Big Bang Nucleosynthesis
Time = ~2 minutes

kT ~ B_D/20, where B_D = 2.2MeV = the binding energy of a deuterium nucleus

The baryon-to-photon ratio

At temperature T, the number density of photons in a blackbody distribution is n~0.244(kT/hc)³. With the present day temperature T_CMB = 2.73K, one can show that the ratio of baryons to photons (n_b / n) = 6.0×10^-10 ( Ω _bh²/0.0222) = 6 x 10^-10 ( Ω _bh₇₀²/0.045) = ~1:10⁹ [where Ω_b is the ratio of the mean density of baryons to the critical density (Hubble constant written as h ≡ H₀/100kms^-1 Mpc^-1 = 0.7h₇₀ and scaled Ω_bh² to the current observational estimate). The photon density itself is 413 photons cm^-3. Roughly speaking, there is one baryon (proton or neutron) for every billion CMB photons.]

Age vs. Temperature

In the earlier radiation-dominated universe, the relation between age and temperature is T(t) ≈ 10¹⁰K(t/ls)^-1/2kT(t) ≈ 1MeV (t/ls)^-1/2. It is interesting to note that for a present-day baryon density of 2 × 10^-7 atoms cm^-3, the number density of baryons when the universe was 1s old was ~ (10¹⁰)³ × 2 × 10^-7 ~ 10²³cm^-3, very roughly the density of water (recall that the proton mass is 1.67 × 10^-24 g). The energy density is much higher than that of water, because it is dominated by radiation, not by matter. A typical photon energy is 1MeV ~ 10^-3m_pc², but with n_γ/n_b ≈ 10⁹ we get Q_γ ~ 10⁶p_bc².

The neutron-to-proton ratio

When the universe is about one second old, the particle species present are photons, neutrinos, and (at an abundance smaller by a factor ~ 10⁹) protons, neutrons, and electrons. Presumably there are also dark matter particles, but they do not matter for BBN. The universe is still dense and hot enough that weak interactions involving neutrinos can convert neutrons to protons and vice versa. Since neutrons are more massive than protons, they are less abundant — conversion of a neutron to a proton is less probable than conversion of a proton to a neutron. In thermal equilibrium n_n/n_p =e^-Q/kT, Q≡(m_n -m_p)c² = 1.2934 MeV. The conversion reactions become slow compared to the age of the universe at t ~ 3 seconds, kT ~ 0.7 MeV. Since the interaction rate is dropping quickly as the density and temperature of the universe decline, there are no subsequent conversions. The neutron-to-proton ratio “freezes in” at n_n/n_p ≈ e^-1.2934/0.7 ≈ 1/6.

Deuterium synthesis

Synthesis of elements heavier than hydrogen has to start with deuterium formation (n+p → D+γ). The binding energy of deuterium is B_D = 2.22MeV. Naively one expects deuterium synthesis to begin when the temperature falls to kT ~ B_D. However, the baryon-to-photon ratio is η ~ 5 × 10^-10, so the exponential tail of the blackbody distribution can still dissociate deuterium even when kT is significantly below B_D. The frequency distribution of photons in a blackbody distribution is dN/dν ∝ ν²/(e^hν/kT– 1) where the last approximation is for hν/kT ≫ 1. Synthesis of deuterium actually begins when kT ~ B_D/-ln η ~ 0.1 MeV, at time t ~2 minutes. The decay time for free neutrons is ~ 900 sec, so in two minutes a small but non-negligible fraction of the neutrons left over from “freeze-out” have decayed. The neutron-to-proton ratio at the time of deuterium synthesis is n/p ~ 1/7.

The ⁴He fraction

The reaction rate for n + p → D + γ is fast, so when the temperature falls below 0.1 MeV, all neutrons are quickly processed into deuterium. The reactions that process D into ⁴He are also fast, and to first order all neutrons go into ⁴He. A robust prediction of the standard big bang model is that 24 ± 1% of the baryonic mass of the universe is in ⁴He , with almost all of the rest in hydrogen. There is a weak dependence of the predicted helium fraction on η, and uncertainty in η is the main contribution to the ±1% uncertainty. Just as the mass fraction of heavy elements is usually written Z, the mass fraction of helium is usually written Y (and hydrogen X). The primordial helium abundance, which is the mass fraction of helium produced in the big bang, not including later contributions from stars, is written Y_P. The Observations: The helium fraction in the interstellar gas of the most metal-poor galaxies is Y = 0.24 ± 0.01. It is difficult to reduce the systematic uncertainties in the observations below 0.01.

Deuterium

A small fraction (~ 10^-4 - 10^-5) of D “escapes” and is never processed into heavier nuclei. This fraction is sensitive to η. Higher η =⇒ less chance of escape, hence smaller deuterium abundance. The primordial deuterium abundance is therefore a good way to determine the mean baryon density of the universe. Stars can destroy deuterium by fusing it into heavier elements, but as far as we know they cannot make it because there are no conditions under which they will make deuterium and not fuse it into heavier elements. The Observations: The measured deuterium-to-hydrogen ratio in the interstellar medium of distant, metal-poor gas absorbing the light of background quasars is (D/H) = (2.6 ± 0.5) × 10^-5. Taking this to be the primordial ratio implies Ω_bh² = 0.021 ± 0.002.

³He and ⁷Li

Incomplete processing leaves a ³He fraction similar to that of D (~ 10-^-5). A small amount (~ 10^-10) of ⁷Li is produced by ⁴He + ³He → ⁷Li. ³He and ⁷Li can both be destroyed and produced in stars, which makes it challenging to determine the primordial value. The Observations: Accounting for the uncertainties in stellar production and destruction, which are a factor of several, measurements are consistent with the BBN (Big Bang Nucleosynthesis) predictions.

The Roadblock

There are no stable elements of atomic number 5 or 8. Starting with protons and neutrons, there is no way to bridge the gap at atomic number 8 to build heavier nuclei. Stars do it by the “triple-α”reaction: ⁴He+⁴He+⁴He-→ 12C*,but this requires high-temperature and density. Consequence: only light elements are made in the early universe. All elements heavier than ⁷Li are made in stars.

The BBN Bottom Line

The standard big bang model, which assumes GR, a homogeneous and isotropic universe, and the standard model of nuclear and particle physics, makes quantitative predictions for the abundances of ⁴He, D, ³He, and ⁷Li produced in the early universe. In the simplest formulation, this theory has one free parameter, the baryon-to-photon ratio η, which has small impact on the predicted ⁴He abundance and substantial impact on the other predictions. There are uncertainties in measuring chemical abundances and in correcting the present-day abundances to primordial values. Within these uncertainties, the observationally inferred primordial abundances of ⁴He, D, ³He, and ⁷Li are consistent with the predictions of standard BBN, for ?η ≈ 5 × 10^-10. The ⁴He test has a precision of ~ 10%, while the others have a precision of a factor of several. This agreement between predictions and observations is strong evidence that the standard big bang model applies back to at least t ~ 1s and T ~ 10¹⁰K. The primordial D/H ratio constrains the baryon density to Ω_bh² = 0.021±0.002, assuming standard BBN. Recently, CMB anisotropies have provided an entirely independent way to constrain the baryon density, yielding Ω_bh² = 0.022 ± 0.001. The agreement of these two independent determinations is further strong evidence for the standard model. This agreement rules out many possible variations on the standard model of particle physics, such as extra light neutrino species, or time-variation of the fundamental constants that control the strength of the nuclear, electromagnetic, and weak forces.