DARK MATTER 101

DARK MATTER 101

From production to detection

David G. Cerdeño
CONTENTS

1 Motivation for dark matter 1

1.1 Motivation for Dark Matter 1
1.1.1 Galactic scale 2
1.1.2 Galaxy Clusters 3
1.1.3 Cosmological scale 4
1.2 Dark Matter properties 5
1.2.1 NonBaryonic 5
1.2.2 Neutral 6
1.2.3 Nonrelativistic 6
1.2.4 Long-lived 6
1.2.5 Collisionless 7

2 Freeze Out of Massive Species 9

2.1 Cosmological Preliminaries 9
2.2 Time evolution of the number density 13
2.2.1 Freeze out of relativistic species 16
2.2.2 Freeze out of non-relativistic species 16
2.2.3 WIMPs 17
2.3 Computing the DM annihilation cross section 18
2.3.1 Special cases 19
2.4 Freeze-in of dark matter 22
v
vi CONTENTS

2.4.1 DM production from decays of heavier bath particles 23

2.5 Late decays of unstable particles 24

3 Direct DM detection 27
3.1 Preliminaries 27
3.1.1 DM flux 27
3.1.2 Kinematics 27
3.2 The master formula for direct DM detection 28
3.2.1 The scattering cross section 28
3.2.2 The importance of the threshold 29
3.2.3 Velocity distribution function 30
3.2.4 Energy resolution, threshold energy and experimental efficiency 30
3.3 Exponential spectrum 30
3.4 Annual modulation 30
3.5 Directional detection 31
3.6 Coherent neutrino scattering 31
3.7 Inelastic 31

4 Axions 33
4.1 The Strong QCD Problem 33
4.2 Axions production 33
References 37
CHAPTER 1

MOTIVATION FOR DARK MATTER

The existence of a vast amount of dark matter (DM) in the Universe is supported by many
astrophysical and cosmological observations. The latest measurements indicate that ap-
proximately a 27% of the Universe energy density is in form of a new type of non-baryonic
cold DM. Given that the Standard Model (SM) of particle physics does not contain any vi-
able candidate to account for it, DM can be regarded as one of the clearest hints of new
physics.

1.1 Motivation for Dark Matter

Astrophysical and cosmological observations have provided substantial evidence that point
towards the existence of vast amounts of a new type of matter, that does not emit or absorb
light. All astrophysical evidence for DM is solely based on gravitational effects (either
trough the observation of dynamical effects, deflection of light by gravitational lensing or
measurements of the gravitational potential of galaxy clusters), which cannot be accounted
for by just the observed luminous matter. The simplest way to solve these problems is the
inclusion of more matter (which does not emit light - and is therefore dark in the astro-
nomical sense1 ). Modifications in the Newtonian equation relating force and accelerations
have also been suggested to address the problem at galactic scales, but this hypothesis is

1 Since dark matter does not absorb light, a more adequate name would have been transparent matter.

Dark Stuff. 1
By D. G. Cerdeño, IPPP, University of Durham
2 MOTIVATION FOR DARK MATTER

Figure 1.1 Left) Vera Rubin. Right) Rotation curve of a spiral galaxy, where the contribution from
the luminous disc and dark matter halo is shown by means of solid lines.

insufficient to account for effects at other scales (e.g., cluster of galaxies) or reproduce the
anisotropies in the CMB.
No known particle can play the role of the DM (we will later argue that neutrinos con-
tribute to a small part of the DM). Thus, this is one of the clearest hints for Physics Beyond
the Standard Model and provides a window to new particle physics models. In the follow-
ing I summarise some of the main pieces of evidence for DM at different scales.
I recommend completing this section with the first chapters of Ref. [1] and the recent
article [2].

1.1.1 Galactic scale

Rotation curves of spiral galaxies Rotation curves of spiral galaxies are probably the
best-known examples of how the dynamical properties of astrophysical objects are affected
by DM. Applying Gauss Law to a spiral galaxy (one can safely ignore the contribution
from the spiral arms and assume a spherical distribution of matter in the bulge) leads to a
simple relation between the rotation velocity of objects which are gravitationally bound to
the galaxy and their distance to the galactic centre:
r
GM (r)
v= , (1.1)
r
where M (r) is the mass contained within the radius r. In the outskirts of the galaxy,
where we expect that M does not increase any more, we would therefore expect a decay
vrot ∝ r−1/2 .
Vera Rubin’s observations of rotation curves of spiral galaxies [3, 4] showed a very slow
decrease with the galactic radius. The careful work of Bosma [5], van Albada and Sancisi
[6] showed that this flatness could not be accounted for by simply modifying the relative
weight of the diverse galactic components (bulge, disc, gas), a new component was needed
with a different spatial distribution (see Fig. 1.1).
Notice that the flatness of rotation curves can be obtained if a new mass component is
introduced, whose mass distribution satisfies M (r) ∝ r in eq.(1.1). This is precisely the
 MOTIVATION FOR DARK MATTER 3

Figure 1.2 Left) Coma cluster and F. Zwicky, who carried out measurements of the peculiar
velocities of this object. Right) Modern techniques [7], based on gravitational lensing, allow for a
much more precise determination of the total mass of this object.

relation that one expects for a self-gravitational gas of non-interacting particles. This halo
of DM can extend up to ten times the size of the galactic disc and contains approximately
an 80% of the total mass of the galaxy.
Since then, flat rotation curves have been found in spiral galaxies, further strengthening
the DM hypothesis. Of course, our own galaxy, the Milky Way is no exception. N-body
simulations have proved to be very important tools in determining the properties of DM
haloes. These can be characterised in terms of their density profile ρ(r) and the velocity
distribution function f (v). Observations of the local dynamics provide a measurement of
the DM density at our position in the Galaxy. Up to substantial uncertainties, the local
DM density can vary in a range ρ0 = 0.2 − 1 GeV cm−3 . It is customary to describe
the DM halo in terms of a Spherical Isothermal Halo, in which the velocity distribution
follows a Maxwell-Boltzmann law, but deviations from this are also expected. Finally, due
to numerical limitations, current N-body simulations cannot predict the DM distribution at
the centre of the galaxy. Whereas some results suggest the existence of a cusp of DM in
the galactic centre, other simulations seem to favour a core. Finally, the effect of baryons
is not easy to simulate, although substantial improvements have been recently made.

Local probes

1.1.2 Galaxy Clusters

Peculiar motion of clusters. Fritz Zwicky studied the peculiar motions of galaxies in
the Coma cluster [8, 9]. The aim was to measure the total mass of the system through a
method that did not rely only on the information from visible objects, and thus included
also the faint and non-luminous components). Assuming that the galaxy cluster is an iso-
lated system, the virial theorem can be used to relate the average velocity of objects with
the gravitational potential (or the total mass of the system).
As in the case of galaxies, this determination of the mass is insensitive to whether ob-
jects emit any light or not. This can then be contrasted with other determinations that are
based on the luminosity. The results showed an extremely large mass-to-light ratio, indica-
4 MOTIVATION FOR DARK MATTER

Figure 1.3 Left) Deep Chandra image of the Bullet cluster. Green lines represent mass contours
from weak lensing. Right) Dark filament in the system Abell 222/223, reconstructed using weak
lensing.

tive of the existence of large amounts of missing mass, which can be attributed to a DM
component.
Modern determinations through weak lensing techniques provide a better gravitational
determination of the cluster masses [10, 7] (see Fig. 1.2). I recommend reading through
Ref.[9] for a derivation of the virial theorem in the context of Galaxy clusters.

Dynamical systems. The Bullet Cluster (1E 0657-558) is a paradigmatic example of

the effect of dark matter in dynamical systems. It consists of two galaxy clusters which
underwent a collision. The visible components of the cluster, observed by the Chandra X-
ray satellite, display a characteristic shock wave (which gives name to the whole system).
On the other hand, weak-lensing analyses, which make use of data from the Hubble Space
Telescope, have revealed that most of the mass of the system is displaced from the visible
components. The accepted interpretation is that the dark matter components of the clusters
have crossed without interacting significantly (see e.g., Ref. [11, 12]).
The Bullet Cluster is considered one of the best arguments against MOND theories
(since the gravitational effects occur where there is no visible matter). It also sets an upper
bound on the self-interaction strength of dark matter particles.

DM filaments. Observations of the distribution of luminous matter at large scales have

shown that it follows a filamentary structure. Numerical simulations of structure formation
with cold DM have been able to reproduce this feature. To date, it is well understood
that DM plays a fundamental role in creating that filamentary network, gravitationally
trapping the luminous matter. Recently, the comparison of the distribution of luminous
matter in the Abell 222/223 supercluster with weak-lensing data has shown the existence
of a dark filament joining the two clusters of the system. That filament, having no visible
counterpart, is believed to be made of DM.

