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The early (0.6 gtapprox 1 ms), hot, dense Universe is filled with radiation (gammas, e ± pairs, nus of all flavors), along with dynamically and numerically insignificant amounts of baryons (nucleons) and dark matter particles. Nuclear and weak interactions are occurring among the neutrons, protons, e ± , and nus (e.g., n + p <--> D + gamma; p + e- <--> n + nue) at rates fast compared to the universal expansion rate. At such high temperatures (T gtapprox 3 MeV), in an environment where the nucleon to photon ratio is very small (eta10 approx 3 - 10), the abundances of complex nuclei (D, 3He,4He, 7Li) are tiny in comparison to those of the free nucleons (neutrons and protons). At the same time, the charged-current weak interactions are regulating the neutron to proton ratio, initially keeping it close to its equilibrium value

Equation 13 (13)

where Deltam is the neutron - proton mass (energy) difference. In this context it is worth noting that if there is an asymmetry between the numbers of nue and bar{nu}e the equilibrium neutron-to-proton ratio is modified to (n / p)eq = exp(-Deltam / T - µe / T) = e-xie(n / p)eq0.

As the Universe expands and cools, the lighter protons are favored over the heavier neutrons and the neutron-to-proton ratio decreases, tracking the equilibrium form in eq. 13. But, as the temperature decreases below T ~ 0.8 MeV, when the Universe is ~ 1 second old, the weak interactions are too slow to maintain equilibrium and the neutron-to-proton ratio, while continuing to fall, deviates from (exceeds) the equilibrium value. Since the n / p ratio depends on the competition between the weak interaction rates and the early-Universe expansion rate (as well as on a possible neutrino asymmetry), deviations from the standard model (e.g., rhoR -> rhoR + rhoX or xie neq 0) will change the relative numbers of neutrons and protons available for building the complex nuclides.

As noted above, while neutrons and protons are interconverting, they are also colliding among themselves creating complex nuclides, e.g., deuterons. However, at early times, when the density and average energy of the CBR photons are very high, the newly formed deuterons find themselves bathed in a background of high-energy gamma rays capable of photodissociating them. Since there are more than a billion CBR photons for every nucleon in the Universe, the deuteron is photodissociated before it can capture a neutron (or a proton, or another deuteron) to build the heavier nuclides. This bottleneck to BBN persists until the temperature drops sufficiently below the binding energy of the deuteron, when there are too few photons energetic enough to photodissociate them before they capture nucleons, launching BBN. This transition (smooth, but rapid) occurs after e ± annihilation, when the Universe is a few minutes old and the temperature has dropped below ~ 80 keV.

Once BBN begins in earnest, neutrons and protons quickly combine to build D, 3H, 3He, and 4He. Since there are no stable mass-5 nuclides, a new bottleneck appears at 4He. Nuclear reactions quickly incorporate all available neutrons into 4He, the most strongly bound of the light nuclides. Jumping the gap at mass-5 requires Coulomb suppressed reactions of 4He with D, or 3H, or 3He, guaranteeing that the abundances of the heavier nuclides are severely depressed below that of 4He (and even of D and 3He), and that the 4He abundance is determined by the neutron abundance when BBN begins. The few reactions that manage to bridge the mass-5 gap lead mainly to mass-7 (7Li or, to 7Be which, later, when the Universe has cooled further, will capture an electron and decay to 7Li); for the range of etaB of interest, the BBN-predicted abundance of 6Li is more than 3 orders of magnitude below that of the more tightly bound 7Li. Finally, there is another gap at mass-8, ensuring that there is no astrophysically significant production of heavier nuclides.

The primordial nuclear reactor is short-lived. As the temperature drops below T ltapprox 30 keV, when the Universe is ~ 20 minutes old, Coulomb barriers abruptly suppress all nuclear reactions. Afterwards, until the first stars form, no pre-existing, primordial nuclides are destroyed (except for those like 3H and 7Be that are unstable and decay) and no new nuclides are created. In ~ 1000 seconds BBN has run its course.

With this as background, the trends of the SBBN-predicted primordial abundances of the light nuclides with baryon abundance shown in Figure 1 can be understood. The reactions burning D and 3He (along with 3H) to 4He are very fast (compared to the universal expansion rate) once the deuterium bottleneck is breached, ensuring that almost all neutrons present at that time are incorporated into 4He. As a result, since 4He production is not rate limited, its primordial abundance is very insensitive (only logarithmically) to the baryon abundance. The very slight increase in YP with increasing etaB reflects the fact that for a higher baryon abundance BBN begins slightly earlier, when slightly more neutrons are available. The thickness of the YP curve in Fig. 1 reflects the very small uncertainty in the BBN prediction; the uncertainty in YP (~ 0.2%; sigmaY ~ 0.0005) is dominated by the very small error in the weak interaction rates which are normalized by the neutron lifetime (taun = 885.7 ± 0.8 s). The differences among the YP predictions from independent BBN codes are typically no larger than DeltaYP ~ 0.0002.

