PNNL

Abstract

Observations suggest a coevolution of SMBHs and hosts. 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 from the bulge center and the virial equilibrium for permeated gases, a key parameter $\varepsilon_b$ was identified that quantifies the mass and energy flow in gases and the efficiency of gas cooling and thus regulates the coevolution of SMBHs and hosts. With the help of Illustris simulations and observations, we determined the redshift variation $\varepsilon_b\propto (1+z)^{5/2}$. A higher $\varepsilon_b$ in the early universe means a more efficient gas cooling that allows rapid evolution of SMBHs and hosts. This simple theory, characterized by a single parameter $\varepsilon_b$, provides the dominant mean cosmic evolution of SMBHs and hosts. All other transient phenomena may only contribute to the dispersion around the mean evolution. Based on this theory and relevant assumptions, scaling laws involving $\varepsilon_b$ were identified for the evolution of SMBHs and their hosts. For host galaxies, the mass-size relation $M_b\propto \varepsilon_b^{2/3}r_b^{5/3}G^{-1}$, the dispersion-size relation $\sigma_b^2\propto(\varepsilon_b r_b)^{2/3}\propto (1+z)$, or the mass-dispersion relation $M_b\propto \varepsilon_b^{-1}G^{-1}\sigma_b^5$ were identified, where $r_b\propto (1+z)^{-1}$ is the bulge size. For SMBHs, three evolution phases were found involving an initial rapid growth stage with a rising luminosity $L_B\propto (\varepsilon_b M_{BH})^{4/5}$, a transition stage with a declining $L_B\propto \varepsilon_b^2 M_{BH} \propto (1+z)^5$, and a dormant stage with $L_B\propto (\varepsilon_b M_{BH})^{4/3}$. The results suggest a rapid initial super-Eddington growth in a short period with a new redshift-dependent luminosity limit $L_X=\varepsilon_b^{4/5}M_{BH}^{4/5}G^{-1/5}c$, in contrast to the Eddington limit. Analytical solutions were formulated for BH mass function $\Phi_{BH}$, AGN mass function $\Phi_{AGN}$, and duty cycle $U$ that predict $\Phi_L\propto L^{-1/5}$ for the faint-end luminosity function, $\Phi_{AGN}\propto M^{-1/5}$ for small-mass-end AGN mass function, and $U\propto M^{-1/5}$ at high redshift.