Fermi–Boltzmann statistics of neutrinos and relativistic effective degrees of freedom in the early universe

Abstract

We investigate the effect of the presence of non-pure fermionic neutrinos on the relativistic effective degrees of freedom in the early universe. The statistics of neutrinos is transformed continuously from Fermi–Dirac to Maxwell–Boltzmann statistics. We find that the relativistic degrees of freedom decreases with the deviation from pure Fermi–Dirac statistics of neutrinos if there are constant and large lepton asymmetries. Additionally, we confirm that the change of the statistics of neutrinos from Fermi–Dirac to Maxwell–Boltzmann is not sufficient to cover the excess of the effective number of neutrinos.