Big Bang nucleosynthesis

Big Bang nucleosynthesis formed the light nuclei during the first minutes of cosmic expansion after weak freeze-out and the deuterium bottleneck.

With negligible lepton chemical potentials, weak equilibrium gives $n_n/n_p\simeq e^{-Q/(k_{B}T)}$, where $Q=(m_n-m_p)c^2$.

Neglecting chemical potentials, the equilibrium abundance ratio is $n_n/n_p\simeq \exp [-(m_n-m_p)c^2/(k_{B}T)]$. Weak freeze-out fixes an approximate starting ratio, which is then reduced by free-neutron decay after freeze-out before most remaining neutrons bind into helium.

Weak neutron--proton conversion freezes out when its interaction rate falls below the Hubble expansion rate.

If an interaction-to-expansion ratio scales as $(T/\kappa {)}^3$, increasing $\kappa$ raises the temperature at which the ratio drops through one.

Light nuclei cannot accumulate until cooling makes deuterium sufficiently resistant to photodissociation, delaying nucleosynthesis after weak freeze-out.

For chemical equilibrium $D\leftrightarrow n+p$, Maxwell--Boltzmann densities and ${\mu }_D={\mu }_n+{\mu }_p$ give, after taking $m_n\simeq m_p$ and $m_D\simeq 2m_p$,Writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$, and $n_{\gamma }=16\pi \zeta (3)(k_{B}T{)}^3/(hc{)}^3$ givesThe very small baryon-to-photon ratio suppresses deuterium until $T$ is far below the binding-energy scale.

Increasing the baryon-to-photon ratio raises the equilibrium deuterium abundance at fixed temperature, so the deuterium bottleneck ends earlier. Fewer neutrons decay before nuclear burning, and neutron-limited helium synthesis produces a larger primordial helium mass fraction if weak freeze-out is unchanged.

Between weak freeze-out and deuterium formation, beta decay lowers the surviving free-neutron 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.