7.3 Inflationary models7 Asymptotics of Expanding Cosmological 7.1 Lessons from homogeneous solutions

7.2 Inhomogeneous solutions with tex2html_wrap_inline2093

For inhomogeneous models with vanishing cosmological constant there is little information available about what happens in general. Fischer and Moncrief [100] 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 [236], degeneration of families of metrics with bounded curvature [2], and the Ricci flow [126]. 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. 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 [170, 43]. 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 [123].

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. Criteria for the development of shocks (or their absence) should be developed, based on the techniques of classical hydrodynamics.

In the case of polarized Gowdy spacetimes there is a description of the late-time asymptotics in the literature [87], although the proofs have unfortunately never been published. The central object in the analysis of these spacetimes is a function P that satisfies the equation tex2html_wrap_inline2097 . The picture that emerges is that the leading asymptotics are given by tex2html_wrap_inline2099 for constants A and B, this being the form taken by this function in a general Kasner model, while the next order correction consists of waves whose amplitude decays like tex2html_wrap_inline2105, where t is the usual Gowdy time coordinate. The entire spacetime can be reconstructed from P by integration. It turns out that the generalized Kasner exponents converge to (1, 0, 0) for inhomogeneous models. This shows that if it is stated that these models are approximated by Kasner models at late times it is necessary to be careful in what sense the approximation is supposed to hold. 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.

There are not too many results on future geodesic completeness for inhomogeneous cosmological models. A general criterion for geodesic completeness is given in [62]. 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.

7.3 Inflationary models7 Asymptotics of Expanding Cosmological 7.1 Lessons from homogeneous solutions

image Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de