Photon mean free path in an ionized universe (${\ell }_{\gamma }=(n_e{\sigma }_T{)}^{-1},t_{\mathrm{sc}}={\ell }_{\gamma }/c$)

For a dilute ionized plasma with nonrelativistic electrons and photon energies well below $m_e{c}^2$, independent Thomson scattering events give photon survival probability $\exp (-n_e{\sigma }_{T}s)$ over proper distance $s$. Its mean gives the displayed mean free path. The scattering rate is ${\Gamma }_{\gamma }=n_e{\sigma }_{T}c$. Photons are tightly coupled when this rate greatly exceeds the Hubble parameter; photon decoupling becomes possible as the free-electron fraction falls. Frequent elastic scattering does not by itself imply chemical equilibration of the photon number.