Recombination temperature ($T_{\mathrm{rec}}$)

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.

For hydrogen ionization fraction $X_e$ of order one, the hydrogen-only Saha equation gives $1\sim n_b(2\pi /(m_{e}T){)}^{3/2}{e}^{B/T}$. The baryon-to-photon ratio and photon number density imply $n_b\sim \eta T^3$. Therefore $1\sim \eta (T/m_e{)}^{3/2}{e}^{B/T}$ and the displayed logarithmic balance follows. A small $\eta$ supplies many ionizing photons per baryon, so neutral hydrogen forms at a temperature much smaller than its Hydrogen binding energy. The numerical temperature depends on the ionization-fraction convention and on whether equilibrium remains valid.

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.