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BIASESINVIRIALBLACKHOLEMASSES:ANSDSSPERSPECTIVE
YUESHEN1,JENNYE.GREENE1,2,MICHAELA.STRAUSS1,GORDONT.RICHARDS3,DONALDP.SCHNEIDER4
SubmittedtoApJonSeptember19,2007
ABSTRACT
We compile black hole (BH) masses for ∼60,000quasarsin the redshiftrange 0.1.z.4.5 includedin
the Fifth Data Release of the Sloan Digital Sky Survey (SDSS), using virial BH mass estimators based on
the Hβ, MgII, and CIV 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
8 redshift;(2)theMgII-andHβ-estimatedBHmassesareconsistentwithoneanother;and(3)theCIVBHmass
0 estimatormaybemoreseverelyaffectedbyadiskwindcomponentthantheMgIIandHβ estimators, giving
0 apositivebiasinmasscorrelatedwiththeCIV-MgIIblueshift. MostSDSSquasarshavevirialBH massesin
2 therange108- 1010M⊙. Thereisaclearuppermasslimitof∼1010M⊙ foractiveBHsatz&2,decreasingat
n lowerredshifts. MakingthereasonableassumptionsthattheunderlyingBHmassdistributiondecreaseswith
a mass and that the Eddington ratio distribution at fixed true BH mass has non-zero width, we show that the
J measuredvirialBHmassdistributionandEddingtonratiodistributionwithinfiniteluminositybinsaresubject
3 toMalmquistbiasifthescatterinluminosityatfixedtrueBHmassisuncorrelatedwiththescatterinlinewidth.
Giventhecurrentversionsofvirialcalibrationsandtheiruncertainties,wepresentamodelwhichreproduces
] theobservedvirialmassdistribution,quasarluminosityfunction,andlinewidthdistributionofoursample;it
h hasanunderlyingBHmassdistributionwhichisapower-lawwithslopeγ ∼- 2.6,andatrueEddingtonratio
p distributionat fixed BH mass which is a log-normalwith mean dependenMton BH mass (∼10- 1.2 for typical
-
o 108 M⊙ BHs) andwithdispersion0.4dex. Inthismodel,theobservedvirialmassdistributionfortheSDSS
r sampleisbiasedhighby∼0.6dexwithinfiniteluminositybins,andtheEddingtonratiodistributionisbiased
t
s lowbythesameamount. A radioquasarsubsample(with1.5.z.2.3)hasmeanvirialBH masslargerby
a ∼0.12 dex than the radio-quietsample matched in luminosity and redshift. A broad absorption line (BAL)
[
quasarsubsample(with 1.7.z.2.2)has a virialmass distributionidenticalto thatof the non-BALquasar
2 samplematchedinluminosityandredshift,withnomeanoffset.
v Subjectheadings:blackholephysics–galaxies: active–galaxies: fundamentalparameters–galaxies: high-
8 redshift–quasars:general–surveys
9
0
3 1. INTRODUCTION Tremaine2002). Thisisparticularlyencouraging,sincewith
. ever larger AGN/quasar samples in modern surveys, we can
9 Therehasbeenincreasinginterestinrecentyearsintherole
hope to understand the cosmic evolution of SMBHs within
0 thatsupermassiveblackholes(SMBHs)playingalaxyforma-
7 tion, primarilybecauseofthe discoverythatmost, if notall, the framework of hierarchical structure formation, and shed
0 present-daymassivegalaxiesharbora SMBH in their nuclei lightonthecoevolutionofSMBHsandtheirhostgalaxies.A
: central issue in this regard is to measure the masses of both
v (e.g., Kormendy & Richstone 1995; Richstone et al. 1998),
inactiveSMBHsandactiveAGNs/quasars.
i andthatthemassofthenuclearSMBHisrelatedtothebulge
X Perhaps the most reliable way to measure the black hole
mass/luminosity(e.g.,Magorrianetal.1998)andevenmore
massisviastellar/gasdynamicsinthevicinityoftheSMBH
r tightlytothebulgestellarvelocitydispersion(e.g.,Ferrarese
a whereits gravitydominatesthe dynamics(e.g.,Richstoneet
&Merritt2000;Gebhardtetal.2000a;Tremaineetal.2002).
al.1998),buttherelevantscalescanberesolvedonlyforthe
These tightcorrelationsimply thatthe formationofgalaxies
nearestgalaxies. Amoreindirectmassindicatorusestheob-
andthegrowthofthecentralSMBHsareultimatelyconnected
served tight correlations between the SMBH mass M and
(e.g.,Silk&Rees1998;Kauffmann&Haehnelt2000;Wyithe BH
the stellar bulge velocity dispersion σ (the M- σ relation),
&Loeb2003;DiMatteoetal.2005;Hopkinsetal.2006).
or the bulge luminosity (the M- L relation). Accurate mea-
On the other hand, it has long been suggested that active
surementsofthebulgevelocitydispersionorbulgeluminos-
galactic nuclei (AGN) or quasars are SMBHs in the process
ity are difficult in luminous quasars, where the AGN light
of accretion and growth (e.g., Salpeter 1964; Zel’dovich &
overwhelmsthatofthehostgalaxy;theserelationsaremostly
Novikov 1964; Lynden-Bell 1969). Indeed, the integrated
applied to measure the local dormantSMBH mass function.
luminositydensityof optically-selectedAGNs, which repre-
Moreseriously, the M- σ andM- L relationshavenotbeen
sents the accretion history of black holes, is consistent with
measureddirectlyatthemostmassiveend,andthehigh-mass
themassdensityinthelocaldormantSMBHpopulation(the
endoftheBHmassfunctionsestimatedusingtheM- σ and
Sołtan [1982] argument; e.g., Salucci et al. 1999; Yu &
M- L relationsdifferby asmuchas one orderofmagnitude
1PrincetonUniversityObservatory,Princeton,NJ08544. (Tundo et al. 2007; Lauer et al. 2007). Thus there are still
2HubbleFellowandPrinceton-CarnegieFellow. unsettledissuesontheusageofthesetechniques.
