Black Hole Growth in Dark Matter and theMBH‐σ Relation

In this article we consider the growth of seed black holes immersed in darkmatter halos. We first investigate the adiabatic growth in various initialdistribution functions (isothermal, power law, and NFW) and find the resultingdensity, radial velocity, and anisotropy profiles. In addition we estimate thegrowth rate for a given black hole mass in the corresponding adiabaticallymodified dark matter distribution function. Only in the isothermal case isthere a convincing black hole mass-age relation. By calculating the line ofsight velocity dispersion for the various cases as a function of the black holemass, we find the predicted adiabatic $M_{bh}-\sigma$ relation; this neverapproaches the recently observed power law. We conclude by abandoningadiabaticity, suggesting that the black hole grows proportionally to the darkmatter halo itself on a dynamic time scale. This allows us to relate theobserved $M_{bh}-\sigma$ relation to the cosmological power spectrum ongalactic scales by using dimensional scaling arguments.Comment: 23 pages, 5 figures. Accepted by the Astrophysical Journa