The dark clump near Abell 1942: dark matter halo or statistical fluke?

1.3 MB·English

by  A. von der Linden

Astronomy&Astrophysicsmanuscriptno.aaclump.small February2,2008
(DOI:willbeinsertedbyhandlater)
The dark clump near Abell 1942:
dark matter halo or statistical fluke? ⋆
A.vonderLinden1,2,T.Erben1,P.Schneider1,andF.J.Castander3
1 Institutfu¨rAstrophysikundExtraterrestrischeForschung,Universita¨tBonn,AufdemHu¨gel71,53121Bonn,Germany
5 2 MaxPlanckInstitutfu¨rAstrophysik,Karl-Schwarzschild-Str.1,Postfach1317,85741Garching,Germany
0 3 Institutd’EstudisEspacialsdeCatalunya/CSIC,GranCapita2-4,08034Barcelona,Spain
0
2 Received20January2005;accepted??

n
a Abstract. Weaklensingsurveysprovidethepossibilityofidentifying darkmatter halosbased ontheirtotalmattercontent
J ratherthanjusttheluminousmattercontent.OnthebasisoftwosetsofobservationscarriedoutwiththeCFHT,Erbenetal.

0 (2000)presentedthefirstcandidatedarkclump,i.e.adarkmatterconcentrationidentifiedbyitssignificantweaklensingsignal
2 withoutacorrespondinggalaxyoverdensityorX-rayemission.
WepresentasetofHSTmosaicobservationswhichconfirmsthepresenceofanalignmentsignalatthedarkclumpposition.

2 Thesignalstrength,however,isweakerthanintheground-baseddata.Itisthereforestillunclearwhetherthesignaliscaused
v
byalensingmassorisjustachancealignment.WealsopresentChandraobservationsofthedarkclump,whichfailtoreveal
2
anysignificantextendedemission.
4
A comparison of the ellipticity measurements from the space-based HST data and the ground-based CFHT data shows a
4
remarkableagreementonaverage,demonstratingthatweaklensingstudiesfromhigh-qualityground-basedobservationsyield
1
reliableresults.
0
5
0 Keywords. gravitationallensing–darkmatter–galaxies:clusters:general
/
h
p
- 1. Introduction formation, this implies that the large galaxies we see today
o
formed from mergers of protogalaxies. Observations support
r Inthecurrentlyfavoredcosmologicalmodel,structureforma-
t this theory: we see more irregular, small galaxies at higher
s tion in the universe is dominated by collisionless Cold Dark
a redshifts and many merger systems and galaxies showing ev-
Matter (CDM). The model of structure formation by gravita-
: idence for recent mergers. This bottom-up scenario also calls
v tional collapse in a pressure-less fluid is able to successfully
for galaxy clusters to build up through the merger of smaller
i
X reproducethefilamentarylargescalestructureobservedinthe
halos.
universe (e.g. Peacock 1999). However, for the formation of
r While it may be possible to (temporarily) drive gas from
a galaxies and galaxy clusters, gas dynamics play an important
galaxy-sized halos, when such halos merge to form cluster-
role.Itseemsobviousthatgalaxyformationistriggeredwhen
sized objects, the majority of them should contain galaxies
gasfallsintothepotentialwellsofdarkmatterconcentrations.
and/or hot gas, so that the resulting massive halo is expected
We thereforeexpectto findgalaxiesatthe high-densitypeaks
tocontainasubstantialamountofluminousmatter.Acluster-
ofthedarkmatterdistribution.IntheCDMscenario,smallha-
sized halo very poor of luminous matter (dark clump) would
loscollapseearlierandmergetolargerhaloslater.Forgalaxy
requireamechanismtodrivethegasoutofallthesmallerha-
Sendoffprintrequeststo:A.vonderLinden los from which it assembled or from the massive halo itself.
e-mail:[email protected] Bothcasesarehighlyunlikely:thefirstisveryimprobable,the
⋆ BasedonobservationsmadewiththeNASA/ESAHubbleSpace secondverydifficultduetothehighmassoftheobject.Atthe
Telescope, obtained at the Space Telescope Science Institute, which moment,therearenowell-motivatedphysicalprocessestoex-
is operated by the Association of Universities for Research in plaineitherscenario.
Astronomy, Inc., under NASA contract NAS 5-26555; on observa-
The discovery of a dark clump would therefore call for a
tions made with the Chandra X-ray Observatory, operated by the
critical reevaluation of our current understandingof structure
Smithsonian Astrophysical Observatory for and on behalf of the
formationintheuniverse.Currently,theonlytoolavailableto
National Aeronautics Space Administration under contract NAS8-
searchforsuchobjectsisgravitationallensing,asitprobesmat-
03060; and on observations made with the Canada-France-Hawaii
terconcentrationsindependentoftheirnature.
