CFHTLenS: Combined probe cosmological model comparison using 2D weak gravitational lensing

0.8 MB·English

by  Martin Kilbinger

Mon.Not.R.Astron.Soc.000,1–18(2009) Printed8January2013 (MNLATEXstylefilev2.2)
CFHTLenS: Combined probe cosmological model comparison using
2D weak gravitational lensing
Martin Kilbinger1,2,3,4⋆, Liping Fu5, Catherine Heymans6, Fergus Simpson6, Jonathan
Benjamin7, Thomas Erben8, Joachim Harnois-De´raps9,10, Henk Hoekstra11,12, Hen-
3 drik Hildebrandt7,8, Thomas D. Kitching6, Yannick Mellier4,1, Lance Miller13, Ludovic
1
Van Waerbeke7, Karim Benabed4, Christopher Bonnett14, Jean Coupon15, Michael J.
0
2 Hudson16,17, Konrad Kuijken11, Barnaby Rowe18,19, Tim Schrabback8,11,20, Elisabetta

n
Semboloni11, Sanaz Vafaei7, Malin Velander13,11
a
J
1CEA/Irfu/SApSaclay,LaboratoireAIM,91191Gif-sur-Yvette,France
7
2ExcellenceClusterUniverse,Boltzmannstr.2,85748Garching,Germany

3Universita¨ts-SternwarteMu¨nchen,Scheinerstr.1,81679Mu¨nchen,Germany
]
O 4Institutd’AstrophysiquedeParis,UMR7095CNRS,Universite´Pierre&MarieCurie,98bisboulevardArago,75014Paris,France
5KeyLabforAstrophysics,ShanghaiNormalUniversity,100GuilinRoad,200234,Shanghai,China
C 6ScottishUniversitiesPhysicsAlliance,InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,Edinburgh,EH93HJ,UK
. 7UniversityofBritishColumbia,DepartmentofPhysicsandAstronomy,6224AgriculturalRoad,Vancouver,B.C.V6T1Z1,Canada
h
8Argelander-Institutfu¨rAstronomie,Universita¨tBonn,AufdemHu¨gel71,53121Bonn,Germany
p
9CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,M5S3H8,Ontario,Canada
-
o 10DepartmentofPhysics,UniversityofToronto,M5S1A7,Ontario,Canada
r 11LeidenObservatory,LeidenUniversity,NielsBohrweg2,2333CALeiden,TheNetherlands
t 12DepartmentofPhysicsandAstronomy,UniversityofVictoria,Victoria,BCV8P5C2,Canada
s
a 13Dept.ofPhysics,OxfordUniversity,KebleRoad,OxfordOX13RH,UK
[ 14InstitutdeCienciesdelEspai,CSIC/IEEC,F.deCiencies,TorreC5par-2,Barcelona08193,Spain
15InstituteofAstronomyandAstrophysics,AcademiaSinica,P.O.Box23-141,Taipei10617,Taiwan
2 16DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,ON,N2L3G1,Canada
v 17PerimeterInstituteforTheoreticalPhysics,31CarolineStreetN,Waterloo,ON,N2L1Y5,Canada
8 18DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,GowerStreet,LondonWC1E6BT,UK
3 19CaliforniaInstituteofTechnology,1200ECaliforniaBoulevard,PasadenaCA91125,USA
3 20KavliInstituteforParticleAstrophysicsandCosmology,StanfordUniversity,382ViaPuebloMall,Stanford,CA94305-4060,USA
3
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2 8January2013
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ABSTRACT
i
X
We present cosmological constraints from 2D weak gravitational lensing by the large-
r
a scale structurein theCanada-FranceHawaiiTelescopeLensingSurvey(CFHTLenS)which
spans 154 square degrees in five optical bands. Using accurate photometric redshifts and
measured shapes for4.2 million galaxiesbetween redshifts of 0.2 and 1.3, we compute the
2D cosmic shear correlation function over angular scales ranging between 0.8 and 350 ar-
cmin.Using non-linearmodelsofthe dark-matterpowerspectrum,we constraincosmolog-
ical parameters by exploring the parameter space with Population Monte Carlo sampling.
Thebestconstraintsfromlensingaloneareobtainedforthe small-scaledensity-fluctuations
amplitude σ8 scaled with the total matter density Ωm. For a flat ΛCDM model we obtain
σ8(Ωm/0.27)0.6 =0.79±0.03.
WecombinetheCFHTLenSdatawithWMAP7,BOSSandanHSTdistance-ladderprior
on the Hubble constant to get joint constraints. For a flat ΛCDM model, we find Ωm =
0.283±0.010andσ8 = 0.813±0.014.InthecaseofacurvedwCDMuniverse,weobtain
Ωm =0.27±0.03,σ8 =0.83±0.04,w0 =−1.10±0.15andΩK =0.006+−00..000064.
WecalculatetheBayesianevidencetocompareflatandcurvedΛCDManddark-energy
CDM models. From the combination of all four probes, we find models with curvature to
be at moderately disfavoured with respect to the flat case. A simple dark-energy model is
indistinguishablefromΛCDM. Ourresultsthereforedonotnecessitate anydeviationsfrom
thestandardcosmologicalmodel.
Keywords: cosmologicalparameters–methods:statistical
(cid:13)c 2009RAS
2 Kilbingeret al.
1 INTRODUCTION amixtureofnumericalsimulationsonsmallscalesandanalytical
predictionsonlargescales.
Weakgravitationallensingisconsideredtobeoneofthemostpow-
erfultoolsofcosmology. Itsabilitytomeasureboththegeometry The reliabilityand accuracy of our photometric redshifts al-
oftheUniverseandthegrowthofstructureoffersgreatpotentialto lows for three-dimensional weak lensing analyses. The measure-
obtainconstraintsondarkenergyandmodifiedgravity.Moreover, mentofthelensingcorrelationsfordifferentredshiftcombinations
tofirst order, weak lensing does not rely on the relationbetween allowsustoobtaininformationonthegrowthofstructure(e.g. Hu
galaxiesanddarkmatter(bias),andisthereforeakeyprobeofthe 1999),andhasagreatpotentialtoconstraindarkenergyandmod-
darkUniverse. ifiedgravitymodels(e.g.Albrechtetal.2006;Uzan2010).Insev-
Cosmic shear, denotes the distortion of images of distant eralcompanion papers, weperform3Dcosmicshearanalyses by
galaxies due to the continuous deflection of light bundles prop- splitting up galaxies into redshift bins using correlation function
agating through the inhomogeneous Universe. The induced cor- methodspresentedhere(lensingtomography:Simpsonetal.2013;
relations between shapes of galaxies are directly related to the Heymansetal.2013;Benjaminetal.2013).
statistical properties of the total (dark + luminous) large-scale Inthispaper,weperforma2Dlensinganalysisusingasingle
matter distribution. With an estimate of the redshift distribu- redshiftdistribution.Despitethefactthattheredshiftinformation
tion of the lensed galaxies, theoretical predictions of weak- isnotusedinanoptimalway,ouranalysishasseveraladvantages.
lensing observables can be tested to obtain constraints on cos- First, it yields the highest signal-to-noise ratio (S/N) for a sin-
mological parameters and models. Recent reviews which also gle measurement. This is particularly important on large angular
summarizepastobservationalresultsareBartelmann&Schneider scales,wheretheS/N istoolowtobeusedfortomography.These
(2001); VanWaerbeke&Mellier (2003); Munshietal. (2008); large scales probe the linear regime, where non-linear and bary-
Hoekstra&Jain(2008);Bartelmann(2010). oniceffectsdonot playarole,andonecanthereforeobtain very
The Canada-France Hawaii Telescope Legacy Survey1 robustconstraintsoncosmology(Sembolonietal.2011).Second,
(CFHTLS)isalargeimagingsurvey, offering aunique combina- we can include low-redshift galaxies without having to consider
tionofdepth(i < 24.5at5σpointsourcelimitingmagnitude) intrinsic alignments (IA; Hirata&Seljak 2004). For a broad red-
AB
andarea(154squar∼edegrees).Itisthelargestsurveyvolumeover shiftdistribution,IAisexpectedtobeasub-dominantcontribution
which cosmicshear has everbeen measured. Thispaper presents tothecosmological shear-shearcorrelationwithanexpected bias
thefirstcosmologicalanalysisofthecompleteCFHTLegacySur- for σ which is well within the statistical uncertainty (Kirketal.