1.1.3 Cosmological scale

Finally, DM has also left its footprint in the anisotropies of the Cosmic Microwave Back-
ground (CMB). The analysis of the CMB constitutes a primary tool to determine the cos-
mological parameters of the Universe. The data obtained by dedicated satellites in the past
 DARK MATTER PROPERTIES 5

Figure 1.4 Left) Contribution to the energy density for each of the components of the Universe.
Right) Planck temperature map.

decades has confirmed that we live in a flat Universe (COBE), dominated by dark matter
and dark energy (WMAP), whose cosmological abundances have been determined with
great precision (Planck).
The abundance of DM is normally expressed in terms of the cosmological density pa-
rameter, defined as ΩDM h2 = ρDM /ρc where ρc is the critical density necessary to re-
cover a flat Universe and h = 0.7 is the normalised Hubble parameter. The most recent
measurements by the Planck satellite, combined with data obtained from Supernovae (that
trace the Universe expansion) yield
ΩCDM h2 = 0.1196 ± 0.0031 . (1.2)
Given that Ω ≈ 1, this means that dark matter is responsible for approximately a 26% of
the Universe energy density nowadays. Even more surprising is the fact that another exotic
component is needed, dark energy, which makes up approximately the 69% of the total
energy density (see Fig. 1.4).

1.2 Dark Matter properties

1.2.1 NonBaryonic

The results of the CMB, together with the predictions from Big Bang nucleosynthesis,
suggest that only 4 − 5% of the total energy budget of the universe is made out of ordi-
nary (baryonic) matter. Given the mismatch of this with the total matter content, we must
conclude that DM is non-baryonic.

Neutrinos. Neutrinos deserve special mention in this section, being the only viable non-
baryonic DM candidate within the SM. Neutrinos are very abundant particles in the Uni-
verse and they are known to have a (very small) mass. Given that they also interact very
feebly with ordinary matter (only through the electroweak force) they are in fact a com-
ponent of the DM. There are, however various arguments that show that they contribute in
fact to a very small part.
First, neutrinos are too light. Through the study of the decoupling of neutrinos in the
early universe we can compute their thermal relic abundance. Since neutrinos are relativis-
tic particles at the time of decoupling, this is in fact a very easy computation (we will come
back to this in Section 2.2.1), and yields
P
2 mi
Ων h ≈ i . (1.3)
91 eV
6 MOTIVATION FOR DARK MATTER

Using current upper bounds on the neutrino mass, we obtain Ων h2 < 0.003, a small
fraction of the total DM abundance.
Second, neutrinos are relativistic (hot) at the epoch of structure formation. As men-
tioned above, hot DM leads to a different hierarchy of structure formation at large scales,
with large objects forming first and small ones occurring only after fragmentation. This is
inconsistent with observations.

1.2.2 Neutral
It is generally argued that DM particles must be electrically neutral. Otherwise they would
scatter light and thus not be dark. Similarly, constrains on charged DM particles can be
extracted from unsuccessful searches for exotic atoms. Constraints on heavy millicharged
particles are inferred from cosmological and astrophysical observations as well as direct
laboratory tests [13, 14, 15]. Millicharged DM particles scatter off electrons and protons
at the recombination epoch via Rutherford-like interactions. If millicharged particles cou-
ple tightly to the baryonphoton plasma during the recombination epoch, they behave like
baryons thus affecting the CMB power spectrum in several ways [13, 14]. For particles
much heavier than the proton, this results in an upper bound of its charge  [14]
 ≤ 2.24 × 10−4 (M/1 TeV)1/2 . (1.4)
Similarly, direct detection places upper bounds on the charge of the DM particle [16]
 ≤ 7.6 × 10−4 (M/1 TeV)1/2 . (1.5)

1.2.3 Nonrelativistic
Numerical simulations of structure formation in the Early Universe have become a very
useful tool to understand some of the properties of dark matter. In particular, it was soon
found that dark matter has to be non-relativistic (cold) at the epoch of structure forma-
tion. Relativistic (hot) dark matter has a larger free-streaming length (the average distance
traveled by a dark matter particle before it falls into a potential well). This leads to incon-
sistencies with observations.
However, at the Galactic scale, cold dark matter simulations lead to the occurrence of
too much substructure in dark matter haloes. Apparently this could lead to a large number
of subhaloes (observable through the luminous matter that falls into their potential wells).
It was argued that if dark matter was warm (having a mass of approximately 2 − 3 keV)
this problem would be alleviated.
Modern simulations, where the effect of baryons is included, are fundamental in order
to fully understand structure formation in our Galaxy and determine whether dark matter
is cold or warm.

1.2.4 Long-lived
Possibly the most obvious observation is that DM is a long-lived (if not stable) particle.
The footprint of DM can be observed in the CMB anisotropies, its presence is essential
for structure formation and we can feel its gravitational effects in clusters of galaxies and
galaxies nowadays.
Stable DM candidates are common in models in which a new discrete symmetry is
imposed by ensuring that the DM particle is the lightest with an exotic charge (and there-
 DARK MATTER PROPERTIES 7

fore its decay is forbidden). This is the case, e.g., in Supersymmetry (when R-parity is
imposed), Kaluza-Klein scenarios (K-parity) or little Higgs models.
However, stability is not required by observation. DM particles can decay, as long
as their lifetime is longer than the age of the universe. Long-lived DM particles feature
very small couplings. Characteristic examples are gravitinos (whose decay channels are
gravitationally suppressed) or axinos (which decays through the axion coupling).

1.2.5 Collisionless
Dynamical systems, such as cluster collisions, set an upper bound to the self-interactions
of DM particles. Observations seem to suggest that the DM component in these objects is
mostly collision-less, thus behaving very differently than ordinary matter. Dark matter’s
lack of deceleration in the bullet cluster constrains its self-interaction cross-section σ/m <
1.25 cm2 g−1 ≈ 2 barn GeV−1 .
Notice however, that self-interacting dark matter with a cross section in the range 0.1 <
σ/m < 1 cm2 g−1 can be very beneficial in order to alleviate the problems with the amount
of substructure in numerical simulations of DM haloes.
[17]
CHAPTER 2

FREEZE OUT OF MASSIVE SPECIES

In this chapter we will address the computation of the relic abundance of dark matter
particles, making special emphasis in the case of thermal production in the Early Universe.

2.1 Cosmological Preliminaries

This section does not intend to be a comprehensive review on Cosmology, but only an
introduction to some of the elements that we will need for the calculation of Dark Matter
freeze-out.
We can describe our isotropic and homogeneous Universe in terms of the Friedman-
Lemaı̂tre-Robertson-Walker (FLRW) metric, which is exact solution of Einstein’s field
equations of general relativity

dr2
 
ds2 = dt2 − a2 (t) + r 2
(dθ 2
+ sin θdφ2
) = gµν dxµ dxν . (2.1)
1 − kr2

The constant k = {−1, 0, +1} corresponds to the spatial curvature, with k = 0 corre-
sponding to a flat Universe (the choice we will be making in these notes). The affine
connection, used to connect nearby tangent spaces (thus enabling the differentiation of
tangent vector fields), is defined as

1 µσ
Γµνλ = g (gσν,λ + gσλ,ν − gνλ,σ ) , (2.2)
2
Dark Stuff. 9
By D. G. Cerdeño, IPPP, University of Durham
10 FREEZE OUT OF MASSIVE SPECIES

These can also be found in the literature as Christoffel symbols, used in the definition of a
covariant derivative. They are greatly simplified in the case of a FLRW metric, since most
of the derivatives vanish.
The expansion of the Universe is controlled by the scale parameter a(t). More specif-
ically, we can define the Hubble parameter, H ≡ ȧ(t)/a(t) (where the dot denotes time
derivation), which encodes the rate at which space is expanding. In the following, we are
going to work with a radiation-dominated Universe. Notice that matter-radiation equality
only occurs very late (when the Universe is approximately 60 kyr) and dark matter freeze-
out occurs before BBN. The Hubble parameter for a radiation-dominated Universe reads

1/2 T2
H = 1.66 g∗ , (2.3)
MP
where MP = 1.22 × 1019 GeV.
It is customary to define the dimensionless parameter x = m/T (where m is a mass pa-
rameter that we will later associate to the DM mass) and extract the explicit x dependence
from the Hubble parameter to define H(m) as follows

1/2 m2
H(m) = 1.66 g∗ = Hx2 . (2.4)
MP

In this section we will try to compute the time evolution of the number density of dark
matter particles, in order to be able to compute their relic abundance today and what this
implies in the interaction strength of dark matter particles. The phase space distribution
function f describes the occupancy number in phase space for a given particle in kinetic
equilibrium, and distinguishes between fermions and bosons.

