= Solution
<Photon decoupling> occurs when the Thomson interaction rate falls below the expansion rate,
$$
\Gamma_\gamma=n_e\sigma_T\simeq H.
$$
At decoupling the universe is approximately matter dominated, so
$$
H(T)=H_0\sqrt{\Omega_{m,0}}
\left(\frac{T}{T_0}\right)^{3/2}.
$$
Using $n_e=X_e\eta(2\zeta(3)/\pi^2)T^3$ in $\Gamma_\gamma=H$ gives
$$
\boxed{X_e(T_{\rm dec})T_{\rm dec}^{3/2}
\simeq\frac{\pi^2H_0\sqrt{\Omega_{m,0}}}
{2\zeta(3)\eta\sigma_TT_0^{3/2}}}.
$$
For $X_e\ll1$, 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 $T_{\rm dec}$, explaining why $T_{\rm dec}\simeq T_{\rm rec}$ is a good approximation.
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