Could cold dark matter be made of large numbers of ultra-cold neutrinos?

Question

Could cold dark matter be made of huge numbers of ultra-cold ( say $10^{-30}\,\rm K$ ) neutrinos?

The number could be huge enough, say $10^{11}$ times higher than the number of baryons, to produce the required mass density.

Their cold temperature and low kinetic energy would make them invisible, or "dark". Their low kinetic energy would not make them relativistic, but slow. So it would genuinely be cold dark matter.

Is there any reason why this is not possible?

There is quite some recent research literature about ultra-light bosonic dark matter that is also ultra-cold, but it does not apply here because neutrinos are fermions, not bosons.

Comments

If "invisible", then why cold neutrinos? Anything invisible will do.

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Neutrinos have an advantage: they are known to exist. Other invisible stuff is speculative. That is why the question asks about neutrinos.

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As neutrinos have too small mass, then any interaction - gravitational in you case - will make them relativistic. They will never stay still (i.e., at rest).

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No, gravitational interactions do not make neutrinos relativistic. To do achieve this a simplified estimate is that  $mgh$ would have to be of the order of $mc^2$. It seems almost impossible to achieve this with gravity. And anyway, if this would be possible, then this would work with all particles, as the $m$ cancels out. No, if neutrinos are ultra-cold, they do not become relativistic via gravity.

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The value of $m$ does not matter in weak fields like on the Earth, when you play with a football ball. In the Universe it is differtent.

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Why different? Galactic gravitational accelerations are much smaller than the one on Earth by many orders of magnitude. And Jupiter-like planets or stars, or even black holes, which have bigger values, are very rare in the volume of the universe. Almost the full volume of the universe has weak gravitational fields. 

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You are reasoning it terms of some mean field values whereas there are so many massive objects to change the neutrino velocities. Even ultra cold, the neutrino may form Fermi-like systems with the neutrino velocity distributions (Fermi distributions at $T\approx zero$). Have you checked this possibility?