I Why can you not describe massive neutrinos with a temperature?

[Phys12]

TL;DR Summary

Why can you describe massless neutrinos with a temperature but not massive neutrinos?

In the Wikipedia article for CvB, it mentions the following: "The above discussion is valid for massless neutrinos, which are always relativistic. For neutrinos with a non-zero rest mass, the description in terms of a temperature is no longer appropriate after they become non-relativistic; i.e., when their thermal energy 3/2 kTν falls below the rest mass energy mνc2. Instead, in this case one should rather track their energy density, which remains well-defined."

(Link: https://en.wikipedia.org/wiki/Cosmic_neutrino_background)

Why is that? My guess would be that if you have massless neutrinos, you can model them as photon gas (which, if I understand correctly, would involve having the individual photons to have no mass), but that isn't true for massive neutrinos. But then can you not model the massive neutrinos as a different kind of gas?

 

[PeterDonis]

the Wikipedia article for CvB

Can you give a link?

 

[Phys12]

Can you give a link?

I've edited my original post to include the link

 

[PeterDonis]

I've edited my original post to include the link

Looking at the link, I think the statement you're asking about is somewhat misleading. You can certainly define a temperature for a system of massive neutrinos even if they are non-relativistic; but the method given in the article for estimating what the temperature of a cosmic neutrino background will be won't work for massive neutrinos after a certain time in the universe's expansion, because that method depends on the ratio of temperatures of the cosmic neutrino background and the cosmic photon background (the CMB) being constant as the universe expands, and that's only true if the neutrinos are relativistic at all times, i.e., massless.

If the neutrinos are massive, what will happen is that their temperature, once they become non-relativistic, will drop faster as the universe expands than the formula in the article would lead you to expect, so that it will become negligibly different from zero in a fairly short time; after that time, tracking their temperature as the universe expands is meaningless. You have to track their energy density as the universe expands. To put it another way, while the CMB, or a background of massless neutrinos, can be treated as radiation forever, a background of massive neutrinos can't be treated as radiation any more after a certain time in the universe's expansion; it has to be treated as matter instead.