A&A585,A129(2016) Astronomy DOI:10.1051/0004-6361/201527353 & (cid:2)c ESO2016 Astrophysics Pan-STARRS1 variability of XMM-COSMOS AGN II. Physical correlations and power spectrum analysis T.Simm1,M.Salvato1,R.Saglia1,2,G.Ponti1,G.Lanzuisi3,4 B.Trakhtenbrot5,(cid:2),K.Nandra1,andR.Bender1,2 1 Max-PlanckInstituteforExtraterrestrialPhysics,Giessenbachstrasse,Postfach1312,85741Garching,Germany e-mail:[email protected] 2 UniversityObservatoryMunich,Ludwig-MaximiliansUniversitaet,Scheinerstrasse1,81679Munich,Germany 3 INAF–OsservatorioAstronomicodiBologna,viaRanzani1,40127Bologna,Italy 4 DipartimentodiFisicaeAstronomiaUniversitàa˘diBologna,vialeBertiPichat6/2,40127Bologna,Italy 5 DepartmentofPhysics,InstituteofAstronomy,ETHZurich,Wolfgang-Pauli-Strasse27,8093Zürich,Switzerland Received13September2015/Accepted20October2015 ABSTRACT Aims.Thegoalofthisworkistobetterunderstandthecorrelationsbetweentherest-frameUV/opticalvariabilityamplitudeofquasi- stellarobjects(QSOs)andphysicalquantitiessuchasredshift,luminosity,blackholemass,andEddingtonratio.Previousanalyses ofthesametypefoundevidenceforcorrelationsbetweenthevariabilityamplitudeandtheseactivegalacticnucleus(AGN)param- eters.However,mostoftherelationsexhibitconsiderablescatter,andthetrendsobtainedbyvariousauthorsareoftencontradictory. Moreover,theshapeoftheopticalpowerspectraldensity(PSD)iscurrentlyavailableforonlyahandfulofobjects. Methods.Wesearchedforscalingrelationsbetweenthefundamental AGNparametersandrest-frameUV/opticalvariabilityprop- ertiesforasampleof∼90X-rayselectedAGNscoveringawideredshiftrangefromtheXMM-COSMOSsurvey,withopticallight curvesinfourbands(g ,r ,i ,z )providedbythePan-STARRS1(PS1)MediumDeepField04survey.Toestimatethevariability P1 P1 P1 P1 amplitude,weusedthenormalizedexcessvariance(σ2 )andprobedvariabilityonrest-frametimescalesofseveralmonthsandyears rms bycalculatingσ2 fromdifferentpartsofourlightcurves.Inaddition,wederivedtherest-frameopticalPSDforoursourcesusing rms continuous-timeautoregressivemovingaverage(CARMA)models. Results.WeobservethattheexcessvarianceandthePSDamplitudearestronglyanticorrelatedwithwavelength,bolometriclumi- nosity,andEddingtonratio.Thereisnoevidenceforadependencyofthevariabilityamplitudeonblackholemassandredshift.These resultssuggestthattheaccretionrateisthefundamentalphysicalquantitydeterminingtherest-frameUV/opticalvariabilityamplitude ofquasarsontimescalesofmonthsandyears.TheopticalPSDofallofoursourcesisconsistentwithabrokenpowerlawshowinga characteristicbendatrest-frametimescalesrangingbetween∼100and∼300days.Thebreaktimescaleexhibitsnosignificantcorre- lationwithanyofthefundamentalAGNparameters.Thelow-frequencyslopeofthePSDisconsistentwithavalueof−1formostof ourobjects,whereasthehigh-frequencyslopeischaracterizedbyabroaddistributionofvaluesbetween∼–2and∼–4.Thesefindings unveilsignificantdeviationsfromthesimpledampedrandomwalkmodelthathasfrequentlybeenusedinpreviousopticalvariability studies.WefindaweaktendencyforAGNswithhigherblackholemasstohavesteeperhigh-frequencyPSDslopes. Keywords.accretion,accretiondisks–methods:dataanalysis–blackholephysics–galaxies:active–quasars:general– X-rays:galaxies 1. Introduction strong in the X-ray, UV/optical, and radio bands (Ulrich et al. 1997). The X-ray band shows very rapid variations, typically Albeitthequestionhasbeenpuzzledoverformanydecades,the withlargeramplitudethanopticalvariabilityonshorttimescales physical origin of active galactic nucleus (AGN) variability is of days to weeks. However, optical light curves exhibit larger still unknown. Several mechanisms have been proposed to ex- variabilityamplitudesonlongertimescalesofmonthstoyearson plain the notoriousflux variations, but to date, there is no pre- thelevelof∼10−20%influx(Gaskell&Klimek2003;Uttley& ferredmodelthat is able to predictall the observedfeaturesof Casella2014).OpticalvariabilityofAGNshasbeenstudiedex- AGN variability in a self-consistent way (Cid Fernandes et al. tensivelyinthelastyears,providingausefultoolforquasarse- 2000;Hawkins2002;Pereyraetal.2006).Unveilingthesource lectionaswellasaprobeforphysicalmodelsdescribingAGNs ofAGNvariabilitypromisesbetterunderstandingofthephysical (Kelly et al. 2009, 2011, 2013; Kozłowski et al. 2010, 2011, processesthatpowertheseluminousobjects.AGNvariabilityis 2012, 2013; MacLeod et al. 2010, 2011, 2012; Schmidt et al. characterizedbynon-periodicrandomfluctuationsinflux,which 2010,2012;Palanque-Delabrouilleetal.2011;Butler&Bloom occur with different amplitudes on timescales of hours, days, 2011;Kimetal.2011;Ruanetal.2012;Zuoetal.2012;Andrae months,years,andevendecades(Gaskell&Klimek2003).Very etal.2013;Zuetal.2013;Morgansonetal.2014;Grahametal. strongvariabilitymayalsobepresentonmuchlongertimescales 2014; De Cicco et al. 2015; Falocco et al. 2015; Cartier et al. of 105–106 yr (Hickox et al. 2014; Schawinski et al. 2015). 2015). Thevariabilityisobservedacross-wavelengthandisparticularly Since the opticalcontinuumradiationisbelievedto be pre- dominantlyproducedbytheaccretiondisk,itisverylikelythat (cid:2) ZwickyFellow. opticalvariabilityoriginatesfromprocessesintrinsictothedisk. ArticlepublishedbyEDPSciences A129,page1of23 A&A585,A129(2016) Onepossiblemechanismmaybefluctuationsoftheglobalmass aboutthe value of the low-frequencyslope of the opticalPSD. accretionrate,providingapossibleexplanationfortheobserved Using a sample of ∼9000 spectroscopically confirmed quasars largevariabilityamplitudes(Pereyraetal.2006;Li&Cao2008; in SDSS Stripe 82, MacLeod et al. (2010) were unable to dis- Sakata et al. 2011; Zuo et al. 2012; Gu & Li 2013). However, tinguish between γ = −1 and γ = 0 (“white noise”) for the consideringthecomparablyshorttimescalesofopticalvariabil- low-frequency slope. Considering the optical break timescale, ity, a superposition of several smaller, independently fluctuat- typicalvaluesbetween10–100daysbutevenupto∼10yrhave ing zones of different temperature at various radii, associated been reported (Collier & Peterson 2001; Kelly et al. 2009). withdiskinhomogeneitiesthatarepropagatinginward,maybe The spreadin the characteristic variabilitytimescale is thought a preferable alternative solution (Lyubarskii1997;Kotov et al. to be connected with the fundamental AGN parameters driv- 2001; Arévalo & Uttley 2006; Dexter & Agol 2011). Such lo- ingthevariability.Theopticalbreaktimescalewasobservedto calized temperature fluctuations are known to describe several scalepositivelywithblackholemassandluminosity(Collier& characteristics of AGN optical variability (Meusinger & Weiss Peterson2001;Kellyetal.2009;MacLeodetal.2010). 