Figure 1

Figure 1. The SBBN-predicted primordial abundances of D, 3He, 7Li (relative to hydrogen by number), and the 4He mass fraction (YP), as functions of the baryon abundance parameter eta10. The widths of the bands reflect the uncertainties in the nuclear and weak interaction rates.

Nuclear reactions burn D, 3H, and 3He to 4He, the most tightly bound of the light nuclides, at a rate which increases with increasing nucleon density, accounting for the decrease in the abundances of D and 3He (the latter receives a contribution from the beta-decay of 3H) with higher values of etaB. The behavior of 7Li is more interesting, reflecting two pathways to mass-7. At the relatively low values of eta10 ltapprox 3, mass-7 is largely synthesized as 7Li by 3H(alpha, gamma)7Li reactions. 7Li is easily destroyed in collisions with protons. So, for low nucleon abundance, as etaB increases, destruction is faster than production and 7Li/H decreases. In contrast, at relatively high values of eta10 gtapprox 3, mass-7 is largely synthesized as 7Be via 3He(alpha, gamma)7Be reactions. 7Be is more tightly bound than 7Li and, therefore, harder to destroy. As etaB increases at high nucleon abundance, the primordial abundance of 7Be increases. Later in the evolution of the Universe, when it is cooler and neutral atoms begin to form, 7Be captures an electron and beta-decays to 7Li. These two paths to mass-7 account for the valley shape of the 7Li abundance curve in Fig. 1.

Not shown on Figure 1 are the BBN-predicted relic abundances of 6Li, 9Be, 10B, and 11B. Their production is suppressed by the gap at mass-8. For the same range in etaB, all of them lie offscale, in the range 10-20 - 10-13.

For SBBN the relic abundances of the light nuclides depend on only one free parameter, the nucleon abundance parameter etaB. As Figure 1 reveals, for the "interesting" range (see below) of 4 ltapprox eta10 ltapprox 8, the 4He mass fraction is expected to be YP approx 0.25, with negligible dependence on etaB while D/H and 3He/H decrease from approx 10-4 to approx 10-5, and 7Li/H increases from approx 10-10 to approx 10-9. The light nuclide relic abundances span some nine orders of magnitude, yet if SBBN is correct, one choice of etaB (within the errors) should yield predictions consistent with observations. Before confronting the theory with data, it is useful to consider a few generic examples of BBN in the presence of nonstandard physics and/or cosmology.

2.1. Nonstandard BBN

The variety of modifications to the standard models of particle physics and of cosmology is very broad, limited only by the creativity of theorists. Many nonstandard models introduce several, new, free parameters in addition to the baryon abundance parameter etaB. Since there are only four nuclides whose relic abundance is large enough to be astrophysically interesting and, as will be explained below in more detail, only three for which data directly relating to their primordial abundances exist at present (D, 4He, 7Li), nonstandard models with two or more additional parameters may well be unconstrained by BBN. Furthermore, as discussed in the Introduction (see Section 1.2 and Section 1.3), there already exist two additional parameters with claims to relevance: the expansion rate parameter S (or, Delta Nnu; see eqs. 8, 11) and the lepton asymmetry parameter L (or, xi; see eq. 12).

The primordial abundance of 4He depends sensitively on the pre- and the post-e ± annihilation early universe expansion rate (the Hubble parameter H) and on the magnitude of a nue - bar{nu}e asymmetry because each will affect the n/p ratio at BBN (see, e.g., Steigman, Schramm & Gunn 1977 (SSG) [6]; for recent results see Kneller & Steigman 2004 (KS) [9]) A faster expansion (S > 1; Delta Nnu > 0) leaves less time for neutrons to convert into protons and the higher neutron abundance results in increased production of 4He. For small changes at fixed etaB, DeltaYP approx 0.16(S - 1) approx 0.013Delta Nnu (KS). Although the relic abundances of D and 3He do depend on the competition between the nuclear reaction rates and the post-e ± annihilation expansion rate (faster expansion -> less D and 3He destruction -> more D and 3He), they are much less sensitive to relatively small deviations from S = 1 (Delta Nnu = 0) [9]. For mass-7 the effect of a nonstandard expansion rate is different at low and high values of etaB. At low baryon abundance (eta10 ltapprox 3), a faster expansion leaves less time for 7Li destruction and the relic abundance of mass-7 increases. In contrast, at high baryon abundance (eta10 gtapprox 3), S > 1 leaves less time for 7Be production and the relic abundance of mass-7 decreases. As for D and 3He, the quantitative change in the 7Li abundance is small for small deviations from SBBN.

