Low redshift AGN in the Hamburg/ESO Survey

19 Pages·2010·0.72 MB·English

A&A516,A87(2010) Astronomy
DOI:10.1051/0004-6361/201014193 &
(cid:2)c ESO2010 Astrophysics
Low redshift AGN in the Hamburg/ESO Survey
II. The active black hole mass function and the distribution function
(cid:2)
of Eddington ratios
A.SchulzeandL.Wisotzki
AstrophysikalischesInstitutPotsdam,AnderSternwarte16,14482Potsdam,Germany
e-mail:[email protected]
Received3February2010/Accepted12April2010
ABSTRACT
WeestimatedblackholemassesandEddingtonratios(L/L )forawelldefinedsampleoflocal(z<0.3)broadlineAGNfromthe
Edd
Hamburg/ESOSurvey(HES),basedontheHβlineandstandardrecipesassumingvirialequilibriumforthebroadlineregion.The
samplerepresentsthelow-redshiftAGNpopulationoverawiderangeofluminosities,fromSeyfert1galaxiestoluminousquasars.
Fromthedistributionofblackholemasseswederivedtheactiveblackholemassfunction(BHMF)andtheEddingtonratiodistribution
function (ERDF) in the local universe, exploiting the fact that the HES has a well-defined selection function. While the directly
determinedERDFturnsoveraroundL/L ∼0.1,similartowhathasbeenseeninpreviousanalyses,wearguethatthisisanartefact
Edd
ofthesampleselection.Weemployedamaximumlikelihoodapproachtoestimatetheintrinsicdistributionfunctionsofblackhole
massesandEddingtonratiossimultaneouslyinanunbiasedway,takingthesampleselectionfunctionfullyintoaccount.Theresulting
ERDFiswelldescribedbyaSchechterfunction, withevidenceforasteadyincreasetowardslowerEddingtonratios,qualitatively
similartowhathasbeenfoundfortype2AGNfromtheSDSS.
Comparing our best-fit active BHMF with the mass function of inactive black holes we obtained an estimate of the fraction of
activeblackholes,i.e.anestimateoftheAGNdutycycle.Theactivefractiondecreasesstronglywithincreasingblackholemass.A
comparisonwiththeBHMFathigherredshiftsalsoindicatesthat,atthehighmassend,blackholesarenowinalessactivestagethan
atearliercosmicepochs.Ourresultssupportthenotionofanti-hierarchicalgrowthofblackholes,andareconsistentwithapicture
where the most massive black holes grew at early cosmic times, whereas at present mainly smaller mass black holes accrete at a
significantrate.
Keywords.galaxies:active–galaxies:nuclei–quasars:general
1. Introduction AGNLF has been used to gain insight into the growth history
ofblackholes(e.g.Soltan1982;Yu&Tremaine2002;Marconi
The observed relations between the black hole mass and the
et al. 2004; Merloni 2004; Shankar et al. 2009), and it be-
properties of the spheroidal galaxy component imply a close
cameclearthatmostoftheaccretionoccursduringbrightQSO
connection between the growth of supermassive black holes
phases.But,usingtheAGNLFaloneusuallyrequiressomead-
(SMBH)andtheevolutionoftheirhostgalaxies.Forlocalgalax-
ditionalassumptions,e.g. for the mean accretionrate, andthus
iesastrongcorrelationbetweenthemassoftheSMBHandthe isaffectedbyuncertaintiesanddegeneracies.Disentanglingthe
luminosity or mass of the bulge component (Magorrian et al.
AGNLF into the underlying distribution functions, namely the
1998; Marconi & Hunt 2003; Häring & Rix 2004), as well as
active black hole mass function (BHMF) and the distribution
with the stellar velocity dispersion (e.g. Ferrarese & Merritt
function of Eddington ratios (ERDF), is able to provide addi-
2000; Gebhardt et al. 2000; Tremaine et al. 2002; Gültekin
tionalessentialconstraintsonthegrowthofSMBHs.
et al. 2009)have been established. Semi-analyticaland numer-
Tounderstandtheinfluenceofblackholegrowthongalaxy
ical simulations also show the importance of black hole activ-
evolutionovercosmictime,firstthepropertiesofgrowingblack
ity and their correspondingSMBH feedback for galaxy evolu-
holesin the localuniversehaveto be wellunderstood.Thus, it
tion (e.g. Di Matteo et al. 2005; Springel et al. 2005; Croton
isimportanttoderiveblackholemassesandaccretionratesfor
et al. 2006;Cattaneo et al. 2006;Khalatyan et al. 2008;Booth
large,welldefinedsamplesofAGN.However,measuringblack
& Schaye 2009). It became clear that the central SMBH of a
holemassesismuchmoredifficultthanmeasuringluminosities.
galaxy and especially its growth is an important ingredientfor
Black hole masses for large samples of AGN can not be mea-
ourunderstandingofgalaxyformationandevolution.
sureddirectly,butonlyestimated,usinglocallyestablishedscal-
Therefore a complete census of the black hole population
ingrelations.
and its properties is required. Active black holes that will be
The best method to estimate M for type 1 AGN is rever-
observable as AGN are particularly important to study black BH
berationmappingofthebroadlineregion(Blandford&McKee
hole growth. A useful tool to study the AGN population is the
1982; Peterson 1993). Assuming virial equilibrium black hole
luminosity function (AGNLF). The observed evolution of the
massescan beestimatedby M = fR ΔV2/G,whereR
BH BLR BLR
(cid:3) Based on observations collected at the European Southern is thesize ofthe broadlineregion(BLR),ΔV isthebroadline
Observatory,Chile(Proposal145.B-0009). width in kms−1 and f is a scaling factor of order unity, which
ArticlepublishedbyEDPSciences Page1of19
A&A516,A87(2010)
dependsonthestructure,kinematicsandorientationoftheBLR. been identified, based on their peculiar spectral energy distri-
AlthoughthephysicsoftheBLRisstillnotwellunderstoodand bution.Follow-upspectroscopyhasbeencarriedouttoconfirm
thus a source of uncertainty (e.g. Krolik 2001), the validity of their QSO nature.The HES picksup quasarswith B <∼ 17.5at
the virial assumption has been shown by the measurement of redshiftsofuptoz(cid:4)3.2.
time lags and line widths for different broad lines in the same The Hamburg/ESO Survey yields a well-defined, flux-
spectrum (Peterson & Wandel 2000; Onken & Peterson 2002; limitedsamplewithahighdegreeofcompleteness.Thesurvey
Kollatschny2003). coversalargeareaontheskyandthequasarcandidateselection
However, reverberation mapping requires extensive and takes care to ensure that low redshift, low luminosity objects,
meticulous observations and thus is not appropriate for large i.e AGN with prominent host galaxies, are not systematically
samples.Fortunately,ascalingrelationshiphasbeenestablished missed.AsinPaperI,wewanttousethiswideluminosityrange
betweenR andcontinuumluminosityoftheAGN,R ∝Lγ atlowredshift,whichisuniqueforopticalsurveys,tostudythe
BLR BLR
(Kaspietal.2000,2005;Bentzetal.2006).Thusitbecamepos- low-redshiftAGNpopulation.
sible to estimate M from single-epochspectra for largesam- ToconstructsuchalocalAGNsampleweselectedallAGN
BH
ples, and has been used extensively in the previous years for fromthefinalHEScatalogue(Wisotzkietal.,inprep.)thatbe-
large AGN samples (e.g.McLure& Dunlop2004;Vestergaard long to the “completesample” and that are located at redshifts
2004;Kollmeieretal.2006;Netzer&Trakhtenbrot2007;Shen z<0.3.Thesamplecontains329type1AGN.Spectraareavail-
etal.2008b;Fineetal.2008;Gavignaudetal.2008;Trumpetal. ableformostoftheobjectsfromthefollow-upobservations.For
2009). fiveobjects,spectrawereeithermissinginourdatabaseorthey
Forthemeasurementofthelinewidth,differentmeasuresare wereofsuchpoorqualitythattheyweredeemednotusablefor
commonlyused,anditisunclearifoneissuperiortotheothers ourpurposes.Thusoursampleis324/329≈98.5%completein
for estimating black hole masses. Most commonly used is the termsofspectroscopiccoverage.
