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Radiative efficiencies of QSOs and its distribution encode rich informationon the evolution of both masses and spins of massive black holes (MBHs) acrosscosmic time. In this paper, we develop a maximum likelihood method tostatistically extract the intrinsic relation between radiative efficiency($\epsilon$) and mass ($M_{\bullet}$) of QSOs from their distribution on theluminosity-(empirically estimated virial) mass plane. By using mock samples, wefind that strong constraint can be put on the $\epsilon-M_{\bullet}$ relationat redshift $z\lesssim 0.4$ from uniform QSO samples similar to those in SloanDigital Sky Survey, and from QSO samples at $z \sim 0.6$ (or $\lesssim 1.0$) ifthe magnitude limit of the survey can be $\sim 1-2$ (or $2-3$) magnitudedeeper. Applying this method to the SDSS DR7 QSOs with $z\lesssim 0.7$, we find$\epsilon \propto M_{\bullet}^{0\sim 1.1}$ (or $\epsilon \proptoM_{\bullet}^{-1.0\sim 0}$) correlation for QSOs with the masses obtainedaccording to the H$\beta$ (or Mg II) empirical mass estimator. Thesecontradictory results may be due to the unknown systematic errors in the twomass estimators, preventing an accurate constraint on the$\epsilon-M_{\bullet}$ relation by using current available QSO samples. We findthat both the estimates of MBH mass and Eddington ratio distribution functionscan be affected by the $\epsilon-M_\bullet$ relation, suggesting that thedetermination of this relation is important for understanding the accretion andgrowth history of MBHs. In future, the intrinsic $\epsilon-M_{\bullet}$relation is expected to be strongly constrained by using QSO samples obtainedfrom surveys deeper than SDSS if the host galaxy contamination and systematicerrors of the mass estimator(s) can be well modeled or removed.

astrophysical journal/the astrophysical journal

Extracting the Possible Intrinsic Relation between Radiative Efficiency  and Mass of QSOs: a Maximum Likelihood Method and its Application to the SDSS  DR7 QSOs