MNRAS000,1–16(2017) Preprint22December 2016 CompiledusingMNRASLATEXstylefilev3.0 Revisiting CFHTLenS cosmic shear: Optimal E/B mode decomposition using COSEBIs and compressed COSEBIs Marika Asgari,1? Catherine Heymans1, Chris Blake2, Joachim Harnois-Deraps3, Peter Schneider4, Ludovic Van Waerbeke3 1ScottishUniversitiesPhysicsAlliance,InstituteforAstronomy,UniversityofEdinburgh,RoyalObservatory,BlackfordHill,Edinburgh, EH93HJ,U.K. 2CentreforAstrophysics&Supercomputing,SwinburneUniversityofTechnology,P.OBox218,Hawthorn,VIC3122,Australia. 6 3DepartmentofPhysicsandAstronomy,UniversityofBritishColumbia,6224AgriculturalRoad,Vancouver,V7T1Z1,B.C,Canada. 1 4Argelander-InstitutfürAstronomie,UniversitätBonn,AufdemHügel71,D-53121Bonn,Germany 0 2 c Accepted2016October4.Received2016August30;inoriginalform2015December21 e D 1 ABSTRACT 2 Wepresentare-analysisoftheCFHTLenSweakgravitationallensingsurveyusingComplete Orthogonal Sets of E/B-mode Integrals, known as COSEBIs. COSEBIs provide a complete ] set of functions to efficiently separate E-modes from B-modes and hence allow for robust O andstringenttestsforsystematicerrorsinthedata.ThisanalysisrevealssignificantB-modes C onlargeangular scalesthatwerenotpreviouslyseen usingthestandardE/Bdecomposition . analyses.WefindthatthesignificanceoftheB-modesisenhancedwhenthedataissplitby h galaxytypeandanalysedintomographicredshiftbins.Addingtomographicbinstotheanal- p - ysisincreasesthenumberofCOSEBIsmodes,whichresultsinalessaccurateestimationof o thecovariancematrixfromasetofsimulations.Wethereforealsopresentthefirstcompressed r COSEBIsanalysisofsurveydata,wheretheCOSEBIsmodesareoptimallycombinedbased t s ontheirsensitivitytocosmologicalparameters.InthistomographicCCOSEBIsanalysiswe a findtheB-modestobeconsistentwithzerowhenthefullrangeofangularscalesareconsid- [ ered. 2 Keywords: Gravitationallensing:weakmethod:dataanalysiscosmology:observations v 5 1 1 0 1 INTRODUCTION laborationetal.2015a)hasbeenwidelyreported.Ithasbeeninter- 0 pretedindifferentwaysasasignfornewphysics(seeforexample . Observationsofweakgravitationallensingbythelarge-scalestruc- 1 Dossett et al. 2015; Planck Collaboration et al. 2015b; Battye & 0 tureintheUniverseprovidesapowerfulprobeofdarkmatter,dark Moss2014),thecombinedeffectsofbaryonicfeedbackandneu- 6 energy and modified gravity theories. The underlying physics of trinos(Harnois-Dérapsetal.2015;Köhlingeretal.2015),orpre- 1 lensingiswellunderstood,leavingthenon-trivialmeasurementit- viouslyunknownsystematicerrors(seeforexampleSpergeletal. : selfasthemainchallengeinreachingthefullpotentialofthiscos- v 2015;Verdeetal.2013;Raveri2015;Addisonetal.2016).Inthis mological tool. Three major new weak lensing surveys are under i paper we address the question of systematic errors by subjecting X way, with the Kilo-Degree Survey (KiDS), the Dark Energy Sur- theCFHTLenSdatatoarigoroustestforshearsystematicsusing r vey(DES),andtheHyper-SuprimeCameraSurvey(HSC).KiDS “CompleteOrthogonalSetsofE/B-modeIntegrals”alsoknownas a and DES recently presented their first ‘cosmic shear’ measure- “COSEBIs”.GravitationallensingcanonlyproduceE-modesand ments (Kuijken et al. 2015; Becker et al. 2015). These new sur- anydetectedB-modesareduetoeithersystematicerrorsorother veysalreadycoverseveralhundredsofsquaredegrees,butfornow physicaleffects.1 they still lack statistical precision in comparison to their deeper TheformalismforCOSEBIswasdevelopedinSchneideretal. butsmallerareapredecessor,theCanada-France-HawaiiTelescope (2010).COSEBIsprovideacompletesetoffunctionsforefficiently Lensing Survey, CFHTLenS (Heymans et al. 2012). As such this separatingE-modesfromB-modesandhenceallowforrobustsys- survey still provides the tightest cosmological constraints from tematicstestsusingtheB-modesandafairlycompresseddataset. weakgravitationallensing. The tension between the results of the CFHTLenS tomo- graphicanalysis(Heymansetal.2013)andthecosmologicalmea- surements from the cosmic microwave background (Planck Col- 1 Whereassourceclusteringandlens-lenscouplingcaninprinciplegener- ateB-modesfromlensing(Schneideretal.2002;Hilbertetal.2009),their amplitudeistoosmalltobesignificantlydetectedincurrentandfuturesur- ? E-mail:[email protected] veys. (cid:13)c 2017TheAuthors 2 M.Asgarietal. Schneideretal.(2010)andEifler(2011)showedthatasmallnum- showninKilbinger(2015)thereisexcellentconsistencybetween berofCOSEBIsmodesareenoughtoessentiallycapturethefull thedifferentcosmologicalconstraintsderivedbythesevariedsta- cosmologicalinformationusingnumericalanalysisandmockdata, tisticalanalysesoftheCFHTLenSsurvey.Themoststringentone, respectively. Asgari et al. (2012) extended the method to tomo- andalsothemostintensionwiththeCMBresultsisthe6-binto- graphicbinsandshowedthatalthoughasmallnumberofCOSE- mographicanalysisofHeymansetal.