation, we made

3 Notice that v0 is the most probable velocity, whereas the mean velocity is v 2  = 32 v02 .
4 Remark that (v + vE )2 = v 2 + vE
2 + 2vv cos θ.
E
4.2 Velocity Distribution of Dark Matter: f (v) 245

v2 f(v)
0.00035 v0= 180 km/s
0.00030 v0= 220 km/s
v0= 260 km/s
0.00025

0.00020

0.00015

0.00010

0.00005

0.00000
0 100 200 300 400 500 600
v (km/s)
v2
e−v /v0
2 2
Fig. 4.1 Examples of distribution function (πv02 )3/2
for different values of v0

the assumption that the dark matter in the halo has a “sufficiently” Maxwellian
velocity distribution. The Maxwell–Boltzmann distribution ∝ e−mχ v /2kT describes
2

the velocities of particles that move freely up to few collisions. We assumed that the
WIMPs are isothermal and isotropically distributed in phase space. It is important
to notice that this assumption is not exact and there can exist deviations from this
Maxwellian distribution. Noticing that

1
mχ v02 = kT ⇒ v0  220 km/s  0.75 × 10−3 c, (4.10)
2
which shows that dark matter is highly non-relativistic in the Milky Way (and
justifies using the Maxwell–Boltzmann distribution instead of Fermi–Dirac or
Bose–Einstein one).
Usually, estimation of the local density ρ0 is in the range 0.2 GeV cm−3 < ρ0 <
0.4 GeV cm−3 leading to the adoption of ρ0 = 0.3 GeV cm−3 as the central value.
However, the main result of a more recent analysis has given a novel determination
of the local dark matter halo density, which assumes spherical symmetry and either
an Einasto or NFW density profile is found to be around 0.39 GeV cm−3 with a
1σ error bar of about 7%; more precisely, it was found ρ0 = 0.385 ± 0.027 GeV
cm−3 for the Einasto profile and ρ0 = 0.389 ± 0.025 GeV cm−3 for the NFW.
This is in contrast to the standard assumption that ρ0 is about 0.3 GeV cm−3 with
an uncertainty of factors of 2 to 3. In any case, the difference in the event rates
corresponds to a simple rescaling of ρ0 .
246 4 Direct Detection [T0 ]

dσ
4.3 Measuring a Differential Rate: d|q|2

4.3.1 Kinematics

One should compute at the first place the energy transferred to a nucleon of mass
MN , at rest, from a collision with a dark matter particle of mass mχ hitting the
nucleus with a velocity vχ . We first compute in the center of mass frame with
velocity (see Fig. 4.2 for illustration):

mχ v χ + 0 μ
vCM = = vχ (4.11)
mχ + MN MN
μ
⇒ VN = vN − vCM = 0 − vCM = − vχ
MN
μ
Vχ = vχ − vCM = vχ
mχ

with μ = mχχ+MNN is the reduced mass and Vi (V i ) the velocity of the nucleus
m M

before (after) the collision in the center of mass frame. In this frame, Pχ + PN =
0 = P χ + P N , the last equality being the conservation of momentum implying
V χ = − M N 
mχ V N . The kinetic energy conservation gives

1 1 μ 1 1 MN
mχ V2χ + MN V2N = vχ2 = mχ V χ + MN V N = [mχ + MN ]V N
2 2 2
2 2 2 2 2 2mχ
μ2 2 μ2
⇒ (VN )2 = vχ  2
⇒ (vN ) = |V N − vCM |2 = 2vχ2 (1 − cos θ ∗ ) (4.12)
MN2 MN2

Fig. 4.2 Collision between

the nucleus N and the WIMP
V’
N
χ in the center of mass frame
VN T*

VF

V’
F
J
 dσ
4.3 Measuring a Differential Rate: d|q|2
247

with θ ∗ the angle of scattering in the rest frame. From Eq. (4.12), one can deduce
the nucleus recoil energy in the laboratory frame (the measured one),

|pN |2  |2
|MN vN μ2 vχ2
ER = = = (1 − cos θ ∗ ). (4.13)
2MN 2MN MN

The maximal recoil energy that can be generated by the elastic collision is
when the nucleus moves backward after the hit (cos θ ∗ = −1) and is ERmax =
2μ2 vχ2
MN . Another way of understanding the phenomenon is to observe that for a
fixed (measured) energy of recoil ER the minimum velocity able to deposit ER
corresponds to cos θ ∗ = −1, which gives

ER MN
vχmin = . (4.14)
2μ2

Usually, each experiment possesses a threshold energy Et h , i.e., a minimum energy

that it can
 measure. The minimum velocity required to measure an event is thus
vχ = Eth
min MN
2μ2
.
As a summary, we obtained

μ2 vχ2
P N = −μvχ ; EN

= ER = (1 − cos θ ∗ ) ;
MN
μ 2
Et ot = Et ot = v ; |q|2 = 2μ2 vχ2 (1 − cos θ ∗ ), (4.15)
2 χ
√
|q| = 2MN ER being the momentum transferred. One can write Eq. (4.13) as

mχ MN
ER = 2 Eχ (1 − cos θ ∗ ) (4.16)
(mχ + MN )2

m M
with Eχ = 12 mχ vχ2 the initial energy of the collision. The function (m χ+MN )2 is
χ N
maximum (for a given mχ ) for MN = mχ (classical ping-pong ball versus bowling
ball situation). It means that for a 100 GeV WIMP, the energy of recoil will be
maximized if the nucleus is also of the order of  100 GeV. This explains the
strategy of the present experiments for the detection of WIMPs, to use relatively
heavy nuclear targets.
248 4 Direct Detection [T0 ]

4.3.2 Differential Rate

dσ
For a given recoil energy, one needs to compute dE R
. Notice that we are in non-
relativistic conditions, which means that we can apply the Fermi’s golden rule:5

df i = 2π ρ(Ef ) |Mf i |2 (4.17)

with

dn dPN d 3 PN (PN )2 dPN d

ρ(Ef ) = and dn = = ,
dPN dEf (2π)3 (2π)3

where f i is the number of transitions6 per unit of time from initial state |i to final
state |f , Mf i the transition matrix element, and ρ(Ef ) the density of final state,
 + E  . From Eq. (4.15) one deduces d π d|q|2
Ef = EN χ = 2πd(cos θ ∗ ) = μ2 vχ2
, PN =
(P  )2
MN VN = μvχ , Ef = μ2 vχ2 = 2μ N
, and so dEf = μ−1 PN dPN . Combining all
these relations with Eq. (4.17) we obtain

|Mf i |2 d|q|2
dif = . (4.18)
4π vχ

dif being the rate of the interaction equals the cross section times the velocity:

dσ |Mf i |2
dif = vχ × dσ ⇒ 2
= , (4.19)
d|q| 4π vχ2

which is a less conventional (but more practical in our case) form of the Fermi’s
golden rule. A quantum field theory treatment can be found in Eq. (B.154).

4.4 Structure Function of the Nucleus: F (q)

One has to keep in mind, when using Eq. (4.19), that the transition matrix element
should be taken between two nuclei final states. However, a WIMP hits protons and
neutrons of the nucleus (in fact quarks at a more microscopical level as we will
explain later on). For a nucleus of nuclear mass number A of density ρ(r), with a
coupling WIMP-nucleons f and nucleon wave functions ψi,f (r) = e−ipi,f .r , one

5 SeeSect. B.4.1 and Eq. (B.75) for details.

6 Due to the very low velocities in consideration for direct detection processes (vχ  200 km s−1 ),
we will work in classical quantum mechanics framework. We could work in relativistic limits with
the changes |M|2 → |M|2 /2Eχ 2EN and 2πd 3 P /(2π)3 → (2π)4 d 3 Pχ d 3 PN /(2π)6 2Eχ 2EN  .
4.4 Structure Function of the Nucleus: F (q) 249

should then write


Mf i = d 3 rψf∗ (r)V (r)ψi (r) with V (r) = Aρ(r)f

⇒ Mf i = Af d 3 rρ(r)e−iq.r = Af F (q) (4.20)

with q = pf − pi and F (q) the form factor of the nucleus, defined as the Fourier
transform of wave function of the target nucleus, normalized to unity when no
momentum is transferred,
  ∞
4π
F (q) = ρ(r)eiq.r d 3 r = r sin(qr)ρ(r)dr, (4.21)
q 0

where we supposed a spherical distribution for ρ.

Exercise Recover the expression (4.21).

If protons and neutrons have different coupling coefficients, the transition matrix
element has two parts and can be written as

Mf i = Zfp + (A − Z)fn F (q). (4.22)

The form factor represents the effect of the coherence

√ of the interaction with the
nucleon. When the momentum transfer |q| = 2MN ER becomes large enough
such that the wavelength becomes similar to the size of the nucleus rN = (aN A1/3 +
bN ) (aN and bN being parameters experimentally determined), the structure of the
nucleus becomes important. In other words when

q rN dimensionless  (2AER )1/2 (aN A1/3 + bN )

 6.92 × 10−3 A1/2 ER (aN A1/3 + bN )
1/2
(4.23)

with ER in keV, and aN and bN expressed in fermi. To find this form factor, one
needs to know the distribution function ρ(r) of the nucleons in the nucleus. There
exist different options:

• The simplest option is to consider a sum of delta Dirac functions, assuming point-
like nucleons ρ(r) ∝ i δ 3 (r − ri ).
• A more (intermediate) realistic distribution is a thin shell of effective radius
rN : ρ(r) = δ(r−r N)
4πr 2
. In this case, it is equivalent to imagine that the nucleons
are homogeneously dispersed around the nucleus. Implementing the density in
250 4 Direct Detection [T0 ]

Eq. (4.21) we then obtain

sin(qrN )
F (qrN ) = . (4.24)
qrN

• Another commonly used approximation is

F (qrN ) = e−(qrn )
2 /10
, (4.25)

which is the exact

√ form factor for a Gaussian density distribution ρ(r) with mean
square radius (3/5) rN , rN = 1.2A1/3 fm.
• Helm found a more realistic form of the nucleons in the nucleus after studying the
scattering of electrons from different nuclear targets [2]. Helm has approximated
the nucleus as a solid core with a nearly constant density and a surface of
thickness s:

ρ(r) = d 3 r ρ0 (r ) ρ1 (r − r ). (4.26)

ρ0 defines the radius of the target nuclei and is given by a constant inside a
sphere of radius rN  1.2A1/3 fm and is zero outside the sphere. ρ1 defines
of 1 fm by ρ1 (r) = e−|r| /2s . ρ0 and ρ1
2 2
the surface thickness of the
 order
are normalized such that d 3 r ρ0,1 (r ) = 1. After implementing the density
function in Eq. (4.21) one obtains the Woods–Saxon form

3j1 (qrN ) −(qs)2/2

F (q) = e , (4.27)
qrN

where j1 is a spherical Bessel function of the first kind. A least squares fit from
muon scattering data gives

7
rN2 = c2 + π 2 a 2 − 5s 2 , c  1.23A1/3 − 0.60 fm, a  0.52 fm, s  0.9 fm.
3
This result is quite similar to the Fermi distribution:
−1

ρF ermi (r) = ρ0 1 + e(r−c)/a . (4.28)

We show in Fig. 4.3 the evolution of the coherence factor F as a function of

the recoil energy ER = q 2 /2MN for different nuclei used in several direct
detection experiments (Sodium for DAMA, Germanium for CDMS, and Xenon for
XENON1T, LZ, or PANDAX). We notice that the effect of the form factor becomes
stronger for higher energies of recoil, where the condition qrN  1 becomes not
valid anymore. The exchanged wavelength begins to reach the size of the nucleus.
4.5 Computing a Rate 251

F2(ER)
1.0

0.8

0.6 A=31 (Si)

0.4
A= 73 (Ge)

0.2
A=131 (Xe)
0.0
0 100 200 300
ER(keV)
Fig. 4.3 Woods–Saxon form factor as a function of the energy of recoil of the nucleus for several
types of nuclei used in different experiments: Silicium (dotted blue), Germanium (dashed green),
and Xenon (-full red)

We also clearly see the influence of the number of nucleons in the process: the
larger is the mass number A, the larger is the radius of the nucleus and the stronger
becomes the decoherence effect. For a Xenon nucleus, the form factor already
vanishes for an energy of recoil of 50–100 keV.

4.5 Computing a Rate

After defining the differential cross section at zero momentum transfer σ0 and
remembering that F(q=0) = 1, we can rewrite Eq. (4.19) as a function of σ0 :
 2 =4μ2 v 2
qmax χ dσ 2 (μAf )2 dσ F 2 (ER )
σ0 = d|q| = ⇒ = σ0 , (4.29)
0 d|q|2 π d|q|2 4μ2 vχ2

and when we combine Eqs. (4.2), (4.19), and (4.20), we can write

dR N0 MN ρ0 σ0 F 2 (ER )
= f (vχ , vE )d 3 vχ . (4.30)
dER A mχ 2μ2 vχ

To have an idea of the evolution of the rate as a function of the recoil energy, one
can build a toy model with vE = 0, vesc = ∞. In this case, assuming a Maxwellian
252 4 Direct Detection [T0 ]


ER MN
velocity distribution and integrating on the WIMP velocity from vmin = 2μ2
to
∞ using Eq. (A.122), one obtains

1
e−v
2 /v 2
f (v) = 0 (4.31)
(v02 π)3/2
dR N0 MN ρ0 σ0 F 2 (ER ) −ER MN /2μ2 v 2 R0 −ER /E0 k
⇒ = e 0 ≡ e ,
dER A mχ μ2 v0 π 1/2 E0 k

which is the form of Eq. (4.1) that we have reconstructed. If moreover we neglect the
influence of the form factor (F (ER )  1), we can compute an analytical expression
of the total rate after integrating on the recoil energy ER from the threshold energy
to an energy of recoil ER and we have

2 N0
ρ0 σ0 v0 e−Eth MN /2μ v0 − e−ER MN /2μ v0
2 2 2 2
R(ER ) =
π 1/2 Amχ

= R0 e−Eth MN /2μ − e−ER MN /2μ

2v2 2v2
0 0 with (4.32)

5 × 10−6 100 GeV  ρ0  σ0  v0 
R0 
A mχ 0.4 GeVcm−3 10−42 cm2 220 kms−1
events
×
day kg

R0 corresponding to the total number of events expected (integration on ER from

0 to ∞). In fact, the majority of the events appear at low energy of recoil due to the
Boltzmann suppression. Lowering threshold energy is thus a fundamental keypoint
in the experimental process to increase the number of events, especially for light
dark matter candidates. A first glance at this number shows that the rate is very weak
and the detection is thus a difficult task from the experimental front. If we take as
an example a 100 GeV WIMP hitting a 100 kg experience of Xenon (A = 131)
with a cross section of 1 pb (10−36 cm2 ), we expect around 5 events per day. An
important point is that σ0 is the WIMP-nucleus cross section. However, to be able to
compare different experiments, the interesting cross section is the WIMP-nucleon
one (proton or neutron) that we will note σ0n . By rescaling, we can easily show that
σ0 can be written as

μ2 2 n
σ0 = A σ0 , (4.33)
μ2n
4.5 Computing a Rate 253

n
V0 (cm2)
1. 10−41
Nevents = 5
A = 131 (Xe)
8. 10−42

6. 10−42 Eth = 50 keV

4. 10−42 Eth = 20 keV

2. 10−42 Eth = 5 keV

0 200 400 600 800 1000

mF (GeV)

Fig. 4.4 Example of limits obtained from Eq. (4.32) if we suppose 5 events observation on 100 kg
of Xenon for different threshold energies: 5 keV (dotted blue), 20 keV (dashed green), and 50 keV
(full red)

with μn = Mn mχ /(Mn + mχ ), “n” standing for nucleon. A 1 pb cross section with

a nucleus of Xenon corresponds to a 1.7 × 10−6 pb cross section with a nucleon.
The exclusion limits given by the different experiences are based on a number of
events expected but finally not observed. We can plot the (total) iso-events expected
from a collision with a nucleus of mass A, above a threshold energy Et h —taking
ER = ∞ in Eq. (4.32)—in a plane (μχ , σ0n ) values of A and Et h . We show the
result in Fig. 4.4. All the points lying on the line predict 5 events for a 100 kg
Xenon-like experiment. Even if the model we were working on is a “toy” model
(without taking into account the velocity of the Earth or the form factor of the
nucleus), it is interesting to see that the behavior and the order of magnitude we
obtained are quite reasonable. The behavior for large mass of dark matter is mainly
governed by R0 in Eq. (4.32) as the exponential factor is suppressed. The rate thus
evolves proportionally to 1/mχ as we can see in Fig. 4.4. For light dark matter mass
however, the Boltzmann suppression becomes dominant. It is indeed more difficult
to find high velocities able to generate sufficient kinetic energy and compensate the
254 4 Direct Detection [T0 ]

low mass of the dark matter. In this region, the queue of the Boltzmann distribution
and the different effects of the Earth velocities become important.

4.6 Being More Realistic

4.6.1 Taking into Account the Earth Velocity

We now have developed all the necessary tools to compute the differential rate
taking into account the form factor of the nucleus and the kinematics of the Earth
in the galactic plane. If we do not neglect the velocity of the Earth, we can have
a more precise analytical expression of the differential rate dR/dER noticing that
|vχ + vE |2 = vχ2 + vE2 + 2v v cos θ
χ E

 ∞  
dR N0 MN ρ0 σ0 F 2 (ER )
e(v−vE ) /v0 − e−(v+vE ) /v0 dv
2 2 2 2
= √
dER A 2mχ μ2 v0 vE π vχmin
 
N0 MN ρ0 σ0 F 2 (ER ) vχmin + vE vχmin − vE
= erf − erf , (4.34)
A 4mχ μ2 vE v0 v0

where vχmin , given by Eq. (4.14), is the minimum WIMP velocity able to induce
a nuclear recoil energy above the threshold. We show in Fig. 4.5 the differential
rate as a function of the energy of recoil for different dark matter masses (10 and
100 GeV). We notice that a 10 GeV dark matter gives a measurable spectrum at very
low energy of recoil (10 keV), whereas the spectrum is harder for heavier WIMPs.
For light WIMP, the shape of the spectrum is mainly independent of the form factor
F (ER ) because at the energy considered (10 keV) F (E  R )  1. The differential
vχmin +vE vχmin −vE
rate depends mainly on the factor erf v0 − erf v0 of Eq. (4.34),
which converges rapidly to 0 as vmin vE for ER  10 keV. Physically speaking,
it means that the number of WIMPs able to give a 10 keV energy of recoil to the
nucleus is very low because of the low mass of the dark matter: the population of
WIMP with high velocity is Boltzmann suppressed, as we have seen in Eq. (3.28).
We do not see such a strong phenomenon in the case of a 100 GeV WIMP on the
right panel of Fig. 4.5. We also plotted the differential rate without taking into
account the form factor. As expected, the effect of F (ER ) is much stronger for
heavier nuclear targets (due to the decoherence effect q rN  1 on “bigger” targets)
and is even the dominant process in the case of the Xenon nucleus above 20 keV,
whereas the effect can be neglected for Silicium targets.
4.6 Being More Realistic 255

dR/dER (keV−1 s−1 kg−1)

1
A=131 (Xe)
0.1

0.01

0.001
A=31 (Si)
10−4

A=73 (Ge)
10−5
5 10 15 20
ER (keV)
dR/dER (keV−1 s−1 kg−1)
1

0.1
A=131 (Xe)
0.01

A=73 (Ge)
0.001
A=31 (Si)

10−4

10−5
20 40 60 80 100
ER (keV)
Fig. 4.5 Examples of differential fluxes obtained for a 10 GeV WIMP (top) and a 100 GeV WIMP
(bottom) hitting different nuclei: Silicium (blue), Germanium (green), and Xenon (red). The effect
of the form factor is shown in the bottom panel where the differential rate is shown without taking
into account the form factor in dotted lines
256 4 Direct Detection [T0 ]

4.6.2 Annual Modulation of the Signal

Up to now we did not take into account that the Earth is rotating annually around
the Sun. As we can see in the left panel of Fig. 4.6 and using Eq. (4.5), the absolute
velocity of the Earth with respect to the Local Standard of Rest—see Sect. 5.1.3—
varies from 217 km/s to 247 km/s being maximum on the 2nd of June. This variation
induces a modulation of 3% in the signal measurable by experiments like DAMA.
We represented the case of a Germanium target in the right panel of Fig. 4.6.

Galactic Center

December

Sun 220 km/s

Earth June

dR/dER (keV−1 s−1 kg−1)

0.1

0.01

Target :Germanium June

0.001 m = 10 GeV

December
10−4

10−5
2 6 10 14
ER (keV)

Fig. 4.6 Top: Rotation of the Earth around the galactic center. Bottom: Spectrum as a function of
the energy of recoil on a Germanium target for a 10 GeV WIMP in June and December
4.7 Influence of the Structure of the Nucleons 257

4.7 Influence of the Structure of the Nucleons∗∗

Since several years, it is known that the uncertainties generated by the quark
contents of the nucleons can be as important (if not more) than astrophysical
uncertainties. Some authors pointed out this issue and analyzed it to supersymmetric
models, in effective operator approach or even in the scalar extension of the SM, but
rarely taking into account the latest lattice results. Indeed, due to its large Yukawa
coupling, the strange quark and its content in the nucleon are of particular interest
in the elastic scattering of the dark matter on the proton. We learnt in the previous
section that the spin-independent part of the cross section at zero momentum (4.29)
can be written as
μ2  2
σ0 = Zfp + (A − Z)fn . (4.35)
π
The couplings appearing in Eq. (4.35) are the effective couplings of dark matter
to nucleons, whereas we know at the microscopic level the dark matter couplings to
the partons (quarks or gluons). Due to the low energy scale of the interaction in the
elastic scattering (the velocity of the dark matter χ is around 300 km/s  10−3 c),
we can define the effective dark matter–quark couplings as Lqqχχ = fq q̄q χ̄χ,
fq being a dimensionful coefficient proportional to 1/m2h in the case of t-channel
Higgs exchange for instance.7 The mean interaction of the dark matter particle with
a nucleon N of mass mN can then be written as fq N|q̄q χ̄χ|N = fq N|q̄q|Nχ̄ χ.
If one defines N|q̄q|N = m N N
mq fq , the effective quark–nucleon coupling becomes

eff mN
fχN = fq fqN . (4.36)
mq

This effective coupling can be understood as the probability for a particle χ of

hitting a quark q (fq ) multiplied by the probability to find a quark q in the nucleon
N (fqN ). Summing on all the quarks constituting the nucleon, we can write8 the
χ
fN=p,n appearing in Eq. (4.35)

⎛ ⎞
χ χ
Aq 2 Aq
fN = mN ⎝ ⎠
χ
fqN + fHN (4.37)
mq 27 mq
q=u,d,s q=c,b,t

7 This corresponds to the Higgs propagator for a t = 0 exchanged momentum, which corresponds

to the definition of σ0 .
8 The partons content of a nucleon can be derived from computations of anomalies of the trace of
μν
energy–momentum tensor in QCD: MN N|N = N| q=u,d,s mq q̄q − 8π 9
αs Gaμν Ga |N, Gaμν
2
being the gluon tensor [3]. That is from where is extracted the factor 27 in Eq. (4.37).
258 4 Direct Detection [T0 ]

χ
with Aq the scattering amplitude on a single quark q and fqN = (mq /mN ) N|q̄q|N
is the reduced (dimensionless) sigma terms of the nucleon N, and fHN =
1− q=u,d,s fqN is the Heavy quark content of the nucleon (through loop interaction
with the gluon content of the nucleon), see [3] and Sect. 4.9.3 for details. The
superscript χ indicates that the coupling of the dark matter to quarks q can be
dependent on the nature of q.
There are different ways of extracting the reduced dimensionless nucleon sigma
terms fqN ≡ (mq /mN ) N|q̄q|N. They can be derived by phenomenological
estimates of the π −N scattering πN (see [4,5] and references therein for a review):

πN ≡ mN fl = ml N|ūu + d̄d|N (4.38)

with ml = (mu + md )/2. While an early experimental extraction gave πN =

45 ± 8 MeV, a more recent determination obtained πN = 64 ± 7 MeV. A complete
summary of the different values of πN can be found in Table 1 of [5], where they
also give a weighted mean of 21 determinations:

πN = 46.1 ± 2.2 MeV. (4.39)

On the other hand, the study of the breaking of SU (3) within the baryon octet
and the observation of the spectrum lead to derive a constraint on the non-singlet
combination σ̃ = ml N|ūu + d̄d − 2s̄s|N. σ̃ can be extracted phenomenologically
from the octet baryon mass splittings, normalized to corrections that can be
calculated in chiral effective field theory. Several studies lead to a value σ̃ =
36 ± 7 MeV, whereas some others obtained σ̃ = 50 MeV, still in the limit allowed
by chiral perturbation theories [5]. By introducing

N|ūu − s̄s|N
z= = 1.49, (4.40)
N|d̄d − s̄s|N

one deduces
md πN y(z − 1) + 2
fdN =
mN mu + md 1+z
mu πN y(1 − z) + 2z
fuN =
mN mu + md 1+z
ms πN
fsN = y, (4.41)
mN mu + md

where

N|s̄s|N σ̃
y=2 =1− (4.42)
N|ūu + d̄d|N πN
4.7 Influence of the Structure of the Nucleons 259

45 64

1.0
fTd
c,t,b (0.8751)
0.8 fTu

0.6 fTs

fTb
0.4

0.2
s (0.0689)
u,d (0.0280) 0.0
40 50 60 70 80 90 100
6 SN
m
Fig. 4.7 Sigma commutator of the proton (fqP = mpq P |q̄q|P ) with two different phenomeno-
logical measurements of πN = 45±8 MeV and 64±7 MeV. We also showed the mean evaluation
from lattice results (points in the left)

represents the strange fraction in the nucleon. We show in Fig. 4.7 the dependence
of fqN as a function of πN . The two extreme values are obtained with the lower
bound of πN at 1σ (37 MeV) and the higher bounds (71 MeV), which give
for (mu , md , ms , mc , mb , mt , mp ) = (2.76, 5., 94.5, 1250, 4200, 171400, 938.3)
[MeV]:

fumin = 0.016 fumax = 0.030

fdmin = 0.020 fdmax = 0.044
fsmin = 0.013 fsmax = 0.454. (4.43)

A quick look at the figure shows us how important is to know the strange content
of the nucleon because it varies a lot in the phenomenologically viable region of
πN that can affect drastically the direct detection prediction. For the heaviest
mN
quarks, the factor mq=c,b,t reduces considerably their influence on the coupling fN
as we can see in Eq. (4.37).
These limitations on the phenomenological estimation of the strange structure of
the nucleon clearly open the way for lattice QCD to offer significant improvements.
Using the Feynman–Hellman relation fqN = (mq /mN )∂mN /∂mq , different authors
have extracted the light quark and strangeness sigma terms (see [4] for a clear
review). The last results obtained by the authors (labeled “Young” from now on)
provide stringent limits on the strange quark sigma terms. The lattice results for fsN
agree that the size is substantially smaller than that has been previously thought:

Young
fs = 0.033 ± 0.022 (4.44)
260 4 Direct Detection [T0 ]

45 64
SI (pb)

Pheno.

10−8
Young et al.

MILC

10−9
20 50 90
N (MeV)
Fig. 4.8 Example of spin-independent elastic scattering cross section as a function of the pion–
nucleon sigma term πN for a scalar dark matter mχ = 90 GeV. We also represented the central
values of the cross section for the lattice simulations [6] labeled “MILC” and “Young”

This result tends to favor the smaller phenomenological evaluation of πN . In the
following, we will consider the central values of fqN extracted from the Young et al.
analysis and referred it to the “lattice” one: fuN = fdN = flN = 0.050, and fsN =
0.033, and the maximum and minimum values for fqN given by phenomenological
references (Eq. 4.43).
As we can see in Fig. 4.8, these uncertainties have a strong impact on the direct
detection cross section on the nucleon, σSI (SI for Spin Independent) up to one
order of magnitude. We also plotted the values of σSI obtained by the two lattice
groups corresponding to the central values (πN , σSI ) = (26 MeV, 2.84 × 10−9
pb) and (47 MeV; 2.95 × 10−9 pb) [6]. We clearly see that the lattice results are in
πN = 37 MeV) =
much more accordance with the lower bound on πN : σSI min (

1.93 × 10−9 pb, whereas σSI max (

πN = 71 MeV) = 1.05 × 10 −8 pb. We compiled

all the necessary values of fi in the following table (see Sect. 4.9.3 and Eq. (4.86)
for a detailed calculation of fHN and fN = flN + 3 × fHN = 79 flN + 29 ).

fiN Lattice Min Max

fuN 0.050 0.016 0.030
fdN 0.050 0.020 0.044
fsN 0.033 0.012 0.454
fN = flN + 3 × fHN 0.326 0.260 0.629
4.8 Spinorial Effect 261

In the rest of the section, we will always present our results with the evaluation
of fsN given by the maximum and minimum allowed values for πN and the lattice
extraction of Young et al. More recently, a general result, taking a mean over more
than 20 analyses, gave these values for fiN [5]:

Nucleon fuN fdN fsN fGN fcN fbN ftN

Proton 0.018(5) 0.027(7) 0.037(17) 0.917(19) 0.078(2) 0.072(2) 0.069(1)
Neutron 0.013(3) 0.040(10) 0.037(17) 0.910(20) 0.078(2) 0.071(2) 0.068(2)

which is quite in accordance with the values we obtained in Eq. (4.43).

4.8 Spinorial Effect

When the dark matter candidate has a spin, there is possibility to measure spin–
spin interaction between the dark matter and the nucleus.9 Indeed, experimentalists
cannot control the energy spectrum of the incident WIMPs but can control the target
material. The choice of the target nuclei has a major impact on the type of WIMP
scatterings allowed: the interaction can be spin-dependent or spin-independent. The
spin-independent interaction occurs when a WIMP scatters off all the nucleons in a
nucleus, for a sufficiently long exchanged wavelength. On the contrary, for a spin-
dependent interaction, the incident WIMP must have a spin, and the target nucleus
must also have net spin from an unpaired nucleon: the unpaired nucleon and the
WIMP interact. As we already discussed, the spin-independent interaction will be
obviously enhanced by a coherence factor of A2 if qR  1 (Heisenberg uncertainty
principle, in other words when the wavelength of the exchanged energy is larger
than the size of the nucleus—not nucleons of course), which is not the case in spin-
dependent interactions if we suppose that the dark matter interacts only with the
unpaired nucleons (the independent single-particle shell model). In this section, we
will extract the spin-dependent interaction of the dark matter on the nucleus step
by step: from the partons level to the nucleons one and then the nucleus level. At
the partons level, the operator responsible for spin-dependent dark matter–nucleus
(χ − N) interactions is generated by the Lagrangian

LSD = αq (χ̄γ μ γ 5 χ)(q̄γμ γ 5 q), (4.45)

q=u,d,s

9 One of the clearest presentations of the phenomenon is present in the first article proposing the
direct detection principle [7] that is also a very pedagogical work.
262 4 Direct Detection [T0 ]

where αq is the coupling determined by the model under study. We can then deduce
from Sect. B.3.3, especially Eq. (B.22)

n|q̄γμ γ 5 q|n = 2sμ(n) q (n) (4.46)

(n)
n standing for nucleon (p or n), and where sμ is the spin of the nucleon, whereas
the q (n)’s are extracted from polarized deep inelastic scatterings measurements.
They represent the fractional spin of the proton carried by quarks of type q.
Assuming flavor SU(3) symmetry, one can extract two independent combinations
out from three q, denoted by F and D, from the observed semileptonic decay
rates of baryon in the lowest octet. Fits to all the decays fix F and D to

1
F = (u − s) = 0.47 ± 0.04,
2
1
D = (u − 2d + s) = 0.81 ± 0.03. (4.47)
2

A third combination, 19 (4u + d + s) = 0.175 ± 0.018, has been measured

in polarized electron and muon scattering experiments, but this result should be
regarded with caution due to the lack of precision of the experiment. Combining the
three measurements gives us

u(p) = d (n) = 0.78 ± 0.08,

d (p) = u(n) = −0.50 ± 0.08,
s (p) = s (n) = −0.16 ± 0.08. (4.48)

The conventional non-relativistic QCD partons model, however, predicts instead

u = 0.97, d = −0.28, s = 0, though with reasonable modification the model
can be made consistent with the scattering data. Global QCD analysis for the g1
structure function including O(αs3 ) corrections corresponds to the following values
of spin nucleon parameters:

u(p) = d (n) = 0.78 ± 0.02,

d (p) = u(n) = −0.48 ± 0.02,
s (p) = s (n) = −0.15 ± 0.02. (4.49)

The coefficient of the effective dark matter–nucleon interaction can thus be

written as
√ αq
LSD = 2 2GF an (χ̄γ μ γ 5 χ)(n̄sμ(n) n) with an = √ q (n).
q=u,d,s
2GF
(4.50)
4.8 Spinorial Effect 263

After summing on all the nucleons n composing the nucleus N and noticing that,
μ
in the non-relativistic limit, χ̄γ μ γ 5 χ = 2sχ , sχ being the spin of the dark matter,
we obtain
√
M = 4 2GF sχ . N||ap Sp + an Sn ||N (4.51)

with Sp,n = i spi ,ni . After applying the Wigner–Eckart theorem to the reduced
matrix element of the spin N||S||N
√
J (2J + 1)(J + 1)
N; J ||S||N; J  = N; J, MJ |S|N; J, MJ  (4.52)
MJ

and in accordance with the convention where the z component of the angular
momentum J and spin operators S are evaluated in the maximal MJ state, e.g.,
S = N|S|N = J, MJ = J |Sz |J, MJ = J , we finally can write

√ (2J + 1)(J + 1)  
M = 4 2GF sχ . ap N|Sp |N + an N|Sn |N (4.53)
J

from the Fermi’s golden rule and after averaging over the initial spin/momentum

dσSD dσ 2MN |M|2

= 2MN =
dER d|q| 2 πv 2 (2J + 1)
2
16MN G2F J (J + 1) N|Sp |N N|Sn |N
= a p + a n
π v2 J J
16MN G2F J (J + 1) 2 0
σSD
=  = 2MN (4.54)
π v2 4μ2 v 2

with

N|Sp |N N|Sn |N 0 32G2F 2

 = ap + an and σSD = μ J (J + 1)2 . (4.55)
J J π

A typical cross section for weakly interacting particles is 10−4 pb. From
the experimental point of view, the PICASSO (Project in Canada to Search for
Supersymmetric Objects) experiment is a direct dark matter search experiment
that is located at SNOLAB in Canada. It uses bubble detectors with Freon as the
active mass. PICASSO is predominantly sensitive to spin-dependent interactions of
WIMPs with the fluorine atoms in the Freon.
264 4 Direct Detection [T0 ]

4.9 More About the Effective Approach

4.9.1 Validity of the Approach

To compute a differential or absolute rate in Sect. 4.7 we took an effective approach

where we integrated out the massive fields exchanged in the interaction process. This
is only possible if the energy Q transferred from the dark matter to the nucleus in
the interaction is negligible compared the mass scale of the intermediate particle. We
2μ2 vχ2
computed in Eq. (4.13) the maximum recoiled energy transferred, ERmax = MN ,
which gives for the maximum momentum of the intermediate state
 MN mχ
Qmax = 2MN × ERmax = 2μvχ = 2 vχ  mχ vχ  100, MeV
MN + mχ
(4.56)

where we took for the nucleus mass MN = mχ = 100 GeV and vχ = 300 km/s
for a numerical approximation. We thus see that any mediator M at the electroweak
scale (MM  100 GeV like a Z or Higgs boson for instance) is sufficiently heavy
to justify the approximation in the propagator (Q2 − MM 2 )2  M 4 : the effective
M
approximation is valid. We can also notice that the energy of recoil of a 100 GeV
nucleus is then of the order of
1 1  mχ 
ERmax  mχ vχ2  keV, (4.57)
2 2 1 GeV
which means about 50 keV for a 100 GeV dark matter particle (still considering
vχ  300 km/s), justifying the low threshold needed in direct detection experiments
(around the keV scale).

4.9.2 Effective Operators

In this section, we propose to calculate scattering amplitudes for couplings of

different nature, at the effective level. It is then easy to adapt our results in the
framework of microscopic models and then to use the results of Appendix B.4.5.

4.9.2.1 Generalities
It is possible to write the more general form of a Lagrangian for a fermion field in
the effective approach

LF = SF χ̄χ f¯f + VF χ̄γ μ χ f¯γμ f + AF χ̄γ μ γ 5 χ f¯γμ γ 5 f − CF χ̄σ μν χ f¯σμν f,

(4.58)

f being the Standard Model fermions (quarks and leptons), χ the dark matter field,
and σ μν = 12 (γ μ γ ν − γ ν γ μ ) is the antisymmetric current tensor. A lot of operators
4.9 More About the Effective Approach 265

are absent because they disappear in the limit of null momentum transfer. Indeed,
ū(p2 )γ 5 u(p1 ) is proportional to p2 − p1 and so is negligible in direct detection
prospects. Let us demonstrate it with the use of Eq. (B.24):

/p2 5 /p1
ū(p2 )γ 5 u(p1 ) = ū(p2 ) γ u(p1 ) + ū(p2 )γ 5 u(p1 )
2mχ 2mχ
(p2 − p1 )ν
= ū(p2 )γ ν γ 5 u(p1 ) ∝ (p2 − p1)ν γ μ  0. (4.59)
2mχ

With the same argument we can show that ū(p2 )γ μ u(p1 ) is not null only for the
time component (μ = 0), whereas ū(p2 )γ μ γ 5 u(p1 ) is null for the time component.
As a consequence, mixing terms of the form ū(p2 )γ μ γ 5 u(p1 ) × ū(p2 )γμ u(p1 ) are
absent. Let us demonstrate it:
/p2 μ /p1
ū(p2 )γ μ u(p1 ) = ū(p2 ) γ u(p1 ) + ū(p2 )γ μ u(p1 )
2mχ 2mχ
(p2 )ν (p1 )ν
= ū(p2 )γ ν γ μ u(p1 ) + ū(p2 )γ μ γ ν u(p1 )
2mχ 2mχ
(p2 )ν (p1 )ν
= ū(p2 )(−γ μ γ ν + 2ημν )u(p1 ) + ū(p2 )γ μ γ ν u(p1 )
2mχ 2mχ
μ μ
(p1 − p2 )ν p p
= ū(p2 )γ μ γ ν u(p1 ) + 2 ū(p2 )u(p1 )  2 ū(p2 )u(p1 ). (4.60)
2mχ mχ mχ

Only the time component is non-zero in the non-relativistic case (pi Ei ). In

the case of the operator ū(p2 )γ μ γ 5 u(p1 ), one can write

(p1 + p2 )ν (p1 + p2 )μ
ū(p2 )γ μ γ 5 u(p1 ) = [..] = ū(p2 )γ μ γ 5 γ ν u(p1 ) + ū(p2 )γ 5 u(p1 )
2mχ 2mχ
E1 + E2
 ū(p2 )γ μ γ 5 γ 0 u(p1 )  −ū(p2 )γ μ γ 0 γ 5 u(p1 ) (4.61)
2mχ

which vanishes in the zero momentum transfer limit for the time component μ = 0.
This result shows that terms of the form ū(p2 )γ μ γ 5 u(p1 ) × ū(p2 )γ μ u(p1 ) can be
neglected in the computation of direct detection processes.
266 4 Direct Detection [T0 ]

4.9.2.2 Scalar Coefficient: Generalities

As we discussed in the previous section, the scalar coefficient of Eq. (4.58),
SF χ̄χ f¯f , is in fact a sum over partonic interactions

LS = SF χ̄χ f¯f = λq χ̄χ q̄q (4.62)

q
mN
⇒ N|LS |N = λq N|q̄q|Nχ̄χ = λq fq N̄ N χ̄ χ
q q
mq

m σ
with fq = mNq N|q̄q|N ≡ mqN that we already introduced in Eq. (4.37) and
which computation10 is reserved for Sect. 4.9.3. We can then compare Eq. (4.62) to
Eq. (4.58) to express the effective scalar coupling to the nucleon SF = q λq mN
mq fq .
From this effective Lagrangian, we can compute the amplitude square |MN |2 and
the cross scattering cross section of χ on the nucleon N:


|MN |2 = T r (/
pN + mN )(/
pN + mN ) T r (/ pχ + mχ )
pχ + mχ )(/
 2  2
mN mN
× λq fq  64 λq fq m2N m2χ . (4.63)
q
mq q
mq

The cross section is then (using Eq. B.154)

 2
|M|2 1 mN m2N m2χ
SI
σχN = = λq fq .
4 16π(mN + mχ )2 q
mq π(mN + mχ )2
(4.64)

If the particle is a Majorana fermion, one needs to multiply the amplitude by

2 (and thus the cross section by 4) because one needs to add the two symmetric
diagrams by the exchange of ingoing/outgoing dark matter particle, and the factor
1/4 comes from the average over the incoming spin particles. To obtain the
scattering amplitude on a nucleus made of Z protons p and (A − Z) neutron n,

10 The physical interpretation of N|q̄q|N is the creation of a quark–antiquark pair from the
nucleons N. This is a measurement of the “number” or probability of presence of the sea quarks
in the nucleons and does not take into account the valence quarks (contrarily to the vectorial
interaction as we will see later).
4.9 More About the Effective Approach 267

one obtains (considering mp  mn = mN )

 2
|M|2 = 64m2χ m2N λp Z + λn (A − Z) (4.65)

mq
with λN = q λq fqN m
mq and fq =
N N
mN N|q̄q|N.

4.9.2.3 Scalar Coefficient: Application

Let us apply the previous result to the computation of the cross section in models
with Higgs (scalar) portal in the case of a spin 0 or spin 1/2 dark matter. If one
includes the Higgs propagator in an effective Lagrangian at the quark level, we can
write after the electroweak symmetry breaking11

LS = yχ H χ̄χ + yq H q̄q
q
yq yχ yq yχ
S ∼
→ Leff χ̄χ q̄q  − χ̄χ q̄q
q (p2 − m2H ) q m2H
y q y χ mN
S =−
⇒ LN fq χ̄χ N̄ N, (4.66)
q m2H mq

where yi are the Yukawa-like couplings and p = pq − pq the transferred energy
in the process, negligible compared to the Higgs mass. We can consider the Higgs
as a general extra-scalar particle. √
In the case of a Standard Model Higgs, we know
m gm
that yq = vq = 2MWq ( H  = v/ 2). We will compute the elastic scattering cross
section for a scalar and a fermionic dark matter by first calculating the amplitude
squared in the case of a scalar dark matter,
 2
yχ2 m2N  yq
|MS | = (4) ×
2
Tr pN + mN )(/
(/ pN + mN ) fq =
m4H q
mq
 2
8m4N yχ2 yq
(4) × fq (4.67)
m4H q
mq

/N − p
in the limit p 
/N MN . The first factor (4) is a symmetry factor that should
be present if χ̄ = χ (a factor 2 from the amplitude if χ is real). From Eq. (B.154)

11 Be careful, in our notation H is the physical Higgs field after the symmetry breaking, which
√
means H → (H + v)/ 2, compared with the definition of the Yukawa coupling. The yq is the
√
Standard Model Yukawa coupling (defined by L = yqSM ( H√+v ) divided by 2). In other words,
2
√ y SM
the 2 has been absorbed in a new definition of yq : yq = √q .
2
268 4 Direct Detection [T0 ]

we can compute the spin-independent cross section on the nucleon N

 2
SI 1 |MS |2 (4) × m4N yχ2 yq
σχN = = fq (4.68)
2 16π(mN + mχ )2 4π(mN + mχ )2 m4H q
mq

the first 1/2 factor comes from the mean on the nucleon spin.
In a microscopic (gauge invariant) model, the effective coupling yχ H χ̄χ is gen-
erated through the invariant term (before the symmetry breaking) − 14 λhSS H † H χ̄χ,
√
which gives after the symmetry breaking H → (v + H )/ 2 → − 14 vλhSS H χ̄χ =
− M2gW λhSS H χ̄χ after the SU (2) × U (1) breaking. We then obtain using mq = yq v

 2
SI (4) × m4N λ2hSS
σχN = fq (4.69)
64π(mN + mχ )2 m4H q

(4) × m4N λ2hSS

⇒ σχA
SI
= (fp Z + (A − Z)fn )2 ,
64π(mN + mχ )2 m4H

N=p,n = q fq =
where σχASI represents the cross section on the nucleus and f N
mq
q mN N|q̄q|N. You can see another computation in Sect. B.4.5. For a 100 GeV
dark matter and a typical coupling λhSS = 10−2 , we obtain a nucleon cross section
SI
of the order of σχN  2 × 10−19 GeV−2  10−46cm2  10−22 barns = 10−10 pb,
which is in the limits of actual experiments. In the case of a fermionic dark matter
we have

yχ2 MN2
|MF |2 = (4) × pχ + mχ )
pχ + mχ )(/
T r (/
m4H


× T r (/
pN + mN )(/
pN + mN )
 2  2
yq yχ2 MN4 m2χ yq
× fq = (4) × 64 fq (4.70)
q
mq m4H q
mq

(4) |MF |2
⇒ σNχ
SI
= = (4)
4 16π(MN + mχ )2
 2
yχ2 MN4 m2χ yq
× fq
πm4H (MN + mχ )2 q
mq
4.9 More About the Effective Approach 269

the factor (4)× being present if we have a Majorana dark matter. In the case of
gm
a microscopic model, using yq = 2MWq and yχ = − 14 λhff

2MW
g generated by the
λhff
gauge invariant Lagrangian − 4 H † H χ̄χ, with  an effective BSM scale, we
obtain
 2
λ2hff MN4 m2χ
SI
σχN = (4) × fq . (4.71)
16π2 m4H (MN + mχ )2 q

The sum fq is performed on all the heavy and light quarks and is explicitly
computed in Sect. 4.9.3, Eq. (4.86).

4.9.2.4 Vector Coefficient: Generalities

The operator χ̄γ μ χ is odd under the charge conjugation and is not present if the
dark matter is a Majorana particle (see Sect. 4.9.2.6). However, if the dark matter is a
Dirac one, the operator χ̄γ μ χ does not vanish and contributes to the direct detection
process. It is typically the case for the exchange of a vectorial particle (Z or Z  ).
Indeed, if the particle is not a Majorana particle, its charge conjugate is not itself
and the detector detects both signals without distinguishing them. Using the Dirac
equation (B.24) and γ 0 (γ ν )† γ 0 = γ ν , we have seen that that the operators χ̄γ μ χ
or q̄γ μ q are in fact proportional to the momentum, see Eq. (4.60). If the interacting
particles are non-relativistic, only the time component is non-vanishing. We can
then write q̄γ 0 q = q † q, and after the decomposition on creation and annihilation
operators (see the Peskin and Schroeder [8] pages 60–62 for details), we obtain

d 3p 1
q(x) =  (aps us (p)e−ip.x + (bps )† v s (p)eip.x ) (4.72)
(2π)3/2 2Ep s
 
d 3p
⇒ d 3 xq †(x)q(x) = ((aps )† aps + b−p
s s
(b−p )† )
(2π)3 s

d 3p
= ((aps )† aps − (bps )† bps ),
(2π)3 s

and the minus sign in the last equation comes from the anticommutation relation
between two fermionic states. aps [(aps )† ] being the annihilation (creation) operator
of a particle of momentum p and spin s, and bps [(bps )† ] the annihilation (creation)
operator of an antiparticle of momentum p and spin s, see (B.77) for details. For
q q
simplification in the following, we will forget the sum on the spin and note ap (bp )
the annihilation operator of (anti)quark q (q̄) of momentum p. We can then define
a nucleon state as a normalized sum of states composed by valence quarks, valence
quarks plus a pair of quark–antiquark of the sea, valence quarks plus 2 pairs of
quark–antiquark. . . We will take the case of a proton and up-type quarks in the sea.
The generalization for a neutron and a complete sea of quarks is straightforward. If
270 4 Direct Detection [T0 ]

we write12
 
|P  = |up1 up2 dp3  1 + |ūp4 up5  + |ūp4 up5 ūp6 up7  + . . . with P |P  = 1,
(4.73)

where the first state represents the valence-quark states and the states in parenthesis
are the sea quarks,

|up1 up2 dp3  = (apu1 )† (apu2 )† (apd 3 )† |0 and |ūp4 up5  = (bpu4 )† (apu5 )† |0.
(4.74)

Using the anticommutation relation apu (apui )† = −(apui )† apu + (2π)3 δ 3 (p − pi ),

we can easily show (do not forget there is an integration on the momentum on
Eq. (4.72) eliminating the δ function and using anticommutation relations to reorder
p1 , p2 , p3

d 3p u † u u † u † d †
(a ) a (a ) (ap2 ) (ap3 ) |0
(2π)3 p p p1

d 3p u † u † u † u † d †
= (apu1 )† (apu2 )† (apd 3 )† |0 − (a ) (ap1 ) (ap ) (ap2 ) (ap3 ) |0
(2π)3 p
u † u † d †
= |up1 up2 dp3  − (ap2 ) (ap1 ) (ap3 ) |0 − 0 = 2 |up1 up2 dp3  (4.75)

d 3p
⇒ (apu )† apu − (bpu )† bpu |up1 up2 dp3  = 2|up1 up2 dp3 .
(2π)3

With the same method, we can easily show that


d 3p u † u
(a ) a |up up dp |ūp4 up5 
(2π)3 p p 1 2 3
= 2|up1 up2 dp3 |ūp4 up5  + |up1 up2 dp3 |ūp4 up5 

d 3p u † u
(b ) b |up up dp |ūp4 up5 
(2π)3 p p 1 2 3
= |up1 up2 dp3 |ūp4 up5  (4.76)

d 3p
⇒ (apu )† apu − (bpu )† bpu |up1 up2 dp3 |ūp4 up5  = 2|up1 up2 dp3 |ūp4 up5 .
(2π)3

12 We will not include the gluonic contribution in the expression of the proton state as the
creation/annihilation operators in q † q do not create or annihilate gluons.
4.9 More About the Effective Approach 271

Combining the previous results, we can deduce

 
d 3p
(apu )† apu − (bpu )† bpu |P  = 2|P  ⇒ d 3 x P |u† u|P  = 2.
(2π)3

We can then obviously prove, following the same procedure that

  
d 3 x P |d † d|P  = 1, d 3 x Ne|u† u|Ne = 1, d 3 x Ne|d † d|Ne = 2,

|Ne being the neutron state. As a conclusion, we have proved that the operator
N|q † q|N measures the contents of the valence quark q in the nucleon N. We can
then deduce the effective dark matter–nucleon coupling from the dark matter–quark
coupling

χ
LV = fq χ̄γ μ χ q̄γμ q ⇒ N|LV |N = fq N|q̄γ μ q|Nχ̄γμ χ
q q
p p
= fq fqN χ̄γ μ χ N̄ γμ N, with fu = fdn = 2 and fd = fun = 1.
q

The particle χ being a Dirac particle, its anti-matter χ̃, has also interaction with
the nucleus. However, contrarily to the scalar interaction, we can easily show using
¯ μ χ̃ = −χ̄γ μ χ
Eq. (B.48) that the operator χ̃γ

χ̃
LV = fq fqN χ̄ γ μ χ N̄ γμ N. (4.77)
q

We notice that the interaction Lagrangian for the anti-dark matter particle is
the opposite to the one with the dark matter interaction. However, these two
processes are independent without interference between them. When one computes
an interaction cross section, one should add both processes, proportional to the local
density of (anti)dark matter, respectively. This happens typically for asymmetric
dark matter where the density of the anti-dark matter is much less important that the
dark matter one.

4.9.2.5 Vector Coefficient: Application

We can also apply the previous calculation to compute the effective Lagrangian in
the case of the exchange of a massive vectorial particle Vμ

yχ yq
L = yχ Vμ χ̄γ μ χ + yq Vν q̄γ ν q ⇒ Leff = χ̄γ μ χ q̄γμ q. (4.78)
q MV2

We will concentrate on fermionic dark matter candidates; we let the reader make
the computation for a scalar dark matter. The amplitude squared is then (we let the
272 4 Direct Detection [T0 ]

reader check it with the trace formulae in the appendix)

|M|2 = |ū(pχ )γ μ u(pχ ) ū(pN )γμ u(pN )|2 = 64 m2χ MN2 , (4.79)

which is the same amplitude squared than in the case of the exchange of a massive
scalar we computed in Eq. (4.70). We can then use the results already obtained.

Exercise Redo the same calculation for axial-vector-type coefficients (γ μ γ 5 ) and

magnetic (σ μν = 12 [γ μ , γ ν ]) interactions.

4.9.2.6 Majorana Case

In the case of a Majorana dark matter particles, the anti-matter field χ̃ = χ. The
expressions are even simpler as some operators vanish. The operator χ̄γ μ χ for
instance can be written as
1 μ   
¯ μ χ̃ = 1 χ̄ γ μ χ − χ̄γ μ χ = 0,
χ̄γ μ χ = χ̄γ χ + χ̃γ
2 2
where we used the relation Eq. (B.48). We also obtain a null contribution for the σμν
term. The only operators that we need to take into consideration are then

SF χ̄ χ f¯f + AF χ̄γ μ γ 5 χ f¯γμ γ 5 f, (4.80)

which gives, respectively, the spin-independent and spin-dependent amplitudes.

4.9.3 Gluons and Heavy Quarks Contributions∗

We can go into a more detailed study of the quark content of the nucleon. Indeed,
in equation (4.58) the fermion f entering in the interaction should be the nucleon.
However, as we discussed, the effective interaction at a microscopic level is on the
partons (quarks and gluons) and not directly on the nucleon. Concentrating on the
scalar case first, one needs to compute

Leff = λf N; χ|χ̄χ f¯f |χ; N = λf χ|χ̄ χ|χ N|f¯f |N

f f

= λq χ̄χ N|q̄q|N + λQ χ̄χ N|Q̄Q|N, (4.81)

q=u,d,s Q=c,b,t

where we have separated light (q = u, d, s) and heavy (Q = c, b, t) quark states.

The values of N|q̄q|N can be deduced directly by lattice QCD or perturbative
QCD approaches. By these methods, one can usually extract the quantity σq =
m
mq N|q̄q|N (in MeV) or equivalently fq = MNq N|q̄q|N (dimensionless).
However, there is no equivalent method to compute N|Q̄Q|N. In order to obtain
4.9 More About the Effective Approach 273

such a quantity, one needs first to compute the trace of the energy–momentum tensor
θμμ in the nucleon state [3]:

MN N|N
β(αs ) a μν
= N|θμμ |N = N| mq q̄q + mQ Q̄Q + G G |N
4αs μν a
q=u,d,s Q=c,b,t

αs 2
= N| mq q̄q + mQ Q̄Q − 11 − nf =q+Q Gaμν Gμν
a |N
8π 3
q=u,d,s Q=c,b,t
(4.82)

for nf quark family, where Gaμν is the gluonic field tensor and β(αs ) = μ ∂α
∂μ =
s

α2
− 2πs 11 − 23 nf is the β-function of the strong SU (3)c gauge coupling, μ being
μν
the energy scale. The contribution proportional to Gaμν Ga in Eq. (4.82) comes from
the triangle anomalies in θμμ [3]. We will separate the energy momentum into heavy
quark states and light quark states. Indeed, one can make one step further and get rid
of the heavy quarks in Eq. (4.82) because there are no valence heavy quarks in the
nucleon and they can enter only via a virtual state, at short distances of order 2m1Q .
1
It is equivalent to develop a quark propagator P/ −m in the regime where mQ P.
Q
The heavy quark term at the heavy mass limit gives (at the first order) [9]

2 αs
mQ Q̄Q → − a + O(αs ),
nQ Gaμν Gμν 2
(4.83)
3 8π
Q=c,b,t

nQ being the number of heavy quarks. As we can see, the limit obtained in
Eq. (4.83) cancels exactly the anomaly contribution of the heavy quarks in the
energy–momentum tensor in Eq. (4.82) and is independent of the mass of the quark
(at first order). We are then left with only the light quark contribution; in other words
# $
9αs a μν
MN N|N = MN = N| mq q̄q − G G |N . (4.84)
8π μν a
q=u,d,s

Combining Eqs. (4.83) and (4.84), we deduce

2nQ MN 2nQ
N| mQ Q̄Q|N = − N| mq q̄q|N
27 27
Q=c,b,t q=u,d,s

mQ 2
⇒ fQ = N|Q̄Q|N = 1− fq , (4.85)
MN 27
q=u,d,s
274 4 Direct Detection [T0 ]

which is the same contribution for all heavy quarks as their mass never appears in the
limit mQ → ∞. We can also understand it as their contributions are generated by
triangle anomalies that are mass-independent (topological) quantities. The useful
quantity in the computation of the direct detection cross section is fq (see
Eq. 4.70 for instance). We then have

2 7 2
fN = fq + fQ = fq + 3 × 1− fq = fq + .
q
27 q
9 q
9
q=u;d;s Q=c,b,t
(4.86)

References
1. G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo,
F.S. Queiroz, Eur. Phys. J. C 78(3), 203 (2018). https://doi.org/10.1140/epjc/s10052-018-5662-
y [arXiv:1703.07364 [hep-ph]]
2. R.H. Helm, Phys. Rev. 104, 1466–1475 (1956). https://doi.org/10.1103/PhysRev.104.1466
3. M.A. Shifman, A. Vainshtein, V.I. Zakharov, Phys. Lett. B 78, 443–446 (1978). https://doi.org/
10.1016/0370-2693(78)90481-1
4. R.D. Young, A.W. Thomas, Nucl. Phys. A 844, 266C–271C (2010)
5. J. Ellis, N. Nagata, K.A. Olive, Eur. Phys. J. C 78(7), 569 (2018). https://doi.org/10.1140/epjc/
s10052-018-6047-y [arXiv:1805.09795 [hep-ph]]
6. R.D. Young, A.W. Thomas, Phys. Rev. D 81, 014503 (2010). https://doi.org/10.1103/PhysRevD.
81.014503 [arXiv:0901.3310 [hep-lat]]
7. M.W. Goodman, E. Witten, Phys. Rev. D 31, 3059 (1985). https://doi.org/10.1103/PhysRevD.
31.3059
8. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Westview Press,
Boulder, 1995)
9. A.I. Vainshtein, V.I. Zakharov, M.A. Shifman, JETP Lett. 22, 55–56 (1975)
In the Galaxies [T0 ]
5

Abstract

As we have seen, the direct detection experiments, which are based in large
majority on the neutrino detectors principles, have not yet given any hints of
dark matter presence in the neighborhood of the Earth. Another possibility is to
look for indirect effects of dark matter particles in our galaxy, or dwarf galaxies
surrounding the Milky Way. The main idea underlying this kind of research is the
possibility for dark matter to annihilate (or decay) in the interstellar medium, or
generally near sources of gravity like the dynamical galactic centers, where the
population of dark matter is the largest due to the gravitational well. The products
of annihilation can be charged (like an electron-positron pair) or neutral (neutrino
or photon final states). In this case, satellites or telescopes look directly for the
product of annihilation. Another possibility is to observe the radiation emitted by
these products, like synchrotron or bremsstrahlung effects for instance.

The indirect effects of the presence of dark matter in the sky range from keV (or X-
ray) spectrum or lines when it concerns radiative processes, to GeV scale for WIMP
annihilation. It can even reach PeV scale (106 GeV) or even EeV scale (109 GeV) for
superheavy dark matter candidates that can be produced in the very early stage of the
Universe as we saw in first part of the book. These kinds of signals are observable
by the Icecube collaboration or the ANITA balloon in the South Pole, which track
ultra-high energy neutrinos. But the final products at such energies are limited by
their interaction with the CMB or their loss of energies during their propagation in
the interstellar medium. For instance, we will see that a ∼ GeV positron loses 90%
of its energy after a little bit more than a 3 light-years journey in the Milky Way,
whereas a 100 TeV photon is stopped by the CMB, through the production onshell
of e+ e− pairs.

