2.1. Weak Equilibrium and the he Abundance

Consider now those early epochs when the Universe was only a few tenths of a second old and the radiation filling it was at a temperature (thermal energy) of a few MeV. According to the standard model, at those early times the Universe was a hot, dense ``soup'' of relativistic particles (photons, e<sup>±</sup> pairs, 3 ``flavors'' (e, µ, ) of neutrino-antineutrino pairs) along with a trace amount (at the level of a few parts in 10¹⁰) of neutrons and protons. At such high temperatures and densities both the weak and nuclear reaction rates are sufficiently rapid (compared to the early Universe expansion rate) that all particles have come to equilibrium. A key consequence of equilibrium is that the earlier history of the evolution of the Universe is irrelevant for an understanding of BBN. When the temperature drops below a few MeV the weakly interacting neutrinos effectively decouple from the photons and e^± pairs, but they still play an important role in regulating the neutron-to-proton ratio.

At high temperatures, neutrons and protons are continuously interconverting via the weak interactions: n + e⁺ <-> p + _e, n + _e <-> p + e^-, and n <-> p + e^- + _e. When the interconversion rate is faster than the expansion rate, the neutron-to-proton ratio tracks its equilibrium value, decreasing exponentially with temperature (n / p = e^{-m / T}, where m = 1.29 MeV is the neutron-proton mass difference). A comparison of the weak rates with the universal expansion rate reveals that equilibrium may be maintained until the temperature drops below ~ 0.8 MeV. When the interconversion rate becomes less than the expansion rate, the n/p ratio effectively ``freezes-out'' (at a value of 1/6), thereafter decreasing slowly, mainly due to free neutron decay.

Although n/p freeze-out occurs at a temperature below the deuterium binding energy, E_B = 2.2 MeV, the first link in the nucleosynthetic chain, p + n -> D + , is ineffective in jump-starting BBN since the photodestruction rate of deuterium ( n e^{-E_B / T}) is much larger than the deuterium production rate ( n_B) due to the very large universal photon-to-baryon ratio ( 10⁹). Thus, the Universe must ``wait'' until there are so few sufficiently energetic photons that deuterium becomes effectively stable against photodissociation. This occurs for temperatures 80 keV, at which time neutrons are rapidly incorporated into he with an efficiency of 99.99%. This efficiency is driven by the tight binding of the ⁴He nucleus, along with the roadblock to further nucleosynthesis imposed by the absence of a stable nucleus at mass-5. By this time (T 80 keV), the n/p ratio has dropped to ~ 1/7, and simple counting (2 neutrons in every ⁴He nucleus) yields an estimated primordial ⁴He mass fraction

(8)

As a result of its large binding energy and the gap at mass-5, the primordial abundance of ⁴He is relatively insensitive to the nuclear reaction rates and, therefore, to the baryon abundance (). As may be seen in Figure 1, while varies by orders of magnitude, the predicted ⁴He mass fraction, Y_P, changes by factors of only a few. Indeed, for 1 ₁₀ 10, 0.22 Y_P 0.25. As may be seen in Figures 1 and 2, there is a very slight increase in Y_P with . This is mainly due to BBN beginning earlier, when there are more neutrons available to form ⁴He, if the baryon-to-photon ratio is higher. The increase in Y_P with is logarithmic; over most of the interesting range in , Y_P 0.01 / .

Figure 2. The predicted 4He abundance (solid curve) and the 2 theoretical uncertainty [3]. The horizontal lines show the range indicated by the observational data.

The ⁴He abundance is, however, sensitive to the competition between the universal expansion rate (H) and the weak interaction rate (interconverting neutrons and protons). If the early Universe should expand faster than predicted for the standard model, the weak interactions will drop out of equilibrium earlier, at a higher temperature, when the n/p ratio is higher. In this case, more neutrons will be available to be incorporated into ⁴He and Y_P will increase. Numerical calculations show that for a modest speed-up (N 1), Y_P 0.013 N. Hence, constraints on Y_P (and ) lead directly to bounds on N and, on particle physics beyond the standard model [1].

It should be noted that the uncertainty in the BBN-predicted mass fraction of ⁴He is very small and almost entirely dominated by the (small) uncertainty in the n - p interconversion rates. These rates may be ``normalized'' through the neutron lifetime, _n, whose current standard value is 887 ± 2 s (actually, 886.7 ± 1.9 s). To very good accuracy, a 1 s uncertainty in _n corresponds to an uncertainty in Y_P of order 2 x 10^-4. At this tiny level of uncertainty it is important to include finite mass, zero- and finite-temperature radiative corrections, and Coulomb corrections to the weak rates. However, within the last few years it emerged that the largest error in the BBN-prediction of Y_P was due to a too large time-step in the numerical code. With this now under control, it is estimated that the residual theoretical uncertainty (in addition to that from the uncertainty in _n) is of the order of 2 parts in 10⁴. Indeed, a comparison of two major, independent BBN codes reveals agreement in the predicted values of Y_P to 0.0001 ± 0.0001 over the entire range 1 ₁₀ 10. In Figure 2 is shown the BBN-predicted he mass fraction, Y_P, as a function of ; the thickness of the band is the ±2 theoretical uncertainty. For ₁₀ 2 the 1 theoretical uncertainty in Y_p is 6 x 10^-4. As we will soon see, the current observational uncertainties in Y_P are much larger (see, also, Fig. 2).