Big Bang Nucleosynthesis

BBN is a working tool, not a museum piece. Its sharpest edge is the baryometer: astronomers hunt for pristine, near-zero-metallicity gas clouds seen in absorption against distant quasars, where the deuterium Lyman-series lines sit blueward of hydrogen's, and measure D/H to a few percent. The current sample gives $\Omega_b h^2$ in striking concordance with Planck's CMB value — a cross-check between the universe at three minutes and at 380,000 years that constrains any new physics altering the expansion in between. On the particle side, precise $Y_p$ measurements from metal-poor extragalactic H II regions bound $N_{\mathrm{eff}}$, dark radiation, sterile neutrinos, and time-varying constants. Underpinning all of it, laboratory efforts to resolve the $\sim 1\%$ "beam versus bottle" discrepancy in the neutron lifetime, and reaction-rate measurements at facilities like LUNA deep under Gran Sasso, feed directly into the predictions. And the lithium problem remains genuinely open: standard BBN overpredicts $^7$Li by a factor of three, and whether the culprit is stellar depletion, an uncertain $^7$Be destruction rate, or beyond-standard-model physics is a live question the field is still chasing.

Everything in BBN is a race between a reaction rate $\Gamma$ and the Hubble expansion rate $H$. The nuclear reactions want to lock every proton and neutron into $^4$He, by far the most tightly bound light nucleus. But the referee is the expanding universe: whenever a rate falls below $H$, that reaction freezes and strands its abundance away from equilibrium. Two such freeze-outs, in sequence, set the whole outcome. The neutrons are the ammunition, and they are the scarce reagent. Above weak freeze-out the reactions $n+\nu_e\leftrightarrow p+e^-$ and $n+e^+\leftrightarrow p+\bar\nu_e$ enforce $n/p=e^{-\Delta m c^2/k_BT}$ with $\Delta m c^2=1.293\,\mathrm{MeV}$. As the weak rate $\Gamma\propto T^5$ plunges past $H\propto T^2$ near $T\approx 0.8\,\mathrm{MeV}$, the ratio sticks at $\approx 1/6$. Free-neutron decay ($\tau_n\approx 879\,\mathrm{s}$) then slowly trims it toward $\approx 1/7$ over the next few minutes. Whatever neutrons survive to the moment synthesis begins are the entire helium budget. The first figure tells this two-stage story.

The deuterium bottleneck is the second, decisive delay. Naively D should assemble once $k_BT$ falls below its binding energy $B_D=2.22\,\mathrm{MeV}$ — but photons outnumber baryons a billion-fold, so the exponential tail of the Planck spectrum keeps enough energetic $\gamma$'s around to photodissociate every deuteron until $T\approx 0.07\,\mathrm{MeV}$. Only then can the network vault the mass gaps at $A=5$ and $A=8$ (there are no stable nuclei there) and pile nucleons up at $^4$He. Because two neutrons go into each helium nucleus, $Y_p$ is fixed almost entirely by the frozen $n/p$ ratio and depends only weakly on $\eta$. The leftover trace species are "incomplete-burning" relics — and it is exactly their sensitivity to $\eta$ that turns the second figure, the abundances versus $\eta$, into cosmology's most precise baryometer.

Standard BBN has exactly one free parameter — the ratio of baryons to photons, set once and for all by whatever baryogenesis occurred earlier. Every abundance is a function of this number.

where: $n_b$ — baryon number density; $n_\gamma$ — photon number density; $\eta$ — the single free parameter of standard BBN. Its tininess, $\eta\sim 10^{-9}$, is the numerical engine of the deuterium bottleneck. Above weak freeze-out, neutrons and protons sit in mutual chemical equilibrium, so their ratio is a Boltzmann factor in the neutron–proton mass splitting. This is the initial condition for everything downstream.

where: $\Delta m$ — neutron–proton mass difference $m_n-m_p$; $T$ — plasma temperature; $k_B$ — Boltzmann constant. Reading it: at $T\gg 1\,\mathrm{MeV}$ the ratio is near unity; by $T\approx 0.8\,\mathrm{MeV}$ it has fallen to $\approx 1/6$. The expansion rate is fixed by the total relativistic energy density. In the radiation era this gives the famous "one second at ten billion kelvin" clock that dates the epoch.

where: $H$ — Hubble parameter; $\rho$ — total energy density; $g_*\approx 10.75$ before $e^\pm$ annihilation; $t$ — cosmic time. The key point for BBN: $H\propto\sqrt{g_*}$, so any extra relativistic species speeds the clock and shifts every freeze-out. Weak freeze-out is the competition between the neutron–proton interconversion rate and $H$. The steep temperature scalings make the crossover sharp.

