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7. NONSTANDARD BBN: S neq 1, Nnu neq 3

As outlined in Section 3, for fixed eta as S increases the BBN-predicted abundances of D, 3He, and 4He increase (less time to destroy D and 3He, more neutrons available for 4He), while that of 7Li decreases (less time to produce 7Li). Since it is the 4He mass fraction that is most sensitive to changes in the early Universe expansion rate and, since the SBBN-predicted value of YP is too large when compared to the data, S < 1 (Nnu < 3) is required. For a slower than standard expansion rate the predicted abundances of D and 3He decrease compared to their SBBN values (at fixed eta) while that of 7Li increases. Since the BBN-predicted abundance of D increases with decreasing baryon density, a decrease in S can be compensated for by a decrease in eta. For eta10 approx 6 and S - 1 << 1, a good approximation (for fixed D) is Delta eta10 approx 6(S - 1) (Kneller & Steigman 2003). In Figure 8 are shown the 4He - D (YP versus D/H) relations for three values of the expansion rate parameterized by Nnu. To first order, the combination of eta and S that recovers the SBBN deuterium abundance will leave the 3He abundance prediction unchanged as well, preserving its good agreement with the observational data. However, the consequences for 7Li are not so favorable. The BBN abundance of 7Li increases with decreasing S but decreases with a smaller eta; the two effects nearly cancel leaving essentially the same discrepancy as for SBBN. For 7Li, a nonstandard expansion rate cannot relieve the tension between the BBN prediction and the observational data.

Figure 8

Figure 8. The BBN-predicted relation between the 4He mass fraction YP and the deuterium abundance yD for three, early-Universe expansion rates corresponding to Nnu = 2, 3, 4. The filled circle with error bars is for the D and 4He primordial abundances adopted here.

Setting aside 7Li, it is of interest to consider the simultaneous constraints from BBN on the baryon density parameter and the expansion rate factor from the abundances of D and 4He; it has already been noted that for this nonstandard case, D and 3He will remain consistent. In Figure 9 are shown the 1sigma, 2sigma, and 3sigma contours in the Delta Nnu - eta plane derived from BBN and the D and 4He relic abundances. As expected from the discussion above, the best fit value of eta (the cross in Figure 9) has shifted downward to eta10 = 5.7 (omegaB = 0.021). While the best fit is for Delta Nnu = - 0.7 (S = 0.94), it should be noted that the standard case of Nnu = 3 is entirely compatible with the data at the ~ 2sigma level.

Figure 9

Figure 9. The 1sigma, 2sigma, and 3sigma contours in the Delta Nnu - eta10 plane from BBN and the relic D and 4He abundances. The best fit values of Delta Nnu and eta10 are marked by the cross.

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