8
veywithweakgravitationallensing.Wemeasuretheseond-order 2010; Mandelbaumetal. 2011; Joachimietal. 2011), see also a
cosmic-shearfunctionsfromtheCFHTLenS(CFHTLensingSur- joint lensing and IA tomography analysis over the full available
vey2), which comprises the final CFHTLS data. Earlier analyses redshift range (Heymansetal. 2013). Therefore, despite the fact
of theCFHTLSused the firstdatarelease (T0001) with4square thata2Dlensingismorelimitedthantomography,itislessnoisy
degreesoftheDeepsurvey(Sembolonietal.2005)and22square andmoreimmunetoprimaryastrophysicalsystematics.Itisthere-
degreesoftheWidepartofthesurvey(Hoekstraetal.2006), fol- foreanecessarybasicstepandputsanyfurthercosmological ex-
lowedbythethirddatarelease(T0003)comprising55square de- ploitationofCFHTLenSusingmoreadvancedtomographicorfull
grees (Benjaminetal. 2007; Fuetal. 2008, hereafter F08). The 3D lensing techniques on solid grounds. Such analyses are pre-
T0003lensingdatawassubsequently employedinfurtherstudies sentedintheCFHTLenScompanionpapers.
(Dore´etal. 2008; Terenoetal. 2009; Kilbingeretal. 2009). Only This paper is organized as follows. Section 2 provides the
CFHTLS-Widei′-banddatawereusedforthoselensinganalyses, expressions for the second-order lensing observables used in this
andtheredshiftdistributionwasinferredfromthephotometricred- analysis,bothobtainedfromtheoreticalpredictionsandestimated
shifts from the Deep survey (Ilbertetal. 2006). Photometric red- fromdata.Themeasuredshearfunctionsandcovariancesare pre-
shiftsintheWidewereobtainedsubsequentlywiththeT0004re- sentedinSect.3.Cosmologicalmodelsandsamplingmethodsare
lease (Couponetal. 2009). The current series of papers uses the introducedinSect.4.Theresultsoncosmologicalparametersand
finalCFHTLenSdatareleaseof154squaredegreesinthefiveop- models are presented in Sect. 5, followed by consistency tests in
ticalbandsu∗,g′,r′,i′,z′.Theanalysisimprovedsignificantlyin Sect.6.ThepaperisconcludedwithadiscussioninSect.7.
severalways:
TheCFHTLSdatahavebeenreanalysedwithanewpipeline
• 2 WEAKCOSMOLOGICALLENSING
(Erbenetal.2009,2013).
Photometric redshifts have been obtained for each individ- In this section the main relations between second-order weak
•
ualgalaxyinthelensingcataloguefromPSF-homogenizedimages lensing observables and cosmological quantities are given. See
(Hildebrandtetal.2012).Theaccuracyhasbeenverifiedindetail Bartelmann(2010);Hoekstra&Jain(2008)forrecentreviews.
byanangularcross-correlationtechnique(Benjaminetal.2013).
PSF modeling and galaxy shape measurement have been
• 2.1 Theoreticalbackground
performed with the forward model-fitting method lensfit, which
has been thoroughly tested on simulations and improved for Weak lensing by the large-scale structure measures the conver-
CFHTLenS(Milleretal.2013). gence powerspectrum P , whichcanbe relatedtothetotalmat-
κ
Systematicstestshavebeenperformedinablindway,toyield ter power spectrum P via a projection using Limber’s equation
• δ
unbiasedcosmologicalresults(Heymansetal.2012). (Kaiser1992):
The cosmology-dependent covariance matrix is obtained by
• χlim ℓ
P (ℓ)= dχG2(χ)P k= ;χ . (1)
κ Z0 δ(cid:18) fK(χ) (cid:19)
1 http://www.cfht.hawaii.edu/Science/CFHTLS Theprojection integral iscarriedout overcomoving distances χ,
2 http://www.cfhtlens.org fromtheobserver out tothelimitingdistance χ of thesurvey.
lim
(cid:13)c 2009RAS,MNRAS000,1–18
CFHTLenS: cosmologicalmodel comparisonusing2D weaklensing 3
ThelensefficiencyGisgivenby separatetheE-andtheB-modeinamoreorlesspureway,aswill
G(χ)= 3 H0 2 Ωm χlimdχ′p(χ′)fK(χ′−χ), (2) nteonwasbeindteesgcrrailbseodv.eTrhtehedefirlitveereddseccoorrnedla-otirodnerfufunncctitoionns.sTcahneybecawnrbite-
2(cid:18) c (cid:19) a(χ)Zχ fK(χ′) estimatedasfollows:
whereH0istheHubbleconstant,cthespeedoflight,Ωmthetotal 1
matterdensity,and a(χ)thescalefactoratcomovingdistance χ. XE,B = 2 ϑi∆ϑi[F+(ϑi)ξ+(ϑi)±F−(ϑi)ξ−(ϑi)]. (6)
Thecomovingangulardistanceisdenotedwithf anddependson Xi
K
thecurvatureKoftheUniverse; Here,∆ϑiisthebinwidth,whichcanvarywithi,forexamplein
thecaseoflogarithmicbins.WithsuitablefilterfunctionsF and
K−1/2sin K1/2w for K >0 +
 (cid:16) (cid:17) F−(App.A),theestimatorXE(XB)issensitivetotheE-mode(B-
fK(w)= w( K)−1/2sinh ( K)1/2w ffoorr KK =<00. moroddere)oobnselyrv.aTbhleesfiilnteTrafubnlect1ioannsdaArepdpeefinndeixdAfo.rthevarioussecond-
 − (cid:16) − (cid:17) (3) Allderivedsecond-orderfunctionsarecalculatedforafamily
offilterfunctions.Fortheaperture-massdispersion,theoptimized
The 3D power spectrum is evaluated at the wave number k =
ringstatistic,and thetop-hat shearroot meansquare (rms),these
ℓ/f (χ),whereℓdenotestheprojected2Dwavemode.Thefunc-
K
tionprepresentstheweighteddistributionofsourcegalaxies. are given for a continuous parameter θ which can be interpreted
asthesmoothingscale.Hereandinthefollowingwewillusethe
notation’ϑ’asthescaleforthe2PCFs,and’θ’asthesmoothing
2.2 Flavoursofreal-spacesecond-orderfunctions scaleforderivedfunctions.
ForCOSEBIs, the filterfunctions are a discrete set of func-
Fromanobservationalpointofview,themostdirectmeasurement
tions. The latter exist in two flavours, Lin-COSEBIs and Log-
of weakcosmological lensing isinreal space, byusingtheweak
gravitational shear signal as derived from galaxy ellipticity mea- COSEBIs,defined through filterfunctions F± which arepolyno-
mialsonlinearandlogarithmicangularscales, respectively. Here
surements. Thetwo-point shearcorrelation functions(2PCFs)ξ
+ weuseLog-COSEBIs,forwhichmanyfewermodesarerequired
andξ− areestimatedinanunbiasedwaybyaveragingoverpairs
tocapturethesameinformationasLin-COSEBIs(Schneideretal.
ofgalaxies(Schneideretal.2002),
2010;Asgarietal.2012).SeeApp.Aformoredetails.