1
f= , (2.5)
e(E−µ)/T ±1

where the (−) sign corresponds to bosons and the (+) sign to fermions. E is the energy
and µ the chemical potential. For species in chemical equilibrium, the chemical potential
is conserved in the interactions. Thus, for processes such as i + j ↔ c + d we have
µi +µj = µc +µd . Notice then that all chemical potentials can be expressed in terms of the
chemical potentials of conserved quantities, such as the baryon chemical potential µB . The
number of independent chemical potentials corresponds to conserved particle numbers.
This implies, for example, that given a particle with µi , the corresponding antiparticle
would have the opposite chemical potential −µi . For the same reason, since the number of
photons is not conserved in interactions, µγ = 0
Using the expression of the phase space distribution function (2.5), and integrating in
phase space, we can compute a series of observables in the Universe. In particular, the
number density of particles, n, the energy density, ρ, and pressure, p, for a dilute and
weakly-interacting gas of particles with g internal degrees of freedom read
Z
g
n = f (p) d3 p, (2.6)
(2π)3
Z
g
ρ = E(p) f (p) d3 p, (2.7)
(2π)3
|p|2
Z
g
p = 3
f (p) d3 p. (2.8)
(2π) 3E(p)
 COSMOLOGICAL PRELIMINARIES 11

Figure 2.1 In the absence of number changing processes, the comoving number density of a species
is preserved.

It is customary (and very convenient) to define densities normalised by the time depen-
dent volume V (t) = a(t)3 . The reason for this is that in the absence of number changing
processes, the comoving number density remains constant with time evolution (or red-
shift) as exemplified in Fig. 2.1. An expanding Universe is a closed system and in thermal
equilibrium the total entropy is conserved.
T dS = d(ρV ) + pdV = d((ρ + p)V ) − V dp = 0 , (2.9)
where we have used that d((ρ + p)V ) = V dp. The entropy density is therefore s =
S/V = (ρ + p)/V . Notice that since the evolution of the Universe is isoentropic, the
entropy density s = S/a3 has precisely that dependence. Applying this prescription to the
number density of particles, we define the yield as a fraction of the number density and the
entropy density as
n
Y = . (2.10)
s
Notice that, in the absence of number-changing processes, the yield remains constant.
The evolution of the entropy density as a function of the temperature is given by 1
2π 2
s= g∗s T 3 , (2.11)
45
where the effective number of relativistic degrees of freedom for entropy is
X  Ti 3 7 X  3
Ti
g∗s = g + g . (2.12)
T 8 T
bosons fermions

Remember also that we can express the energy density as

π2
g∗ T 4 ,
ρ= (2.13)
30
in terms of the relativistic number of degrees of freedom
X  Ti 4 7 X  4
Ti
g∗ = g + g . (2.14)
T 8 T
bosons fermions

1 To arrive at this equation, one can calculate s = (p + ρ)/T for fermions and bosons, using the corresponding

expression for the phase space distribution function.

12 FREEZE OUT OF MASSIVE SPECIES

In these two equations, T is the temperature of the plasma (in equilibrium) and Ti is the
effective temperature of each species.
Solving the integral in eq. (2.6) explicitly for relativistic and non-relativistic particles,
and expressing the results in terms of the Yield results in the following expressions.

relativistic species
gef f
n= ζ(3)T 3 , (2.15)
π2
where gef f = g for bosons and gef f = 43 g for fermions2 . Then, using eq. (2.10), the
Yield at equilibrium reads
45 gef f gef f
Yeq = 4
ζ(3) ≈ 0.278 . (2.16)
2π g∗s g∗s

non-relativistic species
 3/2
mT
n=g e−m/T . (2.17)
2π
Then the Yield at equilibrium reads
45  π 1/2 g  m 3/2 −m/T
Yeq = e . (2.18)
2π 4 8 g∗s T

EXAMPLE 2.1

It is easy to estimate the value of the Yield that we need in order to reproduce the
correct DM relic abundance, Ωh2 ≈ 0.1, since
ρχ 2 mχ nχh2 mχ Y∞ s0 h2
Ωh2 = h = = , (2.19)
ρc ρc ρc
where Y∞ corresponds to the DM Yield today and s0 is todays entropy density. We
can assume that the Yield did not change since DM freeze-out and therefore
mχ Yf s0 h2
Ωh2 = . (2.20)
ρc
Using the measured value s0 = 2970 cm−3 , and the value of the critical density ρc =
1.054 × 10−5 h2 GeV cm−3 , as well as Planck’s result on the DM relic abundance,
Ωh2 ≈ 0.1, we arrive at
 
−10 1 GeV
Yf ≈ 3.55 × 10 . (2.21)
mχ
In Figure 2.2 represent the yield as a function of x for non-relativistic particles, using
expression (2.18). As we can observe, the above range of viable values for Yf cor-
respond to xf ≈ 20. Notice that this is a crude approximation and we will soon be
making a more careful quantitative treatment.

2 We p| in the relativistic limit, and the integrals 0∞ p2 /(ep − 1)dp =

R
are using here the approximation E ≈ |~
R∞ 2 p
2ζ(3), and 0 p /(e + 1)dp = 3ζ(3)/2, in terms or Riemann’s Zeta function. Remember also that ζ(3) ≈
1.202.
 TIME EVOLUTION OF THE NUMBER DENSITY 13

Figure 2.2 Equilibrium yield as a function of the dimensionless variable, x, for non-relativistic
particles. The green band represents the freeze-out value, Yf , for which the correct thermal relic
abundance is achieved (for masses of order 1-1000 GeV.

2.2 Time evolution of the number density

The evolution of the number density operator can be computed by applying the covariant
form of Liuvilles operator to the corresponding phase space distribution function. Formally
speaking, we have
L̂[f ] = C[f ], (2.22)
where L̂ is the Liouville operator, defined as

∂ ∂
L̂ = pµ − Γµσρ pσ pρ µ , (2.23)
∂xµ ∂p

and C[f ] is the collisional operator, which takes into account processes which change the
number of particles (e.g., annihilations or decays). In this expression, we have used the
geodesic equation dpµ /dτ = d2 xµ /dτ 2 = −Γµρσ dxσ /dτ dxρ /dτ = −Γµρσ pσ pρ . In the
expression above, gravity enters through the affine connection, Γµσρ .
One can show that in the case of a FRW Universe, for which f (xµ , pµ ) = f (t, E), we
have
∂ ∂
L̂ = E − Γ0σρ pσ pρ
∂t ∂E
∂ 2 ∂
= E − H|p| . (2.24)
∂t ∂E
14 FREEZE OUT OF MASSIVE SPECIES

Integrating over the phase space we can relate this to the time evolution of the number
density
Z Z
g L̂[f ] 3 g C[f ] 3
d p= d p, (2.25)
(2π)3 E (2π)3 E
The integral on the left-hand side can be easily done by parts, which results in
Z
dn g C[f ] 3
+ 3Hn = 3
d p, (2.26)
dt (2π) E

Regarding the collisional operator, it encodes the microphysical description in terms

of Particle Physics, and incorporates all number-changing processes that create or deplete
particles in the thermal bath. For simplicity, let us concentrate in annihilation processes,
where SM particles (A, B) can annihilate to form a pair of DM particles (labelled 1, 2), or
vice-versa (A, B ↔ 1, 2). The phase space corresponding to each particle is defined as

gi d3 pi
dΠi = , (2.27)
(2π)3 2Ei

from where
Z Z
g C[f ] 3
d p = − dΠA dΠB dΠ1 dΠ2 (2π)4 δ(pA + pB − p1 − p2 )
(2π)3 E
|M12→AB |2 f1 f2 (1 ± fA )(1 ± fB )


−|MAB→12 |2 fA fB (1 ± f1 )(a ± f2 )

Z
= − dΠA dΠB dΠ1 dΠ2 (2π)4 δ(pA + pB − p1 − p2 )

|M12→AB |2 f1 f2 − |MAB→12 |2 fA fB .
 
(2.28)

The terms (1 ± fi ) account for the viable phase space of the produced particles, taking
into account whether they are fermions (−) or bosons (+). Assuming no CP violation
in the DM sector (T invariance) |M12→AB |2 = |MAB→12 |2 ≡ |M|2 . Also, energy
conservation in the annihilation process allows us to write EA + EB = E1 + E2 , thus,
EA +EB E1 +E1
fA fB = fAeq fBeq = e− T = e− T = f1eq f2eq . (2.29)

In the first equality we have just used the fact that SM particles are in equilibrium. This
eventually leads to
Z
g C[f ] 3
d p = −hσvi n2 − n2eq ,


3
(2.30)
(2π) E

where we have defined the thermally-averaged cross-section as

Z
1
hσvi ≡ 2 dΠA dΠB dΠ1 dΠ2 (2π)4 δ(pA + pB − p1 − p2 )|M|2 f1eq f2eq . (2.31)
neq

Collider enthusiasts would realise that this expression is similar to that of a cross-section,
but we have to consider that the “initial conditions” do not correspond to a well-defined
energy, but rather we have to integrate to the possible energies that the particles in the
thermal bath may have. This explains the extra integrals in the phase space of incident
 TIME EVOLUTION OF THE NUMBER DENSITY 15

particles with a distribution function given by f1eq f2eq . We are thus left with the familiar
form of Boltzmann equation,
dn
+ 3Hn = −hσvi n2 − n2eq .