(2013).Wethereforefocus BIsmodesisenoughforeachredshiftbinpair,inthepresenceof oursystematicsanalysisonthistomographicdataset. many redshift bins the total number of COSEBIs needed is rela- Thispaperisstructuredasfollows:Sect.2outlinesthestatisti- tivelyhigh.Thisisalsotrueforalltheotherconventionallyused calmethods,COSEBIsandCCOSEBIs,thatareusedinthisanaly- cosmic shear observables such as the two-point correlation func- sis.Sect.3containsthemainresults,whereweshowthemeasured tionsortheconvergencepowerspectrum. COSEBIsandCCOSEBIs.WequantifythemeasuredB-modesus- Themostcommonapproachtoestimatecovariancematrices ing a χ2 analysis and finally conclude in Sect.4. We verify our istousemockdatafromnumericalsimulations,buttheprecision pipelinetestsonmockdataintheAppendix. withwhichthiscanbemeasureddecreaseswiththenumberofob- servables(Hartlapetal.2007;Taylor&Joachimi2014;Sellentin &Heavens2015).Therequirementtominimizethenumberofob- servablespromptedAsgari&Schneider(2015)todevelopacom- 2 METHODS:COSEBISANDCCOSEBIS pressionmethodwhichreducesthisnumbersubstantially,without significantlossofinformation.Inthispaperweshowthefirstmea- Converting a measured gravitational lensing shear field to a con- surementofthesecompressedCOSEBIs,whicharecalledCCOSE- vergencefielddoesnotnecessarilyresultintherealprojectedmass BIs.WealsopresentthefirstmeasurementoftomographicCOSE- field expected from gravitational lensing theory (see Bartelmann BIs. &Schneider2001,forareviewofweakgravitationallensing).The CFHTLenSisa3Dweaklensingsurvey,analysingu∗,g0,r0, reasonisthatasidefromfirstorderlensingeffectsthereareotherin- i0,z0 multi-banddataspanning154deg2 fromtheCFHTLegacy fluentialfactors.Theseotherfactorsfallintotwocategoriesaccord- ingtowhethertheiroriginisphysicalornon-physical.Theformer SurveyWideProgramme.Observedinsub-arcsecondseeingcon- mayarisefromhigher-orderlensingeffects(contributionsbeyond ditions,thissurveywasoptimisedforweaklensingscience.Pixel- theBornapproximation,seeSchneideretal.1998),andsourcered- leveldataprocessingusedthelensing-qualityTHELIdatareduc- shift clustering (Schneider et al. 2002), or intrinsic galaxy align- tion package (Erben et al. 2013). PSF Gaussianised photometry ments (see Blazek et al. 2011, and references therein); The latter provided precise photometric redshift distributions (Hildebrandt caseinvolvesnoisecontributionsandremainingsystematiceffects, et al. 2012) with a reasonable level of accuracy as scrutinised in forexample,ingalaxyshapemeasurements.Firstorderweakgrav- Choi et al. (2015) using a spectroscopic galaxy cross-correlation itationallensingcanonlyproducemodeswhicharecommonlyre- clustering analysis. Weak lensing shear measurements were de- ferred to as E-modes, whereas, the modes which arise from the rivedandcalibratedusingthelensfitBayesianmodel-fittingmethod imaginarypartoftheestimatedconvergencefield,κ,arecalledB- (Milleretal.2013).Aseriesofdetailedsystematicsanalyseswere modes.Thesemodesaresonamedbecauseofthesimilarmathe- applied to the full data set, resulting in the rejection of a quarter maticalpropertiesoftheshearfieldandthepolarizationofanelec- ofthesurveyareainordertosatisfystrictsystematiccriteria(Hey- tromagneticradiationfield(bothofthemarepolars).B-modecon- mansetal.2012). tributions from physical effects are expected to be negligible for Anumberofdifferentcosmologicalanalyseshavebeencar- a survey like CFHTLenS. Hence any detection of a B-mode will riedoutusingCFHTLenS.Kilbingeretal.(2013)performedatwo- arise from either inaccuracies in the shape measurements and/or dimensionalanalysisofthedatausingseveralcosmicshearestima- selection biases. Since the physical contributions to the B-modes tors,includingCOSEBIs,thestatisticthatformsthefocusofthis areverysmall,measuringastatisticallyzeroB-mode,suggests(but work.This2DanalysiswasextendedbyFuetal.(2014)whoused doesnotguarantee)asatisfactoryPSFcorrection.Separatingthese COSEBIsinconjunctionwiththethirdorderaperturemassstatis- modesisessentialtotestforsystematicerrors. tictoconstraincosmologicalparameters.Asidefromtheanalysis Anyobservable(statistic)whichseparatesE-modesfromB- ofCFHTLenS,Huffetal.(2014)appliedCOSEBIsonSloanDigi- talSkySurvey(SDSS)datatoconstrainσ andΩ h2 . modesatthetwo-pointstatisticslevel,canbewritteninthefollow- 8 m ingform, Analyses of CFHTLenS that incorporated the redshift- dependence of the weak lensing signal started with a two-bin to- 1Z ∞ E = dϑϑ[T (ϑ)ξ (ϑ)+T (ϑ)ξ (ϑ)], (1) mographic analysis in Benjamin et al. (2013) and Simpson et al. 2 + + − − 0 (2013).Thiswasfollowedbyafinersix-bintomographicanalysis 1Z ∞ inHeymansetal.(2013),wherethedatawasmodelledasacombi- B = dϑϑ[T (ϑ)ξ (ϑ)−T (ϑ)ξ (ϑ)], 2 + + − − nationofacosmologicalsignalandacontaminatingsignalfromthe 0 presenceofintrinsicgalaxyalignments(seealsoMacCrannetal. whereξ (ϑ)arethetwo-pointcorrelationfunctions(2PCFs)ofthe ± 2015;TheDarkEnergySurveyCollaborationetal.2015;Joudaki shearfield,ϑistheangulardistancebetweenpairsofgalaxieson etal.2016,forre-analysesofthisdataset).Thesestatisticalanaly- theskyandT (ϑ)arefilterfunctions,thatarechosentoproduce ± seswerebasedonmeasurementsofthetwo-pointshearcorrelation pureE/B-modes,correspondingtoE/B,respectively.InSchneider functions(2PCFs).Usingonlybluegalaxies,forwhichtheintrin- &Kilbinger(2007),conditionsforsuchfilterswereobtained, sicalignmentcontaminationisexpectedtobenegligible,Kitching etal.(2014)carriedoutafull3-Dpowerspectrumanalysisofthe Z ϑmax dϑ Z ϑmax dϑ T (ϑ)=0= T (ϑ), (2) survey. This power spectrum analysis was restricted to relatively ϑ − ϑ3 − ϑmin ϑmin largephysicalscalestominimisetheeffectsofbaryonfeedbackon Z ϑmax Z ϑmax thenon-linearmatterpowerspectrum(seeSembolonietal.2013; dϑϑT (ϑ)=0= dϑϑ3T (ϑ), (3) + + Mead et al. 2015; Harnois-Déraps et al. 2015, for example). As ϑmin ϑmin MNRAS000,1–16(2017) RevisitingCFHTLenSwithCOSEBIsandCCOSEBIs 3 whereϑ >0andϑ isfinite.UsingtheseconditionsSchnei- 2.2 CompressedCOSEBIs:CCOSEBIs min max deretal.