Telescope (CFHT) operated by the National Research Council of
Canada,theInstitutdesSciencesdel’UniversoftheCentreNational In the course of a weak lensing study, Erbenetal. (2000)
delaRechercheScientifiqueandtheUniversityofHawaii. announced the possible discovery of a dark clump, about 7
′
2 A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke?
southofthegalaxyclusterAbell1942.Thisassertionisbased distortions of a foreground lensing mass are larger; addition-
onsignificantalignmentsignalsseenintwoindependenthigh- ally,duetothelack ofseeing,its shapemeasurementsshould
qualityimages,takenwiththeMOCAMandUH8Kcamerasat be more reliable. If the alignment signal seen in the ground-
theCFHT.Thereisnoassociatedapparentgalaxyoverdensity baseddataisduetoalensingmass,itshouldthusbeevenmore
visibleintheseimagesnorindeepH-bandimagesanalyzedby significantintheHSTdata.
Grayetal.(2001).ThereisfaintX-rayemissionabout1′from The structure of this paper is as follows. In Sect. 2 we
the lensing centroid detected by the ROSAT survey, but it is presenttheopticaldataavailabletous,namelytheCFHT im-
unclearwhetherthiscouldbeassociatedwithalensingobject. ages of the original detection and the HST data, along with
If the alignment signal is due to a lensing mass at a similar our data reductionmethodsto extract objectcatalogs suitable
redshiftasthecluster,z = 0.223,thishalowouldhaveamass forlensingstudies.Sect.3givesabriefoverviewoftheweak
of the order of 1014h−1M . At a higher redshift (0.8 - 1.0), it lensingmethodsemployedinthispaper.InSect.4wepresent
wouldrequireamassofth⊙eorderof1015h 1M . are-analysisoftheI-bandimageoftheCFHTdata.Ourweak
−
⊙
Therearecurrentlythreemoresuchdarkclumpcandidates lensinganalysisoftheHSTdata,whichconfirmsthealignment
intheliterature: signal,butnotitsstrength,isdescribedinSect.5.InSect.6we
use a deep Chandra image to show that the ROSAT source is
- Umetsu&Futamase(2000)finda candidateintheirweak likelytobeaspuriousdetection.Theappendicesillustratevar-
lensinganalysisofthegalaxyclusterCL1604+4304using ioustestsforsystematics(App.A)andacomparisonofthein-
datafromtheWFPC2 cameraoftheHST.Intwoseparate dividualshapemeasurementsofobjectscommontotheCFHT
datasets, they find a peak 1.′7 southwest from the cluster andHSTdatasets(App.B).
center,whichcorrespondstoabout830h 1 kpcatthered-
−
shiftofthecluster(z = 0.897).Theyestimatethe massof
the objectto beabout4.8 1014h 1M , assumingitislo- 2. Opticaldata
−
× ⊙
catedatasimilarredshiftasthecluster.
Thegoalofthisworkistounderstandtheoriginofthelensing
- Mirallesetal.(2002)foundaconspicuoustangentialalign-
signalseenbyErbenetal.(2000).We thereforeconsiderboth
ment of galaxies in an image taken by the STIS camera
theground-baseddatasetusedintheoriginaldiscoveryaswell
aboardtheHSTasaparallelobservation.However,follow-
astheHSTdata.Suchatreatmentalsoallowsforadirectcom-
up wide-fieldobservationswith theVLT failed todetecta
parison of the ellipticity measurements of objects detected in
weak lensing signal (Erbenetal. 2003), so that a chance
bothdatasets.
alignmentof52galaxiesintheoriginalSTISanalysisisat
thispointconsideredthemostplausibleexplanationforthis
candidate. 2.1.Ground-baseddata
- Dahleetal. (2003) identify a dark clump candidate about
Our analysis concentrates on the same I-band image as used
6 southwest of the galaxy cluster Abell 959 (z = 0.286)
′
in Erbenetal. (2000), as this covers most of the area imaged
inimagestakenwith theUH8KcameraattheCFHT with
by the HST. We use Chip 3 of a mosaic observationof Abell
evidencethatthisisadarksub-clumpofthecluster.Ifthis
1942takenwiththeUH8Kcamera,withapixelscaleof0.206.
isindeedanobjectattheredshiftofthecluster,theydeduce ′′
amassof(1.1 0.3) 1014h 1M . 9 exposures of 1200 s went into the final image, which has
−
± × ⊙ a seeing of 0.74. Unfortunately, a photometric calibration is
′′
Weinberg&Kamionkowski (2002) argue that about one missing.
outoffiveDarkMatterhalosidentifiedbyweaklensingshould
be a non-virialized halo, i.e. a halo which is in the process
2.2.Space-baseddata
of collapsing and has not yet reached dynamicalequilibrium.
Such objects should have only very little X-ray emission and Our HST datais a WFPC2 mosaic(approximately5 4) of
′ ′
×
abouthalftheprojectedgalaxydensityofvirializedhalos.The sixpointings,eachconsistingof12ditheredexposureswithan
luminosity of such objects would therefore be very difficult exposuretimeof400seach,takenbetweenMay20thandJune
to determine. Accordingly, distinguishing between pure Dark 1st,2001.ThepositionofthemosaicwithrespecttotheI-band
Matter halos and normal, non-virializedhalos may be almost image fromthe CFHT is shownin Fig. 1 The filter employed
impossibleinthesecases. wasF702W.