3Department of Physics, Drexel University, 3141 Chestnut Street, Athirdmethod,reliantonAGNphysicsratherthangalaxy
Philadelphia,PA19104. properties, is reverberationmapping, which uses the tempo-
4Department of Astronomy and Astrophysics, 525 Davey Laboratory, raloffsetbetweencontinuumandemissionlinevariabilityto
PennsylvaniaStateUniversity,UniversityPark,PA16802.
determinethedistanceRfromthecentralenginetothebroad
2 SHENETAL.
emission line region (BLR; e.g., Blandford& McKee 1982; BH mass estimates. Indeed,using a sampleof ∼80z≤0.5
Peterson1993; Kaspiet al. 2000; Petersonet al. 2004). Us- Palomar-Green (Green, Schmidt, & Liebert 1986) quasars,
ing the observed line width V and assuming that the BLR Baskin&Laor(2005)showedthattheCIVFWHMispoorly
is virialized allows determination of the black hole mass: correlatedwiththeHβ FWHM(e.g.,seetheirfigure3),sug-
M ⋍G- 1RV2.Inahandfulofcases,measurementsofV and gestingthatthetwolineshavedifferentorigins. Ontheother
BH
RforseverallinesinthesameobjecthaveshownV ∝R- 1/2, hand,theMgIIFWHMiswell-correlatedwiththatofHβ,and
consistentwiththevirialhypothesis(e.g.,Peterson&Wandel these two low-ionizationline estimatorsusuallygive consis-
2000).Furthermore,forthefewcasesinwhichwehavemore tentvirialmasses(e.g.,McLure&Dunlop2004;Salvianderet
than one mass indicator, the reverberation mapping mass is al.2007).Oneofthepurposesofthispaperistoexplorehow
alsoconsistentwiththedynamicalmassorthatderivedfrom the CIV estimator might be biased relative to low-ionization
theMBH- σrelation(e.g.,Gebhardtetal.2000b;Ferrareseet lineestimatorssuchas MgIIusinga largesampleofquasars
al.2001;Nelsonetal.2004;Onkenetal.2004,2007;Greene fromtheSloanDigitalSkySurvey(SDSS;Yorketal.2000).
&Ho2006;Daviesetal.2006). Thereverberationmapping Despite the caveats in these virial estimators, virial BH
methodisverytimeconsuming,andwehavereliablemasses masses have been measured for various AGN/quasar sam-
withthismethodforonlyaboutthreedozenAGNs. ples covering a wide range of redshifts and luminosities.
However, reverberation mapping data has revealed a cor- For example, McLure & Dunlop(2004)measuredvirial BH
relation between BLR size R and luminosity (Kaspi et al. massesfor12698quasarswith0.1≤z≤2.1fromtheSDSS
2000,2005),whichallowsustoestimateBHmassesbasedon DR1 quasar catalog (Schneider et al. 2003) using Hβ and
single-epochspectra. Using thisR- L correlationand rever- MgII.Vestergaard(2004)usedHβ andCIVtomeasurevirial
berationmappingmasses,variousempiricalscalingrelations BH masses for a hybridsample including 87 z≤0.5 Bright
(whichwetermvirialBHmassestimatorsfromnowon)have Quasar Survey (BQS, Schmidt & Green 1983) quasars, 114
beenderivedusingHβ (Kaspietal.2000;Vestergaard2002; 1.5 . z . 3.5 quasars, and ∼ 150 z > 3.5 SDSS quasars.
McLure & Jarvis 2002; Vestergaard & Peterson 2006), Hα Greene&Ho(2007)presenttheBHmassfunctionfor∼8500
(Greene&Ho2005),MgII(McLure&Jarvis2002;McLure broad-line AGNs with z<0.35 from the SDSS. Fine et al.
& Dunlop 2004), and CIV (e.g., Vestergaard 2002; Vester- (2006)usedcompositespectratomeasuretheredshiftevolu-
gaard&Peterson2006).Thisistheonlypracticalmethodfor tionofthemeanBHmassforthe2QZquasarsample(Croom
measuringblackholemassesforlargenumbersofobjectsat et al. 2004) from z ∼ 0.5 to z ∼ 2.5, and Kollmeier et al.
high redshifts and luminosities (where variability timescales (2006)presentedvirialBHmassesforasampleof407AGN
are too long to measure easily; see the discussion in Kaspi with z∼0.3- 4 selected from the AGN and Galaxy Evolu-
et al. 2007), but the R- L relation has only been established tion Survey (AGES; Kochanek et al. 2004) and focused on
intheluminosityrange∼1042- 1046ergs- 1andforredshifts the distribution of Eddington ratios. Yet by far the largest
z<0.3. quasar sample is the recently published SDSS DR5 quasar
ThevirialBHmassestimatorsareusuallyexpressedas: catalog(Schneideretal.2007),containingmorethan77,000
quasars, about half of which are homogeneously selected
M λL FWHM
log BH,vir =a+blog λ +2log , (e.g.,Richardsetal.2006a;Shenetal.2007).Inthispaperwe
(cid:18) M⊙ (cid:19) (cid:18)1044ergs- 1(cid:19) (cid:18) kms- 1 (cid:19) presentvirialBHmassestimatesforthisopticalquasarsam-
(1) ple and explore possible biases in their estimation. Though
whereλLλ,thesurrogatefortheBLRsizeR,isthecontinuum thedynamicalrangeinluminosityandcolorofSDSSquasars
luminosityneartheemissionline(5100ÅforHβ,3000Åfor is limited, the largesize of the sample providesunprecedent
MgII and 1350Å for CIV), and FWHM is the full-width-at- statistics.
half-maximumof the line. Some authorspreferto use other The paperis organizedas follows: in § 2 we describeour
quantities in their virial estimates. For example, Wu et al. quasarsample.Wepresentourproceduresofestimatingvirial
(2004)suggestusingrecombinationlineluminositiesinstead BH masses in § 3, where we compareresults from different
of continuum luminosities because the latter might be con- linesindetail.ThedistributionsofblackholemassesandEd-
taminated by jet emission or host galaxy starlight (also see dingtonratiosaredescribedin§4,andwediscussourresults
Greene& Ho 2005). Peterson etal. (2004)and Collin etal. aswellassomegeneralissueswithvirialestimatorsin§5.We
(2006)arguethatthesecondmomentofthelineprofileσ is summarizeourresultsin§6. Throughoutthispaperweadopt
line
abetterquantitytocharacterizetheemission-linewidthsthan aflatΛCDMcosmology:ΩM =0.26,ΩΛ=0.74andh=0.71
the FWHM. As we will show below, differentmeasurement (Spergeletal.2007).
