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\DOI{10.5802/crmeca.277}
\datereceived{2024-10-11}
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\dateposted{2025-01-08}
\begin{document}


\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{Cosmology, the big bang and the BKL conjecture}

\alttitle{Cosmologie, big bang et conjecture BKL}

\author{\firstname{Hans} \lastname{Ringstr\"{o}m}\CDRorcid{0000-0001-9383-0748}}
\address{Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden}
\email{hansr@kth.se}

\begin{abstract} 
This is a review article of mathematical results in cosmology, written
in honor of Yvonne Choquet-Bruhat's 100th birthday. It starts with a
brief description of some of the essential questions: strong cosmic
censorship; the relation between the future asymptotics and
geometrization in the vacuum setting; the cosmic no-hair conjecture;
and the BKL-proposal. It then turns to results, starting with ones
obtained in situations with symmetry. It continues with a review of
future global non-linear stability results, stable big bang formation
results, results concerning solutions to linear equations on
cosmological backgrounds and numerical results. The article contains a
substantial, but very incomplete, list of references to the literature.
\end{abstract}

\begin{altabstract} 
Il s'agit d'un article de revue sur les r\'{e}sultats math\'{e}matiques en
cosmologie, \'{e}crit en l'honneur du 100\`{e}me anniversaire d'Yvonne
Choquet-Bruhat. Il commence par une br\`{e}ve description de certaines
questions essentielles : la censure cosmique forte, la relation entre
l'asymptotique future et la g\'{e}om\'{e}trisation dans le cas du vide, la
conjecture de l'absence de cheveux dans le cadre cosmique et la
proposition BKL. Il aborde ensuite les r\'{e}sultats, en commen\c{c}ant par
ceux obtenus dans des situations avec sym\'{e}trie. Il se poursuit par un
examen des r\'{e}sultats de stabilit\'{e} future non lin\'{e}aire globale, des
r\'{e}sultats de formation stable du big bang, des r\'{e}sultats concernant
les solutions d'\'{e}quations lin\'{e}aires sur les espaces-temps
cosmologiques et des r\'{e}sultats num\'{e}riques. L'article contient une
liste substantielle, mais tr\`{e}s incompl\`{e}te, de r\'{e}f\'{e}rences \`{a} la
litt\'{e}rature.
\end{altabstract}


\keywords{\kwd{Global analysis}
\kwd{General relativity}
\kwd{Lorentzian geometry}
\kwd{Nonlinear evolution equations}
\kwd{Cosmology}
\kwd{The big bang and the BKL proposal}}

\altkeywords{\kwd{Analyse globale}
\kwd{Relativit\'{e} g\'{e}n\'{e}rale}
\kwd{G\'{e}om\'{e}trie lorentzienne}
\kwd{\'{E}quations d'\'{e}volution non lin\'{e}aires}
\kwd{Cosmologie}
\kwd{Big bang et proposition BKL}}

\thanks{Vetenskapsr\aa det, the Swedish Research Council, dnr. 2022-03053.} 

\maketitle

\end{noXML}

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{\vspace*{-6pt}}

\section{Introduction}

From Einstein's attempt to find a static model of the universe; to
Friedmann's discovery of expanding solutions; Lema\^{i}tre's conclusion
that the universe is expanding, based on a combination of observations
and expanding solutions; and Hubble's observations, the initial history
of general relativistic cosmology was long and complicated, and we
refer the interested reader to \cite{NussAndBieri} for a detailed
discussion. Once it becomes clear that the universe is expanding, it is
natural to trace that expansion backward in order to find an origin.
However, even if one finds models with a corresponding cosmological, or
``big bang'', singularity, it is not clear if that is the only
\mbox{explanation,} nor is it clear that such a singularity would persist when
relaxing the symmetry assumptions associated with the model. Hawking's
theorem gives one indication that cosmological singularities are a
general feature of solutions to Einstein's equations; see
\cite{MR0208978,MR0208979} for original references and
\cite[pp.~431--434]{MR0719023} for a textbook presentation. Moreover,
the discovery of the cosmic microwave background (CMB) radiation by
Penzias and Wilson \cite{PaW} gives clear observational corroboration
to the idea of a big bang. However, in spite of the importance of
Hawking's result, it does not say much about the nature of
singularities. Does the gravitational field generically blow up in the
direction of the singularity? Is the spacetime inextendible? In order
to answer these questions, it would seem to be necessary to carry out a
more detailed analysis of solutions. Moreover, the high degree of
isotropy of the CMB cannot be explained by a thermal equilibrium having
been established prior to the surface of last scattering due to the
causal structure of the standard models of the universe. For this
reason, either more sophisticated models of the universe or more exotic
models of matter (inducing, e.g., inflation) would seem to be
necessary.

Already at this point, the necessity for a deeper understanding of the
space of cosmological solutions to Einstein's equations is apparent.
This can be viewed as a mathematical problem, and the purpose of this
review article is to give a rough idea of the developments in this
subject since the appearance of Yvonne Choquet-Bruhat's seminal paper
\cite{MR0053338}. Due to the number of results that have been obtained,
giving a complete description is not possible. In fact, this review
article describes a small subset of the results, and the limited
selection is due to the knowledge (or absence thereof) and interests of
the author. We begin the discussion in the next section by highlighting
some of the questions. In the following sections, we describe results. 


\section{Cosmology, conjectures}

Before describing the objects of study, it is natural to note that
there are at least two fundamental problems in the subject of
cosmology: we cannot carry out experiments; and we only observe a small
part of the universe. Deducing the global properties of the universe on
the basis of observations might therefore even in principle be
impossible. In cosmology, it is common to compensate for the lack of
knowledge by imposing philosophical principles. One example is the
\textit{Copernican principle}, a strong version of which states that
there are no privileged observers in the universe. This principle does
not require the universe to appear the same to all observers.
Nevertheless, the currently preferred so called \textit{cosmological
principle}, states that the universe can be modeled by a spatially
homogeneous and isotropic solution. In such solutions, one cannot
distinguish between two points in space, nor between two directions.
This quite extreme assumption reduces the possibilities for the
geometry to a choice of a $3$-dimensional Riemannian manifold of
constant curvature and a scalar function of time, describing the
expansion and contraction of space. Physicists largely restrict
themselves to this situation and are more likely, when problems arise,
to introduce exotic matter models than to consider more general
geometries. In the mathematical study of cosmology, we do want to
consider more general geometries. On the other hand, we are not
interested in, e.g., asymptotically flat solutions, in which there are
clearly preferred asymptotic regimes. Using the fact that there is a
well posed Cauchy problem for Einstein's equations, see e.g.
\cite{MR0053338,MR0250640,MR0105294,MR0269263,MR0270699,MR0337248,MR0356790},
it is natural to describe the conditions in terms of initial data. We
do not attempt to give a formal definition of the meaning of a
cosmological spacetime here (where by a \textit{spacetime}, we mean a
time oriented Lorentz manifold).  However, we have a spacetime in mind
in which the geometry and matter fields in different parts of the
initial hypersurface are qualitatively similar. In particular, initial
data on a closed manifold such that the mean curvature is negative on
some parts and positive on others is not of the type we wish to
consider here; this would mean that some observers experience expansion
and some contraction. In particular, that the initial manifold is
closed is neither a necessary nor a sufficient condition for the
corresponding \mbox{spacetime} to be cosmological.  In fact, we are mainly
interested in initial data sets for which the mean curvature is bounded
away from zero by a non-zero constant, in analogy with the conditions
of Hawking's theorem. 

Next, we describe some of the questions of interest. 

\subsection{Strong cosmic censorship, curvature blow up}

As mentioned in the introduction, Hawking's singularity theorem, see
e.g.\ \cite[Theorem~55A, p.~431]{MR0719023}, ensures that singularities
appear naturally in general relativity. On the other hand, the only
conclusion provided is that there are incomplete timelike geodesics.
Moreover, letting $N$ be the timelike future of a point in Minkowski
space, letting $g$ denote the Minkowski metric restricted to $N$, and
foliating $(N,g)$ with hyperboloids, the conditions of Hawking's
theorem are satisfied (and indicate the occurrence of a singularity),
even though $(N,g)$ is simply a subset of Minkowski space. Clearly, it
is desirable to obtain more information. At the very least, it would be
desirable to prove that if one traces a causal geodesic in an
incomplete direction, then a curvature invariant, such as the
Kretschmann scalar (i.e., the Riemann curvature tensor contracted with
itself), becomes unbounded. Unfortunately, such a statement cannot hold
for arbitrary initial data. However, one can conjecture that for
generic initial data in an appropriately chosen class, the Kretschmann
scalar is unbounded in the incomplete directions of causal geodesics in
the corresponding maximal Cauchy development; this is the
\textit{curvature blow up conjecture}. It is natural to try to prove
the corresponding statement in symmetry classes of solutions. This
conjecture is related to the \textit{strong cosmic censorship
conjecture}, stating that for generic initial data (in an appropriate
class), the corresponding maximal Cauchy development is inextendible.
This would mean that, even though Einstein's equations are not
deterministic (in the sense that one cannot determine what spacetime
one is in on the basis of initial data) they are generically
deterministic. In order to obtain mathematical results, one of course
has to give a precise definition of the class of data considered and
the meaning of the word inextentible. 

