Theoretical Cosmology

A. A. Coley:
arXiv:1909.05346v4 [gr-qc] 9 Feb 2020

Department of Mathematics and Statistics, Dalhousie University,

Halifax, Nova Scotia, B3H 4R2, Canada
email: Alan.Coley@Dal.ca

G. F. R. Ellis:
Mathematics Department, University of Cape Town,
Rondebosch, Cape Town 7701, South Africa
email: george.ellis@uct.ac.za
February 11, 2020

Abstract
We review current theoretical cosmology, including fundamental and
mathematical cosmology and physical cosmology (as well as cosmology in
the quantum realm), with an emphasis on open questions.

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Contents
1 Introduction 4
1.0.1 The uniqueness of the Universe . . . . . . . . . . . . . . . 4
1.0.2 The background model . . . . . . . . . . . . . . . . . . . . 4
1.0.3 Inhomogeneous models . . . . . . . . . . . . . . . . . . . . 5
1.0.4 Perturbed models . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 Fundamental issues . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 Open problems and GR . . . . . . . . . . . . . . . . . . . 7
1.1.2 Philosophical issues . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 Underlying theory . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.5 Homogeneity scale . . . . . . . . . . . . . . . . . . . . . . 9
1.1.6 Local and global coordinates . . . . . . . . . . . . . . . . 10
1.1.7 Periodic boundary conditions in structure formation studies 10
1.1.8 Weak field approach . . . . . . . . . . . . . . . . . . . . . 11
1.1.9 Quantum realm and multiverse . . . . . . . . . . . . . . . 11
1.2 Definition of a cosmological model . . . . . . . . . . . . . . . . . 11
1.3 Problems in mathematical cosmology . . . . . . . . . . . . . . . . 13
1.3.1 Singularity theorems . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Bouncing models . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Mathematical results . . . . . . . . . . . . . . . . . . . . . 15
1.3.4 Extension to cosmology . . . . . . . . . . . . . . . . . . . 16
1.3.5 Computational cosmology . . . . . . . . . . . . . . . . . . 16
1.4 Cosmological observations . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Tension in the Hubble constant . . . . . . . . . . . . . . . 18

2 Problems in theoretical cosmology 20

2.1 Acceleration: dark energy . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Acceleration: inflation . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Alternatives to inflation . . . . . . . . . . . . . . . . . . . 22
2.2.2 Bouncing models revisited . . . . . . . . . . . . . . . . . . 22
2.3 The physics horizon and Synge’s g-method . . . . . . . . . . . . . 23
2.4 Dynamical behaviour of cosmological solutions . . . . . . . . . . 23
2.4.1 Stability of cosmological solutions . . . . . . . . . . . . . 24
2.4.2 Stability of de Sitter spacetime . . . . . . . . . . . . . . . 24
2.4.3 The nature of cosmological singularities . . . . . . . . . . 25
2.4.4 Isotropic singularity . . . . . . . . . . . . . . . . . . . . . 26

3 Problems in physical cosmology 27

3.1 Origin of structure . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Large scale structure of the Universe . . . . . . . . . . . . 27
3.1.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Non-linear perturbations . . . . . . . . . . . . . . . . . . . 29
3.1.4 Non-linear regime . . . . . . . . . . . . . . . . . . . . . . 30

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 3.1.5 Non-Gaussianities . . . . . . . . . . . . . . . . . . . . . . 31
3.1.6 Simulations and post-Newtonian cosmological perturbations 33
3.2 Black holes and gravitational waves . . . . . . . . . . . . . . . . . 33
3.2.1 Gravitational waves . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Primordial gravitational waves . . . . . . . . . . . . . . . 34
3.2.3 Primordial black holes . . . . . . . . . . . . . . . . . . . . 35
3.3 Effects of structure on observations: Gravitational lensing . . . . 36
3.4 Backreaction and averaging . . . . . . . . . . . . . . . . . . . . . 37
3.4.1 Backreaction magnitude . . . . . . . . . . . . . . . . . . . 38
3.5 Spatial curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Problems from the quantum realm 43

4.1 The problem of quantum gravity: . . . . . . . . . . . . . . . . . . 43
4.1.1 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . 44
4.2 Singularity resolution . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Singularity resolution and a quantum singularity theorem 45
4.2.2 Cosmological singularity resolution . . . . . . . . . . . . . 45
4.3 Quantum gravity and inflation . . . . . . . . . . . . . . . . . . . 46
4.3.1 String inflation . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Concluding remarks 49

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1 Introduction
Cosmology concerns the study of the large scale behavior of the Universe within
a theory of gravity, which is usually assumed to be General Relativity (GR). 1
It has a unique nature that makes it a distinctive science in terms of its relation
to both scientific explanation and testing.

1.0.1 The uniqueness of the Universe

There is only one Universe which we effectively see from one spacetime point
(because it is so large) [1]. This is the foundational constraint in terms of
both scientific theory (how do we distinguish laws from initial conditions?) and
observational testing of our models:
• We can only observe our Universe on one past light cone.
• We have to deduce four dimensional (4D) spacetime structure from a 2D
image; distance estimations are therefore key.
• We can’t see many copies of the universe to deduce laws governing how
universes operate. Therefore, we have to compare the one universe with
simulations of what might have been.
This consequently leads to the important question of what variations from our
model need explaining and what are statistical anomalies that do not need any
explanation (i.e., cosmic variance). This question arises, for example, regarding
some cosmic microwave background (CMB) anomalies.

1.0.2 The background model

Cosmology is the study of the behaviour of the Universe when small-scale struc-
tures (such as, for example, stars and galaxies) can be neglected. The “Cosmo-
logical Principle”, which can be regarded as a generalization of the Copernican
Principle, is often assumed to be valid. This principle asserts that: On large
scales the Universe can be well–modeled by a solution to Einstein’s equations
which is spatially homogeneous and isotropic. This implies that a preferred no-
tion of cosmological time exists 2 such that at each instant of time, space appears
the same in all directions (isotropy) and at all places (spatial homogeneity) on
the largest scales. This is, of course, certainly not true on smaller scales such
1 Commonly used Acronyms used in this paper include: Cold dark matter (CDM). Cos-

mic microwave background (CMB). Einstein field equations (EFE). Friedmann-Lemaı́tre-

Robertson-Walker (FLRW). Gravitational waves (GW). General Relativity (GR). Large
scale structure (LSS). Linear perturbation theory (LPT). Loop quantum cosmology (LQC).
Loop quantum gravity (LQG). Primordial gravitational waves (PGW). Primordial non-
Gaussianities (PNG). Quantum gravity (QG).
2 Except in the degenerate cases of spacetimes of constant curvature (de Sitter, anti-de Sitter

and Minkowski spacetimes). Such universe models do not correspond to the real Universe,
which has preferred world lines everywhere [1].

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as the astrophysical scales of galaxies, and it would thus be better if the cosmo-
logical principle could be deduced rather than assumed a priori (i.e., could late
time spatial homogeneity and isotropy be derived as a dynamical consequence
of the Einstein Field Equations (EFE) under suitable physical conditions and
for appropriate initial data). This has been addressed, in part, within the in-
flationary paradigm, when scalar fields are dynamically important in the early
Universe.
The Cosmological Principle leads to a background Friedmann-Lemaı́tre-
Robertson-Walker (FLRW) model, and the EFE determine its dynamics. The
concordance spatially homogeneous and isotropic FLRW model (with a three-
dimensional comoving spatial section of constant curvature which is assumed
simply connected) with a cosmological constant, Λ, representing the cosmolog-
ical constant as an interpretation of dark energy, and CDM is the acronym
for cold dark matter (or so-called ΛCDM cosmology or standard cosmology for
short), has been very successful in describing current observations. Early uni-
verse inflation is often regarded as a part of the concordance model. The back-
ground spatial curvature of the universe, often characterized by the normalized
curvature parameter (Ωk ), is predicted to be negligible by most of simple in-
flationary models. Regardless of whether inflation is regarded as part of the
standard model, spatial curvature is often assumed zero.

1.0.3 Inhomogeneous models

One of the greatest challenges in cosmology is understanding the origin of the
structure of the Universe. An essential feature in structure formation is the
study of inhomogeneities and anisotropies in cosmology. There are three ap-
proaches:
• Using exact solutions and properties where possible [2]. In particular,
the Lemaı̂tre-Tolman-Bondi (“LTB”) spherically symmetric dust model
has been widely used, while the “Strenge Losungen” approach of Ehlers,
Kundt, Sachs, and Trümper at Hamburg provides a powerful method of
examining generic properties of fluid solutions.
• Perturbed models where structure formation can be investigated, as pio-
neered by Lifschitz, Peebles, Sachs and Wolfe, Bardeen (see below).
• Numerical simulations, mainly Newtonian, but now being extended to the
GR by various groups.
In particular, this enables an investigation of the scalar-tensor ratio and CMB
polarization, and redshift space distortions and non-Gaussianities, which are
key to testing inflationary universe models.
It is also important to consider the averaging, backreaction, and fitting prob-
lems relating the perturbed and background models. The main point here is
that the same spacetime domain can be modeled at different averaging scales
to obtain, for example, models representing galactic scales L1 , galaxy cluster

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scales L2 , large scale structure scales L3 , and cosmological scales L4 , with cor-
responding metrics, Ricci tensors, and matter tensors; the issue then is, firstly,
how the FE at different scales are related [3] and, secondly, how observations at
these different scales are related [4].

1.0.4 Perturbed models

In particular, the structure of the Universe can be investigated in cosmology
via perturbed FLRW models. A technical issue that arises is the gauge issue:
how do we map the background model (smooth) to a more realistic (lumpy)
model? One must either handle gauge freedom by very carefully delineating
what freedom remains at each stage of coordinate specialisation, or use gauge
covariant variables (see later).
Cosmic inflation provides a causal mechanism for the generation of pri-
mordial cosmological perturbations in our Universe, through the generation of
quantum fluctuations in the inflaton field which act as seeds for the observed
anisotropies in the CMB and large scale structure (LSS) of our Universe. Al-
though inflationary cosmology is not the only game in town, it is the simplest
and perhaps the only scenario which is currently self-consistent from the point
of view of low energy effective field theory. The recent Planck observations
confirm that the primordial curvature perturbations are almost scale-invariant
and Gaussian. In the standard cosmology, the primordial perturbations, corre-
sponding to the seeds for the LSS, are chosen from a Gaussian distribution with
random phases. This assumption is justified based on experimental evidence,
regardless of whether or not inflation is assumed.
Predictions arising for matter power spectra and CMB anisotropy power
spectra can then be compared with observations; this is a central feature of cos-
mology today. Together with comparisons of element abundance observations
with primordial nucleosynthesis predictions, this has turned cosmology from
philosophy to a solid physical theory. Finally, quantum fluctuations of the met-
ric during inflation, imprinted in primordial B-mode perturbations of the CMB,
are perhaps the most vivid evidence conceivable for the reality of quantum grav-
ity (QG) in the early history of our Universe. Indeed, any direct detection of
primordial gravitational waves (PGW) and primordial non-Gaussianities (PNG)
with the specific features predicted by inflation would provide strong indepen-
dent support to this framework.

In this article we review the philosophical, mathematical, theoretical, phys-

ical (and quantum) challenges to the standard cosmology. For the most part,
important non-theoretical issues (such as, for example, experiments and data
analysis) are not discussed.

1.1 Fundamental issues

Cosmology is a strange beast. On the one hand it has evolved into a mature
science, complete with observations, data analysis and numerical methods. On

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the other hand it contains philosophical assumptions that are not always sci-
entific; this includes, e.g., the assumption of spatial homogeneity and isotropy
at large scales outside our particle horizon and issues regarding inflation and
the multiverse. As well as philosophical questions, there are fundamental phys-
ical problems (e.g., what us the appropriate model for matter, and what is
the applicability of coarse graining) as well as mathematical issues (e.g., the
gauge invariance problem in cosmology). Indeed, many of the open problems
in theoretical cosmology involve the nature of the origin and details of cosmic
inflation.

1.1.1 Open problems and GR

Noted problems have always been of importance and part of the culture in math-
ematics [5]. The twenty-three problems by Hilbert [6] are perhaps the most
well known problems in mathematics. In addition, the set of fifteen problems
presented in [7] nicely illustrate a number of open problems in mathematical
physics. The most important and interesting unsolved problems in fundamental
theoretical physics include foundational problems of quantum mechanics and
the unification of particles and forces and the fine tuning problem in the quan-
tum regime, the problem of quantum gravity and, of course, the problem of
“cosmological mysteries” [8]. However, it should be noted that some of them
are in fact philsophical problems, in that they are not dealing with any conflict
with observations.
We are primarily interested here in problems which we shall refer to as prob-
lems in theoretical cosmology, and particularly those that are susceptible to a
rigorous treatment within mathematical cosmology. Problems in GR have been
discussed elsewhere [9]. There are some problems in GR that are relevant in cos-
mology, and theorems can be extended into the cosmological regime by including
models with matter. For example, generic spacelike singularities, traditionally
regarded as being cosmological singularities, have been studied in detail [10].
It is also of interest to extend mathematical stability results to the case of a
non-zero cosmological constant [11].

1.1.2 Philosophical issues

Philosophical problems have always played an important role in cosmology [12];
e.g., are we situated at the center of the Universe or not and, even in the earliest
days of Einstein, is the Universe static or evolving. In addition, in cosmology
the dynamical laws governing the evolution of the universe, the classical EFE,
require boundary conditions to yield solutions. But in cosmology, by definition,
there is no rest of the Universe to pass their specification off to. The cosmological
boundary conditions must be one of the fundamental laws of physics.
There are a number of important philosophical issues that include the fol-
lowing: There is only one Universe. Consistency of one model does not rule out
alternative models. What can a statistical analysis with only one data point
tell us? What is observable? Due to the existence of horizons, the Universe is

7
only observed on or within our past light cone. A typical question in cosmology
is: Why is the Universe so smooth. Must a suitable explanation be in terms of
‘genericity’ (of possible initial conditions), or can specialness lead to a possible
explanation. There is no physical law that is violated by fine tuning. Indeed,
perhaps the Universe is fine-tuned due to anthropic reasons. However, there
are many caveats in describing physical processes (e.g., inflation) in terms of
naturalness. Indeed, in cosmology the whole concept of ‘naturalness’ is suspect.
Let us discuss some of these issues in a little more detail.
In observational cosmology, the amount of information that can be expected
to be collected via astronomical observations is limited since we occupy a par-
ticular vantage point in the Universe; we are limited in what we can observe
by visual and causal horizons (see discussion below). It can be argued that
the observational limit may be approached in the foreseeable future, at least
regarding some specific scientific hypotheses [12]. There is no certainty that the
amount and types of information that can be collected will be sufficient to test
all reasonable postulated hypotheses statistically. There is under-determination
both in principle and in practice [13, 12]. This consequently leads to a natural
view of model inference as inference to the best explanation/model, since some
degree of explanatory ambiguity appears unavoidable in principle; inference in
cosmology is based on a Bayesian interpretation of probability which includes
a priori assumptions explicitly.
In physical cosmology, we are gravely compromised because we can only test
physics directly up to the highest energies attainable by collisions at facilities
such as the LHC, or from what we can deduce indirectly by cosmic ray obser-
vations. Hence we have to guess what extrapolation from known physics into
the unknown is most likely to be correct; different extrapolations (e.g., string
theory or loop quantum gravity) give different outcomes. As we cannot test
directly the physics of inflation or of dark energy, theorists in fact rely mainly
on Synge’s g-method discussed below: we conclude matter has the properties
we would like it to have, in order to fit with astronomical observations.

