  If the universe is slightly non-extensive, and the distribution functions are
not exactly given by those of Boltzmann-Gibbs, the primordial production of
light elements will be non-trivially modified. In particular, the principle of
detailed balance (PDB), of fundamental importance in the standard analytical
analysis, is no longer valid, and a non-extensive correction appears. This
correction is computed and its influence is studied and compared with previous
works, where, even when the universe was considered as an slightly
non-extensive system, the PDB was assumed valid. We analytically track the
formation of Helium and Deuterium, and study the kind of deviation one could
expect from the standard regime. The correction to the capture time, the moment
in which Deuterium can no longer be substantially photo-disintegrated, is also
presented. This allows us to take into account the process of the free decay of
neutrons, which was absent in all previous treatments of the topic. We show
that even when considering a first (linear) order correction in the quantum
distribution functions, the final output on the primordial nucleosynthesis
yields cannot be reduced to a linear correction in the abundances. We finally
obtain new bounds upon the non-extensive parameter, both comparing the range of
physical viability of the theory, and using the latest observational data.