FWHM, butithasbeensuggestedthatthelinedispersionσ ,
line
i.e.thesecondcentralmomentofthelineprofile,isabettermea-
3. Measurementofemissionlinewidths
sureofthelinewidth(Petersonetal. 2004;Collin etal.2006).
The linedispersionis moresensitivetothe wingsof aline and For the estimation of M for our low redshift AGN sample,
BH
lesstothecore,whereasfortheFWHMtheoppositeisthecase. the broad line width of the Hβ, or alternatively the Hα, emis-
Anadditionalmeasureoflinewidthusedistheinter-percentile sionlinehastobedetermined.Forthemeasurementoftheline
value(IPV,Fineetal.2008). widths of the Hα and Hβ emission lines we fitted the spectral
The applicationof the virialmethodto largeAGN samples regionaroundthese linesby analyticfunctions,i.e. by a multi-
allowedtheestimationoftheactiveBHMF(McLure&Dunlop component Gaussian model plus continuum. Over this short
2004;Shenetal.2008b;Greene&Ho2007;Vestergaardetal. wavelength range we approximated the underlying continuum
2008;Kellyetal.2009;Vestergaard&Osmer2009).Adataset asastraightline.TheHαandHβlinesarefittedbyone,twoor,
that is perfectly suited to study especially low redshiftAGN is ifrequired,byuptothreeGaussians,basedonvisualinspection
providedbytheHamburg/ESOSurvey(HES).Inthispaperwe ofthefits.Duetothelimitedresolutionofthespectrathenarrow
usealocalAGNsample,drawnfromtheHES,toestimatetheir linecomponentcouldonlybesubtractedforafewlines,ifaclear
blackholemassesandEddingtonratios,andconstructtheactive attributionofonefittingcomponenttoanarrowlinecomponent
blackholemassfunctionaswellasthedistributionfunctionof was possible. Thusa narrowcomponentwasonlysubtractedif
Eddingtonratios. clearlyidentifiedinthefit.Carehasbeentakentoavoidcontami-
We first present our data and our treatment of the spectra. nationofthelinesbycontributionfromthe[Oiii]λλ4959,5007
We estimate black hole masses and Eddington ratios from the Å lines and the Feii emission to the Hβ line, as well as from
spectra using the virial method. Next, we determine the active [Nii] and [Sii] to the Hα line, by fitting them simultaneously
BHMF, taking care to account for sample selection effects, in- with theBalmerlines.Fordetailsonthelinefitting wereferto
ducingabiasontheBHMF.Thereby,wenotonlyconstrainthe PaperI.
local active BHMF but also put constraintson the intrinsic un- We use two differentline width measurements,the FWHM
derlying distribution function of Eddington ratios. Finally, we and the line dispersion for comparison, because there is at the
discuss our results in the contextof the local quiescent BHMF momentnoconsensuswhichisthemostappropriateforthees-
aswellasthatofothersurveys. timationofblackholemasses. Bothcanbeeasilyderivedfrom
ThoughoutthispaperweassumeaHubbleconstantofH0 = thefit.Wethencorrectedthelinewidthsforthefiniteresolution
70kms−1Mpc−1andcosmologicaldensityparametersΩm =0.3 ofthespectrograph.Wemeasuredthecontinuumfluxat5100Å
andΩΛ =0.7. from the continuumfit to the Hβ line region. We correctedthe
fluxforGalacticextinction,usingthedustmapsofSchlegeletal.
(1998),andtheextinctionlawofCardellietal.(1989)andcom-
2. Thesample
putedthecontinuumluminosityλLλ(5100Å),hereafterL5100.
The sample of low redshift AGN used in this study is drawn For the estimation of errors we constructed artificial spec-
fromtheQSOcatalogueoftheHamburg/ESOSurvey(Wisotzki traforeachobject,usingthefittedmodelandGaussianrandom
et al. 2000). For a more detailed description of the survey and noise,correspondingtothemeasuredS/N.Weused500realiza-
thesampleused,seeourcompanionpaper(Schulzeetal.2009, tionsforeachspectrum.Wefittedtheseartificialspectra,fitting
hereafterPaperI).Hereweonlygiveashortsummary. the line and the continuum and measured the FWHM, the line
The HES is a wide-angle, slitless spectroscopy survey, dispersion and the line flux. The error was then simply taken
mainlyforbrightQSOs,carriedoutinthesouthernhemisphere, asthedispersionbetweenthevariousrealizations.Thismethod
utilising photographic objective prism plates. The HES covers providesaformalerror,takingintoaccountfittinguncertainties
a formalarea of ∼9500deg2 in the sky. After digitisation, slit- causedbythenoise.Therebyweassumethatourmulti-Gaussian
less spectra in the range 3200Å <∼ λ <∼ 5200Å have been ex- fittingmodelprovidesasufficientlyprecisemodelofthetrueline
tracted from the plates. From these spectra type 1 AGN have shape.Intrinsicdeviationsofthelineshapefromthemodelwill
Page2of19
A.SchulzeandL.Wisotzki:Thelocalactiveblackholemassfunction
Fig.1. Leftpanel: Correlationbetweenσ (Hβ)andσ (Hα).Rightpanel: CorrelationbetweenFWHM(Hβ)and FWHM(Hα).Thesolidline
line line
showstheregressionresultusingtheFITEXYmethodandthedashedlineisaone-to-onecorrespondence.
increase the error. A remaining Feii contribution at Hβ might followingTremaineetal.(2002)byincreasingtheuncertainties
alsoincreasetheerror. untilaχ2perdegreeoffreedomofunitywasobtained.
Forasubsampleof21AGNalsoincludedintheSDSSData Wefoundthefollowingrelationsforthelinewidths:
Release5(DR5; Adelman-McCarthyetal.2007),wecompared
ourresultsto thehigherresolutionSDSSspectra.We fittedthe logσHα =0.96logσHβ−0.08, (1)
SDSSspectrainthesamemannerastheHESspectra.Thecor-
relationforσline istight(wefoundascatterof0.07dexforHβ), logFWHMHα =1.10logFWHMHβ−0.17. (2)
whereasthescatterinthemeasurementoftheFWHMissignif-
icantlylarger(0.18dexforHβ).Thisisatleastpartiallycaused The rms scatter around the best fits are 0.11 dex for σ and
line
bythenarrowcomponentthatcanbedisentangledbetterinthe 0.16dexfortheFWHMrespectively.TherelationsbetweenHα
SDSSspectra.Incontrast,σline islesssusceptibletothenarrow andHβlinepropertiesusingtheFITEXYmethodareshownin
line contribution and thus provides a more precise width mea- Fig.1.
surementforoursample.Wealsoseeevidenceforansmallun- TherelationobtainedfortheFWHM slightlydeviatesfrom
derestimationofthelinewidthcomparedtotheSDSSspectrum, the relations obtained by Greene & Ho (2005) and Shen et al.
especiallyfornarrowerlines,withameandeviationof0.03dex. (2008a),showingastrongerdeviationfromaone-to-onecorre-
ThismightbecausedbythelowerresolutionoftheHESspectra lation.Thismightbeduetothelowerresolutionofourdata,thus
comparedtotheSDSS spectraandthereforethestrongerinflu- theresolutioncorrectionhasastrongereffectonthelinewidth.
enceoftheresolutioncorrection. This is supportedby the slightly larger scatter for our relation.
