Asymptotic behaviour of future-complete cosmological spacetimes

Abstract

This work discusses the apriori possible asymptotic behavior to the future, for (vacuum) space-times which are geodesically complete to the future, and which admit a foliation by constant mean curvature compact Cauchy surfaces. Introduction.Let (M, g) be a CMC cosmological space-time, i.e. a space-time with a compact constant mean curvature Cauchy surface (Σ, g, K). The main focus will be on the vacuum case in 3 + 1 dimensions although we will occasionally consider generalizations to non-negative energy conditions and higher dimensions.A fundamental issue in general relativity is to understand the global structure of (M, g), and in particular the evolution of the geometry of the CMC foliation Σ τ generated by the CMC slice Σ. Singularities of (M, g) will generally form in finite proper time, both to the future and to the past of Σ. Roughly, these may correspond either to big bang or big crunch singularities of the space-time as a whole, or to localized gravitational collapse within only parts of the space-time. The understanding of the mechanism and structure of such singularity formation is of course a central issue in general relativity.Here we concentrate instead on the simpler situation where there is no singularity formation (in finite proper time), say to the future of Σ. Thus, assume (M, g) is geodesically complete (time-like and null) to the future of Σ. The basic issue then is to understand the asymptotic or long-time future behavior of the geometry of the CMC slices Σ τ and of the full space-time (M, g).Similar issues arise, and have been more fully investigated, in connection with globally hyperbolic, geodesically complete space-times with an asymptotically flat Cauchy surface, (purely radiative space-times). Thus, following Penrose, starting from an asymptotically flat initial data surface, one would like to understand the asymptotic structure of the resulting maximal globally hyperbolic space-time in terms of a conformal compactification leading to the structures of Scri; I, I + and I 0 . A great deal of work has been done and is ongoing in this direction, cf. [15], [21], [24] and references therein.The issues we consider then are cosmological analogues of this question. In principle, this should be a simpler situation to analyse, since there is no spatial infinity; compact slices are easier to deal with than non-compact slices.Vince Moncrief, together with Arthur Fischer, has begun a program to understand this issue, cf.[16]-[20] and further references therein. Independently, the author has also initiated such a program, cf. [1]. Both programs have a number of features in common and agree in certain special cases. In the end, they should also lead to the same overall description of the asymptotic behavior at future infinity. However, both the techniques and the point of view of these two approaches are Partially supported by NSF Grant DMS 0072591.

Theorems on Existence and Global Dynamics for the Einstein Equations

“…Fischer and Moncrief [139] have made an interesting proposal that attempts to establish connections between the evolution of a suitably conformally rescaled version of the spatial metric in an expanding cosmological model and themes in Riemannian geometry such as the Thurston geometrization conjecture [341], degeneration of families of metrics with bounded curvature [2], and the Ricci flow [169]. For further related work see [5]. A key element of this picture is the theorem on the stability of the Milne model discussed in Section 5.3.…”