On the 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 due to the cooling of baryonic component, where σ_b is bulge velocity dispersion. The bulge mass-size (M_b-r_b) relation can be expressed as M_b ≈ε_b^2/3r_b^5/3G^-1, or a bulge density-size relation ρ_b ≈ ε_b^2/3r_b^-4/3G^-1, with ε_b≈ a^-5/210^-4m²/s³, as confirmed by the galaxy survey, where a is the scale factor and G is the gravitational constant. For SMBH with a mass M_B, bolometric luminosity L_B, energy cascade leads to a “cascade” force that must be balanced by the BH radiation force in its early life, i.e. L_B/c ~ M_Bε_b, where c is light speed. Since ε_b is much larger in the early universe, BH accretion can be super-Eddington with L_B exceeding the Eddington limit. In addition, the BH mass-dispersion relation (M_B~σ_b⁵) is a natural result of the cascade theory. A three stage mathematical model is proposed for SMBH evolution involving co-evolution, transitional, and dormant stages, respectively. Models are finally compared against the BH accretion history from quasar luminosity function from 2dF Redshift Survey, local galaxy and SMBHs data, and high redshift quasars from SDSS DR7 and CFHQS surveys. Accompanying slides and datasets for this work can be found at https://doi.org/10.5281/zenodo.6569901.