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To conclude, there have been considerable progress over the past several decades on the development of BH weighing methods for quasars. We now have a working technique based on reverberation mapping of broad line AGNs that can measure active virial BH masses with an accuracy of a factor of a few (~ 0.5 dex). Rooted in the RM technique, efficient SE virial mass estimators have been developed to measure BH mass for large statistical samples of broad line quasars based on single-epoch spectroscopy. These methods greatly facilitate quasar studies in the era of modern, large-scale spectroscopic surveys.

However, there are outstanding issues regarding the reliability of these virial (RM or SE) mass estimators, and consequences of their significant uncertainties, which is the focus of this review. Specifically I have the following highlighting remarks.

  1. There are genuine concerns that the current RM sample does not represent the whole quasar/AGN population, and the limited sample size and luminosity/redshift ranges of the RM sample, as well as the poorly understood BLR structure and dynamics, may impact the applicability of these locally-calibrated SE relations to high-z and/or high-luminosity quasars (Section 3.1). Due to limitations of the current RM sample and the uncertainty in the average virial coefficient < f >, systematic biases on the order of a factor of a few are likely present due to these physical caveats.

  2. Even when the extrapolation of these SE estimators to high-z quasars is justified, rigorous statistical biases will arise from the uncertainties (scatter) in these SE masses (Section 3.3). In particular I demonstrated the conceptual difference between errors in SE masses and the distribution of SE masses within restricted luminosity ranges. Since luminosity is used in the estimation of SE masses, the variance in SE mass is reduced when luminosity is constrained. I also derived the luminosity-dependent bias (e.g., Shen et al. 2008a, Shen & Kelly 2010, 2012) that at fixed true mass, the SE masses are over(under)-estimated in the mean when luminosity is higher (lower) than the mean luminosity at this fixed true mass, due to the stochastic variations between luminosity and line width that contribute to the uncertainty of SE masses. Simple simulations were performed to demonstrate these effects, and suggest that sample biases on the order of a factor of a few are present in flux-limited bright quasar sample. Thus these error-induced biases are as significant as the unknown systematic biases in SE masses, and cannot be ignored. More importantly, even when we eliminate all systematic biases (zero-point uncertainty) of these SE estimators in the future, these error-induced statistical biases will largely remain given the imperfect nature of these estimators. The formalism in Section 3.3.2 provides guidance on how to quantify these error-induced sample biases with simulations.

  3. Properly accounting for the selection effect due to the sample flux limit, and statistical biases arising from errors in SE masses, are crucial to interpreting the observed distributions of quasars in statistical samples (Section 4). I discussed how the "observed" distributions of BH mass and Eddington ratio for threshold data and with SE masses differ from the "true" distributions (Section 4.2), and cautioned on some recent claims based directly on the observed distributions. I further commented on the impact of the error-induced biases in SE masses on studies of the evolution of the BH-host scaling relations (Section 4.3), and concluded that the current observations are inconclusive for the claimed evolution.

Looking forward, there is an urgent need to expand the current RM sample with lag measurements, and, to acquire exquisite (velocity-resolved) RM data to utilize the full power of this technique. Only with a substantially larger RM sample that properly probes the AGN parameter space, and with a much better understanding of the BLR geometry and dynamics (for different lines) based on these RM data and ancillary data, can we improve these virial BH mass estimators further. Since resource-wise, RM is a consuming exercise, it would be interesting to explore the possibilities of more efficient RM with wide-field multi-object imaging and spectroscopy, as well as dedicated facilities for single-object-mode monitoring.

There are a few recent innovative proposals regarding RM that are worth mentioning. Zu et al. (2011) proposed an alternative method to measure RM lags, by fitting the observed continuum and emission line light curves with recently-developed statistical models to describe quasar variability (e.g., the "damped random walk" model developed by Kelly et al. 2009b, Kozlowski et al. 2010). Compared with the traditional cross-correlation method in measuring RM lags (e.g., Gaskell & Peterson 1987), this new method can improve lag measurements by simultaneously fitting multiple lines and quantifying error correlations. In addition, there have been efforts to build dynamical BLR models to directly model the RM light curves in the time domain (e.g., Brewer et al. 2011, Pancoast et al. 2012); such forward modeling (as discussed in, e.g., Netzer & Peterson 1997) can in principle provide direct constraints on the geometry of the BLR and the mass of the BH, and so it is worthwhile to explore its potential further. Finally, alternative strategies in RM experiments with no or few spectroscopic data (e.g., Haas et al. 2011, Chelouche & Daniel 2012, Fine et al. 2012), while not as good and reliable as traditional spectroscopic RM, may speed up the process of probing the diversity of AGN parameters in the context of RM.


I thank all my collaborators and colleagues for many stimulating and enlightening discussions on this subject over the past a few years. In writing this review I benefited a lot from inspiring conversations with and/or feedback from Brandon Kelly, Luis Ho, Brad Peterson, Gordon Richards, Roberto Assef, Aaron Barth, Neil Brandt, Kelly Denney, Jenny Greene, Pat Hall, Zotan Haiman, Chris Kochanek, Juna Kollmeier, Tod Lauer, Xin Liu, Youjun Lu, Chien Peng, Alireza Rafiee, Andreas Schulze, Charles Steinhardt, Michael Strauss, Benny Trakhtenbrot, Scott Tremaine, Marianne Vestergaard, David Weinberg, Jong-Hak Woo, and Qingjuan Yu. I acknowledge support from the Carnegie Observatories through a Hubble Fellowship from Space Telescope Science Institute. Support for Program number HST-HF-51314.01-A was provided by NASA through a Hubble Fellowship grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.

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