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Deuterium and big-bang nucleosynthesis: implications for the baryon density
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Planetary and Space Science 50 (2002) 1245 – 1250 www.elsevier.com/locate/pss
Deuterium and big-bang nucleosynthesis: implications for the baryon density Scott Burles Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA Received 26 November 2001; received in revised form 27 May 2002; accepted 30 May 2002
Abstract A new Monte Carlo technique is introduced to extract abundances and uncertainties in the framework of standard big-bang nucleosynthesis. Both very direct and less conservative, this method yields uncertainties in light element abundance predictions which are almost a factor of 3 smaller than that found with previous techniques. Key nuclear reaction rates are identi4ed for which future nuclear cross-section measurements will further signi4cantly reduce the abundance uncertainties. Finally, constraints on the baryon density inferred from BBN when combined with recent measurements of extragalactic deuterium yield b h2 = 0:020 ± 0:001 which agrees precisely with the latest constraints from power spectrum measurements of cosmic microwave background anisotropies. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
2. Sharpening big-bang nucleosynthesis
Deuterium plays a central role in the study of the light elements and the epoch of primordial nucleosynthesis. The observed abundance of deuterium together with other light isotopes of helium and lithium are naturally explained with a primordial origin and production via standard big-bang nucleosynthesis (BBN). The successful predictions of BBN have lent it to serve as a cornerstone of modern cosmological models. The precision of BBN predictions depend decisively on both the observational uncertainties of light element abundance measurements and input uncertainties on a handful of critical nuclear cross-sections. Dramatic improvements (or sharpening) of direct constraints drawn from BBN rely on reducing uncertainties from both observational and nuclear sources. For my contribution to these proceedings, I would like to detail the source and size of the nuclear input uncertainties in the current prediction of the primordial deuterium abundance. These will be compared with previous estimates and techniques, as well as with the recent observational constraints on primordial deuterium.
To continually test and apply constraints derived from BBN, we must both tighten the observational limits on the primordial light element abundances as well as sharpen the predictions based on nuclear input reaction rates. Dear to the theme of this conference, I will take as a case study the standard BBN prediction for deuterium and the extracted limits on the baryon density. In Fig. 1, I outline the relevant uncertainties for this purpose. In order to extract sharp limits on the baryon density, we need to characterize and combine the limits from the astrophysical observations and the nuclear input data. In this summary, I will focus on the propagation of the nuclear input uncertainties. For studies and reviews pertaining to the best limits on primordial deuterium measurements, the reader is politely directed to York (2002), Tytler et al. (2000), Burles et al. (2001a), and Olive et al. (2000). For argument’s sake, I will adopt a 95% cl limit of
E-mail address: [email protected] (S. Burles).
(D=H )p = (3:0 ± 0:4) × 10−5 95% cl;
(1)
based on a weighted treatment of the high redshift deuterium abundances observed with large ground-based telescopes and spectrographs (Burles and Tytler, 1998a, b; O’Meara et al., 2001; Levshakov et al., 2002). This limit is depicted as the left most error-bar in Fig. 1.
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 2 ) 0 0 0 9 0 - 9
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S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 1. From Burles et al. (2001a): 95% con4dence limits on the deuterium abundance associated with standard BBN. This example is given for = 5:6 × 10−10 . The left-most error bar is the current best combined limits from high redshift measurements of D/H (O’Meara et al., 2001; Burles et al., 2001b). For comparison, the corresponding limits on the nucleosynthesis predictions of Smith et al. (1993). is shown next. The full limits on deuterium from a full Monte Carlo analysis (Nollett and Burles, 2000) and the relative individual reaction contributions (computed by adopting SKM rates for all but the reaction in question) 4ll out the remaining the limits.
