Evolution of cosmological dark matter perturbations

We discuss the propagation of dark matter perturbations with non-zero velocity dispersion in cosmological models. In particular a non-zero massive neutrino component may well have a significant effect on the matter power spectrum and cosmic microwave background anisotropy. We present a covariant analysis of the evolution of a dark matter distribution via a two-dimensional momentum-integrated hierarchy of multipole equations. This can be expanded in the velocity weight to provide accurate approximate equations if the matter is non-relativistic, and we also perform an expansion in the mass to study the propagation of relativistic matter perturbations. We suggest an approximation to the exact hierarchy that can be used to calculate efficiently the effect of the massive neutrinos on the CMB power spectra. We implement the corresponding scalar mode equations numerically achieving a considerable reduction in computation time compared with previous approaches.