Energy cascade for distribution and evolution of supermassive black holes and host galaxies

Strong correlations exist between supermassive black holes (SMBHs) and their host galaxies. These correlations suggest a missing component in our current understanding: the role of energy cascade in SMBH-bulge coevolution. In this picture, energy is continuously cascaded from bulge scale r_b down to the BH scale (Schwarzschild radius r_s). Energy cascade has a scale-independent, but decreasing rate ε_b(t)~σ_b³/r_b that regulates the coevolution, where σ_b is bulge velocity dispersion. The host galaxy mass-size relation follows a 5/3 law with M_b ~ ε_b^2/3r_b^5/3G^-1, or an equivalent density-size relation ρ_b~r_b^-4/3, with rate of cascade ε_b ~ a^-5/2x10^-4m²/s³, as confirmed by the galaxy survey. Here a is the scale factor and G is the gravitational constant. Since ε_b is much larger in the early universe, cascade theory predicts that BH accretion can be super-Eddington with luminosity L_B exceeding the Eddington limit. In addition, the BH mass-velocity dispersion relation (M_B ~σ_b⁵) is a natural result of the cascade theory. A three stage model is proposed for SMBH-host evolution involving co-evolution, transitional, and dormant stages, respectively. Models are finally compared against the BH accretion history from quasar luminosity function of 2dF Redshift Survey, local galaxy and SMBHs data, and high redshift quasars from SDSS DR7 and CFHQS surveys.