Critical-density bound on the sum of relic neutrino masses (${\sum }_i{m}_i\le {\rho }_c/(n_{{\nu }_i}+n_{{\bar{\nu }}_i})$)

For three standard thermal relic mass eigenstates, each neutrino-antineutrino pair has number density $3n_{\gamma }/11$ after pair annihilation. Their rest-mass density is this common per-species abundance times the mass sum. Requiring it not to exceed the near-critical total density gives the displayed conservative bound. Dividing instead by the total abundance over three species bounds the mean mass. With $T_{\gamma 0}=2.725$ K and $H_0=72$ km s${}^{-1}$ Mpc${}^{-1}$ the instantaneous-decoupling estimate is about $49$ eV/$c^2$ for the sum, conditional on standard abundances and nonnegative other energy components.