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We study the decoupling process of neutrinos in the early universe in thepresence of three-flavour oscillations. The evolution of the neutrino spectrais found by solving the corresponding momentum-dependent kinetic equations forthe neutrino density matrix, including for the first time the proper collisionintegrals for both diagonal and off-diagonal elements. This improvedcalculation modifies the evolution of the off-diagonal elements of the neutrinodensity matrix and changes the deviation from equilibrium of the frozenneutrino spectra. However, it does not vary the contribution of neutrinos tothe cosmological energy density in the form of radiation, usually expressed interms of the effective number of neutrinos, N_eff. We find a value ofN_eff=3.045, in agreement with previous theoretical calculations and consistentwith the latest analysis of Planck data. This result does not depend on theordering of neutrino masses. We also consider the effect of non-standardneutrino-electron interactions (NSI), predicted in many theoretical modelswhere neutrinos acquire mass. For two sets of NSI parameters allowed by presentdata, we find that N_eff can be reduced down to 3.040 or enhanced up to 3.059.

Relic neutrino decoupling with flavour oscillations revisited