\subsection{Vacuum evolution and geometrization}\label{ssection:geometrisation}

In \cite{MR0640355}, Eardley and Moncrief suggested one way to address
the strong cosmic censorship conjecture: by proving global existence
results for constant mean curvature (CMC) foliations. Their hope was
that the CMC foliations should avoid the singularities and cover the
spacetimes outside the singularities. More specifically, and focusing
on the expanding direction, there are, due to the work of Fischer and
Moncrief on the one hand, see
\cite{MR1939933,MR1919839,MR1894914,MR1889326,MR1919239,MR1870137,MR1843034,MR1729952,MR1480193},
and Anderson on the other, see \cite{MR2052996,MR1888088}, quite
detailed conjectures concerning the future asymptotics of solutions to
Einstein's vacuum equations (with a vanishing cosmological constant) in
the cosmological setting. In these conjectures, the initial manifold is
assumed to be closed and to not admit a Riemannian metric of positive
scalar curvature. Moreover, the authors focus on CMC foliations. Since
the volume of the leaves of the CMC foliation is expected to tend to
infinity in the expanding direction, it is natural to carry out an
expansion normalization of the metric. In the work of Fischer and
Moncrief, the authors rescale the Riemannian metrics on the leaves by
multiplying them by the mean curvature squared. Anderson rescales by
dividing by the proper time squared (where the proper time is measured
to a fixed Cauchy hypersurface). Considering the rescaled metrics as a
family of Riemannian metrics on a fixed closed manifold, the idea is
that in the limit, this family implements a geometrization of the
manifold. In particular, it converges to complete hyperbolic metrics of
finite volume on the hyperbolic pieces and collapses on the Seifert
fibred pieces. In this sense the evolution leads to an isotropization,
on average, in the expanding direction. As will become clear below,
there are many results confirming this picture. However, the results
are \mbox{either} in symmetric settings or concern perturbations of symmetric
settings. This means that the initial manifold must admit a symmetric
metric, so that the geometrization must be trivial. In other words,
confirming the conjectured picture in a non-trivial setting can be
expected to be quite difficult. 

{\vspace*{-1pt}}

\subsection{Cosmic no-hair conjecture}

{\vspace*{-1pt}}

Turning to cosmological solutions with a positive cosmological
constant, the \textit{cosmic no-hair conjecture} states that, in the
expanding direction, spacetimes should asymptote to de Sitter space.
More specifically, fixing, e.g., a future complete timelike geodesic,
say $\gamma$, then the behavior in $J^{-}(\gamma)$ should
asymptotically be indistinguishable from de Sitter space. For a more
formal definition, see \cite[Definition~1.8, pp.~1573--1574]{MR3506608}
and \cite[Conjecture~1.11, p.~1574]{MR3506608}. Again, there are many
results confirming this proposal in specific situations. 

{\vspace*{-1pt}}

\subsection{The BKL proposal}

{\vspace*{-1pt}}

This review article is focused on mathematical results. However, the
questions that are addressed in the mathematical literature are at
least partially inspired by heuristic arguments and proposals arising
in the physics literature. One such proposal is due to Belinski\v{\i},
Khalatnikov and Lifschitz, BKL for short, and dates back to the 60's,
70's and 80's; see, e.g.,
\cite{LifshitzE.M.1963Iirc,LifshitzEM1964PORC,BelinskiiV.A.1970Oatt,KhalatnikovI.M.1970Gcso,cdi_informaworld_taylorfrancis_310_1080_00018737000101171,BelinskiiV.A.1982Agso}.
The rough statement of the BKL proposal is that generic cosmological
singularities are spacelike, local and oscillatory (with the exception
of matter models such as scalar fields and stiff fluids). More
specifically, distinct causal curves typically lose the ability to
communicate in the direction of the singularity; each causal curve
evolves as its own universe in the direction of the singularity; and
the local behavior is modeled by the dynamics of Bianchi type VIII or
IX (mixmaster) spactimes, which are expected to exhibit (chaotic)
oscillatory behavior in the direction of the singularity. A related
idea arising from the BKL proposal is that spatial derivatives should
be dominated by time derivatives in the approach to the singularity.
From this perspective, it is natural to drop the spatial derivatives in
the equations, which leads to an interpretation of Einstein's equations
as a family of ODE's (one ODE for each spatial point). Two examples of
attempts to make this perspective more precise are given by
\cite{EardleyLiangSachs,MR1048674}. See also
Section~\ref{ssection:id on sing} below for a more detailed
discussion. The ideas of BKL have later been developed further in the
physics literature, see, e.g.,
\cite{MR1981434,MR2154191,MR2481269,HeinzleJ.Mark2012So}. More
mathematical results, deriving conclusions from a priori assumptions,
can be found in, e.g.,
\cite{MR4209186,MR4086822,ringstrom2024geometry}. In the next
subsection, we try to make the ideas of BKL more precise in the
non-oscillatory setting. 

{\vspace*{-1pt}}

\subsection{Initial data on the singularity}\label{ssection:id on sing}

{\vspace*{-1pt}}

One of the ideas of BKL is that the dynamics of nearby observers
decouple and that time derivative terms (or ``kinetic terms'')
asymptotically dominate over spatial derivative terms (or spatial
curvature terms) in the direction of the singularity. In
\cite{EardleyLiangSachs}, the authors discuss ideas of this type in
terms of asymptotics of the geometry, and they speak of
\textit{velocity-dominated singularities}. A related perspective is
developed in \cite{MR1048674}, in which the authors introduce the
terminology that a spacetime can be \textit{asymptotically velocity
term dominated (AVTD) near the singularity}. The idea in
\cite{MR1048674} is to introduce a truncated set of equations, the
\textit{velocity term dominated (VTD) system}, and to say that a
solution is AVTD if it asymptotes to a solution to the VTD system; see
\cite{MR1048674} for details. Unfortunately, the VTD system has no
covariant formulation. On the other hand, the evolutionary part of the
VTD system consists of ODE's for each spatial point. This means that
the VTD system is significantly easier to solve than the original
equations. Moreover, in \cite{MR1828850} the authors prove that, given
a real analytic solution to the VTD system in the $3+1$-dimensional
Einstein-scalar field or Einstein-stiff fluid setting (satisfying
certain conditions), there is a unique corresponding solution to the
actual equations. The authors of \cite{MR1828850} call the resulting
singularities \textit{quiescent}, a terminology that goes back to
\cite{Barrow}.  In the $3+1$-dimensional vacuum setting, the behavior
in the direction of the singularity is expected to be oscillatory (in
accordance with the BKL proposal), in which case it is, at this time,
not clear how to specify initial data on the singularity. However, in
higher dimensions (more specifically, for $n+1$-dimensions with $n\geq
10$), quiescence is expected even in the vacuum setting; see
\cite{MR0815631}. Accordingly, \cite{MR1828850} is generalized in
\cite{MR1957378} to higher dimensions and more general matter fields.

The solutions constructed in \cite{MR1828850} and \cite{MR1957378} are
expected to be robust. In other words, perturbations of the
corresponding initial data are expected to lead to maximal Cauchy
developments with similar singularities; see Section~\ref{section:Big
bang formation} below for a partial justification of this expectation.
However, in situations with symmetry, e.g., even singularities in
$3+1$-dimensional vacuum spacetimes can be quiescent; see, e.g,
\cite{MR1623091,MR1657863,MR1779512,MR1710663,MR1926244,MR3085923,MR3705380,MR1939922,MR2136898}.
The reason for this is that the symmetry type can eliminate the
mechanism causing the oscillations. One can also impose conditions on
data on the singularity that eliminate this mechanism. One
corresponding result in the real analytic setting, but in the absence
of symmetries, is obtained in \cite{MR3424515}. A more general result
in the smooth setting is contained in \cite{MR4621384}. 

