Dark matter in the solar system I: The distribution function of WIMPs at the Earth from solar capture

The next generation of dark matter (DM) direct detection experiments and neutrino telescopes will probe large swaths of dark matter parameter space. In order to interpret the signals in these experiments, it is necessary to have good models of both the halo DM streaming through the solar system and the population of DM bound to the solar system. In this paper, the first in a series of three on DM in the solar system, we present simulations of orbits of DM bound to the solar system by solar capture in a toy solar system consisting of only the Sun and Jupiter, assuming that DM consists of a single species of weakly interacting massive particle (WIMP). We describe how the size of the bound WIMP population depends on the WIMP mass, spin-independent cross section, and spin-dependent cross section. Using a standard description of the Galactic DM halo, we find that the maximum enhancement to the direct detection event rate, consistent with current experimental constraints on the WIMP-nucleon cross section, is < 1% relative to the event rate from halo WIMPs, while the event rate from neutrinos from WIMP annihilation in the center of the Earth is unlikely to meet the threshold of next-generation, km^3-sized (IceCube, KM3NeT) neutrino telescopes.