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

First-order perturbations of homogeneous and hypersurface orthogonal 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 in which 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 subsystems, one for five and one for three coefficients. As previously known, vorticity perturbations cannot be generated to any order in a barotropic perfect fluid. Hence, the time development of existing first-order vorticity perturbations is 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.