In discussing homogeneous models in the following we restrict to Bianchi types other than type IX. Then a general theorem of Wald  states that any model whose matter content satisfies the strong and dominant energy conditions and which expands for an infinite proper time is such that all generalized Kasner exponents tend to as . A positive cosmological constant leads to isotropization. The mean curvature tends to the constant value as , while the scale factors increase exponentially.
Wald’s result is only dependent on energy conditions and uses no details of the matter field equations. The question remains whether solutions corresponding to initial data for the Einstein equations with positive cosmological constant, coupled to reasonable matter, exist globally in time under the sole condition that the model is originally expanding. It can be shown that this is true for various matter models using the techniques of [290, 287]. This has been worked out in detail for the case of collisionless matter by Lee . For the case of a perfect fluid with linear equation of state see . Once global existence is known and a specific matter model has been chosen, details of the asymptotic behaviour of the matter fields can be determined and this was done in [224, 303]. For instance, it was shown that the solutions of the Vlasov equation behave like dust asymptotically.
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