Revisiting CFHTLenS cosmic shear: Optimal E/B mode decomposition using COSEBIs and compressed COSEBIs

1.8 MB·

by  Marika Asgari

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)