The VTD solutions in \cite{MR1828850,MR1957378} can be interpreted as
initial data on the singularity. However, there are many other notions
of initial data on the singularity arising when studying symmetry
classes of solutions as well as in results such as
\cite{MR3424515,MR4621384}. For this reason, it is of interest to try
to find a unifying perspective. This is the goal of
\cite{ringstrom2022initial}. In particular, \cite[Definition~10,
p.~5]{ringstrom2022initial} yields the following definition of initial
data on the singularity in the Einstein-scalar field setting with a
cosmological constant:

\begin{definition}\label{def:ndsfidonbbs}
Let $3\leq n\in\nn{}$, $\Lambda\in\rn{}$, $(\Sigma,\mH)$ be a smooth
$n$-dimensional Riemannian manifold, $\mK$ be a smooth $(1,1)$-tensor
field on $\Sigma$ and $\Phi_{0}$ and $\Phi_{1}$ be smooth functions on
$\Sigma$. Then $(\Sigma,\mH,\mK,\Phi_{0},\Phi_{1})$ are
\textit{non-degenerate quiescent initial data on the singularity for
the Einstein-scalar field equations with a cosmological constant
$\Lambda$} if
\begin{enumerate}[(1)]
\item $\tr\mK=1$ and $\mK$ is symmetric with respect to $\mH$.
\item $\tr\mK^{2}+\Phi_{1}^{2}=1$ and $\mathrm{div}_{\mH}\mK=\Phi_{1}d\Phi_{0}$.
\item The eigenvalues of $\mK$, say $\{p_A\}$, are distinct.
\item $\mO^{A}_{BC}$ vanishes in a neighborhood of $\bx\in\Sigma$ if
$1+p_{A}(\bx)-p_{B}(\bx)-p_{C}(\bx)\leq 0$, where
$\mO^{A}_{BC}:=(\gamma^{A}_{BC})^{2}$; $\gamma^{A}_{BC}$ is given by
$[\mX_{B},\mX_{C}]=\gamma^{A}_{BC}\mX_{A}$; and $\mX_{A}$ is an
eigenvector field corresponding to $p_{A}$ such that
$|\mX_{A}|_{\mH}=1$. 
\end{enumerate}
\end{definition}
The reader interested in a justification of why this is a natural
notion of initial data on the singularity is referred to
\cite{ringstrom2022initial,ringstrom2024geometry}.  That the solutions
to the VTD equations that are used in \cite{MR1828850} as a substitute
for data on the singularity are equivalent to data on the singularity
in the sense of Definition~\ref{def:ndsfidonbbs} is demonstrated in
\cite[Section~1.5, pp.~7--9]{ringstrom2022initial}. That the notion
of initial data used in \cite{MR4621384} is a special case of
Definition~\ref{def:ndsfidonbbs} is demonstrated in 
\cite[Section~1.5, pp.~7--9]{ringstrom2022initial}. That the notion
of initial data on the singularity used in \cite{MR1623091,MR1779512}
in the case of $\mathbb{T}^3$-Gowdy is a special case of
Definition~\ref{def:ndsfidonbbs} is demonstrated in
\cite{ringstrom2022initialsymm}. It would be of interest to show that
all the notions of initial data on the singularity used in the
literature (in the Einstein-scalar field setting) are special cases of
Definition~\ref{def:ndsfidonbbs}. In
\cite{francogrisales2024developmentsinitialdatabig}, Andr\'{e}s Franco
Grisales demonstrates that data as in Definition~\ref{def:ndsfidonbbs}
give rise to unique developments in the $3+1$-dimensional
Einstein-non-linear scalar field setting. In particular, ignoring the
condition of non-degeneracy (i.e., the requirement that the eigenvalues
be distinct), \cite{francogrisales2024developmentsinitialdatabig}
simultaneously generalizes \cite{MR1828850} and \cite{MR4621384}. 

In $3+1$-dimensions, one way to avoid the condition of non-degeneracy
in Definition~\ref{def:ndsfidonbbs} is given in \cite[Remark~26,
p.~9]{ringstrom2022initial}. It is also of great interest to generalize
Definition~\ref{def:ndsfidonbbs} to other matter fields. Partly, this
has already been done in \cite{MR1957378}; the VTD solutions in
\cite{MR1957378} can presumably be used to this end. However, there are
many matter models of interest that are not covered in
\cite{MR1957378}. 

The relation between solutions and data on the singularity are the
following. Assume $(M,g,\phi)$ to be a solution to the Einstein-scalar
field equations with a cosmological constant $\Lambda$. Assume,
moreover, that there is a partial foliation of $(M,g,\phi)$ by
spacelike Cauchy hypersurfaces, say $\{\Sigma_t\}$ for $t\in (0,t_+)$
and assume that the mean curvature of $\Sigma_t$, say $\theta_t$ is
such that $\theta_t$ converges uniformly to $\infty$ as $t\downarrow
0$. Define $\mK_t$ to be the \textit{expansion normalized Weingarten
map} of $\Sigma_t$, i.e., the Weingarten map of $\Sigma_t$ divided by
$\theta_t$. Define
$\mH_t:=\bar{g}_t(\theta_t^{\mK_t}\cdot,\theta_t^{\mK_t}\cdot)$, where
$\bar{g}_t$ is the Riemannian metric induced on $\Sigma_t$. Note that
$\mK_t$ is an operator on the tangent space of $\Sigma_t$ and that
$\theta_t$ can be assumed to be strictly positive, so that 
{\[
\theta_t^{\mK_t}:=\sum_{l=0}^\infty \tfrac{(\ln\theta_t)^l\mK_t^l}{l!}.
\]}\unskip
Finally, let $\phi_t$ be the scalar field induced on $\Sigma_t$,
$\dot{\phi}_t$ be the future unit normal derivative of $\phi$ at
$\Sigma_t$, $\Phi_{1,t}:=\dot{\phi}_t/\theta_t$ and
$\Phi_{0,t}:=\phi_t+\Phi_{1,t}\ln\theta_t$. Then $(M,g,\phi)$ is said
to be a \textit{development of initial data
$(\Sigma,\mH,\mK,\Phi_{0},\Phi_{1})$ on the singularity} if
{\begin{equation}\label{eq:limit}
  \mH_t\rightarrow\mH, \quad  
  \mK_t\rightarrow\mK, \quad  
  \Phi_{0,t}\rightarrow\Phi_{0}, \quad  
  \Phi_{1,t}\rightarrow\Phi_1
\end{equation}}\unskip
as $t\downarrow 0$, where the topology defining the convergence should
be adapted to the situation. 

The notion of initial data on the singularity serves many purposes.
First, the definition is covariant. Second, that a solution is a
development of initial data on the singularity can be interpreted as
saying that it is AVTD, in accordance with the BKL proposal and
\cite{MR1048674}. Third, if there are initial data on the singularity
in a certain symmetry class, then it is natural to conjecture that the
behavior is generically quiescent in this symmetry class, and that
generic solutions are developments of initial data on the singularity.
Conversely, if there are no initial data on the singularity, it is
natural to conjecture that the generic behavior is oscillatory. Fourth,
in quiescent settings, initial data on the singularity can potentially
be used to parameterize solutions. In such settings, they thus fill a
double role: they both provide as detailed information concerning the
asymptotics as one could wish and parameterize solutions. However,
there are at least two caveats to keep in mind. First, the formulation
of the limits in (\ref{eq:limit}) introduces a foliation dependence. It
is natural to interpret this dependence as a lack of covariance. On the
other hand, in many situations there are foliations that are uniquely
determined by the spacetime. Assume that $(M,g)$ is a globally
hyperbolic spacetime with closed spacelike Cauchy hypersurfaces;
satisfies the timelike convergence condition (this holds for the
Einstein-scalar field equations with a vanishing cosmological
constant); and has a crushing singularity (in the sense that there is a
foliation by a family of Cauchy hypersurfaces whose mean curvatures
diverge to infinity uniformly). Then $(M,g)$ has a unique CMC crushing
foliation in a neighborhood of the singularity. One natural covariant
relation between the data on the singularity and the development is
then to require that (\ref{eq:limit}) holds for the uniquely determined
CMC foliation. The notion of a cosmological time function, see
\cite{MR1606594}, can be used similarly. Nevertheless, it is of
interest to ask if the existence of one type of foliation inducing
initial data $\mathfrak{I}$ on the singularity leads to the existence
of other types of foliations inducing the same data $\mathfrak{I}$. 

Second, even though a certain combination of matter model and symmetry
class admits initial data on the singularity, it is unlikely that all
solutions are developments of initial data on the singularity. In
particular, a generalization of the notion of spikes appearing in the
$\mathbb{T}^3$-Gowdy symmetric setting, see
Section~\ref{sssection:Gowdyetc}, can be expected to be relevant in
the spatially inhomogeneous setting. Nevertheless, one can hope that
these exceptional features only occur along hypersurfaces on the
singularity; see \cite{ringstrom2022initialsymm} for an example.


\section{Solutions with symmetry}

One approach to the study of cosmological solutions to Einstein's
equations is to start by studying solutions with a high degree of
symmetry and then to gradually relax the symmetry conditions. In view
of the fact that the starting point in standard cosmology is the
cosmological principle, this is a quite natural way to proceed. The
disadvantage of studying classes of symmetric solutions is that they
are automatically non-generic. On the other hand, within a symmetry
class, it might be possible to obtain a global understanding of the
dynamics and to discover unexpected phenomena. Most, though not all,
results in the absence of symmetry are stability results. In this case,
the advantage is that one obtains results for open sets of initial
data. On the other hand, in order to obtain results, one normally has
to prove that the perturbed solutions remain close to the background.
This means that one is unlikely to discover new phenomena.

