We show that the observed scatter in the relations between the mass of supermassive black holes ( SMBHs ) , M _ { bh } , and the velocity dispersion \sigma _ { sph } or mass M _ { sph } of their host spheroid , place 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 } – \sigma _ { 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 } \approx 6 { kpc } ( \sigma _ { sph } / 200 \mbox { km } \mbox { s } ^ { -1 } ) ^ { 1 % .5 } with a scatter of \sim 0.2dex , in agreement with the corresponding projection of the fundamental plane for early type galaxies in SDSS .
At the redshift 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 } – \sigma _ { sph } and M _ { bh } – M _ { sph } relations preclude 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 } – \sigma _ { sph } and M _ { bh } – M _ { sph } relations , only if the SMBH growth is limited by momentum or energy feedback over the dynamical time of the host spheroid .