A comparison of the continuum luminosity, FWHM and Thescatterintherelationbetweenthelinedispersionsislower
black hole mass with the quasar sample of Shen et al. (2008b) thanbetweentheFWHMs,againfavouringσ overFWHMfor
line
is in general agreement with our values for the few objects in ourdata.
common. The Hβ lines are on average broader than Hα with
Theline dispersionismoresensitiveto thewingsofa line, (cid:6)FWHM(Hβ)/FWHM(Hα)(cid:7) = 1.54 and (cid:6)σ (Hβ)/σ (Hα)(cid:7)
thus to the subtractionof the contaminatinglines, i.e. Feii and = 1.29 respectively.This is larger than founldinein otherlisnaemples
[Oiii] for Hβ and [Nii] and [Sii] for Hα. On the other hand, (e.g.Osterbrock&Shuder1982;Greene&Ho2005)butingen-
theFWHMismoresusceptibletothelinecore,thustoaproper eral agreement with the physical expectation of an increasing
subtractionofthenarrowcomponent(see Denneyetal.2009). density orionisationparameterof the BLRwith decreasingra-
For our data the latter seems to exhibit the larger uncertainty. dius.
Togetherwiththeindicationthatσ isapreferablewidthesti-
line
mateovertheFWHM(Petersonetal.2004;Collinetal.2006),
wedecidedtouseσ toestimateblackholemasses,andonly 4. Results
line
givetheresultsusingtheFWHMforcomparison.
4.1.Estimationofblackholemasses
3.1.RelationsbetweenHβandHαlinewidths WeestimatedblackholemassesfortheAGNusingthecommon
scalingrelationship.Thesampleofquasarsanalysediswellin-
Weseeawell-definedcorrelationbetweenthelinewidths(both sidetherangesinredshift,withz < 0.3,andinluminosity,with
FWHM and σ ) of the Hβ and Hα emission lines, as shown 1042 ≤ L ≤ 1046ergs−1,overwhichthescalingrelationship
line 5100
in Fig. 1. To quantify this relation, we applied a linear regres- basedonreverberationmappinghasbeenestablished.Sothees-
sionbetweenHαandHβinlogarithmicunits,usingtheFITEXY timatedblackholemassesdonotsufferfromanextrapolationof
method (Press et al. 1992), that accounts for errors in both thisrelationshipoutsidetherangeforwhichitisobservationally
coordinates. We accounted for intrinsic scatter in the relation tested.
Page3of19
A&A516,A87(2010)
For the scaling relationship between BLR size and contin-
uumluminosityweusethevaluesofBentzetal.(2009):
logR =−21.3+0.519log(L ), (3)
BLR 5100
withL giveninergs−1andR inlightdays.
5100 BLR
Theblackholemassisthuscomputedby
(cid:2) (cid:3) (cid:2) (cid:3)
L 0.52 ΔV 2
MBH =6.7 f 10445e1r0g0s−1 kms−1 M(cid:9) (4)
where f is the scale factor andΔV the line width used, i.e. the
FWHMorσ .Wecomputedblackholemassesbasedonboth
line
widthmeasurements.Weprefertheblackholemassesusingthe
line dispersion and give the FWHM based black hole masses
as a reference. For the line dispersion we used a scale factor
f = 3.85, following Collin et al. (2006). This factor has been
determinedbysettingtheblackholemasses,computedfromthe
mean spectrumofthe reverberationmappingsample andusing
the line dispersion, to the local MBH − σ∗ relation for quies-
centgalaxies,similartotheworkofOnkenetal.(2004).Forthe
FWHMwe usedthe commonscale factor f = 3/4(e.g.Netzer
etal.1990)appropriateforasphericalBLR.
WehavenotcorrectedourcontinuumluminositiesL for
5100 Fig.2. Median composite spectra for 3 luminosity bins, showing the
their host galaxy contribution. This might lead to an overesti- ii
Ca Klineregion.Theuppercompositeshowsthehighluminositybin
mationof MBHforlowerluminosityAGN, wherethe hostcon- (logL5100 > 44.5),themiddlecompositeisforthemediumluminosity
tribution becomes significant. To disentangle the host contri- bin(43.6<logL <44.5)andthelowercompositespectrumshows
5100
bution high resolution HST imaging is required, which is not thelowluminositybin(logL <43.6).TheCaiiKlineat3934Åis
5100
available forour sample. However,the narrow slit used for the indicatedasthedashedline.
spectroscopy and the AGN selection technique already reduce
theexpectedhostcontribution.Thus,thebiasintroducedbyhost
galaxycontaminationisexpectedtobesmall,butwillleadtoa contribution to L of 35−40%. This would reduce our black
5100
systematiceffect. hole mass estimate by 0.10−0.12 dex. The upper limit we can
To estimate the degree of galaxy contribution to our AGN putonthehostcontributionis∼50%,implying0.16dexforthe
ii
spectra we used the equivalent width (EW) of the Ca K line M estimation. The medium luminosity subsample shows an
BH
at3934Å,becausethisistheonlyprominentgalaxyabsorption averagehostcontributionof15−20%,correspondingtoanover-
featurenotconfusedbyAGNemissionfeatureswithinourspec- estimationofM by0.04−0.05dex.
BH
tral range. As this feature is only prominent in evolved stellar Weusedtheseestimatestoapplyaveragehostcorrectionsto
populations,thecontributionfroma veryyoungstellar popula- the continuum luminosities and thus to the black hole masses.
tion might be neglected. However, low luminosity AGN hosts Althoughthesecorrectionsmightbewronginindividualcases,
areknowntohavenotparticularlybluecoloursandgenerallydo for the sample as a whole the host contribution is thereby ac-
notshowextremelyyoungstellarpopulations,butratherindica- countedforasgoodas possibleforthesedata.We verifiedthat
tionsforpoststarbursts(VandenBerk etal. 2006;Daviesetal. ourresultsarenotqualitativelyaffectedbyapplyingorneglect-
2007). ingthiscorrection.Thequantitativechangeintheresultsiscer-
Since the mean signal-to-noise in our spectra is not suffi- tainlyverysmall.
ii
cient to accuratelymeasure the Ca EW in individualspectra, ThedistributionsofblackholemassesusingtheFWHMand
weconstructedcompositespectraforthreeluminositybins,de- thelinedispersionareshowninFig.3.TheusageoftheFWHM
pendingon L5100,showninFig.2.WhilenoCaiiabsorptionis instead of σline slightly shifts the distribution to lower values,
detectedforthehighestluminositycomposite(logL5100 >44.5), with (cid:6)logMBH(cid:7) decreasing from 7.90 to 7.77, and also broad-
itisclearlypresentinthelowerluminositycomposites.Wemea- ens the distribution,with the standarddeviationchangingfrom
sureEWsof0.8Å inthemediumluminosity(43.6<logL < 0.65dexforσ to0.70dexforFWHM.