However,thenoiseinweaklensinganalysesduetointrin-
sic ellipticities of galaxies can have a profound effect on the
2.2.1. Datareduction
statistics of the number of halos detected per area. Intrinsic
ellipticities may mimic tangential alignment, thereby causing OurreductionoftheHSTdataisbasedlargelyonthedither
falsepeaksorboostingthesignificanceoflensingsignals(e.g. package (Fruchter&Hook 2002) for IRAF. Each of the six
Hamanaetal.2003). pointingswas reducedseparately.Simultaneousprocessingof
TodeterminethenatureofthedarkclumpnearAbell1942, thefourindividualchipsisdoneautomaticallybythedither
we obtained a set of HST observations of the field (General routines.Duetotheirbettersignal-to-noisebehavior,onlythe
Observer Program, Proposal ID 9132, PI Erben). The HST chips of the Wide Field Camera, namelyChips 2, 3, and 4 of
probes fainter and thus more distant galaxies, for which the WFPC2,wereusedforthelateranalysis.Theditherpatternof
A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke? 3
Mediancoaddition: Using the previously determined offsets,
theoriginalimagesaremediancombined.Theyaremapped
viathedrizzlealgorithmontoanoutputgridwithpixels
ofhalftheoriginalsize.
Maskcreation: The median image is mapped back onto the
original resolution and offset of the original frames using
the blot algorithm, the inverse of drizzle.The original
frameisthencomparedtothemedianimagetoidentifycos-
micraysviathederivanddriz crtasks.Thus,foreach
frame a mask is created identifying cosmic rays. This is
combinedwithamaskidentifyingdefectorpossiblyprob-
lematicpixels,whichissuppliedwitheachrawframe.
Offsetdetermination: For each frame, those pixels that are
flagged in the mask are substituted by their value in the
blottedimage (the transformedmedianimage).These im-
agesarethencross-correlatedtodeterminetheoffsetsmore
precisely. Possible small rotation angles have to be found
manually.
Coaddition: The images are drizzled onto an output grid of
half the pixel size (pix-frac= 0.5), i.e. twice the origi-
nal resolution. The value of each output pixel is obtained
viaaveraging,wherepixelswhichareflaggedinthemask
imageareomitted.Thedrop-size(i.e.ascalingapplieto
theinputpixelsbeforebeingmapped)usedis0.6.
Performing this routine on all of the six pointings, we obtain
18single-chipimages.AmosaicoftheseisshowninFigure2.
2.3.Catalogextraction
Forbothdatasets,weusedasimilarprocesstobuildtheobject
catalogs. Differences in the procedure arise mainly from the
smallfield-of-viewandmosaicnatureoftheHSTdata.
2.3.1. Preparatorysteps
Fig.1.TheoutlinesofthesixHSTpointingssuperposedonthe We manually updated the astrometric information of the
CFHT I-bandimage.The centerof Abell1942liesin the top ground-basedimageuntilitsbrightobjectscoincidewiththeir
half of the CFHT image, the HST mosaic is centered on the positionsasgivenintheUSNO-A2catalog(Monetetal.1998).
darkclumpposition. Thisstepisnotnecessaryforthelensinganalysisoftheground-
based image, but providesus with a reference catalog for the
astrometriccalibrationoftheHSTmosaic.
TheHSTimagesarealignedroughlywithskycoordinates,
theimagesallowsustoachieveahigherresolutioninthecoad- but since each chip is read out along a different chip border,
dedimageviathedrizzlealgorithm(Gonzagaetal.1998). theindividualimagesarerotatedwithrespecttoeachother.To
Thestepsofthereductionareoutlinedinthefollowing: avoid confusion, we first rotate the images of Chips 2 and 4
by 90 ,respectively,suchthatnorthisupandeasttotheleft
◦
∓
Roughcosmicrayremoval: Tofindtheoffsetsbetweentheim- (approximately).This is performedsolely as a rearrangement
ages, they first have to be cleaned of cosmic rays, which ofpixelvaluestoavoidanyadditionalresamplingprocess.
would otherwise falsify a cross-correlation.Each frameis
cleanedusingtheprecortask.Thisleavesonlyobjectsofa
2.3.2. Masking
minimalsizeintheimage,whichshouldbestarsandgalax-
ies,withlittlecontaminationbycosmicrays. In order to avoid noise signals and distorted ellipticity mea-
Offsetestimation: As the individual frames are dithered with surements,wemaskoutbrightstarsandtheartefactstheycause
respect to each other, it is necessary to find their relative (diffractionspikes,blooming,CTEtrails,aripple-likestructure
offsets. This is done by performing a cross correlation of attheeasternedgeoftheI-bandimage,andbrighteningofthe
the cosmic ray cleaned images produced in the previous backgroundalongcolumnsoftheHSTchipsintwocases).In
step. theHSTimage,wealsomaskthebrightgalaxyinthefield.
4 A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke?
Fig.2.The HST mosaicwith an illustrationof the variousdarkclumpcentroidscited in thetext:the centroidwe foundinthe
ground-baseddata(⋆);theoriginalpositiongivenbyErbenetal.(2000)(+);thepeakpositionintheHSTdata( );thecenter
△
of the X-ray emission as found by ROSAT ( ) and by Chandra ( ); the position of the peak found in the 200 filter scale
′′
∗ ×
forthefaintground-basedgalaxies( );andthegalaxynumberoverdensityofmedium-brightHSTgalaxies(^).Additionally,
◦
we plota circle of 120 radiusaroundthe first position.For our M filter functionof this radius,the maximalweightis then
′′ ap
assignedalongthedashedcircle.Assumingatypicaluncertaintyof1 forcentroidsfoundviaweaklensing,allcitedpositionare
′
compatiblewiththedarkclump.
