Big Bang Nucleosynthesis

Between one second and roughly twenty minutes after the Big Bang, the universe passed through the only epoch in its history when free nucleons could assemble into nuclei in bulk. The outcome is fixed by nuclear and weak physics measured in the laboratory, and it depends on essentially one cosmological number, the ratio of baryons to photons. Big Bang nucleosynthesis (BBN) therefore does two things at once: it predicts the primordial abundances of the light elements, and it weighs the baryonic content of the universe. This lesson derives the neutron-to-proton freeze-out, the deuterium bottleneck that delays nuclear assembly, the resulting abundances as functions of the baryon-to-photon ratio, and the concordance with observation together with its one sore point.

The neutron-to-proton ratio

At temperatures above $k_{B}T\sim 1\text{MeV}$ the weak interactions interconvert neutrons and protons rapidly,

and keep the two in chemical equilibrium. In equilibrium the ratio of number densities follows the Boltzmann factor set by the neutron-proton mass difference $Q=(m_n-m_p)c^2=1.293\text{MeV}$,

At high temperature the exponent is small and neutrons and protons are nearly equal in number. As the universe cools the ratio falls, favoring the lighter proton. If equilibrium held indefinitely the neutrons would disappear entirely; they survive because the weak interactions freeze out.

The weak rate per nucleon scales as ${\Gamma }_{\text{weak}}\propto G_F^2(k_{B}T{)}^5$, while the expansion rate is $H\propto \sqrt{g_{\ast }}{T}^2$, so $\Gamma /H\propto T^3$ and the interactions decouple at a freeze-out temperature $k_B{T}_{\text{fr}}\approx 0.8\text{MeV}$. At that moment the equilibrium ratio is frozen in:

Freeze-out is not the whole story. Free neutrons are unstable, with a mean lifetime ${\tau }_n=880\text{s}$, so between freeze-out and the start of nuclear assembly the ratio continues to fall by free decay,

By the time nucleosynthesis begins at $t\approx 200\text{s}$ the ratio has dropped from $1/5$ to about $1/7$. This residual neutron fraction is what sets the helium yield.

The deuterium bottleneck

Building helium requires deuterium as the first step, $p+n\to d+\gamma$, and every heavier nucleus is assembled from deuterium. The binding energy of deuterium is only $B_D=2.22\text{MeV}$, small on the scale of the photon bath. Even well below $k_{B}T=B_D$, the blackbody spectrum has a vast number of photons in its high-energy tail, because photons outnumber baryons by the enormous factor ${\eta }^{-1}\sim 10^9$. Deuterium is photodissociated, $d+\gamma \to p+n$, as fast as it forms until the temperature drops far enough that even the tail no longer contains a dissociating photon per deuteron.

The condition for deuterium to survive is that the number of photons above $B_D$ per baryon falls below unity. With a photon-to-baryon ratio $\eta$, the relevant estimate sets

corresponding to $T_{\text{nuc}}\approx 0.8\times 10^9\text{K}$ and $t\approx 200\text{s}$. The delay from $k_{B}T=B_D$ down to $k_B{T}_{\text{nuc}}$ — the deuterium bottleneck — is entirely due to the large photon-to-baryon ratio, and it costs about 20% of the frozen neutrons to decay in the meantime.

The reaction network and the helium yield

Once deuterium survives, a fast chain of two-body reactions burns it into helium-4, the most tightly bound of the light nuclei. The dominant paths are

with $t$ the triton (${}^3\text{H}$). Because ${}^4\text{He}$ has by far the largest binding energy per nucleon among the light species, and because there is no stable nucleus of mass number 5 or 8 to bridge to heavier elements, the chain piles essentially all available neutrons into ${}^4\text{He}$ and then stalls.

