4 The Cosmological Cauchy Problem

In this section we discuss the Einstein–Vlasov system for cosmological spacetimes, i.e., spacetimes that possess a compact Cauchy surface. The “particles” in the kinetic description are in this case galaxies or even clusters of galaxies. The main goal is to determine the global properties of the solutions to the Einstein–Vlasov system for initial data given on a compact 3-manifold. In order to do so, a global time coordinate t must be found and the asymptotic behavior of the solutions when t tends to its limiting values has to be analyzed. This might correspond to approaching a singularity, e.g., the big bang singularity, or to a phase of unending expansion.

Presently, the aim of most of the studies of the cosmological Cauchy problem has been to show existence for unrestricted initial data and the results that have been obtained are in cases with symmetry (see, however, [27], where to some extent global properties are shown in the case without symmetry). These studies will be reviewed below. A recent and very extensive work by Ringström has, on the other hand, a different aim, i.e., to show stability of homogeneous cosmological models, and concerns the general case without symmetry. The size of the Cauchy data is in this case very restricted but, since Ringström allows general perturbations, there are no symmetries available to reduce the complexity of the Einstein–Vlasov system. This result will be reviewed at the end of this section.

 4.1 Spatially-homogeneous spacetimes
 4.2 Inhomogeneous models with symmetry
  4.2.1 Surface symmetric spacetimes
  4.2.2 Gowdy and T2 symmetric spacetimes
 4.3 Cosmological models with a scalar field
 4.4 Stability of some cosmological models

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