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Mathematical cosmology is that branch of theoretical physics where some of the most intricate, complex, and deeply unresolved issues lie. Beginning with the Einstein static universe in 1917, in this brief paper we freely float above all major developments that shaped the field until today. We discuss highlights that are further documented in the authors’ recent survey ‘100 years of mathematical cosmology’ scheduled to appear in the Theme Issue ‘The Future of Mathematical Cosmology’. This Theme Issue is to be published in two parts by the Philosophical Transactions of the Royal Society A, and contain a number of important contributions by key researchers in the field.

We discuss qualitative features of the conformal relation between certain classes of gravity theories and general relativity, common to different themes such as $f(R)$, Brans-Dicke-type, and string theories. We focus primarily on the frame relations of the fields involved, slice energy, traceless and Palatini extensions, and selected cosmological applications., v2: 39 pages, more references added, matches version to appear in the inaugural issue of new mdpi journal `Astronomy'

We continue our overview of mathematical cosmology with a survey of the third and fourth periods of the development of the subject. The first Part includes the first two periods and is published separately. The third period (1980–2000) continues here with brief descriptions of the main ideas of inflation, the multiverse, quantum, Kaluza–Klein, and string cosmologies, wormholes and baby universes, cosmological stability and modified gravity. The last period, which ends today, includes various more advanced topics such as M-theoretic cosmology, braneworlds, the landscape, topological issues, the measure problem, genericity, dynamical singularities and dark energy. We emphasize certain threads that run throughout the whole period of development of theoretical cosmology and underline their importance in the overall structure of the field. We end this outline with an inclusion of the abstracts of all papers contributed to the second part of thePhilosophical Transactions of the Royal Society A, theme issue ‘The future of mathematical cosmology’.This article is part of the theme issue ‘The future of mathematical cosmology, Volume 2’.

An elementary survey of mathematical cosmology is presented. We cover certain key ideas and developments in a qualitative way, from the time of the Einstein static universe in 1917 until today. We divide our presentation into four main parts, the first part containing important cosmologies discovered until 1960. The second period (1960-80) contains discussions of geometric extensions of the standard cosmology, singularities, chaotic behaviour, and the initial input of particle physics ideas into cosmology. Our survey for the third period (1980-2000) continues with brief descriptions of the main ideas of inflation, the multiverse, quantum, Kaluza-Klein, and string cosmologies, wormholes and baby universes, cosmological stability, and modified gravity. The last period which ends today includes various more advanced topics such as M-theoretic cosmology, braneworlds, the landscape, topological issues, the measure problem, genericity, dynamical singularities, and dark energy. We emphasize certain threads that run throughout the whole period of development of theoretical cosmology and underline their importance in the overall structure of the field. We end this outline with an inclusion of the abstracts of all papers contributed to the Philosophical Transactions of the Royal Society A, Theme Issue `The Future of Mathematical Cosmology'., Comment: 91 pages

We study the issue of length renormalization in the context of fully covariant gravity theories as well as non-relativistic ones such as Ho\v{r}ava-Lifshitz gravity. The difference of their symmetry groups implies a relation between the lengths of paths in spacetime in the two types of theory. Provided that certain asymptotic conditions hold, this relation allows us to transfer analytic criteria for the standard spacetime length to be finite to the Perelman length to be likewise finite, and therefore formulate conditions for geodesic incompleteness in partially covariant theories. We also discuss implications of this result for the issue of singularities in the context of such theories., Comment: 16 pages

We construct a regular five-dimensional brane-world with localised gravity on a flat 3-brane. The matter content in the bulk is parametrised by an analog of a non-linear fluid with equation of state $p=\gamma\rho^\lambda$ between the `pressure' $p$ and the `density' $\rho$ dependent on the 5th dimension. For $\gamma$ negative and $\lambda>1$, the null energy condition is satisfied, and the geometry is free of singularities within finite distance from the brane, while the induced four-dimensional Planck mass is finite., Comment: 37 pages, 6 figures, final version matches published one

We review recent work on the possibility of primordial synchronization of different Mixmaster regions in generic inhomogeneous spacetime. It is shown that inhomogeneous domains undergoing chaotic oscillations may synchronize themselves exponentially fast and proceed in perfect symphony asymptotically in the past direction. Implications of this result for the structure and evolution of an early phase of the universe are briefly discussed., Comment: 6 pages, uses special world scientific style; invited talk in the 16th Marcel-Grossmann virtual Conference, parallel session EU2

