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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 eta. 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 7Li 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 7Li likelihood functions is due to the presence of a minimum in the predicted lithium abundance in the expected range for eta. For a given observed value of 7Li, there are two likely values of eta. 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 4He abundances discussed above. As one can see, at this level there is clearly concordance between 4He, 7Li and BBN.

Figure 9

Figure 9. Likelihood distributions for four values of primordial 7Li/H (1010 x 7Li = 1.9 (dashed), 1.6 (dotted), 1.23 (solid), and 0.9 (dash-dotted)), and for 4He (shaded) for which we adopt YP = 0.238 ± 0.005 (Eq. (1)). Also shown by the long dashed curve is the likelihood function based on the 4He abundance from Eq. (2).

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 7Li in Eqs. (7) and (8). For 7LiP = 1.6 x 10-10 (shown as the dashed curve), the 95% CL region covers the range 1.55 < eta10 < 4.45, with the two peaks occurring at eta10 = 1.9 and 3.5. This range corresponds to values of OmegaB between

Equation 10 (10)

For 7LiP = 1.23 x 10-10 (shown as the solid curve), the 95% CL region covers the range 1.75 < eta10 < 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 eta10 appeq 2.5. The corresponding values of OmegaB in this case are between

Equation 11 (11)

Figure 10

Figure 10. Combined likelihood distributions for two values of primordial 7Li/H (1010 x 7Li = 1.6 (dashed), 1.23 (solid)), and 4He with YP = 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 appeq (3.4 ± 0.3) x 10-5 [27], there remains an observation with D/H nearly an order of magnitude higher D/H appeq (2.0 ± 0.5) x 10-4 [28].

Figure 11

Figure 11. Likelihood distributions for two values of primordial 7Li/H (1010 x 7Li = 1.6 (dashed) and 1.23 (solid)), and 4He with YP = 0.238 ± 0.002 ± 0.005 from Eq. (1) (shaded) and YP = 0.244 ± 0.002 ± 0.005 from Eq. (2) (long dashed). Also shown are the two likelihood functions for high and low D/H as marked.

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 eta which is robust. Thus the ISM measurements [29] of D/H = 1.6 x 10-5 imply an upper bound eta10 < 9.

It is interesting to compare the results from the likelihood functions of 4He and 7Li 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 4He, 7Li and D/H. The predicted range for eta now becomes

Equation 12 (12)

with the peak likelihood value at eta10 = 2.1, 4He and 7Li abundances from eqs. (1) and (8) respectively. This corresponds to OmegaBh2 = 0.008+.004-.002. The higher 7Li abundance of eq. (7) drops the peak value down slightly to eta10 = 1.8 and broadens the range to 1.5 - 3.4. The higher 4He abundance shifts the peak and range (relative to eq. (12)) up to 2.2 and 1.7 - 3.5.

Figure 12

Figure 12. 50%, 68% & 95% C.L. contours of L47 and L247 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 7Li, particularly for the lower value of 7Li from eq. (8). There is also very limited overlap between D/H and 4He, though because of the flatness of the 4He abundance with respect to eta, as one can see, the likelihood function for the larger value of 4He from eq. (2) is very broad. In this case, D/H is just compatible (at the 2sigma level) with the other light elements, and the peak of the likelihood function occurs at roughly eta10 = 4.8 and with a range of 4.2 - 5.6.

Figure 13

Figure 13. 50%, 68% & 95% C.L. contours of L47 and L247 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 eta10 appeq 4, and there would be a near perfect overlap with the high eta 7Li peak and since the 4He 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 Nnu to vary as well [7, 31, 32, 4]. The likelihood functions now become functions of two parameters curlyL(eta , Nnu) . The peaks of the distribution as well as the allowed ranges of eta and Nnu are easily discerned in the contour plots of Figures 12 and 13 which show the 50%, 68% and 95% confidence level contours in L47 and L247 projected onto the eta-Nnu plane, for high and low D/H as indicated. L47 corresponds to the likelihood function based on 4He and 7Li only, whereas L247 includes D/H as well. The crosses show the location of the peaks of the likelihood functions. L47 peaks at Nnu = 3.2, eta10 = 1.85 and at Nnu = 2.6, eta10 = 3.6. The 95% confidence level allows the following ranges in eta and Nnu

Equation 13 (13)

Note however that the ranges in eta and Nnu are strongly correlated as is evident in Figure 12.

With high D/H, L247 peaks at Nnu, and also at eta10 = 1.85. In this case the 95% contour gives the ranges

Equation 14 (14)

Note that within the 95% CL range, there is also a small area with eta10 = 3.2-3.5 and Nnu = 2.5-2.9.

Similarly, for low D/H, L247 peaks at Nnu = 2.4, and eta10 = 4.55. The 95% CL upper limit is now Nnu < 3.2, and the range for eta is 3.9 < eta10 < 5.4. It is important to stress that these abundances are now consistent with the standard model value of Nnu = 3 at the 2sigma level.


This work was supported in part by DoE grant DE-FG02-94ER-40823 at the University of Minnesota.

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