 Solution

Use the observed present photon temperature $T_{\gamma 0}\simeq 2.725\mathrm{K}$. This is an additional observational input, consistent with ^[ref]. Standard neutrino decoupling and electron-positron entropy transfer giveFor nonrelativistic relic neutrinos this is the temperature parameter of the redshifted momentum distribution, not the temperature of a newly thermalized gas of massive particles.

The photon number density from part b isOne neutrino mass eigenstate and its antineutrino have two populated helicity states. ThusThe total over three species is about $336\mathrm{c}{\mathrm{m}}^{-3}$, including antineutrinos. Comoving number conservation preserves these abundances after the particles become nonrelativistic.

For the supplied $H_0=72\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}=2.33\times 10^{-18}{\mathrm{s}}^{-1}$,Assuming nonnegative other energy components, relic neutrino rest-mass density cannot exceed the near-critical total density. With the common per-species abundance $n_{\ast }=n_{{\nu }_i}+n_{{\bar{\nu }}_i}$,Using the total $336\mathrm{c}{\mathrm{m}}^{-3}$ would instead bound the mean mass, not the sum. This critical-density bound on the sum of relic neutrino masses is deliberately conservative: baryons, dark matter and dark energy leave less density available. It assumes the standard thermal abundance, negligible chemical potentials, no later dilution or extra populated states, and the idealized decoupling count. Small non-instantaneous-decoupling corrections do not matter for this estimate.