Cosmology with Independently Varying Neutrino Temperature and Number

We consider Big Bang nucleosynthesis and the cosmic microwave background when both the neutrino temperature and neutrino number are allowed to vary from their standard values. The neutrino temperature is assumed to differ from its standard model value by a fixed factor from Big Bang nucleosynthesis up to the present. In this scenario, the effective number of relativistic degrees of freedom, $N_{\rm eff}^{\rm CMB}$, derived from observations of the cosmic microwave background is not equal to the true number of neutrinos, $N_\nu$. We determine the element abundances predicted by Big Bang nucleosynthesis as a function of the neutrino number and temperature, converting the latter to the equivalent value of $N_{\rm eff}^{\rm CMB}$. We find that a value of $N_{\rm eff}^{\rm CMB} \approx 3$ can be made consistent with $N_\nu = 4$ with a decrease in the neutrino temperature of $\sim 5\%$, while $N_\nu = 5$ is excluded for any value of $N_{\rm eff}^{\rm CMB}$. No observationally-allowed values for $N_{\rm eff}^{\rm CMB}$ and $N_\nu$ can solve the lithium problem.