Structure of solutions near the initial singularity for the surface-symmetric Einstein-Vlasov system

TL;DR

This paper investigates the behavior of cosmological solutions with collisionless matter near the initial singularity, showing conditions under which the area radius vanishes and describing the asymptotic Kasner-like behavior.

Contribution

It provides new results on the initial singularity structure for the Einstein-Vlasov system with various symmetries and cosmological constants, including existence and asymptotic properties.

Findings

Area radius tends to zero at the initial singularity under certain symmetries.

Past global existence is proved for hyperbolic symmetry with negative or positive cosmological constant.

Early-time behavior is Kasner-like for small initial data.

Abstract

Results on the behaviour in the past time direction of cosmological models with collisionless matter and a cosmological constant $\Lambda$ are presented. It is shown that under the assumption of non-positive $\Lambda$ and spherical or plane symmetry the area radius goes to zero at the initial singularity. Under a smallness assumption on the initial data, these properties hold in the case of hyperbolic symmetry and negative $\Lambda$ as well as in the positive $\Lambda$ case. Furthermore in the latter cases past global existence of spatially homogeneous solutions is proved for generic initial data. The early-time asymptotics is shown to be Kasner-like for small data.