Growth of massive black holes ( MBHs ) in galactic centers comes mainly from gas accretion during their QSO/AGN phases .
In this paper we apply an extended Sołtan argument , connecting the local MBH mass function with the time-integral of the QSO luminosity function , to the demography of MBHs and QSOs from recent optical and X-ray surveys , and obtain robust constraints on the luminosity evolution ( or mass growth history ) of individual QSOs ( or MBHs ) .
We find that the luminosity evolution probably involves two phases : an initial exponentially increasing phase set by the Eddington limit and a following phase in which the luminosity declines with time as a power law ( with a slope of \sim - 1.2 — -1.3 ) set by a self-similar long-term evolution of disk accretion .
Neither an evolution involving only the increasing phase with a single Eddington ratio nor an exponentially declining pattern in the second phase is likely .
The period of a QSO radiating at a luminosity higher than 10 % of its peak value is about 2–3 \times 10 ^ { 8 } yr , during which the MBH obtains \sim 80 \% of its mass .
The mass-to-energy conversion efficiency is \simeq 0.16 \pm 0.04 ^ { +0.05 } _ { -0 } , with the latter error accounting for the maximum uncertainty due to Compton-thick AGNs .
The expected Eddington ratios in QSOs from the constrained luminosity evolution cluster around a single value close to 0.5–1 for high-luminosity QSOs and extend to a wide range of lower values for low-luminosity ones .
The Eddington ratios for high luminosity QSOs appear to conflict with those estimated from observations ( \sim 0.25 ) by using some virial mass estimators for MBHs in QSOs unless the estimators systematically over-estimate MBH masses by a factor of 2–4 .
We also infer the fraction of optically obscured QSOs \sim 60 - 80 \% .
The constraints obtained above are not affected significantly by MBH mergers and multiple-times of nuclear activity ( e.g. , triggered by multiple times of galaxy wet major mergers ) in the MBH growth history .
We discuss further applications of the luminosity evolution of individual QSOs to obtaining the MBH mass function at high redshifts and the cosmic evolution of triggering rates of nuclear activity .