© Springer Nature Switzerland AG 2021 275

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2_5
276 5 In the Galaxies [T0 ]

Exercise Recover the Greisen–Zatsepin–Kuzmin limit (GZK cut) of 5 × 1010 GeV

corresponding to a proton hitting a photon from the CMB producing a + onshell.
With the same kind of reasoning, compute the mean free path of a photon in the
CMB medium (production of e+ e− pair onshell) and of a neutrino in the CMB
(production of a e− W + pair onshell) as a function of their energies.

In order to observe indirectly particles in the Universe, one needs to understand

how such particle populates the Universe, how they are produced, and how they
propagate. The density of dark matter in the Milky Way, especially its spatial
distribution function, is one of the keypoints of its understanding. We will describe
in the first sections of this chapter the different elements one should take into
consideration when studying indirect detection of dark matter, especially the
astrophysical knowledge (and uncertainties) needed to lead such analysis. We will
then enter in more detail into the possible detection modes and strategy set up by
experimentalists and theoreticians to actually observe such particles indirectly.

5.1 The Anatomy of the Milky Way

5.1.1 Internal Characteristics

The unit of distance commonly used in astronomy is the parsec. The parsec (parallax
of one arcsecond, symbol: pc) is a unit of length, equal to just under 31 trillion
kilometers (30.857 × 1015 meters), or about 3.26 light-years.1 It is defined as
the length of the adjacent side of an imaginary right triangle in space. The two
dimensions that specify this triangle are the parallax angle (defined as 1 arcsecond)
and the opposite side (which is defined as 1 astronomical unit (AU), the distance
from the Earth to the Sun). Given these two measurements, along with the rules of
trigonometry, the length of the adjacent side (the parsec) can be found.

Exercise Recover the value of 1 parsec with the geometrical definition given above.

The stellar disk of the Milky Way galaxy is approximately 100,000 light-years
(30 kpc, 9.5 × 1017 km) in diameter and is considered to be, on average, about
1000 light-years (0.3 kpc, 9.5 × 1015 km) thick, see Fig. 5.1. It is estimated to
contain at least 200 billion stars and possibly up to 400 billion stars, the exact
figure depending on the number of very low-mass stars, which is highly uncertain.

1A light-year or light year (symbol: ly) is a unit of length, equal to just under 10 trillion (i.e. 1013 )
kilometers. As defined by the International Astronomical Union (IAU), a light-year is the distance
that light travels in a vacuum in one Julian year.
5.1 The Anatomy of the Milky Way 277

30 kpc, 100 000 ly

650 km/s

8.5 kpc = 27700 ly

1kpc = 3262 ly
SUN GALACTIC CENTER
220 km/s

Fig. 5.1 Anatomy of the Milky Way

The center of the Milky Way is formed by a population of old star, in a spherical
distribution, called the bulge. The Sun (and therefore the Earth and the solar system)
lies close to the inner rim of the Milky Way galaxy’s Orion Arm at a distance of 7.5–
8.5 kpc (25,000–28,000 light-years) from the galactic center. The Earth is turning
around the Sun at about 30 km/s, whereas the orbital speed of the solar system
around the center of the galaxy (hosting its supermassive black hole Sagittarius
A∗ ) is approximately 251 km/s. In the meantime, the Milky Way converges toward
its twin galaxy Andromeda, situated at about 2.5 million light-years from us, at a
speed of 650 km/s. A striking point when dealing with these numbers is the relative
homogeneity of the astrophysical velocities. Indeed, a 300 km/s proton possesses
2
a keV kinetic energy (∼ 12 mp × vc2 GeV). This explains why typical energies due
to interactions in the interstellar medium turn around this scale. As we also learnt
when we treated the case of the direct detection experiments, it is also the typical
energy of recoil we expect, due to the motion of our solar system in the Galactic
halo.

5.1.2 The Color of the Sky: The Diffuse Gamma Ray Background

When we look up on a clear night, we see the beautiful twinkling of stars and planets
amid a black sky. If we could see the same sight with X-ray or gamma-ray eyes, we
would still see bright points, but the sky would no longer be dark. Instead, it would
glow faintly. This is the diffuse high energy background: X-ray and gamma-ray light
from all over the sky. By looking at different wavelengths of X-rays and gamma rays
to different energies, we can find out what causes this background glow.

5.1.2.1 X-Ray Diffuse Background

At low X-ray energies ( 14 keV), the sky glows with radiation from hot gas filling
some of the space between the stars. This gas has a temperature of about 1 million
degrees and is heated in two ways: by supernovae, which leave shining remnants of
hot gas behind, and by the hot winds of massive young stars, which heat surrounding
278 5 In the Galaxies [T0 ]

gas forms stellar wind bubbles. At higher X-ray energies ( 12 keV), the source of
the diffuse background changes considerably. While the emission from supernova
remnants and stellar wind bubbles is still visible, it is less dominant than at lower
energies. Much of the background radiation becomes isotropic (i.e. it looks the same
in all directions). Scientists believe the radiation comes from outside our Milky Way
galaxy, since radiation from within the galaxy would be brighter in some places and
dimmer in others, due to our galaxy’s shape.
Above 1 keV, most of the “diffuse” background is not truly diffuse in origin at
all but comes from many distant extragalactic objects. We know this from “deep”
observations of the diffuse X-ray background. In astronomy, a “deep” observation
means that a detector points to a given point in space for a very long time. Using
the deepest ROSAT observations, over 60% of the 1–2 keV diffuse background has
been resolved into very distant, separate sources, typically quasars.

5.1.2.2 Gamma-Ray Diffuse Background

While there are many individual point sources of gamma rays, there is also a
significant amount of gamma-ray light from gas in the Milky Way. Gamma-ray light
from this gas stretches out in a band across the sky, comprising much of the diffuse
gamma-ray background. The remaining gamma-ray background light we see comes
from far outside our galaxy: it is the faint glow of the rest of the Universe, covering
all the sky beyond the Milky Way.
Yet another source of diffuse gamma-ray radiation in our galaxy is the violent
interaction of matter and anti-matter. Certain galactic phenomena produce positrons.
When positrons and electrons come into contact, they annihilate and release a
burst of gamma-ray energy of exactly 511 keV, corresponding to the rest mass of
the electron (and positron). These gamma rays are part of the diffuse gamma-ray
background in our galaxy.
The diffuse gamma-ray emission from outside our galaxy may be due to the
combined light of far-away individual objects. The objects are most likely active
galactic nuclei (AGN) like blazars, starburst galaxies, and millisecond pulsars.
These are very similar to quasars, the main source of the X-ray background outside
our galaxy. The diffuse gamma-ray background was forecast to be one of the
more robust constraints of annihilating WIMP dark matter, and determining its
spectrum/amplitude is a challenge to restrict the dark matter parameter space.

5.1.3 Galactic Coordinates, Velocity of the Sun and of the Earth

To observe a position in the sky, or when satellites like COBE, WMAP, FERMI, or
PLANCK give a map of the sky, they usually work in galactic coordinates: galactic
longitude (l) and galactic latitude (b). The longitude measures the angular distance
of an object from the Sun eastward along the galactic equator from the galactic
center, whereas the latitude measures the angular distance of an object perpendicular
5.1 The Anatomy of the Milky Way 279

to the galactic equator, positive to the north, negative to the south. This is well
illustrated in Fig. 5.2.
The Milky Way is at his densest near Sagittarius as it is where the galactic center
lies. The center of Sagittarius, a black hole called Sagittarius A∗ , is often considered

as the real galactic center. Its coordinates are in fact:2 (l = 3590 56 39.5", b =

−00 2 46.3"), which is in fact a little bit on the south-east of the galactic center. We
list in the following Table 5.1 some coordinates of most common objects, especially
dwarf galaxies surrounding the Milky Way and important source of observation for
dark matter candidates due to their few amount of baryonic matter.

Exercise Find in this list, the nearest dwarf galaxy from the galactic center, and the
farthest one.

The velocity of the Sun is defined from the Local Standard of Rest (LSR) and is
usually designated by U , V , and W , given in km/s, with U positive in the direction
of the galactic center, V positive in the direction of galactic rotation, and W positive
in the direction of the North Galactic Pole. If one defines a point in space that is
moving on a perfectly circular orbit around the center of the galaxy at the Sun’s
galactocentric distance, all velocities of stars are described relative to this point,
which is known as the Local Standard of Rest. The velocity of the Local Standard of
Rest is then given by (ULSR = 0; VLSR = 0; WLSR = 0). The peculiar motion of the
Sun with respect to the LSR is then measured and gives (U , V , W ) = (10.00 ± 0.36,
5.23±0.62, 7.17±0.38) km/s. We show an illustration of this system of coordinates
in Fig. 5.3. The Sun is thus deriving slowly toward the galactic center, moving a little
bit faster than the entire disk, with some components toward the north pole of the
Milky Way.
We can then deduce the absolute velocity of the Earth in the Milky Way (and as
a consequence, the velocity of the Earth relative to the dark halo):

vE = vr + vS + ve (5.1)

with, in galactic coordinates:

• the velocity of the rotation of the galaxy at the solar neighborhood

vr = (0, 230, 0) km s−1 ;
• the Sun’s proper motion (motion relative to nearby stars)
vS = (10, 5, 7) km s−1 ;
• the Earth’s orbital
 velocity relative
 to the Sun.
−152.5 days −1 .
ve = 15 cos 2π t 365.25 days km s


2 10 = 60 (arcminutes) = 3600" (arcseconds).
280 5 In the Galaxies [T0 ]

Fig. 5.2 The galactic coordinates use the Sun as the origin. Galactic longitude (l) is measured with
primary direction from the Sun to the center of the galaxy in the galactic plane, while the galactic
latitude (b) measures the angle of the object above the galactic plane (Brews ohare licensed under
the Creative Commons Attribution-Share Alike 3.0 Unported license; NASA/JPL-Caltech/ESO/R.
Hurt)
5.2 Computation of a Flux 281

Table 5.1 Galactic coordinates of some dwarf spheroidal galaxies used to constrain dark matter
interaction
Object Longitude (l) Latitude (b) Distance (from the Sun/kpc)
Carina 260.1 −22.2 103 ± 4
Draco 86.4 34.7 84 ± 8
Fornax 237.1 −65.7 138 ± 9
LeoI 226.0 49.1 247 ± 19
LeoII 220.2 67.2 216 ± 9
Sculptor 287.5 −83.2 87 ± 5
Sextans 243.5 42.3 88 ± 4
Ursa Minor 105.0 44.8 74 ± 12

Galaxy rotation
W

GALACTIC CENTER
SUN U

Fig. 5.3 (U , V , W ) system of coordinates to define the peculiar motion of the Sun and stars
relatively to the Local Standard of Rest (LSR). Sometimes U is defined pointing out from the
galactic center (given negative values for the Sun’s velocity U -component)

5.2 Computation of a Flux

To describe the distribution of dark matter in the Milky Way, observations of the sky
are clearly not sufficient. Not only because we still did not observe any dark matter
signal, but also because N-body simulations are much more efficient for that. In two
words, the principle of a simulation is to put all the known ingredients (and there are
a lot: physical laws, proportion between dark and baryonic matter, initial velocities
or orbital momentum, self-interaction of the dark matter or not, self-interaction of
baryonic matter, feedback reactions of the gravitational well formed by the dark
matter halo, entropy injection due to dark matter motions, etc.); let run a code
during days, months, years. . . , look at the results, extract any kind of simulated
galaxies that are similar to the Milky Way in terms of mass, components, size,
orbital momentum, and look at the profile of dark matter obtained in these selected
galaxies. Then, the physicists make a mean of all these distributions and try to fit
them with a usable universal function.
282 5 In the Galaxies [T0 ]

Table 5.2 NFW and Moore et al. density profiles without and with adiabatic compression (NFWc
and Moorec , respectively) with the corresponding parameters, and values of J¯( ) (mean of J
over the solid angle, see Sect. 5.8 for details)
a (kpc) α β γ J¯(10−3 sr) J¯(10−5 sr)
NFW 20 1 3 1 1.214 × 103 1.264 × 104
NFWc 20 0.8 2.7 1.45 1.755 × 105 1.205 × 107
Moore et al. 28 1.5 3 1.5 1.603 × 105 5.531 × 106
Moorec 28 0.8 2.7 1.65 1.242 × 107 5.262 × 108

Considering all the ingredients that appear in the recipe, one can easily under-
stand why different simulation teams will give different distribution functions. All
depend on which kinds of ingredients were thrown in the “soup.” As a simple
example, in the earliest times of simulations, the analysis showed a tendency for the
dark matter to concentrate a lot near the galactic center, which was in tension with
certain observations of Sun motions around Sagittarius. Nowadays, when taking
into account the injection of entropy by the baryons populating the galactic center,
the profiles are much less steeper, and the dark matter concentration much more
in accordance with the observations. We will see another example in Sect. 5.8.3
concerning the adiabatic compression effect.
Various profiles have been proposed in the literature and can be parameterized as

ρ0 [1 + (R0 /a)α ](β−γ )/α

ρ(r) = , (5.2)
(r/R0 )γ [1 + (r/a)α ](β−γ )/α

where ρ0 is the local (solar neighborhood) measured halo density ( 0.3 GeV/cm3 ),
R0 the Sun to galactic center distance ( 8.5 kpc), and a is a characteristic length
given by the resulting simulation, as the powers α, β, and γ . Some results of
simulations are given in Table 5.2. The name of the profile corresponds usually to
the name of the author of the simulations (or their initials, NFW for Navarro, Frenk,
and White).
The calculation of the flux produced by the annihilation of dark matter at a
distance r from the galactic center along the line of sight forming an angle ψ with
the (Sun)-(galactic-center) axe (see Fig. 5.4) necessitates to know the integral3
 2
1 1
J (ψ) = ρ 2 (r(l, ψ))dl (5.3)
8.5kpc 0.3 GeV cm−3 line of sight

in a solid angle .

3 In the case of a dark matter decay, one needs to replace ρ 2 by ρ in the integral.
5.2 Computation of a Flux 283

Fig. 5.4 Topology for the

measurement of a flux from
the galactic halo. In this
example, the experience is
pointed at an angle  from
the galactic center, inside a
cone defined by the solid
angle  = 0 along with is
r
the line of sight l

R0
0


One can write r = l 2 + R02 − 2R0 l cos ψ re-expressing Eq. (5.2) as
ρ(l, cos ψ), and for a solid angle one has d = dφ sin ψdψ ⇒ cos ψ0 =
1 − /2π, which gives
  cos ψ  R0 −10−5 kpc

2
ρ (r)dl = d cos ψ dlρ 2 (l, cos ψ  )
line of sight cos(ψ+ψ0 ) 0
 cos ψ  100 kpc
+ d cos ψ  dlρ 2 (l, cos ψ  ), (5.4)
cos(ψ+ψ0 ) R0 +10−5 kpc

10−5 kpc being the limit radius of the integral, where one cannot say anything about
the pure galactic center where lies a possible black hole (for ψ = 0), and the upper
bound of the integral, 100 kpc, corresponds to the end of the halo core. Here you
can find an example of mathematica code in the case of the galactic center in the
NFW model within a solid angle of 10−3 .
rho0 = 0.3;
alpha = 1;
beta = 3;
gamma = 1;
a = 20.;
284 5 In the Galaxies [T0 ]

R0 = 8.;
r = Sqrt[l^2 + R0^2 - 2 R0 l cpsi];
fr = rho0 *(1 + (R0/a)^alpha)^((beta - gamma)/alpha)/((r/R0)^
gamma (1 + (r/a)^alpha)^((beta - gamma)/alpha));
fr /. r -> Sqrt[l^2 + R0^2 - 2 R0 l cpsi];
fl = %
rhol[l_, cpsi_] = fl
dOmega = 0.001;
cpsi0 = 1 - dOmega/(2 Pi);
ftointegrate = fl^2/(1 - cpsi0)/8.5*(1/0.3)^2;
Jave = NIntegrate[
ftointegrate, {cpsi, cpsi0, 1.}, {l, 0, R0 - 0.00001}] +
NIntegrate[
ftointegrate, {cpsi, cpsi0, 1.}, {l, R0 + 0.00001, 100}] +
0.00002*rhol[0.00001, 0]

Out = 1223.31

The third contribution in the integral is just the center part supposing that the density
is constant in the galactic center region. This hypothesis is not really important as
the 2 × 10−5 kpc region is very small. We give some values of the flux parameter J
for different profiles and solid angle in Table 5.2.

5.3 Example of the Isothermal Profile

It is legitimate to ask how such a form of equation as Eq. (5.2) can appear from
simulations or arise from N-body simulations based on primeval hypothesis. To
illustrate it, we will take a concrete example, which was one of the more popular
profiles in the 80s–90s before more complex simulations leading to NFW types
emerged. This profile is named isothermal profile,

A
ρiso (r) = , (5.5)
(r 2 + rc2 )

where rc is a core radius and A a constant, both determined by the observations

and/or simulations. Considering a cold dark matter profile as a set of collisionless
particles, we cannot treat them as we did when we studied the thermal bath, but we
can consider that statistically they follow a Boltzmann velocity distribution. We can
then write for a set of particles in a potential (r)

ρ(r) = d 3 vf (v, r) (5.6)

with f (v, r) = Ce− kT , C a constant, and

E

1 2
E= mv + m(r). (5.7)
2
5.3 Example of the Isothermal Profile 285

Supposing that the potential does not depend on v, we deduce


m(r) kT ρ
ρ = ρ0 e− kT ⇒ (r) = − ln . (5.8)
m ρ0

From the Poisson’s equation (see the box below),

1 ∂ 2 ∂
 = r  = 4πGρ, (5.9)
r 2 ∂r ∂r
we deduce, using Eq. (A.137),

∂ 2 ∂ m
r ln ρ = − 4πGr 2 ρ (5.10)
∂r ∂r kT
or
kT
ρ(r) = ρiso (r) = . (5.11)
2πGmr 2

Exercise Supposing ρ = A
r2
, recover the result above showing that A = kT
2πGm .

The density ρiso being singular at r = 0, we cure the pathology by introducing a

cutoff at a radius rc , which value is chosen to fit with the observations:

kT
ρiso (r) = . (5.12)
2πGm(r 2 + rc2 )

This profile is thus constant for r rc and follows ρ ∝ 1/r 2 for r rc , recovering
the property of a constant velocity behavior far away from the core of the structure
that we discussed after Eq. (1.3). Notice that the profile is of the form (5.2) with
(α, β, γ ) = (2, 2, 0).
The name isothermal came back from a series of lectures Lord Kelvin gave at
Baltimore in 18844 where he observed that if you consider an ideal gas in a sphere
of density ρ, the force exerted on a surface dS of the external surface can be written
as
GM × ρ × dS × dr
dF = = dP × dS, (5.13)
r2

4 Also published in 1907 by Robert Emden (brother-in-law of Karl Schwarzschild) in a book called
«Gaskugeln».
286 5 In the Galaxies [T0 ]

or
dP GM × ρ
= (5.14)
dr r2
P being the corresponding pressure. Considering a gas at a constant temperature T
(isothermal) we also know that
ρ
P V = nRT = NkT ⇒ P = kT , (5.15)
m
where N is the total number of particles in the sphere. Combining Eqs. (5.14)
and (5.15), with M = 43 πr 3 ρ, we obtain

dρ 4π mG
2
= rdr, (5.16)
ρ 3 kT

which gives

3kT 1
ρ(r) = ∝ 2, (5.17)
2πGmr 2 r

hence the name given to such profiles where ρ ∝ 1/r 2 .

The Poisson’s Equation

The Poisson equation is a consequence of the Gauss’s law for gravity that
stated that the gravitational flux through any closed surface is proportional to
the enclosed mass. The proof is straightforward, once we consider the Newton
law of gravity for an acceleration g:

−GM r
g(r) = , (5.18)
r2 r
which gives in a continuous distribution of mass

r − ri 3
g(r) = −G ρ(ri ) d ri . (5.19)
|r − ri |3

Using

r
∇. = 4πδ(r), (5.20)
|r|3

(continued)
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 287

we obtain

∇.g = −4πGρ(r), (5.21)

which is the differential formulation of the Gauss’s law for gravity. The
gravitational field being conservative can be expressed as a function of a scalar
potential :

g = −∇, (5.22)

which gives, combined with the Gauss’s law,

 = 4πGρ, (5.23)

which is the Poisson’s equation.

5.4 Radiative Processes in Astrophysics Part I: The

Non-Relativistic Case

To compute dark matter effects in the galaxies, and more precisely the effects
generated by their products of annihilation or decay, one first needs to understand
the properties of charged particles (product of the annihilation) propagating in a
charged medium (the interstellar gas for instance). We present in this section the
main processes involved in the loss of energy undergone by a charged particle or
radiation from its interactions with a medium. The typical example is of course
the interaction of a cosmic ray—charged particles—on the CMB or the radiation
emitted by the crossing of such particle in the galactic magnetic field. We will
first describe such phenomena in the non-relativistic approximation (v c) and
then apply the formulae in the relativistic limit. We refer the reader to the very
complete Jackson [1] or Longair [2] books for more details. For nucleon or electron
propagation in the neutral matter of the interstellar medium (90% H and 10% He)
the relevant energy losses are due to electromagnetic and nuclear effects depending
on the specific type of interaction. The most relevant processes of the first group
are ionization and Coulomb scattering, while for the second the relevant ones are
spallation, fragmentation, and radioactive decay. For electromagnetic processes that
involve electrons, even bremsstrahlung in the neutral and ionized medium, as well as
Compton and synchrotron losses, becomes important. Although all these processes
are well known and are often explained during academic courses, the formulae for
the different cases are rather scattered throughout the literature. We try to present in
this section the main results concerning the energy losses in the interstellar medium
and give formulae that can be used in any study concerning the subject.
288 5 In the Galaxies [T0 ]

5.4.1 Maxwell Equations

The Maxwell equations are the first ingredients we need to understand the principle
underlying the propagation and interaction of charged particles. They were discov-
ered by Maxwell between 1861 and 1862 and consist of a set of 4 linear equations
describing the space-time evolution of the electromagnetic fields E and B:
ρ
∇.E = [Maxwell − Gauss] ; ∇.B = 0 [Maxwell − Thomson] (5.24)
0
∂B
∇×E =− [Maxwell − Faraday];
∂t
∂E
∇ × B = μ0 j + μ0 0 [Maxwell − Ampere]
∂t

with c2 = 1/μ0 0 , ρ the density of charges, and j = ei vi , the density of current.

Combining cleverly the equations, one can write
 
∂ 0 2 1 E∧B
j.E + |E| + |B|2 = −∇. = −∇. (5.25)
∂t 2 2μ0 μ0

with = (E×B)/μ0 is the Poynting vector whose module | | gives the power flux
(energy emitted per second per m2 ). j.E = ei vi .E is the density of mechanical
energy (energy emitted by a moving charged particle in an electric field E), whereas
0
2 |E| + 2μ0 |B| is the density of energy of the radiated electric and magnetic field.
2 1 2

These equations are written in the SI system of units (International System).

Sometimes one can find expressions of the fields in another popular system of units,
the Gaussian units, that is part of the CGS (Centimeter–Gram–Second) system.
When using CGS units, it is conventional to use a slightly different definition of
electric field ECGS = c−1 ESI . This implies that the modified electric and magnetic
fields have the same units (in the SI convention this is not the case). Then it
uses a unit of charge defined in such a way that the permittivity of the vacuum
0 = 1/(4πc), hence μ0 = 4π/c. Using these different conventions, the Maxwell
equations become

∇.E = 4πρ ; ∇.B = 0 (5.26)


1 ∂B 1 ∂E
∇×E=− ; ∇×B = 4πj + .
c ∂t c ∂t

5.4.2 Loss of Energy of a Moving Charged Particle

When a charged particle travels in a medium, it can lose energy by transferring part
of its kinetic energy to a nearby atom, causing ionization or excitation (Coulomb and
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 289

ionization losses). Another possibility for the particle is to be excited by an external

radiation (Thomson scattering). The particle can also radiate energy in the presence
of a magnetic field (cyclotron radiation) or electric field (bremsstrahlung radiation).
We study each of these cases in the following sections, in a non-relativistic treatment
first. We let the relativistic treatment to Sect. 5.6. We summarized the processes in
Fig. 5.5.

5.4.2.1 Larmor’s Formula

We will not dig too much into the detail study of radiative fields (see the Jackson
book [1] for more details). A particle of charge e and velocity v generates an electric
field E and a magnetic field B

e (n − β)(1 − β 2 ) e n
E(r, t) = + × [(n − β) × β̇] = Er + Eθ
4π0 κ 3 R2 4π0 c κ 3 R
1
B(r, t) = n×E , (5.27)
c

with n = R/R, β ≡ v/c and κ ≡ 1 − n.β (see Fig. 5.7). Notice in this expression
the presence of the retarded effect illustrated by n − β. Indeed, the direction of the
radiated field at a time t is not exactly radial when a charge is moving: the field was
produced at a time trad < t when the charge was at a position cβ(t −trad ) compared
to its position Rn at t.

Exercise Recover the very first term of Eq. (5.27) and the retarded effect.

The electric field can be divided into the first term, the velocity field, or radial
field Er , which falls off as 1/R 2 and is just the generalization of the Coulomb law
to moving particles. The second term the acceleration field Eθ , which falls off as
1/R, is proportional to the particle’s acceleration and is perpendicular to n. This
electric field, along with the corresponding magnetic field, constitutes the radiation
field. For large values of R, the dominant term is Eθ , the acceleration field. If we
look at the radiated field in a region during a time t at a distance R v × t, we
can neglect the effect of the retarded emission and consider that the retarded effects
are negligible: R  constant during the process. In this limit, and considering the
non-relativistic case (β 1), we can simplify Eq. (5.27) to
e
Erad = n × (n × v̇)
4πR0 c2
1
Brad = n × Erad . (5.28)
c
We illustrate the dependence of the radiated field strength as a function of the
direction of the acceleration in Fig. 5.6. Notice that the value of the radiated field
cancels for θ = 0 and is maximal for θ = π2 , as a direct consequence of n ∧ v̇
290 5 In the Galaxies [T0 ]

Synchrotron radiation
(non rel. : cyclotron)

Coulomb radiation
(Ionization loss)

Ionized Gas

Brehmsstrahlung
(free−free scattering)

CMB
Coulomb radiation

Inverse Compton
(non rel. : Thomson)

Fig. 5.5 Summary of energy loss of a charged particle in an astrophysical medium

in Eq. (5.28). It is also in this direction (θ = π2 ) that the radiated energy will be
maximal.
The energy W radiated during a time dt is then the density of energy radiated
by the charge multiplied by the volume dV of the radiation covered during the time
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 291

Fig. 5.6 Radiated field

strength of an accelerated v
particle

T
.
v

dt, dV = R 2 d × cdt, d being the solid angle from where is seen the radiating
particle at the retarded time (see Fig. 5.7) is:

0 1
d 2W = |Erad |2 + |Brad |2 × R 2 d cdt, (5.29)
2 2μ0

which gives using Eq. (5.28)

d 2W e2 sin2 θ |v̇|2
= , (5.30)
dtd 16π 2 0 c3

θ being the angle between v̇ and n. After integration over d = 2πd cos θ , one
obtains the power P emitted by the radiating charge:

dW e2 |v̇|2
P = = , (5.31)
dt 6π0 c3

which is known under the name of Larmor’s formula.

We can make some remarks about the morphology of radiated field. Notice first
that the acceleration is the proper acceleration of the particle and the loss rate is
measured in its instantaneous rest frame. The electric field strength varies as sin θ ,
and the power radiated per solid angle is proportional to sin2 θ , where θ is the angle
with respect to the acceleration vector of the particle. As a consequence, there is no
radiation along the acceleration vector and the field strength is greater at right angle
292 5 In the Galaxies [T0 ]

E1

(r,t)
E2

cle
rti
R pa
d
. rge
ha
E d:
hec Particle at t
oft
ry
T cto
n aje
Tr

Particle at t=t(retarded)
Fig. 5.7 Example of the electric field E = E1 + E2 radiated by a charged particle at a distance R
from the position of the particle at the retarded time

to the acceleration vector. The radiation is polarized with the electric field vector
lying in the direction of the acceleration vector of the particle, as projected onto
a sphere at a distance r from the charged particle, see Fig. 5.6. In the CGS units
system we would have obtained

2e2|v̇|2
PCGS = . (5.32)
3c3

5.4.2.2 Case of a Rotating Particle

We can then compute the frequency of the radiated field Erad in a specific example
of a charged particle rotating around an axe. With the convention of Fig. 5.8 and
using Eq. (5.28) one can express

r = a(cos ωt i + sin ωt j) ⇒ v̇ = −v ω(cos ωt i + sin ωt j)

evω
⇒ Erad = − (cos ωt a1 + cos θ sin ωt a2 ). (5.33)
4πR0 c2
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 293

Fig. 5.8 Geometry for

polarization decomposition of E
n
radiation emitted by a
circulating charge. The
unitary vectors a1 and a2 are
in the plane perpendicular to
n and form an orthonormal a2 a1 = − i
basis with n (a1 = −i,
a2 = − cos θj + sin θ k,
n = sin θ j + cos θ k)

T R

j
v
i

We then can conclude (which is an important and no so-intuitive result) that a

circulating particle will radiate a monochromatic electromagnetic field of the same
frequency as that of the one of the circulating charged particles. This will be a useful
result when studying cyclotron and synchrotron radiation. This result is not exact
anymore when radiation of higher order than dipole is included.

5.4.3 Coulomb and Ionization Losses

When a high energy charged particle passes through a medium or near other charged
particles like electrons bounded in atoms, it transfers part of its kinetic energy to the
electrons, causing ionization or excitation of the atoms. Indeed, the electromagnetic
force of the traveling particle can give to the electron orbiting on atoms a sufficiently
large momentum to extract it from its orbit (or to make it pass to an upper level) and
ionize the gas (or excite it). From an historical perspective, in 1895, the passing
of electrons (called “cathode rays” at this time) through a tube of Crookes was the
source of the first X-ray observed by Rontgen through the ionization created on the
target. In this section we will concentrate on the influence on the propagation of a
charged particle in mediums made of electrons, and its energy lost along its way.
The result we obtain is quite general and can be applied to several types of radiative
processes. Technically speaking, we call Coulomb scattering when the interaction
is on free electron, and ionization losses when the interaction takes place on bound
electrons, the binding energy corresponding to the ionization potential.
294 5 In the Galaxies [T0 ]

Consider first the collision of a high energy nucleus of mass M, charge Ze,
and velocity v with a stationary electron of mass me . Only a small fraction of the
nucleus kinetic energy is transferred to the electron. We can understand it easily,
remembering that the maximal velocity of an electron after being hit by the nucleus,
vemax , is when all the momentum is transferred to the electron, which escapes at an
angle of 0 degree.

P2 P 2 P 2
Exercise Using the conservation of energy and momentum, 2M1 = 2M 1
+ 2m2 e and
P1 = P1 − P2 , show that V2 = ve = me +M v cos θ , with θ the angle between P1 and
2M

P2 and v = V1 the initial velocity of the nucleus of mass M (hints: compute first
P1 .P1 ).

If one considers solid spheres collision, vemax = m2M

e +M
v, which tends to 2v for me
M. The fractional kinetic energy loss is then less than 12 me (2v)2 / 12 Mv 2 = 4me /M.
We can then consider that the trajectory of the hitting nucleus is not influenced by
the presence of the electron.
To compute the energy transferred to an electron bounded in an atom, we
compute first the momentum received by the electron from the electromagnetic field
created by the traveling particle. The force can be repulsive or attractive depending
on the charge of the moving particle; it does not affect the result as in both cases, the
electron is expelled from the atom. From Fig. 5.9 and remarking that by symmetry
only the component of the force perpendicular to the motion contributes to the
electronic momentum,
 +∞  −π 
Ze2 1 b Ze2
p = |F |dt = sin θ d = ,
−∞ 0 4π0 (b sin θ )2 v tan θ 2π0 bv
(5.34)

which gives for the kinetic energy transferred to the electron

p2 Z 2 e4
Ec = = . (5.35)
2me 8π 2 02 b2 v 2 me

v 2b = duration of collision * v
x


Ze, M
b
r

e, me

Fig. 5.9 Interaction of a high energy particle of charge Ze with an electron at rest
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 295

dx

db

Moving ionizing particle

b
v

Interstellar medium (gaz of atoms)

of density Ne

Fig. 5.10 Moving particle in an interstellar medium of density Ne

The moving particle loses obviously the equivalent energy. When traveling in a
medium like the interstellar medium, one has to sum over all the atoms in the
medium. The number of electrons in a gas of density Ne at a distance between b
and b + db from the axe of the moving particle (called impact parameter) is along
a length dx (see Fig. 5.10) 2πbdbdx × Ne . We can then compute the total loss of
energy on the way

loss  
dEZe Z 2 e4 Ne bmax db Z 2 e4 Ne bmax
=− =− ln , (5.36)
dx 4π02 v 2 me bmin b 4π02 v 2 me bmin

the minus sign representing the fact that the energy is lost by the traveling particle.
The computing of bmin and bmax requires a special treatment. bmax represents the
distance at which the influence of the traveling particle on the electron is negligible.
It corresponds roughly to the time when the orbital period is lower than the typical
interaction time. In other words, if the electron takes more time to move around the
nucleus than to interact with the moving particle, the electromagnetic influence of
the later becomes weak. If one write τ the interacting time and ν0 the frequency of
the rotating electron in the atom (ν0 = ω0 /2π), it corresponds to

2b 1 v
τ < ⇒ b< = bmax . (5.37)
v ν0 2ν0

The lower limit bmin can be obtained if we suppose, by a quantum treatment and
the application of the uncertainty principle, that the maximum energy transfer is
pmax = 2me v (because as we discussed earlier, the maximum velocity transferred
296 5 In the Galaxies [T0 ]

to the electron is 2v) from px  h̄ (Heisenberg principle) we have x 

h̄/2me v. We can then write

h̄
bmin = , (5.38)
2me v

which implies
 
dE Z 2 e4 Ne 2πme v 2 Z 2 e4 Ne πme v 2
=− ln =− ln , (5.39)
dx 4π02 v 2 me h̄ω0 4π02 v 2 me I

where I is the ground state binding energy of the electron in the atom (I =
1
2 h̄ω0 ). But the system being more complex, it usually contains different atoms
with different kinds of excited levels, so one should take into account a mean
ionization potential I¯ instead of I . Integrating the π factor and a factor 2 for physical
interpretation (see below) in the definition of I we finally obtain
 
dE Z 2 e 4 Ne 2me v 2 dE dE Z 2 e 4 Ne 2me v 2
=− ln ⇒ P = =v =− ln .
dx 4π02 v 2 me I¯ dt dx 4π02 v me I¯
(5.40)

The value of I¯ cannot be calculated explicitly (except for very simple models
of atoms) and should thus be measured by experiment. Typical values for the
logarithmic factor, called Coulomb logarithm, are around 20. We recognize in
Eq. (5.40) the ratio of the maximum kinetic energy (2me v 2 ) on the binding energy,
justifying the normalization of I¯. A careful inspection of Eq. (5.40) reveals that the
energy loss of an energetic moving particle depends on the charge and the velocity
of the particle, but not on its mass (except for the Coulomb logarithm), and that low
energy particles lose their energy more rapidly than high energy particles.

5.4.4 Thomson Scattering

The Thomson scattering is the process in which a free charge radiates in response
to an incident electromagnetic wave. If the charge oscillates at non-relativistic
velocities, v c, then we may directly apply the Larmor’s formula computed
in the previous section. This effect arises in astroparticle typically when a photon
radiated by annihilation of dark matter for instance interacts with cosmic rays. If
one supposes an incident linearly polarized wave, the force of the electromagnetic
wave F on the charged particle is

e4 E02
F = e E0 sin ω0 t ⇒ mv̇ = e E0 sin ω0 t × ⇒ P = , (5.41)
12π0 m2 c3
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 297

Fig. 5.11 Thomson

scattering of a polarized
radiation by a charged
particle

e−

where is the normalized E-field direction (see Fig. 5.11). To obtain the above
e2
equation, we just implemented the value of the mean acceleration |v̇|2  = 2m 2
2 E0
into the Larmor’s formula (5.31). Defining the Thomson cross section σT :
  
0
P  = σT × [initial flux of energy] = σT c |E0 |2 = σT cUrad , (5.42)
2

where Urad = 20 |E0 |2 is the energy density of radiation in which the electron is
located,5 we obtain
 2
8π e2 8π 2
σT = = r = 6.65 × 10−29 m2 = 1714 GeV−2 , (5.43)
3 4π0 mc2 3 0
2
with r0 = 4πe mc2 is the classical radius of the charged particle.6 σT is the Thomson
0
scattering cross section and represents the fraction of the initial wave scattered by
the charged particle.
Thomson scattering is one of the most important processes that impedes the escape
of photons from any region. Indeed, notice that there is no change in the energy
of the photons in the process: the energy is first transferred to the electron, which
then by its acceleration radiates back the energy it received. After the scattering, the
photons follow a random walk, each step between a mean free path depending on

5 Indeed, in the interstellar medium, the density of radiation depends on the localization and the

morphology of the gazes.

6 The radius r of a particle of mass m is defined by mc2 = V (r ) = e2
0 0 4π0 r0 .
298 5 In the Galaxies [T0 ]

the density of the electrons. The electrons then prevent the photons to escape from
their own plasma. The Thomson free path is then λT = (σT Ne )−1 , Ne being the
number of electrons per unit volume. This process is one of the more fundamental
ones in particle physics as well as astroparticle or cosmology. It is for instance the
dominant cross section to determine the last scattering surface, i.e. the decoupling
time as it is described in Sect. 3.3.4.2.

5.4.5 Cyclotron Radiation

Cyclotron radiation corresponds to the radiation emitted by a non-relativistic

charged particle moving in a magnetic field B. The Lorentz force acting perpendic-
ular to the motion of the particle creates an acceleration that in turn transforms into
loss of energy by radiation. Cyclotron effect becomes a synchrotron effect when
the particle becomes relativistic (see Fig. 5.5) and is treated in Sect. 5.6.3. The
fundamental principle of dynamics applied to the charged particle of mass m and
charge e at a distance r from the source of the magnetic field gives us

mv 2 v eB
= eB v ⇒ ωL = 2π × νL = 2π × =
r 2πr m
  
eB e B 1 GeV
⇒ νL =  14 MHz −19
(5.44)
2πm 1.6 × 10 1 Gauss m

ωL (νL ) being the cyclotron (Larmor) pulsation (frequency). Notice that the
frequency is independent of the position of the particle or its velocity. Moreover,
as we saw in Eq. (5.33), the radiated field is monochromatic and has the same
frequency as that of the moving particle itself. As all the particles in rotation around
B possess the same frequency, independently of their rotation radius, they will all
contribute to the electromagnetic field at the same frequency. For an electron in a
typical 10 µG magnetic field from galactic center, we expect a frequency around the
MHz, which is the range of measurement of several experiments. In the relativistic
case treated below, we should add the Doppler effect given by Eq. (5.68) with a
mean on cos θ , which gives a frequency multiplied by the factor γ =  1 . In
2
1− v2
c
other words, a relativistic electron of 1 GeV would give a synchrotron radiation at a
frequency around 2000 times larger than that in the non-relativistic case, so between
100 MHz and 1 GHz.
Combining Eq. (5.31) with Eq. (5.44), we can then compute the energy lost per
second by a particle of mass m and charge q × e moving around a magnetic field of
5.4 Radiative Processes in Astrophysics Part I: The Non-Relativistic Case 299

intensity B, noticing that in the non-relativistic case, E = 12 mv 2 :

dE e4 B 2 E c 3 m3
=− 3 3
⇒ E = E0 e−t /τ , with τ = 3π0 4 2 . (5.45)
dt 3π0 c m e B

v2
Exercise Noticing that |v̇| = r , recover the expression (5.45).

Implementing some more natural units for particle physicists, we can express
  2 3
1 4
1 Tesla 1.88 × 10−27 m
τ = 3π(3 × 10 ) 8 3
q × 1.6 × 10−19 B 1 GeV

2 × 1018 1 Gauss 2  m 3
 s.
q4 B 1 GeV

To give an example, an electron in typical local magnetic fields of a galaxy (around

1 µG, see Sect. 5.10.4.1 for a more detailed modelization), we obtain τ  8 × 1012
years, much larger than the age of the Universe. For a proton, the lifetime is even
greater. In other words, cyclotron radiation is not a significant cooling mechanism
for interstellar gas.

5.4.6 Bremsstrahlung Radiation

In 1927, Carl Anderson found that the ionization rate given by the Bethe–Bloch
formulae (5.39) and (5.75) underestimates the energy loss rate for relativistic
electrons. The additional loss was associated to the acceleration of the electron
in the electromagnetic field. Indeed, we computed in Eq. (5.31) the energy lost
by an accelerated particle. Then, a particle passing through the magnetic field
generated by a gas of ionized particle will lose energy by exchanging momentum
with constituents of the gas (Coulomb/ionization scattering) but also from its
own acceleration under the impact of the surrounding electromagnetic field. This
phenomena was in fact already observed by Nikola Tesla in 1880 in another context
and was called bremsstrahlung. The process is identical to what is called free–
free emission in the language of atomic physics, in the sense that the radiation
corresponds to transitions between unbound states of the electron in the field of the
nucleus. In 1934, computations of the spectrum of non-relativistic and relativistic
bremsstrahlung were carried out by Bethe and Heitler.
The bremsstrahlung, from the German bremsen “to brake” and Sthralung
“radiation,” i.e. “braking radiation” or “deceleration radiation,” is the electromag-
netic radiation produced by the deceleration of a charged particle when deflected
by another charged particle, typically an electron by an atomic nucleus (see
Fig. 5.5). The moving particle loses kinetic energy, which is transformed into an
outgoing photon because energy is conserved. Strictly speaking, braking radiation
300 5 In the Galaxies [T0 ]

is any radiation due to the acceleration of charged particles. However, the term is
frequently used in the more narrow sense of radiation from electrons slowing in
matter.

5.5 Notions of Relativity

5.5.1 Main Idea

In a lot of astrophysical processes, the charged particle radiating electromagnetic

field is produced with a relativistic velocity. It is for instance the case for GeV-like
dark matter annihilating into leptons. We would like to apply our preceding results
to such an energetic particle. The main idea is to move to an instantaneous rest frame
K  such that the particle has zero velocity at a certain time and we then can apply all
the previous results we obtained. During an infinitesimal lapse of time the particle
moves non-relativistically and we can use the Larmor’s formula.

5.5.2 Lorentz Transformations

We recall in this section the main results concerning the transformations induced by
special relativity. Consider two frames K and K  that are moving with respect to
each other with a relative velocity v along the x-axis. The origins are assumed to
coincide at t = 0. If a pulse of light is emitted at t = 0, each observer will see an
expanding sphere centered on his own origin:

x 2 + y 2 + z2 − c2 t 2 = 0, x 2 + y 2 + z2 − c2 t 2 = 0.

The actual relations between (x, y, z, t) and (x  , y  , z , t  ) are called the Lorentz
transf ormations:
 v 
x  = γ (x − vt); y  = y, z = z, t  = γ t − 2 x , (5.46)
c

with γ =  1 . The inverse relations xi = f (xi ) have obviously the same form
2
1− v2
c
with v → −v. It is common to define quadrivectors xμ = (ct, x, y, z), μ being
the Lorentz index. Any vectors respecting the Lorentz transformation laws are then
called quadrivectors. As we will see, it is also the case for the energy–momentum
quadrivector pμ = ( Ec , px , py , pz ) for a particle of energy E and momentum p.
There are multiple ways to demonstrate the Lorentz transformations (5.46). Some
with arguments are based purely on physics of transformations, some others taking
into account the ether and then defining a virtual time t  where the equations of
Maxwell are invariant (this was the original method of Poincaré and Lorentz). I give
an example based on physics argument in the following frame. However, the one I
5.5 Notions of Relativity 301

use to teach is (to my taste) more elegant at the price of being more mathematical.
Here is how I do.
If we suppose a constant velocity of light, and we need to define an invariant of a
transformation, we can, by default, set this invariant cdτ to 0 for a particle moving
at the speed of light. There exist two possibilities, x = ct or x = −ct. We can then
define

c2 dτ 2 = (cdt − dx)(cdt + dx) = c2 dt 2 − dx 2. (5.47)

dτ is called proper time. Then, restricting ourselves to a translation along the x-

direction (generalization is straightforward) we can write the transformation

x  = 10 t + 11 x
t  = 00 t + 01 x. (5.48)

Take an observer at O who does not move (dx = 0); the observer O  moving with
a velocity v with respect to him will feel the change of coordinate

dx  = 10 dt (5.49)
dx 
dt  = 00 dt ⇒ 10 = 00 = v00 , (5.50)
dt 
and demanding for dτ to be constant, we have

c2 dt 2 − dx 2 = c2 dt 2 − dx 2 =
     
c2 (00 )2 − (10 )2 dt 2 + c2 (01 )2 − (11 )2 dx 2 + 2 c2 00 01 − 10 11 dxdt,

which gives

c2 (00 )2 − (10 )2 = c2
c2 (01 )2 − (11 )2 = −1
c2 00 01 − 10 11 = 0. (5.51)

Solving this set of equations combined with (5.50), we obtain easily

1
00 = 
v2
1− c2
v
10 = 
v2
1− c2
302 5 In the Galaxies [T0 ]

v
c2
01 = 
v2
1− c2
1
11 =  , (5.52)
v2
1− c2

which are the set of Eq. (5.46), up to a sign on v corresponding to the direction of
the translation.

Deriving the Lorentz Transformations

A complete section is devoted to the Lorentz transformation in
Appendix A.1.1. I will give here just some fundamentals. Suppose a light
signal proceeding through the x-axis in the reference (lab) frame K, and one
should have

x − ct = 0. (5.53)

On the other hand, the hypothesis of constant velocity of light c in any frame
should impose, in a system K  moving with a velocity v relative to K,

x  − ct  = 0. (5.54)

Combining the two previous equations, and adding the situation of a light
traveling toward negative x-axis, one can write

x  − ct  = λ(x − ct), x  + ct  = μ(x + ct), (5.55)

which gives by combination

x  = ax − bct, ct  = act − bx, (5.56)

with a = λ+μ λ−μ 

2 and b = 2 . For the origin of the reference frame K , we
always have x  = 0, which gives x = a ct. Remembering that K  travels with
b

a velocity v relative to K, this gives v = ab c. Now, if one makes a snapshot at

t = 0 of a unit length x = 1 , the observatory in the lab frame will observe

x = x a = a . On the other hand, the unit length
1
x = 1 will be observed
 2
  2

 
in the moving frame at t = 0 as x = x a − ba = ax 1 − vc2 =
 2

a 1 − vc2 . Imposing the principle of relativity (the observed length is the

(continued)
5.5 Notions of Relativity 303

same, independent of the reference frame, x = x  ) we obtain

v
1
a= , and then b =  c
. (5.57)
v2 v2
1− c2
1− c2

The transformation (5.56) then becomes the Lorentz transformation

x − vt ct − vc x
x =  , ct  =  . (5.58)
2 2
1 − vc2 1 − vc2

We can then easily check that x 2 −ct 2 = x 2 −ct 2 confirming that the velocity
of light is the same in both frames.

Writing the Lorentz transformation into a differential form, one can deduce the
transformation laws for velocities:
 v 
dx = γ (dx  + vdt  ); dy = dy  , dz = dz , dt = γ dt  + 2 dx  ⇒
c
dx γ (dx  + vdt  ) ux + v
ux = = = vu
,
dt γ (dt  + c2 dx  )
v
1 + 2x c
uy
uy = vux
,
γ (1 + c2
)
uz
uz = vux
. (5.59)
γ (1 + c2
)

The generalization of such transformations to an axis not parallel to the x-axis can be
written by decomposing u into its parallel (u ) and perpendicular (u⊥ ) components
with respect to v:

u + v u⊥
u = , u⊥ = . (5.60)
vu vu
1+ c2
γ (1 + c2
)

For u = c (light emitted in K  ) we can notice

u⊥ c
tan θ = = , (5.61)
u γv
304 5 In the Galaxies [T0 ]

hZ

h Z
T


e− V e− T

S S’
Fig. 5.12 Doppler effect and aberration (angular contraction)

which means that if the distribution of light is isotropic in the K  frame, the observer,
for highly relativistic speeds (γ 1), will see half of the photons lying within a
cone of half angle 1/γ . This is called the beaming effect that we have illustrated in
Fig. 5.12.
We can also compute explicitly the angle (see Fig. 5.12) θ  under which is seen
an incident radiation from a moving electron with a velocity v in its rest frame as a
function of the incident angle θ measured in the galactic (laboratory) frame:

vy sin θ vx cos θ − vc

sin θ  = =  , cos θ  = = (5.62)
c γ 1 − vc cos θ c (1 − vc cos θ )

with sin θ = vy /c, cos θ = vx /c. Notice that v → −v in the formula compared
to Eq. (5.59) because we express the measured value in S  as a function of the ones
in S.
It becomes then straightforward to deduce the Lorentz transformations for the
acceleration:

ax
ax =
γ (1 + vu
3
c2
x 3
)
ay uy v ax
ay = −
γ 2 (1 + vuc2
x 2
) c2 γ 2 (1 + vuc2
x 3
)
az uz v ax
az = − . (5.63)
γ (1 + vu
2
c2
x 2
) c γ (1 + vu
2 2
c2
x 3
)

If the particle is at rest in K  , we obtain

a = γ 3 a , 
a⊥ = γ 2 a⊥ . (5.64)
5.5 Notions of Relativity 305

Exercise Recover the transformation laws for the acceleration.

We can complete the list with the Lorentz transformation for the electromagnetic
fields

E = E , B = B
 
 1  1
E⊥ = γ E⊥ + v ∧ B , B⊥ = γ B⊥ − v ∧ E . (5.65)
c c

Exercise Recover the Lorentz transformation for the electric and magnetic fields,
and deduce the equations for E and B in the moving frame K  as a function of (x  ,
y  , z ). Compute the time t  necessary in K  to recover Maxwell equations “as if” the
moving observer is at rest.7 Give your conclusion (this was the first step to relativity
proposed by Lorentz in 1894).

5.5.3 Relativistic Larmor’s Formula

From Eq. (5.64) one can then deduce the loss of energy of a relativistic particle in
the laboratory frame. We just need to apply the Larmor’s formula (5.31) in the rest
frame of the particle and then make a Lorentz boost. It gives

e2 γ 4
P = (|a⊥ |2 + γ 2 |a |2 ), (5.66)
6π0 c3

where P is the power emitted by the radiating charge.

5.5.4 Doppler Effect

Any periodic phenomenon in the moving frame K  will appear to have a longer
period by a factor γ when viewed by local observers in frame K. We also have to
add the additional effect on the period due to the delay time for light propagation
between the journey 1 → observer and 2 → observer. The joint effect is called
Doppler effect. Suppose that the moving source emits one period of radiation as it
moves from point 1 to point 2 at velocity v. If the frequency of the radiation in the
rest frame of the source is ω , then the time to move from point 1 to point 2 in the

7 To respect the Galilean principle.

306 5 In the Galaxies [T0 ]

observer’s frame is given by the time dilatation effect

2πγ
t = . (5.67)
ω
Adding the classical Doppler effect taking into account the fact that the journey of
the light is shorter8 by a time d/c with d = vt cos θ , θ being the angle between
the gamma ray and v in the observer referential at the time of emission, one obtains

ω
ω=  . (5.68)
γ 1 − vc cos θ

This is the relativistic Doppler formula, where ω is the frequency observed by the
fixed observer and is illustrated by Fig. 5.12.

5.5.5 Transformations on the Energies

Another quadrivector that we can build and that respects the Lorentz transfor-
mation (5.46) is the energy–momentum one, pμ = (E/c, p). In this case the
transformation can be written as:
 
 vE   E E v
px = γ px + ; py = py , pz = pz , =γ + px (5.69)
c c c c c

with for a particle at rest in K, p = 0 and E = mc2 . Notice that comparing to (5.46),
the transformation corresponds to a reference frame K’ moving with a velocity −v
compared to the K frame where the particle is at rest (p = 0). We then deduce
the energy and momentum of a particle of velocity +v compared to the laboratory
frame:

mc2 mv
E =  = γ mc2 , px =  = γ mv, py = py , pz = pz .
v2 v2
1− c2
1− c2
(5.70)

We remark that, by expanding this expression at low velocities v c, one obtains

1
E  ∼ mc2 + mv 2 ,
2

8 This convention applies if the source at K  moves toward the observer at rest at K. If opposite,
just replace v by −v.
5.5 Notions of Relativity 307

and we find back the expression of the energy as potential (mass) energy plus kinetic
energy. We can also rephrase (5.70) in a useful form

|p |c2 |p |

v= = . (5.71)
E γm

Note also that if a particle of rest mass m moves with a velocity v and loses or gains
the energy E0 (through absorption or emission of a photon  for instance), its energy
2
in the laboratory frame is increased (decreased) by ±E0 / 1 − vc2 , which means that

2
the total energy in the laboratory frame is now (m ± Ec20 )c2 / 1 − vc2 . Its behavior
is then equivalent of a particle of mass m ± E0 /c2 . This is the famous formulae
proposed by Einstein in addenda of his article published in October 1905: a particle
at velocity v receiving (losing) an amount of energy E0 without altering its velocity
receives a mass contribution m = ±E0 /c2 (the famous formula “E = mc2 ”).
This way of “proving” relativity was the one preferred by Einstein himself, but one
has to wait until the work of Cockcroft and Walton in 1932 who observed the first
transmutation mechanism, shooting beam of protons at very high speeds. Firing
protons like bullets, into metal targets, they converted some of the atoms’ mass into
energy.

5.5.6 Fizeau Experiment

One of the first proof of special relativity was made by Fizeau in 1851, even before
Einstein was born,9 when he measured the relative speed of light in moving water of
velocity v. Fizeau used a special interferometer arrangement to measure the effect of
movement of a medium upon the speed of light. For a light of velocity w = c/n, n
being the refraction index of water, the law of composition should give the velocity
in the lab frame u (5.59):
  
w+v w.v  w2 1
u=  (w + v) 1 −  w + v 1 − =w+v 1− 2 ,
1 + w.v
c2
c2 c2 n
(5.72)

which is exactly the experimental result measured by Fizeau in his 1851 experiment.

9 Einstein
himself used to tell that the Fizeau experiment was his first motivation to establish his
own laws of transformation.
308 5 In the Galaxies [T0 ]

5.6 Radiative Processes in Astrophysics Part II: The Relativistic

Case

5.6.1 Relativistic Coulomb Scattering or Ionization Losses

5.6.1.1 Ionization Loss

For the treatment of relativistic Coulomb loss we use Eqs. (5.59) and (5.65) to
compute the transfer of momentum from the moving frame (rest frame of the
moving particle where the treatment is non-relativistic) to the rest frame of the
observer.10 If we write γ = E/mc2 , we have in the rest frame

p = p⊥ = F⊥ dt = F⊥ dt  = p⊥


(5.73)

where F⊥ (E⊥ ) represents the component of the force (electric field) perpendicular
to the direction of motion, which is the only active one as we discussed in Sect. 5.4.3.
The transferred momentum is the same in the relativistic frame than in the rest
frame. It is logical because only the perpendicular component of the momentum
is transferred in the Coulomb interaction, and perpendicular components are never
affected by transformations in relativity. Equation (5.36) is then valid even for a
relativistic moving particle. The difference is then due to the factors bmin and bmax .
bmax is greater by a factor γ because the duration of the impulse is shorter by this
factor. In the case of bmin , the transverse momentum of the electron is greater by a
factor γ , and hence, because of the Heisenberg uncertainty principle,

h̄
x  bmin = ∝ γ −1 , (5.74)
p

where we used the Lorentz transformation for the momentum (velocity) obtained in
Eq. (5.60). Thus, we expect the logarithmic term to have the form ln(2γ 2me v 2 /I¯).
A more precise computation taking into account relativistic quantum effects is given
by the Bethe–Bloch formula
 2
dE Z 2 e4 Ne 2γ me v 2 v2
=− ln − .
dx 4π02 v 2 me I¯ c2

We derived all the expressions above, except for the factor v 2 /c2 , which is always
practically very small compared to ln(2γ 2 me v 2 /I¯), which is  20 in the majority of
physical processes in the galactic medium. As discussed earlier, I¯ is a parameter that
should be obtained experimentally. We usually express the energy loss as a function

10 It
can be the solar system or even the galaxy itself that is considered at rest compared to the
moving relativistic particles.
5.6 Radiative Processes in Astrophysics Part II: The Relativistic Case 309

of the density of mass of the material crossed by the high energy particle using11
 2
dE 4πZ 2 r02 me c3 ZNA 2γ me c2 β 2
=− ln − β2 × ρ
dt βA I¯

with r0 = e2 /4π0 me c2 the classical radius of the electron, β = vc , and we

substitute x → vt. The exact treatment of the ionization energy loss of heavy
particles, taking into account that the interstellar medium is mainly composed of
hydrogen and helium, leads to

dE 4πZ 2 r02 me c3 ZNA 1 2γ 2 me c2 β 2 Emax δ
=− ln − β2 − ×ρ
dt βA 2 ¯
Ii2 2
i=H,H e

4πZ 2 r02 me c3 ZNs 1 2γ 2 me c2 β 2 Emax δ
=− ln − β2 − , (5.75)
β 2 ¯
Ii2 2
i=H,H e

where δ represents a screening effect due to the density of atoms of the medium
crossed by the particle and Emax = 2me c2 β 2 γ 2 /(1 + 2γ me /M + (me /M)2 ), M
being the nucleon mass, is the maximum kinetic energy transferred to one atom and
Ns the number density of the medium. The density correction can be neglected in
our case, and the values of IH = 19 eV and IH e = 44 eV are commonly used. A
more complete treatment in quantum field theory gives the Bethe–Bloch formula for
the ionization loss
 
dE 1 (γ − 1)β 2 E 2 1
= −2πr02 me c3 Zs ns ln + . (5.76)
dt I β 2Is2 8
s=H,H e

5.6.1.2 Coulomb Scattering

The Coulomb collisions in a completely ionized plasma are dominated by scattering
off the thermal electrons. The corresponding energy losses are given by
 
dE 2 3 1 Eme c2 3
= −2πr0 me c Zn ln 2
− (5.77)
dt Coul β 2
4πr0 h̄ c Zn 4
  
Zn 1 1 E/me c2
 −5.5 × 10−16 −3
1 + ln GeV s−1
1 cm 1 − (mc /E)
2 2 74 Zn/(1 cm−3 )

11 Remembering that the Avogadro number N

A is by definition the number of carbon12 atoms in 12
grams of carbon, or equivalently, NA /A represents the number of atoms of nuclei A in one gram,
and thus Z × NA /A represents the number of protons (and then electrons) per gram of material.
We can then deduce Ne = ρZNA /A, ρ being the mass density of material A.
310 5 In the Galaxies [T0 ]

n being the density of ions of charge Z in the ionized plasma and m the mass of the
traveling particle. The values of the densities ne , nH , or nH e depend on the position
in the galactic plane. As the Sun reference point (8 kpc) they are ne = 0.0024 cm−3 ,
nH = 0.6 cm−3 , and nH e = 0.144 cm−3 .

