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

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.