As data analysis improved, theoretical studies of BBN moved toward a more rigorous approach using Monte-Carlo techniques in likelihood analyses [108, 109, 110, 111]. Thus, in order to make quantitative statements about the light element predictions and convolutions with CMB constraints, we need probability distributions for our BBN predictions, for the light element observations and CMB-constrained parameters. We discuss here how we propagate nuclear reaction rate uncertainties into the BBN light element abundance predictions, how we determine the CMB-parameter likelihood distributions, and how we combine them to make stronger constraints.
4.1. Monte-Carlo Predictions for the Light Elements
In standard BBN, the baryon-to-photon ratio (η) is the only free parameter of the theory. Our Monte Carlo error propagation is summarized in Figure 1, which plots the light element abundances as a function of the baryon density (upper scale) and η (lower scale). The abundance for He is shown as the mass fraction Y, while the abundances of the remaining isotopes of D, ^{3}He, and ^{7}Li are shown as abundances by number relative to H. The thickness of the curves show the ± 1 σ spread in the predicted abundances. These results assume N_{ν}= 3 and the current measurement of the neutron lifetime τ_{n} = 880.3 ± 1.1 s.
Using a Monte Carlo approach also allows us to extract sensitivities of the light element predictions to reaction rates and other parameters. The sensitivities are defined as the logarithmic derivatives of the light element abundances with respect to each variation about our fiducial model parameters [112], yielding a simple relation for extrapolating about the fiducial model:
(12) |
where X_{i} represents either the helium mass fraction or the abundances of the other light elements by number. The p_{n} represent input quantities to the BBN calculations (η, N_{ν}, τ_{n}) and the gravitational constant G_{n} as well key nuclear rates which affect the abundance X_{i}. p_{n,0} refers to our standard input value. The information contained in Eqs. (13-17) are neatly summarized in Table 3.
(13) |
(14) |
(15) |
(16) |
(17) |
Variant | Y_{p} | D/H | ^{3}He/H | ^{7}Li/H | ^{6}Li/H |
η (6.1 × 10^{−10}) | 0.039 | -1.598 | -0.585 | 2.113 | -1.512 |
N_{ν} (3.0) | 0.163 | 0.395 | 0.140 | -0.284 | 0.603 |
G_{n} | 0.354 | 0.948 | 0.335 | -0.727 | 1.400 |
n-decay | 0.729 | 0.409 | 0.145 | 0.429 | 1.372 |
p(n,γ)d | 0.005 | -0.194 | 0.088 | 1.339 | -0.189 |
^{3}He(n,p)t | 0.000 | 0.023 | -0.170 | -0.267 | 0.023 |
^{7}Be(n,p)^{7}Li | 0.000 | 0.000 | 0.000 | -0.705 | 0.000 |
d(p,γ)^{3}He | 0.000 | -0.312 | 0.375 | 0.589 | -0.311 |
d(d,γ)^{4}He | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
^{7}Li(p,α)^{4}He | 0.000 | 0.000 | 0.000 | -0.056 | 0.000 |
d(α,γ)^{6}Li | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 |
t(α,γ)^{7}Li | 0.000 | 0.000 | 0.000 | 0.030 | 0.000 |
^{3}He(α,γ)^{7}Be | 0.000 | 0.000 | 0.000 | 0.963 | 0.000 |
d(d,n)^{3}He | 0.006 | -0.529 | 0.213 | 0.698 | -0.522 |
d(d,p)t | 0.005 | -0.470 | -0.265 | 0.065 | -0.462 |
t(d,n)^{4}He | 0.000 | 0.000 | -0.009 | -0.023 | 0.000 |
^{3}He(d,p)^{4}He | 0.000 | -0.012 | -0.762 | -0.752 | -0.012 |
As noted in the introduction, the value of the neutron mean life has had a turbulent history. Unfortunately, the predictions of SBBN remain sensitive to this quantity. This sensitivity is displayed in the scatter plot of our Monte Carlo error propagation with fixed η = 6.10 × 10^{−10} in Figure 2. The correlation between the neutron mean lifetime and ^{4}He abundance prediction is clear. The correlation is not infinitesimally narrow because other reaction rate uncertainties significantly contribute to the total uncertainty in ^{4}He.
Figure 2. The sensitivity of the ^{4}He abundance to the neutron mean life, as shown through a scatter plot of our Monte Carlo error propagation. |
4.3. Planck Likelihood Functions
For this paper, we will need to consider two sets of Planck Markov Chain data, one for standard BBN (SBBN) and one for non-standard BBN (NBBN). Using the Planck Markov chain data [113], we have constructed the multi-dimensional likelihoods for the following extended parameter chains, base_yhe and base_nnu_yhe, for the plikHM_TTTEEE_lowTEB dataset. As noted earlier, we do not use the Planck base chain, as it assumes a BBN relationship between the helium abundance and the baryon density.
