Scalar fields on SL(2,R) and H^2 x R geometric spacetimes and linear perturbations

TL;DR

This paper analyzes the asymptotic behavior of massless scalar fields on specific cosmological vacuum spacetimes with SL(2,R) and H^2 x R geometries, revealing similarities in their solutions and implications for linear perturbations.

Contribution

It introduces harmonics for circle-fibered Bianchi VIII spacetimes and derives asymptotic solutions, highlighting a similarity with Bianchi III models and explaining it via fiber term dominated behavior.

Findings

Asymptotic solutions for scalar fields on SL(2,R) and H^2 x R geometries

Similarity between Bianchi VIII and Bianchi III scalar field behavior

Explanation of similarity through fiber term dominated dynamics

Abstract

Using appropriate harmonics, we study the future asymptotic behavior of massless scalar fields on a class of cosmological vacuum spacetimes. The spatial manifold is assumed to be a circle bundle over a higher genus surface with a locally homogeneous metric. Such a manifold corresponds to the SL(2,R)-geometry (Bianchi VIII type) or the H^2 x R-geometry (Bianchi III type). After a technical preparation including an introduction of suitable harmonics for the circle-fibered Bianchi VIII to separate variables, we derive systems of ordinary differential equations for the scalar field. We present future asymptotic solutions for these equations in a special case, and find that there is a close similarity with those on the circle-fibered Bianchi III spacetime. We discuss implications of this similarity, especially to (gravitational) linear perturbations. We also point out that this similarity can be explained by the "fiber term dominated behavior" of the two models.