1.1.3 Underlying theory

It has been argued [14] that the measure problem, and hence model inference,
is ill defined due to ambiguity in the concepts of probability, in the situation
where additional empirical observations cannot add any significant new infor-
mation. However, inference in cosmological models can be made conceptually
well-defined by extending the concept of probability to general valuations (using
principles of uniformity and consistency) [14].
For example, an important area is empirical tests of the inflationary paradigm
which necessitates, in principle, the specification or derivation of an a priori
probability of inflation occurring (“the measure problem”). The weakness of
all models of inflation is consequently in the initial conditions [15]. To assert
that the flatness of the Universe or the expected value for Λ is predicted by
inflation is absolutely meaningless without such an appropriate measure (this is
particularly true in the case of the multiverse [16]).

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 The fundamental problem is that the theory of inflation cannot be proven to
be correct. Falsifying a “bad theory” (such as the the multiverse solution to the
cosmological constant problem [17]) may be impossible [16] 3 , since parameters
can be added without limit. But it should be possible to falsify a “good theory”,
like inflation [18]. Perhaps the best way to make progress may be to probe the
falsification of inflation, for which there is a robust predicted CMB polarization
signal (induced by GW at the onset of inflation) [16].

1.1.4 Assumptions
It is necessary to make assumptions to derive models to be used for cosmologi-
cal predictions and check with observational data. But what precisely are these
assumptions and how do they affect the results that come out; e.g., is the rea-
son that small backreaction effects are obtained in computations because of the
assumptions that are put in by hand at the beginning? We can only confirm
the consistency of assumptions; we cannot rule out alternative explanations.
The assumption of a FLRW background (cosmological principle) on cosmologi-
cal scales presents a number of problems. There is no solid way to test spatial
homogeneity, even in principle, by direct tests such as (redshift, distance obser-
vations), because we cannot control the possible time evolution of sources and
so cannot be confident they are good standard candles (we do not, for example,
have a solid understanding of supernova explosions and how they might depend
on metallicity, or of radio source evolution). However, observations of struc-
ture growth on the one hand and matter-light interactions via the kinematic
Sunyaev-Zeldovich effect on the other do indeed give rise to solid constraints on
inhomogeneity [19, 20], and indicate that approximate spatial homogeneity does
indeed hold within our past light cone. Due to the existence of horizons, we
can only observe the Universe on or within our past light cone (on cosmological
scales). Assumptions beyond the horizon (Hubble scales) are impossible to test
and so are, in effect, unscientific.

1.1.5 Homogeneity scale

The homogeneity scale is not actually theoretically determined, even in princi-
ple, in the standard cosmological model. It is just “pasted in” to the standard
model a postieri to help fit observations. Even then, what is the derived homo-
geneity scale implied (from the statistics observed). This question is important
in the backreaction question.
There are a number of different approaches to the definition of a scale of
statistical homogeneity. Even if we consider the standard model setting, then
the homogeneity scale depends on the statistical measure used. But there are
arguments that such a definition is not met and will never be met observation-
3 The commonly accepted solution to the mass hierarchy problem at the Planck scale ne-

cessitates an anti-de Sitter space-time and a negative Λ. However, if the sign of Λ is allowed
to have anthropic freedom, the concept of using Bayesian constraints to yield a non-zero value
for Λ from below must be discarded [16].

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ally [21]. Perhaps there is a different notion (e.g., using ergodicity) of statistical
homogeneity in terms of an average positive density. But, any practical mea-
sure of statistical homogeneity is not directly based on a fundemental relation,
but rather on the scale dependence of galaxy-galaxy correlation functions in
observations [22].
Observationally, and based on the 2-point correlation function, the smallest
scale at which any measure of statistical homogeneity can emerge by the current
epoch is in the range 70-120 h−1 M pc. Indeed, if all N-point correlations of
the galaxy distribution are considered, then the homogeneity scale can only be
reached, if at all, on scales above 700 h−1 M pc [21] (also see [23]).

1.1.6 Local and global coordinates

Perhaps, most importantly, what are the assumptions that underscore the use of
an inertial coordinate system over a Hubble scale ‘background’ patch in which
to do perturbation computations or in specifying initial conditions for numerical
GR evolution. In particular, what are the assumptions necessary for the exis-
tence of Gaussian normal coordinates and hence a ‘global’ time and a ‘global’
inertial (Cartesian and orthogonal) spatial coordinate system (and thus a 1 + 3
split) on the ‘background’ patch. This necessitates an irrotational congruence
of fluid-comoving observers and, of course, is related to a choice of lapse and
shift and hence a well defined gauge. And it essentially amounts to assuming
that fluctuations propagate on a fixed absolute Newtonian background (with
post-Newtonian corrections). Global inertial coordinates on a dynamical FLRW
background in a GR framework are not conceptually possible [24]. A collection
of spatially contiguous but causally disconnected regions which evolve according
to GR on small scales do not generally evolve as a single collective background
solution of GR on large cosmological scales.

1.1.7 Periodic boundary conditions in structure formation studies

In addition, what are the assumptions necessary for periodic boundary con-
ditions (appropriate on scales comparable to the homogeneity scale) used in
structure formation studies and numerical simulations? In particular, periodic
boundary conditions impose a constraint on the global spatial curvature and
force it to vanish [25, 26]. Strictly speaking, the (average) spatial curvature is
only zero in the standard cosmology in which the FLRW universe possesses the
space-time structure R × M 3 , in which M 3 is a three-dimensional spatial co-
moving simply connected infinite Euclidean 3-space of constant curvature. The
EFE govern the local properties of space-time but not the global geometry or the
topology of the Universe at large. Nonstandard models with a compact spatial
topology (or small universes) which are periodic due to topological identifica-
tions (and are hence not necessarily spatially flat) are also of interest and have
observational consequences [27]. In particular, it has been shown that CMB
data are compatible with the possibility that we live in a small Universe having
the shape of a flat 3-torus with a sufficiently large volume [28].

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1.1.8 Weak field approach
Finally, what are the assumptions behind the weak field approach, the appli-
cability of perturbation theory (and use of Fourier analysis), Gaussian initial
conditions, averaging and the neglect of backreaction? To different degrees they
all assume a small (or zero) spatial curvature. In particular, all global averages
of spatial curvature are expected to coincide with that in the corresponding ex-
act FLRW model to a high degree of accuracy when averaging linear Gaussian
perturbations. In addition, in cosmology we can observe directions, redshifts,
fluxes, but not distances. To infer a distance from observations in cosmology,
we always use a model. Hence the real space correlation function and its Fourier
transform, the power spectrum, are model dependent.
Essentially we conclude that within standard cosmology the spatial curvature
is assumed to be zero (or at least very small and below the order of other
approximations) in order for the analysis to be valid. In any case, the standard
model cannot be used to predict a small spatial curvature. We will revisit the
issue of spatial curvature later.

1.1.9 Quantum realm and multiverse

Are there possible differences from GR at very small scales that result from a
theory of QG? In particular, do they have any relevance in the cosmological
realm, and conversely what is the impact of cosmology on quantum mechanics
[29]. For example, are there any fundamental particles that have yet to be
observed and, if so, what are their properties? Do they (or the recently observed
Higgs boson) have any relevance for cosmology. There is also the issue of whether
singularities can be resolved in GR by quantum effects and whether singularity
theorems are possible in higher dimensions, that are relevant in cosmology.
Both QG and inflation motivate the idea of a multiverse, in which there
exists a wide range of fundamental theories (or, at least, different versions of
the same fundamental theory with different physical parameters) and our own
Universe is but one possibility [30]. In this scenario the question then arises as
to why our own particular Universe has such finely tuned properties that allow
for the existence of life. This has led to an explanation in terms of the so-called
anthropic principle, which asserts that our Universe must have the properties
it does because otherwise we would not be here to ask such a question. The
cosmology of a multiverse leads to a number of philosophical questions. For
example, is the multiverse even a scientific theory.

1.2 Definition of a cosmological model

A cosmological model has the following components [1].
Spacetime geometry: The spacetime geometry (M, g) is defined by a smooth
Lorentzian metric g (characterizing the macroscopic gravitational field) defined
on a smooth differentiable manifold M [31]. The scale over which the cosmo-
logical model is valid should be specified.

11
Field equations and equations of motion: To complete the definition of a cosmo-
logical model, we must specify the physical relationship (interaction) between
the macroscopic geometry and the matter fields, including how the matter re-
sponds to the macroscopic geometry. We also need to know the trajectories
along which the cosmological matter and light moves. In standard theory, the
space-time metric, g, is determined by the matter present via the EFE:
1
Gab := Rab − Rgab = κTab − Λgab (1)
2
where the total energy momentum tensor, Tab , is the sum of the stress tensors of
P (i)
all matter components present: Tab = (i) Tab , κ is essentially the gravitational
constant, and Λ is the cosmological constant. In colloquial terms: Matter curves
spacetime. Because of the Bianchi identities, Rab[cd;e] = 0, the definition on the
left of (1) implies the identity Gab;b = 0 and hence, provided Λ is indeed constant,
that:
Gab;b = 0 ⇒ T ab;b = 0; (2)
that is, energy-momentum conservation follows identically from the FE (1). The
covariant derivatives in (2) depend on the space-time geometry, so in colloquial
terms: Space-time tells matter how to move. The key non-linearity of GR follows
from the combination of these two statements, and the fact that Rab is a highly
non-linear function of gab (xi ).
In GR a test particle follows a timelike or null geodesic. But a system that
behaves as point particles on small scales may not necessarily do so on larger
scales. That is, if the particles traverse timelike geodesics in the microgeom-
etry, in principle, the macroscopic (averaged) matter need not follow timelike
geodesics of the macrogeometry. However, the fundamental congruence is, in
essence, the average of the timelike congruences along which particles move
in the microgeometry, and defining the effective conserved energy-momentum
tensor T ab through the EFE ensures timelike geodesic motion. In addition, the
(average) motion of a photon is not necessarily on a null geodesic in the averaged
macrogeometry, which will affect observations.
Matter: We require a consistent model for the matter on the characteristic cos-
mological (e.g., averaging) scale, and its appropriate (averaged) physical prop-
erties. Differentiation between the gravitational field and the matter fields is
known not to be scale invariant and, in particular, a perfect fluid is not a scale
invariant phenomenon [32]; averaging in the “mean field theory” in the presence
of gravity changes the equation of state of the matter [33]. In this framework all
of the qualitative effects of averaging are absorbed into the redefined effective
energy-momentum tensor T ab and the redefined effective equation state of the
macro-matter, where T ab is conserved relative to the macrogeometery. The defi-
nition of the Landau frame for any combination of matter fields and radiation is
invariant when matter and matter-radiation interactions take place due to local
momentum conservation.
Timelike congruence: There is a preferred unit timelike congruence u (ua ua =
−1), defined locally at each event, associated with a family of fundamental

12
observers (at late times) or the average motion of energy (at earlier times). In
the case that there is more than one matter component, implying the existence
of more than one fundamental macroscopic timelike congruence, we can always
identify a fundamental macroscopic timelike congruence represented by the 4-
velocity of the averaged matter in the model; i.e., the matter fields admit a
formulation in terms of an averaged matter content which defines an average
(macroscopic) timelike congruence. This then leads to a covariant 1 + 3 split of
spacetime [1]. Mathematically this implies that the spacetime is topologically
restricted and is I-non-degenerate, and consequently the spacetime is uniquely
characterized by its scalar curvature invariants [34]. For example, for a perfect
fluid u is the timelike eigenfunction of the Ricci tensor.
Observationally, this cosmological rest frame is determined as the frame
wherein the CBR dipole is eliminated (the Solar System is moving at about
370km/sec relative to this rest frame). Note that the existence of this preferred
rest frame is an important case of a broken symmetry: while the underlying
theory is Lorentz invariant, it’s cosmologically relevant solutions are not (in
particular, at no point in the history of the universe is it actually de-Sitter –
with its 10-dimensional symmetry group – much less anti-de Sitter).
A note on modified theories of gravity: Let us make a brief comment here. A
key issue is whether GR is, in fact, the correct theory of gravity, especially
on galactic and cosmological scales. Recent developments in testing GR on
cosmological scales within modified theories of gravity were reviewed in [35, 36].
In particular, modified gravity theories have played an important role in the
dark energy problem. Many questions can be posed in the context of modified
gravity theories which include, for example, the general applicability of the
BKL behaviour in the neighborhood of a cosmological singularity. We will
not discuss such questions here, except for the particular question of whether
isotropic singularities are typical in modified gravity theories.

1.3 Problems in mathematical cosmology

In GR, a sufficiently differentiable 4-dimensional Lorentzian manifold is assumed
[31]. The Lorentz metric, g, which characterizes the causal structure of M, is
assumed to obey the EFE, which constitute a hyperbolic system of quasi-linear
partial differential equations which are coupled to additional partial differential
equations describing the matter content [37]. The Cauchy problem is of par-
ticular interest, in which the unknown variables in the constraint equations of
the governing EFE, consisting of a three-dimensional Riemannian metric and
a symmetric tensor (in addition to initial data for any matter fields present),
constitute the initial data for the remaining EFEs. Primarily the vacuum case
is considered in attempting to prove theorems in GR, but this is not the case of
relevance in cosmology. Viable cosmological models contain both matter and ra-
diation, which in physically realistic cases then define a geometrically preferred
timelike 4-velocity field [1] which, because of (1), is related to an eigenvector of
the matter stress tensor (which is unique if we assume realistic energy conditions
[31]).

13
 The EFE are invariant under an arbitrary change of coordinates (general
covariance), which complicates the way they should be formulated in order for
global properties to be investigated [38]. The vacuum EFEs are not hyperbolic
in the normal sense. But utilizing general covariance, in harmonic coordinates
the vacuum EFEs do represent a quasilinear hyperbolic system and thus the
Cauchy problem is indeed well posed and local existence is guaranteed by stan-
dard results [39]. It can also be shown that if the constraints (and any gauge
conditions) are satisfied initially, they are preserved by the evolution. Many
analogues of the results in the vacuum case are known for the EFE coupled
to different kinds of matter, including perfect fluids, gases governed by kinetic
theory, scalar fields, Maxwell fields, Yang-Mills fields, and various combinations
of these. Any results obtained for (perfect) fluids are generally only applicable
in restricted circumstances such as, for example, when the energy density is uni-
formly bounded away from zero (in the region of interest) [37]. The existence
of global solutions for models with more exotic matter, such as stringy matter,
has also been studied [40].

