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.

中文翻译：

---

提取 QSO 的辐射效率和质量之间可能的内在关系：最大似然法及其在 SDSS DR7 QSO 中的应用

QSO 的辐射效率及其分布编码了关于大质量黑洞 (MBH) 质量和自旋在整个宇宙时间内演化的丰富信息。在本文中，我们开发了一种最大似然方法，从 QSO 的光度分布（经验估计的维里尔）中统计提取 QSO 的辐射效率（$\epsilon$）和质量（$M_{\bullet}$）之间的内在关系质量平面。通过使用模拟样本，我们发现可以在红移 $z\lesssim 0.4$ 处的 $\epsilon-M_{\bullet}$ 关系上施加强约束，这些关系来自类似于 Sloan Digital Sky Survey 中的均匀 QSO 样本，以及来自 QSO如果调查的震级限制可以更深 $\sim 1-2$（或 $2-3$），则样本在 $z\sim 0.6$（或 $\lesssim 1.0$）。将此方法应用于 $z\lesssim 0.7$ 的 SDSS DR7 QSO，我们发现 QSO 的 $\epsilon \propto M_{\bullet}^{0\sim 1.1}$（或 $\epsilon \propto M_{\bullet}^{-1.0\sim 0}$）相关性与根据获得的质量到 H$\beta$（或 Mg II）经验质量估计器。这些相互矛盾的结果可能是由于两个质量估计器中未知的系统误差，阻止了使用当前可用 QSO 样本对 $\epsilon-M_{\bullet}$ 关系的准确约束。我们发现 MBH 质量和 Eddington 比率分布函数的估计值都会受到 $\epsilon-M_\bullet$ 关系的影响，表明这种关系的确定对于理解 MBH 的吸积和生长历史很重要。在未来，