From these 2 parameter sets we have the following 2- and 3-dimensional likelihoods from the CMB: _{PLA−base_yhe}(ω_{b}, Y_{p}) and _{PLA−base_nnu_yhe}(ω_{b}, Y_{p}, N_{ν}). The 2-dimensional base_yhe likelihood is well-represented by a 2D correlated gaussian distribution, with means and standard deviations for the baryon density and ^{4}He mass fraction
(18) (19) |
and a correlation coefficient r ≡ cov(ω_{b}, Y_{p}) / √var(ω_{b}) var(Y_{p}) = +0.7200.
The two parameter data can be marginalized to yield 1-dimensional likelihood functions for η. The peak and 1-σ spread in η is given in the first row of Table 4. The following rows correspond to different determinations of η. In the second-fourth rows, no CMB data is used. That is, we fix η only from the observed abundances of ^{4}He, D or both. Notice for example, in row 2, the value for η is low and has a huge uncertainty. This is due to the slightly low value for the observational abundance (7) and the logarithmic dependence of Y_{p} on η. We see again that BBN + Y_{p} is a poor baryometer. This will be described in more detail in the following subsection. Row 5, uses the BBN relation between η and Y_{p}, but no observational input from Y_{p} is used. This is closest to the Planck determination found in [6], though here Y_{p} was taken to be free and the value of η in the Table is a result of marginalization over Y_{p}. This accounts for the very small difference in the results for η: η_{10} = 6.09 (Planck); η_{10} = 6.10 (Table 4). Rows 6-8 add the observational determinations of ^{4}He, D and the combination. As one can see, the inclusion of the observational data does very little to affect the determination of η and thus we use η_{10} = 6.10 as our fiducial baryon-to-photon ratio.
Constraints Used | η × 10^{10} |
CMB-only | 6.108 ± 0.060 |
BBN+Y_{p} | 4.87_{−1.54}^{+2.46} |
BBN+D | 6.180 ± 0.195 |
BBN+Y_{p}+D | 6.172 ± 0.195 |
CMB+BBN | 6.098 ± 0.042 |
CMB+BBN+Y_{p} | 6.098 ± 0.042 |
CMB+BBN+D | 6.102 ± 0.041 |
CMB+BBN+Y_{p}+D | 6.101 ± 0.041 |
The 3-dimensional base_nnu_yhe likelihood is not well-represented by a simple 3D correlated gaussian distribution, but since these distributions are single-peaked we can correct for the non-gaussianity via a 3D Hermite expansion about a 3D correlated gaussian base distribution. Details of this prescription will be given in the Appendix.
The calculated mean values and standard deviations for these distributions are:
(20) (21) (22) |
These values correspond to the peak of the likelihood distribution using CMB data alone. That is, no use is made of the correlation between the baryon density and the helium abundance through BBN. For this reason, the helium mass fraction is found to be rather high. Our value of Y_{p} = 0.261 ± 0.36 (2σ) can be compared with the value given by the Planck collaboration [6] of Y_{p} = 0.263_{−0.37}^{+0.34}.
In this case, we marginalize to form a 2-d likelihood function to determine both η and N_{eff}. As in the 1-d case discussed above, we can determine η and N_{ν} using CMB data alone. This result is shown in row 1 of Table 5 and does not use any correlation between η and Y_{p}. Note that the value of N_{ν} given here differs from that in Eq. (21) since the value in the Table comes from a marginalized likelihood function, where as the value in the equation does not. Row 2, uses only BBN and the observed abundances of ^{4}He and D with no direct information from the CMB. Rows 3-6 use the combination of the CMB data, together with the BBN relation between η and Y_{p} with and without the observational abundances as denoted. As one can see, opening up the parameter space to allow N_{ν} to float induces a relatively small drop η (by a fraction of 1 σ) and the peak for N_{ν} is below the Standard Model value of 3 though consistent with that value within 1 σ.
Constraints Used | η_{10} | N_{ν} |
CMB-only | 6.08±0.07 | 2.67_{−0.27}^{+0.30} |
BBN+Y_{p}+D | 6.10±0.23 | 2.85±0.28 |
CMB+BBN | 6.08±0.07 | 2.91±0.20 |
CMB+BBN+Y_{p} | 6.07± 0.06 | 2.89± 0.16 |
CMB+BBN+D | 6.07±0.07 | 2.90±0.19 |
CMB+BBN+Y_{p}+D | 6.07± 0.06 | 2.88±0.16 |
We note that we have been careful to use the appropriate relation between η and ω_{b} via Eq. 11. Also, in our NBBN calculations we formally use the number of neutrinos, not the effective number of neutrinos, thus demanding the relation: N_{eff} = 1.015333N_{ν}. For the 2D base_yhe CMB likelihoods, we include the higher order skewness and kurtosis terms to more accurately reproduce the tails of the distributions.
4.4. Results: The Likelihood Functions
Applying the formalism described above, we derive the likelihood functions for SBBN and NBBN that are our central results. Turning first to SBBN, we fix N_{ν} = 3 and use the Planck determination of η as the sole input to BBN in order to derive CMB+BBN predictions for each light element. That is, for each light element species X_{i} we evaluate the likelihood
(23) |
where _{bbn}(η; {X_{i}}) comes from our BBN Monte Carlo, and where we use the η − ω_{b} relation in eq. (11). In the case of ^{4}He, we use only the CMB η to determine the X_{i} = Y_{p,BBN} prediction and compare this to the CMB-only prediction.