1.3.1 Singularity theorems

The concepts of geodesic incompleteness (to characterize singularities) and closed
trapped surfaces [41] were first introduced in the singularity theorem due to Pen-
rose [42]. Hawking and Ellis [43] then proved that closed trapped surfaces will in-
deed exist in the reversed direction of time in cosmology, due to the gravitational
effect of the CMB. Hawking subsequently realized that closed trapped surfaces
will also be present in any expanding Universe in its past, which would then
inevitability lead to an initial singularity under reasonable conditions within
GR [44]. This led to the famous Hawking and Penrose singularity theorem [45].
The singularity theorems prove the inevitability of spacetime singularities
in GR under rather general conditions [42, 45], but they say very little about
the actual nature of generic singularities. We should note that there are generic
spacetimes which do not have singularities [46]. In particular, the proof of the
Penrose singularity theorem does not guarantee that a trapped surface will occur
in the evolution. It was proven [47] that for vacuum spacetimes a trapped sur-
face can, indeed, form dynamically from regular initial data free of any trapped
surfaces. This result was subsequently generalized in [48, 49]. A number of ques-
tions still exist, which include proving more general singularity theorems with
weaker energy conditions and with weaker differentiability, and determining any
relationship between geodesic incompleteness and the divergence of curvature
[46]. But perhaps the most important open problem within GR is cosmic cen-
sorship [9].

1.3.2 Bouncing models

Using exotic matter, or alternative modified theories of gravity, can classically
lead to the initial cosmological (or big bang) singularity being replaced by a
big bounce, a smooth transition from contraction to an expanding universe [50],

14
which may help to resolve some fundamental problems in cosmology. Bounce
models have utilized ideas like branes and extra dimensions [51], Penrose’s con-
formal cyclic cosmology [52] (which leads to an interest in an isotropic singular-
ity), string gas [53], and others [50, 54].
The matter bounce scenario faces significant problems. In particular, the
contracting phase is unstable against anisotropies [55] and inhomogeneities [56].
In addition, there is no suppression of GW compared to cosmological pertur-
bations, and hence the amplitude of GW (as well as possible induced non-
Gaussianities) may be in excess of the observational bounds. In a computa-
tional study of the evolution of adiabatic perturbations in a nonsingular bounce
within the ekpyrotic cosmological scenario [57], it was shown that the bounce
is disrupted in regions with significant spatial inhomogeneity and anisotropy
compared with the background energy density, but is achieved in regions that
are relatively spatially homogeneous and isotropic.
The specially fine-tuned and simple examples studied to date, particularly
those based on three spatial dimensions, scalar fields and, most importantly, a
non-singular bounce that occurs at densities well below the Planck scale where
QG effects are small [58], are arguably instructive in pointing to more physical
bouncing cosmological models, and may present realistic alternatives to inflation
to obtain successful structure formation (which we will discuss below).
The precise properties of a cosmic bounce depend upon the way in which it
is generated, and many mechanisms have been proposed for this both classically
and non-classically. Bounces can occur due to QG effects associated with string
theory [59] and loop quantum gravity [60, 61]. In particular, in loop quantum
cosmology there is a bounce when the energy density reaches a maximum value
of approximately one half of the Planck density (although it is also possible
that bounces occur without a QG regime ever occurring [62], because if inflation
occurs, the inflaton field violates the energy conditions needed for the classical
singularity theorems to be applicable). We will discuss this in more detail later.

1.3.3 Mathematical results

Some applications in GR can be studied via Einstein-Yang-Mills (EYM) theory
(which is relevant to cosmological models containing Maxwell fields and form
fields and is perhaps a prototype to studying fields in, for example, string the-
ory). Mathematical results when generalized to Maxwell and YM matter in 4D
[63] are known (and have been studied in two dimensions less by wave maps
with values on spheres [64, 65]).
Global existence in Minkowski spacetime, assuming initial data of sufficiently
high differentiability, was first investigated in [66]. The uniqueness theorem for
the 4D Schwarzschild spacetime was presented in [67]. The uniqueness theorem
for the Kerr spacetime was proven in [68]. In the non-vacuum case the unique-
ness of the rotating electrically charged black hole solution of Kerr-Newman has
not yet been generally proven [69]. Once uniqueness has been established, the
next step is to prove stability under perturbations. Minkowski spacetime has
been shown to be globally stable [70, 71].

15
1.3.4 Extension to cosmology
Many of these problems in GR can be extended to the cosmological realm [9].
The uniqueness and stability of solutions to the EFE in GR are important, 4 and
can be generalized to cosmological spacetimes (with a cosmological constant).
Generic spacelike singularities are traditionally referred to as being cosmological
singularities [10]. In particular, the stability of de Sitter spacetime will be
discussed later. There are also a number of questions in the quantum realm [5],
such as singularity resolution in GR by quantum effects and higher dimensional
models, which are of interest in cosmology.
In essence the perturbation studies leading to theories of structure formation
are stability studies of FLRW models. With ordinary equations of state, initial
instabilities will grow but with a rate that depends on the background model
expansion. Thus if there is no expansion, inhomogeneity will grow exponentially
with time; with power law expansion, they will grow as a fractional power of
time; and with exponential expansion, they will tend to freeze out. However,
the way this happens depends on the comoving wavelength of the perturbation
relative to the scale set by the Hubble expansion rate at that time.5 These
studies hold while the perturbation is linear, and have been heroically extended
to the non-linear case (see later). However numerical simulations are required
for the strongly non-linear case [25].

1.3.5 Computational cosmology

Numerical calculations have always played a central role in GR. Indeed, numeri-
cal computations support many of the conjectures in GR and their counterparts
in cosmology and have led to a number of very important theoretical advances
[9]. For example, the investigation of the mathematical stability of AdS space-
time includes fundamental numerical work and cosmic censorship is supported
by numerical computations. In addition, the role of numerics in the under-
standing of the BKL dynamics, and in various other problems in cosmology
and higher dimensional gravity, has been important. In fact, numerical compu-
tations are now commonly used to address fundamental issues within full GR
cosmology[77, 78, 79, 25, 26, 80].

1.4 Cosmological observations

What turns cosmology from a mathematical endeavour to a scientific theory
is its ability to produce observational predictions that can be tested. Since
the initiation of cosmology as a science by Lemaı̈tre in 1927 [81], telescopes
4 A full proof of the linear stability of Schwarzschild spacetime has recently been established

[72]. The non-linear stability of the Schwarzschild spacetime is still elusive [73] (however,
see [74]). Proving the non-linear stability of Kerr has become one of the primary areas of
mathematical work in GR [71, 75]). All numerical results, and current observational data,
provide evidence that the Kerr (and Kerr-Newman) black holes are non-linearly stable [76].
5 Often erroneously called the ‘Horizon’. It has nothing to do with causality, i.e. with

effects related to the speed of light.

16
of ever increasing power, covering all wavelengths and both Earth-based and
in satellites, have led to a plethora of detailed tests of the models leading to
the era of “precision cosmology”. The tests are essentially of two kinds: direct
tests of the background models based on some kind of “standard candle” or
“standard ruler”, and indirect tests based on studying the statistics both of
structures (inhomogeneities) on the one hand, and their effects on the CMB on
the other. Both kinds of results produce broadly concordant results, but the
latter give tighter restrictions on the background model than the former, because
what kinds of structures can form depends on the dynamics of the background
model.
The basic restricton: The basic observational restriction in cosmology is that
given the scales involved, we can only observe the Universe from one space-time
event (“here and now”) [1]. This would not be the case if the Universe were
say the size of the Solar System, but that is not the case: a key discovery has
been the immense size of the Universe, dwarfing the scales of galaxies which
themselves dwarf the scale of the Solar System. This leads to major limits
on what is observable, because of visual horizons for each kind of radiation or
particle: for example, the CMB is observed on single surface (two-sphere) of
last scattering. The furthest matter we can observe can be influenced by matter
even further out, but such indirect effects are limited by the particle horizon: the
furthest matter that can have had causal influence on us by influences travelling
to us at speeds limited by the speed of light since the start of the Universe.

1.4.1 Anomalies
Within theoretical cosmology there needs to be an adequate explanation of ob-
servational anomalies, which are bound to occur as we make ever more detailed
models of the structures and their effects on the CMB. Geometric optics must
be utilized and model independent observations are sought. In general, data
analysis and statistical methods are not discussed here. However, observations
do, of course, lead to theoretical questions. Are there important neglected selec-
tion/detection effects [82]; i.e., what else can exist that we have not yet seen or
detected? Observations sometimes lead to ridiculous predictions (e.g., w < −1;
phantom matter); care must be taken not to be led into unphysical parameter
space. Appropriate explanations of observational anomalies may well lead to
new fundamental physics and questions.
The standard cosmology has been extremely successful in describing current
observations, up to various possible anomalies and tensions [83], and particularly
some statistical features in the CMB [84] and the existence of structures on
gigaparsec scales such as the cold spot and some super-voids [85].
Although primordial nucleosynthesis has been very successful in accounting
for the abundances of helium and deuterium, lithium has been found to be
overpredicted by a factor of about three [86]. Lithium, along with deuterium, is
destroyed in stars, and consequently it’s observation constitutes evidence (and a
measure) of the primordial abundance after any appropriate corrections. To date
there has been some claims of relief in this tension, but there is no satisfactory

17
resolution of the lithium problem.
A seldom asked question is whether the CMB and matter dipoles are in
agreement [87]. Tests of differential cosmic expansion on such scales rely on
very large distance and redshift catalogues, which are noisy and are subject to
numerous observational biases which must be accounted for. In addition, ideally
any test should be performed in a model independent manner, which requires
removing the FLRW assumptions that are often taken for granted in many
investigations. To date, such a model independent test has been performed for
full sky spherical averages of local expansion [88], using the COMPOSITE and
Cosmicflows-II catalogues; it was found with very strong Bayesian evidence that
the spherically averaged expansion is significantly more uniform in the rest frame
of the Local Group (LG) of galaxies than in the standard CMB rest frame. It
was subsequently shown by that this result is consistent with Newtonian N-body
simulations in the standard cosmology framework [89]. The future of such tests
is discussed in [90], concluding that the amplitude of the matter dipole can
be significantly larger than that of the CMB dipole. Its redshift dependence
encodes information on the evolution of the Universe and on the tracers.
Perhaps more controversially, it has also been suggested that a “dark flow”
may be responsible for part of the motion of large objects that has been observed.
An analysis of the local bulk flow of galaxies indicates a lack of convergence to
the CMB frame beyond 100 Mpc [91], which contradicts standard cosmological
expectations. Indeed, there is an anomalously high and approximately constant
bulk flow of roughly 250 km/s extending all the way out to the Shapley super-
cluster at approximately 260 Mpc, as indicated by low redshift supernova data.
Furthermore, there is a discrepancy which has been confirmed by 6dF galaxy
redshift data [92].

1.4.2 Tension in the Hubble constant

The recent determination of the local value of the Hubble constant based on
direct measurements of supernovae made with the Hubble Space Telescope [93]
is now 3.3 standard deviations higher than the value derived from the most
recent data on the power spectrum temperature features in the CMB provided
by the Planck satellite in a ΛCDM model. Although it is unlikely that there
are no systematic errors (since the value of the Hubble constant has historically
been a source of controversy), the difference might be a pointer towards new
physics [94]. So this is perhaps the most important anomaly that needs to be
addressed.
Although a large number of authors have proposed several different mech-
anisms to explain this tension, after three years of improved analyses and
data sets, the tension in the Hubble constant between the various cosmolog-
ical datasets not only persists but is even more statistically significant. The
recent analysis of [93] found no compelling arguments to question the validity
of the dataset used. Indeed, the recent determination of the local value of the
Hubble constant by Riess et al. in 2016 [93] of H0 = 73.24 ± 1.74kms−1 M pc−1
at 68% confidence level is now about 3 standard deviations higher than the

18
(global) value derived from the earlier 2015 CMB anisotropy data provided by
the Planck satellite assuming a ΛCDM model [95]. This tension only gets worse
when we compare the Riess et al. 2018 value of H0 = 73.52 ± 1.62kms−1M pc−1
[165] to the Planck 2018 value of H0 = 67.27 ± 0.60kms−1M pc−1 [84].
In order to investigate possible solutions to the Hubble constant tension a
number of proposals have been made [166]. For example, in [171] it was shown
that the best-fit to current experimental results includes an additional fourth,
sterile, neutrino family with a mass of an eV order suggested by flavour oscilla-
tions. This would imply an additional relativistic degree of freedom (Nef f = 4)
in the standard model, which may alleviate the H0 tension. Recently it was
argued that GW could represent a new kind of standard “sirens” that will al-
low for H0 to be constrained in a model independent way [207]. It is unlikely
that inhomogeneitites and cosmic variance can resolve the tension [26]. How-
ever, there are suggestions that the emergence of spatial curvature may alleviate
the tension [156, 164, 160, 26, 202]. Any definitive measurement of a non-zero
spatial curvature would be crucial in cosmology. We will revisit this later.

19
2 Problems in theoretical cosmology
2.1 Acceleration: dark energy
The most fundamental questions in cosmology, perhaps, concern dark matter
and dark energy, both of which are ‘detected’ by their gravitational interactions
but can not be directly observed [204].
Indeed, the dark energy problem is believed to be one of the major obstacles
to progress in theoretical physics [128, 129]. Weinberg discussed the cosmologi-
cal constant problem in detail [130]. Conventional quantum field theory (QFT)
predicts an enormous energy density for the vacuum. However, the GR equiva-
lence principle asserts that all forms of mass and energy gravitate in an identical
manner, which then implies that the vacuum energy is gravitationally equiva-
lent to a cosmological constant and would consequently have a huge effect on
the spacetime curvature. But the observed value for the effective cosmologi-
cal constant is so very tiny (in comparison to the predictions of QFT) that a
“bare” cosmological constant, whose origin is currently not known, is necessary
to cancel out this enormous vacuum energy to at least 10−120 . This impossibly
difficult fine-tuning problem becomes even worse if we include higher order LQG
corrections [131].
A number of authors, including Weinberg, have offered the opinion that of
all of the possible solutions to the dark energy problem, perhaps the most rea-
sonable is the anthropic bound, which is itself very controversial [17]. However,
another possibility is that the quantum vacuum does not gravitate. This will
be true if the real gravitational theory is unimodular gravity, leading to the
trace-free EFE [168].
Furthermore, the expansion of the Universe has been increasing for the last
few billion years [132, 133]. Within the paradigm of standard cosmology, it
is usually proposed that this acceleration is caused by a so-called dark energy,
which effectively has the same properties as a very small cosmological constant
(which is a repulsive gravitational force in GR). This cosmological coincidence
problem, which necessitates a possible explanation for why the particular small
observed valued of the cosmological constant currently is of a similar magni-
tude to that of the matter density in the Universe, is an additional problem. In
particular, it is often postulated that dark energy is not due to a pure cosmo-
logical constant but that dynamical models such as, for example, quintessence
and phantom energy scalar field models, are more reasonable. Alternative ex-
planations for these gravitational effects have been proposed within theories
with modified gravity on large scales, which consequently do not necessitate
new forms of matter. The possibility of an effective acceleration of the Universe
due to backreaction has also been discussed.