2.1. What is the di1erence? How important is the sharpening of the BBN predictions? The answer depends on the isotope in question. For 4 He, the current state of the nuclear uncertainties is quite small compared to the statistical and intrinsic uncertainties in the astrophysical data sets. On the other extreme, for 7 Li, the combined BBN nuclear uncertainty signi4cantly outweighs the statistical uncertainty of the observed level of the Spite plateau (Ryan et al., 1999; Bonifacio and Molaro, 1997). And in between, the BBN uncertainty related to deuterium is split rather fairly between observations and nuclear inputs. If this is the case, constraints derived from deuterium can only be as stringent as both sources of uncertainty allow. In other words, progress can only be gained by reducing both sources of uncertainty: one without the other will not lead to signi4cant improvements. Continuing with Fig. 1, the next two error-bars show the limits on the deuterium predictions from two recent Monte Carlo investigations of the nuclear input rates and standard BBN. The last four smaller error-bars denote the relevant single reaction rate uncertainty contribution to the “full MC” result (just add the last four limits in quadrature to calculate the full result). The diIerence between the limits from the earlier work of Smith et al. (SKM) (1993) and Krauss and Romanelli (1990), and a more recent study described 4rst in Burles et al. (1999) (full MC, hereafter cited as “direct Monte Carlo”) is dramatic and important to appreciate. The more recent limits are reduced by almost a factor of 3 over the earlier work. This can be simply explained with
two reasons: (1) New data became available in the interim, which lead to better and tighter characterization of the cross sections important for deuterium, and (2) even more signi4cant, a new technique is introduced to directly relate the 4nal abundance uncertainty (as shown in Fig. 1) to the original nuclear cross-section data at the energies relevant for standard BBN. The highlights of the direct Monte Carlo technique are (1) correct treatment of the random and correlated errors in nuclear data sets, (2) an aggressive assessment of the statistical uncertainties extracted with a direct Monte Carlo of the individual data points, (3) Jexible spline 4tting to allow for possible variations in the cross-section uncertainties as a function of energy, and (4) the reaction rates are numerically integrated from the splined cross-section realizations, which complements functional derivative studies of the cross-sections versus energy. In the remainder of this section, I will describe in detail the diIerences in reaction rates which determine the deuterium abundance. For other opinions on the subject, the reader is directed to recent analyses of nuclear input data using similar Monte Carlo techniques (Vangioni-Flam et al., 2001; Cyburt et al., 2001). The four individual reactions shown in Fig. 1 are critical to the accurate prediction of the deuterium in high-entropy primordial nucleosynthesis. For a synopsis of the 12 critical reactions relevant to light element productions up to mass seven as well as an in-depth description of the direct Monte Carlo technique, the reader is invited to review Nollett and Burles (2000) and references therein. 2.2. The four important reactions The 4rst reaction example is d(d; n) 3 He. Fig. 2 displays the appropriate S-factor (the cross-section normalized with the Coulomb barrier and energy) as a function of center of mass energy. The energy range is chosen to encompass the important range for BBN. The data are shown as symbols with 67% total error bars. Seven nuclear experiments are included in Fig. 2; this is eIectively all of the extant available data on this reaction which has (1) some energy coverage in the displayed range and (2) clearly separates and reports both random and correlated statistical uncertainties. For comparison, I show cross section 4ts to the data from three separate studies. The solid lines show the result of our “direct Monte Carlo”, the central line is the most likely 4t as a function of energy, with 95% cl represented by the two solid outer lines. The light dash-dotted line represents the analytic 4t by SKM and the associated limits are shown as dot-dot-dot-dashed lines. And the darker dashed line represents the suggested best 4t using both theory and data of the cross-section working group (ENDF/B-VI) (ENDF/B-VI Summary Documentation, 1991). Some important points to note about the 4ts and this reaction: (1) all three central 4ts agree very well, (2) the biggest diIerence is the uncertainty level as a function of energy between the two Monte Carlo methods.
S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 2. S-factor and sensitivity curves for d(d,n) at the center of mass energies relevant for standard BBN. See text for description and details.