(2010)constructedtwocompleteorthogonalsetsoffilter Data compression is a challenge that will become increasingly functions,T whichformthebasisoftheCOSEBIs. ± moreimportantforfuturelargescalesurveyssuchasEuclid2 and LSST3.Themainreasondatacompressionisessentialisthatthe numberofsimulationsneededtoestimatethedatacovariancema- trixaccurately,dependsonthenumberofobservables.Therefore, havingasmallersetofobservablesreducesthenumberofcosmo- logicalsimulationsneeded. Asgari&Schneider(2015)developedacompressionmethod whichisbasedonthesensitivityofobservables(statistics)tothe 2.1 COSEBIs parameterstobemeasured.Thismethodreliesonourunderstand- The two sets of COSEBIs basis functions are the Lin- and Log- ingoftheseparameters,sincethecompressedobservablesdepend COSEBIs,whicharewrittenintermsofpolynomialsinϑandln(ϑ) on the covariance and derivatives of the parent observable to the inrealspace,respectively.InadditiontoSchneideretal.(2010),Fu parametersattheirfiducialvalue.Theassumptionbehindthiscom- &Kilbinger(2010)constructedfilterswhichmaximizedthesignal- pressionmethodisthatwehavearelativelygoodideaofthevalue to-noiseratioforaspecificangularrange,ormaximizedtheinfor- oftheparametersthatwewanttomeasure(forexamplefrompre- mationcontentofE statisticsviaFisheranalysis.Inthisanalysis viousobservations),whichiscorrectformostofthecosmological weusetheLog-COSEBIs,astheyrequirefewermodescompared parameters. One might expect to lose a significant portion of the totheLin-COSEBIstoessentiallycapturealltheinformation(see informationabouttheparametersifthefiducialcovariancematrix Schneideretal.2010forasingleredshiftbinandAsgarietal.2012 usedforconstructingtheparametersisnotclosetothetruth.How- forthetomographiccase). ever,Asgari&Schneider(2015)appliedthiscompressionmethod The COSEBIs can be written in terms of the 2PCFs in real totomographicCOSEBIsandshowedthattheweaklensinginfor- space, mationlostduethiscompressionissmallevenforveryinaccurate COSEBIscovariancematrices.Thisimpliesthatthiscompression E(ij) = 1Z θmaxdϑϑ[T (ϑ)ξ(ij)(ϑ)+T (ϑ)ξ(ij)(ϑ)], isinsensitivetotheinaccuraciesintheestimatedcovariancematrix n 2 +n + −n − oftheparentobservables,whichmeansthatusingthiscompression θmin (4) allowsforthesameaccuracyinestimationswithfewercosmologi- 1Z θmax calsimulations. B(ij) = dϑϑ[T (ϑ)ξ(ij)(ϑ)−T (ϑ)ξ(ij)(ϑ)], Here we will also use compressed COSEBIs (CCOSEBIs) n 2 +n + −n − θmin for the analysis of the CFHTLenS data. The CCOSEBIs are lin- (5) earcombinationsoftheCOSEBIs.Thecoefficientsoftheselinear combinationsarewrittenintermsofthecovarianceandthederiva- whereE(ij)andB(ij)aretheEandB-modeCOSEBIsforredshift n n tivesoftheCOSEBIswithrespecttocosmologicalparameters, binsiandj,T (ϑ)aretheCOSEBIsfilterfunctionsandn,anat- ±n uralnumber,istheorderoftheCOSEBIsmodes.Themodeswith Ec =ΓE , (9) largernvaluesaretypicallymoresensitivetosmall-scalevariations where Ec is the E-mode CCOSEBIs vector, E is the Eij vector intheshear2PCFs,whilethemodeswithsmallnaresensitiveto n andΓisthecompressionmatrixdefinedas, large-scalevariations.ThisisbecauseT areoscillatoryfunctions ±n withn+1rootsintheirrangeofsupport.Alternatively,theE/B- Γ≡HC−1 , (10) COSEBIscanbeexpressedasafunctionoftheconvergencepower whereHisamatrixformedofbothfirstandsecondderivativesof spectra: theCOSEBIswithrespecttothecosmologicalparametersandC E(ij) =Z ∞ d‘‘P(ij)(‘)W (‘), (6) is the covariance matrix of COSEBIs (see section 2 of Asgari & n 2π E n Schneider2015,forthedetailsoftheformalism).Thenumberof 0 Z ∞ d‘‘ CCOSEBIsmodesforconstrainingP cosmologicalparametersis Bn(ij) = 2π PB(ij)(‘)Wn(‘), (7) P(P +3)/2, regardless of the number of COSEBIs used. For a 0 total of N COSEBIs modes and P parameters, Γ is a matrix max whereP(ij) aretheE(B)-modeconvergencepowerspectraandthe withP(P+3)/2rowsandNmaxcolumns,wherethefirstP rows E(B) aretheCOSEBIsfirstorderderivativeswhilethelastP(P +1)/2 W (‘)aretheHankeltransformofT (ϑ) n ±n rowsare thesecondorderderivative ofCOSEBIswithrespect to Z ϑmax theparameters. W (‘)= dϑϑT (ϑ)J (‘ϑ) n +n 0 ϑmin Z ϑmax = dϑϑT−n(ϑ)J4(‘ϑ), (8) 3 RESULTS ϑmin Inthissectionweapplytwoanalysismethods,basedonCOSEBIs withJ andJ astheordinaryBesselfunctionsofzerothandfourth 0 4 and CCOSEBIs respectively, to measure the cosmic shear signal order. fromCFHTLenSdata.Beforeapplyingourmethodsonthedatawe WeuseEq.(6)tofindthetheoryvalueoftheE-modeCOSE- performedanumberoftestsincludingblindtestsonmockdata,as BIs as most theories provide us with an input power spectrum. However,inpracticetheshear2PCFsaremorestraightforwardto measurefromdata,hence,Eq.(4)andEq.(5)areusedtocalculate 2 http://sci.esa.int/euclid/,Laureijsetal.(2011) theE/B-modeCOSEBIsfromdataandsimulations. 3 http://www.lsst.org/lsst/ MNRAS000,1–16(2017) 4 M.Asgarietal. Table1.Effectivenumberdensityofgalaxies,neff (arcmin−2),ineach Table2.CosmologicalparametersforaflatΛCDMcosmology.Thefirst redshiftbinforlate-type(Blue)andall(All)Galaxies. rowcorrespondstoCFHTLenS+WMAP7bestfitvalues,thesecondrow belongstoPlanckbestfitvaluesforTT+lowPandthefinalrowshowsthe valuesfortheSLICSsimulations. z-bin Blue:neff All:neff [0.2,0.39] 1.507 1.811 σ8 Ωm ns h Ωb CF+WM 0.794 0.255 0.967 0.717 0.0437 [0.39,0.58] 1.265 1.646 Planck 0.829 0.315 0.9655 0.6731 0.0490 [0.58,0.72] 1.560 1.907 SLICS 0.826 0.2905 0.969 0.6898 0.0473 [0.72,0.86] 1.366 1.788 [0.86,1.02] 1.440 1.729 subjectofthiswork,isessentiallyunaffectedbytheredshiftmea- [1.02,1.3] 1.395 1.708 surementbiases. [0.2,1.3] 8.533 10.589 3.2 Cosmologicalmodels explainedintheAppendix.TheAppendixalsodetailsthetechnical The cosmological models we compare our results to are two flat aspectsofcalculatingtheCOSEBIsfromsheartwo-pointcorrela- ΛCDMmodels,withparameterscorrespondingtothebestfitval- tionfunctions. ues of CFHTLenS+WMAP7 (Heymans et al. 2013) and Planck TT+ lowP (Planck Collaboration et al. 2015a). We assume a pri- mordialpower-lawpowerspectrumandusetheBond&Efstathiou 3.1 Analysis (1984)transferfunctiontocalculatethelinearmatterpowerspec- InordertocompareourresultswiththepreviousCFHTLenSanal- trum. The non-linear power spectrum is estimated using the halo ysis as well as to test the data for systematic errors in a compre- fit formula of Smith et al. (2003). MacCrann et al. (2015) show hensivemanner,weanalysethedatainseveraldifferentways.We thatthischoiceofnon-linearfittingfunctiondoesnotsignificantly choosethethreeangularranges,[10,400],[400,1000]and[10,1000] change cosmological parameter constraints with CFHTLenS, in correspondingtosmall,largeandthecombinationofbothangular comparison to analyses that use improved non-linear correction scales. We also consider two sets of galaxy populations, all and schemes(Takahashietal.2012;Meadetal.2015). blue galaxies only. The blue galaxies are late-type galaxies and ThecosmologicalparametersaregiveninTable2,wherewe are expected to have a negligible intrinsic galaxy alignment sig- also show the parameters for the simulation products which are nal (see Heymans et al. 2013). This population is selected using usedforpipelineverificationsaswellasestimatingthecovariances. theirBayesianphotometricredshiftspectraltype,T >2(seeVe- The cosmological parameters which are presented in Table2 are, B lander et al. 2014, for the definition). In addition, we compare a σ8,thenormalizationofthematterpowerspectrum,Ωm,themean 2D,non-tomographic,analysiswitha6redshiftbinstomographic matterdensityparameter,ns,thespectralindex,h,thedimension- analysis. Table1 shows the redshift bins and their corresponding lessHubbleparameterandΩb,thebaryonicmatterdensityparam- effectivenumberdensityofgalaxiesfortheblueandallgalaxies. eter. Spatial flatness is assumed throughout out this work, which TheredshiftdistributionoftheCFHTLenSdataismeasuredusing meansthatΩΛ = 1−Ωm,whereΩΛ isthedarkenergydensity photometric redshift estimates as explained in Hildebrandt et al. parameter. (2012). Listed below are the configurations we used in this paper 3.3 Covariance whichbestresembletheprevioustwo-pointstatisticscosmicshear analysisofCFHTLenS. The covariance matrix of the COSEBIs is measured from mock galaxycataloguesconstructedfromtheSLICS,asuiteofN-body • Heymans et al. (2013) performed an analysis with a set-up, simulationsdescribedinHarnois-Déraps&vanWaerbeke(2015). whichcorrespondstothetomographic[10,400]angularrangewith The mock galaxy population algorithm, detailed in Joudaki et al. allgalaxies.Theymodelledgalaxyintrinsicalignmentswithasin- (2016),isdesignedtoreproducethepropertiesoftheCFHTLenS gleparameter,astheintrinsic-shearsignalisnon-negligiblewhen catalogues. These new mock catalogues are updated versions of allgalaxiesareconsideredintomographicbins. thoseusedinthepreviousanalysisoftheCFHTLenS,whichoffer • Kitchingetal.(2014)usedlargescales(roughlythe[400,1000] betterprecisionespeciallyatlargeangularscales,sincetheboxsize range) with blue galaxies. They used 3D cosmic shear analysis ofthesimulationsisL=505Mpc/h;whichissignificantlylarger in Fourier space which is approximately equivalent to our tomo- thanthesimulationsetusedformodellingtheearlierCFHTLenS graphicanalysis. measurements(L=(147,231)Mpc/h),hence,thenewsimulation • Kilbinger et al. (2013) used a large range of scales for their set is less affected by suppression of the large-scale variance by analysis which is close to the [10,1000] range we consider. Their finiteboxsizeeffects.Furthermore,weuse497incomparisonto analysisconsideredallgalaxieswithoutanyredshiftbinning. the184independentsimulationsusedintheearlierwork. InthisanalysiswechoosetoignoretheCFHTLenSphotomet- Estimatingcovariancesfromafinitenumberofsimulationsis ricredshiftbiasesanduncertaintiespresentedinChoietal.(2015) noisy which causes biases in the inverse covariance (see Hartlap inordertobeabletodirectlycompareourresultstotheCFHTLenS etal.2007).AssumingGaussianerrorsontheestimatedcovariance analyseslistedabove.Joudakietal.(2016)investigatedtheeffect matrix,Cˆ,theinversecovariancematrixisgivenby of the redshift biases and showed that the effect is small on the n −n −2 C−1 = sim obs Cˆ−1 , (11) cosmologicalinformation.TheB-modeanalysis,whichisthemain n −1 sim MNRAS000,1–16(2017) RevisitingCFHTLenSwithCOSEBIsandCCOSEBIs 5 where n and n are the number of simulations and observ- allpairsofgalaxieswithaseparationfallingwithintheϑbin.Each sim obs ables,respectively.Forn /n < 0.8,theaboveformulapro- galaxyhasaninversevarianceweightassociatedwithit.Lessnoisy obs sim duces an unbiased inverse covariance according to Hartlap et al. galaxyshapeshavealargerweightvalue,ergotheyaremoreim- (2007). It will however still have noise associated with it, which portantintheanalysis.ThedefinitionofwcanbefoundinMiller dependsontheratioofthenumberofobservablestothenumber etal.(2013). ofsimulations.Taylor&Joachimi(2014)extendedthisanalysisby Theestimated2PCFs,fromtheinputellipticitiesandtheiras- providingamoreaccuratecorrectionfortheparametercovariance sociatedweights,w,forredshiftbinsiandj,aregivenby matrixas, Pw w (cid:2)(cid:15)i(xxx )(cid:15)i(xxx )±(cid:15)i(xxx )(cid:15)i(xxx )(cid:3) C = nsim−nobs−2 Cˆ , (12) ξˆ±ij(ϑ)= a b t aPtwbw x a x b , (15) par n −n +n −1 par a b sim obs par where(cid:15) (xxx )arethetangential/crossellipticitiesatpositionxxx , where C is the parameter covariance matrix and n is the t/x a a par par withrespecttothereferenceframeconnectingthepairsofgalaxies numberofparameterstobeestimated.Applyingthiscorrectionto involved.ξˆij(ϑ)isthendividedby1+K(ϑ)tofindanunbiased C resultsinaslightlysmallercovariancematrixincomparison ± par estimate. totheHartlapetal.