2.2 Acceleration: inflation

Inflation is a central part of modern theoretical cosmology. The assumption of
zero spatial curvature (k = 0) is certainly well motivated in the standard model

20
by inflation.
Before the development of inflation, it was already known that a scale in-
variant (Harrison-Zeldovitch) power spectrum is a good fit to the data. But
its origin was mysterious and there was no convincing physical mechanism to
explain it. However, inflation naturally implies this property as a result of cos-
mological perturbations of quantum mechanical origin. Moreover, it allows a
bridge to be built between theoretical considerations and actual astrophysical
measurements. One fundamental assumption of inflation is that, initially, the
quantum perturbations are placed in the vacuum state [159].
As noted earlier, models with a positive cosmological constant are asymptotic
at late times to the inflationary de Sitter spacetime [136, 137]. Scalar field
models with an increasing rate of (volume) expansion are also future inflationary.
For models with an exponential potential, global asymptotic results can be
obtained [138, 139]. Inflationary behavior is also possible in scalar field models
with a power law potential, but typically occurs during an intermediate epoch
rather than asymptotically to the future. Local results in this case are possible,
but they are difficult to obtain and this problem is usually studied numerically.
There are a number of fundamental questions, which include the following.
What exactly is the conjectured inflaton? What is the precise physical details
of cosmic inflation? If inflation is self-sustaining due to the amplification of
fluctuations in the quantum regime, is it still taking place in some (distant)
regions of the Universe? And, if so, does inflation consequently give rise to
an infinite number of “bubble universes”? In this case, under what (initial)
conditions can such a multiverse exist? An investigation of “bubble universes”,
in which our own Universe is but one of many that nucleate and grow within an
ever-expanding false vacuum, has been undertaken (primarily computationally).
For example, the interactions between such bubbles were investigated in [134].
Cosmological inflation is usually taken as a reasonable explanation for the
fact that the Universe is apparently more uniform on larger scales than is an-
ticipated within the standard cosmology (the horizon problem). However, there
are other possible explanations. But how does inflation start? And, perhaps
most importantly, what is the generality for the onset of inflation for generic
spatially inhomogeneous initial data? We note that a rigorous formulation of
this question is problematic due to the fact that there are so many different
inflationary theories and since there are no “natural” conditions for the initial
data. However, any such natural initial conditions are expected to contain some
degree of inhomogeneity 6 . Unfortunately, such initial data does not necessarily
lead to inflation. Although it is known that large field inflation can occur for
simple inhomogeneous initial data (at least for energies with substantial initial
gradients and when the inflaton field is on the inflation supporting portion of
the potential to begin with), it has also been shown that small field inflation is
significantly less robust in the presence of inhomogeneities [135] (also see [134]
and [206]).
6 Note that preliminary calculations in quantum field theory suggest that vacuum fluctua-

tions could induce an enormous cosmological constant [208].

21
2.2.1 Alternatives to inflation
Although inflation is the most widely acceptable mechanism for the generation
of almost scale invariant (and nearly Gaussian adiabatic density) fluctuations
to explain the origin of structure on large scales, possible alternatives include
GR spikes [140], conformal cyclic cosmology [52] and QG fluctuations [158]. In
particular, Penrose has argued that since inflation fails to take fully into account
the huge gravitational entropy that would be associated with black holes in a
generic spacetime, inflation is incredibly unlikely to start, and smooth out the
universe, if its initial state is generic [52]. In addition, in the approach of [158] re-
sults from non-perturbative studies of QG regarding the large distance behavior
of gravitational and matter two-point functions are utilized; non-trivial scal-
ing dimensions exist due to a nontrivial ultraviolet renormalization group fixed
point in 4D, motivating an explanation for the galaxy power spectrum based
on the non-perturbative quantum field-theoretical treatment of GR. Perhaps
the most widely accepted alternative to inflation to obtain successful structure
formation and which is consistent with current observations [96] is the matter
bounce scenario, in which new physics resolves the cosmological singularity.

2.2.2 Bouncing models revisited

Bouncing models include the ekpyrotic and emergent string gas scenarios [96].
The ekpyrotic scenarios [51] are bouncing cosmologies which avoid the problems
of the anisotropy and overproduction of GW in the matter bounce scenario,
since the dynamics of the contracting phase is governed by a matter field (e.g.,
a scalar field with negative exponential potential) whose energy density increases
faster than the contribution of anisotropies. In ekpyrotic scenarios, in which the
bounce is not necessarily symmetric, fluctuations on all currently observable
scales start inside the Hubble radius at earlier times, leading to structure that
is formed causally and hence a solution of the horizon problem in the same
way as in standard big bang cosmology and as in the usual matter bounce.
But, contrary to the matter bounce scenario, during contraction the growth
of fluctuations on super-Hubble scales is too weak to produce a scale-invariant
spectrum from an initial vacuum state, leading to a subsequent blue spectrum
of curvature fluctuations and GW [96]. Therefore, initial vacuum perturbations
cannot describe the observed structures in the Universe. In addition, a negligible
amplitude of GW is predicted on cosmological scales. However, a scale invariant
spectrum of curvature fluctuations can be obtained by using primordial vacuum
fluctuations in a second scalar field in the ekpyrotic scenario [97].
Another alternative to cosmological inflation is the emergent string gas sce-
nario [53], based on a possible extended quasi-static period in the very early
Universe dynamically dominated by a thermal gas of fundamental strings, after
which there is a transition to the expanding radiation phase of standard cos-
mology. The thermal fluctuations of a gas of closed strings on a compact space
with toroidal topology, which do not originate quantum mechanically (unlike in
most models of inflation), then produce a scale-invariant spectrum of curvature

22
fluctuations and GW. The tilt of the spectrum of curvature fluctuations is pre-
dicted to be red as in inflation, but that of the GW is slightly blue, in contrast
to what is obtained in inflation.
We should note that although some of the alternatives to inflation are sug-
gested by ideas motivated by QG, it is also of interest to know whether inflation
occurs naturally within QG. We will discuss this later.

2.3 The physics horizon and Synge’s g-method

The Physics horizon: The basic problem as regards inflation and any attempts
to model what happened at earlier times in the history of the Universe is that
we run into the physics horizon: we simply do not know what the relevant
physics was at those early times. The reason is that we cannot construct par-
ticle colliders that extend to such high energies. Thus we are forced either to
extrapolate tested physics at lower energies to these higher energies, with the
outcome depending on what aspect of lower energy physics we decide to ex-
trapolate (because we believe it is more fundamental than other aspects), or to
make a phenomenological model of the relevant physics.
Synge’s g-method: A very common phenomenological method used is Synge’s
g-method : running the EFE backwards [20]. That is, in eqn. (1), instead of
trying to solve it from right to left (given a matter source, find a metric g that
corresponds to that matter source), rather choose the metric and then find the
matter source that fits. That is, select a metric g with some desirable properties,
calculate the corresponding Ricci tensor Rab and Einstein tensor Gab and then
use (1) to find the matter tensor Tab so that (1) is identically satisfied, and voila!
we have an exact solution of the EFE that has the desired geometric properties.
No differential equations have to be solved. The logic is: via (1),

{gab } ⇒ {Rab } ⇒ {Gab } ⇒ {Tab }. (3)

One classic example is choosing an inflationary scale factor a(t) that leads to
structure formation in the early Universe that agrees with observations. We can
then run the EFE backward as in (3) to find a potential V (φ) for an effective
scalar field φ that will give the desired evolution a(t). It is a theorem that
one almost always can find such a potential [98], essentially because the energy
momentum conservation equations are in that case equivalent to the Klein-
Gordon equation for the field φ; but there is no real physics behind claims of
the existence of such a scalar field. It has not been related to any matter or
field that has been demonstrated to exist in any other context.

2.4 Dynamical behaviour of cosmological solutions

The dynamical laws governing the evolution of the universe are the classical
EFEs. It is of interest to study exact cosmological solutions and especially
spatially inhomogeneous cosmologies [2], and their qualitative and numerical
behaviour. Dynamical systems representations of the evolution of cosmological

23
solutions are very useful [99, 100]. In particular, it is of interest to extend
stability results to the study of cosmological models with matter and in the
case of a non-zero cosmological constant [11].

2.4.1 Stability of cosmological solutions

This concerns the question of whether the evolution of the EFE under small
perturbations is qualitatively similar to the evolution of the underlying exact
cosmological solution (e.g., by including small-scale fluctuations). This prob-
lem involves the investigation of the (late time) behavior of a complex set of
partial differential equations about a specific cosmological solution [101]. The
asymptotic behaviour of solutions in cosmology was reviewed in [100].
We note that for a vanishing cosmological constant and matter that satis-
fies the usual energy conditions, spatially homogeneous spacetimes of (general)
Bianchi type IX recollapse and consequently do not expand for ever. This re-
sult is formalized in the so-called closed universe recollapse conjecture [102],
which was proven in [103]. However, Bianchi type IX spacetimes need not rec-
ollapse in the case that a positive cosmological constant is present. The study
of the stability of de Sitter spacetime for generic initial data is very important,
particularly within the context of inflation (although, as noted earlier, precise
statements concerning the generality of inflation are problematic).

2.4.2 Stability of de Sitter spacetime

A stability result for de Sitter spacetime (vacuum and a positive cosmological
constant) for small generic initial data was proven in [136]. Therefore, de Sitter
spacetime is a local attractor for expanding cosmologies containing a positive
cosmological constant. In addition, it was proven that any expanding spatially
homogeneous model (in which the matter obeys the strong and dominant energy
conditions) that does not recollapse is future asymptotic to an isotropic de Sitter
spacetime [137]. This so-called “cosmic no hair” theorem is independent of the
particular matter fields present. The remaining question is whether general,
initially expanding, cosmological solutions corresponding to initial data for the
EFE with a positive cosmological constant and physical matter exist globally
in time. It is known that this is indeed the case for a variety of matter models
(utilizing the methods of [104]). Global stability results have also been proven
for inflationary models with a scalar field with an exponential potential [138,
139]. It is, of course, of considerable interest to investigate the cosmic no–hair
theorem in the inhomogeneous case. A number of partial results are known in
the case of a positive cosmological constant [105].
The possible quantum instability of de Sitter spacetime has also been investi-
gated. In a semi-classical analysis of backreaction in an expanding universe with
a conformally coupled scalar field and a cosmological constant, it was advocated
that de Sitter spacetime is unstable to quantum corrections and might, in fact,
decay. In principle, this could consequently provide a mechanism that might
alleviate the cosmological constant problem and also, perhaps, the fine-tuning

24
problems that occur for the very flat inflationary potentials that are necessitated
by observations.
In particular, it has been suggested that de Sitter spacetime is unstable due
to infrared effects, in that the backreaction of super-Hubble scale GW could
contribute negatively to the effective cosmological constant and thereby cause
the latter to diminish. Indeed, from an investigation of the backreaction effect of
long wavelength cosmological perturbations it was found that at one LQG order
super-Hubble cosmological perturbations do give rise to a negative contribution
to the cosmological constant [106]. It has consequently been proposed that
this backreaction could then lead to a late time scaling solution for which the
contribution of the cosmological constant tracks the contribution of the matter
to the total energy density; that is, the cosmological constant obtains a negative
contribution from infrared fluctuations whose magnitude increases with time
[160].

2.4.3 The nature of cosmological singularities

Although the singularity theorems imply that singularities occur generally in
GR, they say very little about their nature [46]. For example, singularities
can occur in tilted Bianchi cosmologies in which all of the scalar quantities
remain finite [107]. However, such cosmological models are likely not generic.
Belinskii, Khalatnikov and Lifshitz (BKL) [108] have conjectured that within
GR, and for a generic inhomogeneous cosmology, the approach to the spacelike
singularity into the past is vacuum dominated, local and oscillatory, obeying
the the so-called BKL or mixmaster dynamics. In particular, due to the non-
linearity of the EFE, if the matter is not an effective massless scalar field, then
sufficiently close to the singularity all matter terms can be neglected in the FE
relative to the dynamical anisotropy. BKL have confirmed that the assumptions
they utilized are consistent with the EFE. However, that doesn’t imply that
their assumptions are always valid in general situations of physical interest.
Numerical simulations have recently been used to verify the BKL dynamics in
special classes of spacetimes [109, 110]. Rigorous mathematical results on the
dynamical behaviour of Bianchi type VIII and IX cosmological models have also
been presented [111].
Up to now there have essentially been three main approaches to investigate
the structure of generic singularities, including the original heuristic BKL met-
ric approach and the so-called Hamiltonian approach. The dynamical systems
approach [100], in which the EFE are reformulated as a scale invariant asymp-
totically regularized dynamical system (i.e., a first order system of autonomous
ordinary or partial differential equations) in the approach towards a generic
spacelike singularity, allows for a more mathematically rigorous study of cos-
mological singularities. A dynamical systems formulation of the EFE (in which
no symmetries were assumed a priori) was presented in [112], which resulted in
a detailed description of the generic attractor, precisely formulated conjectures
concerning the asymptotic dynamical behavior toward a generic spacelike sin-
gularity, and a well-defined framework for the numerical study of cosmological

25
singularities [113]. It should be noted that these studies assume that the singu-
larity is spacelike, but there is no reason that this has to be so (this is not, in
fact, generic). The effect of GR spikes on the BKL dynamics and on the initial
cosmological singularity was reviewed in [9].

