Cosmological recombination

Cosmological recombination is the epoch in which cooling allowed free electrons and ions to form neutral atoms, sharply reducing the free-electron density.

When the effective recombination rate per Electron becomes smaller than the expansion rate, the small remaining ionized fraction can no longer track chemical equilibrium. A nonzero Electron fraction remains, exceeding the exponentially small Saha equation prediction. The effective rate includes atomic radiative bottlenecks; chemical freeze-out is distinct from photon decoupling by scattering.

In hydrogen-only cosmology, charge neutrality gives $n_e=n_p$ and baryon conservation gives $n_b=n_p+n_H$. The fraction $X_e$ lies between zero and one, with $n_H=(1-X_e)n_b$. With helium present, Electron abundance per hydrogen nucleus and per total baryon are different conventions and must be specified. The hydrogen-only Saha equation gives the equilibrium fraction.

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.

The recombination temperature is the photon temperature at which the free-electron fraction falls sharply as neutral atoms form. For hydrogen it is about $0.3\mathrm{eV}$ because the photon-to-baryon ratio is very large.

With the rounded hydrogen Saha equation coefficient, the condition $X_e=1/2$ gives the displayed equation for $y=\mathcal{B}/T$. Its low-temperature root is about $42$, hence $T\simeq 0.32$ eV for hydrogen binding energy $13.6$ eV. Dropping the slowly varying logarithm gives the rough leading-log estimate $T\sim 0.4$ eV. This equilibrium convention is distinct from photon decoupling and from a full kinetic recombination calculation.

Chemical equilibrium for $A^{+}+e^{-}\leftrightarrow A^0+\gamma$ givesafter neglecting the ion--atom mass difference in the translational prefactor.

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_{B}T)}$.

For $p+e^{-}\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect

With singly ionized helium and helium mass fraction $Y_p$, charge neutrality gives

Writing $\mathcal{F}=n_e/n_B$ couples the two equilibrium equations:

When $Y_p=0$, $\mathcal{F}=X_{H^{+}}$, and