Weak lensing measurement of galaxy clusters in the CFHTLS-Wide survey

1.4 MB·English

by  HuanYuan Shan

Draftversion February1,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
WEAK LENSING MEASUREMENT OF GALAXY CLUSTERS IN THE CFHTLS-WIDE SURVEY
HuanYuan Shan†1,2,3, Jean-Paul Kneib2, Charling Tao1,3, Zuhui Fan4, Mathilde Jauzac2, Marceau Limousin2,5,
Richard Massey6, Jason Rhodes7,8, Karun Thanjavur9,10,11 and Henry J. McCracken12
Draft versionFebruary 1, 2012
ABSTRACT
We present the first weak gravitational lensing analysis of the completed Canada-France-Hawaii
2 Telescope Legacy Survey (CFHTLS). We study the 64 deg2 W1 field, the largest of the CFHTLS-
1
Wide survey fields, and present the largest contiguous weak lensing convergence “mass map” yet
0
made.
2
2.66 million galaxy shapes are measured, using a Kaiser, Squires and Broadhurst (KSB) pipeline

n verified against high-resolution Hubble Space Telescope imaging that covers part of the CFHTLS.
a Our i′-band measurements are also consistent with an analysis of independent r′-band imaging. The
J
reconstructedlensingconvergencemapcontains301peakswithsignal-to-noiseratioν >3.5,consistent

1 with predictions of a ΛCDM model. Of these peaks, 126 lie within 3′.0 of a brightest central galaxy
3 identified from multicolor optical imaging in an independent, red sequence survey. We also identify
seven counterparts for massive clusters previously seen in X-ray emission within 6 deg2 XMM-LSS
]
survey.
O
With photometric redshift estimates for the source galaxies, we use a tomographic lensing method
C
to fit the redshift and mass of each convergencepeak. Matching these to the optical observations,we
. confirm85 groups/clusterswith χ2 <3.0, ata mean redshift hz i=0.36 and velocity dispersion
h reduced c
p hσci = 658.8 km s−1. Future surveys, such as DES, LSST, KDUST and EUCLID, will be able to
- applythesetechniquestomapclustersinmuchlargervolumesandthustightlyconstraincosmological
o models.
r Subject headings: cosmology: observations, - galaxies: clusters: general, -gravitational lensing: weak,
t
s -X-rays: galaxies: clusters
a
[

5 1. INTRODUCTION & Tormen1999),and the dynamics of dark energy (e.g.,
Bartelmann et al. 2006; Francis et al. 2009; Grossi &
v Clusters of galaxies are the largest gravitationally
1 bound structures inthe universe. The number and mass Springel2009). Understanding the properties of clusters
8 ofthebiggestclustersarehighlysensitivetocosmological isvitaltotesttheoriesofstructureformationandtomap
9 parameters including the mass density Ω , the normal- thedistributionofcosmicmatteronscalesof∼1–10Mpc.
m
1 ization of the mass power spectrum σ (e.g., Press & Theoreticalpredictions of structure formation deal di-
. 8 rectly with the total mass of clusters; measurements
8 Schechter 1974;Frenk et al. 1990;Eke et al. 1996;Sheth
are restricted to indirect proxies that can be observed.
0
Contaminating the translation between theory and ob-
1 1Department of Physics and Tsinghua Center for Astro-
servation are large uncertainties in the interpretation
1 physics,TsinghuaUniversity,Beijing,100084, China
: 2Laboratoired’AstrophysiquedeMarseille,CNRS-Universit´e of galaxy richness, X-ray luminosity/temperature and
v de Provence, 38 rue Fr´ed´eric Joliot-Curie, 13388 Marseille the Sunyaev-Zeldovichdecrement (e.g. Bode et al. 2007;
i Cedex13,France Leauthaud et al. 2010). Weak gravitational lensing, the
X 3Centre de Physique des Particules de Marseille,
CNRS/IN2P3-Luminy and Universit´e de la M´editerran´ee, coherent distortion of galaxies behind a cluster, can po-
r Case907,F-13288MarseilleCedex9,France tentially provide direct measurements of the total mass
a
4Department of Astronomy, Peking University, Beijing, regardless of its baryon content, dynamical state, and
100871, China
5Dark Cosmology Centre, Niels Bohr Institute, University star formation history.
of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, By measuring the shear (coherent elongation) of
Denmark many background galaxies, we can reconstruct the two-
6Institute forAstronomy, Royal Observatory, BlackfordHill, dimensional (2D) weak lensing convergence map, which
EdinburghEH93HJ,UK
7California Institute of Technology, MC 350-17, 1200 East is proportional to density projected along each line of
CaliforniaBoulevard,Pasadena,CA91125,USA sight. Peaks in the convergence map with high signal-
8Jet Propulsion Laboratory, California Institute of Technol- to-noise ratio ν generally correspondto massive clusters
ogy,Pasadena, CA91109, USA (Hamana et al. 2004; Haiman et al. 2004). Since the
9CanadaFranceHawaiiTelescope,65-1238MamalahoaHwy,
three-dimensional (3D) shear signal should increase be-
Kamuela,HI96743,USA
10Department of Physics & Astronomy, University of Victo- hind those clusters in a predictable way that depends
ria,Victoria,BC,V8P1A1,Canada upononlythelens-sourcegeometry,wecanalsousepho-
11National Research Council of Canada, Herzberg Institute tometric redshift estimates of the background galaxies
of Astrophysics, 5071 West Saanich Road, Victoria, BC, V9E
(from multi-band imaging) to measure the redshift and
2E7,Canada
12Institude d’Astrophysique de Paris, UMR 7095, 98 bis mass of each foreground cluster (Wittman et al. 2001,
BoulevardArago,75014Paris,France 2003;Hennawi&Spergel2005;Gavazzi&Soucail2007).
