Continuing on from the cool physics Q&A'd on the threads Where are all the slow neutrinos?, Is it possible that all "spontaneous nuclear decay" is actually "slow neutrino" induced?, and What does the cosmic neutrino background look like today, given that neutrinos possess mass?, I have a follow-up question that stems from this answer to that last question.
More specifically, the Big Bang must have produced a bunch of neutrinos and antineutrinos as a side-effect of the creation of all that matter, and past a certain point these would decouple from matter and just fly on unimpeded through space. Over time, these neutrinos would get redshifted just like the cosmic microwave background did, to form a Cosmic Neutrino Background at a temperature of around $1.9\:\mathrm K$.
Depending on the neutrino masses, this could mean a range of velocities, but if the neutrinos are relatively massive, to quote rob's answer, they
would have typical speeds of under 100 km/s, slower than the escape velocity of some stars. Cold neutrinos might therefore accumulate in gravitational wells, resulting in substantial density enhancement over the 100 ν/cm³ average you expect over intergalactic space.
Now, about this accumulation in gravitational wells, I have a question similar to rob's later comment,
(I'm just the tiniest bit murky on how the cold neutrinos get trapped without scattering, but I'm prepared to believe it's discussed in the literature.)
In contrast, I'm completely in the dark on how the cold neutrinos get trapped without scattering. You've got this small particle, that only interacts gravitationally, coming in towards a star with a velocity at infinity of the order of (but smaller than) the escape velocity. Under normal circumstances, the particle will approach the star... and whizz off again, in a hyperbolic trajectory that ends up with the same asymptotic velocity it came in with. If there's a third body to interact with, it might get deflected into a bound orbit, but with no meaningful interactions but gravity this seems very unlikely to me to work on astronomically large numbers of neutrinos.
What are the rough physics behind the capture of these massive cold neutrinos in a gravitational well?
Answer
It requires that they lose energy somehow (to drop hyperbolic orbits into periodic ones).
There are two basic mechanisms available: gravitational scattering and weak scattering. In both cases we expect the interaction to be elastic, but that doesn't mean the neutrino has as much kinetic energy in the star's frame afterward the interaction as before: it could donate some to the other participants. This effect is necessarily very, very slow.
For gravitational scattering think gravity boost, but in the energy losing direction rather then the energy gaining direction as we usually apply it to spacecraft.
For weak scattering the same basic idea applies, As long as the center of momentum is moving with the neutrino's initial direction in the frame of the star the neutrino will have less energy in the stellar frame after the interaction than before.
In either case the baryonic matter member of the scattering gains energy, but it can radiate that energy by mundane means which is something the neutrino couldn't do by themselves. So the neutrinos can cool by transferring energy to the baryonic matter which cools in the usual way. Needless to say the neutrinos cool more slowly than the ordinary matter as their coupling channel is very weak.
The big question is "Shouldn't the neutrino gain energy roughly as often as it loses it?", and I think the answer is yes. But that results in a distribution of energy gains and loses and we consider the differential fate of the two tails of that distribution. The ones that are pick up energy will leave the vicinity of the star, while some those that lose energy may be tipped from "barely hyperbolic" to "barely elliptical" or if already captured will become slightly more bound. As usual the system cools as much by ejecting it's most energetic members by donating energy to the baryonic component of the system to be radiated by electromagnetic means.
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