### 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 [162] 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 [163] and [164]. 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.