†Emailaddress: [email protected]
2 Shan et al.
Systematic weak lensing cluster searcheshave only re- 2D lensing convergence “mass map” signal, and extract
cently become practicable. Miyazaki et al. (2002) used a catalog of local maxima that represent cluster candi-
Subaru/Suprime-Cam in excellent seeing conditions to dates. InSection5,wesearchfor opticalcounterpartsof
find an excess of 4.9±2.3 convergencepeaks with ν >5 these candidates, dramatically cleaning the catalog. In
in an area of 2.1 deg2. Dahle et al. (2003)and Schirmer Section6,weinvestigatethefull3Dlensingsignalaround
et al. (2003) each identified several shear-selected clus- each cluster, further cleaning the catalog when the lens-
ters with redshifts z ∼ 0.5 determined from two-color ing signal behind spurious peaks does not increase as
photometry. Hetterscheidt et al.(2005)reportedthe de- expected with redshift - but obtaining an independent
tection of five cluster candidates over a set of 50 discon- estimate of the cluster redshift when it does. We finally
nected Very Large Telescope/FORS images covering an exploreglobalscaling relationsbetween cluster mass ob-
effective area of 0.64 deg2, while Wittman et al. (2006) servables, then conclude in Section 7.
found eight detections in the first 8.6 deg2 of the Blanco
2. DATA
Deep Lens Survey. Gavazzi & Soucail (2007) presented
a weak lensing analysis of initial Canada-France-Hawaii 2.1. CFHTLS-Wide T0006 imaging
TelescopeLegacySurvey(CFHTLS) Deepdatacovering The CFHT Legacy Survey is a joint Canadian-
4 deg2. They demonstrated that the image quality at French program to make efficient use of the CFHT
CFHT is easily sufficient for cluster finding. Miyazaki wide field imager MegaPrime, simultaneously address-
et al. (2007) presented the first large sample of weak ing several fundamental questions in astronomy. Each
lensing-selected clusters in the Subaru weak lensing sur- MegaPrime/MegaCamimageconsistsofanarrayof9×4
vey,with100significantconvergencepeaksina16.7deg2 e2v CCDs with a pixel scale of 0′′.187 and a total field
effective survey area. Hamana et al. (2009) reported re- of view of ∼ 1 deg2. The survey used most of the tele-
sults from a multi-object spectroscopic campaignto tar- scopedarkandgraytimefrom2003to2008. Weanalyze
get 36 of these cluster candidates, of which 28 were con- CFHTLS-WideimagingfromtheTerapixT0006process-
firmed (and 6 were projections along a line of sight of ingrun,whichis the firsttoinclude the complete survey
multiple, small groups). and was publicly released on 2010 November 15 (Gora-
The main astrophysical systematic effect afflicting nova et al. 2009). These data cover ∼ 171 deg2 in four
weak lensing cluster surveys is the projection of large-
fields (W1, W2, W3, and W4) of which the 72-pointing,
scale structure along the line of sight. Random noise
∼ 64 deg2 W1 field is the largest, and in five passbands
is also added due to the finite density of resolved source (u′,g′,r′,i′,and z′) down to i∼24.5 and r ∼25.0.
galaxiesandthescatteroftheirintrinsicshapes. Numeri- Fu et al. (2008) showed that the i′-band exposures,
calstudies (White etal.2002;Hamanaetal.2004)show
taken in sub-arcsecond seeing conditions, provide the
that these contaminants significantly reduce the purity
best image quality and resolve the galaxy population
of cluster detection. To improve our analysis, we shall
with highest median redshift. Resolving the shapes of
combine our weak lensing results with multi-wavelength
more distant galaxies is vital for weak lensing analysis,
imaging. Simultaneous detection of a weak lensing sig-
since the strength of the shear signal is proportional to
nature plus an overdensity of galaxies with a single red
the ratio of the Lens–Source and Observer–Source dis-
sequenceprovidesanunambiguousclusteridentification. tances. We therefore choose to analyze the i′-band im-
Furthermore, 3D lensing tomography using photometric
ages(meanseeing0”.73)inthe contiguousW1field. We
redshiftsfromthemulti-wavelengthdatacanremovethe also analyze the independent r′-band imaging to check
otherpotentialhurdlesof: lensingsignaldilutionbyclus-
the calibration of our shear measurements. We also use
termembergalaxies,andidentifyingtheredshiftofweak
photometric redshift estimates for source galaxies ob-
lensing peaks when no correspondinggalaxyoverdensity
tained from the multicolor imaging (Ilbert et al. 2006;
is apparent.
Coupon et al. 2009; Arnouts et al. 2010).
