= Critical-density bound on massless neutrino degeneracy
{title2=$|\xi_\nu|\lesssim[8\pi^2\rho_{\rm crit}/T_{\nu,0}^4]^{1/4}$}
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.
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