There are three categories of mathematical results concerning quiescent big bang singularities: the derivation of asymptotics in a symmetry class; the construction of spacetimes given initial data on the singularity; and the proof of big bang formation in the absence of symmetries, including the proof of stable big bang formation. In a recent article, the first author demonstrated the existence of developments corresponding to a geometric notion of initial data on a big bang singularity. Moreover, this article, combined with previous articles by the second author, gives a unified and geometric perspective on large classes of seemingly disparate results in the first two categories. Concerning the third category, Oude Groeniger et al recently formulated a general condition on initial data ensuring big bang formation, including curvature blow up. This result, among other things, generalises previous results on stable big bang formation. However, it does not include a statement saying that the solutions induce initial data on the singularity. Here we tie all three categories of results together by demonstrating that the solutions of Oude Groeniger et al induce data on the singularity. However, the results are more general and can potentially be used to derive similar conclusions in other gauges.

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.

The subject of these notes is the asymptotic behaviour of solutions to linear systems of wave equations in the vicinity of big bang singularities. In particular, we are interested in the case of crushing singularities (cf. Definition 2.1) with silent and anisotropic asymptotics. Beyond studying wave equations, we here develop a geometric framework for understanding such singularities, and in a companion article (H. Ringstro & uml;m, On the geometry of silent and anisotropic big bang singularities, Preprint, 2025), we combine this framework with Einstein's equations in order to deduce additional information.

In a recent article, we propose a general geometric notion of initial data on big bang singularities. This notion is of interest in its own right. However, it also serves the purpose of giving a unified perspective on many of the results in the literature. In the present article, we give a partial justification of this statement by rephrasing the results concerning Bianchi class A orthogonal stiff fluid solutions and solutions in the T-3-Gowdy symmetric vacuum setting in terms of our general geometric notion of initial data on the big bang singularity.

Hawking's singularity theorem says that cosmological solutions arising from initial data with positive mean curvature have a past singularity. However, the nature of the singularity remains unclear. We therefore ask: If the initial hypersurface has sufficiently large mean curvature, does the curvature necessarily blow up towards the singularity? In case the eigenvalues of the expansion-normalized Weingarten map are everywhere distinct and satisfy a certain algebraic condition (which in 3+1 dimensions is equivalent to them being positive), we prove that this is the case in the CMC Einstein-non-linear scalar field setting. More specifically, we associate a set of geometric expansion-normalized quantities to any initial data set with positive mean curvature. These quantities are expected to converge, in the quiescent setting, in the direction of crushing big bang singularities. Our main result says that if the mean curvature is large enough, relative to an appropriate Sobolev norm of these geometric quantities, and if the algebraic condition is satisfied, then a quiescent (as opposed to oscillatory) big bang singularity with curvature blow-up forms. This provides a stable regime of big bang formation without requiring proximity to any particular class of background solutions. An important recent result by Fournodavlos, Rodnianski and Speck demonstrates stable big bang formation for all the spatially flat and spatially homogeneous solutions to the Einstein-scalar field equations satisfying the algebraic condition. Here we obtain analogous stability results for any solution inducing data at the singularity, in the sense introduced by the third author, in particular generalizing the aforementioned result. Moreover, we are able to prove both future and past global non-linear stability of a large class of spatially locally homogeneous solutions.

This article is the first of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. The results of the present article concern the asymptotic behaviour of solutions to linear systems of wave equations on the corresponding backgrounds. The main features are the following: The assumptions do not involve any symmetry requirements and are weak enough to be consistent with most big bang singularities for which the asymptotic geometry is understood. The asymptotic rate of growth/decay of solutions to linear systems of wave equations along causal curves going into the singularity is determined by model systems of ODE's (depending on the causal curve). Moreover, the model systems are essentially obtained by dropping spatial derivatives and localising the coefficients along the causal curve. This is in accordance with the BKL proposal. Note, however, that we here prove this statement, we do not assume it. If the coefficients of the unknown and its expansion normalised normal derivatives converge sufficiently quickly along a causal curve, we obtain leading order asymptotics (along the causal curve) and prove that the localised energy estimate (along the causal curve) is optimal. In this setting, it is also possible to specify the leading order asymptotics of solutions along the causal curve. On the other hand, the localised energy estimate typically entails a substantial loss of derivatives. In the companion article, we deduce geometric conclusions by combining the framework with Einstein's equations. In particular, the combination reproduces the Kasner map and yields partial bootstrap arguments.

