Thermal decoupling in cosmology

A particle species thermally decouples when its interaction rate falls below the Hubble parameter. Its collisionless relativistic distribution subsequently redshifts with temperature proportional to $a^{-1}$, while annihilations among still-coupled species can heat the remaining plasma.

If a relativistic species decouples when the coupled plasma has $g_{\ast s,\mathrm{dec}}$ entropy degrees of freedom, then after the plasma reaches $g_{\ast s}$ degrees of freedom,The ratio follows from separate redshifting of the decoupled species and cosmological entropy conservation in the coupled plasma.