\subsection{Spatially homogeneous solutions}

For a solution to be spatially homogeneous, it has to arise from
homogeneous initial data. This means that the isometry group of the
initial data has to be transitive. If the initial manifold $\Sigma$ is
$3$-dimensional, this means that the universal covering manifold of
$\Sigma$ is either a $3$-dimensional Lie group, or it is
$\mathbb{S}^2\times\mathbb{R}$. In the latter case, which is referred
to as \textit{Kantowski--Sachs}, the isometry group of simply connected
initial data is the isometry group of the standard Riemannian metric on
$\mathbb{S}^2\times\mathbb{R}$ (note that this isometry group does not
have a $3$-dimensional transitive subgroup). If $G$ is a
$3$-dimensional Lie group and $\mathfrak{g}$ is the corresponding Lie
algebra, let, for $X\in\mathfrak{g}$,
$\mathrm{ad}_X:\mathfrak{g}\rightarrow\mathfrak{g}$ be defined by
$\mathrm{ad}_X(Y)=[X,Y]$. Define $\eta_G\in\mathfrak{g}^*$ by
$\eta_G(X):=\mathrm{tr}\ \mathrm{ad}_X$. If $\eta_G=0$, then the Lie
group is said to be \textit{unimodular} (or of \textit{Bianchi class A}
in the cosmology literature). If $\eta_G\neq 0$, the Lie group is said
to be \textit{non-unimodular} (or of \textit{Bianchi class B} in the
cosmology literature). \textit{Bianchi class A/B initial data} are left
invariant initial data on a Bianchi class A/B Lie group.
\textit{Bianchi class A/B solutions} are solutions arising from Bianchi
class A/B initial data. In what follows, we do not discuss the
Kantowski--Sachs setting but rather focus on the Bianchi solutions.
Imposing spatial homogeneity typically means that the combined
Einstein-matter equations reduce to a system of ODE's. However, in the
case of kinetic theory (the Einstein--Vlasov and Einstein--Boltzmann
equations, e.g.), the equations are still PDE's.

As far as formulations of the equations are concerned, an important
step in obtaining a unified framework for discussing Bianchi spacetimes
was taken in \cite{MR0241088}, see also \cite{MR1964375}. Many
subsequent formulations, such as \cite{MR1014971,MR1200591}, used to
prove several of the results mentioned below, are based on the ideas of
\cite{MR0241088}.

Since the literature in the spatially homogeneous setting is vast, we
can here only mention an extremely limited selection of the results.
Nevertheless, we refer to
\cite{MR0241088,MR1964375,MR1448187,MR0810832} and the references cited
therein for a description of the state of the art prior to the mid
90's; here our main focus will be on results obtained thereafter. 

\subsubsection{The direction of the singularity}

Once the occurrence of a big bang and the high degree of isotropy of
the CMB was established, a fundamental question immediately presented
itself:
\begin{quote}
\ldots\ if the 3~\textdegree K background radiation was last scattered at a
redshift $z = 7$, then the radiation coming to us from two directions
in the sky separated by more than about 30\textdegree\  was last
scattered by regions of plasma whose prior histories had no causal
relationship. [\ldots] Robertson--Walker models therefore give no insight
into why the observed microwave radiation from widely different angles
in the sky has very precisely (0.2\%) the same temperature;
\end{quote}
see \cite{Misner}. In order to address this problem, Misner suggested
considering Bianchi type IX spacetimes. These are special examples of
Bianchi class A spacetimes arising from left invariant initial data on
a Lie group with a Lie algebra given by that of $\mathrm{SU}(2)$.
Misner hoped that such spacetimes would have a different causal
structure, allowing the establishment of a thermal equilibrium prior to
the surface of last scattering. Beyond having the maximal number of
degrees of freedom in the spatially homogeneous setting, and beyond
playing a central role in the BKL proposal, Misner's expectation was
thus that the Bianchi type IX spacetimes should play a central role in
resolving the problem with high degree of isotropy of the CMB. In
addition, there are vacuum Bianchi type IX solutions (namely the
locally rotationally symmetric ones) such that the maximal Cauchy
development can be extended through Cauchy horizons. In fact, there are
two inequivalent maximal extensions, indicating that the initial data
do not determine the spacetime; see \cite{MisnerTaub,MR1236815} for the
original articles and \cite[Theorem~25.4, p.~262]{MR2527641} for a
textbook presentation. Proving that inextendibility holds for initial
data that are not locally rotationally symmetric is thus a natural goal
and would correspond to proving strong cosmic censorship in the
corresponding symmetry class.

\paragraph*{The oscillatory setting.}  
In the BKL proposal, the vacuum
Bianchi type VIII and IX solutions are models for the expected
oscillatory behavior (Bianchi type VIII solutions arise from left
invariant initial data on Lie groups with a Lie algebra equal to that
of $\mathrm{Sl}(2,\mathbb{R})$). That oscillations are to be expected
in the vacuum Bianchi type VIII and IX settings can also be seen
directly from Definition~\ref{def:ndsfidonbbs}; there are no vacuum
Bianchi type VIII or IX initial data on the singularity in the sense of
Definition~\ref{def:ndsfidonbbs}, since the Lie algebras in these cases
are such that the $p_D$ occurring in Definition~\ref{def:ndsfidonbbs}
have to satisfy $2p_A=1+p_A-p_B-p_C>0$ for all distinct $A$, $B$ and
$C$, which is not consistent with the requirements that the sum of the
$p_D$ equals one and the sum of the $p_D^2$ equals one. In spite of
extensive discussions in the physics literature, the first mathematical
result concerning the singularity in Bianchi type IX spacetimes was
obtained in \cite{MR1744051} (to the best of our knowledge). In
particular, \cite{MR1744051} contains a proof of the fact that, with
the exception of the known locally rotationally solutions that give
rise to Cauchy horizons through which the spacetimes can be extended,
all Bianchi type VIII and IX vacuum initial data give rise to
developments such that the Kretschmann scalar is unbounded in the
incomplete directions of causal geodesics. This verifies both the
curvature blow up conjecture and the $C^2$-version of the strong cosmic
censorship conjecture in this setting. Moreover, the expansion
normalized eigenvalues of the Weingarten map do not converge. In this
sense, the solutions exhibit the oscillatory behavior conjectured by
BKL. On the other hand, the result does not resolve the issue of the
causal structure. Moreover, it does not yield the conclusion that
solutions asymptote to an orbit of the BKL map. 

Next, consider Einstein's equations coupled to an orthogonal perfect
fluid with a linear equation of state $p=(\gamma-1)\rho$, where $\rho$
denotes the \textit{energy density}, $p$ the \textit{pressure} and
$\gamma$ is a constant (here assumed to satisfy $\gamma\in (2/3,2)$);
the orthogonality assumption corresponds to the condition that fluid
vector field be perpendicular to the hypersurfaces of spatial
homogeneity. In this setting, generic Bianchi type IX solutions
converge to an attractor on which the dynamics are determined by the
BKL map; see \cite{MR1846852} as well as \cite{MR2512707,MR2512706} for
an overview of the state of the art in 2009. This is an indication that
the BKL map is a good model for the dynamics. Unfortunately, the
methods of \cite{MR1846852} do not seem to be well suited to proving a
similar result for Bianchi type VIII solutions. Nevertheless, in
\cite{brehm2016bianchi}, the author proves that generic Bianchi type
VIII and IX vacuum solutions converge to an attractor. Moreover, he
proves that there is a set of initial data of full measure such that
the corresponding solutions have particle horizons. On the other hand,
he conjectures that for a Baire generic set of initial data, the
corresponding solutions do not have particle horizons. The issue of the
causality is thus quite subtle. 

Turning to the question of whether solutions asymptote to an orbit of
the Kasner map, the first results were obtained in
\cite{MR2673789,MR2802299}; see also \cite{reiterer2010bkl}. In these
results, one fixes an orbit and then proves that there are solutions
converging to the orbit. In fact, in the first two results, the authors
prove that there is a co-dimension one Lipschitz submanifold of the
state space such that initial data in the submanifold yield solutions
converging to the given orbit. Similar results were later obtained in
the magnetic Bianchi type $\mathrm{VI}_0$ setting in \cite{MR3070746}. These
results are certainly very interesting. However, they do not imply that
there is a set of initial data of positive measure such that the
corresponding solutions asymptote to an orbit of the Kasner map.
However, in \cite{MR4576760}, the authors prove that there is a subset
of the Kasner circle, say $S$, of full measure such that for each orbit
originating with a point in $S$, there is a co-dimension one Lipschitz
submanifold of the state space such that initial data in the
submanifold yield solutions converging to the given orbit. Moreover,
they prove that the union of all these Lipschitz submanifolds yields a
subset of the state space with positive measure. The argument is long
and technical, but in the end, one concludes that the set of initial
data such that the corresponding solutions asymptote to an orbit of the
Kasner map has positive measure. Clearly, this is an important step
forward. However, it is still unclear if a similar statement holds for
a subset of the state space of full measure. 

Even though much attention has been devoted to Bianchi types VIII and
IX, Bianchi type $\mathrm{VI}_{-1/9}$ is an equally general class of spatially
homogeneous spacetimes. In fact, it can be argued that it is a more
relevant model for oscillatory behavior in lower degrees of symmetry.
However, due to the difficulty of analyzing the asymptotics of
solutions to the relevant systems of ODE's, so far there are, to the
best of our knowledge, no results. See, however, \cite{MR1981447} for a
formulation of the equations and a heuristic discussion of the
asymptotics. 

