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Some experiments search for dark matter interactions at the center of bodies like the Sun and Earth, assuming that if there is some interaction with ordinary matter that allows dark matter to lose energy, it can collect in a gravity well (and then decay, or annihilate, or do something resulting in observable electromagnetic signal).

Supposedly, with only gravity, the dark matter would never collect. It would fly straight through the Earth and escape at the same speed with which it approached. But isn't there some minuscule energy loss radiated through gravity in this process, due to the acceleration of mass?

In other words, if a dark matter particle existed for an eternity, could it eventually be trapped by an ordinary object through gravitational losses alone?

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There is emission of gravitational radiation whenever mass is accelerated so that the quadrupole moment changes. A particle doing a hyperbolic trajectory around another mass does change quadrupole moment, so there should be emission. Even if it never amounts to much for the Earth, the elliptic orbits of dark matter around the galactic halo would tend to emit gravitational waves that make particles lose energy.

I calculated that a 30 GeV particle orbiting a galaxy mass black hole at 10 kpc will fall in over $10^{88}$ years, while a 2 eV particle takes $10^{98}$ years. So this is a very slow process.

Note that the above calculation does not depend on the black hole being a black hole, only on its mass. So were it instead a normal object it would also eventually capture dark matter... but even more slowly.

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