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Let $T_{-}$ be the residual gas temperature immediately before decay. This is the $T_d$ that enters the target abundance formula, not the higher reheating temperature after decay. Before the decay, separately conserved relic number and gas entropy give the conserved cosmological relic abundance $Y=n_{\varphi }/s_{-}$. The cold nonrelativistic relic has negligible kinetic energy, henceAssume its density dominates over that of the residual gas and that curvature and dark energy are negligible in the Friedmann equation. Using the reduced Planck mass $M_{\mathrm{Pl}}^2=(8\pi G{)}^{-1}$ gives $3M_{\mathrm{Pl}}^2{H}_d^2\simeq {\rho }_{\varphi }(t_d^{-})$. Under the allowed sudden decay of a dominant nonrelativistic relic convention $H_d\simeq \Gamma =t_d^{-1}$, the temperature immediately before decay isThe coefficient inherits the approximation $H_d\simeq 1/t_d$: the result is a sudden-decay estimate, not an exact age relation for a matter-only universe. If $t_d$ were instead the exact cosmic age in an uninterrupted spatially flat matter era, $H_d=2/(3t_d)$ would replace the numerator $3M_{\mathrm{Pl}}^2$ by $4M_{\mathrm{Pl}}^2/3$.

The assumptions also require thermal decoupling in cosmology and negligible prior decay so that $Y$ remains conserved, a fixed relativistic count $g_{\star }$, and rapid thermalization after the energy release. In the same approximation, the reheating temperature $T_{+}$ follows from $c_2{g}_{\star }{T}_{+}^4\simeq 3M_{\mathrm{Pl}}^2/t_d^2$:Its absence of $Y$ at fixed decay rate makes clear why the two temperatures must be distinguished.