Big Bang Nucleosynthesis

Setting the stage: the universe at one second

Run the cosmological clock back to t = 1 second after the Big Bang. The universe is a uniform plasma of photons, neutrinos, electrons and positrons, with a much smaller concentration of free protons and neutrons drifting in the radiation soup. The temperature is about 10¹⁰ K — high enough that the average photon carries roughly 1 MeV of energy, comparable to the rest mass of the electron. Densities are extreme but the expansion is fast: the Hubble rate is about 1 s⁻¹, so the local environment changes on the same timescale as a single nuclear reaction. Whatever happens next has only a few minutes to finish.

At t = 1 s the weak interactions that interconvert protons and neutrons have just frozen out. While the temperature was high, the reactions n + ν_e ↔ p + e⁻ and n + e⁺ ↔ p + ν̄_e kept the neutron-to-proton ratio in equilibrium, with a Boltzmann factor that depends on the small mass difference Δm = 1.293 MeV between the neutron and the proton. As the universe cooled the rates fell faster than the expansion rate, and at kT ≈ 0.8 MeV the n/p ratio froze at about 1/6. From that moment onward, free neutrons were on a clock: their 880 s lifetime gave them only a few half-lives to find a proton before they decayed into hydrogen and an antineutrino.

The next two-and-a-bit minutes are the entire arena of Big Bang nucleosynthesis. The decisive parameter is η, the baryon-to-photon ratio — about 6×10⁻¹⁰ in the universe we live in. Photons outnumber baryons by a factor of a billion, and the high-energy tail of the photon spectrum is what sets the timing of every nuclear reaction.

The deuterium bottleneck

Every path to a heavier nucleus runs through deuterium. The reaction p + n → ²H + γ proceeds quickly, but its reverse — photodissociation by an energetic photon — is even faster as long as the photon bath is hot. Deuterium's binding energy is only 2.22 MeV, weak by nuclear standards. A photon with energy above 2.22 MeV can blow it apart, and although the average CMB photon is far cooler than that, the high-energy Wien tail of the blackbody spectrum is long enough that there are plenty of dissociating photons even when kT is well below 2.22 MeV. With η = 6×10⁻¹⁰ and a Planck distribution, the photons in the tail outnumber baryons enough to keep deuterium broken until the temperature falls below roughly 7×10⁸ K — three minutes into the universe.

Until that moment the entire fusion network is locked. Protons and neutrons collide, briefly form deuterium, get blasted apart and start over. As soon as the bottleneck breaks the network ignites all at once. Within a few additional minutes deuterium fuses to ³He and ³H (tritium), tritium decays to ³He, ³He combines with deuterium to make ⁴He, and ⁴He pairs with ³H or ³He to make ⁷Li and ⁷Be (the ⁷Be later electron-captures to ⁷Li in stars). Helium-4 is by far the most tightly bound of these light nuclei (28.3 MeV), so the network preferentially funnels material into ⁴He.

The endpoint of all this nuclear book-balancing is dictated by two numbers: the n/p ratio at the time of bottleneck breaking, and the requirement that essentially every neutron ends up bound in ⁴He. By t ≈ 3 minutes the n/p ratio has fallen from 1/6 to about 1/7 because of free neutron decay during the wait. With one neutron for every seven protons, every neutron pairs with a proton inside a ⁴He nucleus that contains two of each. So out of every 16 nucleons (2 neutrons and 14 protons), 4 end up bound in one ⁴He nucleus and the remaining 12 are free protons. Mass fraction of ⁴He: 4/16 = 25%. Mass fraction of H: 12/16 = 75%. The arithmetic is essentially that simple.

Final abundances and what they encode

Once the temperature falls below 3×10⁸ K (about 20 minutes in) the Coulomb barriers shut down all further fusion and the abundances freeze. The numbers that emerge from a careful BBN calculation, given Ω_b h² = 0.022 (the value pinned down by Planck), are:

Species	Mass / number ratio	What measures it observationally	Sensitivity to η
¹H	X_p ≈ 0.75 (mass fraction)	By difference; emission lines in HII regions	Weak
⁴He	Y_p ≈ 0.2453 (mass fraction)	Recombination lines in metal-poor HII regions	Weak (logarithmic)
²H (D)	D/H ≈ 2.5×10⁻⁵	Lyman-α absorption in z ≈ 3 quasar sightlines	Strong (η⁻¹·⁶)
³He	³He/H ≈ 1×10⁻⁵	Hyperfine 8.7 GHz line in galactic HII regions	Moderate
⁷Li	⁷Li/H ≈ 5×10⁻¹⁰	Spite plateau in metal-poor halo stars	Strong but non-monotonic
⁶Li	⁶Li/H ≲ 10⁻¹⁴	Difficult; only upper limits	Strong, but post-BBN routes dominate