2.3.3. Sourceextraction The astrometric calibration plays an important role for a
mosaicdatasetsuchasourHSTimages,asitgivestheposition
oftheimageswithrespecttoeachother.Indeed,thepositions
WeusedSExtractor(Bertin&Arnouts1996)toidentifyob-
on the sky will be used later on in the lensing analysis rather
jectsintheimage.SExtractorconsidersN connectedpixels
thanxandypositiononthechip.
thatareatalevelk σ abovetheskybackgroundasanobject,
sky
·
withσ beingthestandarddeviationofthebackgroundnoise.
sky
For the CFHT image,we used N = 3 andk = 1.0.These are Forareferencecatalog,wehadextractedacatalogofbright
verylowthresholds,butsincewelaterwanttocorrelateobjects objects from the astrometrically calibrated ground-based I-
presentinbothdatasets,westrivetoobtainahighnumberden- bandimage(seeSect.2.3.1).Wematchtheentriesoftherefer-
sityofobjects.FortheHSTimagesweusedN =3andk=1.5. encecatalogto objectsfoundinthe HST images.Thisallows
thedeterminationofthepointingandthedistortionoftheim-
age.Objectsthatare detectedbothinthe I-bandandtheHST
HST:astrometricandphotometriccalibration imagecanthenbeidentifiedbyskycoordinates.
A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke? 5
The photometric calibration is done based on the relevant 2.3.7. Shearestimation
keywordsoftheHSTimageheaders.
Aftertheanisotropycorrection,thesecondstepinretrievingan
estimateofthelocalshearfromellipticitymeasurementsisthe
2.3.4. Ellipticitymeasurement correctionforthePg tensor.ItisacombinationofPsmandthe
shearpolarizibilitytensorPsh:
FortheobjectsintheSExtractedcatalogs,we measuretheel-
lipticitiesusingamodifiedversionoftheimcatsoftware,fol- Pg =Psh Psm(P⋆sm)−1P⋆sh , (4)
−
lowing the method of Kaiseretal. (1995). We used the half- where the starred quantities are the corresponding tensors as
light radius as measured by SExtractoras the radius of the measuredonstellar-sizedobjects.Notethattheweightfunction
weightfunctionwithwhichthebrightnessprofileisweighted. withwhichP⋆shandP⋆smaremeasuredshouldbethesameas
The measured image ellipticity χ is related to the source thatusedfortherespectiveobject.
ellipticityχ(s)by Thebasicassumptionofweaklensingstudiesisthattheav-
eragesourceellipticitiesvanish,i.e. χ(s) =0.Wealsoassume
χ=χ(s)+Pgg+Psmq⋆ , (1) that (Pg) 1χs =0,sobyaveragingheq.(i1)weobtain:
−
h i
g= (Pg) 1χaniso . (5)
wheregisthereducedshearinducedbyalensingmasswhich h − i
we ultimately want to determine. q⋆ is the stellar anisotropy Thus the expectation value of the quantity (Pg)−1χaniso is the
kernel, i.e. the anisotropy of the PSF for point-like objects. reducedsheargattherespectivepoint.
Pg and Psm (smear polarizibility tensor) are tensors describ- Pg is an almost diagonal tensor with similar elements on
ing a galaxy’s susceptibility to the two distorting effects. thediagonal.Infact,intheabsenceofaweightfunctionanda
Theyarealsomeasuredfromagalaxy’slightdistribution(see PSFitselementswouldbe:Pg = Pg =2,Pg = Pg =0.We
11 22 12 21
Bartelmann&Schneider2001,foraderivation). canthereforeapproximatethetensorPgbyascalarquantity:
1
Pg = tr(Pg). (6)
s 2
2.3.5. Anisotropycorrection
IthasbeenshownthatwhilethefullPgtensorcanoverestimate
Tocorrectfortheanisotropyinducedbythetelescope–detector theshear,thisapproximationismoreconservativeandwillonly
system, we measurethe ellipticities ofthe stars presentin the underestimatethetrueshear(Erbenetal.2001).
field;fortheseeq.(1)simplifiesto Wethereforeuseavariantofeq.(5)asourestimateofthe
localshearateachgalaxy’sposition:
χ⋆ =P⋆smq⋆ . (2)
χaniso
ε= . (7)
Pg
Stars are selected from a magnitude vs. radius plot. We fit a s
third-order polynomial in chip position (x,y) to the quantity Atthispoint,werejectthoseobjectsfromthecatalogthathave
χ⋆ aradiusequaltoorsmallerthanthestellarlocus,aresaturated
(x,y)toestimateq⋆(x,y).
(cid:18)0.5tr(P⋆sm)(cid:19) stars,orhaveafinalellipticityof ε >1.