ξˆ±(ϑ)= Pijwiwj[εt(ϑi)εt(ϑwj)w±ε×(ϑi)ε×(ϑj)]. (4) conveArgllenocfethpeowabeorvsepefcutnrucmtio.nTshecagnenbeeraelxrperleastsioendiisn terms of the
ij i j
ThesumisperformedoverallPgalaxypairs (ij)withangulardis- X = 1 ∞dℓℓP (ℓ)Uˆ2(ℓ). (7)
tance |ϑi −ϑj| within some bin around ϑ. With εt and ε× we E,B 2π Z0 E,B
denotethetangentialandcross-componentofthegalaxyellipticity,
respectively. The weights w are obtained from the lensfit shape The functions F± and Uˆ2 are Hankel-transform pairs, their
i
relation is given by Crittendenetal. (2002) and Schneideretal.
measurementpipeline(Milleretal.2013).The2PCFsaretheHan-
(2002)as
keltransformsoftheconvergencepowerspectrumP or,morepre-
κ
cisely,oflinearcombinationsoftheE-andB-modespectra,P and ∞
PB,respectively.Namely, E F±(x)=Z0 dttJ0,4(xt)Uˆ2(t). (8)
1 ∞
ξ±(ϑ)= 2π Z0 dℓℓ [PE(ℓ)±PB(ℓ)]J0,4(ℓϑ), (5) 2.2.1 Finitesupport
whereJ andJ arethefirst-kindBesselfunctionsoforder0and
0 4 Themeasuredshearcorrelationfunctionisavailableonlyonafinite
4,andcorrespondtothecomponentsξ+andξ−,respectively. interval[ϑ ;ϑ ].Theupperlimitisgivenbythefinitesurvey
Itisdesirabletoobtainobservableswhichonlydependonthe min max
size.Wechoose460arcminutes,whichisroughlythelargestscale
E-mode and B-mode, respectively. Weak gravitational lensing, to
for which a sufficient number of galaxy pairs are available on at
firstorder,onlygivesrisetoanE-modepowerspectrumandthere-
leastthreeofthefourWidepatches3.Thelowerlimitcomesfrom
fore,anon-detectionoftheB-modeisanimportantsanitycheckof
thefactthatforclosegalaxypairs,shapescannotbemeasuredreli-
thedata.Tothisend,wecalculatethefollowingsecond-ordershear
ably.Forlensfit,galaxieswithseparationssmallerthanthepostage
quantities which can be derived from the correlation functions:
stampsizeof48pixels 9arcsectendtohaveashapebiasalong
The aperture-mass dispersion M2 (Schneideretal. 1998), the ≃
h api theirconnectingdirection(Milleretal.2013).Sincethisdirection
sheartop-hatrms γ 2 (Kaiser1992),theoptimizedringstatistic
h| | i is randomly orientated to a very good approximation, we do not
(Fu&Kilbinger2010),andCOSEBIs(CompleteOrthogonal
RE have to remove those galaxies altogether; setting a minimum an-
SetsofE-/B-modeIntegrals;Schneideretal.2010).Theoptimized
gularseparationofϑ = 9arcsecdoesavoidthecorrelationof
ring statistic was introduced as a generalisation of the so-called min
thosepairs.Bothgalaxiesfromaclosepaircanbeindependently
ringstatistic(Schneider&Kilbinger2007).Thecorrespondingfil-
correlated with other, more distant galaxies, for which the close-
terfunctionshavebeenobtainedtomaximisethefigure-of-meritof
pairshapebiasactsasasecond-ordereffectandcansafelybene-
Ω andσ foraCFHTLS-T0003-likesurvey.Weusethesefunc-
m 8 glected.
tionsforCFHTLenSwhich,despitethelargerarea,hassimilarsur-
veycharacteristics.COSEBIsrepresentyetanothergeneralisation Ifthesupport of thefilterfunctions F+ andF− exceeds the
observablerange,eq.(6)leadstobiasedresults,andapureE-and
andcontainallinformationabouttheE-andB-modeweak-lensing
B-modeseparationisnolongerguaranteed.
fieldfromtheshearcorrelationfunctiononafiniteangularrange.
Being quantities obtained from the 2PCFs by non-invertible
relations,thesederivedfunctionsdonot containthefullinforma- 3 W2asthesmallestpatchextendsto400arcminutes,whereasW1probes
tionabouttheconvergencepowerspectrum(Eifleretal.2008),but scalesaslargeas685arcminutes.
(cid:13)c 2009RAS,MNRAS000,1–18
4 Kilbingeret al.
Table1.E-andB-modeseparatingsecond-orderfunctions.Theyareestimatedasintegrals,orsumsinthediscretecase,overthe2PCFsξ±multipliedwith
thefilterfunctionsF±withtheformalintegrationrange[ϑmin;ϑmax],seeeq.(6).Theargumentϑdenotestheintegrationvariable.
Name XE,B F±(eq.6) Uˆ(eq.7) ϑmin ϑmax Reference
Aperture-massdispersion hMa2p,×i(θ=ϑmax/2) T±(ϑ/θ)/θ2(eq.A1) eq.(A2) 0 2θ Schneideretal.(1998)
Top-hatshearrms h|γ|2i(θ) S±(ϑ/θ)/θ2(eq.A3) eq.(A4) 0 ∞ Kaiser(1992)
Optimizedringstatistic RE,B(θ=ϑmax) T±(ϑ)(eq.A6) n/a ϑmin ϑmax Fu&Kilbinger(2010)
COSEBIs En,Bn T±lo,gn(ϑ)(eq.A8) n/a ϑmin ϑmax Schneideretal.(2010)
Thisisthecasefortheaperture-massdispersionandthetop-
hatshearrms.Fortheformer,onlythelowerangularlimitisprob- 0.14 sum of pdf(zp)
lematicandcausesleakageoftheB-modeintotheE-modesignal best-fit zp
onsmallsmoothing scales.Onscaleslargerthan θ = 5.5arc 0.12
min
minuteshowever,thisleakageisbelow1.5centforϑ =9arc-
min 0.1
sec (Kilbingeretal. 2006). We therefore choose θ = 5.5 ar-
min y
comnisncatloesbsemthaellesrmthalalnesϑtmsminowotehisnegttshceacleorfroerlahtMiona2pfiu(nθc)t.ioNnottoeztehraot quenc 0.08
e 0.06
anddonotuseatheoreticalmodeltoextrapolatethedataontothis fr
range,toavoidacosmology-dependentbias. 0.04
TheB-modeleakageforthetop-hatshearrms γ2 isafunc-
tionofboththelowerandupperavailableangulars(cid:10)c|ale|s(cid:11).Overour 0.02
rangeofscales,thepredictedleakedB-modefortheWMAP7cos-
0
mologyisnearlyconstantwithavalueof5.3 10−7.
× 0 0.5 1 1.5 2 2.5 3 3.5
The first pure E-/B-mode separating function for which the
z
corresponding filter functions have finite support was introduced
inSchneider&Kilbinger(2007).Followingthatapproach,theop-
Figure1.Theredshiftdistributionp(zp)histogram,estimatedasthesum
timizedring statisticand COSEBIswere constructed ina similar
ofpdfsfor0.2 < zp < 1.3(thickredcurve).Forcomparison,thehis-
waytonotsufferfromanE-/B-modeleakage. togramofthebest-fittingphoto-zsisshownasthinbluecurve.Errorbars
An additional bias arises from the removal of close galaxy correspondtothevariancebetweenthefourWidepatches.
pairsinthelensinganalysis,aswasfirstreportedbyHartlapetal.
