  Dark Matter (DM) properties at small scales remain uncertain. Recent
theoretical and observational advances have provided the tools to narrow them
down. Here, we show for the first time that the correlation between internal
velocities and sizes of dwarf galaxies is a sharp probe of small-scale DM
properties. We study modified DM power spectra, motivated by DM production
during inflation. Using semi-analytic models and scaling relations, we show
that such models can change the kinematics and structure of dwarf galaxies
without strongly affecting their total abundance. We analyze data from Milky
Way classical satellite galaxies and those discovered with the Sloan Digital
Sky Survey (SDSS), finding that the DM power spectrum at comoving scales ${4\,
\mathrm{Mpc}^{-1} < k < 37\,\mathrm{Mpc}^{-1}}$ cannot deviate by more than a
factor of 2 from scale invariance. Our results are robust against baryonic
uncertainties such as the stellar mass-halo mass relation, halo occupation
fraction, and subhalo tidal disruption; allowing us to independently constrain
them. This work thus opens a window to probe both dwarf galaxy formation models
and small-scale DM properties.