5100 line
44.5)andof2.0Å inthelowluminosity(logL < 43.6)me- Inthefollowingwewillonlyrefertotheblackholemasses
5100
diancompositespectrum,respectively. using σ . This widthestimate providesa morereliable width
line
Toestimatethecorrespondinggalaxycontribution,weused measurementforourdatacomparedtotheFWHM,asdiscussed
modelspectrafromsinglestellarpopulationmodelswithdiffer- inSection3.Wehaveverifiedthatourresultsarefullyconsistent
ent ages and metallicities (Bruzual & Charlot 2003). The low whenusingtheFWHMinstead.
luminosity AGN, which will show the strongest host contribu-
tion,arepreferentiallyspiralgalaxies.Wemodeledthembystel-
4.2.Eddingtonratios
lar populations with ages between 900 Myr and 2.5 Gyr. We
added various constant AGN contributions to the spectra and To compute the Eddington ratio λ = L /L , which can
measuredtheresultingEWsoftheAGN+galaxyspectra.Tode- be understood as a normalised accretionborlate,Edwde estimated
rivethegalaxycontributionat5100Å weassumeda fluxratio thebolometricluminosityfromtheopticalcontinuumluminos-
of the AGN of f /f = 0.64. An EW of 2.0 Å, as mea- ity L = f L , applying a bolometric correction factor of
5100 3934 bol L 5100
sured for our low luminosity subsample, correspondsto a host f =9,asproposedbyKaspietal.(2000).Themeanbolometric
L
Page4of19
A.SchulzeandL.Wisotzki:Thelocalactiveblackholemassfunction
in agreementwith our low redshift result. We will discuss this
issuefurtherinSect.6.
InFig.4weplotblackholemass,Eddingtonratioandbolo-
metricluminosityagainsteachother.Thefirstthingwehaveto
beawareofwheninterpretingtheseplotsaretheimplicitunder-
lyingcorrelationsbetweenthesequantities.Whatweeffectively
alwaysshowisacombinationofcontinuumluminosityL and
5100
line width σ . Their underlying relation is shown in Fig. 5.
line
ThereisonlysomeweekcorrelationpresentbetweenL and
5100
linewidth.
Physically these plots can be understoodfromthe shape of
the underlying black hole mass function and distribution func-
tionofEddingtonratiosincombinationwiththeselectionfunc-
tion of the survey, as we will explicitly show in Sect. 6. The
Eddingtonratioλspanstherange0.01<λ <1,similartoother
opticalstudies(Woo&Urry2002;Kollmeieretal.2006;Greene
Fig.3. Left panel: Distribution of black hole masses. The black solid & Ho 2007; Shen et al. 2008b). At high values, observations
histogramshowsthedistributionofblackholemassesestimatedfrom have shown that the Eddington rate represents an approximate
σline,thereddashedhistogramshowsMBHestimatedusingtheFWHM upper boundaryto the Eddingtonratio distribution, implying a
(andaconstantscalefactorof f = 0.75).Rightpanel:Distributionof
steep decrease of the Eddington ratio distribution function to-
Eddingtonratios.Thehistogramsarethesameasintheleftpanel. wardSuper-Eddingtonvalues.Atlowλ thesamplesuffersfrom
incompleteness due to the selection effects of the survey. This
canexplaintheobservedrangeofEddingtonratiosandtherough
correlation between M and L , shown in the left panel of
correction factor is still somewhat uncertain, ranging from 7 BH bol
Fig.4.
(e.g. Netzer & Trakhtenbrot 2007) to values around 10 (e.g.
Richards et al. 2006a; Marconi et al. 2004), and also seems to No strong correlation is seen between λand L for this
bol
bedependentonluminosity(Marconietal.2004;Hopkinsetal. low redshift sample. There is a lack of objects in the lower
2007), black hole mass (Kelly et al. 2008) or Eddington ratio right corner of the middle panel of Fig. 4, thus a lack of ob-
(Vasudevan & Fabian 2009; Lusso et al. 2010). However, as- jectswithlow-λandhighluminosity.Theseobjectswouldhave
suming a constant value fL at 5100 Å is a good approxima- MBH > 2 × 109 M(cid:9) and are rare objects due to the steep de-
tion.Thevalueof f = 9isalso in generalagreementwiththe creaseoftheblackholemassfunctionatthehighmassend(see
L
value obtainedby integratingoverthe mean SED presentedby Sects.5.1and6.2).Thusitisnotsurprisingtoseealackofthese
Richards et al. (2006a). The Eddington luminosity is given by objects in the sample. The same applies to the lack of objects
LEdd (cid:2)1.3×1038(M/M(cid:9)) ergs−1. seenintheupperrightcorneroftherightpanelofFig.4.These
The distribution of Eddingtonratios, using the FWHM and wouldbeobjectswithrelativehighMBHandhighEddingtonra-
σ ,areshownintherightpanelofFig.3.ThemeanEddington tio.Thislackisalsocausedbytherarityoftheseobjects,dueto
line
ratio of thissample is(cid:6)logλ(cid:7) = −0.92with standarddeviation the steep decreaseofthe blackhole massfunctionincombina-
of0.46dexusingσ ,and(cid:6)logλ(cid:7)=−0.79with0.56dexdevi- tion with the decrease of the Eddingtonratio distribution func-
line
ationforFWHM. tiontowardtheEddingtonrate.Therefore,inthelocaluniverse
massiveblackholes,accretingclosetotheEddingtonlimit,are
Thisdispersionishigherthanthatfoundbyotherauthorsin
exceedinglyrare.
higherredshiftandhigherluminositysamples(Kollmeieretal.
2006;Shenetal.2008b).Indeed,theshapeoftheobserveddis- In the right panel of Fig. 4, there is an absence of objects
tribution does depend on the underlying distribution function in the lower left corner, i.e. objects with low black hole mass
andtheselectionfunctionofthesurvey.Thustheobserveddistri- andlowEddingtonratio.Theseobjectsarevictimsofthesurvey
butionofEddingtonratiosisaffectedbythefluxlimitationofthe selection.Theywouldhavelowluminositiesandthereforeonly
surveyandisnotaquantityindependentof thespecific survey. theclosestwouldbedetectableinafluxlimitedsample.Anad-
TheEddingtonratiodistributionwillchangeinmeananddisper- ditional effect is that the AGN selection in the HES inevitably
sion with luminosity (Babic´ et al. 2007; Hopkins & Hernquist becomesincompleteat MBJ (cid:4) −19,becausethecontributionof
2009) due to this selection effect. Usually it will broaden with thehostgalaxylighteventotheHESnuclearextractionscheme
decreasingtypicalluminosity. willbecomesubstantial,andtheobjectwillnolongerbedistin-
ThistrendisalsoclearlyvisibleinthesampleofSDSSAGN guishedfromanormalgalaxy,duetoamoregalaxylikeSEDor
presentedbyShenetal.(2008b).Aredshiftdependenceisalso duetoanolongerdetectablebroademissionline.Therefore,no
srinaimdnigicleaart0e.td1o(cid:3)btyhzet(cid:3)hsea4im.r5p,dltaehtaeo.yfFfKoouornlltdmheaeiirteyrwpiehctoallaeld.iss(ap2me0rp0sl6ieo),ntchooafvt∼ecr0oi.nv3gedrtsehxae, s−Ae1Gle8Nct(cid:4)iwoniMtheBffJMe(cid:4)cBtJ.−(cid:4)N1o9−te1is8thaaalrrteelaidndeeytsescoetferiedoquiunsaltlyhleuaffmseuicnrvtoeesdyit,ibeaysndtihntihtsheseurrarinvggehyet
similarredshiftrangeandincludesrelativelyhighluminosityob- panel of Fig. 4 are diagonals from the upper left to the lower
jects. Restricting the sample of Shen et al. (2008b) to z ≤ 0.3 right.Thisselectioneffectexplainstheabsenceofobservedob-
givesadeviationof0.43dex,similartoourresults,butalower jects in this region and results in the apparent anti-correlation
mean Eddingtonratio of −1.17in logarithmicunits. Thistrend betweenEddingtonratioandblackholemass,alsoseeninother
isalsopresentindeepersurveysthatcoverawideredshiftrange. samples (e.g. McLure & Dunlop 2004; Netzer & Trakhtenbrot
In the VVDS a value for the dispersion of ∼0.33 dex has been 2007).
