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

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?

• Rob indicates in an answer about the CMB due to neutrino mass that they travel pretty slowly physics.stackexchange.com/a/268134 Commented Jan 23 at 15:24

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). $$\left = 160 \left(\frac{m_{\nu} c^2}{{\rm eV}}\right)^{-1} \ (1+z)\ \ \ {\rm km/s}$$

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

• If they travel so slowly, can they constitute dark matter? Or are there just not enough of them, considering the fact that for every electron, proton, and neutron there is just one neutrino, and dark matter contains five times the amount of visible mass? Commented Jan 25 at 0:07
• @IlGuercio no they can't, because they were ultra relativistic during the important phases of cosmic structure formation and because there are (in theory) not enough of them. Commented Jan 25 at 6:53
• Just as an aside, do you know how big the total universe was, after inflation? Was most of space already there then, assuming it's a 3-sphere? In other words how big was the balloon, compared to nowadays? Commented Jan 25 at 11:51