techniquesyieldsystematicdifferencesinderivedquantities,
and so it is important to use the original definitions of line 2. THESAMPLE
widthsandluminositiesforwhichevercalibrationisused. The SDSS uses a dedicated 2.5-m wide-field telescope
The UV CIV virial mass estimator can be used for z&2 (Gunn et al. 2006) with a drift-scan camera with 30 2048×
quasars, as the CIV line is the only relevant line available 2048CCDs(Gunnetal.1998)toimagetheskyinfivebroad
in optical spectra in this redshift range. However, it has bands (ugriz; Fukugita et al. 1996). The imaging data are
been suggested by a few authors that the CIV line might be taken on dark photometric nights of good seeing (Hogg et
a less secure mass estimatorthan MgIIorHβ. The CIV line al. 2001), are calibrated photometrically (Smith et al. 2002;
tendstobeasymmetricandblueshiftedwithrespecttolower- Ivezic´etal.2004;Tuckeretal.2006)andastrometrically(Pier
ionizationlinessuchasHβorMgII(e.g.,Gaskell1982;Tytler etal.2003),andobjectparametersaremeasured(Luptonetal.
&Fan1992),andthemostblueshiftedobjectstendtohavethe 2001;Stoughtonetal.2002). Quasarcandidates(Richardset
largestFWHMs(Richardsetal.2002b). Thesefeaturessug- al. 2002a) for follow-up spectroscopy are selected from the
gestthatCIVmightbemoreseverelyaffectedthan theother imaging data using their colors, and are arranged in spec-
lines by a non-virialized gas component, and the measured troscopic plates (Blanton et al. 2003) to be observed with a
line width could depend on the viewing angle, biasing the pairof doublespectrographs. The quasarsobservedthrough
QUASARBLACKHOLEMASSES 3
the Third Data Release (Abazajian et al. 2005) have been
cataloged by Schneider et al. (2005), while Schneider et al.
(2007)extendthiscatalogtotheDR5(Adelman-McCarthyet
al.2007a).
OurparentsampleisthepublishedSDSSDR5quasarcat-
alog(Schneideretal.2007),whichcontains77,429bonafide
quasars that have luminosities larger than M = - 22.0 (us-
i
ing a slightly different cosmology in that paper) and have
at least one broad emission line (FWHM>1000kms- 1) or
have interesting/complexabsorption features. About half of
the quasars in this catalog were selected from a uniform al-
gorithm(asdescribedinRichardsetal.2002a),whichisflux
limited to5 i= 19.1 at z.3 and i= 20.2 at z&3 (magni-
tudesarecorrectedforGalacticextinctionusingtheSchlegel,
Finkbeiner & Davis 1998 map). These objects are flagged
with the UNIFORMFLAG= 1 in the DR5 quasar catalog,
andtheycanbeusedtoconstructastatisticallyhomogeneous
sample(e.g.,Richardsetal.2006a;Shenetal.2007).
There are also flags indicating whether or not a quasar is
detectedintheFIRST(Beckeretal.1995)orROSAT(Voges
et al. 1999) surveys, which we will use to define our radio
quasarsubsample(see§4.6). Mostofouranalysiswillavoid
broadabsorptionlinequasars(BALs),whoselinewidthsare
problematic to measure; we will come back to the BALs in
§4.6. We have identified ∼4200 BALs (with 1.7≤z≤4.2
forCIV and 0.5≤z≤1.9for MgII;Shenet al. 2008)in the
DR5quasarcatalogusingtraditionalBALcriteria(Weymann
etal.1991). ThislistofBALsisbynomeanscomplete,and
an official DR5 BAL catalog is forthcoming (Hall et al., in
preparation;seeTrumpetal.2006fortheDR3BALcatalog).
All spectra are reducedwith the new versionof the SDSS
spectroscopicreductionpipeline,asdescribedintheDR6pa-
per(Adelman-McCarthyetal.2007b);thefluxscaleofthese
spectra is higher than that of previous releases by roughly
38%.Thusinthispaperallluminositiesmeasuredfromspec-
trawillbetypicallylargerby∼0.14dexthanpreviousvalues
forthesameobjects.
Acomprehensivecatalogofourmeasurementsofthespec-
tralpropertiesofeachquasarisgivenintheelectronicversion
ofthispaper;thecolumnsaredescribedinTable1. Thiscat-
alog can be regarded as an extension to the published DR5
quasarcatalog.
3. BLACKHOLEMASSESTIMATES
FIG. 1.—Examplesofspectralfits. Fromtoptobottom:Hβ,MgII,CIV.
To estimate BH masses using virial estimators one needs Ineachpanel,theupperandlowerblacksolidlinesshowstheoriginaland
twoparameters: thewidthofanemissionlineandthecorre- continuum+Fesubtractedspectra;thegraylinesshowthefluxdensityerrors;
theredlines showthefullfits; theorangelines showthefittedpower-law
sponding continuum luminosity. However, different authors
continuum;thelowerbluelinesintheupperandmiddlepanelsshowtheiron
have used differentdefinitionsof line width in their calibra- templatefits;themagentalinesshowthefitsfortheemissionlinesandgreen
tionsofthe virialmass estimators. In whatfollows, we take lines show the Gaussian components. Theblue horizontal segment inthe
care to define line width in a way consistent with each cali- bottompanelmarkstheFWHMforCIV.
bration we use. We also note that the assumed virial coeffi- 4.5becauseoftheambiguitiesoflinewidthmeasurementfor
cient f,whichaccountsforourignoranceoftheBLRgeom- BALs;wereturntotheissueofBALsin§4.6.
etry,isdifferentin differentcalibrations. Forinstance, f =1
in McLure& Dunlop(2004)and f ∼5.5 (e.g., Onken etal. 3.1. Linewidthsandcontinuumluminosities
2004)inVestergaard& Peterson(2006). Thusdifferentver-
3.1.1. HβandMgII
sionsofcalibrationforthesamelinedonotnecessarilyyield
the same results, especially when extrapolatedto high lumi- For the Hβ and MgII estimators, we have adopted the
nosityquasars.Wewilldiscussthisfurtherin§5.1. calibrations in McLure & Dunlop (2004), hence a=0.672,
We focus on nonBALs throughout this section and §4.1- b=0.61forHβ anda=0.505,b=0.62forMgIIinequation
(1).TheHβestimatorhasbeencalibrateddirectlywithrever-
5 There are a few i>19.1 quasars at z.3 which were selected by berationmapping, while the RBLR- λLλ relation for MgII at
thehigh-z(griz)branchofthetargetingalgorithm (Richards etal.2002b). 3000ÅisempiricallydeterminedinMcLure&Jarvis(2002)
Thefraction oftheseobjects istiny(.2%). Althoughtheseobjects have
basedonHβreverberationmappingmasses. Wenotethatthe
UNIFORMFLAG=1inTable1,theywillberejectedwhenconstructinga
flux-limitedsample(see§4.3). calibration for MgII in McLure & Dunlop (2004) is an up-
