---
title: Big-Bang Nucleosynthesis
module: Fusion and Nucleosynthesis
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  In the first three minutes the expanding universe forged the light elements. The weak
  interaction froze the neutron-to-proton ratio near one in six when the reaction rate fell
  below the expansion rate, and free-neutron decay lowered it to about one in seven before
  the deuterium bottleneck broke. Almost every surviving neutron ended in helium-4, fixing
  the primordial helium mass fraction near 0.25, with trace deuterium, helium-3, and
  lithium-7. The deuterium abundance measures the cosmic baryon density.
topics: [Fusion and Nucleosynthesis]
sources:
  - book: Krane
    ref: "Ch. 19 — Nuclear Physics and Cosmology; primordial nucleosynthesis and the early universe"
  - book: Tipler & Llewellyn
    ref: "Ch. 13 — Astrophysics and Cosmology; §13-7 The Early Universe and Nucleosynthesis"
draft: false
---

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.[^krane-bbn]

## Thermal history and the weak equilibrium

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

$$
n + \nu_e \rightleftharpoons p + e^-,
\qquad
n + e^+ \rightleftharpoons p + \bar\nu_e,
\qquad
n \rightleftharpoons p + e^- + \bar\nu_e.
$$

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 $\Delta m\,c^2 = 1.293\ \mathrm{MeV}$,

$$
\frac{n_n}{n_p} = \exp\!\left(-\frac{\Delta m\,c^2}{kT}\right).
$$

At $kT \gg 1\ \mathrm{MeV}$ 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, $\Gamma_{\text{weak}} \propto T^5$, 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**

$$
\Gamma_{\text{weak}}(T_f) \approx H(T_f)
\quad\Longrightarrow\quad
kT_f \approx 0.8\ \mathrm{MeV}\ \ (t \approx 1\ \mathrm{s}).
$$

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

$$
\left.\frac{n_n}{n_p}\right|_{f} = \exp\!\left(-\frac{1.293}{0.8}\right) \approx \frac{1}{6}.
$$

The ratio is not quite constant afterward: free neutrons $\beta^-$-decay with mean life
$\tau_n = 879\ \mathrm{s}$, 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.[^krane-bbn]

$$
% caption: The neutron-to-proton ratio tracks its weak-equilibrium value until the weak
% rate falls below the expansion rate near 0.8 MeV; it then freezes near one in six and
% drifts down to one in seven by neutron decay before nucleosynthesis.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {time (cooling)};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {n:p ratio};
  % equilibrium falling curve (dashed) then departing
  \draw[black, thick, dashed]
    (0.5,3.7) .. controls (1.6,2.9) and (2.5,2.1) .. (3.4,1.5);
  \node[black, anchor=south west, font=\scriptsize] at (1.4,3.0) {equilibrium};
  % actual: follows then freezes into a near-plateau, slight decline
  \draw[acc, very thick]
    (0.5,3.7) .. controls (1.6,2.9) and (2.5,2.1) .. (3.4,1.55)
    .. controls (4.2,1.25) and (4.8,1.2) .. (5.6,1.12)
    -- (8.0,0.85);
  \draw[black, dashed] (3.4,0) -- (3.4,1.55);
  \node[anchor=north, black!70, font=\scriptsize] at (3.4,-0.05) {freeze-out};
  \node[acc, anchor=south west, font=\scriptsize] at (5.4,1.15) {frozen, decaying};
  \node[black, anchor=west, font=\scriptsize] at (7.2,0.7) {onset};
\end{tikzpicture}
$$

## The deuterium bottleneck

Building any nucleus starts with deuterium, formed by radiative capture

$$
p + n \to {}^{2}\mathrm{H} + \gamma, \qquad B_d = 2.22\ \mathrm{MeV}.
$$

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 $2.22\ \mathrm{MeV}$ 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

$$
kT_{\text{bbn}} \approx 0.07\ \mathrm{MeV}\ \ (t \approx 3\ \mathrm{min}).
$$

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:

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

Because ${}^{4}\mathrm{He}$ 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 ${}^{2}\mathrm{H}$
and ${}^{3}\mathrm{He}$ survive unburned, and a trace of ${}^{7}\mathrm{Li}$ forms through
${}^{4}\mathrm{He}({}^{3}\mathrm{H},\gamma){}^{7}\mathrm{Li}$ and
${}^{4}\mathrm{He}({}^{3}\mathrm{He},\gamma){}^{7}\mathrm{Be}(e^-,\nu){}^{7}\mathrm{Li}$.
Nothing heavier is made in appreciable quantity; the heavy elements wait for stars.

$$
% caption: The primordial network starts from free protons and neutrons, forms deuterium as
% the bottleneck breaks, and funnels almost every neutron into helium-4, leaving trace
% deuterium, helium-3, and lithium-7 and stopping at the mass-8 gap.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \tikzset{nuc/.style={draw=black, thick, minimum width=8mm, minimum height=6mm, inner sep=2pt, font=\footnotesize}}
  \node[nuc] (p) at (0,1.0) {p};
  \node[nuc] (n) at (0,-1.0) {n};
  \node[nuc] (d) at (2.2,0) {d};
  \node[nuc] (he3) at (4.6,1.0) {He3};
  \node[nuc] (t) at (4.6,-1.0) {H3};
  \node[nuc, draw=acc, fill=acc!10] (he4) at (7.0,0) {He4};
  \node[nuc] (li7) at (9.2,0) {Li7};
  \draw[->, black, thick] (p) -- (d);
  \draw[->, black, thick] (n) -- (d);
  \draw[->, black, thick] (d) -- (he3);
  \draw[->, black, thick] (d) -- (t);
  \draw[->, black, thick] (he3) -- (he4);
  \draw[->, black, thick] (t) -- (he4);
  \draw[->, black, thick, densely dotted] (he4) -- (li7) node[midway, above, black, font=\scriptsize] {trace};
  \node[black, anchor=west, font=\scriptsize] at (9.4,-0.8) {mass-8 gap};
\end{tikzpicture}
$$

