Supermassive Black Holes in Galactic Bulges
  Growing evidence indicate supermassive black holes (SMBHs) in a mass range of $M_{\rm BH}$$\sim 106-10^{10}M_{\odot}$ lurking in central stellar bulges of galaxies.Extensive observations reveal fairly tight power laws of $M_{\rm BH}$ versus the mean stellar velocity dispersion $\sigma$ of the host stellar bulge.Together with evidence for correlations between $M_{\rm BH}$ and other properties of host bulges, the dynamic evolution of a bulge and the formation of a central SMBH should be linked. In this Letter, we reproduce the empirical $M_{\rm BH}-\sigma$ power laws based on our recent theoretical analyses (Lou & Wang; Wang & Lou; Lou, Jiang & Jin) for a self-similar general polytropic quasi-static dynamic evolution of bulges with self-gravity and spherical symmetry and present a sensible criterion of forming a central SMBH. The key result is $M_{\rm BH}={\cal L}\sigma^{1/(1-n)}$ where $2/3<n<1$ and ${\cal L}$ is a proportional coefficient characteristic of different classes of host bulges. By fitting and comparing several empirical $M_{\rm BH}-\sigma$ power laws, we conclude that SMBHs and galactic bulges grow and evolve in a coeval manner and most likely there exist several classes of galactic bulge systems in quasi-static self-similar evolution and that to mix them together can lead to an unrealistic fitting. Based on our bulge-SMBH model, we provide explanations for intrinsic scatter in the relation and a unified scenario for the formation and evolution of SMBHs in different classes of host bulges. 
Yu-Qing Lou, Yan-Fei Jiang