On the Cosmological Evolution of the Luminosity Function and the Accretion Rate of Quasars

We consider a class of models for the redshift evolution (between $0\lsim z \lsim 4$) of the observed optical and X-ray quasar luminosity functions (LFs), with the following assumptions: (i) the mass-function of dark matter halos follows the Press-Schechter theory, (ii) the black hole (BH) mass scales linearly with the halo mass, (iii) quasars have a constant universal lifetime, and (iv) a thin accretion disk provides the optical luminosity of quasars, while the X-ray/optical flux ratio is calibrated from a sample of observed quasars. The mass accretion rate $\dot{M}$ onto quasar BHs is a free parameter of the models, that we constrain using the observed LFs. The accretion rate $\dot M$ inferred from either the optical or X-ray data under these assumptions generally decreases as a function of cosmic time from $z \simeq 4$ to $z \simeq 0$. We find that a comparable accretion rate is inferred from the X-ray and optical LF only if the X-ray/optical flux ratio decreases with BH mass. Near $z\simeq 0$, $\dot M$ drops to substantially sub-Eddington values at which advection-dominated accretion flows (ADAFs) exist. Such a decline of $\dot M$, possibly followed by a transition to radiatively inefficient ADAFs, could explain both the absence of bright quasars in the local universe and the faintness of accreting BHs at the centers of nearby galaxies. We argue that a decline of the accretion rate of the quasar population is indeed expected in cosmological structure formation models.