Here we present a weak gravitational lensing analysis
Early releases of smaller regions of the CFHTLS have
of the 64 deg2 CFHTLS-Wide W1 field, which is suffi-
alsobeenusedtomeasurethe weaklensingcosmicshear
ciently large to contain several hundred galaxy clusters.
signal(Sembolonietal.2006;Hoekstraetal.2006;Fuet
Compared to the analysis of the CFHTLS-Deep survey
al.2008). Asthesurveysizehasincreased,thestatistical
by Gavazzi & Soucail (2007), our shallower CFHTLS-
errors have shrunk, and difficulty measuring shapes at
Wide imaging (and lower source galaxy density) will fa-
a precision better than the statistical error has so far
vor the detection of higher mass, nearby clusters. The
prevented publication of a cosmic shear analysis of the
huge increase in survey area over any previous survey is
complete survey. However,weak lensing cluster searches
expected to yield many more systems overall. In this
are restricted by construction to regions of the survey
paper, we shall primarily study the properties of the de-
where the signalis strongest, and the circular symmetry
tected clusters,rather thanthe cosmologyinwhich they
of our analysis removes the negative impact of additive
areembedded. For this purpose,we adopta default cos-
shear measurement errors (cf. Mandelbaum et al. 2005).
mologicalmodelwithΩ =0.27,Ω =0.73,σ =0.809,
m Λ 8
H0 =100 h km s−1 Mpc−1, and h=0.71. 2.2. HST COSMOS Imaging
This paper is organized in the following way. In Sec-
The HSTCOSMOSsurvey(Scoville et al.2007)is the
tion2,wedescribetheCFHTandHubbleSpaceTelescope
largestcontiguousopticalimagingsurveyeverconducted
(HST) data used. In Section 3, we present the measure-
from space. High resolution (0′′.12) imaging in the
ment of galaxy shapes in the CFHT imaging, and their
I band was obtained during 2003–2005 across an
calibration against measurements of the same galaxies F814W
area of 1.64 deg2 that also corresponds to the CFHTLS
in the HST imaging. In Section 4, we reconstruct the
D2 deep field. Any galaxies resolved by CFHT are very
Weak lensing in CFHTLS-Wide 3
Table 1
SExtractor ConfigurationParameters
Parameter Value
DETECT MINAREA 3
DETECT THRESH 1.0
DEBLEND NTHRESH 32
DEBLEND MINCONT 0.002
CLEAN PARAM 1.0
BACK SIZE 512
BACK FILTERSIZE 9
BACKPHOTO TYPE local
BACKPHOTO THICK 30
Figure 1. “W1+2+3”pointingfromtheCFHTLS-WideW1field
ini′band,showingmaskedregions. Thispointingisrepresentative
ofthosewithfairlypoorimagequality: theseeingof0′′.78isworse
inonly24of72(1in3)pointings. Inourautomatedalgorithmfor
masking diffraction spikes around bright stars, the basic shape of
thestarmaskispredefined,anditssizeisscaledwiththeobserved
majoraxisofeachstar.
Figure 2. Star selection (redpoints) inthe planes ofmagnitude
easily resolved by HST, which therefore provides highly vs. flux radius (top) and magnitude vs. peak surface brightness
(bottom). We findthe latter morerobust. The redpoints denote
accurate shape measurements almost without the need
objects selected as stars for PSF modeling. The blue objects are
forpoint-spreadfunction(PSF)correction. Weshallcal- spuriousdetection.
ibrate our CFHTLS shape measurements against those
We conduct shape measurement in both CFHTLS i′
from COSMOS by Leauthaud et al. (2010).
andr′ bands. We detect astronomicalsourcesin the im-
Note that measurements of the shapes of individual
galaxiesfromground-basedandspace-basedobservations ages using SExtractor (Bertin & Arnouts 1996). Our
need not necessarily match exactly, even without shape choiceofthe mainSExtractorparametersislistedinTa-
measurement errors, because the different noise proper- ble 1, and the data are filtered prior to detection by a 3
ties of the data sets may make them most sensitive to pixel Gaussian kernel.
differentisophotes,whichcanbe twistedrelativeto each Nearsaturatedstars,many spuriousobjects are found
other. The slightly different passbands may also empha- duetodetectoreffectsandopticalghosting. Itwouldalso
sizedifferentregionsofagalaxy’smorphology. However, bedifficulttomeasuretheshapesofrealstarsorgalaxies
across a large population of galaxies, these differences in these regions, because of the steep background gradi-
shouldaverageout,andacomparisonofsuccessfulshear ents. We havedevelopedanautomaticpipeline to define
measurements between the two data sets should agree. polygonal-shaped masks around saturated stars, and all
objects inside the masks are removed from our catalog.
