As outlined in Section 3, for fixed as S increases the BBN-predicted abundances of D, ^{3}He, and ^{4}He increase (less time to destroy D and ^{3}He, more neutrons available for ^{4}He), while that of ^{7}Li decreases (less time to produce ^{7}Li). Since it is the ^{4}He mass fraction that is most sensitive to changes in the early Universe expansion rate and, since the SBBN-predicted value of Y_{P} is too large when compared to the data, S < 1 (N_{} < 3) is required. For a slower than standard expansion rate the predicted abundances of D and ^{3}He decrease compared to their SBBN values (at fixed ) while that of ^{7}Li 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 . For _{10} 6 and S - 1 << 1, a good approximation (for fixed D) is _{10} 6(S - 1) (Kneller & Steigman 2003). In Figure 8 are shown the ^{4}He - D (Y_{P} versus D/H) relations for three values of the expansion rate parameterized by N_{}. To first order, the combination of and S that recovers the SBBN deuterium abundance will leave the ^{3}He abundance prediction unchanged as well, preserving its good agreement with the observational data. However, the consequences for ^{7}Li are not so favorable. The BBN abundance of ^{7}Li increases with decreasing S but decreases with a smaller ; the two effects nearly cancel leaving essentially the same discrepancy as for SBBN. For ^{7}Li, a nonstandard expansion rate cannot relieve the tension between the BBN prediction and the observational data.
Setting aside ^{7}Li, 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 ^{4}He; it has already been noted that for this nonstandard case, D and ^{3}He will remain consistent. In Figure 9 are shown the 1, 2, and 3 contours in the N_{} - plane derived from BBN and the D and ^{4}He relic abundances. As expected from the discussion above, the best fit value of (the cross in Figure 9) has shifted downward to _{10} = 5.7 (_{B} = 0.021). While the best fit is for N_{} = - 0.7 (S = 0.94), it should be noted that the standard case of N_{} = 3 is entirely compatible with the data at the ~ 2 level.
Figure 9. The 1, 2, and 3 contours in the N_{} - _{10} plane from BBN and the relic D and ^{4}He abundances. The best fit values of N_{} and _{10} are marked by the cross. |