(2.32)
dt
Notice that this is an equilibrium-restoring equation. If the right-hand-side of the equation
dominates, then n traces its equilibrium value n ≈ neq . However, when Hn > hσvin2 ,
then the right-hand-side can be neglected and the resulting differential equation dn/n =
−3da/a implies that n ∝ a−3 . This is equivalent to saying that DM particles do not
annihilate anymore and their number density decreases only because the scale factor of the
Universe increases.
It is also customary to define the dimensionless variable 3
m
x= . (2.33)
T

EXAMPLE 2.2

Using the yield defined in equation (2.10) we can simplify Boltzmann equation. No-
tice that
d a3 n
     
dY d n 1 2 3 dn 1 dn
= = = 3a ȧn + a = 3Hn + . (2.34)
dt dt s dt a3 s a3 s dt s dt

Here we have used that the expansion of the Universe is iso-entropic and thus a3 s
remains constant. Also we use the definition of the Hubble parameter H = aȧ . This
allows us to rewrite Boltzmann equation as follows
dY
= −shσvi Y 2 − Yeq
2


. (2.35)
dt
Now, since a ∝ T −1 and s ∝ T 3 ,
d 3 d d a
(a s) = 0 → (aT ) = 0 → =0, (2.36)
dt dt dt x
which in turns leads to
dx
= Hx , (2.37)
dt
and thus
dY dY dx dY
= = Hx . (2.38)
dt dx dt dx
Using the results of Example (2.2) we can express Boltzmann equation (2.32) as
dY −sxhσvi
Y 2 − Yeq2


=
dx H(m)
−λhσvi
Y 2 − Yeq
2


= 2
, (2.39)
x

3 It
is important to point that this definition of x is not universal; some authors use T /m and care should be taken
when comparing results from different sources in the literature.
16 FREEZE OUT OF MASSIVE SPECIES

where we have used the expression of the entropy density (2.11) in the last line and defined

2π 2 MP g∗s
λ ≡ m
45 1.66 g∗1/2
g∗s
≈ 0.26 1/2 MP m . (2.40)
g∗
Eq. (2.39) is a Riccati equation, without closed analytical form. Thus, to calculate its
solutions we have to rely on numerical methods. However, it is possible to solve it approx-
imately.

2.2.1 Freeze out of relativistic species

The freeze-out of relativistic species is easy to compute, since the yield (2.16) has no
dependence on xf . Neutrinos are a paradigmatic example of relativistic particles and one
must in principle consider their contribution to the total amount of dark matter (after all,
they are dark).
Since neutrinos decouple while they are still relativistic, their yield reads
gef f
Yeq ≈ 0.278 . (2.41)
g∗s

Neutrinos decouple at a few MeV, when the species that were still relativistic are e± , γ, ν
and ν̄. Thus, the number of relativistic degrees of freedom is g∗ = g∗s = 10.75. For one
neutrino family, the effective number of degrees of freedom is gef f = 3g/4 = 3/2. Using
these values, the relic density today an be written as
2
P
2 i mνi Y∞ s0 h
Ωh =
ρc
P
i mν i
≈ . (2.42)
91 eV
P
Notice that in order for neutrinos to be the bulk of dark matter, we would need i mνi ≈
9 eV , which is much bigger than current upper limits (for example, obtainedP from cos-
mological observations). Notice, indeed, that if we consider the current bound i mνi ≤
0.3 eV we can quantify the contribution of neutrinos to the total amount of dark matter,
resulting in Ωh2 ≤ 0.003. This is less than a 3% of the total dark matter density.

2.2.2 Freeze out of non-relativistic species

The case of non-relativistic species is far more interesting. Once We can define the quantity

∆Y ≡ Y − Yeq . (2.43)

Boltzmann equation (2.39) is now easier to solve, at least approximately, as follows

For early times, 1 < x  xf , the yield follows closely its equilibrium value, Y ≈
Yeq , and we can assume that d∆Y /dx = 0. We then find
dYeq
dx x2
∆Y = − . (2.44)
Yeq 2λhσvi
 TIME EVOLUTION OF THE NUMBER DENSITY 17

Thus, at freeze-out we obtain

x2f
∆ Yf ≈ , (2.45)
2λhσvi

where in the last line we have used that for large enough x, using eq. (2.18) implies
dYeq
dx ≈ −Yeq .

For late times, x  xf , we can assume that Y  Yeq , and thus ∆Y∞ ≈ Y∞ , leading
to the following expression,

d∆Y λhσvi
≈ − 2 ∆2Y , (2.46)
dx x
This is a separable equation that we integrate from the freeze-out time up to nowa-
days. In doing so, it is customary to expand the thermally averaged annihilation cross
section in powers of x−1 as hσvi = a + xb .
Z ∆Y ∞ Z x∞
d∆Y λhσvi
=− dx . (2.47)
∆Yf ∆2Y xf x2

Taking into account that x∞  xf , this leads to

 
1 1 λ b
= + a+ . (2.48)
∆ Y∞ ∆ Yf xf 2xf

The term 1/∆Yf is generally ignored (if we are only aiming at a precision up to a few
per cent [18]) . We can check that this is a good approximation using the previously
derived (2.45) for xf ≈ 20 (which, as we saw in Fig. 2.2 is the value for which the
equilibrium Yield has the right value). This leads to
xf
∆Y∞ = Y∞ =  . (2.49)
b
λ a+ 2xf

The relic density can now be expressed in terms of this result as follows

mχ Y∞ s0 h2
Ωh2 =
ρc
10 −10
GeV−2
≈ b
a + 40
3 × 10−27 cm3 s−1
≈ b
. (2.50)
a + 40

This expression explicitly shows that for larger values of the annihilation cross sec-
tion, smaller values of the relic density are obtained.

2.2.3 WIMPs

Equation (2.50) implies that in order to reproduce the correct relic abundance, dark matter
particles must have a thermally averaged annihilation cross section (from now on we will
18 FREEZE OUT OF MASSIVE SPECIES

shorten this to simply annihilation cross section when referring to hσvi) of the order of
hσvi ≈ 3 × 10−26 cm3 s−1 .
We can now consider a simple case in which dark matter particles self-annihilate into
Standard Model ones through the exchange (e.g., in an s-channel) of a gauge boson. It is
easy to see that if the annihilation cross section is of order hσvi ∼ G2F m2W IM P , where
GF = 1.16 × 10−5 GeV−2 , then the correct relic density is obtained for masses of the
order of ∼ GeV.

2.3 Computing the DM annihilation cross section

In the previous sections we have derived a relation between the thermally averaged annihi-
lation cross section and the corresponding dark matter relic abundance. This is very useful,
since it provides an explicit link with particle physics. A central point in that calculation
was the expansion in velocities of the thermally averaged annihilation cross section.

3 b0 15 c
hσvi = ha + bv 2 + cv 4 + . . .i = a + + + ... . (2.51)
2x 8 x2
Notice that in the expressions of the previous section we have defined b ≡ 3b0 /2. As we
also mentioned before, DM candidates tend to decouple when xf ≈ 20. For this value, the
rms velocity of the particles is about c/4, thus corrections of order x−1 can in general not
be ignored (they can be of order 5 − 10%). Moreover, some selection rules can actually
lead to a = 0 for some particular annihilation channels and in that case hσvi is purely
velocity-dependent.
It is important to define correctly the relative velocity that enters the above equation. In
Ref. [18] an explicitly Lorentz-invariant formalism is introduced where

d3 p1
Z Z
g1 C[f1 ] 3 =− hσviMøl (dn1 dn2 − dneq eq
1 dn2 ) , (2.52)
2π E1

where hσviMøl n1 n2 is invariant under Lorentz transformations and equals vlab n1,lab n2,lab
in the rest frame of one of the incoming particles. In our case the densities and Møller
velocity refer to the cosmic comoving frame. In terms of the particle velocities ~vi = p~i /Ei ,
1/2
vMøl = |~v1 − ~v2 |2 + |~v1 × ~v2 |2

. (2.53)
The thermally-averaged product of the dark matter pair-annihilation cross section and their
relative velocity hσvMøl i is most properly defined in terms of separate thermal baths for
both annihilating particles [18, 19],

d p1 d3 p2 σvMøl e−E1 /T e−E2 /T

R 3
hσvMøl i(T ) = R , (2.54)
d3 p1 d3 p2 e−E1 /T e−E2 /T

where p1 = (E1 , p1 ) and p2 = (E2 , p2 ) are the 4-momenta of the two colliding particles,
and T is the temperature of the bath. The above expression can be reduced to a one-
dimensional integral which can be written in a Lorentz-invariant form as [18]
Z ∞ √ 
1 2
√ s
hσvMøl i(T ) = ds σ(s)(s − 4mχ ) sK1 , (2.55)
8m4χ T K22 (mχ /T ) 4m2χ T
 COMPUTING THE DM ANNIHILATION CROSS SECTION 19

where s = (p1 + p2 )2 and Ki denote the modified Bessel function of order i. In comput-
ing the relic abundance [20] one first evaluates eq. (2.55) and then uses this to solve the
Boltzmann equation. The freeze out temperature can be computed by solving iteratively
the equation  r 
mχ 45 −1/2
xf = ln hσv Møl i(x f ) xf (2.56)
2π 3 2g∗ GN
√
where g∗ represents the effective number of degrees of freedom at freeze-out ( g∗ ≈ 9).
As explained in the previous section, one finds that the freeze-out point xf ≡ mχ /Tf is
approximately xf ∼ 20.
The procedure can be simplified if we consider that the annihilation cross section can
be expanded in plane waves. For example, consider the dark matter annihilation process
χχ → ij and assume that the thermally averaged annihilation cross section can be ex-
pressed as hσviij ≈ aij + bij x. It can then be shown that the coefficients aij and bij can
be computed from the corresponding matrix element. For example,

1 1
 Z 
1 Nc
aij = 2 β(s, mi , mj ) d cos θCM |Mχχ→ij |2 , (2.57)
mχ 32π 2 −1 s=4m2 χ

where θCM denotes the scattering angle in the CM frame, N c = 3 for q̄q final states and 1
otherwise, and
1/2  1/2
(mi + mj )2 (mi − mj )2

β(s, mi , mj ) = 1− 1− (2.58)
s s

The contribution for each final state is calculated separately.