3. GALAXYSHAPEMEASUREMENT The masks in all images are then visually inspected; our
automatedpipelinefailsinafewcases(mainlyverysatu-
3.1. Object Detection and Masking
ratedstarsforwhichthecentroidofthestarmeasuredby
4 Shan et al.
Figure 4. Projectionofthestellarellipticitiesinthe(e1,e2)plane
before(black)andafter(red)PSFanisotropycorrection. Thepost-
correctionresidualsareconsistentwithfeaturelesswhitenoise.
constrained locus within the size-magnitude plane (Fig-
ure 2(a)). Heymans et al. (2006) suggest using the full
widthathalfmaximum(FWHM).However,wefindthat
FWHM is not robustly measured by SExtractor,so we
instead use the µ -magnitude plane (Bardeau et al.
max
2005, 2007; Leauthaud et al. 2007), where µ is the
max
peak surface brightness (Figure 2(b)). The red points in
Figure 2 indicate the selected stars; our chosen locus re-
flectsacarefulbalancebetweenobtainingsufficientstars
tomodelthesmall-scalevariationsthatweobserveinthe
PSF pattern, and introducing spurious noise by includ-
Figure 3. Spatialvariationofmeasuredstellarellipticitiesinthe ing faint stars. The blue points are spurious detections
representativeCFHTLS-WideW1+2+3field,before(top)andaf- of noise, cosmic rays, etc. (cf. Leauthaud et al. 2007).
ter (bottom) PSF anisotropy correction. The longest tick marks
We then measure the Gaussian-weighted shape mo-
representellipticitiesof∼11%. Themeanabsoluteellipticityafter
correctionis0.62%. ments of the stars, and construct their ellipticity. In
additionto cuts in µ andmagnitude, wealsoexclude
SExtractorwaswidelyoffsetfromthediffractionspikes) max
noisyoutlierswithsignal-to-noiseν <100orabsoluteel-
and those stellar masks are corrected by hand. An ex- lipticitye∗ morethan2σawayfromthemeanlocalvalue,
ample of the masks for one CFHT pointing is shown in
and we iteratively remove objects very different from
Figure 1. This pointing has slightly worse than average
neighboring stars. In 15 pointings with the worst im-
imagequality,soweshalluseitthroughoutthispaperas
agequality,including W1+2+3,the PSFbecomeslarger
a conservative representation of our analysis. After ap- than r ∼ 0′′.5 in the corners of the field of view, so we
g
plying all ofour masks acrossthe entire survey,the final
finallyaddtheseregionstothesurveymask(andexclude
effective sky coveragedrops from64deg2 to ∼51.3deg2
galaxies in them from our weak lensing analysis).
and 55.0 deg2 for i′ and r′ bands, respectively.
Having obtained our clean sample of stars, we con-
We shall employ the popular KSB method for galaxy
struct a spatially varying model of the PSF across the
shearmeasurement(Kaiseretal.1995;Luppino&Kaiser
field of view. In most pointings, we fit the ∼30 stars in
1997;Hoekstraetal.1998). Inthismethod,theobserved
eachofthe36individualCCDscomposingtheMegaCam
galaxyshapeismodeledasaconvolutionofthe(sheared)
focal plane, using a polynomial of second order in x and
galaxy with the PSF, which is modeled as an isotropic,
y. For stacked data with large dithers, we use a higher
circular profile convolved with a small anisotropy.
orderpolynomial. Figures 3 and 4 show the stellar ellip-
ticitybeforeandaftercorrectionfortheW1+2+3point-
3.2. PSF Modelling
ing,usingaweightfunctionofdefaultsizer tomeasure
g
To measure the shapes of galaxies,it is first necessary the PSF shape moments. The residual stellar ellipticity
tocorrectthemforconvolutionwiththePSFimposedby after correction is a consistent random scatter around
thetelescopeopticsandEarth’satmosphere. Thechang- zero, of width σ ∼0.01.
ei
ingsizeandshapeofthePSFacrossthefieldofviewand TheellipticityofthePSFchangesfromthecoretothe
between exposures can be traced from stars, which are wings. WemeasurethePSFshapeusingdifferently-sized
intrinsically point sources. We identify stars from their weight functions and, when correcting each galaxy, use
Weak lensing in CFHTLS-Wide 5
Figure 5. Ellipticity of the PSF changes from the core to the
wings. This shows the mean PSF ellipticity in the i′-band of the
CFHTLS-WideW1pointingW1+2+3 asafunctionofthe sizeof
the Gaussian weight function with which it is measured, before
(black) andafter (red)PSFanisotropycorrection. Theerrorbars
showthermsscatter throughoutthatpointing.
the same size weight function to measure both the PSF
andgalaxyshapes. Figure5showsthevariationofmean
stellarellipticityasafunctionoftheweightfunctionsize
rg, before and after PSF anisotropy correction. Figure 6. Galaxy magnitude and redshift distributions. Top:
number counts of galaxies per magnitude bin immediately after
3.3. Galaxy Shape Measurement star-galaxy separation (solid histograms) and after all the lensing
cuts(dottedhistograms)forimagingini′ (blacklines)andr′ (red
Galaxies are selected as those objects with half light lines) bands. Bottom: the redshift distribution of galaxies used
radius 1.1rPSF < r < 4 pixels, where rPSF is the size fromthei′ (blacklines)andr′ (redlines)bands.
h h h
of the largest star, signal-to-noise ν > 10, magnitude
21.5<i′ <24.5andSExtractorflagFLAGS=0. Toalso ically tested on simulated images containing a known
exclude blended or close pairs that could bias ellipticity shear signal as part of the Shear Testing Programme
measurements,wealsocutobjectswithcorrectedelliptic- (STEP; Heymans et al. 2006; Massey et al. 2007) and
ity|e |>1andpairsofgalaxieswithin3′′. Aftersurvey the Gravitational lensing Accuracy Testing (GREAT08;
cor
masking and catalog cuts, the galaxy number density is Bridleetal.2010)challenge. Inallcases,themethodwas
n ∼ 11.5 arcmin−2 in an area of A ∼ 51.3 deg2 of i′- found to have small and repeatable systematic errors.