For similar reasons, YP is sensitive to an asymmetry in the electron neutrinos which, through the charged current weak interactions, help to regulate the n/p ratio. For xie > 0, there are more neutrinos than antineutrinos, so that reactions such as n + nue -> p + e-, drive down the n/p ratio. For small asymmetry at fixed etaB, KS find DeltaYP approx -0.23xie. The primordial abundances of D, 3He, and 7Li, while not entirely insensitive to neutrino degeneracy, are much less affected by a nonzero xie than is 4He (e.g., Kang & Steigman 1992 [10]).

Each of these nonstandard cases (S neq 1, xi neq 0) will be considered below. While certainly not exhaustive of the nonstandard models proposed in the literature, they actually have the potential to provide semi-quantitative, if not quantitative, understanding of BBN in a large class of nonstandard models. Note that data constraining the primordial abundances of at least two different relic nuclei (one of which should be 4He) are required to break the degeneracy between the baryon density and the additional parameter resulting from new physics or cosmology. 4He is a poor baryometer but a very good chronometer and/or, leptometer; D, 3He, 7Li have the potential to be good baryometers.

2.2. Simple - But Accurate - Fits To The Primordial Abundances

While BBN involves only a limited number of coupled differential equations, they are non-linear and not easily solved analytically. As a result, detailed comparisons of the theoretical predictions with the inferred relic abundances of the light nuclei requires numerical calculations, which may obscure key relations between abundances and parameters, as well as the underlying physics. In particular, the connection between the cosmological parameter set {etaB, S, xie} and the abundance data set {yD, YP, yLi} 2 may be blurred, especially when attempting to formulate a quantitative understanding of how the latter constrains the former. However, it is clear from Figure 1 that the relic, light nuclide abundances are smoothly varying, monotonic functions of etaB over a limited but substantial range. While the BBN-predicted primordial abundances are certainly not linearly related to the baryon density (nor to the other parameters S and xie), over the restricted ranges identified above, KS [9] found linear fits to the predicted abundances (or, to powers of them) which work very well indeed. Introducing them here enables and simplifies the comparison of theory with data (below) and permits a quick, reasonably accurate, back of the envelope, identification of the successes of and challenges to BBN.

For the adopted range of etaB, yD = yD(etaB) is well fit by a power law,

Equation 14 (14)

While the true yD - etaB relation is not precisely a power law, this fit (for 4 ltapprox eta10 ltapprox 8) is accurate (compared to a numerical calculation) to better than 1%, three times smaller than the ~ 3% BBN uncertainty estimated by Burles, Nollett, Turner 2001 (BNT) [11]; this fit and the numerical calculation agree with the BNT result to 2% or better over the adopted range in etaB. Note that since different BBN codes are largely independent and often use somewhat different nuclear reaction data sets, the differences among their predicted abundances may provide estimates of the overall uncertainties. It is convenient to introduce a "deuterium baryon density parameter" etaD, the value of eta10 corresponding to an observationally determined primordial D abundance.

Equation 15 (15)

Generalizing this to include the two other parameters, KS find

Equation 16 (16)

This fit works quite well for 2 ltapprox yD ltapprox 4, corresponding to 5 ltapprox etaD ltapprox 7. In Figure 2 the deuterium isoabundance curves are shown in the S - eta10 plane, while Figure 3 shows the same isoabundance contours in the xie - eta10 plane. It is clear from Figures 2 and 3 that D is a sensitive baryometer since, for these ranges of S and xie, etaD approx eta10.

Figure 2

Figure 2. Isoabundance curves for Deuterium (dashed lines) and Helium-4 (solid lines) in the expansion rate factor (S) - baryon abundance (eta10) plane. The 4He curves, from bottom to top, are for YP = 0.23, 0.24, 0.25. The D curves, from left to right, are for yD = 4.0, 3.0, 2.0.

Figure 3

Figure 3. As in Figure 2, in the neutrino asymmetry (xie) - baryon abundance (eta10) plane. The 4He curves, from bottom to top, are for YP = 0.25, 0.24, 0.23. The D curves, from left to right, are for yD = 4.0, 3.0, 2.0.

Next, consider 4He. While over a much larger range in eta10, YP varies nearly logarthmically with the baryon density parameter, a linear fit to the YP versus eta10 relation is actually remarkably accurate over the restricted range considered here.

Equation 17 (17)

Over the same range in eta10 this fit agrees with the numerical calculation and with the BNT [11] predictions for YP to within 0.0002 (ltapprox 0.1%), or better. Any differences between this fit and independent, numerical calculations are smaller (much smaller) than current estimates of the errors in the observationally inferred primordial value of YP. The following linear fits, including the total error estimate, to the YP - S and YP - xie relations from KS work very well over the adopted parameter ranges (see Figures 2 & 3).

