4. THE LIKELIHOOD ANALYSIS

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

The dominant source of uncertainty in the BBN light-element predictions stem from experimental uncertainties of nuclear reaction rates. We propagate these uncertainties by randomly drawing rates according their adopted probability distributions for each BBN evaluation. We choose a Monte Carlo of size N = 10000, keeping the error in the mean and error in the error at the 1% level. It is important that we use the same random numbers for each set of parameters (η, N_ν). This helps remove any extra noise from the Monte Carlo predictions and allows for smooth interpolations between parameter points.

For each grid point of parameter values we calculate the means and covariances of the light element abundance predictions. We add the 1/√N errors in quadrature to our evaluated uncertainties on the light element predictions. We have examined the light element abundance distributions, by calculating higher order statistics (skewness and kurtosis), and by histogramming the resultant Monte Carlo points and verified that they are well-approximated with log-normal or gaussian distributions.

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, ³He, and ⁷Li 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.

Figure 1. Primordial abundances of the light nuclides as a function of cosmic baryon content, as predicted by SBBN (“Schramm plot”). These results assume Nν = 3 and the current measurement of the neutron lifetime τn = 880.3 ± 1.1 s. Curve widths show 1−σ errors.

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)

4.2. The Neutron Mean Life

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 ⁴He abundance prediction is clear. The correlation is not infinitesimally narrow because other reaction rate uncertainties significantly contribute to the total uncertainty in ⁴He.

Figure 2. The sensitivity of the 4He 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 ⁴He 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 ⁴He, 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: η₁₀ = 6.09 (Planck); η₁₀ = 6.10 (Table 4). Rows 6-8 add the observational determinations of ⁴He, 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 η₁₀ = 6.10 as our fiducial baryon-to-photon ratio.

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 ⁴He 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.

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 ⁴He, 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 ⁴He 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 ³He prediction, for which there is no reliable observational test at present. Finally, panel (d) reveals a sharp discord between the BBN+CMB prediction for ⁷Li and the observed primordial abundance–the two likelihoods are essentially disjoint.

Figure 3. Light element predictions using the CMB determination of the cosmic baryon density. Shown are likelihoods for each of the light nuclides, normalized to show a maximum value of 1. The solid-lined, dark-shaded (purple) curves are the BBN+CMB predictions, based on Planck inputs as discussed in the text. The dashed-lined, light-shaded (yellow) curves show astronomical measurements of the primordial abundances, for all but 3He where reliable primordial abundance measures do not exist. For 4He, the dotted-lined, medium-shaded (cyan) curve shows the CMB determination of 4He.

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 ∼ 10¹⁰ and z ∼ 1000. The consistency among all three Y_p determinations is similarly remarkable, and the joint concordance between D and ⁴He represents a non-trivial success of the hot big bang model. Yet this concordance is not complete: the pronounced discrepancy in ⁷Li 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 Nν = 3 (points). The correlation between Yp 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 Yp(η) relation significantly reduces the uncertainty in η due to the CMB Yp − 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, ⁴He 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.