2.3.1 Special cases

The derivation of equation (2.50) relied on the expansion of hσvi in terms of plane waves.
This expansion can be done when hσvi varies slowly with the energy (we can express this
in terms of the centre of mass energy s). However, there are some special cases in which
this does not happen and which deserve further attention.

Annihilation thresholds
A new annihilation channel χ + χ → A + B opens up when 2mχ ≈ mA + mB . In
this case the expansion in velocities of hσvi diverges (at the threshold energy) and it
is no longer a good approximation [18]. Notice in particular that below the threshold,
the expression of aij in Equation (2.57) is equal to zero (as it is only evaluated for
s > 4m2χ ). A qualitative way of understanding this is of course that DM particles have
a small velocity, which is here approximated to zero. In the limit of zero velocity, the
total energy available is determined by the DM mass.
However, we are here ignoring that a fraction of DM particles (given by their thermal
distribution in the Early Universe) have a kinetic energy sufficient to annihilate into
heavier particles (above the threshold). In other words, hσvi is different from zero
below the corresponding thresholds. A very good illustration of this effect is shown
in Ref. [18] and is here reproduced in Fig. 2.3.
The thin solid line corresponds to the approximate expansion in velocities and shows
that not only hσvi is zero below the threshold, but also diverges at the threshold,
20 FREEZE OUT OF MASSIVE SPECIES

Figure 2.3 Relativistic thermal average near a threshold (thick solid line) compared to the result
fro the expansion in powers of x−1 (thin line). Figure from Ref. [18].

thereby not leading to a good solution. Expression (2.55), represented by a thick solid
line, still provides a good solution .
Resonances
The annihilation cross section is not a smooth function of s in the vicinity of an s-
channel resonance. Thus, the velocity expansion of hσvi will fail (although once
more, expression (2.55) still provides a good solution). For a Breit-Wigner resonance
(due to a particle φ) we have
4πw m2φ Γ2φ
σ= B i B f , (2.59)
p2 (s − m2φ )2 + m2φ Γ2φ

in terms of the centre of mass momentum p = 1/2(s − 4m2 )1/2 and the statistical
factor w = (2J + 1)/(2S + 1)2 . The quantities Bi,f correspond to the branching
fractions of the resonance into the initial and final channel.
We can define the kinetic energy per unit mass in the lab frame, , as
(E1,lab − m) + (E2,lab − m) 2 − 4m2
= = , (2.60)
2m 4m2
and rewrite the expression for σ in the lab frame (we want to use Equation (3.21)
in Ref. [18] to compute hσvMøl i). Summing to all final states, and using vlab =
21/2 (1 + )1/2 /(1 + 2), we obtain

8πw γφ2
σvlab = b φ () , (2.61)
m2 ( − φ )2 + γφ2
2

with the definitions b() = Bi (1 − Bi )(1 + )1/2 /(1/2 (1 + 2), γφ = mφ Γφ /4m2 ,

and φ = (m2φ − 4m2 )/4m2 .
 COMPUTING THE DM ANNIHILATION CROSS SECTION 21

Figure 2.4 Relativistic thermal average in a resonance (thick solid line) compared to the result fro
the expansion in powers of x−1 (thin line). Figure from Ref. [18].

It can be shown that in the case of a very narrow resonance, γφ  1, the expression
above can be approximated as

8πw
σvlab = bφ ()πγφ δ( − φ ) , (2.62)
m2
the relativistic formula for the thermal average then reads [18]

16πw x 1/2 p
hσvMøl i = πγφ φ (1 + 2eφ )K1 (2x 1 + φ )bφ (eφ )θ(φ ) . (2.63)
m2 K22 (x)

Notice that φ > 0 when m < 2mφ , i.e., when the mass of the DM is not enough
to enter the resonance. The reason is easy to understand. Only through the extra
kinetic energy provided by the thermal bath, the resonance condition can be satisfied.
However, when the mass of the DM exceeds the resonance condition, the kinetic
energy only takes us further away from the resonant condition and the thermalised
cross section tends to vanish. In other words, the centre of mass rest energy exceeds
mφ /2. This can be seen in Figure 2.4.
For a large width the expression has to be computed numerically and can be found in
Ref. [18].

Coannihilations
When deriving Boltzmann equation (2.32) we have only considered one exotic species,
but this needs not be the case. In fact, in most particle models for DM, there are more
exotic species that we need to take into account. Notice that, in principle, this would
lead to a system of coupled Boltzmann equations. If we label exotic species as χi ,
with i = 0, 1 . . . k, and SM particles as A, B, we have to consider all number chang-
22 FREEZE OUT OF MASSIVE SPECIES

ing processes for each species,

(i) χi + χj → A + B
(ii) χi + A → χj + B
(iii) χj → χi + A

If we consider the (usual) case in which the DM is protected by a symmetry (e.g.,

in the case of Supersymmetric theories) and that the exotic particles all must decay
eventually into the lightest one χ0 , then, we must only trace the evolution of the total
Pk
number density of exotic species, n = i=0 ni . Under this assumption, processes
(ii) and (iii) do not need to be considered, as they do not change the number of exotics.
This is correct as long as the rate of these is faster than the expansion of the Universe.

Regarding process (i) we have to be aware that the cross section σij is going to appear
multiplied by the corresponding number densities, ni nj . Now, we are considering
the case in which both particles i and j are non-relativistic and as a consequence, ni,j
are Boltzmann suppressed, ni,j /e−mi,j /T . Thus, unless mj ≈ mi , the abundance of
χj is negligible and only the process χi + χj → A + B is important (and we are back
to the case of a single exotic).

However, when mj ≈ mi , there can be coannihilation effects and particle j may serve
as a channel through which particles i can be more effectively depleted. This is the
case, e.g., of the stau and the neutralino in supersymmetric theories.

2.4 Freeze-in of dark matter

In the previous section, we have explained in full detail how DM particles can be produced
in the early Universe through pair-annihilation processes. As we discussed earlier, if the
annihilation cross section happens to be of the order to the Electroweak scale, the resulting
relic density is of the right order to reproduce the observed DM abundance. However, this
WIMP paradigm is by no means the only way in which DM particles can be produced
in the right amount. In this section, we will address another interesting possibility that is
applicable to particles with a much smaller interaction scale.
Let us begin by assuming that the DM particles, χ, has extremely weak interactions, and
that its initial abundance is zero. An implicit assumption in all of this is that the reheating
temperature of the Universe after inflation was not high enough for χ to be in thermal
equilibrium. Notice that DM particles can still be produced by interactions of particles in
the thermal bath (following the notation of Ref. [21], we will refer to bath particles as Bi ).
The production rate is small, as a consequence of the small DM coupling, however, since
they are produced out of equilibrium, these DM particles do not annihilate (and of course
do not decay). As a consequence, a relic density builds up. The final DM density depends
on the specific interactions with bath particles.
In order to carry out the computation, notice that we can make use of Boltzmann equa-
tion, as formulated in eq. (2.26), but now we have to write the collisional operator accord-
ing to the interactions of DM particles with those of the bath. We will here consider one
characteristic example, that should serve as a guide to consider other possibilities.
 FREEZE-IN OF DARK MATTER 23

2.4.1 DM production from decays of heavier bath particles

Consider the decay of a heavy bath particle into a lighter one and a DM particle, B2 →
B1 χ. If mB2 > mB1 + mχ this process will dominate DM production. The collisional
operator is easy to write,
Z
dn g C[f ] 3
+ 3Hn = d p
dt (2π)3 E
Z
= dΠB1 dΠB2 dΠχ (2π)4 δ 4 (pB2 − pB1 − pχ ) ×
h i
2 2
|MB2 →B1 χ | fB2 (1 ± fB1 )(1 ± fχ ) − |MB1 χ→B2 | fB1 fχ (1 ± fB2 )
Z
= dΠB2 ΓB2 2gB2 mB2 fB2 . (2.64)

