Matter and dynamics in closed cosmologies

TL;DR

This paper extends dynamical systems methods to analyze the matter content in closed cosmologies, focusing on specific models to understand their expansion, recollapse, and asymptotic behaviors.

Contribution

It generalizes previous approaches to include perfect fluids with arbitrary equations of state and provides a comprehensive dynamical analysis of Bianchi type IX and Kantowski-Sachs models.

Findings

Models expand from a singularity and recollapse under strong energy condition.

The presence of Einstein's static model influences local dynamics.

Criteria for collapse or eternal expansion are established.

Abstract

To systematically analyze the dynamical implications of the matter content in cosmology, we generalize earlier dynamical systems approaches so that perfect fluids with a general barotropic equation of state can be treated. We focus on locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal perfect fluid models, since such models exhibit a particularly rich dynamical structure and also illustrate typical features of more general cases. For these models, we recast Einstein's field equations into a regular system on a compact state space, which is the basis for our analysis. We prove that models expand from a singularity and recollapse to a singularity when the perfect fluid satisfies the strong energy condition. When the matter source admits Einstein's static model, we present a comprehensive dynamical description, which includes asymptotic behavior, of models in the neighborhood of the Einstein model; these results make earlier claims about ``homoclinic phenomena and chaos'' highly questionable. We also discuss aspects of the global asymptotic dynamics, in particular, we give criteria for the collapse to a singularity, and we describe when models expand forever to a state of infinite dilution; possible initial and final states are analyzed. Numerical investigations complement the analytical results.