4.4 Stability of some cosmological models

In standard cosmology, the universe is taken to be spatially homogeneous and isotropic. This is a strong assumption leading to severe restrictions of the possible geometries as well as of the topologies of the universe. Thus, it is natural to ask if small perturbations of an initial data set, which corresponds to an expanding model of the standard type, give rise to solutions that are similar globally to the future?

In a recent work, Ringström [162Jump To The Next Citation Point] considers the Einstein–Vlasov system and he gives an affirmative answer to the stability question for some of the standard cosmologies.

The standard model of the universe is spatially homogeneous and isotropic, has flat spatial hypersurfaces of homogeneity, a positive cosmological constant and the matter content consists of a radiation fluid and dust. Hence, to investigate the question on stability it is natural to consider cosmological solutions with perfect fluid matter and a positive cosmological constant. However, as is shown by Ringström, the standard model can be well approximated by a solution of the Einstein–Vlasov system with a positive cosmological constant. Approximating dust with Vlasov matter is straightforward, whereas approximating a radiation fluid is not. By choosing the initial support of the distribution function suitably, Ringström shows that Vlasov matter can be made to mimic a radiation fluid for a prescribed amount of time; sooner or later the matter will behave like dust, but the time at which the approximation breaks down can be chosen to be large enough that the radiation is irrelevant to the future of that time in the standard picture.

The main results in [162] are stability of expanding, spatially compact, spatially locally homogeneous solutions to the Einstein–Vlasov system with a positive cosmological constant as well as a construction of solutions with arbitrary compact spatial topology. In other words, the assumption of almost spatial homogeneity and isotropy does not seem to impose a restriction on the allowed spatial topologies.

Let us mention here some related works although these do not concern the Einstein–Vlasov system. Ringström considers the case where the matter model is a non-linear scalar field in [163Jump To The Next Citation Point] and [164Jump To The Next Citation Point]. The background solutions, which Ringström perturb and which are shown to be stable, have accelerated expansion. In [163] the expansion is exponential and in [164] it is of power law type. The corresponding problem for a fluid has been treated in [165] and [175], and the Newtonian case is investigated in [109] and [40] for Vlasov and fluid matter respectively.

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