Oscillatory behavior can arise for two reasons; either because there
are enough degrees of freedom (as in the case of Bianchi types
$\mathrm{VI}_{-1/9}$, VIII and IX) or because of the presence of appropriate
matter models. For examples of the latter, the reader is referred to,
e.g., \cite{MR1739422,MR2736319,MR3070746}.

\paragraph*{The convergent (or quiescent) setting.}
The behavior in the
direction of the singularity can be convergent/quiescent mainly for two
reasons. One possibility is that the matter fields are such that
solutions naturally become convergent; this happens, e.g., in the
presence of a scalar field or a stiff fluid. Another possibility is
that the geometry is such that the mechanism causing oscillations is
absent; this corresponds to condition (4) in
Definition~\ref{def:ndsfidonbbs}. In the case of orthogonal perfect
fluids and Bianchi class A types different from VIII and IX, the
situation was quite well understood already in
\cite{MR1448187,MR1468587}. On the other hand,
\cite{MR1200591,MR3916961} gives a good understanding of Bianchi class
B orthogonal perfect fluid solutions which are not of type
$\mathrm{VI}_{-1/9}$. In particular,
\cite{MR1448187,MR1744051,MR1846852,MR3916961,MR4134481} yield the
$C^{2}$-version of strong cosmic censorship for non-stiff Bianchi class
A and B spacetimes different from $\mathrm{VI}_{-1/9}$. The stiff fluid case
is addressed in
\cite{MR1846852,radermacher2017orthogonal,groeniger2023quiescence}. It
is reasonable to expect convergent/quiescent solutions to arise from
initial data on the singularity; see \cite{ringstrom2022initialsymm}
for a justification of this statement in the Bianchi class A orthogonal
stiff fluid setting. 

\paragraph*{Recollapse.} 
Under certain circumstances, spacetimes are
expected to recollapse, meaning that there is a singularity both to the
future and to the past. More specifically, if the spatial topology of
the spacetime is $\mathbb{S}^3$ or $\mathbb{S}^2\times\mathbb{S}^1$,
then the universe is in \cite{BGT} conjectured to recollapse. This
expectation goes under the name of the \textit{closed universe
recollapse conjecture}. In \cite{MR1024729}, the authors prove the
closed universe recollapse conjecture in the case of diagonal Bianchi
type IX universes satisfying the dominant energy condition and with
non-negative average principal pressures. See also \cite{MR1888088} for
a discussion of this conjecture in more general settings. 

\subsubsection{The expanding direction}
Turning to the expanding direction, it is natural to distinguish
between non-accelerated expansion and accelerated expansion.

\paragraph*{Non-accelerated expansion.}
For simple matter models with
non-negative pressure and satisfying the dominant energy condition, one
expects recollapse in the case of Kantowski--Sachs and Bianchi type IX.
What remains to be considered is thus Bianchi class A types different
from IX as well as Bianchi class B types. The Bianchi class A vacuum
setting is discussed in \cite{MR1856645,MR1984954,MR2519770}. Note that
\cite{MR1984954,MR2519770} contains a discussion of the relation
between the asymptotics and the conjectures formulated in
Section~\ref{ssection:geometrisation}. Since the relevant manifolds
in the Bianchi class A setting are Seifert fibred, one expects collapse
of the fibres after appropriate expansion normalization, and this is
what happens. 

It is also of interest to study the future asymptotics of solutions in
the presence of matter fields. There is a very extensive literature on
this subject. We here give a few examples. A fundamental question in
cosmology is if one can deduce isotropy of the universe from the
isotropy of the CMB. There is a result of this nature, due to Ehlers,
Geren and Sachs, and referred to as the EGS theorem, see
\cite{EhlersJ.1968ISot}. However, the theorem states that if the
universe has a perfectly isotropic CMB, then the universe is spatially
homogeneous and isotropic. It would be more interesting to know that if
the CMB is almost isotropic, then the universe is almost isotropic.
Such a result would be referred to as an almost-EGS theorem. However,
that such a result cannot hold follows from
\cite{MR1709125,NilssonU.S1999AAIC,MR1779503} in which the authors
study Bianchi type $\mathrm{VII}_0$ solutions; see also \cite{MR4420830}. In
the above discussions, we have focused on orthogonal perfect fluids.
However, it is also of interest to consider tilted perfect fluids; see
\cite{MR1999502,MR2056311,MR2100437,MR2115363,MR2341005,MR2378267,MR2438991,MR2673790}
for some examples of results in this setting. For results in the
presence of kinetic matter, see, e.g.,
\cite{MR1797966,MR2738261,MR3046462,MR3097256,MR3114223,MR2736319}. 

\paragraph*{Accelerated expansion.}
The fundamental paper \cite{MR0720996}
on the asymptotics of Bianchi solutions to Einstein's equations with a
positive cosmological constant establishes that, with the exception of
Bianchi type IX, solutions asymptote to de Sitter space. Concerning the
matter, it is only necessary to assume that the stress energy tensor
obey the strong and dominant energy conditions. In the case of Bianchi
type IX, a similar conclusion holds, assuming the cosmological constant
is initially large enough relative to the spatial scalar curvature.
This would seem to  establish the cosmic no-hair conjecture in the
spatially homogeneous setting. However, such a conclusion presupposes
that the solution exists globally to the future. In the case of a
specific matter model, this has to be established; see, e.g.,
\cite{MR2092074}. After the appearance of observational support for
accelerated expansion in the late universe, interest in mechanisms
yielding accelerated expansion increased. In the mathematical
literature, some corresponding examples of studies are given in
\cite{MR2056321,MR2136118,MR2204696}.

\subsection{Spacetimes with a $2$-dimensional isometry group}\label{ssection: two dim symm}

A natural next step is to consider spacetimes with a $2$-dimensional
group of isometries. In this setting, even the issue of global
existence of solutions is non-trivial, and the corresponding problem is
a natural starting point. Once global existence has been established,
it is then natural to try to analyze the asymptotics, prove strong
cosmic censorship in the corresponding symmetry class etc. In what
follows, when we speak of strong cosmic censorship, we always have the
$C^2$-version in mind. 

\subsubsection{Global existence, strong cosmic censorship}\label{sssection:Gowdyetc}

One of the first examples of global existence for arbitrary initial
data (in a symmetry class of the type of interest here), is the one
concerning Gowdy spacetimes obtained in \cite{MR0640355}. However, it
was followed by a large collection of results, proving global existence
of geometrically natural foliations for several different matter
fields; see, e.g.,
\cite{MR0679197,MR1067565,MR1143281,MR1344287,MR1435993,MR1474313,MR1603033,MR1478534,MR1722133,MR2001695,MR1981217,MR2038152}.
The symmetry classes studied are Gowdy symmetry,
$\mathbb{T}^2$-symmetry and surface symmetry. In the first two cases,
the basic symmetry assumption is that there is a compact and connected
$2$-dimensional Lie group of isometries acting effectively on the set
of initial data on a closed $3$-manifold. Due to
\cite{MR0085460,MR0095897}, this means that the group has to be
$\mathbb{T}^2$. Moreover, the initial hypersurface must be
$\mathbb{T}^3$, $\mathbb{S}^2\times\mathbb{S}^1$, $\mathbb{S}^3$ or one
of the Lens spaces (which means that a covering space of it is
$\mathbb{S}^3$, so that we ignore Lens spaces from now on); see
\cite{MR1067565}. The $2$-torus action on the initial hypersurface
gives rise to two Killing vector fields of the corresponding
development, say $X_i$, $i=1,2$. Then
$\mathcal{T}_i:=\epsilon_{\alpha\beta\gamma\delta}X^{\alpha}_1X^{\beta}_2\nabla^{\gamma}X_i^\delta$ are the associated
\textit{twist constants}, where $\epsilon_{\alpha\beta\gamma\delta}$
are the components of the volume form and $\nabla$ is the Levi-Civita
connection of the development. In the vacuum setting, the twist
constants are constants, but for general matter models they are
functions. The Gowdy setting corresponds to demanding that the twist
constants vanish (this happens automatically in the case of
$\mathbb{S}^2\times\mathbb{S}^1$ and $\mathbb{S}^3$ spatial topology);
see \cite{MR0339764} for the original article by Gowdy. The possible
topologies (ignoring the Lens spaces) are thus $\mathbb{T}^3$,
$\mathbb{S}^2\times\mathbb{S}^1$ and $\mathbb{S}^3$ in the Gowdy
setting. The general case that the twist constants do not vanish is
referred to as the $\mathbb{T}^2$-symmetric setting. In this case, the
relevant spatial topology is $\mathbb{T}^3$. One can take a slightly
more general perspective, insisting on only local Killing vector
fields; see \cite{MR1478534,MR1814465}. In the surface symmetric
setting, the topology of the spacetime is
$\mathbb{R}\times\mathbb{S}^1\times\mathbb{F}$, where $\mathbb{F}$ is a
closed $2$-manifold. Letting $\mathbb{G}$ denote the universal cover of
$\mathbb{F}$, the isometry group of one of the standard metrics on
$2$-dimensional Euclidean, spherical or hyperbolic space should act on
the last factor of $\mathbb{R}\times\mathbb{S}^1\times\mathbb{G}$ by
isometries of the lifted spacetime metric. The surface symmetric
setting is only interesting outside of vacuum. Even though it is of
course interesting to know that there is a global constant mean
curvature foliation, for example, the corresponding results should be
understood as a first step to analyzing the asymptotics in general and
proving strong cosmic censorship in particular.