where: $\Gamma_{n\leftrightarrow p}$ — weak interconversion rate; $G_F$ — Fermi constant; $T_{\mathrm{fr}}\approx 0.8\,\mathrm{MeV}$ — the freeze-out temperature where $\Gamma\simeq H$. Because $\Gamma/H\propto T^3$, the reactions switch off within a factor of two in temperature. Deuterium formation is governed by a nuclear Saha equation — the equilibrium balance between $p+n\leftrightarrow D+\gamma$. The exponential $e^{B_D/k_BT}$ is what must eventually beat the small $\eta$.

where: $n_D$ — deuterium number density; $B_D$ — deuteron binding energy; $m_N$ — nucleon mass; the factor $3/4=g_D/(g_n g_p)=3/(2\cdot 2)$ collects spin statistics. Reading it: D is suppressed by density but exponentially favoured as $T$ drops. Once the bottleneck opens, the helium mass fraction follows from simple neutron counting — two neutrons and two protons per $^4$He, with neutrons the limiting reagent.

where: $Y_p$ — primordial $^4$He mass fraction; $n/p$ — the ratio at deuterium formation, $\approx 1/7$. Note $Y_p$ is a mass fraction, unlike the number ratios D/H and Li/H. Solving the full reaction network at $\eta\approx 6\times10^{-10}$ yields the standard trace abundances — the numbers a survey actually tries to measure.

where: all are number ratios relative to hydrogen. The predicted $^7$Li exceeds the observed halo-star value ($\approx 1.6\times10^{-10}$) by a factor $\sim 3$ — the lithium problem. Finally, deuterium's steep dependence on $\eta$ converts a measured D/H into a baryon density, the quantity the CMB also measures.

where: $\Omega_b$ — baryon density parameter; $h$ — Hubble constant in units of $100\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. For $\eta\approx 6.1\times10^{-10}$ this gives $\Omega_b h^2\approx 0.0224$, matching the CMB.

BBN outcomes track dimensionless ratios of rates and the value of $\eta$; the useful limiting behaviours are:

1 · The helium mass fraction — the two-minute calculation (confidence builder)

Predict $Y_p$ from the frozen neutron-to-proton ratio, given $k_BT_{\mathrm{fr}}\approx 0.8\,\mathrm{MeV}$, $\Delta m c^2=1.293\,\mathrm{MeV}$, $\tau_n=879\,\mathrm{s}$, and a nucleosynthesis onset at $t_{\mathrm{nuc}}\approx 260\,\mathrm{s}$.

Step 1. Freeze-out value: $(n/p)_{\mathrm{fr}}=e^{-1.293/0.8}=e^{-1.616}=\mathbf{0.199}$, essentially $1/5$. This is the neutron budget the weak reactions leave behind.

Step 2. Erode it by neutron decay over the wait for the bottleneck to open: $(n/p)_{\mathrm{nuc}}=0.199\,e^{-260/879}=0.199\times e^{-0.296}=0.199\times 0.744=\mathbf{0.148}\approx 1/7$. The delay costs about a quarter of the neutrons.

Step 3. Feed into equation (6): $Y_p=\dfrac{2(0.148)}{1+0.148}=\dfrac{0.296}{1.148}=\mathbf{0.258}$. Using the round $n/p=1/7$ gives exactly $Y_p=0.25$; a full reaction network, which keeps letting neutrons decay during the burning, trims this to the standard $Y_p\approx 0.247$. The remarkable result: roughly a quarter of all baryonic mass emerges from the Big Bang as helium, fixed almost entirely by one Boltzmann factor and the neutron lifetime.

2 · The bottleneck temperature and the age of the universe at synthesis

Estimate the temperature and cosmic time at which deuterium finally forms, taking $\eta=6.1\times10^{-10}$, $B_D=2.22\,\mathrm{MeV}$, and the radiation-era clock $t\approx (T/10^{10}\,\mathrm{K})^{-2}\,\mathrm{s}$.

Step 1. The bottleneck opens when the equilibrium deuterium fraction reaches order unity, i.e. when $B_D/k_BT\approx\ln\eta^{-1}$. Compute the logarithm: $\ln(1/6.1\times10^{-10})=\ln(1.64\times10^{9})=\mathbf{21.2}$.

