 Primordial helium response to extra relativistic species

With a weak conversion time proportional to $T^{-5}$ and a radiation-era age proportional to $g_{\ast }^{-1/2}{T}^{-2}$, cosmological weak freeze-out occurs at $T_f\propto g_{\ast }^{1/6}$. 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.