Extracting the Possible Intrinsic Relation between the Radiative Efficiency and Mass of QSOs: A Maximum Likelihood Method and its Application to the SDSS DR7 QSOs
Radiative efficiencies of QSOs and its distribution encode rich information on the evolution of both masses and spins of massive black holes (MBHs) across cosmic time. In this paper, we develop a maximum likelihood method to statistically extract the intrinsic relation between radiative efficiency ($\epsilon$) and mass ($M_{\bullet}$) of QSOs from their distribution on the luminosity-(empirically estimated virial) mass plane. By using mock samples, we find that strong constraint can be put on the $\epsilon-M_{\bullet}$ relation at redshift $z\lesssim 0.4$ from uniform QSO samples similar to those in Sloan Digital Sky Survey, and from QSO samples at $z \sim 0.6$ (or $\lesssim 1.0$) if the magnitude limit of the survey can be $\sim 1-2$ (or $2-3$) magnitude deeper. Applying this method to the SDSS DR7 QSOs with $z\lesssim 0.7$, we find $\epsilon \propto M_{\bullet}^{0\sim 1.1}$ (or $\epsilon \propto M_{\bullet}^{-1.0\sim 0}$) correlation for QSOs with the masses obtained according to the H$\beta$ (or Mg II) empirical mass estimator. These contradictory results may be due to the unknown systematic errors in the two mass estimators, preventing an accurate constraint on the $\epsilon-M_{\bullet}$ relation by using current available QSO samples. We find that both the estimates of MBH mass and Eddington ratio distribution functions can be affected by the $\epsilon-M_\bullet$ relation, suggesting that the determination of this relation is important for understanding the accretion and growth history of MBHs. In future, the intrinsic $\epsilon-M_{\bullet}$ relation is expected to be strongly constrained by using QSO samples obtained from surveys deeper than SDSS if the host galaxy contamination and systematic errors of the mass estimator(s) can be well modeled or removed.