Cosmic quenching and scaling laws for evolution of galaxies and SMBHs

Abstract Observations suggest a tight coupling between spuermassive black hole (SMBH) growth and star formation in the host galaxy. Based on Illustris simulations and astronomical observations, a cosmic quenching over the cosmic time scale is proposed to regulate the synchronized evolution of both SMBH and host. A central quantity of cosmic quenching is the parameter 𝜀𝑏 (unit: 𝑚2/𝑠3) that describes the rate of mass and energy flow of gases along the bulge radial direction. The parameter 𝜀𝑏 also controls the efficiency of gas cooling and the supply of cold gas. The value 𝜀𝑏 ≈ 10−4 (1 + 𝑧)5/2𝑚2/𝑠3 can be determined which decreases with time. For a larger 𝜀𝑏 in the early universe, gas cooling is more efficient in providing a richer supply of cold gas for a rapid initial evolution of both SMBH and host. At lower redshifts, a smaller 𝜀𝑏 means less efficient gas cooling, less cold gas supply, and slower star formation and black hole growth (quenched). The rate of mass accretion naturally exhibits a peak at 𝑧 ≈ 2 due to the decrease in 𝜀𝑏. Scaling laws involving 𝜀𝑏 are identified that govern the evolution of both SMBHs and their hosts. For host galaxies, we identify the mass-size relation 𝑀𝑏 ∝ 𝜀2/3 𝑏 𝑟5/3 𝑏 𝐺−1 and the dispersion-size relation 𝜎2 𝑏 ∝ (𝜀𝑏𝑟𝑏)2/3 ∝ (1 + 𝑧), where 𝑟𝑏 ∝ (1 + 𝑧)−1 is the bulge size. For SMBH, an initial rapid growth is identified with a sharp increase in luminosity 𝐿𝐵 ∝ (𝜀𝑏𝑀𝐵𝐻)4/5𝐺−1/5𝑐, followed by a transition stage with a decreasing luminosity 𝐿𝐵 ∝ 𝜀2𝑏𝑀𝐵𝐻 ∝ (1 + 𝑧)5, and a dormant stage with 𝐿𝐵 ∝ (𝜀𝑏𝑀𝐵𝐻)4/3𝐺1/3𝑐−5/3. Here 𝐺 is the gravitational constant, 𝑐 is the speed of light, 𝑀𝐵𝐻 is the mass of BH. For SMBH-galaxy co-evolution, the observed M-σ correlation can be analytically derived as 𝑀𝐵𝐻 ∝ 𝜎5 𝑏 /(𝜀𝑏𝐺). Using these scaling laws, analytical solutions are formulated for the evolution of the SMBH mass function Φ𝐵𝐻 (𝑀, 𝑧), the AGN mass function Φ𝐴𝐺𝑁 (𝑀, 𝑧), duty cycle 𝑈(𝑀, 𝑧), and the Eddington ratio distribution. The model predicts Φ𝐿 ∝ 𝐿−1/5 for the faint-end quasar luminosity function, Φ𝐴𝐺𝑁 ∝ 𝑀−1/5 for a small-mass-end AGN mass function, and 𝑈 ∝ 𝑀−1/5 for the duty cycle at high redshift. Finally, the complete redshift evolution of some observed high-redshift SMBHs can be modeled. The results suggest an initial super-Eddington growth in a short period when the SMBHs are still small, followed by a slow growth due to cosmic quenching when the SMBHs become large.