The bottom two panels represent the intrinsic sensitivity of the 4nal abundances (for D and 7 Li) to a cross-section perturbation at the speci4c energy (b =0:02; 0:01 for solid and dashed lines, respectively). These can actually be de4ned as functional derivatives and they directly quantify, relatively, the important range of energy for each reaction in standard BBN. In addition, the horizontal bar (at 30 keV b) with vertical tick marks displays extreme limits in energy. Disregarding the cross-section outside of the inner tick marks in the reaction rate integral modi4es the 4nal abundances by less than 1/10 of a standard deviation. And disregarding the cross-section outside of the outer ticks changes the 4nal abundances by less than 1 part in 105 ! The uncertainty contribution due to the diIerent Monte Carlo treatments can be assessed by eye. At the peak (or valley) of the sensitivity function at Ecm ≈ 0:14 MeV, the resulting uncertainties are at least a factor of two smaller by applying the direct Monte Carlo technique. The next three 4gures (Figs. 3–5) show the best available nuclear data and similar 4ts as explained for d(d,n). The derived errors on d(d,p) are small and well characterized across the entire energy range shown in Fig. 3. Out of the four reactions, this contributes the least amount of uncertainty. On the other extreme, the experimental data sets for d(p; ) and p(n; ) are sparse at best. The direct Monte
Fig. 3. Same as Fig. 2, but for the reaction d(d,p).
Fig. 4. Same as Fig. 2, but for the reaction d(p; ).
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S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 5. Same as Fig. 2, but for the reaction p(n; ). The “data” points of Chen are representative eIective 4eld theory calculations (Chen and Savage, 1999). Even more precise eIective 4eld theoretical results have been presented by Rupak (2000).
Carlo procedure does converge for d(p; ) but just barely. There are no overlapping data sets, and the critical range near 100 keV is covered only by one recent experiment (Ma et al., 1997). For p(n; ) shown in Fig. 5, the direct Monte Carlo method breaks down entirely. The shape of the neutron capture cross section cannot be recovered from the extant data at the energies in question. To proceed, and gainfully include the uncertainties introduced from p(n; ), we adopt the theoretical 4t (Hale et al., 1997) with a suggested overall normalization uncertainty of 5%. The theoretical 4ts is well motivated and, for comparison, agrees with predictions from eIective 4eld theory (shown as dots with errors). Fig. 6 is simply a two-dimensional version of Fig. 1. Now the fractional uncertainties of the direct Monte Carlo, as well as the four critical reactions below, are plotted as a function of baryon-to-photon ratio. The zero-point in each case is shown as a horizontal dashed line and represent the abundances of deuterium calculated by SKM. The shaded envelopes show the fractional uncertainties at 95% cl, and oIsets, of the direct Monte Carlo abundance yields relative to SKM abundances. The dash-dotted lines enclosing the combined 2-sigma uncertainty in the SKM comparison.
Fig. 6. The relative BBN deuterium abundance 95% uncertainties as a function of baryon to photon ratio (normalized to 10−10 ) and spread among the critical reactions. The error bar on the far right-hand side shows the current 67% cl observational uncertainties for comparison.
3. Implications for the baryon density In all models of primordial nucleosynthesis, the 4nal deuterium abundance is very sensitive to the speci4c entropy during nucleosynthesis. For standard BBN, this is usually parameterized as the inverse speci4c entropy: the present-day baryon to photon ratio, . In standard cosmology with adiabatic expansion and no decaying relic particles, this ratio remains essentially unchanged since the epoch of electron positron annihilation (age of the universe ≈ 100 s). The present-day baryon energy density b is then directly proportional to the present-day photon number density inferred from an extremely accurate measurement of the temperature of the cosmic microwave background (CMB) Mather et al. (1999). This physical baryon density can be expressed as a fraction of the critical cosmic density with the expected Hubble constant dependence: b h2 = (3:650 ± 0:008) × 107 ;
(2)
where h represents a Hubble parameter normalized to the 100 km=s=Mpc and the uncertainty is a 95% limit due to the small temperature uncertainty of CMB.
S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 7. Predicted big-bang abundances shown as 95% con4dence bands (Burles et al., 2001b) with recent light element abundance determinations shown as boxes and arrows (also 95%). See text for details.
