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ASTRONOMY A&Amanuscriptno. (willbeinsertedbyhandlater) AND ASTROPHYSICS Yourthesauruscodesare: 2(12.03.3;12.12.1) February1,2008 Clustering of galaxies at faint magnitudes J.U.Fynbo1,2,W.Freudling2,3,andP.Møller2 1 InstituteofPhysicsandAstronomy,A˚rhusUniversity,DK-8000A˚rhusC.,Denmark 2 EuropeanSouthernObservatory,Karl-Schwarzschild-Straße2,D-85748,GarchingbeiMu¨nchen,Germany 3 SpaceTelescope–EuropeanCoordinatingFacility,Karl-Schwarzschild-Straße2,D-85748,GarchingbeiMu¨nchen,Germany 0 0 Received28May1999/Accepted21December1999 0 2 n Abstract. Significantuncertaintiesexistin the measuredam- clusteringathighredshift. While a moredirectmeasurement a plitudeoftheangulartwo-pointcorrelationfunctionofgalax- ofthetwopointcorrelationissignificantlymorepowerful(see J ies at magnitudes I ≈ 26 and fainter. Published results from e.g.LeFe`vreetal.1996),itscomputationrequiresdeepmagni- 7 HST and ground-based galaxy catalogs seem to differ by as tudelimitedredshiftsurveys,whichcurrentlyarenotfeasable much as a factor of 3, and it is not clear whether the corre- atthefaintestreachablemagnitudelimits.Bycontrast,thean- 1 lation amplitude as a function of magnitude increases or de- gularcorrelationfunctioncaneasilybecomputedfromphoto- v 5 creases in the faintest magnitude bins. In order to clarify the metricgalaxycatalogsalone.WiththeavailabilityoftheNorth- 0 situation,we presentnewresultsfrombothground-basedand ern Hubble Deep Fields (HDF-N, Williams et al. 1996), the 1 HST galaxycatalogs.Theangulartwo-pointcorrelationfunc- angularcorrelationfunctionhasbeencomputedtoanRmagni- 1 tionasafunctionoflimitingRandImagnitudeswascomputed tudeof29(Villumsenetal.1996,hereafterVFC,Colleyetal. 0 from a galaxy catalog created from the Hubble Deep Field - 1996).Thesestudiessuggestthattheamplitudeofthecorrela- 0 0 South (HDF-S) WFPC2 image. The measured amplitudes of tionfunctioncontinuouslydecreaseswiththemagnitudelimit / thecorrelationatanangularseparationof1arcsecareconsis- ofthesampleoveramagnituderangeofmorethan10magni- h tent with those measured in the Northern counter part of the tudesdowntothefaintestmagnitudelimitsprobedsofar.The p - field.Theflankingfields(FFfields) ofthe Hubbledeepfields amplitudeatanangularseparationofonearcsec,w(1′′),seems o wereusedtoextendthemagnituderangeforwhichwecompute tofollowcloselyapowerlaww(1′′)∝10−0.27R(e.g.Brainerd r t correlation amplitudes towards brighter magnitude bins. This etal.1995). s a allows easier comparison of the amplitudes to ground based Despite the apparent simplicity of computing the angular : data. The ESO NTT Deep Field catalog was used to measure correlationfunction,asignificantcontroversyhasarisenabout v i thecorrelationatsimilarmagnitudesfromgroundbaseddata. themagnitudelimitsatwhichthecorrelationfunctionflattens. X We findthatthe measuredcorrelationfromboththeHST and Measuring the correlation amplitude at a separation of 1 ar- r ground based samples are consistent with a continuously de- cmin,Brainerd&Smail(1998)foundthatthecorrelationam- a creasing clustering amplitude down to the faintest magnitude plitude reaches a minimum for samples with limiting I mag- limits.Finally,thenewlymeasuredcorrelationamplitudesasa nitude of about 23 and stays flat for even deeper