(2011).Aspreviouslydiscussed, lensfitproducesashapebiasfor
galaxiesseparatedbylessthan9arcsec,butthisbiasisrandomand 3.1 Redshiftdistribution
theclosepairsarethereforeusedintheanalysis, tobecorrelated
Adetailedstudyofthereliabilityofourphotometricredshifts,the
with other galaxies at larger distances. That said, for very close
contaminationsbetweenredshiftbins,andthecosmologicalimpli-
blendedgalaxies,whereitisnon-trivialtodetermineifthereisone
cations is performed in Benjaminetal. (2013). This work shows
ortwogalaxiesobserved,galaxyshapescannotevenbeattempted.
thatthetrueredshiftdistributionp(z)iswellapproximatedbythe
These blended pairs are therefore not reliably detected and lost
sumoftheprobabilitydistributionfunctions(pdfs)forallgalaxies.
fromouranalysis.Thiscausesapotentialbias,sincethesegalaxies
The pdfs are output by BPZ (Bayesian Photometric Redshift Es-
areremovedpreferentiallyatlowredshift,wheregalaxysizesare
timation; Ben´ıtez 2000) as a function of photometric redshift z ,
larger,andfromhigh-densityregionscomparedtovoids,because p
andhavebeenobtainedbyHildebrandtetal.(2012).Theresulting
galaxiestracethelarge-scalestructure.FromFig.3ofHartlapetal.
p(z)isconsistentwiththecontaminationbetweenredshiftbinsas
(2011)weinferthatthemagnitudeofthiseffectisatthepercent
estimatedbyanangularcross-correlationanalysis(Benjaminetal.
levelonscaleslargerthan0.8arcminforthe2PCFs.
2010). The contamination is relatively low for galaxies selected
with 0.2 < z < 1.3, which is confirmed by a galaxy-galaxy-
p
lensingredshiftscalinganalysisinHeymansetal.(2012).There-
sultingp(z)isshowninFig.1.Themeanredshiftisz¯=0.748.In
contrast,themeanredshiftofthebest-fittingz histogramisbiased
p
lowwithz¯=0.69.
3 CFHTLENSSHEARCORRELATIONDATAAND
COVARIANCE
3.2 Angularcorrelationfunctions
The CFHTLenS data are described in several companion papers;
forafullsummaryseeHeymansetal.(2012).Stringentsystemat- Wecalculatethetwo-pointshearcorrelationfunctionsbyaverag-
icstestshavebeenperformedinHeymansetal.(2012)whichflag ingoverpairsofgalaxies,usingthetreecode ATHENA4.Galaxies
andremoveanydatainwhichsignificantresidualsystematicsare arepartitionedintonestedbranchesofatree,formingrectangular
detected.Itisthiscleanedsample,spanning 75percentofthe boxesinrightascensionαanddeclinationδ.Fortwobranchesat
∼
totalCFHTLenSsurveyarea,thatweuseinthisanalysis.Thiscor- angulardistanceϑandboxsizesd (i=1,2alongtheαandδdi-
i
respondsto129outof171MegaCampointings.Inthispaper,we rection,respectively)ifbothopeninganglesω d /ϑaresmaller
i i
≡
complementthosetestsbymeasuringtheB-modeuptolargescales
(Sect.3.5).ComparingthistothepreviousanalysisofF08shows
theimprovedqualityofthelensinganalysisbyCFHTLenS. 4 http://www2.iap.fr/users/kilbinge/athena
(cid:13)c 2009RAS,MNRAS000,1–18
CFHTLenS: cosmologicalmodel comparisonusing2D weaklensing 5
Northpole(δ=π/2)
φ2 C
β2 10-8 ++
D
-9
α 1 10 M
−
α 2 al -10 V (Gauss)
n 10
o V (Clone)
δ2 ϑ diag 10-11 Cm
e 10-12
c
β1 n
a -13
φ1 ri 10
a
v -14
δ1 o 10
e q u ator (δ = 0) α2 C 10-15
-16
10
α1 1 10 100
ϑ [arcmin]
Figure2.Angles andcoordinates onaspherefortwogalaxies i = 1,2
locatedat(αi,δi).Greatcirclesegmentsaredrawnasboldlines.
C
-8 --
10
-9
10
thanathresholdangleω ,thetreeisnotfollowedfurtherdownby
th l
a -10
descendingintosub-branches.Instead,theweightedaverageshear n 10
o
in each branch is used for the 2PCF estimator (eq. 4). We found g -11
thatavalueofωth = 0.03givessufficientaccuracycomparedto dia 10
thebrute-forceapproach. e 10-12
We use distances and angles on the sphere to calculate the nc
shearcorrelationfunctions. Fortwogalaxies i = 1,2atrightas- ia 10-13
r
censionanddeclination(α ,δ ),wecalculatethegreat-circledis- a
i i v -14
tanceϑwith o 10
C
-15
10
cosϑ=cos(α α )cosδ cosδ +sinδ sinδ . (9)
2 1 1 2 1 2
− -16
10
Eachgalaxy’sellipticityismeasuredinalocalCartesiancoordinate 1 10 100
systemwiththex-axisgoingalongthelineofconstantdeclination
ϑ [arcmin]
andthey-axispointingtotheNorthpole.Weprojectthisellipticity
tothetangentialandradialcomponentwithrespecttotheconnect-
ing great circle. For that, we calculate the angle β between the
great circlesegments ϑ and α . Then, the projectioin or so-called Figure3.DiagonalofthecovarianceC++(toppanel)andC−−(bottom),
i split up into various terms: Shot-noise D (solid red line), mixed term M
courseanglesareφi =π/2 βi.Withthesineandcosineruleson (dashedgreen),cosmicvarianceV(dottedbluelineandcrosses)andshear
−
thesphere,weget calibrationcovarianceCm(seeSect.3.4).
sin(α α )cosδ
cosφ1 = 2−sinϑ1 2;
sinφ1 =cosδ2sinδ1−sinsδi2ncϑosδ1cos(α2−α1), (10) 3.3 Datacovariance
Tomodelandinterprettheobservedsecond-ordershearfunctions,
andcorrespondingexpressionsforφ byexchangingindices. weneed toestimatethedatacovariance and itsinverse. Thecos-
2
To estimate the smoothed second-order quantities, we com- mic shear covariance C is composed of the shot-noise D, which
putethe2PCFson10,000linearangularbins.Thisislargeenough only appears on the diagonal, a cosmic-variance contribution V,
not to cause a significant E-/B-mode leakage due to the approxi- andamixedtermM(Schneideretal.2002).Thecovarianceofthe
mationoftheintegralsoverthecorrelationfunctionsbythedirect 2PCFsiscomprisedoffourblockmatrices.Thediagonalconsists
sum(eq.6).WeverifiedthisusingCFHTLenSnumericalsimula- ofC++andC−−whicharetheauto-correlationcovariancematri-
tionswithnoB-mode(seenextSection),seealsoBecker(2012). cesof ξ+ andξ−,respectively. Theoff-diagonal blocksare C+−
Wechoose thesmallestangulardistancebetween twogalaxiesto and C−+ = Ct+− which denote the cross-correlation covariance
be9arcsec,correspondingtothefirstbincentretobe10.4arcsec. betweenξ+andξ−.