Fig. 7 shows the most recent light element abundance predictions as a function of both b h2 (above) and (below). The vertical thickness of the abundance predictions represents the 95% limits derived from direct Monte Carlo. The vertical placement and height of each box represents a single measurement in the case of deuterium and large statistical samples in the case of 4 He (Izotov and Thuan, 1998; Olive et al., 1998) and 7 Li (Ryan et al., 1999). The horizontal placement of the boxes are set to overlap the 95% limits of the predicted abundances. The vertical solid band is the weighted deuterium constraint cited above, which combines the measurements of the four overlapping boxes. This version of light-element abundances versus baryon density may be considered enlightening, confusing, or both. For concordance with standard BBN, all primordial abundances should agree on one value of the baryon density. There is one sample of 4 He measurements which is at odds with the deuterium measurements (Olive et al., 1998). And even more striking, the Spite plateau Lithium abundance is about a factor of two lower than expected (Ryan et al., 1999). And the latest deuterium measurement by Pettini and Bowen (2001) gives a central value equal to the interstel-
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lar medium today and about a factor of two lower than the weighted mean of other measurements. So what should one take away from Fig. 7? I believe the deuterium measurements are converging on a unique cosmological value. It is and will continue to be a diMcult measurement, and known systematics will contribute additional scatter in the sample. Lithium may yet have some astrophysical processing during the main sequence lifetime of the representative halo stars in the plateau. Helium measurements, of both isotopes, are diMcult and will require further attention. The real excitement is still to come. The CMB anisotropy power spectrum is sensitive to the baryon density on scales of 1/2 degree and smaller. Current and upcoming interferometric measurements by DASI (Pryke et al., 2002, submitted for publication) and balloon-borne measurements by BOOMERANG and MAXIMA-1 (JaIe et al., 2001) will place tighter and accurate independent constraints on the physical baryon density. Combined these will approach the precision of the deuterium inferred constraints with standard BBN. So in the next year, we can expect to compare the baryon density from independent measures to the level of √ 2 × 10% (Schramm and Turner, 1998). It gets better: the arrival of the MAP satellite at the second Lagrange point (Page, 2001) and the acquisition of a full-sky high resolution map, open the possibility for an even more precise comparison. A full-sky map of this quality has the potential to constrain the baryon density to much better than 5%. In order to take full advantage of this upcoming test, the observations of light elements must continue to improve, and predictions of BBN must be made as sharp as possible.
Acknowledgements I am delighted to thank the Observatoire de Meudon and the local organizing committee for a unique and exciting conference. I also am deeply indebted to my collaborators in the work described here: D. Kirkman, M. Lemoine, K. Nollett, J. Truran, M. Turner, and D. Tytler.
References Bonifacio, P., Molaro, P., 1997. Mon. Not. R. Astron. Soc. 285, 847. Burles, S., Tytler, D., 1998a. Astrophys. J. 499, 699. Burles, S., Tytler, D., 1998b. Astrophys. J. 507, 732. Burles, S., et al., 1999. Phys. Rev. Lett. 82, 4176. Burles, S., Nollett, K.M., Turner, M.S., 2001a. Phys. Rev. D 63, 063512. Burles, S., Nollett, K.M., Turner, M.S., 2001b. Astrophys. J. 552, L1. Chen, J.-W., Savage, M.J., 1999. Phys. Rev. C 60, 065205. Cyburt, R.H., Fields, B.D., Olive, K.A., 2001. New Astron. 6, 215. ENDF/B-VI Summary Documentation, 1991. Cross Section Evaluation Working Group, NNDC, Brookhaven National Laboratory. Hale, G.M., et al, 1997. ENDF/B-VI evaluation, Mat #125, Rev. 2.
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Izotov, Y.I., Thuan, T.X., 1998. Astrophys. J. 500, 188. JaIe, A.H., et al., 2001. Phys. Rev. Lett. 86, 3475. Krauss, L.M., Romanelli, P., 1990. Astrophys. J. 358, 47. Levshakov, S.A., Dessauges-Zavadsky, M., D’Odorico, S., Molaro, P., 2002. Astrophys. J. 565, 696. Ma, L., et al., 1997. Phys. Rev. C 55, 588. Mather, J., et al., 1999. Astrophys. J. 512, 511. Nollett, K.M., Burles, S., 2000. Phys. Rev. D 61, 123505. Olive, K.A., Steigman, G., Skillman, E., 1998. Astrophys. J. 483, 788. Olive, K.A., Steigman, G., Walker, T.P., 2000. Phys. Rep. 333, 389. O’Meara, J.M., Tytler, D., Kirkman, D., Suzuki, N., Prochaska, J.X., Lubin, D., Wolfe, A.M., 2001. Astrophys. J. 552, 718.
