Next Contents Previous

2.4. Nonstandard BBN

The predictions of the primordial abundance of 4He depend sensitively on the early expansion rate (the Hubble parameter H) and on the amount - if any - of a nue - bar{nu}e asymmetry (the nue chemical potential µe or the neutrino degeneracy parameter xie). In contrast to 4He, the BBN-predicted abundances of D, 3He and 7Li are determined by the competition between the various two-body production/destruction rates and the universal expansion rate. As a result, the D, 3He, and 7Li abundances are sensitive to the post-e± annihilation expansion rate, while that of 4He depends on both the pre- and post-e± annihilation expansion rates; the former determines the "freeze-in" and the latter modulates the importance of beta-decay (see, e.g., Kneller & Steigman 2003). Also, the primordial abundances of D, 3He, and 7Li, while not entirely insensitive to neutrino degeneracy, are much less affected by a nonzero xie (e.g., Kang & Steigman 1992). Each of these nonstandard cases will be considered below. Note that the abundances of at least two different relic nuclei are needed to break the degeneracy between the baryon density and a possible nonstandard expansion rate resulting from new physics or cosmology, and/or a neutrino asymmetry.

2.4.1. Additional Relativistic Energy Density

The most straightforward variation of SBBN is to consider the effect of a nonstandard expansion rate H' neq H. To quantify the deviation from the standard model it is convenient to introduce the "expansion rate factor" (or speedup/slowdown factor) S, where

Equation 7 (7)

Such a nonstandard expansion rate might result from the presence of "extra" energy contributed by new, light (relativistic at BBN) particles "X". These might, but need not, be additional flavors of active or sterile neutrinos. For X particles that are decoupled, in the sense that they do not share in the energy released by e± annihilation, it is convenient to account for the extra contribution to the standard-model energy density by normalizing it to that of an "equivalent" neutrino flavor (Steigman et al. 1977),

Equation 8 (8)

For SBBN, Delta Nnu = 0 (Nnu ident 3 + Delta Nnu) and for each such additional "neutrino-like" particle (i.e. any two-component fermion), if TX = Tnu, then Delta Nnu = 1; if X should be a scalar, Delta Nnu = 4/7. However, it may well be that the X have decoupled even earlier in the evolution of the Universe and have failed to profit from the heating when various other particle-antiparticle pairs annihilated (or unstable particles decayed). In this case, the contribution to Delta Nnu from each such particle will be < 1 (< 4/7). Henceforth we drop the X subscript. Note that, in principle, we are considering any term in the energy density that scales like "radiation" (i.e. decreases with the expansion of the Universe as the fourth power of the scale factor). In this sense, the modification to the usual Friedman equation due to higher dimensional effects, as in the Randall-Sundrum model (Randall & Sundrum 1999a, b; see also Cline, Grojean, & Servant 1999; Binetruy et al. 2000; Bratt et al. 2002), may be included as well. The interest in this latter case is that it permits the possibility of an apparent negative contribution to the radiation density (Delta Nnu < 0; S < 1). For such a modification to the energy density, the pre-e± annihilation energy density in Equation 1 is changed to

Equation 9 (9)

Since any extra energy density (Delta Nnu > 0) speeds up the expansion of the Universe (S > 1), the right-hand side of the time-temperature relation in Equation 3 is smaller by the square root of the factor in parentheses in Equation 9.

Equation 10 (10)

In the post-e± annihilation Universe the extra energy density is diluted by the heating of the photons, so that

Equation 11 (11)


Equation 12 (12)

