 Thermal relic density bound for a tunable fermion

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.