Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 9B (Cosmology)

For bosons the occupation number is $n_{\mathbf{p}}=0,1,2,\dots$. Summing the geometric series for the single-mode grand canonical ensemble gives the Bose-Einstein distribution

For fermions the Pauli exclusion principle permits only $n_{\mathbf{p}}=0,1$. The two-term partition sum gives the Fermi-Dirac distribution

Write $n_b=n_e+n_H\simeq \eta n_{\gamma }$. Charge neutrality gives $n_p=n_e$, and thereforeSubstitution into the Saha ionization equation yieldsUsing the given photon density,

The baryon-to-photon ratio is extremely small. Even when $k_{B}T\ll I$, the high-energy tail of the enormous photon population contains enough ionizing photons per baryon to delay cosmological recombination. The exponential factor must become large enough to overcome both $\eta \sim 10^{-9}$ and the phase-space factor, which occurs near $k_{B}T\simeq 0.3\mathrm{eV}$ rather than near $I$.