Thus, we obtain anisotropy-corrected ellipticity measure- | |
ments:
2.3.8. Rotationofellipticities
χaniso =χ Psmq⋆ . (3)
− The ellipticities were measured with respect to the x-axis of
eachimage,whichforallimagesrunsapproximatelyalongthe
2.3.6. Modificationfor theHSTimages eastaxis.However,thelensinganalysesaredoneinskycoor-
dinates.Wethereforeneedtotransformtheellipticitymeasure-
Since the PSF cannot be described analytically across chip- ments so that their position angle is measured relative to the
borders, the anisotropy correction for a mosaic has to be ap- rightascensionaxis.Thisisdonewiththetransformation
plied to the single chips. With the small field of view of the
ε εe2ıϕ
WF chips, we have the added difficulty that for each image, →
thereareonlyaboutfivestarsthatcouldbeusedforthepolyno- whereϕistheangleofrotationoftheimage.
mialfitting,obviouslynotenough.Sincetheimagesweretaken After this step, the 18 single-image catalogs of the HST
consecutively, we can assume that the PSF does not change dataaremergedintoonecatalog.
considerablybetweenthesixpointings.Wethereforeapplyan
anisotropycorrectionforeachchipbaseduponallthestarsthat
2.3.9. Weighting
werefoundinthesiximagestakenbythatchip.
The18single-chipcatalogsarecombinedtothreecatalogs, To describe the reliability of its shape measurement,we want
oneforeachchip.Becausetherearestillfewstarseveninthese to assign a weight to each galaxy, based on its noise proper-
catalogs, the stellar sequence is selected manually. For each ties. Since our shear estimates are gained from averages over
catalog,athird-orderpolynomialisfitted. ellipticities,agoodweightestimateis
TheanisotropycorrectionoftheHSTimagesisfurtherdis- 1
w . (8)
cussedinAppendixA.1. i ∝ σ2
i
6 A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke?
whereσ isthe(two-dimensional)dispersionofimageelliptic- whereφisthepolarangleofthevectorϑ.Theweightfunction
i
ities in the ensemble over which is averaged.For intrinsic el- Qisdeterminedintermsofw.
lipticities,σε 0.3 0.4(measuredfromdata).Measurement Eq. (9) is intuitively clear: a lens most often deforms im-
≈ −
errorscauseσi tobehigherthanthis.Wefollowtheweighting agessotheyaligntangentiallytothecenterofmass.Anaver-
schemeofErbenetal.(2001),andselectastheensembletoav- ageoverthetangentialcomponentsofgalaxyellipticitiesmust
erageoverthe20closestneighborsoftherespectivegalaxyin therefore be a measure of the surface mass. With this inter-
theparameterspacespannedbyhalf-lightradiusrgandmagni- pretation, Map is a useful quantityin its own righteven if the
tudem. weak lensing assumption, γ = ε , breaks down or if part of
h i
theweightfunctionliesoutsidethefield.
2.3.10. Finalcuts Theimaginaryshearcomponentisthecrosscomponent:
After the weighting, we remove objects with ε > 0.8. Such
objects would dominate the shear signal, but| t|hese are also γ (θ0;ϑ)= m γ(θ0+ϑ)e−2ıφ . (11)
× −I
the objects that are most afflicted by noise in the Pg ten- (cid:16) (cid:17)
sor. Additionally, we use only objects for which Pg > 0.3. Substituting it for the tangential shear in eq. (9) yields M
a×p
This leaves about 2000 objects in both catalogs, which cor- whose expectation value vanishes. Evaluating it analogous to
responds to 20 galaxies/arcmin2 for the I-band image and 65 M can thus be used as a method to check the quality of the
ap
galaxies/arcmin2fortheHSTimage. dataset.
3. Weaklensingmethods 3.2.1. Applicationtorealdata
Weaklensinganalysesarebasedonusingestimatesofthelocal – InordertoapplytheweightfunctionQtofinitedatafields,
shearγtoreconstructinformationontheconvergenceκ,which a cut-offradiusθ shouldbe used, beyondwhich the fil-
out
isadimensionlessmeasureofthesurfacemassdensity.Inthe terfunctionvanishes.Otherwise,thearea ofintegrationis
weaklensingregime,κ 1,sothat ε =g=γ/(1 κ) γ. not well sampled by galaxy images. A compensated filter
≪ h i − ≈
w(ϑ), for which w(ϑ) 0 for ϑ > θ yields a weight
out
| | | | ≡ | |
functionQ(ϑ)whichvanishesbeyondthesamecut-offra-
3.1.Massreconstruction
| |
dius. We use filter and weight function as introduced in
Both the shear γ and the convergence κ are linear combina- Schneideretal.(1998),withl=1.
tions of second derivatives of the lensing potential, so that it
is possible to express κ as an integral over γ via the Kaiser–
SquiresInversion(Kaiser&Squires1993).Thismethodisusu- 1
w(x)
allynotapplieddirectly,astheshotnoiseintroducedbysum- Q(x)
0.8
mingoverindividualgalaxies(shearmeasurements)produces
infinitenoise.Thiscanbeavoidedbyfirstsmoothingtheshear
0.6
measurements; however, such a smoothing scale introduces
correlated errors. Another problem arises from the limited 0.4
field-of-view of any observations. Seitz&Schneider (2001)
expressthis as a von Neumann boundaryproblem,leading to 0.2
theso-calledfinite–fieldinversion.Werelyonthismethodfor
0
massreconstructionsthroughoutthepaper.