g i′
bandimaging;andn ∼7.9arcmin−2 inA ∼55.0deg2 If the PSF anisotropy is small, the shear γ can be re-
ofr′-banddata. Notegthatbotharelowerthra′nthegalaxy coveredtofirst-orderfromtheobservedellipticityeobs of
density n ∼ 38 arcmin−2 obtained in CFHTLS-Deep each galaxies via
g
imaging by Gavazzi & Soucail (2007). This will restrict Psm
our detections to generally more massive clusters. Fig- γ =Pγ−1(cid:18)eobs− Psm∗e∗(cid:19), (1)
ure 6 shows the magnitude distribution of the galaxies,
andtheredshiftdistributionofthe72%(76%)ofgalax- where asterisks indicate quantities that should be mea-
iesselectedinthei′(r′)bandsthatalsohavephotometric sured from the PSF model interpolated to the position
redshift estimates by Arnouts et al. (2010). of the galaxy, Psm is the smear polarizability, and P is
γ
Wethenmeasuretheshapesofalltheselectedgalaxies. the correction to the shear polarizability that includes
Our implementation of KSB is based on the KSBf9014 smearing with the isotropic component of the PSF. The
pipeline (Heymans et al. 2006). This has been gener- ellipticities are constructed from a combination of each
object’s weighted quadrupole moments, and the other
14 http://www.roe.ac.uk/∼heymans/KSBf90/Home.html quantities involve higher order shape moments. All def-
6 Shan et al.
1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
1 1
0.5 0.5
0 0
-0.5 -0.5
-1 -1
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Figure 7. Choices for the way shear polarizability Pγ can be Figure 8. Robustnessofthecalibrationofourshearmeasurement
fittedtoagalaxypopulationintheCFHTKSBf90pipeline,tore- as a function of image quality. Linearly-spaced contours compare
duce noise and bias in individual measurements. Linearly spaced our shear measurements of galaxies in subsets of CFHTLS-Deep
contours compare our shear measurements of galaxies in a sub- imagingwithvaryingmeanseeing(blacksolid: 0′′.90,reddashed:
set of the CFHTLS-Deep imaging, stacked to the depth of the 0′′.69,bluedotted: 0′′.57)tomeasurementsfromtheHubbleSpace
CFHTLS-Wide survey, against measurements from the Hubble Telescope. Shearsareconsistentlyunderestimatedbyourpipeline,
Space Telescope. Dashedlinesshowthebest-fitrelationγCFHT= butthecalibrationisremarkablyrobust.
1
(m−1)γHST+c. Thefourpanelsillustratevariousfittingschemes.
1
Top-left: raw(noisy)Pγ measurements fromeachgalaxy, without
fitting. Top-right: fitted as a polynomial in galaxy size Pγ(rh). by CFHT is very well resolved by HST, and imaged to
Bottom-left: fitting function Pγ(rh,mag) from Fu et al. (2008). very high signal-to-noise ratio by the COSMOS survey,
Bottom-right: best-fitrational function Pγ(rh,mag), asdescribed the space-based shear measurements require only negli-
inthetext.
giblePSFcorrectionandsufferfromonly negligibleshot
noise. A consistent shear measurement between ground
initions are taken from Luppino & Kaiser (1997). Note and space for this subset of galaxies would therefore in-
that we approximate the matrix P by a scalar equal to dicate a robust shear measurement across the CFHTLS.
γ
half its trace. Since measurements of Tr P from indi- Multiplicative shear measurement biases m are the
γ
vidual galaxies are noisy, we follow Fu et al. (2008) and most problematic for circularly-symmetric cluster mea-
fit it as a function of galaxy size and magnitude, which surements. Multiplicative biases cannot be internally
are more robustly observable galaxy properties. diagnosed within a shear catalog, so our comparison
Following Hoekstra et al. (2000), we weight the shear against external data is most useful for checking that
contribution from each galaxy as m is sufficiently small that it corresponds to a bias
smaller than our statistical errors. Within the KSB
1 Pγ2 framework, difficulties in shear calibration mainly rest
w = = , (2)
σ2 σ2P2+σ2 in measurement of the shear polarizability P , so we
e,i 0 γ e,i γ
firstinvestigatedifferentpossibilitiesforfittingP across
γ
whereσe,iistheerrorinanindividualellipticitymeasure- a galaxy population. Figure 7 compares shear mea-
ment obtained via the formula in Appendix A of Hoek- surements from a subset of the CFHTLS-Deep imag-
stra et al. (2000), and σ0 ∼ 0.278 is the dispersion in ing with mean seeing 0′′.69 (similar to the mean see-
galaxies’ intrinsic ellipticities. ing in our survey)stackedto the depth of the CFHTLS-
Wideimagingagainstshearmeasurementsobtainedfrom
3.4. Calibration of Multiplicative Shear Measurement HST. The dashed lines show the best-fit linear relations
Biases γCFHT = (1 + m)γHST + c, which are obtained using
1 1
WeexploittheopportunitythattheCFHTLS-DeepD2 a total least-squares fitting method (e.g. Kasliwal et al.
