Saha-Thomson decoupling estimate ($X_e{n}_H{\sigma }_{T}c\sim H$)

Combining the hydrogen-only Saha equation with the Thomson decoupling rate criterion gives an elementary estimate of photon decoupling. With present baryon density about $0.25{\mathrm{m}}^{-3}$, hydrogen fraction about $0.75$, present photon temperature $2.725\mathrm{K}$ and a matter-era normalization $H_0\sqrt{{\Omega }_m}\simeq 38\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, the coupled estimate gives a temperature near $3100\mathrm{K}$. Hydrogen chemical equilibrium fails during the actual kinetic recombination history, and the local scattering-rate criterion differs from the maximum of the visibility function; the estimate is not a precision prediction.