The Demographics of Broad-Line Quasars in the Mass-Luminosity Plane. I. Testing FWHM-Based Virial Black Hole Masses

1.7 MB·English

by  Yue Shen

DRAFTVERSIONJANUARY30,2012
PreprinttypesetusingLATEXstyleemulateapjv.05/04/06
THEDEMOGRAPHICSOFBROAD-LINEQUASARSINTHEMASS-LUMINOSITYPLANE.I.TESTING
FWHM-BASEDVIRIALBLACKHOLEMASSES
YUESHENANDBRANDONC.KELLY1
HARVARD-SMITHSONIANCENTERFORASTROPHYSICS,60GARDENSTREET,CAMBRIDGE,MA02138,USA.
DraftversionJanuary30,2012
ABSTRACT
2 We jointly constrain the luminosity function (LF) and black hole mass function (BHMF) of broad-line
1
quasarswithforwardBayesianmodelinginthequasarmass-luminosityplane,basedonahomogeneoussam-
0 ple of ∼58,000 SDSS DR7 quasars at z∼0.3- 5. We take into account the selection effect of the sample
2
flux limit; more importantly, we deal with the statistical scatter between true BH masses and FWHM-based

n single-epochvirialmassestimates, aswellas potentialluminosity-dependentbiasesof these massestimates.
a TheLFistightlyconstrainedintheregimesampledbySDSS,andmakesreasonablepredictionswhenextrap-
J
olatedto∼3magnitudesfainter. DownsizingisseeninthemodelLF.Ontheotherhand,wefinditdifficultto

7 constraintheBHMFtowithinafactorofafewatz&0.7(withMgIIandCIV-basedvirialBHmasses). This
2 ismainlydrivenbytheunknownluminosity-dependentbiasofthesemassestimatorsanditsdegeneracywith
othermodelparameters,andsecondlydrivenbythefactthatSDSSquasarsonlysamplethetipoftheactiveBH
]
populationathighredshift.Nevertheless,themostlikelymodelsfavorapositiveluminosity-dependentbiasfor
O
MgIIandpossiblyforCIV,suchthatatfixedtrueBHmass,objectswithhigher-than-averageluminositieshave
C
over-estimatedFWHM-basedvirialmasses. Thereistentativeevidencethatdownsizingalsomanifestsitselfin
. theactiveBHMF,andtheBHmassdensityinbroad-linequasarscontributesaninsignificantamounttothetotal
h
p BH massdensityatall times. Within ourmodeluncertainties,we do notfind a strongBH massdependence
- ofthemeanEddingtonratio;butthereisevidencethatthemeanEddingtonratio(atfixedBHmass)increases
o withredshift.
r Subjectheadings:blackholephysics—galaxies:active—quasars:general—surveys
t
s
a
[
1. INTRODUCTION 2002; Yu&Lu 2004, 2008; Shankaretal. 2004, 2009;
2 One major effort in modern galaxy formation studies is Marconietal. 2004; Merloni 2004; Hopkinsetal. 2007;
v to understand the cosmic evolution of supermassive black Merloni&Heinz2008).TheagreementbetweentherelicBH
2 holes (SMBHs), given their ubiquitous existence in almost mass density and the accreted mass density provides com-
7 pelling evidence that these two populations are ultimately
everylocalbulge-dominantgalaxy,andpossiblerolesduring
3 connected. Thereforeit is of imminentimportance to quan-
theirco-evolutionwiththehostgalaxy(e.g.,Magorrianetal.
4 tifytheabundanceofactiveSMBHsasafunctionofredshift.
1998; Gebhardtetal. 2000; Ferrarese&Merritt 2000;
. ThedemographicsoftheactiveSMBHpopulationhasbeen
7 Gültekinetal. 2009; Hopkinsetal. 2008; Somervilleetal.
0 2008). Over the past decade, the rapidly growing body the central topic for quasar studies since the first discov-
1 of observational data and numerical simulations have led ery of quasars (Schmidt 1963, 1968). Traditionally this is
1 to a coherent picture of the cosmic formation and evolu- doneintermsoftheluminosityfunction(LF),i.e.,theabun-
v: tion of SMBHs within the hierarchical ΛCDM paradigm danceof objectsatdifferentluminosities. Measuringthe LF
andits evolutionhasbeen the mostimportantgoalfor mod-
i (e.g., Haiman&Loeb 1998; Kauffmann&Haehnelt 2000;
X ernquasarsurveys(e.g.,Schmidt&Green1983;Greenetal.
Wyithe&Loeb 2003; Volonterietal. 2003; Hopkinsetal.
1986).InthelastdecadetheLFhasbeenmeasuredfordiffer-
r 2006,2008;Shankaretal.2009,2010;Shen2009).Although
a entpopulationsofactiveSMBHsandindifferentbands(e.g.,
many fundamental issues regarding SMBH growth still
Fanetal. 2001, 2004; Boyleetal. 2000; Wolfetal. 2003;
remain unclear (such as BH seeds, fueling and feedback
Croometal. 2004, 2009; Haoetal. 2005; Richardsetal.
mechanisms),thesecosmologicalSMBHmodelsarestarting
2005, 2006; Jiangetal. 2006, 2008, 2009; Fontanotetal.
to reproduce a variety of observed SMBH statistics in an
2007;Bongiornoetal.2007;Willottetal.2010b;Uedaetal.
unprecedentedmanner.
2003; Hasingeretal. 2005; Silvermanetal. 2005, 2008;
It is now widely appreciated that SMBHs grow by gas
Bargeretal. 2005), and it constitutesa crucial observational
accretion in the past, during which they are witnessed as
componentinallcosmologicalSMBHmodels.
quasars and active galactic nuclei (AGNs) (e.g., Salpeter
AmoreimportantphysicalquantityofSMBHsisBHmass.
1964;Zel’dovich&Novikov1964;Lynden-Bell1969).Inthe
BH mass is directly related to growth, and when the BH is
localUniverse,themassfunctionofdormantSMBHsisesti-
˙
mated by convolving the galaxy distribution functions with active, it determines the accretion efficiency (M/MBH) via
various scaling relations between galaxy properties and BH theEddingtonratioandanassumedradiativeefficiency(e.g.,
mass. This relic SMBH population has been used to con- such as the average value constrained by the Soltan argu-
strain the accretion history of their active counterparts, us- ment).ThusknowingthemassfunctionofSMBHsasafunc-
ingtheSołtanargumentanditsextensions(e.g.,Soltan1982; tion of redshift adds significantly to our understandings of
Small&Blandford1992;Saluccietal.1999;Yu&Tremaine theircosmicevolution.
It remains challenging to directly measure the dormant
1HubbleFellow. BHMF at high redshift. This is not only because the
2 SHEN&KELLY
galaxy distribution functions are less well-constrained at sions. Thusitisimportanttoconsidertheseeffectswhenthe
high redshift, but also because the evolution of the scal- statisticsisbecominggoodenough.
ing relations (both the mean relation and the scatter) be- Kellyetal. (2009)developedaBayesian frameworktoes-
tween galaxy propertiesand BH mass is poorly understood. timatetheBHMF/LFforbroad-linequasars,whichaccounts
On the other hand, it has become possible to measure the for the uncertainty in virial BH mass estimates, as well as
active BHMF of broad-line quasars2, using the so-called the selection incompleteness in BH mass (since the sample
virial BH mass estimators based on their broad emission is selected in luminosity). This method was subsequently
line and continuum properties measured from single-epoch applied to the SDSS DR3 quasar sample (Kellyetal. 2010),
spectra (e.g., Wandeletal. 1999; McLure&Dunlop 2004; basedonvirialmassestimatesfromVestergaardetal.(2008).
