= Mass-dependent fermion freeze-out
{title2=$m_{\rm nr}=\frac{\alpha_Y^4m_{\rm pl}}{1.66\sqrt{g_*}},\quad T_D=m_Y(m_Y/m_{\rm nr})^{1/3}$}
In a radiation-dominated toy model with $\Gamma=\alpha_Y^4T^5/m_Y^4$, setting the interaction rate equal to the <Hubble parameter> gives the displayed decoupling scale. The transition $T_D=m_Y$ defines $m_{\rm nr}$. For smaller masses, $x_D=m_Y/T_D=(m_{\rm nr}/m_Y)^{1/3}\gg1$. An instantaneous chemically equilibrated <cosmological particle freeze-out> estimate gives $n_Y/s=[45g_Y/(2\pi^2g_{*s})(2\pi)^{-3/2}]x_D^{3/2}e^{-x_D}$. This estimate presupposes number-changing <thermal equilibrium>, <radiation domination> and the assumed rate law, rather than following from elastic thermalization alone.
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