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

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.

General Perturbations of Homogeneous and Orthogonal Locally Rotationally Symmetric Class II Cosmologies with Applications to Dissipative Fluids

“…With R a = q a , the heat flux, the expressions below are consistent with the second law of thermodynamics as stated in equation (15). The bulk viscosity is given by…”

“…Perturbations on the backgrounds of section IV were studied in [13][14][15]. In these works the energy-momentum tensor was assumed to be that of a barotropic perfect fluid, both for the background and the perturbations.…”

“…by δ a G and Ĝ, which are zero on the homogeneous background. On performing a harmonic decomposition of the first order quantities, it is seen that the Ĝ derivatives may be expressed in terms of the G a ≡ δ a G [13][14][15]. Hence, the perturbations of the background quantities will be represented by the G a…”

“…As a first generalization away from isotropy, without introducing the full complexity of general anisotropic models, we will in the following consider backgrounds with one direction of anisotropy. In some earlier papers [13][14][15] we studied perfect fluid perturbations of homogeneous and Locally Rotationally Symmetric (LRS) cosmologies of class II using a perturbation method based on the 1+1+2 covariant split of spacetime [16][17][18], which is an extension of the 1+3 covariant split [19][20][21][22][23].…”

On the Interaction Between Electromagnetic, Gravitational, and Plasma Related Perturbations on LRS Class II Spacetimes

“…However, not all quantities are zero on the background. Thus, to represent the perturbations of the quantities in S (0) , we will follow [11][12][13][14] and make use of the spatial gradients of these quantities, as these gradients vanish on the background due to the assumed homogeneity. We therefore introduce the following gauge invariant variables…”