Vestergaard&Peterson 2006), a technique rooted on rever- This Bayesian framework is a more rigorous and quantita-
beration mapping (RM) studies of local broad-line AGNs tive treatment than the simple forward modeling performed
(e.g., Blandford&McKee 1982; Peterson 1993; Kaspietal. inShenetal.(2008),andallowsamorereliablemeasurement
2000; Petersonetal. 2004; Bentzetal. 2006, 2009a). These ofthetrueactiveBHMFanditsuncertaintyforquasars.
single-epoch virial BH mass estimators are calibrated em- EquippedwithanimprovedversionofthisBayesianframe-
pirically using the RM AGN sample to yield on average work, in this paper we measure the active BHMF and LF
consistent BH mass estimates compared with RM masses, based on a homogeneoussample of ∼58,000 quasars from
which are further tied to the BH masses predicted using SDSS DR7 with FWHM-based virial mass estimates from
theM - σ relation(e.g.,Tremaineetal. 2002;Onkenetal. Shenetal.(2011). Themuchimprovedstatisticsnowallows
BH
2004). Thenominalscatterbetweenthesesingle-epochvirial a detailed examination of the joint distribution in the mass-
estimates and the RM masses is on the order of ∼ 0.4 luminosity plane, and provides better constraints on BH ac-
dex (e.g., McLure&Jarvis 2002; McLure&Dunlop 2004; cretionproperties.
Vestergaard&Peterson2006). Akeydifferenceinourapproachcomparedwithmostear-
A couple of recent studies have applied this technique to lierworkistheattempttoaccountfortheuncertainty(error)
measure the active BHMF with statistical quasar and AGN in these virialmass estimates. We distinguishthree typesof
samples (e.g. Greene&Ho 2007; Vestergaardetal. 2008; errorsinsingle-epochvirialBHmassestimates:
Vestergaard&Osmer 2009; Schulze&Wisotzki 2010). A
robust determination of the active BHMF constitutes an im- • measurement error, which is propagated from the un-
portant building block of cosmological SMBH models, in certainties of FWHM and continuumluminosity mea-
addition to the luminosity function. These virial mass es- surementsfromthespectra;themeasurementerrorsare
timators also enable statistical studies on the Eddington ra- typically ≪ 0.3 dex for our sample (see Fig. 1) and
tiosofbroad-linequasarsandAGNs(e.g.,Vestergaard2004; hencearenegligible;however,measurementerrormay
McLure&Dunlop2004;Kollmeieretal.2006;Sulenticetal. become importantfor other samples with low spectral
2006; Babic´ etal. 2007; Jiangetal. 2007; Kurketal. 2007; quality.
Netzeretal. 2007; Shenetal. 2008; Gavignaudetal. 2008;
Labita 2009; Trumpetal. 2009, 2011; Willottetal. 2010a; • statisticalerror,whichisthescatterofvirialBHmasses
Trakhtenbrotetal. 2011), over a wide range of luminosities around RM masses when these virial estimators were
andredshifts,andthereforeprovideconstraintsontheaccre- calibrated against local RM AGN sample; the statis-
tionefficiencyoftheseactiveSMBHs. tical error is & 0.3 dex (e.g., McLure&Jarvis 2002;
With the development of these virial mass estimators, we McLure&Dunlop 2004; Vestergaard&Peterson
nowhavebothBHmassestimatesandluminositiesforbroad- 2006), which will be taken into account in our
linequasarsamples. GiventheintimaterelationbetweenBH Bayesianapproach.
mass and luminosity, it is important and necessary to study
• systematic biases, which may result from the virial
their joint distribution and evolution in the mass-luminosity
assumption, the usage of RM masses as true masses
plane(e.g.,Steinhardt&Elvis2010a;Steinhardtetal.2011).
during calibration, the usage of a particular definition
Thisrepresentsasignificantstepforwardtostudythedemog-
of line width as the surrogate for the virial velocity,
raphyofquasarsthanusingLFalone,andoffersnewinsights
the extrapolation of the virial calibrations to high lu-
on the propertiesand evolutionof the active SMBH popula-
minosity/redhift, as well as other possible systemat-
tion.
ics (e.g., Krolik 2001; Collinetal. 2006; Shenetal.
However, the importance of distinguishing between virial
2008; Marconietal. 2008; Fineetal. 2008, 2010;
mass estimates and true BH masses can hardly be over-
Netzer 2009; Denneyetal. 2009; Wangetal. 2009;
stressed. While these virial estimators currently are the
Grahametal. 2011; Rafiee&Hall 2011b; Steinhardt
only practical way to estimate BH masses for large sam-
2011).
ples of broad-line quasars and AGNs, the nontrivial uncer-
tainty of these imperfect estimators has severe impact on We generally neglect systematic biases in the current
the mass distribution under study. The difference between study, as they are poorly understood at present. That
virialmassesandtruemassesnotonlymodifiestheunderly- means we assume on average these virial mass esti-
ing true distribution, but also introducesMalmquist-type bi- mators give unbiased mass estimates (see §3.2.1 for
ases (e.g., Shenetal. 2008; Kellyetal. 2009; Shen&Kelly themeaningof“unbiased”). However,wedoconsider
2010). These effects tend to dilute any potential mass- a possible luminosity-dependentbias (e.g., Shenetal.
dependent trends or correlations (e.g., Kelly&Bechtold 2008;Shen&Kelly2010),whichwedescribeindetail
2007; Shenetal. 2009), and may lead to unreliable conclu- in§3.2.1. Thisisnotonlybecauseluminosityisanex-
plicitterminallvirialestimators,butalsobecausethat
2 Fromnowon,unlessotherwisespecified, weusetheterm“quasar”to manystudieswithvirialBHmassesarerestrictedtofi-
refertobroad-line(type1)quasarsforsimplicity.