(2007)method,forn <<(n −n ),but par sim obs To estimate the 2PCFs, we use Athena4(see Kilbinger et al. thereisstillnoiseassociatedwithit.Sellentin&Heavens(2015) 2014)atreecodethatcalculatessecond-ordercorrelationfunctions extendedthisanalysisfurthertomitigatecovariancematrixestima- frominputgalaxycatalogues.Theopeninganglethatweuseis0.02 tionuncertaintiesbymarginalisingoverthetruecovariancematrix radians,whichshowsnosignificantdifferenceswithabruteforce givenitsestimatedvalue.TheyshowimprovementsovertheHart- (openingangle=0)estimation. lap et al. (2007) and Taylor & Joachimi (2014) estimate, by not- Theestimated2PCFsaretheninsertedintoEqs.(4)and(5)to ingthattheircorrectedcovariancematrixdistributionisnolonger determinetheCOSEBIsEandB-modes,respectively(thedetailsof Gaussian. whichareexplainedintheAppendix).ThetheoryvaluesofCOSE- Inouranalysis,themaximumnumberofobservablesthatwe BIs are estimated using Eqs.(6) and (7) which relate the COSE- use is 7 × 21 = 147 COSEBIs modes, where 7 is the number BIs to the convergence power spectrum directly. In this analysis of COSEBIs modes in each redshift pair and 21 is the number weusethefirst7COSEBIsmodes,sinceAsgarietal.(2012)have of redshift pairs for the tomographic case. As a result the ratio shownthattheseareenoughtoessentiallycapturethefullinforma- n /n ≈ 0.3, which can cause about 7% errors in the esti- obs sim tionforupto7cosmologicalparameters5.Assumingtomographic matedinversecovarianceusingtheHartlapetal.(2007)correction. binseachredshiftbinpairwillhave7COSEBIsmodeswhichadds Thisvaluefortheerroronthecovariancematrixisacceptablefor upto147modesintotal.UsingthecompressionmethodinAsgari analysingCFHTLenSdataaroundthemaximumlikelihoodpoint. &Schneider(2015)wedecreasethisnumberto20. However,aroundthetailsofthelikelihooddistributiontheSellentin Fig.1showsthemeasuredCOSEBIsforasingleredshiftbin & Heavens (2015) correction becomes significant. Therefore, we usingallgalaxies.Thepanelsshowtheresultsforthethreeangu- applythiscorrectioninSect.3.5,wherewecalculatethep-values, lar ranges, [10,1000], [10,400] and [400,1000]. The symbols show primarilytoassessthesignificanceofthedetectedB-modes. theCOSEBIsmodesestimatedfromthedatawhilethetheoryval- ues are shown as curves. The COSEBIs modes are discrete and 3.4 Measurements thecurvesaredrawntoaidtheviewer.TheE-modeCOSEBIsare shownbyblacksquareswhiletheredcirclesaretheB-modes.The FollowingHeymansetal.(2012),weanalysethe129CFHTLenS B-modesareshiftedtotherighttoaidtheviewer.Theerrorsonthe fieldsthatpassedthesystematictests,representing75%ofthetotal dataareestimatedfromthesimulationsandarecorrelated(seethe observedarea. covarianceinFig.A3).AswewillseeinSect.3.5,theB-modesin Wecalibratethedatacorrectingforadditiveandmultiplicative thisplotareonlysignificantfortheangularscale[400,1000].The biasesbetweentheobserved,(cid:15) ,andthetrueellipticities,(cid:15) , obs true theoryE-modecurvesbelongtoCFHTLenS+WMAP7andPlanck modelledas bestfitvalueslistedinTable2.Wealsoseethatthehighestsignal- (cid:15) =(1+m)(cid:15) +c, (13) to-noiseratiocomesfromsmallscalesasexpected(seeAsgarietal. obs true 2012,forexample). where(cid:15)isacomplexqunatitydefinedas(cid:15) = (cid:15) +i(cid:15) ,where(cid:15) 1 2 1 Fig.2showstheestimatedCOSEBIsforthetomographiccase and(cid:15) arerealquantities. 2 with blue galaxies. The E/B-modes are separated into the upper In CFHTLenS analyses, c was measured to be zero and and lower triangle of the plot. Each panel belongs to a redshift 2×10−3 onaveragefor(cid:15) and(cid:15) respectively.Theoriginofthe 1 2 bin pair indicated at its corner. Similar to Fig.1 the measured E additivebiasisunknownanditsvalueiscalibratedfromthedata andB-modesareshownasblacksquaresandredcircles,respec- empirically. It is likely that the multiplicative bias, m, originates tively. The curves show the theory values of the E-modes for the from the effect of noise in shape measurements (see for example CFHTLenS+WMAP7andPlanckcosmologiesinTable2.Thean- Melchior&Viola2012).Itisestimatedfromgalaxyimagesimu- lations.Whiletheadditivebiasissubtractedfromtheobserved(cid:15) 2 directly,theeffectofthemultiplicativebiasisappliedgloballyas explainedinMilleretal.(2013).Themeasured2PCFsaredivided bythecalibrationfunction, 4 http://www.cosmostat.org/software/athena/ 5 DependingontheoriginoftheB-modesystematic,7COSEBIsmodes P 1+K(ϑ)= abwawPb(1+wmwa)(1+mb) , (14) mFuarythneortwboerkenisouregqhutiorecdatpotuterestadllifofefrtehnetsinyfsotermmaattiiocnscienntahreioBs-amndodheowsigtnhaely. ab a b impactthedifferentCOSEBIs.Forthepurposeofthispaper,however,we where wa and ma are the weight and the multiplicative bias as- match our B-mode analysis to the 7 modes that are optimal for E-mode sociated with a galaxy at position a. The sum is carried out over measurements. MNRAS000,1–16(2017) 6 M.Asgarietal. 5 Planckcosmologiesareshownasthebluesolidcurveandthered 4 EnCFHTLenS EnPlanck [10,1000] dashedcurve.NotethattheCCOSEBIsmodesarediscreteandthe theoryvaluesareconnectedforaneasiercomparison.TheB-modes 3 areshownonthesamescaleastheE-modes.TheCCOSEBIsare 2 designed to be sensitive to cosmological information about these 1 parameters.Therefore,theymaynotbeassensitivetotheB-modes in the data. As we will see for most cases that we have studied, 0 eveniftherearesignificantB-modespickedupbyCOSEBIs,the 1 B-modesarenotalwayssignificantwiththeCCOSEBIs.Theex- 1.5 ceptionisthe[400,1000]angularrangewhichshowssignificantB- En Bn [10,400] modeseitherway. 