General Perfect Fluid Perturbations of Homogeneous and Orthogonal Locally Rotationally Symmetric Class II Cosmologies

TL;DR

This paper analyzes first-order perfect fluid perturbations in LRS class II cosmologies with a cosmological constant, deriving evolution equations for scalar, vector, and tensor modes, including vorticity effects, and exploring their behavior as gravitational and sonic waves.

Contribution

It extends previous work by including vorticity perturbations and provides a detailed harmonic decomposition and decoupled evolution equations for these cosmological models.

Findings

Vorticity perturbations are not generated but influence other perturbations.

Weyl tensor perturbations behave as gravitational waves unaffected by vorticity.

Vorticity introduces first-order disturbances in sonic waves.

Abstract

First order perturbations of homogeneous and hypersurface orthogonal LRS (Locally Rotationally Symmetric) class II cosmologies with a cosmological constant are considered in the framework of the 1+1+2 covariant decomposition of spacetime. The perturbations, which are of perfect fluid type, include general scalar, vector and tensor modes and extend some previous works where vorticity perturbations were excluded. A harmonic decomposition is performed and the field equations are then reduced to a set of eight evolution equations for eight harmonic coefficients, representing perturbations in density, shear, vorticity and the Weyl tensor, in terms of which all other variables can be expressed algebraically. This system decouples into two sub-systems, one for five and one for three coefficients. As previously known, vorticity perturbations cannot be generated to any order in a barytopic perfect fluid. Hence the time development of existing first order vorticity perturbations are seen to be completely determined by the background. However, an already existing vorticity will act as source terms in the evolution equations for the other quantities. In the high frequency approximation the four independent Weyl tensor harmonics evolve as gravitational waves on the anisotropic background in the same manner as in the case without vorticity, whereas vorticity gives a first order disturbance of sonic waves.