Black hole growth in dark matter and the MBH-σ relation

In this paper, we consider the growth of seed black holes immersed in dark matter halos. We investigate first the adiabatic growth in various initial distribution functions (isothermal; power law; and Navarro, Frenk, and White) 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, which 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 timescale. This allows us to relate the observed M-BH-sigma relation to the cosmological power spectrum on galactic scales by using dimensional scaling arguments.