2013; Ruan et al. 2014; Sun et al. 2014) and may arise from Alternatively to performing a PSD analysis, which in gen- thermal or magnetorotational instabilities in a turbulent accre- eral requires well-sampled and uninterrupted light curves, it is tion flow, as suggested by modern numericalsimulations (e.g., customary to use simpler variability estimators that allow in- Hiroseetal.2009;Jiangetal.2013). ferring certain properties of the PSD for large samples of ob- The strong temporal correlation of optical and X-ray vari- jects and sparsely sampled light curves. Convenientvariability ability observed in simultaneous light curves on timescales of toolsarestructurefunctions(e.g.,Schmidtetal.2010;MacLeod months to years indicates that inward-moving disk inhomo- etal.2010;Morgansonetal.2014)ortheexcessvariance(e.g., geneitiesmaydrivethelong-termX-rayvariability(Uttleyetal. Nandraetal. 1997;Pontietal. 2012;Lanzuisietal. 2014).On 2003; Arévalo et al. 2008, 2009; Breedt et al. 2009, 2010; timescales shorter than the break timescale, the X-ray excess Connollyetal.2015).Ontheotherhand,theshorttimelagsof variancewasfoundtobeanticorrelatedwiththeblackholemass afewdaysbetweendifferentopticalbands(Wandersetal.1997; andtheX-rayluminosity,whereasthereiscurrentlynoconsen- Sergeevetal.2005)areinfavorofamodelinwhichX-rayvari- sus regardingthe correlation with the Eddington ratio (Nandra ability is driving the optical variability approximately on light et al. 1997; Turner et al. 1999; Leighly 1999; George et al. traveltimesbyirradiatingandtherebyheatingtheaccretiondisk 2000;Papadakis2004;O’Neilletal.2005;Nikołajuketal.2006; (Cackettetal.2007).Whichevermechanismactuallydominates, Miniutti et al. 2009; Zhou et al. 2010; González-Martín et al. it is important to compare the properties of optical and X-ray 2011;Caballero-Garciaet al. 2012;Ponti et al. 2012;Lanzuisi variability,becauseunderstandingtheircouplingprovidesade- et al. 2014; McHardy 2013). Considering the optical variabil- tailed view of the physical system at work that can hardly be ityamplitude,ananticorrelationwithluminosityandrest-frame obtainedbyothermethodsthantiminganalysis. wavelength is well established on timescales of ∼years (Hook Thepowerspectraldensity(PSD)statesthevariabilitypower etal.1994;Giveonetal.1999;VandenBerketal.2004;Wilhite per temporal frequency ν. The X-ray PSDs of AGNs are ob- etal.2008;Baueretal.2009;Kellyetal.2009;MacLeodetal. servedtobewelldescribedbyabrokenpowerlawPSD(ν)∝νγ 2010; Zuo et al. 2012). Conflicting results have been obtained withγ = −2forfrequenciesabovethebreakfrequencyν and regarding a dependence of the optical variability amplitude on br γ=−1forfrequenciesbelowν (Lawrence&Papadakis1993; the black hole mass, because some authors found positive cor- br Greenetal.1993;Nandraetal.1997;Edelson&Nandra1999; relations, others negative correlationsor almost no correlation, Uttleyetal.2002;Markowitzetal.2003;Markowitz&Edelson althoughtheyprobedsimilar variabilitytimescales(Woldetal. 2004;McHardyetal.2004;González-Martín&Vaughan2012). 2007;Wilhiteetal.2008;Kellyetal.2009;MacLeodetal.2010; SuchPSDsaremodeledbya stochasticprocessconsistingofa Zuoetal.2012).Finally,ananticorrelationbetweenopticalvari- seriesofindependentsuperimposedeventsandaretermed“red abilityandtheEddingtonratiohasbeenreportedbyseveralau- noise” or “flicker noise” PSDs, because low frequencies con- thors on timescales of several months (Kelly et al. 2013) and tributethemostvariabilitypower,whereashigh-frequencyvari- several years (Wilhite et al. 2008; Bauer et al. 2009; Ai et al. ability is increasingly suppressed (Press 1978). The character- 2010;MacLeodetal.2010;Zuoetal.2012;Meusinger&Weiss istic frequency ν was found to scale inversely with the black 2013).However,theobservedtrendswiththe AGNparameters br hole mass and linearly with the accretion rate (McHardy et al. show large scatter, with the derived slopes often suggesting a 2006).However,theactualdependencyonthe accretionrate is veryweakdependence. lessclearandwasnotrecoveredbyGonzález-Martín&Vaughan In this workwe aim to investigatethe correlationsbetween (2012). the optical variability amplitude, quantified by the normalized Because optical light curves are not continuous and gener- excess variance, and the fundamentalAGN physicalproperties allysufferfromirregularsampling,standardFouriertechniques byusingawell-studiedsampleofX-rayselectedAGNsfromthe used in the X-rays cannot be applied, and therefore the shape XMM-COSMOSsurveywith opticallightcurvesin five bands of the optical PSD of AGNs is not well known to date. But available from the Pan-STARRS1 Medium Deep Field 04 sur- thereisevidencethattheopticalPSDresemblesabrokenpower vey.Inaddition,weperformaPSDanalysisofouropticallight law as well. For example, the high-frequency part of the op- curves using the CARMA approach introduced by Kelly et al. tical PSD has been found to be described reasonably well by (2014)toderivetheopticalPSDshapeforalargesampleofob- a power law of the form PSD(ν) ∝ ν−2 (Giveon et al. 1999; jects,includingthecharacteristicbreakfrequency,thePSDnor- Collier &Peterson2001;Czernyetal.2003;Kelly etal.2009, malization,andthePSDslopesathighandlowfrequencies.The 2013;Kozłowskietal.2010;MacLeodetal.2010;Andraeetal. paperisorganizedasfollows:inSect.2wedescribeoursample 2013;Zuetal.2013).However,recentPSDanalysesperformed of variable AGNs; the methodsused to quantifythe variability using high-quality Kepler light curves suggest that the high- amplitude and to modelthe PSD are introducedin Sect. 3; the frequency optical PSD may be characterized by steeper slopes correlationsbetweenthevariabilityamplitudeandtheAGNpa- ofbetween−2.5and−4(Mushotzkyetal.2011;Edelsonetal. rametersarepresentedinSect.4;theresultsofthepowerspec- 2014; Kasliwal et al. 2015). Likewise, there is still confusion trum analysis are depicted in Sect. 5; we discuss our findings A129,page2of23 T.Simmetal.:Pan-STARRS1variabilityofXMM-COSMOSAGN.II. in Sect. 6, and Sect. 7 summarizes the most important results. 3. Method:variabilityamplitudeandpower AdditionalinformationaboutthesampleandthePSDfitresults spectrummodel in different wavelength bands are provided in Appendices A andB,respectively. 3.1.Normalizedexcessvariance To quantifythe variabilityamplitude we measuredthe normal- izedexcessvariance(Nandraetal.1997)givenby 2. SampleofvariableAGNs ⎛ (cid:4) (cid:5) ⎞ TashrdoeufignheoduitnthSisimwmorketwael.u(s2e0t1h5e,shaemreeasftaemrpSl1e5o)f.vTahriisabslaemApGleNiss σ2rms =(cid:2)s2−σ2err(cid:3)/(cid:4)f¯(cid:5)2 = (cid:4)f1¯(cid:5)2 ⎜⎜⎜⎜⎜⎜⎜⎜⎝(cid:9)i=N1 (fNi−−f1¯)2 −(cid:9)i=N1 σN2err,i⎟⎟⎟⎟⎟⎟⎟⎟⎠ (1) drawnfromthecatalogofBrusaetal.