Statistics of neutrinos and relativistic effective degrees of freedom in the early universe

We study the effects of the presence of non-pure fermionic neutrinos on the relativistic effective degrees of freedom $g_{\ast }$ in the early universe. The statistics of neutrinos is transformed from Fermi–Dirac (FD) to Bose–Einstein (BE) via Maxwell–Boltzmann (MB) statistics. The equilibrium energy density of pure bosonic neutrinos is larger than the energy density of pure fermionic neutrinos. One may expect that the relation $g_{\ast }^{\mathrm{FD}}<g_{\ast }^{\mathrm{MB}}<g_{\ast }^{\mathrm{BE}}$. We show that this relation is not always satisfied with degenerate neutrinos. We discuss briefly the cosmological consequences of this transformation for dark matter problem as well as the baryon–photon ratio in the universe.