Interpreting dark matter direct detection independently of the local velocity and density distribution

We demonstrate precisely what particle physics information can be extracted from a single direct detection observation of dark matter while making absolutely no assumptions about the local velocity distribution and local density of dark matter. Our central conclusions follow from a very simple observation: the velocity distribution of dark matter is positive definite, f(v) >= 0. We demonstrate the utility of this result in several ways. First, we show a falling deconvoluted recoil spectrum (deconvoluted of the nuclear form factor), such as from ordinary elastic scattering, can be mocked up by any mass of dark matter above a kinematic minimum. As an example, we show that dark matter much heavier than previously considered can explain the CoGeNT excess. Specifically, m(chi) < m(Ge) can be in just as good agreement as light dark matter, while m(chi) > m(Ge) depends on understanding the sensitivity of xenon to dark matter at very low recoil energies, ER less than or similar to 6 keVnr. Second, we show that any rise in the deconvoluted recoil spectrum represents distinct particle physics information that cannot be faked by an arbitrary f(v). As examples of resulting nontrivial particle physics, we show that inelastic dark matter and dark matter with a form factor can both yield such a rise.