(2010),whichpresentsthe multiwavelength counterparts to the XMM-COSMOS sources fromthelightcurveconsistingofNmeasuredfluxes f withindi- (Hasingeretal.2007;Cappellutietal.2009).Wehaveselected i vidualerrorsσ andarithmeticmean f¯.Thenormalizedexcess the X-ray sources that have a pointlike and isolated counter- err,i part in HST/ACS images and that are detected in single Pan- variance,orjustexcessvariance,quotestheresidualvarianceaf- STARRS1 (PS1) exposures. In addition, we focused on the tersubtractingtheaveragestatisticalerrorσ2 fromthesample err bands for which the observationaldata are of high quality and variance s2 of the light-curve flux. The σ2 values calculated rms available for most of our objects. Thus, the sample comprises from the total light curves of our AGNs were used in S15 and 184(g ),181(r ),162(i ),131(z )variablesourcesdetected are available at the CDS (see Appendix C of S15 for details). P1 P1 P1 P1 inthePS1MediumDeepField04(MDF04)survey.Inthefol- TheerrorontheexcessvariancecausedbyPoissonnoisealone lowing we refer to this sample as the “total sample”. We note (Vaughanetal.2003)iswelldescribedby taiTthnhnaaadbatnltemg9hi6i2ovs%reoesnafotmPhfSSapo1n1lu5er9bfco7aoobn%rnjdetdacoeaitftnrsaesaiallnireldeodescuonnlputauiprsfimcseeeirbdfiseleiarhmdssaiavivtnsiadntreegyiatapeMcbechlte1DiPoiFAnnS0Gs1t4hoNbfilssaivg1nbahdaart)nina.cddbMu9ir(l2vosite%eryees, err(cid:4)σ2rms(cid:5)= (cid:13)(cid:14)(cid:14)(cid:14)(cid:14)(cid:14)(cid:15)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝(cid:16)N2 · (cid:4)σf¯2e(cid:5)r2r⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠2+⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝(cid:17)σN2err · 2(cid:4)Ff¯v(cid:5)ar⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠2, (2) haveaspecifiedspectroscopicredshift(Trumpetal.2007;Lilly (cid:18) etal.2009).Theremainingsourcesonlyhavephotometricred- where F = σ2 is the fractionalvariability(Edelsonetal. shiftsdeterminedinSalvatoetal.(2011).However,forthe92% var rms 1990).AsdemonstratedbyAllevatoetal.(2013),therearead- with knownspectroscopicredshifts, the accuracyof the photo- ditional error sources associated with the stochastic nature of metric redshifts is σ = 0.009 with a fraction of outliers NMAD AGNvariability,red-noiseleakage,thesamplingpattern,andthe of5.9%.Thereforewedonotdistinguishbetweensourceswith signal-to-noise ratio of the light curves. In particular, these bi- spectroscopicandphotometricredshiftsinthefollowing. asesdependontheshapeofthePSD(seee.g.Table2inAllevato During the whole analysis we only consider the objects etal.2013),andthereforeanexcessvariancemeasurementcan classified as type 1 AGN when investigating correlations be- systematicallyover-orunderestimatetheintrinsicvarianceofa tween the physical AGN parameters and variability. Of the light curve by a factor of a few (we refer to the discussion in type 1 objects of the total sample, 95 (g ), 97 (r ), 90 (i ), P1 P1 P1 Sect.5.4). 75 (z ) have known spectroscopic redshifts, SED-fitted bolo- P1 Followingthe procedurein S15, we considereda sourceas metric luminosities L (Lusso et al. 2012), and black hole bol variableinagivenbandif masses M (Rosarioetal.2013).Theblackholemasseswere BH (cid:4) (cid:5) all derived with the same method described in Trakhtenbrot& σ2 −err σ2 >0. (3) Netzer (2012)from the line width of broad emission lines (Hβ rms rms and MgIIλ2798Å), using virial relations that were calibrated withreverberationmappingresultsoflocalAGNs.Forthesame We emphasize that this is only a 1σ detection of variability. sources we therefore also possess the Eddington ratio defined However,in this work we aim to investigatethe relation of the by λ = L /L , where L is the Eddington luminosity. amplitudeof variabilitywith AGN physicalpropertiesdownto Edd bol Edd Edd Thissample,hereaftertermed“MBHsample”,coversaredshift the lowest achievable level of variability. Using a more strin- rangefrom0.3to2.5.Westressthatthisisalargesampleofob- gentvariabilitythresholdwould dramaticallylimitthe parame- jectswithhomogeneouslymeasuredAGNparameters,spanning ter space of MBH, Lbol and λEdd values we can probe. Finally, a wide redshift range, for which we can study the connection thequalityoftheσ2 measurementsofoursampleisgenerally rms of rest-frame UV/optical variability with fundamentalphysical high,aspresentedinAppendixAofS15. properties of AGNs in four wavelength bands. As detailed in Theintrinsicvarianceofalightcurveisdefinedtomeasure Appendix A, our sample does not suffer from strong selection theintegralofthePSDoverthefrequencyrangeprobedbythe effects, which could significantly bias any detected correlation timeseries.Sincetheexcessvarianceisanestimatorofthefrac- betweenvariabilityandtheAGNparameters.However,sinceour tionalintrinsicvariance,itisrelatedtothePSDby sampleisdrawnfromaflux-limitedX-rayparentsample,thereis (cid:19) atendencyforhigherredshiftsourcestobemoreluminous.We νmax foundthatthiseffecthasonlynegligibleimpactontheresulting σ2rms ≈ PSD(ν)dν, (4) correlationsbetweenvariabilityandluminosity,however. νmin withν = 1/T andtheNyquistfrequencyν = 1/(2Δt)for min max 1 Therearesevenvariabletype2AGNsinoursamplethatwereclas- a light curve of length T and bin size Δt, with the PSD nor- sifiedeitherspectroscopically (sixobjects) or on thebasisof thebest malized to the squared mean of the flux (Vaughan et al. 2003; SEDfittingtemplate(oneobject). González-Martínetal.2011;Allevatoetal.2013). A129,page3of23 A&A585,A129(2016) 3.2.Measuringσ2 ondifferenttimescales rms Although the excess variance is a variability estimator that is measured from the light-curve fluxes and the individual ob- serving times do not appear explicitly in the calculation, the total temporal length and the sampling frequency of the light curve affect the resulting σ2 value. As described in the pre- rms vious section, the excess variance estimates the integral of the variabilitypowerspectrumovertheminimalandmaximaltem- poral frequency covered by the light curve. Therefore we can probe different variability timescales by measuring the excess variancefromdifferentpartsofthelightcurves.Thetotalsam- ple only contains σ2 values computed from the nightly aver- rms agedtotallightcurveswhichtypicallyconsistof∼70–80points and cover a period of aboutfour years. The light curves split into severalsegmentswith observationsperformedaboutevery onetothreedays over a period of aboutthreetofourmonths, interrupted by gaps of aboutseventonine months without ob- servations. Correspondingly, the