Step 2. The $(k_BT/m_Nc^2)^{3/2}$ phase-space factor adds roughly $5$ to the required exponent, so take $B_D/k_BT\approx 26$. Then $k_BT\approx 2.22/26=\mathbf{0.085\ MeV}\approx 0.07$–$0.09\,\mathrm{MeV}$.

Step 3. Convert to kelvin: $0.085\,\mathrm{MeV}\times 1.16\times10^{10}\,\mathrm{K/MeV}\approx 9.9\times10^{8}\,\mathrm{K}$. The simple clock then gives $t\approx (9.9\times10^{8}/10^{10})^{-2}=(0.099)^{-2}\approx\mathbf{100\ s}$; the precise treatment, accounting for the drop in $g_*$ after $e^\pm$ annihilation, stretches this to $t\approx 200$–$300\,\mathrm{s}$. Synthesis happens at "about three minutes" and $T\sim 10^{9}\,\mathrm{K}$ — the celebrated first three minutes — not at the seconds-old, MeV-scale epoch a naive $B_D$ argument would suggest.

3 · Reading the baryon density off deuterium, and counting neutrinos

A quasar-absorption system gives primordial $\mathrm{D/H}=2.5\times10^{-5}$. Use $\Omega_b h^2=3.66\times10^{7}\,\eta$ to find the baryon density, then estimate how much a hypothetical fourth neutrino would raise $Y_p$.

Step 1. The measured D/H corresponds to $\eta\approx 6.1\times10^{-10}$ (from the steep $\mathrm{D/H}\propto\eta^{-1.6}$ calibration). Then $\Omega_b h^2=3.66\times10^{7}\times 6.1\times10^{-10}=\mathbf{0.0223}$ — agreeing with the CMB value $0.0224$ to better than a percent, a spectacular independent check spanning physics at $3\,\mathrm{min}$ and $380{,}000\,\mathrm{yr}$.

Step 2. A fourth light neutrino raises $N_{\mathrm{eff}}$ from $3.046$ to $\approx 4.05$, so $\Delta N_{\mathrm{eff}}\approx 1$. Using the sensitivity $\Delta Y_p\approx 0.013\,\Delta N_{\mathrm{eff}}$ gives $\Delta Y_p\approx\mathbf{0.013}$, lifting $Y_p$ from $0.247$ to $\approx 0.260$. This is why BBN bounded the number of light neutrino species to $\approx 3$ years before LEP measured $N_\nu=2.984\pm 0.008$ from the $Z$ width.

BBN is the point where nuclear physics, particle physics, and cosmology fuse into a single quantitative prediction — and its story belongs to a young calculator who worked out the first light-element network and, along the way, predicted the microwave background a full seventeen years before it was found.

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 10 · Dodelson, Modern Cosmology, 2nd ed., Ch. 3 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 30 · Kolb & Turner, The Early Universe, Ch. 4 · Weinberg, Cosmology, §3.2 · Particle Data Group Review, Big Bang Nucleosynthesis. The key numerical anchors are: (1) $(n/p)_{\mathrm{fr}}=e^{-1.293/0.8}=0.199$, decaying to $\approx 1/7$ with $\tau_n=879\,\mathrm{s}$ over $t_{\mathrm{nuc}}\approx 200$–$300\,\mathrm{s}$; (2) $Y_p=2(n/p)/(1+n/p)=0.25$ at $n/p=1/7$, with full networks giving $Y_p\approx 0.247$; (3) the bottleneck condition $B_D/k_BT\approx\ln\eta^{-1}\approx 21$, with the phase-space factor pushing the effective exponent to $\approx 26$, so $k_BT\approx 0.07$–$0.09\,\mathrm{MeV}$; (4) $\Omega_b h^2=3.66\times10^{7}\,\eta\approx 0.0223$ for $\eta=6.1\times10^{-10}$; (5) the radiation-era clock $t\approx (T/10^{10}\,\mathrm{K})^{-2}\,\mathrm{s}$.

Prerequisites. Thermal History of the Universe · Equilibrium Thermodynamics of the Early Universe · Decoupling and Freeze-Out · Neutrino Decoupling and the Cosmic Neutrino Background · Effective Number of Relativistic Degrees of Freedom · Baryon Asymmetry and Baryogenesis. Leads to. Primordial Helium Abundance and the Deuterium Bottleneck · The Lithium Problem · Recombination and the Saha Equation · Last Scattering Surface and Photon Decoupling.

Ask the tutor: Want to walk through why the deuterium bottleneck — not the binding energy — sets the clock for the first three minutes, or work a D/H measurement all the way to $\Omega_b h^2$?