In the last line we have assumed no Pauli blocking to approximate (1 ± fB1 ) ≈ 1 and we
have neglected the initial abundance of DM particles, fχ ≈ 0. We have also expressed
2
|MB2 →B1 χ | in terms of the decay width, ΓB2 , the number of degrees of freedom ob
B2 and its mass. If we write the phase space element explicitly, and we consider that for
particles in thermal equilibrium we can approximate fB2 = 1/(eEB2 /T ± 1) ≈ e−EB2 /T ,
we are left with
Z ∞ 3
dn d pB2 fB2 ΓB2 mB2
+ 3Hn = gB2 3
. (2.65)
dt mB2 (2π) EB2

The integral on the right-hand side is easy to solve, as it can be reduced to the first modified
Bessel function of the second kind, K1 (mB2 /T ), resulting in

dn gB2 ΓB2 m2B2

+ 3Hn = T K1 (mB2 /T ) . (2.66)
dt 2π 2
As we did in the previous section, it is much simpler to rewrite this expression in terms of
the derivative of the yield Y = n/s in terms of the dimensionless variable x = mB2 /T ,
which leads to
Z ∞
45gB2 Mp ΓB2
Y = √ K1 (x) x3 dx . (2.67)
4π 4 (1.66)m2B2 g∗S g∗ xmin

Finally, solving for xmin = 0, yields

 
135 gB2 ΓB2 Mp
Y ≈ √ . (2.68)
8π 3 (1.66)g∗S g∗ m2B2
Rx
The function K1 (x) x3 has a maximum around x ≈ 2.4 and its integral 0 max K1 (x) x3 dx
stabilises above xmax ≈ 8, we have plotted this behaviour in Fig. 2.5. As we can observe,
the final yield is proportional to the bath’s particle partial decay width. This is very interest-
ing, as the decay width is directly proportional to the DM coupling square. Thus, the final
yield (and DM relic abundance) increases if the DM coupling increases. This behaviour is
the opposite as observed for WIMPs in eq.(2.49). This behaviour holds as long as the DM
coupling is small. If we increase the DM coupling, there comes a point at which the DM
particles produced reach thermal equilibrium and then we have to go back to the freeze-out
computation of the previous section.
24 FREEZE OUT OF MASSIVE SPECIES

2 4 6 8 10 12

Figure 2.5 Yield of a freeze-in species (in arbitrary units) as a function of x = m/T .

Finally, from eq.(2.68) we can use the explicit expression for the partial decay width
in a two-body final state and compute the resulting relic density. It can be seen that in
order to reproduce the correct relic abundance, the coupling needed is of the order of
λ ≈ 10−13 . Interestingly, the final value of the Yield is also sensitive to the initial value of
xmin from which we integrate. Notice that xmin will be given by the temperature at which
the Universe reheated after inflation. Thus, the frreeze-in mechanism has a very interesting
connection to inflation.
A similar computation can be made for other possible production channels, for example,
scattering of bath particles B1 B2 → B3 χ. In this case, the Boltzmann equation (2.64) has
to be modified accordingly taking into account the matrix elements of the process and the
number densities of the particles involved.
The freeze in mechanism has been used for example to argue that gravitinos (the super-
symmetric companion of the graviton) and axinos (the supersymmetric companion of the
axion) can be viable candidates for DM.

2.5 Late decays of unstable particles

As we have just seen in the freeze-in mechanism of the previous section, it is conceivable
that particles with a small coupling to SM ones are produced out of equilibrium due to
either scattering or decays of particles in the thermal bath. The frozen-in particles need
not be absolutely stable, but given their small couplings their lifetime can be large. If the
lifetime is larger than the age of the Universe (1017 s), we should not worry as the compu-
tation of the relic density is not altered and this simply corresponds to a case of decaying
DM (very interesting from the point of view of indirect detection). However, if the lifetime
is smaller, then it is obvious that this particle cannot be the DM. Late-decaying particles
can however contribute to the (non-thermal) production of DM. A possible scenario is as
follows.
Consider a canonical WIMP DM candidate, χ1 , which decoupled at x = mχ1 /T ≈ 20
via a freeze-out mechanism as described in Section 2.2.2, which leads to to a thermal
relic abundance Yχth1 . Simultaneously, a semi-stable particle χ2 , with very small couplings,
freezes-in via the mechanism explained in Section 2.4, with a yield Yχ2 . If particle χ2
 LATE DECAYS OF UNSTABLE PARTICLES 25

can decay into particle χ1 (after the latter has frozen-out), then eventually all the number
density of the heavy particle, is translated into the lighter one leading to a non-thermal
contribution, Yχ2 = Yχnt1
. The total yield for the lighter particle is therefore the sum of
both contributions
Yχ1 = Yχth1 + Yχnt1
. (2.69)
This exotic scenario can occur in supersymmetric models, where χ2 is the gravitino
and χ1 is the lightest neutralino. It should be emphasized that the late decay of exotic
particles (and the associated injection of electromagnetic and hadronic particles) can ruin
the predictions of BBN and also alter the black body shape of the CMB spectrum. Stringent
constraints exist if these decays occur after BBN, but in general the model is safe if the
lifetime of χ2 is smaller than approximately 1 minute.
CHAPTER 3

DIRECT DM DETECTION

3.1 Preliminaries

3.1.1 DM flux
We can easily estimate the flux of DM particles through the Earth. The DM typical velocity
is of the order of 300 km s−1 ∼ 10−3 c. Also, the local DM density is ρ0 = 0.3 GeV cm−3 ,
thus, the DM number density is n = ρ/m.
vρ 107
φ= ≈ cm−2 s−1 (3.1)
m m
These particles interact very weakly with SM particles.
Assuming a typical WIMP cross section σ

3.1.2 Kinematics

Direct DM detection is based on the search of the scattering between DM particles and
nuclei in a detector. This process is obly observable through the recoiling nucleus, with an
energy ER . DM particles move at non-relativistic speeds in the DM halo. Thus, the dy-
namics of their elastic scattering off nuclei are easily calculated. In particular, the recoiling
energy of the nucleus is given by
1 4mχ mN 1 + cos θ
ER = mχ v 2 (3.2)
2 (mχ + mN )2 2
Dark Stuff. 27
By D. G. Cerdeño, IPPP, University of Durham
28 DIRECT DM DETECTION

It can be checked that for DM particles with a mass of the order of 100 GeV, this leads
to recoil energies of approximately ER ∼ 100 keV. Notice also that the maximal energy
transfer occurs on a head-on-collision and when the DM mass is equal to the target mass.
In such a case
1 1 1  mχ 
ERmax
= mχ v 2 = mχ × 10−6 = keV (3.3)
2 2 2 1 GeV
where we have used that in a DM halo the typical velocity is v ∼ 10−3 c.
Experiments must therefore be very sensitive and be able to remove an overwhelming
background of ordinary processes which lead to nuclear recoils of the same energies.

3.2 The master formula for direct DM detection

The total number of detected DM particles, N , can be understood as the product of the DM
flux (which is equal to the DM number density, n, times its speed, v), times the effective
area of the target (i.e., the number of targets NT times the scattering cross-section, σ), all
of this multiplied by the observation time, t,

N = t n v NT σ . (3.4)

We will be interested in determining the spectrum of DM recoils, i.e., the energy depen-
dence of the number of detected DM particles. Thus,
dN dσ
= t n v NT . (3.5)
dER dER
Now, the DM velocity is not unique, and in fact DM particles are described by a local
velocity distribution, f (~v ), where ~v is the DM velocity in the reference frame of the detec-
tor. We therefore have to integrate to all possible DM velocities, with their corresponding
probability density, Z
dN dσ
= t n NT vf (~v ) d~v , (3.6)
dER vmin dE R

where q
vmin = mχ ER /2µ2χN (3.7)
is the minimum speed necessary to produce a DM recoil of energy ER , in terms of the
WIMP-nucleus reduced mass, µχN . Using n = ρ/mχ and NT = MT /mN (where MT
is the total detector mass and mN is the mass of the target nuclei), and defining the exper-
imental exposure  = t MT , we arrive at the usual expression for the DM detection rate
Z
dN ρ dσ
= vf (~v ) d~v . (3.8)
dER mχ mN vmin dER

3.2.1 The scattering cross section

The scattering takes place in the non-relativistic limit. The cross section is therefore ap-
proximately isotropic (angular terms being suppressed by v 2 /c2 ∼ 10−6 . This implies
that
dσ σ
∗
= constant = (3.9)
d cos θ 2
 THE MASTER FORMULA FOR DIRECT DM DETECTION 29

On the other hand,

max 1 + cos θ∗ dER max

ER
ER = ER → = (3.10)
2 d cos θ∗ 2
From this, we can see that
dσ dσ d cos θ∗ σ mN σ
= = max = (3.11)
dER d cos θ∗ dER ER 2µ2χN v 2

Notice finally that the momentum transfer from WIMP interactions reads (remember
that we are considering non-relativistic processes and thus we neglect the kinetic energy of
the nucleus) p
q = 2 mN ER (3.12)
and is typically of the order of the MeV. The equivalent de Broglie length would be λ ∼
2π~/p ∼ 10 − 100 fm. For light nuclei, the DM particle sees the nucleus as a whole,
without substructure, only for heavier nuclei we have to take into account a suppression
form factor. The nuclear form factor, F 2 (ER ), accounts for the loss of coherence
dσ mN σ0 2
= F (ER ) (3.13)
dER 2µ2χN v 2

Finally, the scattering cross section receives different contributions, depending on the mi-
croscopic description of the interaction.
In the end, we can
Z
dN ρ 2 f (~v )
= 2 σ0 F (ER ) d~v . (3.14)
dER 2 mχ µχN vmin v

The inverse mean velocity

Z
f (~v )
η(vmin ) = d~v . (3.15)
vmin v
is the main Astrophysical input.