We study the problem of the instability of inhomogeneous radiation universes in quadratic lagrangian theories of gravity written as a system of evolution equations with constraints. We construct formal series expansions and show that the resulting solutions have a smaller number of arbitrary functions than that required in a general solution. These results continue to hold for more general polynomial extensions of general relativity., v2: 34 pages, references added, version to match the published one

We consider the problem of asymptotic synchronization of different spatial points coupled to each other in inhomogeneous spacetime and undergoing chaotic Mixmaster oscillations towards the singularity. We demonstrate that for couplings larger than some threshold value, two Mixmaster spatial points $A,B$, with $A$ in the past of $B$, synchronize and thereby proceed in perfect unison towards the initial singularity. We further show that there is a Lyapunov function for the synchronization dynamics that makes different spatial points able to synchronize exponentially fast in the past direction. We provide an elementary proof of how an arbitrary spatial point responds to the mean field created by the oscillators, leading to their direct interaction through spontaneous synchronization. These results ascribe a clear physical meaning of early-time synchronization leading to a resetting effect for the two BKL maps corresponding to two distinct oscillating spatial points, as the two maps converge to each other to become indistinguishable at the end of synchronization. Our results imply that the universe generically organizes itself through simpler, synchronized, states as it approaches the initial singularity. A discussion of further implications of early-time inhomogeneous Mixmaster synchronization is also provided., Comment: 40 pages, version to appear in the Phil. Trans. Roy. Soc. A

This article gives a review of a recent construction, the ambient cosmological metric, and its implications for the global geometry of the universe. According to this proposal, the universe is a bounding hypersurface carrying a conformal structure and lying at the (conformal) infinity of a new, 5-geometry which satisfies the Einstein equations with fluid sources. We discuss our main results about the implied topological nature of the conformal infinity of the ambient metric, the non-existence of spacetime singularities on the boundary, as well as the validity of cosmic censorship as a direct consequence of a new principle on the boundary., Comment: v2: 5 pages, downsized version to appear in the MG14 Proceedings

We consider theories of gravity that include many coupled scalar fields with arbitrary couplings, in the geometric framework of wave maps. We examine the possibility of obtaining acceptable cosmological solutions without the inclusion of a potential term to the scalar fields. To illustrate the theory, we study two simple models and compare their solutions to those in General Relativity. We also address the issue of the conditions that must be satisfied by the wave maps for an accelerated phase of the Universe., 13 pages, 2 figures. To appear in IJMPA

We study spacetime singularities in a general five-dimensional braneworld with curved branes satisfying four-dimensional maximal symmetry. The bulk is supported by an analog of perfect fluid with the time replaced by the extra coordinate. We show that contrary to the existence of finite distance singularities from the brane location in any solution with flat (Minkowski) branes, in the case of curved branes there are singularity-free solutions for a range of equations of state compatible with the null energy condition., 22 pages, 3 figures

We analyse the causal structure of the ambient boundary, the conformal infinity of the ambient (Poincar\'e) metric. Using topological tools we show that the only causal relation compatible with the global topology of the boundary spacetime is the horismos order. This has important consequences for the notion of time in the conformal geometry of the ambient boundary., Comment: 7 pages

We consider the problem of describing the asymptotic behaviour of \textsc{FRW} universes near their spacetime singularities in general relativity. We find that the Bel-Robinson energy of these universes in conjunction with the Hubble expansion rate and the scale factor proves to be an appropriate measure leading to a complete classification of the possible singularities. We show how our scheme covers all known cases of cosmological asymptotics possible in these universes and also predicts new and distinct types of singularities. We further prove that various asymptotic forms met in flat cosmologies continue to hold true in their curved counterparts. These include phantom universes with their recently discovered big rips, sudden singularities as well as others belonging to graduated inflationary models., 20 pages, latex; v2: 21 pages, additions to comply with final version, to appear in the J. Geom. Phys

We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric topology but with the Zeeman fine topology. This then places severe constraints to the convergence of a sequence of causal curves and the extraction of a limit curve, and also to our ability to formulate conditions for singularity formation., 8 pages