field includes the HST COSMOS survey field, and ver- 2008) that accounts for the noise present in both shear
ify the calibrationofour shearmeasurementpipeline for catalogs. The top-left panel shows the CFHT shear
ground-baseddataagainstanindependentanalysisofthe measurements with Pγ na¨ıvely obtained from each raw,
much higher resolution space-based data (Leauthaud et noisy galaxy without any fitting. This results in a large
al. 2007, 2010). We stack subsets of the CFHTLS-Deep bias on shear measurements and a large amount of ex-
D2 imaging to the same depth as the CFHTLS-Wide tra noise. The top-right panel shows the shear mea-
survey and analyze it using the same pipeline applied surements if Pγ is fitted as a function of galaxy size,
to the CFHTLS-Wide W1 field. Since any galaxy seen P (r )=a +a r +a r2. The bottom-left panelshows
γ h 0 1 h 2 h
Weak lensing in CFHTLS-Wide 7
Figure 9. Meanshearmeasurementsfromgalaxiesini′-bandob-
servations of the entire CFHTLS-Wide W1 field. In the absence
of additive systematics, these should be consistent with zero. In
practice,theyalwaysremainwithinthedashedlinesthanindicate
anorderofmagnitudelowerthanthe1%–10%shearsignalaround
clusters. Upperandlowerpanels showcomponents γ1 andγ2,re-
spectively. Left,middle,andrightpanelsshowtrendsasafunction
ofgalaxymagnitude,size,anddetection signal-to-noise.
Figure 10. The cross-correlation between shear measurements
and stellar ellipticities, as a function of the separation between
shearsifPγ(rh,mag)=a0+a1rh+a2rh2+a3mν (Fuetal. galaxies and stars, averaged throughout the CFHTLS-Wide W1
2008). The bottom-rightpanelshowsthe matchedshear field. If all residual influence of the observational PSF has been
successfullyremovedfromthegalaxyshapemeasurements,thered
with P (r ,mag) the best-fit rational function
γ h (lower)pointsshouldbeconsistentwithzero.
a +a m +a m2 +a r
P = 0 1 ν 2 ν 3 h . (3)
γ 1+a4mν +a5m2ν +a6rh+a7rh2 tion. Figure 10 shows the correlation ξsys between the
corrected shapes of galaxies and the uncorrected shapes
In this example, the coefficients are a0 = 25.07, a1 = of stars. Following Bacon et al. (2003) and Massey et
−2.19, a2 = 0.045, a3 = 0.53, a4 = 0.58, a5 = −0.022, al.(2005),we normalize the star-galaxyellipticity corre-
a6 = −0.85, and a7 = 0.14. The more sophisticated fits lation by the uncorrected star-star ellipticity correlation
produceashearcatalogthatisamarginallybettermatch to assess its impact on shear measurements
tothereliableHSTmeasurements,andthis isevenmore
trueifweredothe analysisusingthe fullCFHTLS-Deep <e∗(x)γ(x+θ)>2
ξ (θ)= , (4)
depth, in which galaxies are fainter and smaller. We sys <e∗(x)e∗(x+θ)>
henceforth choose to adopt the rational function fit to
P for all subsequent analyses, obtaining new best-fit where e∗ is the ellipticity of the stars before PSF cor-
γ
coefficients for each pointing. rection and γ is the shear estimate from galaxies. We
Toquantifytheperformanceofourshearmeasurement find that our PSF correctionis well within requirements
pipeline as a function of image quality, we stack subsets for our analysis because on cluster scales 1–5 arcmin,
of the CFHTLS-Deep D2 imaging with low-, medium-, the amplitude of ξsys is at least one order of magnitude
andhigh-seeingtothesamedepthastheCFHTLS-Wide smaller than the cosmic shear signal
survey, and analyze each separately (Figure 8). We find 1 ∞
that our CFHTLS pipeline consistently underestimates ξ± =ξtt(θ)±ξxx(θ)= ℓPκ(ℓ)J0,4(ℓθ)dℓ , (5)
shear, but that the calibration is remarkably robust to 2π Z0
seeing conditions. We can therefore simply recalibrate where ξ (θ) (ξ (θ)) are the correlation functions be-
tt xx
our pipeline for all images by multiplying all measured tween components of shear rotated tangentially (at 45◦)
shears by 1/(1−0.21). to the line between pairs of galaxiesseparatedby an an-
gle θ and J , J are Bessel functions of the first kind.
3.5. Assessment of Residual Additive Shear Systematics 0 4
Because gravitational lensing is achromatic while sys-
Additive shear measurement systematics c generally tematicsaretypicallynot, wecanalsoassessthe robust-
cancelout in circularly-symmetriccluster measurements nessofourmeasurementsbycomparingshearsmeasured
(Mandelbaum et al. 2006). However, to double-check from independent imaging acquired in multiple bands.
for significant additive systematics, we first measure the The first attempt at comparing multicolor shear mea-
mean shears hγi across all 72 pointings of the CFHTLS- surements was made by Kaiser et al. (2000) using the
Wide W1 field. Figure 9 demonstrates that the mean CFHT12K camera. The I and V bands showed signif-
shear is consistent with zero as expected, for galaxies of icantly different signals that were inconsistent with the
all sizes, magnitudes, and signal-to-noise ratios. change in redshift distribution between the two filters.
