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7.2 Inhomogeneous solutions with /\ = 0

For inhomogeneous models with vanishing cosmological constant there is little information available about what happens in general. 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. More generally, the rescaled metric is supposed to converge to a hyperbolic metric (metric of constant negative curvature) on a region that is large in the sense that the volume of its complement tends to zero. If the topology of the Cauchy surface is such that it is consistent with a metric of some Bianchi type, then the hyperbolic region will be missing and the volume of the entire rescaled metric will tend to zero. In this situation it might be expected that the metric converges to a (locally) homogeneous metric in some sense. Evidently the study of the nonlinear stability of Bianchi models is very relevant to developing this picture further.

Independently of the Fischer-Moncrief picture the study of small (but finite) perturbations of Bianchi models is an avenue for making progress in understanding expanding cosmological models. There is a large literature on linear perturbations of cosmological models and it would be desirable to determine what insights the results of this work might suggest for the full nonlinear dynamics. There has recently been important progress in understanding linear vacuum perturbations of various Bianchi models due to Tanimoto [330331332]. Just as it is interesting to know under what circumstances homogeneous cosmological models become isotropic in the course of expansion, it is interesting to know when more general models become homogeneous. This does happen in the case of small perturbations of the Milne model. On the other hand, there is an apparent obstruction in other cases. This is the Jeans instability [23764]. A linear analysis indicates that under certain circumstances (e.g., perturbations of a flat Friedmann model) inhomogeneities grow with time. As yet there are no results on this available for the fully nonlinear case. A comparison that should be useful is that with Landau damping in plasma physics, where rigorous results are available [166].

The most popular matter model for spatially homogeneous cosmological models is the perfect fluid. Generalizing this to inhomogeneous models is problematic since formation of shocks or (in the case of dust) shell-crossing must be expected to occur. These signal an end to the interval of evolution of the cosmological model, which can be treated mathematically with known techniques. Initial steps have been taken to handle shocks in solutions of the Einstein-Euler equations, based on the techniques of classical hydrodynamics. The global existence (but not uniqueness) of plane symmetric weak solutions of a type which can accomodate shocks was proved in [32], while criteria proving the occurrence of shocks in plane symmetry were established in unpublished work of F. Ståhl and the author.

There are not too many results on future geodesic completeness for inhomogeneous cosmological models. A general criterion for geodesic completeness is given in [90]. It does not apply to cases like the Kasner solution but is well-suited to the case where the second fundamental form is eventually negative definite. It is part of the conclusions of [316Jump To The Next Citation Point] that Gowdy spacetimes on a torus are future geodesically complete. Information on the asymptotics is also available in the case of small but finite perturbations of the Milne model and the Bianchi type III form of flat spacetime, as discussed in Sections 5.3 and 5.4, respectively.

For solutions of the Einstein-Vlasov system with hyperbolic symmetry it has been shown by Rein [277] that future geodesic completeness holds for a certain open set of initial data. For solutions of the Einstein equations coupled to a massless linear scalar field with plane symmetry, future geodesic completeness has been shown by Tegankong [339].


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