Why can you not describe massive neutrinos with a temperature?

In summary, the Wikipedia article for CvB discusses how the description of neutrinos in terms of temperature is not appropriate for non-relativistic massive neutrinos, as their thermal energy falls below their rest mass energy. Instead, their energy density must be tracked, as they can no longer be treated as radiation and must be considered as matter. This is due to the ratio of temperatures between the cosmic neutrino background and the cosmic photon background changing as the universe expands.
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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?
 
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Phys12 said:
the Wikipedia article for CvB

Can you give a link?
 
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PeterDonis said:
Can you give a link?
I've edited my original post to include the link
 
  • #4
Phys12 said:
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.
 
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1. Why can you not describe massive neutrinos with a temperature?

The reason for this is because temperature is a measure of the average kinetic energy of particles, and massive neutrinos have very low kinetic energy due to their small mass. Therefore, they do not have a well-defined temperature.

2. Can neutrinos be described with a temperature at all?

Neutrinos can be described with a temperature in certain situations, such as in the early universe when they were highly energetic and abundant. However, as they travel through space and their energy decreases, their temperature becomes less relevant.

3. How do scientists study neutrinos if they cannot be described with a temperature?

Scientists study neutrinos by measuring their properties, such as mass, spin, and flavor. They also study their interactions with other particles to understand their behavior and role in the universe.

4. Are there any other ways to describe neutrinos besides temperature?

Yes, besides temperature, scientists also use other physical quantities such as energy, momentum, and cross-sections to describe neutrinos. These quantities are more relevant and useful in studying neutrinos than temperature.

5. Could there be a way to measure the temperature of massive neutrinos?

Currently, there is no way to directly measure the temperature of massive neutrinos. However, scientists are constantly researching and developing new techniques to better understand these elusive particles, so it is possible that in the future, we may be able to indirectly measure their temperature through advanced experiments and observations.

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