Constraints on the process that regulates the growth of supermassive black holes based on the intrinsic scatter in the M_bh-σ_sph relation

We show that the observed scatter in the relations between the mass of supermassive black holes (SMBHs), M_bh, and the velocity dispersion σ_sph or mass M_sph of their host spheroid places interesting constraints on the process that regulates SMBH growth in galaxies. When combined with the observed properties of early-type SDSS galaxies, the observed intrinsic scatters imply that SMBH growth is regulated by the spheroid velocity dispersion rather than its mass. The M_bh-M_sph relation is therefore a by-product of a more fundamental M_bh- σ_sph relation. We construct a theoretical model for the scatter among baryon-modified dark matter halo profiles, out of which we generate a population of spheroid hosts and show that these naturally lead to a relation between effective radius and velocity dispersion of the form R_sph ≈ 6 kpc(σ_sph/200 km s^-1)^1.5 with a scatter of ∼0.2 dex, in agreement with the corresponding projection of the fundamental plane for early-type galaxies in the SDSS. At the redshitt of formation, our model predicts the minimum scatter that SMBHs can have at fixed velocity dispersion or spheroid mass under different formation scenarios. We also estimate the additional scatter that is introduced into these relations through collisionless mergers of purely stellar spheroids at z < 1. We find that the observed scatter in the M_bh-σ_sph and M _bh-M_sph relations precludes the properties of dark matter halos from being the governing factor in SMBH growth. The apparent relation between halo and SMBH mass is merely a reflection of the fact that massive halos tend to host massive stellar spheroids (albeit with a large scatter owing to the variance in formation histories). Finally, we show that SMBH growth governed by the properties of the host spheroid can lead to the observed values of scatter in the M_bh-σ_sph and M_bh-M _sph relations, in cases where the SMBH growth is limited by momentum or energy feedback over the dynamical time of the host spheroid.,