Photon decoupling

Photon decoupling occurs when the Thomson-scattering rate drops below the Hubble expansion rate. Photons then free stream and form the cosmic microwave background.

The physical Thomson scattering rate equals the inverse photon mean scattering time. Comparing it with the Hubble parameter gives a local approximate decoupling criterion. A more precise last-scattering temperature is obtained from optical depth and the visibility maximum; these conventions need not give exactly the same temperature.

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.