The number density of a non-relativistic species in thermal equilibrium is given by

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

Suppose that thermal and chemical equilibrium is maintained between protons p (mass $m_{\mathrm{p}}$, degeneracy $g_{s}=2$ ), neutrons $\mathrm{n}$ (mass $m_{\mathrm{n}} \approx m_{\mathrm{p}}$, degeneracy $g_{s}=2$ ) and helium-4 nuclei ${ }^{4} \mathrm{He}\left(\right.$ mass $m_{\mathrm{He}} \approx 4 m_{\mathrm{p}}$, degeneracy $g_{s}=1$ ) via the interaction

$$2 \mathrm{p}+2 \mathrm{n} \leftrightarrow{ }^{4} \mathrm{He}+\gamma$$

where you may assume the photons $\gamma$ have zero chemical potential $\mu_{\gamma}=0$. Given that the binding energy of helium-4 obeys $B_{\mathrm{He}} / c^{2} \equiv 2 m_{\mathrm{p}}+2 n_{\mathrm{n}}-m_{\mathrm{He}} \ll m_{\mathrm{He}}$, show that the ratio of the number densities can be written as

$$\frac{n_{\mathrm{p}}^{2} n_{\mathrm{n}}^{2}}{n_{\mathrm{He}}}=2\left(\frac{2 \pi m_{\mathrm{p}} k T}{h^{2}}\right)^{9 / 2} \exp \left(-B_{\mathrm{He}} / k T\right)$$

Explain briefly why the baryon-to-photon ratio $\eta \equiv n_{B} / n_{\gamma}$ remains constant during the expansion of the universe, where $n_{B} \approx n_{\mathrm{p}}+n_{\mathrm{n}}+4 n_{\mathrm{He}}$ and $n_{\gamma} \approx\left(16 \pi /(h c)^{3}\right)(k T)^{3}$.

By considering the fractional densities $X_{i} \equiv n_{i} / n_{B}$ of the species $i$, re-express the ratio ( $\uparrow$ ) in the form

$$\frac{X_{\mathrm{p}}^{2} X_{\mathrm{n}}^{2}}{X_{\mathrm{He}}}=\eta^{-3} \frac{1}{32}\left(\frac{\pi}{2}\right)^{3 / 2}\left(\frac{m_{\mathrm{p}} c^{2}}{k T}\right)^{9 / 2} \exp \left(-B_{\mathrm{He}} / k T\right)$$

Given that $B_{\mathrm{He}} \approx 30 \mathrm{MeV}$, verify (very approximately) that this ratio approaches unity when $k T \approx 0.3 \mathrm{MeV}$. In reality, helium-4 is not formed until after deuterium production at a considerably lower temperature. Explain briefly the reason for this delay.