Asymmetric Velocity Distributions from Halo Density Profiles in the Eddington Approach

We show how to obtain the energy distribution f(E) in our vicinity starting from WIMP density profiles in a self-consistent way by employing the Eddington approach and adding reasonable angular momentum dependent terms in the expression of the energy. We then show how we can obtain the velocity dispersions and the asymmetry parameter beta in terms of the parameters describing the angular momentum dependence. From this expression, for f(E), we proceed to construct an axially symmetric WIMP a velocity distribution, which, for a gravitationally bound system, automatically has a velocity upper bound and is characterized by the same asymmetriy beta. This approach is tested and clarified by constructing analytic expressions in a simple model, with adequate structure. We then show how such velocity distributions can be used in determining the event rates, including modulation, in both the standard and the directional WIMP searches.