2.4.4 Isotropic singularity

Penrose [52] has utilized entropy considerations to motivate the “Weyl curvature
hypothesis” that asserts that on approach to an initial cosmological singular-
ity the Weyl curvature tensor should tend to zero or at least remain bounded
(this conjecture subsequently led to the conformal cyclic cosmology proposal).
It is difficult to represent this proposal mathematically but the clearly formu-
lated geometric condition presented in [114], that the conformal structure should
remain regular at the singularity, is closely related to the original Penrose pro-
posal. Such singularities are called isotropic or conformal singularities. It is
known [115] that solutions of the EFE for a radiation perfect fluid that admit
an isotropic singularity are uniquely characterized by particular free data spec-
ified at the singularity. The required data is essentially half as much as the
data necessary in the case of a regular Cauchy hypersurface. This result was
generalized to the case of a perfect fluid with a linear equation of state[116],
and can be further extended to more general matter models (e.g., more general
fluids and a collisionless gas of massless particles) [37].
As noted earlier, we do not aim to discuss alternative theories of gravity
in this review. However, it is of cosmological interest to determine whether
isotropic singularities are typical in any modified theories of gravity. For ex-
ample, the past stability of the isotropic FLRW vacuum solution, on approach
to an initial cosmological singularity, in the class of theories of gravity con-
taining higher–order curvature terms in the GR Lagrangian, has been inves-
tigated [117]. In particular, a special isotropic vacuum solution was found to
exist, which behaves like a radiative FLRW model, that is past stable to small
anisotropies and inhomogeneities (which is not the case in GR). Exact solutions
with an isotropic singularity for specific values of the perfect fluid equation of
state parameter have also been obtained in a higher dimensional flat anisotropic
Universe in Gauss-Bonnet gravity [118]. A number of simplistic cosmological
solutions of theories of gravity containing a quadratic Ricci curvature term in
the Einstein-Hilbert Lagrangian have also been investigated [119].

26
3 Problems in physical cosmology
The predicted distribution of dark matter in the Universe is based on obser-
vations of galaxy rotation curves, nucleosynthesis estimates and computations
of structure formation [120]. The nature of the missing dark matter is not yet
known (e.g., whether it is due to a particle or whether the dark matter phenom-
ena is not characterized by any type of matter but rather by a modification of
GR). But it is, in general, anticipated that this particular problem will be ex-
plained within conventional physics. More recently primordial black holes have
been invoked to explain the missing dark matter and to alleviate some of the
problems associated with the CDM scenario (see later) [196].

3.1 Origin of structure

The CMB anisotropies and structure observed on large angular scales are com-
puted using linear perturbations about the standard background cosmological
model. However, such large scale structure could never have been in causal con-
tact within conventional cosmology and hence its origin cannot be explained by
it without invoking inflation. In general, the testable predictions of inflationary
models are scale-invariant and nearly Gaussian adiabatic density fluctuations
and almost, but not exactly, a scale-invariant stochastic background of relic
GW. However, and as noted earlier, possible alternatives to inflation to obtain
successful structure formation consistent with current observations [96] exist,
including the popular matter bounce cosmologies.

3.1.1 Large scale structure of the Universe

In the standard cosmology it is assumed that cosmic structure at sufficiently
large scales grew out of small initial fluctuations at early times, and we can
study their evolution within (cosmological) linear perturbation theory (LPT)
[203]. We assume that on large scales there is a well defined mean density and
on intermediate scales, the density differs little from it. This is a highly non-
trivial assumption, which is perhaps justified by the isotropy of the CMB. It is
usual to use a fluid model for matter and a kinetic theory model for radiation.
At late times and sufficiently small scales fluctuations of the cosmic density
are not small. The density inside a galaxy is about two orders of magnitude
greater than the mean density of the Universe, and LPT is then not adequate
to study structure formation on galaxy-cluster scales of a few Mpc and less. It
is necessary to treat clustering non-linearly using N-body simulations. Since
this is mainly relevant on scales much smaller than the Hubble scale, it has
usually been studied in the past with non-relativistic N-body simulations. On
intermediate to small scales, density perturbations can become large. Inside a
galaxy they are small, and even inside a galaxy cluster the motion of galaxies
is essentially decoupled from the Hubble flow (i.e., clusters do not expand).
Therefore, the gravitational potential of a galaxy remains small, and in the
Newtonian (longitudinal) gauge, metric perturbations remain small. In the

27
past, this together with the smallness of peculiar velocities has been used to
argue that Newtonian N-body simulations are sufficient.
In the adiabatic case, the last scattering surface is a surface of constant
baryon density, so the observed CMB fluctuations do not represent density
fluctuations, as is often stated [163]. Thus, in standard perturbation theory
language, this shows that in the uniform density gauge (which for adiabatic
perturbation is the same as the uniform temperature gauge) the density fluctu-
ations are given exactly by the redshift fluctuations. In the non-adiabatic case
this will no longer be true. The main shortcoming of the conventional analysis
is, of course, the instantaneous recombination approximation (accurate to a few
percent only for multipoles with ℓ < 100); to go beyond this one has to use
a Boltzmann approach [163] (although nothing changes conceptually). Also,
in principle we cannot neglect radiation or neutrino (even massive) velocities.
In addition, Newtonian simulations only consider 1 (of in general 6) degrees of
freedom, and observations are made on the relativistic, perturbed light cone.
Hence relativistic calculations are needed.

3.1.2 Perturbation theory

The complexity of the distribution of the actual matter and energy in our ob-
served Universe, consisting of stars and galaxies that form clusters and super-
clusters of galaxies across a broad range of scales, cannot be described within
the standard spatially homogeneous model. To do this we must to be able to
describe spatial inhomogeneity and anisotropy using a perturbative approach
starting from the uniform FLRW model as a background solution [183]. The
perturbations live on the four-dimensional background spacetime, which is split
into three-dimensional spatial hypersurfaces utilizing a (1+3) decomposition.
Within the standard cosmology a flat background spatial metric (k = 0) in
LPT is assumed, which is consistent with current observations. For generalisa-
tions to spatially hyperbolic or spherical FLRW models see, e.g., [170].
The introduction of a spatially homogeneous background spacetime to de-
scribe the inhomogeneous Universe leads to an ambiguity in the choice of co-
ordinates. Selecting a set of coordinates in the (real) inhomogeneous Universe,
which will then be described by an FLRW model plus perturbations, essen-
tially amounts to the assignation of a mapping between spacetime points in
the inhomogeneous Universe and the spatially homogeneous background model.
The freedom in this selection is the gauge freedom, or gauge problem, in GR
perturbation theory. Either the gauge freedom must be handled very carefully
by delineating what freedom remains at each stage of coordinate specialisation
[200], by using gauge covariant variables [198], or utilizing 1+3 gauge invariant
and covariant variables [199].
Indeed, gauge-invariant variables are widely utilized since they constitute
a theoretically effective way to extract predictions from a gravitational field
theory applied to the Universe for large-scale linear evolution [170]. In addition,
by using gauge-invariant variables the analysis is reduced to the study of only
three decoupled second order ordinary differential equations, and they represent

28
physical quantities that can be immediately connected to observations. In the
review [183] the focus was on how to construct a variety of gauge invariant
variables to deal with perturbations in different cosmological models at first
order and beyond. Most work to date has been done only to linear order where
the perturbations obey linear FE.
As a theoretical application the origin of primordial curvature and isocur-
vature perturbations from field perturbations during inflation in the very early
Universe can be considered. LPT allows the primordial spectra to be related
to quantum fluctuations in the metric and matter fields at considerably higher
energies. In the most simple single field inflationary models it is, in fact, pos-
sible to equate the primordial density perturbation with the curvature pertur-
bation during inflation, which essentially remains constant on very large scales
for adiabatic density perturbations. The observed power spectrum of primor-
dial perturbations revealed by the CMB and LSS is thus a powerful probe of
inflationary models of the very early Universe.
The outstanding problems within LPT are mostly technical issues and, in
particular, include the important questions of the physical cut off to the short
and long wavelength modes and the convergence of the perturbations (and hence
the validity of the perturbative approach itself).

3.1.3 Non-linear perturbations

The new frontier in cosmological perturbation theory is the investigation of
non-linear primordial perturbations, at second-order and beyond. Although the
simple evolution equations obtained at linear order can be extended to non-linear
order [183], the non-linearity of the EFE becomes evident and consequently the
resulting definitions of gauge invariant quantities at second order clearly become
more complicated than those at first order. Recently, perturbations at second
order [186] and, more generally, non perturbative effects have been studied,
where there are certainly more foundational problems.
Perturbative methods allow quantitative statements but have limited do-
mains of validity. Recently, several groups have started to develop relativistic
simulations [77]. Numerical relativistic N-body simulations are a unique tool
to study more realistic scenarios, and appear to compare well to numerical rel-
ativity fluid simulations [167]. However, assumptions are still made that need
to verified. In particular, care must be taken in applying Newtonian intuition
to GR. For example, [25] do not solve the full EFE and use the fact that the
gravitational potential is very small, but spatial derivatives, and second deriva-
tives, are not small. Therefore, when computing the Einstein tensor they go
only to first order in the gravitational potentials and their time derivatives (but
also include quadratic terms of first spatial derivatives and all orders for second
spatial derivatives).
New qualitatively effects occur beyond linear order. The non-linearity of
the FE inevitably leads to mixing between scalar, vector and tensor modes and
the existence of primordial density perturbations consequently generate vector
and tensor modes. Non-linearities then permit additional information to be de-

29
termined from the primordial perturbations. A lot of effort is currently being
devoted to the investigation of higher order correlations (and issues of gauge
dependence). Non-Gaussianity in the primordial density perturbation distri-
bution would uncover interactions beyond the linear theory. Such interactions
are minimal (suppressed by slow-roll parameters) in the simplest single field
inflation models, so any detection of primordial non-Gaussianity would cause a
major reassessment about our knowledge of the very early Universe. In prin-
ciple, this approach can be easily extended to higher-orders, although large
primordial non-Gaussianity is expected to dominate over non-linearity in the
transfer functions.
However, cosmological perturbation theory based on a cosmological 1+3 split
is ill-suited to address important questions concerning non-linear dynamics or to
evaluate the viability of scenarios based on classical modifications of GR. A new
formulation of a fully non-perturbative approach has been advocated [169], along
with a gauge fixing protocol that enables the study of these issues (and especially
the linear mode stability in spatially homogeneous and nearly homogeneous
backgrounds) in a wide range of cosmological scenarios, based on a method
that has been successful in analyzing dynamical systems in mathematical and
numerical GR based on the generalized harmonic formulation of the EFE.

3.1.4 Non-linear regime

At the non-linear order a variety of different effects come into play, including
gravitational lensing of the source by the intervening matter and the fact that
redshift is affected by peculiar motion, both of which have relatively simple
Newtonian counterparts. But there are a host of complicated relativistic cor-
rections once light propagation is worked out in more detail. There are selection
effects too: we are much more likely to observe sources in halos, some objects
are obscured from view by bright clusters, and so on.
Within the context of perturbation theory it is relatively easy to predict
the expectation value of the bias in the Hubble diagram for a random direction
[191]. The full second-order correction to the distance-redshift relation has
been calculated within cosmological perturbation theory, yielding the observed
redshift and the lensing magnification to second order appropriate for most
investigations of dark energy models [189]. These results were used in [187]
to calculate the impact of second-order perturbations on the measurement of
the distance to the last-scattering surface, where relativistic effects can lead to
significantly biased measurements of the cosmological parameters at the sub-
percent to percent level if they are neglected.
The somewhat unexpected percent level amplitude of this correction was
discussed in [188], but the focus therein was on on the effect of gravitational
lensing only and thus did not consider the perturbations of the observed redshift,
notably due to peculiar velocities, which can lead to a further bias in parameter
estimation. In addition, [190] noted that the notion of average is adapted to
the observation of the Hubble diagram and may differ from the most common
angular or ensemble averages, and suggested a possible non-perturbative way

30
for computing the effects of inhomogeneities on observations based on light-
like signals using the geodesic light-cone gauge to explicitly solve the geodetic-
deviation equation.
In order to comprehensively address the issue of the bias of the distance-
redshift relation, previous work was improved upon by fully evaluating the ef-
fect of second-order perturbations on the Hubble diagram [191]. In particular,
the notion of average which affects bias in observations of the Hubble diagram
for inhomogeneity of the Universe was carefully derived, and its bias at second-
order in cosmological perturbations was calculated. It was found that this bias
considerably affects direct estimations of the evolution of the cosmological pa-
rameters, and particularly the equation of state of dark-energy. Despite the
fact that the bias effects can reach the percent level on some parameters, er-
rors in the standard inference of cosmological parameters remain less than the
uncertainties in observations [191].
In further work [80], a non-perturbative and fully relativistic numerical cal-
culation of the observed luminosity distance and redshift for a realistic cosmo-
logical source catalog in a standard cosmology was undertaken to investigate the
bias and scatter, mainly due to gravitational lensing and peculiar velocities, in
the presence of cosmic structures. The numerical experiments provide conclu-
sive evidence that the non-linear relativistic evolution of inhomogeneities, once
consistently combined with the kinematics of light propagation on the inhomo-
geneous spacetime geometry, does not lead to an unexpectedly large bias on the
distance-redshift correlation in an ensemble of cosmological sources. However,
inhomogeneities introduce a significant non-Gaussian scatter that can give a
large standard error on the mean when only a small sample of sources is avail-
able. But even for large, high-quality supernovae samples this scatter can bias
the inferred cosmological parameters at the percent level [80].
It was argued in [192], using a fully relativistic treatment, that cosmic vari-
ance (i.e., the effects of the local structures such as galaxy clusters and voids) is
of a similar order of magnitude to current observational errors and consequently
needs to be taken into consideration in local measurements of the Hubble expan-
sion rate within the standard cosmology. In addition, the constraint equation
relating metric and density perturbations in GR is inherently non-linear, and
leads to an effective and intrinsic non-Gaussianity in the large-scale dark matter
density field on large scales (even when the primordial metric perturbation is
itself Gaussian) [194].

