Self-regulated black hole accretion, the M–σ relation and the growth of bulges in galaxies

We argue that the velocity dispersions and masses of galactic bulges and spheroids are byproducts of the feedback that regulates rapid black hole growth in protogalaxies. We suggest that the feedback energy liberated by accretion must pass through the accreting material, in an energyconserving flux near the black hole and a momentum-conserving flux farther away from the black hole. If the inflowing gas dominates the gravitational potential outside the Bondi radius, feedback from Eddington-limited accretion drives the density profile of the gas to that of a singular isothermal sphere. We find that the velocity dispersion associated with the isothermal potential, σ , increases with time as the black hole mass M grows, in such a way that M ∝ σ 4. The coefficient of this proportionality depends on the radius at which the flow switches from energy conserving to momentum conserving, and gives the observed M–σ relation if the transition occurs at ∼100 Schwarzschild radii. We associate this transition with radiative cooling and show that bremsstrahlung, strongly boosted by inverse Compton scattering in a two-temperature (T p T e) plasma, leads to a transition at the desired radius. According to this picture, bulge masses Mb are insensitive to the virial masses of their dark matter haloes, but correlate linearly with black hole mass. Our analytic model also explains the M b–σ (Faber–Jackson) relation as a relic of black hole accretion. The model naturally explains why the M–σ relation has less scatter than either the M–M b (Magorrian) or the Faber–Jackson relation. It suggests that the M–σ relation could extend down to very low velocity dispersions, and predicts that the relation should not evolve with redshift.