Dynamics of compact homogeneous universes

TL;DR

This paper provides a comprehensive analysis of the dynamics of compact homogeneous universes, including explicit calculations of Teichmüller deformations and degrees of freedom, with examples for various Bianchi types.

Contribution

It introduces a method to explicitly construct compact homogeneous universes by reducing spacetime identifications to homogeneous sections and deriving necessary conditions.

Findings

Explicit formulas for Teichmüller deformations

Counting of dynamical degrees of freedom

Examples for Bianchi types II, VI0, VII0, and I

Abstract

A complete description of dynamics of compact locally homogeneous universes is given, which, in particular, includes explicit calculations of Teichm\"uller deformations and careful counting of dynamical degrees of freedom. We regard each of the universes as a simply connected four dimensional spacetime with identifications by the action of a discrete subgroup of the isometry group. We then reduce the identifications defined by the spacetime isometries to ones in a homogeneous section, and find a condition that such spatial identifications must satisfy. This is essential for explicit construction of compact homogenoeus universes. Some examples are demonstrated for Bianchi II, VI${}_0$, VII${}_0$, and I universal covers.