The equilibrium number density of fermions at temperature $T$ is

$$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(\epsilon(p)-\mu) / k T]+1}$$

where $g_{s}$ is the spin degeneracy and $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$. For a non-relativistic gas with $p c \ll m c^{2}$ and $k T \ll m c^{2}-\mu$, show that the number density becomes

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

[You may assume that $\int_{0}^{\infty} d x x^{2} e^{-x^{2} / \alpha}=(\sqrt{\pi} / 4) \alpha^{3 / 2}$ for $\alpha>0$.]

Before recombination, equilibrium is maintained between neutral hydrogen, free electrons, protons and photons through the interaction

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

Using the non-relativistic number density $(*)$, deduce Saha's equation relating the electron and hydrogen number densities,

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

where $I=\left(m_{p}+m_{e}-m_{H}\right) c^{2}$ is the ionization energy of hydrogen. State clearly any assumptions you have made.