Clustering of galaxies at faint magnitudes

9 Pages·0.14 MB·

by  J. U. Fynbo

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 am-
plitude 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-NgalaxybyFerna´ndez-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 pre-
sented 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
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whichsignificantlyimprovedthemanuscript.Thisworkmade
useoftheESO/ST-ECFScienceArchiveFacility.
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