Big Bang Nucleosynthesis (BBN)

• Big Bang Nucleosynthesis (BBN) is the process that formed light elements like D, 3He, 4He, and 7Li through nuclear reactions in the early, radiation-dominated universe.
• It employs a complex network of nuclear reactions, weak and electromagnetic interactions, and precise thermodynamic conditions to predict elemental abundances.
• Observational concordance in D/H and 4He, alongside the unresolved lithium problem, makes BBN a stringent test of the standard cosmological model and new physics scenarios.

Big Bang Nucleosynthesis (BBN), or primordial nucleosynthesis, describes the synthesis of the lightest nuclides—principally D, $^3$He, $^4$He, and $^7$Li—via a sequence of nuclear reactions in the expanding radiation-dominated Universe during the first few minutes after the Big Bang. BBN remains a cornerstone quantitative probe of early-universe cosmology, providing insights into particle physics, nuclear processes, and the interplay of all known fundamental forces within a cosmological setting. The measured abundances of these light elements, coupled with high-precision calculations, form a critical test of the standard cosmological model and tightly constrain new physics scenarios.

1. Cosmological and Physical Framework

BBN occurs in a spatially homogeneous, isotropic Friedmann–Robertson–Walker (FRW) Universe dominated by relativistic species. The expansion is described by the first Friedmann equation: $$H^2(t) = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho(t)$$ where $a(t)$ is the scale factor, $H$ is the Hubble parameter, $G$ is Newton's constant, and $\rho$ is the total energy density. During BBN,

$\rho_R(T) = \frac{\pi^2}{30}\,g_*(T)\,T^4$

where $g_*(T)$ counts the effective relativistic degrees of freedom (photons, $e^\pm$, neutrinos, …).

The temperature–time relation in the radiation-dominated regime is

$$t \simeq \frac{0.301}{\sqrt{g_*}}\,\frac{M_{\text{Pl}}}{T^2} \simeq 1~\text{s}~\left(\frac{T}{1~\text{MeV}}\right)^{-2}$$

where $M_{\text{Pl}}\simeq1.22\times10^{19}$~GeV (Cooke, 2024).

The microphysical evolution is set by the baryon-to-photon ratio $\eta \equiv n_b/n_\gamma \sim 10^{-9}$, fixed with high precision by CMB observations, and the number of effective neutrino species $N_{\rm eff}$, which controls the expansion rate via $g_*$.

2. Nuclear Reaction Network and Evolution Equations

The light-element yields are determined by a stiff network of coupled differential equations for the abundance $Y_i \equiv n_i/n_b$ of each nuclide $i$: $$\frac{dY_i}{dt} = -\sum_{j,k} n_b\,Y_j\,Y_k\langle \sigma v \rangle_{jk\rightarrow i} + \sum_{\ell, m} n_b\,Y_\ell\,Y_m\langle \sigma v \rangle_{\ell m\rightarrow i}$$ where $\langle \sigma v \rangle$ denotes the thermally averaged cross section for each reaction, evaluated over the nuclear Maxwell–Boltzmann velocity distributions (Cooke, 2024). Modern BBN codes solve these equations numerically, incorporating the full suite of weak, strong, and electromagnetic rates, finite-temperature QED corrections, and neutrino decoupling.

Principal Light-Element Reactions

The dominant BBN network up to $A=7$ includes:

• Weak interactions: $n \leftrightarrow p$
• $p(n,\gamma) \text{D}$ ($Q=2.224~\text{MeV}$)
• $\text{D}(p,\gamma)^3\text{He}$, $\text{D}(d,n)^3\text{He}$, $\text{D}(d,p)t$
• $t(d,n)^4\text{He}$, $^3\text{He}(d,p)^4\text{He}$
• $t(\alpha, \gamma)^7\text{Li}$, $^3\text{He}(\alpha, \gamma)^7\text{Be}$
• $^7\text{Li}(p, \alpha)^4\text{He}$, $^7\text{Be}(n, p)^7\text{Li}$

Kinetic equilibrium is maintained by rapid Coulomb scattering of ions off $e^\pm$ pairs. Monte Carlo and Fokker–Planck analyses confirm that the nuclei are described to $\lesssim1\%$ precision by the Maxwell–Boltzmann distribution under standard BBN conditions (Sasankan et al., 2019).

