Non-equilibrium effects on particle freeze-out in the early universe

Deviations from thermodynamic equilibrium are of fundamental importance in the early universe. The examples are plentiful, for example the freeze-out of photon thermodynamic equilibrium due to hydrogen recombination [1] or the equivalent freeze-out of neutrinos slightly prior to electron-positron annihilation [2]. These phenomena have been investigated many times in the literature, but almost always in the context of deviations from chemical equilibrium, that is, kinetic or scattering equilibrium is assumed to hold at all times. In the limit of Boltzmann statistics the single particle distribution then takes the form