Cosmic neutrino background ($C_{\nu}B$)

Question

Usually when you calculate the total density of radiation of the universe you use the formula:

$$\rho_{R}=\rho_{\gamma}+\rho_{\nu}$$

Where $\rho_{\gamma}$ is the density of photons and $\rho_{\nu}$ is the density of neutrinos. I understand that this formula is valid for masless neutrinos or when you have a high enough temperature. However at the present time the neutrinos of the $C_{\nu}B$ don't have enough temperature to remain relativist, then why we still use the above equation?

Thank you. I think that the answer is in "The neutrinos maintain their relativistic Fermi-Dirac distribution as they cool, with much lower occupation of high energy states. The distribution does not depend on the neutrino mass" — Nothing, Apr 18 '20 at 02:34

ProfRob · Accepted Answer · 2020-04-19T10:34:35.693

The equation refers to energy densities, not number or mass densities, so it is valid at all epochs (if you just want to sum the contribution of photons and neutrinos) and doesn't change if the particles are non-relativistic.

What is true though is that you just wouldn't class the cosmic neutrino background as "radiation" in the present-day universe, because at least some of the neutrinos have mass and are moving much slower than $c$.

score -1 · Answer 2 · answered Apr 19 '20 at 04:55

I don't think we do use this equation. It was relevant in the early, radiation dominated, universe, but the current universe is matter dominated and Friedmann solutions ignore the radiation term. If we want to make a study of the current form of the cosmic neutrino background, then we need to consider whether neutrinos are massless (and the formula remains valid) or whether neutrinos have mass and can become gravitationally bound to galaxies or to galaxy clusters.

Cosmic neutrino background ($C_{\nu}B$)

2 Answers2