magnitude functionofmagnitudelimitwerecomparedtopreviouslypub- limited samples. On the other hand, the deeper HDF-N mea- lishedmeasurementsatlargerseparations.Forthiscomparison, surementsataseparationof1arcsecfoundacontinuouslyde- thecorrelationfunctionwasapproximatedbyapowerlawwith creasing clustering amplitude down to the faintest magnitude anindexof0.8.Thescatterinthecorrelationamplitudesistoo limits.Thisdiscrepancycouldbeduetoanumberofdifferent largetobeexplainedbyrandomerrors.Wearguethatthemost reasonssuch asbiasedfield selection,differentredshiftselec- likelycauseistheassumptionthattheshapeofthecorrelation tionthroughdifferencesinthebandpasses,systematicerrorsin functiondoesnotdependonthemagnitudelimit. thedeterminationoftheclusteringamplitude(e.g.duetogra- dientsinthesensitivityofthedetector),orsimplyrandomfluc- Key words: Cosmology : observations – Cosmology : large tuations in the correlation amplitudes. However, it could also scalestructureofUniverse meanatruediscontinuityoftheshapeand/oramplitudeofthe correlationfunctionatamagnitudeofI ≈26orR≈27. Before suggesting the existence of a discontinuity of the 1. Introduction correlation amplitudes, several tests of the results should be carried out.First of all, both the groundbased results andthe The angular correlation function of galaxies w(θ) is still the HDF-Nresultsshouldbeverifiedbyindependentsamples.Sec- most commonly used tool for investigating the evolution of ondly,anyapparentdiscontinuityshouldbetestedwithasingle Sendoffprintrequeststo:J.U.Fynbo samplewhichcoverstherelevantmagnituderange.Uptonow, Correspondenceto:[email protected] availablegroundbasedhavenotbeennotdeepenoughtoover- 2 J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes lapwiththeHDFresults,whichareonlyavailableformagni- Table1.Surveydataforthedeepfieldsusedinthedetermina- tudesfainterthanR≈26.Thecurrentworkaddressesbothof tionofthetwopointcorrelationfunction thesepoints.WehaveusedthesouthernHDFfieldtoobtainan independentverificationofthepreviousHDF-Nresults.Inad- field Area I(complete) R(complete) dition,wehaveusedthe“flankingfields”oftheNorthernand arcsec2 HDF-S 3.7 28.0 28.5 SouthernHDFs(Williamsetal.1998)toderiveacatalogovera HDF-SFF 33.2 25.5 - largerareaandthusallowingustocomputethew(θ)atbrighter HDF-N 4.2 28.0 28.5 magnitudesfromanHDF-likesample.Finally,wehaveuseda HDF-NFF 8.7 25.5 - sampleofgalaxiesdetectedinadeepgroundbasedimage,the NTT 4.4 25.5 25.5 so-calledESONTTdeepfield(Arnoutsetal.1999)tocompute w(θ) from ground-based data at faint magnitudes. The com- bineddatasetsallowustoderivecorrelationamplitudesfrom both HST and groundbased data at identicalmagnitudebins, the numberof stars expectedfor a field with the galactic lati- andtherebysearchfordiscontinuitiesinthecorrelationampli- tudeanddepthastheHDF-S.Alsoweexcludesaturatedstars tudesasafunctionofmagnitude. foundbyvisualinspectionfromthegalaxycatalog.Thereare InSect.2,wepresenttheresultsfromtheHDFfieldswhich notverybrightstarsintheHDF-Ssowedidnotmaskoutany includetheNorthernHDF,theSouthernHDFandtheflanking regionswithintheareashowninFig.1.WechosetousetheAu- fields. In Sect. 3, we present new results from the ESO NTT tomaticApertureMagnitudes(AAM)insteadoftheCorrected deepfield.InSect.4,weinvestigatethecorrelationamplitude IsophotalMagnitudes(CIM),since thelatter provedunstable. asafunctionofmagnitudebycombiningthenewdatawithpre- As wellAAM asCIM are intendedto give anestimate ofthe viousestimatesintheliterature.Finally,inSect.5,wediscuss totalfluxofanobject.TheAAMismeasuredusinganelliptical theresultsandpresentourconclusions. aperturewithminoraxisb=2.5r1ǫandmajoraxisa=2.5rǫ1, where r is the first moment of the light distribution and ǫ is 1 theellipticity. 