We also look for residual systematics left in the weak After a great deal of algorithmic progress, Semboloni et
lensing cosmicshearsignaldue to imperfect PSFcorrec- al. (2006) obtained consistent shear measurements from
8 Shan et al.
where the partial derivatives ∂ are with respect to θ .
i i
The convergence field κ(θ), which is proportional to the
massprojectedalongalineofsight,canalsobeexpressed
in terms of the lensing potential as
1
κ= (∂2+∂2)φ. (8)
2 1 2
We shall reconstruct the convergence field from our
shear measurements via the Kaiser & Squires (1993)
(KS93) method. This is obtained by inverting Equa-
tions (6) and (7) in Fourier space:
γˆ =Pˆκˆ (9)
i i
for i=1,2,where the hatsymboldenotes Fourier trans-
forms, we define k2 =k2+k2 and
1 2
k2−k2
Pˆ (k)= 1 2, (10)
Figure 11. Gravitationallensingisachromatic,someasurements 1 k2
ofgalaxyshapesfromimagingindifferentcolorsshouldonaverage
be consistent. This shows a comparison of shear measurements 2k k
obtainedfromCFHTLS-Wider′-bandandi′-bandimagingofthe Pˆ (k)= 1 2. (11)
wholeW1field. Thedashed lineshows the best-fitlinear relation 2 k2
γi =(m−1)γr+c.
1 1 This inversion is non-local, so we deal with masked re-
gions of the shear field by masking out the same area in
theconvergencefield,plus a1′.5border. Weshallignore
i′-band and r′-band CFHTLS-Deep data. Gavazzi &
any signal within these regions, and set the convergence
Soucail(2007)extractedconsistentshearfromthe g′, r′,
to zero in relevant figures.
z′ and i′ bands of CFHTLS-Deep. Gavazzi et al. (2009)
For the finite density of source galaxies resolved by
also measured consistent values of the PSF-correctedel-
CFHT, the scatter of their intrinsic ellipticities means
lipticities of central Coma cluster galaxies in MegaCam-
that a raw, unsmoothed convergence map κ(θ) will be
u∗ and CFH12k-I bands. Our analysis pipeline mea-
very noisy. Following Miyazaki et al. (2002), we smooth
sures shears in the independent CFHTLS r′-band and
the convergence map by convolving it (while still in
i′-band imaging that are consistent within the ∼ 0.1
Fourier space) with a Gaussian window function,
rms noise (Figure 11). To maximize the total number
of galaxies in our shear catalog, we therefore combine 1 θ2
r′-band and i′-band measurements. Only the unique r′- WG(θ)= πθ2 exp(cid:18)−θ2 (cid:19), (12)
band galaxies are added to the i′-band catalog. The G G
combined catalog includes ∼ 2.66 million galaxies, with As shown by van Waerbeke (2000), if different galax-
ng ∼14.5 arcmin−2 for lensing measurements. ies’ intrinsic ellipticities are uncorrelated, the statistical
We shallcontinue testingfor systematicsateachstage propertiesoftheresultingnoisefieldcanbedescribedby
ofouranalysis,bycheckingthattheshearsignalbehaves Gaussianrandomfieldtheory(Bardeenetal.1986;Bond
asexpected,andisconsistentwithexternaldatasets. An & Efstathiou 1987) on scales where the discreteness ef-
important example of this is the (nonphysical) B-mode fectofsourcegalaxiescanbeignored. TheGaussianfield
signal, which we shall compute wherever we reconstruct is uniquely specified by the variance of the noise, which
the(physical)E-mode. Puregravitationalfieldsproduce is in turn controlled by the number of galaxies within a
zeroB-modeforisolatedclustersandonlytiny B-modes smoothing aperture (Kaiser & Squires 1993; Van Waer-
through coupling between multiple systems along adja- beke 2000)
cent lines of sight (Schneider et al. 2002). The B-mode σ2 1
signalcorrespondstotheimaginarycomponentofP (ℓ); σ2 = e , (13)
κ noise 2 2πθ2n
it can be conveniently measured by rotating all galaxy G g
shears through 45◦ then remeasuring the E-mode signal
where σ is the rms amplitude of the intrinsic elliptic-
e
(Crittenden et al. 2002).
ity distribution and n is the density of source galaxies.
g
We define the signal-to-noise ratio for weak lensing de-
4. MASSRECONSTRUCTIONS
tections by
4.1. Kaiser-Squires Inversion and Masking κ
ν ≡ . (14)
The shear field γi(θ) is sparsely and noisily sampled σnoise
bymeasurementsoftheshapesofgalaxiesatpositionsθ.
To define the noise level in theoretical calculations of ν,
The smooth, underlying shear field γ (θ) can be written
i we adopt a constant effective density of galaxies equal
in terms of the lensing potential φ(θ) as
to the mean within our survey. For observational cal-
1 culations of ν, we use the mean galaxy density in each
γ1 = 2(∂12−∂22)φ, (6) pointing—butdonotconsiderthenon-uniformityofthe
density within eachfield due to masks or galaxy cluster-
γ =∂ ∂ φ, (7) ing.
2 1 2
Weak lensing in CFHTLS-Wide 9
Figure 12. Distribution of foreground mass in the W1+2+3 pointing, reconstructed from shear measurements via the KS93 method.
