Neutrino degeneracy parameter (${\xi }_{\nu }$)

The neutrino degeneracy parameter is the dimensionless chemical potentialIn thermal equilibrium the corresponding antineutrino has chemical potential $-{\mu }_{\nu }$.

Combining the degenerate neutrino and antineutrino energy density with a critical density bound constrains the neutrino degeneracy parameter through the fourth root of the available energy budget. The numerical bound depends on $T_{\nu ,0}/T_{\gamma ,0}$ and the number of degenerate flavors. The standard ratio $(4/11{)}^{1/3}$ assumes neutrino decoupling before electron-positron annihilation in cosmology; it should be stated as a thermal-history assumption if the degeneracy is very large. The massless formula is inappropriate for a species that is nonrelativistic today.

For one massless neutrino flavor with one helicity per member and $|{\mu }_{\nu }|\gg T_{\nu }$, the Fermi-Dirac distribution fills momentum states up to the positive chemical potential. If ${\mu }_{\nu }>0$, the neutrino density is ${\mu }_{\nu }^4/(8{\pi }^2)$ and the antineutrino contribution is approximately $3T_{\nu }^4{e}^{-{\mu }_{\nu }/T_{\nu }}/{\pi }^2$. For negative chemical potential the roles reverse. Thus the leading total energy density is even in ${\mu }_{\nu }$ and has no factor of two from the two species. Extra populated internal degrees of freedom do multiply it. This distinction prevents using a positive-degeneracy approximation for an exponentially dilute negative-chemical-potential distribution.

A degenerate relic neutrino has ${\xi }_{\nu }\gg 1$. In the ultrarelativistic limit its Fermi-Dirac distribution approaches a filled momentum sphere, giving for one internal statewhile the antineutrino density is exponentially suppressed.

A nonzero neutrino chemical potential produces unequal neutrino and antineutrino densities and hence a cosmological lepton asymmetry. It changes the radiation density, and an electron-neutrino asymmetry also changes neutron--proton weak equilibrium during Big Bang nucleosynthesis.