2. HDFsamples The resulting number of galaxies in the galaxy catalog is 1346with456intheWF2field,530intheWF3fieldand360 2.1. HDFSouth:Thegalaxycatalog intheWF4field. We used the “drizzled” HDF-S images distributed by the As zero-points for the photometry we use those released Space Telescope – European Coordinating Facility archive. by the STScI with the WFPC2 HDF-S data for the VEGA- The galaxy catalog was generated using the SExtractor pro- MAG system. In the following we shall refer to the F606W gram (Bertin & Arnouts 1996). The catalog of Clements & andF814WasRandIrespectively.Fig.2comparesthegalaxy Couch (1996) was used by VFC to compute w(θ) from the countsforHDF-NandHDF-S.Itisseenthatthedistributions HDF-Nimages.Inordertobeabletocompareourresultswith aresimilar.Inparticular,themagnitudebinsatwhichthenum- theresultsobtainedfromtheHDF-Nweusetheexactsameex- bercountsdropsignificantlyduetoincompletenessinthetwo tractionparametersasinputtoSExtractorasthosedescribedin samplesarealmostidentical.Thebasicpropertiesofthegalaxy Clements&Couch.We useaminimumobjectextractionarea catalogsfromthe two fieldsas wellas fromotherfields to be of30pixelsandadetectionthresholdof1.3σ abovetheback- discussedbelow,arelistedinTable1. ground.Asadetectionimageweuseamasterframecalculated asthesumofthecombinedimagestakenwiththeF606Wand 2.2. CorrelationfunctionfromtheHDFsouthcatalog F814Wfilters, which are similar to the R and I passbandsre- spectively. As a detection filter we use a top-hat filter of 30 WeextractfromthegalaxycatalogeightR-magnitude-limited connected pixels. We checked the reliability of the extraction samples with limiting magnitudes from R=25.5 to R=29.0 in parametersbycomparingthemasterframewiththeoutputob- 0.5magsteps.AsfortheHDF-N,galaxiesbrighterthanR=23 ject image, which SExtractor optionally provides. We could areexcluded. not detect any tendency to find spurious objects in the vicin- The angular correlation function w(θ) is estimated using ity of bright objects. The useful part of the combinedmosaic theoptimalestimatordescribedbyLandy&Szalay(1993): frames are differentbetween the HDF-N and HDF-S because theWFPC2fieldsofviewareslightlyrotatedrelativetothemo- DD−2DR+RR saiced image which is aligned with the north/south direction. w(θ)= , (1) RR Therefore,were-definedforeachoftheWF2,WF3andWF4 CCDsusefulareasfromvisualinspectionofthemasterframe. whereDDisthenumberofgalaxy-galaxypairs,DRthenum- Fig.1showsthechosenareas.Onlyobjectsdetectedwithinthe ber of galaxy-random pairs and RR the number of random- shown boundaries were used in the analysis. In order to ex- random pairs at the separation θ. We generate a set of 32000 cluded stars from the galaxy catalog we set an upper limit to randompointsdistributedaccordingtothePoissondistribution. theneuralnetworkstarparameterintheoutputSExtractorcat- Thenumberofgalaxy-randompairsandrandom-randompairs