Wecalculatethe2PCFs(eq.4)astheweightedmeanoverthefour Since the cosmic shear field is non-Gaussian on small and
Widepatches,usingthenumberofgalaxypairsasweightforeach medium angular scales, the cosmic variance involves four-point
bin. functions. Neglecting those can yield overly optimistic cosmo-
(cid:13)c 2009RAS,MNRAS000,1–18
6 Kilbingeret al.
−1 −0.8 −0.6 −0.4 −0.2 −0 0.2 0.4 0.6 0.8 1 thelasttwodatapointsshowanincreasedvariance,whichisdueto
thefiniteClonefieldgeometry.WhencomparingtheClonemean
correlation function to a theoretical prediction with cut-off scale
40 C+− C−− k = (2π/147) h−1Mpc,weget aroughagreement betweenthe
two,indicatingthatthelackofpowerisindeedcausedbythefinite
box. Wedraw similarconclusions forthecosmic variance of ξ−,
showninthelowerpanelofFig.3.Further,weverifiedthataJack-
Clone knifeestimateofthevariancebysub-dividingtheCHFLTenSdata
30 into129subfieldsgivesconsistentresults.
3.3.1 Graftingthecovariancematrix
j 20 C++ C−+ We construct the total covariance out to ϑ = 350 arcmin
by grafting the Clone covariance V to the analytical Gaus-
cl
sian prediction. For the latter, we use the method developed in
Kilbinger&Schneider (2004), which takes into account the dis-
cretenatureofthegalaxydistributionandthefieldgeometry.First,
10 Gaussian
weaddtheClonecovarianceV totheGaussiancosmiccovariance
cl
termV .Thecombinedcosmiccovarianceis
G
V =g V +(1 g )V , (11)
ij ij cl,ij ij G,ij
−
0 wherethemodulationfunctiong alleviatesdiscontinuitiesinthe
ij
0 10 20 30 40 combinedmatrix.Wechooseg tobeabi-levelstepfunction,with
ij
g = 1/2;g = 1ifbothindicesi,j aresmallerthanthestep
i s,s ij
indexs;andg =0ifatleastoneoftheindicesiorjislargerthan
ij
orequaltos.Thestepindexsischosensuchthatϑ isthescale
s
Figure 4. Correlation coefficient of the total covariance, shot-noise plus closest to 30 arcmin. Equation (11) is applied to all covariances
mixedpluscosmicvariance.Thelowerrighttriangle(withintheredbound-
between the two shear correlation functions, i.e. V++,V+− and
ary)istheGaussianprediction,theupperleft(blue)triangleshowstheto-
V−−.
tal covariance with non-Gaussian parts from the Clone. The four blocks
The Clone covariance also contains an additional variance
areC+−,C−−,C++andC−+,respectively,asindicatedintheplot.The
term, which was discovered recently (Satoetal. 2009). This so-
axeslabelsarethematrixindicesiandj.
called halo sample variance (HSV) stems from density fluctua-
tions on scales larger than the (finite) survey size that are corre-
logical constraints (Sembolonietal. 2007; Takada&Jain 2009; lated with fluctuations on smaller scales. For example, the num-
Hilbertetal.2011). ber of halos in the survey depend on the large-scale modes out-
To account for non-Gaussianity, we use N-body simula- sidethesurvey, sincehalosareclusteredanddonotjustfollowa
tions from Harnois-De´rapsetal. (2012). From these simulations, Poissondistribution.Thisintroducesanextravariancetothemea-
a‘Clone’oftheCFHTLenSdatahasbeenproducedwiththesame suredpowerspectrum.Thehalosamplevarianceisproportionalto
galaxy redshift distribution, galaxy clustering, masks, and noise thermsdensityfluctuationsatthesurvey scale(Satoetal.2009).
properties. The cosmological lensing signal is added using ray- Sinceoursimulatedlight-conesarecut-outsfromlargerboxesof
tracing through the light cones. The Clone cosmology is a flat sizeL=147Mpc/h(L=231Mpc/h)atredshiftbelow(above)
ΛCDMmodelwithΩ = 0.279,Ω = 0.046,n = 0.96,σ = unity,theydocontainFourierscalesoutsidethesurveyvolumeand
m b s 8
0.817andh = 0.701. Thelensingsignalforeachgalaxyiscon- theircoupling to smallerscales. The halo sample variance isim-
structed by ray-shooting through the simulated dark-matter dis- portant on small scales, where our cosmic variance is dominated
tribution. Each simulated line of sight spans a field of view of bytheClonecovariance.FollowingSatoetal.(2009),weestimate
3.5 3.5squaredegrees.Wefitcloseto 4 4MegaCampoint- thehalosamplevariancetodominatetheCFHTLenStotalcovari-
ings×on each line of sight, which is possible×because of overlap- anceatℓ 2 103,correspondingto5arcminuteswhichisthe
≈ ×
pingareasbetweenpointings.Atotalof184independent linesof Clonecovarianceregime.
sightareusedtocalculatethefield-to-fieldcovariancematrix.The Themissinglarge-scaleFouriermodesinthesimulationbox
finalmatrixisscaledwiththeratioof theeffectiveareas(includ- cause the HSV to be underestimated. A further underestima-
ing masks) of 0.11 which corresponds to 90 per cent of the area tion comes from the rescaling of the Clone lines of sight to the
of 16 MegaCam pointing that fit into each line of sight, divided CFHTLenS area since, in contrast to the other covariance terms,
by129MegaCampointingsusedinthisanalysis.Weaverageover theHSVtermdecreaseslessstronglythantheinversesurvey area
threedifferentsamplesofthegalaxyredshiftprobability distribu- (Satoetal. 2009). According to Kayoetal. (2012), when naively
tion, where galaxy redshifts were drawn from the corresponding rescalingfroma25deg2-surveyto1500deg2,thesignal-to-noise
pdf. ratioistoooptimisticbynotmorethan10percent.Forare-scaling
As shown in the upper panel of Fig. 3, the Gaussian pre- to the smaller CFHTLenS area, this bias is expected to be much
diction forthe cosmic variance (Kilbinger&Schneider 2004) for less.
ξ provide a good match to the Clone covariance on intermedi-
+
ate scales, 10 arcmin < ϑ < 30 arcmin. On larger scales, up
3.3.2 Cosmology-dependentcovariance
to ϑ < 200 arcmin, the∼nume∼rical simulations under-predict the
powerduetothefiniteboxsize(e.g.Power&Knebe2006).Only Ourgraftedcovarianceofthetwo-pointcorrelationfunctionises-
(cid:13)c 2009RAS,MNRAS000,1–18
CFHTLenS: cosmologicalmodel comparisonusing2D weaklensing 7
timated for a fiducial cosmological model, which is given by the
1.6⋅1013
N-body simulations. In order not to bias the likelihood function
ofthedata(Sect.4.2)atpointsotherthanthatfiducialmodel,we 1.4⋅1013
need to account for the fact that the covariance depends on cos-
mological parameters. We model the cosmology-dependence of 1.2⋅1013
thecovariancematrixfollowingEifleretal.(2009),whosuggested
13
approximativeschemesforthemixedtermMandthecosmicvari- 10
ance term V. Accordingly, for the cosmic-variance term, we as- e
sumeaquadraticscalingwiththeshearcorrelationfunction.This rac 8⋅1012
T
istrue on large scales, where the shear fieldis close to Gaussian 6⋅1012
andthecovarianceisindeedproportionaltothesquareofthecor- 1022⋅tr C
relationfunction.Wecalibratethesmall-scaleClonecovariancein 4⋅1012 tr C-1
mthiegshatmsceawleaayr,ealsikaenlyydtoiffbeeresnmcaelsl.inthewaythenon-Gaussianpart 2⋅1012 α(p=10)⋅tr C-1
ForthemixedtermM,weusethefittingformulaprovidedby α(peff=6)⋅tr C-1
0
Eifleretal. (2009). They approximate the variation with Ω and
m 0.1 1
σ , leaving the matrix fixed for other parameters. The shot-noise
8
p/n
termDdoesnotdependoncosmology.Thefinalexpressionforthe
covariancematrixis
Figure5.Traceofthecovariance andinversegraftedcosmiccovariance,
C (p)=D δ δ plotted against the ratio of the number of bins p to the number of real-
µν,ij i ij µν
isations n.Shownare thecases forcovariance (redsquares), theinverse
+M (p ) Ωm α(µ,ν,i,j) σ8 β(µ,ν,i,j) (greenfilledcircles),andtwocaseswheretheinversehasbeenmultiplied
µν,ij 0 (cid:18)0.25(cid:19) (cid:16)0.9(cid:17) withtheAnderson-Hartlapfactorα,correspondingtothenumberofbins
ξ (ϑ ,p) ξ (ϑ ,p) p=10(blueopencircles),andp=6(magentatriangles),causinganover-
+Vµν,ij(p0)ξµ(ϑi,p )ξν(ϑj,p ), (12) correctioninbothcases.Thisshowsthatwecanusetheinversecovariance
µ i 0 ν j 0 estimatorwithoutcorrection(greencurve).
