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Dark matter interacts with the gravitational force right? Well, unlike black holes, neutron stars are actually visible, and they're an enormous gravitational sink, so dark matter should collect to them.
But if all that is true, which it seems to be, why haven't astronomers detected or used neutron stars to detect dark matter?
Yes, neutron stars might actually accumulate weakly interacting dark matter and this allows some observational constraints on its nature. Basically, the temperature and continued existence of neutron stars places bounds on the density and interaction cross-section of dark matter.
A dark matter particle that does not interact with matter will just have its trajectory bent by the gravity field of a heavy object, so most unbound particles will just swoop past on a hyperbolic trajectory. But as discussed in (Adams & Laughlin 1997), if there is some matter-dark matter interaction then the particle may scatter from a matter particle and now have less than escape velocity. This way white dwarfs and neutron stars would indeed accumulate dark matter in their cores. The rate of accumulation is proportional to $\rho v \sigma$ where $\rho$ is the dark matter density, $v$ the average relative velocity and $\sigma$ the cross section. Adams & Laughlin estimate that a white dwarf star would accumulate its own mass in $10^{25}$ years, but this is going to be depend on the cross section (if it is too small the dark matter will pass through) which is at present unknown.
Were this accumulation the only thing happening it would eventually make white dwarfs and later neutron stars implode. However, dark matter is plausibly a mix of particles and antiparticles that annihilate each other at a rate $\sim \rho^2$; in an enriched environment like a white dwarf core this would produce energy from emitted photons heating things up. Adams & Laughlin estimate the luminosity as to about $10^{-12}L_\odot$, which is imperceptible in the current era but would eventually keep white dwarfs at 63 K in the far future (until the dark matter halo runs out).
Other, more elaborate, calculations lead to accretion estimates that are higher. If the rate were high enough, then we would not see any cool dense objects - so white dwarf and neutron star cooling gives some bounds on the possible density and cross sections, albeit not very strict ones. For example, one model suggests that neutron stars would level out at 10,000 K. Cool star observations can also already rule out some dark matter models.
There are even some arguments that super-earth planets in dense dark matter halos might be heated significantly, although this may require unrealistically dense halos and big cross sections. The current heath flow of Earth does give some constraints on how strongly it can interact.
So neutron stars are not directly giving us dark matter detection, but they (and planets and white dwarfs) are giving us some information.
I want to clarify a part of this question that some people may not understand. If dark matter literally were only affected by gravity, then you would not expect to see it collecting at the center of neutron stars. As a dark matter falls towards the center of a neutron star, it picks up speed until it passes on through the neutron star and starts to slow down. But when it leaves the vicinity of the neutron star it will have the same speed that it had when entering the vicinity. In order to collect dark matter, the neutron star has to slow it down somehow. This is what Anders Sandberg meant when he mentioned the interaction cross-section of dark matter. That refers to the probability of interactions that might slow down the dark matter particles enough for them to be trapped.
To put it a little differently, suppose a particle were far enough away from the neutron star so that the neutron star's gravity is can basically be ignored. Imagine that the particle is drifting towards the neutron star, so that eventually it will pass through it. Then, by the time the particle reaches the neutron star, it is mathematically guaranteed to be above escape velocity. The definition of escape velocity is the velocity that a particle will have if it falls from infinitely far away. The neutron star cannot trap anything because anything that drifts by is guaranteed to pick up precisely enough speed falling in so that, by the time it gets back out to the same distance it was before, it's moving away at the same speed it was moving in before.
To be fair, there is one caveat here. If the particles had significant gravity, so they could strongly influence each other, then they could interact in such a way that one particle flies out even faster, and the other one is trapped in orbit around (maybe also passing through) the neutron star. Some of Jupiter's moons may have been captured that way. But dark matter particles are thought to have negligible mass.
I am not saying anything different from Anders Sandberg here, but I just want to emphasize the importance of his statement that "most particles will just swoop on past."
I have never seen any discussion of the interaction between any of the various Dark Matter candidates and neutron star matter. But we can still say something useful about the prospect.
First, remember that we don't know what Dark Matter (DM) is. We do have a number of theories that are reasonable extensions of the Standard Model which contains particles which behave sorta-kinda like we think DM behaves, but not only do we not have any good evidence for any of them, we have looked for most of them and have failed to find anything. The negative evidence falls well short of certain, but also suggests that there is something important we don't know yet.
At any rate, you're correct that DM ought to be attracted by the neutron star's (NS's) gravity, and it seems plausible that the DM would react with the NS's dense matter. But the only interactions that I'm aware of would release a bit of heat and a bit of electromagnetic radiation at the point of interaction. (DM particles aren't hugely energetic, and DM isn't very dense.) This would be promptly absorbed and result in an ultra-minuscule heating of the NS.
And neutron stars are far away. It is very difficult to see how we could hope to observe any effects of whatever interaction may be taking place.
Why aren't neutron stars full of dark matter?
Because dark matter doesn't consist of particles. There's something of a myth that it does, which I think comes from particle physicists who have never actually read Einstein's original material. I also think science is something of a competitive business, and there's a tendency for advocates to promote their own theory (eg WIMPs) and claim that a competitor theory (eg MOND) is flawed.
Dark matter interacts with the gravitational force right? Well, unlike black holes, neutron stars are actually visible, and they're an enormous gravitational sink, so dark matter should collect to them.
Remember that we have good scientific evidence for flat galactic rotation curves and other phenomena. These suggest that either a) there's some unseen "dark matter" around somewhere, or b) that gravity doesn't work quite the way that people think. However the evidence does not actually say dark matter is made out of particles and falls down.
But if all that is true, which it seems to be, why haven't astronomers detected or used neutron stars to detect dark matter?
Because we do not live in some Chicken Little world where the sky is falling in. I'm referring to Gullstrand-Painlevé coordinates which model a gravitational field as a place where space is falling down. Einstein rejected the idea, but some contemporary physicists take it seriously, see this for example.
Image credit Andrew Hamilton
Why is this relevant? Because in his 1916 Foundation of General Relativity Einstein said “the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy”. This is spatial energy, and it isn't made of particles. The energy density of space near the Earth is greater than the energy density of space further away from the Earth. Because of this, there's a gravitational effect. This is why "gravity gravitates". Einstein also described a gravitational field as a place where space is "neither homogeneous nor isotropic". So dark matter might simply be inhomogeneous space. Don't forget that as per the raisin-cake analogy, the space between the galaxies expands whilst the space between the galaxies does not. Conservation of energy tells me that this will surely lead to an inhomogeneous spatial energy density. And that an older galaxy will be surrounded by a bigger/steeper halo of inhomogeneous space than a younger galaxy, so it will look like there's more dark matter present.
What Einstein said means that there's "dark matter" of sorts in the room you're in, right in front of your face. Only it isn't made of particles, and it isn't falling down. Instead it's made of space. Don't forget that space is dark, and there's a lot of about.
