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Due to the metric expansion of space, light emitted from earlier times in faraway parts of the universe and travels for a long time to reach us get progressively redshifted, decreasing the energy of the photon. (e.g. light from faraway galaxies, cosmic microwave background radiation, etc) And the earlier and further away it was emitted, the more redshifted it becomes.

What is the analogous thing that happens to particles with rest mass (e.g. neutrinos)? When you decrease the energy of a particle with rest mass, its speed decreases. (Or looked at another way, if the thing emitting it is moving away, then the particle is traveling slower towards us relative to us.) So does that mean that particles emitted from earlier times from far away traveling through empty space to reach us get slowed down (and more slowed down the earlier and further away it came from)?

If so, could this slowing down cause neutrinos to turn from "hot dark matter" into "warm dark matter" or "cold dark matter"?

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The de Broglie wavelengths of freely propagating particles (i.e. forget interactions) are redshifted by the expansion of the universe. Another way of saying this is that their peculiar momenta with respect to a co-moving local volume decrease as the inverse of the scale factor.

As neutrinos have a non-zero mass (perhaps 0.1 eV - see Battye & Moss 2014, but at least one neutrino must have mass $>0.05$ eV). They decouple from the rest of the Universe at about 1 second and freely propagate. The expansion then reduces their momenta to the extent that they should have a temperatures $<2$K in the present-day Universe, typical kinetic energies of 0.2 meV (e.g. see Kaboth et al. 2010 ) and may have speeds of only (depending on their actual masses) perhaps $\sim 10^3-10^4$ km/s and so are non-relativistic.

So indeed, (massive) neutrinos started off a "hot" dark matter and have cooled as the universe ages. Neutrinos behave as radiation-like particles in the early universe (and beyond the epoch of recombination), but once $kT$ drops below their rest mass energy then they are not "hot".

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