Mean square number fluctuation for a fermion source and its dependence on neutrino mass for the universal cosmic neutrino background

Using the general formulation for obtaining chemical potentialμ of an ideal Fermi gas of particles at temperature T, with particle rest mass m0 and average density 〈N〉/V, the dependence of the mean square number fluctuation 〈ΔN 2〉/V on the particle mass m0 has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energym 0c2 is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy kBT. Application of our results to the detection of the universal very low energy cosmic neutrino background (CNB), from any of the three species of neutrinos, shows that it is possible to estimate the neutrino mass of these species if from approximate experimental measurements of their momentum distribution one can extract, someday, not only the density 〈N v〉/V but also the mean square fluctuation 〈Δ v 2 〉/V. If at the present epoch, the universe is expanding much faster than thermalization rate for CNB, it is shown that our analysis leads to a scaled neutrino massm v instead of the actual massm 0v .