niteluminositybinsorflux-limitedsamples. Moreover,
QUASARDEMOGRAPHICS 3
understandinganypotentialluminosity-dependentbias
is crucial to probe the true distribution in the mass- TABLE1
luminosityplane. SUMMARYOFZBINS
¯
This paper is organizedas follows. In §2 we describe the zbin zrange NQ/Nvir Mi,lim[z=2]
data;wepresentthetraditionalbinnedLF/BHMFin§3.1and Hβ
describetheBayesianapproachin§3.2.Wepresentourmodel 1... [0.3,0.5] 4298/4149 - 22.94
results in §4, discuss the results in §5 and conclude in §6. 2... [0.5,0.7] 4206/4027 - 23.84
ThroughoutthepaperweadoptaflatΛCDMcosmologywith M3.g.I.I [0.7,0.9] 3955/3873 - 24.61
cosmologicalparametersΩΛ=0.7,Ω0=0.3,h=0.7,tomatch 4... [0.9,1.1] 4871/4772 - 25.06
most of the recentquasardemographicswork. Volume is in 5... [1.1,1.3] 5872/5789 - 25.39
comovingunitsunlessotherwisestated. Wedistinguishvirial 6... [1.3,1.5] 5925/5855 - 25.73
7... [1.5,1.7] 6459/6340 - 25.99
massesfromtruemasseswithasubscriptvir ore. Quasarlu- 8... [1.7,1.9] 5839/5566 - 26.29
minosityisexpressedintermsoftherest-frame2500Åcon- CIV
tinuum luminosity (L≡λL or l ≡logL for short), and we 9... [1.9,2.4] 7761/7545 - 26.83
λ 10.. [2.4,2.9] 1695/1641 - 27.34
adoptaconstantbolometriccorrectionCbol=Lbol/L=5. 11.. [2.9,3.5] 4317/4003 - 26.66
12.. [3.5,4.0] 1830/1666 - 27.00
2. THEDATA 13.. [4.0,4.5] 661/518 - 27.36
14.. [4.5,5.0] 270/152 - 27.45
Our parent sample is the SDSS DR7 quasar catalog
(Schneideretal. 2010), which contains 105,783 bona fide
NOTE.—Thesecondcolumnliststhebound-
quasars with i-band absolute magnitude Mi <- 22 and have aries of each zbin. The third column lists
at least one broad emission line (FWHM>1000kms- 1) or the total number of quasars and those with
have interesting/complex absorption features. Among these measurablevirialmasses(measurementerror<
0.5 dex) in each zbin. The fourth column
quasars, about half were targeted using the final quasar tar-
liststhelimitingluminosityintermsoftheab-
getalgorithmdescribedinRichardsetal.(2002),andforma solute i-band magnitude normalized at z = 2
homogeneous, statistical quasar sample (e.g., Richardsetal. (Richardsetal.2006),whichcorrespondstothe
2006;Shenetal.2007b),whichweadoptinthecurrentstudy. flux limit (i=19.1 and 20.2 for z<2.9 and
z>2.9)andisestimatedatthemedianredshift
Quasars in this homogeneoussample are flux-limited to i= foreachzbin.
19.1 below z=2.9 and to i=20.2 beyond3. There is also a
brightlimitofi=15forSDSSquasartargets,whichonlybe-
comesimportantforthemostluminousquasarsatthelowest TABLE2
redshift (see Fig. 1 in Shenetal. 2011). We have used the BINNEDDR7VIRIALBHMF
continuumandemissionlineK-correctionsinRichardsetal.
(2006) to compute the absolute i-band magnitude normal- ¯z logMBH,vir logΦ(MBH,vir) logσ(MBH,vir)
ized at z = 2, Mi[z = 2]. At z < 0.5, host contamination (M⊙) (Mpc- 3logMB- 1H,vir) (Mpc- 3logMB- 1H,vir)
becomes more and more prominent towards lower redshift 0.4 7.50 - 6.378 - 7.370
(e.g.,Shenetal. 2011), so we restrictoursample to z≥0.3. 0.4 7.75 - 5.957 - 7.156
Our final sample includes 57,959 quasars at 0.3 ≤ z ≤ 5. 0.4 8.00 - 5.813 - 7.110
The sky coverageof this uniformquasar sample is carefully
NOTE. —Thefulltableisavailableintheelectronicversion
determined, using the approach detailed in the appendix in
ofthepaper.
Shenetal.(2007b),tobe6248deg2.
Threelineestimatorswereused:Hβ(Vestergaard&Peterson
Thevirialmassestimatesandmeasurementerrorsforthese
quasars were taken from Shenetal. (2011). We refer the 2006, z<0.7); MgII (Shenetal. 2011, 0.7≤z<1.9); CIV
(Vestergaard&Peterson 2006, z > 1.9). Virial BH masses
readerto Shenetal. (2011) fordetails regardingthe spectral
based on two estimators are smoothlybridgedacrossthe di-
measurementsandvirialmassestimates. Inshort,thespectral
vidingredshift,i.e.,thereisnosystematicoffsetbetweentwo
regionaroundeachofthethreelines(Hβ, MgII,andCIV) is
fit by a power-law continuum plus iron template4, and a set differentestimators. Fig.1 (left)showstheredshiftdistribu-
tionofvirialmassestimatesinoursample,wherethevertical
of Gaussians for the line emission. Narrow line emission is
dashed lines mark the divisions between two estimators and
modeled for Hβ and MgII but not for CIV. We use the con-
thegridweuse tocomputethe BHMF(seebelow)isshown
tinuum luminosity and line FWHM from the spectral fits to
ingray.Werejectobjectswithameasurementerror>0.5dex
compute a virial mass using one of the fiducial virial cali-
invirialmassestimatesincomputingtheBHMF,andwewill
brations adopted in Shenetal. (2011, eqns. 5,6,8). >95%
correctforthisincompletenessinmassestimatesinSec3.
of the 57,959 quasars have measurable virial BH masses.
Fig. 1 (right) shows the distribution of measurement errors
3. THEQUASARLFANDBHMF
(propagatedfrom the FWHM and continuum luminosity er-
rors) of these virial mass estimates. The vast majority of 3.1. TheTraditionalApproach
virialestimateshaveameasurementerrorfarbelow0.3- 0.4 Following the common practice in the literature (e.g.,
dex, the nominal statistical uncertainty of virial estimators. Fanetal. 2001; Richardsetal. 2006; Greene&Ho 2007;
Vestergaardetal. 2008; Vestergaard&Osmer 2009;
3 Thereareatinyfractionofuniformly-selected quasarstargetedbythe Schulze&Wisotzki 2010), we use the 1/V method
max
HiZbranchofthetargetselectionalgorithm(Richardsetal.2002)atz<2.9
(e.g.,Schmidt1968)toestimatetheLFandactiveBHMF:
downtoi=20.2. Wehaverejectedthesequasarsinourflux-limitedsample
(see Shenetal.2011,formoredetails). Ω zmax dV
4ExceptforCIV,whereweonlyfitapower-lawcontinuumwithnoiron V = Θ(L,z) cdz, (1)
max
templateapplied. 4πZ dz
zmin
4 SHEN&KELLY
FIG. 1.—Left: Redshiftdistribution ofvirialBHmassesinoursample. Right: DistributionofmeasurementerrorsofthevirialBHmassestimates. The
vastmajorityofvirialmassestimateshavenegligiblemeasurementerrorscomparedwiththenominalstatisticaluncertaintyofvirialBHmassestimatorsσvir∼
0.3- 0.4dex.
thetabulatedselectionfunction5inRichardsetal.(2006)with
interpolationto estimate Θ(L,z), andcalculateV for each
max
quasarinaredshift-luminosity(virialmass)bin. Thebinned
LF,Φ(M[z=2])≡dn/dM[z=2]isthen:
i i
N
1 1
Φ(M[z=2])= , (2)
i ∆M[z=2] (cid:18)V (cid:19)
i Xj=1 max,j
withaPoissonstatisticaluncertainty
1 N 1 2 1/2
σ(Φ)= , (3)
∆M[z=2](cid:20) (cid:18)V (cid:19) (cid:21)
i Xj=1 max,j
where the summation is over all quasars within a redshift-
magnitudebin.
