Primordial helium mass fraction

If almost all neutrons enter helium-4 and $r=n_n/n_p$, its primordial mass fraction is $Y_p\simeq 2r/(1+r)$.

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.

Chemical equilibrium of $n+{\nu }_e\leftrightarrow p+e^{-}$, with negligible electron chemical potential, gives the stated neutron-to-proton ratio. Positive electron-neutrino degeneracy reduces the neutron fraction, whereas negative degeneracy increases it. The primordial helium mass fraction then responds through neutron-limited helium synthesis while protons outnumber neutrons; a sufficiently neutron-rich mixture instead becomes proton limited, so that trend must not be extrapolated indefinitely. Extra neutrino energy density also accelerates cosmic expansion, changing weak freeze-out and the time available for neutron decay. These effects can compete; a quantitative Big Bang nucleosynthesis calculation must account for the altered reaction rates as well as the altered expansion rate.

When protons outnumber neutrons, helium-4 production is neutron-limited and produces about one helium nucleus per two neutrons.

Increasing primordial helium at the expense of hydrogen reduces fuel for long-lived main-sequence hydrogen burning and changes subsequent stellar evolution.