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5.1. Degenerate BBN

There is another alternative to SBBN which, although currently less favored, does have a venerable history: BBN in the presence of a background of degenerate neutrinos. First, a brief diversion to provide some perspective. In the very early universe there were a large number of particle-antiparticle pairs of all kinds. As the baryon-antibaryon pairs or, their quark-antiquark precursors, annihilated, only the baryon excess survived. This baryon number excess, proportional to eta, is very small (eta ltapprox 10-9). It is reasonable, but by no means compulsory, to assume that the lepton number asymmetry (between leptons and antileptons) is also very small. Charge neutrality of the universe ensures that the electron asymmetry is of the same order as the baryon asymmetry. But, what of the asymmetry among the several neutrino flavors?

Since the relic neutrino background has never been observed directly, not much can be said about its asymmetry. However, if there is an excess in the number of neutrinos compared to antineutrinos (or, vice-versa), "neutrino degeneracy", the total energy density in neutrinos (plus antineutrinos) is increased. As a result, during the early, radiation-dominated evolution of the universe, rho -> rho' > rho, and the universal expansion rate increases (S > 1). Constraints on how large S can be do lead to some weak bounds on neutrino degeneracy (see, e.g. Kang & Steigman 1992 and references therein). This effect occurs for degeneracy in all neutrino flavors (nue, nuµ, and nutau). For fixed baryon density, S > 1 leads to an increase in D/H (less time to destroy D), more 4He (less time to transform neutrons into protons), and a decrease in lithium (at high eta there is less time to produce 7Be). Recall that for S = 1 (SBBN), an increase in eta results in less D (more rapid destruction), which can compensate for S > 1. Similarly, an increase in baryon density will increase the lithium yield (more rapid production of 7Be), also tending to compensate for S > 1. But, at higher eta, more 4He is produced, further exacerbating the effect of a more rapidly expanding universe.

However, electron-type neutrinos play a unique role in BBN, mediating the neutron-proton transformations via the weak interactions (see eq. 2.24). Suppose, for example, there are more nue than bar{nu}e. If µe is the nue chemical potential, then xie ident µe / kT is the "neutrino degeneracy parameter"; in this case, xie > 0. The excess of nue will drive down the neutron-to-proton ratio, leading to a reduction in the primordial 4He mass fraction. Thus, a combination of three adjustable parameters, eta, Delta Nnu, and xie may be varied to "tune" the primordial abundances of D, 4He, and 7Li. In Kneller et al. (2001; KSSW), we chose a range of primordial abundances similar to those adopted here (2 leq 105(D/H)P leq 5; 0.23 leq YP leq 0.25; 1 leq 1010(Li/H)P leq 4) and explored the consistent ranges of eta, xie geq 0, and Delta Nnu geq 0. Our results are shown in Figure 16.

Figure 16

Figure 16. Isoabundance contours for D, 4He, and 7Li in the Delta Nnu - eta10 plane for four choices of the nue degeneracy parameter xie. The shaded areas identify the ranges of parameters consistent with the abundances indicated (see the text).

It is clear from Figure 16 that for a large range in eta, a combination of Delta Nnu and xie can be found so that the BBN-predicted abundances will lie within our adopted primordial abundance ranges. However, there are constraints on eta and Delta Nnu from the CMB temperature fluctuation spectrum (see KSSW for details and further references). Although the CMB temperature fluctuation spectrum is insensitive to xie, it will be modified by any changes in the universal expansion rate. While SBBN (Delta Nnu = 0) is consistent with the combined constraints from BBN and the CMB (see Section 4.5) for eta10 approx 5.8 (OmegaB h2 approx 0.021), values of Delta Nnu as large as Delta Nnu ltapprox 6 are also allowed (KSSW).

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