\paragraph*{Strong cosmic censorship in the Gowdy setting.}
In
\cite{MR1075858} (see also \cite{MR1048674}), the authors prove strong
cosmic censorship for polarized Gowdy spacetimes. The Gowdy spacetimes
have a two dimensional group of isometries, but the additional
requirement of polarization means that the essential equations that
need to be solved are linear. This significantly simplifies the
analysis. Nevertheless, there are infinite dimensional sets of initial
data leading to maximal Cauchy developments that are extendible.
Proving that there is an open and dense subset of initial data whose
maximal Cauchy developments are inextendible is therefore non-trivial.

A proof of strong cosmic censorship in the general $\mathbb{T}^3$-Gowdy
symmetric vacuum setting can be found in
\cite{MR2032917,MR2042268,MR2053013,MR2222842,MR2600872} (these results
can also be used to deduce strong cosmic censorship in the polarized
$\mathbb{T}^3$-symmetric Einstein--Maxwell setting, see
\cite{MR2532056}). In this case, the essential equations can be thought
of as wave map equations with hyperbolic space as a target. In
particular, the equations are non-linear, and this causes substantial
complications and rich dynamics. In particular, spikes appear. These
features were first discovered numerically, see below, and later
understood analytically, see \cite{MR1847897} and \cite{MR2388199}.
Moreover, they are expected to be important more generally.
Nevertheless, it is possible to prove that there is an open and dense
set of initial data such that the corresponding maximal Cauchy
developments are inextendible. In fact, quite detailed information can
be derived concerning the asymptotics; see also
\cite{ringstrom2022initialsymm} for an interpretation of the
asymptotics in the direction of the singularity in terms of data on the
singularity. Note, however, that strong cosmic censorship has neither
been demonstrated in the $\mathbb{S}^2\times\mathbb{S}^1$ nor in the
$\mathbb{S}^3$ vacuum Gowdy-settings.

\paragraph*{Strong cosmic censorship in the $\mathbb{T}^2$- and surface
symmetric settings.}
In the vacuum $\mathbb{T}^2$-symmetric setting
there are no results concerning strong cosmic censorship. This is due
to the expectation that the behavior in the direction of the
singularity should be oscillatory and therefore much more complicated;
see the discussion of numerical work below for details. However, in the
expanding direction there are some partial results as well as numerical
simulations, see below. On the other hand, including matter sometimes
improves the situation. In order to understand why, it is useful to
recall that in the case of Bianchi type IX vacuum spacetimes, e.g., the
locally rotationally symmetric solutions are such that the maximal
Cauchy development can be extended in inequivalent ways. They are
therefore counterexamples to determinism. Due to the existence of these
examples, proving strong cosmic censorship in the vacuum Bianchi type
IX setting is subtle. Considering, on the other hand, Bianchi type IX
solutions with non-trivial orthogonal perfect fluid matter, it can
easily be demonstrated that the Ricci tensor contracted with itself
blows up in the direction of the singularity for all solutions. Proving
strong cosmic censorship in this setting is therefore trivial. In this
sense, including matter can simplify the situation. Similarly, the
reason proving strong cosmic censorship in the $\mathbb{T}^3$-Gowdy
symmetric vacuum setting is non-trivial is that there are
counterexamples to extendibility in that setting as well. Results
concerning strong cosmic censorship in the surface symmetric setting
with collisionless matter can be found in \cite{MR3481283}, see also
\cite{MR1362953,MR2281415}. See also \cite{MR2194327} for a future
inextendibility result which covers, e.g., the Einstein--Vlasov
equations in the $\mathbb{T}^2$-symmetric setting, including vacuum. A
proof of strong cosmic censorship in the $\mathbb{T}^2$-symmetric
Einstein--Vlasov setting with non-trivial Vlasov matter can be found in
\cite{dafermos2006strong}.   Finally, \cite{MR2465730} extends this
result to the case of a positive cosmological constant. Note, however,
that \cite{MR2465730} actually also covers the vacuum setting, assuming
the area element of the group action does not vanish in the limit (in
the direction of the singularity). 

\paragraph*{One bounce in the $\mathbb{T}^3$-Gowdy symmetric vacuum
setting.}
To the best of our knowledge, there are no results in the
oscillatory and spatially inhomogeneous setting. However, Warren Li
recently proved a result describing one BKL bounce in the
$\mathbb{T}^3$-Gowdy symmetric vacuum setting; see
\cite{li2024bklbouncesoutsidehomogeneity}. 

\subsubsection{The expanding direction}

Beyond proving strong cosmic censorship, it is of interest to derive
detailed asymptotics in the expanding direction; in particular, it is
of interest to verify the conjectures formulated in
Section~\ref{ssection:geometrisation}. In the case of polarized
$\mathbb{T}^3$-Gowdy symmetric vacuum spacetimes, such an analysis can
be found in \cite{MR1955788,MR2136117}. In fact, the asymptotics
derived in \cite{MR1955788} are detailed enough that they uniquely
determine the solution, see \cite{MR2136117}. The reason the analysis
is so complete in this setting is that the essential equation is linear
in this case. Note also that the conclusions follow as a very special
case of the theory developed in \cite{MR4163813}. In the case of
general vacuum $\mathbb{T}^3$-Gowdy spacetimes, the future asymptotics
are analyzed in \cite{MR2032917,MR2217283} (see also \cite{MR2532056}
for corresponding conclusion in the polarized $\mathbb{T}^3$-symmetric
Einstein--Maxwell setting). In the $\mathbb{T}^2$-symmetric vacuum
setting, there are general results that do, however, not provide very
detailed information, see \cite{MR3312437}. In the polarized
$\mathbb{T}^2$-symmetric vacuum setting, there are results providing
more detailed information, assuming one starts out close enough to an
asymptotic regime, see \cite{MR3513138}. There are also numerical
studies of this situation, see below.

Including matter, there are results in the case of the Einstein--Vlasov
equations with hyperbolic symmetry, see \cite{MR1981217,MR2075051}. In
the case of the Einstein--Vlasov equations with surface symmetry and a
positive cosmological constant, there are results in
\cite{MR1991190,MR2138579}. In \cite{MR3506608}, the authors derive
detailed information concerning the future asymptotics in the
$\mathbb{T}^3$-Gowdy symmetric Einstein--Vlasov setting. In particular,
the authors prove the cosmic no-hair conjecture in this setting.

\subsubsection{Specifying data on the singularity}

There is an extensive literature on specifying data on the singularity
for classes of spacetimes with a $2$-dimensional group of isometries,
see, e.g.,
\cite{MR1623091,MR1657863,MR1779512,MR1710663,MR1926244,MR3085923,MR3705380}.
It would be interesting to relate these results to
Definition~\ref{def:ndsfidonbbs}. This has been done in
\cite{ringstrom2022initialsymm} in the case of $\mathbb{T}^3$-symmetric
Gowdy spacetimes. However, it remains to be done for the other cases.
Related results concerning isotropic singularities can be found in
\cite{MR1603037,MR1710659,MR1764130,MR1766545}. 

\subsection{The $\mathrm{U}(1)$-symmetric setting}\label{ssection: one dim symm}

Finally, it is of interest to discuss the $\mathrm{U}(1)$-symmetric
setting; see \cite{MR3248565} for a general review. 


\subsubsection{Existence results and future asymptotics}

For general ideas concerning the reduction of the equations in the
$\mathrm{U}(1)$-symmetric setting, the reader is referred to
\cite{MR1038613}. The question of local existence, using variables that
can be expected to be useful to treat global issues, is discussed in
\cite{MR1288395,MR1392984}.  Global existence is then discussed in
\cite{MR1813771}. In \cite{MR1877233} (see also \cite{MR1957625}), the
authors prove a small data future global existence result which
includes a proof of future causal geodesic completeness; the existence
of a CMC foliation covering the causal future of the initial
hypersurface; and the conclusion that the geometry asymptotes to that
of a model spacetime. This result treats the so-called polarized case.
An extension to the non-polarized case is given in
\cite{MR1998845,Yvonne-non-polarised}.

\subsubsection{Data on the singularity}

In the direction of the singularity, it is in the generic vacuum
setting natural to expect oscillations, see the below discussion of
numerical work. However, in the polarized setting, it is natural to
expect quiescent asymptotics. In \cite{MR2136898}, the authors
construct real analytic solutions satisfying these expectations using
Fuchsian techniques. See also \cite{MR1939922}. 