5.6.2 Inverse Compton Scattering

The Inverse Compton interaction corresponds to the relativistic limit of the Thomson
scattering (see Fig. 5.5). The strategy to compute it is quite straightforward: to place
ourselves in the framework where the charged particle is at rest, and then to apply
a Lorentz transformation to obtain the result in the galactic (observer) coordinate
frame. The energy loss per unit of time being Lorentz invariant, P  = dE  /dt  =
dE/dt = P , we need to compute dE  /dt  = σT cUrad  (Eq. 5.42). For that we need
 
to compute Urad = N h̄ω as a function of Urad = N h̄ω, N being the density of
photons. The frequency ω → ω follows the classical Doppler effect described
through Eq. (5.68). However, there is another subtlety, which is equivalent to a
second Doppler effect in a sense: cUrad  is a f lux of energy. One then needs to

compute the interval of time t between the arrival of two pulses on the electron
in S  as a function of the interval of the same event t measured in S. After a look
at Fig. 5.13 we can write

x2 − x1
t = t2 − t1 + cos θ = (t2 − t1 )γ [1 + (v/c) cos θ ]. (5.78)
c

Fig. 5.13 Inverse Compton S

scattering geometry

e−

V t’2
t1 = t’1

t2
d = V(t2−t1) cos T
5.6 Radiative Processes in Astrophysics Part II: The Relativistic Case 311

One obtains, exactly as for the Doppler effect, t = t  × γ [1 + (v/c) cos θ ], that
is, the time interval between the arrival of photons from the direction θ is shorter by
a factor γ [1 + (v/c) cos θ ] in S  than in S. Thus, the rate of arrival of photons, and
correspondingly their number density, is greater by this factor. Then, as observed in
S  , the energy of the beam is therefore

Urad = γ 2 [1 + (v/c) cos θ ]2 × Urad . (5.79)

After a mean on the angle θ and using the relation γ 2 − 1 = (v/c)2 γ 2 we can write
 π 
 1 4 1
Urad = γ 2 [1 + (v/c) cos θ ]2 2π sin θ dθ = Urad γ 2 −
4π 0 3 4
 2
4 v
⇒ P I C = P   = σT cUrad γ2
3 c2
 2
−17 E 511 keV 2
 2.56 × 10 GeV s−1 , (5.80)
1 GeV m

where we have used v  c, σT form (5.43) and Urad = 2.4 × 10−10 GeV cm−3
from Eq. (3.47).

5.6.3 Synchrotron Radiation

Synchrotron radiation is the relativistic limit of the cyclotron mechanism (see

Fig. 5.5). We will compute the synchrotron emission with both methods: first
directly in the observer (fixed) frame using the relativistic Larmor’s formula. We
will then recover this result computing the emission in the particle frame (particle
will be at rest) but feeling the electric field generated by the relativistic boost.

5.6.3.1 From the Observer Point of View

To obtain the formulation for the cyclotron radiation produced by a relativistic
particle (called synchrotron radiation) we first need to compute the acceleration of
the particle. The relativistic dynamics principle gives us

d
γ mv = e(v × B). (5.81)
dt
We then should recover the same result than for the synchrotron replacing m → γ m
and applying the relativistic Larmor’s formula, noticing that there is no acceleration
parallel to B and that the acceleration is perpendicular to the velocity vector
(Eq. 5.81), and we deduce |a |2 = 0. We then obtain

2
e4 γ 2 B 2 v⊥ e4 γ 2 B 2 v 2 sin2 θ
P = 2 3
= . (5.82)
6π0 m c 6π0 m2 c3
312 5 In the Galaxies [T0 ]

If one averages on the velocity angle:


v2 2 2 e4 γ 2 B 2 v2
2
v⊥  = sin2 θ d = v ⇒ P  = P synch = (5.83)
4π 3 9π0 m2 c3

If the particle is relativist, one can consider E = pc with p = γ mv, which implies
v 2 = E 2 /(γ 2 m2 c2 ) and
 2
dE e4 B 2 E 2 2e4 E 2 B
P synch = − = =
dt 9π0 m4 c5 9π02 m4 c7 2μ0
  
−18 E 2 B 511 keV 4
 2.51 × 10 GeV s−1 . (5.84)
GeV μG m

5.6.3.2 From the Particle Point of View

In the rest frame of the particle, the fundamental principle of dynamic can be written
as

d e v

mv = e(E + v × B ) = eE = γ vB sin θ  (5.85)
dt c v

because the particle is at rest, v = 0, θ being the angle between v and B . We used
Eq. (5.65) for the Lorentz transformation on E . We then obtain
   v 2
dE e2 |v̇ |2 e4 γ 2 B 2 v 2 sin2 θ
P =− = 3
= 2 3
= 2σT cUB γ 2 sin2 θ
dt 6π0 c 6π0 m c c

which implies

4  v 2
P synch = σT cUB γ2 (5.86)
3 c

with UB = B 2 /2μ0 , recovering Eq. (5.83).

5.6.4 Relativistic Bremsstrahlung

When a charged particle passes through an electric field, it can be accelerated or

decelerated, depending on its charge. In both cases, the change in the velocity
will generate a loss of energy, following the Larmor’s formula. In the presence of
magnetic field, this loss of energy corresponds to the synchrotron radiation, whereas
in the electric field case we use to call it bremsstrahlung. In this case, we define a
radiation length X by the thickness of the medium over which the energy of the
5.6 Radiative Processes in Astrophysics Part II: The Relativistic Case 313

incident particle is reduced by a factor e:

E = E0 e − X .
x
(5.87)

X can be expressed in cm, but it is more common to write it in g.cm−2 dividing it

by the mass density

X
X→ (5.88)
nM
with n the density of the ionized particles in the gas and M its atomic mass. We can
then express the loss of energy by bremsstrahlung as

dE nM
− =cE , (5.89)
dt brem X

where c is the velocity of light.

5.6.5 Energy Losses: Summary

As a summary, we can write the total energy loss for an electron in a galactic
medium from synchrotron (synch) (Eq. 5.86), Inverse Compton (IC) (Eq. 5.80),
Coulomb (Coul), ionization (I), and bremsstrahlung (brem) radiation in the rela-
tivistic case (v  c) summarized in Fig. 5.5:

P  = P synch + P I C + P Coul + P I + P brem


4 4 1 Eme c2 3
= σT cUB + σT cUrad + 2πr02 me c3 Zn ln −
3 3 β 4πr0 h̄2 c2 nZ 4

1 (γ − 1)β 2 E 2 1 ns Ms
+2πr02 me c3 Zs ns ln 2
+ + cE
β 2Is 8 Xs
s=H,H e s=H,H e
 2  4
E B 511 keV
= 2.51 × 10−18
GeV μG m
 2
−17 E 511 keV 2
+2.56 × 10
GeV m
  
−16 Zn 1 1 E/me c2
+5.5 × 10 1+ ln
1 cm−3 β 74 Zn/(1 cm−3 )
  
1 Zs ns (γ − 1)β 2 E 2 1
+7.4 × 10−18 −3
ln 2
+
β 1 cm 2Is 8
s=H,H e
314 5 In the Galaxies [T0 ]

Table 5.3 Number densities of atomic hydrogen (HI ), molecular hydrogen (H2 ), ionized hydro-
gen (HI I ), and helium (H e) as a function of the galactic radius R
R (kpc) nH I (atom cm−3 ) 2nH 2 (mol cm−3 ) nH I I (atom cm−3 ) nH e (atom cm−3 )
0 0.16411 0. 0.0275 0.01805
1 0.16411 0.06211 0.04226 0.02488
2 0.20349 0.12422 0.08674 0.03605
3 0.24616 0.47352 0.15635 0.07917
4 0.36104 0.76053 0.19357 0.12337
5 0.51694 1.04366 0.15542 0.17167
6 0.51694 0.99151 0.08488 0.16593
7 0.54976 0.88814 0.03951 0.15817
8 0.60720 0.71078 0.02390 0.14498
9 0.50280 0.39278 0.02024 0.09851
10 0.65116 0.10128 0.01901 0.08277
11 0.40338 0. 0.01802 0.04437
12 0.57451 0. 0.01701 0.06320
13 0.48365 0. 0.01598 0.05320
14 0.22138 0. 0.01494 0.02435
15 0.14515 0. 0.01389 0.01597
16 0.09448 0. 0.01286 0.01039
17 0.06782 0. 0.01184 0.00746
18 0.04867 0. 0.01085 0.00535
19 0.03491 0. 0.00989 0.00384
20 0.02504 0. 0.00897 0.00275


E ns Ms
+3 × 10 10
GeV s−1 (5.90)
GeV Xs
s=H,H e

Ms being the atomic mass of species s and Xs its radiation length (XH 
62.8 g cm−2 , XH e  93.1 g cm−2 ). The local density of different species at a certain
distance from the Milky Way center can be found in Table 5.3. The electron species
can be identified with the ionized hydrogen H I I species). We show in Fig. 5.14 the
summary of the losses as a function of the charged particle energy.

5.7 Ultra-High Energetic (UHE) Processes

If a very high energetic particle like a proton for instance interacts with the photons
of the Cosmological Microwave Background (CMB), pions or other particles can be
created onshell. As a consequence, it exists a maximum energy (and thus free path)
above which these particles (protons or photons) cannot propagate in the Universe
5.7 Ultra-High Energetic (UHE) Processes 315

dE
( ) GeV s−1
dt

10−8 Synchrotron(B=100 PG)

Inverse Compton
Bremstrahlung
Ionization
10−12
Coulomb

10−16

10−20

0.01 0.1 1 10 100 1000 10000

E (GeV)
Fig. 5.14 Combining the energy loss mechanisms for B = 100µG

anymore. Their spectrum should then possess a typical cut that should be observable
by dedicated experiences (called GZK cut for Greisen–Zatsepin–Kuzmin in the
case of protons). In other words, if ultra-high energy events are observed by such
experiments, the origin would be local (like dark matter origin for instance), or at
least relatively near the Earth, corresponding to a short mean free path.

5.7.1 Cosmic Rays Case

Photons of the CMB act on accelerated protons through the + resonance

γ + p → + → n + π + → n + μ+ + νμ
γ + p → + → p + π 0 → p + γ + γ
γ + p → p + Nπ, (5.91)
316 5 In the Galaxies [T0 ]

the charged pions decaying into ultra-high energy muons and muons neutrinos.12
These high energy neutrinos can be detected by ground based detectors like Icecube.
Computing the maximum energy allowed by the proton before the reaction occurs
is equivalent to compute the energy necessary for the reaction to occur in the rest
frame of the proton and then to make a Lorentz transformation back in the galactic
(laboratory) frame.
In the proton rest frame R, the threshold energy EγR of the photon is obviously the
mass of the pion EγR > 200 MeV. Using the formulae computed for the Doppler
effect (Eq. 5.68) we deduce for the energy of the photon in the laboratory frame Eγ0
 v 
200 MeV  EγR = Eγ0 γ 1 + cos θ < 2γ Eγ0 (5.92)
c

with Eγ0 = 6 × 10−4 eV (see Sect. E.1.1), which gives

γ  1.7 × 1011 ⇒ Ep0 = γ EpR = γ mp c2  1.7 × 1020 eV = Epmax . (5.93)

A more detailed treatment, taking into account the integration over the Planck
spectrum of the CMB, decreases this value to Epmax = 5 × 1019 eV. Any proton
of energy Ep  5 × 1019 eV reacts then with the CMB to produce pions and then
muons and muon neutrinos. This limit is called the GZK cutoff computed for the
first time by Greisen and independently by Kuzmin and Zatsepin in 1966. The mean
free path λ for a single scattering is then given by

λ × σpπ n0γ = 1 ⇒ λ = (σpπ n0γ )−1  1023m  3 Mpc (5.94)

σpπ  2.5 × 10−4 barns being the interaction cross section of the reactions (5.91)
and nγ = 411 cm−3 (Eq. 3.42) the density of photons of the CMB. After each
reaction described in (5.91), the proton in the galactic frame loses the kinetic energy
Ep = γ mπ c2 . Its initial energy being Ep0 = γ mp c2 , the relative loss of energy
Ep /Ep0  1/10. As a consequence, if the highest energy cosmic rays are protons,
they cannot have originated from farther than about 30 Mpc from our galaxy.

5.7.2 Photons and Neutrinos Cases

The phenomena of a GZK cutoff exist also for photons and neutrinos. Indeed, a
very highly energetic photon can hit a photon from the CMB, producing a e+ e−

12 For the reader not used to the particle physics contents, the pions are composed of π 0 = √1 (ūu+
2
d̄d), π + = d̄u and π − = d ū. The Delta baryons s also called Delta resonances are made of
++ = uuu, + = uud, 0 = udd, and − = ddd. Their masses are about 1.232 GeV and can
be considered (in the case of + and 0 ) as excited states of the proton and neutron, respectively.
5.8 Indirect Detection of Gamma Ray 317

pair onshell via a t-channel exchange of a virtual electron. Taking into account that
the CMB temperature is T0  2.34 × 10−4 eV,

(P1 + P2 )2 = 2P1 .P2  4ECMB Eγ = 4m2e ⇒ Eγ  400 TeV. (5.95)

In fact, when taking into account the mean free path, a photon of 100 TeV will
lose 90% of its energy in the Milky Way before crossing it. The same exercise can
be done for neutrinos; considering the process on the CMB through the t-channel
MW2
exchange of an electron with a photon + W final state, we obtain Eνcut  4ECMB 
5 × 1015 GeV.

Exercise Compute the mean free path for a photon and neutrino as a function of its
energy in the Milky Way.

5.8 Indirect Detection of Gamma Ray

5.8.1 The Principle

The main principle of the indirect detection of gamma rays is to observe the sky,
in different ranges of energy, depending on the satellites or telescopes. It can range
from the keV scale for XMM Newton satellite to more than 10 TeV scale for the
HESS telescope in Namibia. The main advantage of terrestrial detectors concerns
its ability to measure higher energetic photons, whereas the size of the satellites
limits drastically the sensitivity of the instruments. The FERMI satellite, however,
is able to give limits on the flux up to 300 GeV.
For the continuum of gamma rays, the observed differential flux at the Earth coming
from a direction forming an angle ψ with respect to the galactic center is


dγ (Eγ , ψ) dNγi 1
= σi v ρ 2 dl , (5.96)
dEγ
i
dEγ 8πm2DM line of sight

where the discrete sum is over all dark matter annihilation channels, dNγi /dEγ is
the differential gamma-ray yield, σi v is the annihilation cross section averaged
over its velocity distribution, mDM is the mass of the dark matter particle, and ρ
is the dark matter density. Assuming a spherical halo, one has ρ = ρ(r) with
the galactocentric distance r 2 = l 2 + R02 − 2lR0 cos ψ, where R0 is the solar
distance to the galactic center ( 8 kpc). It is customary to separate in the above
equation the particle physics part from the halo model dependence introducing the
318 5 In the Galaxies [T0 ]

(dimensionless) quantity J (ψ) we already discussed in Eq. (5.3):

 2
1 1
J (ψ) = ρ 2 (r(l, ψ)) dl . (5.97)
8.5 kpc 0.3 GeV/cm3 line of sight

We can explain the origin of Eq. (5.96) in easily. The differential flux (number of
particles arriving per unit of surface per unit of time and per unit of energy) is the
cross section of a given final state i (σi ) multiplied by the density of the target on
its way in one second ( mρDM × v) multiplied by the number of particles per unit of
surface ( mρDM dl) giving a spectrum dN i
dE . If one looks per solid angle, one also needs
to divide by 4π, adding 12 factors, because we can detect only one of the two final
states, the second one flies in the opposite direction. One can then write

dγ (Eγ , ψ)
= 0.94 × 10−13 cm−2 s−1 GeV−1 sr−1
dEγ
 
dNγi σi v 100 GeV 2
× −29 3 −1
J (ψ) . (5.98)
dEγ 10 cm s mDM
i

Actually, when comparing to experimental data, one must consider the integral of
J (ψ) over the spherical region of solid angle  given by the angular acceptance
of the detector that is pointing toward the galactic center, i.e. the quantity J¯( )
with

¯ 1
J ( ) ≡ J (ψ) d , (5.99)
 

must be used in the above equation. For example, for experiments like FERMI,
MAGIC, or HESS   10−5 sr. The gamma-ray flux can now be expressed as

γ (Et hr ) = 0.94 × 10−13 cm−2 s−1

 mχ  
dNγi σi v 100 GeV 2
× dEγ J¯( ) , (5.100)
Ethr dEγ 10 cm3 s−1
−29 mDM
i

where Et hr is the lower threshold energy of the detector. Concerning the upper
limit of the integral, note that the dark matter moves at galactic velocity and
therefore its annihilation occurs (almost) at rest. We also notice that for typical
values of J¯( ) ∼ 1 − 103 (see Table 5.2), and WIMP cross section σ v 
10−26 cm3 s−1 (see Eq. 3.160) we expect a flux of 1 GeV photons around 10−2 − 10
per m2 per second. That is a very low flux, and the sensitivity of the experiments
dedicated to indirect detection searches should be extremely high.
5.8 Indirect Detection of Gamma Ray 319

5.8.2 Galactic Halo

A crucial ingredient for the calculation of J¯ in (5.99), and therefore for the
calculation of the flux of gamma rays, is the dark matter density profile of our galaxy.
As we discussed after Eq. (5.2), the different profiles that have been proposed in the
literature can be parameterized as

ρ0 [1 + (R0 /a)α ](β−γ )/α

ρ(r) = , (5.101)
(r/R0 )γ [1 + (r/a)α ](β−γ )/α

where ρ0 is the local (solar neighborhood) halo density, R0 the distance from the
Sun to the galactic center ( 8.5 kpc) and a is a characteristic length. The measured
value ρ0 = 0.3 GeV/cm3 is commonly used in analysis, since this is just a scaling
factor, and modifications to its value can be straightforwardly taken into account.
As we discussed in Sect. 5.2, the values of α, β, and γ are determined by N-body
simulations.

5.8.3 Adiabatic Compression Mechanism

Highly cusped profiles are deduced from N-body simulations. In particular,

Navarro, Frenk, and White obtained in 1996 a profile with a behavior ρ(r) ∝ r −1 at
small distances. A more singular behavior, ρ(r) ∝ r −1.5 , was obtained by Moore et
al. in 1999. To show how sensitive are these results to the ingredients you consider
in the simulation, let us take the example of the adiabatic compression process.
Indeed, these predictions are valid only for halos without baryons. One can improve
simulations in a more realistic way by taking into account the effect of the normal
gas (baryons). While they lose their energy through radiative processes falling
to the central region of the galaxy, they modify slowly (adiabatically) the mass
distribution. The resulting gravitational potential is then deeper, and the dark matter
must move closer to the center increasing its density, in other words, and adiabatic
compression.
The increase in the dark matter density is often treated using adiabatic invariants.
We assume spherical symmetry, circular orbit for the particles, and conservation of
the angular momentum M(r)r = constant, where M(r) is the total mass enclosed
within radius r. The mass distributions in the initial (r) and final (rf ) configurations
are therefore related by [M(r) + Mb (r)]r = [Mb (rf ) + M(r)]rf , where M(r) and
Mb (r) are, respectively, the mass profile of the galactic halo before the cooling of
the baryons at a distance r (obtained through N-body simulations), and the baryonic
composition of the Milky Way observed now. From this expression, one can deduce
rf as a function of r and then the new distribution of dark matter. A more precise
approximation can be obtained including the possibility of elongated orbits. In this
case, the mass inside the orbit, M(r), is smaller than the real mass, the one the
particle “feels” during its revolution around the galactic center. As a consequence,
320 5 In the Galaxies [T0 ]

Table 5.4 NFW and Moore et al. density profiles without and with adiabatic compression (NFWc
and Moorec , respectively) with the corresponding parameters, and values of J¯( )
a (kpc) α β γ J¯(10−3 sr) J¯(10−5 sr)
NFW 20 1 3 1 1.214 × 103 1.264 × 104
NFWc 20 0.8 2.7 1.45 1.755 × 105 1.205 × 107
Moore et al. 28 1.5 3 1.5 1.603 × 105 5.531 × 106
Moorec 28 0.8 2.7 1.65 1.242 × 107 5.262 × 108

the modified compression model is based on the conservation of the product M(r)r,
where r is the averaged radius of the orbit. The time average radii are given by
x ∼ 1.72x 0.82/(1 + 5x)0.085, with x ≡ r/rs , and rs the characteristic radius of the
assumed approximation.
The models and constraints applied for the Milky Way are summarized in
Table 5.4. There we label the resulting NFW and Moore et al. profiles with adiabatic
compression by NFWc and Moorec , respectively. As one can see, at small r, the dark
matter density profile following the adiabatic cooling of the baryonic fraction is a
steep power law ρ ∝ r −γc with γc ≈ 1.45(1.65) for a NFWc (Moorec) compressed
model. We observe for example that the effect of the adiabatic compression on an
NFW profile is basically to transform it into a Moore et al. one. The important
point we wanted to underline here was not specifically on the adiabatic compression
mechanism, but just to give an idea to the reader of why so many simulations can
give so many different results for the dark matter distribution in the Milky Way,
generating orders of magnitude of differences in the theoretical predictions of fluxes.
Of course, these results have important implications for the computation of the
gamma-ray fluxes from the galactic center. In particular, in Table 5.4 we see that
for  = 10−3 sr one has J¯NFWc /J¯NFW  145 and J¯Moorec /J¯Moore et al.  77.
Thus the effect of the adiabatic compression is very strong, increasing the gamma-
ray fluxes about two orders of magnitude. Similarly, for  = 10−5 sr one obtains
J¯NFWc /J¯NFW  953 and J¯Moorec /J¯Moore et al.  95.
Let us finally remark that the J¯ calculation is usually regulated by assuming a
constant density for r < 10−5 kpc. This procedure has consequences essentially for
divergent J¯ when γ ≥ 1.5. We use a cutoff in density, ρ(r < Rs ) = 0, Rs being
the Schwarzschild radius of a 3 × 106 M black hole. In addition we also consider
dark matter annihilation and estimate its effect by requiring an upper value for dark
matter density, ρ(r < rmax ) = ρ(rmax ) = mχ /( σ v.tBH ) ( tBH ∼ 1010 yr, being
the age of the black hole). We also apply the same procedure with a more realistic
cut-off value ρ(r < 10−6 pc) = 0 suggested by dark matter particle scatterings on
stars. This kind of effects can be significant only for the Moorec halo model and
increase the J¯ value by less than an order of magnitude for σ v.
5.9 The Tricky Case of the Galactic Center 321

5.9 The Tricky Case of the Galactic Center

5.9.1 The Idea

As we understood, astrophysical searches for dark matter are a fundamental part

of the experimental efforts to explore the dark sector. The strategy is to search
for dark matter annihilation products in preferred regions of the sky, i.e., those
with the highest expected dark matter concentrations and still close enough to
yield high induced fluxes at the Earth. For that reason, the galactic center, nearby
dwarf spheroidal galaxy (dSphs) satellites of the Milky Way, as well as local
galaxy clusters, are thought to be among the most promising objects for the
searches. In particular, dSphs represent very attractive targets because they are
highly dark matter dominated systems and are expected to be free from any other
astrophysical gamma-ray emitters that might contaminate any potential dark matter
signal. Although the expected signal cannot be as large as that from the galactic
center, dSphs may produce a larger signal-to-noise (S/N) ratio. This fact allows
us to place very competitive upper limits on the gamma-ray signal from dark
matter annihilation, using data collected by the Large Area Telescope onboard the
Fermi gamma-ray observatory for instance. These are often referred to as the most
stringent limits on dark matter annihilation cross section obtained so far.
Despite these interesting limits derived from dSphs, the galactic center is still
expected to be the brightest source of dark matter annihilations in the gamma-ray
sky by several orders of magnitude. Although several astrophysical processes at
work in this crowded region make it extremely difficult to disentangle the signal
from conventional emissions, the induced gamma-ray emission is expected to be so
large there that the search is still worthwhile. Furthermore, the dark matter density
in the galactic center may be larger than what is typically obtained in N-body
cosmological simulations. Ordinary matter (baryons) dominates the central region
of our galaxy. Thus, baryons may significantly affect the distribution as we just saw
when discussing the adiabatic compression phenomenon. It is also observed in many
cosmological simulations that include hydrodynamics and stars formation. If this is
the only effect of baryons, then the expected annihilation signal will substantially
increase.
In this section, we analyze in detail the constraints that can be obtained for
generic dark candidates from Fermi-LAT inner galaxy gamma-ray measurements
assuming some specific (and well motivated) dark matter distributions. This should
be seen as a concrete application of the theoretical approach we had in the previous
sections. Our approach is conservative, requiring simply that the expected dark
matter signal does not exceed the gamma-ray emission observed by the Fermi-LAT
in an optimized region around the galactic center. The region is chosen in such a
way that the S/N ratio is maximized. This kind of analysis is quite common in the
field of indirect detection searches.
322 5 In the Galaxies [T0 ]

5.9.2 Dark Matter Density Profiles

As we discussed in the preceding section, cosmological N-body simulations provide

important results regarding the expected dark matter density in the central region of
our galaxy. Simulations suggest the existence of a universal dark matter density
profile, valid for all masses and cosmological epochs. It is convenient to use the
following parameterization for the dark matter halo density, which covers different
approximations for dark matter density and is a condensed form of Eq. (5.101),
easier to manipulate analytically:
ρs
ρ(r) =     α  β−γ , (5.102)
γ α
r
rs 1 + rs
r

where ρs and rs represent a characteristic density and a scale radius, respectively.

The NFW density profile, with (α,β,γ ) = (1,3,1), is by far the most widely used in
the literature. Another approximation is the so-called Einasto profile
%  α &
2 r
ρEin (r) = ρs exp − −1 , (5.103)
α rs

which provides a better fit than NFW to numerical results. Finally, we will also
consider dark matter density profiles that possess a core at the center, such as the
purely phenomenologically motivated Burkert profile:

ρs rs3
ρBurkert (r) = . (5.104)
(r + rs ) (r 2 + rs2 )

Early results on the central slopes of the dark matter profiles showed some
significant disagreements between the estimates, with values ranging from γ = 1.5
to γ = 1. As the accuracy of the simulations improves, the disagreement became
smaller. For the Via Lactea II (VLII) simulation, the slope was estimated to be
γ = 1.24. A re-analysis of the VLII simulation and new simulations performed by
the same group gives the slope γ = 0.8 − 1.0, which is consistent with the Aquarius
simulation. Another improvement comes from the fact that the simulations now
resolve the cusp down to a radius of ∼ 100 pc, which means that less extrapolation
is required for the density of the central region.
Yet, there is an additional ingredient that is expected to play a prominent role in
the centers of dark matter halos: baryons. Although only a very small fraction of the
total matter content in the Universe is due to baryons, they represent the dominant
component at the very centers of galaxies like the Milky Way. Actually, the fact
that current N-body simulations do not resolve the innermost regions of the halos
is a minor consideration relative to the uncertainties due to the interplay between
baryons and dark matter. As we pointed out, the effect of the baryonic adiabatic
compression might be crucial for indirect dark matter searches, as it increases by
5.9 The Tricky Case of the Galactic Center 323

several orders of magnitude the gamma-ray flux from dark matter annihilation in
the inner regions, and therefore its detectability.
There is, however, another possible effect related to baryons that tends to
decrease the dark matter density and flatten the central cusp The mechanism relies
on numerous episodes of baryon infall followed by a strong burst of star formation,
which expels the baryons. At the beginning of each episode the baryons dominate
the gravitational potential. The dark matter contracts to respond to the changed
potential. A sudden onset of star formation drives the baryons out. The dark
matter also moves out because of the shallower potential. Each episode produces
a relatively small effect on the dark matter, but a large number of them results in a
significant decline of its density. Indeed, cosmological simulations that implement
this process show a strong decline of the dark matter density. Whether the process
happens in reality is still unclear. Simulations with the cycles of infall-burst-
expansion process require that the gas during the burst stage does not lose energy
through radiation, which is not realistic. Still, the strong energy release needed by
the mechanism may be provided by other processes and the flattening of the central
cusp may occur. If this happened to our galaxy, then the dark matter density within
the central ∼ 500 pc may become constant. This would reduce the annihilation
signal by orders of magnitude. We note that this mechanism would wipe out the
dark matter cusp also in centers of dwarf galaxies. Yet, recent works that also include
stellar feedback offer a much more complicated picture in which galaxies may retain
or not their dark matter cusps depending on the ratio between their stellar-to-halo
masses.
We show in Table 5.5 the values obtained after simulations for the different
parameters entering in the profiles, as well as an illustration of the density profiles
in Fig. 5.15.
The effect of baryonic adiabatic compression is clearly noticed in Table 5.5,
where at small r as a steep power law ρ ∝ 1/r γ with γ = 1.37 for NFWc [3],
which is in contrast to the standard NFW value, γ = 1 [4].
A value of γ = 1.37 is indeed perfectly consistent with what has been
found in recent hydrodynamic simulations, and it is also compatible with current
observational constraints (mainly derived from microlensing and dynamics) on the
slope of the DM density profile in the central regions of the Milky Way. Some
studies actually allow for even steeper adiabatically contracted profiles. Finally, for
the Burkert profile, we decided to choose a core radius of 2 kpc. This core size is
in line with that suggested by recent hydrodynamic simulations of Milky Way size
halos. For the normalization of the profile researchers choose the value of the local

Table 5.5 Dark matter Profile α β γ ρs [GeV cm−3 ] rs [kpc]

density profiles, following the
Burkert – – – 37.76 2
notation of
Eqs. (5.102–5.104) Einasto 0.22 – – 0.08 19.7
NFW 1 3 1 0.14 23.8
NFWc 0.76 3.3 1.37 0.23 18.5
324 5 In the Galaxies [T0 ]

106
NFW
105 Einasto
NFWc
104 Burkert
ρ [GeV cm–3]

103

102
101
100
10–1
10–2
0.001 0.01 0.1 1 10
r [kpc]

10 25

10 24
[GeV2 cm–5 sr]

10 23

10 22
NFW
Einasto
10 21
J(

NFWc
Burkert

1020
0 5 10 15 20
[deg]

Fig. 5.15 Top panel: dark matter density profiles, with the parameters given in Table 5.5. Bottom
panel: The J¯( ) quantity integrated on a ring with an inner radius of 0.5◦ (∼ 0.07 kpc) and
an external radius of  (R tan ) for the DM density profiles given in Table 5.5. Blue (solid),
red (long-dashed), green (short-dashed) and yellow (dot-dashed) lines correspond to NFW, NFWc ,
Einasto, and Burkert profiles, respectively. The four dark matter density profiles are compatible
with current observational data

density suggested by analysis of local motions of stars, giving ρ(r = 8.5 kpc) 
0.3 GeV cm−3 for all the profiles. The resulting profiles are all compatible with
current observational constraints. Note, however, that some recent work favors a
substantially larger core radius and a slightly higher normalization for Burkert-
like profiles. Let us finally point out that there are other possible effects driven
by baryons that might steepen the dark matter density profiles in the centers of
5.9 The Tricky Case of the Galactic Center 325

dark matter halos, such as central black holes, which we will not consider in this
textbook.

5.9.3 Gamma-Ray Flux from Dark Matter Annihilation

The gamma-ray flux from dark matter annihilation in the galactic halo has two main
contributions: prompt photons and photons induced via Inverse Compton scattering
(see Sect. 5.6.2). The former are produced indirectly through hadronization, frag-
mentation, and decays of the annihilation products or by internal bremsstrahlung, or
directly through one-loop processes (but these are typically suppressed in most dark
matter models). The second contribution originates from electrons and positrons
produced in dark matter annihilations, via Inverse Compton scattering of the
ambient photon background. The other two possible contributions to the gamma-
ray flux from dark matter annihilation can be neglected at the first approximation:
radiation from bremsstrahlung is expected to be subdominant with respect to Inverse
Compton scattering in the energy range considered (1–100 GeV) and a few degrees
of the galactic plane (see Fig. 5.14), and synchrotron radiation is only relevant
at radio frequencies, below the threshold of the main satellites or telescopes.
Thus the gamma-ray differential flux from dark matter annihilation from a given
observational region  in the galactic halo can be written as follows:
 
dγ dγ dγ
(Eγ ,  ) = + . (5.105)
dEγ dEγ prompt dEγ I CS

We discuss in detail both components in the next subsections.

5.9.3.1 Prompt Gamma Rays

A continuous spectrum of gamma rays is produced mainly by the decays of
π 0 ’s generated in the cascading of annihilation products and also by internal
bremsstrahlung. While the former process is completely determined for each given
final state of annihilation (bb̄, τ + τ − , μ+ μ− , and W + W − channels), the latter
depends in general on the details of the dark matter model such as the dark
matter particle spin and the properties of the mediating particle. For the prompt
contribution, we can directly take our Eq. (5.96)

dγ dNγi σi v ¯
= J ( ) . (5.106)
dEγ prompt i
dEγ 8πm2DM

This equation has to be multiplied by an additional factor of 1/2 if the DM particle

studied is its own antiparticle. But be careful, the Majorana case has a lot of
subtleties as we explain in detail in Sect. B.2.2. The discrete sum is over all DM
annihilation channels. As we underlined in the previous section, dNγi /dEγ is the
326 5 In the Galaxies [T0 ]

differential gamma-ray yield, σi v is the annihilation cross section averaged over

the velocity distribution of the dark matter, mDM is the mass of the dark matter
particle, and the quantity J¯( ) (commonly known as the J-factor) was defined in
Eq. (5.99):
 
1
J¯( ) ≡ d ρ 2 (r(l, )) dl . (5.107)
 l.o.s.

The J-factor accounts for both the dark matter distribution and the geometry
of the system. The integral of the square of the dark matter density ρ 2 in the
direction of observation  is along the line of sight (l.o.s), and r and l represent
the galactocentric distance and the distance to the Earth, respectively. Indeed, in
Eq. (5.106), all the dependences on astrophysical parameters are encoded in the J-
factor itself, whereas the rest of the terms encode the particle physics input.13 The
most crucial aspect in the calculation of J¯( ) is related to the modeling of the
DM distribution in the GC.
In the right panel of Fig. 5.15, the J¯( ) quantity corresponding to each
of the four profiles discussed in Sect. 5.9.2 is shown as a function of the angle 
from the GC. The associated observational regions  to each  are taken around
the GC. The angular integration is over a ring with an inner radius of 0.5◦ and an
external radius of . We have assumed a r = 0.1 pc constant density core for both
NFW and NFWc , although the results are almost insensitive to any core size below
∼1 pc. Remarkably, the adiabatic compression increases the DM annihilation flux
by several orders of magnitude in the inner regions. This effect can turn out to be
especially relevant when deriving limits on the DM annihilation cross section. We
also note that for the Burkert profile the value of J¯( ) is larger than for the
NFW and Einasto profiles. This is so because of the relative high normalization
used for this profile compared to the others and, especially, due to the annular
region around the GC where most studies are focused on, which excludes the GC
itself (where such cored profiles would certainly give much less annihilation flux
compared to cuspy profiles, see left panel of Fig. 5.15). We note, however, that the
use of another Burkert-like profile with a larger DM core than the one used here may
lead to substantially lower J¯( ) values, and thus to weaker DM constraints.
Notice finally that the NFWc profile reaches a constant value of J¯( ) for a
value of  smaller than the other profiles. This is relevant for optimization of the
region of interest for DM searches, since we see that for NFWc a larger region
of analysis will not increase the DM flux significantly as for NFW, Einasto, and
Burkert profiles. We show in Fig. 5.16 a typical region that is used to analyze data

13 Strictly
speaking, both terms are not completely independent of each other, as the minimum
predicted mass for halos is set by the properties of the dark matter particle and is expected to
play an important role also in the J-factor when substructures are taken into account. At a first
approximation, we do not consider the effect of substructures on the annihilation flux, as large
substructure boosts are only expected for the outskirts of dark matter halos, and thus they should
have a very small impact on inner galaxy studies.
5.9 The Tricky Case of the Galactic Center 327

Fig. 5.16 Example of a

region that we need to focus
on around the galactic center
(GC) to look for possible
traces of dark matter
annihilation

from the galactic center. Notice how we exclude the Milky Way bar and the pure
galactic center to maximize the signal-to-noise ratio. The aim of indirect searches
of dark matter is to see, in the white region, if any extra-photons can be observed,
and then constrain the annihilation cross section (or even exclude some models)
applying directly the theoretical prediction (5.106).

5.9.3.2 Gamma Rays from Inverse Compton Scattering

Electron and positron (e± ) fluxes are generated in dark matter annihilations mainly
through the hadronization, fragmentation, and decays of the annihilation products,
since direct production of e+ e− is suppressed by small couplings in most dark
matter models. These e± propagate in the galaxy and produce high energy gamma
rays via ICS of the ambient photon background. The differential flux produced by
ICS from a given observational region  in the galactic halo is given by
 
dIγCS σi v mDM dEI dNei± II C (Eγ , EI ; )
= (EI ) d ,
dEγ
i
8πm2DM me Eγ dEe Eγ

(5.108)

where EI is the e± injection energy,  corresponds to the angular position where

the ICS gamma rays are produced, and the function II C (Eγ , EI ; ) is given by
  EI PI C (Eγ , Ee ; x)
II C (Eγ , EI ; ) = 2Eγ dl dEe I˜(Ee , EI ; x) .
l.o.s. me bT (Ee ; x)
(5.109)
328 5 In the Galaxies [T0 ]

Here x = (l, ) and bT ∝ E 2 is the energy loss rate of the electron in the Thomson
limit. The function PI C is the photon emission power for ICS, and it depends on the
interstellar radiation (ISR) densities for each of the species composing the photon
background. It is known that the ISR in the inner galactic region can be well modeled
as a sum of separate black-body radiation components corresponding to star-light
(SL), infrared radiation (IR), and cosmic microwave background (CMB).
The last ingredient in Eq. (5.109) is the I˜(Ee , EI ; x) function, which can be given
in terms of the well-known halo function,

I (E, EI ; x) = I˜(E, EI ; x)[(bT (E)/b(E, x))(ρ(x)/ρ )2 ]−1 , (5.110)

where ρ is the DM density at Sun’s position and b(E, x) encodes the energy loss
of the e± . The I˜(Ee , EI ; x) function obeys the diffusion-loss equation

Ee2 ∂ b(Ee ; x) ˜
∇ 2 I˜(Ee , EI ; x) + I (Ee , EI ; x) = 0 , (5.111)
K(Ee ; x) ∂Ee Ee2

and is commonly solved by modeling the diffusion region as a cylinder with radius
Rmax =20 kpc, height z equal to 2L, and vanishing boundary conditions. Also the
diffusion coefficient K(E; x) can be considered homogeneous inside the cylinder
with an energy dependence following a power law K(E) = K0 (E/1GeV)δ . For
these three parameters L, K0 , and δ, the so-called diffusion coefficients, one usually
adopts three sets referred to as MIN, MED, and MAX models, which account for the
degeneracy given by the local observations of the cosmic rays at the Earth including
the boron to carbon ratio, B/C. The use of the different diffusion models, MIN,
MED, MAX, does not introduce a large variation in the dark matter constraints.
Let us finally make some remarks concerning the importance of the energy loss
function b(E; x). We develop the phenomenon of energy loss in the section devoted
to the synchrotron radiation (5.10) but we can say few words here. The two main
energy loss mechanisms of e± in the galaxy are the ICS and synchrotron radiation
produced by interaction with the galactic magnetic field. The former is the only
contribution to the energy losses that is usually considered, since it is the most
important one in studies of sources far from the GC. But when the e± energy reaches
several hundreds of GeV, synchrotron radiation can dominate the energy loss rate
due to the suppression factor in the ICS contribution in the Klein–Nishina regime.
By contrast, synchrotron radiation losses do not have this suppression and are driven
by the magnetic field energy density uB (x) = B 2 /2. Although the strength and exact
shape of the galactic magnetic field is not well known, in the literature it is broadly
described by the from

r − 8.5 kpc z
B(r, z) = B0 exp − − , (5.112)
10 kpc 2 kpc
5.9 The Tricky Case of the Galactic Center 329

Fig. 5.17 Galactic plane as observed by the FERMI telescope, and our region of analysis (left);
limit obtained on the annihilation cross section for different types of halo profiles (right) where we
considered a τ + τ − final state

normalized with the strength of the magnetic field around the solar system, B0 ,
which is known to be in the range of 1 to 10 µG. This field grows toward the
GC, and therefore one should expect that the energy losses are dominated by
synchrotron radiation in the inner part of the galaxy. On the other hand, we can
expect that when the magnetic field is stronger, the energy of the injected e±
is more efficiently liberated in the form of synchrotron emission resulting in a
softer spectrum, and producing therefore smaller constraints on the dark matter
annihilation cross section.
We present the result of a typical analysis in Fig. 5.17. On the left, we show
the galactic plane as observed by the FERMI telescope. The brightest regions
correspond to the highest fluxes of photons. We have superimposed on it the
drawing of the region of interest (ROI) that was used for the analysis. We have
excluded the galactic bar of the Milky Way. On the right side, the non-observation
of gamma-ray excess over the background gives limit on the annihilation cross
section of the dark matter (and, as a consequence, on the model) when comparing
the data with Eq. (5.106). We find that cross sections above  10−24 cm3 s−1 are
excluded for a 100 GeV dark matter in the case of an NFW halo, whereas a
compressed NFW halo excludes σ v  5 × 10−27cm3 s−1 for the same mass.
Notice that in the later case, thermal dark matter would be excluded because it
predicts σ v  3 × 10−26cm3 s−1 (3.160, and horizontal line in the figure). The
final state was chosen to be τ + τ − and results can slightly change for different final
states, due to their different shapes in the spectrum. Indeed, some final states have
tendency to radiate more photons than other ones and can then put stronger limits
on the annihilation cross section σ v. We also see that, in this specific final state,
the ICS is clearly subdominant and begins to restrict the parameter space only for
dark matter masses above the TeV scale.
330 5 In the Galaxies [T0 ]

5.10 Dark Matter and Synchrotron Radiation

As we discussed in great detail in Sect. 5.6.3, an electron moving in the magnetic

field(s) generated inside our galaxy will lose energy by synchrotron (cyclotron in the
non-relativistic case) radiation. We can then easily understand that, if dark matter
particles annihilate into charged particles, the e+ e− pair being the simplest of the
possible final states, experiments can observe the effects of these charged particles
that should produce synchrotron emissions. The task is far to be easy but reveals
to be promising. For that, one needs to understand in detail the behavior of an
electron in a magnetic field. Generically speaking, for an electron propagating in
the interstellar medium, the diffusion-loss equation for its density Ne can be written
as:

∂ dNe ∂ dNe
= b(Ee , r) + Q(Ee , r), (5.113)
∂t dEe ∂Ee dEe

where
 2
1 ρ(r) dNe
Q(Ee , r) = σ v (5.114)
2 mχ dEe

is the electrons injected by dark matter annihilation at a constant rate (per second
per GeV) and b(E) represents the energy loss rate, i.e. the energy lost per electron
in 1 s at the energy E. It is time to explain and analyze this equation step by step.

5.10.1 Neglecting Diffusion

The number of electrons with energies lower than Ee is given by

 Ee dNe
Ne (Ee ) = dEe .
0 dEe

During 1 s, the electrons of energy E( Ee ) lose an amount of energy dE dt (E) =

b(E). One can thus deduce that all electrons having an energy between Ee and
Ee + b(Ee )dt will have their energy below Ee after 1 s. This number is dN
dE × dt =
e dE
dNe dNe
b(Ee ) dE . In other words, N(Ee ) → N(Ee ) + b(Ee )dt dEe during the time dt. Per
GeV and per second, the number of particles
 “lost”
 at energy Ee (i.e. having their
dNe
energy dropped below Ee ) is then ∂Ee b(Ee ) dEe , which represents the loss energy
term in Eq. (5.113).
If we suppose that the transport timescale of the particle (which will fly away
from its production point) is much longer than the cooling timescale responsible
for the loss of energy, a steady state equation (“steady state” in the sense that we
5.10 Dark Matter and Synchrotron Radiation 331

supposed that the galaxy has reached an equilibrium) can be written as:

Number of particles created between Ee and Emax + number of particles lost by

cooling effects = 0,

or after integration of Eq. (5.113) and reminding that dNe /dEe (∞, r) = 0.
 ∞ dNe
dEe Q(Ee , r) + 0 − b(Ee ) (Ee , r)
Ee dEe
 mDM dNe
= dEe Q(Ee , r) − b(Ee ) (Ee , r) = 0
Ee dEe

implying
 mDM
dNe Ee dEe Q(Ee , r)
(Ee , r) = . (5.115)
dEe b(Ee )

b(E) represents the loss energy rate and can have different origins: synchrotron
radiation (discussed below), Inverse Compton, bremsstrahlung, or Coulomb. We
will note them bsyn , bI C , bbrem , and bcom , respectively: b(E) = bsyn (E)+bI C (E)+
bbrem (E) + bcom (E).

5.10.2 Synchrotron Loss of Energy

As we just saw, among the indirect dark matter detection channels, gamma-ray
emission represents one of the most promising opportunities due to the very low
attenuation in the interstellar medium and to its high detection efficiency. Positrons
and protons strongly interact with gas, radiation, and magnetic field in the galaxy
and thus the expected signal sensibly depends on the assumed propagation model.
However, during the process of thermalization in the galactic medium the high
energies e+ and e− release secondary low energy radiation, especially in the
radio and X-ray band, which, hence, can represent a chance to look for dark
matter annihilation. A synchrotron emission is the relativistic limit of the cyclotron
radiation given by the Larmor’s formula (see Sect. 5.4.2 for more details concerning
radiative loss of energy), which gives the total power radiated by a non-relativistic
point charge as it accelerates (or decelerates):

e2 2 e2 2
SI
PLarmor = |a|2; CGS
PLarmor = |a| , (5.116)
6π0 c3 3 c3

where a stands for v̇ and SI or CGS (Centimeter–Gram–Second system) refers

to the metric system (Longair versus Rybicki convention, see Eq. (5.31) for more
332 5 In the Galaxies [T0 ]

details). We will give the results in both conventions as different authors use freely
different types of conventions through their works. One can express the Larmor’s
formula as a function of the B-field after applying the equation of motion

dv ev⊥ B
m = ev × B ⇒ v = cste; v̇⊥ = a⊥ = [SI]
dt m
v e ev⊥ B
m = v × B ⇒ v = cste; v̇⊥ = a⊥ = [CGS]. (5.117)
dt c mc
Implementing Eq. (5.117) in Eq. (5.116) one obtains

e4 B 2 v⊥
2
2 e4 B 2 v⊥
2
SI
PLarmor = ; CGS
PLarmor = . (5.118)
6π0 m2 c3 3 m2 c 5

The cyclotron Larmor’s formula transforms into a synchrotron radiation for very
relativistic electrons, with the replacement14

m → γm ; (|a |2 + |a⊥ |2 ) → γ 4 (γ 2 |a |2 + |a⊥ |2 ). (5.119)

Implementing in Eq. (5.118) one can write

 2
e4 B 2 γ 2 v⊥
2
B
P = = σ SI
T 0  γ 2 2 2 3
β ⊥ B c = 2σT
SI 2 2
γ β ⊥ c [SI]
6π0 m2 c3 2μ0
4 SI 2 2 SI
⇒ P = σ cγ β UB , (5.120)
3 T
where we used

1 2
2
β⊥  = β 2 sin2 θ  = sin2 θ dφd cos θ = . (5.121)
4π 3

For the CGS system,

2  2
2e4 B 2 γ 2 v⊥ 1 CGS 2 2 2 B
P = 2 5
= σT cγ B v⊥ = 2σT
CGS 2 2
γ β ⊥ c [CGS]
3m c 4π 8π
4 CGS 2 2 CGS
⇒ P = σ cγ β UB , (5.122)
3 T

14 Remember that in special relativity the Lorentz transformation on accelerations gives a = γ 3 a
and 
a⊥ = γ 2 a⊥ ,see Eq. (5.63).
5.10 Dark Matter and Synchrotron Radiation 333

which at the end can be finally written universally as

4 i 2 2 i
P = σ cγ β UB (5.123)
3 T
 2  2 2
e2
with σTSI = 8π 3 4π0 mc 2 , σT
CGS
= 8π
3
e
mc2
, UBSI = B/2μ0 and UBCGS =
B/8π. σT is the Thompson cross section (see Sects. 3.3.4.2 and 5.4.4), equal to
0.665 × 10−24cm2 for electrons. In a third step, one needs to transform Eq. (5.120)
as a function of the energy of the electron Ee . If it is considered as relativistic, this
energy can be written as Ee2 = p2 c2 , with p = γ mv  γ mc, and one obtains
Ee2 = γ 2 m2 v 2 c2  γ 2 m2 c4 , which gives us

2  2 2
e 4 B 2 E⊥ 2 e4 B 2 Ee2 Ee B
P = ⇒ P  = = P0 (5.124)
6π0 m4 c5 3 6π0 m4 c5 GeV μG

with P0 = 2.51 × 10−18 GeV s−1 . One can also write Eq. (5.124) with a more
explicit dependence on the energy density of the magnetic field B:
  2
4 e4 Ee2 B2 4 e4 Ee2 Ee UB
P = = UB = P̃0
3 6π02 m4e c7 2μ0 3 6π02 m4e c7 GeV ev/cm3
(5.125)

with P̃0 = 1.02 × 10−16 GeV s−1 . One can apply Eq. (5.125) to the Inverse
Compton loss of energy process. Indeed, this process is generated by the interaction
of an electron with a photon from the background of energy density Urad . One can
 2  
Ee Urad
then write bI C (Ee ) = P̃0 GeV ev/cm 3 . The relevant radiation background for
the Inverse Compton scattering is given by an extragalactic uniform contribution
consisting of the CMB, with UCMB = 8π 5 (kT )4 /15(hc)3  0.26 eV/cm3 ,
the optical/infrared extragalactic background, and the analogous spatially varying
galactic contribution, the Interstellar Radiation Field (ISRF). The ISRF intensity
near the solar position is about 5 eV/cm3 and reaches values as large as 50
eV/cm3 in the inner kpc’s. Combining the synchrotron and Inverse Compton loss of
energy processes, and forgetting in the first approximation the bremsstrahlung and
Compton effects, one obtains

b(E) = bsyn (E) + bI C (E) = (5.126)

    
2
E 2 B 2
E Urad
2.51 × 10−18 + 1.02 × 10−16 GeV s−1 .
GeV μG GeV eV/cm3
334 5 In the Galaxies [T0 ]

One can see from Eq. (5.126) that the Inverse Compton scattering dominates the loss
energy process up to a magnetic field B  50µG.
To compare our prediction with the experiment, one needs to convert the power
spectrum, which is a function of the energy of the electron, into a spectrum in
frequencies ν. We do so considering that the total energy emitted between ν and
ν + dν is equal to the energy emitted between E and E + dE: P (ν)dν =
n(E, r)P (E)dE, or P (ν) = n(E, r)P (E)dE/dν. P (ν) is called the emissivity
and n(E, r) = dN/dE is the density number of particles with energy comprised
between E and E + dE. We thus see that we need to know the “conversion”
factor dE/dν to compute the emissivity; in other words, we need to express the
frequency of the synchrotron radiation as a function of the electron energy. In
fact, the synchrotron spectrum of an electron gyrating in a magnetic field has its
prominent peak at the resonance frequency νc
  2   −1/2
3 3eE 2 B B E ν B
νc = γ 2 νL = = ν0 ⇒ E(ν)  GeV,
2 4πm3e c4 µG GeV ν0 μG
(5.127)

where νL is the Larmor’s frequency (5.44) and ν0 = 16.02 × 106 Hz.

Exercise Derive the expression above, taking the relativistic limit of the cyclotron
frequency Eq. (5.44) and applying a mean on the solid angle.

This implies that, in practice, a δ−approximation around the peaks ν  νc works

extremely well. To compute the total fluxes Fν (Watt per meter square per second
per Hertz), one needs to integrate the expression (5.115) along the line of sight (los)
and the solid angle d :
 
dE 1 dN
Fν = P (E) × × d ds (r). (5.128)
dν 4π los dE

Combining Eqs. (5.115), (5.124), (5.127), and (5.128) one obtains for the flux,
expressed in Jansky (10−26 Wm−2 Hz−1 ):
⎛ ⎞
 ν −1/2  B −1/2
⎜ 1 ⎟
Fν = 2.92 × 10 Jy × ⎝
7
 2 ⎠ × ×
P̃0 GHz μG
1+ P0
B
μG
 mDM   m 
dNsource σv DM −2
× dE × ×
Eν dE 3.10−26cm3 s−1 GeV
 2  
ρ0 1 ds(r, θ ) 2
× 3
× d f (r), (5.129)
GeV/cm 4π kpc
5.10 Dark Matter and Synchrotron Radiation 335

where we have defined ρ(r) = ρ0 × f (r), ρ0 = 0.3 GeV cm−3 being the local dark
matter density. The integral on the solid angle and line of sight can be developed
using
   θ  D(1−sin θ  )
1 1
d ds(r, θ ) = sin θ  dθ  dl, (5.130)
4π 2 0 0

where D = 8.5 kpc is the distance from the observer (on Earth) to the galactic
center. The angle θ is the one formed between the√direction of D and the line of
sight, and l is related to r through the relation r = D 2 + l 2 − 2Dl cos θ .
Concerning the loss of energy, we only considered the synchrotron and Inverse
Compton process. If one takes into account Coulomb losses and Bremsstrahlung,
one would obtain
 2 2  2
E B E
b(E) = P0 + P̃0 (5.131)
GeV μG GeV
+b0Coul n(1 + log(γ /n)/75) + b0Brem n(0.36 + log(γ /n)), (5.132)

where n is the mean number density of electrons in cm−3 , and the average over
space gives about n  1.3 × 10−3 , γ = E/me and b0Coul  6.13 × 10−16, b0Brem 
1.51 × 10−16 in GeVs−1 units.

5.10.3 Taking into Account Spatial Diffusion∗∗

If we take into account the spatial diffusion, Eq. (5.113) becomes

∂n(x, E) ∂
= ∇ · [K(x, E) ∇n(x, E)] + [b(x, E)n(x, E)] = q(x, E) ,
∂t ∂E
(5.133)

where b(x, E) encodes the energy loss rate, n(x, E) = dN/dE is the number
density of electron per energy, and q(x, E) is the source term. The problem is more
complex by the dependence of the processes on the position x. As we discussed
above, cosmic ray electrons lose energy mainly through synchrotron radiation and
Inverse Compton scattering (IC), with a rate b(x, E) that at the galactic medium is
typically of the order 10−16 GeV s−1 . Additional bremsstrahlung losses of electron
energies in the interstellar medium are neglected in our approach, although they
may have a very small effect, since for the synchrotron frequency we work with, the
electron energies of interest are around 1 GeV.
Assuming a steady state, Eq. (5.133) can be re-expressed as

∂ ñ(x, E)
− K0 ñ(x, E) = q̃(x, E) (5.134)
∂ t˜
336 5 In the Galaxies [T0 ]

with K(x, E) = K0 E δ and in which the derivative with respect to energy has been
parameterized in terms of the parameter t˜ ≡ − dE(E δ /b(x, E)). If at the level
of propagation one considers that energy losses have an average value over all the
diffusion regions, i.e. b(x, E) ≈ b(E) = E 2 /τ GeV s−1 , as in Eq. (5.126), then
in (5.134) ñ(x, E) = E 2 n(x, E) and q̃(x, E) = E 2−δ q(x, E). The solution of this
equation can be found in the Green function formalism to be
 ∞
1
n(x, E) = dEs (5.135)
b(E) E

× d 3 xG(x, E ← xs , Es )q(xs , Es ),
DZ

where the volume integral is over the diffusion zone (DZ). The Green function
G(x, E ← xs , Es ) gives the probability for an electron injected at xs with energy
Es to reach x with energy E < Es and has a general solution of the form
τ
G(x, E ← xs , Es ) = G(x, t˜ ← xs , t˜s ) (5.136)
E2
 3/2 2
1 − (x)
G(x, t˜ ← xs , t˜s ) = e 4K0 t˜ , (5.137)
4πK0 t˜

where t˜ = t˜ − t˜s and (x)2 = (x − xs )2 . The unique argument of the Green
function is actually the diffusion length λ = 4K0 t˜ because the energy dependence
enters only in this combination. This is the characteristic length of an electron
traveling during its propagation. Here we are going to focus on diffusion models
for which the half-thickness Lh ∼ 4 kpc is small compared to the radius of the disk
Rh , such that in practice the radial boundary has negligible effect on propagation.
The Green function for which n(x, E) vanishes at z = ±Lh may be expressed as
n=∞
1
(−1)n exp−(2nL+(−1)
n z )2 /λ2
G̃(λ, L) = √ s
. (5.138)
( πλ)3 n=−∞

As for the source term, in the case of production from DM annihilation with cross
section σ v, it can be expressed as
% &2
ρ(x) dN
q(x, E) = η σ v . (5.139)
mDM dE

Here ρ(x) is the DM profile, given in units of GeV/cm3 , and η = 1/4 or η =

1/2 depending on the Dirac or Majorana nature of DM. The injection spectrum of
electrons is given by dN/dE.
5.10 Dark Matter and Synchrotron Radiation 337

With all these ingredients we can express the electron number density as
 2  mDM
ρ 1 dN
n(x, E) = η σ v dEs Ihalo . (5.140)
mDM b(E) E dEs

Here I (λ) is the so-called halo function, which is defined as

 % &2
ρ(xs )
Ihalo = d xs G̃(x, E ← xs , Es )
3
. (5.141)
DZ ρ

When electrons (and positrons) are created in the galaxy, they are accelerated by
the local magnetic field and produce synchrotron radiation with an energy flux per
unit frequency ν per solid angle (or spectral energy distribution) given by
  mDM
1
Fν = dl dE P (x, ν, E) n(x, E), (5.142)
4π los me

where the integration is performed along the line of sight (los) and on the electron
energies. In this relation P (x, ν, E) is the synchrotron power (per unit frequency)
emitted at ν by an electron of energy E that, for an energy distribution of electrons
n(x, E), must be integrated over all the electron energies E that lead to synchrotron
radiation at the same frequency ν. In practice we saw that the radiation power as a
function of ν is strongly peaked near a so-called critical frequency νc , defined as
 2
3eBE 2 E B
νc = ≡ 16MHz (5.143)
4πm3e GeV µG

in natural units, so that there is a near one-to-one correspondence between ν and E

that, we take to be such that

4 e3 B ν
P (ν, E) ≈ δ − 0.29 . (5.144)
27 me νc (E)

Indeed, the peak of the synchrotron radiation, when numerically solved, is not
exactly at ν = νc but more precisely at ν = 0.29 νc .
Using this handy approximation, which, in one form or another, is often adopted
in the literature, the flux of Eq. (5.142) takes a simple form,

1 E
Fν = dl Psync (E) n(x, E), (5.145)
4π los 2ν

where
1/2   1/2
ν µG
E ≡ E(ν) ≈ GeV, (5.146)
4.7MHz B
338 5 In the Galaxies [T0 ]

which stems from ν = 0.29 νc (E), and where Psync is the total synchrotron energy
loss rate of an electron of energy E,15

e4 B 2 E 2
Psync (E) =
ˆ dνP (ν, E) =
9πm4e
 
−18 E 2 B 2
 2.5 · 10 GeV s−1 . (5.147)
GeV µG

Finally, we may express the synchrotron energy flux as

% 
8 Jy
η σ v
Fν = 1.21 × 10 (5.148)
sr 2 3.1 × 10−26cm3 /s
  2
1GeV 2 ρ μG 1/2
×
mDM GeV/cm3 B
 1/2   mDM
GHz Psync (E) 1 dl dN
× dEs Ihalo ,
ν b(E) 4π kpc E dEs

using jansky flux units (Jy), 1 Jy = 10−26 W·m−2 ·Hz−1 . The total energy loss
assumed here is given by (see also Eq. (5.90))

b(E) = Psync (E)(1 + rIC/sync ), (5.149)

where
  −2
2 Urad Urad B
rIC/sync =  2 (5.150)
3 B 2 /2 8 eV/cm3 10 µG

is the ratio between IC and synchrotron energy loss and Urad is the total radiation
density.