shortest sampled timescale is on the order of a few days for the MDF04 survey, depending on weather constraints during the survey, whereas the longest Fig.1. Top panel: histogram of the rest-frame observation length of timescaleisaboutfouryears.However,thesamplingpatternof the total light curve for the year timescale MBH sample (g band). P1 the MDF04 light curves additionally allows measuring the ex- Bottompanel:histogramoftherest-frameobservationlength(average cessvariancefromthewell-sampledindividualsegmentsofthe valueofthelight-curvesegments)forthemonthtimescaleMBHsample light curves, consisting of typically 10–20 points that span a (g band). P1 time interval of aboutthreetofourmonths. For each AGN we additionally calculated an excess variance value measured on as measured on timescales of years and months whenever ap- timescalesofmonthsbyaveragingtheσ2 valuesofthelight- rms plicable. For referencewe display the propertiesof the various curve segments and propagating the err(σ2 ) values of each rms samples used throughout this work and how they are selected considered segment. To avoid effects by sparsely sampled seg- fromtheparentsampleinFig.2. ments,whichwouldlowerthequalityofthevariabilityestima- tion, we includedonly the segments with more than ten obser- vations in the averaging. The sample of variable type 1 AGNs 3.3.CARMAmodelingofthepowerspectraldensity with knownphysicalparametersforthis shortertimescale, that Considering Eq. (4), modeling the PSD of a light curve pro- is, the MBH sample on timescales of months fulfilling σ2 − rms vides more fundamental variability information than the inte- err(σ2 )>0,comprises76(g ),63(r ),41(i ),and43(z ) rms P1 P1 P1 P1 grated σ2 quantity. The shape of the PSD potentially allows sources,respectively.Theconsiderablysmallersamplesizefol- rms gaininginsightintotheunderlyingphysicalprocessesconnected lowsfromthefactthatthelightcurvesegmentsofmanyAGNs to variability (Lyubarskii 1997; Titarchuk et al. 2007). To es- eitherhavefewerthantenmeasurementsorarealmostflat,lead- timatethePSDsofourlightcurves,weappliedthecontinuous- ingtoverylowandevennegativeσ2 values.We observethat rms timeautoregressivemovingaverage(CARMA)modelpresented the variability amplitude on timescales of years is on average inKellyetal.(2014).Thisstochasticvariabilitymodelfullyac- aboutanorderofmagnitudelargerthanontimescalesofmonths. countsforirregularsamplingandGaussianmeasurementerrors. Althoughtheobserver-frametimescalescoveredbythelight Italsoallowsforinterpolationandforecastingoflightcurvesby curves of our sample are very similar for each AGN, the wide modelingthelatterasacontinuous-timeprocess. redshift range encompassed by our sources leads to a variety Azero-meanCARMA(p,q)processforatimeseriesy(t)is of different rest-frame timescales. This is illustrated in Fig. 1, definedasthesolutionofthestochasticdifferentialequation showing the distribution of the rest-frame observation length dpy(t) dp−1y(t) Tcurovfetsheegmtoetnalts,liogbhttacinuerdvebyanddivtihdeingavtehreagoebsvearlvueer-forfamtheevlaigluhet dtp +αp−1 dtp−1 +...+α0y(t)= by1+z toaccountforcosmologicaltimedilation.Thedataof dq(cid:9)(t) dq−1(cid:9)(t) theMBHsample(gP1 band)ontimescalesofyearsandmonths βq dtq +βq−1 dtq−1 +...+(cid:9)(t). (5) are displayed. From this we note that the rest-frame length of It is assumed that the variability is driven by a Gaussian thetotallightcurvecomprisesofaboutonetothreeyearsforour continuous-time white noise process (cid:9)(t) with zero mean and sources, whereas the rest-frame length of the light-curve seg- variance σ2. Apart from σ2, the free parameters of the model ments corresponds to timescales of aboutonetothreemonths. To reduce possible biases introduced by the spread in redshift, are the autoregressive coefficients α0,..., αp−1 and the moving averagecoefficientsβ ,..., β . In practice, the mean of the time we additionally considered the sources of the MBH sample 1 q series μ is also a freeparameter,and the likelihoodfunctionof with redshifts between 1 < z ≤ 2 in our investigations, re- thetimeseriessampledfromaCARMAprocessiscalculatedon ferredtoasthe“1z2_MBHsample”.Onvariabilitytimescalesof thecenteredvaluesy˜ =y −μforeachlight-curvepointi. years,the1z2_MBHsamplecontains72(g ),74(r ),69(i ), i i P1 P1 P1 ThePSDofaCARMA(p,q)processisgivenby and 56 (zP1) AGNs. The corresponding 1z2_MBH sample on (cid:20) tainmde3sc1al(ezs )ofobmjeocnttsh.sIncothmepfroislleosw6in1g(egxP1c)e,ss49va(rriaPn1)c,e3a0na(liyPs1i)s, PSD(ν)=σ2|(cid:20)qj=0βj(2πiν)j|2, (6) P1 | p α (2πiν)k|2 (Sect. 4) we compare the variability properties of our sources k=0 k A129,page4of23 T.Simmetal.:Pan-STARRS1variabilityofXMM-COSMOSAGN.II. (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:1)(cid:2)(cid:3)(cid:4)(cid:1)(cid:2)(cid:1)(cid:4)(cid:1)(cid:5)(cid:6)(cid:4)(cid:1)(cid:7)(cid:1) (cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:4)(cid:13)(cid:14)(cid:15)(cid:14)(cid:16)(cid:17)(cid:14)(cid:18)(cid:19)(cid:4)(cid:20)(cid:21)(cid:22)(cid:23)(cid:17)(cid:15)(cid:22)(cid:24)(cid:14)(cid:19)(cid:4)(cid:22)(cid:13)(cid:21)(cid:15)(cid:11)(cid:17)(cid:14)(cid:18) (cid:11)(cid:12)(cid:13)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:1)(cid:25)(cid:5)(cid:4)(cid:1)(cid:3)(cid:3)(cid:4)(cid:1)(cid:6)(cid:3)(cid:4)(cid:26)(cid:7) (cid:27)(cid:28)(cid:29)(cid:30)(cid:28)(cid:4)(cid:31)(cid:21)(cid:18)(cid:14)(cid:15)(cid:4)(cid:10)(cid:14)(cid:13)(cid:22)(cid:18)(cid:32)(cid:11)(cid:15)(cid:13)(cid:4)(cid:21)(cid:24)(cid:4) (cid:33)(cid:10)(cid:21)(cid:24)(cid:14)(cid:23)(cid:4)(cid:20)(cid:21)(cid:34)(cid:14)(cid:10)(cid:4)(cid:15)(cid:11)(cid:34)(cid:4)(cid:35)(cid:22)(cid:17)(cid:4)(cid:21)(cid:24) (cid:14)(cid:10)(cid:4)(cid:15)(cid:6)(cid:3)(cid:16)(cid:8)(cid:10)(cid:7)(cid:17)(cid:4)(cid:5)(cid:10) (cid:18)(cid:2)(cid:21)(cid:3)(cid:22)(cid:6)(cid:3)(cid:16)(cid:8)(cid:10)(cid:7)(cid:17)(cid:4)(cid:5)(cid:10) (cid:18)(cid:19)(cid:20)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:11)(cid:12)(cid:13)(cid:23)(cid:18)(cid:19)(cid:20)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:18)(cid:19)(cid:20)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:26)(cid:25)(cid:4)(cid:26)(cid:36)(cid:4)(cid:26)(cid:37)(cid:4)(cid:36)(cid:25) (cid:2)(cid:26)(cid:4)(cid:36)(cid:26)(cid:4)(cid:36)(cid:6)(cid:4)(cid:25)(cid:25) (cid:36)(cid:5)(cid:4)(cid:5)(cid:7)(cid:4)(cid:3)(cid:1)(cid:4)(cid:3)(cid:7) (cid:38)(cid:12)(cid:20)(cid:14)(cid:9)(cid:1)(cid:19)(cid:4)(cid:13)(cid:20)(cid:14)(cid:16)(cid:9)(cid:39)(cid:19)(cid:4)(cid:30)(cid:33)(cid:40)(cid:19)(cid:4)(cid:41)(cid:42)(cid:21)(cid:15) (cid:38)(cid:12)(cid:20)(cid:14)(cid:9)(cid:1)(cid:19)(cid:4)(cid:13)(cid:20)(cid:14)(cid:16)(cid:9)(cid:39)(cid:19)(cid:4)(cid:30)(cid:33)(cid:40)(cid:19)(cid:4)(cid:41)(cid:42)(cid:21)(cid:15) (cid:38)(cid:12)(cid:20)(cid:14)(cid:9)(cid:1)(cid:19)(cid:4)(cid:13)(cid:20)(cid:14)(cid:16)(cid:9)(cid:39)(cid:19)(cid:4)(cid:30)(cid:33)(cid:40)(cid:19)(cid:4)(cid:41)(cid:42)(cid:21)(cid:15) (cid:43)(cid:4)(cid:1)(cid:37)(cid:4)(cid:20)(cid:21)(cid:22)(cid:23)(cid:17)(cid:13)(cid:4)(cid:20)(cid:14)(cid:10)(cid:4)(cid:13)(cid:14)(cid:44)(cid:31)(cid:14)(cid:23)(cid:17) Fig.2.Flowchart