2013MNRAS.428..421S

We develop semi-empirical models of the supermassive black hole and active galactic nucleus (AGN) populations, which incorporate the black hole growth implied by the observed AGN luminosity function assuming a radiative efficiency ε and a distribution of Eddington ratios λ. By generalizing these continuity-equation models to allow a distribution P(λ{verbar}MBH, z), we are able to draw on constraints from observationally estimated λ distributions and active galaxy fractions while accounting for the luminosity thresholds of observational samples. We consider models with a Gaussian distribution of logλ and Gaussians augmented with a power-law tail to low λ. Within our framework, reproducing the high observed AGN fractions at low redshift requires a characteristic Eddington ratio λc that declines at late times, and matching observed Eddington ratio distributions requires a P(λ) that broadens at low redshift. To reproduce the observed increase of AGN fraction with black hole or galaxy mass, we also require a λc that decreases with increasing black hole mass, reducing the AGN luminosity associated with the most massive black holes. Finally, achieving a good match to the high-mass end of the local black hole mass function requires an increased radiative efficiency at high black hole mass. We discuss the potential impact of black hole mergers or a λ-dependent bolometric correction, and we compute evolutionary predictions for black hole and galaxy specific accretion rates. Despite the flexibility of our framework, no one model provides a good fit to all the data we consider; it is particularly difficult to reconcile the relatively narrow λ distributions and low duty cycles estimated for luminous broad-line AGN with the broader λ distributions and higher duty cycles found in more widely selected AGN samples, which typically have lower luminosity thresholds.