5.10.4 General Astrophysical Setup∗

5.10.4.1 Astrophysical Uncertainties

As we discussed when analyzing gamma ray from the galactic center in Sect. 5.9.2,
studies coming from N-body simulations have led to popular expressions for the
distribution of DM in the galactic halo, like the Navarro–Frenk–White (NFW)

15 In these expressions, we are a bit loose regarding the definition of the magnetic field. In principle
B should be the effective magnetic field felt by the electron, B⊥ . In practice, assuming an isotropic
distribution of electron velocities, B⊥ 2  = 2/3B 2 . This is explicit in Eq. (5.147). For convenience,

this factor of 2/3 has been included in the definition of Eq. (5.144).
5.10 Dark Matter and Synchrotron Radiation 339

density profile
ρs
ρNFW (r) =  2
. (5.151)
r r
1+
rs rs

That is a specific case of Eq. (5.102) or the Einasto profile (5.103)

%  α &
2 r
ρEIN (r) = ρs exp − −1 , (5.152)
α rs

where r is the radial distance from the center of the DM halo, and (ρs , rs ) are
parameters that are fitted in the simulation to recover astronomical observables.
On the other hand, observations of galactic rotation curves as well as some of the
simulations that include also baryonic feedback on DM density find DM density
profiles that are more cored toward the inner regions of the galaxy. One example of
such profile is the isothermal profile

rs2
ρiso (r) = ρs (5.153)
r 2 + rs2

or modified Einasto profile (in this case the parameter α is smaller when compared
to the parameters found in simulations that contain only DM component). While
the parameter α for the Einasto profile is fixed from a fit to the simulations, the
values of parameters ρs , a typical scale density, and rs , a typical scale radius for
the Milky Way, are determined from astrophysical observations (e.g. local stellar
surface brightness, stellar rotational curves, total Milky Way mass within a given
distance. . . ). There exist other alternatives like Burkert-like profile of Eq. (5.104)
that we will, however, not consider in this section. As we will see in Sect. 5.10.4.2,
the ratio of the synchrotron signal calculated with DM density of these three profiles
gets smaller at higher latitudes, as at those distances (closer to the solar position)
DM density is better constrained. As the rotational curve measurements are poor
at distances smaller than 2 kpc (or ∼ 10◦) from the galactic center and those
regions of the galaxy are baryon dominated, the chosen region of interest (ROI) for
analysis usually spans |b| ∈ (10 ± 3)◦ in galactic latitudes, and |l| < ◦
∼ 3 in galactic
longitudes.
Together with DM density profile, the cosmic ray propagation parameters
pose one of the main uncertainties in prediction of the synchrotron signal. This
corresponds, for instance, to different values of the width of the Milky Way where
the diffusion takes place (Lh ) or the spatial diffusion parameter K0 in Eq. (5.134).
Three typical models used in the literature are called MIN/MED/MAX and featured
in Table 5.6. They are used to probe the uncertainty in cosmic ray propagation
parameters. Originally, those parameters were derived to produce the maximal,
median, and minimal anti-proton flux from dark matter, while being compatible
with the cosmic ray secondary to primary B/C ratio measurement. Therefore, by
340 5 In the Galaxies [T0 ]

Table 5.6 Upper three rows: parameter sets derived using a semi-analytical approach, to lead to
MIN/MED/MAX anti-proton fluxes at Earth from an exotic galactic component. Lower four rows:
parameters consistent with a fit to cosmic ray data, and shown to reproduce the gamma-ray diffuse
data well. Last row: plain diffusion model, shown to be consistent with the radio data at 22 MHz–94
GHz frequencies
Model Lh [kpc] K0 [cm2 s−1 ] δ va [km s−1 ]
MIN 1 4.8 1026 0.85 0
MED 4 3.4 1027 0.70 0
MAX 15 2.3 1028 0.46 0
1a 4 6.6 1028 0.26 34.2
1b 4 6.6 1028 0.35 42.7
2a 10 1.2 1029 0.3 39.2
2b 10 1.05 1029 0.3 39.2
PD 4 3.4 1028 0.5 0

construction, they do not necessarily capture the uncertainty in the electron fluxes in
the inner galaxy, which is of interest here. In Sect. 5.10.4.3 we will comment in more
detail on the impact of a choice of cosmic ray parameters, exploring additional sets
consistent with the cosmic ray data and which were i) derived numerically and ii)
shown to reproduce the observed whole sky gamma ray or radio emission, therefore
probing more directly the signals in the inner galaxy.
The galactic magnetic field (GMF) is considered possibly the most important
ingredient when dealing with synchrotron radiation. In the diffuse interstellar
medium it has a large scale regular component as well as a small scale random
part, both having a strength of order micro-Gauss. The best available constraints
in determining the large scale GMF are Faraday rotation measures and polarized
synchrotron radiation, while random component is deduced mainly based on the
synchrotron emission. Several 3D models of the large scale magnetic field have
been developed the recent years. It is common to use a simple parameterization for
a total magnetic field as customary in the literature:

r −r |z|
B(ρ, z) ∝ exp − − . (5.154)
Rm Lm

The parameters Rm and Lm should in principle depend on the diffusion model

assumed (or vice versa), since the propagation in the galactic medium is intimately
related with the magnetic field. We took Lm = δ · Lh and Rm = δ · Rh in Table 5.6.
An actual extent of Lm and Rm does not play a critical role (the difference between a
constant magnetic field and the exponential form defined above is < ∼ 30%), as long
as the field extends into the region of interest (i.e. Lm >
∼ 1 kpc). The normalization at
Sun’s position B is more or less well constrained and is evaluated to B  6 µG,
consistent with the present measurements. However, the value of the field in the
5.10 Dark Matter and Synchrotron Radiation 341

inner galaxy is considerably less known and we can rewrite the normalization
of (5.154) as

B0 ≡ B [1 + K (RI G − r)], (5.155)


where (RI G − r) is the unit step function as a function of r = ρ 2 + z2 . With
this change, while leaving B unchanged locally we allow for the magnetic field to
have a higher effective normalization B (1 + K) in the inner galaxy (IG). A typical
value of RI G is RI G = 2 kpc. Now that we have fixed the spatial dependence of
the B-field, we will explore the impact of overall normalization B (1 + K) in the
inner galaxy, on synchrotron fluxes in Sect. 5.10.4.4.
Contributions to the energy losses for electrons are assumed to come only from
synchrotron and IC processes in our semi-analytical approach, as commented above.
In principle, the radiation density Urad (see Eq. (5.150)) has a spatial profile, which
affects the synchrotron flux estimations. However, in the semi-analytical estimations
used here we take Urad to be constant, which turns out to be a good approximation
for any analysis, as is justified in the next section. A value of Urad 8 eV cm−3 is
usually considered in the literature.16 Concerning the frequency of observation we
focus on the data taken at 45 MHz. Indeed, in a wide range of frequencies (22 to
1420 MHz) the change in the measured synchrotron flux is very small, while going
to lower frequencies maximizes a synchrotron signal of low-mass WIMPs.

5.10.4.2 Synchrotron Signal for Different Choices of DM Density Profile

In the remainder of the text we will focus on two DM profiles, NFW (Eq. 5.151) and
isothermal (ISO) (Eq. 5.153), using the following values of parameters, consistent
with observations: ρs = 0.31 GeV cm−3 , rs = 21 kpc, for an NFW profile, and ρs =
1.53 GeV cm−3 and rs = 5 kpc for isothermal profile.17 However, in this section we
also show the prediction for the synchrotron signal in the case of a modified Einasto
profile. In particular, this profile is modified to have a shallower inner slope than the
usual Einasto profile found in DM-only simulations and it describes better results
of simulations that include baryonic feedback (parameters we use are α = 0.11,
rs = 35.24 kpc, ρs = 0.041 GeV cm−3 ). As this profile is “bulkier” at distances 1
kpc from the GC, the DM signals are generally higher than those of NFW in that
region.

16 Note, however, that the conditions in the inner galaxy might be quite different from a simple CR

propagation setup assumed here. In particular, observations of the bubble-like structures centered
at the galactic center and extending to 50 deg in latitudes in gamma rays and WMAP haze at
microwave frequencies, observed by the WMAP satellite and Planck satellite, witness of possibly
more complicated configuration of the magnetic fields and CR propagation parameters in that
region. While bubble-like structures appear subdominant with respect to the standard components
of the diffuse emission, we caution that before their origin is understood, the actual structure of
magnetic fields or the CR propagation conditions cannot be reliably modeled.
17 Note that we make a conservative choice by choosing a rather extended core. Smaller values of

rs would result in fluxes more similar to those obtained with the NFW profile.
342 5 In the Galaxies [T0 ]

45 MHz, ,m 10 GeV 45 MHz, bb, m 10 GeV

1 109
5 108
5 108
'z 4 kpc, 1a', EinB 'z 4 kpc, 1a', EinB
'z 4 kpc, 1a', NFW 'z 4 kpc, 1a', NFW
8 'z 4 kpc, 1a', ISO 'z 4 kpc, 1a', ISO
1 10 1 108
K

K
5 107 5 107
2.5

2.5
MHz

MHz
1 107
1 107
T

T
6
5 106
5 10

1 106
1 106

0 20 40 60 80 0 20 40 60 80
b deg b deg

Fig. 5.18 Comparison between synchrotron signals for a DM mass of 10 GeV annihilating to
muons (left) and b quarks (right) for three DM density profiles: modified Einasto, NFW, and
ISO thermal profile. ρ = 0.43 GeV cm−3 is assumed for this plot (see text for the remaining
parameters) and propagation of electrons is done using a CR propagation setup as shown in
Table 5.6

The local value of DM density is set to ρ = 0.43 GeV cm−3 . One should also
keep in mind that the overall normalization of DM distribution ρ is uncertain,
being in the (0.43 ± 0.113 ± 0.096) GeV cm−3 range18. Therefore, in addition to the
differences in the signal caused by the DM profile shape, and shown in Fig. 5.18,
synchrotron signals scale with ρ 2 . We usually express the flux Fν in terms of the
2hν 3
brightness temperature of the radiation, T , where Fν = c2 hν
1
is the usual
e kT −1
black-body relation.

5.10.4.3 Synchrotron Signal for Different Choices of Cosmic Ray

Parameters
As discussed above, MED/MIN/MAX sets of CR propagation parameters were
derived using a semi-analytical description of CR propagation, and a fit to B/C
measurement, with a requirement to produce minimal, medium, and maximal
DM generated anti-proton fluxes, at a solar position. We show a comparison of
synchrotron signal calculated with these choices of CR propagation parameters
in Fig. 5.19. We see that the propagation parameters can generate one order
of magnitude of uncertainties in the synchrotron flux produced by dark matter
annihilation.

5.10.4.4 Synchrotron Signal for Different Choices of Magnetic Field∗∗

In this section we study how dark matter limits change depending on the assump-
tions of the overall normalization of the magnetic field in our ROI. It has already
been noticed that for a fixed electron injection spectrum there exists an optimum
value for the magnetic field that maximizes the synchrotron flux at a given

18 However,
depending on the analysis, the uncertainty window on this value can vary in the 0.2–0.8
GeV cm−3 range.
5.11 Sommerfeld Enhancement 343

45 MHz, ,m 10 GeV 45 MHz, bb, m 10 GeV

109
109 MAX
MAX
MED
MED
MIN
MIN
z 4 kpc, model 1a 8
10 z 4 kpc, model 1a
108 z 4 kpc, model 1b z 10 kpc, model 2b
z 10 kpc, model 2a
K

K
PD
z 10 kpc, model 2b
2.5

2.5
PD
MHz

MHz
107
107
T

T
106
106

0 20 40 60 80 0 20 40 60 80
b deg b deg

Fig. 5.19 Comparison between synchrotron signals for a DM mass of 10 GeV annihilating to
muons (top) and b quarks (bottom figure) for different CR propagation setups, detailed in Table 5.6.
NFW DM profile is assumed here

synchrotron frequency. However, in this section we want to understand this fact

in more detail. As can be seen in Eq. (5.148), the magnetic field influences the flux
through the energy losses, and through the electron energy. Indeed, given a specific
frequency ν, and a given annihilation channel, it can be expressed in the following
form:

B2 1
F (ν, B) ∝ √ , (5.156)
α + B2 B

where α represents here the rest of energy losses, here assumed to be only IC.
Note that since both synchrotron and IC losses scale with energy as E 2 , the energy
dependence cancels in this particular analysis. From (5.156) one observes two
extremal cases: one in which synchrotron loss is negligible with respect to the rest
of energy losses, for which the flux scales with B as F ∼ B 3/2 , thus increasing as
B increases, and the other, in which synchrotron
√ is actually the dominant energy
loss, after which the flux scales as F ∼ 1/ B, thus decreasing as B increases.
In other words, there will be an intermediate value of the magnetic field for which
synchrotron becomes the dominant energy loss, and this value is actually the one
maximizing the flux.
Figure 5.20 shows the shape of synchrotron flux as a function of the value of
magnetic field at GC. Taking into account only IC (apart from synchrotron of
course), one can have an idea about the maximum of the flux already by direct
differentiation of (5.156), assuming the values of α correspondent
√ to this case. The
value of B for which the flux is maximal scales as BGC max
∝ Urad . For Urad = 8
eV/cm3 , BGCmax
 26 µG.
344 5 In the Galaxies [T0 ]

1e+09
Synchrotron temperature [K ( /MHz)2.5]

1e+08

1e+07

1e+06

mX = 10 GeV
mX = 100 GeV
100000
2 4 6 8 10 20 40 60 80 100
BGC (PG)

Fig. 5.20 Flux predicted at 10 deg off the GC, as a function of the value of magnetic field at
the GC (roughly, at our ROI, we have BROI ≈ 0.5BGC ). Lines represent the results in the semi-
analytical approach. We assume a DM annihilating directly to electrons, using the NFW profile
and the MED diffusion model

5.11 Sommerfeld Enhancement

5.11.1 Generalities

The limit on the annihilation cross section we just computed, from γ -ray, anti-
matter, or synchrotron radiation, can be largely modified by an effect called the
“Sommerfeld enhancement.” The Sommerfeld enhancement is an elementary effect
in non-relativistic quantum mechanics, which accounts for the effect of a potential
on the interaction cross section. This enhancement is named after Sommerfeld
who proposed it in 1931. More technical details are developed in Sect. B.6. The
application of this effect has been used extensively in the dark matter sector to
explain (for instance) positron excesses observed by a satellite named PAMELA
some years ago (and not confirmed by other experiments since).
We can find a gravitational classical analogy of this quantum effect. Indeed,
if a meteor of mass m approaches the Earth of mass M with a velocity v, the
lower is the velocity, the more chance it has to enter into collision with it as the
gravitational attraction will compensate the kinetic energy of the meteor as we can
see in Fig. 5.21. Without any force of gravity, the cross section is simply σ0 = πR,
5.11 Sommerfeld Enhancement 345

M
h

R M1
R v1

Fig. 5.21 Classical manifestation of the Sommerfeld enhancement in the case of a meteor hitting
the Earth

R being the radius of the Earth. By conservation of angular momentum we can write
Li = Lf with

vh
Li = |OM × mv| = mhv, |OM1 × mv1 | = mRv1 ⇒ v1 = (5.157)
R
where h is the impact parameter and R is the radius of the Earth (see the figure).
The conservation of energy gives

1 2 1 GN Mm 2GN M
mv = mv12 − ⇒ v12 = v 2 + (5.158)
2 2 R R
GN being the Newton gravitational constant. Combining Eqs. (5.157) and (5.158)
we obtain
 
2GN M 2GN M
h2 = R 2 1 + ⇒ σ = πh 2
= σ0 1+ . (5.159)
v2 R v2 R

For very small values of v, there is a large enhancement of the cross section due to
the gravity. Even if the correction vanishes as gravity is switched off (GN → 0), the
expansion parameter is 2GN M/(Rv 2 ), which can become large at small velocity.

5.11.2 Solving the Schrodinger’s Equation

Before reading this paragraph, we advice the reader to read Sect. B.6 to remind some
basic structures of the solutions of Schrodinger’s equation. We are considering the
solution to Schrodinger’s equation for scattering of an incoming plane wave in the
z-direction eikz = eikr cos θ by a potential localized in a region near the origin, so
346 5 In the Galaxies [T0 ]

that the total wave function beyond the range of the potential has the form19

eikr
ψk = eikr cos θ + f (θ ) . (5.160)
r
The overall normalization is of no concern, as we are only interested in the fraction
of the ingoing wave that is scattered. Since the interaction or the annihilation
is taking place locally, near r = 0, the only effect of a perturbation V to the
Schrodinger’s equation described in Sect. B.6 is to change the value of the modulus
of the wave function at the origin relative to its unperturbed value. Then we can
write

σ = σ0 Sk , (5.161)

where the Sommerfeld enhancement factor S is simply

|ψk (0)|2
Sk = (0)
= |ψk (0)|2 . (5.162)
|ψk (0)|2

One can wonder why using ingoing plane wave following the z-axis and not directly
a spherical plane wave. Indeed, many potentials in nature are spherically symmetric,
or nearly so, and from a theorist point of view it would be nice if the experimentalists
could exploit this symmetry by arranging to send in spherical waves corresponding
to different angular momenta rather than breaking the symmetry by choosing a
particular direction. Unfortunately, this is difficult to arrange, and we must be
satisfied with the remaining azimuthal symmetry of rotations about the ingoing
beam direction. In fact, though, a full analysis of the outgoing scattered waves
from an ingoing plane wave yields the same information as would spherical wave
scattering. This is because a generic wave can actually be written as a sum over
spherical waves:

ψk = Al Pl (cos θ ) Rkl (r). (5.163)

l

Note that the Al coefficients are independent on φ as they correspond to the m =

0 solution of the Ylm (θ, φ) eigenvector of the lˆ operator (see Sect. B.6 for more
details). To determine the coefficients Al , we use the fact that the combination Al ×
Pl (cos θ ) is independent of the radius r. Thus, we can compute them for large values
of r where the influence of the potential V (r) is negligible and ψk  eikz can be
approximated by a plane wave. We will then look at the influence of V (r) for small
values of the radius r, keeping the values of Al obtained for r → ∞. We use
the asymptotic expansion of eikz and identify it with the asymptotic Schrodinger

19 The symmetry around the azimuthal axis imposes f (θ, φ) = f (θ).

5.11 Sommerfeld Enhancement 347

solution for ψk with a null potential20 for Rkl (r) = Rkl

0
(r) obtained21 in Eq. (B.208)
and using the fact that for large r, δl kr

1
eikz = (2l + 1)Pl (cos θ )[eikr − (−1)l e−ikr ] (5.164)
2ikr
l
1  π  1
 Al Pl (cos θ ) sin kr − l + δl ⇒ Al = i l (2l + 1)e−iδl .
r 2 k
l

It is now very simple to determine ψk (0) and then the Sommerfeld factor (5.162).
If the potential V (r) does not blow faster than 1/r near r  0, then we can ignore
it relative to the kinetic terms, and we have Rkl (r) ∝ r l → 0 as we showed in
Eq. (B.206). As a consequence, all but the term with a null momentum l = 0 vanish
at the origin.
' '
' Rk,0 (0) '2
Sk = '' ' (5.165)
k '

with Rk0 (r) = χk (r)/r solution of Eq. (B.209) with l = 0

d 2 χk (r)
+ k 2 − 2MV (r) χk (r) = 0 (5.166)
dr 2

with the normalization condition χk (r) → sin(kr + δ0 )  sin(kr) when r → ∞,

corresponding to the solution of a free particle obtained in Eq. (B.204) as V (r) has
no influence anymore when r → ∞. As we noticed, Rk0 → constant when r → 0,
one deduces χk (r) → 0 when r → 0, implying χk (r)  r dχdr k (r)
when r  0. We
then can express the Sommerfeld factor

' dχk '

' (0) '2
Sk = '' dr ' (5.167)
k '

with the boundary conditions χ(r)  sin kr, r → ∞ and χ(0) = 0. We can check
that for a null potential, χk0 = A sin(kr), with A = 1 by boundary condition. Then
(χk0 ) = k, and Sk = 1 as we expected.

20 Suppose the potential V (r) → 0 when r → ∞.

21 Note 0
the change or normalization by multiplying Rkl by the momentum k.
348 5 In the Galaxies [T0 ]

5.11.3 The Coulomb Potential

Let us begin to compute the Sommerfeld enhancement for the attractive Coulomb
potential V (r) = − αr . Equation (5.166) can then be written as

d 2χ v2 2 l(l + 1)
2
+ 2
χ+ χ− χ =0 (5.168)
dx α x x2
with x = r × αM. The solutions for this equation are usually expressed in terms
of confluent hypergeometric functions. Its solution is the regular Coulomb Wave
function that plays an important role in various problems in quantum mechanics
and can be expressed, for l = 0:

−2π αv v
χ(x) = x 1+ Ai x i (5.169)
(e−2π v
α
− 1) α i

Ai being coefficients of the order of unity, which does not have influence for
the Sommerfeld enhancement as x, r → 0. Applying the solution of (5.169) to
Eq. (5.167) we obtain

' '2
' 2π αv '
Sk = '' ' .
' (5.170)
1 − e−2π v
α

Note that as v → ∞, Sk → 1, which means that the high velocity regime does
not affect the interaction cross section. However, as v → 0 Sk → 2π αv and
what is named Sommerfeld enhancement. We show in Fig. 5.22 the evolution of
the Sommerfeld enhancement in the case of the attractive Coulomb interaction for
different values of the coupling α. We clearly see that the effect can enhance the
interaction or annihilation cross section of orders of magnitude. We also notice that
the enhancement is independent on the mass of the interacting particle.

5.11.4 The Yukawa Interaction

In the preceding Coulomb potential, generated by electromagnetic force, the

particles exchanged are massless photons. In general, the mediating particles φ can
be massive. This interaction is described by a Yukawa potential
α
V (r) = − e−mφ r , (5.171)
r
5.11 Sommerfeld Enhancement 349

Sk
100

80 D 

60

D 
40

D 
20

D 
0
0.0 0.2 0.4 0.6 0.8 1.0
v/c
Fig. 5.22 Sommerfeld enhancement as a function of v/c in the case of an attractive Coulomb
potential V (r) = − αr for different values of α

where mφ is the mass of the exchanged particle. The Yukawa potential is clearly
a generalization of the Coulomb potential that we recover for mφ = 0. The
Schrodinger equation then becomes
mφ x
d 2χ v2 2e− αM l(l + 1)
2
+ 2
χ + χ− χ = 0. (5.172)
dx α x x2
The above Eq. (5.172) does not possess analytical solution. A numerical solution
exists and has been studied extensively in the literature. However, it is possible to
give an approximate analytical expression if one approximates the Yukawa potential
by the Hulthen potential:

αδe−δr
V (r)  VH (r) = . (5.173)
1 − e−δr

One can find the best choice of δ to best reproduce the Yukawa potential,

π 2 mφ
δ= . (5.174)
6
Exercise Recover analytically this value for δ.
350 5 In the Galaxies [T0 ]

Sk
106
v=10 −4
105

104

v=10 −3
1000

100

10 v=10 −2

1 v=0.1
10 50 100 500 1000 5000
M (GeV)
Fig. 5.23 Sommerfeld enhancement as a function of the mass of the dark matter M in the case
−mφ r
of an attractive Yukawa potential V (r) = − αe r for α = 0.1 and mφ = 10 GeV for different
values of v/c

l(l+1) l(l+1) δ 2 e−δr

Approximating also Mr 2
by M (1−e−δr )2 we can derivate an analytical solution
of Eq. (5.172):

 
 πα  sinh 12vM
πmφ
Sk      . (5.175)
v 36v 2 M 2
cosh 12vM
2m −
− cos 2π π6αM
πmφ φ π 4 m2φ

We want to underline that the obtention of such result is far from being straight-
forward and need a solid algebraic work. We propose to the reader to do it as
an exercise. The advantage of having expressed Eq. (5.175) analytically is that its
utilization is straightforward and avoids any computing consuming time. We can
notice that we recover the Coulomb limit in the case mφ → 0 (Fig. 5.23).

5.12 Structure Formation Constraints

Other constraints can be extracted from the observation of the sky, not only related
to annihilation processes, as it is the case for the observation of γ -rays, anti-matter,
or synchrotron radiation. For instance, the formation of large scale structures (LSS)
5.12 Structure Formation Constraints 351

is a fundamental keypoint in any strategy to unveil the nature of the dark matter.
Indeed, like clouds of smokes in a room, energetic movements of the hands will
destroy any formation of small structures; the dark matter, if too hot, will prevent
the formation of the first structures. This constraint on the temperature of the dark
matter is for instance one of the main arguments that excludes light neutrinos as the
main component of the Universe. It is often said that “a dark matter mass below 1
keV is excluded by LSS,” and we will recover this result in the following section,
before looking in more detail the mechanism that leads to galactic scale structure
and the influence of the dark matter on it.

5.12.1 Free Streaming

If dark matter is composed of collisionless particles (like neutrino for instance), after
they are kinematically decoupled from the bath (see Sect. 3.2), they are subject to
Landau damping , also known as collisionless phase mixing or free streaming. Until
perturbations become Jeans unstable (see Sect. 5.12.2) and begin to grow after the
decoupling time, collisionless particles can stream out of overdense regions and into
underdense regions, smoothing out inhomogeneities. We will estimate the scale of
collisionless damping in this section.22
Once a species decouples from the plasma, it simply travels in free fall in the
expanding Universe. We thus may choose the particle motion to be along dφ =
dθ = 0 so that the motion of a freely propagating particle is simply given by
a(t)dr = v(t)dt. The distance λF S traversed by a free streaming particle at a time t
can then be written
  t
v(t  ) 
λF S (t) = dr = 
dt . (5.176)
0 a(t )

While the particles are relativistic, v = c, and later, v ∝ a −1 because the momentum
is redshifted. We thus need to introduce a new scale, the scale factor anr where the
dark matter particles become non-relativistic. It is defined by the condition kTnr =
−1 kT . Following Eq. (A.25) we have the link between temperature and scaling
anr 0
factor: T ∝ a −1 . We thus can deduce the temperature when a particle χ becomes
non- relativistic: mχ  Tnr ⇒ anr  mT0χ . In the case of the neutrino χ = ν,
Eq. (3.89) gave us mν ≈ νh
2 × 92 eV implying

T0 2.35 × 10−4  −1  −1

anr = = νh
2
= 2.6 × 10−6 νh
2
. (5.177)
mν 92

22 In
order to take this effect into account properly, one must integrate the Boltzmann equation that
describes the collisionless component.
352 5 In the Galaxies [T0 ]

We will then integrate the free streaming equation in the 3 regimes independently: t
between 0 and tnr , between tnr and tEQ , and finally between tEQ and t0 :

• Between t = 0 and t = tnr , the neutrino is relativistic and thus v = c and the
expansion factor is proportional to t 1/2 [Eq. (A.27)], a(t) = α t 1/2
 t dt  2 a(t)tnr
λ0−nr
F S (t) = = t 1/2 = 2 2 [0 < t < tnr ]. (5.178)
0 a(t  ) α anr

• Between tnr and tEQ , the velocity of the particle v(t) is proportional to a −1 (t)
anr anr
by redshift, which means v(t) = c a(t ) = a(t ) , which gives

nr−EQ nr−EQ nr−EQ

λF S (t) − λF S (tnr ) = λF S (t) − λ0−nr
F S (tnr ) (5.179)
 
t anr dt  tnr t
= = ln
tnr a 2 (t  ) anr tnr

nr−EQ tnr tnr a(t)
⇒ λF S (t) = 2 +2 ln
anr anr anr

tnr a(t)
=2 1 + ln [tnr < t < tEQ ].
anr anr

• Between t = tEQ and t = t0 , the dependence on the velocity is of course

still redshifted, but the relation between time and scale factor from Eq. (A.27)
is a(t) = α t 2/3 . Remarking tnr = tEQ (anr /aEQ )2 we then obtain

EQ−t0 nr−EQ
tdt 
λF S (t) = λF S (tEQ ) + anr 2 
(5.180)
tEQ a (t )
 
tnr aEQ 3tnr aEQ 1/2
=2 1 + ln + 1−
anr anr anr a(t)
  1/2
EQ−t0 2tnr 5 aEQ 3 aEQ
⇒ λF S (t) = + ln − [tEQ < t < t0 ].
anr 2 anr 2 a(t)

We can see that λF S (t) quickly reaches and asymptotic value λ∞

F S . With the values
obtained in Eq. (3.52) and Eqs. (A.22–A.27) we can deduce tnr and aEQ and thus
compute the asymptotic limit λ∞ FS:

 
MP 45 5 ρ0R Tnr 1 eV 1 eV
λ∞
FS = + ln ≈ 70 Mpc ≈ 210 Mpc
T0 Tnr 4π 3 gρ 2 ρ0M T0 Tnr mν
(5.181)
5.12 Structure Formation Constraints 353

O FS (Mpc) aEQ

m Q = 1 eV

a nr (100 eV)
100
a nr (1 keV) m Q = 10 eV
10
m Q = 100 eV

1
m Q = 1 keV

0.1

0.01
10−8 10−6 10−4 0.01 1
a(t)
Fig. 5.24 Free streaming distance (in Mpc) as a function of the scaling parameter a(t) for different
masses of neutrino (from 1 eV to 1 keV)

if we suppose Tnr  mν /3 (an exact solution of the Boltzmann solution is needed

here). It means that structures of smaller scales than 70 Mpc should have been
destroyed by neutrino of masses mν < 1 eV. If they were the main matter constituent
of the density of the Universe. We plotted a numerical solution of λF S in Fig. 5.24.
The limit mν = 1 keV is known as the warm dark matter limit. Indeed, the
free streaming of such particle is around 1 Mpc, which is the typical size of
protogalaxies. Hot dark matter are particles still relativistic at the recombination
time (mν  10 eV as we can see from Fig. 5.24). In fact, more generically
speaking, candidate particles can be grouped into three categories on the basis
of their effect on the fluctuation spectrum. If the dark matter is composed of
abundant light particles that remain relativistic until shortly before recombination,
then it may be called “hot.” The best candidate for hot dark matter is a neutrino.
A second possibility is for the dark matter particles to interact more weakly than
neutrinos, to be less abundant, and to have a mass of order 1 keV. Such particles are
dubbed “warm dark matter” because they have lower thermal velocities than massive
neutrinos. There are few candidate particles that fit this description. Particles from
supersymmetric frameworks like gravitinos and photinos or even sterile neutrinos
have been suggested. Any particles that became non-relativistic very early, and so
were able to diffuse a negligible distance, are termed “cold” dark matter (CDM) .
354 5 In the Galaxies [T0 ]

5.12.2 Jeans Radius and Mass

The structure formation through gravitational collapses is a highly non-trivial

science, treated in numerous textbooks. We will try in this very short notice to
give the main ideas and results following a (false but approximate) linear treatment.
In two words, if one contracts a volume V of a gas of density ρ in equilibrium,
an increase of pressure will counterbalance the contraction. However, if the sound
speed in the gas cs is not sufficient, the time for the information to cross the radius
R (ts = cRs ) will be too long, and the gain of the gravitational potential generated by
smaller distances between bodies can dominate before the pressure has time to act.
In other words, if

R 1
ts =  tG = √ , (5.182)
cs Gρ

where tG is the gravitational free-fall characteristic time (obtained from the virial
theorem v 2 ∼ R 2 Gρ) the collapse occurs. We can then extract from Eq. (5.182) the
Jeans radius RJ (Jeans mass MJ ) corresponding to the maximal radius (mass) of a
gas of constant density ρ that can keep its equilibrium state:

cs cs2
RJ = √ ; MJ ∼ RJ3 ρ = 3 √ , (5.183)
Gρ G2 ρ

which gives

 c   10−3 GeV/cm3 1
2
s
MJ = 2 × 10 M 12
, (5.184)
148 ρ


T
where we used the sound velocity cs = 1480 km/s 108g K , Tg being the temperature
of the gas. Above MJ , the volume of gas begins to contract under gravity. These
numbers (Tg  106 K, M = 2 × 1012M ) correspond roughly to the characteristics
of the Milky Way, the mean mass density, ρ, being mainly from dark matter source,
being less well measured. We then understand that a too small deviation from a mean
density δρ will not induce a sufficiently efficient gravitational collapse to generate
galactic- or cluster-sized objects.
5.12 Structure Formation Constraints 355

The Speed of Sound

The speed of sound cs appearing in Eq. (5.182) is a generalization of the
distance travelled by a sound wave while propagating through an elastic
medium. This fundamental notion appears in structure formations, as well
as primordial baryonic oscillation (BAO) or even aspects of dark matter
production. In an ideal gas, it depends only on its temperature and its
composition. Sound wave is then generalized to any kind of fluid, dark one, or
the primordial plasma. Consider the sound wave propagating at the velocity cs
in an elastic medium of density ρ. Per unit of time, the mass passing through
a surface A along the direction z is

dz
ṁ = ρ × × A = ρ × cs × A. (5.185)
dt
The flux being the same at the entrance and the exit of the volume, we should
have
d dρ dcs
(ρcs ) = 0 ⇒ cs = −ρ . (5.186)
dt dt dt
From Newton second’s law
dcs
ρ × A × dz × = −dP × A, (5.187)
dt
where we introduced the pressure, as the force acting per unit of surface on
A; combining Eqs. (5.186) and (5.187) we obtain

dP
cs2 = , (5.188)
dρ

where we used dz dt = cs . Considering the equation of state P = wρc , we

2

obtain cs = 0 for a dust-type gas, and cs = √c

for a relativistic gas. As one
3
can see, the speed of sound in the plasma is different from the speed of light.
If we are in the presence of a plasma made of non-relativistic baryons and
photons, noticing that δρb δT δρR
ρb = 3 T and ρR = 4 T , we can write
δT


δρb 3 ρb 3 ρb
= ⇒ δρ = δρR 1 + (5.189)
δρR 4 ρR 4 ρR

(continued)
356 5 In the Galaxies [T0 ]

and

δP δPR 1 δρR c2
cs2 = = = =  . (5.190)
δρ δρ 3 δρ 3 1 + 34 Rb

Exercise Show that for a plasma made of an admixture of radiation (ρR )

ρb
and non-relativistic baryons (ρb ), w = ρc
P
2 = 3(1+Rb ) , with Rb = ρR .
1

Decomposing dP
dρ = dP da
da dρ , recover Eq. (5.190).

The influence of the baryon is then to “slow down” the photons in the plasma.
In other words, the measurement of the horizon at CMB time gives a hint
concerning Rb , the baryonic content of the plasma, see Sect. 3.7.2.4. Notice
that we supposed an isolated system with constant entropy. If not, the real
expression should be read
 
∂P ∂P
cs2 = + . (5.191)
∂ρ S ∂S P

It is also interesting to ask what is the timescale needed for collapsing a structure
of density ρ that generates itself a gravitational potential . Using the Poisson
law (5.23),

 = 4πGρ, (5.192)

and the conservation of mass passing through a surface S, δM ∝ ρSdr:

dδM ∂ρ ∂ρ
∝ S.dr + dv.S ρ = 0 ⇒ = −ρ × ∇.v. (5.193)
dt ∂t ∂t
If ρ0 is the mean density, we can write ρ = ρ0 (1 + δ(t)), and Eq. (5.193) becomes
at first order in δ

δ̇ = −∇.v ⇒ ∇.v̇ = −δ̈ = −, (5.194)

where, for the last expression, we used the Newtonian equation of motion

v̇ = −∇, (5.195)

and supposed to simplify a spherical symmetry. Solving Eq. (5.194) gives then
√
δ̈ = 4πρ0 Gδ ⇒ δ = δ0 e 4πρ0 Gt
∝ et /tG , (5.196)
5.12 Structure Formation Constraints 357

where δ0 is the original seed of the perturbation. The exponential growth is the
characteristic of a Newtonian static Universe. A more complex analysis, taking into
account the dilution effect due to an expanding Universe, was performed by Evgenii
Lifshitz in 1946, showing that the growth is much softer and follows a power law:

δ ∝ t 2/3 . (5.197)

It was noticed very early after the discovering of the CMB that structure
formation was in tension with the data. Indeed, if one supposes that the decoupling
time happens at a redshift of ∼ 1100, and that the matter density nowadays is
0  2.5×10−30 g cm−3 , the mass included in a volume V = 4π (R CMB )3 ,
roughly ρm 3 H
with RHCMB
= ctCMB the horizon size at decoupling time, is:

4π
MH = (ctCMB )3 ρm
0
(1 + zCMB )3 (5.198)
3
with tCMB =380,000 years and zCMB = 1100; we obtain

MH  3.3 × 1017M , (5.199)

which is much larger than the galactic structures we observe nowadays. In a top-
down scenario, where the galactic substructures would have been formed from
fragmentation of huge gravitationally bound systems that would be acceptable. This
scenario was the one preferred by the Soviet cosmologist Yakov Zeldovich in 1977.
However, even the first serious simulations running in the 80s immediately revealed
that in this case, the galaxy-mass objects form rather late, at a redshift z ∼ 2,
whereas we know that galaxies existed long before, from their direct observation
at much higher redshift. The bottom-up scenario is then largely favored by the data,
which necessitates the presence of structures below the galactic scale  1012M
shortly after the decoupling epoch. The presence of a dark matter component can
solve this problem.

5.12.3 The Influence of Dark Matter

As we discussed above, the large mass trapped inside the horizon size at the
decoupling epoch would have generated too large structures. However, if we
introduce a massive component mdm that decoupled from the thermal bath at a
temperature Tdm ∼ mdm (corresponding to a time tdm ∼ H1 ∼ M2P ), we deduce
Tdm
that the mass included in the dark matter horizon from the origin to tdm is

 2
4π 1 keV
Mdm ∼ (ctdm )3 ρ(Tdm = mdm ) ∼ 109 M . (5.200)
3 mdm
358 5 In the Galaxies [T0 ]

Dark matter masses above the keV scale can then generate structures of the size
of dwarf galaxies. Notice how this simple argument is in agreement with the
more complex free-streaming constraints of Fig. 5.24 or even the Tremaine–Gunn
bound (3.98). However, the fact that inhomogeneous systems can be formed thanks
to the presence of a dark component is far to be sufficient to explain the structure of
the Universe observed today at the cluster scale.
When the physics community left the steady state paradigm (proposed by Bondi,
Gold and Hoyle) to adopt the Big Bang option, it became obvious that the baryonic
fraction ηb deduced from measurements in the galaxies and cluster of galaxies (see
Fig. 1.5) is too small to be compatible with an Einstein–de Sitter model where
m = 1. That implies the need for dark “something” that could be dark matter, dark
energy, or both. James Peebles, one of the first post-CMB cosmologists, noticed it
already in 1965. But he was also one of the first to look at the influence of the dark
matter in the structure formation. Indeed, whereas in the steady state paradigm, the
Universe being here forever, the time needed to produce the first galaxies does not
really matter. However, once Big Bang appears as the dominant paradigm, quickly
James Peebles understood that the time to produce the first complex structures is
quite short because it should begin after the recombination time, around 380,000
years after the Big Bang, when the baryons have decoupled from the thermal plasma
and are allowed to collapse gravitationally. This was even more problematic if one
considers a Universe only dominated by a baryonic component b , in which density
is strongly limited by the measured light elements abundance (see Fig. 1.5).
What is the level of inhomogeneity in our present Universe? We observe that at
scales below R  8 h−1 ∼ 12 Mpc, roughly the cluster scale, the inhomogeneity
began to be larger than the mean mass density, in other words,

δρ
(R  12 Mpc)  1. (5.201)
ρ

From the Lifshitz relation (5.197), one deduces that

' '
δρ '' δρ '' 1 1
= ∼ (5.202)
ρ 'z ρ '0 1 + z 1+z

in a matter dominated Universe where a ∝ t 2/3 . We should then expect, at the

decoupling time tCMB , where zCMB = 1100, an amplitude of fluctuation
' '
δρ '' δT ''
∼ 10−3 ⇒  2.5 × 10−4 , (5.203)
ρ 'CMB T 'CMB

where we have supposed, in the last equality, that the Universe is only composed of
δT 4
baryons, and that, at decoupling time, δρ
ρ ∼ T 4 ∼ 4 T . This level of fluctuations
δT

should appear at the decoupling time at a scale corresponding to ∼ 12 Mpc

presently, which means ∼ 10 kpc at z = 1100.
5.12 Structure Formation Constraints 359

Exercise Show that a present cluster scale R = 12 Mpc corresponds to an angle of

the order of arcminute in the CMB map.

However, already in 1984, experimental limits on the CMB anisotropies were

δT
T  10−4 on an angular scale corresponding to regions containing the masses of
galaxies or clusters of galaxies at z = 1100. Nothing being observed, researchers
like James Peebles or Joseph Silk proposed that a massive hypothetical particle
could resolve this contradiction. Indeed, if a dark matter fluid has decoupled from
the plasma at z ∼ 10,000, the dark matter fluctuations have grown to δρ −4 at
ρ ∼ 10
the decoupling time, while the photon–baryon sound-wave fluctuations are stuck to
10−5 , consistent with observations.
One can then ask what is the value of δT T expected at the decoupling time if one
supposes δρ = 1 at a cluster scale of 8 h−1 Mpc. Locally, fluctuations produce a
ρ
gravitational potential δ such as
   2
GδM R2 ρ δρ δρ Rh δρ
δ = ∼ Gδρ R 2 ∼ ∼ 3R 2 H 2 ∼3 .
R MP2 ρ R ρ R 3000 Mpc ρ R

The observation of a cutoff at R ∼ 8h−1 Mpc implies


δρ
δ  2.1 × 10−5 = 2.1 × 10−5 . (5.204)
ρ 8 Mpc

From (5.197), we see that δ ∝ δρR 2 is scale invariant for a matter dominated
Universe. We then deduce

δ|CMB = 10−5 . (5.205)

To find the relation between δ and δT T one should use the Sachs–Wolfe method .
From General Relativity, we know that the invariant measure in the presence of a
gravitational perturbation can be written in the Newtonian gauge, in the weak field
limit,

ds 2 = (1 + 2δ)dt 2 − R 2 (t)(1 − 2δ)dχ 2 , (5.206)

where χ is the comoving coordinate (see Eq. 2.20). The effect of the potential is to
redshift the energy of photons emitted from the regions where δ = 0.
√
p 1 − 2δ
= √  1 − 2δ, (5.207)
p 1 + 2δ
360 5 In the Galaxies [T0 ]

where p (p) is the momentum23 with (without) δ. This corresponds to a photon
redshift

'
E − E δE p − p δT δT ''
z= = = = = 2δ ⇒ ∼ 2 × 10−5 .
E E p T T 'CMB
(5.208)

We have seen that a present amplitude of mass fluctuation of the order δρ

ρ = 1 at
a scale R = 8 h−1 Mpc implies, at the decoupling time, a variation in the CMB
−5
T  2 × 10 , which is effectively what has
temperature fluctuation of the order δT
been observed by the satellites WMAP and PLANCK. That is one of the greatest
achievements of the CDM model.

5.12.4 Correlation Function∗

Two-point correlation function of galaxies is the simplest estimator of the galaxies

distribution at large scale and defines the amplitude of fluctuations. The two-
point correlation function quantifies the excess of probability to find a galaxy 2
at a distance r from a galaxy 1 selected randomly, compared to a mean uniform
distribution. For instance, if the galaxies are distributed following a Poisson
distribution with a mean density ρ0 , the probability dN(r) to find a galaxy in a
volume dV2 at a distance r from the galaxy 1 in the volume dV1 is the fraction of
galaxies in dV1 × the fraction of galaxies in dV2 :

dN(r) = ρ02 dV1 dV2 . (5.209)

When we write Poissonian distribution, we mean random stationary process (iden-

tical mean properties whatever is the size or the place of the sample considered), i.e.
the galaxies are placed by chance and independently one with respect to each other,
with an homogeneous spatial probability. The spectrum of such a distribution is
thus flat: this is the distribution of a “white noise.” If the distribution of the galaxies
differs slightly from a Poissonian distribution, the probability will be

dN(r) = ρ02 (1 + ζ(r))dV1 dV2 , (5.210)

where ζ (r) is the two-point correlation function of the galaxies. We used r instead
of r with the hypothesis of the isotropy of the distribution. ζ(r) > 1 indicates a
surdensity, ζ (r) = 0 indicates a galaxy density equals to the mean density, and

23 To see it in the non-relativistic limit, just consider p = mv  , and v 2 = 1−2δ 2

1+2δ v .
5.12 Structure Formation Constraints 361

ζ (r) < 0 indicates an underdensity. The probability of finding a galaxy at a distance

r from another one can thus be written with Eq. (5.209)

dN(x, r) = ρ(x)dV 1 ρ(x + r)dV2 . (5.211)

We can then introduce the contrast of density δρ

ρ(x) − ρ0
δρ (x) = ⇒ ρ(x) = ρ0 (1 + δρ (x))
ρ0
⇒ dN(r) = dN(x, r) = ρ02 (1 + δρ (x)δρ (x + r))dV1 dV2
= ρ02 (1 + ζ(r))dV1 dV2

with ζ (r) = δρ (x)δρ (x + r) is called the function of autocorrelation of the matter
distribution.

5.12.5 Power Spectrum P (k)∗

We can define a power spectrum function P (k) by


V
ζ(r) = eik.r P (k)d 3 k. (5.212)
(2π)3

P (k) is then the Fourier transform of the two-point correlation function ζ(r).
In other words, P (k) represents the probability of having distribution of matter
periodically distributed in the close Universe with a wave vector k  1/λ. In the
isotrope case, Eq. (5.212) can be written as
 ∞
V sin(kr) 2
ζ(r) = P (k) k dk. (5.213)
2π 2 0 kr

r represents a comoving scale (or wavelength) corresponding to the wave number

sin(kr)
k = 2πr . The function kr acts like a windows letting only the modes with k <
2π
r to contribute to the fluctuation at the scale r. For these reasons, fluctuations
with k > 2πr (i.e. small scales compared to r) do not contribute at the scale r. As
a conclusion the correlation function at a scale r will be sensible only to power
spectrum corresponding to scales > r.
To have an idea, recent measurements with some simulation hypotheses give
 −γ
r
ζ(r)  (5.214)
r0

with γ  1.77 and r0  5h−1 Mpc for scales from 0.1 to 10 h−1 Mpc. Higher is r0
larger is ζ (r) and so the structure features. r0 quantifies the clustering.
362 5 In the Galaxies [T0 ]

References
1. J.D. Jackson, Classical Electrodynamics (Wiley, New York). ISBN 978-0-471-30932-1. OCLC
925677836
2. M.S. Longair, High Energy Astrophysics: Volume 1, Particles, Photons and their Detection
(Cambridge University Press, Cambridge, 2011) (ISBN 0521756189), 1992, 440pp., (ISBN
0521387736)
3. O.Y. Gnedin, A.V. Kravtsov, A.A. Klypin, D. Nagai, Astrophys. J. 616, 16–26 (2004). https://
doi.org/10.1086/424914. [arXiv:astro-ph/0406247 [astro-ph]]
4. J.F. Navarro, C.S. Frenk, S.D.M. White, Astrophys. J. 462, 563–575 (1996). https://doi.org/10.
1086/177173. [arXiv:astro-ph/9508025 [astro-ph]]
Cosmology and Astrophysics
A

A.1 Useful Cosmology

A.1.1 Lorentz Transformation

In this section, I will review very briefly the Lorentz transformations, more from an
historical perspective that one can read in classical textbook on the subject. I urge
the reader to have a look at the original Einstein’s article of 1905 which is very clear
and pedagogical. The first step made by Einstein is to consider a space of what he
called himself “events.” Let us suppose the event as being a ray of light leaving one
observer in movement, reaching a mirror and going back to him. The fact that the
light comes back to the observer is fundamental, as the observer can define (and
then measure) the time that the light takes to make a round trip. If one considers
only a travel from the observer A to another observer B, it would not be possible to
define a traveling time (or an interval of time), independent of the unknown distance
between A and B. This is deeply linked with the concept of simultaneity, i.e.
“what does it mean to A that the light arrives to B and how does he know it if it
exists a maximal velocity in Nature”. In a Newtonian world where instantaneous
signals are possible, this question makes no sense. However, this is not the case in
the Einstein world, that is the reason why the first 5 pages of his original article are
uniquely dedicated to the definition of simultaneity.
As we saw, it is important for A to being able to measure (and thus define) a time
delay, which means, he has to see for himself the ray of light coming back. That
will be two events in space-time : (xA , t1 ) when the light leaves A and (xA , t2 ) when
the light reaches A again after being reflected by the mirror. If one supposes that
A moves with a velocity v [in its own referential of events (x  ,t  )] with respect to
a static1 reference frame (x, t) where an observer B stays. Let us suppose that the

1 Obviously, the notion of static and moving reference frames has no meaning in the Einstein

approach where absolute space does not exist (contrarily to the Newtonian approach). These

© Springer Nature Switzerland AG 2021 363

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
364 A Cosmology and Astrophysics

distance between the two mirrors is l  in the moving referential (x  , t  ) and l in the
static referential (x, t). The round trip time will be for A

2l 
t = , (A.1)
c
where c is the velocity of light, whereas for the static observer B, it will be

l l 2l 1
t= + = . (A.2)
v−c v+c c 1 − v22
c

Being conscious of the Lorentz proposition of lengths contraction (following the

null result of the Michelson–Morley experiment2), l  =  l and combining
2
1− v2
c
Eqs. (A.1) and (A.2), we obtain

t
t=  . (A.3)
v2
1− c2

It is interesting to notice that in the articles of Lorentz (and Fitzgerald), the authors
supposed a real physical contraction of objects (and not of space) due to atoms
being closer in the direction of motion induced by the effects of the electromagnetic
forces.
We can then use this result to compute the transformation laws and express the
events in the moving frame with respect to events in the static frame. l can be written
as x, distance between two points in the static frame, that can be translated, in the
moving frame using Lorentz contraction by x  =  x , which allows us to write
2
1− v2
c

x
x =  + αt (A.4)
v2
1− c2

giving for the point x = vt (x  = 0), α = −  v

, or in other words
2
1− v2
c

x − vt
x =  . (A.5)
2
1 − vc2

denominations are just here to help visualizing the situation of the relative motion of two observers.
Any static frame is a moving frame in another referential.
2 To be more precise, the first proposition of lengths contraction was made by Fitzgerald in 1889,

whereas the Lorentz paper was published in 1892.

A.1 Useful Cosmology 365

If we apply the same techniques to find the transformation law for the time, we can
write

t  = at + bx. (A.6)

Considering the point x = 0, it travels at velocity −vwith respect to the moving

frame, so one can write, for this point, t (x = 0) = t  1 − vc2 . Indeed, the time of
2

the static referential is diluted in this case because the “static” frame is the moving
frame view from the observer B. Implementing this value of t and x = 0 in Eq. (A.6)
we have
1
a=  , (A.7)
v2
1− c2

whereas
 if one considers the point x  = 0 (x = vt), applying this time t  (x = vt) =
2
1 − vc2 t we obtain
⎛ ⎞
2 v
⎝ 1 − ⎠t =  1
v
t + b x(= vt) ⇒ b =  c2
. (A.8)
c2 1− v2
1− v2
c2 c2

Finally, combining all the preceding equations, we can write

x − vt
x =  (A.9)
2
1 − vc2


t − cv2 x
t =  (A.10)
2
1 − vc2

which is the set of equations usually called Lorentz transformations.3

We let the reader check that the relations (A.9) and (A.10) keep constant the
quantity c2 t 2 − x 2 (or c2 t 2 − x 2 − y 2 − z2 in the 3 dimensional space). At a classical
Galilean level, where the time does not transform, it is both the quantities ct 2 and
x 2 + y 2 + z2 that are conserved independently. This cannot be the case in relativity
for the obvious reason that time and position are tightly related. However, as noticed

3 Even if it is Poincaré who wrote them for the first time two weeks before the Einstein paper in
1905, Lorentz having not seen a group structure and the symmetry x ↔ t in the transformations
had a different asymmetric solution for t  . It is Poincaré, with respect to the Dutch physicist who
insisted to call them itself “Lorentz transformations”.
366 A Cosmology and Astrophysics

by Poincaré, the structure of group theory still preserves one quantity in the Lorentz
transformations. It is a length, completely independent of the referential of study,
such that all the observers will agree on, moving or not, and it can be written

ds 2 = c2 dt 2 − dx 2 − dy 2 − dz2 = c2 dt 2 − dl 2 . (A.11)

In fact, as it is presented in some textbook, demanding for ds to be an invariant

of the theory leads to the Lorentz transformations (A.9)–(A.10).

A.1.2 Friedmann Equation

The scale factor of the Universe (his size) is governed by Friedmann’s equation,
 2
ȧ 8πG K 8π K
= ρ− 2 = 2
ρ − 2, (A.12)
a 3 a 3MPl a

where we have ignored the cosmological constant for simplicity. Neglecting in the
first step the curvature term (as one can do at early time), and approximating ȧ ∼
a/t, Eq. (A.12) implies the familiar result that the expansion timescale t ∝ ρ −1/2 .
We choose the boundary condition a(t0 ) = 1 at present time and a(0) = 0 at the
early stage of the Universe. The Hubble constant H0 is the present time value of the
Hubble parameter H0 = (ȧ/a)t0 . K is the value of hypersurfaces of space–time, and
ρ is the total matter density. We define the quantity

3H02
ρcr = ≈ 2 × 10−29 h2 g cm−3 = 1.1 × 10−5 GeV cm−3 (A.13)
8πG
which is called the critical density for reasons that will become clear below. The
density 0 is the current total matter density ρ(t0 ) = ρ0 in unit of ρcr ,
ρ0
0 = . (A.14)
ρcr

The Hubble constant is commonly written as

H0 = 100 h km s−1 Mpc−1 ≈ 3.2 × 10−18 h s−1 , (A.15)

where 0.5 < h < 1 express our ignorance of H0 . We can then extract from the
Friedmann’s equation (A.12) remembering that a(t0 ) = 1,

K = H02 ( 0 − 1) (A.16)

In other words, a Universe which contains the critical matter density is flat.
A.1 Useful Cosmology 367

The first law of thermodynamics with conservation of entropy, dS = dU + pdV

can be written as

d(ρa 3 ) da 3
+p =0 (A.17)
dt dt

For ordinary matter, ρ p ∼ 0, hence ρ ∝ a −3 . For relativistic matter, p = ρ/3,

−4
hence ρ ∝ a . Therefore, the matter density changes with a as

ρ(a) = ρ0 a −n = ρcr 0a
−n
, (A.18)

where n = 3 for ordinary matter (“dust”), and n = 4 for relativistic matter

(“radiation”).
Summarizing, we can rewrite Friedmann’s equation into the form
−n
H 2 (a) = H02 [ 0a −( 0 − 1)a −2 ]. (A.19)

Since the Universe expands, a < 1 for t < t0 , and so the expansion rate H (a) was
larger in the past. At very early times, a 1, the first term in Eq. (A.19) dominates
because n ≥ 3 and we can write
1/2 −n/2
H (a) = H0 0 a . (A.20)

This is called the Einstein–de Sitter limit of Friedmann’s equation.

A.1.3 The Horizon

The size of casually connected regions of the Universe is called the horizon size. It
is given by the distance a photon can travel in the time since the Big Bang. Since the
appropriate timescale is provided by the inverse Hubble parameter, the horizon size
 = cH −1 (a), and the comoving horizon size is (reminding that the velocity of
is dH
light c = 1)

c 1 1 −1/2 n/2−1
dH (a) = = = 0 a , (A.21)
aH (a) aH (a) H0

where we have inserted the Einstein–de Sitter limit of Friedmann’s equation.

cH0−1 = 3 h−1 Gpc is called the Hubble radius.
The previous calculations show that the matter density today is completely
dominated by ordinary rather than relativistic matter. But since the relativistic matter
density grows faster with decreasing scale factor a than the ordinary matter density,
there had to be a time aeq 1 before which relativistic matter dominated. Using
368 A Cosmology and Astrophysics

Eq. (3.50), the condition

R −4 M −3
0 aeq = 0 aeq (A.22)

yields4

3.7 × 10−5
aeq =  2.7 × 10−4 (A.23)
0.15
which means that the Universe was ten thousands time smaller at the epoch when
radiation and matter were in equilibrium. We have seen in Sect. 5.12 that aeq plays
a big role in structure formation constraints. If we suppose that matter dominated
completely for all a > aeq (i.e. ignoring the contribution from radiation to the matter
density), Eq. (A.21) yields

−1/2
0 1/2
dH (aeq ) = aeq ≈ 90 Mpc. (A.24)
H0

As we noticed in Eq. (A.18) the photon temperature evolves as a −4 . We also have

demonstrated in Eq. (3.27) that the energy density of radiative particles evolves as
Tγ4 which implies that Tγ ∝ a −1 . We can then compute that temperature of the
photon at the time when a = aeq :

−1 2.4 × 10−4
Tγ (aeq ) = Tγ0 × aeq =  0.9 eV. (A.25)
2.7 × 10−4

We recover the results obtained in the Sect. 3.1.6.

We can also compute the time dependance of the evolution of the scale factor.
Indeed, from ȧ/a = H ∝ ρ0 a −n/2 which gives after integration
1/2

t ∝ a (n−2)/2 (A.26)

which means

a ∝ t 1/2 [radiation dominated] a ∝ t 2/3 [matter dominated] (A.27)

ȧ 1 2
⇒H = = [radiation dominated] [matter dominated].
a 2t 3t
From the previous section, we computed the energy density of relativistic particle
ργ ν = π3g2 eμ/T T 4 , which means ργ ν ∝ T 4 for a null chemical potential μ.

4 For another way of computing the equilibrium time or temperature, see Sect. 3.1.6.
A.2 Basics of General Relativity 369

Moreover, if we write p = adr/dt, and knowing that t scales as a −2 in radiation

time, one can deduce that p scales as a −1 in radiation dominated epoch.

p ∝ a −1 , (A.28)

which means that for a non-relativistic particle p = mv implies v ∝ a −1 .

Reminding that the TOTAL entropy of the Universe S = sa 3 = g∗S T 3 a 3 is
constant, one can deduce

a(T ) ∝ g∗S T −1 [in matter AND radiation dominated epoch ]

1/3
(A.29)

implying

2.4 × 10−4 eV 0.3 eV

a(T ) = a0 = aDEC . (A.30)
T T
Combining with the results from the thermodynamics equilibrium of the previous
section and supposing that during the early radiation dominated epoch ρ ∼ ρR ,
Eq. (3.27) combined with Eq. (A.12), one obtains

8πG 1/2 T2 T2
H = ρR = 1.66gρ1/2 = 0.32gρ1/2 . (A.31)
3 MP l MP

Moreover, if a ∝ t n , one observes that H = ȧ/a = nt −1 implying

 −2
−1/2 MP l T
t = 0.301g∗ ∼ [radiation dominated : n=1/2] (A.32)
T2 MeV

t ∝ T −3/2 [matter dominated: n = 2/3 ], (A.33)

where gρ = gboson + 78 gf ermion counts the total number of massless degrees of

freedom (m T ).

A.2 Basics of General Relativity

A.2.1 The Context

“Gμν = κTμν ”. This equation was on my wall, next to the photo of the 1927 Solvay
congress, during a large part of my study. As we discussed in the Sect. A.1.1,
the special relativity transformations and physics can be reconstructed from the
basic concept of the invariance of the length scale ds 2 = c2 dt 2 − dl 2 between
referentials in relative movement with constant velocity. This is obviously not the
case in classical Galilean transformations which impose dt 2 and dl 2 independent
370 A Cosmology and Astrophysics

Fig. A.1 Illustration of a 2d

surface

on the reference frame, separately. In the same manner, the Einstein transformation
laws of General Relativity concern the independence of the physics on the referential
frame, in any metric, including curved space-time. It can be derived from the
conservation of a unit of volume, as the Lorentz transformation can be deduced
from the conservation of ds 2 .

