Anatomy of Eddington-like inversion methods in the context of dark
  matter searches
  Irrespective of the dark matter (DM) candidate, several potentially
observable signatures derive from the velocity distribution of DM in halos, in
particular in the Milky Way (MW) halo. Examples include direct searches for
weakly-interacting massive particles (WIMPs), $p$-wave suppressed or
Sommerfeld-enhanced annihilation signals, microlensing events of primordial
black holes (PBHs), {\em etc}. Most current predictions are based on the
Maxwellian approximation which is not only theoretically inconsistent in
bounded systems, but also not supported by cosmological simulations. A more
consistent method sometimes used in calculations for direct WIMP searches
relies on the so-called Eddington inversion method, which relates the DM
phase-space distribution function (DF) to its mass density profile and the
total gravitational potential of the system. Originally built upon the isotropy
assumption, this method can be extended to anisotropic systems. We investigate
these inversion methods in the context of Galactic DM searches, motivated by
the fact that the MW is a strongly constrained system, and should be even more
so with the ongoing Gaia survey. We still draw conclusions that apply to the
general case. In particular, we illustrate how neglecting the radial boundary
of the DM halo leads to theoretical inconsistencies. We also show that several
realistic configurations of the DM halo and the MW baryonic content entail
ill-defined DFs, significantly restricting the configuration space over which
these inversion methods can apply. We propose consistent solutions to these
issues. Finally, we compute several observables inferred from constrained
Galactic mass models relevant to DM searches (WIMPs or PBHs), {\em e.g.}
moments and inverse moments of the DM speed and relative speed distributions.
