Unified Halo-Independent Formalism Derived From Convex Hulls

Using the Fenchel-Eggleston theorem for convex hulls (an extension of the Caratheodory theorem), we prove that any likelihood can be maximized by either a dark matter 1- speed distribution F(v) in Earth's frame or 2- Galactic velocity distribution fgal(), consisting of a sum of delta functions. The former case applies only to time-averaged rate measurements and the maximum number of delta functions is (−1), where is the total number of data entries. The second case applies to any harmonic expansion coefficient of the time-dependent rate and the maximum number of terms is . Using time-averaged rates, the aforementioned form of F(v) results in a piecewise constant unmodulated halo function 0BF(vmin) (which is an integral of the speed distribution) with at most (-1) downward steps. The authors had previously proven this result for likelihoods comprised of at least one extended likelihood, and found the best-fit halo function to be unique. This uniqueness, however, cannot be guaranteed in the more general analysis applied to arbitrary likelihoods. Thus we introduce a method for determining whether there exists a unique best-fit halo function, and provide a procedure for constructing either a pointwise confidence band, if the best-fit halo function is unique, or a degeneracy band, if it is not. Using measurements of modulation amplitudes, the aforementioned form of fgal(), which is a sum of Galactic streams, yields a periodic time-dependent halo function BF(vmin, t) which at any fixed time is a piecewise constant function of vmin with at most downward steps. In this case, we explain how to construct pointwise confidence and degeneracy bands from the time-averaged halo function. Finally, we show that requiring an isotropic Galactic velocity distribution leads to a Galactic speed distribution F(u) that is once again a sum of delta functions, and produces a time-dependent BF(vmin, t) function (and a time-averaged 0BF(vmin)) that is piecewise linear, differing significantly from best-fit halo functions obtained without the assumption of isotropy.