Big bang nucleosynthesis (BBN) is one of few probes of the very early Universe with direct experimental or observational consequences [1, 2, 3]. In the context of the Standard Models of cosmology and of nuclear and particle physics, BBN is an effectively parameter-free theory . Namely, standard BBN (SBBN) assumes spacetime characterized by General Relativity and the ΛCDM cosmology, and microphysics characterized by Standard Model particle content and interactions, with three light neutrino species, with negligible effects due to dark matter and dark energy.
In SBBN, the abundances of the four light nuclei are usually parameterized by the baryon-to-photon ratio η ≡ nb / nγ, or equivalently the present baryon density, Ωb h2 ≡ ωb. This quantity has been fixed by a series of precise measurements of microwave background anisotropies, most recently by Planck yielding η = 6.10 ± 0.04 . Thus the success or failure of SBBN rests solely on the comparison of theoretical predictions with observational determinations.
While precise predictions from SBBN are feasible, they rely on well-measured cross sections and a well-measured neutron lifetime. Indeed, even prior to the WMAP era, theoretical predictions for D, 3He, and 4He were reasonably accurate, however, uncertainties in nuclear cross sections leading to 7Be and 7Li were relatively large. Many modern analyses of nuclear rates for BBN were based on the NACRE compilation  and recent BBN calculations by several groups are in good agreement [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Recent remeasurements of the 3He(α, γ)7Be cross section  did improve the theoretical accuracy of the prediction but exacerbated the discrepancy between theory and observation . Very recently, the NACRE collaboration has issued an update (NACRE-II) of its nuclear rate tabulation . These were first used in  and we incorporate the new rates in the results discussed below.
The neutron mean life has had a rather sordid history. Because it scales the weak interaction rates between n ↔ p, the neutron mean life controls the neutron-to-proton ratio at freeze-out and directly affects the 4He abundance (and the other light elements to a lesser extent). The value τn = 918 ± 14 s reported by Cristensen et al. in 1972  dominated the weighted mean for the accepted value through mid 1980s. Despite the low value of 877 ± 8 s reported by Bondarenko et al. in 1978 , the ‘accepted' mean value (as reported in the “Review of Particle Physics”) remained high, and the high value was reinforced by a measurement by Byrne et al.  in 1980 of 937 ± 18 s. The range 877 s - 937 s was used by Olive et al.  to explore the sensitivity of BBN predictions to this apparently uncertain quantity treated then as an uncertain input parameter to BBN calculations (along with the number of light neutrino flavors Nν and the baryon-to-photon ration, η). In the late 1980's a number of lower measurements began to surface, and in 1989 Mampe et al.  claimed to measure a mean life of 877.6 ± 3 s (remarkably consistent with the current world average). Subsequently, it appeared that questions regarding the neutron mean life had been resolved, as the mean value varied very little between 1990 and 2010, settling at 885.6 ± 0.8. However, in 2005, there was already a sign that the mean life was about to shift to lower values once again. Serebrov et al.  reported a very precise measurement of 878.5 ± 0.8 which was used in BBN calculations in . This was followed by several more recent measurements and reanalyses tending to lower values so that the current world average is  τn = 880.3 ± 1.1. We will explore the impact of the new lifetime on the light element abundances.
On the observational side, the 4He abundance determination comes most directly from measurements of emission lines in highly ionized gas in nearby low-metallicity dwarf galaxies (extragalactic HII regions). The helium abundance uncertainties are dominated by systematic effects [30, 31, 32]. The model used to determine the 4He abundance contains eight physical parameters, including the 4He abundance, that is used to predict a set of ten H and He emission line ratios, which can be compared with observations [33, 34, 35, 36]. Unfortunately, there are only a dozen or so observations for which the data and/or model is reliable, and even in those cases, degeneracies among the parameters often lead to relatively large uncertainties for each system as well as a large uncertainty in the regression to zero metallicity. Newly calculated 4He emissivities  and the addition of a new near infra-red line  have led to lower abundance determinations [38, 39], bringing the central value of the 4He abundance determination into good agreement with the SBBN prediction. Moreover, Planck measurements of CMB anisotropies are now precise enough to give interesting measures of primordial helium via its effect on the anisotropy damping tail.
