Let us now to turn to the question of concordance between the BBN predictions and the observations discussed above. This is best summarized in a comparison of likelihood functions as a function of the one free parameter of BBN, namely the baryon-to-photon ratio . By combining the theoretical predictions (and their uncertainties) with the observationally determined abundances discussed above, we can produce individual likelihood functions [7] which are shown in Figure 9. A range of primordial ^{7}Li values are chosen based on the the abundances in Eqs. (7) and (8) as well as a higher and lower value. The double peaked nature of the ^{7}Li likelihood functions is due to the presence of a minimum in the predicted lithium abundance in the expected range for . For a given observed value of ^{7}Li, there are two likely values of . As the lithium abundance is lowered, one tends toward the minimum of the BBN prediction, and the two peaks merge. Also shown are both values of the primordial ^{4}He abundances discussed above. As one can see, at this level there is clearly concordance between ^{4}He, ^{7}Li and BBN.
The combined likelihood, for fitting both elements simultaneously, is given by the product of two of the functions in Figure 9. The combined likelihood is shown in Figure 10, for the two primordial values of ^{7}Li in Eqs. (7) and (8). For ^{7}Li_{P} = 1.6 x 10^{-10} (shown as the dashed curve), the 95% CL region covers the range 1.55 < _{10} < 4.45, with the two peaks occurring at _{10} = 1.9 and 3.5. This range corresponds to values of _{B} between
For ^{7}Li_{P} = 1.23 x 10^{-10} (shown as the solid curve), the 95% CL region covers the range 1.75 < _{10} < 3.90. In this case, the primordial value is low enough that the two lithium peaks are more or less merged as is the total likelihood function giving one broad peak centered at _{10} 2.5. The corresponding values of _{B} in this case are between
Figure 10. Combined likelihood distributions for two values of primordial ^{7}Li/H (10^{10} x ^{7}Li = 1.6 (dashed), 1.23 (solid)), and ^{4}He with Y_{P} = 0.238 ± 0.005 ± 0.005 (Eq. (1)). |
When deuterium is folded into the mix, the situation becomes more complicated. Although there are several good measurements of deuterium in quasar absorption systems [26], and many of them giving a low value of D/H (3.4 ± 0.3) x 10^{-5} [27], there remains an observation with D/H nearly an order of magnitude higher D/H (2.0 ± 0.5) x 10^{-4} [28].
Because there are no known astrophysical sites for the production of deuterium, all observed D is assumed to be primordial. As a result, any firm determination of a deuterium abundance establishes an upper bound on which is robust. Thus the ISM measurements [29] of D/H = 1.6 x 10^{-5} imply an upper bound _{10} < 9.
It is interesting to compare the results from the likelihood functions of ^{4}He and ^{7}Li with that of D/H. This comparison is shown in Figure 11. Using the higher value of D/H = (2.0 ± 0.5), we would find excellent agreement between ^{4}He, ^{7}Li and D/H. The predicted range for now becomes
with the peak likelihood value at _{10} = 2.1, ^{4}He and ^{7}Li abundances from eqs. (1) and (8) respectively. This corresponds to _{B}h^{2} = 0.008^{+.004}_{-.002}. The higher ^{7}Li abundance of eq. (7) drops the peak value down slightly to _{10} = 1.8 and broadens the range to 1.5 - 3.4. The higher ^{4}He abundance shifts the peak and range (relative to eq. (12)) up to 2.2 and 1.7 - 3.5.
Figure 12. 50%, 68% & 95% C.L. contours of L_{47} and L_{247} where observed abundances are given by eqs. (1 and 7), and high D/H. |
If instead, we assume that the low value of D/H = (3.4 ± 0.3) x 10^{-5} [27] is the primordial abundance, there is hardly any overlap between the D and ^{7}Li, particularly for the lower value of ^{7}Li from eq. (8). There is also very limited overlap between D/H and ^{4}He, though because of the flatness of the ^{4}He abundance with respect to , as one can see, the likelihood function for the larger value of ^{4}He from eq. (2) is very broad. In this case, D/H is just compatible (at the 2 level) with the other light elements, and the peak of the likelihood function occurs at roughly _{10} = 4.8 and with a range of 4.2 - 5.6.
Figure 13. 50%, 68% & 95% C.L. contours of L_{47} and L_{247} where observed abundances are given by eqs. (1 and 7), and low D/H. |
It is important to recall however, that the true uncertainty in the low D/H systems might be somewhat larger. Mesoturbulence effects [30] allow D/H to be as large as 5 x 10^{-5}. In this case, the peak of the D/H likelihood function shifts down to _{10} 4, and there would be a near perfect overlap with the high ^{7}Li peak and since the ^{4}He distribution function is very broad, this would be a highly compatible solution.
We can obtain still more information regarding the compatibility of the observed abundance and BBN by considering generalized likelihood functions where we allow N_{} to vary as well [7, 31, 32, 4]. The likelihood functions now become functions of two parameters ( , N_{}) . The peaks of the distribution as well as the allowed ranges of and N_{} are easily discerned in the contour plots of Figures 12 and 13 which show the 50%, 68% and 95% confidence level contours in L_{47} and L_{247} projected onto the -N_{} plane, for high and low D/H as indicated. L_{47} corresponds to the likelihood function based on ^{4}He and ^{7}Li only, whereas L_{247} includes D/H as well. The crosses show the location of the peaks of the likelihood functions. L_{47} peaks at N_{} = 3.2, _{10} = 1.85 and at N_{} = 2.6, _{10} = 3.6. The 95% confidence level allows the following ranges in and N_{}
Note however that the ranges in and N_{} are strongly correlated as is evident in Figure 12.
With high D/H, L_{247} peaks at N_{}, and also at _{10} = 1.85. In this case the 95% contour gives the ranges
Note that within the 95% CL range, there is also a small area with _{10} = 3.2-3.5 and N_{} = 2.5-2.9.
Similarly, for low D/H, L_{247} peaks at N_{} = 2.4, and _{10} = 4.55. The 95% CL upper limit is now N_{} < 3.2, and the range for is 3.9 < _{10} < 5.4. It is important to stress that these abundances are now consistent with the standard model value of N_{} = 3 at the 2 level.
Acknowledgments
This work was supported in part by DoE grant DE-FG02-94ER-40823 at the University of Minnesota.