Page, L., 2001. Proceedings of the IAU Symposium, p. 201. Pettini, M., Bowen, D., 2001. Astrophys. J. 560, 41. Pryke, C., Halverson, N.W., Leitch, E.M., Kovac, J., Carlstrom, J.E., Holzapfel, W.L., Dragovan, M., 2002. Astrophys. J. 568, 46. Rupak, G., 2000. Nucl. Phys. A 678, 405. Ryan, S.G., Norris, J.E., Beers, T.C., 1999. Astrophys. J. 523, 654. Schramm, D.N., Turner, M.S., 1998. Rev. Mod. Phys. 70, 303. Smith, M., Kawano, L., Malaney, R., 1993. Astrophys. J. Suppl. Ser. 85, 219. Tytler, D., et al., 2000. Phys. Scr. T85, 12. Vangioni-Flam, E., Coc, A., Casse, M., 2001. Nucl. Phys. A 688, 393. York, D.G., 2002. Planet. Space. Sci., 50, 1251–1255.
Planetary and Space Science 50 (2002) 1245 – 1250 www.elsevier.com/locate/pss
Deuterium and big-bang nucleosynthesis: implications for the baryon density Scott Burles Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA Received 26 November 2001; received in revised form 27 May 2002; accepted 30 May 2002
Abstract A new Monte Carlo technique is introduced to extract abundances and uncertainties in the framework of standard big-bang nucleosynthesis. Both very direct and less conservative, this method yields uncertainties in light element abundance predictions which are almost a factor of 3 smaller than that found with previous techniques. Key nuclear reaction rates are identi4ed for which future nuclear cross-section measurements will further signi4cantly reduce the abundance uncertainties. Finally, constraints on the baryon density inferred from BBN when combined with recent measurements of extragalactic deuterium yield b h2 = 0:020 ± 0:001 which agrees precisely with the latest constraints from power spectrum measurements of cosmic microwave background anisotropies. ? 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
2. Sharpening big-bang nucleosynthesis
Deuterium plays a central role in the study of the light elements and the epoch of primordial nucleosynthesis. The observed abundance of deuterium together with other light isotopes of helium and lithium are naturally explained with a primordial origin and production via standard big-bang nucleosynthesis (BBN). The successful predictions of BBN have lent it to serve as a cornerstone of modern cosmological models. The precision of BBN predictions depend decisively on both the observational uncertainties of light element abundance measurements and input uncertainties on a handful of critical nuclear cross-sections. Dramatic improvements (or sharpening) of direct constraints drawn from BBN rely on reducing uncertainties from both observational and nuclear sources. For my contribution to these proceedings, I would like to detail the source and size of the nuclear input uncertainties in the current prediction of the primordial deuterium abundance. These will be compared with previous estimates and techniques, as well as with the recent observational constraints on primordial deuterium.
To continually test and apply constraints derived from BBN, we must both tighten the observational limits on the primordial light element abundances as well as sharpen the predictions based on nuclear input reaction rates. Dear to the theme of this conference, I will take as a case study the standard BBN prediction for deuterium and the extracted limits on the baryon density. In Fig. 1, I outline the relevant uncertainties for this purpose. In order to extract sharp limits on the baryon density, we need to characterize and combine the limits from the astrophysical observations and the nuclear input data. In this summary, I will focus on the propagation of the nuclear input uncertainties. For studies and reviews pertaining to the best limits on primordial deuterium measurements, the reader is politely directed to York (2002), Tytler et al. (2000), Burles et al. (2001a), and Olive et al. (2000). For argument’s sake, I will adopt a 95% cl limit of
E-mail address: [email protected] (S. Burles).
