Prior to a time $t \sim 100,000$ years, the Universe was filled with a gas of photons and non-relativistic free electrons and protons maintained in equilibrium by Thomson scattering. At around $t \sim 400,000$ years, the protons and electrons began combining to form neutral hydrogen,

$$p+e^{-} \leftrightarrow H+\gamma$$

[You may assume that the equilibrium number density of a non-relativistic species $\left(k T \ll m c^{2}\right)$ is given by

$$n=g_{s}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left(\left(\mu-m c^{2}\right) / k T\right)$$

while the photon number density is

$$n_{\gamma}=16 \pi \zeta(3)\left(\frac{k T}{h c}\right)^{3}, \quad(\zeta(3) \approx 1.20 \ldots)$$

Deduce Saha's equation for the recombination process $(*)$ stating clearly your assumptions and the steps made in the calculation,

$$\frac{n_{e}^{2}}{n_{H}}=\left(\frac{2 \pi m_{e} k T}{h^{2}}\right)^{3 / 2} \exp (-I / k T)$$

where $I$ is the ionization energy of hydrogen.

Consider now the fractional ionization $X_{e}=n_{e} / n_{B}$ where $n_{B}=n_{p}+n_{H}=\eta n_{\gamma}$ is the baryon number of the Universe and $\eta$ is the baryon to photon ratio. Find an expression for the ratio

$$\left(1-X_{e}\right) / X_{e}^{2}$$

in terms only of $k T$ and constants such as $\eta$ and $I$.

Suggest a reason why neutral hydrogen forms at a temperature $k T \approx 0.3 \mathrm{eV}$ which is much lower than the hydrogen ionization temperature $k T=I \approx 13 \mathrm{eV}$.