Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 2

Chemical equilibrium for $e^{-}+p\leftrightarrow H+\gamma$ requires ${\mu }_e+{\mu }_p={\mu }_H$, since the photon chemical potential vanishes. Inserting the nonrelativistic Maxwell-Boltzmann distribution and using charge neutrality $n_p=n_e$ giveswhere the proton-to-hydrogen mass ratio and the stated degeneracy factors have been approximated by one. Thus

With $X_e=n_e/n_b$, charge neutrality and $n_b=n_p+n_H$ giveMultiplying part a by $n_b=\eta n_{\gamma }$ and using $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ yields the Saha ionization equationAlthough $T\ll B_H$ suppresses ionization of an individual atom, the baryon-to-photon ratio is only about $10^{-9}$. The enormous number of photons per baryon leaves enough photons in the high-energy thermal tail to ionize hydrogen until $T\simeq 0.3\mathrm{eV}$, far below $B_H$.

Photon decoupling occurs when the Thomson interaction rate falls below the expansion rate,At decoupling the universe is approximately matter dominated, soUsing $n_e=X_e\eta (2\zeta (3)/{\pi }^2)T^3$ in ${\Gamma }_{\gamma }=H$ givesFor $X_e\ll 1$, the Saha equation makes $X_e(T)$ exponentially steep through $e^{-B_H/(2T)}$. Power-law changes of $H_0$ or ${\Omega }_{m,0}$ therefore cause only a small shift in $T_{\mathrm{dec}}$, explaining why $T_{\mathrm{dec}}\simeq T_{\mathrm{rec}}$ is a good approximation.

At fixed $\eta$, the recombination temperature is determined by atomic physics and the Saha equation, not by today's CMB temperature. Under the stated approximation,Since $1+z_{\mathrm{dec}}=T_{\mathrm{dec}}/T_0$,and the same ratio holds for the large redshifts themselves to excellent accuracy. The comoving distance ${\chi }_{\mathrm{dec}}={\int }_0^{z_{\mathrm{dec}}}dz/H(z)$ can nevertheless be held fixed by adjusting the late-time expansion history, for example $H_0$, ${\Omega }_{m,0}$, spatial curvature, or dark-energy density and equation-of-state parameters.