WIMPs at the center of the Earth

Assume cold dark matter does exist in the form of WIMPs: Weakly Interacting Massive Particles.

WIMPs feel the force of gravity, so they should become trapped in gravity wells such as the earth or other massive bodies.

How much of the earth's mass is composed of dark matter?

One way to experimentally confirm the existence of dark matter is to measure the gravitational mass of the earth using any of the standard methods, and compare it with the mass obtained by an alternate method assuming the earth is completely composed of regular matter.

Measuring the seismic propagation of earthquakes gives us a density profile of the earth. Integrating the density gives us the total amount of regular matter in the Earth. WIMP matter does not affect the seismic propagation of earthquakes.

The earth's active internal structure might be too complicated to determine its absolute density profile, in which case other astronomical bodies are available. We could go to the moon, which is geologically dead, and set off explosives. For a small asteroid, we could drill a core sample all the way to its core. Neutron stars undergo starquakes which might reveal its internal structure. Observing and modeling the Sun may determine how much of the Sun is made of regular matter.

The dark matter surface of the earth need not coincide with the regular matter surface. If the dark matter surface is higher than the ground we see, then satellites would orbit with unexpected periods depending on how much dark matter was below the satellite. This probably would have been observed already (elliptical orbits would behave very weird) so the dark matter surface is underground.

It might be too difficult to decouple gravitational effects from the seismic density profile to get an absolute density profile. If so, we can obtain a pure gravitational density profile to compare it against.

One can directly measure the gravitational density profile of the earth by shooting artificial or natural neutrinos through the earth and measuring how much they deflect based on how much earth they passed through. Supernova neutrinos are conveniently all parallel. Of course, neutrinos are horrendously difficult to detect, and measuring their angle is even more difficult.

There may be other ways of measuring the earth's internal gravitational mass distribution: satellites perhaps, or somehow use the Earth-Moon dipole.