wheretheindicesµ,ν standforthecomponents‘+’and‘ ’,and
−
i,jaretheangular-scaleindices.Here,p=(Ω ,σ ,...)denotes
m 8
thecosmologicalparameterforwhichthecovarianceisevaluated, de-biasedbymultiplicationwiththeAnderson-Hartlapfactorα=
with p being the fiducial model of the Clone simulation (Sect. (n p 2)/(n 1)(Hartlapetal.2007).
0 − − −
3.3).Theshot-noise termD andthemixedterm M arees- Ourfinal2PCFcovariance,however,isthemixtureofanML
i µν,i,j
timated using the method of Kilbinger&Schneider (2004). The estimate and analytical expressions. The ML estimator is modu-
cosmic-variancetermV isgivenineq.(11).Thepower-lawin- latedwiththeGaussiancosmicvarianceviaeq.(11),towhichwe
µν,ij
dicesαandβdependontheangularscalesandthecovariancecom- addtheshot-noise andmixedterms,eq.(12).Wequantifyapos-
ponent(µ,ν) ‘+’,‘ ’ ,andhavebeenobtainedinEifleretal. sible bias of the inverse covariance (C++)−1 by varying n for a
(2009).Wenot∈et{hatthe−in}versecovarianceisverysensitivetotwo fixedp = 10.Forastepindexs = 7,correspondingtoϑs = 37
apparentoutliersofαandβfortheC+−-part.Toavoidnumerical arcmin, Fig.5shows that thetraceof (C++)−1 does not depend
issues,wereplacethesetwonumbersbythemeanoftheirneigh- ontheratiop/nforourgraftedcosmiccovariancematrix.Multi-
bouring values. In our case the mixed term of the covariance is plicationwithαfor p = 10resultsinanovercorrection, causing
importantandcannotbeneglected(seealsoKilbinger&Schneider a strong decrease of tr(C++)−1 with p/n. A similar albeit less
2004;Vafaeietal.2010),incontrasttorecentfindingsbyJeeetal. strongdecrease isseen when naively takingintoaccount the fact
(2012). thattheClonecovarianceonlycontributestoaneffectivenumber
DuringtheMonte-Carlosampling(Sect.4.2),thecovariance of scales peff = 6, according to eq. (11) with s = 7. The three
is updated at each sample point p using eq. (12). We make sure curvesfortheinverseseemtoconvergefor p/n 0.Therefore,
→
that each calculation of the covariance resulted in a numerically wehavereasontobeconfidentthatanybiasoftheunalteredinverse
positive-definite matrix, and discard the (rare) sample points for ofeq.(12)issmall,andhencewedonotneedtoapplythescalar
whichthisisnotthecase. correctionfactor α.Theadditionofadeterministiccomponent to
InFig.4weshowthetotalcovarianceC = D+M+Vand theMLcovarianceseemstobesufficienttorendertheestimateof
compareittotheGaussianprediction C = D+M+V .Both theinversetobeunbiased.
G G
casesaresimilaronmostscales.Onsmallscalesthegraftedcovari-
ance shows strongercross-correlations between scales, indicating
non-Gaussian effects. We find that the additional covariance C 3.3.4 Covarianceofderivedsecond-orderfunctions
m
duetotheshearcalibration(seeSect.3.4)canbeneglected,ascan
Expressionsforthecovarianceofthederivedsecond-order statis-
beseeninFig.3andSect.6.2.
tics (eq. 6) are straight-forward to obtain, and can be calculated
byintegratingthecovarianceofthe2PCFs(Schneideretal.2002).
However,thenecessaryprecisionforthenumericalintegrationre-
3.3.3 Inversecovarianceestimator
quiresalargenumberofangularbinsforwhichthe2PCFcovari-
IthasbeenshowninAnderson(2003)andHartlapetal.(2007)that ance hasto becalculated, which isvery time-consuming. Conse-
themaximum-likelihood(ML)estimatoroftheinversecovariance quently, for all derived second-order functions we choose not to
isbiasedhigh.Thefield-to-fieldcovariancefromtheCloneissuch grafttheClonecovariancetotheGaussiancovariance,butinstead
an ML estimate. The bias depends on the number of realisations onlyusetheClonetocalculatethetotalcovarianceofthederived
or fields n, and the number of bins p. The ML estimator can be functions.Toincludeshotnoise,weaddtoeachgalaxys’shearan
(cid:13)c 2009RAS,MNRAS000,1–18
8 Kilbingeret al.
intrinsicellipticityasaGaussianrandomvariablewithzeromean
anddispersionσε=0.38.Thelatteriscalculatedasσε2 = iεiε∗i, ξξ+((ϑϑ))
where the sum goes over all CFHTLenS galaxies in ourPredshift 10-4 −
range. Therefore, the covariance between the 184 Clone lines of
sightgivesusthetotalcovarianceD+M+V.Contrarytothecase
n
ofthe2PCFs(previoussection),thiscovariancestemsfromapure atio 10-5
ML estimate, and therefore the inverse needs to be de-biased by el
theAnderson-Hartlap factor α. Withatypical number of angular orr
c
scales of p = 10 to 15 the corresponding α is of order 0.9. We ar
showthatourcosmologicalresultsareindependentofthenumber She 10-6
ofrealisationsinSect.6.2.Notethatfortheallderivedestimators,
thecosmology-dependenceofthecovarianceisneglected.
ForupcomingandfuturetomographicsurveyssuchasKiDS5, 10-7
DES6, HSC7, Euclid8 (Laureijsetal. 2011) or LSST9, a much
larger suite of simulations will be necessary. The number of re- 1 10 100
alisationsnhastobesubstantiallylargerthanthenumberofbins ϑ [arcmin]
p(Hartlapetal.2007).Foramulti-bintomographic shearsurvey,
p can easily be of the order of several hundreds or more if other Figure6.Themeasuredshearcorrelationfunctionsξ+(blacksquares)and
probesarejointlymeasuredsuchasgalaxyclusteringormagnifi- ξ−(bluecircles),combinedfromallfourWidepatches.Theerrorbarscor-
respondtothetotalcovariancediagonal.Negativevaluesareshownasthin
cation.Thisnecessitatesontheorderofathousandandmoreinde-
pointswithdottederrorbars.Thelinesarethetheoretical predictionusing
pendentlinesofsight.Thisnumberhastobemultipliedbymany
theWMAP7best-fittingcosmologyandthenon-linearmodeldescribedin
ifapropertreatmentofthecosmology-dependence istobetaken Sect.4.3.ThedatapointsanderrorbarsarelistedinTableB1.
intoaccount. Moreover, asimpleup-scaling of smallersimulated
fieldstofullsurveysizemightnotbeeasybecauseofthedifferent
area-scalingoftheHSVterm.
