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

Abstract

Using the general formulation for obtaining chemical potential µ 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 <ΔN²>/V on the particle mass m₀ has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energy m₀c² is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy k_BT. 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_v²>/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 mass mo_v.