Unified Halo-Independent Formalism From Convex Hulls for Direct Dark Matter Searches

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 $f^{\rm gal}(\vec{u})$, 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 $({\mathcal N}-1)$, where ${\mathcal N}$ 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 ${\mathcal N}$. Using time-averaged rates, the aforementioned form of $F(v)$ results in a piecewise constant unmodulated halo function $\tilde\eta^0_{BF}(v_{\rm min})$ (which is an integral of the speed distribution) with at most $({\mathcal N}-1)$ 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 $f^{\rm gal}(\vec{u})$, which is a sum of Galactic streams, yields a periodic time-dependent halo function $\tilde\eta_{BF}(v_{\rm min}, t)$ which at any fixed time is a piecewise constant function of $v_{\rm min}$ with at most ${\mathcal N}$ 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 ...