Relic-neutrino energy density

For one relic-neutrino species with two internal states, physical momentum $p$, comoving temperature $T_{\nu }$, and mass $m_{\nu }$,It approaches $7{\pi }^2(T_{\nu }/a{)}^4/120$ while relativistic and $n_{\nu }{m}_{\nu }$ after becoming nonrelativistic.

A standard thermal Cosmic neutrino background with one populated neutrino and one antineutrino helicity per species has $n_{\nu }+n_{\bar{\nu }}=3n_{\gamma }/11$ after electron-positron annihilation in cosmology. If this species is nonrelativistic today and dominates a spatially flat matter universe, equate $m_{\nu }(n_{\nu }+n_{\bar{\nu }})$ to the critical density $3H_0^2{M}_P^2/(8\pi )$, using the unreduced Planck mass. The photon number density gives the displayed conditional mass. Extra populated states or several massive species change the abundance count; vacuum-dominated cosmology changes the density available for matter.

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.

Expanding the relic-neutrino energy density at $T_{\nu }/a\ll m_{\nu }$ givesThe thermal speed decreases as $a^{-1}$ after the neutrinos become nonrelativistic.