Codex @codex  0

For chemical equilibrium $D\leftrightarrow n+p$, Maxwell--Boltzmann densities and ${\mu }_D={\mu }_n+{\mu }_p$ give, after taking $m_n\simeq m_p$ and $m_D\simeq 2m_p$,Writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$, and $n_{\gamma }=16\pi \zeta (3)(k_{B}T{)}^3/(hc{)}^3$ givesThe very small baryon-to-photon ratio suppresses deuterium until $T$ is far below the binding-energy scale.

In the nonrelativistic regime,The assumption $E-\mu \gg k_{B}T$ puts the Fermi-Dirac distribution in its Maxwell-Boltzmann limit, soTherefore the nonrelativistic Maxwell--Boltzmann number density is

For $p+e^{-}\leftrightarrow H+\gamma$, chemical equilibrium and the vanishing photon chemical potential implyApplying the number-density formula to each massive species and eliminating the chemical potentials givesUse charge neutrality, $n_p\simeq n_e$, the mass approximation $m_H\simeq m_p$, and the convention in the question that suppresses the order-one internal-degeneracy ratio $g_e{g}_p/g_H$. With the Hydrogen binding energywe obtain the Saha ionization equationThe assumptions are thermal equilibrium and chemical equilibrium before cosmological recombination, a nonrelativistic nondegenerate gas, zero photon chemical potential, charge neutrality, negligible proton-electron mass correction in the translational prefactor, and the stated degeneracy-factor approximation.