Abstract
In the first part of this paper we consider expanding vacuum cosmological spacetimes with a free TN-action. Among them, we give evidence that Gowdy spacetimes have AVTD (asymptotic velocity term dominated) behavior for their initial geometry, in any dimension. We then give sufficient conditions to reach a similar conclusion about a T2-invariant four dimensional nonGowdy spacetime. In the second part of the paper we consider vacuum cosmological spacetimes with crushing singularities. We introduce a monotonic quantity to characterize Kasner spacetimes. Assuming scale-invariant curvature bounds and local volume bounds, we give results about causal pasts.
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