The 1/V binned BHMF, Φ(M )≡dn/dlogM ,
FIG. 2.—ComparisonbetweentheDR3(Richardsetal.2006)andDR7 max BH,vir BH,vir
binnedLF(thiswork)forthesameluminosity-redshiftgrid.TheDR7results isthen:
areingoodagreementwithearlierDR3results.
N
1 1
Φ(M )= , (4)
BH,vir ∆logM (cid:18)V (cid:19)
BH,virXj=1 max,j
TABLE3
BINNEDDR7LF withaPoissonstatisticaluncertainty
¯z Mi[z=2] logΦ(Mi[z=2]) logσ(Mi[z=2]) 1 N 1 2 1/2
(Mpc- 3mag- 1) (Mpc- 3mag- 1) σ(Φ)= , (5)
∆logM (cid:20) (cid:18)V (cid:19) (cid:21)
0.4 - 22.65 - 5.669 - 6.920 BH,vir Xj=1 max,j
0.4 - 22.95 - 5.643 - 7.078
0.4 - 23.25 - 5.858 - 7.350 where the summation is over all quasars within a redshift-
massbin.
Asasanity check,we computedtheDR7 quasarluminos-
NOTE.—Thefulltableisavailableintheelectronic
versionofthepaper. ity function (LF) in the same L- z grid as in Richardsetal.
(2006),andfounditinexcellentagreementwiththeDR3re-
sultswithsmallerstatisticalerrorbars(Fig.2).
whereΩistheskycoverageofoursample,dV /dzisthedif- TocomputethebinnedLFandBHMFwechoosearedshift
c
ferential comoving volume, z and z are the minimum grid(zbins)thatavoidsstraddlingtwomassestimators,with
min max
and maximum redshift within a redshift-luminosity (virial
mass) bin that is accessible for a quasar with luminosity L, 5 Thereisnodifferenceinthetargetselectioncompletenessbetweenthe
and Θ(L,z) is the luminosity selection function mapped on uniform DR3 quasars usedin Richardsetal. (2006)andthe uniform DR7
quasarsusedhere,sincethefinalquasartargetalgorithmwasimplemented
a two-dimensional grid of luminosity and redshift. We use afterDR1.
QUASARDEMOGRAPHICS 5
boundaries of 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, 2.4, (e.g.,seeEqn.8below). Forclarity,weuse p(x|y)todenote
2.9, 3.5, 4.0, 4.5, 5.0. Withineach ofthe 14 zbinswe use the conditionalprobabilitydistributionof quantity x at fixed
a mass grid with a bin size of ∆logM =0.25 starting y,andx|ytodenotearandomvalueofxatfixedydrawnfrom
BH,vir
fromlogM =7.375,andaluminositygridwithabinsize p(x|y).
BH,vir
of ∆M[z=2]=0.3 starting from M[z=2]=- 22.5. Table For the local RM AGN sample (which has a dispersion
i i
1 summarizesinformationfor each zbin. Fig. 3 shows the of∼1 dexinluminosity), single-epochvirialBH mass esti-
binnedvirialBHMFusingthe1/V technique. Ourbinned mateswerecalibratedagainstRMmasses(assumedtobetrue
max
virial BHMF results are similar to the binned virial BHMF masses)tohavetherightmean,andascatter(uncertainty)of
estimatedinVestergaardetal.(2008)basedonDR3quasars, ∼0.4 dex around RM masses (e.g., McLure&Jarvis 2002;
withmuchbetterstatisticsduetoincreasedsamplesize. McLure&Dunlop 2004; Vestergaard&Peterson 2006). To
Two important facts limit the application of the binned account for the effects of the uncertainty in virial BH mass
virial BHMF. First, it is inappropriate to use the selection estimates, we first must understand the origin of this uncer-
function upon luminosity selection for the BHMF, i.e., BHs tainty.Itisnaturaltoascribethisuncertaintytotwofacts(e.g.,
with instantaneous luminosity fainter than the flux limit of Shenetal.2008;Shen&Kelly2010):a)luminosityisanim-
the survey will be missed regardless of their masses. As a perfecttracer of the BLR size; b) line width is an imperfect
result, thebinnedBHMFsuffersfromincompleteness,espe- tracerof the virialvelocity. Takentogether,some portionof
cially at the low-mass end, and the turn-over of the BHMF thevariationsinluminosityandlinewidthareindependentof
at low masses seen in Fig. 3 is not real. Second, virial BH eachother,causingthevirialmassestimatestoscatteraround
massesarenottruemasses. Substantialscatterbetweenvirial the true value; the remaining portion of the variations in lu-
mass estimates and the true masses changes the underlying minosity and line width cancel with each other, and do not
BHmassdistribution,andmayleadtosignificantMalmquist- contributetothescatterinthevirialmassestimates.
type biases (e.g., Shenetal. 2008; Kellyetal. 2009, 2010; Tobetter understandthis, considerthe followingexample.
Shen&Kelly2010).Thelattereffectisparticularlyimportant Take a population of N BHs with the same true mass m≡
atthehigh-massend(wherethecontaminationfromintrinsi- logM ,andassuming:a)TheFWHMandluminosityfollow
BH
cally lighter BHs can dominate over the indigenouspopula- lognormaldistributionsatthisfixedtrueBHmass;b)amean
tion)andathighredshift(wherethevirialBHmassestimator luminosity-radius(R- L)relationR∝L0.5,andalinearmean
switches to the more problematic CIV line, e.g., Shenetal. relationbetweenFWHMandthevirialvelocityv;andc)the
2008). The Bayesian framework developed in Kellyetal. virialmassesareunbiasedonaverage. Forthispopulationof
(2009)anddescribedin§3.2remediestheseissues, andpro- BHs,theluminosityl≡logLofindividualobjectisgivenby:
videsmorereliableestimatesfortheintrinsicBHMF. l|m=hli +G (0|σ′)+G (0|σ ), (6)
Bearingin mindthe limitationsofthe binnedBHMF, Fig. m 1 l 0 corr
3showsacoherentevolutionforthemostmassive(M & wherel|m is theindividualobjectluminosityatthisfixedm,
BH,vir
3×109 M⊙) BHs: their abundance rises from high redshift Gi(µ|σ) is a Gaussian random deviate with mean µ and dis-
andreachesmaximumaroundz∼2, thendecreasestowards persionσ, and hlim is the expectationvalue of luminosityat
lowerredshift. Thistrendislikelyamanifestationoftherise thistrueBHmass. Similarlywecangenerateindividualline
andfallofbrightquasarsseenintheLF,andwewilltestthis widthw≡logFWHMas:
trendwiththeBayesianapproachdescribedinSec3.2. w|m=hwi +G (0|σ )- 0.25G (0|σ ), (7)
m 2 w 0 corr
For future comparison purposes only, we tabulated the
binnedvirialBHMFinTable2;butweremindthereaderthat wherehwim istheexpectationvalueoflinewidthatthistrue
it should be interpreted with caution. We also tabulated the BHmass.Theindividualvirialmassestimateme≡logMBH,vir
binnedLFinTable3. atthisfixedMBHisthen
3.2. TheBayesianApproach me|m=m+0.5G1(0|σl′)+2G2(0|σw), (8)
Asdiscussedearlier,thecausalconnectionbetweentheLF which implies that the virial BH mass estimates follow a
andBHMFnaturallyrequiresadeterminationofthejointdis- lognormal distribution around the correct mean (i.e., m),
tributioninthemass-luminosityplane.Indoingso,oneneeds but have a lognormal scatter (virial uncertainty) σvir =
toaccountforselectioneffectsofthefluxlimitofthesample, (0.5σ′)2+(2σ )2 around the mean (e.g., Shenetal. 2008;
l w
andtodistinguishbetweenvirialmassesandtruemasses. The pShen&Kelly2010). TheG0termsofvariationinluminosity
bestapproachisaforwardmodeling,inwhichwespecifyan and FWHM exactly cancel with each other and do not con-
underlying distribution of true masses and luminosities and tribute to the virial uncertainty, and were referred to as the
map to the observedmass-luminosityplane by imposingthe “correlatedvariations”inFWHMandluminosityintheabove
fluxlimitandrelationsbetweenvirialmassesandtruemasses, papers; while the G and G terms were referred to as the
1 2
and compare with the observeddistribution (e.g., Shenetal. “uncorrelatedvariations”inFWHMandluminosity,andthey
2008; Kellyetal. 2009, 2010). This is a complicated and contributetothevirialuncertaintyinquadraticsum. Theap-
model-dependentproblem. Below we first demonstrate our proachinKellyetal.(2009,2010)implicitlyassumedσ′=0,
l
bestunderstandingsoftherelationshipbetweenvirialmasses while the approach in Shenetal. (2008) and Shen&Kelly
and true masses, then we describe our model parameteriza- (2010)istosetσ =0andconsidernon-zeroσ′. Thelatter
corr l
tionsandtheimplementationoftheBayesianframework.We choiceismotivatedbythe factthattheobserveddistribution
deferthecaveatsinourmodelto§5.3. of FWHM for SDSS quasar samples is already narrow (dis-
persion.0.15dex)andthepremisethatthevirialuncertainty
3.2.1. Preliminaries
σ shouldbenolessthan∼0.3dex.
