Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 3 / v / a / Solution

{title2=$T_f\propto g_*^{1/6}$}

With a weak conversion time proportional to $T^{-5}$ and a radiation-era age proportional to $g_*^{-1/2}T^{-2}$, <cosmological weak freeze-out> occurs at $T_f\propto g_*^{1/6}$. Extra radiation therefore raises the freeze-out temperature and the <neutron-to-proton ratio> $r_f=e^{-Q/T_f}$. Before decay, the <primordial helium mass fraction> is $Y_f=2r_f/(1+r_f)$, giving $\delta Y_f/Y_f=Q\delta\log T_f/[T_f(1+r_f)]$. A shorter delay through the <deuterium bottleneck> also leaves more neutrons. After freeze-out, conserve baryon number and evolve the neutron fraction to obtain $Y_p\simeq Y_f e^{-(t_N-t_f)/\tau_n}$. A precise abundance requires the weak rates and nuclear-burning history, not merely the number of relativistic species.

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 Incoming links: Primordial helium response to extra relativistic species