3.1.5 Non-Gaussianities
In standard cosmology, the primordial perturbations corresponding to the seeds
for the LSS are selected from a Gaussian distribution with random phases, jus-
tified primarily from the fact that primordial non-Gaussianity (PNG) has not
yet been observed and also theoretically (e.g., the central limit theorem); thus a
Gaussian random field constitutes a satisfactory representation of the properties
of density fluctuations. However, any deviation from perfect Gaussianity will,
in principle, reveal important information on the early Universe, and an investi-

31
gation of PNG is especially relevant if these initial conditions were generated by
some dynamical process such as, for example, inflation. In particular, a direct
measurement of non-Gaussianity would permit us to move beyond the free-field
limit, yielding important information about the degrees of freedom, the possi-
ble symmetries and the interactions characterizing the inflationary action. The
current status of the modelling of, and the searching for, PNG of cosmological
perturbations was reviewed in [179].
In order to evaluate PNG from the early Universe to the present time, it
is necessary to self-consistently calculate non-Gaussianity during inflation. We
must then evolve scalar and tensor perturbations to second order outside the
horizon, matching conserved second-order gauge-invariant variables to their val-
ues at the end of inflation (appropriately taking into account reheating). Finally,
we need to investigate the evolution of the perturbations after they re-entered
the Hubble radius, by computing the second-order radiation transfer function
and matter transfer function for the CMB and LSS, respectively. Although these
calculations are very complicated, PNG represents an important tool to probe
fundamental physics during inflation at energies from the grand unified scale,
since different inflationary models predict different amplitudes and shapes of the
bispectrum, which complements the search for primordial gravitational-waves
(PGW) (via a stochastic GW background).
The Planck satellite has produced good measurements of higher-order CMB
correlations, resulting in considerable stringent constraints on PNG. The latest
data regarding non-Gaussianity tested the local, equilateral, orthogonal (and
various other) shapes for the bispectrum and led to new constraints on the pri-
mordial trispectrum parameter [84]. The most extreme possibilities have been
excluded by CMB and LSS observations, and now primarily the detection of (or
constraints from) mild or weak deviations from primordial Gaussian initial con-
ditions are sought, characterized by a small parameter, fN L , compatible with
observations. Even though the sensitivity is not comparable to CMB data [84],
the bispectra for redshift catalogues can be determined (e.g., the three-point
correlation functions for the WiggleZ and BAO spectroscopic surveys) [182],
and interesting observational bounds on the local fN L from current constraints
on the power spectrum can be obtained (see [179] and references within). Ne-
glecting complications arising from the breaking of statistical isotropy (such as
sky-cut, noise, etc.) the procedure is, in general, to fit the theoretical bispec-
trum template, and fN L is found to be approximately 0.01 in generic inflation
[173].
PNG is certainly the best way of practically investigating the only guaran-
teed prediction of inflation [16]. Indeed, even though standard models of slow-
roll inflation only predict tiny deviations from Gaussianity (consistent with the
Planck results), specific oscillatory PNG features can be indicative of particular
string-theory models. Therefore, the search for PNG is of interest for theoreti-
cally well-motivated models of inflation and the Planck results can potentially
severely constrain a variety of classes of inflationary models beyond the sim-
plest paradigm. However, only the failure to find any such evidence for PNG
can falsify inflation.

32
 There are some outstanding issues regarding non-Gaussianity [179]. First,
it has been argued that the consistency relation is certainly not observable for
single field inflation since, in the strictly squeezed limit, this term can be gauged
away by an appropriate coordinate tranformation (so that the only residual term
is proportional to the same order of the amplitude of tensor modes). Second,
in the non-linear evolution of the matter perturbations in GR the second order
dark matter dynamics leads to post-Newtonian-like contributions which mimic
local PNG. A recent estimate of the effective non-Gaussianity due to GR light
cone effects comparable to a PNG signal were discussed in [179], which would
correspond in the comoving gauge to an fN L in the pure squeezed limit. There-
fore, such a GR PNG signature may not be detectable via any cosmological
observables.

3.1.6 Simulations and post-Newtonian cosmological perturbations

There has been a lot of recent interest in testing the validity of GR using cos-
mological observables related to structure formation. Since the physics involved
in horizon-sized cosmological perturbations is quite different to that which oc-
curs on smaller scales, where galaxies and clusters of galaxies are present, this
is challenging. LPT [183] is not suitable for investigating gravitational fields
associated with structures that have highly non-linear density contrasts (which
necessarily have to be small in order for the perturbative expansion to be well
defined). GR numerical simulations using, for example, the gevolution code
developed by Adamek, Durrer and co-workers [25], have proven to be an im-
portant new tool for studying structure formation. Targeted fully relativistic
non-linear simulations with an evolving non-zero spatial curvature have also
been developed [156].
Alternatively, 2-parameter post-Newtonian cosmological perturbation schemes
have been proposed [185]. Indeed, recent progress [184] has been made in ap-
plying the techniques from post-Newtonian expansions of the gravitational FE
into cosmology in the presence of highly non-linear structures to relate the func-
tions that parameterize gravity on non-linear scales to those that parameterize
it on very large scales. This so-called parameterized post-Newtonian cosmology
(PPNC) has been used to analyse alternative theories of gravity [184]. This was
achieved by simultaneously expanding all of the relevant equations in terms of
two parameters; the first associated with the expansion parameter of LPT, and
the second characterizing the order-of-smallness from post-Newtonian theory
[185]. An alternative Lagrangian-coordinates based approximation scheme to
provide a unified treatment for the two leading-order regimes was presented in
[193].

3.2 Black holes and gravitational waves

3.2.1 Gravitational waves
Recent progress in numerical GR has allowed for a detailed investigation of the
collision of two compact objects (such as, e.g., black holes and neutron stars). In

33
such a violent inspiral an enormous amount of gravitational radiation is emit-
ted. The detection and subsequent analysis of the gravitational wave (GW)
signals produced by black hole mergers necessitate extremely accurate theoret-
ical predictions that can be utilized as template waveforms that can then be
used to cross-correlate with the output of GW detectors. This is, of course, of
fundamental import in view of the recent LIGO observations [152]. Indeed, such
an analysis led to the direct detection of GW by the LIGO-Virgo collaboration
[153]. To a large extent the numerical problem has been solved in the case of
a black-hole merger, although the relatively simple properties of the two-body
non-linear gravity waveforms [154] have not been fully understood mathemati-
cally. There is also the recent binary neutron star merger event, which is much
more difficult to model within GR. There are a number of open problems, par-
ticularly concerning the physical nature of the recently observed merger events
[155]. GW astronomy will potentially play an increasingly important role within
cosmology [209]. For example, there is a promise that they will allow very good
direct estimates of the distance of colliding black holes, avoiding the need for
the usual cosmic distance ladder.

3.2.2 Primordial gravitational waves

Primordial GW (PGW) add to the relativistic degrees of freedom of the cos-
mological fluid. Any change in the particle physics content, perhaps due to a
change of phase or freeze-out of a species, will leave a characteristic imprint on
an otherwise featureless spectrum of PGW. The existence of a stochastic PGW
background at a detectable level would then probe new physics beyond the stan-
dard cosmological model, and this may be possible with the Laser Interferometer
Space Antenna (LISA) [178].
Recently, a class of early-Universe scenarios has been theoretically identified
which produce a strongly amplified, blue-tilted spectrum of GW [177]. Detec-
tion of GW over a broad range of frequencies can provide important information
concerning the underlying source [177], and also may well be of relevance for
the spectrum of GW emitted by other scaling sources. In addition, a population
of massive primordial black holes (PBHs) would be anticipated to generate a
stochastic background of GW [197], regardless of whether they form binaries or
not. The focus is usually on the GW generated by either stellar black holes (ob-
servable by LIGO) or supermassive black holes (observable by LISA). However,
with an extended PBH mass function, the GW background ought to encompass
both of these limits and also every intermediate frequency. Many supermassive
black holes are in binary pairs that orbit together and eventually merge, emit-
ting GW in the process. The LISA detection window includes mergers of black
holes in the mass range of 104 − 107 solar masses [174]. Due to the possibility
that the coalescing black holes observed by LIGO [153] could be of primordial
origin, black holes in the intermediate mass range of 10 − 103 solar masses are
of particular interest since such PBHs might contribute to the dark matter (see
below).
The primary goal of CMB observations is the polarization signal induced

34
by GW at the start of inflation. There is a considerable effort underway to
obtain stricter limits on the tensor-to-scalar ratio, r, the quantitative measure
of the ratio of the primordial amplitude of the B-mode (or shearing) polar-
ization component due to GW to the scalar (or compressive) mode of CMB
temperature fluctuations associated with the density fluctuations that seeded
structure formation. While PGW have not yet been detected, the upper limit
on r from the BICEP2/Keck CMB polarization experiments [84] (in conjunction
with Planck temperature measurements and other data) is less than or equal to
approximately 0.07 at the 95% confidence level. However, the tensor amplitude
predicted depends on the (fourth power of the) energy scale of inflation, and
so the primordial polarization signal could, in principle, be unmeasurably small
[16].

3.2.3 Primordial black holes

The possibility of 10 − 103 solar mass objects is of particular interest in view
of the recent detection of black-hole mergers by LIGO which has, in particular,
revitalized the interest in stellar mass black holes of around thirty solar masses
(which are larger than initially expected) [153], and especially non-evaporating
primordial black holes (PBHs). In particular, it has been suggested that massive
PBHs could provide the dark matter [180] or the supermassive black holes which
reside in galactic nuclei and power quasars [181].
The most natural mechanism for PBH formation involves the collapse of
primordial inhomogeneities, such as might arise from inflation (or spontaneously
at some kind of phase transition). Interest in PBH increased due to the discovery
that black holes radiate [195], since only PBH could be small enough for this
to be relevant cosmologically. Indeed, evaporating PBHs have been invoked to
explain several cosmological features [196]. Since it was initially believed that
PBHs would grow as fast as the Universe during the radiation-dominated era
and consequently attain a huge mass by the present time, it was thought that
PBH never formed and could thus be excluded. However, such an argument is
essentially Newtonian and neglects the cosmological expansion, and in [175] it
was shown that there is no self-similar solution in which a black hole can grow
as fast as the Universe. Therefore, once formed, their contribution to the dark
matter of the Universe grows with time (the mass of non-evaporating PBH is
unchanged after formation and can only grow if they accrete matter) [176].
PBH would have the particle horizon mass at formation and could form
as early as the Planck epoch, when QG forces are comparable to gravitational
forces that at later epochs are far too weak on particle scales. However, as
the Universe expands and cools, tiny black holes of Planck mass all quickly
disappear. More massive black holes live longer and should survive until today
as early Universe relics [175]. Attention has consequently shifted to larger PBHs,
which are unaffected by Hawking radiation. Such PBHs might have important
cosmological consequences.
Perhaps the most exciting possibility is that PBH larger than 103 solar
masses could provide the dark matter which comprises 25% of the critical den-

35
sity [196]. 7 Since PBHs formed in the radiation-dominated era, they are not
subject to the well-known cosmological nucleosynthesis constraint that baryons
can contribute at most 5% to the critical density. PBH should thus be classified
as non-baryonic and behave like any other form of cold dark matter (CDM).
The subject has consequently become very popular and non-evaporating PBHs
may turn out to play a more important cosmological role than evaporating ones.
PBHs could provide the dark matter but a number of constraints restrict
their possible mass ranges [180], including those arising from gravitational mi-
crolensing, but PBHs at a level of 10% of the dark matter are still possible over
a wide range of masses. The PBH density might be much less than the dark
matter density, but the PBHs are not necessarily required to provide all of the
dark matter [196]. For intermediate mass black holes of 103 solar masses a dark
matter mass fraction of only 0.1% still allows for important consequences for
structure formation. Cosmological structures could be generated either individ-
ually through a ‘seed’ effect or collectively through the ‘Poisson’ effect (fluctu-
ations in the black hole population generates an initial density perturbation for
PBH dark matter), consequently alleviating some of the possible problems as-
sociated with the standard CDM scenario (even when they may only contribute
a small portion of the dark matter density). Both mechanisms for generating
fluctuations then amplify through gravitational instability to bind massive re-
gions [181] and have been considered as either alternatives or in conjunction
with other CDM scenarios.

3.3 Effects of structure on observations: Gravitational

lensing
A particularly important cosmological question is whether gravitational lensing
significantly alters the distance-redshift relation D(z) to the CMB last scattering
surface or the mean flux density of sources. Any such D(z) bias could change
CMB cosmology, and the corresponding bias in the mean flux density could
alter supernova cosmology.
In spatially homogeneous and isotropic cosmologies the ratio between the
proper size of a source and the angular diameter distance is a function of redshift
only. In an inhomogeneous Universe, lensing by intervening metric fluctuations
can cause magnification of the angular size, with a corresponding change of
flux density, since surface brightness is not affected by gravitational lensing.
Therefore, the apparent distance to objects at a given redshift can effectively
become a randomly fluctuating quantity.
Using conservation of photons (i.e., flux conservation), Weinberg [161] argued
that in the case of transparent lenses there is no mean flux density amplification,
so that the uniform universe formula for D(z) remains unchanged (where the
averaging is over sources, and the result relies on the implicit assumption that
the area of a constant-z surface is unaffected by gravitational lensing). This
7 An alternative to PBHs includes persistent (or “pre-big-bang”) black holes occurring in

bouncing cosmologies [172].

36
issue has recently been revisited [162], and it was argued that in an ensemble
averaged (and more appropriate cosmological) sense, the perturbation to the
area of a surface of constant redshift is in reality a very small (approximately
one part in one million) effect, supporting Weinberg’s argument and validating
the usual treatment of gravitational lensing in the analysis of CMB anisotropies.
However, Weinberg’s argument regarding the mean flux density appears to
contradict well-known theorems of gravitational lensing, such as the focusing
theorem. Non-linear relativistic perturbation theory to second order indicates
that there is bias in the area of a surface of constant redshift and in the mean
distance to the CMB last scattering surface. Indeed, a lot of investigations
of gravitational lensing continue to advocate significant effects in the mean.
Bolejko [157] (also see references in [162]) has provided a comprehensive review
of such studies, some of which claim large effects, some of which obtain effects at
the level of a few percent (which would still be important), while others argue
that the effects are exceedingly small. A non-vanishing perturbation to the
mean flux densities of distant sources caused by intervening structures, at least
for sources that are viewed along lines of sight that avoid mass concentrations,
effectively contradict Weinberg’s result. Recent non-linear analysis does suggest
that non-linear effects have not been proven to be negligible [80, 191, 163].