3. Key Physical Stages: Freeze-Out and the Deuterium Bottleneck

Neutron–Proton Freeze-Out

At high temperatures ($T\gg1$~MeV), $n$ and $p$ rapidly interconvert via the weak charged-current processes: $$n + \nu_e \leftrightarrow p + e^{-}, \qquad n + e^+ \leftrightarrow p + \bar{\nu}_e, \qquad n \leftrightarrow p + e^{-} + \bar{\nu}_e$$ The neutron-to-proton ratio tracks the equilibrium Boltzmann factor: $$\frac{n_n}{n_p} = \exp\left(-\frac{\Delta m}{T}\right), \quad \Delta m = 1.293~\text{MeV}$$ Weak rates $\Gamma \sim G_F^2 T^5$ fall below the Hubble rate $H \sim T^2$ at $T_f\simeq0.8$~MeV, yielding $X_n(T_f) \simeq 0.20$ (Cooke, 2024). Subsequent neutron $\beta$ decay ($\tau_n\sim880$~s) reduces the surviving neutron fraction by $T\simeq0.06$~MeV.

The Deuterium Bottleneck

The onset of nucleosynthesis is delayed by the high entropy (large $n_\gamma/n_b$), which suppresses composite nuclei via photodissociation. D is strongly photodissociated until

$$\eta n_\gamma e^{-B_D/T} \simeq n_b \implies T_D \sim \frac{B_D}{\ln \eta^{-1}} \approx 0.06~\text{MeV}$$

where $B_D=2.224$~MeV. Once D survives, rapid sequences produce $^3$He, $t$, $^4$He, and the $A=7$ nuclides (Cooke, 2024, Turner et al., 2021). This bottleneck is entropy-driven, not simply due to the D binding energy (Turner et al., 2021).

4. Analytic Estimates and Scaling with Baryon Density

$^4$He mass fraction $Y_p$ can be estimated: $$Y_p = \frac{4 n_{^4\text{He}}}{n_b} = \frac{2 X_n(T_D)}{1 + X_n(T_D)} \approx 0.25$$

with $X_n(T_D)\sim0.14$ (post-decay). Other light abundances show power-law dependence on $\eta$:

• $$\frac{\text{D}}{\text{H}} \approx 2.6\times10^{-5}\left(\frac{6}{\eta_{10}}\right)^{1.6}$$
• $$\frac{^3\text{He}}{\text{H}} \sim 1\times10^{-5} \left(\frac{6}{\eta_{10}}\right)^{0.6}$$
• $$\frac{^7\text{Li}}{\text{H}} \sim 4\times10^{-10}\left(\frac{\eta_{10}}{6}\right)^2$$ where $\eta_{10} \equiv 10^{10}\eta$ (Cooke, 2024, Fields et al., 2019). $Y_p$ is weakly dependent on $\eta$, but D/H and $^7$Li/H vary strongly, making them sensitive baryometers.

5. Modern Numerical Results, Observational Tests, and Uncertainties

Numerical Predictions versus Observations

BBN predictions, using $\eta_{10}^\text{CMB}=6.10\pm0.04$ [Planck], are:

• $Y_p = 0.247 \pm 0.0003$ ~(mass fraction $^4$He)
• D/H$_{\text{pred}} = (2.50 \pm 0.05)\times10^{-5}$
• $^3$He/H$_\text{pred} \lesssim 1\times10^{-5}$
• $^7$Li/H$_\text{pred} = (4.7 \pm 0.7)\times10^{-10}$

The corresponding observational measurements are:

• D/H$= (2.527 \pm 0.030)\times10^{-5}$ (high-$z$ QSO absorbers)
• $Y_p = 0.245 \pm 0.003$ (H\,II regions in dwarf galaxies)
• $^3$He/H$\lesssim 1\times10^{-5}$ (Galactic H\,II)
• $^7$Li/H$= (1.6 \pm 0.3)\times10^{-10}$ ("Spite plateau" in halo stars) (Cooke, 2024).

