Mass-dependent fermion freeze-out ($m_{\mathrm{nr}}=\frac{{\alpha }_Y^4{m}_{\mathrm{pl}}}{1.66\sqrt{g_{\ast }}},T_D=m_Y(m_Y/m_{\mathrm{nr}}{)}^{1/3}$)

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.

For a stable nonrelativistic thermal relic, compare energy densities rather than number densities. If $m_Y=m_{\mathrm{nr}}/x^3$ and $Y_Y=C{x}^{3/2}{e}^{-x}$, its density relative to baryons is $(C{m}_{\mathrm{nr}}/m_B{Y}_B)x^{-3/2}{e}^{-x}$. Requiring this ratio below an allowed dark-matter-to-baryon ratio $R$ means $x+\frac{3}{2}\log x\ge \log [C{m}_{\mathrm{nr}}/(R{m}_B{Y}_B)]$. With fixed rate parameters this excludes the weakly interacting large-mass branch, while the sufficiently Boltzmann-suppressed smaller-mass branch can satisfy the density bound. Decay or a nonthermal initial abundance requires a different constraint.