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 epsilon(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 epsilon(b) proportional to (1+z)(5/2). A higher epsilon(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 epsilon(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 epsilon(b) were identified for the evolution of SMBHs and hosts. For host galaxies, the mass-size relation M-b proportional to epsilon(2/3)(b)r(b)(5/3)G(-1), the dispersion-size relation sigma(2 )(b)proportional to (epsilon(b)r(b))(2/3 )proportional to (1+z), or the mass-dispersion relation M-b proportional to epsilon(-1)(b)G(-1)sigma(5)(b) were identified, where r(b) proportional to (1+z)-(1) is the bulge size. For SMBHs, three evolution phases were found involving an initial rapid growth stage with a rising luminosity L-B proportional to (epsilon M-b(BH))(4/5), a transition stage with a declining L-B proportional to epsilon M-2(b)BH proportional to (1+z)(5), and a dormant stage with L-B proportional to (epsilon M-b(BH))(4/3). Our results suggest a rapid initial super-Eddington growth in a short period with a new redshift-dependent luminosity limit L-X proportional to epsilon(4/5)(b)M(BH)(4/5)G(-1/5)c, in contrast to the Eddington limit. Analytical solutions were formulated for the BH mass function Phi(BH), active galactic nucleus (AGN) mass function Phi(AGN), and duty cycle U that predict Phi(L )proportional to L-1/5 for the faint-end luminosity function, Phi(AGN )proportional to M-1/5 for small-mass-end AGN mass function Phi(L), and U proportional to M-1/5 at high redshift.