\section{Future global non-linear stability in the expanding direction}

In the cosmological setting, spacelike Cauchy hypersurfaces with mean
curvature bounded away from zero appear naturally (in the absence of an
asymptotically flat regime, hypersurfaces with vanishing mean curvature
are, for many matter models, expected to be surrounded by hypersurfaces
with mean curvature bounded away from zero, and hypersurfaces with
significant positive mean curvature on some parts and significant
negative mean curvature on other parts are unnatural in cosmology for
the reasons given in the introduction). For a hypersurface satisfying
this condition, the volume form is increasing (or decreasing)
everywhere on the hypersurface. Moreover, due to Hawking's singularity
theorem, the corresponding spacetime is either past or future timelike
geodesically incomplete (assuming appropriate matter content). For
these reasons, it is, when studying stability, natural to consider
future expanding and future causally geodesically complete spacetimes
and past contracting and past causally geodesically incomplete
spacetimes in cosmology. When we speak of stability in the expanding
direction, we typically assume that the background solution is future
causally geodesically complete and has a foliation of the causal future
of a Cauchy hypersurface such that each leaf of the foliation has
strictly positive mean curvature. It is natural to divide the results
that have been obtained according to the rate of the expansion, in
particular according to whether the expansion is accelerated or not.


\subsection{Non-accelerated expansion}

In the case of non-accelerated expansion, it is, in accordance with the
ideas described in Section~\ref{ssection:geometrisation}, to be
expected that the most favorable situation is when the spatial geometry
is close to hyperbolic. Accordingly, the first future global non-linear
stability result is the stability of the Milne model in the vacuum
setting, see \cite{MR2098919}; the metric of the Milne model is
$-dt^2+t^2\gamma$ on $(0,\infty)\times\Sigma$, where $(\Sigma,\gamma)$
is a closed hyperbolic $3$-manifold. The argument is based on
considering energies arising from the Bel-Robinson tensor. This result
was later generalized to higher dimensions in \cite{MR2863911} using
rougher energy estimates, and to different matter models, see, e.g.,
\cite{MR4124987,MR4193643,MR4062701}. It is also worth noting that
while asymptotic stability is obtained in \cite{MR2098919}, only
stability is obtained in \cite{MR2863911}, since there in higher
dimensions is a non-trivial moduli space of negative Einstein
manifolds; this adds an additional complication to the problem. 

\subsection{Accelerated expansion}\label{ssection:acc exp}

The first result on future global non-linear stability in the
cosmological setting is due to Helmut Friedrich, see \cite{MR0868737}.
In \cite{MR0868737}, the author proves, among other things, the global
non-linear stability of de Sitter space. The method of proof is to
derive a set of conformal field equations with respect to which the
global stability problem becomes a local problem. However, this method
is based on conformal invariance of the equations. The stability result
\cite{MR0868737} is extended to the Einstein--Maxwell--Yang--Mills setting
in \cite{MR1131434}. Similar methods have been used to obtain, e.g.,
the results \cite{MR3385978,MR3607462,MR3008768}. 

The methods developed in \cite{MR0868737} apply to specific matter
models and specific spacetime dimensions. However, future global
non-linear stability is expected to hold more generally. In
\cite{MR2403395}, future global non-linear stability is demonstrated in
the Einstein-non-linear scalar field setting. The particular situation
considered in \cite{MR2403395} is that of a potential with a strictly
positive non-degenerate lower bound. However, the methods apply in any
spacetime dimension $\geq 4$. Moreover, they are based on coordinates
similar to those used by Choquet-Bruhat to prove local existence in
\cite{MR0053338}. However, modifications are necessary; the contracted
Christoffel symbols are replaced by appropriately chosen so-called
\textit{gauge source functions} and additional correction terms are
added to the equations. Using the resulting equations, stability can be
demonstrated using a bootstrap argument. Moreover, due to the causal
structure of solutions, it is possible to prove stability results that
are future global in time given only local assumptions in space.
Consequently, it is possible to prove future global non-linear
stability results for all spatially locally homogeneous solutions that
exhibit de Sitter like expansion at the same time. The results in
\cite{MR2403395} were later generalized in \cite{MR2520511} to the case
of an exponential potential using similar methods. After
\cite{MR2403395} appeared, related results using similar methods (with
additional complications due to the matter fields) appeared in, e.g.,
\cite{MR2802385,MR2960029,MR3063248,MR3120746,MR3186493,MR3335528,MR3506608}.
In most of these references, the authors prove future global non-linear
stability of specific spatially locally homogeneous solutions. However,
in \cite{MR3506608} the authors prove future global non-linear
stability of arbitrary $\mathbb{T}^3$-Gowdy symmetric solutions to the
Einstein--Vlasov system with a positive cosmological constant. In other
words, the background solutions in this case do not only exhibit
substantial anisotropies initially, they also exhibit substantial
spatial inhomogeneities. Next, \cite{MR3186493} contains, beyond
stability results, a proof of the fact that for any choice of closed
manifold for the spatial topology and any given choice of currently
preferred standard model, there is a solution to the Einstein--Vlasov
equations with a positive cosmological constant and the prescribed
spatial topology such that observers in the corresponding spacetime
cannot distinguish between their universe and the prescribed standard
model; see \cite{MR3186493} for details. In other words, given the
currently preferred standard models of the universe, it is not possible
to determine the global shape of the universe on the basis of
observations. 

Later, different methods were developed to prove future global
non-linear stability in the presence of a positive cosmological
constant, see \cite{MR3528422}. In \cite{MR3528422}, the arguments rely
on a conformal rescaling of the metric, but are still based on a
particular choice of wave gauge, in analogy with \cite{MR2403395}. The
advantage of the methods developed in \cite{MR3528422} is that the
proof becomes substantially shorter. Moreover, the methods apply to
perfect fluids with a linear equation of state of the form $p=K\rho$,
where $0<K\leq 1/3$. In other words, as opposed to earlier results
concerning perfect fluids, the methods immediately include the
radiation case. The methods developed in \cite{MR3528422} have then
been applied in, e.g., \cite{MR3875826,MR3816202,MR4226449}. 

\section{Big bang formation in the absence of symmetries}\label{section:Big bang formation}

Prior to the work of Rodnianski and Speck, results concerning big bang
singularities in the absence of symmetries were restricted to the ones
obtained by specifying initial data on the singularity, see
Section~\ref{ssection:id on sing}. The first stability result was
obtained in \cite{MR3874696}, based on a study of the linearized
equations in \cite{MR3739229}. In \cite{MR3874696}, the authors prove
that the occurrence of big bang singularities in spatially flat and
isotropic solutions to the Einstein-scalar field equations (and to the
Einstein-stiff fluid equations) is a past globally non-linearly stable
phenomenon. The gauge used in the proof is determined by setting the
shift vector field to zero and demanding that the time coordinate $t$
equal $1/\theta$, where $\theta$ is the mean curvature of the
constant-$t$ hypersurface (we here use the opposite sign convention to
that of \cite{MR3874696} concerning the mean curvature). However, the
geometric objects are expressed with respect to coordinate vector
fields. The result \cite{MR3874696} was later extended to prove past
(future) global non-linear stability of the big bang (big crunch) in
the case of spatially homogeneous and isotropic solutions with
spherical spatial geometry and scalar field matter, see
\cite{MR3875820}. In \cite{fajman2023cosmic}, stable big bang formation
is obtained in the case of hyperbolic spatial geometry. Moreover, the
authors in this case also prove future global non-linear stability.  In
\cite{MR4375450}, the authors extend the results of \cite{MR3874696} to
cover the case of moderate anisotropies. In particular,
\cite{MR4375450} yields stable big bang formation for solutions to
Einstein's vacuum equations with $\mathbb{T}^{D}$ spatial topology,
assuming $D\geq 38$.  In \cite{MR3874696,MR3875820,MR4375450}, the
arguments are based on using a CMC foliation. This is very useful since
it synchronizes the singularity. On the other hand, the CMC condition
leads to an elliptic equation for the lapse function. In this sense,
the gauge is non-local. On the other hand, the causal structure is such
that particle horizons appear. In other words, two causal curves going
into the singularity typically asymptotically lose the ability to
communicate. This means that, from a causal point of view, the behavior
localizes. It would therefore be of interest to prove a past global
non-linear stability result which localizes in the spatial directions.
This is done in \cite{MR4675129}. In this paper, the authors use a
different gauge which makes it possible to localize the analysis. The
foliation the authors use in this case is determined by the condition
that the scalar field be constant along the leaves of the foliation. In
particular, the result requires the presence of a scalar field and
therefore does  not apply in the vacuum setting. In
\cite{beyer2023past}, the authors generalize this result to the
presence of a perfect fluid with a linear equation of state
$p=c_s^2\rho$, where $1/n<c_s^2<1$ and $n$ is the spatial dimension
(note that for $n=3$, the case $0\leq c_s^2\leq 1/n$ is expected to be
unstable in the direction of the singularity due to
\cite{beyer2024past}). From these results, it is clear that the
borderline case of a radiation fluid is of particular importance. It is
therefore of interest to note that \cite{fajman2024past} yields stable
big bang formation of the FLRW solutions to the Einstein--Vlasov-scalar
field equations (the result simultaneously covers the spatially flat,
hyperbolic and spherical settings); note that Vlasov matter behaves as
a radiation fluid in the direction of the singularity. 