New observations and analyses of quasar absorption systems have dramatically improved the observational determination of D/H. Using a a handful of systems where accurate determinations can be made, Cooke et al.  not only significantly lowered the uncertainty in the mean D/H abundance, but the dispersion present in old data has all but been erased. Because of its sensitivity to the baryon density, D/H is a powerful probe of SBBN and now the small uncertainties in both the data and prediction become an excellent test of concordance between SBBN and the CMB determination of the baryon density.
In contrast to the predicted abundances of 4He and D/H, the 7Li abundance shows a definite discrepancy with all observational determinations from halo dwarf stars. To date, there is no solution that is either not tuned or requires substantial departures from Standard Model physics. Attempts at solutions include modifications of the nuclear rates [17, 41, 42] or the inclusion of new resonant interactions [43, 44, 45]; stellar depletion ; lithium diffusion in the post-recombination universe ; new (non-standard model) particles decaying around the time of BBN [48, 49, 50, 51]; axion cooling ; or variations in the fundamental constants [53, 54].
In this review, we survey the current status of SBBN theory and its compatibility with observation. Using an up-to-date nuclear network, we present new Monte-Carlo estimates of the theoretical predictions based on a full set of nuclear cross sections and their uncertainties. We will highlight those reactions that still carry the greatest uncertainties and how those rates affect the light element abundances. We will also highlight the effect of the new determination of the neutron mean life on the 4He abundance and the abundances of the other light elements as well.
Compatibility with observation is demonstrated by the construction of likelihood functions for each of the light elements  by convolving individual theoretical and observational likelihood functions. Our BBN calculations are also convolved with data chains provided by the 2015 Planck data release . This allows us to construct 2-dimensional (η, Yp) and 3-dimensional (η, Yp, Nν) likelihood distributions. Such an analysis is timely and important given the recent advances in the 4He and D/H observational landscape. In fact, despite the accuracy of the SBBN prediction for D/H, the tiny uncertainty in observed D/H now leads to a likelihood dominated by theory errors. The tight agreement between D/H prediction and observation is in sharp contrast to the discrepancy in 7Li/H. We briefly discuss this “lithium problem,” and discuss recent nuclear measurements that rule out a nuclear fix to this problem, leaving as explanations either astrophysical systematics or new physics. While the 6Li was shown to be an artifact of astrophysical systematics [55, 56], for now, the 7Li problem persists.
As a tool for modern cosmology and astro-particle physics, SBBN is a powerful probe for constraining physics beyond the Standard Model . Often, these constraints can be parameterized by the effect of new physics on the speed-up of the expansion rate of the Universe and subsequently translated into a limit on the number of equivalent neutrino flavors or effective number of relativistic degrees of freedom, Neff. We update these constraints and compare them to recent limits on Neff from the microwave background anisotropy and large scale structure. We will show that for the first time, D/H provides a more stringent constraint on Neff than the 4He mass fraction.
The structure of this review is as follows: In the next section, we discuss the relevant updates to the nuclear rates used in the calculation of the light element abundances. We also discuss the sensitivity to the neutron mean life. In this section, we present our base-line results for SBBN assuming the Planck value for η, the RPP value for τn and Nν = 3. In section 3, we briefly review the values of the observational abundance determinations and their uncertainties that we adopt for comparison with the SBBN calculations from the previous section. In section 4, we discuss our Monte Carlo methods, and construct likelihood functions for each of the light elements, and we extend these methods to discuss limits on Nν in section 5. The lithium problem is summarized in section 6. A discussion of these results and the future outlook appears in section 7.