  Astrophysical observations spanning dwarf galaxies to galaxy clusters
indicate that dark matter (DM) halos are less dense in their central regions
compared to expectations from collisionless DM N-body simulations. Using
detailed fits to DM halos of galaxies and clusters, we show that
self-interacting DM (SIDM) may provide a consistent solution to the DM deficit
problem across all scales, even though individual systems exhibit a wide
diversity in halo properties. Since the characteristic velocity of DM particles
varies across these systems, we are able to measure the self-interaction cross
section as a function of kinetic energy and thereby deduce the SIDM particle
physics model parameters. Our results prefer a mildly velocity-dependent cross
section, from $\sigma/m \simeq 2\; {\rm cm^2/g}$ on galaxy scales to $\sigma/m
\simeq 0.1\; {\rm cm^2/g}$ on cluster scales, consistent with the upper limits
from merging clusters. Our results dramatically improve the constraints on SIDM
models and may allow the masses of both DM and dark mediator particles to be
measured even if the dark sector is completely hidden from the Standard Model,
which we illustrate for the dark photon model.
