Big-Bang Nucleosynthesis

The lightest nuclei were made not in stars but in the first minutes of the expanding universe, while it was hot and dense enough to sustain nuclear reactions everywhere at once. Big-bang nucleosynthesis is the best-tested application of nuclear physics to cosmology: from a single parameter, the ratio of baryons to photons, it predicts the primordial abundances of deuterium, helium-3, helium-4, and lithium-7, and the predictions match observation across nine orders of magnitude in abundance. The physics is the same barrier-suppressed fusion of the previous lessons, run in reverse of a star — a plasma that starts hot and cools, rather than one that contracts and heats.¹

Thermal history and the weak equilibrium

At times earlier than one second, temperatures exceeded $1 0^{10} K$ ( $k T ≳ 1 MeV$ ) and the universe held a thermal bath of photons, electron-positron pairs, neutrinos, and a trace of nucleons — about one baryon per $1 0^{9}$ photons. Neutrons and protons interconverted freely through the weak interaction,

n + ν_{e} ⇌ p + e^{-}, n + e^{+} ⇌ p + \overset{ν}{ˉ}_{e}, n ⇌ p + e^{-} + \overset{ν}{ˉ}_{e} .

While these reactions are fast compared with the expansion, the two species stay in chemical equilibrium, and their number ratio follows the Boltzmann factor of the neutron-proton mass difference $Δ m c^{2} = 1.293 MeV$ ,

\frac{n _{n}}{n _{p}} = exp (- \frac{Δ m c ^{2}}{k T}) .

At $k T ≫ 1 MeV$ the ratio is near unity; as the universe cools it drops, because converting a proton to the heavier neutron costs energy the thermal bath increasingly lacks.

Freeze-out of the neutron fraction

The weak reactions cannot maintain equilibrium forever. Their rate per nucleon scales steeply with temperature, $Γ_{weak} \propto T^{5}$ , while the Hubble expansion rate in the radiation era scales as $H \propto T^{2}$ . The two cross when the reactions can no longer keep up with the expansion, at the freeze-out temperature

Γ_{weak} (T_{f}) \approx H (T_{f}) ⟹ k T_{f} \approx 0.8 MeV (t \approx 1 s) .

Below $T_{f}$ the interconversion effectively stops and the neutron-to-proton ratio is frozen at its equilibrium value there,

\frac{n _{n}}{n _{p}}_{f} = exp (- \frac{1.293}{0.8}) \approx \frac{1}{6} .

The ratio is not quite constant afterward: free neutrons $β^{-}$ -decay with mean life $τ_{n} = 879 s$ , so over the few minutes until nucleosynthesis begins the ratio falls further, to about $1/7$ . That the neutron happens to live several minutes — long compared with the one-second freeze-out but short compared with cosmic times — is what leaves any neutrons at all to build nuclei.¹

The neutron-to-proton ratio tracks its weak-equilibrium value until the weak rate falls below the expansion rate near 0.8 MeV; it then freezes near one in six and drifts down to one in seven by neutron decay before nucleosynthesis.

The deuterium bottleneck

Building any nucleus starts with deuterium, formed by radiative capture

p + n \to^{2} H + γ, B_{d} = 2.22 MeV .

Deuterium is only weakly bound, and the photon bath outnumbers baryons by a billion to one. Even after the mean photon energy drops below $B_{d}$ , the high-energy tail of the Planck spectrum still contains enough photons above $2.22 MeV$ to photodisintegrate every deuteron as fast as it forms. Nucleosynthesis cannot proceed until the temperature falls far enough that even the tail is depleted, which because of the huge photon-to-baryon ratio happens well below $B_{d} / k$ , at

k T_{bbn} \approx 0.07 MeV (t \approx 3 min) .

This delay is the deuterium bottleneck. Its length is set by the baryon-to-photon ratio: fewer baryons per photon means a longer wait, and the wait is what lets a fraction of the neutrons decay before they are locked into nuclei, lowering the final helium yield. Once deuterium survives, it does so suddenly, and the pent-up reactions run to completion in minutes.

The reaction network and the helium yield

When the bottleneck breaks, a fast network converts the surviving neutrons into helium-4, the most tightly bound light nucleus. Deuterium fuses to helium-3 and tritium, which fuse onward to helium-4:

^{2} H +^{2} H \to^{3} He + n,^{3} H + p;^{2} H +^{3} H \to^{4} He + n;^{2} H +^{3} He \to^{4} He + p .

