Energy spectrum of thermalizing high energy decay products in the early universe

We revisit the Boltzmann equation governing spectrum of energetic particles originating from decay massive progenitors during process thermalization. assume that these decays occur when background temperature $T$ is much less than mass $M$ progenitor. pay special attention to IR cutoff provided by thermal bath, and include suppression resulting interference multiple scattering reactions (LPM effect). solve integral numerically, construct an accurate analytical fit solutions.