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

$$
Y_p = \frac{4\,n_{\mathrm{He}}}{n_{\mathrm{nucleons}}}
= \frac{2 n_n}{n_n + n_p}
= \frac{2r}{1 + r}.
$$

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.[^krane-bbn]

> **Result.** The primordial helium mass fraction is set almost entirely by the frozen
> neutron-to-proton ratio, $Y_p = 2r/(1+r)$. Because $r$ depends on the freeze-out
> temperature, the neutron lifetime, and the expansion rate, the measured $Y_p \approx 0.25$
> constrains the number of light neutrino species: extra relativistic species speed the
> expansion, raise $T_f$ and $r$, and would push $Y_p$ higher than observed.

## 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** $\eta = n_b/n_\gamma$.
Deuterium is the sharpest probe: a higher baryon density means more efficient burning of
deuterium into helium, so the surviving deuterium falls steeply as $\eta$ 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 $\eta$, and they do,
near

$$
\eta \approx 6\times10^{-10},
\qquad
\left(\frac{\mathrm{D}}{\mathrm{H}}\right) \approx 2.5\times10^{-5}.
$$

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 ${}^{7}\mathrm{Li}$ abundance is about three times the value
observed in old stars, the **lithium problem**, still unresolved.[^krane-bbn]

$$
% caption: Predicted primordial abundances against the baryon-to-photon ratio: helium-4
% rises weakly, deuterium and helium-3 fall, lithium-7 dips and rises, and the vertical band
% marks the single baryon density where all measurements agree.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {baryon-to-photon ratio};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {abundance};
  % observed concordance band
  \fill[acc!12] (4.0,0) rectangle (4.7,4.4);
  \node[acc, anchor=south, font=\scriptsize, rotate=90] at (4.35,2.6) {observed};
  % He4 nearly flat (solid, top)
  \draw[black, very thick] (0.6,3.7) .. controls (3.0,3.85) and (5.5,3.95) .. (8.0,4.05);
  \node[black, anchor=south, font=\scriptsize] at (1.4,3.75) {He4};
  % D falling (dashed)
  \draw[black, thick, dashed] (0.6,3.4) .. controls (2.6,2.4) and (4.6,1.4) .. (8.0,0.5);
  \node[black, anchor=south west, font=\scriptsize] at (6.4,0.7) {D};
  % He3 gentle fall (densely dotted)
  \draw[black, thick, densely dotted] (0.6,2.5) .. controls (3.0,2.1) and (5.5,1.75) .. (8.0,1.5);
  \node[black, anchor=south, font=\scriptsize] at (2.0,2.35) {He3};
  % Li7 valley (thin solid)
  \draw[black, thick] (0.6,2.2) .. controls (2.4,1.1) and (3.6,0.85) .. (4.6,1.15)
    .. controls (5.8,1.5) and (7.0,2.1) .. (8.0,2.6);
  \node[black, anchor=north east, font=\scriptsize] at (8.0,2.5) {Li7};
\end{tikzpicture}
$$

## Timeline

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

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

$$
% caption: The nucleosynthesis timeline: neutron-proton freeze-out at one second, a wait
% through the deuterium bottleneck governed by neutron decay, and the burst of helium
% formation near three minutes, after which the abundances are fixed.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[very thick, ->] (0,0) -- (8.6,0) node[right, black!70] {time};
  % markers
  \fill[black] (1.0,0) circle (2pt);
  \draw[black] (1.0,0) -- (1.0,0.9);
  \node[black!70, anchor=south, font=\scriptsize, align=center] at (1.0,0.95) {freeze-out\\1 s};
  \fill[black] (4.4,0) circle (2pt);
  \draw[black] (4.4,0) -- (4.4,1.5);
  \node[black!70, anchor=south, font=\scriptsize, align=center] at (4.4,1.55) {bottleneck breaks\\3 min};
  \fill[black] (6.4,0) circle (2pt);
  \draw[black] (6.4,0) -- (6.4,0.9);
  \node[black!70, anchor=south, font=\scriptsize, align=center] at (6.4,0.95) {He4 formed};
  \fill[black] (8.0,0) circle (2pt);
  \draw[black] (8.0,0) -- (8.0,1.5);
  \node[black!70, anchor=south, font=\scriptsize, align=center] at (8.0,1.55) {frozen\\20 min};
  % bottleneck span under axis
  \draw[black, <->] (1.0,-0.55) -- (4.4,-0.55);
  \node[black, anchor=north, font=\scriptsize] at (2.7,-0.55) {neutrons decaying};
\end{tikzpicture}
$$

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](/nuclear-physics/fusion-nucleosynthesis/stellar-nucleosynthesis), and
the same Coulomb-barrier tunneling that runs
[stellar fusion](/nuclear-physics/fusion-nucleosynthesis/fusion-reactions-confinement) 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-bbn]: **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 $kT_f \approx
0.8\ \mathrm{MeV}$, the deuterium bottleneck, the helium yield $Y_p = 2r/(1+r)$, 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/](https://pdg.lbl.gov/); the neutron lifetime $\tau_n = 879\ \mathrm{s}$
and $\Delta m\,c^2 = 1.293\ \mathrm{MeV}$ are the CODATA/NIST values,
[https://physics.nist.gov/cuu/Constants/](https://physics.nist.gov/cuu/Constants/). See also
**Tipler & Llewellyn**, _Modern Physics_, §13-7.