3.4 Backreaction and averaging

Averaging in GR is a fundamental problem within mathematical cosmology [3].
The cosmological FE on the largest scales are derived by averaging or coarse
graining the EFE of GR. A solution of this problem is critical for the correct
interpretation of cosmological data [24] (on the largest scales the dynamical
behavior can be significantly different from the dynamics in the standard cos-
mology; e.g., the expansion rate can be greatly affected [145]).
First, it is of great importance to provide a rigorous mathematical defini-
tion for averaging (tensors on a differential manifold) in GR. A spacetime or
space volume averaging approach must be well defined and generally covariant
[141, 143], and produce the structure equations for the averaged macroscopic
geometry (and give a prescription for the correlation functions in the macro-
scopic FE which emerge in the averaging of the non-linear FE), which do not
necessarily take on precisely the same mathematical form as the original FE
[143]. It is straightforward to average scalar quantities and since, in general,
a spacetime is determined entirely by its scalar curvature invariants, a specific
spacetime averaging scheme based on scalar invariants only has been proposed
[142]. In addition, only scalar quantities are (space volume) averaged within the
(1 + 3) cosmological spacetime splitting approach of Buchert [145].
Although the standard FLRW ΛCDM cosmology has, to date, been very
successful in explaining all of the observational data (up to a number of potential
anomalies and tensions [83]) it does require, as yet undetected, sources of dark
energy density that currently dominate the dynamics of the Universe. More
importantly, the actual Universe is neither isotropic nor spatially homogeneous
on small scales. Indeed, observations of the current late epoch Universe uncovers

37
a very complicated picture in which the largest gravitationally bound structures,
consisting of clusters of galaxies of different sizes, form, in turn, “knots, filaments
and sheets that thread and surround very underdense voids” [146]. An enormous
fraction of the volume of the current Universe is, in fact, contained within voids
of a single characteristic size of about 30 megaparsecs [147] with an almost
“empty” density contrast [148].
In principle, a number of coarse grainings over different scales is required
to reasonably model the observed complicated gravitationally bound large scale
structures [22]. In standard cosmology it is implicitly taken that the matter
distribution on the largest scale can be modeled by an “effective averaged out”
stress-energy tensor, regardless of the physical details of the actual coarse grain-
ing at each scale. However, based on the two-point galaxy correlation function,
the very smallest scale on which there can be a reasonable definition of sta-
tistical homogeneity is 70–120 megaparsecs [149], and even then variations for
the number density of galaxies on the order of several percent still arise in the
largest possible survey volumes [150, 21]. It is fair to say that it is not at all
clear what the largest scale is that matter and geometry on smaller scales can
be coarse-grained such that the average evolution is still an exact solution of
the EFE.
A smooth macroscopic geometry (with macroscopic matter fields), applicable
on cosmological scales, is obtained after an appropriate averaging. The coarse
graining of the EFE for local inhomogeneities on small scales can generally
lead to important backreaction effects (consisting of not just the mean cosmic
variance) [151] on the average dynamics of the Universe [145]. In addition, all
cosmological observations are deduced from null geodesics (the paths of photons)
which travel enormous distances, preferentially traversing the underdense voids
of the actual Universe. But inhomogeneities perturb curved null geodesics, so
that observed luminosity distances can be significantly affected.
A consistent approach to cosmology is consequently to treat GR as a meso-
scopic theory, which is applicable only on the mesoscopic scales for which it has
actually been verified, containing a mesoscopic metric field and a mesoscopic ge-
ometry. The effective macroscopic dynamical equations on cosmological scales
are then obtained by averaging. It had originally been hoped that such a back-
reaction approach might help resolve the dark energy and dark matter prob-
lems. However, it now seems unlikely that backreaction can replace dark energy
(although large effects are theoretically possible from inhomogeneities and av-
eraging [24]). But it can certainly affect precision cosmology at the level of 1
% [26] and may offer a better understanding of some issues in cosmology (such
as the emergence of a homogeneity scale and non-zero spatial curvature due to
non-linear evolution of cosmic structure).

3.4.1 Backreaction magnitude

This last point is very important, and since it has been a source of some con-
troversy let us summarize briefly here. The Universe is very inhomogeneous
on small scales at the present time but smooth on large scales. It must be re-

38
membered that density pertubations (δρ/ρ) ≃ 1028 on Earth, but the metric is
very close to Minkowski. To establish the backreaction effects we need approx-
imation methods to deal with metric perturbations (δh/h) ≃ 10−5 but second
derivatives ≃ 1028 . Various approaches have been tried:

• Zalaletdinov [143] developed a very complex bimetric averaging formalism

that can, in principle, be applied in general; the effect of such averaging on
cosmological observations was estimated to be of the order of about 1 %
[144]. A global Ricci deformation flow for the metric, which is generically
applicable in cosmology, was introduced by Carfora.
• Buchert [145] developed an explicit (1 + 3) spatial averaging scheme, al-
though the scheme is not fully deterministic and depends on some ad hoc
phenomenological assumptions. Models based on this scheme, and partic-
ularly the “timescape cosmology” of Wiltshire which utilizes time-dilation
effects in voids, can predict very large effects, and there have been claims
that the results are sufficient to explain dark energy [24].
• There have been various approximation schemes that have claimed that
the backreaction effects are negligible (≃ 10−5 ), including a scheme by
Green and Wald [205, 151] which uses distributional methods that does
not involve explicit averaging.
• Durrer and collaborators have developed detailed second order calcula-
tions that predict percent level changes (i.e., that are sufficient to be of
significance in GR precision cosmology studies), and they (using the “gevo-
lution” numerical code) and others have subsequently confirmed this with
N-body simulations.

The most reasonable outcome of this debate, at least in our view and particularly
in light of the latter results, is that observable differences caused by backreaction
effects will be of the order of 1%.

3.5 Spatial curvature

Current constraints on the background spatial curvature, characterized by Ωk ,
within the standard cosmology are often used to “demonstrate” that it is dy-
namically negligible: Ωk ∼ 5 × 10−3 (95% confidence level) [95]. However, in
standard cosmology the spatial curvature is assumed to be zero (or at least very
small and below the order of other approximations) for the analysis to be valid.
Therefore, strictly speaking, the standard model cannot be used to predict a
small spatial curvature.
In general, Ωk is assumed to be constrained to be very small primarily based
on CMB data. However, the recently measured temperature and polarization
power spectra of the CMB provides a 99% confidence level detection of a negative
Ωk = −0.044 (+0.018, −0.015), which corresponds to a positive spatial curva-
ture [84]. Direct measurements of the spatial curvature Ωk using low-redshift

39
data such as supernovae, baryon acoustic oscillations (BAO) and Hubble con-
stant observations (as opposed to fitting the FLRW model to the data) do not
place tight constraints on the spatial curvature and allow for a large range of
possible values (but do include spatial flatness). Low-redshift observations often
rely on some CMB priors [201] and, in addition, are sensitive to the assumptions
about the nature of dark energy. 8
Attempts at a consistent analysis of CMB anisotropy data in the non-flat
case suggest a closed model with Ωk ∼ 1% [122, 202]. Including low redshift
data, Ωk = −0.086 ± 0.078 was obtained [122], which provides weak evidence in
favor of a closed spatial geometry (at the level of 1.1σ), with stronger evidence
for closed spatial hypersurfaces (at a significantly higher σ level) coming from
dynamical dark energy models [202] (see also [123]). The inclusion of CMB
lensing reconstruction and low redshift observations, and especially BAO data,
gives a model dependent constraint of Ωk = −0.0007 ± 0.0019 [84].
As an illustration, constraints on the phenomenological two curvature model
(which has a simple parametrized backreaction contribution [144] leading to
decoupled spatial curvature parameters Ωkg , Ωkd in the metric and the Fried-
mann equation, respectively, and which reduces to the standard cosmology when
Ωkg = Ωkd ), were investigated in [164]. It was found that the constraints on the
two spatial curvature parameters are significantly weaker than in the standard
model, with constraints on Ωkg an order of magnitude tighter than those on Ωkd ,
and there are tantalizing hints from Bayesian model selection statistics that the
data favor Ωkd 6= Ωkg at a high level of confidence.
Observations on recently emerged, present-day (large-scale mean) average
negative curvature are weak and not easy to measure [124]. Local inhomo-
geneities and perturbations to the distance-redshift relation at second-order
contribute a monopole at the sub-percent level, leading to a shift in the apparent
value of the spatial curvature (as do other GR curvature effects in inhomoge-
neous spacetimes). Indeed, in an investigation of how future measurements of
Ωk are affected by GR effects, it was shown that constraints on the curvature
parameter may be strongly biased if cosmic magnification is not included in the
analysis [125].
Given that current curvature upper limits are at least one order of mag-
nitude away from the level required to probe most of these effects, there is
an imperative to continue pushing the curvature parameter, Ωk , constraints to
greater precision (i.e., to about the 0.01% level). These will become increasingly
measurable in future surveys such as the Euclid satellite. In addition, the cur-
rent curvature parameter estimations are not yet at the cosmic variance limit
(beyond which constraints cannot be meaningfully improved due to the cosmic
variance of horizon scale perturbations); indeed, the current measurements are
more than one order of magnitude away from the limiting threshold [125]. The
prospects for further improving measurements of spatial curvature are discussed
in [126]. Most importantly, we are interested in model independent [127] and
8 For late Universe observables there is significant degeneracy between Ω and dark en-
k
ergy parameters; the standard approach is to treat Ωk and these parameters as independent
quantities, and to marginalize over the dark energy parameters [121].

40
explicitly CMB-independent [121] checks of the cosmic flatness.
However, currently there is no fully independent constraint with an appro-
priate accuracy for a value of Ωk of approximately less than 0.01 on the cosmic
flatness from cosmological probes. In principle, a small non-zero measurement
of Ωk perhaps indicates that the assumptions in the standard model are not
met, thereby motivating models with curvature at the level of a few percent.
Such models are certainly not consistent with simple inflationary models in
which Ωk is expected to be negligible [204]. We remark that an observation of
non-zero spatial curvature, even at the level of a percent or so, could be the re-
sult of backreaction effects and be a signal of non-trivial averaging effects [143].
Note that calculations imply a small positive spatial curvature [144] (although
backreaction estimates have tended to give a negative mean curvature [124]).
If the geometry of the universe does indeed deviate slightly from the standard
FLRW geometry (for example, due to the evolution of cosmic structures), then
the spatial curvature will no longer necessarily be constrained to be constant
and any effective spatial flatness may not be preserved. An investigation of a
small emerging spatial curvature can be undertaken by relativistic cosmological
simulations [77, 25]. However, such simulations need to include all relativistic
corrections and can suffer from gauge issues [210, 25]. In particular, using
a fully inhomogeneous, anisotropic cosmological numerical simulation, it was
shown that [26]: (i) On small scales, below the measured homogeneity scale
of the standard cosmology, deviations in cosmological parameters of 6 - 31%
were found (in general agreement with LPT and with deviations depending on
an observer’s physical location). (ii) On the approximate homogeneity scale
of the Universe mean cosmological parameters consistent to about 1% with
the corresponding standard cosmology were found (although the parameters
can deviate from these mean values by 4-9% again depending on the physical
location in the simulation domain). (iii) Above the homogeneity scale of the
Universe, 2 - 3% variations in mean spatial curvature and backreaction were
found.
As noted above, attempts to study relativistic models of inhomogeneities
rely upon metric forms that are designed to be “close to” the spatially homoge-
neous and isotropic metric form. However, these can not also be used to address
the cosmological backreaction problem; backreaction can only be present if the
structure–emerging average spatial curvature, and hence the large–scale aver-
age of cosmological variables, are allowed to evolve [211]. A dynamical coupling
of matter and geometry on small scales which allows spatial curvature to vary
is a natural feature of GR. Indeed, the requirement that spatial curvature re-
mains constant as in an FLRW model on arbitrarily large scales of cosmological
averaging is not a natural consequence of any principles of GR. Schemes that
suppress average curvature evolution (e.g., by employing periodic boundary con-
ditions as in Newtonian models and neglecting global curvature evolution) can
not describe global backreaction but only cosmic variance [24]. Moreover, within
standard cosmology, spatial fluctuations are conceived to evolve on an assumed
background FLRW geometry, but this description only makes sense with respect
to their spatial average distribution and its evolution. We note that even small

41
fluctuations within averaging schemes are also subject to gauge issues [212]. In
principle, large effects are possible from inhomogeneities and averaging [22, 24].
Recently, a relativistic (Simsilun) simulation based on the approximation of
a ‘silent universe’ was presented [156]. The simulation begins with perturba-
tions around a (flat) standard model (with initial conditions set up using the
Planck data). The perturbations are allowed to have non-zero spatial curvature.
Initially, the negative curvature of underdense regions is compensated by the
positive curvature of overdense regions [213, 144]. But once the evolution enters
the non-linear regime, this symmetry is broken and the mean spatial curvature
of the universe slowly drifts from zero towards negative curvature induced by
cosmic voids (which occupy more volume than other regions). The results of
the Simsilun simulation indicate that the present-day curvature of our Universe
is Ωk ∼ 0.1, as compared to the spatial flatness of the early universe.
It should be emphasised that the fact that structure formation implies that
the present-day Universe (is volume-dominated by voids and) is characterized
by on average negative curvature is a subtle issue that follows from the result
that intrinsic curvature does not obey a conservation law [24, 23]. Indeed, it
dispels the naive expectation that on large scales the distribution of positive
spatial curvature for high-density regions and negative spatial curvature for the
voids, averages out to the almost or exactly zero spatial curvature assumed.

42
4 Problems from the quantum realm
There are a number of very fundamental problems in the quantum regime, cul-
minating in the question of whether there is a single unified theory of quantum
gravity (QG). And, in particular, is this “theory of everything” string theory?
Some problems in the quantum realm are relevant for cosmology. For example,
do there exist any fundamental particles that are predicted by QG that have
not yet been observed and, if so, what are their properties and are they of im-
portance in cosmology? In particular, the detection of the Higgs boson seems to
complete the standard model, but with additional new physics that is needed to
protect the particle mass from quantum corrections (that could increase it by
14 orders of magnitude). It is believed that supersymmetry is the most reason-
able solution to this naturalness problem, but the most simple supersymmetric
models have not proved successful and, to date, there is no convincing mecha-
nism to break supersymmetry nor to determine the multiple parameters of the
supersymmetric theory. In addition, does a theory of QG lead to a multiverse in
cosmology? And, perhaps most importantly, do theories of QG naturally lead
to inflation?

4.1 The problem of quantum gravity:

The standard model of particle physics concerns only 3 forces: namely, elec-
tromagnetism and the strong and weak nuclear forces. A primary goal of the-
oretical physics is to derive a theory of QG in which all 4 forces, including
that of gravitation, are unified within a single field theory. Up to now, no at-
tempt at such a unification has been successful. In particular, it is of interest
to know whether QG can be formulated for cosmology and whether there is
any extension of quantum mechanics required for QG, and especially quantum
cosmology? Quantum cosmology gives rise to a number questions concerning
a possible theory for the initial cosmological state [29], which include: what
laws or principles might characterize the initial conditions of the Universe and
what are the subsequent predictions of the initial conditions for the Universe on
macro-, meso- and micro-scopic scales?
Let us first briefly discuss two cosmological problems that originate in QG
and have a very distinct mathematical formulation of particular interest here.

4.1.1 Higher dimensions

Ordinary spacetime is 4D, but additional dimensions are possible in, for ex-
ample, string theory [214]. At the classical level, gravity has a much richer
structure in higher dimensions than in 4D. In particular, there has been a lot of
work done on the uniqueness and stability of black holes in arbitrary dimensions
[215]. For example, closed trapped surfaces and singularity theorems in higher
dimensions have been discussed [216] and the positive mass theorem has been
proven in arbitrary dimension [217]. However, the problem of stability in higher
dimensions is much more difficult. Indeed, there is evidence from numerical

43
simulations to indicate that there are higher dimensional black holes that are
not stable [215]. In addition, the question of cosmic censorship in higher dimen-
sions is extremely difficult and is perhaps not even well posed. In fact, there
is numerical evidence that suggests that cosmic censorship does not hold [218]
and that black holes are not necessarily stable to gravitational perturbations in
higher dimensions [219]. Indeed, black holes become highly deformed at very
large angular momenta and resemble black branes, and in spacetime dimensions
greater than six exhibit an “ultraspinning instability” [220].
Higher dimensional spacetime manifolds are also considered in a number
of cosmological scenarios. For example, in the cosmological context all known
mathematical results can be investigated in models with a non-zero cosmological
constant. In addition, theoretical results, such as the dynamical stability of
higher dimensional cosmological models, are of interest. In particular, spatially
homogeneous cosmologies in higher dimensions, and especially extensions of the
BKL analysis, have been investigated [221].

