  The cosmic energy density in the form of radiation before and during Big Bang
Nucleosynthesis (BBN) is typically parameterized in terms of the effective
number of neutrinos N_eff. This quantity, in case of no extra degrees of
freedom, depends upon the chemical potential and the temperature characterizing
the three active neutrino distributions, as well as by their possible
non-thermal features. In the present analysis we determine the upper bounds
that BBN places on N_eff from primordial neutrino--antineutrino asymmetries,
with a careful treatment of the dynamics of neutrino oscillations. We consider
quite a wide range for the total lepton number in the neutrino sector, eta_nu=
eta_{nu_e}+eta_{nu_mu}+eta_{nu_tau} and the initial electron neutrino asymmetry
eta_{nu_e}^in, solving the corresponding kinetic equations which rule the
dynamics of neutrino (antineutrino) distributions in phase space due to
collisions, pair processes and flavor oscillations. New bounds on both the
total lepton number in the neutrino sector and the nu_e -bar{nu}_e asymmetry at
the onset of BBN are obtained fully exploiting the time evolution of neutrino
distributions, as well as the most recent determinations of primordial 2H/H
density ratio and 4He mass fraction. Note that taking the baryon fraction as
measured by WMAP, the 2H/H abundance plays a relevant role in constraining the
allowed regions in the eta_nu -eta_{nu_e}^in plane. These bounds fix the
maximum contribution of neutrinos with primordial asymmetries to N_eff as a
function of the mixing parameter theta_13, and point out the upper bound N_eff
< 3.4. Comparing these results with the forthcoming measurement of N_eff by the
Planck satellite will likely provide insight on the nature of the radiation
content of the universe.
