The correlation between black hole mass and bulge velocity dispersion in hierarchical galaxy formation models

Recent work has demonstrated that there is a tight correlation between the mass of a black hole and the velocity dispersion of the bulge of its host galaxy. We show that the model of Kauffmann & Haehnelt, in which bulges and supermassive black holes both form during major mergers, produces a correlation between M-bh and sigma with a slope and scatter comparable to the observed relation. In the model the M-bh-sigma relation is significantly tighter than the correlation between black hole mass and bulge luminosity, or the correlation between bulge luminosity and velocity dispersion. There are two reasons for this: (i) the gas masses of bulge progenitors depend on the velocity dispersion but not on the formation epoch of the bulge, whereas the stellar masses of the progenitors depend on both; (ii) mergers between galaxies move black holes along the observed M-bh-sigma relation, even at late times when the galaxies are gas-poor and black holes grow mainly the by merging of pre-existing black holes. We conclude that the small scatter in the observed M-bh-sigma relation is consistent with a picture in which bulges and black holes form over a wide range in redshift.