  We use cosmological simulations to study the effects of self-interacting dark
matter (SIDM) on the density profiles and substructure counts of dark matter
halos from the scales of spiral galaxies to galaxy clusters, focusing
explicitly on models with cross sections over dark matter particle mass
\sigma/m = 1 and 0.1 cm^2/g. Our simulations rely on a new SIDM N-body
algorithm that is derived self-consistently from the Boltzmann equation and
that reproduces analytic expectations in controlled numerical experiments. We
find that well-resolved SIDM halos have constant-density cores, with
significantly lower central densities than their CDM counterparts. In contrast,
the subhalo content of SIDM halos is only modestly reduced compared to CDM,
with the suppression greatest for large hosts and small halo-centric distances.
Moreover, the large-scale clustering and halo circular velocity functions in
SIDM are effectively identical to CDM, meaning that all of the large-scale
successes of CDM are equally well matched by SIDM. From our largest cross
section runs we are able to extract scaling relations for core sizes and
central densities over a range of halo sizes and find a strong correlation
between the core radius of an SIDM halo and the NFW scale radius of its CDM
counterpart. We construct a simple analytic model, based on CDM scaling
relations, that captures all aspects of the scaling relations for SIDM halos.
Our results show that halo core densities in \sigma/m = 1 cm^2/g models are too
low to match observations of galaxy clusters, low surface brightness spirals
(LSBs), and dwarf spheroidal galaxies. However, SIDM with \sigma/m ~ 0.1 cm^2/g
appears capable of reproducing reported core sizes and central densities of
dwarfs, LSBs, and galaxy clusters without the need for velocity dependence.
(abridged)
