Bianchi cosmologies in a Thurston-based theory of gravity

TL;DR

This paper explores how Thurston geometries influence Bianchi cosmologies within a gravity theory that depends on topology, revealing new isotropization and recollapse properties differing from General Relativity.

Contribution

It demonstrates that certain Bianchi solutions exist for all topologies and shows how topology affects isotropization and recollapse in a Thurston-based gravity theory.

Findings

Shear-free solutions with perfect fluid exist for all topologies.

Most BKS metrics isotropize with positive cosmological constant, except some Bianchi II models.

Recollapse is impossible under the weak energy condition in this framework.

Abstract

The strong interplay between Bianchi--Kantowski--Sachs (BKS) spacetimes and Thurston geometries motivates the exploration of the role of topology in our understanding of gravity. As such, we study non-tilted BKS solutions of a theory of gravity that explicitly depends on Thurston geometries. We show that shear-free solutions with perfect fluid, as well as static vacuum solutions, exist for all topologies. Moreover, we prove that, aside from non-rotationally-symmetric Bianchi II models, all BKS metrics isotropize in the presence of a positive cosmological constant, and that recollapse is never possible when the weak energy condition is satisfied. This contrasts with General Relativity (GR), where these two properties fail for Bianchi IX and KS metrics. No additional parameters compared to GR are required for these results. We discuss, in particular, how this framework might allow for simple inflationary models in any topology.