How fast do the neutrinos in the neutrino background radiation move through the universe?

Question

The expansion of space drains energy from particles. That's why the CMBR contains less energy nowadays than in the early stages of the universe. What does this mean for the neutrinos present in the neutrino background radiation? How fast (or slow) do they travel?

Answer 1

Self-plagiarising part of my own answer to a related question.

It does of course depend on the neutrino mass that you assume, but not in the way you might naively expect.

If you assume a rest mass-energy of 0.1 eV, use the 1.95K cosmic neutrino background temperature and the Maxwell-Boltzmann distribution, you get an incorrect rms speed of 21,000 km/s.

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. $$F(p,T) = \frac{1}{\exp(pc/kT) + 1}$$ As the universe expands, the de Broglie wavelength of particles is stretched by a factor equivalent to the scale factor of the universe $a \propto (1+z)^{-1}$. Thus the momentum $p \propto (1+z)$. The energy of relativistic particles also goes as $(1+z)$, but once neutrinos become non-relativistic (see below), their energies ($=p^2/2m_{\nu}$) fall as $(1+z)^{2}$ (see Rahvar 2006).

The net effect of this is that the average speed of the neutrinos at redshift $z$ is given by (see Safdi et al. 2014). $$\left<v\right> = 160 \left(\frac{m_{\nu} c^2}{{\rm eV}}\right)^{-1} \ (1+z)\ \ \ {\rm km/s}$$

Neutrino masses are not fully constrained. At least two of the three flavours must have masses $0.05<m_{\nu}c^2 <2$ eV that make them non-relativistic at the current epoch. The total neutrino mass (all three flavours) is probably less than 2 eV from beta decay experiments; but some cosmological constraints using galaxy clustering data and the cosmic microwave background suggest this could be as low as $<0.5$ eV (Guisarma et al. 2013).