Massive Free-Streaming Neutrinos and Rise of $N_\nu$ at Recombination

We present the Einstein-Vlasov solution for the momentum distribution of the relic free-streaming neutrinos. We show that it is possible to explain a rise in the effective number of neutrinos ($N_\nu$) from those present at the end of big bang nucleosynthesis (BBN) $N_\nu(T_{BBN})=3.046$ (theoretical) or $N_\nu(T_{ BBN})=3.71^{+0.47}_{-0.45}$ (measured) towards $N_\nu(T_{r})=4.34^{+.086}_{-0.88}$ (measured) at the time of electron-ion recombination (r). The effect is due to the ambient temperature, $T_r=0.253$ eV, being near to the neutrino mass. If a thermal equilibrium distribution is inadvertently used, one instead expects a decrease in $N_\nu$ between BBN and recombination. We present explicit values for $m_\nu$ needed to account for the observed increase in $N_\nu$. The smaller the number of dominant mass neutrinos and the larger the change in $N_\nu$ needed between BBN and recombination, the larger is the value of $m_\nu$ we find. If no new mechanism is discovered to increase the theoretical value $N_\nu(T_{BBN})=3.046$ then the relic neutrinos are predicted to have $0.528\le \sum m_{\nu_i} \le 2.26$ eV and will contribute between $5\%$ and $22\%$ of the matter inventory in the Universe.