The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, we also derive asymptotics and demonstrate that the leading order asymptotics can be specified. It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then the spatial derivatives can be ignored. This line of reasoning is incorrect: we give examples of equations such that 1) the factors multiplying the spatial derivatives decay exponentially, 2) the factors multiplying the time derivatives are constants, 3) the energies of individual modes of solutions asymptotically decay exponentially, and 4) the energies of generic solutions grow as $e^{e^{t}}$ as $t\rightarrow \infty$. When the factors multiplying the spatial derivatives grow exponentially, the Fourier modes of solutions oscillate with a frequency that grows exponentially. To obtain asymptotics, we fix a mode and consider the net evolution over one period. Moreover, we replace the evolution (over one period) with a matrix multiplication. We cannot calculate the matrices, but we approximate them. To obtain the asymptotics we need to calculate a matrix product where there is no bound on the number of factors, and where each factor can only be approximated. Nevertheless, we obtain detailed asymptotics. In fact, it is possible to isolate an overall behaviour (growth/decay) from the (increasingly violent) oscillatory behaviour. Moreover, we are also in a position to specify the leading order asymptotics.

In this paper, we study solutions to the Klein–Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein–Gordon equation, there are smooth functions ui, i = 0,1, on the Lie group under consideration, such that $${u_{\sigma}(\cdot,\sigma)-u_{1}}$$ and $${u(\cdot,\sigma)-u_{1}\sigma-u_{0}}$$ asymptotically converge to zero in the direction of the singularity (where $${\sigma}$$ is a geometrically defined time coordinate such that the singularity corresponds to $${\sigma\rightarrow-\infty}$$). Here ui, i = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein–Gordon equation on a flat Kasner background. In that setting, $${u_{\sigma}}$$ does, generically, not converge.

The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions: the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T-3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T-2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T-3-Gowdy symmetry to this list of requirements, we obtain C-0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter.

In the late 90's, the standard perspective on how to model the universe changed dramatically; observational data concerning supernovae, obtained in 98-99, indicate that our universe is expanding at an accelerated rate. As a consequence, it is of interest to prove that cosmological solutions to Einstein's equations with accelerated expansion are future stable. That is the topic of the present contribution. The current standard models of the universe include different types of matter, but it turns out that many of the essential difficulties appear already in the vacuum setting. As a consequence, we here focus on giving a rough outline of how to prove future stability in the case of Einstein's vacuum equations with a positive cosmological constant. However, we also wish to give an overview of the stability results that have been obtained more generally, and to give an idea of how to arrive at the conclusion that the universe is expanding at an accelerated rate on the basis of observations.

In the subject of cosmology, spatially homogeneous solutions are often used to model the universe. It is therefore of interest to ask what happens when perturbing into the spatially inhomogeneous regime. To this end, we, in the present paper, study the future asymptotics of solutions to Einstein’s vacuum equations in the case of \({\mathbb{T}^{2}}\)-symmetry. It turns out that in this setting, whether the solution is spatially homogeneous or not can be characterized in terms of the asymptotics of one variable appearing in the equations; there is a monotonic function such that if its limit is finite, then the solution is spatially homogeneous and if the limit is infinite, then the solution is spatially inhomogeneous. In particular, regardless of how small the initial perturbation away from spatial homogeneity is, the resulting asymptotics are very different. Using spatially homogeneous solutions as models is therefore, in this class, hard to justify.

The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely related to the issue of singularities in general relativity. Finally, we discuss the topic of stability of solutions to Einstein's equations. This is not only an important and active area of research, it is also one that only became meaningful thanks to the work of Yvonne Choquet-Bruhat.

The solutions of Einstein's equations used by physicists to model the universe have a high degree of symmetry. In order to verify that they are reasonable models, it is therefore necessary to demonstrate that they are future stable under small perturbations of the corresponding initial data. The purpose of this contribution is to describe mathematical results that have been obtained on this topic. A question which turns out to be related concerns the topology of the universe: what limitations do the observations impose? Using methods similar to ones arising in the proof of future stability, it is possible to construct solutions with arbitrary closed spatial topology. The existence of these solutions indicate that the observations might not impose any limitations at all.

The solutions of Einstein's equations used by physicists to model the universe have a high degree of symmetry. In order to verify that they are reasonable models, it is therefore necessary to demon ...

After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

Abstract When proving local existence of solutions to nonlinear wave equations, estimates of Moser type play an important role. With respect to local coordinates, the Einstein–Vlasov system is a system of integro-partial differential equations. In order to prove local existence of solutions to this system, it is of interest to derive estimates for such equations which are analogous to the Moser estimates. This is the subject of Chapter 16.

Abstract The standard models of the universe are spatially homogeneous, isotropic and have matter content consisting of perfect fluids. However, it is possible to approximate dust and radiation fluids (that are normally used) by matter of Vlasov type. We demonstrate that this is the case in Chapter 28.

Abstract One of the main results of the book is the future stability of a large class of spatially locally homogeneous solutions. However, in order to obtain such a result, we first need to analyse the spatially homogeneous solutions. In Chapter 25, we begin this analysis by formulating a symmetry reduced version of the equations in the spatially homogeneous setting. We also provide a first continuation criterion.

Abstract In Chapter 21, we discuss the preservation of regularity of solutions to the Vlasov equation in the manifold setting. In particular, we demonstrate that the function spaces we defined previously fit together naturally with the Vlasov equation.