Thermalization temperature for a massless gauge mediator ($\Gamma /H={\alpha }_Y^2{m}_{\mathrm{pl}}/(1.66\sqrt{g_{\ast }}T)$)

A rate proportional to $T$ competes with radiation-era expansion proportional to $T^2$. The interaction-to-expansion ratio therefore grows as the universe cools, crossing unity near $T_{\mathrm{eq}}={\alpha }_Y^2{m}_{\mathrm{pl}}/(1.66\sqrt{g_{\ast }})$. If a phase transition occurs at $T_c\gg T_{\mathrm{eq}}$, this high-temperature interaction never establishes thermal equilibrium before the transition. Subsequent faster number-changing processes may still establish a thermal distribution; the initial state is not automatically Fermi-Dirac just because the particles are relativistic.