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 [5]. 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 ≡ *n*_{b} /
*n*_{γ}, or equivalently the present
baryon density, Ω_{b} *h*^{2} ≡
ω_{b}.
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
[6].
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, ^{3}He, and ^{4}He were
reasonably accurate, however, uncertainties in nuclear cross sections
leading to ^{7}Be and ^{7}Li were relatively
large. Many modern analyses of nuclear rates for BBN were based on the
NACRE compilation
[7]
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
^{3}He(α, γ)^{7}Be cross section
[18]
did improve the theoretical accuracy of the prediction but exacerbated
the discrepancy between theory and observation
[19].
Very recently, the NACRE collaboration
has issued an update (NACRE-II) of its nuclear rate tabulation
[20].
These were first used in
[21]
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 ^{4}He abundance
(and the other light elements to a lesser extent). The value
τ_{n} = 918 ± 14 *s*
reported by Cristensen et al. in 1972
[22]
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
[23],
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.
[24]
in 1980 of 937 ± 18 s. The range 877 s - 937 s was used by Olive et al.
[25]
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 *N*_{ν} 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.
[26]
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.
[27]
reported a very precise measurement of
878.5 ± 0.8 which was used in BBN calculations in
[28].
This was followed by several more recent measurements and reanalyses
tending to lower values so that the current world average is
[29]
τ_{n} = 880.3 ± 1.1. We will explore the impact of
the new lifetime on the light element abundances.

On the observational side, the ^{4}He 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 ^{4}He abundance contains eight
physical parameters, including the ^{4}He 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 ^{4}He
emissivities
[37]
and the addition of a new near infra-red line
[36]
have led to lower abundance determinations
[38,
39],
bringing the central value of the ^{4}He 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. [40] 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 ^{4}He and D/H, the
^{7}Li 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
[46];
lithium diffusion in the post-recombination universe
[47];
new (non-standard model) particles decaying around the time of BBN
[48,
49,
50,
51];
axion cooling
[52];
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 ^{4}He
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
[10]
by convolving individual theoretical and observational likelihood
functions. Our BBN calculations are also convolved
with data chains provided by the 2015 *Planck* data release
[6].
This allows us to construct 2-dimensional (η,
*Y*_{p}) and 3-dimensional (η, *Y*_{p},
*N*_{ν}) likelihood distributions.
Such an analysis is timely and important given the recent
advances in the ^{4}He 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 ^{7}Li/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 ^{6}Li was shown to be an artifact of astrophysical
systematics
[55,
56],
for now, the ^{7}Li problem persists.

As a tool for modern cosmology and astro-particle physics, SBBN is a
powerful probe for constraining physics beyond the Standard Model
[57].
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, *N*_{eff}. We update
these constraints and compare them to recent limits on
*N*_{eff}
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 *N*_{eff} than the ^{4}He 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 τ_{n} and
*N*_{ν} = 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
*N*_{ν} in section 5.
The lithium problem is summarized in section 6.
A discussion of these results and the future
outlook appears in section 7.