A.2.2 Measuring a Length, a Surface, or a Volume

Before entering in detail, dealing with volume in the 4-dimensional space-time

means dealing with volume. We remind that under a change of variable, x → x  =
f (x), the unit of length dx transforms to

∂f (x)
dx  = dx. (A.34)
∂x
This can be generalized in a 2-dimensional space to compute an element of surface.
Indeed, the area A of a parallelepiped with sides dx and dy is the base |dx|
multiplied by the height h = |dy| × sin θ , θ being the angle between dx and dy, see
Fig. A.1, or

A = |dx| × |dy| × sin θ = |dx ∧ dy|. (A.35)

Under a change of variables x = f (u, v) and y = g(u, v), we can write

∂ ∂ ∂ ∂
dx = f (x, y) du + f (u, v) dv ; dy = g(x, y) du + g(u, v) dv
∂u ∂v ∂u ∂v
' '
' ∂f ∂g ∂f ∂g ''
⇒ A = |dx ∧ dy| = '' − |du ∧ dv| = J × |du ∧ dv|, (A.36)
∂u ∂v ∂v ∂u '

J being the Jacobian of the transformation.5 We can generalize Eq. (A.36) to

any manifold and dimensions, defining the transformation (x μ → x μ (x ν )) of an

' μ'
' ∂x '
5A Jacobian of a transformation x μ → x μ with x μ = x μ (x ν ) is given by J = det ' ∂x ν '.
A.2 Basics of General Relativity 371

element of volume dτ into dτ  by

dτ = J × dτ  ; dτ  = J −1 × dτ (A.37)
μ
with J = | ∂x −1 = | ∂x |. ν
∂x ν | and J ∂x μ
I hope this little demonstration will answer the typical question I had in my lectures
on the subject: “ why the element  ∂ of surface dxdy does not simply transform
to ∂u∂
f (u, v) + ∂v∂
f (u, v) × ∂u g(u, v) + ∂v
∂
g(u, v) dudv”. The negative sign
generated by the wedge product translates the obvious fact that a surface under two
parallel vectors is null.

A.2.3 The Einstein–Hilbert Action (I)

We remind the reader that a contravariant vector Aμ and a covariant vector Aμ are
defined by their transformation laws under a change of coordinates x μ → x μ :

∂x μ α ∂x α
Aμ = A ; and Aμ = Aα . (A.38)
∂x α ∂x μ

This definition ensures the conservation of the norm Aμ Aμ , especially the square of
the invariant distance ds 2 = dx μ dxμ .

Exercise Prove this statement.6

More, precisely, writing the element of length

ds 2 = gμν (x)dx μ dx ν (A.39)

we let the reader check that if one defines

dxμ = gμν (x)dx ν ,

dxμ possesses the transformation properties of a covariant vector (dxμ = ∂x α

∂x μ dxα ),
and we can rewrite Eq. (A.39)

ds 2 = dxμ dx μ .

Notice that, if one needs to deal with fermionic fields, it is more convenient to work
in the tetrad (vierbein in German) framework. This allows to place ourselves in the
Minkowski flat space Xa , with the metric ηab , where the γ a matrices take their
Lorentz form (B.16), and then “bend” them in the physical metric g μν . In other

β ∂x β ∂x β (x  ) ∂x μ ∂x β
6 You will need for this to notice that δα = ∂x α = ∂x α = ∂x α × ∂x μ .
372 A Cosmology and Astrophysics

words, one introduces locally the flat metric ηab , where the vector basis includes
four orthonormal vectors ea such that ea .eb = ηab . Here, the ea are 4-dimensional
vectors in the tangent Minkowski space. With respect to the physical space x μ , we
μ μ
can write ea = ea eμ , where eμ is a corresponding local basis, and ea serves as the
intermediate coefficients between the two basis, which means

∂x μ ∂Xa
eaμ = a
, eμa = . (A.40)
∂X ∂x μ
and from

∂Xa ∂Xb μ ν
ηab dXa dXb = ηab dx dx , (A.41)
∂x μ ∂x ν
one deduces

gμν = eμa eνb ηab . (A.42)

and

γ μ = eaμ γ a . (A.43)

With the same philosophy, one can define a contravariant and covariant tensor by

∂x μ ∂x ν αβ ∂x α ∂x β
T μν = T ; and T 
= Tαβ . (A.44)
∂x α ∂x β μν
∂x μ ∂x ν
If one defines a space-time dependant metric gμν (x), it is straightforward to
prove that the condition

ds 2 = gμν (x)dx μ dx ν = constant (A.45)

∂x μ
under a change of coordinates x μ → x μ (x), or dx μ = ∂x α dx
α implies

 ∂x α ∂x β
gμν (x  ) = gαβ (x). (A.46)
∂x μ ∂x ν

It then follows that metric gμν (x) is a covariant tensor of second rank. Another very
interesting consequence of the relation (A.46) is that we can write, in the matrix
form

g (x) = J × g × JT ⇒ det[g ] = (det[J])2 . det[g], (A.47)

A.2 Basics of General Relativity 373

in other words, combining Eqs. (A.37) √

and (A.47),√
we notice that under coordinate
transformations dτ →= J −1 dτ and |g| → J |g|, with g = det gμν , which
means that the product
 √
|g|d 4 x = −gd 4 x (A.48)

defines the invariant 4-volume element of the theory. The |g| = −g relation arises
from the fact that the matrix gμν has three negatives and one positive eigenvalues,7
√
the determinant g is then negative, and −g is real. One can then define a
first invariant gravitational action, which should be written in terms of covariant
quantities and invariant under the general change of coordinates. We can express
it as function of a scalar Lagrangian, integrated over an invariant 4-volume. The
simplest Scalar Lagrangian is obviously a constant . S is then given by


1 √
S = −  d 4 x −g, (A.49)
8πG

G being the Newton constant of gravity. This action will play a fundamental role
in cosmology, because of describing the cosmological constant action. The 8πG
factor is a convention that will simplify the Einstein equations of fields as we will
see later on.

A.2.4 Tooling with the Metric

When dealing with fundamental fields, one needs to deal with derivatives of the
field with respect to the coordinate system, especially derivative of vector fields.
Whereas it is obvious that for a scalar field φ, ∂ μ φ (∂μ φ) behaves like a contravariant
(covariant) vector fields, it is not the case for ∂μ Aν . Indeed, under a change of
coordinate

∂x ν ∂x σ ∂Aρ ∂x σ ∂ 2 x ν ρ
∂μ Aν → ∂μ Aν = + A . (A.50)
∂x ρ ∂x μ ∂x σ ∂x μ ∂x ρ ∂x σ
Whereas the first term corresponds indeed to a tensor transformation of the type
Eq. (A.44), the second part should be eliminated, the same way covariant derivatives
are defined in gauge theory, defining

Dμ Aν = ∂μ Aν + αμ
ν
Aα , (A.51)

7 Also called a Lorentzian signature.

374 A Cosmology and Astrophysics

ν , the analog of the gauge field, is called a Christoffel symbol or spin

where αμ
connection. Imposing that Dμ Aν transforms as a tensor,

∂x α ∂x ν
Dμ Aν = Dα Aβ ,
∂x μ ∂x β
one obtains

ν ∂x ν ∂x ρ ∂x σ β ∂x ν ∂ 2 x ρ
αμ =  + . (A.52)
∂x β ∂x α ∂x μ ρσ ∂x ρ ∂x α ∂x μ

Exercise Demonstrate this relation.

As we can notice, the first term in the right-hand side of (A.52) corresponds to
ν . The
a classical covariant-contravariant transformation for the rank-3 tensor αμ
second term is the one needed to cancel the extra-contribution generated by the
derivative in the right-hand side of (A.50). Notice that we can also derive the
expression of the derivative for a covariant vector by solving

∂x α ∂x β
Dν Aμ = Dα Aβ . (A.53)
∂x ν ∂x μ

Another possibility is to use the Leibniz formula8

Dμ (Aν Aν ) = Aν Dμ Aν + (Dμ Aν )Aν = Dμ |A|2 = ∂μ |A|2 (A.54)

which gives

Dμ Aν = ∂μ Aν − νμ
α
Aα

At first sight, we can imagine thousands (and much more) of tensors of rank-
3 satisfying the condition (A.52). However, one needs two extra constraints. First
of all, to preserve the metric in the differentiation process, one needs the covariant
derivative to commute with the metric tensor, i.e.
 
gμν (x).Dα Aν = Dα gμν (x).Aν (A.55)

which gives, using Leibniz relation

Dα gμν = 0 = ∂α gμν − μα

β
gβν − να
β
gμβ (A.56)

8 As an exercise, show that Dμ (AB) = A(Dμ B) + (Dμ A)B, whatever are the tensors A and B.
A.2 Basics of General Relativity 375

that we can reduce to

∂α gμν = μα
β
gβν + να
β
gμβ . (A.57)

α
Adding the symmetry condition μν = νμ
α , one can derive uniquely the

connection tensor

1 αρ  
α
μν = g ∂μ gαν + ∂ν gαμ − ∂α gμν . (A.58)
2

The Covariant Derivative for Spinor Fields

The expression (A.51) is valid for the derivative of a vector field. For a
scalar, it is straightforward to see that Dμ φ = ∂μ φ. For a spinor field ψ,
the construction of a covariant derivative is a little bit more involved. If we
define
1
Dμ ψ = ∂μ ψ − ωμab [γa γb − γb γa ]ψ, (A.59)
4
in order to obtain the equation for spinor connections (A.51),

Dμ (ψ̄γ ν ψ) = ∂μ (ψ̄γ ν ψ) + αμ

ν
ψ̄γ α ψ, (A.60)

using the vierbein conventions (A.40) and (A.43), we let the reader prove that

1 b βa 1
ωμab = (eα e − eαa eβb )βμ
α
+ (eαb ∂μ eαa − eαa ∂μ eαb ). (A.61)
4 4

Just a remark, when developing Dμ (ψ̄γ α ψ), do not forget to include the term
ψ̄(Dμ γ ν )ψ with γ ν = eaν γ a .

A.2.5 A Geometrical Approach

We can also recover the expression (A.52) with a pure geometrical approach, as it is
done in several very good textbooks. The idea is quite elegant. Instead of imposing
a covariant constraint (A.53), the Christoffel symbol is in fact defined by the change
376 A Cosmology and Astrophysics

induced on a vector Aμ when traveling from the point x to x̃ with

μ
x̃ μ = x μ + dx μ ; Aμ (x) → Ã(x̃) = Aμ (x) − αβ Aα dx β . (A.62)

μ
We will recover the transformation law on αβ (A.52) (which was obtained by
asking Dμ Aν to behave as a tensor) demanding this time that the variation of the
vector field Aμ at any position transforms as a vector. For that, one needs to compute
Aμ (x̃  ) and Ãμ (x̃  ).9 We then can write from (A.62)

μ ∂x μ α μ ∂x
α ∂x β σ
Ãμ (x̃  ) = Aμ (x) − αβ Aα (x)dx β = A (x) − αβ A ρ
(x) dx
∂x α ∂x ρ ∂x σ
(A.63)

and from the definition of Ãμ as a vector

∂ x̃ μ α ∂x μ ∂ 2 x μ  
Ãμ (x̃  ) = α
à ( x̃) = α
+ α σ
dx σ × Aα (x) − ρσ
α
Aρ dx σ ,
∂x ∂x ∂x ∂x
(A.64)

where we used x̃ μ = x μ +dx μ . Comparing Eqs. (A.63) and (A.64) and developing
at the first order in dx μ , we obtain

μ ∂x μ ∂x ρ ∂x σ ν ∂x ρ ∂x σ ∂ 2 x μ
αβ =  − (A.65)
∂x ν ∂x α ∂x β ρσ ∂x α ∂x β ∂x ρ ∂x σ

which is equivalent to Eq. (A.52) (exercise). Notice also that the definition of Dν Aμ
(A.51) has also a geometrical interpretation. Indeed, it is straightforward to check
that

Aμ (x̃) − Ãμ (x̃) = Dν Aμ .dx ν . (A.66)

Dν Aμ can then be interpreted as the differential vector at the point x̃, or in other
words, between the vector at x̃ and the transported vector from x to x̃ (see the box
below). The last (but not the least) tensor one should take into account while looking
for invariant in general space-time transformations is the Riemann tensor.

9 Notice that both A and à should be computed in the same point x̃  to be able to compare their
difference. That a subtle but fundamental point.
A.2 Basics of General Relativity 377

The Parallel Transport

The parallel transport is a fundamental notion in General Relativity, especially
when comes the time to measure movement in gravitational potential and
definitions of geodesics. It is in fact the essence of the General Relativity
principle. It answers to the question : “how do the coordinates of a tensor
evolve, staying parallel to itself, during its motion on a trajectory in space-
time?”. We illustrate the situation in Fig. A.2, where we plotted the tangential
μ
vector along a path s, dx ds (in blue) and the parallel transported vector (in
red). From Eq. (A.66) the transported vector along ds should follow Ãμ (x̃) =
Aμ (x̃), or

dAμ μ dx
ν
Dν Aμ dx ν = 0 ⇒ + αν Aα = 0. (A.67)
ds ds
dx μ
Notice that if one takes Aμ = ds , the tangential vector itself, we obtain the
equation

d 2xμ α
μ dx dx
ν
+ αν = 0, (A.68)
ds 2 ds ds
which is the equation of a geodesic, or as we can feel after a look at Fig. A.2,
the equation of a trajectory that keeps constant its tangential vector, i.e. the
shorter way from a point A to a point B (straight line in the case of flat space,
great radius in the case of a sphere). These are the trajectories that follow a
particle under a space-time deformed by gravity.

Fig. A.2 Illustration of a parallel transported vector along a path s

378 A Cosmology and Astrophysics

A.2.6 The Riemann Tensor

The geometry of a manifold is not determined by how a tensor evolves under

the change of coordinates, but how a vector evolves along a closed curve on the
manifold. In other way, how the covariant derivative (Dν Aμ ∝ dAμ ) behaves in one
direction of the curve, and then in the other direction. This behavior is determined
by the commutation relation Dν Dμ − Dμ Dν , defining the Riemann tensor Rσλ μν by

Dμ Dν Aρ − Dν Dμ Aρ = Rαμν
ρ
Aα . (A.69)

This is reminiscent to the commutation relation in gauge theories, where one can
ρ
think Rαμν as the structure function of the gauge transformation. Applying (A.51)
twice, it is straightforward to check that
ρ
Rαμν = ∂μ αν
ρ
− ∂ν αμ
ρ
+ σρ μ αν
σ
− σρ ν αμ
σ
. (A.70)

Exercise Check the relation (A.70). Then, take a vector Aμ at x ν and parallel
transport it to x̃ ν = x ν + dy ν + dzν following two different ways: moving along dy
and then dz, or along dz and then dy. Show that

Ãρ (dy → dz) − Ãρ (dz → dy) = Aα Rαμν

ρ
dzμ dy ν , (A.71)

expressing the fact that the Riemannian tensor is an intrinsic characteristic of the
curvature.

The interesting point is to be able to define a scalar from the Riemann tensor.
Indeed, as the proper length ds 2 = c2 dt 2 − dl 2 defines the invariant of any
coordinates transformation in the Minkowski space-time, asking for the invariance
of the scalar which is related to the curvature will naturally lead to the Einstein
equation of General Relativity. Defining as a first step the Ricci tensor Rμν

Rμν = Rμσ
σ
ν = ∂σ μν − ∂μ σ ν + ασ μν − αμ νσ ,
σ σ σ α σ α
(A.72)

we obtain the scalar curvature R by contracting the Ricci indices,

R = g μν Rμν . (A.73)
A.2 Basics of General Relativity 379

We can construct an invariant volume from R, as we did for the cosmological

constant, defining this way the Einstein–Hilbert action SEH

 
1 √ M2 √
SEH =− d x −gR = − P
4
d 4 x −gR. (A.74)
16πG 2

A.2.7 The Einstein Equation of Fields in Vacuum

We have now everything in the hand to derive the Einstein equation of fields. Let us
derive them in vacuum first. Adding to the cosmological constant action (A.49) the
curvature scalar, one can write the total action S


1 √
S = S + SEH =− d 4 x −g (R + 2). (A.75)
16πG

To obtain the equations of fields, one needs to minimize the action, or in other
words, to satisfy the condition δS = 0. To do that, we will first minimize the
cosmological constant part S part in Eq. (A.75):
 
 δ(−g)  √
δS = − d 4x √ =− d 4 x −gg μν δgμν , (A.76)
16πG −g 16πG

where we used (A.80)

δg = gg μν δgμν . (A.77)

How to Differentiate a Determinant

We recall here the basics of matrix computation. The determinant of a matrix
aij can be written as det a = i,j aij A
j i where Aij is the comatrix of
ij
aij . A is obtained by taking first the matrix of the Minors (a Minor mi,j
is the determinant of the matrix where the row i and column j have been
suppressed), then multiplying each Minor by (−1)i+j . The resulting matrix
is the matrix of the cofactors Aij . The inverse of the matrix aij can then be
written as (to demonstrate)

1 1
bij = (Aij )! = Aj i . (A.78)
det a det a
(continued)
380 A Cosmology and Astrophysics

Then, from

det a = aij Aj i , (A.79)

we can deduce,

∂ det a
= Aj i = det a × bij . (A.80)
∂aij

For the Einstein–Hilbert part, we can write


1  √ √ √ 
δSEH =− d 4 x δ( −g)R + −gδg μν Rμν + −gg μν δRμν
16πG
= δS1 + δS2 + δS3 . (A.81)

The δS1 part is straightforward as it corresponds exactly to the same kind of

variation we computed for the cosmological constant part δS :

1 √
δS1 = − d 4 x −gRg μν δgμν . (A.82)
32πG

To compute δS2 , we notice that

g μα gαμ = δνμ ⇒ δ(g μα gαν ) = 0 ⇒ δg μν = −g μα δgαβ g βν (A.83)

we obtain
 
1 √ 1 √
δS2 = d 4 x −gg μα δgαβ g βν Rμν = d 4 x −gR μν δgμν .
16πG 16πG
(A.84)

For δS3 , we let the reader check that, using δRμν = Dα δμν
α − D  α , we can
ν μα
compute

1 √
δS3 = − d 4 x −gDα [g μν δμν
α
− g μα δμα
α
]
16πG

1 √ 
=− d 4 x∂α −g[g μν δμνα
− g μα δμα
α
] =0
16πG
A.2 Basics of General Relativity 381

because δS3 is the integral of a total derivative. At the end, combining (A.76),
(A.82), and (A.84) and asking for δS + δS1 + δS2 = 0, we obtain

1
Gμν = R μν − Rg μν = g μν (A.85)
2

which are the Einstein equation of fields in the vacuum.

A.2.8 Adding Matter Fields

Obviously, the Universe is far to be empty. One then needs to introduce a matter
Lagrangian Lm and the matter action Sm
 
√ √
Sm = d 4 x −gLfields = d 4 x −gLm , (A.86)
fields

where the sum runs over all the fields of the theory under consideration. We can then
define the stress–energy–momentum tensor Tμν by
 
1 √
4 1 √
δSm = d x −gTμν δg μν = − d 4 x −gT μν δgμν , (A.87)
2 2

where we used (A.83) for the negative sign. The Einstein equations then become

1
Gμν = R μν − Rg μν = 8πG T μν + g μν , (A.88)
2

where T μν is the energy tensor and G = 1/MP2 l the gravitational coupling (MP l
is the Planck mass = 1.22 × 1019 GeV). Different T μν are computed in the main
text (see Sect. 2.2.3 for a scalar field for instance). In the following, we compute
explicitly the stress–energy tensor for a perfect fluid.

A.2.9 The Perfect Fluid Stress–Energy–Momentum Tensor

To solve the set of equations (A.85), one needs to look into more detail the stress–
energy–momentum tensor T μν . In Sect. 2.2.3 we computed it for a scalar field,
which is very useful when dealing with inflation if we consider a scalar inflation.
However, after inflation occurred, the Universe is filled with a relativistic gas and
then dominated by non-relativistic matter (dust). To understand the evolution of the
382 A Cosmology and Astrophysics

scale parameter R during the reheating (and post reheating) epoch, one needs to find
the energy content of a set of N particles in a given volume V . We will neglect the
effects of gravity on these particles at a first approximation.
When we consider a set of N particles (or equivalently a set of N charges in the
electromagnetic theory of fields), it is useful to define a quadrivector describing the
inflow and outflow of particles (or charges) in a given volume V , or more precisely
the flow of the density of particles. If a volume V contains N particles, we can define
in the rest frame a number density n = N V . For an observer moving at a velocity v
with respect to the rest frame (under any direction x 1 , x 2 , or x 3 ), the contraction
of length (A.9) tells us that n →  n . The dilatation of density in a moving
2
1− v2
c
referential compared to a rest frame can be interpreted as the behavior of 
the time
√
v2
component of a quadrivector. If one writes n = n0 dτdt
with dτ = ds
c = 1− c2
(from A.11), n0 being a constant, we can build a four-vector

dx μ
nμ = n0 , (A.89)
dτ
where n0 and dτ are constant, invariant under Lorentz transformations. It is then
obvious that nμ has the properties of a vector, whose time component is the density
i
of particles. How should we then interpret ni = n0 dx dτ ? If we develop (A.89) as
function of the local time t we obtain
⎛ ⎞ ⎛ ⎞
1 1
⎜ dx 1 ⎟ ⎜ 1⎟
n0 ⎜ dt ⎟ n0 v
⎜ ⎟,
nμ =  ⎜ dx 2 ⎟ =  ⎝ 2⎠ (A.90)
1 − vc2 ⎝ dt3 ⎠ 1 − vc2 v 3
2 2

dx v
dt

where n =  n0 v is called the number density current and is the equivalent of the
2
1− v2
c
current J μ in electrodynamics. In a sense, it corresponds to the number of particles
(or charges) that penetrate the volume due to its movement in space.
How to build a tensor from the number density vector? The easiest way is to
suppose that the energy (momentum) is proportional to the density (current density)
of the particles in the volume V , or in other words

pμ = T μν nν V . (A.91)

It is indeed easy to check that in the rest frame, the components are

p0  pi
T 00 = = and T i0 = = πi (A.92)
n0 V n0 n0 V
A.2 Basics of General Relativity 383

if we set n0 = 1 by simplicity,10 T 00 =  represents then the energy density whereas

T i0 = π i is the momentum density as it would be measured by an observer at rest
in the inertial frame under discussion.
The computation (and interpretation) of the components T 0i can be done directly
from T i0 . Indeed, from (A.91) we can write along the direction x 1 (direction
perpendicular to the plane (x 2 , x 3 )):

E
p0 = T 01 x 2 x 3 x 0 ⇒ T 01 = (A.93)
St

with S = x 2 x 3 . T 0i can then be interpreted by the flux of energy passing

through a surface perpendicular to the ith direction, which is equivalent to the flux
of momenta T i0 :

T 0i = T i0 .

Finally, for the spatial–spatial component, by the same reasoning

pi F i
T ij = = = P ij (A.94)
Sj t Sj

with Sj the element of surface perpendicular to the j th direction and F i the
force exerted on the element of surface Sj , in other words, P ij = T ij is the
pressure oriented in the ith direction, exerted on the surface perpendicular to the j th
direction, which is also called the stress tensor. In summary,

Energy density | Energy flow
T μν
= (A.95)
Momentum density | Stress tensor

which can be written, in the rest frame and considering a perfect fluid (homogeneous
and isotrope), as
⎛ ⎞
ρ 0 0 0
⎜0 P 0 0⎟
=⎜ ⎟
μν
T0 ⎝0 (A.96)
0 P 0⎠
0 0 0 P

To generalize T μν in any referential frame, it should be in a tensor form with

μν
T0 as a limit in the rest frame. The only covariant form can be written T μν =
μ
Auμ uν + Bημν , with uμ = dxdτ . Demanding to recover (A.96) in the rest frame, we

10 The tensor T μν can always been defined up to a constant.

384 A Cosmology and Astrophysics

obtain

dx μ dx ν dx μ dx ν
T μν = (ρ + P ) − ημν P = (ρ + P ) − g μν P , (A.97)
dτ dτ dτ dτ

where, to generalize ημν → g μν , we have used the fact that any curved metric
can be viewed as a Minkowski metric locally.

A.2.10 Deflection Angle

As it is well known, the general theory of relativity has been confirmed in 1919 by
Eddington when he measured a deviation in the light emitted by stars and passing
near the Sun during the 1919 eclipse. It is obviously beyond the scope of this book
(and this appendix) to give a complete mathematical description of the theory of
General Relativity (I let the reader jump to [2] for a nice introduction in the subject).
The basics of deflected light, or even the computation of the perihelion of Mercury
is based on the fact that light as particles move on a timeline geodesic, and so we
classically study some of the geodesics of the Schwarzschild vacuum solution:

2K = (1 − 2m/r)t˙2 − (1 − 2m/r)−1 ṙ 2 − r 2 θ̇ 2 − r 2 sin2 θ φ̇ 2 (A.98)

with 2K = 0, +1 or −1 depending on whether the tangent vector is null, or

has positive or negative length, respectively. In other words, when computing the
Mercury perihelion, 2K = 1 whereas the light travels on a null geodesic and thus
2Kγ = 0. For the light, the Eq. (A.98) then becomes (you can check it as an
exercise)

d 2u
+ u = 3mu2 (A.99)
dφ 2

with u = 1/r. In the limit of special relativity, m vanishes and the equation becomes

d 2u
+ u = 0, (A.100)
dφ 2

the general solution of which can be written in the form

1
u= sin(φ − φ0 ), (A.101)
b
where b is the impact parameter, the distance of closest approach to the origin (see
Fig. A.3 for illustration). This is the equation of a straight line as φ goes from φ0 to
φ0 + π. The straight line motion is the same as is predicted by Newtonian theory.
A.2 Basics of General Relativity 385

r
I
b
I0 x
0
apparent star

G deflection angle

light ray
r I
H1 H2
sun
real star

Fig. A.3 Illustration of the deflected angle due to gravitational field in General Relativity

The equation of a light ray in Schwarzschild space-time can be thought of as a

perturbation of the classical equation, treating m/r as small. We therefore look for
a solution of the form

u = u0 + 3mu1 , (A.102)

where u0 is the solution of (A.101). Taking φ0 = 0 for convenience, we obtain

sin2 φ sin φ m(1 + C cos φ + cos2 φ)

u1 + u1 = u20 = 2
⇒ u + , (A.103)
b b b2
where C is an arbitrary constant of integration for u1 . Since m/b is small, this is
clearly a perturbation from straight line motion. We are interested in determining
the angle of deflection δ for a light ray in the presence of a spherically symmetric
source, such as the Sun. A long way from the source r → ∞ and hence u → 0,
which requires the right-hand side of the Eq. (A.103) to vanish. Let us take the
values of φ for which r → ∞, that is the angles of the asymptote, to be −1 and
386 A Cosmology and Astrophysics

π + 2 , respectively, as shown in Fig. A.3. Using the small angle approximation for
1 and 2 , we get
1 m 2 m
− + 2 (2 + C) = 0, − + 2 (2 − C) = 0. (A.104)
b b b b
Adding, we find

4m
δ = 1 + 2 = (A.105)
b
or, in non-relativistic units,

4GM
δ= (A.106)
c2 b

A.3 Matter/Radiation Domination

As we have already seen , the radiation11 (γ ν) and matter (M) densities are directly
linked to the scale R of the Universe by (see the preceding paragraph)

ργ ∝ a −4 ρν ∝ a −4 ρM ∝ a −3 . (A.107)

From this relation, one can understand that with time, and the expansion of
the Universe, the radiation density decreases faster than the matter density.
Nowadays, the last measurements of the present densities give ργ0 ν 
7.8 × 10−34(T0 /2.725)4 g/cm3 and ρM 0  1.88 × 10−29 2 3
M h g/cm . From
Eq. (A.107) we obtain ργ ν (t)/ρM (t) ∝ a0 /a(t) = 1 + z one can deduce the
radius/redshift/temperature of equal matter and radiation densities (aEQ , zEQ , TEQ )
with T0 = 2.725 K and M h2  0.146:

1 + zEQ = a0 /aEQ = 2.3 × 103 (A.108)

TEQ = T0 (1 + zEQ ) = 0.77 eV = 9000K (A.109)
tEQ = 1.4 × 105 years. (A.110)

TEQ
One can also calculate directly TEQ from T0 = m
γν
with T0 = 2.75 K, γ νh
2 =
4.15 × 10−5 and m h2 = 0.146 gives TEQ = 9000 K/ 0.81 eV/80,000 years.

11 We will note ργ ν the energy density of relativistic particle (CMB + ν) and ργ the energy density
of the CMB.
A.5 Classical Thermodynamic: The Laplace’s Law 387

A.4 Thermodynamical Fundamental Relations

gρ (today) = 3.36; gs (today) = 3.91

π2
ρR = g∗ T 4 = 8.09 × 10−34g cm−3 today
30
2π 2
s= g∗s T 3  2909 cm−3 today
45
2ζ(3) 3
nγ = T = 411 cm−3 today (A.111)
π2
Average energy per particles:

ρ π2
EE m ≡ = T  2.701 T (BOSE) (A.112)
n 30ζ(3)
7π 4
EE m = T  3.151 T (FERMI)
180ζ(3)
3
EE m =m+ T
2
with ζ (3)  1.20206.
nB
= η = 2.68 × 10−8 2
Bh . (A.113)
nγ

A.5 Classical Thermodynamic: The Laplace’s Law

The laws that are used in thermodynamics of the early Universe are the ones
corresponding to a gas of particle transforming adiabatically. We demonstrate in this
section the Laplace law P V γ = constant, valid in such transformations. The first
principle of thermodynamics affirms that “ the change in the internal energy dU of
a closed system is equal to the amount of heat δQ supplied to the system, plus the
amount of work δW received by the system from its surroundings” (or equivalently
“minus the amount of work done by the system on its surroundings”). In the case of
a thermodynamical system, only the internal energy varies

dU = δW + δQ.

The mechanical work is the product of the external pressure P multiplied by the
variation of volume dV , or, if we consider the work received by the system, one
should write

δW = −P dV .
388 A Cosmology and Astrophysics

Moreover, if the process is adiabatic it is generated without exchange of heat, δQ =

0. One then obtains

dU = −P dV .

If one considers the enthalpy of the system, H = U + P V , one can write for an
adiabatic transformation

dH = V dP .

Supposing that the gas is perfect, the enthalpy and internal energy depend only on
the temperature T

dU = Cv dT , dH = Cp dT ,

where Cv and Cp (in units of Joule per Kelvin) are, respectively, the specific heat at
constant volume and pressure.12 We can then deduce

Cp dT = V dP
Cv dT = −P dV

which gives after combining both equations

dV dP
γ + = 0, ⇒ P V γ = constant (A.114)
V P
C
with γ = Cpv . This is the Laplace’s law. Another interesting relation is between
internal energy and pressure, called the equation of state of the gas. If one defines
α by P = αu, u being the density of internal energy (kinetic one in the absence
of other sources and in adiabatic processes), one can write P V = αU . Combining
with Eq. (A.114) and remembering that dU = −P dV one obtains

αdU = P dV +V dP = P dV −γ P dV = P dV (1−γ ) = −αP dV ⇒ α = γ −1.

(A.115)

12 A specific heat, or heat capacity is the heat required to raise a unit mass of the material by one

degree. This can be done at constant volume or at constant pressure and the corresponding symbols
are Cv and Cp .
A.6 Tooling with Math 389

A.6 Tooling with Math

A.6.1 Function (z)

A.6.1.1 Definition, Propriety

 ∞
(z) = t z−1 e−t dt
0
(z + 1) = z(z)
(n) = (n − 1)! (A.116)

A.6.1.2 Some Values

4√
(−3/2) = π ≈ 2.36
3
√
(−1/2) = −2 π ≈ −3.54
√
(1/2) = π ≈ 1.77
(1) = 0! = 1
1√
(3/2) = π ≈ 0.89
2
(2) = 1! = 1
3√
(5/2) = π ≈ 1.33
4
(3) = 2! = 2
15 √
(7/2) = π ≈ 3.32
8
(4) = 3! = 6. (A.117)

A.6.2 The Riemann Zeta Function ζ(z)

A.6.2.1 Definition, Propriety

∞
1 1 1 1 1
ζ (z) = n−z = + z + z + z + z + ... (A.118)
1z 2 3 4 5
1

A.6.2.2 Some Values

1
ζ(0) = −
2
ζ(1/2) ≈ −1.46
390 A Cosmology and Astrophysics

ζ(1) = ∞
ζ(3/2) ≈ 2.61
π2
ζ(2) ≈ 1.64 =
6
ζ(3) ≈ 1.2
π4
ζ(4) ≈ 1.08 = (A.119)
90
π6
ζ(6) =
945
π8
ζ(8) =
9450
π 10
ζ(10) = .
93555

A.6.3 Modified Bessel Function of the 2nd Kind Kn (z)

A.6.3.1 Definition, Propriety

z2 Kn (z) + zKn (z) − (z2 + n2 )Kn (z) = 0

√ n  ∞ 
πz −zt 2 n− 12 (n + 12 )(2z)n ∞ cos tdt
Kn (z) = e (t − 1) dt = √
2 (n + 2 ) 1
n 1 1
π 0 (t + z2 )n+ 2
2

n
Kn (z) − Kn (z) = Kn+1 (z)
z
K0 (z) ∼ − log z (A.120)
 ∞ 
K1 (z) = z e−zt t 2 − 1dt
1
 ∞  ∞ 
z2 −zt 1
K2 (z) = e (t − 1)
2 3/2
dt = K1 (z) − K1 (z) = z te−zt t 2 − 1dt
3 1 z 1

π −z
z → ∞, Kn (z)  e .
2z

A.6.3.2 Some Values

K1 (0.01) = 99.97; K1 (0.1) = 9.84; K1 (1) = 0.60;

K1 (10) = 1.9 × 10−5 ; K1 (100) = 4.7 × 10−45
A.6 Tooling with Math 391

K2 (0.01) = 20000; K2 (0.1) = 200; K2 (1) = 1.63;

K2 (10) = 2. × 10−5 ; K2 (100) = 4.7 × 10−45
 ∞  ∞
z2 K1 (z)dz = 2; z4 K1 (z)dz = 16;
0 0
 ∞  ∞
z3 K2 (z)dz = 8; z5 K2 (z)dz = 96;
0 0

A.6.3.3 Some Approximations

For x  10, we can write
 π 1  
2 −x 3 5 21
K1 (x)  e 1+ 1− 1− . (A.121)
2x 8x 16x 24x

A.6.4 Useful Integrals

 
−cx 2 π √
e = erf( cx) erf being the error function
4c

1 −cx 2
xe−cx = −
2
e
2c
 
π √ 1
x 2 e−cx erf( cx) − xe−cx
2 2
= 3
16c 2c

x 2 −cx 2 1
x 3 e−cx = − − 2 e−cx
2 2
e
2c 2c
 
e−cx
2
3 π √
x 4 e−cx
2
= erf( cx) − [3x + 2cx 3]
8 c5 4c2

e−cx
2
c2
x 5 e−cx = −
2
3
1 + cx 2 + x 4
c 2
 
15 π √ 15 5 1
x 6 e−cx erf( cx) − e−cx
2 2
= 7
x + 2 x3 + x5
16 c 8c3 4c 2c
 ∞
ecx (cx)n
dx = ln |x| +
x n.n!
n=1
 cx  
e 1 ecx ecx
= − + c (for n = 1)
xn n−1 x n−1 x n−1
 +∞ 
1 π
e−ax dx =
2

0 2 a
392 A Cosmology and Astrophysics

 +∞

2 −ax 2 1 π
x e dx =
0 4 a3
 +∞

3 π
x 4 e−ax dx =
2

0 8a 2 a
 +∞

n −ax 2 (2k − 1)!! π
x e dx = k+1 k (n = 2k, k integer)
0 2 a a
 ∞ 1
t (t 2 − 1)1/2 e−zt dt = K2 (z)
1 z
 ∞ 
xn
dx = (n + 1)ζ(n + 1)y(δ) (A.122)
0 ex − δ

with y(δ) = 1 if δ = 1 and 1 − 21n if δ = −1. n!! is the double factorial defined as
n!! = n × (n − 2) × (n − 4) × ..
 x
t a−1 e−t dt = γ (a, x), (A.123)
0

γ (a, x) being the incomplete gamma function that can be approximate

xa
γ (a, x)  , for x 1. (A.124)
a
And finally,
 ∞  ∞
dx dx
= log(2) ; =∞ (A.125)
0 ex + 1 0 ex − 1

1 x 1  
dx = − ln aeλx + b . (A.126)
aeλx +b b bλ

A.6.4.1 Euler–Masheroni Constant γ

γ = 0.577 (A.127)

A.6.4.2 Gauss Error Function erf

 x
2
e−t dt.
2
erf(x) = √ (A.128)
π 0
A.6 Tooling with Math 393

A.6.4.3 Delta Dirac δ

One useful definition of the Dirac-δ function is
 +∞
1
δ(x − α) = eip(x−α)dp (A.129)
2π −∞
 +∞  +∞
δ(x)dx = 1, f (x)δ(x − α)dx = f (α) (A.130)
−∞ −∞

δ(x − x0 )
δ(g(x)) = , (A.131)
|g  (x0 )|

where x0 is the root of the function g(x). If g(x) possesses several roots, one has to
sum over them.

A.6.5 Laplace Operator

In mathematics, the Laplace operator , or Laplacian, is given by the divergence of

the gradient of a function f :

f = ∇ 2 f = ∇.∇f, (A.132)

where, in the n Euclidian space ∇ is defined by


∂ ∂
∇= ,..., , (A.133)
∂x1 ∂xn

which gives, in the Cartesian coordinates xi :

n
∂ 2f
f = . (A.134)
i
∂xi2

In (r, θ ) polar coordinates, we can write


1 ∂ ∂f 1 ∂ 2f ∂ 2f 1 ∂f 1 ∂ 2f
f = r + 2 2
= 2
+ + 2 2, (A.135)
r ∂r ∂r r ∂θ ∂r r ∂r r ∂θ

whereas in (r, θ, z) cylindrical coordinates, we obtain


1 ∂ ∂f 1 ∂ 2f ∂ 2f
f = ρ + 2 2
+ 2, (A.136)
ρ ∂ρ ∂ρ ρ ∂φ ∂z
394 A Cosmology and Astrophysics

and in (r, θ, φ) spherical coordinates

 
1 ∂ ∂f 1 ∂ ∂f 1 ∂ 2f
f = 2 r2 + 2 sin θ + . (A.137)
r ∂r ∂r r sin θ ∂θ ∂θ r 2 sin2 θ ∂φ 2
Particle Physics
B

B.1 Feynman Rules

In this section1 we summarize point-by-point how to compute a decay rate and a

cross section, from the amplitude of the process M. We then make explicit the rules
to compute the amplitude M in any microscopic model.

B.1.1 Decay Rates and Cross Sections

To compute a differential rate, with n particles in the final state, one should follow
similar rules if you consider a decay rate or a scattering cross section. In the former
case, for a decaying particle of energy E, the differential width dn is obtained by
multiplying the following factors:

• A factor of (2π)4 δ 4 (Pf − Pi ) where Pf is the total four-momentum of the

n decay products and Pi is the four-momentum of the decaying particle,
representing the condition of energy–momentum conservation;

1 This section is freely adapted from the excellent book by Franz Gross “Relativistic Quantum

Mechanics and Field Theory”, Wiley Eds.

© Springer Nature Switzerland AG 2021 395

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
396 B Particle Physics

• A factor

d 3 pi
(2π)3 Ei

for each particle in the final state where pi and Ei are the momentum and
energy of the ith particle;
1
• A factor 2E for the initial particle which is decaying;
• And the absolute square of the M-matrix.

The differential width is then (we let the reader go through the Sect. B.4.1 for
a more detailed analysis of the cross section computation, inspired by the Fermi’s
golden rule)

1 ! d 3 pi
i=n
dn = (2π)4 δ 4 (Pf − Pi ) |M|2 . (B.1)
2E (2π)3 2Ei
i=1

The total decay rate is obtained by integrating Eq. (B.1) over all outgoing momenta,
summing over all final spins and averaging over the initial spin, S, of the decaying
particle.

1
= dn . (B.2)
2S + 1
spins

We usually note |M̄|2 the mean of the amplitude square, summed over the final
spins, averaged over the initial spin.
The differential cross section for the production of n particles is obtained
multiplying the following factors:

• A factor of (2π)4 δ 4 (Pf − Pi )where Pf is the total four-momentum of

the n final states and Pi is the four-momentum of the two initial particles,
representing the condition of energy–momentum conservation;
• A factor

d 3 pi
(2π)3 Ei

for each particle in the final state where pi and Ei are the momentum and
energy of the ith particle;
B.2 Feynman Rules 397

• A factor 4EE 1 
 where E and E are the energies of the two particles in the
initial states;
• A factor of 1/v where v is the flux, or relative velocity of the two (colinear)
colliding particles, equal to

p p
v= +  (B.3)
E E

where p and p are the magnitudes of their momenta, and

• The absolute square of the M-matrix.

The differential cross section can then be written as

1 ! d 3 pi
i=n
dσ = (2π)4 δ 4 (Pf − Pi ) |M|2 . (B.4)
4EE  v (2π)3 2Ei
i=1

The unpolarized cross section for scattering into some final state in the phase volume
 is therefore obtained by integrating Eq. (B.4) over all outgoing momenta in  ,
summing over all final spins and averaging over initial spins:

1
σ = dσ , (B.5)
 (2S + 1)(2S  + 1)
spins

where S and S  are the spins of the initial particles. Finally, in calculating both decay
rates and cross sections, for each set of m identical particles in the final states, the
integral over momenta must either be divided by m! or limited to the restricted cone
θ1 < θ2 < . . . < θm .

B.2 Feynman Rules

B.2.1 General Rules

We give in this section the generic rules to compute the amplitude M.

• The diagrams consist of lines and vertices.

• Each internal line represents the propagation of a virtual particle, between two
vertices which represent points in space-time. Each vertex is determined by
an interaction term in the Lagrangian, where particles can be destroyed and/or
created.
398 B Particle Physics

• Energy conservation fixes the four-momentum of each internal lines. If diagrams

contain loops, there exists at least one momentum for each loop which cannot be
determined, and should be integrated over.

Each diagram should be associated with a number with the following rules:
Rule 0
Multiplication by an overall factor of i, which is reminiscent of the form of the
S-matrix S = ei T , see Eq. (B.83).
Rule 1
An operator for each vertex, determined by the form of the Lagrangian.
Rule 2
A propagator for each internal line with four-momentum k , the precise form of
which depends on the particle propagating. For spin zero bosons with isospin
indices (i,j ), for fermions with Dirac indices (α,β), and for photons or massive
vector bosons with polarization indices (μ,ν), they are written as

iδij
Spin 0 : ij (k) =
− k 2 − i
m2
i(m + k/)αβ
Spin 1/2 : αβ (k) = 2
m − k 2 − i
−i kμ kν
photon or gluon : μν (k) = ημν − 2 (1 − α)
−k 2 − i k
−i[ημν − kμ kν /m2 ]
vector boson : μν (k) =
m2 − k 2 − i

energy momentum conservation determines k as function of the external

momenta, whereas α appearing in the case of massless gauge boson (photon
and gluon) depends on the chosen gauge : α = 0 in the Landau Gauge whereas
α = 1 in the Feynman gauge for instance.
Rule 3

• Multiply from the left by ū(p3 , s3 ) for each outgoing fermion with momentum
p3 and spin s3 .
• Multiply from the right by u(p1 , s1 ) for each incoming fermion with momen-
tum p1 and spin s1 .
• Multiply from the right by v(p4 , s4 ) for each outgoing antifermions with
momentum p4 and spin s4 .
• Multiply from the left by v̄(p2 , s2 ) for each incoming antifermion with
momentum p2 and spin s2 .
• Multiplying by μ∗ for each outgoing vector with polarization index μ.
• Multiplying by μ for each incoming vector with polarization index μ.
B.2 Feynman Rules 399

These last polarization vectors should be contracted with the γ μ matrices

emerging from the vertex implying fermions.
Rule 4
Symmetrize between identical bosons in the initial or final state, antisymmetrize
between identical fermions in the initial or final state. Concretely speaking, it
means that if we have 2 identical bosons in the final state, one should divide
the amplitude square by 2, as well as for Majorana final states (see below), the
antisymmetrization cancelling the minus sign arising from the exchange of two
spinors. In other words, |M|2 = 12 |M1 + M2 |2 , where Mi are the amplitudes of
the 2 identical bosons (or fermions with a minus sign).
Rule 5
Multiply by the momentum conservation factor (2π)4 δ (4) ( i pi ), pi being
the external momenta and integrate over each internal four-momentum k left
undetermined

d 4k
.
(2π)4

Rule 6
Add a minus sign to each closed fermion loop.
Rule 7
Multiply by n!1 for bubbles with n! identical neutral bosons.
Rule 8 √
Include the field renormalization factors Zi for each external particle i.
Rule 9
M is the sum over all topologically distinct diagrams with equal weight except
for a relative (-1) if two diagrams differ by the permutation of two fermion
operators. This gives for instance a relative sign between the two diagrams
contributing to e− e− scattering and similarly the two contributing to Bhabha
scattering.
Rule 10
For each particle with a mass which could be shifted by self-interactions, m →
m + δm, a mass counter-term iδm is added to remove the mass shift.

B.2.2 Majorana Rules

In the case of Majorana fermions, the Feynman rules have to be adapted. Indeed,
Majorana flows violate the fermion number, and we cannot compute the amplitude
exactly the same way. We compute explicit examples in Sect. B.4.8. Please, read also
the Sect. B.3.8 before this one to recall the clear definition of a Majorana fermion
and its Lagrangian (which differs from a Dirac one by a factor 1/2). There are so
many factors 2 in the calculation, that it is easy to be lost, and computation should
be made carefully to avoid double counting.
400 B Particle Physics

Our point of departure will be the Lagrangian (B.62). We will take the specific
example of the Majorana coupling to a Higgs-like field φ. The Majorana part is

1
yλ φ λ̄λ, (B.6)
2
where we took  = 1, but the reasoning is valid for any kind of coupling structure.
While treating Majorana fermions, the Feynman rules for identical particles in the
final states apply: one should multiply by 2 the coupling which appears in the
lagrangian (so a factor 4 in the amplitude square) and divide by 2 the physical
process (antisymmetrisation because 2 identical fermions in the final state). We will
explain the former factor “2” in detail in this section.
Indeed, to compute an amplitude, one needs to develop λ into the creation and
destruction operators2 b† and b.

d 3k
λ= (b(k)ue−ikx + b(k)†veikx ) (B.7)
(2π)3 2E

with u and v the Dirac spinors studied in Sect. B.3.2. If one looks for any process,
the fact of having two “b’s” in the development of λ will generate a factor 2 of
symmetry. This fundamentally comes from the fact that one cannot trace a fermion
flow with Majorana fermion and “entering” or “exiting” flows can combine together.
Let us take a concrete example. A particle φ decaying into two λ’s

1
λ(p1 )λ(p2 )|L|φ = 0|b(p1 )b(p2 ) yλ φλλ aφ† |0
2
yλ
= [ū(p1 )v(p2 ) + ū(p2 )v(p1 )] = yλ ū(p1 )v(p2 ),
2

where we used the identity for Majorana particles in the last equality, u = C v̄ T
and v = C ūT . As one notices, the factor 12 in front of yλ disappeared, and we
recover the same Feynman rule than in the case of the Dirac fermion. That is the
fundamental point, and main result of this section. For a concrete example, have
a look at Sect. B.4.8 where we computed the decay of a scalar into Majorana and
Dirac particles.

B.2.2.1 Another Interpretation

One can find another interpretation in the literature in [1]. This interpretation
released on the fermionic fluxes. Indeed, whereas for general fermion fields χ, a

2 Inthe case of a Dirac fermions, b† should be replaced by d † , because a Dirac fermion ψ does not
follow ψ C = ψ.
B.2 Feynman Rules 401

iΓ iΓ

iΓ iΓ

iΓ iΓ

iΓ iΓ

Fig. B.1 The Feynman rules for fermionic vertices with orientation (thin arrow)

Lagrangian can be written as

¯  χ̃,
χ̄χ = χ̃ (B.8)

where  is the coupling matrix, χ̃ = C χ̄ T (C being the charge conjugate operator

(see Sect. B.3.7)) and   = C T C −1 . This means that the same term in the
Lagrangian will generate a process with particle(p1 )–antiparticle(p2) in the final
state, and the symmetric one, antiparticle(p1)–particle(p2), p1 and p2 being their
respective momentum. Both final states are of course different. For a Majorana final
state χ̃ = χ. Therefore, if both χ  s are Majorana fermions,  =   . In other
way, there is no flow of fermions, and one should compute matrix elements with
Majorana states without distinguishing the possible flows of particles/antiparticles.
This is illustrated more clearly in Fig. B.1 where we show the Feynman rules for
fermionic vertices with orientation. As we can see, the two figures on top are the
same, and one should not add them in a computation as it will be double counting.
On the other hand, the two last figures represent the Dirac case, and two different
states of particle/antiparticle with exchanged momentum. Only one of this diagram
should also contribute to the process. In conclusion, for the computation of the
matrix element, in the Majorana as in the Dirac case, one should count only one
flow, as in one case (Dirac) the other flow correspond to another final state with
exchanged momentum, whereas in Majorana case, that would be double counting.
402 B Particle Physics

Let us be more concrete. In the computation in a rate, implying two Majorana

fermions in the final state, it is important to add a symmetry factor in the final result.
Let us consider for instance the simplest example of the decay of a scalar φ into two
Majorana fermions λ compared to a (Dirac) pair of electron positron final state of
momentum p1 and p2 . The Lagrangian

1
L = yφe φ ēe + yφλ φ λ̄λ (B.9)
2

|M|2
gives, respectively, the width (see Eq. B.165) φi = 16π×sym mφ

+ e− (yφe )2 mφ (yφλ )2 mφ
φe = , φλλ = , (B.10)
8π 16π
where we neglected the final state masses. Indeed, in the Dirac (electron) case, the
amplitude is M = −iyφe ū(p1 )v(p2 ) which implies spin |M|2 = 4(yφe )2 p1 .p2 =
2(yφe )2 m2φ , and no symmetry factor sym is present for Dirac final state because
both particles are different. In the Majorana case λ, as we explained above, only
one flow should be considered to avoid double counting. Then, spin |M|2 =
 
4(yφλ )2 p1 .p2 = 2(yφλ )2 m2φ , exactly as in the Dirac case. The 12 symmetry factor
sym between the two width can be understood by a look at Fig. B.1. For a given final
state angle, the e+ e− system is definite, whereas there are two possibilities for the
λλ final state : both can have p1 and/or p2 . There is then a double counting once
we integrate on the total solid angle, and  one should symmetrize it to take it into
account, multiplying the final result by 12 .

B.2.3 Standard Model Couplings

We show in the following diagrams the Feynman rules in the Standard Model
context and its simplest extension (Feynman rules = iL× symmetry factors). In
the case of the anomalies mediated trivectorial couplings, we are based on the
Lagrangian

LCS = α1  μνρσ Zμ Zν Fρσ

Y
+ α2  μνρσ Zμ Zν Fρσ

, (B.11)

which gives the interactions

μνσ
Z  ZZ (p3 ; p1 , p2 ) = 2α1 sW  μνρσ (p1 − p2 )ρ ,
μνσ
Z  Zγ (p3 ; p1 , p2 ) = 2α1 cW  μνρσ (p2 )ρ ,
μνσ
Z  ZZ  (p3 ; p1 , p2 ) = 2α2  μνρσ (p2 − p3 )ρ , (B.12)
B.2 Feynman Rules 403

with the obvious notation sW = sin θW and cW = cos θW . Notice the decomposition
Yμ = cos θW Aμ − sin θW Zμ and the symmetrization for the identical final state
Zμ Zν (p1 )Zσ (p2 ) μνρσ (p2 )ρ = 12 Zμ Zν (p1 )Zσ (p2 ) μνρσ (p2 − p1 )ρ multiplied by
the symmetry factor (2).

iq e γμ iq e(p + p )μ −2ie2 q 2 ημν

ig 1 1 ig μ
γμ (−4q sin2 θW ± 1) ∓ γ 5 (T3 = + / − ) √ γ (1 − γ 5 )
4 cos θW 2 2 2 2

−ie ημν (p1 − p2 )σ + ηνσ (p2 − p3 )μ + ησμ (p3 − p1 )ν −ig cos θW ημν (p1 − p2 )σ + ηνσ (p2 − p3 )μ + ησμ (p3 − p1 )ν

−i(g cos θW )2 2ημν ηρσ − ημρ ηνσ − ημσ ηνρ −ig 2 sin θW cos θW 2ημν ηρσ − ημρ ηνσ − ημσ ηνρ

−ie2 2ημν ηρσ − ημρ ηνσ − ημσ ηνρ −ig 2 2ημν ηρσ − ημρ ηνσ − ημσ ηνρ
404 B Particle Physics

2
ig i g i 2
ig MW ημν MW ημν ημν g ημν
cos2 θW 2 cos θW 2

2 2
3i gMH 3i gMH Yf mf gmf
− − −i √ = −i = −i
2 MW 4 MW 2 v 2MW
v
mf = Yf H = Yf √
2

μνρσ
2i μνρσ
(p2 )ρ = 2i cos θW (pA Z Z
2 )ρ − 2i sin θW (p2 − p1 )ρ 2i (p2 − p3 )ρ

B.3 Diracology

B.3.1 Matrices

We present in this appendix the necessary tools to compute high energy processes,
in the framework of quantum field theory.

Pauli Matrices
  
01 0 −i 1 0
σ1 = σ2 = σ3 = (B.13)
10 i 0 0 −1
B.3 Diracology 405

Dirac Matrices

{γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2ημν I4 ; γμ = ημν γ ν {γ 5 , γ μ } = 0, γ 0 γμ† γ 0 = γμ

(B.14)

we name the operator σ μν = 2i [γ μ , γ ν ] = 2i (γ μ γ ν − γ ν γ μ ) with ημν the

Minkowski metric with the signature (+,−,−,−) and I4 the 4 × 4 identity matrix
and

0 σi
γ =
i
(B.15)
−σi 0

which gives in explicit form,

⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0010 0 0 01 0 0 0 −i
⎜0 0 0 1⎟ ⎜ 0 0 1 0⎟ ⎜ 0 0i 0 ⎟
γ0 = ⎜ ⎟ ⎜ ⎟ ⎜
⎝ 1 0 0 0 ⎠ γ = ⎝ 0 −1 0 0 ⎠ γ = ⎝ 0 i 0 0 ⎠
1 2 ⎟

0100 −1 0 0 0 −i 0 0 0
⎛ ⎞ ⎛ ⎞
0 01 0 −1 0 0 0
⎜ 0 0 0 −1 ⎟ ⎜ 0 −1 0 0 ⎟
γ3 = ⎜ ⎟ 5 0 1 2 3 ⎜
⎝ −1 0 0 0 ⎠ γ ≡ iγ γ γ γ = ⎝ 0 0 1 0 ⎠ .
⎟ (B.16)
0 10 0 0 0 01

Another possible representation for γ 0 and γ 5 , respecting the same commuta-

tion/anticommutation laws is
⎛ ⎞ ⎛ ⎞
10 0 0 001 0
⎜ 01 0 0 ⎟ ⎜ 1⎟
γ0 = ⎜ ⎟ γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = ⎜ 0 0 0 ⎟. (B.17)
⎝0 0 −1 0 ⎠ ⎝1 0 0 0⎠
00 0 −1 010 0

Depending on the problem, it is sometimes easier to use conventions (B.16).

This is the case when treating processes with interactions distinguishing left and
right helicities for fermions, as in the Standard Model. Indeed, in this case, the
eigenvectors are eigenvectors of γ 5 , which is diagonal, rendering the calculations
easier. On the other hand, when trying to find solutions of the Dirac equations, the
convention (B.17), being diagonal in time derivative, gives simpler solutions. We
will usually work respecting the definitions (B.16), explicitly indicating if we use
convention (B.17).
406 B Particle Physics

B.3.2 Dirac Equation

The Dirac equation for a spinor ψ of mass m and energy E 2 = p2 + m2 should be

written as

(iγ μ ∂μ − m)| = (iγ 0 ∂t − iγ i ∂xi − m) | = 0 (B.18)

Using convention (B.17), the solutions of the equation are then

⎛ ⎞
 1
E+m ⎜
⎜ 0 ⎟
⎟ −ipx
|1  = ⎜ p3 ⎟e = u1 e−ipx ;
2m ⎝ E+m ⎠
p1 +ip2
E+m
⎛ ⎞
 0
E+m ⎜
⎜ 1 ⎟
⎟ −ipx
|2  = ⎜ p1 −ip2 ⎟e = u2 e−ipx ;
2m ⎝ E+m ⎠
−p3
E+m
⎛ p3
⎞
 E+m
E+m ⎜
⎜
p1 +ip2 ⎟
⎟ ipx
|3  = ⎜ E+m ⎟ e = v2 eipx ;
2m ⎝ 1 ⎠
0
⎛p ⎞
1 −ip2
 E+m
E+m ⎜
⎜
−p3 ⎟
⎟ ipx
|4  = ⎜ E+m ⎟ e = v1 eipx . (B.19)
2m ⎝ 0 ⎠
1

It is often easier to work in the rest frame, and then applying a Lorentz transforma-
tion. The solutions for p = 0 are then
⎛ ⎞ ⎛ ⎞
1 0
⎜ 0 ⎟ −imt ⎜ 1 ⎟ −imt
|1  = ⎜
⎝0⎠e
⎟ = u01 e−imt ; |2  = ⎜
⎝0⎠e
⎟ = u02 e−imt ;
0 0
⎛ ⎞ ⎛ ⎞
0 0
⎜ 0 ⎟ imt ⎜ 0 ⎟ imt
|3  = ⎜ ⎟
⎝ 1 ⎠ e = v2 e ;
0 imt
|4  = ⎜ ⎟
⎝ 0 ⎠ e = v1 e .
0 imt
(B.20)
0 1

Exercise Find the eigenvectors of the Dirac equation, for a particle at rest, using
the convention (B.16).
B.3 Diracology 407

B.3.3 The Spin Matrix

The spin matrix i can be written as

⎛ ⎞ ⎛ ⎞
01 00 0 −i 0 0

h̄ σi 0 h̄ ⎜ 10 0 0⎟ h̄ ⎜i 0 0 0 ⎟
i = → 1 = ⎜ ⎝
⎟ ; 2 = ⎜ ⎟;
2 0 σi 2 00 0 1⎠ 2 ⎝0 0 0 −i ⎠
00 10 0 0 i 0
⎛ ⎞
1 0 0 0
h̄ ⎜ 0 −1 0 0 ⎟
3 = ⎜ ⎟. (B.21)
2 ⎝0 0 1 0 ⎠
0 0 0 −1

Exercise Check that, in the case of a particle at rest, 3 |1,3  = 2 |1,3 

h̄
and
3 |2,4 = − h̄2 |2,4.

