# On the initial geometry of a vacuum cosmological spacetime

## TL;DR

This paper investigates the initial geometric structure of vacuum cosmological spacetimes with symmetries, providing evidence for AVTD behavior in Gowdy spacetimes and establishing conditions for similar behavior in certain non-Gowdy models, along with analysis of spacetimes with singularities.

## Contribution

It offers new evidence for AVTD behavior in Gowdy spacetimes across dimensions and establishes conditions for AVTD-like behavior in non-Gowdy, symmetric vacuum spacetimes, also analyzing spacetimes with singularities.

## Key findings

- Gowdy spacetimes exhibit AVTD behavior in any dimension.
- Sufficient conditions are identified for AVTD behavior in certain non-Gowdy spacetimes.
- Results on causal structure under curvature and volume bounds in spacetimes with singularities.

## Abstract

In the first part of this paper we consider expanding vacuum cosmological spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy spacetimes have AVTD (asymptotically velocity term dominated) behavior for their initial geometry, in any dimension. We then give sufficient conditions to reach a similar conclusion about a $T^2$-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.