vir
Here we describe our modeling of the statistical errors of Physically σ′ is unlikely to be zero. If this were true, it
l
virial mass estimates, under the premise that these FWHM- would imply that single-epochluminosity is an unbiased in-
basedvirialmassestimatorsonaveragegivethecorrectmean dicator for the instantaneous BLR radius at fixed BH mass.
6 SHEN&KELLY
FIG. 3.—BinnedvirialBHMFusingthe1/Vmaxtechnique. Ineachpanelthepointswitherrorbarsaretheresultsforeachzbin,andthedottedanddashed
linesarereferenceresultsinzbin1andzbin9.Themeanredshiftineachzbinismarkedontheupper-rightcornerofeachpanel.
TABLE4
MODELLF,BHMFANDEDDINGTONRATIOFUNCTION
logΦ(L) logΦ(MBH) logΦ(MBH,det) logΦ(λ) logΦ(λdet)
¯z Mi[z=2] (erlgogs-L1) Φ0 (Mpc-Φ3+dex-1) Φ- lo(gMM⊙B)H Φ0 (Mpc-Φ3+dex-1) Φ- Φ0 (Mpc-Φ3+dex-1) Φ- logλ Φ0 (MpcΦ-3+dex-1) Φ- Φ0 (Mpc-Φ3+dex-1) Φ-
0.4 -16.904 42.00 -6.299 -6.002 -6.639 6.000 -10.684 -8.179 -12.383 -19.804 -17.507 -21.693 -4.000 -9.482 -9.108 -10.225 -17.622 -16.349 -19.266
0.4 -16.979 42.03 -6.240 -5.950 -6.572 6.025 -10.550 -8.114 -12.201 -19.521 -17.287 -21.363 -3.975 -9.368 -9.002 -10.097 -17.397 -16.152 -18.999
0.4 -17.054 42.06 -6.182 -5.900 -6.506 6.050 -10.415 -8.049 -12.023 -19.240 -17.068 -21.038 -3.950 -9.256 -8.896 -9.970 -17.175 -15.955 -18.736
NOTE.—Thefulltableisavailableintheelectronicversionofthepaper.
Whileinpracticeitismorenaturaltoexpectthereareuncor- indicators(suchasFWHM)mightstillnotresponsetoallthe
relatedrandomscatterinbothLandR,indicatingastochastic variationsin luminosity. For example, considera single BH
terminadditiontothedeterministicterm(whenpredictingR whereitsluminosityvaries(anditsBLRradiusvariesinstan-
withL), whichwilllead tobiasedestimatesforR atfixedL. taneouslyfollowingaperfectR- Lrelation),andsupposethat
The sourcesof thisstochastic term may include: a) the con- thebroadlineiscomposedofanon-virializedcomponentand
tinuum luminosity variation and response of the BLR is not avirializedcomponent.Whenluminosityincreases(BLRex-
synchronized;b)individualquasarshavedifferentBLRprop- pands), the virialized componentreducesline width, but the
erties; c) optical-UV continuum luminosity is not as tightly non-virializedcomponentmayincreaselinewidthifitisorig-
connected to the BLR as the ionizing luminosity. Further- inatedfromaradiativelydrivenwind(inthecaseofCIV,more
more, even if single-epoch luminosity were an unbiased in- blueshiftedCIVtendstohavealargerFWHM,e.g.,Shenetal.
dicator of the instantaneous BLR radius, certain line width 2008): thecombinedlineFWHMmaynotchangeduetothe
QUASARDEMOGRAPHICS 7
twooppositeeffects.Thereforeinthiscasealthoughluminos- dynamicalrange at fixed true BH mass, and hence are good
ityistracingtheBLR sizeperfectly,somevariationinlumi- indicatorsforBLRradiusandvirialvelocity.β=0represents
nosityisnotcompensatedbyvariationsinFWHMandshould theextremesituationwhereFWHMrespondstoallthevaria-
becountedastheuncorrelatedvariationσ′. tioninluminosityatfixedtruemass(plusadditionalscatterin
l
A non-zero σ′ implies that the distribution of virial mass FWHM),andnobiasinvirialmassesisincurredwhenlumi-
l
estimatesatfixedtruemassandfixedluminosity, p(m |m,l), nositydeviatesfromhli . β=0isgenerallyassumedinmost
e m
is different from the distribution of virial mass estimates at studieswithvirialBHmasses.
fixed true mass, p(m |m). In the extreme case where σ = We also notethatoneadvantageofusingEqn.(10) isthat
e corr
0, i.e., FWHM does not change in response to variations in it does not rely on the assumption that the mean R- L rela-
luminosityatall,wehave tion andthe linear mean relationbetween FWHM and virial
m |m,l=m+0.5(l- hli )+2G (0|σ ). (9) velocity used in these virial estimators are correct. In other
e m 2 w words,ifthevirialestimatorsadoptedinthisworkusedincor-
Hence not only is the distribution p(me|m,l) narrower than rectformsfor the mean R- Lrelation and the mean relation
p(me|m), butalso the expectationvalueof me isbiased from betweenFWHM andvirialvelocity,thena negativevalueof
thetrueBHmassforanyfixedluminositiesexceptforl=hlim. β maybeneededtocorrectme atfixedmandl. Ofparticular
Nowconsideramoregeneralformoftheluminositydistri- interesthereiswhetherornotradiationpressureisimportant
butionatfixedtruemassandtheactualslopeintheobserved inthedynamicsoftheBLR(e.g.,Marconietal.2008),which
luminosity-radiusrelation, we can parameterizethe distribu- willindicateanegativeβbasedonthevirialmassesthathave
tionof p(me|m,l)as: not been corrected for radiation pressure. We will test if a
m |m,l=m+β(l- hli )+ǫ , (10) negative β is required to model the observed distribution in
e m ml
ourBayesianapproach.
