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.
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