0.001
W1
3.4 Ellipticitycalibrationcorrections
0.0001 W2
W3
We apply the shear calibration as described in Heymansetal. 1e-05 W4
(m2u0l1t2ip)l,icwahtiivcehbaiacscomun,ts for a potential additive shear bias c and ξϑ()+ 1e-06
1e-07
εobs =(1+m)εtrue+c. (13)
1e-08
Theadditivebiasisfoundtobeconsistentwithzeroforε1.Thesec- 1e-09
ondellipticitycomponent ε showsasignal-to-noiseratio(S/N) 1 10 100
2
and size-dependent bias which we subtract for each galaxy. This ϑ [arcmin]
representsacorrectionwhichisonaverageatthelevelof2 10−3. 0.001
Themultiplicativebiasmismodelledasafunctionofthe×galaxy W1
0.0001 W2
S/N andsizer.Itisfitsimultaneouslyin20binsofS/N andr,
W3
seeMilleretal.(2013).Weusethebest-fittingfunctionm(S/N,r) 1e-05 W4
andperformtheglobalcorrectiontotheshear2PCFs,seeeqs.(19) ϑ()− 1e-06
and(20)ofMilleretal.(2013).Accordingly,wecalculatethecali- ξ
1e-07
brationfactor1+Kastheweightedcorrelationfunctionof1+m,
1e-08
w w (1+m )(1+m )
1+K(ϑ)= ij i j i j . (14) 1e-09
P ijwiwj 1 10 100
P ϑ [arcmin]
Thefinalcalibrated2PCFsareobtainedbydividingξ+andξ− by
1+K.Theamplitudeof1+K isaround0.91onallscales.The
errorsonthecorrelationfunctionfromthefituncertaintyarenegli- Figure7.Themeasuredshearcorrelationfunctionsξ+(toppanel)andξ−
(bottom),forthefourWidepatches.TheerrorbarscorrespondtoPoisson
giblecomparedtoourstatisticalerrors.Furthermore,wecalculate
noise.
thecovariancematrixC forthecorrelationfunctionfromthisun-
m
certainty,andshowinSect.6.2thatthecosmologicalresultsremain
unchangedbyaddingthistermtotheanalysis.
Figure6showsthecombinedandcorrected2PCFs,whichare
theweightedaveragesoverthefourWidepatcheswiththenumber
of pairs as weights. Note that the data points are strongly corre-
lated,inparticularξ onscaleslargerthanabout10arcmin.Cos-
+
mologicalresultsusingthisdatawillbepresentedinSect.5.The
5 kids.strw.leidenuniv.nl
correlationsignalsplitupintothecontributionsfromthefourWide
6 www.darkenergysurvey.org
7 http://www.naoj.org/Projects/HSC/HSCProject.html patchesisplottedinFig.7.Thereisnoapparentoutlierfield.The
8 www.euclid-ec.org scatterislargerthansuggestedbythePoissonnoiseonlargescales,
9 http://www.lsst.org/lsst inagreementwiththeexpectedcosmicvariance.
(cid:13)c 2009RAS,MNRAS000,1–18
CFHTLenS: cosmologicalmodel comparisonusing2D weaklensing 9
3.5 E-andB-modes put lessweight on small scales where thesignal-to-noise ratioin
the2PCFsislarger.
Theaperture-massdispersionisshownintheupperpanelofFig.8.
A further derived second-order quantity are the shear E-/B-
TheB-modeisconsistentwithzeroonallscales.Wequantifythis
modecorrelationfunctions ξ (Crittendenetal.2002;Penetal.
byperforminganullχ2test,takingintoaccounttheB-modePois- E,B
2002), which have been used in F08. Whereas they share the in-
soncovarianceC×asmeasuredontheClone, convenience withthetop-hatshearrmsof aformalupperinfinite
χ2B = hM×i(θi) C−×1 ijhM×i(θj). (15) itnhteergerfaotrieonnoltimbeit,usthedeyinoftfheirswnoorakd.vantage overthelatter, and will
Xij (cid:2) (cid:3)
SinceherethecovarianceisentirelyestimatedfromtheCloneline-
3.6 Conclusiononestimators
of-sight,theinversehastobede-biasedusingtheAnderson-Hartlap
factor.WeconsidertheB-modeovertheangularrange [5.5;140] We compared various second-order real-space shear functions,
arcmin.Asdiscussedbefore,thelowerscaleiswheretheB-mode startingwiththefundamental two-point correlation functions ξ±.
duetoleakageisdowntoafewpercent.Theupperlimitisgiven From the 2PCFs we calculated a number of E-/B-mode separat-
bythelargestscaleaccessibletotheClone,whichismuchsmaller ing functions. The top-hat shear rms γ 2 is of limited use for
thanthelargestCFHTLenSscale:Itis280arcmin,resultinginan cosmologicalanalysisbecauseofthecho|sm|oilogy-dependent E-/B-
u1p4p.9e/r1l5im=it0o.f9h9M,ca2porireosfphoanldfinthgattosacanleo.n-TnhuellrBes-umltoindgepχr2o/bdaboifliotyf misocdoenfilenaekdagtoe.sFmoarllthsecaalpeesr,twurhee-mreaasssthdeisoppetrismioinzehdMria2npgisthtaitsisletiackage
E
of46percent.Evenifweonlytakethehighestsix(positive)data andCOSEBIswereintroducedtoavoidanyleakage.ThedrawbRack
points,wefindtheχ2perdegreeoffreedom(dof)tobeχ2/dof = of the 2PCFsisthat they are sensitive tolarge scales outside the
4.12/6 = 0.69, whichislessthan1σ significance.Thenon-zero surveyarea andthusmay contain anundetectable B-mode signal
B-modesignalataround50-120arcminfromF08isnotdetected (Schneideretal. 2010). COSEBIs capture the E-/B-mode signals
here. inanoptimal wayonafiniteangular-scaleinterval [ϑ ;ϑ ].
min max
Thetop-hat shearrmsB-modeisconsistent withzero onall TheinterpretationofCOSEBIsandthematchingofmodestoan-
measured scales, as shown in the middle panel of Fig. 8. Note, gularscales arenot straightforward sincethe corresponding filter
however,thatofallsecond-orderfunctionsdiscussedinthiswork, functionsarestronglyoscillating.
γ 2 istheonewiththehighestcorrelationbetweendatapoints. For lensing alone, we obtain cosmological parameter con-
h| | i
Thepredicted leakage fromthe B-tothe E-modeissmallerthan straintsonΩ andσ forthedifferentestimatorsdiscussedinthis
m 8
the measured E-mode, but becomes comparable to the latter for section.TheresultsandcomparisonsarepresentedinSect.5.1.