alogof0.98,whichexcluded20objects.Thisisconsistentwith ata givenmagnitudelimitare normalizedto the totalnumber J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes 3 Fig.1.TheWFPC2HDF-Sfieldshowingthetheusedareasof eachofthethreeWideFieldcameraCCDs. ofgalaxy-randomandrandom-randompairsrespectively.The errorsarecalculatedfromPoissonstatistics. Sincethemeandensityofgalaxieshastobeestimatedfrom the sample itself, the integral of the correlation function over Fig.2.Comparisonofthegalaxycountsin0.1magbinsinthe the surveyarea is forcedto zero. This reducesthe correlation HDF-NandHDF-SfieldsintheIandRbands.Thesolidhis- w(θ)functionbyanamountC givenbytheintegral: togramshowsthecountsforHDF-Nandthedashedhistogram 1 showsthecountsinHDF-S.Inthetopplotthefatlongdashed C = Ω2 ZZ dΩ1dΩ2w(θ), (2) line shows the galaxy counts in the HDF flanking fields and thefatsolidline thecountsinNTTSUSI DeepField.Thefat whereΩ isthesolidangleofthesurveyarea.C iscommonly solidlineinthelowerplotshowsthegalaxycountsfortheNTT referredtoasthe’integralconstraint‘.Assumingthatw(θ)isa SUSIDeepField. powerlaw, w(θ)=Aθ−γ+1, γ =1.8, (3) been taken duringthe HDF observingcampaigns.We use the then C = 0.078(0.077,0.080)A for the used parts of the twodeepestoftheHDF-NflankingfieldsandthenineHDF-S WF2(WF3,WF4)areashowninFig.1. flankingfields. Fig.3showstheobservedw(θ)fortheeightmagnitudelim- TheHDFproperfieldswerechosentoavoidbrightgalax- its.Theerrorbarsrepresent1σ Poissonerrors.Theamplitude ies.Itcouldbearguedthatthischoicebiasesthemeasuredcor- ofw(θ)isdeterminedbyfitting relation amplitude (Brainerd & Smail 1998). As the flanking w(θ)=Aθ−γ+1 − C, γ =1.8, (4) fields have been chosen with proximity to the HDF fields as the only criterion, no bias related to the HDF proper field is whichtakesintoaccounttheintegralconstraint. present. TheflankingfieldswereonlyimagedwiththeF814Wfilter. There is a total of 14 fields. Three of the southernfields con- 2.3. Flankingfields tainedverybrightstars.Ratherthanmaskingoutthesestarswe The high resolution deep images of HDF fields are uniquely decidednottousetheseframesintheanalysis.Catalogsfrom suitedtoinvestigatethecorrelationfunctionatthefaintestmag- theremaining11fieldsweregeneratedinthesamemanneras nitudes. However, their small field of view includes too few fortheHDF-Sfield.Theaveragenumbercountsasafunction brightergalaxiesto provideusefuloverlapwith groundbased of magnitudeare shown in the upperplot in Fig. 2. For mag- measurementsofthecorrelationfunction.Thissituationcanbe nitudesuptoan Imagnitudeofabout25thenumberperarea improved by taking advantage of the HDF “flanking fields”. is consistent with the one in the deep fields. The larger area ThesefieldsarecontiguoustotheHDFfieldsproperandhave therefore increases the sample in the brighter magnitude bins 4 J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes Fig.3. Themeasuredangularcorrelationfunctionfortheeightmagnitudebins.Thelimitingmagnitudeforeachbinhasbeen indicatedoneachplot.AlsoshownisthebestfittoafunctionoftheformgiveninEq.