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1.2. Primordial Nucleosynthesis

Nuclear reactions set in when the temperature is down to about 0.1 Mev, since before that deuterium is prevented by photo-disintegration from building up to a sufficient abundance to allow further reactions to occur (the deuterium bottleneck). Physical constants affecting the outcome are, naturally, the relevant nuclear reaction cross-sections and two factors affecting the neutron-proton ratio at the onset of synthesis, namely Nnu and the weak interaction constant GF; GF2 is inversely proportional to the half-life of the neutron, tau1/2, which has to be found from experiment.

Fig. 1 shows the predicted abundances from primordial nucleosynthesis according to Standard (i.e. homogeneous) Big Bang Nucleosynthesis (SBBN) theory as a function of eta, or, equivalently, of Omegab0 h02 where Omegab0 is the fraction of the cosmological closure density 3H02 / (8piG) = 1.96 × 10-29h02 gm cm-3 supplied by baryons, H0 is the Hubble constant and h0 or h100 the same in units of 100 km s-1 Mpc-1 which lies somewhere between 0.5 and 1. eta is related to Omegab0 through the known temperature of the microwave background

Equation 5      (5)

where the suffix zero refers to the present epoch and eta10 is eta in units of 10-10. The trends in fig. 1 arise from the series of nuclear reactions starting with


where the nuclear chain is temporarily halted because there are no stable nuclei with mass numbers 5 and 8. The main effect is to build up 4He with a mass fraction

Equation 6      (6)

where n / p is the neutron-proton ratio at the onset of synthesis, but traces of D and 3He survive because expansion and cooling slow down nuclear reactions before their destruction is complete. The two-body character of the reactions that destroy deuterium leads to the steep decrease in its abundance with eta. Later traces of 7Li and 7Be are also built up, the latter eventually decaying to 7Li by K-capture:


Figure 1

Figure 1. Primordial abundances predicted from SBBN theory after Yang et al. (1984). Olive et al. (1990) and Deliyannis et al. (1990), 7Li from the latter reference being shown with ±2sigma error limits, as functions of eta and Omegab0 h1002. Helium abundances are given for two possible values of the neutron half-life assuming Nnu = 3. Horizontal lines show upper limits (and a lower limit for D) based on observation and on reasonable galactic chemical evolution considerations. Tal vertical lines show the corresponding limits on the density parameters from SBBN while the shorter double vertical lines show approximate limits from midly inhomogeneous models adapted from Kurki-Suonio et al. (1990).

At low densities, the first of these reactions is the main contributor to 7Li and its abundance is a decreasing function of nucleon density because T and 7Li are both destroyed in two-body reactions. At higher densities the second reaction predominates; 3He and 7Be are more robust, so that the 7Li curve turns round and rises at higher densities leading to a minimum in the interesting range.

Helium itself increases only very slowly with eta because virtually all the neutrons initially present are soaked up in its production, as expressed by equation (6). However, its abundance is also significantly affected by Nnu and tau1/2 because of the effects of these constants on the initial n / p ratio. Before (electron) neutrino decoupling, this ratio is kept in thermal equilibrium by weak interactions so that

Equation 7       (7)

whereas afterwards the ratio is virtually frozen at the value given by (7) with T equal to the decoupling temperature Td. Because the weak-interaction cross section is proportional to (energy)2 or T2, the reaction time scale varies as (nnu GF2 T2)-1 or tau1/2 T-5, where nnu is the number density of electron neutrinos, whereas the expansion time scale is ~ (G rho)-1/2 proportional to (11/4 + (7/8)Nnu)-1/2 T-2, so the decoupling temperature varies as tau1/21/3(11/4 + (7/8)Nnu)1/6. This means that larger values of tau1/2 or Nnu, lead to higher Td and hence to larger n / p and larger primordial helium abundances. After decoupling, n / p decreases only slowly through free decay (and residual weak interactions) leading to a very slow dependence on eta given by

Equation 8       (8)

(Olive et al. 1990) for 2.5 leq eta10 leq 10 and with tau1/2 in minutes. Recent experimental values of tau1/2 are 10.25 (Mampe et al. 1989; Gudkov et al. 1990) and 10.32 (Byrne et al. 1990). The sensitivity to Nnu, first noted by Hoyle & Tayler (1964), and the relatively high degree of precision to which Yp can be estimated (see below), have enabled cosmological limits to be placed on the number of neutrino families, first no more than 5 (Steigman, Schramm & Gunn 1977), then 4 (Yang et al. 1984) and finally 3 (Pagel 1988; Pagel & Simonson 1989; Olive et al. 1990) which was confirmed in accelerator experiments on the Zo (e.g. Ellis, Salati & Shaver 1990), but with the provisos that SBBN does not exclude neutrinos massive enough to be non-relativistic at a few Mev, which are excluded up to 45 Gev by Zo decay, and conversely SBBN does exclude hypothetical light particles coupling to photons but not to the Zo. Consistency of SBBN theory also imposes an upper limit on tau1/2 of 10.4 minutes. Upper and lower limits to eta and Omegab0 h02 resulting from a comparison of SBBN predictions with primordial abundances deduced by various means from observations, as discussed below, are shown by the tall vertical lines in fig. 1.

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