Gravitational, shear and matter waves in Kantowski-Sachs cosmologies

Abstract

Abstract. A general treatment of vorticity-free, perfect fluid perturbations of KantowskiSachs 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 π/2 and share the same speed of sound.

General perfect fluid perturbations of homogeneous and orthogonal locally rotationally symmetric class II cosmologies

“…The evolution equations for the remaining variables are consistent with previous research, see e.g. [29,30], with the addition of the vorticity appearing as a source term for all variables except Σ T . These equations are still quite cumbersome to solve analytically though, and so we will proceed to examine them in the high frequency limit in section VII.…”

“…Since there is no appearance of the vorticity in these equations the result is exactly the same as the one obtained in [29,30] where the vorticity vanishes on the perturbed model. Hence, both E T and H T represents decoupled gravitational waves, which will to first order propagate at the speed of light.…”

“…Our results will now be compared to previous results, see [29] and [30], where the only differences should be related to the impact of Ω S and Ω V . As we saw in the previous section though, the evolution of the vorticity completely decouples from the other variables, independent of the magnitude of the frequency.…”

“…In some previous papers we have studied perturbations on the backgrounds given in section III [29,30]. Here these studies will be extended to the most general first order perturbations, consistent with a perfect fluid, by including also vorticity of the fluid.…”

“…In some earlier papers [29][30][31] we studied vorticity free perturbations on a class of anisotropic cosmological models, given by the homogeneous and orthogonal locally rotationally symmetric (LRS) spacetimes of class II [32,33]. The restriction to zero vorticity was partly due to the significant simplifications this implies for the commutator relations, but was also motivated by that vorticity cannot be generated in a perfect fluid with barotropic equation of state, cf.…”