While the abundances of D, 3He, and 7Li are most sensitive to the baryon density (eta), the 4He mass fraction (Y) provides the best probe of the expansion rate. This is illustrated in Figure 2 where, in the Delta Nnu - eta10 plane, are shown isoabundance contours for D/H and YP (the isoabundance curves for 3He/H and for 7Li/H, omitted for clarity, are similar in behavior to that of D/H). The trends illustrated in Figure 2 are easy to understand in the context of the discussion above. The higher the baryon density (eta10), the faster primordial D is destroyed, so the relic abundance of D is anticorrelated with eta10. But, the faster the Universe expands (Delta Nnu > 0), the less time is available for D destruction, so D/H is positively, albeit weakly, correlated with Delta Nnu. In contrast to D (and to 3He and 7Li), since the incorporation of all available neutrons into 4He is not limited by the nuclear reaction rates, the 4He mass fraction is relatively insensitive to the baryon density, but it is very sensitive to both the pre- and post-e± annihilation expansion rates (which control the neutron-to-proton ratio). The faster the Universe expands, the more neutrons are available for 4He. The very slow increase of YP with eta10 is a reflection of the fact that for a higher baryon density, BBN begins earlier, when there are more neutrons. As a result of these complementary correlations, the pair of primordial abundances yD ident 105(D / H)P and YP, the 4He mass fraction, provide observational constraints on both the baryon density (eta) and on the universal expansion rate factor S (or on Delta Nnu) when the Universe was some 20 minutes old. Comparing these to similar constraints from when the Universe was some 380 Kyr old, provided by the WMAP observations of the CBR polarization and the spectrum of temperature fluctuations, provides a test of the consistency of the standard models of cosmology and of particle physics and further constrains the allowed range of the present-Universe baryon density (e.g., Barger et al. 2003a, b; Crotty, Lesgourgues, & Pastor 2003; Hannestad 2003; Pierpaoli 2003).

Figure 2

Figure 2. Isoabundance curves for D and 4He in the Delta Nnu - eta10 plane. The solid curves are for 4He (from top to bottom: Y = 0.25, 0.24, 0.23). The dotted curves are for D (from left to right: yD ident 105(D/H) = 3.0, 2.5, 2.0). The data point with error bars corresponds to yD = 2.6 ± 0.4 and YP = 0.238 ± 0.005; see the text for discussion of these abundances.

2.4.2. Neutrino Degeneracy

The baryon-to-photon ratio provides a dimensionless measure of the universal baryon asymmetry, which is very small (eta ltapprox 10-9). By charge neutrality the asymmetry in the charged leptons must also be of this order. However, there are no observational constraints, save those to be discussed here (see Kang & Steigman 1992; Kneller et al. 2001, and further references therein), on the magnitude of any asymmetry among the neutral leptons (neutrinos). A relatively small asymmetry between electron type neutrinos and antineutrinos (xie gtapprox 10-2) can have a significant impact on the early-Universe ratio of neutrons to protons, thereby affecting the yields of the light nuclides formed during BBN. The strongest effect is on the BBN 4He abundance, which is neutron limited. For xie > 0, there is an excess of neutrinos (nue) over antineutrinos (bar{nu}e), and the two-body reactions regulating the neutron-to-proton ratio (Eq. 5) drive down the neutron abundance; the reverse is true for xie < 0. The effect of a nonzero nue asymmetry on the relic abundances of the other light nuclides is much weaker. This is illustrated in Figure 3, which shows the D and 4He isoabundance curves in the xie - eta10 plane. The nearly horizontal 4He curves reflect the weak dependence of YP on the baryon density, along with its significant dependence on the neutrino asymmetry. In contrast, the nearly vertical D curves reveal the strong dependence of yD on the baryon density and its weak dependence on any neutrino asymmetry (3He/H and 7Li/H behave similarly: strongly dependent on eta, weakly dependent on xie). This complementarity between yD and YP permits the pair {eta, xie} to be determined once the primordial abundances of D and 4He are inferred from the appropriate observational data.

Figure 3

Figure 3. Isoabundance curves for D and 4He in the xie - eta10 plane. The solid curves are for 4He (from top to bottom: YP = 0.23, 0.24, 0.25). The dotted curves are for D (from left to right: yD ident 105(D/H) = 3.0, 2.5, 2.0.) The data point with error bars corresponds to yD = 2.6 ± 0.4 and YP = 0.238 ± 0.005; see the text for discussion of these abundances.

Next Contents Previous