D/H and $Y_p$ show percent-level concordance between BBN and observation. The only persistent anomaly is $^7$Li, which is overproduced in BBN by a factor $\sim$3 compared to stellar determinations ("Cosmic Lithium Problem") (Cooke, 2024, Fields et al., 2019).

Uncertainty Budget and Nuclear Inputs

The leading sources of theoretical uncertainty are:

• The cross sections for $d+d\rightarrow{}^3$He$+n$ and $d+d\rightarrow{}^3$H$+p$ near $E\sim0.05$–0.4 MeV
• The free neutron lifetime ($\tau_n$, now at $\sim0.1\%$ precision but subject to “bottle” vs “beam” tension)
• Finite-temperature QED corrections and the precise value of $N_{\rm eff}$ (Cooke, 2024, Foley et al., 2017).

Monte Carlo propagation of lognormal cross-section uncertainties and baryon density yields robust 1$\sigma$ intervals: $$\begin{aligned} Y_p &= 0.24670 \pm 0.00056 \ \mathrm{D/H} &= (2.54 \pm 0.12)\times10^{-5} \ {}^3\mathrm{He/H} &= (1.04 \pm 0.09)\times10^{-5} \ {}^7\mathrm{Li/H} &= (5.39 \pm 0.73)\times10^{-10} \end{aligned}$$ (Foley et al., 2017). State-of-the-art BBN codes include PArthENoPE, AlterBBN, PRIMAT, PRyMordial, and LINX (Cooke, 2024).

6. BBN as a Probe of Beyond-Standard-Model Physics

BBN is uniquely sensitive to changes in:

• Expansion rate via extra relativistic energy density—parameterized as $\Delta N_{\rm eff}$. The effect on $^4$He is $\Delta Y_p \approx 0.013\,\Delta N_{\rm eff}$, with similar sensitivity in D/H (Pospelov et al., 2010, Cooke, 2024).
• Baryon-to-photon ratio $\eta$ and its time variation.
• Properties of neutrinos: chemical potentials, decoupling, new species (Grohs et al., 2023).
• Decaying or annihilating massive relics: altered light-element yields from energy injection.
• Modifications to gravity or fundamental constants.

Current bounds exclude $\Delta N_{\rm eff} > 0.2$ (Foley et al., 2017, Fields et al., 2019). Models introducing late-time neutron injection to "solve" the Li problem increase D/H beyond observed values, now sharply excluded by $1\%$ deuterium measurements (Coc, 2016).

7. Future Prospects and Open Questions

• The lithium problem remains unresolved; neither revised nuclear physics nor refined stellar modeling has reconciled the BBN $^7$Li prediction with Spite–plateau measurements (Cooke, 2024).
• Future advances in reaction cross-section measurements (notably $d+d$ and $d(p,\gamma)^3$He), improved neutron lifetime measurements, new H\,II region spectroscopy, and higher-precision quasar absorption observations (with 30\,m-class telescopes) will further test and sharpen BBN (Cooke, 2024).
• Next-generation CMB Stage-4 experiments will probe $N_{\rm eff}$ to $\sigma(N_{\rm eff})\lesssim0.02$, enabling the detection or exclusion of even minimal amounts of dark radiation and delivering competitive $Y_p$ measurements (Grohs et al., 2023, Fields et al., 2019, Cooke, 2024).
• The tight concordance of BBN with D/H and $Y_p$ supports the standard cosmological model back to $t\sim1$ s after the Big Bang, positioning BBN and CMB as complementary early-universe laboratories.

BBN stands as the only process currently simultaneously sensitive to all four fundamental forces and as a uniquely cross-disciplinary probe constraining cosmology, particle physics, and fundamental constants with percent-level precision (Cooke, 2024).