For p3 = 0 the result is still valid as a translation in z axis does not change the
projection of the spin momentum on the z axis. However, after a translation on the
x or y axis, a p1 and p2 component appears (relativistic boost). One can also give
another useful expression for the spin matrix:

σi 0 ¯ i γ 5 |,
 †| | =  † |γ 0 γ i γ 5 | = |γ (B.22)
0 σi

where we have used

ψ̄ = ψ † γ 0 . (B.23)

Some useful relations are sometimes used to simplify the calculation of amplitudes.
From the Dirac equation we can write

/p1 u(p1 ) = mu(p1 ); /p1 v(p1 ) = −mv(p1 ); ū(p2 )/p2 = mū(p2 ); v̄(p2 )/p2 = −mv̄(p2 ).
(B.24)

1−γ 5 1+γ 5
The helicity operators PL = 2 and PR = 2 can then be written in the
convention (B.16)
⎛ ⎞ ⎛ ⎞
00 00 1 00 0
⎜0 0 0 0⎟ ⎜0 10 0⎟
PR = ⎜
⎝0 0
⎟ PL = ⎜ ⎟. (B.25)
1 0⎠ ⎝0 00 0⎠
00 01 0 00 0
408 B Particle Physics

We can then decompose a Dirac spinor into its helicity (eigenvalues) component,

L
= . (B.26)
R

B.3.4 Proca Equation

The Proca equation is the equation of movement of spin-1 fields. It can be deduced
from the massive spin-1 Lagrangian

1 1
L = − F μν Fμν + M 2 Aμ Aμ (B.27)
4 2
with

Fμν = ∂μ Aν − ∂ν Aμ . (B.28)

The Euler–Lagrange equation gives

∂L ∂L
− ∂μ = 0 ⇒ ∂μ F μν + M 2 Aν = 0. (B.29)
∂Aν ∂(∂μ Aν )

Acting ∂ν on (B.29), we obtain

∂ν Aν = 0 (B.30)

which transforms (B.29) into

( + M 2 )Aν = 0, (B.31)

which is the Proca equation. We have 3 degrees of freedom, one being eliminated
by (B.30). We then obtain 3 polarizations that we can write, supposing a particle
moving along the z-axis:
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 0 |p|
1 ⎜ 1⎟ 1 ⎜ 1 ⎟ 1 ⎜
⎜ 0 ⎟;
⎟
+1 = √ ⎜ ⎟; −1 = √ ⎜ ⎟; 0 = (B.32)
⎝
2 i⎠ 2 ⎝ −i ⎠ M ⎝ 0 ⎠
0 0 E

where λ represents the polarization vector with the spin projection λ along the z-
axis.
B.3 Diracology 409

Exercise Check that λ are unitary vectors respecting the spin-1 relation (B.30),
μ
pμ λ = 0.

B.3.5 Rarita–Schwinger Equation

Even if the Nature showed us spin-0, spin- 12 , and spin-1 fields, some models
exhibit spin- 32 particles. The gravitino, partner of the graviton in supergravity is one
example. In 1941, Rarita and Schwinger, based on a 1939 work by Fierz and Pauli,
constructed the first spin- 32 Lagrangian. The basic idea is quite simple. Such a field
μ should verify the Dirac and the Proca equation. The resulting wave function
should then be a direct product of the Dirac solution (B.20) and Proca solution
(B.32). In other words, μ should satisfy

(iγ ρ ∂ρ − m)μ = 0; ∂ μ μ = 0. (B.33)

The spin projection on the z-axis can take the values + 32 , + 12 , − 12 , and − 32 . We
can then write, following the Clebsch–Gordan decomposition

+3 1
μ 2 =  + 2 μ+1

+1 1 2 +1 0
= √  − 2 μ+1 +
1
μ 2  2 μ
3 3

−1 1 2 −1 0
= √  + 2 μ−1 +
1
μ 2  2 μ
3 3
−3
μ 2 =  − 2 μ−1 .
1

1 1
Exercise In the rest frame, using for  + 2 , |1  (or |3 ), and for  − 2 , |2  (or
4 ) of Eq. (B.20), and  λ of Eq. (B.32), show that

γ μ μ = 0. (B.34)

This relation being Lorentz invariant, if it is valid in the rest frame, it is also valid
in any Lorentz-transformed frame. With the three relations we have in (B.33) and
(B.34), we see that we can generalize the Dirac Lagrangian, introducing terms of
410 B Particle Physics

the type γ μ μ , ∂ μ μ and all types of combination of these terms. We then obtain
 
¯ μ iημν γ ρ ∂ρ − mημν −iγ μ ∂ ν − iγ ν ∂ μ + iγ μ γ ρ γ ν ∂ρ + mγ μ γ ν ν .
L3/2 = 
(B.35)

Noticing that

1 μ ν
γ μ γ ν = −γ ν γ μ + 2ημν , γ μν = [γ , γ ] = γ μ γ ν − ημν (B.36)
2
and defining

1  μ ν ρ 
γ μνρ = γ γ γ + γ ν γ ρ γ μ + γ ρ γ μγ ν − γ ν γ μγ ρ − γ μγ ρ γ ν − γ ρ γ ν γ μ
3!
= γ μ γ ν γ ρ − ηνρ γ μ − ημν γ ρ + ημρ γ ν , (B.37)

we can then simplify (B.35)

 
¯ μ iγ μρν ∂ρ + mγ μν ν
L3/2 =  (B.38)

which is the Rarita–Schwinger Lagrangian. Notice the combinations of γ μ ∂ ν

factors that we added with a certain coefficient. In the original paper of 1941, it
was a factor 13 that was chosen by the authors.

B.3.6 Parity Operator

The parity operator, P, inverts all space coordinates used in the description of
a physical process. Consider for instance a scalar wave function ψ(x, y, z, t).
Performing the parity operation on this wave function will transform it to
ψ(−x, −y, −z, t), or

Pψ(x, y, z, t) = ψ(−x, −y, −z, t). (B.39)

The parity transformation can be viewed as a mirroring with respect to a plane (for
instance z → −z) followed by a rotation around an axis perpendicular to the plane
(the z-axis). As angular momentum is conserved, physics will be invariant under
the rotation and so the parity operation tests for invariance to mirroring with respect
to a plane of arbitrary orientation. Parity conservation or P-symmetry implies that
any physical process will proceed identically, when viewed in mirror image. This
sounds rather natural. After all, we would not expect a dice for instance to produce
a different distribution of numbers if one swaps the position of the one and the six
of the dice.
B.3 Diracology 411

It is important to underline that we discuss in this section the intrinsic parity

associated with each particle, and not the one associated with the orbital wave
function of the particle. In other words, the parity of the wave function of a particle
or a system of particles is the product of the orbital parity times the product of the
intrinsic parities of the particles involved.
In interactions where parity is conserved (electromagnetic and strong), we can use
this to compute selection rules for the various reactions. We do this aided by the
following two ideas :

• Under parity, the orbital wave function will change by (−1)L where L is the
angular momentum quantum number. This is because the parity operator xi →
−xi , changes θ → θ − π and φ → φ + π. Due to this change in the angle,
the orbital wave function, which always involves spherical harmonics (central
potential) changes as (−1)L. This is actually the reason that the photon has
negative parity and is referred to as a 1− state. Photons are emitted via atomic
dipole transitions where L = ±1. Hence, the atomic parity changes by (−1)
during these transitions, and for the overall parity of the system (atom + photon)
to be conserved (electromagnetic interactions), we must have that the photon has
negative parity.
• The overall wave function must be symmetric for systems of bosons and
antisymmetric for systems of fermions.

In summary, the overall parity of the wave function of a set of particles is given by

P = (−1)L × P1 × P2 × . . . × Ps , (B.40)

where L is the total angular momentum and P1 , P2 , .., Ps are the intrinsic parities
of the particles involved.

B.3.6.1 Fermion Case

For fermions fields respecting the Dirac equation, imposing that the laws of physics
are invariant under the change of space coordinates : xi → x̃i = −xi , Eq. (B.18)
transforms as

(iγ 0 ∂t + iγ i ∂xi − m)|(−x) = 0 ⇒ γ 0 (iγ 0 ∂t + iγ i ∂xi − m)|(−x) = 0

⇒ (iγ 0 ∂t − iγ i ∂xi − m)γ 0 |(−x) = 0 (B.41)

which means that under parity |(x) → |(−x) = eiφ γ 0 | = P|, eiφ being
present as it is a general unitarity transformation:

|(x) →P |(−x) = P| (x) = eiφ γ 0 |(x); P = eiφ γ 0 . (B.42)

412 B Particle Physics


L
Note that under the parity P = the spinor  =
γ 0, transforms into
R

R
= . This is easily understandable as the helicity (projection of the spin on
L
the velocity vector) changes orientation in space under the parity operation: a left-
handed particle becomes a right-handed particle. This was in fact the way taken by
Wu for her experiment in 1956 [3] to show the violation of parity. Indeed, finding
a process which produces a particle with a preferred helicity also proves that P-
symmetry is violated. Wu did it measuring the products of 60 Co decay.
From the definition of the parity operator, we can compute the parity of different
bilinear forms which are useful to compute some processes. We can then question
their existence :

Scalar : ¯
 ¯
→  † (γ 0 )† γ 0 γ 0  = +, P
¯ = +1 (B.43)
Pseudoscalar : ¯ 5  →  † (γ 0 )† γ 0 γ 5 γ 0  = −γ
γ ¯ 5 , Pγ
¯ 5  = −1

Vector : γ i † ¯ ,
¯  →  (γ ) γ γ γ  = −γ
0 † 0 i 0 i
Pγ
¯ i  = −1; Pγ
¯ 0  = +1

Axial : γ ¯ i γ 5 ,
¯ i γ 5  →  † (γ 0 )† γ 0 γ i γ 5 γ 0  = +γ Pγ
¯ i γ 5  = +1; Pγ
¯ 0 γ 5  = −1

Tensor : σ † 0 † 0 ¯ ij ,
¯ ij  →  (γ ) γ σij γ  = +σ 0
Pσ
¯ ij  = +1; Pσ
¯ 0j  = −1

with σμν = [γμ , γν ] = γμ γν − γν γμ . Equation (B.43) summarize all the nature

of couplings one can find in any specific microscopic models. Moreover, in view of
the form of the matrix γ 0 , we can notice that P|1,2  = +|1,2 and P|3,4 =
−|3,4. In other words, the antiparticles states have opposite parity than the particle
states.

B.3.6.2 Boson Case

In the case of a bosonic particle, one names scalar the particle φ with an even
parity, [φ(−xi , t) = φ(xi , t)] and pseudo-scalar the particle φ̃ with an odd parity
[φ̃(−xi , t) = −φ̃(xi , t)]. In the Standard Model, the two kinds of bosons exist: the
Higgs h, which is a scalar particle, and the neutral pion π 0 , a pseudo-scalar one. But
how we can measure the parity of a particle?

B.3.7 The Charge Conjugate Operator

The charge conjugation operator C transforms a particle ψ into its antiparticle ψ c .

We can find this expression as we did for the parity operator, directly from the Dirac
equation. Indeed, in the presence of an electromagnetic field, one should add the
interaction of the photon field Aμ with the fermion of charge e. The Dirac equation
for the field ψ then becomes

iγ μ (∂μ − ieAμ )ψ − mψ = 0 (B.44)

B.3 Diracology 413

which imply after taking the complex conjugate of the equation and multiplying by
γ2

−iγ 2 (γ μ )∗ (∂μ + ieAμ )ψ ∗ − γ 2 mψ ∗ = 0 ⇒ iγ μ (∂μ + ieAμ )(γ 2 ψ ∗ ) − m(γ 2 ψ ∗ ) = 0,

(B.45)

where we used (γ 0,1,3 )∗ = γ 0,1,3 and (γ 2 )∗ = −γ 2 . Defining ψ c as ψ c =

iγ 2 ψ ∗ = C ψ̄ T = iγ 2 γ 0 ψ̄ T (the factor i is added to integrate a phase and to
keep C as a real operator), we notice that the Dirac equation for ψ c is exactly the
same than for ψ, but with an opposite charge: ψ c is the antiparticle of ψ, and C
the charge conjugate operator. We can then find the expression of C in the space of
4-dimensional spinors ψ = (χ, η)T . The operation can be written as

ψ c = C ψ̄ T = iγ 2 γ 0 ψ̄ T = Cγ0T ψ ∗ , (B.46)

where
⎛ ⎞
0 1 0 0
 
⎜ −1 0 0 0 ⎟ iσ2 η∗
C = iγ γ = ⎜
2 0 ⎟ ⇒ ψ= χ →C ψ c = C ψ̄ T = ;
⎝ 0 0 0 −1 ⎠ η −iσ2 χ ∗
0 0 1 0
(B.47)

we can then prove easily the following useful relations:

C −1 = −C ∗ ; C ! = −C; C −1 γμ C = −γμ! ; C † = C −1 . (B.48)

A Majorana particle is a particle which charge conjugate is equal to itself, ψ c = ψ.

We can then write
  
χ iσ2 η∗ χ
ψ= ⇒ ψ = iγ γ ψ̄
c 2 0 T
, ψ =ψ ⇒ ψ =
c
η −iσ2 χ ∗ −iσ2 χ ∗
(B.49)

which is the generic form for a Majorana particle. Have a look at Sect. B.3.8 for
a more detailed definition of a Majorana fermion in the context of Dirac equation
and Sect. B.2.2 for the Feynman rules with Majorana particles. Some operators are
suppressed for Majorana particles: For instance, if ψ c = ψ we can write

ψ¯c γ μ ψ c = ψ ! C † γ μ γ 0 Cψ ∗ = ψ ! C −1 γ μ γ 0 Cψ ∗ = ψ ! (γ μ )! (γ 0 )! ψ ∗ (B.50)
1 μ 
= (ψ ! (γ μ )! (γ 0 )! ψ ∗ )! = −ψ̄γ μ ψ ⇒ ψ̄γ μ ψ = ψ̄γ ψ + ψ¯c γ μ ψ c = 0,
2
414 B Particle Physics

the negative sign is coming for the exchange of two fermions. As a consequence, we
can deduce that a Majorana dark matter does not couple to a Zμ (as the dark matter
has no T 3 charges, no coupling to γ 5 ) or Zμ boson without axial coupling as the
sum of both contributions will cancel. On the other hand, if a Majorana dark matter
has an axial coupling to Z or Z  the annihilation rate is not null. We can also easily
show with the same method that

ψ¯c ψ c = ψ̄ψ (B.51)

showing that a Majorana dark matter can have a coupling to the Higgs boson.
Another useful relation (used in the Sect. C.3) concerning the neutrino is (we let the
reader to prove it)

ψ̄L ψR = ψ̄Rc ψLc . (B.52)

B.3.8 The Majorana Case

B.3.8.1 Definition
The Feynman rules for a Majorana fermion differ from Dirac fermion, the same
manner Feynman rules for real scalar differ from imaginary scalars. All can be,
at the end, summarized by factors 2 or 1/2, but it is important (and interesting) to
understand how to treat the rules accurately. First of all, one should remember the
basic facts of a Dirac equation. If one decomposes a (4-dimension)  Dirac spinor
χ
ψ into its (2-dimension) Weyl spinors χ and η, we can write ψ = which
η
generates the Dirac equation (B.18) if one writes the Lagrangian

LDirac = iχ † σ̄ μ ∂μ χ + iη† σ μ ∂ν η − Mη† χ − M ∗ χ † η = ψ̄(iγ μ ∂μ − M)ψ,

(B.53)

with σ̄ 0 = σ 0 and σ̄ i = −σ i . It is important to keep in mind that the 4-component

Dirac spinor  has been built from Eq. (B.53) which is the Lagrangian leading to
the set of 2 Dirac equations for the 2-component spinors χ and η. We will try now,
to build the same Lagrangian for a Majorana spinor. Before going into the details,
one can have a guess of the result. Indeed, ψ is a 4-component spinors but have only
2 degrees of freedom in a Majorana case. A Majorana spinor to a Dirac one has the
same correspondence than a real field to a scalar one. One should understand then
that there should be a factor 1/2 entering into the game, LMaj orana = 12 LDirac .
This is what we will demonstrate more rigorously.
B.3 Diracology 415

B.3.8.2 Dirac-Like Majorana Equation

As we saw in the previous section, a (4-component) Majorana spinor λ isdefined as
χ
λC = C λ̄T = λ. We can write, by analogy with the Dirac spinor λ = , and
η
using Eq. (B.46)
 
iσ2 η∗  ab ηb∗
λ = C λ̄ = iγ γ λ̄ =
C T 2 0 T
= (B.54)
−iσ2 χ ∗ − ab χb∗

where

0 1
 ab = (B.55)
−1 0

is the Kronecker symbol. The Kronecker symbol is in fact fundamental when

dealing with Grassmann (anti-commuting) variables. One can consider it as a metric
in this case, allowing to write a scalar product for anti-commuting variables. Indeed,
if one defines η.χ = ηa  ab χb , it is straightforward to check that η.χ = χ.η using
the property3 of  ab and χa ηb = −ηb χa .
The condition λC = λ leads to
  
χ
λ= ⇒ λ̄ = λ† 0
γ = −χ T T
 ; χ †
(B.56)
− χ ∗

which implies

λ̄λ = χ T χ − χ † χ ∗ . (B.57)

B.3.8.3 In the “Left–Right” Representation

With the Weyl representation in which we made all the calculation (defining our γ  s
matrices) it is easy to rewrite the mass term using the “left” and “right” eigenvectors
of the projection operators PL and PR (Eq. B.25). If one defines
 
χ 0
λL = PL λ = and λR = (λL )C = , λ = λL + λR , (B.58)
0 −χ ∗

we can then write

λ̄λ = λ̄L λR + λ̄R λL = χ T χ − χ † χ ∗ (B.59)

3 Inthe case of the complex conjugate, the conventional notation is η∗ .χ ∗ = (η∗ )a ab (χ ∗ )b with
ab = − ab . This convention makes the scalar product easier to manipulate as one can see from
Eq. (B.60) and Eq. (B.61).
416 B Particle Physics

(notice the change of sign comparing Eq. (B.57) and Eq. (B.59)). This reminds of
course the definition of a Majorana mass term MM (λL )C λL which is the mass
term we can write from a 4-component Majorana spinor. For more details on the
Majorana mass, and its application to see-saw mechanism, see Sect. C.4.
However, the Dirac equation4 i σ̄μ ∂μ χ = −M ∗ χ ∗ is derived from the
Lagrangian

M M∗ ∗ ∗
L = iχ † σ̄μ ∂μ χ − χ.χ + χ .χ , (B.60)
2 2
where we used the definition of the scalar product defined above. Notice the factor
1/2 in front of the mass term. It corresponds to the same factor for the mass term of
real scalar fields (with respect to complex scalar fields), and appears naturally when
one derives the Dirac equation from the equation of motion ∂∂χ L = ∂ ∂ L . In
μ ∂(∂μ χ)
4-component notation, the Lagrangian (B.60) is then given by (for real mass term)

1 M
LMaj orana = i λ̄γ μ ∂μ λ − λ̄λ (B.61)
2 2
which corresponds indeed to our first guess we made at the beginning of the
section LMaj orana = 12 LDirac . If moreover, one supposes that the mass term is
coming from Yukawa-like couplings with a field φ, as in the Standard Model, one
should write for generic couplings  (which contains γ  s and combinations of γ  s
matrices),

L = LDirac + LMaj orana (B.62)

1 1
= ψ̄(iγ μ ∂μ − Mψ )ψ + λ̄(iγ μ ∂μ − Mλ )λ + yψ φ ψ̄ψ + yλ φ λ̄λ.
2 2

The factor 12 in front of the Yukawa coupling yλ is fundamental when one needs
to compute processes through the Feynman rules. Indeed, if φ is a scalar field
developing a vev φ = √1 (v + a + ib), the mass is then defined with the same
2
y yλ
convention, Mψ = √ψ v, Mλ = √ v. That point is very important once we made any
2 2
computation with Majorana spinors: to check that the convention of the couplings
fits with the mass term in the Lagrangian.

B.3.8.4 Furry’s Theorem

In quantum electrodynamics, the Furry theorem tells that the contribution of a
Feynman diagram, consisting of a closed polygon of fermion lines connected to

4 Remember that the fundamental Dirac equation should be written in 2-component notation. The
4-component one, especially in the Weyl (“Left–Right”) representation is mainly practical in the
Standard Model because the helicities have different quantum number under the SU(2) gauge
group.
B.3 Diracology 417

Fig. B.2 Illustration of the Furry’s theorem, giving a null amplitude for an odd number of external
electromagnetic legs

an odd number of photon lines, vanishes. The demonstration is quite easy if one
considers a closed fermion loop of diagram with 3 external legs illustrated in
Fig. B.2. Remembering that the electromagnetic current changes sign under charge
conjugation i.e.

Cj μ C † = −j μ , (B.63)

where we make use of Eq. (B.48). Remembering that the vacuum should be charge
invariant (C|0 = 0), we obtain

0|j μ |0 = 0|C † Cj μ C † C|0 = − 0|j μ |0 ⇒ 0|j μ |0 = 0. (B.64)

The same argument applies to any vacuum expectation value of an odd number
of electromagnetic currents, and by consequence, the vacuum expectation of all
diagrams with odd number of legs vanishes. This property is known as Furry’s
theorem.

B.3.9 Traces

ημν ημν = 4
T r[γ μ ] = T r[γ 5 ] = 0
T r[γ 5 γ μ ] = T r[γ 5 γ μ γ ν ] = T r[γ 5 γ μ γ ν γ ρ ] = 0
T r[1] = 4
T r[γ μ γ ν ] = 4ημν
418 B Particle Physics

T r[p / 2 ] = 4p1 .p2

/ 1 .p
/ 1 − m1 )(p
T r[(p / 2 − m2 )] = 4p1 .p2 + 4m1 m2
T r[γ μ γ ν γ ρ ] = 0
 
T r[γ μ γ ν γ ρ γ σ ] = 4 ημν ηρσ − ημρ ηνσ + ημσ ηνρ
T r[γ μ γ ν γ ρ γ σ γ 5 ] = −4i μνρσ
 μ ν ν μ

/ 2 γ ] = 4 p1 p2 − η p1 .p2 + p1 p2
μ ν μν
T r[p
/ 1γ p
/ 1p
T r[p / 2p / 4 ] = 4 [(p1 .p2 )(p3 .p4 ) − (p2 .p4 )(p1 .p3 ) + (p2 .p3 )(p1 .p4 )]
/ 3p
/ 2 γ ]T r[p / 4 γν ] = 32 [(p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 )]
μ ν
T r[p
/ 1γ p / 3 γμ p
/ 2 γ ]T r[γμ γν ] = −32p1 .p2
μ ν
T r[p
/ 1γ p
/ 1 − Mχ )γ (p/ 2 + Mχ )γ ]T r[(p / 3 − mf )γμ (p/ 4 + mf )γν ]
μ ν
T r[(p
 
= 32 (p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 ) + 2Mχ2 m2f + (p1 .p2 )m2f + (p3 .p4 )Mχ2

/ 1 + Mχ )γ (p/ 2 + Mχ )γ ]T r[(p / 3 + mf )γμ (p/ 4 + mf )γν ]

μ ν
T r[(p
 
= 32 (p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 ) + 2Mχ2 m2f − (p1 .p2 )m2f − (p3 .p4 )Mχ2

Tr[γ α γ β γ μ γ ν γ ρ γ σ ]
= 4ηαβ (ημν ηρσ − ημρ ηνσ + ημσ ηνρ ) − 4ηαμ (ηβν ηρσ − ηβρ ηνσ + ηβσ ηνρ )
+4ηαν (ηβμ ηρσ − ηβρ ημσ + ηβσ ημρ ) − 4ηαρ (ηβμ ηνσ − ηβν ημσ + ησβ ημν )
+4ηασ (ηβμ ηνρ − ηβν ημρ + ηρβ ημν ) (B.65)

Throughout the book, we will use the notation Pμ for a quadrivector and pi for a
3-dimensional vector.

B.3.10 Mandelstam Variables

In a process 1 + 2 → 3 + 4, one usually defines the Mandelstam variables by

s = (p1 + p2 )2 = (p3 + p4 )2
t = (p3 − p1 )2 = (p4 − p2 )2
u = (p3 − p2 )2 = (p4 − p1 )2 . (B.66)

Developing the expression above, we can express all the scalar products as function
of these Lorentz invariant variables and the masses of the scattering particles. Once
we obtain an expression for an amplitude, or amplitude square as function of s, t
B.3 Diracology 419

and u, we can transpose it in any reference frame. We have

s − m21 − m22 s − m23 − m24

p1 .p2 = ; p3 .p4 =
2 2
t − m21 − m23 t − m42 − m24
p1 .p3 = ; p2 .p4 =
2 2
u − m21 − m24 u − m42 − m23
p1 .p4 = ; p2 .p3 =
2 2
and

s + t + u = m21 + m22 + m23 + m24 . (B.67)

Exercise Recover the expressions above.

B.3.11 The Generators Tia

The generators of the non-Abelian gauge groups SU (2) and SU (3) are given by

For SU (2), T2i = σ2i , σi Being the Pauli Matrices

  
0 1/2 0 −i/2 1/2 0
T2 =
1
T2 =
2
T23 =
1/2 0 i/2 0 0 −1/2

For SU (3), the Gell-Mann Matrices Are

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
01 0 0 −i 0 1 0 0 001
λ1 = ⎝ 1 0 0⎠ λ2 = ⎝ i 0 0 ⎠ λ3 = ⎝ 0 −1 0 ⎠ λ4 = ⎝ 0 0 0 ⎠
00 0 0 0 0 0 0 0 100

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 0 −i 000 00 0 10 0
1 ⎝
λ5 = ⎝ 0 0 0 ⎠ λ6 = ⎝ 0 0 1 ⎠ λ7 = ⎝ 0 0 −i ⎠ λ8 = √ 0 1 0 ⎠.
3
i 0 0 010 0i 0 0 0 −2
420 B Particle Physics

B.4 Lorentz Invariant Scattering Cross Section

and Phase Space

B.4.1 The FERMI’s Golden Rule

B.4.1.1 The Non-relativistic Case

The golden rule, which in fact was first proposed by P. Dirac is in fact a natural
consequence of the Schrodinger equation. Supposing a system of n eigenstates of a
Hamiltonian H0

H0 |n = En |n, (B.68)

we can expand in the plane wave solutions |n the solution of a perturbed system
H = H0 + δH , at a time t

|φ(t) = an (t)e−iEn t |n. (B.69)

n

Projecting on f | the solution of the equation

∂
H |φ(t) = i |φ(t)
∂t
daf (t)
⇒ i = f |δH |nan (t)ei(Ef −En )t . (B.70)
dt n

Supposing an initial state |φi  = e−iEi t , we obtain

 
Ef −Ei
(E −E ) sin 2 t
i f2 i t
iak (t) = 2 f |δH |ie
Ef − Ei

E −E
sin2 f 2 i t
|ak (t)|2 = 4| f |δH |i|2 . (B.71)
|Ef − Ei |2

If one looks an interval of final energies, with a density of state ρ(Ef ) per unit
energy interval, the probability of transition Pf i should be written as
 
 Ef −Ei
sin2 2 t
Pf i = 4 dEf ρ(Ef )| f |δH |i|2 . (B.72)
|Ef − Ei |2

If one supposes that ρ(E) varies slowly with E (by slow we mean that the
observation time t is relatively long) and that f |δH |i is invariant across the final
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 421

states, Pf i becomes
 
 Ef −Ei
∞ sin2 2 t
Pf i = 4ρ(Ef )| f |δH |i|2 dEf = 2πρ(Ef )| f |δH |i|2t,
−∞ |Ef − Ei |2
(B.73)

where we used
 ∞ sin2 αx
dx = απ. (B.74)
−∞ x2

dPf i
The golden rule can then be written, for the transition rate i → f , f i = dt

f i = 2πρ(Ef )|Mf i |2 (B.75)

with Mf i = f |δH |i.

B.4.1.2 Normalization
In classical quantum mechanics, the wave function φ is normalized such as
 
|φ|2 d 3 x = |φ|2 dV = 1. (B.76)

This means that the particle has a probability of presence of 1 in the entire Universe.
However, in field theory, there exists the possibility to create or annihilate particles
from the void. The normalization is then a free parameter and a question of taste,
as long as one stays coherent. Moreover, it is clear that the expression (B.76) is not
Lorentz invariant due to the presence of the volume element of dV = d 3 x. In order
to restore the invariance, it is natural to normalise with a parameter with a dilution
√
factor γ . The usual convention is to 2E particle per unit of volume, E being the
energy of the particle:

1
|ψ|2 dV = 2E, φ = √ ψ.
2E

Once developed in the momentum phase space, as function of its Fourier transfor-
mation,
  
d 3p
ψ(x, t) =  a(p)eiEt −ip.x + a† (p)e−iEt +ip.x , (B.77)
(2π)3 2E
422 B Particle Physics

a and a† being expansion coefficients called destruction and creation operator,

respectively, which commutation relation

[a(p), a(p )] = [a† (p), a† (p )] = 0 (B.78)

and

[a† (p), a(p )] = [a(p), a† (p )] = δ (3) (p − p ) (B.79)

are deduced from the quantum relations

[∂t ψ(x, t), ψ(x , t)] = −iδ (3)(x − x ) (B.80)

and

[∂t ψ(x, t), ∂t ψ(x , t)] = [ψ(x, t), ψ(x , t)] = 0. (B.81)

The √ 1 is the classical coefficient to normalize the wave function. Indeed, the
(2π)3
 ipx
plane wave function φ(x) = √1 e dp is normalized such as |φ(x)|2 = 1.
2π
We let the reader check it, knowing that the delta function δ(x) can be written as
δ(x) = 2π1
eipx dp, see Eq. (A.129).

B.4.1.3 The S-Operator

Let us denote |i and |f  as the initial and final state, respectively, in a Fock space.
We can define the transition between |i and |f  by a S matrix which represents
the dynamics of the process and whose amplitude squared is the probability of
transition:

Sf i = f |S|i.

Hence, the probability for the process |i → |f  is

P (|i → |f ) = |Sf i |2 . (B.82)

In general, we can write

Sf i = δf i + i(2π)4δ (4) (pf − pi ) . Tf i

or

S = I + iT, (B.83)

the first part representing a non-interacting particle, while the second part is the
dynamical part with the condition of conservation of energy–momentum. Moreover,
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 423

put this way, it is a reminiscent of an exponential development, S = ei T . What is

physically of interest for us is the transition probability per unit time

|Sf i |2
wf i = .
T
From Eq. (B.82), we see that we must address the issue of defining the value of
a squared Dirac δ-function. To do this we use a rather pragmatic approach due to
Fermi himself:

i(pf0 −pi0 )t
[2πδ(pf0 − pi0 )]2 = dte 2πδ(pf0 − pi0 ) = T 2πδ(pf0 − pi0 )

[(2π)3 δ (3) (pf − pi )]2 = d 3 xei(pf −pi ).x (2π)3 δ (3) (pf − pi ) = V (2π)3 δ (3) (pf − pi )

|Sf i |2
⇒ wf i = = V (2π)4 δ (4) (pf − pi ).|Tf i |2 . (B.84)
T

To talk about the transition rate, we look at Fock-space with a fixed number of
particles. Experimentally, the angle and energy–momentum is only accessible up
to a given accuracy. We therefore use differential cross section in angle d and
energy–momentum dp near , p, respectively.

B.4.1.4 Computing the Rate

In a cubic box of volume V = L3 with infinitely high potential wells, the authorized
momentum-values are discretely distributed.
 3
2π L L
p= n ⇒ dn = dp ⇒ d 3 n = d 3p (B.85)
L 2π 2π

and hence
nf
! V
dwf i = V (2π)4 δ (4) (pf − pi ) |Tf i |2 d 3 pf (B.86)
(2π)3
f =1

where nf stands for the number of particles in the final state. In order to get rid of
the normalization factors, we define a new matrix element Mf i by,
n ⎛ ⎞
nf
!i
1 ! 1
Tf i = √ ⎝  ⎠ Mf i . (B.87)
i=1
2Ei V f =1
2Ef V

At first sight, the apparition of the energies of both the initial and final states might
be surprising. It is however needed in order to compensate the non-invariance of
the volume, so that EV is a Lorentz invariant quantity as we saw in the previous
424 B Particle Physics

section. This corresponds to the 2E normalization. Substituting the definition (B.87)

in Eq. (B.86), we get the fundamentally important expression for the rate R = dwf i

nf
V 1−ni !ni
1 ! d 3 pf
R = dwf i = (2π) 4 (4)
δ (pf − pi )|Mfi | 2
.
(2π)3nf i=1
2Ei
f =1
2Ef
(B.88)

B.4.1.5 Application
We can apply the expression (B.88) to decay rate and scattering. For the decay of a
particle a into nf particle

a → 1 + 2 + . . . + nf .

We have for the total decay width,


1 1 d 3 p1 d 3 pnf
a = .. (2π)4 δ (4)(pf − pi )|Mf i |2 . (B.89)
2Ea (2π)3nf 2E1 2Enf

We remark that since Ea is not a Lorentz invariant quantity, a also depends on

the reference frame.
For the scattering cross section, the case of two particles interacting via the
reaction

a + b → 1 + 2 + . . . + nf (B.90)

thus getting the scattering cross section σ (a + b → 1 + 2 + . . . + nf ) defined by

# of transition a + b → 1 + 2 + . . . + nf per unit time wf i

σ = = .
# of incoming particles per unit surface and time incoming flux
(B.91)

The denominator can also be stated as

vab
incoming flux = (number density) . (relative velocity) =
V
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 425

we then find

⎛ ⎞
 nf
! nf
1 1 d 3 pf
σi→nf = ⎝ ⎠ (2π)4 δ (4) ( pf − pa − pb )|Mf i |2
4F (2π)3nf 2Ef
f =1 f =1
(B.92)

in which we see once more the Lorentz invariant M/

oller flux factor

F = Ea Eb vab = (pa .pb )2 − m2a m2b
1
= (s − (ma + mb )2 )(s − (ma − mb )2 ). (B.93)
2
We see that the total cross section is manifestly a Lorentz invariant quantity, since it
only depends on Lorentz invariants.

B.4.2 Special Case

In the early Universe, the initial momentum of the scattering particles 1 and 2 is not
uniquely determined as they live in a thermal plasma, with a statistical distribution
of their momentum and energies f1 and f2 . Moreover, it is not possible to define the
relative velocity between the particles as they are all relativistic. One then needs to
use the fundamental expression for the rate R = n1 n2 σ v :

R= f1 f2 d1 d2 (2π)4 δ (4) (P1 + P2 − P3 − P4 )d3 d4 |M1,2→3,4 |2

d 3 p1 d 3 p2 d2
= f1 f2 3 3
(2π)4 |M1,2→3,4|2 (B.94)
(2π) (2π) 2E1 2E2

d pi 3
with di = (2π) 3 2E and where d2 is the two body phase space, computed in the
i
next section and d the solid angle between particle 1 and 3 in the center of mass
frame of (3,4). In the massless case (m3 = m4 = 0), one obtains d2 = 512π
d
6 , see
Eq. (B.101), which gives

d 3 p1 d 3 p2 |M1,2→3,4|2
R= f1 f2 d . (B.95)
(2π)3 (2π)3 128E1E2 π 2

I let the reader go to the Sect. 2.3.4.3 to have a more detailed study of this specific
case.
426 B Particle Physics

B.4.3 Computing the Phase Space

d 3p
We can show that d = (2π)3 2E
is a Lorentz invariant quantity. Indeed, if one
considers a Lorentz boost of parameter5 β in the z direction,

px = px , py = py , pz = γ (pz − βE), E  = γ (E − βpz ). (B.96)

d 3p dp  dpz dpz
One then needs to check that d = (2π)3 2E
= (2π)3 2E 
, or more simply E = E
For that, one can write

dpz γ dpz − βγ dE dpz (1 − β pEz ) dpz

= = = , (B.97)
E γ (E − βpz ) pz
E(1 − β E ) E
 pz
where we have used the fact that E = p 2 + m2 ⇒ dE
dpz = E. That proves the
Lorentz invariant character of d.

B.4.3.1 2-Body Phase Space

One can then√compute the two-body phase space d2 in a reaction with center of
mass energy s and quadri-momentum Pcom

d 3 p1 d 3 p2
d2 = d1 d2 δ (4) (Pcom − P1 − P2 ) = δ (4) (Pcom − P1 − P2 )
(2π)3 2E1 (2π)3 2E2
p22 dp2 d √
= δ( s − E1 − E2 ) (B.98)
4(2π)6 E1 E2
 
with E1 = p22 + m21 and E2 = p22 + m22 in the center of mass frame. We can
indeed compute the phase space in the center of mass frame as we showed in the
previous section that it is Lorentz invariant. Using the relation6 δ(g[x]) = gδ(x)
 (x) , one
can write
√ √   δ(p2 − p2∗ )
δ( s − E1 − E2 ) = δ( s − p22 + m21 − p22 + m22 ) = p2
 +  p2 2
p2 +m1
2 2 p2 +m22
(B.99)


5 Where, as usual, γ = 1/ 1 − β 2 .
6 From now on when using this formula we will note by simplification δ(x) as δ(x − x ∗ ), where x ∗

is a zero of g(x).
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 427

which gives

p2∗ d [s − (m1 − m2 )2 ]1/2 [s − (m1 + m2 )2 ]1/2

d2 = √ = d , (B.100)
4(2π)6 s 512π 6s

where we used

[s − (m1 − m2 )2 ]1/2[s − (m1 + m2 )2 ]1/2

p2∗ = √ .
2 s

In the massless states, one obtains

d
d2 (mi = 0) = . (B.101)
512π 6

B.4.3.2 N-Body Phase Space

One can generalize the previous computation in the case of N-body particles system.
We can rewrite n-body phase space in the center of mass frame

√ !
n
d 3 pi  n
  n
√ 
dn ( s; p1 , .., pn ) = 3
× δ (3) pi δ Ei − s (B.102)
(2π) 2Ei
i=1 i=1 i=1

d 3 pn ! d 3 pi
n−1
 n−1   n−1 √ 
= 3 3
δ (3)
pi − (−pn ) δ Ei − s − (−En )
(2π) 2En (2π) 2Ei
i=1 i=1 i=1

d 3 pn
= dn−1 (; p1 , .., pn−1 )
(2π)3 2En
√
with  2 = ( s−En )2 −pn2 , where we used the Lorentz invariance of the phase space
to compute dn−1 in the center
 √ of mass frame of the system (p1 , .., pn−1 ) passing
√
from ( s − En ; −pn ) to ( ( s − En )2 − pn2 ; 0) after a boost of momentum +pn .
As an example, one can compute the three-body phase space, from the two-body
one we computed already in Eq. (B.100).

d 3 p3 p2 (3 )d
d3 = 3
×
(2π) 2E3 4(2π)6 3
√
with 32 = ( s − E3 )2 − p32 , which gives
√ √
d 3 p3 d [( s − E3 )2 − p32 − (m1 − m2 )2 ]1/2 [( s − E3 )2 − p32 − (m1 + m2 )2 ]1/2
d3 =  √
8 × (2π)9 E3 2 ( s − E )2 − p2 3 3
428 B Particle Physics

which gives in the massless case:


m =0 p3 dp3 d 13 d 12 √
d3 i = s − 2 sp3 . (B.103)
16(2π)9

Another trick to compute a 3-body phase space is shown in Sect. B.4.6.2.

B.4.3.3 Summary in the Massless Case

We can then summarize our result for 2-, 3-, and 4-body final states in massless
cases:

i =0
d 12
dm
2 =
8(2π)6
m =0 p3 dp3 d 13 d 12
d3 i = .
16(2π)9

B.4.3.4 Examples
We will compute in the next section, specific examples of decay or annihilation
rates. They are given by

|M|2
d = (2π)4 dn (P ; p1 , . . . , pn ) (B.104)
2M
with

 n
!
n
d 3 pi
dn (P ; p1 . . . , pn ) = δ 4 P − pi
(2π)3 2Ei
i=1 i=1

for the width of a particle of mass M and amplitude of decay M decaying into n
particles. In the case of annihilation between particles 1 and 2 into n final particles,
the cross section dσ can be written as

(2π)4 |M|2
dσ =  × dn (p1 + p2 ; p3 , . . . , pn+2 ), (B.105)
4 (P1 .P2 )2 − m21 m22

where we used Eq. (B.93) for the Lorentz invariant flux.

B.4 Lorentz Invariant Scattering Cross Section and Phase Space 429

B.4.4 Annihilation

B.4.4.1 General Formulae

As we just saw in the previous section, the annihilation cross section between two-
body states (1, 2) → (3, 4) can be written as

(2π)4 |M|2 d 3 p3 d 3 p4
dσ =  δ 4 (P1 + P2 − P3 − P4 ) .
4 (P1 .P2 )2 − m21 m22 (2π)3 2E3 (2π)3 2E4
(B.106)

Using s = (P1 + P2 )2 = m21 + m22 − 2P1 .P2 and m1 = m2 = Mχ , one has

  
(P1 .P2 )2 − m21 m22 = ( 2s − Mχ2 )2 − Mχ4 = ( 2s − 2Mχ2 ) 2s . After integration on
d 3 p3 δ 3 (p1 + p2 − p3 − p4 ), one can then rewrite Eq. (B.106)

|M|2 |p4 |2 dφd cos θ d|p4 |

dσ =    δ
64π 2 ( 2s − 2Mχ2 ) 2s |p4 |2 + m23 |p4 |2 + m24
  
√
× s − |p4 |2 + m23 − |p4 |2 + m24 , (B.107)

we place ourself in the center of mass frame (p3 = −p4 , |p3 | = |p4 |) and
where
δ(x)
E4 = |p4 |2 + m24 . Reminding that the function delta respects δ[f (x)] = |f  (x)| ,

  
√ δ(|p4 |)
δ s− |p4 |2 + m23 − |p4 |2 + m24 = , (B.108)
 |p4 | +  |p4 |
|p4 |2 +m23 |p4 |2 +m24

one can write Eq. (B.107).

|M|2 |p4 |dφd cos θ

dσ =   
64π 2 ( 2s − 2Mχ2 ) 2s |p4 |2 + m23 + |p4 |2 + m24

|M|2 |p4 |dφd cos θ

=  √ , (B.109)
64π 2 ( 2s − 2Mχ2 ) 2s s

  √
where we used |p4 |2 + m23 + |p4 |2 + m24 = E3 + E4 = E1 + E2 = s.
(s+m24 −m23 )2
Moreover, |p4 = E42 − m24 and E42 =
|2 4s implies |p4 |2 = 4s
1 2
(s + m44 +
m43 − 2sm24 − 2sm23 − 2m23 m24 ). Replacing this expression for |p4 | in Eq. (B.109),
430 B Particle Physics

we obtain, with d = dφd cos θ


dσ |M|2 s 2 − 2m23 s − 2m24 s + (m23 − m24 )2
=  (B.110)
d 128π 2s (P1 .P2 )2 − m21 m22

|M|2 [s − (m3 − m4 )2 ][s − (m3 + m4 )2 ]
=
64π 2 s [s − (m1 − m2 )2 ][s − (m1 + m2 )2 ]

(m23 −m24 )2
|M|2 s − 2m23 − 2m24 + s
=  .
64π 2 s s − 4Mχ2

But it is important to include the symmetric/statistic factors : 1/4 (1/(2S+1)*

1/(2S+1)) for a fermionic dark matter and 1/Sym for a symmetry factor if the
particles 3 and 4 are identical (Z 0 Z 0 for instance but not W + W − or f f ). One
then obtains

(m2 −m2 )2
dσ 1 |M| 2 s − 2m23 − 2m24 + 3 s 4
= 
d (2S1 + 1)(2S2 + 1)Sym 64π 2 s s − 4Mχ2

1 |M|2 2m23 + 2m24 (m23 − m24 )2
 1− + , (B.111)
(2S1 + 1)(2S2 + 1)Sym 32π vs 2 4Mχ2 16Mχ4

where we have developed s  4Mχ2 + Mχ2 v 2 , v being the relative velocity between
the two colliding particles (also called Moller velocity). The simplified formulae for
massless final state and amplitude which does not depend on the diffusion angle
(scalar-like interactions) after integration on the solid angle is

1 |M|2 d cos θ
σv  . (B.112)
(2S1 + 1)(2S2 + 1)Sym 16πs

B.4.4.2 A Shorter Formulation

In the literature, especially when discussing dark matter processes, one sometimes
prefers to write another way Eq. (B.106) :
⎛ ⎞ ⎛ ⎞
1 ! d 3 pf 1
dσ = ⎝ ⎠ (2π) δ ⎝P1 + P2 −
4 (4)
Pf ⎠ |M̄|2 ,
2E1 2E2 |v2 − v1 | (2π)3 2Ef
f inal f inal
(B.113)
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 431

where |M̄| should be understood to mean the spin averaged squared amplitude. The
two-body phase space is
⎛ ⎞ ⎛ ⎞
! d 3 pf 
1 d CM |p3 |
dP S2 (p3 , p4 ) = ⎝ ⎠ δ (4) ⎝P1 + P2 − Pf ⎠ = .
(2π)3 2Ef 256π 6 ECM
f inal f inal
(B.114)

I want to add an important note here. One must be careful when using “v.” In
Eq. (B.111) v represents the relative velocity of particle 1 with respect to particle
2, also noted |v2 − v1 |, and not the absolute velocity or the velocity in the center
of mass (CoM) frame. It is usual in dark matter literature to name v the relative
velocity (in expressions of σ v for instance). However, when one computes cross
section in quantum field theory, we usually do it in the center of mass frame, and
vCoM = v/2. In other words, vdark mat t er = v = |v2 − v1 | = 2 vCoM . This
vdark mat t er = |v2 − v1 | is the one used in Eq. (3.176) for instance. In other words,
one can make all the computation in CoM frame, and then, when implementing
velocities dependance we can substitute vCoM by v/2, v being the relative velocity,
which is the velocity which interests us because it is the one “measured” by the
rotation of the dark matter halo.
Let us illustrate it concretely. When one has to compute s = (P1 + P2 )2 , supposing
m1 = m2 = m, one needs to develop

s = (P1 + P2 )2 = m21 + m22 + 2E1 E1 − 2p1 .p2 = 4m2 + 4|pCoM |2 . (B.115)

Remembering that pCoM = γ mvCoM , one deduces

4m2 4m2
s = 4m2 (1 + γ 2 vCoM
2
)= =  4m2 + m2 v 2 (B.116)
1 − vCoM
2 1 − v 2 /4

justifying the approximation used in Eq. (B.111).

B.4.4.3 A Note on the Symmetry Factor

We want to add a little remark concerning the symmetry factor 1/Sym in
Eq. (B.112). If there are nf identical particles in the final states, the symmetry

factor is Sym = f (nf !) and σ = Sym 1
dσ . We need the symmetry factor
because merely integrating over all the outgoing momenta in the phase space
treats the final state as being labeled by an ordered list of momentum. But if
some outgoing particles are identical, this is not correct; the momentum of the
identical particles should be specified by an unordered list, because for example
the state a1† a2† |0 is identical to the state a2†a1† |0. The symmetry factor provides the
appropriate correction.
432 B Particle Physics

B.4.4.4 The Specific Case of Majorana or Identical Initial Particle

The counting of states in annihilation processes for Majorana fermions is non-trivial,
and has led to a factor-of-two ambiguity in the literature which also propagated into
different codes present in the market. In short, the σ v used when computing a relic
abundance should not be confounded by the σ v used when one needs to deal with
indirect detection rates. In the Dirac case, or complex scalar dark matter, there are
no differences, whereas a factor 1/2 appears when dealing with Majorana/identical
particles, the halo annihilation being twice less than the Boltzmann one.
The clearest way to see the origin of the factor of 1/2 is probably to go back
to the Boltzmann equation. In essence, one can view σ v as the thermal average
(averaged over momentum and angles) of the cross section times velocity in the zero
momentum limit; in this average one integrates over all possible angles. For identical
particles in the initial state, one includes each possible initial state twice, therefore
one needs to compensate by dividing by a factor of 2; the prefactor in the zero
momentum limit becomes then σ v/2. In the Boltzmann equation describing the
time evolution of the dark matter candidate number density, the 1/2 does not appear
as it is compensated by the factor of 2 one has to include because 2 dark matter
particles are depleted per annihilation, but we need to include the factor of 1/2
explicitly in other cases where we need the annihilation rate (like for annihilation in
the halo). Indeed, when integrating on phase space, we count twice χ(p1 )χ(p2 )
annihilation as they will exchange impulsion in the process of integrating on θ
and φ.

B.4.4.5 Unitarity Limit

One interesting feature of the scattering theory is the possibility to extract con-
straints from theoretical bounds, like the unitarity limit. Indeed, asking for the S
matrix (B.83) to respect

S† S = I, (B.117)

we obtain the condition

2 ImT = T† T. (B.118)

The expression above is called the optical theorem. To appreciate its physical
implications, defining the scattering amplitude of a process p1 p2 → p3 p4 by
(B.87)

Mp1 p2 →p3 p4 = p3 p4 |T|p1 p2 ,

B.4 Lorentz Invariant Scattering Cross Section and Phase Space 433

the optical theorem becomes

 
−i Mp1 p2 →p3 p4 − M∗p3 p4 →p1 p2 (B.119)
 !
n
d 3 qk
= Mp1 p2 →qk M∗p3 p4 →qk (2π)4 δ (4) (p3 + p4 − qk ).
n
(2π)3 2Ek
k=1 k

If one looks at the forward scattering amplitude, we can write

 !
n
d 3 qk
2 Im Mp1 p2 = (2π)4 δ (4)(p1 + p2 − qk ) × |Mp1 p2 →qk |2 .
n
(2π)3 2Ek
k=1 k
(B.120)

The optical theorem relates the forward scattering amplitude (on the left) to the total
cross section (on the right). Indeed, we can recognize on the right-hand side of the
equation the sum on the all possible final states onshell (described by the momentum
qk ) for a given initial state (p1 , p2 ). Using Eq. (B.106) or (B.113), we then obtain

2 × Im Mp1 p2 2 × Im Mp1 p2 Im Mp1 p2

σt ot = = = √ (B.121)
4F 2E1 2E2 |v2 − v1 | 2p s

with p = pcom and F = 12 s(s − 4Mχ2 ). It is common to develop amplitudes M
around Legendre polynomials

M= al (2l + 1)Pl (cos θ ) (B.122)

l

with
√
s
al = 8π sin δl eiδl , (B.123)
p

where δl denotes the scattering-phase for the l-th partial wave. In a following
exercise, we will justify this form for the al . In the forward direction, we then obtain
for a non-relativistic particle of mass Mχ

4π(2l + 1) sin2 δl 16π(2l + 1) sin2 δl

σt ot = 2
= ,
p Mχ2 v 2

where v = |v2 − v1 |.
434 B Particle Physics

Exercise Show that asking for h2  0.1 while still respecting the unitarity
constraint, in the case of a WIMP candidate, implies Mχ  400 TeV.

Exercise Considering the development

M(s, θ ) = al (2l + 1)Pl (cos θ ) (B.124)

l

and using the property

 1 2
Pl (x)Pl  (x)dx = δll  , (B.125)
−1 2l + 1

with the help of Eq. (B.109) show that


d 3 p3 d 3 p4
(2π)2 δ 4 (p3 + p4 − p1 − p2 )|M(s, θ )|2 d
(2π)3 2E3 (2π)3 2E4
1 p
= √ |al |2 (2l + 1), (B.126)
4π s
l

where d = dφd cos θ . From the optical theorem (B.120), applied to the extreme
case where σt ot = σelast ic (in other words, q1 = p3 , q2 = p4 , and qk>2 = 0),
deduce that
p
Im al = √ |al |2 (B.127)
8π s

and then Eq. (B.123). We know that these al are the maximum values allowed by
the optical theorem, meaning that the elastic scattering cross section computed with
these values is the maximum cross section allowed by the unitarity constraint S† S =
1, or in other words,

4π(2l + 1) sin2 δl
σscat t ering < . (B.128)
|p|2

B.4.4.6 Scalar Fermi-Like Interaction

—2
¯ ¯
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 435

To illustrate this result, we can compute the annihilation cross section for a scalar–
scalar type interaction of the form :

GS
LS = √ χχf f. (B.129)
2

The amplitude may be written as

GS
M = √ v(pχ̄ )u(pχ )u(pf )v(pf¯ ) ⇒ |M|2
2
G2S
= / χ̄ − Mχ )(p
T r(p / χ + Mχ ) × T r(p
/ f¯ − mf )(p
/ f + mf )
2
G2S
= (4Pχ̄ .Pχ − 4Mχ2 )(4Pf¯ .Pf − 4m2f )
2
G2S
= (2s − 8Mχ2 )(2s − 8m2f ) (B.130)
2
which imply
 
G2S s − 4m2f (s − 4Mχ2 )(s − 4m2f )
σS = cf  (B.131)
32π s − 4Mχ2 s

with cf the color factor, equal to 3 for quarks and 1 for leptons. The annihilation
cross section becomes after developing around v
  3
G2S Mχ2 mf 2 2
σS v  cf 1− v2 . (B.132)
16π Mχ

B.4.4.7 Vector Fermi-Like Interaction

—2
¯ ¯

Another example is a vector–vector-like interaction:

GV
LV = √ χγ μ χf γμ f. (B.133)
2
436 B Particle Physics

One can compute directly |M|2 :

G2V
|M|2 = / 1 − Mχ )γ (p
T r[(p μ ν
/ 2 + Mχ )γ ]T r[(p / 3 − mf )γμ (p/ 4 + mf )γν ]
2
G2  
= V 32 (p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 ) + 2Mχ2 m2f + (p1 .p2 )m2f + (p3 .p4 )Mχ2
2
G2  s 2 
= V 32 (p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 ) + Mχ + m2f . (B.134)
2 2
√
Developing in the center of mass frame,  where p1 = p2 ⇒ E1 =E2 = s/2 =
E3 = E4 , one deduces |p1 | = |p2 | = s/4 − Mχ2 , |p3 | = |p4 | = s/4 − m2f and
p1 .p3 = p2 .p4 = E1 E3 − |p1 ||p3 | cos θ ⇒ (p1 .p3 )(p2 .p4 ) + (p1 .p4 )(p2 .p3 ) =
(p1 .p3 )2 + (p1 .p4 )2 = (E 2 − |p1 ||p3 | cos θ )2 + (E 2 + |p1 ||p3 | cos θ )2 = 2E 4 +
2|p1 |2 |p2 |2 cos2 θ , with E = E1 = E2 = E3 = E4 .
After integrating on the solid angle d cos θ , one obtains
   4 s2 4s  s 
d cos θ (p1 .p3 )2 + (p2 .p4 )2 = 4E 4 + |p1 |2 |p3 |2 = + − Mχ2 − m2f
3 4 3 4 4
(B.135)

which gives finally


s − 4m2f 
G2V 2π 1 s 4 s  s 
σV = 2
cf  32 + − Mχ2 − m2f + Mχ2 + m2f
2 4 64π s − 4Mχ2 4 3s 4 4

G2V s − 4m2f  1
= cf  s + (s − 4Mχ2 )(s − 4m2f ) + 4(Mχ2 + m2f ) . (B.136)
32π s − 4M 2 3s
χ

Usually, when you will compute a cross section, you will use the easiest units
of masses and energy which will be the GeV. However, fluxes are usually measured
in more common units for experimentalists i.e. cm−2 or seconds. The conversion
factor will be : 1 GeV−2 = 1.2 × 10−17cm3 s−1 .

B.4.4.8 Neutrino Interaction

−

—2
¯ +
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 437

GF G2
L = √ ν̄ν ēe ⇒ |M|2 = F 2s(2s − 8m2e ) (B.137)
2 2

with s = (pν + pν̄ )2 = 4Eν2 in the rest frame. Applying directly Eq. (B.111) to our
case (being careful to use a massless neutrino approximation), one obtains

G2F (s − 4m2e )3/2 G2 (E 2 − m2e )3/2

σνν = √ = F ν (B.138)
32π s 8π Eν

which finally gives, using the thermal average of relativistic species Eq. (A.112) :
 2
G2F 7π 4 9.93G2F
σνν  T2  × T 2. (B.139)
8π 180ζ(3) 8π

B.4.4.9 Annihilation into Monochromatic Photons

(1)

1
/2
(2)

For a complex scalar dark matter, this process is effectively generated through the
Lagrangian7

1
L= SS ∗ Fμν F μν (B.140)
2
with  a typical mass scale of particles running in the loop generating the effective
interaction. These particles being charged under the electromagnetic forces they
should be above  500 GeV as no such particles have been found at LHC yet. This
type of process is useful when searching for monochromatic lines in the sky, as it is
still considered as a smoking gun signal for dark matter detection (see Sect. 5.145
for more details). Following the Feynman rule, the coupling can then be written as

2i μ
CSSγ γ = (∂μ A1ν − ∂ν A1μ )(∂ μ Aν2 − ∂ ν A2 ) (B.141)
2
A1 and A2 being the photons fields 1 and 2, respectively, the factor 2 arising from
the symmetry in the exchange of identical particles γ(1) ↔ γ(2) . We then need to

7 Other dimension-6 point-like interactions of the form SS ∗ Fμν F̃ μν can also be generated, see
Sect. (B.3.7) for some examples.
438 B Particle Physics

develop the coupling, contracting it with the polarization of the photons to compute
the amplitude M:

2 μ μ 4
M=− 2
(pμ1 ν1 − pν1 μ1 )(p2 ν2 − p2ν 2 ) = − 2 (p1 .p2 )(1 .2 ) − (p2 .1 )(p1 .2 ) .
 
(B.142)

From M, one can deduce |M|2

16 μ μ μ μ 
|M|2 = (p1 .p2 )μ1 2 − (pμ2 1 )(pν1 2ν ) (p1 .p2 )μ1  2 − (pμ2  1 )(pν1 2ν )
4
μ,ν,μ ,ν 
(B.143)

after developing and using i i

μμ μ μ = −ημμ one obtains

32 8 |M|2 s 2m2s
|M|2 = (p1 .p2 )2 = 4 s 2 ⇒ σ v = =  ,
4  2 × 8πs 2π 4 π4
(B.144)

where we have developed around s  4m2s for low velocity8 and divided by the
symmetry factor 2 for the photons in the final state.

B.4.4.10 Annihilation in the Case of Real Scalar Dark Matter to Pairs of

Fermions

h
¯

The two parts of the Lagrangian one needs to compute the scalar annihilation of
Dark Matter SS → h → f¯f are (see B.236)9

MW λH S MW
LH SS = −λH S hSS → CH SS = −i
2g g
gmf gmf
and LHff =− hf¯f → CH ff = −i (B.145)
2MW 2MW

8 Do not forget that v is the relative velocity of the colliding particles.

9 Notice the factor 2 between LH SS and CH SS coming from the fact that S is real: S = S ∗ (it
corresponds to the 2 possible contractions).
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 439

which gives

λ2H S m2f (s/2 − 2m2f )

|M|2 = (B.146)
(s − MH
2 )2 +  2 M 2
H H

H being the width of the Higgs boson (including its own decay into SS, see next
section). When one implements this value of |M|2 into Eq. (B.111) one obtains after
simplification
( (
) 2 (M 2 − m2 )m2 )
|M|2 ) )
m 2 λ m2f
σ vSf f = *1 − f = HS S f f *
1 − . (B.147)
8πs MS2 16πMS2 (4MS2 − MH
2 )2 MS2

B.4.4.11 Annihilation in the Case of Vectorial Dark Matter to Pairs of

Fermions

h
¯

One can compute this annihilation cross section by the normal procedure or noticing
that a neutral vectorial dark matter of spin 1 corresponds to 3 degrees of freedom.
After averaging on the spin, one can then write σ vV = 3×33
σ vS = 13 σ vS . The
academical computation for Vμ (p1 )Vμ (p2 ) → f f gives

λhV i μ
L⊃− ημν H † H V μ (p1 )V ν (p2 ) ⇒ ChV V = − λhV MW ημν 1 2ν (B.148)
4 g
i gmf
and Chf f = − u(pf )v(pf ) implying
2M W

μ ν λhV mf i
M = −1λ 2ρ ημν u(pf ) v(pf ),
2 (s − m2 ) + ih mh h

α∗  β∗ η λhV mf v(p )
M∗ = 1λ
i
u(pf ).
2ρ αβ f 2
2 (s − m ) − ih mh f

μ
μ α∗ p1 p1α
Using λ 1λ 1λ = −ημα + , one obtains
m2V


2λ2hV m2f (m2V − m2f ) (p1 .p2 )2 p1 .p1 p2 .p2 6λ2hV m2f (m2V − m2f )
|M|2 = 4+ − − =
(s − m2h )2 + h2 m2h m4V m2V m2V (s − m2H )2 + h2 m2h
(B.149)
440 B Particle Physics

implying, after averaging on the spin 1 initial state particles

(
)
λ2hV (m2V − m2f )m2f ) 2
σ vVf f = *1 − mf (B.150)
48πm2V [(4m2V − m2h )2 + h2 m2h ] m2V

B.4.4.12 Exchange of a Vector: The Case of the Vectorial Coupling

c
¯ ¯

We compute in this section the process χ χ̄ → Z  → f f¯.

