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?