Freeze Out of Relativistic Species

For relativistic species, theyield has no dependence on $x_f$. Neutrinos are a paradigmatic example and we must consider their contribution to the total amount of dark matter. Since they decouple while they are still relativistic, their yield reads $Y_{eq}\approx 0.278\frac{g_{eff}}{g_{\ast s}}$.

Neutrinos decouple at a few MeV, when the species that were still relativistic are $e^{\pm },\gamma ,\nu$ and $\bar{\nu }$.

Thus, the number of relativistic degrees of freedom is $g_{\ast }=g_{\ast s}=10.75$. For one neutrino family, the effective number of degrees of freedom is $g_{eff}=3g/4=3/2$.

Using these values, the relic density today can be written as

$\Omega h^2=\frac{{\sum }_i{m}_{{\nu }_i}{Y}_{\infty }{s}_0{h}^2}{{\rho }_c}\approx \frac{{\sum }_i{m}_{{\nu }_i}}{91eV}$.

In order for neutrinos to be the bulk of dark matter, ${\sum }_i{m}_{{\nu }_i}\approx 91eV$, which is much bigger than the current upper limits (obtained from cosmological observations). Even if we consider the current bound ${\sum }_i{m}_{{\nu }_i}\le 0.3eV$, it results in neutrinos contributing to 3% of the total dark matter density.