Helium-4 is a "baryometer" of low precision — its abundance changes only logarithmically with η. Deuterium is the precision baryometer: D/H scales roughly as η⁻¹·⁶, so a 10% measurement of D/H pins down η to about 6%. The ⁴He measurement instead constrains the relativistic energy density at BBN, parameterised as N_eff. With three Standard Model neutrinos, N_eff = 3.046 (slightly above 3 because of incomplete neutrino decoupling). BBN gives N_eff = 2.9 ± 0.3, consistent with three flavours and tightly bounding any extra "dark" radiation.

Worked example: deriving η from D/H

The cleanest empirical input to BBN is the deuterium-to-hydrogen ratio measured in the absorption spectra of distant quasars whose light passes through nearly pristine, high-redshift hydrogen clouds. The current best value is

D/H = (2.547 ± 0.025) × 10⁻⁵

derived from a sample of seven well-resolved damped Lyman-α systems at z ≈ 2.5–3.0. To turn that into η we use the BBN scaling that comes out of the full reaction-network calculation:

D/H ≈ 2.51 × 10⁻⁵ × (η₁₀ / 6.10)⁻¹·⁶
       where η₁₀ = η × 10¹⁰

Solving for η₁₀ given the measured D/H:

(η₁₀ / 6.10)⁻¹·⁶ = 2.547 / 2.51 = 1.0147
 η₁₀ / 6.10       = 1.0147^(−1/1.6) = 0.9909
 η₁₀              = 6.044
 η                = 6.04 × 10⁻¹⁰

Compare this to the value of η inferred from the heights of the acoustic peaks in the Planck 2018 CMB power spectrum:

η_CMB = (6.13 ± 0.04) × 10⁻¹⁰

The two numbers agree to about 1.5%, even though they probe the universe at epochs separated by a factor of 10¹¹ in time (3 minutes vs 380,000 years). That agreement is not a small feat: it requires that the laws of nuclear physics, the Standard Model couplings, and the equation of state of the early universe all behave as we expect them to over an enormous span of cosmic history.

Variants and extensions

• Inhomogeneous BBN. Models in which the early universe had baryon-density fluctuations on scales larger than the diffusion length predict different neutron-to-proton segregation and altered abundances. Tight observational constraints rule out strong inhomogeneity, but mild inhomogeneity is still viable and is often invoked as a possible escape from the lithium problem.
• BBN with extra relativistic species. "Dark radiation" beyond the three Standard Model neutrinos increases N_eff, raises Y_p, and tightens the time available before bottleneck breaking. Sterile neutrinos, axion-like particles and ultra-light dark sectors are all constrained by the joint BBN + CMB fit.
• BBN with varying fundamental constants. Time-varying fine-structure constant α or Newton's G would change Coulomb barriers and the expansion rate. The agreement of BBN with present-day nuclear physics limits |Δα/α| to about 10⁻² over the past 13.8 Gyr.
• BBN with decaying particles. Long-lived massive particles that decay during or just after BBN can dissociate light nuclei and rebuild the abundances. Such scenarios are sometimes invoked to dial down ⁷Li without disturbing D and ⁴He.
• Asymmetric DM and lepton asymmetry. A non-zero electron-neutrino chemical potential ξ_e would shift the n/p equilibrium and change Y_p. BBN bounds |ξ_e| ≲ 0.05, far tighter than any laboratory limit.