4 SHENETAL.
TABLE1
CATALOGFORMAT
Column Format Description
1 A18 SDSSDR5designationhhmmss.ss+ddmmss.s(J2000.0)
2 F11.6 Rightascensionindecimaldegrees(J2000.0)
3 F11.6 Declinationindecimaldegrees(J2000.0)
4 F7.4 Redshift
5 F8.3 Mi(z=2)(h=0.71,ΩM=0.26,ΩΛ=0.74,K-correctedtoz=2,followingRichardsetal.2006a)
6 F7.3 Bolometricluminosity(log(Lbol/ergs- 1))
7 I5 Spectroscopicplatenumber
8 I5 Spectroscopicfibernumber
9 I6 MJDofspectroscopicobservation
10 I12 Targetselectionflagwhenthespectrumwastaken(i.e.,usingTARGETphotometry)
11 I3 FIRSTselectionflag(0or1)
12 I3 ROSATselectionflag(0or1)
13 I3 Uniformselectionflag(0or1)
14 I3 BALflag(0or1)
15 I7 HβFWHM(kms- 1)
16 F8.3 MonochromaticluminosityλLλat5100Å(1044ergs- 1)
17 F7.3 VirialBHmassestimatedusingHβ(log(MBH,vir/M⊙))
18 I7 MgIIFWHM(kms- 1)
19 F8.3 MonochromaticluminosityλLλat3000Å(1044ergs- 1)
20 F7.3 VirialBHmassestimatedusingMgII(log(MBH,vir/M⊙))
21 I7 CIVFWHM(kms- 1)
22 F8.3 MonochromaticluminosityλLλat1350Å(1044ergs- 1)
23 F7.3 VirialBHmassestimatedusingCIV(log(MBH,vir/M⊙))
24 F7.3 VirialBHmass(usingHβforz<0.7;MgIIfor0.7<z<1.9andCIVforz>1.9)
25 I7 CIV-MgIIblueshift(kms- 1)
26 F8.3 MeanspectrumS/N(signal-to-noiseratio)
NOTE.—(1)ObjectsinthistableareinthesameorderasintheDR5quasarcatalog(Schneideretal.2007);(2)K- corrections
arethesameasinRichardsetal.(2006a);(3)BolometricluminositiesarecomputedusingbolometriccorrectionsinRichards
etal.(2006b)usingoneofthe5100Å,3000Å,or1350Åmonochromaticluminositiesdependingonredshift;(4)Entriesreading
- 9999forFWHM,luminosity, BHmassorCIV-MgIIblueshiftmeasurementindicatethatthisquantitywasnotmeasureable
fromtheSDSSspectrum,eitherbecauseitfelloutsideoftheSDSSspectralcoverageorbecauseoflowS/N.
datedversionofthatinMcLure&Jarvis(2002),whereinthe The FWHM of the broad Gaussian-componentof Hβ
former only objects with 1044ergs- 1 < λLλ < 1047ergs- 1 andMgIIis thentakenasthe line widthto beinserted
were included in their fitting. This luminosity range is suit- inequation(1).
ablefortheSDSSquasarsamplestudiedhere.
WefollowMcLure&Dunlop(2004,AppendixA)tomea- It is not entirely clear if there exists a strong narrow line
surethelineFWHMforHβ andMgIIwithslightchangesin component for MgII, thus the way to measure the MgII
details: FWHM in McLure & Dunlop (2004) and here is not justi-
fied. IfwemeasuretheMgIIFWHMfromthetwo-Gaussian
1) apower-lawcontinuumandanironemissionlinetem- fitted flux, it will typically be smaller by ∼0.15 dex. How-
platearesimultaneouslyfittedtothespectrumwiththe ever, in order to use their calibration we have to follow the
Hβ orMgIIemissionlineregionsexcluded.Thefitting sameproceduretomeasuretheMgIIFWHM.Aswewillsee
ranges are [4435,4700]Å and [5100,5535]Å for Hβ, laterin§3.3.1,thefittedbroadlineFWHMsforMgIIandHβ
[2200,2675]Å and [2925,3090]Å for MgII. The iron are quite similar, and as a result, both line estimators yield
templateforHβistakenfromBoroson&Green(1992) consistentvirialmasses.
andtheirontemplateforMgIIisamodifiedversionof For continuum luminosities λLλ at 5100Å for Hβ and at
the Vestergaard& Wilkes (2001)template(Salviander 3000ÅforMgII,wehaveusedthefittedcontinuumfluxden-
etal.2007)whichextendsundertheMgIIlineitself. In sityatthecorrespondingwavelengthandcorrectedforGalac-
the fitting, the normalization and velocity broadening ticextinctionusingtheSchlegel,Finkbeiner&Davis(1998)
oftheirontemplateareleftasfreeparameters. map.
The calibration of the Hβ mass estimator is not unique,
2) Thebest-fitcontinuumandironemissionarethensub- andotherauthorshavedeterminedslightlydifferentversions
tracted from the spectrum, and two Gaussians are fit- (e.g., Vestergaard 2002; Vestergaard & Peterson 2006). We
ted to each emission line, one for the broadline com- find other forms of the Hβ estimator yield comparable but
ponent and the other for the narrow line component. notidenticalresults. Inparticular,theVestergaard&Peterson
The line fitting ranges are [4700,5100]Å for Hβ and (2006) Hβ calibration gives a systematic ∼0.15 dex offset
[2700,2900]Å for MgII. The FWHM of the narrow comparedwiththeMcLure&Dunlop(2004)calibration.We
componentis constrainedto beless than1200kms- 1, willcomebacktothispointin§5.
and that of the broad component is constrained to be
largerthan1200kms- 1 (e.g.,Hao etal. 2005). In the 3.1.2. CIV
case of Hβ, two additionalGaussians whose FWHMs FortheCIVestimator,weusethecalibrationinVestergaard
are tied to that of the narrow Hβ component are fit- &Peterson(2006),whichhasa=0.66andb=0.53inequa-
tedsimultaneouslyfor[OIII]4959Åand[OIII]5007Å. tion(1).
