The Weyl tensor in Spatially Homogeneous Cosmological Models

TL;DR

This paper analyzes the evolution of the Weyl curvature invariant in all spatially homogeneous cosmological models, revealing late-time behaviors and implications for gravitational entropy in universe models with different topologies.

Contribution

It provides a comprehensive classification of the asymptotic evolution of the Weyl curvature in all Bianchi and Thurston types, including effects of topology and late-time behavior.

Findings

Weyl curvature invariant often dominates Ricci invariant at late times.

Late-time behavior classified into five categories.

Gravitational entropy measure increases in all ever-expanding models.

Abstract

We study the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted gamma-law perfect fluid. We investigate all the Bianchi and Thurston type universe models and calculate the asymptotic evolution of Weyl curvature invariant for generic solutions to the Einstein field equations. The influence of compact topology on Bianchi types with hyperbolic space sections is also considered. Special emphasis is placed on the late-time behaviour where several interesting properties of the Weyl curvature invariant occur. The late-time behaviour is classified into five distinctive categories. It is found that for a large class of models, the generic late-time behaviour the Weyl curvature invariant is to dominate the Ricci invariant at late times. This behaviour occurs in universe models which have future attractors that are plane-wave spacetimes, for which all scalar curvature invariants vanish. The overall behaviour of the Weyl curvature invariant is discussed in relation to the proposal that some function of the Weyl tensor or its invariants should play the role of a gravitational 'entropy' for cosmological evolution. In particular, it is found that for all ever-expanding models the measure of gravitational entropy proposed by Gron and Hervik increases at late times.