(D=H )p = (3:0 ± 0:4) × 10−5 95% cl;
(1)
based on a weighted treatment of the high redshift deuterium abundances observed with large ground-based telescopes and spectrographs (Burles and Tytler, 1998a, b; O’Meara et al., 2001; Levshakov et al., 2002). This limit is depicted as the left most error-bar in Fig. 1.
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 2 ) 0 0 0 9 0 - 9
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S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 1. From Burles et al. (2001a): 95% con4dence limits on the deuterium abundance associated with standard BBN. This example is given for = 5:6 × 10−10 . The left-most error bar is the current best combined limits from high redshift measurements of D/H (O’Meara et al., 2001; Burles et al., 2001b). For comparison, the corresponding limits on the nucleosynthesis predictions of Smith et al. (1993). is shown next. The full limits on deuterium from a full Monte Carlo analysis (Nollett and Burles, 2000) and the relative individual reaction contributions (computed by adopting SKM rates for all but the reaction in question) 4ll out the remaining the limits.
2.1. What is the di1erence? How important is the sharpening of the BBN predictions? The answer depends on the isotope in question. For 4 He, the current state of the nuclear uncertainties is quite small compared to the statistical and intrinsic uncertainties in the astrophysical data sets. On the other extreme, for 7 Li, the combined BBN nuclear uncertainty signi4cantly outweighs the statistical uncertainty of the observed level of the Spite plateau (Ryan et al., 1999; Bonifacio and Molaro, 1997). And in between, the BBN uncertainty related to deuterium is split rather fairly between observations and nuclear inputs. If this is the case, constraints derived from deuterium can only be as stringent as both sources of uncertainty allow. In other words, progress can only be gained by reducing both sources of uncertainty: one without the other will not lead to signi4cant improvements. Continuing with Fig. 1, the next two error-bars show the limits on the deuterium predictions from two recent Monte Carlo investigations of the nuclear input rates and standard BBN. The last four smaller error-bars denote the relevant single reaction rate uncertainty contribution to the “full MC” result (just add the last four limits in quadrature to calculate the full result). The diIerence between the limits from the earlier work of Smith et al. (SKM) (1993) and Krauss and Romanelli (1990), and a more recent study described 4rst in Burles et al. (1999) (full MC, hereafter cited as “direct Monte Carlo”) is dramatic and important to appreciate. The more recent limits are reduced by almost a factor of 3 over the earlier work. This can be simply explained with
two reasons: (1) New data became available in the interim, which lead to better and tighter characterization of the cross sections important for deuterium, and (2) even more signi4cant, a new technique is introduced to directly relate the 4nal abundance uncertainty (as shown in Fig. 1) to the original nuclear cross-section data at the energies relevant for standard BBN. The highlights of the direct Monte Carlo technique are (1) correct treatment of the random and correlated errors in nuclear data sets, (2) an aggressive assessment of the statistical uncertainties extracted with a direct Monte Carlo of the individual data points, (3) Jexible spline 4tting to allow for possible variations in the cross-section uncertainties as a function of energy, and (4) the reaction rates are numerically integrated from the splined cross-section realizations, which complements functional derivative studies of the cross-sections versus energy. In the remainder of this section, I will describe in detail the diIerences in reaction rates which determine the deuterium abundance. For other opinions on the subject, the reader is directed to recent analyses of nuclear input data using similar Monte Carlo techniques (Vangioni-Flam et al., 2001; Cyburt et al., 2001). The four individual reactions shown in Fig. 1 are critical to the accurate prediction of the deuterium in high-entropy primordial nucleosynthesis. For a synopsis of the 12 critical reactions relevant to light element productions up to mass seven as well as an in-depth description of the direct Monte Carlo technique, the reader is invited to review Nollett and Burles (2000) and references therein. 2.2. The four important reactions The 4rst reaction example is d(d; n) 3 He. Fig. 2 displays the appropriate S-factor (the cross-section normalized with the Coulomb barrier and energy) as a function of center of mass energy. The energy range is chosen to encompass the important range for BBN. The data are shown as symbols with 67% total error bars. Seven nuclear experiments are included in Fig. 2; this is eIectively all of the extant available data on this reaction which has (1) some energy coverage in the displayed range and (2) clearly separates and reports both random and correlated statistical uncertainties. For comparison, I show cross section 4ts to the data from three separate studies. The solid lines show the result of our “direct Monte Carlo”, the central line is the most likely 4t as a function of energy, with 95% cl represented by the two solid outer lines. The light dash-dotted line represents the analytic 4t by SKM and the associated limits are shown as dot-dot-dot-dashed lines. And the darker dashed line represents the suggested best 4t using both theory and data of the cross-section working group (ENDF/B-VI) (ENDF/B-VI Summary Documentation, 1991). Some important points to note about the 4ts and this reaction: (1) all three central 4ts agree very well, (2) the biggest diIerence is the uncertainty level as a function of energy between the two Monte Carlo methods.