illustratingtheselectionof allsamples considered in this work. Below the sample name (bold face)welistthesamplesizeforeachPS1bandintheor- derg ,r ,i ,z .Wealsostatethedefiningproperties P1 P1 P1 P1 of each sample, such as objects with known AGN type, (cid:14)(cid:10)(cid:4)(cid:15)(cid:6)(cid:3)(cid:16)(cid:8)(cid:10)(cid:7)(cid:17)(cid:4)(cid:5)(cid:10) (cid:18)(cid:2)(cid:21)(cid:3)(cid:22)(cid:6)(cid:3)(cid:16)(cid:8)(cid:10)(cid:7)(cid:17)(cid:4)(cid:5)(cid:10) spectroscopic redshift (spec-z), black holemass (MBH), (cid:24)(cid:25)(cid:26)(cid:23)(cid:18)(cid:19)(cid:20)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) (cid:24)(cid:25)(cid:26)(cid:23)(cid:18)(cid:19)(cid:20)(cid:6)(cid:7)(cid:4)(cid:8)(cid:9)(cid:5)(cid:10) bolometric luminosity (L ), or objects within a certain (cid:36)(cid:6)(cid:4)(cid:36)(cid:3)(cid:4)(cid:5)(cid:26)(cid:4)(cid:25)(cid:5)(cid:4) (cid:5)(cid:1)(cid:4)(cid:3)(cid:26)(cid:4)(cid:7)(cid:37)(cid:4)(cid:7)(cid:1) bol (cid:1)(cid:4)(cid:45)(cid:4)(cid:39)(cid:4)(cid:45)(cid:46)(cid:4)(cid:6) (cid:1)(cid:4)(cid:45)(cid:4)(cid:39)(cid:4)(cid:45)(cid:46)(cid:4)(cid:6) redshift range (see text for details). The two rightmost samplesareintroducedinSect.5.3. which forms a Fourier transform pair with the autocovariance 4. CorrelationsofvariabilityandAGNparameters functionattimelagτ (cid:21)(cid:20) (cid:22)(cid:21)(cid:20) (cid:22) 4.1.Wavelengthdependenceoftheexcessvariance (cid:9)p q βrl q β (−r )l exp(r τ) R(τ)=σ2 l=0 l k(cid:23) l=0 l k (cid:4) k (cid:5), (7) ThemultibandPS1observationsoftheMDF04surveyallowfor k=1 −2Re(rk) lp=1,l(cid:2)k(rl−rk) rl∗+rk aninvestigationofthechromaticnatureofvariability,thatis,the dependenceontheradiationwavelength.Figure3showstheex- wherer∗isthecomplexconjugateandRe(r )therealpartofr , k k k cessvariancesofthetotalsample(variabilitytimescaleofyears) respectively.Thevaluesr ,...,r denotetherootsoftheautore- 1 p forseveralfilterpairs.Theintersectionoftheobjectswithmea- gressivepolynomial sured σ2 values in each of the two consideredPS1 bandsare rms (cid:9)p plotted.Eachsubpaneldisplaysthebluerbandonthey-axisand A(z)= α zk. (8) the redder band on the x-axis, the redshift is given as a color k bar. The σ2 values clearly are strongly correlated, which is k=0 also expresrsmesd by the Spearman rank order correlation coeffi- TheCARMAprocessisstationaryifq < pandRe(r ) < 0for cientρ andthecorrespondingtwo-tailed p-valueP ,givingthe k S S allk. TheautocovariancefunctionofaCARMA processrepre- probabilitythat a ρ value at least as high as the observedone S sents a weighted sum of exponentialdecays and exponentially could arise for an uncorrelated dataset. The ρ values quoted S dampedsinusoidalfunctions.Sincetheautocovariancefunction in each subpanel of Fig. 3 are all very close to +1 and the re- is coupled to the PSD by a Fouriertransform,the latter can be spective P valuesare essentially zero. However,we observea S expressedasaweightedsumofLorentzianfunctions,whichare systematic trend that the bluer bands exhibit larger variability knowntoprovideagooddescriptionofthePSDsofX-raybina- amplitudes than the redder bands, as the respective σ2 val- rms riesandAGNs(Nowak2000;Bellonietal.2002;Belloni2010; ues are shifted upward on the one-to-one relation. The offset DeMarcoetal.2013,2015). increases when a specified blue band is compared with the se- The CARMA model includes the Ornstein-Uhlenbeckpro- ries of bands with longer wavelength, that is, when comparing cess or the “damped random walk”, which is depicted in de- the pairs (g ,r ), (g ,i ) and (g ,z ). The variability am- P1 P1 P1 P1 P1 P1 tail in Kelly et al. (2009)and was foundto accuratelydescribe plitudes, however, seem to approach increasingly similar val- quasar light curves in many subsequent works, as the special uestowardthenear-IRregime.Thedifferencebetweentheσ2 rms caseof p = 1andq = 0.ConsideringEqs.(6)and(7),wenote measurementsof the i and z bandsis less pronouncedthan P1 P1 that CARMA models providea flexible parametricform to es- therespectivevaluesofthepairs(g ,r )and(r ,i ).Noevo- P1 P1 P1 P1 timate the PSDs andautocovariancefunctionsof the stochastic lution of the aforementionedwavelengthdependencewith red- light curvesof AGNs. For further details on the computational shiftis observedbecause thereare noregionsthat are predom- methods,includingthecalculationofthelikelihoodfunctionof inantly occupied by high- or low-redshift sources in any sub- aCARMAprocessandtheBayesianmethodtoinfertheproba- panel.Thesametrendsareobservedwhenusingtheexcessvari- bilitydistributionofthePSDgiventhemeasuredlightcurve,we ance values measured on timescales of months. We emphasize refertoKellyetal.(2014)andthereferencestherein. that these findings agree with previous studies that observed A129,page5of23 A&A585,A129(2016) Fig.3.ComparingtheexcessvariancemeasuredontimescalesofyearsinthedifferentPS1bands.Thedataofallobjectsfromthetotalsample withvariabilityinformationinbothconsideredbandsareshown.TheSpearmancorrelationcoefficientandtherespective p-valuearereportedin eachsubpanel. Theredshift isgivenasacolor bar.Theblacklinecorresponds totheone-to-one relation.Theblackerrorbarsaretheaverage values. local and high-redshift AGNs to be more variable at shorter Table1.Spearmancorrelationcoefficientρ andrespective p-valueP S S wavelength(Edelsonetal. 1990;Kinneyetal. 1991;Paltani& ofσ2 andM . rms BH Courvoisier1994;diClementeetal.1996;CidFernandesetal. 