where again hlim is the expectation value of luminosity at Isthereanyindicationforanon-zeroβ fromthereverber-
fixedtrue mass, ǫml is a randomdeviatewith zero meanand ation mapping AGN sample? There are only ∼3 dozens of
dispersion σml, denoting the scatter of virial mass estimates RMAGNsandwedonothaveenoughobjectswiththesame
atfixedtruemassandfixedluminosity,andtheerrorslopeβ BH mass to test source-by-source variations. Nevertheless,
describesthelevelofluminosity-dependentmassbiasatfixed wecanstilltesttheluminosity-dependentbiasusingrepeated
truemassandluminosity.Bothβandǫmlaretobeconstrained spectraforthesameobjectwhenitsluminositychangesasig-
byourdata. Eqn.(10)impliesthatthevarianceofmassesti- nificantamountovertime. NGC5548isthemostfrequently
matesatfixedtruemassandluminosityisreducedto: monitored RM AGN (Hβ only), and has been observed in
Var(m |m,l)=Var(m |m)(1- ρ2), (11) differentluminosity states with a spread of ∼0.5 dex in lu-
e e
minosity(e.g.,Petersonetal.2004;Bentzetal. 2009b), thus
where ρ2 =β2Var(l|m)/Var(me|m), and Var(...) refers to the providesanidealtestcaseforsingle-sourcevariations.
varianceofadistribution.Theformaluncertaintyofthevirial In Fig. 4 we show the Hβ virial product for NGC 5548,
massestimatoristhen computed using the continuum luminosity and line width
measured at different luminosity states in each monitoring
σ ≡ Var(m |m)= Var(m |m,l)+β2Var(l|m). (12)
vir e e period, as a function of continuum luminosity. The spec-
q
p
Ifwe assumea singlelog-normaldistributionfor p(l|m)and tral measurements were taken from Collinetal. (2006), and
ǫ (withadispersionσ ),theaboveequationreducesto wehavecorrectedthecontinuumluminosityforhoststarlight
ml ml
using the correction provided by Bentzetal. (2009a). The
σ = σ2 +β2σ2 . (13) linewidthsweremeasuredfromboththemeanandrmsspec-
vir q ml l tra7 for each monitoring period. The left and right panels
Notethathereσl isthetotaldispersioninlogLatfixedmass, of Fig. 4 show the virial product computed using FWHM
ratherthantheportionσ′ thatisnotrespondedbyFWHMas and line dispersion, respectively, and its scaling with lumi-
l
inEqns.(6)and(8). nosity is the same as in the virial mass estimators provided
Eqn. (10) is a rather generic form that describes the rela- by Vestergaard&Peterson (2006). The FWHM-based virial
tion between virial masses and true masses and the possible productshows an average trend of increasing with luminos-
luminosity-dependentbiasinvirialmasses6,andisoneofthe ity, which means that FWHM does not fully response to
basicequationsinourBayesianapproach.Thevalueofβ de- the variationsin luminosity, leading to a positive bias in the
pendsontherelativecontributionsfromσl′andσcorrinthelu- virial product (and thus in the virial mass estimate). This
minositydispersionatfixedmass. Undertheassumptionthat trend seems to be slightly weaker when using line disper-
the mean R- L relation and a linear mean relation between sioninstead. AlinearregressionanalysisusingtheBayesian
FWHMandvirialvelocityarecorrectasintheadoptedvirial method of Kelly (2007) yields: β ∼0.65±0.27 (FWHM,
estimators, a non-zero σ′ leads to a positive β. If the value mean);β∼0.51±0.34(FWHM,rms);β∼0.20±0.30(σ ,
l line
ofβ approachestheslopeintheadoptedmeanR- Lrelation, mean); β ∼0.45±0.29 (σ , rms), where uncertaintiesare
line
thenitsuggestseitherluminosityorFWHMisapoorindica- 1σ. Whileitisinconclusivebasedonthissingleobject,there
torforBLRsizeorvirialvelocityoverthenarrowdynamical issomeindicationthata positiveβ isfavored,especiallyfor
rangeatfixedtrueBHmass(althoughtheycouldstillberea- the virial product based on FWHM from the mean spectra,
sonableindicatorsforlargedynamicalrangesinmassandlu- which is the closest to that based on FWHM from single-
minosity).Ontheotherhand,ifβissmall,thenitmeanslumi- epoch spectra. It would be important to test this for more
nosityandFWHMvaryinconcordanceevenoverthenarrow
7 Strictly speaking, for single-epoch virial mass estimates, neither the
6Onecanworkoutasimilarequationforthedistributionofmeatfixedm meannorrmsspectra areavailable. However, thespectral variability dur-
andFWHMw,p(me|m,w)=m+β′(w- hwim)+ǫmw.Anon-zeroσwinEqn. ingeachmonitoringperiodissmallenoughsuchthatthemeanspectrumis
(7)willleadtoanon-zeroβ′andp(me|m,w)6=p(me|m).However,thisisof closetosingle-epochspectrawithinthisperiod.
littlepracticalvaluesincevirialmassesareneverbinnedinFWHM.
8 SHEN&KELLY
FIG. 4.—Thedependenceofthevirialproductcomputedfromluminosityandlinewidthasafunctionofluminosity,forasingleobjectNGC5548andfor
Hβonly. ThedataarefromCollinetal.(2006),andarebasedonbothmeanandrmsspectraduringeachmonitoringperiod. Errorbarsrepresentmeasurement
errors.Theerrorbarsinluminosityhavebeenomittedintheplotforclarity.Thecontinuumluminosityhasbeencorrectedforhoststarlightusingthecorrection
providedbyBentzetal.(2009a).theblackandbluedashedlinesarethebestlinear-regressionfitsusingtheBayesianmethodofKelly (2007),formeasurements
basedonmeanandrmsspectra,respectively. Left: virialproductbasedonFWHM;thedatapointforYear5(JD48954-49255)basedonthermsspectrumhas
beensuppressedduetoproblematicmeasurements(e.g.,Petersonetal.2004;Collinetal.2006).Right:virialproductbasedonlinedispersionσline.
