Toward Precise Constraints on the Growth of Massive Black Holes

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 ~–1.2 to –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) × 10⁸ yr, during which the MBH obtains ~80% of its mass. The mass-to-energy conversion efficiency is ≃0.16 ± 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 (~0.25) by using some virial mass estimators for MBHs in QSOs, unless the estimators systematically overestimate MBH masses by a factor of 2-4. We also infer the fraction of optically obscured QSOs, ~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.