  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 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 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.
