= Critical-density bound on the sum of relic neutrino masses
{title2=$\sum_i m_i\le\rho_c/(n_{\nu_i}+n_{\bar\nu_i})$}
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 $T_{\gamma0}=2.725$ 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.
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