Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 9D (Cosmology)

Chemical equilibrium gives ${\mu }_D={\mu }_p+{\mu }_n$. Substitution of the three Maxwell--Boltzmann densities therefore cancels the chemical potentials and leaves the rest-mass factorUsing $g_p=g_n=2$, $g_D=4$, $m_n\simeq m_p$, and $m_D\simeq 2m_p$, the translational prefactor isHence

Because $X_i=n_i/n_B$,Using $n_B=\eta n_{\gamma }$ and the given photon density in the deuterium equilibrium abundance formula givesAlthough the exponential favors deuterium once $k_{B}T<B_D$, the very small factor $\eta$ expresses the enormous excess of photons. The high-energy tail therefore photodissociates newly formed deuterium until the temperature is well below $B_D/k_B$; this is the deuterium bottleneck.

A larger $\eta$ increases the deuterium fraction at fixed temperature, so the bottleneck ends at a higher temperature and earlier time. With weak decoupling unchanged, fewer free neutrons then beta-decay before nuclear reactions bind them. Since primordial helium production is neutron-limited, the baryon-density effect on primordial helium makes $Y_{\mathrm{He}}$ larger than the standard value.