Big bang cosmology can be said to have gone full circle. The prediction of the CMB was made in the context of the development of BBN and of what became Big Bang Cosmology . Now, the CMB is providing the precision necessary to make accurate prediction of the light element abundances in SBBN. In the Standard Model with Nν = 3, BBN makes relatively accurate predictions of the light element abundance as displayed by the thickness of the bands in Figure 1. These can be compared directly (or convoluted through a likelihood function) to the observational determination of the light element abundances. The agreement between the theoretical predictions and the abundance D/H is stunning. Recent developments in the determination of D/H has produced unparalleled accuracy . This agreement is seen instantly when comparing the likelihood functions of the observations with that of the predictions of BBN using CMB data as seen in the second panel of Figure 3. The helium data has also seen considerable progress. New data utilizing a near infrared emission line  has led to a marked drop in the uncertainty of the extrapolated primordial 4He abundance . While the error remains large compared with the precision of the BBN prediction, the agreement between theory and observation is still impressive.
Is two out of three okay? Despite the success of the BBN predictions for 4He and D/H, there remains a problem with 7Li [19, 114]. The predicted primordial abundance is about a factor of three higher than the abundance determined from absorption lines seen in a population of low metallicity halo stars. The primordial abundance has since 1981 been associated with a narrow plateau  of abundance measurements. Recently, the extent of this plateau has been called into question as a significant amount of downward dispersion is seen at very low metallicity ([Fe/H] < −3) [95, 94]. If stellar depletion is the explanation of the discrepancy between the plateau value and the BBN prediction, it remains to be explained why there are virtually no low metallicity stars with abundances above the plateau for all metallicities below [Fe/H] < −1.5. If depletion is not the answer, then perhaps the lithium discrepancy points to new physics beyond the Standard Model.
In this review, we have presented the latest combined analysis of BBN predictions using raw CMB data provided by Planck [6, 113]. He have constructed a series of likelihood function which include various combinations of the CMB, the BBN relation between the baryon density and the helium abundance, and various combinations of 4He and D/H data. We presented detailed fits and sensitivities of the light element abundances to the various input parameters as well as the dominant input nuclear rates. This allowed us to make relatively precise comparisons between theory and observations in standard BBN. The uncertainty in the prediction of 4He remains dominated by the uncertainty in the neutron mean life. We also considered a one-parameter extension of SBBN, allowing the number of relativistic degrees of freedom characterized by the number of neutrino flavors to differ from the Standard Model value of Nν = 3. Despite the additional freedom, strong constraints on η and Nν were derived. When all abundance data is used in conjunction with BBN and CMB data, we obtain a 95% CL upper limit of Nν < 3.2. As one of the deepest probes in Big Bang Cosmology, BBN continues to thrive.
Going beyond 2015, we expect further improvements in the data which will better test the Standard Model. More high resolution data on 4He emission lines could yield a further drop in the uncertainty in primordial helium. One should recall that there are still only a little over a dozen objects which are well described by models of the emission line regions. That said, there are less than half a dozen quasar absorption systems which yield high precision D/H abundances. Moreover, the nuclear physics uncertainties in D/H now dominate the error budget. Thus there is strong motivation for future measurements of the rates most important for deuterium: d(p,γ)3He, as well as d(d, n)3He, d(d, n)t, and n(p, γ)3He [131, 132]. We can be hopeful that future measurements lead to a reduction in the already small uncertainty in primordial D/H; futuristically, there is hope of detecting cosmological 92 cm deuterium hyperfine lines that would probe D/H at extremely high redshift . Lastly, we can be certain to expect updated results from the CMB data when the Planck collaboration produces its final data release.