The resulting CMB+BBN abundance likelihoods appear as the dark-shaded (purple, solid line) curves in Figure 3, which also shows the observational abundance constraints (Section 3) in the light-shaded (yellow, dashed-line) curves. In panel (a), we see that the ^{4}He BBN+CMB likelihood is markedly more narrow than its observational counterpart, but the two are in near-perfect agreement. The medium-shaded (cyan, dotted line) curve in this panel is the CMB-only Y_{p} prediction, which is the least precise but also completely consistent with the other distributions. Panel (b) displays the dramatic consistency between the CMB+BBN deuterium prediction and the observed high-z abundance. Moreover, we see that the D/H observations are substantially more precise than the theory. Panel (c) shows the primordial ^{3}He prediction, for which there is no reliable observational test at present. Finally, panel (d) reveals a sharp discord between the BBN+CMB prediction for ^{7}Li and the observed primordial abundance–the two likelihoods are essentially disjoint.
Figure 3 represents not only a quantitative assessment of the concordance of BBN, but also a test of the standard big bang cosmology. If we limit our attention to each element in turn, we are struck by the spectacular agreement between D/H observations at z ∼ 3 and the BBN+CMB predictions combining physics at z ∼ 10^{10} and z ∼ 1000. The consistency among all three Y_{p} determinations is similarly remarkable, and the joint concordance between D and ^{4}He represents a non-trivial success of the hot big bang model. Yet this concordance is not complete: the pronounced discrepancy in ^{7}Li measures represents the “lithium problem” discussed below (Section 5). This casts a shadow of doubt over SBBN itself, pending a firm resolution of the lithium problem, and until then the BBN/CMB concordance remains an incomplete success for cosmology.
Quantitatively, the likelihoods in Fig. 3 are summarized by the predicted abundances
(24) (25) (26) (27) (28) |
where the central value give the mean, and the error the 1σ variance. The slightly differences from the values in Table 2 arise due to the Monte Carlo averaging procedure here as opposed to evaluating the abundance using central values of all inputs at a single η.
We see that the BBN/CMB comparison is enriched now that the CMB has achieved an interesting sensitivity to Y_{p} as well as η. This interplay is further illustrated in Figure 4, which shows 2-D likelihood contours in the (η, Y_{p}) plane, still for fixed N_{ν} = 3. The Planck contours show a positive correlation between the CMB-determined baryon density and helium abundance. Also plotted is the BBN relation for Y_{p}(η), which for SBBN is a zero-parameter curve that is very tight even including its small width due to nuclear reaction rate uncertainties. We see that the curve goes through the heart of the CMB predictions, which represents a novel and non-trivial test of SBBN based entirely on CMB data without any astrophysical input. This agreement stands as a triumph for SBBN and the hot big bang, and illustrates the still-growing power of the CMB as a cosmological probe.
Figure 4. The 2D likelihood function contours derived from the Planck Markov Chain Monte Carlo base_yhe [113] with fixed N_{ν} = 3 (points). The correlation between Y_{p} and η is evident. The 3-σ BBN prediction for the helium mass fraction is shown with the colored band. We see that including the BBN Y_{p}(η) relation significantly reduces the uncertainty in η due to the CMB Y_{p} − η correlation. |
Thus far we have used the CMB η as an input to BBN; we conclude this section by studying the constraints on η when jointly using BBN theory, light-element abundances, and the CMB in various combinations. Figure 5 shows the η likelihoods that result from a set of such combinations. Setting aside at first the CMB, the BBN+X curves show the combination of BBN theory and astrophysical abundance observations, _{BBN+X}(η) = ∫_{bbn}(η, X) _{obs}(X) dX, with X ∈ (Y_{p}, D / H). The CMB-only curve marginalizes over the Planck Y_{p} values _{CMB−only}(η) = ∫_{PLA−base_yhe}(ω_{b}, Y_{p}) dY_{p} where we use the η−ω_{b} relation in eq. (11). The BBN+CMB curve adds the BBN Y_{p}(η) relation. Finally, BBN+CMB+D also includes the observed primordial deuterium.
We see in Fig. 5 that of the primordial abundance observations, deuterium is the only useful “baryometer,” due to its strong dependence on η in the Schramm plot (Fig. 1). By contrast, ^{4}He alone offers no useful constraint on η, tracing back to the weak Y_{p}(η) trend in Fig. 1. The CMB alone has now surpassed BBN+D in measuring the cosmic baryon content, but an even stronger limit comes from BBN+CMB. As seen in Fig. 4, this tightens the η constraint due to the CMB correlation between Y_{p} and η Finally BBN+CMB+D provides only negligibly stronger limits. The peaks of the likelihoods correspond to the values in Table 4, and the tightest constraints are all consistent with our adopted central value η = 6.10 × 10^{−10}.
Figure 5. The likelihood distributions of the baryon-to-photon ratio parameter, η, given various CMB and light-element abundance constraints. |