  Primordial nucleosynthesis is one of the three historical evidences for the
big bang model, together with the expansion of the universe and the cosmic
microwave background. Now that the number of neutrino families and the baryonic
densities have been fixed by laboratory measurements or CMB observations, the
model has no free parameter and its predictions are rigid. Departure from its
predictions could provide hints or constraints on new physics or astrophysics
in the early universe. Precision on primordial abundances deduced from
observations have recently been drastically improved and reach the percent
level for both deuterium and helium-4. Accordingly, the BBN predictions should
reach the same level of precision. For most isotopes, the dominant sources of
uncertainty come from those on the laboratory thermonuclear reactions. This
article focuses on helium-4 whose predicted primordial abundance depends
essentially on weak interactions which control the neutron-proton ratio. The
rates of the various weak interaction processes depend on the experimentally
measured neutron lifetime, but also includes numerous corrections that we
thoroughly investigate here. They are the radiative, zero-temperature,
corrections, finite nucleon mass corrections, finite temperature radiative
corrections, weak-magnetism, and QED plasma effects, which are for the first
time all included and calculated in a self consistent way, allowing to take
into account the correlations between them, and verifying that all satisfy
detailed balance. The helium-4 predicted mass fraction is $0.24709\pm0.00017$.
In addition, we provide a Mathematica code (PRIMAT) that incorporates, not only
these corrections but also a full network of reactions, using the best
available thermonuclear reaction rates, allowing the predictions of primordial
abundances up to the CNO region.