(4)andtheplusandminus1σfits. byalmostanorderofmagnitude.Therelevantparametersare andHDF-Sprojectstoobtainourfinalestimateofthecorrela- againsummarizedinTable1. tionamplitudesfromthe HDFfields.InFig.5weperformthe Finally, we measure the two point correlation function in same comparison for the I band, this time also including the thethreebins23<I<24.5,23<I<25.0,and23<I<25.5in measurementsfromtheflankingfields.Themeasurementsare thesamewayasdescribedinSect.2.2. inallbinsconsistentwithin1σ,andwe againcombineallthe availabledatafromtheHDF-NandHDF-Sprojectsincluding the Flanking fields to obtain our final estimate of the I band 2.4. ThecorrelationfunctionfromthecombinedHubbleDeep correlationamplitudesfromtheHDFfields. Fields InFig.4wecomparetheHDF-SRbandresultswiththeresults derived by VFC for the HDF-N. The triangles with 1σ error FortheRbandwedeterminew(θ)inthe8magnitudebins bars show the results for the HDF-S and the squares with 1σ R<25.5, R<26 ..., R<29. In these bins the number counts errorbarsshowtheresultsfromtheHDF-N.Forallmagnitude are not significantly influenced by incompleteness. For the I binsthemeasurementsfromthetwofieldsagreewithin1σ.We band we determine w(θ) in the 8 magnitude bins I<24.5, conclude that any differencesare due to random fluctuations, I<25,...,I<28.Theflankingfieldsarehereincludedinthemea- and thereforecombine all the available data from the HDF-N surementofw(θ)inthefirstthreebins. J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes 5 Thedeterminationofw(θ)andthepowerlawfitisdoneas describedin Sect. 2.2.We detecta positivecorrelationampli- tudeinall8binsinbothRandI. 3. TheESO-NTTdeepfield Inordertoverifythatthecorrelationamplitudeswemeasured intheHSTWFPC2framesarenotinsomewayduetoproper- tiesof theWFPC2 instrumentorHDFobservingorreduction procedure, it is desirable to compare to a catalog of galaxies derivedfromdeephigh-resolutiongroundbasedobservations. TheESO-NTTfield(Arnoutsetal.1999),forwhichtheimages are publicly available, is suitable for this purpose. A catalog of galaxies for this field has been provided by S. D’Odorico at the ESO website at http://www.eso.org/ndf. The field pro- videsdeepmulti-colourimagingtakenwithgoodsamplingof the PSF. The depthof the imagesis about26.7in R and 26.3 in I. The effective seeings of the co-added images are about 0.7-0.8arcsec. Wedeterminetheamplitudeofw(θ)inthemagnitudeinter- vals22–24.5,22–25.0and22–25.5forboththeIandRfilters. Fig.4.ComparisonfortheRbandoftheamplitudeofthean- We use an area of 4.4 arcmin2 confined by the pixel values gularcorrelationfunctioninsevenmagnitudebinsforHDF-N 40 <x,y< 1020 (chosen to avoid edge-effects). For this area (triangles)andHDF-S(squares).ThedatapointsfortheHDF-S we derive an integralconstraintof C = 0.0468A.Due to the havebeenshifted0.1magtothelefttoallowaneasiercompar- small size of the NTT SUSI Deep Field we cannotdetermine ison.Inallbinsdothemeasurementsagreewithin1σ. theamplitudeofw(θ)veryprecisely.Nevertheless,wedode- tectapositivecorrelationsignalinallbutoneofthemagnitude bins.Themeasuredamplitudesofw(θ) arein allbinsconsis- tent with the measurements in the two HDFs within 1σ (see Fig.6andFig.7). Stellar contamination of the sample can reduce the mea- suredcorrelationamplitude.Forasamplewithafractionf star ofgalaxies,themeasuredcorrelationamplitudeisreducedbya factor∝(1−fstar)−2.Fortunately,theexcellentseeingallows a firm galaxy/star discrimination. We follow Arnouts et al. in classifyingasstarsallobjectswithSExtractorclassifierlarger than 0.9 in the I band. This procedure removes about 5% of