  Identifying the true theory of dark matter depends crucially on accurately
characterizing interactions of dark matter (DM) with other species. In the
context of DM direct detection, we present a study of the prospects for
correctly identifying the low-energy effective DM-nucleus scattering operators
connected to UV-complete models of DM-quark interactions. We take a census of
plausible UV-complete interaction models with different low-energy
leading-order DM-nuclear responses. For each model (corresponding to different
spin-, momentum-, and velocity-dependent responses), we create a large number
of realizations of recoil-energy spectra, and use Bayesian methods to
investigate the probability that experiments will be able to select the correct
scattering model within a broad set of competing scattering hypotheses. We
conclude that agnostic analysis of a strong signal (such as Generation-2 would
see if cross sections are just below the current limits) seen on xenon and
germanium experiments is likely to correctly identify momentum dependence of
the dominant response, ruling out models with either "heavy" or "light"
mediators, and enabling downselection of allowed models. However, a unique
determination of the correct UV completion will critically depend on the
availability of measurements from a wider variety of nuclear targets, including
iodine or fluorine. We investigate how model-selection prospects depend on the
energy window available for the analysis. In addition, we discuss accuracy of
the DM particle mass determination under a wide variety of scattering models,
and investigate impact of the specific types of particle-physics uncertainties
on prospects for model selection.