1996;VandenBerketal.2004;Kozłowskietal.2010;MacLeod ΔT ∼years,1z2_MBHsample etal.2010;Zuoetal.2012). Filter ρ P S S g –0.04 7.7×10−1 P1 4.2.Excessvarianceversusblackholemass rP1 0.09 4.6×10−1 i –0.01 9.6×10−1 P1 Determining accurate black hole masses for a large number z –0.06 6.7×10−1 P1 of AGNs across the Universe is observationally expensive. ΔT ∼months,1z2_MBHsample However, recent works probing the high-frequencypart of the Filter ρ P S S PSDdeliveredblackholemassestimateswith∼0.2–0.4dexpre- g –0.31 1.6×10−2 P1 cision based on scaling relationsof black hole mass and X-ray r 0.06 7.1×10−1 P1 variability(Zhouetal.2010;Pontietal.2012;Kellyetal.2011, i –0.03 8.9×10−1 P1 2013). It is therefore important to know whether optical vari- z –0.13 5.0×10−1 P1 ability provides another independent tool for measuring black hole masses of AGNs, since massive time-domain optical sur- Notes. The values of the 1z2_MBH sample are quoted for σ2 mea- rms veys such as PS1 and LSST would then allow deriving black suredontimescalesofyears(top)andmonths(bottom). holemassestimatesfora verylargenumberofquasarsregard- lessoftheAGNtype. In Fig. 4 we plot the g band excess variance measured P1 on timescales of years and months versus the black hole mass we do not report them here. Therefore we conclude that there forthe1z2_MBHsample.Eventhoughtheestimateduncertain- is no significant anticorrelation between optical variability and ties of the black hole masses of our sample are large, typically blackholemassfortheprobedvariabilitytimescalesofourlight ∼0.25 dex, there is little evidence for any correlation between curves. We stress that other optical variability studies found a M andσ2 measuredontimescalesofyears.Atleastforthe correlationofvariabilityandM usingdifferentvariabilityesti- BH rms BH g bandweobservea weakanticorrelationwith M forvari- mators,butinvestigatingvariabilitytimescalesthataresimilarto P1 BH abilitymeasuredontimescalesofmonthswithρ = −0.31and thoseofourwork.However,theseresultsareinconsistentinthe S P = 1.6× 10−2, but the scatter in the relation is quite large. sensethatseveralworksstateapositivecorrelationbetweenthe S Moreover, we do not find any significant anticorrelation relat- variability amplitude and M (e.g., Wold et al. 2007; Wilhite BH ing M withthemonthlytimescaleσ2 valuesoftheremain- etal.2008;MacLeodetal.2010),whereasothersreportananti- BH rms ingPS1 bands.Thecorrelationcoefficientsand p-valuesof the correlationwith M (Kellyetal.2009,2013).Finally,wenote BH 1z2_MBHsamplearesummarizedinTable1forallconsidered thatFig.4showsnoobviousdependenceonredshift,andwedo PS1 bands and for both variability timescales. The correlation notobserve any trendfor the other PS1 bands. This is also the coefficientsofthe MBH sample areverysimilar, whichis why casefortheMBHsample. A129,page6of23 T.Simmetal.:Pan-STARRS1variabilityofXMM-COSMOSAGN.II. Fig.5. Excess variance (g band) measured on timescales of years Fig.4.Excessvariance(g band)measuredontimescalesofyears(top) P1 P1 (top) and months (bottom) versus L in units of 1045ergs−1 for the andmonths(bottom)versusMBHinunitsofM(cid:8)forthe1z2_MBHsam- 1z2_MBH sample. The best-fit powbeorllaw is plotted as a black solid ple.Spearman’srandtherespective p-valuearereportedineachsub- line, the dashed lines show the 1σ errors on the fit parameters. The panel.Theredshiftisgivenasacolorbar.Theblackerrorbarscorre- redshiftisgivenasacolorbar.Theblackerrorbarscorrespondtothe spondtotheaveragevalues. averagevalues. 4.3.Excessvarianceversusluminosity However, the stronger anticorrelation observed for shorter The existence of an anticorrelation between optical variabil- variability timescales might also be merely a selection effect, ity and luminosity has been recognized for many years, but caused by consideringa particular subsample of objects of the it was often difficult to distinguish the relation from a depen- largersampleofAGNsthatarevaryingontimescalesofyears. dency on redshift. We also observe a strong anticorrelation of Forthisreason,weadditionallysearchedfortheanticorrelation the excess variance with bolometric luminosity in our dataset. with L by selecting the same subsample of sources from the bol TherespectiveSpearmancorrelationcoefficientsarereportedin 1z2_MBH sample for both variability timescales. This test re- Table 2. For the variability on timescales of years, the anticor- vealed that the observed difference in the strength of the an- relationishighlysignificantintheg ,r andi bandsforthe ticorrelation for the two variability timescales is still present, P1 P1 P1 1z2_MBHsample.Onshortervariabilitytimescalesofmonths, withρ = −0.45,P = 7.2×10−4 (g band)forvariabilityon S S P1 theanticorrelationisevenstrongerandvisibleinallconsidered timescalesofyears,andρ =−0.60,P =1.9×10−6(g band) S S P1 PS1bands.Furthermore,wenotethattheanticorrelationisgen- for variability on timescales of months. This finding implies erallystrongestfortheg bandandisbecominglesssignificant thatregardlessofthemechanismthatcausestheanticorrelation P1 towardtheredderbands.Westressthattheanticorrelationisalso between the excess variance and the bolometric luminosity, it detected with similar significance considering the MBH sam- must be strongly dependent on the characteristic timescale of ple.Figure5presentstheg bandexcessvarianceasafunction thevariability. P1 of bolometric luminosityfor the 1z2_MBH sample. The figure To estimate the functional dependency of σ2 on L , we rms bol clearlydemonstratesthattheanticorrelationwithbolometriclu- used the Bayesian linear regression method of Kelly (2007), minosity is apparentfor both probedvariability timescales and which considers the measurement uncertainties of the two re- that the relation is much tighter for the shorter timescales of lated quantities. To do this, we fit the linear model logσ2 = rms months. β+αlogL +(cid:9)withL = L /1045ergs−1tothedataset. bol,45 bol,45 bol A129,page7of23 A&A585,A129(2016) Table2.Spearmancorrelationcoefficientρ andrespective p-valueP Table3.Scalingofσ2 withL . S S rms bol ofσ2 andL . rms bol ΔT ∼years,1z2_MBHsample ΔT ∼years,1z2_MBHsample Filter α β (cid:9) Filter ρS PS g −0.84±0.16 −1.16±0.12 0.36±0.04 P1 g –0.57 2.1×10−7 r −0.74±0.17 −1.43±0.13 0.41±0.04 P1 P1 r –0.47 2.9×10−5 i −0.85±0.19 −1.47±0.14 0.43±0.04 P1 P1 iP1 –0.49 1.6×10−5 zP1 −0.55±0.23 −1.73±0.19 0.42±0.04 zP1 –0.27 4.1×10−2 ΔT ∼months,1z2_MBHsample ΔT ∼months,1z2_MBHsample Filter α β (cid:9) Filter ρS PS gP1 −1.29±0.17 −1.66±0.10 0.17±0.05 griPPP111 –––000...766144 261...266×××111000−−−1740 rziPPP111 −−−011...933517±±±000...235253 −−−222...101792±±±000...123405 000...222238±±±000...000578 z –0.60 3.7×10−4 P1 Notes.Fittedvaluesoftherelationlogσ2 = β+αlogL +(cid:9) for rms bol,45 Notes. The values of the 1z2_MBH sample are quoted for σ2 mea- eachconsidered PS1band assuming ΔlogLbol = 0.15. Thevaluesfor suredontimescalesofyears(top)andmonths(bottom). rms the 1z2_MBH sample are quoted for σ2rms measured on timescales of years(top)andmonths(bottom). In additionto the zeropointβ and the logarithmicslope α, this model also fits the intrinsic scatter (cid:9) inherent to the relation. of years. The correspondingvalues for timescale variability of SincethesymmetricerroroftheexcessvariancegivenbyEq.