QUASARBLACKHOLEMASSES 5
FIG.2.—Left:correspondencebetweenMi(z=2)andLbol.Linesshowmedianvaluesandstandarddeviations. TheabsolutemagnitudehasbeenK-corrected
tothei-bandmeasuredatz=2(i.e.,arest-framewavelengthof∼2500Å),followingRichardsetal.(2006a).Right:redshiftdistributionofLbolforoursample,
whereblackpointsareuniformlyselectedquasarswhilegraypointsarequasarsselectedwithspecialtargetingalgorithms.
FIG. 3.—DistributionsofFWHMsasfunctionsofredshiftandluminosity. TheFWHMsforthethreelinesareplottedtogether. Contoursarelocalpoint
densitycontours,drawntoguidetheeye.Intheleftpanel,wealsoshowthemeanand1- σoftheFWHMdistributionasafunctionofredshift. FWHMsshow
littleornodependenceoneitherredshiftorluminosity,andtheirdistributionsfollowalog-normalatallredshiftsandluminosities(seeFig.4).
Iron emission contamination is not a problem for CIV, so gion [1500,1600]Å.The FWHM is then measured from the
we have simply fitted a power law continuum to regions in fitted model flux, providing better measurements of FWHM
thevicinityofCIV,namely[1445,1465]Åand[1700,1705]Å fornoisyspectra. TheCIVline in highredshiftquasarsmay
.Thiscontinuumisthensubtractedfromthespectrumforthe bebiased byassociated absorptionlines. To reducetheir ef-
linefittingwhichfollows. fect,asecondlinefitisperformedwhichexcludespixelsthat
TheFWHM for CIV asdefinedin Vestergaard& Peterson are1.5σbelowthefirstfittedmodel,whereσistheestimated
(2006)isindeedthe“full-width-at-halfmaximum”ofthefull errorinfluxdensity. Themodelisreplacedbythesecondfit
lineprofile,whetherornotthelineprofileisGaussian. Since ifthereducedχ2issmallerthaninthefirstfit. Also,ifoneor
many CIV lines have profiles “peakier” than a Gaussian, the twoofthethreeGaussianshaveintegratedfluxlessthan5%
FWHMobtainedfromasingleGaussianfitoverestimatesits ofthetotalmodelflux,theyareremovedfromthemodelflux;
formaldefinition. Forthisreason,followingthestandardap- thisstepensuresthattherearenounphysicallynarrowpeaks
proachintheliterature(e.g.,Baskin&Laor2005;Fineetal. tobiastheFWHMmeasurement. AsinthecasesofHβ and
2006),we use a set ofthree Gaussiansto fit the CIV line re- MgII,werequireFWHM>1200kms- 1.
6 SHENETAL.
FIG.4.—DistributionsoflogFWHM(inkms- 1)indifferentredshift(horizontallyarranged)andluminosity(verticallyarranged)binsforHβ,MgIIandCIV.
SolidhistogramsshowtheresultswithoutaS/NcutanddashedhistogramsshowtheresultswithS/N>10. Weonlyplotahistogramiftherearemorethan
200objects(noS/Ncut)or50objects(withS/Ncut)ineachbin. ThenumberofquasarsdropsrapidlywhenweimposetheS/Ncut,butthesolidanddashed
histogramsshowalmostidenticaldistributions. Themeansanddispersionsofafittedlog-normaltothedistributionsareshownintheupper-leftcornersofeach
panel.
TheCIVFWHMs measuredinthiswayaresystematically Fig.3showsthedistributionoflogFWHMinbinsofred-
narrowerby∼0.15dexthanwhatresultsfromasingleGaus- shift and bolometric luminosity. The FWHMs of all three
sian fit. Clearly, we would get systematic offsets in the CIV emissionlinesareonlyweaklydependentoneitherredshiftor
basedBHmassesifweusedthesingleGaussianFWHM. luminosity within our sample, and they follow a log-normal
The 1350Å continuum luminosity λL is measured us- distribution with typical dispersion ∼ 0.1- 0.2 dex. This
λ
ing the mean flux density within the wavelength range width is real and is not dominated by noise; we find essen-
[1340,1360]Å,and corrected for Galactic extinction. Given tially identical distributions when restricting our sample to
the wavelength coverage of the SDSS spectra, this implies objectswithS/Nperpixelgreaterthan10,orgreaterthan20.
that we cannot estimate CIV black hole masses for quasars To further demonstrate that the FWHM distributions do not
withredshiftbelowz∼1.8. evolve, Fig. 4 shows the FWHM distributions for the three
linesindifferentredshiftandluminositybins. Ourmeasured
3.2. BolometricLuminosities distributionsof FWHMs are consistent with other investiga-
tions(e.g.,Baskin&Laor2005;Salvianderetal.2007),with
For eachobjectin oursample thathas a measurablevirial
similarwidthsanda∼0.1dexdifferenceinthemeanvalues6
mass, we use its continuum luminosity λL (at 1350Å for
λ (whichmightbecausedbysmalldifferencesinthetechniques
CIV,3000ÅforMgIIand5100ÅforHβ)toestimatethebolo-
todetermineFWHMs). TheobserveddistributionofFWHMs
metricluminosity.Thebolometriccorrectionfactors(BC)are
hasprofoundeffectsonourdeterminationofthedistribution
computed using the composite SED for a sample of SDSS-
ofvirialBHmasses,aswillbefurtherexploredinsection4.3.
DR3quasarsconstructedbyRichardsetal. (2006b),andare
slightlydifferentfromthecommonlyadoptedvaluesinElvis 3.3. ComparisonofEmissionLines
et al. (1994). In particular, BC = 3.81, BC = 5.15,
1350 3000 There are subsamples of quasars for which we have two
BC =9.26. When two continuumluminosities are avail-
5100 lines available; given the SDSS spectroscopic coverage, we
able,theygivequitesimilarbolometricluminosities.Thetyp-
observebothHβ andMgIIfor0.4.z.0.9, andbothMgII
icalerrorinbolometriccorrectionsassumingauniversalSED
andCIVfor1.7.z.2.2.Thereforewecancheckifdifferent
is∼0.1dexforoptically-selectedSDSSquasars(Richardset
BHmassestimatorsgiveconsistentresults.
al.2006b),butitmightbeslightlylargerforBC . Theleft
1350
panelofFig.2showstherelationbetweenLbol andthez=2 3.3.1. HβvsMgII
i- bandabsolutemagnitudecalculatedusingtheK-correction
in Richards et al. (2006a), while the right panel shows L
bol 6 Thedifference of∼0.1dexinFWHMwillresultin∼0.2dexmean
versusredshiftforoursample (thedarkerpointsaretheuni-
offsetinvirialmasses. Wedonotconsiderthissubstantial, sincethezero-
formlyselectedsubsample). pointofthevirialcalibrationshasuncertaintiesofthesamelevel.
