(a) Around $t \approx 1 \mathrm{~s}$ after the big bang $(k T \approx 1 \mathrm{MeV})$, neutrons and protons are kept in equilibrium by weak interactions such as

$$\tag{*} n+\nu_{e} \leftrightarrow p+e^{-}$$

Show that, in equilibrium, the neutron-to-proton ratio is given by

$$\frac{n_{n}}{n_{p}} \approx e^{-Q / k T}$$

where $Q=\left(m_{n}-m_{p}\right) c^{2}=1.29 \mathrm{MeV}$ corresponds to the mass difference between the neutron and the proton. Explain briefly why we can neglect the difference $\mu_{n}-\mu_{p}$ in the chemical potentials.

(b) The ratio of the weak interaction rate $\Gamma_{W} \propto T^{5}$ which maintains (*) to the Hubble expansion rate $H \propto T^{2}$ is given by

$$\tag{†} \frac{\Gamma_{W}}{H} \approx\left(\frac{k T}{0.8 \mathrm{MeV}}\right)^{3}$$

Explain why the neutron-to-proton ratio effectively "freezes out" once $k T<0.8 \mathrm{MeV}$, except for some slow neutron decay. Also explain why almost all neutrons are subsequently captured in ${ }^{4} \mathrm{He}$; estimate the value of the relative mass density $Y_{^{4} \mathrm{He}}=\rho_{^{4}\mathrm{He}} / \rho_{\mathrm{B}}$ (with $\rho_{\mathrm{B}}=\rho_{n}+\rho_{p}$ ) given a final ratio $n_{n} / n_{p} \approx 1 / 8$.

(c) Suppose instead that the weak interaction rate were very much weaker than that described by equation $(†)$. Describe the effect on the relative helium density $Y_{^{4} \mathrm{He}}$. Briefly discuss the wider implications of this primordial helium-to-hydrogen ratio on stellar lifetimes and life on earth.