theobjectsinthefield.Forobjectsfainterthan24theSExtrac- torclassifier isnotefficient,so weassumeconservativelythat about 5% of the objects used to measure the correlation amplitude are stars. In fact, the star countsare shallower at faint magnitudesthanthatofgalaxies(Reidetal.1996,theirFig.1), which meansthat the star fractionat I > 24 mustbe smaller than for I < 24. Therefore,a conservativeupper limit of the errorintroducedbystellarcontaminationofthecatalogis10%, whichissignificantlysmallerthantherandomuncertainties. Fig.5. Comparisonfor the I band of the amplitude of the an- 4. Thecorrelationamplitudeasafunctionofmagnitudes gularcorrelationfunctionforsevenmagnitudebinsforHDF-N The correlation amplitudes presented above overlap neither (triangles),HDF-S(squares)andforthethreefirstbinsalsothe in magnitude nor scale with most such measurements from flankingfields(FF,circles).ThedatapointsfortheHDF-Shave groundbaseddata.Investigationsofthecorrelationamplitude beenshifted0.1magtotheleftandthepointsfortheflanking asafunctionofmagnitudeswhichusethesedatathereforere- fields0.1magtotherighttoallowaneasiercomparison.Inall quireknowledgeofthepowerlawindexδ inordertocompare binsdothemeasurementsagreewithin1σ. amplitudes measured at different separations. The HST data presented here are consistent with the usual δ = 0.8 power 6 J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes law for angular separations between 1 arcsec and 1 arcmin, medianmagnitudeoftwogalaxysampleswithlimitingmagni- butdonotruleoutsteeperofflatterslopes.Ontheotherhand, tude 25 at the faint end and limiting magnitudesat the bright ground based data have been used to derive the shape of the endof20and24isonlyabout0.5mag).Morelikely,however, correlationsfunctionatseparationsbetween0.5and5arcmin- thedisagreementisrelatedtotheextrapolationfrom1arcmin utes (e.g. Postman et al. 1998),but it is notclear whetherthe scales to 1 arcsec scales. In Postman et al. (1998, their Table slopescanbeextrapolatedtosmallerseparations.Inthefollow- 3)itcanbeseenthatthepowerδ seemtodeclineatI > 21to inginvestigation,wehaveadoptedaslopeof0.8inmostcases. smallervaluesthanthecanonicalvalue0.8(δ =0.7atI =22 The uncertainty in the slope is probablythe largest source of and δ = 0.5 at I = 23), especially at the smallest scales 0.5 uncertaintyinthecomparisonofmeasurementsfromdifferent arcsec< θ <5arcmin.AsimilartrendwasnotedbyBenoist surveys. etal.(1999).Brainerd&Smail(1998)findatbestfittingvalue Fig. 6showsthemeasuredamplitudesofw(θ) at1arcsec of δ = 0.7 at I > 24.5. A change in slope from δ = 0.8 to as a function of the limiting magnitudes of surveys in the R δ =0.5amountstofactorof3.4differenceintheextrapolation band from measurements by Brainerd et al. (1995), Roche et from1arcminscalesto1arcsecscales.Duetotheseuncertain- al.(1993),Couchetal.(1993)andWoods&Fahlman(1997). ties, ourcurrentknowledgeof theamplitudeandshapeofthe Wherenecessary(datafromRocheetal.andCouchetal.),we correlationfunctionintherange1arcmin<θ <1arcsecatthe havescaledoriginalcorrelationamplitudesquotedineachpa- faintestmagnitudelimitsreachablefromtheground,I=23–25, per to the amplitudeat 1 arcsec by assuminga powerlaw for isunfortunatelystillpoor. thecorrelationfunctionwithδ = 0.8.Similarly,Fig.7shows the measured amplitudesof w(θ) at 1 arcsec as a function of 5. Summaryandconclusions thelimitingmagnitudesofthesurveysintheIband.Themea- Fromthedatapresentedabovewecandrawthefollowingcon- surementsofPostmanetal.