-0.2
3.2.Mass-aperturestatistics
-0.4
Themass-aperture,or M ,Statistics,developedbySchneider 0 0.2 0.4 0.6 0.8 1
ap
x = θ / θ
(1996), provides a method with defined noise properties to out
identifymassconcentrations.Itisbasedupontherelation Fig.3. The filter function w (solid line) we used and its cor-
respondingweightfunction Q (dashedline)showninunitsof
Map(θ0)=ZIR2d2θκ(θ0+θ)w(|θ|)=ZIR2d2ϑγt(θ0;ϑ)Q(|ϑ|).(9) θout.
The aperturemass M (θ ) presentsa measure of the average
ap 0
convergence,mulitpliedwith a filter functionw, arounda po-
– We can sample the shear field only at discrete points,
sitionθ inthelensplane.Ifw(θ)isacompensatedfilter,M
0 ap
| | namely those where there is a measured galaxy image. In
avoidsthemasssheetdegeneracy.Therightsideofeq.(9)ex-
theweaklensingregime,theimageellipticityisonaverage
presses M in terms of the tangentialshear γ at the position
ap t
θ +ϑwithrespecttoθ : a direct measure of the shear, so that we can use the tan-
0 0
gentialellipticityε asan estimateforthetangentialshear
t
γ(θ ;ϑ)= e γ(θ +ϑ)e 2ıφ , (10) γ (ε definedanalogouslytoγ).
t 0 0 − t t t
−R
(cid:16) (cid:17)
A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke? 7
Wethusestimate M byusing Table1.Summaryofthedifferencesoftheobjectcatalogsused
ap
for the lensing analyses of Erbenetal. (2000) and this work.
πθ2
M (θ ;θ ) = out ε(θ)Q(θ θ ), (12) The brightness key refers to the magnitude bins used to split
ap 0 out t i i 0
N | − | thesample(notethatthemagnitudesarenotcalibrated).
Xi
where i runsover all N galaxieswithin a radius θout from the Erbenetal.(2000) thiswork
pointθ .
0 masking √
With the weighting scheme introduced in eq. (8), this be- −
sourceextraction 6px 1.0σ 3px 1.0σ
comes: ≥ ≥
weighting √
Map(θ0;θout) = πθo2ut Piεt(θi)σ−i2σQ2(|θi−θ0|) . (13) |bεr|ightness m≤&−1.203 ≤a0ll.8
i −i
P m 23.67
≥
3.2.2. Significances 22.54<m<23.67
m 22.54
AnyM valueisincompletewithoutanestimateofitssignifi- ≤
ap
cance,i.e.howitcomparestothetypicalnoiseleveloftheM
ap
estimator.Thesignal-to-noiseratioisgivenby
Withthisinmind,wereconsidereq.(18).Suchrandomiza-
S Map Map tionsrepresentapossibleensembleaverage,whichwedenote
= = , (14)
N σ by ... .Foreachrealization,theellipticitymodulusremains
Map qhMa2pi−hMapi2 thehsamieφ,onlytheorientationchanges.Inthiscase,theweights
where the denominator should represent the case of no lens- alsoremainthesameandwecansimplifytheexpression:
ing. The expectation value M of M then vanishes since
tthheeegxaplaexciteastioanrevaolruieenotefdtancoghmenptailpaeilteellylipratiancpditoiems,lyvaannidshtehsu.sInhεthtie, hMa2piφ = πθo2ut 2 Pi,j hεσt,i2iεσt,2jjiφ 2QiQj = (πθ2o2ut)2 Pi |εσi2i|2 Q22i
(cid:16) (cid:17) 1 1
caseofanon-weightedMapestimator,weobtain: (cid:18) i σ2i(cid:19) (cid:18) i σ2i(cid:19)
P P
2 since,astheellipticitymodulusisfixed,
πθ2
hMa2pi = (cid:16) Nou2t(cid:17) Xi hε2t(θi)iQ2(|θi−θ0|). (15) εt,iεt,j φ = |εi|2 δij .
h i 2
Inthecaseofnolensing, ε2 istheone-dimensionalvariance
h t,ii The signal-to-noise ratio for the weighted estimator is there-
oftheintrinsicellipticities:
fore:
1
hε2t,ii= 2σ2ε (16) S = √2 εt(θi)σ−i2Q(|θi−θ0|) . (19)
N
andthesignal-to-noiseratiobecomes: Xi q i|εi|2σ−i2Q2(|θi−θ0|)
P
To check the validity of the assumptions we made, we com-
S √2 ε(θ)Q(θ θ )
= i t i | i− 0| . (17) pared results from randomizations and this analytic formula
N σε P iQ2(|θi−θ0|) andfoundthemtobeequivalent.
pP
Fortheweightedestimator,onefacestheproblemthatthe
weights1/σ2areingeneralnotcompletelyindependentofthe 4. Re-analysisoftheCFHTdata
i
tangential ellipticity ε . We assign weights to galaxy images
t,i
InTable1,wesummarizethedifferencesbetweentheanalyses
by considering the variance of ellipticities of an ensemble of
ofErbenetal.(2000)andthiswork.