3.2.2 The importance of the threshold

From the kinematics of the DM-nucleus interaction, we see that, for a given recoil energy
ER , we require a minimal velocity of DM particles, given by expression (3.7).
Thus, given that experiments are only sensitive to DM interactions above a certain en-
ergy threshold, ET , this means that we are only probing a part of the WIMP velocity
distribution function (for a given DM mass). Conversely, given that DM particles have a
maximum velocity in the halo (otherwise they become unbound and escape the galaxy),
the experimental energy threshold is a limitation to explore low-mass WIMPs.

EXAMPLE 3.1

Consider a germanium experiment and a xenon experiment with a threshold of 2 keV.

Given the escape velocity in a canonical isothermal halo, vesc = 554 km s−1 , deter-
mine the minimum DM mass that these experiments can probe.
This is the reason that experiments loose sensitivity for small masses.
30 DIRECT DM DETECTION

3.2.3 Velocity distribution function

It is customary to consider the Isothermal Spherical Halo, which assumes that the Milky
Way (MW) halo is an isotropic, isothermal sphere with density profile ρ ∝ r−2 . The
velocity distribution, in the galactic rest frame, for such a halo reads
1 v |2
|~
fgal (~v ) = 3/2
e− 2σ2 , (3.16)
(2πσ)

where the√one-dimensional velocity dispersion, σ, is related to the circular speed, vc , as

σ = vc / 2. The canonical values are vc = 220km s−1 , with a statistical error of order
10% (see references in [22])
Now, in order to use it for direct detection experiments we need to carry out a Galilean
transformation ~v → ~v + ~vE , such that

f (~v ) = fgal (~v + ~vE (t)). (3.17)

where ~vE (t) is the velocity of the Earth with respect to the Galactocentric rest frame.

~vE (t) = ~vLRF + ~v + ~vorbit (t) (3.18)

Notice that vE includes contributions from the speed of the Local Standard of Rest vLSR ,
the peculiar velocity of the Sun with respect to vLSR , and the Earths velocity around the
Sun, which has an explicit time dependence.
Notice that if we work with the SHM, the angular integration in the computation of
direct detection rates can be easily done as follows
Z Z Z Z
f (~v ) 3 1 v |2 +|~
|~ v E |2 |~
v | |~
vE | cos θ
d v = dφ d cos θ dv v 2 3/2
e− 2σ2 e σ2
v (2πσ )
2σ 2
 
|~v | |~vE |
Z
v |2 +|~
|~ v E |2
−
= 2π dv v e 2σ 2 sinh
|v||~vE |(2πσ)3/2 σ2
√  
|~v | |~vE |
Z
2 v |2 +|~
|~ v E |2
= dv √ e− 2σ2 sinh (3.19)
πσ|~vE | σ2

3.2.4 Energy resolution, threshold energy and experimental efficiency

3.3 Exponential spectrum

η(vmin ) ∝ e−αER (3.20)

3.4 Annual modulation

vE (t) = v0 [1.05 + 0.07 cos(2π(t − tp )/1yr)] (3.21)

The amplitude of the modulation in the velocity is very small, approximately a 7% (vorbit cos θorbit /(vLRF +
v ) ≈ 15/220
This eventually implies a modulation in η(vmin ), which means that the amount of DM
particles in the tail of the distribution can change substantially. This, in turn, leads to a
modulation in the detected rate.
 DIRECTIONAL DETECTION 31

3.5 Directional detection

The current experimental situation regarding directional DM detection has been sum-
marised in the recent review article, Ref. [22].

3.6 Coherent neutrino scattering

Solar neutrinos might leave a signal in DD experiments, either through their coherent scat-
tering with the target nuclei or through scattering with the atomic electrons.
In general, the number of recoils per unit energy can be written
Z
dR  dφν dσν
= dEν , (3.22)
dER mT dEν dER
where  is the exposure and mT is the mass of the target electron or nucleus. If several iso-
topes are present, a weighted average must be performed over their respective abundances.
The SM neutrino-electron scattering cross section is
G2F me

dσνe
= (gv + ga )2 + (3.23)
dER 2π
 2 
ER me ER
(gv − ga )2 1 − + (ga2 − gv2 ) ,
Eν Eν2
where GF is the Fermi constant, and
1 1
gv;µ,τ = 2 sin2 θW − ; ga;µ,τ = − , (3.24)
2 2
for muon and tau neutrinos. In the case νe + e → νe + e, the interference between neutral
and charged current interaction leads to a significant enhancement:
1 1
gv;e = 2 sin2 θW + ; ga;e = + . (3.25)
2 2
The neutrino-nucleus cross section in the SM reads
G2
 
dσνN mN ER
= F Q2v mN 1 − F 2 (ER ), (3.26)
dER 4π 2Eν2
where F 2 (ER ) is the nuclear form factor, for which we have taken the parametrisation
given by Helm [23]. Qv parametrises the coherent interaction with protons (Z) and neu-
trons (N = A − Z) in the nucleus:
Qv = N − (1 − 4 sin θW )Z. (3.27)

3.7 Inelastic

[24]
The WIMP needs to have sufficient speed to interact with the nucleus and promote to
an excited state (with energy separation δ)
1
µχN v 2 > δ (3.28)
2
32 DIRECT DM DETECTION

This leads to the condition

r  
1 mN ER
vmin = +δ (3.29)
2mN ER µχN

Therefore, the main effect at a given experiment is to limit the sensitivity only to a part
of the phase space of the halo. This favours heavy nuclei (since they can transfer more
energy to the outgoing WIMP) and can account for observation in targets such as iodine
(DAMA/LIBRA) while avoiding observation in lighter ones such as Ge (CDMS)
CHAPTER 4

AXIONS

4.1 The Strong QCD Problem

The most general gauge invariant QCD Lagrangian up to dimension four reads
1 αS a a,µν
L = − Gaµν Ga,µν + q̄ (iγµ Dµ − Mq ) q − θG G̃ (4.1)
4 8π µν
The zero temperature mass of the axion field is therefore field by non-perturbative QCD,
related to the axion scale, fa , and can be computed to be
10−11
ma (T = 0) = 5.7 × 10−5 . (4.2)
fa
The axion potential is periodic, but it is typical to adopt a harmonic approximation
1 2
V (θ) = m2a (T ) (1 − cos θ) ≈ m (T )θ2 . (4.3)
2 a

4.2 Axions production

The Peccei-Quinn (PQ) symmetry breaks at a high temperature, TP Q , providing a well-

known dynamical solution to the so-called strong-CP problem. In this scenario, the (com-
plex) axion field takes a non-vanishing vacuum expectation value, which fixes one of its
component, leaving just a phase, which we will call axion.
Dark Stuff. 33
By D. G. Cerdeño, IPPP, University of Durham
34 AXIONS

There are various production mechanisms for axions, which can be thermal or non-
thermal. Which of these mechanisms dominate, strongly depends on whether the PQ tem-
perature is higher or lower than the reheating temperature, TR .
The axion mass arises from non-perturbative QCD effects, which are negligible at high
temperatures, but become relevant at a critical time, t1 that satisfies ma t1 ≈ 1, and where
the temperature of the Universe is approximately T1 ≈ 1 GeV.

Thermal production Axions can be produced in the plasma of the Early Universe, mainly
by processes which involve quarks and gluons [25]. These processes can lead to a popu-
lation of hot axions if the Peccei-Quinn axion scale is fa < 1012 GeV [26] (and in fact,
any population of cold axions produced before this period would also thermalise). After
colour confinement, the leading thermal production process is through the coupling to pi-
ons, π + π ↔ π + a [27]. These hot axions would contribute to the universe energy density
and, like in the case of neutrinos, cosmological constraints impose upper bounds on their
mass (see for example Ref. [28]). I, particular, if thermally produced, axions would con-
tribute to the effective number of relativistic degrees of freedom, which is measured to be
Nef f = 3.15 ± 0.23, leading to an upper bound on the axion mass of ma < 0.529 eV [29].
A recent update of the thermal axion production can be found in Ref. ??, which includes
not only axion couplings to gluons, but also contributions from couplings to electroweak
bosons.