found(Gavignaudetal.2008),whileintheCOSMOSsurveya In Sect. 6 we will explicitly show by Monte Carlo simu-
dispersion of ∼0.4 dex has been observed(Trump et al. 2009), lations how the observed distributions arise from an assumed
Page5of19
A&A516,A87(2010)
Fig.4. Left panel: Black hole mass versus bolometric luminosity. Middle panel: Eddington ratio versus bolometric luminosity. Right panel:
Eddingtonratioversusblackholemass.ThethreelinesindicateEddingtonratiosof1(solid),0.1(dashed)and0.01(dotted).
ratioforopticaltype1AGNsampleswouldbeλ (cid:4)0.01,asthis
isapproximatelytheobservedlowervalue.
Bythisdefinition,oursamplesuffersfromincompletenessat
lowblackholemasses,becausesomelowmassandlowλ AGN
willbefainterthantheflux-limit.Thesampleisnotselectedon
blackholemassorEddingtonratiobutonAGNflux.Asalready
mentioned,thesamplebecomesincompleteatM (cid:4)−19.Thus,
BJ
atlowblackholemassonlytheAGNabovethisluminositylimit
willbedetected.Thisintroducesaselectioneffectontheblack
holemassdistributionthatneedstobetakenintoaccountforthe
determination of the BHMF. In the following, we will refer to
thisselectioneffectontheblackholemassandtheEddingtonra-
tiodistributionassamplecensorship,todistinguishitfrommore
direct, for example redshift dependent, selection effects on the
AGNluminositydistribution.
Itisinprinciplepossibletocorrectforthissamplecensorship
byproperuseoftherespectiveselectionfunction.Ifapplyingthe
usualselectionfunction,whichisa functionof luminosity,and
is appropriateforthe determinationof the luminosityfunction,
to the determination of the black hole mass function, incom-
Fig.5. Distributionof theHβ linewidth(σ ) withcontinuum lumi- pleteness is introduced because it has not properly accounted
line for active black holes below the flux limit (Kelly et al. 2009).
nosityL .Thereisonlylittlecorrelationseenbetweenlinewidthand
5100
L . Instead, the selection function has to be derived as a function
5100
ofblackholemassandthisselectionfunctionhastobeapplied
to the constructionof the BHMF. However,to do so would re-
underlyingBHMFandEddingtonratiodistributionfunctionun- quireknowledgeofthe, a prioriunknown,Eddingtonratiodis-
derconsiderationofthesurveyselectioncriteria. tributionfunction.Thusthisapproachisnotfeasiblewithoutad-
ditional assumptions. Nevertheless, it can be useful as a con-
sistency check, as we will show in Sect. 6.3. To avoid such
5. BlackholemassfunctionandEddingtonratio additional assumptions, we used a different approach to deter-
distributionfunction mine the intrinsic underlyingactive BHMF fromour data, tak-
ing into account the effect of sample censorship. These results
5.1.Thelocalactiveblackholemassfunction
arepresentedinSect.6.
The BHMF of quiescent galaxies in the local universe can be However,inthissectionwefirstdeterminetheactiveBHMF,
estimated, based on the relation between M and bulge prop- ignoring the effect of sample censorship on the data. We con-
BH
erties (e.g. Salucci et al. 1999; Yu & Tremaine 2002; Shankar structtheBHMFusingtheusualselectionfunctionalsousedfor
etal.2004;Marconietal.2004).Onlyasmallfractionoflocal the determinationoftheAGNluminosityfunction.However,it
black holes are currently in an active state, accreting at a sig- must be kept in mind that in this case we ignore active black
nificant level and appearearingas an AGN. However, AGN do holeswithluminositiesbelowthe fluxlimitofthesurvey,even
notaccreteatasinglevalueofλ,butrathershowawidedistri- if their Eddington ratio is high enough to call it active by the
bution of Eddingtonratios(e.g.Heckmanet al. 2004;Yu et al. abovedefinition.Thus,thisdeterminedBHMFsuffersfromin-
2005;Merloni&Heinz2008;Ho2009;Kauffmann&Heckman completeness at low mass caused by the sample censorship.
2009).Thereforeitisnotobviouswhatexactlytocallanactive Nevertheless, this exercise is worthwhile, because it does not
blackhole.Apragmaticdefinitionistousealowerlimitforthe require any assumptions on the shape of the mass function or
Eddington ratio. A natural choice for such a lower Eddington anyinformationabouttheEddingtonratiodistributionfunction.
Page6of19
A.SchulzeandL.Wisotzki:Thelocalactiveblackholemassfunction
Table1.Fittingresultsforthelocalactiveblackholemassfunction,correctedforevolutionbutnotforsamplecensorship.
Function φ∗•inMpc−3 logM∗ α β χ2 χ2/d.o.f.
DPL 2.86×10−6 7.86 −0.74 −3.11 7.29 0.81
Schechter 2.73×10−6 8.06 −0.84 – 15.47 1.55
mod.Schechter 4.96×10−6 6.97 −0.25 0.51 13.64 1.52
DPL(FWHM) 9.95×10−7 8.21 −1.16 −3.62 11.66 1.30
Schechter(FWHM) 1.37×10−6 8.19 −1.08 – 16.40 1.64
mod.Schechter(FWHM) 5.15×10−6 6.81 −0.59 0.32 15.33 1.70
Whilethelowmassendclearlywillbeaffectedbysamplecen-
sorship, the high mass end is already well determined by this
approach,providingimportantinformationon this mass range.
Also,thisuncorrectedBHMFcanbebettercomparedwithpre-
vious estimates on the BHMF that usually have not properly
accountedforthesamplecensorship.
