Dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter

TL;DR

This paper studies the long-term behavior of Bianchi type I cosmological models with various anisotropic matter types, establishing an anisotropy classification based on asymptotic behavior near singularities.

Contribution

It introduces a unified anisotropy classification for Bianchi type I solutions with diverse anisotropic matter models, based on a single parameter influencing asymptotic dynamics.

Findings

Solutions either converge to Kasner states or oscillate between them near singularities.

The classification includes convergent and oscillatory types, depending on matter properties.

The analysis applies to multiple matter models like collisionless matter, elastic matter, and magnetic fields.

Abstract

We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an `anisotropy classification' for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type \Aplus: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types \Bplus and \Cplus: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type \Dplus: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions.