Cosmic quenching and scaling laws for the evolution of supermassive black holes and host galaxies

Observations suggest a co-evolution of supermassive black holes (SMBHs) and host galaxies. In this paper, we consider the mass and energy flow in a near-equilibrium bulge suffused by gases of varying temperatures. By assuming the rate of energy flow independent of the distance r from the bulge center and the local virial equilibrium for permeated gases on scale r, a key parameter ε b was identified that quantifies the mass and energy flow in gases and the efficiency of gas cooling (or the "specific" cooling rate per unit mass), and thus regulates the co-evolution of both SMBHs and hosts. With the help of Illustris simulations and observations, we determined the redshift variation ε b∝ (1+z)5/2. A higher ε b in the early Universe means a higher specific cooling rate that allows rapid evolution of SMBHs and hosts. This simple theory, characterized by a single parameter ε b, provides the dominant mean cosmic evolution of SMBHs and hosts. All other transient phenomena may only contribute to the dispersion around this mean evolution. Based on this theory and relevant assumptions, scaling laws involving ε b were identified for the evolution of SMBHs and hosts. For host galaxies, the mass–size relation Mb∝ ε b2/3rb5/3G⁻¹, the dispersion–size relation σ b²∝ (ε b rb)2/3∝ (1+z), or the mass–dispersion relation Mb∝ ε b⁻¹G⁻¹σ b⁵ were identified, where rb∝ (1+z)⁻¹ is the bulge size. For SMBHs, three evolution phases were found involving an initial rapid growth stage with a rising luminosity LB∝ (ε b MBH)4/5, a transition stage with a declining LB∝ ε b² MBH ∝ (1+z)⁵, and a dormant stage with LB∝ (ε b MBH)4/3. Our results suggest a rapid initial super-Eddington growth in a short period with a new redshift-dependent luminosity limit LX∝ε b4/5MBH4/5G-1/5c, in contrast to the Eddington limit. Analytical solutions were formulated for the BH mass function Φ BH, active galactic nucleus (AGN) mass function Φ AGN, and duty cycle U that predict Φ L∝ L-1/5 for the faint-end luminosity function, Φ AGN∝ M-1/5 for small-mass-end AGN mass function Φ L, and U∝ M-1/5 at high redshift.