We compile black hole ( BH ) masses for \sim 60 , 000 quasars in the redshift range 0.1 \lesssim z \lesssim 4.5 included in the Fifth Data Release of the Sloan Digital Sky Survey ( SDSS ) , using virial BH mass estimators based on the H \beta , Mg II , and C IV emission lines .
We find that : ( 1 ) within our sample , the widths of the three lines follow log-normal distributions , with means and dispersions that do not depend strongly on luminosity or redshift ; ( 2 ) the Mg II - and H \beta -estimated BH masses are consistent with one another ; and ( 3 ) the C IV BH mass estimator may be more severely affected by a disk wind component than the Mg II and H \beta estimators , giving a positive bias in mass correlated with the C IV -Mg II blueshift .
Most SDSS quasars have virial BH masses in the range 10 ^ { 8 } -10 ^ { 10 } M _ { \odot } .
There is a clear upper mass limit of \sim 10 ^ { 10 } M _ { \odot } for active BHs at z \gtrsim 2 , decreasing at lower redshifts .
Making the reasonable assumptions that the underlying BH mass distribution decreases with mass and that the Eddington ratio distribution at fixed true BH mass has non-zero width , we show that the measured virial BH mass distribution and Eddington ratio distribution within finite luminosity bins are subject to Malmquist bias if the scatter in luminosity at fixed true BH mass is uncorrelated with the scatter in line width .
Given the current versions of virial calibrations and their uncertainties , we present a model which reproduces the observed virial mass distribution , quasar luminosity function , and line width distribution of our sample ; it has an underlying BH mass distribution which is a power-law with slope \gamma _ { M } \sim - 2.6 , and a true Eddington ratio distribution at fixed BH mass which is a log-normal with mean dependent on BH mass ( \sim 10 ^ { -1.2 } for typical 10 ^ { 8 } M _ { \odot } BHs ) and with dispersion 0.4 dex .
In this model , the observed virial mass distribution for the SDSS sample is biased high by \sim 0.6 dex within finite luminosity bins , and the Eddington ratio distribution is biased low by the same amount .
A radio quasar subsample ( with 1.5 \lesssim z \lesssim 2.3 ) has mean virial BH mass larger by \sim 0.12 dex than the radio-quiet sample matched in luminosity and redshift .
A broad absorption line ( BAL ) quasar subsample ( with 1.7 \lesssim z \lesssim 2.2 ) has a virial mass distribution identical to that of the non-BAL quasar sample matched in luminosity and redshift , with no mean offset .