Halo Independent Direct Detection of Momentum-Dependent Dark Matter

We show that the momentum dependence of dark matter interactions with nuclei can be probed in direct detection experiments without knowledge of the dark matter velocity distribution. This is one of the few properties of DM microphysics that can be determined with direct detection alone, given a signal of dark matter in multiple direct detection experiments with different targets. Long-range interactions arising from the exchange of a light mediator are one example of momentum-dependent DM. For data produced from the exchange of a massless mediator we find for example that the mediator mass can be constrained to be $\lesssim 10$ MeV for DM in the 20-1000 GeV range in a halo-independent manner.