Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 14E (Cosmology)

The Bose--Einstein and Fermi--Dirac occupation factors areIn the stated nonrelativistic dilute regime, $E-\mu$ is dominated by $m{c}^2$, so the exponential is large and either denominator is asymptotic to $\exp ((E-\mu )/(k_{B}T))$. Integrating the resulting Maxwell--Boltzmann occupation over momentum states gives

For $E=m{c}^2+p^2/(2m)$, integration by parts or differentiation of the supplied Gaussian integral givesConsequently

For any nonrelativistic species $j$, part (a) givesChemical equilibrium and zero photon chemical potential implyTaking the appropriate ratios of the number-density formula, approximating the ion and atom masses as equal in the translational prefactors, and using the given degeneracies yieldsandThe factors are $g_e{g}_{H^{+}}/g_{H^0}=1$ and $g_e{g}_{H{e}^{+}}/g_{H{e}^0}=2$.

Baryon number and charge neutrality giveSince $Y_p=4n_{He}/n_B$,It follows thatThus $\alpha =1-Y_p$ and $\beta =Y_p/4$.

DefineUsing $n_e=n_B\mathcal{F}$, $n_{j^{+}}=X_{j^{+}}{n}_j$, and $n_{j^0}=(1-X_{j^{+}})n_j$, the two Saha relations become the closed systemwhereThese are two equations for the two ionization fractions.

When $Y_p\to 0$, $\mathcal{F}=X_{H^{+}}$, and the hydrogen equation reduces tothe usual hydrogen-only Saha equation.