1.0 En(Bn) 0.5 10−10 3.5 Figure-of-meritandfitting 0.0 ToquantifythesignificanceofthemeasuredB-modesweestimate theirχ2valuewithzero, 0.5 X χ2 = BtC−1B, (16) B 3 [400,1000] whereBisavectorcomposedofB ,BtisitstransposeandC−1 2 n istheinverseoftheB-modecovariancematrix,estimatedfromthe 1 SLICSsimulations.Wealsoestimatetheχ2valuesfortheE-modes 0 comparedtothebestfitvaluesofCFHTLenS+WMAP7andPlanck 1 (seeTable2).Therawvalueoftheχ2 isnotparticularlyinforma- tive,evenwhenthedegrees-of-freedomisknown(seeAndraeetal. 2 2010,forexample).Henceinsteadweshowthep-valuesforthees- 3 1 2 3 n4 5 6 7 timatedχ2 values.Thep-valueshowstheprobabilityoffindinga χ2 valuelargerthantheoneestimated.Wechooseasignificance level of 99%, p-value=0.01, which corresponds to a deviation of about2.6σ foranormaldistribution.Recallthataχ2 distribution isskewedtowardssmallervaluesandasymptoticallyreachesanor- Figure1.MeasuredCOSEBIsfromtheCFHTLenSdataforasinglered- maldistributionforlargenumbersofdegrees-of-freedomasillus- shift bin using all galaxies. Three angular ranges are considered here. The dashed line shows the zero B-mode value. The Bn modes (red cir- tratedinFig.A6. Additionally,usinganinvertednoisycovariance cles)areshiftedtotherightforvisualassistance.TheEn(blacksquares) changes a χ2 distribution and hence the derived p-values, which are compared with their theoretical values given the Planck (red dotted weaccountforusingthemethodproposedbySellentin&Heavens curve) and CFHTLenS+WMAP7 (blue solid curve) cosmologies. The (2015). CFHTLenS+WMAP7theoreticalvaluesarebestfitvaluesforthe[10,400] Fig.4showsthep-valuesfortheCOSEBIsB versusn , n max angularrangewithtomography(seeHeymansetal.2013).Thevaluesof the maximum number of COSEBIs used starting from the first thecosmologicalparametersforthetheoreticalcurvesaregiveninTable2. mode.Thep-valuesareshownforthethreeconfigurationswhich NotethattheCOSEBIsmodesarediscreteandthetheoryvaluesarecon- are closest to the previous CFHTLenS analysis described in nectedtoeachotherforvisualinspection.Theerrorsareestimatedfrom Sect.3.1. The grey circles correspond to [10,1000] range without simulateddataexplainedinAppendix.A3.Notethatthedifferentmodes tomographyandwithallgalaxies,whichresemblesKilbingeretal. arecorrelated(seethecovarianceinFig.A3) (2013).Thebluesquaresbelongto[400,1000]angularrangewith tomography and blue galaxies similar to Kitching et al. (2014). ThediamondsconfigurationisthesameasHeymansetal.(2013), gularrangeconsideredis[10,1000].Unlikethesingleredshiftbin whereallgalaxiesintheangularrange[10,400]areconsideredand case,weseestatisticallynon-zeroB-modesinthisFigure. binnedinredshift.Inthisplotweseethatthep-valuesforthesingle Fig.3showsthefirstmeasurementofCCOSEBIsfromdata. redshiftbincasearealwaysabove0.01whichmeansthattheyare We use blue galaxies with 6 tomographic bins and the three an- insignificant. In contrast, on large scales the B-modes are always gular ranges to estimate the CCOSEBIs. Here we choose the 5 below0.01andaresignificant. Inaddition,thetomographicanaly- cosmological parameters in Table2 to compress COSEBIs into 5 sisusingthelowerangularrange,[10,400],alsoshowsinsignificant firstorderand15secondorderCCOSEBIs,usingthePlanckval- Bmodeswithap-valueabove0.01. WhenweuseCCOSEBIsthe uesasourfiducialcosmologytocalculatethecompressionmatrix B-modessignificancedecreases,aswewillseeinTable3. (seeEq.10).TheCCOSEBIsmodesarenamedaftertheparameters Table3 shows the p-values for all the cases that we have whichareusedtodefinethem,shownonthex-axis.Thefirstorder considered. The first four columns indicate the set-up, while the modesonlydependononecosmologicalparameter,whereas,the last six show the p-values for that set-up for, B = 0, E = n n second order CCOSEBIs depend on two parameters which could ECFHTLenS,E = EPlanck,Bc = 0,Ec = Ec,CFHTLenS and n n n bethesame.Forexample,thepointsrelatedtoΩ hshowthevalue Ec = Ec,Planck, respectively. The p-values for CCOSEBIs are m ofthesecondorderCCOSEBIsmodewhichisbasedonthederiva- onlyshownforthetomographiccaseswhereCCOSEBIsoffersa tivesofCOSEBIstoΩ andh.Theorderingofthemodesisarbi- compression. The n column shows the number of COSEBIs m max traryandtheapparentoscillationsinthefigurecanberearranged. modesineachredshiftbinwhichareusedintheanalysis.Weshow ThetheoryvaluesoftheCCOSEBIsforCFHTLenS+WMAP7and the results for both the first 2 and 7 COSEBIs. The p-values are MNRAS000,1–16(2017) RevisitingCFHTLenSwithCOSEBIsandCCOSEBIs 7 n n n n n n 1234 56 7 1234 56 7 1234 56 7 1234 56 7 1234 56 7 1234 56 7 15 z-11 z-12 z-13 z-14 z-15 z-16 Blue Galaxies 10 [1',100'] 5 Bn 6 z-bins 0 10 10 − 5 10 15 15 z-11 z-22 z-23 z-24 z-25 z-26 10 10 En 5 5 Bn 10 10 0 0 10 10 − − 5 5 10 10 15 15 z-12 z-22 z-33 z-34 z-35 z-36 10 10 En 5 5 Bn 10 10 0 0 10 10 − − 5 5 10 10 15 15 z-13 z-23 z-33 z-44 z-45 z-46 10 10 En 5 5 Bn 10 10 0 0 10 10 − − 5 5 10 10 15 15 z-14 z-24 z-34 z-44 z-55 z-56 10 10 En 5 5 Bn 10 10 0 0 10 10 − − 5 5 10 10 15 15 z-15 z-25 z-35 z-45 z-55 z-66 10 10 En 5 5 Bn 10 10 0 0 10 10 − − 5 5 10 10 15 z-16 z-26 z-36 z-46 z-56 z-66 10 En 5 EnCFHTLenS 10−10 0 EPlanck n 5 10 1234 56 7 1234 56 7 1234 56 7 1234 56 7 1234 56 7 1234 56 7 n n n n n n Figure2.MeasuredCOSEBIsfromtheCFHTLenSdatafor6redshiftbinsusingbluegalaxies.Theangularrange[10,1000]isusedhere.TheB-modes(red circles)areshownintheupperrighttriangle,whiletheE-modes(blacksquares)areshowninthelowerlefttrianglefortheredshiftbinpairsindicatedfor eachpanel.ThetheoreticalvaluesofEnareshownforthePlanck(reddottedcurve)andCFHTLenS+WMAP7(bluesolidcurve)cosmologies(seeTable2). NotethattheCOSEBIsmodesarediscreteandthetheoryvaluesareconnectedtoeachotherforvisualinspection.Theerrorsareestimatedfromthemock dataexplainedinAppendix.A3.NotethatthedifferentmodesarecorrelatedasshowninFig.A4. MNRAS000,1–16(2017) 8 M.Asgarietal. 