Equation 18 (18)

As an aside, the dependence of the 4He mass fraction on the neutron lifetime (taun) can be included in eq. 18 by adding a term 0.0002(taun - 887.5), where taun is in seconds. A very recent, new measurement of taun by Serebrov et al. [12] suggests that the currently accepted value (taun = 887.5 s) should be reduced by 7.2 s. If confirmed, this would lead to a slightly smaller BBN-predicted 4He abundance: DeltaYP = -0.0014. The corresponding shift in the 4He inferred baryon density parameter is negligible compared to its range of uncertainty (DeltaetaB / etaB = -0.14), as is that for the shift in the upper bound to Nnu (DeltaNnumax = +0.11). These corrections are ignored here.

In analogy with the deuterium baryon density parameter introduced above, it is convenient to introduce etaHe, defined by

Equation 19 (19)

so that

Equation 20 (20)

For SBBN (S = 1 & xie = 0), etaHe is the value of eta10 corresponding to the adopted value of YP. Once YP is chosen, the resulting value of etaHe provides a linear constraint on the combination of eta10, S, and xie in eq. 20. This fit works well [9] for 0.23 ltapprox YP ltapprox 0.25, corresponding to -5 ltapprox etaHe ltapprox 7. As Figures 2 & 3 reveal, 4He is an excellent chronometer and/or leptometer, since the YP isoabundance curves are nearly horizontal (and very nearly orthogonal to the deuterium isoabundance curves).

As with D, the 7Li abundance 3 is well described by a power law in eta10 over the range in baryon abundance explored here: yLi ident 1010(Li/H) propto eta102. The following KS fit agrees with the BBN predictions to better than 3% over the adopted range in eta10,

Equation 21 (21)

While this fit predicts slightly smaller lithium abundances compared to those of BNT [11], the differences are at the 5-8% level, small compared to the BNT uncertainty estimates as well as those of Hata et al. (1995) [13] (~ 10 - 20%).

In analogy with etaD and etaHe defined above, the lithium baryon abundance parameter etaLi (allowing for a 10% overall uncertainty) is defined by

Equation 22 (22)

The simple, linear relation for etaLi as a function of eta10, S, xie, which KS find fits reasonably well over the adopted parameter ranges is,

Equation 23 (23)

This fit works well for 3 ltapprox yLi ltapprox 5, corresponding to 5 ltapprox etaLi ltapprox 7, but it breaks down for yLi ltapprox 2 (etaLi ltapprox 4); see Fig. 1. As is the case for deuterium, lithium can be an excellent baryometer since, for the restricted ranges of S and xie under consideration here, etaLi approx eta10.

Finally, it may be of interest to note that for 3He the power law y3 - etaB relation, where y3 ident 105(3He/H), which is reasonably accurate for 4 ltapprox eta10 ltapprox 8 is

Equation 24 (24)

The difficulty of using current observational data, limited to chemically evolved regions of the Galaxy, to infer the primordial abundance of 3He, along with the relatively weak dependence of y3 on eta10, limits the utility of this nuclide as a baryometer [14]. 3He can, however, be used as a test of BBN consistency.

2.3. SBBN-Predicted Primordial Abundances

Before discussing the current status of the observationally determined abundances (and their uncertainties) of the light nuclides, it is interesting to assume SBBN and, for the one free parameter, etaB, use the value inferred from non-BBN data such as the CBR (WMAP) and Large Scale Structure (LSS) [2] to predict the relic abundances.

From WMAP alone, Spergel et al. 2003 [2] derive eta10 = 6.3 ± 0.3. Using the fits from Section 2.2, with S = 1 and xie = 1, the SBBN-predicted relic abundances are: yD = 2.45 ± 0.20; y3 = 1.03 ± 0.04; YP = 0.2485 ± 0.0008; yLi = 4.67 ± 0.64 ([Li]P = 2.67 ± 0.06).

When Spergel et al. 2003 [2] combine the WMAP CBR data with those from Large Scale Structure, they derive a consistent, but slightly smaller (slightly more precise) baryon abundance parameter eta10 = 6.14 ± 0.25. For this choice the SBBN-predicted relic abundances are: yD = 2.56 ± 0.18; y3 = 1.04 ± 0.04; YP = 0.2482 ± 0.0007; yLi = 4.44 ± 0.57 ([Li]P = 2.65-0.06+0.05).

2 YP is the 4He mass fraction while the other abundances are measured by number compared to hydrogen. For numerical convenience, yD ident 105(D/H) and yLi ident 1010(Li/H). Back.

3 It is common in the astronomical literature to present the lithium abundance logarithmically: [Li] ident 12 + log(Li/H) = 2 + log(yLi). Back.

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