From the Lagrangian i χ̄γ μ Dμ χ + i f¯γ μ Dμ f , we extract the interaction
gD qχ χ̄ γ μ Zμ χ + qf f¯γ μ Zμ f which gives

 
2q q
gD χ f pμZ pνZ
M = −i ū(pf )γ v(pf¯ ) ημν −
μ
v̄(pχ̄ )γ ν u(pχ ) (B.151)
s − MZ2  MZ2 
4 q2 q2
gD χ f
⇒ |M|2 = 32 [(pf .pχ̄ )(pf¯ .pχ ) + (pf .pχ )(pf¯ .pχ̄ )
(s − MZ2  )2
+m2f (pχ pχ̄ ) + m2χ (pf .pf¯ ) + 2m2χ m2f ]
4 q2 q2
gD χ f
⇒ |M|2 = 64 [E 4 + (E 2 − m2f )(E 2 − m2χ ) cos2 θ + E 2 (m2f + m2χ )].
(s − MZ2  )2

After the average on the entering spins (1/4), we obtain


|M̄|2 s − 4m2f
dσ = Cf  2πd(cos θ )
64π 2 s s − 4m2
χ

4πCf αD 2 q2 q2
χ f
4E 2 − 4m2f
⇒σ =  (2E 2 + m2χ )(2E 2 + m2f )
3E 2 (4E 2 − MZ2  )2 4E 2 − 4m2
χ

4πCf αD 2 q2 q2
χ f
s − 4m2f
=  (s + 2m2χ )(s + 2m2f ) (B.152)
3s(s − MZ2  )2 s − 4m2χ
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 441

2
gD
Cf being the color index of the fermions in the final state and αD = 4π , which
gives
(
8πCf αD2 q2 q2 ) ) m2f
χ χ f *
σ vf f¯ = 1 − 2 (2m2χ + m2f ). (B.153)
(4m2χ − MZ2  )2 mχ

B.4.5 Spin-Independent Diffusion, Elastic Scattering

If one wants to compute the diffusion of a scalar particle of mass mS and momentum
pS on a proton of momentum pp and with momentum pp after the hit, following

Eq. (B.110) (with m3 → m1 and m4 → m2 ) and noticing (P1 .P2 )2 − m21 m22 =

2 s − 2m1 s − 2m2 s + (m1 − m2 ) , we can express
1 2 2 2 2 2 2

SI
dσS−p |M|2 |M|2 |M|2
= ⇒ σS−p
SI
=  (B.154)
d 2
64π s 16πs 16π(mp + mS )2

before averaging on the initial spins and before including the possible symmetry
factors. Using Eq. (B.145), one can write
 
i
M=− λH S mq p|q̄q|pū(pp )u(pp ) . (B.155)
2(PH − MH
2 2)
q
442 B Particle Physics

mq
If one defines10 fq = mp p|q̄q|p, f = q fq , and noticing that PH2 − MH
2 

−MH
2 , Eq. (B.155) becomes

1 m2p λ2H S f 2  m4p λ2H S f 2

|M|2 = 4
(Pp .Pp + m 2
p )  4
2 MH MH
m4p λ2H S f 2
⇒ σS−p
SI
= (B.156)
16π(mp + mS )2 MH
4

because the proton is mainly at rest11 Pp .Pp + m2p  2m2p .

B.4.6 Decaying Particles

We have seen in Eq. (B.104) and more generally in Sect. B.4.3 how to compute the
phase space for a n-body final state. Concentrating on decay processes, there exists
some “tricks” to facilitate such computations that we will develop in this chapter.

B.4.6.1 2-Body Decay

The expression we obtained combining Eqs. (B.101) and (B.104) gives, for a particle
φ of mass Mφ decaying into 2 quantum states P1 = (E1 , p1 ) and P2 = (E2 , p2 ):

(2π)4 d 3 p1 d 3 p2
dφ = δ (4) (P1 + P2 − Pφ )|Mφ→1,2 |2 (B.157)
2Mφ (2π) 2E1 (2π)3 2E2
3

with Pφ = (Mφ , 0, 0, 0). The main difficulty is to express the phase space as
function of the variables that can appear in |Mφ→1,2 |, E1 , E2 and cos θ12 . In
Sect. B.4.3 we took the absolute momentum |p1 | as the main variable, but we could
also have taken E1 . Indeed, defining

d 3 p1 d 3 p2
d2 = δ (4) (P1 + P2 − Pφ ) (B.158)
(2π)3 2E1 (2π)3 2E2

and noticing that E12 = p12 + m21 ⇒ E1 dE1 = p1 dp1 , we can write within a solid
angle d 12 = d cos θ12 dφ12

d 3 p1 p1 dE1 d 12
d2 = 6
δ(E1 + E2 − Mφ ) = δ(E1 + E2 − Mφ ),
(2π) 2E1 2E2 4(2π)6 E2

10 See Sect. 4.9.2 for the details to transform a scattering at the quark level to a scattering at a
nucleon level.
11 The first 1 is coming from the mean on the spin of one fermion, the initial proton.
2
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 443

where E2 is not a variable anymore as it is fixed by the condition δ (3) (p1 + p2 = 0)

which can be translated by

E2 = E12 − m21 + m22

E12 − m21 dE1 d 12 
⇒ d2 =  δ(E1 + E12 − m21 + m22 − Mφ ),
4(2π)6 E12 − m21 + m22

where all the phase space is expressed uniquely as function12 of E1 . Making use of
the identity δ[f (x)] = |f 1(x)| δ(x) one can write

E12 − m21
d2 = d 12 (B.159)
4(2π)6Mφ

which is equivalent to Eq. (B.98) where we took p1 as the fundamental parameter.

There exists a third way to compute the phase space, using a simple trick that
will be useful once one needs to compute decays into 3-body, 4-body, or even more.
The idea is to transform the last integration on the momentum space d 3 p2 into an
integration on the 4-momentum space d 4 P2 . Indeed, noticing that
 
d 3 p2
can be written d 3 p2 dE2 δ(E22 − p22 − m22 )θ (E2 ), (B.160)
2E2

where we used the δ[f (x)] formula and θ (x) is the heaviside function. Eq. (B.158)
then becomes

d 3 p1
d2 = d 4 P2 δ (4) (P1 + P2 − Mφ )θ (E2 )δ(E22 − p22 − m22 ) (B.161)
(2π)6 2E1
 
E12 − m21 dE1 d 12 E12 − m21 d 12
= δ((Mφ − E1 ) − (E1 − m1 ) − m2 ) =
2 2 2 2
2(2π)6 4(2π)6Mφ

with

Mφ2 + m21 − m22

E1 = . (B.162)
2Mφ

Ei
12 It is common to see in the literature expressions as function of xi = Mφ for convenience.
444 B Particle Physics

Combining Eq. (B.157) with Eq. (B.161), the 2-body decay can then be written
as


E12 − m21 Mφ2 + m21 − m22
dφ = d 12 |Mφ→1,2 |
2
with E1 = .
32π 2 Mφ2 2Mφ
(B.163)

We can also write

 [(Mφ2 − (m1 + m2 )2 )(Mφ2 − (m1 − m2 )2 )]1/2
E12 − m21 = . (B.164)
2Mφ

In the case where m1 = m2 , the formula is greatly simplified, after integration on

the phase space to

1 |Mφ→1,2 |2 
= Mφ2 − 4m21 (B.165)
2 16πMφ2
 
the factor 12 coming from the (anti)symmetrization of identical (fermions) bosons
in the exchange 1 ↔ 2. Notice also that |M| should be understood as a mean on the
polarization state, which means care should be taken, notably dividing by the factor
1
2S+1 , S being the spin of the decaying particle.

B.4.6.2 3-Body Decay

One does not gain a lot using the trick explained in the previous section if one has
to deal with 2-body decays only. This idea is more efficient concerning the 3-body
decay. Indeed, rewriting the 3-body phase space and using (B.160) on p3 :

d 3 p1 d 3 p2 d 3 p3
d3 = δ (4) (P − P1 − P2 − P3 )
(2π)3 2E1 (2π)3 2E2 (2π)3 2E3
d 3 p1 d 3 p2 d 3 p1 d 3 p2
= δ(E 2
3 − p3
2
− m 2
3 ) = δ(P32 − m23 )
(2π)9 2E1 2E2 (2π)9 2E1 2E2
d 3 p1 d 3 p2
= δ[(P − P1 − P2 )2 − m23 ]
(2π)9 2E1 2E2
E1 dE1 d 12 E2 dE2 d 23
= δ[Mφ2 − 2Mφ (E1 + E2 ) + 2E1 E2 (1 − cos θ12 )],
4(2π)9
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 445

where, in the last expression, we put m1 = m2 = m3 = 0 for simplicity.

Having no particular axis of references, the integration over d 23 gives 4π, whereas
d 12 = dφ12 d cos θ12 = 2πd cos θ12 . After integrating the δ function over cos θ12 ,
one obtains
dE1 dE2
d3 = (B.166)
4(2π)7

with

Mφ2 − 2Mφ (E1 + E2 ) + 2E1 E2

cos θ12 = . (B.167)
2E1 E2
M
Whereas the limit for E1 is 0 < E1 < 2φ the integration limits for E2 can be
obtained with the two extreme cases : 2 going in opposite direction of 1 and 3 (and
then taking the maximum energy) corresponding to cos θ12 = −1 (which gives from
M
Eq. (B.167) E2 = 2φ ) and 2 going in the same direction of 1, whereas 3 takes all
Mφ
the energy (cos θ = 1, corresponding to E2 = 2 − E1 );
we then can write

 Mφ  Mφ
1 2 2
φ = dE1 dE2 |Mφ→1,2,3 |2
64π 3 Mφ 0
Mφ
2 −E1

Mφ2 − 2Mφ (E1 + E2 ) + 2E1 E2

with cos θ12 = . (B.168)
2E1 E2

B.4.6.3 Application to Muon Decay

We can directly apply our procedure above to compute the muon lifetime in the
Fermi approximation of contact interactions. The Fermi Lagrangian can be written
as
4GF
L = √ (ν μ γ μ PL μ)(eγμ PL νe ) (B.169)
2

which gives for the amplitude

|M|2 = 32G2F (P .P1 )(P2 .P3 ) = 32G2F Mμ2 E1 (Mμ − 2E1 ), (B.170)
446 B Particle Physics

where particle “1” is νμ , “2” νe and “3” the electron. The integration gives

 Mμ
G2 Mμ 2 G2F Mμ5
μ = F 3 E12 (Mμ − 2E1)dE1 = . (B.171)
2π 0 192π 3

B.4.7 Higgs Lifetime

We can apply our generic discussions above to compute the Higgs branching ratios.

B.4.7.1 Higgs Lifetime from H → f (p1 )f¯(p2 )

Combining Eq. (B.165) and the SM Lagrangian (Sect. B.9) one obtains

gmf gmf g 2 m2f

CHf f¯ = −i ⇒ M = ū(p1 ) u(p2 ) ⇒ |M|2 = 2 / 1 + mf )(p
Tr(p / 2 − mf ).
2MW 2MW 4MW
(B.172)

Using Eq. (B.65) one can write

g 2 m2f
H →f f¯ = 2 M2
2
(MH − 4m2f )3/2 . (B.173)
32πMW H

B.4.7.2 Higgs Lifetime from H → Z(p1 )Z(p2 )

h
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 447

In this case, one uses the H ZZ coupling from the Lagrangian given in Eq. (B.234),
we obtain
g g
M∗μν = −i
μ ν α∗ β∗
Mμν = i MZ ημν 1λ 2ρ MZ ηαβ 1λ 2ρ (B.174)
cos θW cos θW

λ and ρ being the polarization of the Z(p1 ) and Z(p2 ), respectively. After the
μ α∗ μ
summation on the polarization, λ 1λ 1λ = −ημα + p1 p1α /MZ2 one obtains
 μ
 β
g2 p1 p1α p1ν p1
|M| = 2
M 2
ημν −η μα
+ ηαβ −η νβ
+
cos2 θW Z MZ2 MZ2

g 2 MZ2 MH
4 MZ2 MZ4
= 1 − 4 + 12 (B.175)
4MZ4 cos2 θW MH2 MH4

which gives
 
3
g 2 MH 4MZ2 MZ2 MZ4
H →ZZ = 2
1− 2
1−4 2
+ 12 4
. (B.176)
128πMW MH MH MH

B.4.7.3 Higgs Lifetime from H → W + W −

−

Similarly, on can compute

 
3
g 2 MH 2
4MW 2
MW 4
MW
H →W W = 2
1− 2
1−4 2
+ 12 4
. (B.177)
64πMW MH MH MH

The difference between the W W and ZZ prefactor comes from the symmetry (Z =
Z ∗ ⇒ 1/2 of W W final state.)
448 B Particle Physics

B.4.7.4 Higgs Lifetime from H → SS with Singlet Scalar Dark Matter

From the simplest singlet extension of the Standard Model, one can extract (see
appendix) the H SS interaction term after the SU (2) × U (1)Y breaking,13

MW2
MW λH S MW
LH SS = −λH S H SS → CH SS = −i ⇒ |M|2 = λ2H S .
2g g g2
(B.178)

From Eq. (B.165) one can compute

2 
λ2H S MW
H →SS = 2
2 − 4M 2 .
MH S (B.179)
32πg 2 MH

B.4.7.5 Higgs Lifetime from H → Xμ Xμ

There are different ways of computing this process. The easiest way would be to see
the vector as a 3-degree of freedom scalar. In this case, the sum on the 3 degrees of
freedom convoluated with the spin averages gives

X→Xμ Xμ = 3 × H →SS . (B.180)

B.4.7.6 General Scalar Width: T → χ χ̄

Generically, a scalar coupling to a fermion can be written as y T χ̄χ. T can be a

Higgs field or a moduli field decaying into 2 dark matter candidates for instance. In

13 Notice the factor 2 between LH SS and CH SS coming from the fact that there are 2 identical
particles in final states (it corresponds to the 2 possible contractions), see Eq. (B.239) for details.
B.4 Lorentz Invariant Scattering Cross Section and Phase Space 449

this case,

M = −iy ū(pχ )v(pχ̄ ), ⇒ |M|2 = 4y 2 (pχ .pχ̄ − m2χ )

|M|2  2 (MT2 − 4m2χ )3/2

⇒ T = MT − 4m 2 = y2
χ . (B.181)
16πMT2 8πMT2

T can also be the inflation field to compute the reheating temperature.

B.4.8 Majorana Case

To compute the decay of the scalar into a pair of Majorana particle, we let the reader
have a look at Sects. B.2.2 and B.3.8 to understand the origin of the factors 12
appearing in the Lagrangian and 2 in the Feynman rules, canceling each other at
the end. Writing the Lagrangian L = yχ T χ̄χ + 12 yλ T λ̄λ, one obtains

Mλ = −iyλ ū(pλ )v(pλ ) ⇒ |Mλ |2 = 4yλ2 (pλ .pλ̄ − m2λ )

 
1 |Mλ |2 (M 2 − 4m2λ )3/2
⇒ Tλ = 2
MT2 − 4m2λ = yλ2 T . (B.182)
2 16πMT 16πMT2
 
The last factor 12 is the symmetry factor (antisymmetrization because of identical
final state). We can also apply it to the decay of the scalar into a spin 3/2 Majorana
(like the gravitino for instance14).

B.4.9 Vector Lifetime

B.4.9.1 Generalities
We can write a general vectorial Lagrangian of a vector Vμ of mass M, momentum
P , and polarization λ coupling to fermions ψi1 and ψi2 of masses m1 , m2 and
momentum p1 and p2 ,

Vi − A i γ 5 1
LV = i ψ̄i2 γ μ (∂μ − igV Vμ )( )ψi (B.183)
2

14 Beaware that the Lagrangian of supergravity exhibit more complex structures that the simplest
one we propose to study here.
450 B Particle Physics

which gives the amplitude

i i
M= gV ū(p1 )γ μ (Vi − Ai γ 5 )v(p2 )μλ , M∗ = − gV v̄(p2 )γ ν (Vi − Ai γ 5 )u(p2 )νλ∗ .
2 2
(B.184)

μ ν∗
After the summation on the polarization, λ 1λ 1λ = −ημν + P μ P ν /MV2 one
obtains

gV2 Pμ Pν
|M|2 = Ci T r (p/1 + m1 )γ μ (Vi − Ai γ 5 )(p/2 − m2 )γ ν (Vi − Ai γ 5 ) −ημν +
4 M2
gV2  
= Ci (|Vi |2 + |Ai |2 ) 2M 2 − m21 − m22 + 6(|Vi |2 − |Ai |2 )m1 m2
2

Ci being internal degrees of freedom like the color charge of the fermions for
instance. After a mean on the spin, |M̄|2 = 2J1+1 |M|2 = 13 |M|2 and using
Eq. (B.163) we can write

gV2 M  
V = (|Vi |2 + |Ai |2 ) 2 − x12 − x22 + 6(|Vi |2 − |Ai |2 )x1 x2
96π
 mi
× [1 − (x1 + x2 )2 ][1 − (x2 − x1 )2 ], with xi = . (B.185)
M

B.4.9.2 W ± and Z Widths

In the Standard Model, the ±
√ Lagrangian involving W is written following
Eq. (B.183) with gV = g/ 2, Vi = Ai = 1 and neglecting the masses of the
final states (x1 = x2 = 0) we obtain

g 2 MW
(W + → e+ νe ) = . (B.186)
48π
In the case of the Z boson gV = g/ cos θW the couplings Ai and Vi depend on the
fermion ψi ; we have

g 2 MZ  
(Z → ψi ψ̄i ) = Ci 2
|ViZ |2 + |AZ
i |
2
(B.187)
48π cos θW

i = Ti and Vi = −2qi sin θW + Ti , qi being the electric charge of the

3 2 3
with AZ Z

particle i and Ti = ±1/2 its isospin.

3
B.5 s-Wave, p-Wave, Helicity Suppression and All That 451

B.5 s-Wave, p-Wave, Helicity Suppression and All That

B.5.1 Velocity Suppression

We used frequently in the analysis of the annihilation cross section the development

σ v = a + bv 2 . (B.188)

This comes from the fact that, on a general ground, a cross section can be
decomposed on the basis of the orbital momentum l : σ v = l (σ v)l ∝ | l (2l +
1)al Pl (cos θ )|2 , Pl being the Legendre polynomial function. The lth partial wave
contribution to the annihilation rate is suppressed as v 2l , where v is the relative
velocity between the annihilating particles. Indeed, the lth partial wave includes a
factor Pl (cos θ ). Since the amplitude is an analytic function of the Lorentz invariants
s, t (cos θ ) and u(cos θ ), the true argument of Pl is proportional to p̄i p̄f cos θ for an
initial particle of impulsion pi and final particle of impulsion pf , the latter scaling
as v̄i v̄f . Integration over singularities picks out the highest power of Pl , and we
get an amplitude scaling as (v̄i v̄f )l . If annihilating particles have a non-zero spin,
j
l is replaced by j and Pl (cos θ ) by the Wigner function dμλ(θ). In the expression
a + bv 2 , a is then called s-wave (l = 0) term and bv 2 p-wave (l = 1) term.
The virial velocity in our galactic halo is about ∼ 300km s−1 ∼ 10−3 c, so even
for l = 1 the suppression is considerable. Thus only the l = 0 partial wave gives
an unsuppressed annihilation rate in today’s Universe. The l > 1 states are too
suppressed to give any observable rate. When computing the relic abundance the
velocity is less suppressed. Indeed, the velocity of a dark matter particle χ of mass
mχ can be expressed as function of the temperature of the Plasma from Eq. (3.162),
v2  = 6 m T
. As we computed in Eq. (3.170), at the freeze out time TF O , Tm FO
≈ 20
implying

TF O
v 2 F O = 6 ≈ 0.3. (B.189)
mχ

B.5.2 Spin Selection

For any kind of process occurring at low velocity (like annihilation of dark
matter in the halo for instance), one can have indications concerning the form
of the annihilation cross section from selection rules. Indeed, fermion bilinears
of momentum L have parity P = (−1)L+1 . The (−1)L factor coming from
the parity of the Legendre polynomials in the wave function decomposition (the
polynomials are odd in cos θ for odd values of L and even for even values of L,
parity transforming cos θ into − cos θ . The extra −1 factor exists for Majorana as for
Dirac fermions (but not for scalar particles of course) but for different reasons. In the
Dirac case, the u and v spinors (equivalently the positive and negative energy states)
452 B Particle Physics

are independent and have opposite parity corresponding to the ±1 eigenvalues of the
parity operator γ 0 (see Sect. B.3.6). Reinterpreting the two spinor types, or positive
and negative energy states, as particle and antiparticle, then leads automatically to
opposite intrinsic parity for the particle–antiparticle pair. In the Majorana case, the
fermion has intrinsic parity ±i, and so the two-particle state has intrinsic parity
(±i)2 = −1.
As we discussed in the previous section, the Lth partial wave contribution
to the annihilation rate is suppressed as v 2L . A Majorana pair is even under
charge-conjugation (particle–antiparticle exchange), and so from the relation C =
(−1)L+S = +1 one infers that L and S must be either both even, or both odd
for the pair. The origin of the C = (−1)L+S rule is as follows: under particle–
antiparticle exchange, the spatial wave function contributes (−1)L , and the spin
wave function contributes (+1) if in symmetric triplet S = 1 state, and (-1) if in
the antisymmetric S = 0 singlet state, i.e. (−1)S+1. In addition, there is an overall
(-1) for anticommutation of the two particle-creation operators b† d † for the Dirac
case, and b†b† for the Majorana case. In the case of a bosonic initial state, the
formula is the same C = (−1)L+S because the singlet/triplet spin cancels the anti-
commutation relation.

B.5.3 Application to Specific Models

We can apply such reasoning to specific models to understand the behavior of the
cross section we obtained. For instance, if a scalar dark matter as well as vectorial
dark matter had no velocity dependence whereas fermionic dark matter is excluded
as the annihilation is dominated by the p-wave and so is excluded by the direct
detection experiments due to the large value of the dark matter coupling to the Higgs
¯
to still realize relic abundance constraint. Indeed, the scalar bilinear of the form 
is even under P (see Eq. B.43) which means for a fermionic dark matter, from P =
(−1)L+1, that L = 1, and the wave function is velocity suppressed. On the contrary,
terms of the form γ ¯ 5  of γ
¯ μ  are P-odd and thus L = 0 is authorized. This
is valid for Majorana as for Dirac cases. If the dark matter is a boson, the parity
is P = (−1)L = +1; the S ∗ S or V ∗ V are thus not velocity suppressed contrarily
to the fermion bilinear . ¯ We then conclude that scalar operators for bosonic
dark matter and pseudo-scalar ones for fermionic dark mater are the only ones not
velocity suppressed.
At the more microscopic level, an amplitude from a Lagrangian containing a  ¯
gives an amplitude proportional to M ∝ v̄u, which when applied to the Eq. (B.19)
we obtain M ∝ |p|/m and is thus velocity suppressed. On the other hand,
since γ 5 is antidiagonal, annihilation through γ ¯ 5  is not velocity suppressed.
Concerning the direct detection, however, the difference is that we have now an
initial state particle and final state particle, the amplitude is proportional to M ∝
ūu which is unsuppressed. On the other hand, the γ ¯ 5  is velocity suppressed
5 ¯ 5
due to the antidiagonal γ . So the operator γ  is very efficient: it induces
B.5 s-Wave, p-Wave, Helicity Suppression and All That 453

Fig. B.3 Illustration of dark 1 2

matter annihilating into a pair fR fL
of Standard Model Dirac
fermions 1 2 1 2
JR JL JR + JL = 0

1 2 2
fR fL mf f R

1 2 1 2
JR JR JR + JR = 0

unsuppressed annihilation and suppressed direct detection.15 We observe this effect

in supersymmetry where the Majorana neutralino annihilates mainly through the
exchange of the pseudo-scalar A as the charge invariant term is χ̄10 γ 5 χ10 A (A being
a pseudo-scalar, A∗ = −A, ensuring that the Lagrangian is neutral with respect
to electromagnetic charge). Terms of the type χ̄10 χ10 H / h exist but are velocity
suppressed as we explained above.

B.5.4 Helicity Suppression

We just saw that only some configurations of the dark matter two-particle wave
function survive from the velocity suppression effect. However, for some of them,
there is another source of suppression called helicity suppression which acts on the
s-wave function. Indeed, in the Majorana case, the operator γ ¯ 5  is even under
charge conjugate ( →  under C for a Majorana). As we discussed in the previous
section, C = (−1)L+S which means that both L and S should be even/odd. In the
s-wave part of the function, L = 0 ⇒ S = 0 ⇒ J = L + S = 0 (another way to
see it is to observe that at a null velocity, for a spinless particle, no direction in space
is preferred so J = 0). The helicity of the initial state is thus null. If the final state
is made of Standard Model fermions, the bilinear operator should be of the form
f¯R fL , particles and antiparticles with opposite helicity, which imply Jfinal = 0, see
Fig. B.3. The only way to conserve the total momentum J in the annihilation is
to flip by mass insertion a flipping operator transforming the left-handed fermion
fL into a right-handed fermion fR . The amplitude has thus to be proportional to
mf /m . This is what we call the helicity suppression mechanism.
It is important to notice that this argument is valid for a Majorana fermion (argument
of J = 0) but is also valid for a spin 0 particle. However, we cannot apply the
conclusion to a Dirac fermion or fermions with vector-like coupling of the style

15 We ¯ 5  is CP-odd and contribute to several processes,

have to be careful as the operator γ
limiting the value of its coupling.
454 B Particle Physics

¯ μ , because the vectorial exchanged particle will discuss only to the left-handed
γ
fermions of the Standard Model.

B.5.5 Summary

We saw that for the annihilation process, it is the C and P quantum numbers that
determine the nature of a final state, given an initial state. If one considers that the
initial state and final state consist of a pair particle/antiparticle, the operators are
given (for fermions) as

C = (−1)L+S P = (−1)L+1 (B.190)

whereas for a bosonic pair

C = (−1)L+S P = (−1)L . (B.191)

The only allowed s-wave states are then L = 0, S = 0 (J = 0); L = 0,

S = 1 (J = 1), and L = 0, S = 2 (J = 2). We list in the Table B.1 below
the value of J , C and P for sermonic and bosonic final state. For any bosonic or
sermonic bilinear, the transformation of the bilinear under rotations determines the
total angular momentum of the state that this bilinear either creates or annihilates.
This information along with the C and P quantum numbers of the bilinear are thus
sufficient to determine the spin and orbital angular momentum of the initial and
final state. The S and L quantum numbers of the states created (annihilated) by
every lowest-dimension bilinear are listed in Table B.2.
One can recover all the results presented in Table B.3 noticing that for the
annihilation, |v̄(p1 )u(p2 )|2 ∝ s − 4m2χ ∝ v 2 and for the direct detection,
|ū(p1 )u(p2 )|2 ∝ s + 4m2χ ; whereas the roles are opposite in the case of pseudo-
scalar couplings |v̄(p1 )γ 5 u(p2 )|2 as the introduction of a γ 5 matrix inverses
the impulsion p2 → −p2 . Similar arguments are valid for ū(p1 )γ μ u(p2 ) =
μ
p2
m ū(p1 )u(p2 ) after applying the Dirac equation.

Table B.1 The C and P S L J C P S L J C P

transformation properties of a
fermion/anti-fermion (left) or 0 0 0 + − 0 0 0 + +
boson/anti-boson (right) state 0 1 1 − + 0 1 1 − −
for a given quantum number 1 0 1 − − 1 0 1 − +
1 1 0,1,2 + + 1 1 0,1,2 + −
1 2 1,2,3 − − 1 2 1,2,3 − +
1 3 2,3,4 + + 2 0 2 + +
2 1 1,2,3 − −
2 2 0,1,2,3,4 + +
2 3 1,2,3,4,5 − −
2 4 2,3,4,5,6 + +
B.6 Schrodinger Equation 455

Table B.2 The C and P and Bilinear C P J State

J quantum numbers of any
ψ̄ψ + + 0 S = 1, L = 1
state that can be either created
or annihilated by the bilinear. i ψ̄γ 5 ψ + − 0 S = 0, L = 0
For each possible state, the S ψ̄γ 0 ψ − + 0 None
and L quantum numbers are ψ̄γ i ψ − − 1 S = 1, L = 0, 2
also given ψ̄γ 0 γ 5 ψ + − 0 S = 0, L = 0
ψ̄γ i γ 5 ψ + + 1 S = 1, L = 1
ψ̄σ 0i ψ − − 1 S = 1, L = 0, 2
ψ̄σ ij ψ − + 1 S = 0, L = 1
φ †φ + + 0 S = 0, L = 0
iI m(φ † ∂ 0 φ) − + 0 None
iI m(φ † ∂ i φ) − − 1 S = 0, L = 1
Bμ† B μ + + 0 S = 0, L = 0; S = 2, L = 2
iI m(Bν† ∂ 0 B ν ) − + 0 None
iI m(Bν† ∂ i B ν ) − − 1 S = 0, L = 1; S = 2, L = 1, 3
i(Bi† Bj − Bj† Bi ) − + 1 S = 1, L = 0, 2
i(Bi† B0 − B0† Bi ) − − 1 S = 0, L = 1; S = 2, L = 1, 3
 0ij k Bi ∂j Bk + − 0 S = 1, L = 1
− 0ij k B0 ∂j Bk + + 1 S = 2, L = 2
B ν ∂ν B0 + + 0 S = 0, L = 0; S = 2, L = 2
B ν ∂ν Bi + − 1 S = 1, L = 1

B.6 Schrodinger Equation

B.6.1 Generalities

The Schrodinger equation for a particle of mass M with an impulsion k described

by a wave function ψ in a potential V is usually written as

Table B.3 Dependance of the annihilation cross section and the scattering cross section for
different type of couplings
Operator Annihilation σ v Direct detection [A,v0 ]
g χ̄χ q̄q ∝ g 2 v 2 [χ̄ χ: S=1, L=1] SI ∝ g 2 A2 × f N (q)
χ
g χ̄γ 5 χ q̄q ∝ g 2 m2q [χ̄ γ 5 χ : S=0, L=0 ⇒ J = 0:h.f.] 0 [∝ (v0 )2 ]
q
g χ̄χ q̄γ 5 q ∝ g 2 v 2 [χ̄ χ: S=1, L=1] 0 [∝ (v0 )2 ]
q χ
g χ̄γ 5 χ q̄γ 5 q ∝ g 2 m2q [χ̄ γ 5 χ : S=0, L=0 ⇒ J = 0:h.f.] 0 [∝ (v0 )2 (v0 )2 ]
g χ̄γ μ χ q̄γμ q ∝ g2 SI ∝ g × (2u + d) × pχ .pq
2
q
g χ̄γ μ χ q̄γμ γ 5 q ∝ g2 0 [∝ (v0 )2 ]
χ
g χ̄γ μ γ 5 χ q̄γμ q ∝ m2q [L=0,S=0(C=1) ⇒ J=0:h.f.] 0 [∝ (v0 )2 ]
g χ̄γ μ γ 5 χ q̄γμ γ 5 q ∝ m2q [L=0,S=0(C=1) ⇒ J=0:h.f.] SD ∝ g 2 × (A − 2Z)2

1 k2
ψ + V ψ = E ψ = ψ (B.192)
2M 2M
456 B Particle Physics

with  the Laplacian which can be expressed in spherical coordinates (r, θ, φ)

 
1 ∂ ∂ψ 1 1 ∂ ∂ψ 1 ∂ 2ψ
ψ = 2 r2 + 2 sin θ + . (B.193)
r ∂r ∂r r sin θ ∂θ ∂θ sin2 θ ∂φ 2

For a radial potential, it is possible to decompose the wave function

ψ = ψkl
m
(r) = Rkl (r) lm (θ ) m (φ). (B.194)

m
In the above equation, we have defined ψkl as the wave function eigenvector of the
momentum k, kinetic moment l, and its projection on z-axis m. The operator lˆz ψ =
mψ = −i ∂ , the solution for (φ) can then be solved directly after normalization
 2π ∗ ∂φ
0 m (φ) m (φ)dφ = δmm
 

∂m (φ) 1
−iRkl (r) lm = m Rkl (r) lm (θ ) m (φ) ⇒ m (φ) = √ eimφ
∂φ 2π
(B.195)

whereas the l dependance is included in the  function from the equation lˆ2 ψ =
 
l(l + 1)ψ with lˆ2 = − sin1 θ ∂θ
∂ ∂
sin θ ∂θ ∂2
+ 12 ∂φ 2 . Implementing Eq. (B.194) in
sin θ
the preceding equation, we obtain

1 d dlm m2
sin θ − lm + l(l + 1)lm = 0. (B.196)
sin θ dθ dθ sin2 θ

This equation is well known in the theory of spherical functions. Corresponding

solutions are called associated Legendre polynomials Plm (cos θ ). After normaliza-
tion, the eigenfunction solutions of the Eq. (B.196) can be written for m > 0,

(2l + 1)(l − m)! m
lm = (−1) i m l
Pl (cos θ ). (B.197)
2(l + m)!

For any m, we combined (θ ) and (φ) in Ylm (θ, φ)

1/2
m+|m| (2l + 1) (l − |m|)! |m|
Ylm (θ, φ) = (−1) 2 il Pl (cos θ ) eimφ . (B.198)
4π (l + |m|)!
B.6 Schrodinger Equation 457

The m = 0 solution is often useful for problems with a z-axis symmetry

(independence on the angle φ)

2l + 1
Yl0 = i l
Pl (cos θ ) (B.199)
4π

with

1 dn
Pn (x) = (x 2 − 1)n . (B.200)
2n n! dx n

We then can summarize that the general solution of Schrodinger equation can be
written as

ψk = Al Pl (cos θ ) Rkl (r), (B.201)

l

where Rkl (r) are the continuum radial functions associated with angular momentum
l satisfying
 
1 d d l(l + 1)
− r 2 Rkl + + 2M V (r) Rkl = k 2 Rkl . (B.202)
r 2 dr dr r2

B.6.2 Solutions

Let us concentrate first on the case of the radial solution Rkl 0 (r) for a free particle
(V (r) = 0) Defining χkl (r) = rRkl0 (r), Eq. (B.202) yields


d 2 χkl (r) l(l + 1)
2
+ k2 − χkl (r) = 0. (B.203)
dr r2

For the simplest case l = 0, the two solutions are χk0 (r) = sin(kr)/ cos(kr) giving
0 (r) the zeroth-order Bessel and Neumann functions, respectively :
for Rk0

sin(kr) cos(kr)
0
Rk0 (r) = [Bessel] or Rk0
0
(r) = − [Neumann]. (B.204)
kr kr
Note that the Bessel function is the one well-behaved at the origin. To obtain the
general solution for any value of the momentum l the idea is to write Rkl 0
(r) =
(kr) χkl (r) and replacing in Eq. (B.202), we can show that the function χ = 1r dr
l d
χkl
follows the equation for the momentum l + 1. We can then write a recursion formula
for generating all the χkl from χk0 :

1 dχkl (r) 1 d l  sin kr 
χk(l+1) (r) = 2 ⇒ 0
Rkl (r) = (−kr) l
. (B.205)
k r dr k 2 r dr kr
458 B Particle Physics

∞ n (kr)2n
Developing16 sin kr
kr = 0 (−1) (2n+1)! , we can show that

(kr)l
0
Rkl (r)  as r → 0. (B.206)
(2l + 1)!!

This r l behavior near the origin is the usual well-behaved solution to Schrodinger’s
equation in the region where the centrifugal term dominates. Now, for large values
or r, it is obvious that the dominant term is generated by differentiating only the
trigonometric function at each step which gives

1  π
0
Rkl (r)  sin kr − l as r → ∞. (B.207)
kr 2
In fact, the real wave function should be a combination of the Bessel and Neumann
wave function. One should thus apply the recursive procedure also to the function
cos kr
kr which is equivalent to introducing an extra phase factor called the phase
shif t, giving at the end the general solution

1  π 
0
Rkl (r)  sin kr − l + δl as r → ∞. (B.208)
kr 2
In the case of a generic potential, the Schrodinger’s equation can then be written as

d 2 χkl (r) l(l + 1)
+ k2 − − 2MV (r) χkl (r) = 0. (B.209)
dr 2 r2

B.7 The Strong-CP Problem

B.7.1 QCD Lagrangian

The strong-CP problem is one of the oldest issue of the minimal versions of the
Standard Model. In the QCD Lagrangian, there is the possibility to add a gauge
invariant term of the form

1 g2
LQCD = − Gaμν Gaμν + i q̄L γ μ Dμ qR + i q̄R γ μ Dμ qL + θ0 3 2 Gaμν G̃aμν ,
4 32π
(B.210)

16 The minus sign appearing in (−kr)l in Eq. (B.205) comes from the standard normalization of
0
the function Rkl .
B.7 The Strong-CP Problem 459

with

Gaμν = ∂μ Gaν − ∂ν Gaν − g3 f abc Gbμ Gcν , [Gaμ , Gbμ ] = if abc Gcμ , (B.211)

1
G̃aμν = μνρσ Gaρσ , (B.212)
2
is its dual tensor, and
√
1 3
f 123 = 1; f 147 = −f 156 = f 246 = f 257 = f 345 = −f 367 = ; f 458 = f 678 = .
2 2

Exercise Show that in a parity P operation (B.39), Gaμ → Gaμ . Deduce then that
Gaμν Gaμν does not violate P .

The term proportional to θ0 in Eq. (B.210) violates P , CP , and T and produces

a large neutron dipole moment which is in contradiction with experimental results
(see the box). It is a total derivative, and therefore does not affect the equation of
motion. However, the non-Abelian structure of the strong interactions, combined to
the divergence of g3 (μ)μ→0 means that it cannot be reduced to a boundary term in
the action (S = d 4 xL). The gauge fields in this case do not necessarily vanish at
infinity.

Exercise Show that the θ0 term of Eq. (B.210) can be written ∂μ K μ , with

1
K μ =  μνρσ Gaν ∂ρ Gaσ + f abc Gaν Gbρ Gcσ . (B.213)
3
The value of the electric dipole moment of the neutron expected from the θ0 -term
is

dnt h = 2.4 × 10−16 θ0 e.cm,

corresponding to an experimental upper bound

|θ0 | < 10−10.

How to justify such a tiny parameter? This is called the strong-CP problem.

Electric Dipole Moment of the Neutron

The electric dipole moment can be roughly estimated by the difference of
charge multiplied by the distance which separates them. If one takes the

(continued)
460 B Particle Physics

molecule of water for instance, H2 0, the oxygen atom is separated from the
two hydrogen atoms by a distance of ∼ 0.1 nm. The dipole moment can then
be approximated by
th
dH 2O
 10−8 e × cm, (B.214)

e being the electric charge. The measured value dH2 O = 0.5 × 10−8 e.cm is
exp

in pretty good accordance with the prediction.

In the case of the neutron, the distance between the two down-type quarks
and the up-type quark can be estimated at  10−15 m, corresponding to a
dipole moment dn  10−13 e.cm. The exact calculation gives

dnt h = 10−15 e.cm. (B.215)

However, no dipole moment has been observed, the measurement giving an

upper bound dn < 10−26 e.cm, more than 10 orders of magnitude below
exp

the prediction.

Notice that, if we neglect the quark mass, a chiral transformation qL → e−iα qL ,

qR → eiα qR will generate in the Lagrangian a term of the form

g32
L ⊃ −2α Ga G̃aμν . (B.216)
32π 2 μν
The reason for this anomalous symmetry is that the measure is not invariant under
this transformation. We clearly see that there exists a shift symmetry θ0 → θ0 − 2α.
Then, θ0 is not physical, and there is no strong CP problem. However, quarks have
masses. The full QCD Lagrangian should be written as

1 g2
LQCD = − Gaμν Gaμν + θ0 3 2 Gaμν G̃aμν (B.217)
4 32π
+i q̄L γ μ Dμ qL + i q̄R γ μ Dμ qR − mq q̄L qR − m∗q q̄R qL . (B.218)

To render the mass matrices mq real, one needs to rotate the spinors as follows:

qL → eiφL qL ; qR → eiφR qR . (B.219)

If m = |m|eiφ , any rotation respecting φL − φR = φ should transform the masses

into real parameters. However, these transformations on the quarks generate at
quantum level a term

g32
LQCD = (φL − φR ) Ga G̃aμν . (B.220)
32π 2 μν
B.7 The Strong-CP Problem 461

We give some arguments in the box below to catch the main argument to understand
the form of LQCD . The absence of observation of neutron dipole moment imposes
an incredibly fine-tuned cancellation between θ0 and φ, two parameters with a
completely different nature : whereas the θ0 contribution appears at tree level, even
without the presence of any quark masses, the φ-term arises through the Yukawa
interactions in the Higgs sector.17 We will denote θ = θ0 + (φL − φR ) from now on.

B.7.2 The Axionic Peccei–Quinn Solution

The idea proposed by Helen Quinn and Roberto Peccei in [8] is to introduce an axion
field a such as, below a breaking scale fa , under a chiral U (1)P Q transformation

a a
q → e−iαγ q ;
5
→ + 2α, (B.221)
fa fa

the Lagrangian

−i faa g32 1
LcP Q = [mq q̄R e qL + h.c.] + θ Ga G̃aμν + ∂ μ a∂μ a (B.222)
32π 2 μν 2
stays invariant at the classical level.

Exercise Check that the Lagrangian (B.222) is invariant under the chiral U (1)P Q
transformation (B.221).

Chiral Anomalies
To understand why transformation (B.219) generates terms of the form
(B.220), we need to look at the figure below. A transformation of the type
5θ
q → eiγ 2 q will generate in the Lagrangian a term of the type imθ q̄γ 5 q. At
quantum level, this correspond to a δLeff

θ d4p 1 1 1
δLeff = −i g32 mT r[T a T b ] Tr γ5 /1 /2 ,
2 (2π)4 p
/ + /
k 1 − m p
/ − m p
/ − /
k 2−m

(continued)

17 Only the up anddown quarks are important here, the heavier quarks giving a much more reduced
contribution. φ should be understood as Arg(Det mq ).
462 B Particle Physics

T i being the SU(3) generators. After some little algebra, the heavy quarks
running in the loop give

μ ρ d4p 1 θg32 a aμν
δLeff = g32 m2 θT r[T a T b ]μνρσ k1 k2ν 1 2σ = G G̃
(2π)4 (p2 − m2 )3 32π 2 μν

p + k1 k1

imθq̄γ 5 q p

p − k2 k2

We saw that, at quantum level, redefining the quarks gives an equivalent

Lagrangian

Q a g32 1
LP Q = [mq q̄R qL + h.c.] + +θ 2
Gaμν G̃aμν + ∂ μ a∂μ a. (B.223)
fa 32π 2

Clearly, the P Q symmetry is broken at the quantum level, and the axion is in fact a
pseudo-Nambu–Goldstone boson. The miracle of this construction is that at a scale
below QCD  200 MeV, the appearance of quark condensates under the form of
mesons q̄q breaks the chiral symmetry and generates an effective potential for the
axion18

 2  2
1 a mu md 1 a
V (a) ∼ +θ q̄q ∼ +θ m2π fπ2 , (B.224)
2 fa mu + md 8 fa

with mπ = 135 MeV, fπ = 93 MeV are pion mass and pion decay constant,
respectively. The potential admits a minimum at faa = −θ , eliminating the term
responsible of the dipole moment in (B.223), solving in the most elegant way the
strong-CP problem. In the meantime, once we develop a around its minimum fa θ ,

 2
18 The potential should be periodic in a 1
fa , the origin of the 2
a
fa +θ term is in fact 1 − cos( faa +
θ).
B.8 Useful Spectrum 463

V (a) generates a mass term for the excitation of the axion field given by

 12
mπ fπ 10 GeV
ma =  6.3 µeV . (B.225)
2fa fa

B.8 Useful Spectrum

B.8.1 Gamma Spectrum

In order to determine these spectral functions, we generated 300,000 events of

Standard Model particles decaying (directly or through secondary decays) into γ -
rays using the PYTHIA [4] package, taking care in order to include all possible
decay channels. Following the method of [5], we fitted the resulting spectra through
functions of the form:

dNγi
= exp [Fi ( ln(x) )] , (B.226)
dx
where i represents the i-th WIMP annihilation channel, i = W W, ZZ, etc.;
x = Eγ /mχ with mχ being the WIMP mass and F are seventh-order polynomial
functions which were found to be the following:

W W (x) = −7.72088528 − 8.30185509 x − 3.28835893 x 2 − 1.12793422 x 3

− 0.266923457 x 4 − 0.0393805951 x 5 − 0.00324965152 x 6
− 0.000113626003 x 7,
ZZ(x) = −7.67132139 − 7.22257853 x − 2.0053556 x 2 − 0.446706623 x 3
− 0.0674006343 x 4 − 0.00639245566 x 5 − 0.000372241746 x 6
− 1.08050617 · 10−5 x 7 ,
b b̄(x) = −11.4735403 − 17.4537277 x − 11.5219269 x 2 − 5.1085887 x 3
− 1.36697042 x 4 − 0.211365134 x 5 − 0.0174275134 x 6
− 0.000594830839 x 7,
uū(x) = −4.56073856 − 8.13061428 x − 4.98080492 x 2 − 2.23044157 x 3
− 0.619205713 x 4 − 0.100954451 x 5 − 0.00879980996 x 6
− 0.00031573695 x 7,
464 B Particle Physics

d d̄(x) = −4.77311611 − 10.6317139 x − 8.33119583 x 2 − 4.35085535 x 3

− 1.33376908 x 4 − 0.232659817 x 5 − 0.0213230457 x 6
− 0.000796017819 x 7,
τ − τ + (x) = −5.64725113 − 10.8949451 x − 7.84473181 x 2 − 3.50611639 x 3
− 0.942047119 x 4 − 0.14691925 x 5 − 0.0122521566 x 6
− 0.000422848301 x 7.

The case of WIMP annihilation into μ+ μ− pairs has a relatively small decay
contribution, to the photon spectrum, coming from the μ → e− ν̄e νμ γ channel,
which has a small branching ratio. e+ e− pair production contributes to the gamma-
ray spectrum through different (not decay) processes, mainly Inverse Compton
scattering and synchrotron radiation. These contributions depend crucially on the
assumptions made concerning the intergalactic medium and will not be analyzed
here. This means, practically, that the e+ e− and μ+ μ− spectral functions are set
equal to zero. A graphical representation of these functions can be seen in Fig. B.4.
These functions can afterward be used in order to generate any gamma-ray flux
according to Eq. (5.96). As we can see, all contributions are quite similar, apart
from the τ − τ + channel which has a characteristic hard form. Nevertheless, at high
energies, the form of all contributions becomes almost identical.

B.8.2 Positron Spectrum

For the positron spectrum at the source, the same procedure gives us

W W (x) = −1.470768 − 1.865343 x − 1.865343 x 2 − 1.940513 x 3

− 0.7538001 x 4 − 01.432966 x 5 − 0.01350088 x 6 − 0.0005048965 x 7 ,
ZZ(x) = −2.909439 − 1.091164 x − 1.184799 x 2 − 1.621001 x 3
− 0.6838962 x 4 − 0.1359553 x 5 − 0.01320953 x 6 − 5.058684 · 10−4 x 7 ,
bb̄(x) = −12.33180 − 27.10877 x − 29.62288 x 2 − 18.59724 x 3
− 6.705565 x 4 − 1.436652 x 5 − 0.1813250 x 6 − 0.01246600 x 7 ,
cc̄(x) = −12.07344 − 17.57427 x − 11.85548 x 2 − 5.235151 x 3
− 1.391299 x 4 − 0.2141368 x 5 − 0.01755747 x 6 − 0.0005935255 x 7 ,
t t¯(x) = −8.90339 − 21.5246 x − 27.6548 x 2 − 18.8567 x 3
− 6.80256 x 4 − 1.32199 x 5 − 0.129489 x 6 − 0.00494683 x 7 ,
B.8 Useful Spectrum 465

10

1
E J · dN J /dE J

0.1
WW
ZZ
bb
WW
0.01 uu m F = 100 GeV
dd

0.001
1 10 100
E J (GeV)

10

1
E J · dN J/dE J

0.1
WW
ZZ
bb
0.01 WW m F = 500 GeV
uu
dd

0.001 1 10 100
E J (GeV)

Fig. B.4 Separate contributions from Standard Model particles decaying into γ -rays for mχ =
100 and 500 GeV. The PYTHIA result points have been suppressed for the sake of clarity
466 B Particle Physics

τ − τ + (x) = −4.004877 − 6.630655 x − 5.779419 x 2 − 3.504474 x 3

− 1.200937 x 4 − 0.2254182 x 5 − 0.02172664 x 6 − 0.0008412294 x 7 .
H H (x) = −8.731713 − 10.64431 x − 6.924815 x 2 − 3.131443 x 3
− 0.7984188 x 4 − 0.1135447 x 5 − 0.0084599666 x 6 − 0.00025748991 x 7 .
((for a 200 GeV Higgs)

B.8.3 Antiproton Spectrum

For the antiproton, we can parameterize the spectrum as follows :

dNpi 1
=  i
, (B.227)
dE mχ p1 x + p2i | log10 (x)|p4
i p3

with

i 1/p1i 1/p2i 1/p3i 1/p4i

W 306 m0.28
χ + 7.4 × 10−4 m2.25
χ 2.32 m0.05
χ −8.5 m−0.31
χ −0.39 m−0.17
χ − 2 × 10−2 m0.23
χ
Z 480 mχ + 9.6 × 10−4 m2.27
0.26
χ 2.17 m0.05
χ −8.5 m−0.31
χ
−0.075
−0.33 mχ − 1.5 × 10−4 m0.71
χ
t 1.35 m1.45
χ 1.18 m0.15
χ −2.22 −0.21
b 1.75 m1.4
χ 1.54 m0.11
χ −2.22 −0.31 m−0.052
χ
c 1.7 m1.4
χ 3.12 m0.04
χ −2.22 −0.39 m−0.076
χ

B.9 Lagrangians

B.9.1 Standard Model

The SM Lagrangian can be written as

1 1 a μνa 1
LSM = − Gaμν Gμνa − Fμν F − Bμν B μν + |Dμ H |2 (B.228)
4 4 4
¯ Dμ  + μH |H | − λH |H |4
+ i γ μ 2 2
(B.229)

with T a = σ a /2 (σ a the Pauli matrices) and where Dμ represents the covariant

derivative,

Y
Dμ = (∂μ − ig3 Ga T3a − igWμa T2a − ig  Bμ ) (B.230)
2
B.9 Lagrangians 467

 
1−γ 5
νL 2 ν
 are fermionic multiplet = 1−γ 5
in the case of the SU (2)L generators
eL
2 e
Wμa T a for instance, and

Fμν = ∂μ Bν − ∂ν Bμ a
Wμν = ∂μ Wνa − ∂ν Wμa + gf abc Wμb Wνc , (B.231)

where under the SU (2) × U (1)Y of parameter α a and β, Wμ , Bμ and H transforms

as

L /H → e−iαa T
a −iβ Y
R → e−iβ 2 R
Y
2 L /H
1 1
Bμ → Bμ − ∂μ β Wμa → Wμa − ∂μ α a − f abc α b Wμc .
g g

B.9.1.1 Higgs Couplings

The minimum of the Higgs potential is obtained for

v = μH / λH and m2h = 2μ2H (B.232)

YH
Dμ H = ∂μ − igWμa T a − ig  Bμ H
2
⎛ ⎞
Wμ3  YH g
∂μ − ig 2 − ig 2 Bμ −i 2 (Wμ − iWμ ) ⎠
1 2
=⎝ W3
H.
−i 2 (Wμ + iWμ ) ∂μ + ig 2μ − ig  Y2H Bμ
g 1 2


0 √
If one develops around H = , |Dμ H |2 =
(v + h)/ 2
⎛ g
⎞2
−i √ (Wμ1 − iWμ2 )(v + h) √
⎝ 2 2
gWμ3
⎠ if one defines Wμ± = (Wμ1 ∓ iWμ2 )/ 2,
∂μ h  YH Bμ
√ + i( √ − g √ )(v + h)
2 2 2 2 2
MZ2
one can identify MW |W ± |2
2 + 2 |Zμ |
2 which gives

gv g 2 + g 2 g
MW = MZ = v= v. (B.233)
2 2 2 cos θW

g2v + μ
For the Higgs coupling, the development of |Dμ H |2 gives L ⊃ 2 H Wμ W− +
g2v
gMW H Wμ+ W−
μ gMZ
4 cos2 θW
H Zμ Zμ = + 2 cos θW H Zμ Zμ. The couplings are then

igMZ2 M2
CH W W = igMW , CH ZZ = × 2 (symmetry factor, Z = Z∗ ) = ig Z .
2MW MW
(B.234)
468 B Particle Physics

B.9.1.2 Vectorial Couplings

Developing the Lagrangian (B.229) using (B.230) one can write

g
¯ L γ μ Wμa T a L = √ g
g ēγ μ Wμ− (1 − γ 5 )ν + √ ν̄γ μ Wμ− (1 − γ 5 )e (B.235)
2 2 2 2

B.9.2 Singlet Scalar

The simplest extension of the SM is the addition of a real singlet scalar field.
Although it is possible to generalize to scenarios with more than one singlet, the
simplest case of a single additional singlet scalar provides a useful framework to
analyze the generic implications of an augmented scalar sector to the SM. The most
general renormalizable potential involving the SM Higgs doublet H and the singlet
S is
1 1
L = LSM + (Dμ H )† (D μ H ) + μ2H H † H − λH |H |4
2 4
1 λS 4 μ2S 2 λH S 2 †
+ ∂μ S∂ μ S − S − S − S H H
2 4 2 4
κ1 κ3
− H † H S − S 3 − V0 , (B.236)
2 3
where Dμ represents the covariant derivative.

YH
Dμ H = (∂μ − igWμa T a − ig  Bμ )H
2
⎛ ⎞
Wμ3  YH B g
∂ − ig − ig −i (W 1 − iW 2 )
=⎝ ⎠H
μ 2 2 μ 2 μ μ
W3
−i g2 (Wμ1 + iWμ2 ) ∂μ + ig 2μ − ig  Y2H Bμ

and we have eliminated a possible linear term in S by a constant shift, absorbing the
resulting S-independent term in the vacuum energy V0 . We require that the minimum
of the potential occur at v = 246 GeV . Fluctuations around this vacuum expectation
value are the SM Higgs boson. For the case of interest here for which S is stable and
may be a dark matter candidate, we impose a Z2 symmetry on the model, thereby
eliminating the κ1 and κ3 terms. We also require that the true vacuum of the theory
satisfies S = 0, thereby precluding mixing of S and the SM Higgs boson and
the existence of cosmologically problematic domain walls. After the electroweak
breaking, the scalar Lagrangian can be written

μ2S 2 1 MW 2
LS = − S − λH S S 2 h2 − λH S hS . (B.237)
2 8 2g
B.9 Lagrangians 469

In this case, the masses of the scalars are

√ 
2μH λH S μ2H
v= √ , MH = μH , mS = μ2S + (B.238)
λH λH 2

and the HSS coupling generated is

λH S MW λH S MW
CH SS : −i ×2 (symmetry factor, S = S∗ ) = −i . (B.239)
2g g

B.9.3 Extra U(1) and Kinetic Mixing

The matter content of any dark U (1)D extension of the SM can be decomposed
into three families of particles:

• The V isible sector is made of particles which are charged under the SM gauge
group SU (3) × SU (2) × UY (1) but not charged under UD (1) (hence the dark
denomination for this gauge group).
• the Dark sector is composed of the particles charged under UD (1) but neutral
with respect of the SM gauge symmetries. The dark matter (ψ0 ) candidate is the
lightest particle of the dark sector.
• The Hybrid sector contains states with SM and UD (1) quantum numbers.
These states are fundamental because they act as a portal between the two
previous sector through the kinetic mixing they induce at loop order.

From these considerations, it is easy to build the effective Lagrangian generated at

one loop :

1 1 δ
L = LSM − B̃μν B̃ μν − X̃μν X̃μν − B̃μν X̃μν
4 4 2
+i ψ̄i γ μ Dμ ψi + i ¯ j γ μ Dμ j ,
 (B.240)
i j

B̃μ being the gauge field for the hypercharge, X̃μ the gauge field of UD (1) and
ψi the particles from the hidden sector, j the particles from the hybrid sector,
Dμ = ∂μ − i(qY g̃Y B̃μ + qD g̃D X̃μ + gT a Wμa ), T a being the SU (2) generators, and

g̃Y g̃D j j m2j
δ= qY qD log (B.241)
16π 2 Mj2
j

with mj and Mj being hybrid mass states.

470 B Particle Physics

Notice that the sum is on all the hybrid states, as they are the only ones which can
contribute to the B̃μ X̃μ propagator. After diagonalization of the current eigenstates
that makes the gauge kinetic terms of Eq. (B.240) diagonal and canonical, we can
write after the SU (2)L × U (1)Y breaking:19

Aμ = sin θW Wμ3 + cos θW Bμ (B.242)

Zμ = cos φ(cos θW Wμ3 − sin θW Bμ ) − sin φXμ

(ZD )μ = sin φ(cos θW Wμ3 − sin θW Bμ ) + cos φXμ

with, at the first order in δ:

α 2δ sin θW
cos φ =  sin φ = 
α 2 + 4δ 2 sin θW
2
α 2 + 4δ 2 sin2 θW
α = 1 − MZ2 D /MZ2 − δ 2 sin2 θW (B.243)

± (1 − MZ2 D /MZ2 − δ 2 sin2 θW )2 + 4δ 2 sin2 θW

and + (−) sign if MZD < (>)MZ . The kinetic mixing parameter δ generates an
effective coupling of SM states ψSM to ZD , and a coupling of ψ0 to the SM Z
boson which induces an interaction on nucleons. Developing the covariant derivative
on SM and ψ0 fermions state, we compute the effective ψSM ψSM ZD and ψ0 ψ0 Z
couplings at first order in δ and obtain

L = qD g̃D (cos φ Zμ ψ̄0 γ μ ψ0 + sin φ Zμ ψ̄0 γ μ ψ0 ). (B.244)

19 Ournotations for the gauge fields are (B̃ μ , X̃ μ ) before the diagonalization, (B μ , X μ ) after
μ
diagonalization, and (Z μ , ZD ) after the electroweak breaking.
Neutrino Physics
C

C.1 Astrophysical and Cosmological Sources of Neutrino

The neutrino arriving on Earth has different origins. Some are produced in the
Sun (solar neutrino) whereas others are produced via the interaction of cosmic
rays, mainly composed of ultra-energetic proton, on the atmosphere (atmospheric
neutrinos). Others can be directly generated via ultra-high energy (UHE) sources
like blazars or Active Galactic Nuclei (AGN). These UHE neutrinos propagate and
diffuse in the Universe, pinpointing their source clearly because, contrarily to the
cosmic rays produced by the same objects, neutrinos are not affected by magnetic
fields or the interstellar medium (ISM). Another production mechanism of neutrino
is the interaction of the cosmic rays on the photons from the CMB cosmogenic
neutrino : p + γCMB →  → n + π + , and then π + → μ+ νμ and μ+ → e+ ν̄μ νe .
This source is one of the dominant one and produces 2 μ-types neutrinos for 1
e-type one. The flavor ratio at the production is then (1:2:0), which will be an
important point to distinguish them on Earth. Finally, the last source of neutrino,
not yet detected, is obviously the relic ones from the decoupling of the primordial
plasma. They have a temperature of 1.95 K and a density nν = 109 cm−3 , see
Sect. C.3, compared to the 2.5 K and nγ = 394 cm−3 of the photons from the
CMB (see Eq. 3.42). We will look into the details of each production and detection
mechanism in this section.