10 Ec,CFHTLenS Ec,Planck µ µ 5 Ec µ 10−10 0 5 [1,100] 0 0 4 Bc 2 µ 0 10−10 2 [[11,,110000]] 00 00 4 6 4 2 Ec µ 0 10−10 2 4 [1,40] 0 0 3 2 Bc 1 µ 0 10−10 1 [1,40] 2 0 0 10 5 Ec µ 0 10−10 5 10 [40,100] 0 0 15 10 5 Bc µ 0 10−10 5 10 [40,100] 0 0 15 σ Ω n h Ω σ σ σ Ω σ n σ h σ Ω Ω Ω Ω n Ω h Ω Ω nn nh nΩ hh hΩ Ω Ω 8 m s b 8 8 8 m 8 s 8 8 b m m m s m m b s s s s b b b b CCOSEBIs mode Figure3.MeasuredCCOSEBIsfromtheCFHTLenSdatafor6redshiftbinsusingbluegalaxies.TheB-modesareshownasgreencircles.Theblackdashed lineshowswherethezerolinefortheB-modeslies.ThemeasuredE-modesareshownasblacksquares,whilethetheoryvaluescorrespondingtothebest fitvaluesforCFHTLenS+WMAP7andPlanck(seeTable2)cosmologiesareshownasbluesolidcurvesandreddottedcurves,respectively.Notethatthe CCOSEBIsmodesarediscreteandthetheoryvaluesareconnectedtoeachotherforvisualinspection.Theerrorsareestimatedfromsimulateddataexplained inAppendix.A3.Notethatthedifferentmodesarecorrelated(seethecovarianceinFig.A5). MNRAS000,1–16(2017) RevisitingCFHTLenSwithCOSEBIsandCCOSEBIs 9 0 (2013).However,forbluegalaxiesthecontributionfromintrinsic 10 alignmentisexpectedtobesmall,henceweexpectandfindagood -1 fittotheCFHTLenSvalues. 10 )max 10-2 3.6 SingleParameterFit 2n χ > 10-3 Weuseaverysimpleparametrizationtofitthetheorytodata,con- 2χ sistingofonefreeparameter.Wefinditsbestfitvaluebyminimiz- ( -4 P 10 All, 1 z-bins, [10,1000] ingitsχ2valueandtheerrortothefitcorrespondstotheparameter Blue, 6 z-bins, [400,1000] valueat∆χ2 = 1aroundtheminimumχ2.FortheB-modesthe 10-5 All, 6 z-bins, [10,400] singleparametermodelweuseisaconstant, 10-6 Bn =KB , and Bc =KBc , (17) 1 2 3 4 5 6 7 n for COSEBIs and CCOSEBIs, respectively. For the E-modes max the models are a constant, times the theory E-modes, with CFHTLenS+WMAP7 and Planck cosmologies. For COSEBIs Figure4.P-valuesforχ2 ofB-modecomparedtozeroversusthenum- theseare ber of COSEBIs modes. nmax denotes the number of COSEBIs modes fromn=1ton=nmax.Thep-valueistheprobabilityoftheχ2value E =KCFHTLenSECFHTLenS , (18) beinglargerthanthevaluefound,assumingBn =0isthemodel.Avery n E n smallp-valueshowsapooragreementbetweenthetheoryandtheestimated and values.Werejectthenullhypothesis(zeroB-modes)forp-valuessmaller than0.01,whichcorrespondstoasignificancelargerthan99%.Theblue En =KEPlanckEnPlanck , (19) squaresshowtheresultsforbluegalaxieswith6redshiftbinsforthelargest whereasforCCOSEBIs angularscales,thelightdiamondbelongtoallgalaxieswith6redshiftbins andsmallangularscales.Finallythegreycirclesshowthep-valuesforall Ec =KECcFHTLenSEc,CFHTLenS, (20) galaxies,asingleredshiftbinandthe[10,1000]range. and Ec =KEPclanckEc,Planck , (21) written in boldface where they are larger than 0.01 which corre- are the two models. The best fit and error values for K , sponds to the significance level within 99%. Looking at the B B columnandthesingleredshiftbincases,weseethattheB-modens KECFHTLenS, KEPlanck, KBc, KECcFHTLenS, KEPclanck are listed in Table4. The format of this table is the same as Table3. Null B- areonlysignificantatlargescales([400,1000]). Whenredshiftbin- modesresultinastatisticallyzeroK ,however,astatisticallyzero ningisconsidered,withtheexceptionofthe[10,400]casethereare B K isnotasufficientconditionforB-modestobezero.Therows B significantB-modesinthedata. TheB-modesarenotalwayscon- forwhichtheCOSEBIsB-modesareconsistentwithzerofromthe sistentbetweenthetwogalaxypopulationswhichhintsatacorre- p-valuetestareshowninboldface.SomeoftheK valueswhich B lationbetweengalaxycolourandresidualsystematics.Alsonotice are consistent with zero in this table correspond to significant B- thatthelargestscalesshowsignificantB-modesforallthedifferent modesfromthep-valuetest.ThisshowsthattheB-modepatternin setsofdataanalysed. thedataisnotalwayswell-modelledbyaconstantvalue. The Bc column shows the CCOSEBIs B-modes which are typically less significant than that of COSEBIs. As discussed be- fore,thisisduetothefactthattheCCOSEBIsarebasedonlinear 3.7 Comparisontopreviousanalyses combinationsofCOSEBIswhicharemostsensitivetocosmolog- icalparameters.Theyarethereforenotnecessarilysensitivetothe TheTables3and4allowustocompareourresultswiththepre- B-modeswhich,forCFHTLenS,appeartocanceltosomedegree vious CFHTLenS cosmic shear analysis. We first consider Hey- withthecompressedformofthestatistic.Consequently,tomeasure mans et al. (2012) who detail a systematics test using an E-B B-modesweneedtouseCOSEBIs,whichprovideacompleteset mode decomposition for three different two-point statistics; the offunctionsforthisanalysis. top-hatshearvariance,the2PCFandthemassaperturestatistics. ComparingtheECFHTLenSandEPlanckcolumnsweseethat Analysing angular scales from [10,600] applying no redshift bin- n n Planck provides a better match to the single redshift bin data ning,theyfoundnosignificantB-modes,whichisconsistentwith for all the cases6. However, when tomography is considered the ourresults. CFHTLenS cosmology provides a better match with the excep- Kilbinger et al. (2013) performed a two-dimensional analy- tionoftheverylargescales.Forbluegalaxiesat[400,1000]thep- sis of the datausing several cosmic shear methods, including the valuesforECFHTLenS andEPlanck arecomparable.Wealsonote COSEBIs.Theaimoftheirworkwastousealargeangularrange n n that for many of the tomographic cases, neither provide a good toestimatecosmologicalparameters,buttheyfaceddifficultieses- match.Whenallgalaxiesareconsideredweneedtoaddintrinsic