Future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI$_0$

TL;DR

This paper proves future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system in Bianchi types II and VI$_0$, extending previous results to new symmetry classes with small data assumptions.

Contribution

It generalizes stability results from locally rotationally symmetric to reflection symmetric Bianchi II and VI$_0$ solutions, including the first such result for Bianchi VI$_0$.

Findings

Solutions are asymptotic to known models (Collins-Stewart and Ellis-MacCallum).

Established stability under small initial data.

Extended stability analysis to previously unaddressed Bianchi VI$_0$ case.

Abstract

Using the methods developed for the Bianchi I case we have shown that a boostrap argument is also suitable to treat the future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI$_0$. These solutions are asymptotic to the Collins-Stewart solution with dust and the Ellis-MacCallum solution respectively. We have thus generalized the results obtained by Rendall and Uggla in the case of locally rotationally symmetric Bianchi II spacetimes to the reflection symmetric case. However we needed to assume small data. For Bianchi VI$_0$ there is no analogous previous result.