1.2 s 1.2 Error Slope b 1.2 a , <L>(cid:181) Ma 1 500 400
1.0 ml 1.0 1.0 1 BH urces 400 urces 300
0.8 0.8 0.8 So So
0.6 0.6 0.6 # of 300 # of 200
d 200 d
0.4 0.4 0.4 bserve 100 bserve 100
0.2 0.2 0.2 O 0 O 0
0.0 0.0 0.0 44.5 45.0 45.5 46.0 46.5 7.0 7.5 8.0 8.5 9.0 9.5 10.010.5
0.15 0.20 0.25 0.30 -0.1 0.0 0.1 0.2 0.3 0.4 0.7 0.8 0.9 1.0 1.1 1.2 log L [erg s-1] log MBH,vir [MO •]
11..02 s l 11..02 <logl > 11..02 s (logl ) urces 1000 urces 1000
0.8 0.8 0.8 So 100 So 100
of of
0.6 0.6 0.6 d # d #
0.4 0.4 0.4 bserve 10 bserve 10
0.2 0.2 0.2 O 1 O 1
0.0 0.0 0.0 44.5 45.0 45.5 46.0 46.5 7.0 7.5 8.0 8.5 9.0 9.5 10.010.5
0.350.400.450.500.550.60 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 0.350.400.450.500.550.60 log L [erg s-1] log MBH,vir [MO •]
FIG.5.—Modelparametersforzbin2.Shownherearetheposteriordis- FIG. 6.—Posteriorchecksforzbin2. Thesolidblackhistogramsshows
tributionsofsomemodelparametersandderivedquantities.Fromtop-leftin theobserveddistributions. Theredpointsanderrorbarsaremedianresults
clockwiseorder: thedispersioninmassestimatesatfixedtruemassandlu- anduncertaintiesfromoursimulatedsamplesusing500randomdrawsfrom
minosity,σml;theerrorslopeβ;theslopeinthemean(true)mass-luminosity theposteriordistributions. Thetopandbottompanelsshowthehistograms
relationforourEddingtonratiomodel,α1;thedispersioninEddingtonratios inlinearandlogarithmicscales,respectively.
forallbroad-linequasars,σ(logλ)whereλ≡Lbol/LEdd;themeanEdding-
tonratioforallbroad-line quasars,hlogλi; thedispersioninluminosityat
fixedtruemassinourEddingtonratiomodel,σl. Now we proceed to describe our model setup and the im-
plementationoftheBayesianframework. Belowwedescribe
objectswithrepeatedspectra,andforMgIIandCIVaswell.
thebasicsofourmodelapproach. Moredetailsregardingthe
To summarize, because luminosity is an explicit term in
Bayesian approach can be found in Kellyetal. (2009) and
virial mass estimators, these virial mass estimates are no
Kellyetal.(2010).
longer independent (and unbiased) estimates of true masses
whenrestrictedtoanarrowluminosityrangeoraflux-limited
a. The BHMF and luminosity distribution model. As in
sample, for cases where β 6= 0. Our view of the uncer-
Kellyetal. (2009, 2010), we use a mixture of log-
tainties in these virial mass estimates (i.e., the scatter in
normaldistributions as our modelfor the true BHMF,
p(m |m), as determined in the calibrations against the RM
e and a single log-normal luminosity (Eddington ratio)
AGNs) is thus different from that in Kollmeieretal. (2006)
distributionatfixedtrueBHmass. Themixtureoflog-
andSteinhardt&Elvis(2010b).
normals is flexible enough to capture the basic shape
ofanyphysicalBHMF,andgreatlysimplifiesthecom-
3.2.2. ImplementingtheBayesianFramework
putationasmanyintegrationscanbedoneanalytically.
QUASARDEMOGRAPHICS 9
ThemodeltrueBHMFreads
47.0
dV - 1 K π (m- µ )2
Φ(m)=N k exp - k , (14)
(cid:18)dz(cid:19) Xk=1 2πσk2 (cid:20) 2σk2 (cid:21)
q 46.5
where m≡logM , N is the total numberof quasars,
BH
dsµiekasnac,nridabneσdkthPaereBKk=tH1hπMekmF=,eaa1sn.waWneddeoduinssopeteKrfisni=odns3ioglfnotighfi-ecnakontrhmtdGailfasfuetsro-- -1L [erg s]bol46.0
encewhenincreasingthenumberoflog-normalsused. og
l
The luminosity distribution at fixed BH mass is mod-
eledas 45.5
1 [l- α - α (m- 9)]2
p(l|m)= exp - 0 1 , (15)
2πσ2 (cid:18) 2σl2 (cid:19)
l 45.0
q 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
wherel≡logL, α0 andα1 describeamass-dependent log MBH/log MBH,vir [MO •]
mean luminosity, and σ is the scatter in luminosityat
l
fixedmass. TheLFistherefore FIG.7.—Posteriorchecksforzbin2inthemass-luminosityplaneabove
thefluxlimit. ThetwoblacklinesindicateEddingtonratiosof0.01and1.
Theblackandredcontoursarefortheobservedandsimulateddistributions
Φ(l)= Φ(m)p(l|m)dm. (16) usingvirialmasses,andthebluedashedcontourshowsthesimulateddistri-
Z butionwithtruemasses. Ourmodelfitstheobserveddistributionwell,and
thedistributionusingtruemassesisdifferentfromthatusingvirialmasses.
b. The virial mass prescription. We assume that virial
massesareunbiasedwhenaveragedoverluminosityat
mass function in mass bins that are severely incomplete, as
fixedtruemass(i.e.,Eqns.8,10),andwegeneratevirial
small errors in the selection function can lead to large de-
masses at fixed true mass andluminosityaccordingto
viations in p(I =1|θ), which appears in the denominator in
Eqn.(10),assumingasingleGaussian(withdispersion
Equation (18). Instead, we use the luminosity derived from
σml)todescribethescatterǫml atfixedmassandlumi- the i-band magnitude to ameliorate this effect, as the selec-
nosity.
tionfunctioniscalculatedintermsofi.
It is necessary to impose some prior constraints on β and
c. Theredshiftdistribution. Becausetheredshiftbinsare
σ basedonthereverberationmappingdataset,asthesepa-
narrow, we approximate the distribution of redshifts ml
rameters are degenerate with some of the other parameters.
acrossthebinasapower-law,wherethepower-lawin-
Unlikemostpreviouswork,wedonotfixthevaluesofβ and
dexγ isafreeparameter:
σ to,say,β=0andσ =0.4dex,butuseapriordistribution
ml ml
(1+γ)zγ whichincorporatesouruncertaintyintheseparameters. This
p(z|γ)= . (17)
z1+γ- z1+γ uncertaintywillbereflectedintheprobabilitydistributionof
max min themassfunction,giventheSDSSDR7dataset. Weapplied
Here,zmaxandzmindefinetheupperandlowerboundary theBayesianlinearregressionmethodofKelly (2007)tothe
oftheredshiftbin,respectively. reverberationmappingsample in orderto estimate the prob-
ability distribution of β and σ based on this sample. The
d. Theposteriordistribution p(θ|m ,l,z)is ml
e methodofKelly (2007)incorporatesthemeasurementerrors
N in the mass estimates, which is important when estimating
p(θ|m ,l,z)∝p(θ)[p(I=1|θ)]- N p(m ,l,z|θ) (18) the amplitude of the scatter in the mass estimates. We set
e e,i i i
Yi=1 hlim equaltothemeanluminosityforthereverberationmap-
ping sample8. We used the values of RM black hole mass
where θ(π ,µ ,σ ,α ,α ,σ,β,σ ,γ) is the set of
k k k 0 1 l ml
(assumed to be true masses) given by Petersonetal. (2004).