C.1.1 Solar Neutrinos

Electron neutrinos are produced in the Sun as a product of nuclear fusion. By far
the largest fraction of neutrinos passing through the Earth are Solar neutrinos. The
main contribution comes from the proton–proton reaction. The reaction is

p + p → d + e+ + νe .

© Springer Nature Switzerland AG 2021 471

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
472 C Neutrino Physics

From this reaction, 86% of all solar neutrinos are produced. The deuterium will then
fuse with another proton to create a 3 He atom and a gamma ray. This reaction can
be seen as

d + p → 3H e + γ . (C.1)

The isotope 4 He is then produced using two 3 He from the previous reaction
3
H e + 3 H e → 4 H e + 2p. (C.2)

With both helium-3 and helium-4 in the system now, beryllium can be fused by the
reaction of one of each helium atom as seen in the reaction:
3
H e + 4 H e → 7 Be + γ . (C.3)

Since there are four protons and only three neutrons, the beryllium can go down two
different paths from here. The beryllium could capture an electron and produce a
lithium-7 atom and an electron neutrino. It can also capture a proton due to the
abundance in a star. This will create boron-8. Both reactions are as seen below
respectively:
7
Be + e− → 7 Li + νe . (C.4)

This reaction produces 14% of the solar neutrinos. The lithium-7 will combine with
a proton to produce 2 atoms of helium-4. On the other hand, the beryllium will
capture a proton to form the boron-8 as follows:
7
Be + p → 8 B + γ . (C.5)

Then, the Boron-8 will beta decay into beryllium-8 due to the extra proton which
can be seen below:
8
B → 8 Be + e+ + νe . (C.6)

The reaction produces about 0.02% of the solar neutrinos, but these few solar
neutrinos have the larger energies. The highest flux of solar neutrinos comes
directly from the proton–proton interaction, and has a low energy, up to 400
keV. From the Earth, the amount of neutrino flux at Earth is enormous (around
7 × 1010 particles cm−2 s−1 ). Moreover, since neutrinos are insignificantly absorbed
by the mass of the Earth, the surface area on the side of the Earth opposite the Sun
receives about the same number of neutrinos as the side facing the Sun.
One should notice that the first solar neutrino generated by proton–proton fusion
(constituting 99.77% of the solar neutrinos) was observed in August 2014 by the
Borexino experiment [7]. The Borexino experiment was a joint collaboration of
several European (Italy, Germany, France, Poland), USA, and Russian institutions,
C.3 Neutrino Mass 473

installed in the Gran Sasso Laboratory in central Italy. The reaction was harder to
observe than the one generated by the Boron because of the very low energy of
the escaping neutrino (around the keV compared to  10 MeV for the 8 B ones).
The fusion reactions occur deep within the Sun, and which then pass right through
the Sun, taking just over eight minutes to reach the Earth. On the other hand, all
solar energy measurements before were based on light from the Sun’s photosphere
(the familiar Sunlight which lights up our skies and warms the Earth). But the
energy carried by this Sunlight was produced in solar fusion reactions about 100,000
years ago.1 Comparison between the Borexino measurement and those of the Sun’s
radiant energy reveals that the solar power has not changed in quite a long time.
The collaboration reported spectral observations of pp neutrinos, demonstrating that
about 99% of the power of the Sun (3.84 × 1033 ergs per second) is generated by
the proton–proton fusion process.

C.1.2 Atmospheric Neutrinos

Atmospheric neutrinos result from the interaction of cosmic rays with atomic nuclei
in the Earth’s atmosphere, creating showers of particles, many of which are unstable
and produce neutrinos when they decay. A collaboration of particle physicists from
Tata Institute of Fundamental Research (India), Osaka City University (Japan), and
Durham University (UK) recorded the first cosmic ray neutrino interaction in an
underground laboratory in Kolar Gold Fields in India in 1965.

C.2 Ultra-High Energetic Neutrinos

In the galaxy, or outer space, there are different sources of ultra-High Energy
neutrinos (> 1000 TeV = 1 PeV). It ranges from cosmic rays interaction with
the Inter Stellar Medium (ISM) to direct production from Super Nova remnants.
In any cases, these Ultra-High energy neutrino interacts with the CMB or the
Earth atmosphere, producing specific phenomena, like a GZK-cut or the Glashow
resonance. We will develop these two effects in this section and show how one can
observe them experimentally.

C.3 Neutrino Mass

C.3.1 Dirac Mass

To give mass to neutrino, we face the same “issue” that we have concerning the mass
term of “up” type quark. Indeed, the Lagrangian Ldown = −λd L̄dR + h.c., where

1 The average time for energy to percolate from the central regions of the Sun and reach its surface.
474 C Neutrino Physics

L represents the left doublet (u, d)TL ,  the Higgs field, and dR the right-handed
part of the spinor (dL , dR )T [Weyl convention
√ see Sect. B.3.1]. However, the Higgs
field develops a vev v,  = (0, (v + φ) 2)T which gives mass to down type
particles like the bottom quarks, strange ones or the electrons or taus, but not to the
top quarks or neutrino that are of the “up” type SU (2)L . The idea is to introduce the
charge conjugate of the Higgs field ,  ˜ c = iσ2 ∗ , σi being the Pauli matrices (see
Sect. B.3.7). With this new field, one easily sees that the coupling −λu L̄˜ c uR +h.c.
generates a mass term for the up-type spinors once the Higgs develops a vev. The
expression of the neutrino mass is then

νL
LDirac
ν = −mD ν̄ν = −mD (ν̄L νR + ν̄R νL ), with ν = (C.7)
νR
√
and mD = λν v 2. We know today that neutrinos (at least some of the neutrinos)
have masses ( 0.1 eV). We can just introduce these very small masses via the Dirac
mechanism, but of course this leaves completely unexplained the smallness of such
masses with respect to other fermion masses. Alternatively, one could introduce the
so-called Majorana mass term as we will see below.

C.3.2 Majorana Mass

C.3.2.1 Without Right-Handed Neutrino

It is also possible to generate mass to the left-handed neutrino through a Majorana
mass term, without needing to ever introduce the right-handed degrees of freedom.
Let us first recall that a fermion field ψ possess a charge conjugate counterpart
ψ c = C ψ̄ T where the role of particles and antiparticles are exchanged.2 A Majorana
fermion is one where ψ c = ψ, i.e. a Majorana fermion is its own antiparticle. Let us
assume that the left-handed neutrinos are particles of this kind. This is, in principle
possible as they are not charged.3 Thus let us define

νLc = C ν̄LT . (C.8)

We cannot impose νL = νLc for the simple reason that νLc is actually a right-
handed field, but we can assume that νL and νLc actually describe the left-handed and
right-handed degrees of freedom of one fermion that is identical to its antifermion,

2 See Sect. B.3.7 for more details.

3 Itis important to underline here that this Majorana mass term can only exist for neutrino as it is
the only neutral fermion in the Standard Model, which is invariant under the electromagnetic U (1)
transformation. Indeed, a term of the form 12 mM e (ēe + ē e) clearly violates the electric charge
c c

conservation.
C.3 Neutrino Mass 475

respectively. We can then construct the Lorentz invariant mass term


1 1 νL
− mL ν̄L νLc + h.c. = − mL ν̄ν with ν = = νc. (C.9)
2 M 2 M νLc

This is the so-called Majorana mass term. It is not possible to make this gauge
invariant with a dimension 4 operator. Indeed, this term changes weak hypercharge
by 2 units—not possible with the standard Higgs interaction, requiring the Higgs
field to be extended to include an extra triplet with weak hypercharge for instance—
whereas for right chirality neutrinos as we will see, no Higgs extensions are
necessary. For both left and right chirality cases, Majorana terms violate lepton
number, but possibly at a level beyond the current sensitivity of experiments to
detect such violations.
It is also interesting to notice that this minimal scheme to give mass to neutrino does
not advocate right-handed states: is it indeed possible to give mass to neutrino (to
the price of non-conserving lepton number and a Majorana nature for the neutrino).
This is the minimal scheme which was the point of view adopted by Gribov and
Pontecorvo [6].
Note that Majorana mass terms have nothing to do with the Majorana representation
in spinor space. One can use any representation for the fields of which Majorana
and Dirac mass terms are composed. Neither do Majorana mass terms imply the
associated particles/fields are Majorana fermions, of which you may have heard.
Majorana fermions are their own antiparticles. More on this in Sect. B.3.7. For now,
we will assume that both Dirac and Majorana mass terms contain only Dirac type
particles.

C.3.2.2 With Right-Handed Neutrino

We just saw that we can write a Majorana mass term for the left-handed neutrino,
at the price to work in dimension 6 operators. However, if one introduces the right-
handed neutrino from the Eq. (C.7), we clearly understand that nothing forbids us
to write gauge invariant dimension 4 mass operators with the right-handed neutrino
of the form:
1  
− mR ν̄R νRc + ν̄Rc νR . (C.10)
2 M
Note that in the expression (C.10), the second term destroys a right-handed particle
and create a right-handed antiparticle (equivalent to a left-handed particle) as in
the case of Eq. (C.7). The lepton number is however not conserved because two
neutrino are created from the vacuum whereas their Dirac coupling attribute to them
a leptonic number. Adding the three contributions (C.7, C.9, C.10) just discussed
476 C Neutrino Physics

above, one obtains

1   1 R  
Lν = −mD (ν̄L νR + ν̄R νL ) − mL
M ν̄L νL + ν̄L νL −
c c
mM ν̄R νRc + ν̄Rc νR
2 2
  L
1 c  ν mM mD
= − ν̄L ν̄R M Lc + h.c. with M = , (C.11)
2 νR mD mR M

where we used ν̄L νR = ν̄Rc νLc (Eq. B.52).

C.4 The See-Saw Mechanism

From the equation (C.11) one can easily diagonalize the mass matrix in the (νLc , νR )
space and then study the different possibilities or hierarchy between mD , mL
M , and
mRM . However, to really understand the subtleties of the mechanism one should first
study simple cases.

C.4.1 A Simple Example

Let us suppose that the Higgs does not couple to the neutrino; in this case, the Dirac
mass is absent of the potential and one can write mD = 0 in (C.11). One then needs
to express the mass eigenstates (ν, N) as function of νL , νLc , νR , νRc from
 L   
1 mM 0 νL 1 mν 0 ν
− (ν̄Lc , ν̄R ) + h.c. = − (ν̄, N̄ ) (C.12)
2 R
0 mM νRc 2 0 mN N

which gives (noticing that ν̄R νR = 0)

 
  νL   νRc
ν = νL + νLc = ; N = νR + νRc = , (C.13)
νLc νR

where we have used the Dirac representation of the spinors ν and N which are
4-dimensional objects. As we can notice, in this case both ν and N are Majorana
particles (ν c = ν and N c = N), which can justify the term “Majorana masses” for
mL R
M and mM . Another important point concerns the notation, which is sometimes

νL
quite difficult to follow in the literature. The notation should be read
νLc
 
1 0
νL + νLc , νL and νLc being 4 dimensional spinors, even if as a left-handed
0 1
spinor (νL ) and right-handed spinor (νLc ), each of them possesses effectively only 2
 
1 0
degrees of freedom. and are the base vectors in the plane of the PL and
0 1
C.4 The See-Saw Mechanism 477


νL
PR projectors. So, in a sense, it is formally correct to write νL = , but on the
0
left side, one should understand νL as a two–dimensional Weyl spinor.

C.4.2 Generalization

Now, if we assume that in the mass term enter the left–handed field νL , the right–
handed sterile neutrino νR , and allowing for a violation of the lepton number L,
from (C.11) we can write to the most general Dirac and Majorana mass term
 
1 νL mL
M mD
LD+M = − n̄L M ncL + h.c. with nL = and M = .
2 νRc mD mR
M

We denoted nL as it is a 2 dimensional vector (in the mass basis) only formed by

left-handed spinors. Noticing that a left-chiral spinor cannot mix with another left-
chiral spinor (νL νL = 0), we can write

1
LD+M = − n̄ M n (C.14)
2

with n = nL + ncL which is a Majorana vector in the mass basis, which components
are the sum of left and right chirality spinors. So we have reduced the problem of
νL + νLc
finding eigenstates to the one of diagonalizing M into the basis n = =
νR + νRc

n1
. M being a complex symmetrical matrix, it can be diagonalized with the help
n2
of one unitary matrix U. We have M = U mU T with

m1 0
m= . (C.15)
0 m2

We skip the computation of the eigenvalues which are obtained solving the
characteristic equation for the matrix M: |M − λI d| = 0, given 2 values for the
second order equation4 for λ, m1 , and m2 :

1
m1 = M + mL ) +
(mL R
M + mM ) + 4(mD − mM mM )
(mL R 2 2 L R
2

1
m2 = M + mL ) −
(mL R
M + mM ) + 4(mD − mM mM ) . (C.16)
(mL R 2 2 L R
2

4 Notice that one eigenvalue is negative, which is not an issue as the field can be redefined by a

global phase transformation.

478 C Neutrino Physics

C.4.3 The Specific Case mL

M
=0

C.4.3.1 The Eigenvalues

We have seen in the preceding section that the Majorana mass term for the left-
handed neutrino should appear only at two loops order, or generated by dimension
6 operator, and so should naturally be considered as much smaller than the right-
handed Majorana term mR M which can be naturally present in the Lagrangian by
gauge invariance. The mass values m1 and m1 then becomes

1 R
m1 = m + M ) + 4mD  mM
(mR 2 2 R
2 L
 m2D
1 R
m2 = m − (mR ) 2 + 4m2  , (C.17)
2 L M D
mR
M

where we have developed at the first order in mD /mR M , supposing mD mRM . This
approximation is justified in a lot of extensions of the Standard Model (but not all)
and can be understood as the fact that the Majorana mass term mR M being a free
mass term, its natural scale should be the GUT √ or PLANCK scale (or at least an
intermediate scale) whereas mD = yLR vH / 2 (vH = 246 GeV being the Higgs
field vev and yLR the Yukawa coupling between νL and νR , see below) is boded by
the electroweak scale.
We then have a clear view on the famous see-saw mechanism (and on its name).
Indeed, a quick glance at Eq. (C.17) shows us that a hierarchy between mD and mR M
will naturally drive one eigenvalue toward zero, keeping the second one relatively
high. In other words, after diagonalization, the left-handed part of the multiplet (νL )
will rotate and acquire a small admixture of νR which will generate its mass term,
proportional to the admixture.

C.4.3.2 The Eigenvectors

If finding the mass terms m1 and m2 is an easy task, one should be careful when
treating the compositions on the new states N1 and N2 , mass eigenstates of the
Lagrangian and thus, the physical states, the only one having really a meaning in
any observable computation. To find them, one has to write down the equations
   
m1 0 N1 0 mD n1
(N̄1 N̄2 ) = (n̄1 n̄2 ) (C.18)
0 m2 N2 mD mR
M n2

with
   
N1 n1 cos θ − sin θ n1
= Rθ = . (C.19)
N2 n2 sin θ cos θ n2
C.4 The See-Saw Mechanism 479

Solving the Eq. (C.18) with the definition (C.19) we obtain

2mD mD
tan 2θ = R
⇒ θ  sin θ  R (C.20)
mM mM

which gives
  
N1 n1 − θ n2 νL + νLc − θ (νR + νRc )
 = . (C.21)
N2 n2 + θ n1 νR + νRc + θ (νL + νLc )

As we can notice, N1 and N2 are Majorana particles (Ni = Nic ). N1 represents the
physical neutrino of the Standard Model, whose mass is constraint to be below 1
eV, whereas N2 would be the (non-observed) heavy states responsible for the see-
saw mechanism.

C.4.4 An Application: Coupling to a Scalar Field (Majoron)

It is always possible to work in a dynamical framework where the masses are given
by the breaking of a symmetry “a la Higgs,” with a new heavy scalar field (majoron
field S) whose vev generates mass to the right-handed neutrino. The Lagrangian can
be written, after the breaking of symmetry (assuming typical Higgs potential in the
heavy scalar sector)

h1 yLR
L = − √ ν̄Rc (vS + S)νR − √ ν̄L (vH + h)νR + h.c.
2 2
h1 yLR
= − √ (ν̄Rc + ν̄R )(vS + S)(νR + νRc ) − √ (ν̄L + ν̄Lc )(vH + h)(νR + νRc )
2 2 2
yLR
− √ (ν̄R + ν̄Rc )(vH + h)(νL + νLc ), (C.22)
2 2

that we can express as function of the physical states N1 and N2

h1 h1 θ 2 h1 θ
L = − √ S N̄2 N2 − √ S N̄1 N1 + √ S(N̄1 N2 + N̄2 N1 ) (C.23)
2 2 2
yLR yLR θ
− √ h(N̄1 N2 + N̄2 N1 ) + √ (N̄2 N2 + N̄1 N1 ) − m1 N̄1 N1 − m2 N̄2 N2
2 2 2 2

y 2 v2 √
with m1  √LR H and m2  2h1 vS .
2 2h1 vS
Such a dynamical model has a lot of interesting characteristics, like for instance the
decay of the heavy scalar S into two (very energetic) light states N1 which can be
measured by the Icecube telescope for instance.
Useful Statistics
D

D.1 5σ and p-Value

D.1.1 5σ

An important example of constructing a confidence interval is when a single variable

x follows a Gaussian distribution. This happens frequently when one needs to work
with a great number ( 100) of events. We then define the probability that a variable
appear within an interval ±δ
 μ+δ 
1 −(x−μ)2 /2σ 2 δ
1−α = √ e dx = erf √ , (D.1)
2πσ μ−δ 2σ

where μ is the mean of the variable x, and σ its variance. If one has to deal with a
few number of events (dozens), it is more precise to use the Poisson distribution,
n e−ν
f (n; ν) = ν n! with n = 1, 2, 3.. and ν > 0. We illustrate the distribution
behavior and its σ limits in the Fig. D.1 and Table D.1. As a remark, the air below
the Gaussian at 5σ correspond to a “null hypothesis” (see definition of the p-value
below) of ∼ 6×10−7 (see table) so a probability ±3×10−7. In other words, an event
observed at 5 σ above a predicted background would have one chance on 3 millions
to be part of the background (and thus 99.99997% of chance to be a “something
else” beyond the Standard Model). We will discuss this case more in detail when
examining the look-elsewhere effect.
Indeed, as we will see later, a 5σ confidence level—expressed in terms of Higgs
search for instance—discovery corresponds to the condition that the probability
of finding an excess rate of the measured size that is caused by Standard Model
particles without Higgs particle is lower than 3 × 10−7 . Setting the limit for
calling the collected data a discovery is, of course, a matter of convention. It is
based on a trade-off between the advantage of calling a viable scientific claim
empirically well-established and the potential damage of endorsing a false scientific

© Springer Nature Switzerland AG 2021 481

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
482 D Useful Statistics

Fig. D.1 Gaussian distribution as function of standard σ deviations (Wolfgang Kowarschick,

under the Creative Commons Attribution-Share Alike 4.0 International license)

Table D.1 Area of the tails Confidence level δ α Probability

α outside ±δ from the mean
of a Gaussian distribution 1σ 0.3173 68.27%
2σ 4.55 × 10−2 95.45%
3σ 2.7 × 10−3 99.73%
4σ 6.3 × 10−5 99.9937%
5σ 5.7 × 10−7 99.99994%
6σ 2.0 × 10−9 99.9999998%

claim. In many scientific fields, a 3σ effect, that is a probability of less than

0.15% that the null hypothesis holds, is considered sufficient for establishing a
phenomenon. In particle physics, the discovery of a new particle is taken to be of
very high importance and is used in the analysis of the background in all future high
energy scattering experiments. Therefore, the risk of erroneously acknowledging a
discovery of a particle should be kept particularly low and a stronger criterion seems
advisable. One should also take into account the look-elsewhere effect that can
reduce considerably the 5σ effect (see below). Indeed, historically the 5σ limit was
established based on largely pragmatic considerations. While statistical fluctuations
at a 4σ level did and do occur from time to time in high energy physics experiments,
no 5σ signal in a particle experiment has up to this point ever turned out to be a
fluctuation. A 5σ limit therefore seemed plausible simply based on historical record.
The fact that a 4σ fluctuations do occur can be statistically explained based on the
number of experiments that are carried out in conjunction with the size of the look-
elsewhere effects which usually apply in those contexts.
D.2 Systematics vs Statistics 483

D.1.2 p-Value

The p-value is defined as the probability of obtaining a statistic at least as extreme

as the one actually observed assuming that the null hypothesis is true. The smaller
is the p-value, the less probable the “null hypothesis” is and so the results are
“probably” a signal above a background. In other words, if the p-value computed
by statistical tools (given by experimentalists) is below 0.05 or 0.01 (depending on
the significance one is asking for), the researcher will reject the “null hypothesis”:
the observed result would be highly unlikely under the null hypothesis. The p-value
can be seen as the probability that “something” observed happens assuming random
coincidence.

D.2 Systematics vs Statistics

Any experimental analysis has to deal with two kind of uncertainties: the statistical
uncertainties and systematics ones. In many cases, the systematics uncertainties
are comparable to the statistical ones. However, consistent definition and practice
are elusive which leads to confusion and in some cases incorrect interference. We
will try to give some practical definitions in this section to help understand the
importance of each of them.
Examples of statistical uncertainties include the finite resolution of an instrument,
the Poisson fluctuations associated with measurements involving finite sample sizes,
and random variations in the system one is examining. Systematic uncertainties,
on the other hand, arise from uncertainties associated with the nature of the
measurement apparatus, assumptions made by the experimenter, or the model used
to make inferences based on observed data. Common examples of systematic
uncertainty include uncertainties that arise from the calibration of the measurement
device, the probability of detection of a given type of interaction (often called the
“acceptance” of the detector), and parameters of the model used to make inferences
that themselves are not precisely known.
Let us illustrate this correlation in a typical example, the measurement of the W
boson cross section in hadronic colliders. The production rate of the W boson σW
in pp̄ collisions is theoretically known in the Standard Model. This cross section
is also measured by a number of observed counts Nc (given some topologies and
specific cuts depending on the channels analyzed by the experimenters), estimating
the number of “background” events in this sample from other process Nb , estimating
the acceptance of the apparatus including all selection requirements used to define
the sample of events , and counting the number of pp̄ annihilation, L. The cross
section for W boson production is then

Nc − Nb
σW = . (D.2)
L
484 D Useful Statistics

For instance, a measurement performed by CDF collaboration where the transverse

mass of a sample of candidate W → νe decays would be quoted as

σW = 2.64 ± 0.01(stat) ± 0.18(syst) nb, (D.3)

where the first uncertainty reflects the statistical uncertainty arising from the
size of the candidate sample (approximately 38,000 candidates) and the second
uncertainty arises from the background subtraction in Eq. (D.2). We can estimate
these uncertainties as

1
σst at = σ0
Nc

  
δNb 2 δ 2 δL 2
σsyst = σ0 + + . (D.4)
Nb  L

where the three terms in σsyst are the uncertainties arising from the background
estimate δNb , the acceptance δ, and the integrated luminosity δL. The parameter
σ0 is the measured value.
The “paradox” that makes the definition of statistics and systematics so incertain
is that in the same sample of events, experimenters also measured the acceptance 
through the Z boson observation from its decay into electron. The level of accuracy
on  becomes also a stochastic variable depending on the size of the sample. One
can then also consider δ as a statistical uncertainty and redefined

 2
1 δ
σst at = σ0 +
Nc 

 2  2
δNb δL
σsyst = σ0 + (D.5)
Nb L

resulting in a different assignment of statistical and systematics uncertainties. This

matter in a sense that as a definition we took for “systematic” uncertainty, it should
not depend on the size of the sample used to make the observation. It is supposed
to come from an external measurement like a calibration for instance, and does not
scale with the sample size. In the specific case of the measurement of the W cross
section,  does not fill this requirement. We can include in this family of systematics
any degree of freedom which depends stochastically on an external measurement
(which means any observable which is dominated by statistics uncertainties). If a
threshold or an energy calibration (systematic) is dominated by a measurement from
another experiment dominated by statistics, these uncertainties could be considered
in fact as statistical uncertainties.
The background systematics however has a different nature. Indeed, the back-
grounds events are gluon or quarks, produced in the collision but which fake an
D.3 Look-Elsewhere Effect (LEE) 485

electron or a muon. A reliable estimate of this background is difficult to make

from first principle, as the rate of such QCD events is many orders of magnitude
larger than the W boson cross section. The way to reject background event is to
discriminate between events having high missing transverse energy (taken by the
neutrino in the W decay) and what experimenters called the “isolation” value:
events from QCD background will have more particles produced at proximity of
the electron or muon candidate. We remember of course also the “faster than
light” neutrino results of OPERA that was claimed in 2011. This “faster-than-light”
neutrino experiment suffered from a systematic error that affected all the data; faulty
cables consistently gave the researchers bad readings. No matter how many times
physicists repeated the experiments, they would get the same yet inaccurate results.

D.3 Look-Elsewhere Effect (LEE)

D.3.1 Generality

The look-elsewhere effect is common in scientific analysis of experiments studying

models with a lot of degrees of freedom. Indeed, the more degrees of freedom a
theory has, the more likely it is to make a statistically significant observation. This
was very useful for the hunting of the Higgs boson in 2011.
What is the look-elsewhere effect in short? It is the main reason why the strong 5σ
approach is applied in particle physics. Indeed, “5σ ” corresponds to 3 events in 10
millions. But having such a signal which can be seen for instance as a bump at a
given energy E in a diet spectrum corresponding to and invariant mass M = E.
Naively speaking, suppose that such a bump is such high above the background
that one has 1 chance on ten thousands for it to be a statistical fluctuation: it seems
natural to claim for a potential discovery, corresponding to the production of a new
particle of mass M. Now, suppose the study has been made in an energy range
between 1 to 100 GeV, and the bump has a width of 2 GeV (see Fig. D.2). To know
what is the real chance to observe such a bump at an energy E, say 60 GeV, one
has to convolute the probability of observing such event in the 60 GeV bin to the
general probability of having such a bump in another bin. In other words, if we do to
“look-elsewhere” the probability of such event to be the background is only 1/10000
(and so we can consider it as an “event”); however, if one considers that this can
statistically happen in any bin, we have 50 chances (100 GeV/2 GeV) that this event
can occur in another bin which mean a total probability of 50 × 1/10,000 = 1/200
which cannot be considered anymore as a “discovery” signal. Put in another way,
the probability of finding a signal exactly at 60 ± 2 GeV is one chance on 10,000,
whereas the probability of finding a signal “somewhere between 1 and 100 GeV” is
one chance on 200. In general, a good rule of thumb is the following: if the signal
has a width W , and if one examines a spectrum spanning a mass range from M1
to M2 , then the “boost factor” due to the LEE is (M2 − M1 )/W . This may easily
reach a factor 10 or 100 depending on the detail of the searches. Of course, this is a
simplified version of the LEE effect because I have supposed that all 50 possibilities
486 D Useful Statistics

Number of events
(arbitrary scale)

Observed events

5

Predicted background

Energy scale
(arbitrary scale)

Number of events
(arbitrary scale)

4

Predicted background

Energy scale
(arbitrary scale)

Fig. D.2 Illustration of the look-elsewhere effect: a 5σ signal observed above the background in
a specific energy range (left figure) can be reduced to a 4σ signal (or even lower) if one take into
consideration that, looking elsewhere, this signal has also a probability of appearing somewhere
else (right figure). If one knows exactly the energy bin where the signal should appear (from other
considerations or other experiments), this signal is effectively at 5σ of confidence level
D.3 Look-Elsewhere Effect (LEE) 487

have the same chance to happen (a flat distribution), which is never the case in any
complex studies. That explains the concrete difficulties to take into account this
effect in modern analysis (especially at the LHC).
One can also find effects of such LEE effect in the common life: a Swedish study
in 1992 tried to determine whether or not power lines caused some kind of poor
health effects. The researchers surveyed everyone living within 300 meters of high-
voltage power lines over a 25-year period and looked for statistically significant
increases in rates of over 800 ailments. The study found that the incidence of
childhood leukemia was four times higher among those that lived closest to the
power lines, and it spurred calls to action by the Swedish government. The problem
with the conclusion, however, was that they failed to compensate for the look-
elsewhere effect; in any collection of 800 random samples, it is likely that at
least one will be 3 standard deviations above the expected value, by chance alone.
Subsequent studies failed to show any links between power lines and childhood
leukemia, neither in causation nor even in correlation.

D.3.2 Applying the LEE Effect to the Higgs Discovery

Let us now look specifically at the Higgs detection at the LHC. The two main
channels analyzed by the ATLAS and CMS collaborations are the H → γ γ and
H → 4 leptons final states, mainly because of the low background level of these
two final states. In December 2011, the significance of the entire excess rate of both
event types was close to 4σ . This corresponds to a probability close to 3 × 10−5
that the observed “signal” would arise as a fluctuation of Standard Model physics
without a Higgs particle. This new boson was looked over a range of about 80 energy
bins at the LHC. This corresponded to a probability of about 2.5 × 10−3 that an
oscillation of the size of the December data could occur at the LHC experiment. This
probability does not even amount a 3σ effect and therefore was clearly insufficient
for establishing the existence of a new particle. The data of July 2012 then had a
significance above the 5σ level, which corresponded to a probability of the null
hypothesis of less than 3 × 10−5 after taking into account the LEE effect. This was
sufficient for declaring the data a discovery.
What happened in December 2011, however, is that a lot of theoreticians
(the author of the present book included) considered that a 4σ limit can be
considered seriously as the look-elsewhere effect can be reduced by “theoretical”
or “philosophical” argument. Indeed, there was strong evidence that (1) the Higgs
field should exist and (2) that this mass range due to perturbative effects should lie
below 1 TeV. Indeed, in this sense one can see the limit of the look-elsewhere effect,
and its part of arbitrariness. Up to which scale one should take the possibilities of
statistical fluctuations? Which kind of data from other experiments, or theoretical
arguments should we take into account in the analysis? Or should we play the
role of completely blind physicists, unaware of other experiments or theoretical
constructions and history? For instance, before the Higgs signal of December 2011,
there was other measurements from LEP or Tevatron, limiting the Higgs mass in
488 D Useful Statistics

the window 115 to 141 GeV, reducing thus the analysis to 26 bins, increasing at the
same time the significance by a factor 3, the probability of a statistical fluctuation
being then below 10−3 . But of course, that one should be taken into account if one
believes that the Higgs exists.

D.3.3 Applying the LEE Effect to the Dark Matter Searches

In April 2012 was reported “A tentative gamma-ray line from Dark Matter anni-
hilation at the Fermi Large Area Telescope” [9] with a statistical significance of
4.6σ at 130 GeV. When taking into account the look elsewhere effect, the signal
dropped down to 3.2σ i.e. taking into account the fact that one should search for
a line in multiple regions of the sky and at multiple energies. In May 2013 the
Fermi LAT collaboration has reported their “Search for Gamma-ray Spectral Lines
with the Fermi Large Area Telescope and Dark Matter Implications”, computing
the statistical significance of this line feature at 130 GeV at a lower statistical
significance of 3.3σ , and only 1.6σ after taking the look-elsewhere effect into
account. As we can see, the LEE effect was important in this case, as it seemed
finally that such signal was not due to dark matter annihilation.

D.4 Bayesian vs Frequentist Approach

Since several decades, the fight exists between people sustaining the frequentist
or Bayesian approach. In the supersymmetric communities several authors used
to claim in every conferences that the Bayesian approach was the one to use in
any supersymmetric study. We will remind you in this section the fundamentals of
Bayesian approach.
To understand the Bayesian approach, we will take a very concrete example.
Suppose a woman believes she may be pregnant after a single sexual encounter,
but she is unsure. So she takes a pregnancy test that is known to be 90% accurate
(meaning it gives positive result to positive cases 90% of the time) and the test
produces a positive result. Ultimately, she would like to know the probability she
is pregnant, given a positive test (p(preg|test +)); however, what she knows is the
probability of obtaining a positive test result if she is pregnant (p(test + | preg)), and
she knows the result of the test. Bayes’ theorem offers a way to reverse conditional
probabilities, and, hence, provides a way to answer this question.
Bayes’ original theorem applied to point probabilities. The basic theorem states
simply:

p(A|B)p(B)
p(B|A) = , (D.6)
p(A)
D.4 Bayesian vs Frequentist Approach 489

which can be translated by the (conditional) probability of having B given A is equal

to the (conditional) probability of having A given B, multiplied by the (marginal)
probability of having B divided by the (marginal) probability of having A.1
Proof: we understand easily that the probability of having A and B is p(A, B) =
p(A|B)p(B). With the same reasoning, we also can write that p(B, A) =
p(B|A)p(A). p(A, B) = p(B, A) one then can write

p(B, A) = p(A, B) ⇒ p(B|A)p(A) = p(A|B)p(B) (D.7)

which gives the Bayes’ result.

Now, back to our example, suppose that, in addition to the 90% accuracy rate, we
also know that the test gives false positive results 50% of the time. In other words,
in cases in which a woman is not pregnant, she will test positive 50% of the time.
Thus p(test+|not preg) = 0.5. With this information, combined with some “prior”
information concerning the probability of becoming pregnant from a single sexual
encounter, Bayes’ theorem provides a prescription for determining the probability
of interest.
The prior information we need, p(B) = p(preg), is the marginal probability
of being pregnant, not knowing anything beyond the fact that the woman has
had a single sexual encounter. This information is considered prior information,
because it is relevant information that exists prior to the test. We may know
from previous research that, without any additional information (e.g. concerning
the date of last menstrual cycle), the probability of conception for any single
sexual encounter is approximately 15%. With this information, we can determine
p(B|A) ≡ p(preg|test+) as

p(test + |preg)p(preg)
p(preg|test+) =
p(test + |preg)p(preg) + p(test + |not preg)p(not preg)
(D.8)

which gives

0.90 × 0.15
p(preg|test+) = = 0.241. (D.9)
0.90 × 0.15 + 0.50 × 0.85

Thus, the probability that the woman is pregnant, given the positive test is only
0.241. Using Bayesian terminology, this probability is called “posterior probability”
because it is the estimated probability of being pregnant, obtained after observing
the data (the positive test). We can be surprised of such a low value given the “90%”
accuracy of the test. This comes mainly from the 50% of probability (high) to obtain
a positive test when the person is not pregnant.

anyone still believes that p(A|B) = p(B|A) [probability of A given B = probability of B given
1 If

A], remind them that the probability of being pregnant, given that the person is female is ∼ 3%,
while the probability of being female, given that they are pregnant, is considerably larger.
490 D Useful Statistics

If the woman is aware of these limitations, she can redo the test. But now, she
can use the “updated” probability of being pregnant (p = 0.241) as the new p(B);
that is the prior probability for being pregnant has now been updated to reflect the
results of the first test. If she repeats the test again and again observes a positive
result, her new “posterior probability” of being pregnant is

0.90 × 0.241
p(preg|test+) = = 0.364. (D.10)
0.90 × 0.241 + 0.50 × 0.759

This result is still not very convincing evidence that she is pregnant, but if she
repeats the test again and finds positive result, her probability increases to 0.507 up
to 0.984 at the tenth positive attempt. From a Bayesian perspective, we begin with
some prior probability for some event, and we update this prior probability with new
information to obtain a posterior probability. The posterior probability can then be
used as a prior probability in a subsequent analysis. From a Bayesian point of view,
this is an appropriate strategy for conducting scientific research.
We can then understand how one should apply such statistical analysis in
supersymmetric models. Indeed, before that the LHC gave any limits, the aim was
to know how a point in supersymmetric parameter space was “probable” or not.
Usually, frequentists (the author of the present book included) make a scan on the
par mater space, and compute how far, or near (from a frequentist approach, i.e.
2 or 3σ ), this point is from the data measured. On the other hand, the bayesian
researchers would take the complementary approach: given experimental data, what
is the probability that a point in the parameter space is around the data.
Numbers
E

I have included in this appendix the main tables, conversion factors, and formulae I
concretely regularly needed in my cosmological/astrophysical/particle computation.
I hope you will enjoy using them as much as we enjoyed (sic!) writing them.

E.1 Useful Formulae

E.1.1 Cosmology

T0 = 2.349 × 10−4 eV = 2.725 0 K; G = 1/MP2 l ; MP l = 1.22 × 1019 GeV

 
8πG √ 4π 3 T 2 T2 T2
H (T ) = ρ(T ) = gρ  1.66gρ1/2 = 0.32gρ1/2
3 45 MP l MP l MP
 g 1/2 T 2 1
ρ
= π MP = √ MP l
90 MP 8π
3H02 ρA
ρc0 = = 10−5 h2 GeV cm−3 = 2 × 10−29h2 g/cm3 ; A =
8πG ρc0
n0γ = 411 cm−3 ; ργ0 = 2.62 × 10−10 GeV/cm3 ; Eγ0  = ργ0 /n0γ = 6 × 10−4 eV;

Tν0 = 1.95 0 K = 1.68 × 10−4 eV; nν (Tν0 ) = 112 cm−3 ; gρ0 = 3.36
2π 2
s(T ) = gs (T )T 3 ; gs0 = 3.91 ; s(T0 ) = 2.2 × 10−38 GeV3 = 2909 cm−3
45
 
ρc0 100 GeV 135 × 10−3 H02 MP2
Y (T0 ) = =
ms0 0.1 m 16π 3 gs0 T03

© Springer Nature Switzerland AG 2021 491

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
492 E Numbers


−10 1 GeV
Y (T0 )  5 × 10
m
2  
dm h ndm (T ) Mdm
= 5.9 × 10 6
[for T  EWSB]
0.1 T3 1 GeV

gA gA
fA = ; nA (T ) =
fA (p)d 3 p
e(E−μA )/ kT ± 1 (2π)3

3 ζ(3)
A
nEQ (T m) gA T 3 (fermion) boson
4 π2

7 π2
A
ρEQ (T m) gA T 4 (fermion) boson
8 30

mT 3/2 −m/T
nA
EQ (T m) = gA e ; ρEQ
A
(T m) = m × nA
EQ (T m)
2π

nχ (T0 )mχ nχ (T0 )mχ 1.6 × 108 nχ (T0 )  mχ 
χ = = = ,
ρc 1.05 × 10−5 h2 GeV.cm−3 h2 T03 1 GeV

where we used 1 cm3 = 1700

(see table below). We then obtain
T03

 
gs0 nχ (TRH )  mχ 
χ h = 1.6 × 10
2 8
(E.1)
gsRH 3
TRH 1 GeV

where we used
3
aRH
nχ (T0 ) = nχ (TRH ) × and gs0 a03 T03 = gsRH aRH
3 3
TRH .
a03

with gs0 = 3.91 and gsRH = 106.75 for TRH > mt .

E.1.2 Particle Physics

√
H  = v/ 2 ⇒ v = 246 GeV (174 GeV if H  = v)

gv g 2 + g 2 g MW
MW = ; MZ = v= v ; MZ =
2 2 2 cos θW cos θW
σEW  |M|2 /64π 2 s  10−9 GeV−2
E.2 Tables 493

g g
cos θW =  ; sin θW =  ; e = g sin θW
g + g 2
2 g 2 + g 2
Aμ = cos θW Bμ + sin θW Wμ3 ; Zμ = − sin θW Bμ + cos θW Wμ3
√
2 g2
GF = 2
 1.166 × 10−5 GeV−2
8 MW
1 1
Wμ+ = √ (Wμ1 − iWμ2 ); Wμ− = √ (Wμ1 + iWμ2 );
2 2
σT = 6.65 × 10−29 m2

g2
Perturbativity: α
4π = (4π)2
< 1 giving g < 4π

E.2 Tables

Fig. E.1 Periodic Table of the Elements (Double sharp, licensed under the Creative Commons
Attribution-Share Alike 4.0 International license)
494 E Numbers

Quantity Symbol Value

Speed of light in vacuum c 299 792 458 m/s
Planck constant h 6.626 ×10−34 J s = 4.14 × 10−24
GeV
Planck constant reduced h̄ ≡ h/2π 1.055 ×10−34 J s
= 6.58 × 10−22 MeV s
Planck Mass MP l 1.2211 × 1019 GeV =
2.1768 × 10−5 g
√
Reduced Planck Mass MP = MP l / 8π 2.43 × 1018 GeV
Planck time tP l 5.4 × 10−44 s
Planck size lP l 1.6 × 10−35 m
Conversion factor h̄c 197.3 MeV fm
1 GeV = 1.6 × 10−10 J
Boltzmann constant kB 1.38 × 10−23 J K−1 =
8.617 × 10−5 eV K−1
Weinberg angle sin θW / sin2 θW 0.481/0.23122
Hypercharge coupling g 0.356
Weak coupling g 0.65
Strong coupling g3 1.126
Electromagnetic coupling (MZ ) e = g(M
 Z ) sin θW 0.313
Electromagnetic coupling (me ) e= 4π
137 0.303
α1−1 (MZ ) (g1 /4π)−1
2 58.98 ± 0.04
α2−1 (MZ ) (g22 /4π)−1 29.57 ± 0.03
α3−1 (MZ ) (g32 /4π)√−1 8.40 ± 0.14
2
Fermi constant GF = 82 Mg 2 1.166 × 10−5 GeV−2
W
Typical EW cross section σEW 10−9 GeV−2
Electron charge e 1.6 × 10−19 C
Electron mass me 0.511 MeV/c2 = 9.11 ×10−31 kg
Proton mass mp 938.27 MeV/c2 = 1.672 ×10−27
kg
Neutron mass mn 939.57 MeV/c2 = 1.675 ×10−27
kg
Electron charge e 1.6 ×10−19 C
e2
Electron radius r0 = 1
4π0 me c2 2.8 × 10−15 m
Vacuum permittivity 0 8.854 ×
10−12 F m−1 [A2 s4 kg−1 m−3 ]
Vacuum permeability μ0 = 1/0 c2 4π ×10−7 = 12.566×10−7 N A−2
Gauss G 1 Gauss = 10−4 Tesla
Muon mass mμ 105.7 MeV/c2
Pion mass mπ 0 135 MeV/c2
Tau mass mτ 1777 MeV/c2
Deuteron mass md 1876 MeV/c2
E.2 Tables 495

Quantity Symbol Value

Higgs vev H  = √v H  = 174 GeV; v = 246 GeV
2
Higgs mass/width mh / h 125 GeV/c2 / 4.07 MeV/c2
W mass/width MW = g2 v / W 80.385 ± 0.015 GeV/c2 /
2.085 ± 0.042 GeV/c2
MW
Z mass/width MZ = cos θW / Z 91.1876 ± 0.0021 GeV/c2 /
2.4952 ± 0.0023 GeV/c2
Up quark mass mu 1.5–4.5 MeV/c2
Down quark mass md 5–8.5 MeV/c2
Strange quark mass ms 80–155 MeV/c2
Charm quark mass mc 1–1.4 GeV/c2
Bottom quark mass mb 4–4.5 GeV/c2
Top quark mass mt 175 GeV/c2
Conversion K/GeV 1K =0.862 × 10−4 eV
Conversion GeV−2 /cm3 s−1 σv 1 GeV−2 =1.2 × 10−17 cm3 s−1
Conversion GeV−1 /L 1 GeV−1 = 1.9733 × 10−14 cm
Conversion GeV−1 /t 1 GeV−1 = 6.67 × 10−25 s
Conversion GeV−2 /barn 1 GeV−2 = 4 × 10−28 cm2 =
4 × 10−4 barn
Conversion cm2 /barn 10−24 cm2 = 1 barn
Conversion GHz/GeV 1GHz =4 × 10−6 eV
Conversion GeV/kg 1GeV =1.78 × 10−27 kg
Conversion barn/cm2 1 barn = 10−24 cm2 / 1 pbarn =
10−36 cm2
Conversion cm2 g−1 /GeV−3 1 cm2 g−1 = 4.4 × 103 GeV−3
Gravitational constant G 6.7 × 10−11 m3 kg−1 s−2
6.7 × 10−39 h̄c(GeV/c2 )−2
Present Hubble expansion rate H0 100 h km/s/Mpc =2.13 × 10−42 h
GeV
Hubble time H0−1 3.08 × 1017 h−1 sec =
9.77 × 109 h−1 yrs
Hubble distance cH0−1 2998 h−1 Mpc = 9.25 × 1027 h−1
cm
Reduced Hubble expansion rate h H0 /(100 km/s/Mpc)
Normalized Hubble expansion rate h 0.677 ± 0.002
Horizon size LH  14 Gpc
Critical density ρc 1.05 × 10−5 h2 GeV
cm−3 = 1.8 × 10−32 h2 kg cm−3
8 × 10−47 h2
GeV4  4 × 10−47 GeV4
Age of the Universe t0 13.8 Gyr = 4.3 × 1017 s.
= 6.6 × 1041 GeV
Mean free path of photon λ0 1026 m
CMB temperature T0 2.725 ±0.001 K = 2.349 × 10−4
eV
(CMB temperature)3 T03 1700 cm−3
496 E Numbers

Quantity Symbol Value

Zeta Riemann ζ(3) 1.20206
Parsec pc 1 pc = 3.086 × 1016 m = 3.262
light-years
Solar velocity with respect to 371 km/s
CMB
Jansky Jy 10−26 W m−2 Hz−1
Watt W J.s−1 =kg m2 s−3
Solar mass M 1.98 × 1030 kg = 1.1 × 1057 GeV
Solar radius R 696 340 km
Mass-to-light ratio M /L 5133 kg/W
1 year 3.15 × 107 s
Light-years lyrs 1 lyr = 9.46 × 1015 m
Binding energy of the Hydrogen BH 13.6 eV
Binding energy of the deuterium BD 2.2 MeV
Deuterium mass mD 1877.62 MeV

Quantity z Energy Temperature time

0.862 × 10−4 eV 1K 1017 s
Nucleosynthesis 0.1 MeV 109 K 100 s
Matter domination 3400 0.81 eV 9000 K 100,000 years
Decoupling (atoms formation) 1100 0.3 eV 3000K 380,000 years
Dark energy domination 0.66 3.9 × 10−4 eV 4.565 K 1010 years
Actual 0 2.35 × 10−4 eV 2.725K 13.8 × 109 years

Cross section Value (in pb)

σe+ e− →ννγ 2.7 × 10−3

Field q = T 3 + Y/2 T3 Y
uL +2/3 +1/2 +1/3
dL −1/3 −1/2 +1/3
uR +2/3 0 +4/3
dR −1/3 0 −2/3
νeL 0 +1/2 −1
eL −1 −1/2 −1
eR −1 0 −2
NR 0 0 0
H 0 −1/2 +1
References 497

Quantity Symbol Value

Hubble Constant H0 ; h 67.77 ± 4 km/s/Mpc; 0.677 ± 0.002
Total density t 1.02 ± 0.02
Critical density ρc0 1.05 × 10−5 h2 GeV cm−3 ; 8 × 10−47 h2 GeV4
Dark matter energy density 0
ρcdm 1.28 × 10−6 GeV cm−3
Dark energy density ρ0 3.32 × 10−6 GeV cm−3
Dark Energy density ; h
2 0.689 ± 0.006; 0.316 ± 0.003
Total matter density m; m h2 0.311 ± 0.006; 0.142 ± 0.0009
CDM density cdm ; cdm h
2 0.265 ± 0.002; 0.119 ± 0.001
Baryonic density b; bh
2 0.050 ± 0.0002; 0.0224 ± 0.0001
Photon density γ; γ h 2 5.5 × 10−5 ; 2.47 × 10−5
 1.4 × 10−3 ; 93.14ν  93.14 = 6.44 × 10−4
2 m 0.06
Neutrino density ν; ν h
Radiation density R R = γ + ν = 1.68 γ
Curvature density K K = 0.0007 ± 0.0019
Baryons to photons ratio ηb 6.12 × 10−10
Age of the Universe t0 13.787 ± 0.020 Gyr
Size of the Universe (radius) R0 ∼ 46 light years (14 Gpc)

References
1. A. Denner, H. Eck, O. Hahn, J. Kublbeck, Nucl. Phys. B 387, 467–481 (1992). https://doi.org/
10.1016/0550-3213(92)90169-C
2. R. d’Inverno, Introducing Einstein’s Relativity (Clarendon, Oxford, 1992), 383 p.
3. C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, R.P. Hudson, Phys. Rev. 105, 1413–1414
(1957). https://doi.org/10.1103/PhysRev.105.1413
4. T. Sjostrand, S. Mrenna, P.Z. Skands, JHEP 05, 026 (2006). https://doi.org/10.1088/1126-6708/
2006/05/026. [arXiv:hep-ph/0603175 [hep-ph]]
5. J. Hisano, S. Matsumoto, O. Saito, M. Senami, Phys. Rev. D 73, 055004 (2006). https://doi.org/
10.1103/PhysRevD.73.055004. [arXiv:hep-ph/0511118 [hep-ph]]
6. V.N. Gribov, B. Pontecorvo, Phys. Lett. B 28, 493 (1969). https://doi.org/10.1016/0370-
2693(69)90525-5
7. G. Bellini et al. [BOREXINO], Nature 512(7515), 383–386 (2014). https://doi.org/10.1038/
nature13702
8. R.D. Peccei, H.R. Quinn, Phys. Rev. Lett. 38, 1440–1443 (1977). https://doi.org/10.1103/
PhysRevLett.38.1440
9. C. Weniger, JCAP 08, 007 (2012). https://doi.org/10.1088/1475-7516/2012/08/007.
[arXiv:1204.2797 [hep-ph]]
Index

Symbols Boltzmann equation (exceptions), 210

Urad , 310 Boltzmann equation (general), 194
ηb , 149 Bremsstrahlung radiation (non–relativistic),
ηb , 48, 186 299
Neff , 179 Bremsstrahlung radiation (relativistic), 312
ρM , 148 Bullet cluster, 14
ργ , 148
ρν , 148, 151
2-point correlation function, 360 C
3-body decay, 444 CGS units, 287, 331
3-body phase space, 427 Charge conjugate operator, 412
5σ , 481 Charge operator, 452
Chemical decoupling, 160
Chiral anomaly, 461
A Christoffel symbol, 374
Adiabatic compression, 319 Cluster of galaxies, 9
Adiabatic transformations, 387 CMB anisotropies, 231
Age of the Universe, 48 Coannihilation, 213
Anderson, Carl, 299 Codistance, 50
Anisotropies (CMB), 231 Cold dark matter, 353
Annihilation (formulae), 429 Comoving coordinate, 28
Atmospheric neutrino, 473 Comoving distance, 50
Avogadro number, 243, 309 Compton scattering, 183, 310
Axion, 461 Coordinates (galactic), 279
Axion (dark matter), 217 Correlation function, 360
Cosmological constant, 371
Cosmological constant era, 148
B Cosmological constant problem, 153
Baryogenesis, 186 Cosmological microwave background (CMB),
Baryonic oscillation, 355 19
Baryon to photon ratio, 186 Cosmological principle, 33
Bayesian statistic, 488 Coulomb scattering, 293
BBN, 185 Coulomb scattering (relativistic), 308
BBN (historical way), 187 Couplings, 402
Beaming effect, 304 Covariant derivative, 375
Bethe Bloch formula, 309 Cowsik-McClelland bound, 174
Boltzmann equation, 199 Creation operator, 269, 422
Boltzmann equation (baryon to photon ratio), Cross section (generalities), 420
187 Curvature, 47

© Springer Nature Switzerland AG 2021 499

Y. Mambrini, Particles in the Dark Universe,
https://doi.org/10.1007/978-3-030-78139-2
500 Index

Cyclotron frequency, 298 Entropy (present), 174

Cyclotron radiation, 298 Entropy (total), 156
Equation of motion (preheating), 62
Equation of state, 44, 63, 137, 388
D Euler-Lagrange equation, 62, 408
Damping (Landau), 351
Dark ages, 184
Dark matter and reheating, 118 F
Dark matter production Fermi’s golden rule, 248, 420
inflaton decay, 101 Feynman gauge, 398
and reheating, 118 Field strength tensor, 459
before thermalization, 102 FIMP (Freeze in Massive Particle), 213
non-thermal phase, 101 Fizeau experiment, 307
Dark radiation, 178 Flatness problem, 56
Decay and non-thermal production, 225 FLRW (Friedmann - Lemaitre - Robertson -
Decay rate, 428, 442 Walker) metric, 33
Deceleration parameter, 31, 42 Foreword, ix
Decoupling, 157 Free streaming, 351
Decoupling (after), 161 Freeze-in, 213
Decoupling temperature, 199 Freeze in mechanism, 216
Deflection angle (GR), 384 Frequentist statistic, 488
Degrees of freedom (computation of), 144 Friedmann equations, 29, 46, 84
Degrees of freedom (summary), 147 Furry’s theorem, 416
Degrees of freedom (today), 155
Delta function δ, 393 G
Density of energy, 288 Galactic coordinates, 279
Destruction operator, 422 Gauss’s law, 286
Determinant (differentiation), 379 Gell-Mann matrices, 419
Deuterium bottleneck, 17 General relativity, 369
Deuterium formation, 191 Goldstino, 222
Dirac equation, 406 Gravitational lensing, 12
Dirac mass, 473 Gravitino, 228
Dirac matrix, 405 Gravitino (dark matter), 222
Dirac traces, 417 Gravitino (decay), 228
Distribution function, 161 Graviton, 60
Distribution function (non-thermal), 94 GZK cutoff, 315
Doppler effect, 305

H
E Helicity suppression, 453
E=mc2 , 307 Higgs vev, 492
Earth velocity, 254 Higgs couplings, 467
Effective operators (direct detection), 264 Higgs lifetime, 446
E-folds, 58 Historical facts, 1
Einstein-de Sitter model, 49 Homogeneous Universe, 26, 33, 44
Einstein equation of field, 379 Hoovering Universe, 50
Einstein-Hilbert action, 371, 379 Horizon problem, 53
Elastic scattering, 441 Hot dark matter, 204, 353
Electric dipole moment of the neutron, 459 Hubble constant (thermal epoch), 107
Electron (loss of energy), 330 Hubble expansion, 46
Electron radius, 297 Hubble law, 37
Energy losses, 313 Hubble law (Newtonian approach), 26
Enthalpy, 387 Hubble parameter and time, 89
Entropy, 154, 387 Hydrogen (binding energy), 182
Index 501

I Matter-antimatter assymmetry, 185

Indirect detection (γ ), 325 Matter domination redshift, 152
Inflation, 53 Matter era, 148
Inflation (slow-roll), 63 Matter field, 381
Inflaton (equation of motion)), 61 Matter radiation equilibrium, 386
Inflaton decay, 121, 449 Maxwell equations, 288
Internal energy, 387 Mean–free path (photon), 185
Inverse Compton, 333 Microlensing, 13
Inverse-Compton scattering, 310 Mirror dark matter, 180
Ionization losses, 293 Misalignment mechanism, 219
Isothermal profile, 284 MoND, 19
Isotropic Universe, 26, 33, 44 Muon decay, 445

J N
Jeans instability, 354 Neff , 149, 178
Nbody simulation, 11
Neutrino (atmospheric), 473
K Neutrino (Cowsik-McClelland bound),
Kinetic decoupling, 160 174
Kinetic energy, 143 Neutrino (density), 148
Neutrino (mass), 473
Neutrino (solar), 471
L Neutrino (sources), 471
Lagrangian (extra U(1)), 469 Neutrino (ultra high energy), 473
Lagrangian (singlet scalar), 468 Neutrino decoupling, 172, 174
Lagrangian (Standard Model), 466 Non-instantaneous reheating, 110, 117
Landau gauge, 398 Nucleon structure, 257
Laplace law, 387 Nucleosynthesis, 185
Laplace operator, 393 Number of baryons, 156
Larmor’s formula, 289, 331 Number of photons, 156
Larmor’s formula (relativistic), 305
Last scattering surface, 183
Lee-Weinberg bound, 208 P
Light elements abundance, 194 Parallel transport, 377
Lithium7, 192 Parametric resonance, 76
Local abundance, 6 Parity operator, 410
Look-elsewhere effect (LEE), 485 Peccei-Quinn symmetry, 461
and the Higgs boson, 488 Perturbations, 71
Lorentz transformations, 300 Perturbativity (limit), 493
Phase-space, 426, 442
Photon (decoupling), 181
M Poisson’s equation, 286
Magnetic field (modelisation), 338 Pole effect (σ v), 210
Majorana (decay to), 449 Power spectrum, 361
Majorana (Lagrangian), 414 Preheating, 76, 110
Majorana (neutrino), 474 Preheating (equation of motion), 62
Majorana (rules), 399 Pressure, 86, 136
Majorana mass, 474 Pressure of a gas, 136
Majorana particle, 412 Proca equation, 408
Majorana particle (annihilation), 432 Profile (isothermal), 284
Majorana particle (direct detection), 272 Profiles (dark matter), 322
Majoron, 479 p-value, 481, 483
Mandelstam variables, 418 p–wave, 451
502 Index

Q Stress-energy tensor (scalar field), 59

QCD, 458 Stress-momentum tensor (general), 60
QCD phase transition, 147 Strong-CP problem, 458
QCD-axion dark matter, 221 Structure constant SU(3), 459
Structure formation, 350
s–wave, 451
R Symmetry factor, 401, 431
Radiation era, 148 Synchrotron radiation, 311, 330
Radiative pressure, 32 Systematic uncertainty, 483
Radiative processes, 287
Rarita-Schwinger equation, 409
Recoil energy, 247 T
Recombination, 181 Temperature (maximale), 110
Redshift, 36 Tensor operator, 412
Redshift (free streaming)), 351 Thermal decoupling, 160
Reheating, 106, 110 Thermalization, 97, 109, 164, 166, 168
Reheating (instantaneous), 109 Thermodynamic, 138
Re-ionization time, 184 Thomson cross section, 333
Relativity transformation, 300 Thomson scattering, 184, 296
Retarded potential, 288 Threshold (Boltzmann equation), 213
Ricci tensor, 378 Time and temperature, 148
Riemann tensor, 378 Time evolution (densities), 90
Rotation curves, 8 Time evolution (reheating), 90
Traces (Dirac matrices), 417
Transfer of energy, 164
S Tremaine-Gunn bound, 176
Sachs-Wolfe effect, 359 Two-body decay, 442
Scalar curvature, 379 Two-point correlation function, 360
Scattering matrix, 422
Schrodinger equation, 455
See-saw mechanism, 476 U
Size of the Universe, 38, 51 Ultra High Energetic Cosmic Rays (UHECR),
Slow-roll (conditions), 66 314
Solar neutrino, 471 Ultra high energetic neutrino, 473
Sommerfeld enhancement, 344 Unitarity limit, 432
Sound speed, 235, 355
Spectrum (antiproton), 466
Spectrum (photon), 463 V
Spectrum (positron), 464 Vectorial couplings, 468
Speed of sound, 235, 355 Velocity suppression, 452
Spin 1, 408 Vierbein, 371
Spin 3/2, 409
Spin 3/2 (decay into), 449
Spin connection, 374 W
Spin dependent direct detection, 261 Warm dark matter, 353
Spin operator, 407 WIMP (in brief), 205
Spin selection, 451 W lifetime, 450
Statistical uncertainty, 483
Statistics, 481
Statistics (gas and radiation), 135 Z
Stress-energy momentum (cosmological Z , 50, 153
constant)), 46 ZEQ , 151, 176
Stress-energy momentum tensor (perfect fluid, Z lifetime, 449, 450
381