QUASARBLACKHOLEMASSES 7
FIG.5.—ComparisonsbetweentheHβandtheMgIIestimators.left:histogramoftheFWHMratios.right:histogramofthevirialBHmassratios.
FIG.6.—Aone-to-onecomparisonbetweentwovirialmassestimators.left:HβversusMgII;right:MgIIversusCIV.Thecorrelationismuchbetterbetween
HβandMgIIthantheonebetweenMgIIandCIV.Contoursarelocalpointdensitycontours,drawntoguidetheeye.
For a subsample of ∼ 8,000 quasars with redshift 0.4. We plot the Hβ against the MgII virial masses in the left-
z.0.9wehavebothHβ andMgIIlinewidthmeasurements. hand panel of Figure 6. The two mass estimators correlate
The distribution of log(Hβ FWHM/MgII FWHM) is shown stronglywithoneotherovertwoordersofmagnitudeinmass.
in the left panel of Fig. 5, which follows a Gaussian with
mean 0.0062 and dispersion 0.11 dex. The distribution of 3.3.2. MgIIvsCIV
log(MHβ/MMgII)isshownintherightpanelofFig. 5,which
BH BH Linewidths of both MgII and CIV are available for a sub-
followsaGaussianwithmean0.034anddispersion0.22dex.
sample of ∼15,000 quasars with redshift 1.7.z.2.2, of
The small discrepancy in the mean offsets between our re-
which∼5,000quasarshavebothMgIIandCIVvirialmasses
sultsandthoseinMcLure&Dunlop(2004,wherethemean
available.However,beforewecomparetheresults,wereview
FWHMratioandBHmassratiooffsetsare- 0.004and0.013
someoftheuniquecharacteristicsoftheCIVline.
dex, respectively)is perhapsdue to differentsamples, but is
First, the definitions of FWHM are quite different for
totallynegligibleconsideringthe∼0.3- 0.4dexuncertainties
MgII and CIV. While the broad componentof MgII is well-
ofeitheroftheBHmassestimators. Therefore,theMgIIand
described as a Gaussian in our fits, the vast majority of CIV
Hβ estimatorsgiveconsistentBHmasses.
linesarebynomeansGaussian. Thedifferenceinlineshapes
AplotoftheHβ FWHM againsttheMgIIFWHM isvery
suggests that the broad MgII and CIV might have different
similartoFig.3inSalvianderetal.(2007);thetwoFWHMs
physicalorigins.Second,itiswellknownthathighionization
arecorrelated,withaslopedifferingslightlyfromunity.
lines such as CIV often show systematic blueshifts with re-
8 SHENETAL.
FIG. 7.—VariouspropertiesagainsttheCIV-MgIIblueshift. Contoursarelocalpointdensitycontours,drawntoguidetheeye. Left: CIVFWHMagainst
blueshift.ThemeanvalueofFWHMsriseswithincreasingblueshift.Forthoseobjectswiththelargestblueshift,theirFWHMsrarelydropbelow4000kms- 1.
Middle: MgIIFWHMagainstblueshift. IncontrarytotheCIVcase,theMgIIFWHMshowsamilddecreasewithincreasingblueshift,butitisalsoconsistent
withnocorrelationatallconsideringthehugescatter. Right: Bolometricluminosityagainstblueshift. Themeanluminosityincreaseswithblueshift,butagain
thedispersionislarge.
specttolowionizationlinessuchasMgII(e.g.,Gaskell1982; showsa mild anti-correlationbetween the MgII FWHM and
Tytler & Fan 1992; Richards et al. 2002b). Richards et al. the CIV-MgII blueshift, and the right panel shows a mild
(2002b) used ∼800 SDSS quasars to show that CIV is sys- correlation between the bolometric luminosity Lbol and the
tematicallyblueshiftedwithrespecttoMgIIby∼900kms- 1 blueshift, which has been found before (e.g., Richards et
on average. They interpreted this apparent blueshift as due al. 2002b). The cause of the anti-correlation between MgII
to the absorptionof the CIV red wing, as mightbe expected FWHM and CIV-MgII blueshift is not clear at this point. It
if the broad-line region gas is actually a radiatively driven, mightreflectinclinationeffectsas well. The differentcorre-
equatorial outflow. Such disk-wind models are compelling lationsoftheCIVandMgIIlinewidthswithblueshiftsuggest
foravarietyofreasons,particularlytheirsuccessinexplain- thatthetwolineshavedifferentorigins,aswearguedabove.
ingthepropertiesofbroadabsorptionlinequasars(e.g.,Mur- Since the CIV estimator is calibrated using Hβ reverbera-
ray et al. 1995; Proga et al. 2000; Elvis 2000). Richards et tionmappingmasses(e.g., Vestergaard2002;Vestergaard&
al. also see a trend of increasing average CIV FWHM with Peterson 2006), there should be no mean offsetin either the
increasingCIV-MgIIblueshift,whichagainsuggeststhatthe FWHMsortheestimatedBHmasses,aswefind(seetheblack
broadCIVlineisdominatedbyadiskwindcomponent;inthis solidhistogramsinthebottompanelsinFigs. 8and9).While
casetheCIV-MgIIblueshiftreflectstheorientationoftheline- the logof the ratioof MgIIto CIV FWHMs followsa Gaus-
of-sight (LOS). There is additional evidence supporting this sianwithmean0.027anddispersion0.18dex,thecorrelation
disk wind/orientation scheme, including the anti-correlation between the two line widths is very weak (top panel of Fig.