We constructed this active BHMF, not corrected for sam-
ple censorship, in an equivalent manner as for the determina-
tionofaluminosityfunction(seePaperI).Wemadeuseofthe
classical1/V estimator(Schmidt1968)toconstructabinned
max
BHMF. ThedifferentialBHMF (spacedensityperlog M )is
BH
thusgivenby:
(cid:4)
1 1
Φ(M•)= ΔlogM• k Vmkax, (5)
where V is the maximalaccessible volumein whichthe ob-
max
ject, with given magnitude could have been found, given the Fig.6. Thedifferential activeblackhole massfunctionfor z = 0,not
flux-limitofthesurveyandtheredshiftbinused.TheAGNsam- correctedforsamplecensorship.FilledblacksymbolsshowtheBHMF
pleusedhasbeenselectedbasedonUVexcessmeasuredinthe using thelinedispersion toestimatetheblack holemass. Thedashed
lineshowsthedoublepowerlawfittotheBHMF,thedottedlinegives
slitless spectraandnoselectionbasedonthepresenceofemis-
theSchechterfunctionfitandthedasheddottedlinerepresentsthefit
sionlinesisapplied.Thus,theV valuesusedareequaltothe
max usingamodifiedSchechterfunction.
onesusedforthedeterminationoftheAGNluminosityfunction,
presentedinPaperI.Welackusablespectrafor5objects,andfor
an additional7 objectswe couldnotfit Hβ due to poorquality where M∗ is a characteristic break mass, φ∗ the normalisation
ofthespectrainthisregionand/orduetoalowHβcontribution. and α and β are the two slopes. A Schechter (1976) function,
Therefore we could not estimate M for 12 objects. We took givenby:
BH
thisintoaccountinthesurveyselectionfunctionbymultiplying (cid:2) (cid:3) (cid:2) (cid:3)
the effective area by a factor of 317/329.The exclusion of the φ(M•)= φ∗ M• αexp −M• · (7)
7 objectswithoutproperHβmeasurementmaypotentiallybias M∗ M∗ M∗
our results. Thereforewe estimated the Hβ width from the Hα
isalsoused.
measurement, using Eq. (1), and then estimated M for these
BH We additionally used a functional form, motivated by the
7objects.IncludingtheseobjectsresultsinaconsistentBHMF.
quiescent BHMF. The quiescent BHMF is given as a convolu-
Thus,thein-orexclusionoftheseobjectsmakesnodifference.
tionofaSchechterfunctionwithaGaussianandcanbeparam-
Toderivethelocal(z=0)BHMFwecorrectedforevolution
eterisedbythefollowingfunction(e.g.Aller&Richstone2002;
within our narrow redshift bin, 0 < z < 0.3, as described in
Shankaretal.2004):
PaperI.Weappliedasimplepuredensityevolutionmodelwithin (cid:2) (cid:3) ⎛ (cid:8) (cid:9) ⎞
tohuerrBedHsMhiFfttboinr,eid.seh.iρf(tzz)e=ro.(1T+hisz)skDpewciifithckvDal=ue5e,nthsuurseasdajursetsiunlgt φ(M•)= φ∗ M• αexp⎜⎜⎜⎜⎝− M• β⎟⎟⎟⎟⎠ . (8)
M∗ M∗ M∗
of the V/V testconsistentwith(cid:6)V/V (cid:7) = 0.5,aswouldbe
max max
expectedinthecaseofnoevolution. This basically corresponds to an ad-hoc modification of the
The differential active BHMF of the HES is computed for Schechter function with an extra parameter β. A value β > 1
binsofΔlogM = 0.25dexintherange106 ≤ M ≤ 109.5. correspondsto a decrease stronger than exponentialand β < 1
TheresultingdiBffHerentiallocalBHMF,notcorrectedBfHorsample correspondstoamilderthanexponentialdecrease.Forβ=1this
functionturnsintotheusualSchechterfunction.Inthefollowing
censorship,isshowninFig.6.
werefertoitasthemodifiedSchechterfunction.
We usedthefollowingfunctionalformstofittheBHMF. A
TheseBHMFsareconnectedtotheexpressioninlogarithmic
doublepowerlaw,givenby: units by Φ(M•) = (M•/log10e)φ(M•). The resulting fitting pa-
rametersofthesethreefunctionstoourbinnedBHMFarelisted
φ(M•)= (M•/M∗)−φα∗/+M(∗M•/M∗)−β, (6) tiinonTapbelrefo1r.mAslplogoirvleyaatcctheepthaibglheefisttsb,lawchkihleoltehemSascsheesc.hHteorwfeuvnecr-,
Page7of19
A&A516,A87(2010)
Table 2. Binned black hole mass function, not corrected for sample
censorship.
σ FWHM
line
logM N logΦ N logΦ
BH M M
5.75 1 −6.27+0.31 4 −5.75+0.20
−∞ −0.36
6.00 1 −6.24+0.30 0
−∞
6.25 4 −5.34+0.21 6 −5.33+0.20
−0.40 −0.41
6.50 5 −5.55+0.20 8 −5.49+0.19
−0.40 −0.33
6.75 6 −5.50+0.15 11 −5.29+0.15
−0.24 −0.22
7.00 20 −5.45+0.12 28 −5.34+0.11
−0.18 −0.15
7.25 27 −5.07+0.12 35 −5.15+0.12
−0.17 −0.16
7.50 38 −5.38+0.10 42 −5.65+0.09
−0.12 −0.12
7.75 47 −5.32+0.13 33 −5.45+0.15
−0.18 −0.22
8.00 42 −5.64+0.10 45 −5.63+0.13
−0.13 −0.19
8.25 42 −5.85+0.17 36 −5.78+0.15
−0.28 −0.24
8.50 45 −6.35+0.16 32 −6.37+0.16 Fig.7.ThedifferentialEddingtonratiodistributionfunctionforz = 0,
−0.25 −0.27
8.75 21 −7.21+0.13 23 −7.20+0.13 not corrected for sample censorship. The dashed line shows the best
−0.19 −0.18 Schechterfunctionfit.
9.00 13 −7.43+0.13 8 −7.62+0.17
−0.20 −0.29
9.25 5 −7.91+0.17 6 −7.83+0.16
−0.29 −0.25 6. ReconstructionoftheintrinsicBHMFandERDF
Notes.Ngivesthenumberofobjectsineachbin,logφandΔlogφgives 6.1.Method
thespacedensityperunitlogarithmicblackholemassinsolarmasses
andits1σerrorrespectively. Asalreadynoted,theBHMFpresentedsofarisbasicallylumi-
nosity limited and thus incomplete at low mass in terms of an
accretion rate limited active BHMF. We now want to constrain
the intrinsic active BHMF by our observations, correcting for
theBHMFislesswellconstrainedathighM duetothesmall thissamplecensorship.Weuselogλ =−2asthelowerlimitfor
BH
numberofobjectsinthesebins. theEddingtonratio;forlogλ≥−2wecallblackholes“active”.
TheshapeoftheBHMFisdescribedbyasteepdecreaseof The selection function of the survey is a function of lumi-
thespacedensitytowardshigherMBHwithβ≈−3inthedouble nosity,andthusoftheproductof MBHandλ.Therefore,there-
powerlaw,andasignificantflatteningatMBH ≈108 M(cid:9)toward constructionoftheactiveBHMFalsorequirestheknowledgeof
lowerM .Thehighmassregimeisnotaffectedbythealready theERDF.Bothdistributionfunctionscannotbedeterminedin-
BH
mentioned sample censorship, while the low mass flattening is dependentlyfromeachother.InSect.6.3,asaconsistencytest,
partiallycausedbythesystematicunderrepresentationoflowλ wewilldeterminetheactiveBHMFassumingaspecificERDF.
objectsatlowmass. Butwithoutsuchanassumptionbothdistributionfunctionshave
tobedeterminedatthesametime.Thisistheapproachwewill
followinthissection.
