Shankar, Francesco, Weinberg, David H. and Miralda-Escude, Jordi
(2009)
Self-consistent models of the AGN and black hole populations: duty cycles, accretion rates, and the mean radiative efficiency.
*Astrophysical Journal*, 690, 20-41.
(doi:10.1088/0004-637X/690/1/20).

## Abstract

We construct evolutionary models of the populations of active galactic nuclei (AGNs) and supermassive black holes, in which the black hole mass function grows at the rate implied by the observed luminosity function, given assumptions about the radiative efficiency and the luminosity in Eddington units. We draw on a variety of recent X-ray and optical measurements to estimate the bolometric AGN luminosity function and compare to X-ray background data and the independent estimate of Hopkins et al. to assess remaining systematic uncertainties. The integrated AGN emissivity closely tracks the cosmic star-formation history, suggesting that star formation and black hole growth are closely linked at all redshifts. We discuss observational uncertainties in the local black hole mass function, which remain substantial, with estimates of the integrated black hole mass density ρ_{•} spanning the range 3-5.5 × 10^{5} M_{☉} Mpc^{–3}. We find good agreement with estimates of the local mass function for a reference model where all active black holes have a fixed efficiency ε = 0.065 and L_{bol}/L_{Edd} ~ 0.4 (shifting to ε = 0.09, L_{bol}/L_{Edd} ~ 0.9 for the Hopkins et al. luminosity function). In our reference model, the duty cycle of 10^{9} M_{☉} black holes declines from 0.07 at z = 3 to 0.004 at z = 1 and 10^{–4} at z = 0. The decline is shallower for less massive black holes, a signature of "downsizing" evolution in which more massive black holes build their mass earlier. The predicted duty cycles and AGN clustering bias in this model are in reasonable accord with observational estimates. If the typical Eddington ratio declines at z < 2, then the "downsizing" of black hole growth is less pronounced. Models with reduced Eddington ratios at low redshift or black hole mass predict fewer low-mass black holes (M_{•} ≲ 10^{8} M_{☉}) in the local universe, while models with black hole mergers predict more black holes at M_{•} > 10^{9} M_{☉}. Matching the integrated AGN emissivity to the local black hole mass density implies ε = 0.075 × (ρ_{•}/4.5 × 10^{5} M_{☉} Mpc^{–3})^{–1} for our standard luminosity function estimate, or 25% higher for Hopkins et al.'s estimate. It is difficult to reconcile current observations with a model in which most black holes have the high efficiencies ε ~ 0.16-0.20 predicted by MHD simulations of disk accretion. We provide electronic tabulations of our bolometric luminosity function and our reference model predictions for black hole mass functions and duty cycles as a function of redshift

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