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 [330, 331, 332]. 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 [237, 64]. 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 .
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accomodate shocks was proved in , while criteria proving the occurrence of shocks in plane symmetry were established in unpublished work of F. Ståhl and the author.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 . 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  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  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 .
© Max Planck Society and the author(s)