between the CIV-MgII blueshift and radio loudness, and the 8);indeedthewidthof0.18dexislargerthanthedispersion
correlationbetweentheCIV-MgIIblueshiftandBALfraction in the MgIIFWHM distribution itself. Correspondingly,the
(seealsoGallagheretal.2005).Thesepropertiesarealsoseen logoftheratioofMgIItoCIVBHmassesfollowsaGaussian
in a sample of low redshiftbroadline AGNs (Sulentic et al. withmean- 0.06dexanddispersion0.34dex.Baskin&Laor
2007). Ofcourse,alternativeinterpretationsofthesefeatures (2005)similarlyfoundweakcorrelationsbetweentheCIVand
arepossible,whichdependmoreonthestructureofthewind Hβ FWHMsatlowredshift,whileNetzeretal.(2007)report
thanontheLOSorientation(seeRichards2006). a similar effect at z≈2 (butsee Kelly & Bechtold 2007 for
IfthebroadCIVisindeedcontaminatedorevendominated an alternate view). Moreover, this scatter is systematic with
by a disk wind component, then the measured CIV FWHM blueshift. ThebluepointsandhistograminFigure8referto
doesnotrepresenttheactualvirialvelocityoftheBLR.Ifthe objectswithblueshiftlargerthan1000kms- 1, whilethered
LOS is along the disk wind, then the FWHM will overesti- pointshavesmallerblueshift.
mate the virial velocity, while if the LOS is along the pole TheCIVandMgIIvirialmassestimatorsareplottedagainst
direction, the FWHM will probably underestimate the virial one another in the right-hand panel of Figure 6. The corre-
velocity. Although the CIV estimator is calibrated in such lation is muchweaker than we foundfor the MgII-Hβ com-
a way that it gives, on average, BH masses consistent with parison. Theratioofthesetwomassesiscorrelatedwiththe
thoseestimatedfromHβ(e.g.,Vestergaard2002;Vestergaard CIV-MgIIblueshift,asis showninFig. 9 (toppanel). Given
&Peterson2006),therewillbebiasesintheCIV-basedvirial thesimilaritybetweenthepropertiesoftheMgIIandHβlines,
massesiftheabovepictureiscorrect. andthe evidencethattheCIVline maybeaffectedbyadisk
We explore these issues by plotting various quantities wind, we argue that Hβ and MgII are better BH mass esti-
againsttheCIV-MgIIblueshiftinFig.7. Theleftpanelshows mators than is CIV. In particular, we emphasize the danger
the correlation between the CIV FWHM and the CIV-MgII of using CIV as a BH mass estimator for individual objects
blueshift. We see the rise of the average CIV FWHM as the andsmallsamples.Forexample,Kurketal.(2007)andJiang
blueshiftincreases,asreportedbyRichardsetal.(2002b). In et al. (2007a) have obtained observations of both MgII and
particular,veryfewobjectswithlargeblueshiftshaveFWHM CIVinfourz∼6quasars. InthethreequasarsinwhichCIV-
less than 4000 kms- 1, while the distribution of FWHM is MgIIblueshiftsareobserved,theCIVvirialmassis∼2times
broader for objects with small blueshifts. The middle panel largerthantheMgIImass, whiletheyareingoodagreement
QUASARBLACKHOLEMASSES 9
FIG.8.—Top:CIVFWHMsagainstMgIIFWHMsforasampleofquasars FIG.9.—Top:DifferenceintheMgIIandCIVbasedvirialBHmassesas
withbothFWHMsavailable. BlueandreddotsshowthosewithCIV-MgII functionoftheCIV-MgIIblueshift. Forthosewithsmall(large)blueshifts,
blueshift greaterandlessthan1000kms- 1; thesolidlineshowstheunity the CIV estimator systematically underestimates (overestimates) the BH
relation. Bottom: histogramsoftheratiosoftheMgIIFWHMtotheCIV massascomparedtotheMgIIbasedvirialmasses. Contoursarelocalpoint
FWHM.Blackhistogramisforthewholesample;blueandredonesarefor densitycontours,drawntoguidetheeye. Bottom:Distributionsoftheratios
objectswithCIV-MgIIblueshiftgreaterandlessthan1000kms- 1. ofthetwovirialmassestimates. Theblackhistogramisfortheentiresam-
ple;blueandredonesareforobjectswithCIV-MgIIblueshiftgreaterand
for the one object with a CIV-MgII redshift, consistent with lessthan1000kms- 1.
the trends we see in Figure 9. Finally we point out that the It is beyond the scope of this paper to carry out a thor-
ratioofMgIIandCIVvirialmassesdoesnotdependonlumi- oughinvestigationoftheBLRanddiskgeometryanddynam-
nosity in the range where both line estimators are available, icsneededtounderstandthephysicsbehindthetrendswe’ve
excludingluminosity-basedbiases. found.Therefore,inthefollowingsections,wesimplyusethe
Ofcoursewecannotmeasuretheblueshift,andtherebycor- CIV estimator to measure BH masses at z>1.9. Our confi-
recttheCIVmassestimateforthebiasofFigure9,ofobjects denceintheCIVestimatorisbuiltonthelargesamplestudied
forwhichourspectralcoverageonlyincludesCIV.However, here and the belief that there is no systematic offsets in the
Richardsetal.(2002b)showthattheCIV-MgIIblueshiftcor- meanusingtheCIVestimator.
relatesbroadlywiththe CIVlineasymmetry,suggestingthat
we might use this asymmetry as a surrogate to estimate the 4. DISTRIBUTIONOFBLACKHOLEMASSES
mass estimate bias. However, the correlation between the We present virial BH masses for the SDSS DR5 quasars
CIV-MgIIblueshiftandtheCIVlineasymmetryshowsalarge withz.4.5.Excludingthoseobjectsforwhichwecannotob-
amount of scatter, making it difficult to correct for this bias tainreliableFWHM orcontinuumluminositymeasurements
usingtheCIVlineasymmetry. due to poor spectra or redshift limitations, we have 56,872
10 SHENETAL.
objects (nonBALs) with measurable virial BH masses. The BHmassdistributionsineachluminosity/redshiftbinshould
masses are based on the Hβ estimator for z<0.7, MgII for be comparable, as is seen. The width of the virial BH mass
0.7<z<1.9,andCIVfor1.9<z<4.5,toavoidusingmea- distributionswithineachluminosity/redshiftbinisthus∼0.3
surementswhicharetooclosetotheredandblueendsofthe dexorless.
