Calculating Neutrino Binding in Gravitational Potentials

[Spinnor]

If we knew the four momentum of a neutrino should we then be in position to calculate whether the gravitational potential of our galaxy might "bind" that neutrino?

Thank you for any thoughts.

 

[Chalnoth]

If we knew the four momentum of a neutrino should we then be in position to calculate whether the gravitational potential of our galaxy might "bind" that neutrino?

Thank you for any thoughts.

Yes.

 

[Vanadium 50]

It is not hard to get an order of magnitude estimate: E \approx kT \approx GM/r \approx G \rho r^2, or r \approx \sqrt{kT/G \rho}.

Now, the temperature of the neutrinos is about 2K instead of 2.7K for photons- I dimly remember a cub root of 4/11 in the problem, but that's not important - so there's only about a 15% difference in the radius at which neutrinos clump vs. where photons clump. Since we don't see significant photon "clumping" - CMBR anisotropies are small - we don't expect significant neutrino clumping on cosmological scales either.

 

[marcus]

If we knew the four momentum of a neutrino should we then be in position to calculate whether the gravitational potential of our galaxy might "bind" that neutrino?

Thank you for any thoughts.

Spinnor,
Check out the Wikipedia article on escape velocity.
http://en.wikipedia.org/wiki/Escape_velocity

It says that for an object to escape the galaxy (starting from as far out from galaxy center as we are) it has to have a speed of roughly 1000 km/second.

Neutrinos come with all different speeds. Some are relativistic, but that wouldn't necessarily be true for all of them. There could well be some neutrinos around here which are traveling less than 1000 km/second and which are bound----trapped in the potential well of the galaxy.

That speed is atypically slow for a neutrino. One typically thinks of them traveling near the speed of light, which is 300,000 km/second.

The reason I'm answering your question this way is because you asked if we knew the momentum of a particular neutrino, what could we say? Another way to put the question is to ask how slow would one have to be going, in order to be trapped?

The figure of 1000 is only ballpark---because there is a cloud of dark matter around our galaxy we don't have a precise estimate of its total mass. The mass further in is easier to estimate. The sun is orbiting the center of galaxy at a speed of about 250 km/second. We can measure that and calculate the total amount of mass inside the sun's roughly circular orbit, but much further out we don't have such a good handle.

Given that it is only a rough estimate one can still say that in order to be trapped a particle (like a neutrino) would need to be slowed down to about 1/300 of the speed of light.
Slowed down from around 300,000 to around 1000 km/second. And we could ask, what could have slowed a neutrino down that much? Is there any mechanism? For example could the expansion of the universe have "cooled" some neutrinos so much that some are traveling only 1/300 of the speed of light?*

Is that where your question seems to point? If you want to pursue that line of questioning then there are more things to ask about.

Vanadium already gave you an answer in terms of the statistical or thermal distribution of speeds. Along those lines you might be able to find out what fraction of the background of early universe neutrinos are by now going less than 1/300 c.
Interesting thought.

(BTW it is said that there is not enough mass in cold neutrinos to account for a significant part of the cold dark matter, so there has to be some other kind of particle. It would still be interesting to try to follow the reasoning.)

*EDIT: Nobody answered, so I'll give it away Yes the expansion does slow neutrinos down.