Abstract :
Deviations from kinetic equilibrium of massive particles caused by the universe expansion are calculated analytically in the Boltzmann approximation. For the case of an energy-independent amplitude of elastic scattering, an exact partial differential equation is derived instead of the usual integro-differential equation. A simple perturbative solution of the former is found. For the case of an energy-dependent amplitude the problem cannot be reduced to a differential equation but the solution of the original integro-differential equation can be found in terms of a Taylor expansion, which in the case of a constant amplitude shows perfect agreement with the perturbative solution of the differential equation. Corrections to the spectrum of (possibly) massive tau-neutrinos are calculated. The method may be of more general interest and can be applied to the calculation of spectrum distortion in other (not necessarily cosmological) non-equilibrium processes.