spectra. Weshouldnotethatthequasarsinthiscatalogwere Atfacevalue,theverynarrowwidthsofthevirialBHmass
notall uniformlytargeted; in particular, special targetingal- distributionineachluminositybinseemsomewhatsurprising,
gorithmswereusedtoselectquasarcandidatestofaintermag- since many studies have quoted significantly larger scatter
nitudesthan the main quasar survey(Richardset al. 2002a). between single-epoch virial masses and both reverberation-
The completeness of these fainter quasars is poor. We start mapped masses on the one hand and stellar velocity disper-
by neglectingthis effect; we take all the quasarsin the DR5 sions on the other. For instance, McLure & Jarvis (2002)
quasar catalog with measurable masses and present the red- find an rms scatter of 0.4 dex of virial MgII masses around
shiftevolutionof virialBH masses(§4.1) andEddingtonra- reverberation-mappingmasses,whileVestergaard&Peterson
tiodistributions(§4.2). Werestrictourselvestothecomplete (2006) find scatters of 0.3 and 0.5 dex of the CIV and Hβ
subsample in §4.3 and §4.4, where we model the observed virialmassesabouttheRMmassesrespectively.Anindepen-
blackholemassdistribution.WedescribetheactiveBHmass dentestimatemaybederivedbycomparingvirialmasseswith
functionin§4.5, anddiscussradio-loudandBAL quasarsin stellarvelocitydispersions;Greene&Ho(2006)findanrms
§4.6. scatterof0.4dexofvirialHαmassesaroundtheexpectation
fromthe M- σ relation. In allofthese cases, itis important
4.1. RedshiftevolutionofvirialBHmasses to remember that the objects in question are typically much
less luminous (and at much lower redshift) than the quasars
Fig.10showsthedependenceofvirialBHmassesonred-
westudyhere,butthesearethemeasurementsatourdisposal
shift, where gray dots represent individual objects and the
atthepresenttime.
filled circles show the mean value at each redshift. Be-
However,theobservedwidthofthevirialmassdistribution
cause of the incompleteness of the sample at the faint end,
within a given luminosity/redshift bin is not the uncertainty
the “mean” BH masses here are weighted towards high lu-
inthevirialmassestimatorsthemselves. Althoughwedonot
minosityobjectsat allredshifts. Nevertheless, the meanBH
haveadeterministicmodelfortheoriginoftheuncertaintyin
mass increases with redshift below z∼2, and then flattens
thevirialrelations,itmustresultbothfromtheimperfectrela-
out above z ∼ 2; the most massive BHs seem to turn off
tionbetweenluminosityandBLRradiusontheonehand,and
aroundthis epoch. There is clear evidenceof a limiting BH
fromthe imperfectrelation between FWHM and BLR virial
mass ∼3×109- 1010 M⊙, consistent with the conclusions velocityontheother. Thustheuncertaintyinthevirialmass
ofMcLure&Dunlop(2004)forz≤2.1SDSSDR1quasars.
estimators comes from the uncorrelatedrms scatters in both
Thetrendofthisfigureisquitesimilartofig. 4ofVestergaard
luminosityandFWHM.Whenweimposeluminositycutsto
(2004),exceptthatwehavefilledthegapsintheirfigurewith
oursample, i.e., eitherbyworkingwithinluminositybinsor
ourmuchlargersample.
because of the flux limits of our sample, we will artificially
narrowtheobserveddistributionofvirialmasses.
4.2. DistributionofEddingtonratios
Atthesametime,iftheunderlyingBHmassdistributionis
The relation between virial BH mass and bolometric lu- suchthattherearemoreBHstowardslowermasses/lowerlu-
minosity is shown in Fig. 11. The figure shows objects us- minosities,thenmoreobjectswillscatterfromlow-massbins
ingthethreevirialestimatorsindifferentcolors(redforHβ, tohighthantheotherwayaround,givingrisetoaMalmquist-
greenforMgIIandblueforCIV),whilethediagonallinesin- typebias(e.g.,Eddington1913;Malmquist1922)intheob-
dicate various Eddington ratios L/L , where L =1.3× servedvirialblackholemassandEddingtonratio.Inthenext
Edd Edd
1038(MBH/M⊙)ergs- 1 istheEddingtonluminosity. Thevast section,weuseMonteCarlosimulationstoinvestigatetheim-
majority of quasars in our sample are bounded by L and pactof luminositycuts and Malmquistbias on ourobserved
Edd
0.01L ,withageometricmeanvalueof∼0.1L . Weare distributionsofluminosity,linewidth,virialBHmasses,and
Edd Edd
not surprised to see that objects using these virial relations Eddingtonratios. The results are shown in § 4.4, where we
occupy a stripe in the mass-luminosity plane, with a slope alsopresentananalyticderivationoftheMalmquistbias.
∼b∼0.5andscatter∼0.2- 0.3dexaroundthemean. This
simplyreflectsthefactthattheFWHMdistributionisalmost
independent on luminosity, and has a very narrow width of 4.3. MonteCarlosimulations
∼0.1- 0.15 dex at fixed luminosity (see Fig. 4). This also
We ask whether we can reproduce the observed distribu-
explainstheremarkablysimilarappearanceoftheM - L
BH bol tionsgivenasetofreasonableassumptionsforthetrueunder-
diagram in several other investigations even though the un-
lyingdistributionsofblackholemassesandEddingtonratios.
derlyingsamplesarequitedifferent(e.g.,Woo&Urry2002;
In particular, our modelshould be able to reproducethe ob-
Kollmeieretal. 2006),simplybecausesimilarvirialestima-
served distributions of FWHMs in each redshift-luminosity
torsareusedtoestimateBHmasses,andtheFWHMdistribu-
bin, and a reasonable amount of uncertainty in virial rela-
tionsofdifferentsamplesaresimilar.
tions. Also, the model should produce a luminosity distri-
Fig. 12 shows the distributions of L /L in five lumi-
bol Edd bution consistent with the observed quasar luminosity func-
nosity and four redshift bins. The distribution of Eddington
tion. Since the BH mass and Eddington ratio distributions
ratiosislog-normalinalmostallofthebins. Noappreciable
may evolve with redshift, we consider two redshift ranges:
redshiftevolutionisseenforobjectswiththesame luminos-
0.7<z<1.0, where the MgIIestimator is used throughout,
ity. Thewidthsofthedistributionsareassmallas∼0.3dex,
and1.9<z<2.1,wheretheCIVestimatorisused. Weonly
similartothoseinKollmeieretal. (2006). Sincevirialmass
considerthesubsetofuniformly-selectedquasarsinthissec-
is approximatelyproportionalto the squarerootof luminos-
ity,thewidthsoftheobservedEddingtonratioandthevirial tion, andwe restrictourselvesto objectswith log erLgbso-l1 >
(cid:16) (cid:17)