timatingtheCOSEBIsfromtheirmockdata,knownastheClone alignmentcorrectionstoourmodelaswasdoneinHeymansetal. simulations.Themainreasonfortheirdifficultieswasthefactthat the accuracy of the simulations for very large angular scales is limited,duetothefiniteboxsize.Consequently,theydidnotuse 6 Hereweusep-valuesasaproxyforχ2 values,whichwouldbeused COSEBIsfortheirfinalanalysisofthedata.Hereweusedupdated insamplingtheparameterlikelihoodinatypicalcosmologicalanalysis.We simulations(SLICS)withbetteraccuracyforlargescalesanddid willnotattempttorejecteitherECFHTLenSorEPlanckusingthismethod notencountersimilarproblems.Wecomparetheirresultswithour n n orquantifytheirtension. [10,1000]angularrangewithallgalaxiesandasingleredshiftbin. MNRAS000,1–16(2017) 10 M.Asgarietal. Table3.P-valuesforχ2 ofBn=0,En=EnCFHTLenS,En=EnPlanck,Bc=0,Ec=Ec,CFHTLenSandEc=Ec,Planck.Thep-valuesdenotethe probabilityoftheχ2valuesbeinglargerthanthevaluesfound,assumingthemodeliscorrect.Eachrowcorrespondstoadifferentangularrange(θrange), groupofgalaxies(Galaxies),numberofredshiftbins(z-bins)andnumberofCOSEBIsmodes(nmax)consideredintheanalysis.TheCCOSEBIsp-values areonlyshownforthetomographiccasewherethenumberofCCOSEBIsmodesissmallerthanthatofCOSEBIs.TheP-valueswhicharelargerthan0.01 areshowninboldfaceandliewithinthe99%confidencelimit.SeeTableB1fortheχ2valuesanddegrees-of-freedomforeachentryinthistable. COSEBIs CCOSEBIs θrange Galaxies z-bins nmax Bn EnCFHTLenS EnPlanck Bc Ec,CFHTLenS Ec,Planck 2 4.5e−01 1.6e−01 3.4e−01 − − − 1 7 2.4e−01 4.5e−02 2.1e−01 − − − All 2 2.0e−03 3.3e−03 4.6e−04 3.5e−02 7.8e−02 1.1e−02 6 7 6.6e−03 6.1e−04 2.0e−04 8.0e−01 4.8e−04 2.5e−04 [10−1000] 2 2.1e−01 6.8e−02 3.5e−01 − − − 1 7 2.2e−01 2.1e−03 1.3e−02 − − − Blue 2 4.4e−04 3.5e−02 5.7e−03 6.0e−02 2.7e−01 6.7e−02 6 7 3.9e−03 9.7e−03 4.9e−03 4.7e−01 1.1e−01 3.5e−02 2 7.4e−01 3.2e−02 5.1e−01 − − − 1 7 7.4e−01 1.2e−01 5.0e−01 − − − All 2 3.6e−02 3.0e−03 1.3e−03 2.5e−01 3.2e−02 1.1e−02 6 7 2.0e−02 6.8e−03 1.9e−03 6.2e−01 4.2e−03 1.3e−03 [10−400] 2 6.5e−01 1.8e−02 8.8e−01 − − − 1 7 2.7e−01 3.2e−03 2.4e−02 − − − Blue 2 3.8e−02 2.4e−02 1.2e−02 7.0e−01 3.2e−01 2.0e−01 6 7 2.7e−01 8.9e−03 2.6e−03 7.6e−02 1.4e−01 5.7e−02 2 4.4e−03 4.0e−02 6.9e−02 − − − 1 7 2.4e−03 6.2e−02 8.7e−02 − − − All 2 1.1e−03 1.2e−02 1.6e−02 4.8e−02 4.5e−03 5.9e−03 6 7 1.8e−06 4.7e−06 5.3e−06 3.5e−02 6.4e−03 8.9e−03 [400−1000] 2 5.5e−03 1.5e−01 2.1e−01 − − − 1 7 2.9e−03 4.4e−02 5.5e−02 − − − Blue 2 3.6e−03 6.7e−02 7.4e−02 7.3e−04 1.1e−01 1.2e−01 6 7 1.2e−06 1.1e−04 1.1e−04 9.6e−04 1.7e−01 1.8e−01 TheyreportedinsignificantB-modeswhichisconsistentwithour modellingoftheintrinsicalignments,therearenootherdifferences results. betweenourstudyandHeymansetal.(2013). Fu et al. (2014) added three-point statistics to the Kilbinger Kitchingetal.(2014)restrictedtheirstudytolargescalesand et al. (2013) 2D analysis and found significant B-modes in their bluegalaxieswithredshiftinformation.Theyreportednosignifi- thirdorderstatistics.Ourfindingscombinedwiththeirsshowthat cantB-modes.AlthoughthescalestheyusedaredefinedinFourier there are still (high-order) residual systematic errors left in the spacewheretheyperformedtheiranalysis,theyroughlycorrespond CFHTLenSdata. tothelargescalesthatwehaveconsideredhere.Incontrasttotheir study we find very significant B-modes in the [400,1000] range. Onereasonforthisinconsistencycouldbethattheirmaskmodel lacks the precision to find the B-modes (see Asgari et al. 2016, 4 CONCLUSIONS formaskmodelling).Incontrasttopowerspectrumanalysis,mask InthispaperwerevisitedtheCFHTLenSdataandfoundevidence modelling has little or no effect on the estimation of COSEBIs. for systematic errors on large scales, and when the data is anal- Alternatively,thisinconsistencycouldbeduetothecomplexityof ysed in tomographic bins. We used COSEBIs, which is a robust translating the angular ranges used in a COSEBIs analysis to the efficientandcompletemethodforE/B-modeseparation.Weexpect Fouriermodesconsideredin3D-lensing. weaklensingtopredominantlyproduceE-modes,makingB-modes OurbestfitCFHTLenS+WMAP7fiducialcosmologycomes undesirable. Although the absence of B-modes does not guaran- fromHeymansetal.(2013),whousedthe[10,400]rangewithto- tee a perfect data analysis, it is a necessary condition for a sur- mography.TheydidnotincorporateanyE/B-modedecomposition vey like CFHTLenS. For future large scale and space based sur- methodsintheiranalysissincetheyused2PCFstofindtheirbest veys, where the measurement errors are significantly smaller, the fittingvalues.Fortheangularrangetheyused,wefindsignificant B-modescouldalsoindicateotherphysicalphenomena.Forexam- B-modeswhenallgalaxiesand7COSEBIsmodesareconsidered. ple we know that some intrinsic alignment models predict these Whenonly2COSEBIsmodesareconsidered,oronlyusingblue modes(seeBlazeketal.2011,andreferencestherein).Beforeper- galaxies, the B-modes are consistent with zero. Considering blue formingouranalysiswecarriedoutanumberofblindtestsoncos- galaxiesonlywhereintrinsicalignmentsarenotimportant,wesee mologicalsimulations,totesttheaccuracyofourpipelineswhich thatourmeasurementsfavourtheirbestfitvaluesincomparisonto arereportedintheAppendix.ThesignificanceoftheB-modeswe Planck.Inparticular,theCCOSEBIs matchestoboththeoretical found is highest for large scales, [400,1000], especially when the valuesforthiscase,howevertheCFHTLenS+WMAP7 isabetter galaxies are divided into redshift bins. They also depend on the match,asexpected.Asidefromourchoiceofobservables andthe galaxy population used in the analysis. We repeated our analysis MNRAS000,1–16(2017)