 Mass-dependent fermion freeze-out

In a radiation-dominated toy model with $\Gamma ={\alpha }_Y^4{T}^5/m_Y^4$, setting the interaction rate equal to the Hubble parameter gives the displayed decoupling scale. The transition $T_D=m_Y$ defines $m_{\mathrm{nr}}$. For smaller masses, $x_D=m_Y/T_D=(m_{\mathrm{nr}}/m_Y{)}^{1/3}\gg 1$. An instantaneous chemically equilibrated cosmological particle freeze-out estimate gives $n_Y/s=[45g_Y/(2{\pi }^2{g}_{\ast s})(2\pi {)}^{-3/2}]x_D^{3/2}{e}^{-x_D}$. This estimate presupposes number-changing thermal equilibrium, radiation domination and the assumed rate law, rather than following from elastic thermalization alone.