  Dark matter particles interacting via the exchange of very light spin-0
mediators can have large self-interaction rates and obtain their relic
abundance from thermal freeze-out. At the same time, these models face strong
bounds from direct and indirect probes of dark matter as well as a number of
constraints on the properties of the mediator. We investigate whether these
constraints can be consistent with having observable effects from dark matter
self-interactions in astrophysical systems. For the case of a mediator with
purely scalar couplings we point out the highly relevant impact of
low-threshold direct detection experiments like CRESST-II, which essentially
rule out the simplest realization of this model. These constraints can be
significantly relaxed if the mediator has CP-violating couplings, but then the
model faces strong constraints from CMB measurements, which can only be avoided
in special regions of parameter space.