5.2.ThelocalEddingtonratiodistributionfunction Knowing both distribution functions, the AGN luminosity
functionisdirectlygivenasaconvolutionofthetwo:
Given the estimates of the Eddington ratio λ for our sample, (cid:13)
∞
we can analogously determine the local Eddingtonratio distri-
bution function(ERDF) for the HES, equivalentto the BHMF. Φ(L)= Pλ(λ)Φ•(M•)dlogM• , (9)
This determination also does not take into account the effect M•,min
of sample censorship.We computedthe local ERDF in bins of where we adopt logM•,min = 6. With Pλ we define the nor-
Δlogλ = 0.25 dex in the range −2.25 ≤ logλ ≤ 0.25. The malisedERDF,thus:
resuTlthinegudnicffoerrreenctteiadlAloGcaNlEspRaDceFdiesnsshiotywdneicnliFneigs.a7t.highaswell Pλ(λ)= (cid:14) Φλ(λ) · (10)
asatlowλ,showingapeakaroundlogλ ≈ −1.0.Wefittedthe Φλ(λ)dlogλ
ERDF by a Schechter function, neglecting the lowest λ point. We determined the BHMF and ERDF together, performing a
The resulting best fit values are φλ∗ = 6.66 × 10−6 Mpc−3, maximum likelihood fit to the data (e.g. Marshall et al. 1983).
logλ∗ =−1.01andαλ =−0.05withavalueofχ2perdegreeof We consider the joint Poisson probability distribution of black
freedomof1.9. hole mass and Eddingtonratio. We minimise the functionS =
However,alsothisERDFisstronglyaffectedbysamplecen- −2lnL,withLbeingthelikelihoodoffindingtheobserveddata,
sorship. While at the highest Eddington ratios (logλ > −1) giventherespectivemodel.Thus,S isgivenby:
the majority of AGN will be detected by the survey, at low (cid:2)
Eddingtonratio(logλ <−1)asignificantnumberofobjectswill (cid:4)N
have a too low luminosity to be detected. Therefore the space S =−2 lnp(M•,i,λi)+2 p(M•,λ)dlogλdlogM• .(11)
density at low λ will be underestimated by the derived ERDF. i=1
In the next section we will reconstructthe intrinsic underlying The sum is over the observed objects and the integral is equal
ERDFaswellastheintrinsicBHMF. to the expected number of objects, given the assumed BHMF
Page8of19
A.SchulzeandL.Wisotzki:Thelocalactiveblackholemassfunction
andERDF.Theprobabilitydistributionp(M•,λ)dlogλdlogM• We decided to model the ERDF by a Schechter function,
givestheprobabilityoffindinganAGNwithblackholemassbe- corresponding to an exponential cutoff close to the Eddington
tweenlogM•andlogM•+dlogM•andEddingtonratiobetween limit and a wide power law-like distribution at low Eddington
logλ andlogλ +dlogλ inanobservedsample.Thetotalnum- ratio.Thisparameterisationdiffersfromtheoftenassumedlog-
ber of objects N is then given by integration of p(M•,λ) over normaldistribution.However,a log-normaldistributionis only
M• andλ. motivated by the observed distribution, not accounting for any
We will now briefly motivate the used probability distribu- selectioneffects.Also,alog-normaldistributionenforcesamax-
tion p(M•,λ) for our sample. The observed numberof objects imum and a turnover at low λ. A Schechter function is more
inasampleisgivenby: flexible,allowingforaturnoveratlowvalues,butnotenforcing
(cid:2) it.Inparticular,itallowsanincreaseofthespacedensityatlow
N = Ωeff(m,z)ΦL(logL,z)ddVz dlogLdz, (12) λAG. TNhi(sYsuhaetpealw. 2o0u0ld5;bHeocpoknisnisste&ntHweirtnhqoubisster2v0a0ti9o;nKsaouffftympaen2n
weffheecrteivΦeLs(ulorvgeLy)aisretaheasAGaNfunlucmtioinnoosiftyapfupnacretinotnmanagdnΩiteuffdeisatnhde &tionHe(Mckemrlaonni2&009H),eiwnzith20e0s8ti)maastewseflolratshewtiothtaml AodGeNl epxoppeuctlaa--
tionsofAGNlightcurvesfromself-regulatedblackholegrowth
redshift,thusthe selection functionforourflux-limitedsurvey.
(Yu & Lu 2008; Hopkins & Hernquist 2009). Aside from the
Thiscan be understoodasa selectionfunctiondependingon z,
HMa•mabnudrgλ/,EtShOusSΩurevffe(my,thze) s=elΩecetffio(Ln,fzu)n=ctiΩoneffw(Mith•i,nλo,uz)r.cFoovrerthede Salclhyetcehstteedrfaulnocgt-inoonrpmaarlaEmRetDerFisaastifounncotfiotnhaelEfoRrDmF.,Twogeeathdedritwiointh-
theSchechterfunctionitcoversawiderangeofpossibleparam-
redshift range is almost independent of redshift. Thus, as dis-
eterisationsfortheERDF.
cussed in PaperI, we canmarginaliseoverredshift.Fordetails
Fromourdatawearenotabletoconstrainadependenceof
ontheselectionfunctionoftheHESseeWisotzkietal.(2000).
the ERDF on M , so we assumed the ERDF to be indepen-
Apart from the flux limit, our sample is incomplete at the BH
lowest luminosities M > −19. For low luminosity AGN dent of MBH, already implicitly assumed in Eq. (9). The nor-
BJ malisationoftheERDFisfixedbytheconditionthattheBHMF
the host galaxy contribution becomes an important factor and
andERDFhavetopredictthesamespacedensityofAGN.This
the objects might no longer be classified as an AGN, due to
theSEDbeingdominatedbystarlight.AsshowninPaperI,the leaves two freeparametersforthe ERDF, the breakλ∗ andthe
sample is highly complete brighterthan M ≈ −19. Thus, we low-λ slopeαλ fortheSchechterfunction,orthemeanλ∗ and
adoptedaluminositylimitof M < −19inBJtheselectionfunc- thewidthσλ forthelog-normaldistribution.However,thesetwo
tion, Ωeff. We also restricted theBoJbservedsample to this lower pbaercaamuseetetrhseidnabtaobthyccaosnesstraurcetinoontnieneddespteondbeenctofnrsoimsteenatcwhitohthtehre,
luminosityforthecomparisonofthesampleproperties.
The AGN luminosity function Φ (logL) is related to the observedluminosityfunction(LF).Thus,foragivenBHMFand
BHMFandtheERDFviaEq.(9).ForLtheredshiftevolution,we afixedvalueforλ∗,αλ isgivenbytheconditionthattheLFde-
rivedfromtheBHMFandERDFbyEq.(9)hastobeconsistent
assumed the simple pure density evolution model of Sect. 5.1.
with the observed LF. Our approach automatically ensures the
In this case the black hole mass function is separable into a
function of M• and a function of z, Φ(M•,z) = Φ(M•)ρ(z) = consistencyoftheBHMFandtheERDFwiththeobservedLF.