Misalignment mechanism Cold axions can be produced in the early Universe through
the so-called misalignment mechanism [30, 31, 32], also called vacuum realignment in the
literature.
We can consider a toy model, where the potential for the complex field φ(x) at high
temperature reads
λ
2
V (φ) = |φ|2 − va2 , (4.4)
4
Then, when the Universe cools to TP Q ∼ va the field φ takes a non-vanishing vacuum
expectation value in each Hubble volume
hφi ≡ va eiθ(x) . (4.5)
The remaining angle is related to what we will define as the axion field, a(x) as
a(x) = va θ(x) . (4.6)
The axion mass plays no role for high temperatures, but when the Universe cools down
to the confinement scale T ∼ Λ, non-perturbative QCD corrections “tilt” the potential,
Ṽ (θ) = m2a (T )fa2 (1 − cos θ) . (4.7)
This forces the axion to “realign” (after TP Q it adopted a given value of θ and now it is
forced to go towards the minimum of the potential, θ = 0). This also gives the axion an
effective mass whose time-dependence can be calculated in terms of the low-temperature
value, ma , to be  4
Λ
ma (T ) = 0.2 ma . (4.8)
fa
The equation of motion fo rhte axion can be computed in a FRW universe, using the
definition of the D’Alambertian
1
θ̈ + 3H θ̇ − 2 ∇2 θ + m2a (T (t)) sin θ = 0 . (4.9)
a (t)
 AXIONS PRODUCTION 35

Notice that we are using a(t) to denote the scale parameter.

If TP Q > TR then the axion reached the PQ minimum before reheating, taking a ran-
dom value for the phase, which is then fixed by cosmic expansion. This means that θ is
homogeneous at QCD confinement. This implies that the spatial derivatives can be ne-
glected and we are left with the equation of a damped harmonic oscillator (using sin θ ≈ θ
for small angles),
θ̈ + 3H θ̇ + m2a θ = 0 . (4.10)
The axion mass is not relevant at early times (and therefore θ remains constant), but after
a time t1 = 1/ma (T1 (t1 )), that is equivalent to a critical temperature of approximately
1 GeV. At this point, the axion field responds by attempting to minimise its potential,
oscillating around minimum (vacuum-realignment). The critical time and temperature read
 1/3
−7 fa
t1 ≈ 2 × 10 s , (4.11)
1012 GeV
and 1/6
1012 GeV

T1 ≈ 1 GeV . (4.12)
fa
One can estimate the momentum of the resulting quanta of the axion field. Assuming a
typical coupling of fa ∼ 10−12 , which corresponds to an axion mass of ma ∼ 6×10−6 eV,
one obtains
1
pa (t1 ) ∼ ∼ 10−9 eV . (4.13)
t1
In other words, classical, spatially coherent oscillating fields are equivalent to a coherent
state of extremely non-relativistic dark matter, i.e. CDM). Notice that these axions, once
produced, are not thermalised and therefore remain as cold dark matter throughout the
evolution of the Universe.
Finally, we can compute the energy density of the axion field

fa 2  2 
ρ= θ̇ + ma (t)2 θ2 . (4.14)
2
The initial energy of the axion before starting oscillating is a function of the original mis-
alignment angle, θ1 ,
fa 2 2
ρ= m (t1 )θ12 . (4.15)
2 a
When converted to a cosmological density parameter, we obtain
 2
2 fa
Ωa h = 0.15 θ12 . (4.16)
1012 GeV

Notice that, according to this equation, an axion with a scale fa ∼ 1012 GeV would
reproduce the cold dark matter relic density observed by the Planck satellite, assuming that
the original mis-alignment is of the order of θ1 ∼ 1 (for naturalness arguments). If one
is willing to abandon naturalness and introduce a degree of fine-tuning in θ1 , the axion
scale can be raised arbitrarily. This solution is normally referred to as “anthropic” for that
reason.
REFERENCES

[1] J. Silk et al., Particle Dark Matter: Observations, Models and Searches. Cambridge Univ.
Press, Cambridge, 2010, 10.1017/CBO9780511770739.
[2] G. Bertone and D. Hooper, A History of Dark Matter, Submitted to: Rev. Mod. Phys. (2016) ,
[1605.04909].
[3] V. C. Rubin and W. K. Ford, Jr., Rotation of the Andromeda Nebula from a Spectroscopic
Survey of Emission Regions, Astrophys. J. 159 (1970) 379–403.
[4] V. C. Rubin, N. Thonnard and W. K. Ford, Jr., Rotational properties of 21 SC galaxies with a
large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/,
Astrophys. J. 238 (1980) 471.
[5] A. Bosma, 21-cm line studies of spiral galaxies. 2. The distribution and kinematics of neutral
hydrogen in spiral galaxies of various morphological types., Astron. J. 86 (1981) 1825.
[6] T. S. van Albada, J. N. Bahcall, K. Begeman and R. Sancisi, The Distribution of Dark Matter
in the Spiral Galaxy NGC-3198, Astrophys. J. 295 (1985) 305–313.
[7] R. Gavazzi, C. Adami, F. Durret, J.-C. Cuillandre, O. Ilbert, A. Mazure et al., A weak lensing
study of the Coma cluster, Astron. Astrophys. 498 (2009) L33, [0904.0220].
[8] F. Zwicky, Die Rotverschiebung von extragalaktischen Nebeln, Helv. Phys. Acta 6 (1933)
110–127.
[9] F. Zwicky, On the Masses of Nebulae and of Clusters of Nebulae, Astrophys. J. 86 (1937)
217–246.
[10] J. M. Kubo, A. Stebbins, J. Annis, I. P. Dell’Antonio, H. Lin, H. Khiabanian et al., The Mass
Of The Coma Cluster From Weak Lensing In The Sloan Digital Sky Survey, Astrophys. J. 671
(2007) 1466–1470, [0709.0506].
[11] M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman et al., Direct
constraints on the dark matter self-interaction cross-section from the merging galaxy cluster
1E0657-56, Astrophys. J. 606 (2004) 819–824, [astro-ph/0309303].

Dark Stuff. 37
By D. G. Cerdeño, IPPP, University of Durham
38 REFERENCES

[12] D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones et al., A direct

empirical proof of the existence of dark matter, Astrophys. J. 648 (2006) L109–L113,
[astro-ph/0608407].
[13] S. D. McDermott, H.-B. Yu and K. M. Zurek, Turning off the Lights: How Dark is Dark
Matter?, Phys. Rev. D83 (2011) 063509, [1011.2907].
[14] C. Kouvaris, Composite Millicharged Dark Matter, Phys. Rev. D88 (2013) 015001,
[1304.7476].
[15] A. D. Dolgov, S. L. Dubovsky, G. I. Rubtsov and I. I. Tkachev, Constraints on millicharged
particles from Planck data, Phys. Rev. D88 (2013) 117701, [1310.2376].
[16] E. Del Nobile, M. Nardecchia and P. Panci, Millicharge or Decay: A Critical Take on Minimal
Dark Matter, JCAP 1604 (2016) 048, [1512.05353].
[17] S. Tulin, H.-B. Yu and K. M. Zurek, Resonant Dark Forces and Small Scale Structure, Phys.
Rev. Lett. 110 (2013) 111301, [1210.0900].
[18] P. Gondolo and G. Gelmini, Cosmic abundances of stable particles: Improved analysis, Nucl.
Phys. B360 (1991) 145–179.
[19] M. Srednicki, R. Watkins and K. A. Olive, Calculations of Relic Densities in the Early
Universe, Nucl. Phys. B310 (1988) 693.
[20] T. Nihei, L. Roszkowski and R. Ruiz de Austri, Exact cross-sections for the neutralino slepton
coannihilation, JHEP 07 (2002) 024, [hep-ph/0206266].
[21] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, Freeze-In Production of FIMP Dark
Matter, JHEP 03 (2010) 080, [0911.1120].
[22] F. Mayet et al., A review of the discovery reach of directional Dark Matter detection, Phys.
Rept. 627 (2016) 1–49, [1602.03781].
[23] R. H. Helm, Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even
Nuclei, Phys. Rev. 104 (1956) 1466–1475.
[24] D. Tucker-Smith and N. Weiner, Inelastic dark matter, Phys. Rev. D64 (2001) 043502,
[hep-ph/0101138].
[25] M. S. Turner, Thermal Production of Not SO Invisible Axions in the Early Universe, Phys. Rev.
Lett. 59 (1987) 2489.
[26] E. Masso, F. Rota and G. Zsembinszki, On axion thermalization in the early universe, Phys.
Rev. D66 (2002) 023004, [hep-ph/0203221].
[27] S. Chang and K. Choi, Hadronic axion window and the big bang nucleosynthesis, Phys. Lett.
B316 (1993) 51–56, [hep-ph/9306216].
[28] A. Melchiorri, O. Mena and A. Slosar, An improved cosmological bound on the thermal axion
mass, Phys. Rev. D76 (2007) 041303, [0705.2695].
[29] E. Di Valentino, E. Giusarma, M. Lattanzi, O. Mena, A. Melchiorri and J. Silk, Cosmological
Axion and neutrino mass constraints from Planck 2015 temperature and polarization data,
Phys. Lett. B752 (2016) 182–185, [1507.08665].
[30] J. Preskill, M. B. Wise and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B120
(1983) 127–132.
[31] L. F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B120
(1983) 133–136.
[32] M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. B120 (1983) 137–141.