(2) monthsreadρ =−0.69andP =2.2×10−5withfittedparame- S S becomesasymmetricinlog-space,weusedasymmetrizederror tersofα=−1.27±0.22,β=−1.57±0.14,and(cid:9) =0.16±0.06. bytakingtheaverageoftheupperandlowererror.Fortheerror Several authors observed an anticorrelationof σ2 and lu- rms ofLbol,Rosarioetal.(2013)observedanrmsscatterof0.11dex minosity and arguedthat this relation may be a byproductof a by comparing a subsample of 63 QSOs with spectra from two morefundamentalanticorrelationof σ2 and M seen at fre- different datasets, whereas Lusso et al. (2011) found a 1σ dis- quencies above ν in X-ray studies, srimnsce the mBHore luminous br persionof0.2dexfortheirSED-fittingmethodforalargersam- sources tend to be the more massive systems (e.g., Papadakis ple.Inthisworkwe performedallfitsassuminga conservative 2004; Ponti et al. 2012). This was also proposed by Lanzuisi averageuncertaintyof0.15dexforeachAGN. et al. (2014), who studied the low-frequencypart of the X-ray The fitted values for the 1z2_MBH sample are listed in PSD, because of the very similar slopes they found for the an- Table3foreachconsideredPS1band,andthebest-fittingmodel ticorrelationsofσ2 with M andX-rayluminosity.Todeter- rms BH is also displayedin Fig. 5.We note thatthe modelfits produce minewhetherthereisasimilartrendinourdata,wedisplaythe thesamelogarithmicslopes,atleastwithinthe1σerrors,forall blackholemassascolorcodeinFig.6,whichotherwiseshows thosePS1bandsshowingasignificantanticorrelationaccording the same information as the upper panel of Fig. 5. The rough to the ρS and PS values. A comparison of the two considered proportionalityofLbol and MBH isapparentinthecolorcodeas variability timescales shows that the determined slopes of the aweaktrendthat M increasesinthe x-axisdirection.Forthe BH σ2 values measured on timescales of months are systemati- y-axisdirection we observe low- and high-masssystems at the rms cally steeper. However, within one or two standard deviations, samelevelofvariabilityamplitude.Thisisalsothecaseforσ2 thefitted slopesareconsistentwith a valueofα ∼ −1 forboth measuredontimescalesofmonths(notshownhere).Howeverr,misf variability timescales, indicating that the relation may be cre- theanticorrelationofσ2 andL werecausedbyahiddenan- rms bol ated by the same physicalprocess2. We stress that the intrinsic ticorrelationwith M ,thenthelessmassiveAGNswouldpre- BH scatter of the relation is only ∼0.2–0.25 dex for variability on dominantlyoccupytheupperregionoftheplot,andviceversa. timescales of months, whereas the scatter is about a factor of Giventhat L ∝ M˙,where M˙ denotesthemassaccretionrate, bol twolargerforvariabilityontimescalesofyears.Fittingthelin- thesefindingssuggestthatthefundamentalAGNparameterde- ear model to the MBH sample, that is, including the full red- terminingtheopticalvariabilityamplitudeisnottheblackhole shift range, results in very similar slopes for variability mea- mass,buttheaccretionrate. sured on timescales of months. But the presence of some high redshift outliers in the larger sample with σ2 measured on rms 4.4.Excessvarianceversusredshift timescales of years drives the fitting routine toward much flat- ter slopes of α ∼ −0.5. Finally, we tested that the anticor- In the relations presented above we do not observe any strong relation between σ2 and L is also recovered when apply- rms bol evolutionwithredshift.Bycorrelatingtheexcessvariancewith ing a 3σ cut in the variability detection (see Eq. (3)). For the the redshift of our AGNs, we find no significant dependency g band 1z2_MBH sample, we then obtain ρ = −0.58 and P1 S in any band; this is summarized in Table 4. However, we can P = 1.3× 10−7 with fitted parameters of α = −0.85±0.16, S predicttheexpectedevolutionofthevariabilityamplitudewith β = −1.15±0.12,and(cid:9) = 0.35±0.04fortimescalevariability redshift in view of the scaling relations outlined in the previ- oussections.Sinceweobserveoursourcesinpassbandswitha 2 Wefoundthattheassumed x-axiserrorstronglyaffectsthederived fixedwavelengthrange,theactualrest-framewavelengthprobed slopeforourfittingroutine.Performingtestfitswiththeg banddata yielded slopes of −0.70,−0.84,−1.00, and −1.74 using PΔ1logL = by each filter is shifted to shorter wavelength for higher red- bol shift. Towards higher redshift we therefore probe UV variabil- 0.01,0.15,0.2,and0.3,respectively.Largerx-axiserrorsthereforesys- tematicallysteepenthefittedslope,andthiseffectisparticularlystrong ityinthebluestPS1bands,whereastheredderbandscoverthe for large errors. However, the bulk of data points clearly suggests a rest-frame optical variability of the AGNs. But we showed in valueof∼–1. Sect.4.1thatthevariabilityamplitudegenerallydecreaseswith A129,page8of23 T.Simmetal.:Pan-STARRS1variabilityofXMM-COSMOSAGN.II. Fig.6. Same as Fig. 5 for σ2 measured on timescales of years, but withM ascolorbar. rms Fig.7. Excess variance (gP1 band) measured on timescales of months BH versusredshiftfortheMBHsample.Thebolometricluminosityisgiven asacolorbar. Table4.Spearmancorrelationcoefficientρ andrespective p-valueP S S ofσ2rmsandz. Table5.SpearmancorrelationcoefficientρSandrespective p-valuePS ofσ2 andλ . rms Edd ΔT ∼years ΔT ∼months Filter ρS PS ρS PS ΔT ∼years,1z2_MBHsample gP1 –0.16 1.3×10−1 –0.22 5.2×10−2 Filter ρS PS rziPPP111 –––000...011545 511...989×××111000−−−111 –––000...022905 421...700×××111000−−−111 griPPP111 –––000...554268 222...679×××111000−−−675 z –0.25 6.5×10−2 P1 Notes.ThevaluesoftheMBHsamplearequotedforσ2 measuredon rms ΔT ∼months,1z2_MBHsample timescalesofyears(leftcolumn)andmonths(rightcolumn). Filter ρ P S S g –0.32 1.2×10−2 P1 r –0.57 1.9×10−5 P1 increasingwavelengthforoursources.Assumingthattheintrin- i –0.47 9.4×10−3 P1 sic variability does not change dramatically from one AGN to z –0.27 1.4×10−1 P1 another,wewouldthereforeexpectto observea positivecorre- lation of the excess variance with redshift for the same band. Notes. The values of the 1z2_MBH sample are quoted for σ2 mea- rms However,wefoundstrongevidencethattheintrinsicvariability suredontimescalesofyears(top)andmonths(bottom). amplitude of AGNs is anticorrelated with bolometric luminos- ity. The weak selection effect apparent in Fig. A.1 shows that weactuallyobservethemostluminousobjectspredominantlyat 4.5.ExcessvarianceversusEddingtonratio higherredshift.Fromthisselectioneffectalonewewouldexpect ananticorrelationbetweentheexcessvarianceandredshift.The The last fundamentalAGN parameter for which we