Are ultra-cold neutrinos an option for cold dark matter?

Question

Nobody hass seen cold dark matter. Are ultra-cold (non-relativistic) neutrinos, below 1 fK (femtokelvin), an option for dark matter?

This is a question about normal neutrinos - electron neutrinos and muon neutrinos and tau neutrinos - and not any additional, invented neutrino types.

This is a question about neutrinos that are not in the cosmic neutrino background, which have a temperature of 1.95K.

The question is about cosmological dark matter, not about galactic dark matter.

Their density could be large enough to produce the observed cosmological dark matter density.

Due to their low temperature and low kinetic energy they would be undetectable.

They would be continuously emitted by the cosmological horizon in the same way as black hole radiation is continuously emitted by a black hole. Alternatively, they would arise automatically throughout the universe, whenever space grows in size.

Why is this not possible? Or why is it possible?

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Given that the answers below imply that ultra-cold neutrinos cannot be dense enough because of their fermion character, could fermion condensation (via Cooper pairs, as in superconductors) solve the problem?

Answer 1

This is not possible because neutrinos are fermions. If the neutrinos are very light, you need a lot of them, and because of the Pauli exclusion principle you can't pack enough in a galaxy. This is the reason for the Tremaine-Gunn bound, which rules out fermionic dark matter lighter than $\sim1 \, \text{eV}$.

However, there are plenty of models of dark matter involving ultralight bosons, such as axions, dilatons, hidden photons, and so on. It's perfectly possible for dark matter to be a lot of ultra-cold, ultra-light particles.

Answer 2

No.

Neutrinos are dark matter: they are massive but don't interact with light. However, they are a very sub-dominant component. As @knzhou already correctly pointed out, the Tremaine-Gunn bound rules out that most of dark matter would be fermions with masses below approximately 1eV (such as neutrinos). Even better, observations from the Cosmic Microwave Background can be used to measure the neutrino density when the universe was 300,000 years old. One finds the density of neutrinos to be less than 0.1%. See e.g. this slightly outdated but easy-to-read discussion leading up to their equation 13.

In the context of what you are discussing, you may want to have a look at e.g. the Wikipedia article on the Cosmic Neutrino Background. This seems to be what you are referring to. That's essentially the equivalent of the Cosmic Microwave Background, but instead of photons emitted when the universe cooled so much that they could free-stream, it's neutrinos emitted when the universe cooled so much that they could free-stream.