  Many models of dark matter include self-interactions beyond gravity. A
variety of astrophysical observations have previously been used to place limits
on the strength of such self-interactions. However, previous works have
generally focused either on short-range interactions resulting in individual
dark matter particles scattering from one another, or on effectively
infinite-range interactions which sum over entire dark matter halos. In this
work, we focus on the intermediate regime: forces with range much larger than
dark matter particles' inter-particle spacing, but still shorter than the
length scales of known halos. We show that gradients in the dark matter density
of such halos would still lead to observable effects. We focus primarily on
effects in the Bullet Cluster, where finite-range forces would lead either to a
modification of the collision velocity of the cluster or to a separation of the
dark matter and the galaxies of each cluster after the collision. We also
consider constraints from the binding of ultrafaint dwarf galaxy halos, and
from gravitational lensing of the Abell 370 cluster. Taken together, these
observations allow us to set the strongest constraints on dark matter
self-interactions over at least five orders of magnitude in range, surpassing
existing limits by many orders of magnitude throughout.
