  The success of primordial nucleosynthesis has been limited by the
long-standing Lithium problem. We use a self-consistent perturbative analysis
of the effects of relevant theoretical parameters on primordial
nucleosynthesis, including variations of nature's fundamental constants, to
explore the problem and its possible solutions, in the context of the latest
observations and theoretical modeling. We quantify the amount of depletion
needed to solve the Lithium problem, and show that transport processes of
chemical elements in stars are able to account for it. Specifically, the
combination of atomic diffusion, rotation and penetrative convection allows us
to reproduce the lithium surface abundances of Population II stars, starting
from the primordial Lithium abundance. We also show that even with this
depletion factor there is a preference for a value of the fine-structure
constant at this epoch that is larger than the current laboratory one by a few
parts per million of relative variation, at the two to three standard
deviations level of statistical significance. This preference is driven by the
recently noticed discrepancy between the best-fit values for the
baryon-to-photon ratio (or equivalently the Deuterium abundance) inferred from
cosmic microwave background and primordial nucleosynthesis analyses, and is
largely insensitive to the Helium-4 abundance. We thus conclude that the
Lithium problem most likely has an astrophysical solution, while the Deuterium
discrepancy provides a possible hint of new physics.