S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 2. S-factor and sensitivity curves for d(d,n) at the center of mass energies relevant for standard BBN. See text for description and details.
The bottom two panels represent the intrinsic sensitivity of the 4nal abundances (for D and 7 Li) to a cross-section perturbation at the speci4c energy (b =0:02; 0:01 for solid and dashed lines, respectively). These can actually be de4ned as functional derivatives and they directly quantify, relatively, the important range of energy for each reaction in standard BBN. In addition, the horizontal bar (at 30 keV b) with vertical tick marks displays extreme limits in energy. Disregarding the cross-section outside of the inner tick marks in the reaction rate integral modi4es the 4nal abundances by less than 1/10 of a standard deviation. And disregarding the cross-section outside of the outer ticks changes the 4nal abundances by less than 1 part in 105 ! The uncertainty contribution due to the diIerent Monte Carlo treatments can be assessed by eye. At the peak (or valley) of the sensitivity function at Ecm ≈ 0:14 MeV, the resulting uncertainties are at least a factor of two smaller by applying the direct Monte Carlo technique. The next three 4gures (Figs. 3–5) show the best available nuclear data and similar 4ts as explained for d(d,n). The derived errors on d(d,p) are small and well characterized across the entire energy range shown in Fig. 3. Out of the four reactions, this contributes the least amount of uncertainty. On the other extreme, the experimental data sets for d(p; ) and p(n; ) are sparse at best. The direct Monte
Fig. 3. Same as Fig. 2, but for the reaction d(d,p).
Fig. 4. Same as Fig. 2, but for the reaction d(p; ).
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S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 5. Same as Fig. 2, but for the reaction p(n; ). The “data” points of Chen are representative eIective 4eld theory calculations (Chen and Savage, 1999). Even more precise eIective 4eld theoretical results have been presented by Rupak (2000).
Carlo procedure does converge for d(p; ) but just barely. There are no overlapping data sets, and the critical range near 100 keV is covered only by one recent experiment (Ma et al., 1997). For p(n; ) shown in Fig. 5, the direct Monte Carlo method breaks down entirely. The shape of the neutron capture cross section cannot be recovered from the extant data at the energies in question. To proceed, and gainfully include the uncertainties introduced from p(n; ), we adopt the theoretical 4t (Hale et al., 1997) with a suggested overall normalization uncertainty of 5%. The theoretical 4ts is well motivated and, for comparison, agrees with predictions from eIective 4eld theory (shown as dots with errors). Fig. 6 is simply a two-dimensional version of Fig. 1. Now the fractional uncertainties of the direct Monte Carlo, as well as the four critical reactions below, are plotted as a function of baryon-to-photon ratio. The zero-point in each case is shown as a horizontal dashed line and represent the abundances of deuterium calculated by SKM. The shaded envelopes show the fractional uncertainties at 95% cl, and oIsets, of the direct Monte Carlo abundance yields relative to SKM abundances. The dash-dotted lines enclosing the combined 2-sigma uncertainty in the SKM comparison.