The helium mass fraction follows from a simple neutron count. Each helium-4 nucleus contains two neutrons and two protons; nearly every neutron present at $T_{\text{nuc}}$ ends up in a helium nucleus, paired with an equal number of protons. With a neutron-to-proton ratio $r=n_n/n_p\approx 1/7$ at onset, the mass fraction in helium is

Roughly a quarter of the baryonic mass emerges as helium-4, with the rest almost entirely hydrogen. This value is insensitive to cosmological parameters, because it depends only on the frozen $n/p$ ratio through $Q$, the neutron lifetime, and $g_{\ast }$ — all laboratory quantities.

is on the expansion rate at freeze-out: a larger $g_{\ast }$ (for instance, more neutrino species) speeds the expansion, raises $T_{\text{fr}}$, freezes in more neutrons, and increases $Y_p$. This is the sensitivity that lets BBN constrain $N_{\text{eff}}$.

Abundances as a function of the baryon-to-photon ratio

The single cosmological input to BBN is the baryon-to-photon ratio,

fixed and conserved once the photon and baryon numbers are set. Because $n_{\gamma }$ is known from the CMB temperature, $\eta$ is equivalent to the baryon density parameter,

Each light-element abundance responds to $\eta$ in a characteristic way, and this is what makes BBN a measurement rather than merely a consistency check:

• Helium-4 ($Y_p\approx 0.247$) rises only logarithmically with $\eta$: a higher baryon density makes deuterium survive slightly earlier, so fewer neutrons decay and $Y_p$ increases weakly. It is the least sensitive probe but the best-measured.
• Deuterium falls steeply with $\eta$, roughly $\text{D/H}\propto {\eta }^{-1.6}$: more baryons burn deuterium more completely into helium, leaving less residual D. This steep slope makes deuterium the sharpest baryometer.
• Helium-3 falls gently with $\eta$, tracking deuterium but partly replenished and partly destroyed in stars, which complicates its use.
• Lithium-7 is non-monotonic, with a minimum near the observed $\eta$: at low $\eta$ it forms directly, at high $\eta$ it forms through ${}^7\text{Be}$ that later captures an electron, and the two channels produce a valley.

The predicted curves cross the observed abundances at a common value of $\eta$ — the defining success of BBN. The deuterium abundance, measured in the light of distant quasars absorbed by unprocessed intergalactic gas, gives $\text{D/H}\approx 2.5\times 10^{-5}$ and pins $\eta$ to a few percent. That $\eta$ agrees with the completely independent value from the CMB acoustic peaks, a concordance across physics separated by 380,000 years of cosmic history.

Concordance and the lithium problem

The measured primordial abundances span nine orders of magnitude, from $Y_p\approx 0.25$ down to $\text{Li/H}\sim 10^{-10}$, and BBN reproduces them from one parameter fixed elsewhere.

The helium and deuterium agreement is the strongest quantitative evidence that the universe was once hot and dense, and that its physics at $k_{B}T\sim 1\text{MeV}$ was the physics measured in the laboratory. The one discrepancy is the lithium problem: the abundance of ${}^7\text{Li}$ measured in the atmospheres of old, metal-poor halo stars — the Spite plateau — is a factor of about three below the BBN prediction for the CMB value of $\eta$. Proposed resolutions fall into three classes: depletion of lithium in the stellar atmospheres over billions of years, systematic errors in the nuclear reaction rates feeding ${}^7\text{Be}$, or new physics altering the expansion or particle content during BBN. None is established, and the lithium problem remains open.¹

BBN also constrains parameters beyond $\eta$. Because the helium yield depends on the expansion rate at freeze-out, the concordance limits the effective number of relativistic neutrino species to $N_{\text{eff}}=2.9\pm 0.3$, consistent with the three Standard Model neutrinos and independent of the CMB determination. Any extra light species — a fourth neutrino, or other relativistic relics — would speed the expansion, raise $Y_p$, and break the agreement.

The baryon density fixed here, ${\Omega }_b\approx 0.05$, is far below the total matter density ${\Omega }_m\approx 0.31$ inferred from galaxy rotation curves and clusters. The difference is the first cosmological argument that most of the matter is non-baryonic, a conclusion the CMB independently confirms in the next lessons on recombination and the acoustic peaks.