(a) In the early universe electrons, protons and neutral hydrogen are in thermal equilibrium and interact via,

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

The non-relativistic number density of particles in thermal equlibrium is

$$n_{i}=g_{i}\left(\frac{2 \pi m_{i} k T}{h^{2}}\right)^{\frac{3}{2}} \exp \left(\frac{\mu_{i}-m_{i} c^{2}}{k T}\right)$$

where, for each species $i, g_{i}$ is the number of degrees of freedom, $m_{i}$ is its mass, and $\mu_{i}$ is its chemical potential. [You may assume $g_{e}=g_{p}=2$ and $g_{H}=4$.]

Stating any assumptions required, use these expressions to derive the Saha equation which governs the relative abundances of electrons, protons and hydrogen,

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

where $I$ is the binding energy of hydrogen, which should be defined.

(b) Naively, we might expect that the majority of electrons and protons combine to form neutral hydrogen once the temperature drops below the binding energy, i.e. $k T \lesssim I$. In fact recombination does not happen until a much lower temperature, when $k T \approx 0.03 I$. Briefly explain why this is.

[Hint: It may help to consider the relative abundances of particles in the early universe.]