galaxy images with similar noise properties. But in general,
large ellipticities are often noise-afflicted, so that they are as-
signed a lower weight. The expression for hMa2pi then cannot 4.1.Massreconstruction
besimplifiedadhoc:
Inordertoapplythefinite-fieldinversion(Sect.3.1),wefurther
M2 = (πθ2 )2 i,jεt,iεt,jσ−i2σ−j2QiQj . (18) cuttheI-bandimagetoavoidtheripple-likereflectionartefact
h api out *P σ 2σ 2 + attheeasternedgealtogether.Thisnarrowstheavailablefield,
i,j −i −j
butavoidsproblemsattheboundaries.Aresultingmassrecon-
P
The significance of a detection is related to the probabil- structionisshowninFig.4.
ity that the observed alignment of tangential ellipticities can Abell 1942 shows up prominently in the top half of the
be mimicked by a random distribution of galaxy ellipticities. field,withthepeakofthemassmapcenteredapproximatelyon
A commonlyused possibility to determine the significance is thecDgalaxy.Inthelowerhalfoftheimage,thereisasecond,
thereforetorandomizethepositionanglesofthegalaxyimages albeitlowerpeakatthesamepositionasdetectedinErbenetal.
andcalculateM ofthese.ThisisrepeatedN times. (2000).RelativetothepeakκofA1942,ourdarkclumpsignal
ap rand
8 A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke?
Abell 1942 is detected only weakly for large filter scales,
a result that is consistent with Erbenetal. (2000). The mass
reconstructionfromthepreviousparagraphdemonstrateswhy
the dark clump reaches a much higher M significance than
ap
thecluster:atthedarkclumpposition,thenegativepartofthe
filterfunction(Fig.3)isevaluatedlargelyatthepositionofthe
hole, thereby boosting the signal. The same in reverse is true
fortheholeitself:itssignificanceisboostedbyitsproximityto
theDarkClump.Yetitssignificanceremainslowerthanthatof
thedarkclump.
4.2.2. Roughredshiftdependence
On average,the moredistanta galaxy,the fainterit is. By in-
troducingmagnitudecutswesplitthegalaxysampleintothree
partsof about660galaxieseach with differentmean redshift.
Thisisaverycruderedshiftdistinction,butshouldrevealany
trendoflensstrengthwithredshift.
The results of this analysis are shown in Fig. 6. We see
that the dark clump signal stems mostly from faint galaxies,
which supports the notion that this is a high-redshift object.
However,these are also those objectsthat are mostsubjectto
noiseeffects.
At the 120 filter scale, there is also a 3σ contribution
′′
frombrightgalaxies.Assumingthatthelensingmassisindeed
a high redshiftobject, these brightgalaxiesare unlikely to be
athigherredshifts.Thus,thisisprobablynotalensingsignal.
Yet, this“contamination”canexplainthe highsignal-to-noise
ratioweseeatthisfilterscale.
Fig.4. Mass reconstruction of the CFHT I-band image ac-
cordingtoSeitz&Schneider(2001).Solid(dashed)linesgive ForAbell1942,thereisa strongsignalatthesmallestfil-
positive (negative) κ contours, starting at 0.02 and in-(de- ter scale, centered on the cD galaxy which exhibits a strong
±
)creasingin0.01intervals.Thesmoothingscaleis60 . lensingarc.Itmaywellbethatattheseradiiwearenotinthe
′′
weaklensingregimeanymoreandthetangentialalignmentis
alreadyratherdistinct.Intheotherfilterradii,thereisnopar-
ticularlystrongsignal.Thismightbeduetothegenericweight
is slightly larger than given by Erbenetal. (2000). However,
functionwhichisnotadaptedtotheNFWprofile.
northofthedarkclump,thereisa“hole”-a regionofsignif-
OurplotdoesnotshowthenegativeM contourstoavoid
ap
icantlynegativeκvalues.Althoughthisisasomewhatdiscon-
overcrowding. Unlike to the dark clump, all three magnitude
certingresult,itmustbestressedthatκisunderestimatedinthe
binscontributetothe“hole”.
wholefieldduetothemasssheetdegeneracyandtheclusterin
thefield(twoofthe threefield boundariesclosetothecluster
and well within its extent display nearly vanishing κ values). 4.2.3. Map crosscomponent
Unfortunately,theoriginalanalysisofErbenetal.(2000)only
In Sect. 3.2, we argued that M , i.e. M calculated with the
investigatesregionsof positive κ, so that this resultcannotbe a×p ap
cross component of the shear instead of the tangential one,
compared.
mustvanish.Bycheckingthevalidityofthisassumptioninthe
dataset,wecanidentifypossibleproblems.
4.2.Mapanalysis Theresultsareshownin Fig.7. Particularlyatsmallfilter
scales,therearesomepositiveandnegativepeakswithsignifi-
4.2.1. Completesample
cances&3σ (negativevaluesareindicatedbynegativesignal-
| |
WeperformtheM analysisasdescribedinSect.3.2andeval- to-noise ratios). However, most of the peaks are at the edge
ap
uate the M statistic on a grid with grid spacings of 2 . The of the field, where a part of the weight function lies outside
ap ′′
resultisshowninFig.5.Thedarkclumpsignalisseensignifi- thefield.While(therealpartof) Map retainsitsjustificationat
cantlyatallfilterscales,butitisparticularlystrongforthe120 theseplacesasbeingsimplyameasureofthetangentialalign-
′′
filter, where it reachesa peaksignificance of 5σ. In the other ment,wecannolongerassumethatMa×pvanishes.
filters, the significance is at the 3.5σ level, as in Erbenetal. Thereis no peakwith a significancelargerthan 2σ in the
(2000). vicinityofthedarkclump,sothedetectionpassesthistestwell.
