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.

For the standard Big Bang nucleosynthesis parameter branch, a larger baryon-to-photon ratio increases the efficiency of nuclear reactions burning deuterium into heavier nuclei, reducing residual primordial D/H. Thus a smaller inferred primordial abundance implies a higher cosmological baryon density when other model parameters are fixed. Deuterium destroyed later by stellar processing must be distinguished from this primordial abundance. Coupling the inference to the cluster baryon-fraction estimate of matter density raises ${\Omega }_m$ at fixed cluster baryon fraction.

For the measured mean cosmic microwave background temperature, the baryon-to-photon ratio and cosmological baryon density obey the displayed conversion. Big Bang nucleosynthesis, especially deuterium, measures this density during the first minutes, while acoustic peak ratios measure it near recombination. Agreement tests the thermal history and approximate conservation of baryon number and photon entropy between these epochs. It is not a claim that every observed light-element abundance has no residual tension.

At the cosmological baryon density favored by deuterium and the cosmic microwave background, standard Big Bang nucleosynthesis predicts more lithium-7 than is directly inferred from the Spite plateau, typically by a factor of a few. Stellar depletion, atmosphere systematics and possible new microscopic physics must be distinguished before interpreting this discrepancy as a failure of the cosmological framework.

Primordial abundances require extrapolation to material minimally processed by stars. Helium is inferred from hydrogen and helium emission in low-metallicity ionized regions, with ionization and temperature corrections. Deuterium is inferred from isotope-shifted absorption in low-metallicity quasar systems. Lithium is measured in old metal-poor stellar atmospheres and requires a correction, or a bound, for stellar depletion. These observables probe different astrophysical environments and systematic errors.

The Spite plateau is the approximately constant lithium abundance in warm, metal-poor halo stars. It provides an important estimate of primordial lithium only after accounting for stellar atmosphere physics and possible transport and nuclear destruction of lithium within the stars.

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.

Use natural units and a positive nuclear binding energy ${\epsilon }_D=m_n+m_p-m_D$. Chemical equilibrium of $n+p\leftrightarrow D+\gamma$ requires ${\mu }_D={\mu }_n+{\mu }_p$. The nonrelativistic Maxwell--Boltzmann number density givesHere $g_D=3$ and $g_n=g_p=2$ count the spin states. Taking $m_n\simeq m_p\simeq m_N$ and $m_D\simeq 2m_N$, then writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$ and using the photon number density yields the displayed abundance. The coefficient is approximately $8.14$. Omitting the spin-state ratio $3/4$ gives the rougher prefactor $16\zeta (3)/\sqrt{\pi }$. A convention $B=m_D-m_n-m_p<0$ instead writes the exponential as $e^{-B/T}$. The small baryon-to-photon ratio suppresses stable deuterium until the temperature is well below the binding scale, producing the deuterium bottleneck.

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 $Y_p\simeq 2r/(1+r)$.

At fixed cosmological weak freeze-out temperature $T_f$ and fixed neutron survival factor, neutron-proton chemical equilibrium gives $r_f=e^{-Q/T_f}$ and neutron-limited helium synthesis gives $Y_p\propto 2r_f/(1+r_f)$. Differentiating yields the displayed sensitivity. A finite change $Q\mapsto (1+\alpha )Q$ gives $Y_p^{\prime }/Y_p=(1+e^{Q/T_f})/(1+e^{(1+\alpha )Q/T_f})$. Increasing the positive mass difference reduces helium production. Changes to the weak conversion rates, neutron lifetime or nuclear history must be supplied separately for a full abundance prediction.

After cosmological weak freeze-out, the surviving neutron fraction is $x_n=r_f(1+r_f{)}^{-1}{e}^{-\Delta t/{\tau }_n}$, where $r_f$ is the initial neutron-to-proton ratio and ${\tau }_n=t_{1/2}/\ln 2$ is the mean lifetime. Baryon number is conserved because a decaying neutron becomes a proton. If almost all surviving neutrons enter helium-4, neutron-limited helium synthesis gives the displayed primordial helium mass fraction. Increasing the half-life increases survival. If the longer lifetime also represents uniformly weaker weak interactions, it raises the temperature of cosmological weak freeze-out and the initial neutron fraction, reinforcing the helium increase. That last conclusion requires specifying how the weak rates change, rather than treating lifetime alone as a complete microscopic model.

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.

During radiation domination, the weak conversion rate scales as $T^5$ whereas the Hubble parameter scales as $\sqrt{G{g}_{\ast }}{T}^2$. Their crossing fixes the cosmological weak freeze-out temperature. A faster expansion raises it, leaving a larger neutron-to-proton ratio, and shortens the interval for free-neutron decay after freeze-out. Both effects increase the helium-4 mass fraction if the extra component does not independently change reaction rates or inject decay products.

If the Friedmann equation remains approximately $H^2=8\pi G\rho /3$, radiation domination and fixed weak microphysics give $H\propto \sqrt{G}{T}^2$ and a neutron-proton conversion rate proportional to $T^5$. Cosmological weak freeze-out then occurs at $T_f\propto G^{1/6}$. Increasing $G$ during this era raises the neutron-to-proton ratio at freeze-out and shortens the time for free-neutron decay after freeze-out. Both increase the primordial helium mass fraction when neutron-limited helium synthesis remains efficient. Arbitrary variation of $G$ over time requires its history and the full gravitational field equations; this scaling is a controlled background approximation, rather than a complete model of gravity with arbitrary time dependence.

A moderately enhanced Hubble parameter raises the primordial helium mass fraction through earlier cosmological weak freeze-out and less free-neutron decay after freeze-out, provided nuclear burning still incorporates nearly all surviving neutrons into helium-4. This provision matters: if expansion carries the gas through the nuclear-burning temperature range faster than the reaction times, neutron-limited helium synthesis no longer applies. More neutrons surviving the bottleneck alone does not prove more final helium in that limit.

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