Recombination and photon decoupling

Let $n_H$ denote the total number density of hydrogen nuclei, whether bound or ionized. Define the free-electron fraction by $x_e=n_e/n_H$; electrical neutrality gives the same free-proton fraction. In chemical equilibrium, the reaction linking an electron, proton, hydrogen atom, and photon fixes the chemical-potential relation. Nondegenerate nonrelativistic matter distributions and a thermal photon bath then give the Saha equation. The ground-state spin multiplicities are included in the conventional prefactor displayed here.

The hydrogen binding energy alone does not set the recombination temperature. Because photons outnumber hydrogen nuclei, the exponential photon tail can keep the gas ionized at temperatures far below the binding energy. The explicit inverse density in the Saha expression matters: at fixed temperature, a denser hydrogen gas favors neutralization. In a real expanding history, both temperature and density change, so one must specify their relation before solving for an epoch.

Direct capture into the ground state emits a photon energetic enough to ionize another atom, so it need not yield a lasting increase in neutral hydrogen. Population must instead pass through excited states and lose radiation without immediately reversing the net recombination. Two important escape routes are the metastable two-photon transition from the second s state and redshifting of resonant Lyman-alpha photons out of the line. These processes introduce timescales absent from an equilibrium population formula.

As expansion outpaces effective atomic relaxation, the electron fraction departs from the Saha value. A simplified effective-level treatment captures the bottleneck qualitatively; precision work follows more levels and radiative transfer. Helium recombination and later astrophysical reionization also alter the electron history. Thus using Saha alone all the way through last scattering cannot justify a precise visibility peak or residual ionization fraction, even if it gives a reasonable initial scale for when atoms begin forming.

At low photon energy, free electrons scatter photons with the Thomson cross section. The rate per photon is free-electron density times cross section times light speed. Compare it with expansion for a local diagnostic, but integrate the rate along a trajectory to obtain optical depth. The probability of no further scattering is an exponential of minus that integrated depth. A particular ionized fraction therefore cannot universally define decoupling independently of density and expansion.

For optical depth defined from an emission epoch to the present, the last-scattering visibility combines the chance to scatter near that epoch with the chance of no scattering afterward. In conformal time it is $g(\eta)=-\tau'(\eta)e^{-\tau}$, since the remaining optical depth decreases toward today. Recombination, the peak of this visibility, and baryon drag decoupling answer different questions. A later reionized population can add optical depth without undoing the earlier atomic history.