Where Big Bang nucleosynthesis shows up

• Quasar absorption spectroscopy. The Keck and VLT high-resolution echelle spectrographs (HIRES, UVES) measure the D I Lyman-α 1215.34 Å line offset by 82 km/s from H I 1215.67 Å in damped Lyman-α absorbers. Resolving the doublet at S/N ≈ 100 yields D/H to 1% precision.
• HII region helium spectroscopy. The He I 4471 Å, 5876 Å and 6678 Å recombination lines in metal-poor extragalactic HII regions (e.g. I Zw 18, SBS 0335-052) give Y_p ≈ 0.2453 ± 0.0034 once corrected for stellar contamination and collisional excitation.
• Spite plateau in halo stars. The Li I 6707.8 Å absorption line in metal-poor F and G dwarfs ([Fe/H] < −1.5) is flat at log(Li/H) ≈ −9.85, the basis of the lithium problem. Modern surveys (e.g. SAGA database) now contain > 1000 plateau stars.
• Planck CMB cross-check. Planck 2018's measurement of the second-to-first acoustic peak ratio fixes Ω_b h² = 0.0224 ± 0.0001 — a baryon density that, plugged into BBN codes such as PArthENoPE or AlterBBN, predicts Y_p and D/H consistent with all measurements except ⁷Li.
• Solar-system meteorite isotopes. Refractory inclusions in primitive chondrites carry deuterium signatures that, after correction for chemical fractionation in the protoplanetary disk, are consistent with the BBN value, anchoring D/H in the local interstellar medium to the cosmic primordial value.

Why the periodic table waits for stars

The remarkable feature of BBN is that it gets only as far as ⁷Li. The reason is the topology of stable nuclei. Beyond mass 4, nuclear stability resumes only at mass 6 (⁶Li) and mass 7 (⁷Li), with no stable nucleus at mass 5 or mass 8. ⁵Li and ⁵He both decay in less than 10⁻²¹ seconds; ⁸Be lives 10⁻¹⁶ seconds. To get to carbon-12 the universe needs the triple-alpha process, in which three ⁴He nuclei meet within ⁸Be's vanishing lifetime, requiring densities and confinement times found only in stellar cores. The Big Bang plasma was dense enough at t ≈ 1 s but too hot, and cool enough at t ≈ 3 min but too dilute. The mass gaps therefore guarantee that the universe emerges from BBN with essentially no carbon, no oxygen, no iron — a chemically empty cosmos waiting 100 million years for the first generation of stars to begin building the rest of the periodic table.

The lithium problem in detail

Of the four BBN observables, three (D, ⁴He, ³He) match the CMB-anchored prediction to better than the measurement uncertainty. The fourth, ⁷Li, is too low by a factor of about three. Standard BBN with η_CMB = 6.13×10⁻¹⁰ predicts ⁷Li/H ≈ 5×10⁻¹⁰, while the Spite plateau gives ⁷Li/H ≈ 1.6×10⁻¹⁰. The discrepancy is more than 4σ given current measurement precisions.

Three classes of explanation remain in play. Astrophysical: the surfaces of metal-poor halo stars may have depleted their initial lithium by atmospheric mixing, gravitational settling or rotational diffusion. Models can recover the BBN value but require fine-tuning of stellar evolution parameters. Nuclear: uncertainties in the cross sections of ³He(α,γ)⁷Be — the dominant production channel for ⁷Be that later becomes ⁷Li — could be larger than quoted, but recent measurements at LUNA have tightened the rate to ±5%. New physics: particles that decay during BBN can dissociate ⁷Li or alter its formation rate. None of these solutions is universally accepted, and the lithium problem remains the one outstanding tension in standard BBN.

Common pitfalls

• Confusing mass fraction with number fraction. Y_p ≈ 0.245 is the mass fraction of ⁴He. The number fraction (atoms of ⁴He per atom of total baryon) is roughly Y_p / 4 ≈ 0.06, since each ⁴He has four nucleons. Reports that mix the two by a factor of four are common in popular sources.
• Quoting "neutron half-life" when the relevant quantity is the mean lifetime. The neutron's mean lifetime τ_n ≈ 880 s sets the BBN clock; the half-life is τ_n × ln 2 ≈ 610 s. Using the wrong one shifts ⁴He predictions by a few tenths of a percent.
• Assuming BBN proves there are exactly three neutrino flavours. BBN bounds N_eff to within ~0.3 of three; it is consistent with three flavours but does not exclude small extra contributions, especially partial ones from new species that decoupled before BBN.
• Calling the cosmic helium "primordial" without a metallicity correction. Today's universe is about 28% helium by mass — a few percent above the BBN value because stars produce helium too. Y_p must be extracted by extrapolating to zero metallicity in HII region observations.
• Treating the lithium problem as solved. Stellar atmospheric depletion is not yet conclusively demonstrated, and new-physics solutions are not yet conclusively excluded. The discrepancy is real and currently active research.