  There is a vast menagerie of plausible candidates for the constituents of
dark matter, both within and beyond extensions of the Standard Model of
particle physics. Each of these candidates may have scattering (and other)
cross section properties that are consistent with the dark matter abundance,
BBN, and the most scales in the matter power spectrum; but which may have
vastly different behavior at sub-galactic "cutoff" scales, below which dark
matter density fluctuations are smoothed out. The only way to quantitatively
measure the power spectrum behavior at sub-galactic scales at distances beyond
the local universe, and indeed over cosmic time, is through probes available in
multiply imaged strong gravitational lenses. Gravitational potential
perturbations by dark matter substructure encode information in the observed
relative magnifications, positions, and time delays in a strong lens. Each of
these is sensitive to a different moment of the substructure mass function and
to different effective mass ranges of the substructure. The time delay
perturbations, in particular, are proving to be largely immune to the
degeneracies and systematic uncertainties that have impacted exploitation of
strong lenses for such studies. There is great potential for a coordinated
theoretical and observational effort to enable a sophisticated exploitation of
strong gravitational lenses as direct probes of dark matter properties. This
opportunity motivates this white paper, and drives the need for: a) strong
support of the theoretical work necessary to understand all astrophysical
consequences for different dark matter candidates; and b) tailored
observational campaigns, and even a fully dedicated mission, to obtain the
requisite data.
