Critical-density bound on massless neutrino degeneracy ($|{\xi }_{\nu }|\lesssim [8{\pi }^2{\rho }_{\mathrm{crit}}/T_{\nu ,0}^4{]}^{1/4}$)

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.