Left: thephysicalE-modeconvergencesignal. Right: theB-modesystematicssignal,createdbyrotatingtheshearsby45◦ thenremaking
themap. Contoursaredrawnatdetectionsignificances of3σ, 4σ,and5σ,withdashedlinesfornegative values.
It turns out that a simple Gaussian filter of width
Table 2
θG ≈ 1′ is close to the optimal linear filter for cluster TheNumberofLocalMaximaandMinimaintheConvergence
detection, and this choice has been extensively studied MapoftheCFHTLS-WideW1Field,asaFunctionofSmoothing
in simulations (White et al. 2002; Hamana et al. 2004; Scale.
Tang & Fan 2005). Because of our relatively low source
Smoothing E-mode E-mode B-mode B-mode
galaxydensity,thegalaxies’randomintrinsicshapeswill ScaleθG ν>3.5 ν<−3.5 ν>3.5 ν<−3.5
produce spurious noise peaks, degrading the complete- 0′.5 1512 1270 1244 1033
ness and purity of our cluster detection. To reduce con- 1′.0 543 445 361 282
tamination,werepeatourmassreconstructionusingtwo 2′.0 281 233 148 126
smoothing scales θ = 1′ and θ = 2′. The map with
G G
greatersmoothingwillbelessnoisy;tohelpremovespu-
(MRLens; Starck et al. 2006), to better display large-
rious peaks from the higher resolution map, we consider
scale features. The MRLens filtering effectively sup-
onlythosepeaksdetectedaboveasignal-to-noisethresh-
presses noise peaks, but results in non-Gaussian noise
old in both maps.
that complicates the peak selection (Jiao et al. 2011),so
Figure 12 shows the reconstructed convergence field
we shall not use it further.
correspondingtoforegroundmassintheW1+2+3point-
Toassessthereliabilityofthismap,weshallfirstinves-
ing. The left panel shows the E-mode reconstruction
tigatethestatisticalpropertiesoflocalmaximaandmin-
with KS93 method after smoothing by a 1arcmin Gaus-
ima. Figure 15 shows the distribution of peak heights,
siankernel. Thiscontainsseveralhighsignal-to-noisera-
as a function of detection signal-to-noise. The bimodal
tiopeaks,whiletheassociatedB-modesystematicsmea-
distribution in both the B-mode and E-mode signals is
surementintherightpanelisstatisticallyconsistentwith
dominated by positive and negative noise fluctuations,
zero, with fewer peaks. As weak lensing produces only
butanasymmetricexcessintheE-modesignalisappar-
curl-free or E-mode distortions, a detection (significant
ent at both ν >3.5 and, at lower significance, ν <−3.5.
above statistical noise) of curl or B-mode signal would
The amplitude, slope and non-Gaussianityof this excess
have indicated contamination from residual systematics,
areallpowerfuldiscriminatorsbetweenvalues ofparam-
e.g. imperfect PSF correction.
etersin cosmologicalmodels (Pireset al.2009). Positive
peaks correspond mainly to dark matter halos around
4.2. Large-scale Lensing Mass Map
galaxyclusters. Localminima couldcorrespondto voids
A reconstructed“darkmatter mass”convergencemap (Jain & van Waerbeke 2000; Miyazaki et al. 2002), but
for the entire 64 deg2 CFHTLS-Wide W1 field is pre- the large size of voids is ill-matched to our θ =1′ filter
G
sented in Figure 13. We detect 301 peaks with ν > 3.5 width, and their density contrast can never be greater
inmapswithbothsmoothingscalesθ =1′andθ =2′. than unity, so this aspect of our data is likely just noise.
G G
The same information is reproduced in Figure 14, with Figure 16 recasts the peak distribution into a cumu-
θ =6′andaftermulti-scaleentropyrestorationfiltering lative density of positive maxima or negative minima.
G
10 Shan et al.
-4
-6
-8
-10
-12
38 36 34 32 30
Figure 13. Reconstructed“darkmattermass”convergencemapfortheentire64deg2 CFHTLS-WideW1field. Thishasbeensmoothed
by a Gaussian filter of width θG = 1′. Black contours are drawn at detection signal-to-noise ratios ν = 3.0, 3.5, 4.0, and red contours
continuethissequencefromν≥4.5.
As expected, we find a non-Gaussian mass distribution cles with error bars. In these theoretical calculations,
with more highly significant positive maxima (corre- we model the population of backgroundgalaxies as hav-
sponding to mass overdensities) than highly significant ing an intrinsic ellipticity dispersion σ =0.278, density
e
negative minima (see Table 2). Analytic predictions of n =14.5arcmin−2andtheredshiftdistributionfromFu
g
peak counts are also overlaid. Following van Waerbeke etal.(2008)inaΛCDMuniverse. Atν >4.5,itappears
(2000), dot-dashed lines show the expected density of that theory may begin to predict more peaks than are
pure noise peaks, and dotted lines show the expected observed. However, these are very small number statis-
number of true dark matter halos. Predictions fromFan tics, and our observations are consistent with analytical
et al. (2010), which also take into account the effect of predictionswithinPoissonnoise. Atverylowν,thenum-
noise on the heights of true peaks and the clustering of ber of peaks is washed out and actually decreases when
noise peaks near dark matter halos, are shown as cir- noiseissuperimposed,becauseitisimpossibletoextract