θ>100arcmin,wheretheleakagereachesupto50percentofthe Wedecided tousethe2PCFstocompute cosmological con-
E-mode. straintsincombinationwiththeotherprobes,forthefollowingrea-
Theoptimized ringstatisticfor η = ϑmin/ϑmax = 1/50 is sons.Thegoalofthispaperistoexplorethelargestscalesavailable
plottedinthelowerpanelofFig.8.EachdatapointshowstheE- forlensinginCFHTLenS.Thisisonlypossiblewithasufficiently
andB-modeontheangularrangebetweenϑminandϑmax,thelat- largesignal-to-noiseratiowhenusingthe2PCFs.Wenotethaton
terofwhichislabelledonthex-axis.TheB-modeisfoundtobe theselargescalesoursystematicstests,thestar-galaxyshapecor-
consistentwithzero,aχ2nulltestyieldsa35percentprobability relation(Heymansetal.2012) andtheE-/B-modedecomposition
ofanon-zeroB-mode. (thiswork),werenotpossible.However,sincebothtestshavere-
Wefirst test ourcalculation of COSEBIson the CFHTLenS vealednosystematicsonsmallerscales,weareconfidentthatthe
Clone with noise, where wemeasure a B-mode of at most afew shear signal up to very large scales is not significantly contam-
10−12forn65andϑmax 6250arcmin.Eventhoughthisisa inated. Moreover, the implementation of a cosmology-dependent
×
fewordersofmagnitudeslargerthantheB-modeduetonumerical covarianceiscurrentlyonlyfeasibleforthe2PCFs.
errorsfromtheestimationfromtheory,itisinsignificantcompared
totheE-modesignal. Whenincludingthelargestavailablescales
fortheClonehowever,ϑ 280arcmin,theB-modeincreases
max
∼ 4 COSMOLOGYSET-UP
tobe of the orderof theE-mode. Thisistrueindependent of the
binningorwhethernoiseisadded.Wepresumethatthisisdue to 4.1 Datasets
insufficient accuracy with which the shear correlation function is
estimatedfromthesimulationontheseverylargescales,fromonly Weusethefollowingdatasetsandpriors:
a small number of galaxy pairs. Further, for n > 5 a similarly
(i) CFHTLenStwo-pointshearcorrelationfunctionsandcovari-
largeB-modeisfoundforsomecasesof(ϑ ,ϑ ).Again,the
min max ance as described in Sect. 3. We choose the smallest and largest
accuracyofthesimulationsisnotsufficienttoallowforprecisenu-
angularbinstobe0.9and300arcmin,respectively.Thisincludes
merical integration over the rapidly-oscillating filter functions of
galaxypairsbetween0.8and350arcmin.
Log-COSEBIsforhighermodes(Becker2012).Wewilltherefore
(ii) Cosmic microwave background (CMB) anisotropies:
restrictourselveston65forthesubsequentcosmologicalanaly-
WMAP7 (Larsonetal. 2011; Komatsuetal. 2011). The released
sis. WMAP code10 is employed to calculate the likelihood, see also
ThemeasuredCOSEBIsmodesareshowninFig.9.Weuse Dunkleyetal.(2009).WeuseCAMB11(Lewisetal.2000)toget
assmallestscaleϑ=10arcsec,andtwocasesofϑ of100and
max the theoretical predictions of CMB temperature and polarisation
250arcmin.InbothcaseswedonotseeasignificantB-mode.The
power-andcross-spectra.
signal-to-noise ratio of the high mode points decreases when the
angularrangeisincreased:Forϑ = 250arcminonlythefirst
max
twomodesaresignificant. Thisisnot unexpected, sincethefilter 10 http://lambda.gsfc.nasa.gov
functionsforϑmax =250arcminsamplelargerangularscalesand 11 http://camb.info
(cid:13)c 2009RAS,MNRAS000,1–18
10 Kilbingeret al.
1.2⋅10-5
E-mode
1.0⋅10-5 B-mode ϑ = 10" ... 100’
n 8.0⋅10-6 WMCAloPn7e 3.5⋅10-10
o E-mode
ersi 6.0⋅10-6 3.0⋅10-10 B-mode
s disp 4.0⋅10-6 2.5⋅10-10 WMCAloPn7e
as 2.0⋅10-6 2.0⋅10-10
m
ure- 0 Bn 1.5⋅10-10
Apert -2.0⋅10-6 1.15⋅⋅1100--66 E, n 1.0⋅10-10
-4.0⋅10-6 5⋅10-7 5.0⋅10-11
0
-6.0⋅10-6 20 50 100 0
1 10 100 -5.0⋅10-11
θ [arcmin] -1.0⋅10-10
1 2 3 4 5
mode n
-4 E-mode
10
B-mode
WMAP7
Clone ϑ = 10" ... 250’
ms 10-5 1.5⋅10-9
ear r 1.0⋅10-9
h
s -6
at 10 5.0⋅10-10
h
p-
o 0
T
-7 n
10 B, n -5.0⋅10-10
E
-1.0⋅10-9
-8
10
1 10 100 -1.5⋅10-9 E-mode
B-mode
θ [arcmin] -2.0⋅10-9 WMAP7
Clone
3.0⋅10-6 -2.5⋅10-9
E-mode 1 2 3 4 5
RB 2.5⋅10-6 WBM-mAoPd7e mode n
R, E 2.0⋅10-6 Clone
c Figure9.COSEBIswithlogarithmicfilterfunctions.Theleft(right)panel
atisti 1.5⋅10-6 correspondtoamaximumangularscaleof100arcmin(250arcmin).The
st filled(open)squarescorrespondtotheCFHTLenSE-mode(B-mode).The
d ring 1.0⋅10-6 eTrhroerdbaasrhseadrelintheeisCtlhoenethfieeolrde-ttioc-afilepldrerdmicsti(ornesfcoarleadWtoMthAePC7F-cHoTsmLeonloSgayr,etah)e.
mize 5.0⋅10-7 dottedcurveshowstheClonemeanCOSEBIs.
pti
O 0
(iii) Baryonic acoustic oscillations (BAO): SDSS-III (BOSS).
-5.0⋅10-7 We use the ratio D /r = 13.67 0.22 of the apparent BAO
V s
10 100 ±
atz = 0.57tothesoundhorizondistance,asaGaussianrandom
θ [arcmin] variable,fromAndersonetal.(2012).
(iv) Hubble constant. Weadd a Gaussian prior forthe Hubble
Figure 8. Smoothed second-order functions: Aperture-mass dispersion constantofh=0.742 0.036fromCepheidsandnearbytype-Ia
±
hM2 i(leftpanel),sheartop-hatrmsh|γ|2i(middle)andoptimizedring supernovaedistancesfromHST(Riessetal.2009,hereafterR09).
ap
statistic RE (right), split into the E-mode (black filled squares) and B-
mode(redopensquares). TheerrorbarsaretheClone field-to-field rms. IncontrasttoKilbingeretal.(2009)wedonotincludesuper-
ThedashedlineisthetheoreticalpredictionforaWMAP7-cosmology(with novaeoftype(SNIa)Ia.BOSSputsatightconstraintontheexpan-
zeroE-/B-modeleakage),thedottedcurveshowstheClonelines-of-sight sionhistoryoftheUniverse,whichisinexcellentagreementwith
meanE-modesignal.ForhMa2piandh|γ|2itheWMAP7-predictionofthe corresponding constraints using the luminosity distance from the
leakedB-modeisshownasreddashedcurve;theshadedregioninthemid-
mostrecentcompilationofSNIa(Conleyetal.2011;Suzukietal.
dlepanelcorrespondstothe95percentWMAP7confidenceintervalofσ8 2012). Both BOSSand SNIa are geometrical probes, and adding
(flatΛCDM).Forthesheartop-hat rms,negative points areplotted with
SNLStoWMAP7+BOSSyieldslittleimprovementoncosmolog-
dashederrorbars.
(cid:13)c 2009RAS,MNRAS000,1–18