Big Bang Nucleosynthesis
The synthesis of light elements in the first few minutes: neutron freeze-out, deuterium bottleneck, helium-4 production, and constraints on new physics from primordial abundances
Overview
Big Bang Nucleosynthesis (BBN) is one of the three pillars of the Big Bang model. During the first three minutes after the Big Bang, the universe cooled through the temperature range \(T \sim 10\) MeV to \(\sim 0.01\) MeV, during which protons and neutrons fused into light nuclei. The predicted abundances of D,\({}^3\text{He}\), \({}^4\text{He}\), and \({}^7\text{Li}\)agree spectacularly with observations (with the lithium problem being a notable exception).
1. Neutron-Proton Equilibrium
At temperatures \(T \gtrsim 1\) MeV, weak interactions maintain equilibrium between neutrons and protons via:
$$n + \nu_e \leftrightarrow p + e^-, \qquad n + e^+ \leftrightarrow p + \bar{\nu}_e, \qquad n \leftrightarrow p + e^- + \bar{\nu}_e$$
In equilibrium, the neutron-to-proton ratio is set by the Boltzmann factor:
Equilibrium n/p Ratio
$$\frac{n}{p}\bigg|_{\text{eq}} = e^{-\Delta m / T}$$
where \(\Delta m = m_n - m_p = 1.293\) MeV is the neutron-proton mass difference.
2. Neutron Freeze-Out
The weak interaction rate scales as \(\Gamma_w \sim G_F^2 T^5\) while the Hubble rate scales as \(H \sim T^2/M_{\text{Pl}}\). Freeze-out occurs when\(\Gamma_w \sim H\), giving:
Freeze-Out Temperature
$$T_f \approx 0.8\;\text{MeV} \approx 10^{10}\;\text{K}$$
At freeze-out the neutron-to-proton ratio is:
$$\frac{n}{p}\bigg|_{T_f} = e^{-1.293/0.8} \approx \frac{1}{5}$$
After freeze-out, neutrons continue to decay via \(\beta\)-decay with lifetime\(\tau_n \approx 880\) s. By the time nucleosynthesis begins at \(t \approx 180\) s, the ratio has dropped to:
$$\frac{n}{p}\bigg|_{\text{BBN}} \approx \frac{1}{5}\,e^{-180/880} \approx \frac{1}{7}$$
3. The Deuterium Bottleneck
Nucleosynthesis must proceed through deuterium: \(n + p \to D + \gamma\). However, deuterium has a small binding energy \(B_D = 2.22\) MeV. Despite \(T < B_D\)well before BBN, the enormous photon-to-baryon ratio \(\eta^{-1} \sim 10^{10}\)means the high-energy tail of the photon distribution photodisintegrates deuterium until \(T \lesssim 0.07\) MeV. The deuterium abundance satisfies:
$$\frac{n_D}{n_b} \propto \eta_b\,\exp\!\left(\frac{B_D}{T}\right)$$
Only when this ratio becomes of order unity can heavier nuclei form. This delay is the deuterium bottleneck.
4. Helium-4 Mass Fraction
Once the bottleneck is overcome, essentially all available neutrons are rapidly incorporated into \({}^4\text{He}\) (the most tightly bound light nucleus). If all neutrons end up in helium, the mass fraction is:
Primordial Helium Mass Fraction
$$\boxed{Y_p = \frac{2(n/p)}{1 + n/p} \approx \frac{2 \times 1/7}{1 + 1/7} = \frac{2/7}{8/7} = \frac{1}{4} \approx 0.25}$$
Observations of metal-poor HII regions give \(Y_p = 0.245 \pm 0.003\), in excellent agreement. This is one of the great quantitative successes of the Big Bang model.
5. Other Light Element Abundances
BBN also produces trace amounts of other light elements. Their abundances are sensitive to the baryon-to-photon ratio \(\eta_b\):
$$\text{D/H} \approx 2.5 \times 10^{-5}, \qquad {}^3\text{He/H} \approx 10^{-5}, \qquad {}^7\text{Li/H} \approx 5 \times 10^{-10}$$
Deuterium is the “baryometer” of BBN: its abundance decreases sharply with increasing \(\eta_b\) because higher baryon density burns D into helium more efficiently. The measured D/H ratio pins down:
Baryon-to-Photon Ratio
$$\boxed{\eta_b = \frac{n_b}{n_\gamma} \approx 6.1 \times 10^{-10}}$$
Equivalently, \(\Omega_b h^2 \approx 0.0224\), in remarkable agreement with the independent determination from the CMB power spectrum.
6. Constraints on New Physics: \(N_{\text{eff}}\)
The expansion rate during BBN depends on \(g_*\), which counts all relativistic degrees of freedom. Any additional light species (e.g., sterile neutrinos, light dark matter, or dark radiation) increases \(g_*\) and hence \(H\), leading to earlier neutron freeze-out and more \({}^4\text{He}\). This is parameterized by:
$$\rho_R = \left[1 + \frac{7}{8}\left(\frac{4}{11}\right)^{4/3} N_{\text{eff}}\right]\rho_\gamma$$
The Standard Model predicts \(N_{\text{eff}} = 3.044\) (slightly above 3 due to non-instantaneous neutrino decoupling). BBN constrains:
BBN Constraint on Extra Radiation
$$N_{\text{eff}} = 2.89 \pm 0.28 \quad (95\%\;\text{C.L.})$$
This rules out a full additional neutrino species and severely constrains any light beyond-Standard-Model particles that were in thermal equilibrium during BBN.
7. The Cosmological Lithium Problem
While the BBN predictions for D, \({}^3\text{He}\), and \({}^4\text{He}\)are in excellent agreement with observations, the predicted primordial \({}^7\text{Li}\)abundance is a factor of \(\sim 3\) higher than what is observed in metal-poor halo stars:
$$\left(\frac{{}^7\text{Li}}{\text{H}}\right)_{\text{BBN}} \approx 5 \times 10^{-10}, \qquad \left(\frac{{}^7\text{Li}}{\text{H}}\right)_{\text{obs}} \approx 1.6 \times 10^{-10}$$
This \(\sim 3\sigma\) discrepancy, known as the cosmological lithium problem, remains unresolved. Proposed solutions include stellar depletion mechanisms (atomic diffusion, turbulent mixing), systematic errors in the nuclear cross sections (particularly\({}^3\text{He}(\alpha,\gamma){}^7\text{Be}\)), and new physics that modifies the BBN reaction network — for example, late-decaying particles that photodisintegrate\({}^7\text{Be}\) before it captures an electron to form \({}^7\text{Li}\). No single explanation has achieved consensus, making it one of the most intriguing open problems at the intersection of nuclear physics and cosmology.