Because $^{4} He$ is a deep energy sink and there are no stable nuclei at mass $5$ or mass $8$ , the chain piles up at helium-4 and nearly stops. Small residues of $^{2} H$ and $^{3} He$ survive unburned, and a trace of $^{7} Li$ forms through $^{4} He (^{3} H, γ)^{7} Li$ and $^{4} He (^{3} He, γ)^{7} Be (e^{-}, ν)^{7} Li$ . Nothing heavier is made in appreciable quantity; the heavy elements wait for stars.

The primordial network starts from free protons and neutrons, forms deuterium as the bottleneck breaks, and funnels almost every neutron into helium-4, leaving trace deuterium, helium-3, and lithium-7 and stopping at the mass-8 gap.

Counting is straightforward once the network is understood to sweep essentially all neutrons into helium. Each helium-4 takes two neutrons and two protons, so if the ratio at onset is $n_{n} / n_{p} = r$ , the helium mass fraction is

Y_{p} = \frac{4 n _{He}}{n _{nucleons}} = \frac{2 n _{n}}{n _{n} + n _{p}} = \frac{2 r}{1 + r} .

With $r \approx 1/7$ this gives $Y_{p} \approx 0.25$ : about a quarter of the baryonic mass of the universe emerges as helium, and nearly all the rest as hydrogen. The observed primordial helium fraction, $Y_{p} = 0.247$ , matches this to within the measurement error, and the prediction depends only on the neutron lifetime, the freeze-out temperature, and the length of the deuterium bottleneck — all fixed by known nuclear and weak physics.¹

Abundances as a baryometer

The trace species are more sensitive to conditions than helium is, and their yields depend on the single free parameter of the theory, the baryon-to-photon ratio $η = n_{b} / n_{γ}$ . Deuterium is the sharpest probe: a higher baryon density means more efficient burning of deuterium into helium, so the surviving deuterium falls steeply as $η$ rises. Helium-3 falls gently, lithium-7 traces a valley with a minimum, and helium-4 rises only logarithmically. The measured abundances must all agree at one value of $η$ , and they do, near

η \approx 6 \times 1 0^{- 10}, (\frac{D}{H}) \approx 2.5 \times 1 0^{- 5} .

This value, derived purely from nuclear abundances in the first three minutes, agrees with the entirely independent baryon density inferred from the cosmic microwave background acoustic peaks — a concordance that is among the strongest evidence for the hot big bang. One residual tension remains: the predicted $^{7} Li$ abundance is about three times the value observed in old stars, the lithium problem, still unresolved.¹

Predicted primordial abundances against the baryon-to-photon ratio: helium-4 rises weakly, deuterium and helium-3 fall, lithium-7 dips and rises, and the vertical band marks the single baryon density where all measurements agree.

Timeline

The whole episode occupies about twenty minutes, ordered by temperature.

$t \approx 1 s$ , $k T \approx 1 MeV$ : weak reactions freeze the neutron-to-proton ratio near $1/6$ .
$1 s \to 3 min$ : free neutrons decay, lowering the ratio toward $1/7$ , while the deuterium bottleneck holds nucleosynthesis back.
$t \approx 3 min$ , $k T \approx 70 keV$ : deuterium survives, the network ignites, and almost all neutrons are bound into helium-4 within minutes.
$t ≳ 20 min$ : expansion has cooled and thinned the plasma below the threshold for further reactions, and the abundances freeze out for good.

The nucleosynthesis timeline: neutron-proton freeze-out at one second, a wait through the deuterium bottleneck governed by neutron decay, and the burst of helium formation near three minutes, after which the abundances are fixed.

Big-bang nucleosynthesis closes the arc of the module: the same weak process that sets the neutron fraction here starts hydrogen burning in the proton-proton chain, and the same Coulomb-barrier tunneling that runs stellar fusion sets the rates of the primordial network. The theory turns nuclear cross sections and the neutron lifetime into a measurement of the baryon content of the universe.

Krane, Introductory Nuclear Physics, Ch. 19 (cosmological context): the weak equilibrium of neutrons and protons, the freeze-out of $n / p$ near $1/6$ at $k T_{f} \approx 0.8 MeV$ , the deuterium bottleneck, the helium yield $Y_{p} = 2 r / (1 + r)$ , and the light-element abundances as a function of the baryon-to-photon ratio. The primordial-abundance and baryon-density review is maintained by the Particle Data Group, https://pdg.lbl.gov/; the neutron lifetime $τ_{n} = 879 s$ and $Δ m c^{2} = 1.293 MeV$ are the CODATA/NIST values, https://physics.nist.gov/cuu/Constants/. See also Tipler & Llewellyn, Modern Physics, §13-7. ↩ ↩² ↩³ ↩⁴