  Big Bang Nucleosynthesis (BBN) relates key cosmological parameters to the
primordial abundance of light elements. In this paper, we point out that the
recent observations of Cosmic Microwave Background anisotropies by the Planck
satellite and by the BICEP2 experiment constrain these parameters with such a
high level of accuracy that the primordial deuterium abundance can be inferred
with remarkable precision. For a given cosmological model, one can obtain
independent information on nuclear processes in the energy range relevant for
BBN, which determine the eventual ^2H/H yield. In particular, assuming the
standard cosmological model, we show that a combined analysis of Planck data
and of recent deuterium abundance measurements in metal-poor damped Lyman-alpha
systems provides independent information on the cross section of the radiative
capture reaction d(p,\gamma)^3He converting deuterium into helium.
Interestingly, the result is higher than the values suggested by a fit of
present experimental data in the BBN energy range (10 - 300 keV), whereas it is
in better agreement with ab initio theoretical calculations, based on models
for the nuclear electromagnetic current derived from realistic interactions.
Due to the correlation between the rate of the above nuclear process and the
effective number of neutrinos Neff, the same analysis points out a Neff>3 as
well. We show how this observation changes when assuming a non-minimal
cosmological scenario. We conclude that further data on the d(p,\gamma)^3He
cross section in the few hundred keV range, that can be collected by
experiments like LUNA, may either confirm the low value of this rate, or rather
give some hint in favour of next-to-minimal cosmological scenarios.
