  Dark matter (DM) capture in stars offers a rich phenomenology that makes it
possible to probe a wide variety of particle DM scenarios in diverse
astrophysical environments. In spite of decades of improvements to refine
predictions of capture-related observables and better quantify astrophysical
and particle-physics uncertainties, the actual impact of the Galactic
phase-space distribution function of DM has been overlooked. In this work, we
tackle this problem by making use of self-consistent equilibrium phase-space
models based on the Eddington inversion formalism and an extension of this
method to a DM halo with some degree of anisotropy in velocity space. We
demonstrate that incorrectly accounting for the variation of the DM velocity
distribution with position in the Galaxy leads to a systematic error between a
factor two and two orders of magnitude, depending in particular on the target
star, the DM candidate mass and the type of interaction involved. Moreover, we
show that underlying phase-space properties, such as the anisotropy of the
velocity tensor, actually play an important part -- previously disregarded --
and can have a sizable impact on predictions of capture rates and subsequent
observables. We argue that Eddington-like methods, which self-consistently
account for kinematic constraints on the components of the Galaxy, actually
provide a reliable next-to-minimal approach to narrow down uncertainties from
phase-space modeling on predictions of observables related to DM capture in
stars.
