= Solution
Use <natural units>, assume negligible neutrino <chemical potentials>, and approximate <neutrino decoupling> as occurring before <electron-positron annihilation in cosmology>. At high <temperature>, <weak interactions> maintain a common temperature for <neutrinos>, <antineutrinos>, <electrons>, <Positrons> and <photons>. Since $\Gamma_\nu/H\propto T^3$, the reaction rate eventually loses against expansion. Decoupled relativistic <neutrinos> then free stream; their momenta redshift as $a^{-1}$, so their distribution retains a temperature parameter satisfying $aT_\nu=\mathrm{constant}$.
The still-coupled electromagnetic bath receives the <entropy> of annihilating <electrons> and <Positrons>. Its <effective number of relativistic entropy degrees of freedom> is initially
$$
g_{*s}^{\rm EM}=2+\frac78(2+2)=\frac{11}{2},
$$
where the two <photon> polarizations count as bosonic states and the four electron-positron spin states have the <fermion> entropy weight $7/8$. After the pairs disappear, $g_{*s}^{\rm EM}=2$. <Cosmological entropy conservation> in this coupled bath gives $g_{*s}^{\rm EM}(aT_\gamma)^3=\mathrm{constant}$. The three already decoupled neutrino species are excluded from this bath's entropy count. At decoupling $T_\nu=T_\gamma$, hence
$$
\boxed{\frac{T_\nu}{T_\gamma}=\left(\frac{2}{11/2}\right)^{1/3}=\left(\frac4{11}\right)^{1/3}\simeq0.714.}
$$
Subsequent expansion preserves this ratio in the idealized history. Residual interactions during annihilation give small corrections to the instantaneous-decoupling approximation. If a relic species later becomes massive, $T_\nu$ here remains the redshifting momentum-distribution parameter; it need not be its nonrelativistic kinetic temperature.
A single ordinary neutrino species includes one populated <neutrino> <helicity> and one <antineutrino> <helicity>. For negligible <chemical potential>, integration of the <Fermi-Dirac distribution> gives
$$
n_\nu+n_{\bar\nu}=\frac{3\zeta(3)}{2\pi^2}T_\nu^3,\qquad n_\gamma=\frac{2\zeta(3)}{\pi^2}T_\gamma^3.
$$
Therefore the <number density of a relativistic neutrino relative to photons>, with the antiparticle included, is
$$
\boxed{\frac{n_\nu+n_{\bar\nu}}{n_\gamma}=\frac34\left(\frac{T_\nu}{T_\gamma}\right)^3=\frac3{11}.}
$$
The factor $3/4$ is the fermionic number-density weight, distinct from the energy and entropy weight $7/8$. This relation continues to hold after this species becomes nonrelativistic if its comoving particle number is conserved.
Under the proposed matter-dominated, spatially flat model, the <Friedmann equation> requires the present <relic-neutrino energy density> to be approximately the <critical density>. Using the unreduced <Planck mass> $M_P=G^{-1/2}$ gives
$$
\rho_{c0}=\frac{3H_0^2M_P^2}{8\pi}\simeq m_\nu(n_\nu+n_{\bar\nu}).
$$
Thus the <critical-density mass of a thermal relic neutrino> is
$$
\boxed{m_\nu\simeq\frac{11\pi}{16\zeta(3)}\frac{H_0^2M_P^2}{T_{\gamma0}^3}.}
$$
Taking the paper's rounded values, $T_{\gamma0}=2\times10^{-13}\ {\rm GeV}$, $H_0=h/(4.5\times10^{41}\ {\rm GeV}^{-1})$ and $M_P\sim10^{19}\ {\rm GeV}$, gives
$$
\boxed{m_\nu\sim1.1\times10^{-7}h^2\ {\rm GeV}\sim10^2h^2\ {\rm eV}.}
$$
For $0.5<h<1$, this is of order $30$--$100\ {\rm eV}$ with these approximate constants. More precise constants change the coefficient, not the requested mass scale. This mass is much larger than the present momentum scale $T_{\nu0}\simeq1.4\times10^{-4}\ {\rm eV}$, validating the nonrelativistic <energy density> approximation. The result assumes that one thermal relic species dominates the matter density; it is conditional on that model rather than a measurement of a neutrino mass.
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