can probe factthatwe do notfinda dependencyof variabilityon redshift correlationswithvariabilityistheEddingtonratio.Thecorrela- for our AGN sample is most likely the result of the superposi- tioncoefficientsandp-valuessuggestananticorrelationbetween tionofthe two aforementionedeffects,whichare actingindif- σ2 andλ withhighsignificanceforbothstudiedvariability rms Edd ferentdirections.Thisexplanationagreeswithwhatweobserve timescales in the MBH and the 1z2_MBH sample. The values inFig.7,displayingtheexcessvarianceversusredshiftandthe for the 1z2_MBH sample are quoted in Table 5. However, the bolometricluminosity as a color bar. The slight anticorrelation relation is not as tight as the one with bolometric luminosity, of σ2 with redshift is counterbalanced by a positive correla- buttheuncertaintyofλ isconsiderablylargerbecausetheer- rms Edd tion, which is visible in various stripes of constant luminosity rorsofL andM bothcontributetoitsvalue.The1σdisper- bol BH showinganincreasingvariabilityamplitude.Thepositivecorre- sion of the black hole masses is 0.24dex accordingto Rosario lationofthevariabilityamplitudewithredshiftasaresultofthe etal.(2013),buttheactualuncertaintymightbeevenlargerdue redshift-dependentwavelengthprobedbyagivenfilterwasalso tosystematicerrors.Theanticorrelationisapparentforallcon- observedinearlierworks(Cristianietal.1990,1996;Hooketal. sidered PS1 bands, although it is less robust for the z band. P1 1994;CidFernandesetal.1996).Ourresultsalsoagreewithre- Moreover,comparingthetwovariabilitytimescales,wefindthe cent studies that did not find any significant evolution of vari- anticorrelation to be more significant for the σ2 values mea- rms ability with redshift or identified an observed correlation to be suredontimescalesof years,in contrastto whatis observedin causedbytheaforementionedselectioneffects(MacLeodetal. therelationwith L .However,giventhecomparablylargeun- bol 2010;Zuoetal.2012;Morgansonetal.2014).Finally,thelow certaintiesoftheλ values,thisdifferenceshouldnotbeover- Edd intrinsicscatterintherelationwithL suggeststhatbiasesdue interpreted. In addition, we checked that the ρ and P values bol S S to the broad redshift distribution of our sample are negligible obtainedforthesamesubsampleofobjectsareverysimilarfor comparedtothestrongdependenceonL . bothvariabilitytimescales. bol A129,page9of23 A&A585,A129(2016) Table6.Scalingofσ2 withλ . rms Edd ΔT ∼years,1z2_MBHsample Filter α β (cid:9) g −1.37±0.38 −2.97±0.34 0.28±0.07 P1 r −1.21±0.33 −3.01±0.30 0.33±0.06 P1 i −1.16±0.41 −3.12±0.38 0.38±0.07 P1 z −0.69±0.54 −2.74±0.47 0.41±0.05 P1 ΔT ∼months,1z2_MBHsample Filter α β (cid:9) g −1.16±0.89 −3.53±0.85 0.33±0.07 P1 r −0.95±0.32 −3.62±0.30 0.19±0.07 P1 i −1.26±0.70 −4.15±0.75 0.28±0.10 P1 z −0.48±0.61 −3.45±0.55 0.36±0.07 P1 Notes.Fittedvaluesoftherelationlogσ2 =β+αlogλ +(cid:9)foreach rms Edd consideredPS1bandassumingΔlogL =0.15andΔlogM =0.25. bol BH Thevaluesof the1z2_MBH samplearequotedforσ2 measured on rms timescalesofyears(top)andmonths(bottom). WeusedthesamefittingtechniqueasdescribedinSect.4.3 withapower-lawmodeloftheformlogσ2 =β+αlogλ +(cid:9) rms Edd to find the scaling of σ2 with λ . For the error of λ we rms Edd Edd assumedΔlogL = 0.15andΔlogM = 0.25foreachAGN, bol BH added in quadrature3. The results are listed in Table 6, and we show the data with the fitted relation for the r band in Fig. 8 P1 forthe1z2_MBHsample.Wenotethatowingtothelargeerror barsoftheEddingtonratioandthelargescatterintheanticorre- lation,the uncertaintiesof the fitted parametersare quitelarge. Considering those PS1 bands that exhibit a significant anticor- relation, that is, the g , r and i bands, we find logarithmic P1 P1 P1 slopes very similar to those of the L relation with α ∼ −1 bol withinthe1σerrorsforbothvariabilitytimescales.Theintrinsic scatteroftherelationbetweenσ2 andλ is∼0.2–0.4dex. rms Edd In contrastto the well-established anticorrelationof optical variability and luminosity, the actual dependency of the vari- ability amplitude on the Eddington ratio is less clear, but ev- Fig.8.Excessvariance(r band)measuredontimescalesofyears(top) P1 idence for an anticorrelation was detected in previous investi- andmonths(bottom)versusλ forthe1z2_MBHsample.Thebest-fit Edd gations (Wilhite et al. 2008; Bauer et al. 2009; Ai et al. 2010; powerlawandothersymbolsaredisplayedasinFig.5. MacLeod et al. 2010; Zuo et al. 2012; Kelly et al. 2013). The highlysignificantanticorrelationsbetweenσ2 andthequanti- andthereforeprovidesinformationaboutthepartofthePSDthat rms tiesλEddandLbolreportedinthisworkstronglysupporttheidea ispredominantlyintegratedbyourσ2rms measurements. thattheaccretionrateisthemaindriverofopticalvariability. 5.1.FittingtheCARMAmodel 5. Powerspectrumanalysis To modelour lightcurvesas a CARMA(p,q) process, we used Wedidnotcorrectourσ2 measurementsfortherangeinred- thesoftwarepackageprovidedbyKellyetal.(2014),whichin- rms shiftcoveredbyoursources,buttheexcessvariancedependson cludesanadaptiveMetropolisMCMCsampler,routinesforob- therest-frametimeintervalssampledbyalightcurve,therefore tainingmaximum-likelihoodestimatesoftheCARMAparame- ourresultsmaybeweaklybiased,althoughwedidnotfindany ters, andtoolsforanalyzingthe outputofthe MCMCsamples. strongtrendwithredshift.Furthermore,theindividualsegments Finding the optimal order of the CARMA process for a given of the MDF04 light curves used in calculating the excess vari- lightcurvecanbedifficult,andthereare severalwaystoselect anceontimescalesofmonthsdonothavethesamelengthingen- pandq.FollowingKellyetal.(2014),wechosetheorderofthe eral,introducingfurtherbiasesonthesetimescales.However,we CARMA model by invoking the corrected Akaike Information can independently verify our results by applying the CARMA Criterion(AICc;Akaike1973;Hurvich&Tsai1989).TheAICc modeling of variability described in Kelly et al. (2014), which foratimeseriesofN valuesy=y ,...,y isdefinedby 1 N does not suffer from the latter problems. What is more, this 2k(k+1) modelallowsanin-depthstudyofthePSDsofourlightcurves AICc(p,q)=2k−2logp(y|θ ,p,q)+ , (9) mle N−k−1 3 WealsoperformedthefitsusinglargeruncertaintiesofΔlogM = BH withkthenumberoffreeparameters,p(y|θ)thelikelihoodfunc- 0.3–0.4.However,becauseofthesystematicsteepeningofthederived slopesforlargerx-axiserrorswereportedinSect.4.3,theseerrorslead tion of the light curve, and θmle the maximum-likelihood es- toslopesthataremuchsteeperthantheoveralldistributionofthedata timate of the CARMA model parameters summarized by the implies. symbol θ. The optimal CARMA model for a given light curve A129,page10of23