We show that globally and regularly hyperbolic future geodesically incomplete isotropic universes, except for the standard all-encompassing `big crunch', can accommodate singularities of only one kind, namely, those having a non-integrable Hubble parameter, $H$. We analyze several examples from recent literature which illustrate this result and show that such behaviour may arise in a number of different ways. We also discuss the existence of new types of lapse singularities in inhomogeneous models, impossible to meet in the isotropic ones., v2: 16 pages, references added, clarrifications throughout, version to appear in the J.Geom.Phys

We review recent work on the use of the slice energy concept in generalized theories of gravitation. We focus on two special features in these theories, namely, the energy exchange between the matter component and the scalar field generated by the conformal transformation to the Einstein frame of such theories and the issue of the physical equivalence of different conformal frame representations. We show that all such conformally-related, generalized theories of gravitation allow for the slice energy to be invariably defined and its fundamental properties be insensitive to conformal transformations., Comment: 16 pages, In: Proceedings of the 11th Greek Relativity Meeting

We show that for generic sliced spacetimes global hyperbolicity is equivalent to space completeness under the assumption that the lapse, shift and spatial metric are uniformly bounded. This leads us to the conclusion that simple sliced spaces are timelike and null geodesically complete if and only if space is a complete Riemannian manifold.

We review recent results on the Cauchy-Kowalevsky structure of theories with higher derivatives in vacuum. We prove genericity of regularity of solutions under the assumption of analyticity. Our approach is framed in the general context of formal series expansions of the metric around a regular point., Comment: 4 pages, contribution for the Proceedings of the MG13 Meeting on General Relativity, Stockholm, July 2012

We present a new approach to the issues of spacetime singularities and cosmic censorship in general relativity. This is based on the idea that standard 4-dimensional spacetime is the conformal infinity of an ambient metric for the 5-dimensional Einstein equations with fluid sources. We then find that the existence of spacetime singularities in four dimensions is constrained by asymptotic properties of the ambient 5-metric, while the non-degeneracy of the latter crucially depends on cosmic censorship holding on the boundary., Comment: v3: 32 pages, longer version to appear in the EPJC, main idea made explicit, various mathematical and physical explanations added

The explicit integrability of second-order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé property in the parent second-order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé property in the construction of quadratic systems from scalar nth-order ordinary differential equations invariant under time invariant under time translation and rescaling.

We examine the variational and conformal structures of higher order theories of gravity which are derived from a metric-connection Lagrangian that is an arbitrary function of the curvature invariants. We show that the constrained first order formalism when applied to these theories may lead consistently to a new method of reduction of order of the associated field equations. We show that the similarity of the field equations which are derived from appropriate actions via this formalism to those produced by Hilbert varying purely metric Lagrangians is not merely formal but is implied by the diffeomorphism covariant property of the associated Lagrangians. We prove that the conformal equivalence theorem of these theories with general relativity plus a scalar field, holds in the extended framework of Weyl geometry with the same forms of field and self-interacting potential but, in addition, there is a new `source term' which plays the role of a stress. We point out how these results may be further exploited and address a number of new issues that arise from this analysis., Comment: 8 pages, LaTeX (REVTeX 3.1), submitted to J. Math. Phys., references added (nothing changed but LaTeX style)

The existence of envelopes is studied for systems of differential equations in connection with the method of asymptotic splittings which allows to determine the singularity structure of the solutions. The result is applied to braneworlds consisting of a 3-brane in a five-dimensional bulk, in the presence of an analog of a bulk perfect fluid parametrizing a generic class of bulk matter. We find that all flat brane solutions suffer from a finite distance singularity contrary to previous claims. We then study the possibility of avoiding finite distance singularities by cutting the bulk and gluing regular solutions at the position of the brane. Further imposing physical conditions such as finite Planck mass on the brane and positive energy conditions on the bulk fluid, excludes however this possibility, as well., Comment: v2: 26 pages, matches version to appear in EPJC

