Relic neutrinos: Physically consistent treatment of effective number of neutrinos and neutrino mass*

It is well known that the effective number of cosmic neutrinos, $N_\nu$, is larger than the standard model number of neutrino flavors $N^f_\nu=3$ due a small flow of entropy into neutrinos from $e\pm$ annihilation. Observational bounds from both BBN and the CMB suggest a value of $N_\nu$ that is larger than the current theoretical prediction of $N_\nu=3.046$. We show in a model independent way how $N_\nu$ relates to the neutrino kinetic freeze-out temperature, $T_k$, which we treat as parameter. We derive the relations that must hold between $N_\nu$, the photon to neutrino temperature ratio, the neutrino fugacity, and $T_k$. Our results imply that measurement of neutrino reheating, as characterized by $N_\nu$, amounts to the determination of $T_k$. We follow the free streaming neutrinos down to a temperature on the order of the neutrino mass and determine how the cosmic neutrino properties i.e. energy density, pressure, particle density, depend in a physically consistent way on both neutrino mass and $N_\nu$. We continue down to the present day temperature and characterize the neutrino distribution in this regime as well. See arXiv:1212.6943, PRD in press.