Φ(M•)(1+z)kD,withkD =5. usedTotwaossdeisffsetrheentgmooedthnoedsss.oTfhfiistifsorretqhueirienddibveidcuaualsemtohdeemlsawxie-
The expected number of objects for a given survey and an
mumlikelihoodmethoddoesnotprovideitsownassessmentof
assumedBHMFandERDFisthengivenby:
(cid:3) the goodness of fit. First, we used a two-dimensional K-S test
dV (Fasano & Franceschini 1987) on the unbinned data. Second,
N = ΩeffPλ(λ)Φ•(M•)(1+z)kD dzdlogλdlogM•dz.(13) we employed a χ2 test, binning the data in bins of 0.5 dex in
M and λ respectively. The results are given with the best fit
BH
Thus, the bivariate probability distribution of black hole mass parametersinTable3.
andEddingtonratioisgivenby:
(cid:13)
p(M•,λ)= ΩeffPλ(λ)Φ•(M•)(1+z)kDddVzdz. (14) 6.2.Results
ThefirstmodelconsistsofadoublepowerlawBHMF,withthe
Given this bivariate distribution for an assumed BHMF and high mass slope fixed to β = −3.11, and a Schechter function
ERDF, we minimise the likelihood function S (Eq. (11)) us- ERDF. The best fit distribution functions are shown as black
ing a downhill simplex algorithm (Nelder & Mead 1965). As dashedlineinFig.8andtheirfitparametersaregiveninTable3.
a lower limit for the fitting we employed a black hole mass of TheBHMFshowsasteep highmassslopewithα ≈ −2and
BH
Mmin = 106 M(cid:9) and an Eddington ratio of λmin = 10−2. The thebreakisconsistentwiththeuncorrectedBHMF.TheERDFis
HESsamplewasrestrictedtotheselimitsaccordingly. increasingtowardslowλ downtotheappliedlimitofλ =0.01.
For the BHMF we assumed three different models. Firstly Thisfunctionprovidesagoodfittothehighmassendofthe
we used a double power law with the high mass slope fixed to uncorrectedBHMF,whichisonlylittleaffectedbysamplecen-
thevalueβ =−3.01,determinedfromtheuncorrectedBHMF sorship. At the low mass end the uncorrected BHMF strongly
BH
inSect.5.1.Thislowerstherequirednumberoffreeparameters underpredictsthe active black hole space density, compared to
andisjustified,becausethehighmassregionintheuncorrected thereconstructedunderlyingactiveBHMF.Thisalsoholdstrue
BHMFisonlyweaklyaffectedbyincompleteness.Secondlywe for all other applied functional forms for the BHMF and the
also used a double power law, but leaving the high mass slope ERDF.ThesamealsoappliestotheuncorrectedERDF.Theun-
as a free parameter,to be determinedin the fit. As thirdmodel corrected ERDF is strongly biased and underestimates the BH
weusedthefunctiongivenbyEq.(8),thusamodifiedSchechter spacedensity.Thebestfittotheuncorrected BHMFandtothe
function.Thestartingvaluesfortheminimisationalgorithmare ERDF is clearly rejected bythe maximumlikelihoodapproach
takenfromthefittotheuncorrectedBHMF. withhighconfidence.Theyarenotabletoproducetheobserved
Page9of19
A&A516,A87(2010)
Table3.FittingresultsfortheactiveblackholemassfunctionandtheEddingtonratiodistributionfunction.
ρ
act
BHMF ERDF φ∗•[Mpc−3] logM∗ αBH βBH logλ∗ αλ/σλ DKS pKS χ2/d.o.f. pχ2 [M(cid:9)Mpc−3]
DPL(β) S 2.97×10−6 7.97 −2.11 −3.11 −0.57 −1.90 0.100 2.8e−2 61.3/25 4.2e−5 1621
DPL S 2.86×10−6 8.01 −2.10 −3.21 −0.56 −1.94 0.101 2.6e−2 63.4/25 2.1e−5 1687
mS S 2.75×10−6 8.11 −2.11 0.50 −0.55 −1.95 0.094 4.8e−2 56.8/25 1.8e−4 1767
mS ln 2.36×10−6 8.07 −2.12 0.48 −1.83 0.49 0.081 1.2e−1 50.8/25 1.1e−3 1388
Notes.ThefirstcolumnindicatesthefunctionusedfortheBHMF.“DPL”isforadoublepowerlaw,withβindicatingthefixingofthehighmass
slopeand“mS”isforamodifiedSchechterfunction.ThesecondcolumnindicatestheERDF.“S”isforaSchechterfunctionand“ln”standsfor
alog-normaldistribution.
-3 -3
-4 -4
-5 -5
-6 -6
-7 -7
-8 -8
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 -2.0 -1.5 -1.0 -0.5 0.0
Fig.8.ResultsforthereconstructedBHMFandERDF.TheleftpanelgivestheBHMFandtherightpaneltheERDFrespectively.Theblackpoints
showthebinneduncorrecteddistributionfunction,withfilledcirclesrepresentingbinsthatdonotsuffersignificantlyfromsamplecensorshipand
opencirclesrepresentbins,biasedbysamplecensorship. TheyareshownforcomparisonwiththereconstructedBHMFandERDF.Theblack
dashed line shows adouble power law BHMF withfixed high mass slope β = −3.11 and Schechter ERDF,theblue dashed dotted line isfor
afreedouble power lawBHMF andSchechter ERDF,theredsolidlinerepresents amodifiedSchechter BHMFand Schechter ERDFand the
greendashedlineisforamodifiedSchechterBHMFandlog-normalERDF.Thegrayareasshowthe1σconfidenceregionsofbothdistribution
functions,takingintoaccountall4parametricmodels.
distributionsofM andλ andarenotconsistentwiththeAGN For illustration, in Fig. 9 we compare these expected distribu-
BH
LF.Thisclearlyshowsthattheusualapproachusedtoconstruct tionswiththeobservedoneswithinagridoffreeparameterα
BH
anuncorrectedBHMFandERDFisstronglybiased. andαλ.ForatoosteepBHMFthenumberoflowmassobjects
We briefly want to illustrate how the maximum likelihood is largerthan observed,while for a too flat BHMF the number
approach is able to reject certain models for the BHMF and oflowmassobjectsislowerthanobserved.AsteepERDFcor-
ERDFandfavourothers.Tocomputetheexpecteddistributions responds to a break of the ERDF close to the Eddingtonlimit,
within a grid of free parameters, we restricted the number of thus more objects above the Eddingtonlimit and less at low λ
parameterstotwo.We fixedthebreakandnormalisationofthe arepredicted,comparedtotheobservations.ForatooflatERDF,
BHMF. Thus, with the high mass slope already fixed, the only the break needsto be at a low value of λ and thus notenough
freeparameterfortheBHMFisthelowmassslopeα .Forthe objectsclosetotheEddingtonlimitarepredicted.
BH
ERDF there are two free parameters, the break and the low-λ We also carried out Monte Carlo simulations for a grid of
sblyoptheeocfotnhsetrSacinhtetcohtreercofuvnecrttihoen.oHbsoewrveevderA,oGnNe oLfF.thWeseetiosofikxαeλd afrbeoevep,aarasmweetellrsasαfBoHr tahnedbαesλt,fiutsminogdethleofsathmeemaasxsiummupmtiolnikselais-
as a free parameter and determined the break by a χ2 minimi- hoodestimation.Hereweproceededasfollows:FirsteachAGN
sation of the LF computedvia Eq. (9) to the observedLF. The getsassignedaredshift,thenitsblackholemassisdrawnfrom
normalisedobserved distributionoflogM• andlogλ aregiven the assumed BHMF, and finally an Eddington ratio is drawn
by: from the ERDF. From these values absolute and apparent B
(cid:13) J
magnitudesarecomputed,applyingabolometriccorrection.By
1
p(logM•)= N p(M•,λ)dlogλ (15) meansoftheapparentmagnitudeBJ itisdecidediftheobjectis
selectedbythesurveyornot,takingintoaccounttheflux-limit.
(cid:13)
1 WeranMonteCarlosimulationsforawiderangeofα and
p(logλ)= N p(M•,λ)dlogM• . (16) αλ and found results consistent with what we discussedBaHbove
Page10of19