Nonequilibrium corrections to energy spectra of massive particles in   expanding universe
  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 one. A simple perturbative solution of the former is found. For the case of an energy-dependent amplitude the problem cannot be reduced to the differential equation but the solution of the original integro-differential equation can be found in terms of the Taylor expansion, which in the case of aconstant amplitude shows a 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) nonequilibrium processes. 
A.D. Dolgov