model parameters, N is the total number of quasars,
p(θ)istheprioronθ, p(I=1|θ)istheprobabilityasa FortheHβ calibration,weusedthevalueof5100Åluminos-
functionofθthatabroad-linequasarisincludedinthe ity given in Bentzetal. (2009a) and value of FWHM given
flux-limited SDSS quasar sample, and the likelihood in Vestergaard&Peterson (2006); when there were multiple
function p(m ,l,z|θ) is determined by Eqns. (10), measurements,weaveragedthemtogether. ForCIVweused
e,i i i
(14),(15)and(17). the values given in Vestergaard&Peterson (2006). For Hβ
wefoundthatβ =0.16±0.1,andthattheposteriordistribu-
Inthiswork,wederivethecontinuumluminosityat2500Å, tionforσ2 iswelldescribedbyascaledinverseχ2 distribu-
ml
l ≡logL, from the i-band magnitude according to the pre- tionwithν≈20degreesoffreedomandscaleparameters2=
scription givenin Richardsetal. (2006). This is a departure 0.1.ForCIVwefoundthatβ=0.15±0.14,ν≈20,s2=0.17.
fromtheapproachtakenbyKellyetal.(2010),whousedthe ForCIVthisissimilartotheusuallyquotedscatterinthemass
continuumluminosityestimatedbyVestergaardetal.(2008). estimatesofs≈0.4dex,butthescatterislessforHβ,s≈0.3
However, as explained in Kellyetal. (2010), because the dex. Thisreduceduncertaintyislikelybecausewehaveused
SDSSselectionfunctionisintermsofthei-bandmagnitude,
they had to assume a modelfor the distribution of i at fixed 8 Ideallyoneshoulduseapopulation ofobjects withthesametrueBH
luminosity,andthencalculatetheselectionfunctioninterms mass to perform linear regression using Eqn. (10), which is not available
giventhelimitedsizeoftheRMsample. SoinsteadweusethewholeRM
ofluminositybyaveragingoverthismodeldistribution.They
sample. Nevertheless,thederivedpriorconstraintonβisweakandourre-
notethatthisapproachcanleadtoinstabilityintheestimated sultsareinsensitiveontheprioronβ.
10 SHEN&KELLY
45.5
45.0
44.5
-1g s] 44.0
L [er
g 43.5
o
l
43.0
42.5
42.0
7.0 7.5 8.0 8.5 9.0 9.5
log MBH/log MBH,vir [MO •]
FIG.8.—Thesimulatedmass-luminosityplaneforzbin2,whichextends
belowthefluxlimit(theblackhorizontalline). Theredcontouristhedis-
tributionbasedontrueBHmasses,andisdeterminedbyourmodelBHMF
andEddingtonratiomodel. Theblackcontouristhedistributionbasedon
virialBHmasses. Thefluxlimitonlyselectsthemostluminousobjectinto
oursample,andthedistributionbasedonvirialBHmassesisflatterthanthe
onebasedontruemassesduebothtothescatterσml andanon-zeroβ(see
Eqn.10).
5100Åluminosityvalueswhicharecorrectedforhostgalaxy
starlight(Bentzetal.2009a).
Based on the reverberation mapping results, we impose a
Cauchy prior distribution on β with mean and scale param-
eters equal to those derived from the reverberationmapping
data,andweimposeascaled-inverseχ2 priordistributionon
σ2 withν=15degreesoffreedomandscaleparametersetto
ml
thatfromthereverberationmappingsample.ForMgIIweuse FIG. 9.—ModelLF(toppanel)andBHMF(bottompanel)forzbin2.
thevaluesderivedforHβsincethesingle-epochvirialmasses ThedatapointsanderrorbarsarethebinnedLFandvirialBHMFestimated
in§3.1. Thecolor shaded regions are the 68%percentile range from our
basedonthetwolinesseemtocorrelatewitheachotherwell
modelLFandBHMF.Inthebottompanel,thegreenshadedregionisforthe
(e.g.,Shenetal.2011). WehaveusedaCauchypriorbecause detectable(i.e.,abovethefluxlimit)trueBHMF,andthemagentaoneisfor
ithassignificantlymoreprobabilityinthetailsthantheusual allthebroad-linequasars.Theturnoverofthemagentalinebelow∼108M⊙
Gaussiandistribution,makingourpriorassumptionsmorero- isafeatureconstrainedbythedataandourmodel,i.e.,ifthereweremore
lowermassBHs,itwouldbedifficulttofittheobserveddistributioninthe
bust. Similarly, we reduced the degrees of freedom for the
mass-luminosityplane(cf.Fig.8).
prior on σ2 compared to the reverberationmapping sample
ml
inordertoincreaseourprioruncertaintyonσ . Thischoice
ml
We use zbin2 as an example to demonstrate the infor-
ofpriorassumesanuncertaintyonσ of≈20%.
ml
As in Kellyetal. (2009) and Kellyetal. (2010), we use a mation that we can retrieve from the posterior distributions.
Markov Chain Monte Carlo (MCMC) sampler algorithm to ThisbinusesthemostreliableHβ linetoestimatevirialBH
obtainrandomdrawsofθaccordingtoEqn.(18)andthusthe masses;italsohasnegligiblehostgalaxycontaminationcom-
posteriordistributionofmodelparametersgiventheobserved paredwithzbin1. Thereforetheconstraintsforthisbinare
data in the virial mass-luminosity plane. Our MCMC sam- expectedtobethemostrobust.
pler employs a combination of Metropolis-Hastingsupdates Fig.5showstheposteriordistributionsofourmodelparam-
eters for zbin2. These parameters are tightly constrained,
withparalleltempering. ThereaderisreferredtoKellyetal.
(2009)andKellyetal.(2010)forfurtherdetails. althoughdegeneracydoesexistamongtheseparameters.
Different from Kellyetal. (2009) and Kellyetal. (2010), Fig. 6 presents the posterior checks of our model against
wemodelthedatainindividualredshiftbinsinsteadofforthe the data. The black histograms show the distribution of ob-
whole sample. We treat each redshiftbin as an independent served luminosities and virial masses. The points and error
dataset. Thisallowsustoexplorepossibleredshiftevolution bars are the median and 68% percentile for simulated sam-
ofourmodelparameters,aswellasthesensitivityonchanges plesgeneratedusingrandomdrawsfromtheposteriordistri-
bution. Thisensuresthatourmodelreproducestheobserved
in the detection luminosity threshold and the specific virial
mass estimator used. The caveat is that the constraints are luminosityandvirialmassdistributions. Fig.7furthershows
generallyweakergivenless data pointsin eachbin, and that thecomparisonbetweenmodelpredictionanddatainthetwo-
theconstrainedparametersdonotnecessarilyvarysmoothly dimensional mass-luminosity plane, where the model is the
acrossadjacentredshiftbins. one that has the maximum posterior probability. The black
and red contours show the observed and model-predicted
4. RESULTSOFTHEBAYESIANAPPROACH joint-distributionsof luminosity and virial mass in our sam-
ple, while the blue contours show that for the true masses.
4.1. zbin2asanexample
The true masses are scattered and biased according to Eqn.