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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 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, 9 Sep 2024).