Accretion-Driven Evolution of Black Holes: Eddington Ratios, Duty
  Cycles, and Active Galaxy Fractions
  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
\epsilon, and a distribution of Eddington ratios \lambda. By generalizing these
continuity-equation models to allow a distribution P(\lambda|mbh,z) we are able
to draw on constraints from observationally estimated P(\lambda) distributions
and active galaxy fractions while accounting for the luminosity thresholds of
observational samples. We consider models with a Gaussian distribution of log
\lambda, and Gaussians augmented with a power-law tail to low \lambda. Within
our framework, reproducing the high observed AGN fractions at low redshift
requires a characteristic Eddington ratio \lambda_c that declines at late
times, and matching observed Eddington ratio distributions requires P(\lambda)
that broadens at low redshift. To reproduce the observed increase of AGN
fraction with black hole or galaxy mass, we also require a \lambda_c 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 \lambda-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.
