Recombination

Remarks

Let $B = 13.6$ eV be the binding energy of hydrogen. In thermal equilibrium, ignoring helium, the Saha equation gives $$\frac{x_e^2}{1 - x_e} = \frac{1}{n_H} \left( \frac{m_e k T}{2 \pi \hbar^2} \right)^{3/2} e^{-B / k T}.$$ Here $k$ is the Boltzmann constant, $m_e$ the electron mass and $\hbar$ the reduced Planck constant. There are about a billion photons per baryon, so $n_H$ is tiny compared with the photon density. Hence the right side stays large until $k T \approx 0.3$ eV, far below $B$. This is a temperature of about $3000$ to $4000$ K. A full treatment out of equilibrium gives a slower recombination. It is largely complete by $z \approx 1100$, when the photons last scatter.

Example: the transition from $x_e \approx 1$ to $x_e \ll 1$ in hydrogen, between redshifts of roughly $1500$ and $1000$, is recombination. Non-example: the earlier capture of electrons by helium nuclei, at higher redshifts, is called helium recombination; it does not satisfy the definition, which concerns hydrogen. The later epoch when the gas became ionised again by the first stars and quasars is reionisation, not recombination.