= Thermal relic density bound for a tunable fermion
{title2=$\frac{\rho_Y}{\rho_B}=\frac{m_Y}{m_B}\frac{Y_Y}{Y_B},\quad Y_i=n_i/s$}
For a stable nonrelativistic <thermal relic>, compare <energy> densities rather than number densities. If $m_Y=m_{\rm nr}/x^3$ and $Y_Y=Cx^{3/2}e^{-x}$, its <density> relative to <baryons> is $(Cm_{\rm nr}/m_BY_B)x^{-3/2}e^{-x}$. Requiring this ratio below an allowed dark-matter-to-baryon ratio $R$ means $x+\tfrac32\log x\ge\log[Cm_{\rm nr}/(Rm_BY_B)]$. 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.
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