Fig. 6. The relative BBN deuterium abundance 95% uncertainties as a function of baryon to photon ratio (normalized to 10−10 ) and spread among the critical reactions. The error bar on the far right-hand side shows the current 67% cl observational uncertainties for comparison.
3. Implications for the baryon density In all models of primordial nucleosynthesis, the 4nal deuterium abundance is very sensitive to the speci4c entropy during nucleosynthesis. For standard BBN, this is usually parameterized as the inverse speci4c entropy: the present-day baryon to photon ratio, . In standard cosmology with adiabatic expansion and no decaying relic particles, this ratio remains essentially unchanged since the epoch of electron positron annihilation (age of the universe ≈ 100 s). The present-day baryon energy density b is then directly proportional to the present-day photon number density inferred from an extremely accurate measurement of the temperature of the cosmic microwave background (CMB) Mather et al. (1999). This physical baryon density can be expressed as a fraction of the critical cosmic density with the expected Hubble constant dependence: b h2 = (3:650 ± 0:008) × 107 ;
(2)
where h represents a Hubble parameter normalized to the 100 km=s=Mpc and the uncertainty is a 95% limit due to the small temperature uncertainty of CMB.
S. Burles / Planetary and Space Science 50 (2002) 1245 – 1250
Fig. 7. Predicted big-bang abundances shown as 95% con4dence bands (Burles et al., 2001b) with recent light element abundance determinations shown as boxes and arrows (also 95%). See text for details.
Fig. 7 shows the most recent light element abundance predictions as a function of both b h2 (above) and (below). The vertical thickness of the abundance predictions represents the 95% limits derived from direct Monte Carlo. The vertical placement and height of each box represents a single measurement in the case of deuterium and large statistical samples in the case of 4 He (Izotov and Thuan, 1998; Olive et al., 1998) and 7 Li (Ryan et al., 1999). The horizontal placement of the boxes are set to overlap the 95% limits of the predicted abundances. The vertical solid band is the weighted deuterium constraint cited above, which combines the measurements of the four overlapping boxes. This version of light-element abundances versus baryon density may be considered enlightening, confusing, or both. For concordance with standard BBN, all primordial abundances should agree on one value of the baryon density. There is one sample of 4 He measurements which is at odds with the deuterium measurements (Olive et al., 1998). And even more striking, the Spite plateau Lithium abundance is about a factor of two lower than expected (Ryan et al., 1999). And the latest deuterium measurement by Pettini and Bowen (2001) gives a central value equal to the interstel-
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lar medium today and about a factor of two lower than the weighted mean of other measurements. So what should one take away from Fig. 7? I believe the deuterium measurements are converging on a unique cosmological value. It is and will continue to be a diMcult measurement, and known systematics will contribute additional scatter in the sample. Lithium may yet have some astrophysical processing during the main sequence lifetime of the representative halo stars in the plateau. Helium measurements, of both isotopes, are diMcult and will require further attention. The real excitement is still to come. The CMB anisotropy power spectrum is sensitive to the baryon density on scales of 1/2 degree and smaller. Current and upcoming interferometric measurements by DASI (Pryke et al., 2002, submitted for publication) and balloon-borne measurements by BOOMERANG and MAXIMA-1 (JaIe et al., 2001) will place tighter and accurate independent constraints on the physical baryon density. Combined these will approach the precision of the deuterium inferred constraints with standard BBN. So in the next year, we can expect to compare the baryon density from independent measures to the level of √ 2 × 10% (Schramm and Turner, 1998). It gets better: the arrival of the MAP satellite at the second Lagrange point (Page, 2001) and the acquisition of a full-sky high resolution map, open the possibility for an even more precise comparison. A full-sky map of this quality has the potential to constrain the baryon density to much better than 5%. In order to take full advantage of this upcoming test, the observations of light elements must continue to improve, and predictions of BBN must be made as sharp as possible.
Acknowledgements I am delighted to thank the Observatoire de Meudon and the local organizing committee for a unique and exciting conference. I also am deeply indebted to my collaborators in the work described here: D. Kirkman, M. Lemoine, K. Nollett, J. Truran, M. Turner, and D. Tytler.
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