We study curved isotropic cosmologies filled with two interacting fluids near their time singularities. We find that a number of these universes asymptote to flat limits in the sense that their asymptotic properties become indistinguishable from those of flat Friedmann-Robertson-Walker models on approach to the singularity along any asymptotic direction. In particular, there are no essential singularities in these models. We discuss connections of this result with possible extensions of the cosmic no hair theorem to the case of two interacting fluids, and also provide links to a quantum cosmological treatment of real and complex Euclidean such solutions., v2: 26 pages, new mathematical appendix, improved readability, version to appear in Phys. Rev. D

We review ongoing research related to the asymptotic dynamics of isotropic universes in theories with higher derivatives, especially near the initial singularity. We treat two major cases, that is universes in vacuum, and also those filled with radiation using the method of asymptotic splittings of vector fields. Our solutions have the form of formal Fuchsian-type series for the basic unknowns and are valid near collapse or big rip regimes asymptotically., 4 pages, contribution for the Proceedings of the MG13 Meeting on General Relativity, Stockholm, July 2012, v2: a minor correction

We analyse the question of whether, in higher derivative gravity theories, the present isotropic state of the universe could have been obtained by starting from “arbitrary” initial conditions. We show that the Collins-Hawking conclusion holds in this extended case, that is, the set of spatially homogeneous cosmologies which can approach isotropy at late times is of measure zero in the space of all spatially homogeneous universe models. This involves an examination of a generalized form of the Raychaudhuri equation valid in a general higher-order theory of gravity in four spacetime dimensions. We show that this equation is conformally equivalent to what we call a Raychaudhuri system, that is, to the usual Raychaudhuri equation in general relativity coupled to a scalar-wave equation satisfied by the scalar field generated by the conformal transformation with the usual effective potential. This result is also used to prove directly in higher-order gravity the known forms of cosmic no-hair theorems in the Sitter and power-law inflation in general relativity. Generalizations to arbitrary dimensionality and also the addition of other curvature invariants are discussed.

We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincar\'e to the dominant part of the vector field on approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces., Comment: v2: 14 pages, more references and more discussion, matches version to appear in Grav. Cosm. arXiv admin note: text overlap with arXiv:1212.6737

We show that a general solution of the Einstein equations that describes approach to an inhomogeneous and anisotropic sudden spacetime singularity does not experience geodesic incompleteness. This generalises the result established for isotropic and homogeneous universes. Further discussion of the weakness of the singularity is also included., Comment: v2: 7 pages, minor improvements, version to appear in PRD

We find various asymptotic limits of universes containing two interacting fluids which interact and exchange energy. We study the finite-time singularities that may develop in these cosmologies and obtain the asymptotic behaviours of the solutions near such blow up regimes., Comment: 4 pages, contribution for the Proceedings of the MG13 Meeting on General Relativity, Stockholm, July 2012; v2: eqn numbering corrected

We construct solutions of the Friedmann equations near a sudden singularity using generalized series expansions for the scale factor, the density, and the pressure of the fluid content. In this way, we are able to arrive at a solution with a sudden singularity containing two free constants, as required for a general solution of the cosmological equations., Comment: 4 pages, contribution for the Proceedings of the MG13 Meeting on General Relativity, Stockholm, July 2012

The relation between graviton creation and conformal noninvariance is explored in the context of higher order gravity theories and some consequences for the amount of the created gravitons in the universe are given and compared to that in general relativity.

We consider the higher-order gravity theory derived from the quadratic lagrangian $R+\epsilon R^2$ in vacuum as a first-order (ADM-type) system with constraints, and build time developments of solutions of an initial value formulation of the theory. We show that all such solutions, if analytic, contain the right number of free functions to qualify as general solutions of the theory. We further show that any regular analytic solution which satisfies the constraints and the evolution equations can be given in the form of an asymptotic formal power series expansion., Comment: v2: 21 pages, new author, various improvements, IJMPA in press

By extending previous analysis of the authors, a systematic study of the singularity structure and possible asymptotic behaviors of five-dimensional braneworld solutions is performed in the case where the bulk is a mixture of an analog of perfect fluid (with a density and pressure depending on the extra coordinate) and a massless scalar field. The two bulk components interact by exchanging energy so that the total energy is conserved. In a particular range of the interaction parameters, we find flat brane general solutions avoiding the singularity at finite distance from the brane, in the same region of the equation of state constant parameter $\gamma=P/\rho$ that we found previously in the absence of the bulk scalar field $(-1, Comment: 48 pages