Gravitational, shear and matter waves in Kantowski-Sachs cosmologies

TL;DR

This paper develops a detailed covariant framework to analyze gravitational, shear, and matter waves in Kantowski-Sachs cosmologies, revealing how perturbations evolve and interact in an anisotropic universe with a positive cosmological constant.

Contribution

It introduces a comprehensive set of evolution equations for perturbations in Kantowski-Sachs models, including the coupling of gravitational and matter perturbations, and explores gravitational wave behavior in this anisotropic setting.

Findings

Gravitational degrees of freedom are represented by gravito-magnetic perturbations.

One gravito-magnetic mode couples to matter density perturbations, unlike in Friedmann models.

Tensorial perturbations evolve as gravitational waves with shared sound speed.

Abstract

A general treatment of vorticity-free, perfect fluid perturbations of Kantowski-Sachs models with a positive cosmological constant are considered within the framework of the 1+1+2 covariant decomposition of spacetime. The dynamics is encompassed in six evolution equations for six harmonic coefficients, describing gravito-magnetic, kinematic and matter perturbations, while a set of algebraic expressions determine the rest of the variables. The six equations further decouple into a set of four equations sourced by the perfect fluid, representing forced oscillations and two uncoupled damped oscillator equations. The two gravitational degrees of freedom are represented by pairs of gravito-magnetic perturbations. In contrast with the Friedmann case one of them is coupled to the matter density perturbations, becoming decoupled only in the geometrical optics limit. In this approximation, the even and odd tensorial perturbations of the Weyl tensor evolve as gravitational waves on the anisotropic Kantowski-Sachs background, while the modes describing the shear and the matter density gradient are out of phase dephased by $\pi /2$ and share the same speed of sound.