4.1.2 AdS/CFT correspondence

Anti de Sitter (AdS) spacetimes are of interest in QG theories formulated in
terms of string theory due to the AdS/CFT (or Maldacena gauge/gravity dual-
ity) correspondence, in which string theory on an asymptotically AdS spacetime
is conjectured to be equivalent to a conformally invariant quantum field theory
(CFT) on its boundary [222, 223]. This holographic paradigm leads to a num-
ber of cosmological questions. In particular, the AdS/CFT conjecture strongly
motivates the dynamical investigation of asymptotically AdS spacetimes. But,
of course, such a spacetime is clearly not this Universe. In addition, recently
it has been conjectured that AdS spacetimes are unstable to arbitrarily small
perturbations [224].
The global non-linear stability of AdS has been investigated in spherically
symmetric massless scalar field models within GR [225]. Numerical evidence
seems to indicate that AdS spacetimes are non-linearly unstable to a “weakly
turbulent mechanism” in which an arbitrarily small black hole is formed whose
mass is determined by the initial energy. Such a non-linear instability appears
to happen for various typical perturbations. However, there are also many
perturbations that don’t lead to an instability, which consequently implies the
existence of “islands of stability” [226, 227]. It is of great interest to study
the non-linear stability of AdS with no assumptions on symmetry; however,
such a study is currently intractable both analytically and numerically. But the
general gravitational case is clearly richer than the case of spherical symmetry
analysed to date [226]. Therefore, it is of great significance to determine if
the conjectured non-linear instability in AdS spacetime in more general models
behaves in a similar or a different way to that in spherically symmetric scalar
field collapse [224].

44
4.2 Singularity resolution
4.2.1 Singularity resolution and a quantum singularity theorem
The existence of singularities indicates a failure of GR when the classical space-
time curvature is sufficiently large. This is exactly when QG effects are antici-
pated to be important. Therefore, the problem of if, and when, QG can extend
solutions of classical GR beyond the singularities is crucial [228]. It is, of course,
pertinent to determine whether all singularities can be removed in QG. How-
ever, it is certainly not true that all singularities can be resolved within string
theory; for example, it is known that the string in an exact plane wave back-
ground does not propagate through the curvature singularity in a well-behaved
manner [229].
Gauge/gravity duality, which can be regarded as providing an indirect for-
mulation of string theory [230], has been utilized to study singularities in the
quantum realm and investigate cosmic censorship with asymptotically AdS ini-
tial data. The existence of a quantum version of cosmic censorship was suggested
from holographic QG [231]. It has been deduced that a large class of bounces
through cosmological singularities are forbidden. Consequently, although some
singularities can indeed be resolved, a novel singularity theorem is possible.
Therefore, it is important to determine whether a quantum mechanical gener-
alization of any of the singularity theorems exists, which would subsequently
imply that singularities are inevitable even in quantum settings. In particular,
it has been shown that a fine-grained generalized second law of horizon thermo-
dynamics can be used to prove the inevitability of singularities [232], thereby
extending the classical singularity theorem of Penrose [41] to the semi-classical
regime. It is plausible that this result, which was constructed in the context of
semiclassical gravity, will still hold in a complete theory of QG [232]. Therefore,
not all singularities can be resolved within QG.

4.2.2 Cosmological singularity resolution

Cosmological singularity resolution can be investigated within loop quantum
gravity (LQG) and string theory. (Black hole singularity resolution was reviewed
in [5].) LQG is a non-perturbative canonical quantization of gravity. It has been
suggested that singularities may be generically resolved within LQG as a result
of QG effects [233]. In particular, the classical big bang singularity is replaced by
a symmetric quantum big bounce when the energy density is of the order of the
Planck density, which occurs without any violation of the energy conditions or
fine tuning. The resulting quantum big bounce connects the currently expanding
universe to a pre-bounce contracting classical universe.
The application of LQG in the context of cosmology is referred to as loop
quantum cosmology (LQC). In LQC the infinite degrees of freedom reduce to a
finite number due to spatial homogeneity. A variety of spatially homogeneous
cosmologies have been investigated [234]. In particular, solutions to the effective
equations for the general class of Bianchi type IX cosmological spacetimes has
been investigated within LQC computationally, wherein the big bang singular-

45
ity was shown to be resolved [235]. The reduction of symmetries within LQC
involves a very considerable simplification, and consequently crucial aspects of
the dynamics may be neglected. However, partly due to evidence supporting
the BKL conjecture, it is believed that the singularity resolution in spatially
homogeneous cosmologies does capture important features of singularity reso-
lution in more general spatially inhomogeneous cosmological models [234, 236].
There are ongoing attempts to include spatial inhomogeneities in the analysis
[237].
Various singularities have been investigated within standard LQC. It has
been conjectured that all curvature singularities which result in geodesic in-
completeness are so-called strong singularities (such as the big bang in GR). In
recent years a number of other types of cosmological singularities have been ob-
tained, which include the big rip and the big freeze, and sudden and generalized
sudden singularities. Of these, the big rip and big freeze are strong singularities
within GR, whereas sudden and generalized sudden singularities are weak singu-
larities. Using a phenomenological matter model in GR, it has been established
that strong singularities are, in general, resolved in LQC, whereas quantum
geometry does not usually affect weak singularities [238]. A comprehensive in-
vestigation of the resolution of a variety of singularities within modified LQC
models, in which the bounce can be asymmetric and the bounce density can be
affected, was performed using an effective spacetime description and compared
with the analysis in standard LQC [238].

4.3 Quantum gravity and inflation

Although some of the alternatives to inflation alluded to earlier are suggested by
ideas motivated by QG, it is also of interest to determine whether inflation occurs
naturally within QG. For example, it appears to be difficult to get inflation
within string theory [239]. In particular, so-called swampland criteria constrain
inflationary models and there are no-go theorems for the existence of de-Sitter
vacua in critical string theory. The fact that exact de Sitter solutions with a
positive cosmological constant cannot describe the late-time behaviour of the
Universe [239] is often interpreted as “bad news” for string theory.
The observations of Planck 2018 (of the almost scale-invariant and Gaussian
primordial curvature perturbations) [84] are compatible with the predictions
of simple single scalar field inflation models with a canonical kinetic term and
an appropriately flat self-interaction potential minimally coupled to gravity.
However, despite the success of the single-field slow-roll inflation model, it is
not straightforward to embed such a model within a fundamental theory [240].
However, the so-called α-attractor models and, in particular, the KKLMMT
model [241], have been actively studied. The most attractive theoretical proper-
ties of these models is their conformal symmetry and their successful embedding
into supergravity via hyperbolic geometry. The KKLMMT model is often ac-
knowledged as the first to discuss the origin of D-brane inflation within string
theory [241], and provides the motivation for more general string inspired cos-
mological models. These models predict values for the spectral index and the

46
tensor-to-scalar ratio which match observational data well. Thus, phenomeno-
logical D-brane inflation has attained renewed importance, independent of its
string theory origin, since Planck 2018 [84]. Indeed, it has been shown [242]
that further phenomenological models of D-brane inflation can be derived within
the string theory approach (see also [240]). Because scalar fields (such as, for
example, moduli fields) occur ubiquitously in fundamental theories such as su-
pergravity and string/M theory, multi-field generalizations of the α-attractor
models have also been considered [243].
A number of inflationary cosmologies have been suggested within the context
of string/M-theory [241, 239]. However, very few models exist that can be
embedded within LQC [244]. In particular, there are a number of approaches
to QG which include bouncing regimes. In resolving the initial singularity, it
is of interest to determine whether slow-roll inflation is subsequently allowed
(or is even natural). Inflation within the context of LQC, and how the bounce
affects the evolution of the inflaton (as compared to the normal scenario with
no bounce), was investigated in [245]. The evolution of the inflaton from the
initial bounce was studied analytically for a number of important potentials in
the case that the inflaton is taken to be the same scalar field that gives rise to
the LQC bounce. It was found [245] that LQC, or any bouncing model in which
the total energy density of the inflaton field is bounded at the transition, does
provide a viable description of the pre-inflationary epoch and the subsequent
smooth evolution to the standard inflationary era. The results were particularly
encouraging in that the bounds obtained theoretically (on the critical bounce
value for the inflaton field in order for there to subsequently be an appropriate
slow roll inflationary regime) match (where appropriate) the known results from
the numerical dynamics of the fully non-linear LQC.

4.3.1 String inflation

Cosmological inflation and its realization within QG and, in particular, in string
theory, was reviewed in [240]. Examples of string inflation include brane and
axion inflation. There are also string inspired effective field theories. Since
string theory is considerably more constrained, some effective field theories that
are apparently consistent at low energies do not, in fact, admit ultraviolet QG
completions (leading to improved predictivity). However, there are indications
that it might not be possible to embed simple inflationary models in string
theory [239, 240]. One problem is that in order to obtain a period of slow-
roll inflation from simple scalar field potentials, field values in excess of the
Planck mass are required. But for sufficiently large values of the fields string
effects on the shape of the potential must be included, which tend to destroy
its required flatness except perhaps in the case of special field symmetries [96]
(however, even then string theoretical arguments such as the so-called “Weak
Gravity Conjecture” [246] can lead to the effective field theory analysis being
invalidated).
In addition, generating effective theories from string theory can also lead to
different ideas as to what a natural (or a minimal) inflationary model might be.

47
Indeed, a comprehensive understanding of naturalness within string theory is
elusive. However, a general feature of all stringy constructions is the existence
of a number of light scalar fields, so while multiple ‘unnecessary’ fields might
be considered non-minimal in many field theory models, they are ubiquitous
within string theory. Time-dependent solutions with string scale curvatures are
crucial for any further comprehension, especially if we hope to progress from
the paradigm of an effective theory for the massless modes.
To date, it is fair to say that there have not been any convincing realizations
of inflation in the context of superstring theory. Making predictions in string
theory is made exceedingly difficult by the landscape problem that string theory
has an enormous number of vacua. Despite the fact that dynamics within the
landscape is not well understood, it appears that false vacuum eternal inflation
is an unavoidable consequence. In addition, all 4D de Sitter vacua in super-
symmetric string theories are metastable, since 10D supersymmetric Minkowski
spacetime has zero energy, but de Sitter spacetime has positive vacuum energy.
In particular, there are well-known no-go theorems for the existence of stable
de-Sitter vacua in critical string theory [239]. This is a real problem for inflation
should string theory be the final theory of QG.
The so-called string swampland criteria constrain inflationary models [247].
In addition, the second of the swampland conjectures implies, as noted above,
that exact de Sitter solutions with a positive cosmological constant cannot de-
scribe the fate of the Universe at late times within string theory [239]. Dynam-
ical dark energy scalar field models must also satisfy particular criteria so as
to avoid the swampland. The observational implications of such string-theory
criteria on quintessence models and the accompanying constraints on the dark
energy were studied in [248]. However, since string theory does not naturally
lead to scalar fields with an appropriate energy scale to be a reasonable candi-
date for quintessence, novel physics from string theory must be introduced to
explain dark energy. In some very special models it is possible to characterize
the Planck-suppressed corrections to the string theory inflatonary action, lead-
ing to the first indications for inflation within string theory [240]. But many
critical challenges still remain. Indeed, the ‘simple’ cosmological observations
(of the almost scale-invariant and Gaussian primordial curvature perturbations
measured by Planck) to date are often interpreted as an argument against com-
plex models of inflation in string theory (however, see [240]).

48
5 Concluding remarks
We have reviewed recent developments and described a number of open ques-
tions in the field of theoretical cosmology. We described the concordance cos-
mological model and the standard paradigms of modern cosmology, and then
discussed a number of fundamental issues and open theoretical questions, em-
phasizing the various assumptions made and identifying which results are in-
dependent of these assumptions. Indeed, standard cosmology contains a num-
ber of philosophical assumptions that are not always scientific, including the
assumption of spatial homogeneity and isotropy at large scales outside our par-
ticle horizon. Perhaps a more tangible fundamental issue concerns the measure
problem and the issue of initial conditions in inflation. Many of the fundamental
problems arise due to the inhomogeneities in the Universe. However, this is also
one of the great strengths of present day cosmology: our models predict what
structure will occur, and consequently the astounding development of observa-
tional projects determining in great detail the characteristics of such structure
that serve to give strong limits on cosmological parameters.
Cosmology is not only a mathematical endeavour, but it is a testable sci-
entific theory due to its ability to produce observational predictions. In recent
times there has been a plethora of such detailed tests, leading to the so-called
era of precision cosmology. Perhaps fundamental questions are less relevant for
current working cosmologists, who are more concerned with physical cosmology
and data and statistical analysis. But as the modern emphasis changes to more
physical and observational issues, theoretical cosmology is still important and
fundamental questions persist. In some sense, we hope to record here the state
of the art as it now exists.
A qualitative analysis of the properties of cosmological models and the prob-
lems of the stability of cosmological solutions and of singularities is important in
mathematical cosmology. A number of open problems in theoretical cosmology
involve the nature of the origin and details of cosmic inflation, and its relation
to fundamental physics. Perhaps the most urgent open problems of theoretical
cosmology include the early and late time accelerated expansion of the universe
and the role of the cosmological constant Λ. As we have emphasized, computa-
tional cosmology is becoming an increasingly important tool in the investigation
of theoretical and physical cosmology.
We then reviewed a number of open problems in physical cosmology, with
particular focus on perturbation theory (and gauge issues) and the formation
and distribution of large scale structure in the Universe at present times (and
especially in the non-linear regime). Backreaction is still an important issue,
although perhaps the more formal mathematical averaging problem is currently
more relevant. Finally, gravitational wave astronomy will potentially play an in-
creasingly important role within cosmology. Indeed, there is a robust prediction
within inflation for a gravitational wave induced CMB polarization signal.
We also discussed cosmological problems in quantum gravity, including the
possible resolution of cosmological singularities and the crucial issue of the role
of inflation within quantum gravity.

49
 Finally we have emphasized that, given the uniqueness of the Universe and
the limitations on the domain we can explore by any conceivable observations,
it is key to carry out all possible consistency tests of our models. For example,
the first and foremost is the age of the universe: is the Universe older than its
stellar and galactic content? If not, cosmology is in deep trouble. Fortunately
this consistency test seems to be satisfied at present (thanks to the cosmological
constant). Another consistency test is that all number counts must display a
dipole aligned with the CMB dipole; this is presently being contested.

50
Acknowledgements
We would like to acknowledge Bernard Carr for his helpful comments, and to
thank Timothy Clifton and Julian Adamek for fruitful discussions. Financial
support was provided by NSERC of Canada (AAC) and NRF of South Africa
(GFRE).

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