A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke? 9
Fig.5.ResultsofaweightedM analysisoftheI-banddata.Shownarepositive(negative)S/Ncontoursassolid(dashed)lines
ap
starting at 2.0σ in 0.5σ intervals.The filter scales from leftto rightare 80 , 120 , 160 , and 200 . The dark clump signal
′′ ′′ ′′ ′′
±
showsupstrongestinthe120 filter,withapeaksignificanceof4.9σ.
′′
Fig.6. The same as the previous figure (Fig. 5), but with the galaxies divided into three magnitude bins and only positive
contours.Whitecontourscorrespondtothebrightestgalaxies,graycontourstothoseofmediumbrightness,andblackcontours
tothefaintest.
4.3.Radialprofile pendenceofthe meantangentialellipticityshowninFig.9 of
Erbenetal.(2000).
Despite several differences to the lensing analysis of Wedeterminethepositionofthedarkclumpfromthe M
ap
Erbenetal.(2000),the M resultsagreeatleastqualitatively. peakatthe120 filterscale,wherethesignalisstrongest.We
ap ′′
We are curious whether we can also reproduce the radial de- find itto be α = 14h38m21.6s, δ = 3 32 43. , which is ata
◦ ′ ′′
10 A.vonderLindenetal.:ThedarkclumpnearAbell1942:darkmatterhaloorstatisticalfluke?
Fig.7.Resultsofa M analysisoftheI-bandimage.Shownarebothpositive(white)andnegative(black)contours,startingat
a×p
2σin0.5σincrements,forfilterscalesof80 ,100 ,160 ,and200 (lefttoright).
′′ ′′ ′′ ′′
±
mean is calculated for each bin. To estimate the standard de-
viation, we randomize the position angles of these galaxies
1000 times and calculate the mean tangential ellipticity each
time,thusgaininganestimateforthestandarddeviation.This
analysisisdoneforthecompletegalaxysampleaswellasthe
threesamplessplitaccordingtobrightness.Theresultsofitare
showninFig.8.
Particularlyforthefaintestgalaxies,wefindpositivevalues
out to 120 . This agrees well with the strong shear signal
′′
seen for these galaxies. We can also identify the cause of the
signalseenforbrightgalaxiesatthe120 filterscale:thetwo
′′
significantlypositive binsat90 and110 (the filter function
′′ ′′
employedassignsthehighestweightaroundaradiusθ /√2).
out
For the mediumbrightgalaxies ε is largelyconsistentwith
t
h i
zero.
ComparedtoErbenetal.(2000),whomeasure ε 0.06
t
h i≈
at 100 , we find a higher value ( 0.1). On the other hand,
′′
≈
wefindpositivevaluesonlyoutto120 ratherthan150 .And
′′ ′′
sincethecentroidpositionsdonotcoincide,theinnertwobins
arenotcomparable.Yet,wecanalsobeconfidentthatthesig-
Fig.8.Themeantangentialellipticity ε relativetotheposi-
h ti nalisnotcausedjustbyafewgalaxies.
tionoftheDarkClumpasafunctionofdistancefromit(from
the ground-based data). Each point corresponds to the mean
of a 20 wide bin, where filled circles denote the complete
′′ 4.4.Summary
sample,crossesthefaintgalaxies,trianglesthemediumbright
ones,andsquaresthebrightgalaxies.Theerrorbarsrepresent We have successfully confirmedthe weak lensing signalseen
thestandarddeviationof εt estimatedfromrandomizationof in two sets of CFHT observations (our re-analysis of the V-
h i
thepositionanglesofthegalaxies.Notethatthefirstbinisvery banddata are notshownhere but agreewell with Erbenetal.
sensitivetothechoiceofcentroid-itsdeviationfromtheshear (2000)). We show that the alignment signal comes from faint
profileinthelatterbinsisthusnotproblematic. galaxies, which supports the hypothesisthat it is caused by a
lensing mass at high redshifts. One must keep in mind, how-
ever,thatthesearealsothoseobjectsmostaffectedbynoise.
distanceof18.6fromthepositiongivenbyErbenetal.(2000), With several variations of the catalog that enters the M
′′ ap
andthusjustatthe1σleveltheygivefortheuncertaintyofthe analysis,wetestedthatthedetectionofthedarkclumpisresis-
centroid’sposition. tantagainsttheseandconsistentlyrecoveredatallfilterscales.
The tangential ellipticity relative to this position is calcu- It reaches a peak significance of about 5σ, although this sig-
lated for each galaxy within 200 . They are then binned ac- naliscontaminatedbyatangentialalignmentofbrightobjects,
′′
cordingtotheirdistancefromthedarkclumpandtheweighted whichisunlikelytobealensingeffect.