(1998),Brainerd&Smail(1998), clusions. Woods&Fahlman(1997),Benoistetal.(1999),Benoistetal. (in prep.) and Lidman & Peterson (1996) are included. Here, – The clustering amplitudes measured in the HDF-N field alltheamplitudesexceptthosefromWoods&Fahlman(1997) have been confirmed with the independent sample from andPostmanetal.(1998)wereextrapolatedtoaseparationof the Southern HDF. This rules out that the low amplitudes 1arcsecwithδ = 0.8.ForthePostmanetal.(1998)measure- measuredin the HDF-N field are dueto some cosmologi- ments, the data at 1 arcmin and 0.5 arcmin were extrapolated cal fluctuations. The HDF-N field seems to provide a fair usingtheslopemeasuredbytheauthorsintherange0.5arcmin sampleofhigh-redshiftgalaxies. –5arcmin. – TheclusteringintheHDFflankingfieldsisconsistentwith BothinFig.6andFig.7,weshowtheoreticalpredictionsof theonemeasuredfromgroundbasedsamplesatthebright the expectedcorrelationamplitudes.The modelsare identical endofthemagnituderange,andwiththeonesmeasuredin tothoseshownbyVFC,butwiththeassumedredshiftdistribu- theHDFsatthefaintend.Thisprovidesadditionalsupport tion in the faintmagnitudebinsderivedfromthe photometric totheviewthattheHDFfieldsrepresentindeedafairview redshiftdistributionofHDF-NgalaxybyFernańdez-Sotoetal. ofthehigh-redshiftuniverse. (1999). The median redshifts adopted at magnitudes brighter – The measured correlation amplitudes in the ESO-NTT than26intheRbandarethesameasthoseusedbyVFC.The galaxysampleinmagnitudebinswhichoverlapthoseofthe modelsarespecifiedbyapresentclusteringlengthr0 andand HDF sample are consistent with the HDF measurements. evolutionparameterǫ(seeVFCfordetails).Themodelsshown This observation verifies that correlations measured with havea presentclusteringlengthofr0 = 4.0h−1 Mpc andas- WFPC2arenotduetosomeinstrumentaleffects. sumenoevolutionofclustering(dottedline,ǫ=0),linearevolu- – Theresults plottedin Fig. 6 show a continuousdecline of tionofclustering(solidline,ǫ=0.8)ornon-linearevolutionof the amplitude of the correlation function in R. We do not clustering(dashedline,ǫ=1.6). confirmtheBrainerd&Smail(1998)detectionofaflatten- Several data sets shown in Fig. 6 and Fig. 7 are in con- ingintheIbandatI=22–23. flictwitheachotheratthe5-7σlevel.Thisreflectsthefactthat – There is some indication of a flattening of the correlation themainsourceoferrorsaresystematicerrors,whiletheerror amplitudesatthefaintestlevels(R≈27orI ≈26).Using barsinmostcasesonlyincludestatisticalerrors.Inparticularat models for the redshift distribution, VFC interpreted this faintmagnitudes,systematicuncertaintiesarehardtoquantify. flatteningasevidenceforlinearevolutionoftheclustering ThedisagreementsappeartobelargerintheI-band.Thismight of a galaxy population which at present has a correlation beduetothefactthatintheIband,variationsintheeffective lengthofabout4h−1Mpc.Thedatapresentedinthispaper filterbandpassesresultinrelativelylargedifferencesinthered- arestillingoodagreementwiththisinterpretation. shiftdistributionof extractedgalaxysamples. Also, the limit- – Interpretationof Fig. 6 and 7 is difficultgiven the signifi- ing magnitude on the abscissa is somewhat