Origin of the Correlations Between Supermassive Black Holes and Their Host Galaxies

Abstract: Observations have shown that supermassive black holes in nearby elliptical galaxies correlate tightly with the stellar velocity dispersion (the $\MBH - \sigma$ relation) and the stellar mass (the $\MBH - \Mhost$ relation) of their host spheroids. However, the origin of these correlations remains ambiguous. In a previous paper by Zhu et al., we proposed a model which links the M-$\sigma$ relation to the the dynamical state of the system and the $\MBH - \Mhost$ relation to the self-regulation of galaxy growth. To test this model, we compile a sample of observed galaxies with different properties and examine the dependence of the above correlations on these parameters. We find that galaxies that satisfy the the $\MBH - \sigma$ correlation appear to have reached virial equilibrium, as indicated by the ratio between kinetic energy and gravitational potential, 2K/U $\sim$ 1. Furthermore, the ratio of black hole accretion rate to star formation rate remains nearly constant, BHAR /SFR $\sim$ $10^{-3}$, in active galaxies over a wide range of mass in the redshift range z=0 - 3. These results confirm our theoretical model that the observed correlations have different origins: the $\MBH - \sigma$ relation may result from galaxy relaxation, while the $\MBH$ - $\Mhost$ relation may be due to self-regulated black hole accretion and star formation in galaxies.