Could cosmological cold dark matter be a neutrino condensate?

Question

What is wrong with the following? (Note that the question is not about galactic dark matter, but about cosmological dark matter.)

1. Neutrinos are dark matter.

2. A neutrino condensate would be cold. (Often, neutrinos are dismissed as being automatically hot dark matter.)

3. Cold neutrinos could be generated continuously by the horizon. Their number would increase with time. So their density could be significant.

4. Their temperature would be much lower than the cosmological neutrino background. The neutrino condensate would be a separate neutrino bath, much colder than the 1.95K of the CNB.

5. A condensate (in Cooper pairs) would not encounter any density limit (in contrast to free fermions, such a warm or hot netrinos).

6. And a cold condensate would not wash out early fluctuations (in contrast to hot neutrinos).

(7. They could also form galactic dark matter. - No, they could not, as several answers and comments pointed out.)

Answer 1

A neutrino condensate would involve lowering the energy of a pair of neutrinos. The only particles that could participate in such interactions would be those that are already at the top of a degenerate "Fermi sea" of neutrinos. In other words the formation of pairs would have the effect of "squaring off" the occupation index distribution and leaving a small gap between the top of the Fermi sea and a small number of higher energy neutrinos; in much the same way that pair-forming in neutron stars only acts on the small fraction of neutrons with the highest energies.

As such, there would be almost no effect on the bulk dynamics of the neutrino fluid and the Tremaine-Gunn (1979) restriction on forming galaxy halos from light leptons would still be a problem. The bulk of the neutrino fluid would still be degenerate and it would be impossible to pack enough light neutrinos into the available phase space.

Aside from this - what is the long-range pair-forming interaction that can work between particles that only interact via the short-range weak force. Gravity? Two neutrinos separated by $10^{-5}$ m (from the number density of ~0.1 eV neutrinos needed to explain galaxy halos) have a gravitational potential energy of $\sim 10^{-23}$ eV. So they would have to be colder than $10^{-19}$ K to avoid the pairs being thermally broken.

If the neutrinos are just meant to contribute to some general background rather than to galaxy halos then the question arises - why wouldn't they concentrate in galaxy halos in the same way? In any case the universal average density of dark matter would suggest a neutrino number density (assuming the same rest mass) about 3-4 orders of magnitude lower than the concentrations in galactic halos. This changes the average separation to $10^{-4}$ m and thus the pairs would have to be colder than $10^{-18}$ K.

I don't see the merit of this hypothesis if the proposed dark matter doesn't actually explain most of the problems that dark matter is required for - i.e. galaxy and cluster dynamics and point #5 is incorrect (as well as #7 that was originally part of the question I answered).