In spite of the interest of the results mentioned above, they do not
cover the regime one would naturally expect from, e.g.,
Definition~\ref{def:ndsfidonbbs}; i.e., that a sufficient condition
should be that the eigenvalues, say $\ell_A$, of the expansion
normalized Weingarten map satisfy $1+\ell_A-\ell_B-\ell_C>0$ for $B\neq
C$. However, this is remedied in \cite{MR4583776}. In this article, the
authors prove stable big bang formation for the full range of spatially
homogeneous and spatially flat solutions to the Einstein-scalar field
equations for which one expects stability (in the polarized
$U(1)$-symmetric setting, the conditions can even be relaxed, in
accordance with condition (4) in Definition~\ref{def:ndsfidonbbs}).
Again, the authors use a CMC gauge with vanishing shift vector field.
However, in \cite{MR4583776} the authors use a Fermi--Walker propagated
frame as opposed to considering the components of the geometry with
respect to coordinate vector fields. Moreover, the equations are for
the components of the second fundamental form with respect to the
Fermi--Walker propagated frame, the connection coefficients of the
frame, the components of the frame (and co-frame) with respect to
coordinate vector fields, the scalar field and the lapse function. The
unknowns and the equations are therefore quite different from the ones
in \cite{MR3874696}. The advantage of the formulation in
\cite{MR4583776} is that the variables are such that one can formulate
(and improve) quite mild bootstrap assumptions. Since the bootstrap
assumptions are not very strong, it is easier to improve them. On the
other hand, the conclusion that the bootstrap assumptions are satisfied
does not yield that much information. Nevertheless, the information is
sufficient to conclude that the eigenvalues of the expansion normalized
Weingarten map converge, that the expansion normalized quantities
associated with the scalar field converge and that the curvature blows
up.

All the results mentioned above are stability results. In other words,
the starting point is a given background solution (or family of
background solutions). The conclusions of the results are then that
perturbing initial data corresponding to the relevant background
solution yields solutions that also have big bang singularities. In
analogy with Hakwing's theorem, it would be desirable to prove that
there is a general condition, which does not refer to a specific
background solution, ensuring big bang formation with curvature blow
up. Such a result is obtained in \cite{groeniger2023formation}. The
result is in the Einstein-non-linear scalar field setting and the
assumptions involve: bounds on the potential; the expected bounds on
the eigenvalues of the expansion normalized Weingarten map; and, for a
fixed bound on suitable Sobolev norms on the expansion normalized
quantities (introduced in connection with (\ref{eq:limit})), a
corresponding lower bound on the mean curvature (the conditions are
such that, in particular, they are satisfied for all solutions with a
CMC foliation that induce data on the singularity). The conclusions
include past causal geodesic incompleteness and curvature blow up; see
\cite{groeniger2023formation} for details. Since the results of
\cite{groeniger2023formation}  hold in the case of the
Einstein-non-linear scalar field equations, they can be combined with
the future global non-linear stability results in this setting,
described in Section~\ref{ssection:acc exp}, in order to conclude
past and future global non-linear stability of a large class of
spatially locally homogeneous solutions. In particular,
\cite{groeniger2023formation} applies in all the Einstein-scalar field
settings in which stable big bang results were previously obtained.
Many of the methods used in \cite{groeniger2023formation}  are inspired
by those developed in \cite{MR4583776}. However, there are also
significant differences. In particular, in \cite{MR4583776} there is a
background solution to prove stability of, and in the main theorem of
\cite{groeniger2023formation} there is not. One important step in
\cite{groeniger2023formation} is therefore to construct a substitute
for a background solution. 

\section{Linear equations on cosmological backgrounds}

There is a growing body of work concerning linear systems of wave
equations on cosmological backgrounds. There is of course a vast
physics literature on this subject, but we here focus on mathematical
results. Some very early work is contained \cite{MR0650132}.  Since the
central equation in the case of polarized Gowdy is linear,
\cite{MR1075858,MR1955788,MR2136117} are, simultaneously, studies of
linear equations on cosmological backgrounds are directly relevant for
the behavior of solutions to Einstein's equations. In \cite{MR2659736},
the authors study scalar perturbations of the Einstein--Euler equations.
Further studies, mainly of the Klein--Gordon equation on spatially
homogeneous backgrounds, are contained in, e.g.,
\cite{MR2666964,MR2563211,MR3575683,MR3995117,MR3612079,MR3942084,MR4032875,MR3975860,MR4134481,MR4170295,MR4412386}.
Most of the results concern the equations on background geometries with
convergent asymptotics. However, it is worth noting that
\cite{MR4032875} contains results in the case that the background
geometry has chaotic dynamics. In \cite{MR4163813}, a general theory
for linear systems of wave equations on cosmological backgrounds is
developed. However, the geometries have to be such that the resulting
equations are separable. The corresponding results cover many, though
not all, of the previously mentioned results. For example, geometries
which are similar to those of Kasner spacetimes are included (and
solutions to systems of wave equations on such backgrounds are
analyzed, both as far as past and future asymptotics are concerned).
The main goal in \cite{MR4163813} is to derive optimal energy
estimates. This is done both with and without derivative losses.
Moreover, in some situations it is demonstrated that there is a
homeomorphism between the regular initial data and asymptotic data. In
\cite{ringstrom2021wave}, the case of spatially inhomogeneous
backgrounds is considered. In this case, a priori assumptions
concerning the geometry in the direction of a crushing singularity are
made. Systems of linear wave equations on the corresponding backgrounds
are then studied. In particular, optimal energy estimates and
expansions of the solutions in the direction of the singularity are
derived. However, the assumptions are general enough that the limits of
the coefficients of the equations need not be continuous. This means
that one has to localize along the causal future of a causal curve
going into the singularity in order to derive detailed asymptotics.

In \cite{MR4070941}, conformal scattering of the Maxwell-scalar field
system on de Sitter space is studied.  Some aspects of \cite{MR4163813}
are improved in \cite{grisales2023asymptotics} and then applied to
Maxwell's equations on a Kasner background (in the direction of the
singularity). As already mentioned, the linearized Einstein-scalar
field equations are studied in \cite{MR3739229}. A scattering theory
for the linearized Einstein-scalar field equations on a generalized
Kasner background is developed in \cite{li2024scattering}; see also
\cite{cicortas2024scattering} for a result on scattering for the wave
equation on de Sitter backgrounds in even spacetime dimensions.  

\section{Numerical work}

There is a substantial literature of numerical results concerning
cosmological solutions to Einstein's equations. It is not realistic to
do it justice here. However, we wish to give some references that can
be used as a starting point for those who wish to acquaint themselves
with the literature.

\subsection{Results in symmetric settings}

Some of the early numerical work was carried out in \cite{MR1042881},
concerning Bianchi type IX solutions, and \cite{MR1247900}, concerning
solutions with a $1$- or $2$-dimensional symmetry group. In fact, the
numerical studies of solutions with symmetries ran in parallel with the
mathematical studies described in Section~\ref{ssection: two dim
symm} and \ref{ssection: one dim symm}. Moreover, the mathematical
studies benefited from the numerical studies and vice versa. In
\cite{MR1437421}, the algorithms in the Bianchi IX setting are improved
and in \cite{MR1633408} the authors probe the BKL proposal with
numerical techniques. More detailed studies of quiescent and
oscillatory behavior for cosmological singularities can be found in,
e.g.,
\cite{MR1669443,MR1682356,MR1790740,MR1813870,MR1858721,MR2053000}. 
Some studies concerning the future asymptotics in the
$\mathbb{T}^2$-symmetric setting are to be found in
\cite{MR4681793,MR4062458}.

The idea that spikes, appearing, e.g., in $\mathbb{T}^3$-symmetric
Gowdy spacetimes, occur naturally in the direction of the singularity
arose due to numerical studies (the importance of spikes was later
confirmed in mathematical results; cf., e.g.,
\cite{MR1847897,MR2600872}); see
\cite{MR1682356,MR2010197,MR2388199,MR3129295,MR3435414,MR3562679,MR3730895,MR4386165}
for some references concerning spikes. 

\subsection{Results in the absence of symmetries}

By now there is also a substantial literature in the absence of
symmetries. Some of the early examples can be found in
\cite{MR2007938,MR2116846,MR2124097,MR2333165}.   See
\cite{MR4197034,MR4604360} for two more recent examples. 


\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}
The author would like to acknowledge the support of Vetenskapsr\aa det,
the Swedish Research Council, dnr. 2022-03053. 

\section*{Acknowledgments}
This article was in part written while the author was enjoying the
hospitality of the Erwin Schr\"{o}dinger Institute in Vienna. 

\CDRGrant[VR]{2022-03053}

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