 Solution

Photon decoupling occurs when the Thomson interaction rate falls below the expansion rate,At decoupling the universe is approximately matter dominated, soUsing $n_e=X_e\eta (2\zeta (3)/{\pi }^2)T^3$ in ${\Gamma }_{\gamma }=H$ givesFor $X_e\ll 1$, the Saha equation makes $X_e(T)$ exponentially steep through $e^{-B_H/(2T)}$. Power-law changes of $H_0$ or ${\Omega }_{m,0}$ therefore cause only a small shift in $T_{\mathrm{dec}}$, explaining why $T_{\mathrm{dec}}\simeq T_{\mathrm{rec}}$ is a good approximation.