We consider the astrophysical consequences of the self-interacting dark matter (SIDM) scenario for a general velocity-dependent cross section per unit mass that varies as some power of velocity: σDM = σ0 -a. Accretion of SIDM onto seed black holes can produce supermassive black holes that are too large for certain combinations of σ0v and a, a fact that is used to obtain a new constraint on the dark matter interaction. Constraints due to other astrophysical considerations are presented and previous constraints for a constant cross section are generalized. The black hole constraint is extremely sensitive to the cusp slope, α, of the inner density profile ρ ~ r-α of dark halos. For the most probable value of α = 1.3, we find that there exists a tiny region in the parameter space for SIDM properties, with a ≈ 0.5 and a ≈ 0.5, such that all constraints are satisfied. However, the adiabatic compression of the dark halo by baryons as they cool and contract in normal galaxies yields a steeper cusp, ρ ~ r. We find that in both the highly collisional and collisionless limits invariance arguments require α' = (6 - α) (4 - α), where α and α' are the (4α) inner profile slope of the dark halo before and after compression, respectively. This gives the tighter constraint a 0.02, which would exclude SIDM as a possible solution to the purported problems with cold dark matter (CDM) on subgalactic scales in the absence of other dynamical processes. Nevertheless, SIDM with parameters consistent with this stronger constraint, can explain the ubiquity of supermassive black holes in the centers of galaxies. A "best-fit" model is presented with a = 0 and = 0.02, which reproduces the supermassive black hole masses and their observed correlations with the velocity dispersion of the host bulges. Specifically, the approximately fourth-power dependence of black hole mass on galactic velocity dispersion is a direct consequence of the power spectrum of primeval perturbations having an index of n ≈ -2 and the value of a. Although the dark matter collision rates for this model are too small to directly remedy problems with CDM, mergers between dark halos harboring supermassive black holes at high redshift could ameliorate the cuspy halo problem. This scenario also explains the lack of comparable supermassive black holes in bulgeless galaxies like M33.
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