Black Hole Growth in Dark Matter and the M_{bh} - \sigma Relation

In this article we consider the growth of seed black holes immersed in dark matter halos. We first investigate the adiabatic growth in various initial distribution functions (isothermal, power law, and NFW) and find the resulting density, radial velocity, and anisotropy profiles. In addition we estimate the growth rate for a given black hole mass in the corresponding adiabatically modified dark matter distribution function. Only in the isothermal case is there a convincing black hole mass-age relation. By calculating the line of sight velocity dispersion for the various cases as a function of the black hole mass, we find the predicted adiabatic $M_{bh}-\sigma$ relation; this never approaches the recently observed power law. We conclude by abandoning adiabaticity, suggesting that the black hole grows proportionally to the dark matter halo itself on a dynamic time scale. This allows us to relate the observed $M_{bh}-\sigma$ relation to the cosmological power spectrum on galactic scales by using dimensional scaling arguments.