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Signal predictions for galactic dark matter (DM) searches often rely onassumptions on the DM phase-space distribution function (DF) in halos. Thisapplies to both particle (e.g. $p$-wave suppressed or Sommerfeld-enhancedannihilation, scattering off atoms, etc.) and macroscopic DM candidates (e.g.microlensing of primordial black holes). As experiments and observationsimprove in precision, better assessing theoretical uncertainties becomespressing in the prospect of deriving reliable constraints on DM candidates ortrustworthy hints for detection. Most reliable predictions of DFs in halos arebased on solving the steady-state collisionless Boltzmann equation (e.g.Eddington-like inversions, action-angle methods, etc.) consistently withobservational constraints. One can do so starting from maximal symmetries and aminimal set of degrees of freedom, and then increasing complexity. Key issuesare then whether adding complexity, which is computationally costy, improvespredictions, and if so where to stop. Clues can be obtained by makingpredictions for zoomed-in hydrodynamical cosmological simulations in which onecan access the true (coarse-grained) phase-space information. Here, we test anaxisymmetric extension of the Eddington inversion to predict the full DM DFfrom its density profile and the total gravitational potential of the system.This permits to go beyond spherical symmetry, and is a priori well suited forspiral galaxies. We show that axisymmetry does not necessarily improve overspherical symmetry because the (observationally unconstrained) angular momentumof the DM halo is not generically aligned with the baryonic one. Theoreticalerrors are similar to those of the Eddington inversion though, at the 10-20%level for velocity-dependent predictions related to particle DM searches inspiral galaxies. We extensively describe the approach and comment on theresults.

Testing the predictions of axisymmetric distribution functions of galactic dark matter with hydrodynamical simulations