dependingon the cantdisagreementbetweendatapointsfromdifferentstud- limitingmagnitudesatthebrightend.However,asthegalaxy ies. This disagreement could be caused by a change of δ counts(N)increasessteeplywiththelimitingmagnitudeatthe as a functionof magnitudeandscale. To reach a finalun- faintend (mag , logN ∝ 0.6 mag ), the uncertaintydue derstandingofthetwo-pointcorrelationfunctiondatacom- lim lim todifferentlimitingmagnitudesissmall(thedifferenceinthe piledin this work,it thereforeappearsto be essential first J.U.Fynboetal.:Clusteringofgalaxiesatfaintmagnitudes 7 HDF fields HDF fields NTT SUSI Deep Field NTT SUSI Deep Field Benoist et al. Roche et al. Brainerd & Smail Couch et al. Woods & Fahlman Woods & Fahlman Lidman & Peterson Brainerd et al. Postman et al. Fig.6. Theamplitudeof w(θ) at 1 arcsec as a functionof the Fig.7. Theamplitudeof w(θ) at 1 arcsec as a functionofthe limitingRbandmagnitudeforanumberofgroundbasedsur- limiting I band magnitudefor a number of groundbased sur- veysand the measurementsfrom the HubbleDeep fields pre- veysand the measurementsfrom the Hubble Deep fields presented in this paper. The superimposed curves are theoretical sented in this paper. The superimposed curves are again the predictionsforapopulationofgalaxieswithapresentcluster- modelswith a presentclustering lengthof r0 = 4.0h−1 Mpc inglengthofr0 = 4.0h−1 Mpcandassumingnoevolutionof and assuming no evolution of clustering (dotted line), linear clustering(dottedline),linearevolution(solidline)aswellas evolution(solidline)aswellasnon-linearevolutionofcluster- non-linearevolutionofclustering(dashedline). ing(dashedline). tounderstandthedetailedbehavioroftheshapeofthecor- LandyS.D.,SzalayA.S.,1993,ApJ412,64 relationfunction. LeFe`vreO.,HudonD.,LillyS.J.,etal.,1996,ApJ460,L1 LidmanC.E.,PetersonB.A.,1996,MNRAS279,1357 PostmanM.,LauerT.R.,SzapudiI.,OegerleW.,1998,ApJ506,33 ReidI.N.,YanL.,MajewskiS.,ThompsonI.,SmailI.,1996,AJ112, Acknowledgments 1472 RocheN.,ShanksT.,MetcalfeN.,FongR.,1993,MNRAS263,360 We wish tothankC. BenoistandB. Thomsenforhelpfuldis- Villumsen J.V., Freudling W., da Costa L.N., 1996, ApJ 481, 578 cussions and C. Benoist for giving giving us access to some (VFC). ofhisresultsfromtheEISdatapriortopublication.Wethank WilliamsR.E.,BrettB.,DickinsonM.,etal.,1996,AJ112,1335 S. D’Odorico for making available the catalog from the NTT WilliamsR.E.,BaumS.A.,BergeronL.E.,etal.,1998,AAS193,7501 Deep Field. We thank the referee, T. Brainerd, for comments WoodsD.,FahlmanG.G.,1997,ApJ490,11 whichsignificantlyimprovedthemanuscript.Thisworkmade useoftheESO/ST-ECFScienceArchiveFacility. References Arnouts S., D’Odorico S., Cristiani S., Fontana A., Giallongo E., 1999,A&A341,641 BenoistC.,daCostaL.N.,OlsenL.F.,etal.,1999,A&A346,58 BertinE,ArnoutsS,1996,A&AS117,393 BrainerdT.G.,SmailI.,1998,ApJ496,L137 BrainerdT.G.,SmailI.R.,MouldJ.R.,1995,MNRAS275,781 ClementsD.L.,CouchW.J.,1996,MNRAS280,43L ColleyW.N.,RhoadsJ.E.,OstrikerJ.P.,Spergel,D.N.,1996,ApJ473, L63 CouchW.J.,JurcevicJ.S.,BoyleB.J.,1993,MNRAS260,241 Fernańdez-SotoA.,LanzettaK.M.,YahilA.1999,ApJ513,34 This figure "h1572_f1.gif" is available in "gif"(cid:10) format from: http://arXiv.org/ps/astro-ph/0001105v1 This figure "h1572_f3.gif" is available in "gif"(cid:10) format from: http://arXiv.org/ps/astro-ph/0001105v1