arX

iv:a

stro

-ph/

0702

359v

1 1

3 Fe

b 20

07

Weak Gravitational Lensing with COSMOS: Galaxy Selection and

Shape Measurements

Alexie Leauthaud1, Richard Massey2, Jean-Paul Kneib1,2, Jason Rhodes2,3,

David E. Johnston3, Peter Capak2, Catherine Heymans4, Richard S. Ellis2,

Anton M. Koekemoer5, Oliver Le Fevre1, Yannick Mellier6,7, Alexandre Refregier8, Annie

C. Robin9, Nick Scoville2, Lidia Tasca1, James E. Taylor10 & Ludovic Van Waerbeke4

alexie.leauthaud@oamp.fr

ABSTRACT

With a primary goal of conducting precision weak lensing measurements from space, theCOSMOS⋆ survey has imaged the largest contiguous area observed by the Hubble Space Telescope(HST) to date using the Advanced Camera for Surveys (ACS). This is the first paper in a serieswhere we describe our strategy for addressing the various technical challenges in the productionof weak lensing measurements from the COSMOS data. We first construct a source catalog from575 ACS/WFC tiles (1.64 degrees2) sub-sampled at a pixel scale of 0.03′′. Defects and diffractionspikes are carefully removed, leaving a total of 1.2 × 106 objects to a limiting magnitude ofF814W = 26.5. This catalog is made publicly available. Multi-wavelength follow-up observationsof the COSMOS field provide photometric redshifts for 73% of the source galaxies in the lensingcatalog. We analyze and discuss the COSMOS redshift distribution and show broad agreementwith other surveys to z ∼ 1. Our next step is to measure the shapes of galaxies and to correctthem for the distortion induced by the time varying ACS Point Spread Function and for ChargeTransfer Efficiency effects. Simulated images are used to derive the shear susceptibility factorsthat are necessary in order to transform shape measurements into unbiased shear estimators. Forevery galaxy we derive a shape measurement error and utilize this quantity to extract the intrinsicshape noise of the galaxy sample. Interestingly, our results indicate that the intrinsic shape noisevaries little with either size, magnitude or redshift. Representing a number density of 66 galaxiesper arcminute2, the final COSMOS weak lensing catalog contains 3.9×105 galaxies with accurateshape measurements. The properties of the COSMOS weak lensing catalog described throughoutthis paper will provide key input numbers for the preparation and design of next-generation widefield space missions.

Subject headings: cosmology: observations – gravitational lensing – large-scale structure of Universe

1Laboratoire d’Astrophysique de Marseille, BP 8, Tra-verse du Siphon, 13376 Marseille Cedex 12, France.

2California Institute of Technology, MC 105-24, 1200East California Boulevard, Pasadena, CA 91125, U.S.A.

3Jet Propulsion Laboratory, Pasadena, CA 91109.4Department of Physics & Astronomy, University of

British Columbia, 6224 Agricultural Road, Vancouver,B.C. V6T 1Z1, Canada.

5Space Telescope Science Institute, 3700 San MartinDrive, Baltimore, MD 21218, U.S.A.

6Institut d’Astrophysique de Paris, UMR7095 CNRS,

Universite Pierre & Marie Curie - Paris, 98 bis bd Arago,75014 Paris, France.

7Observatoire de Paris, LERMA, 61, avenue del’Observatoire, 75014 Paris, France.

8Service d’Astrophysique, CEA/Saclay, 91191 Gif-sur-Yvette, France

9Observatoire de Besancon, BP1615, 25010 BesanconCedex, France

10Department of Physics and Astronomy, University ofWaterloo, 200 University Avenue West, Waterloo, Ontario,Canada N2L 3G1

⋆Based on observations with the NASA/ESA Hubble

1

1. Introduction

As we look towards distant galaxies, fluctua-tions in the intervening mass distribution cause aslight, coherent distortion of their intrinsic shapes.This effect, known as weak gravitational lens-ing, has been used for more than a decade toprobe the cosmography and the growth of struc-ture (for a review, see Bartelmann & Schneider2001). Although technically challenging becausethe weak lensing signal is minuscule and buriedin a considerable amount of noise, this field hasshown substantial progress due to the advent ofhigh resolution space based imaging, the prolifer-ation of wide-field multi-color surveys, and a de-termined effort to improve image analysis meth-ods and minimize systematic errors through theShear TEsting Program (STEP; Heymans et al.2006; Massey et al. 2006).

Historically first observed only around clus-ter cores (Tyson et al. 1990), weak lensing hasemerged as a versatile and effective techniqueto probe the mass distribution of clusters (e.g.,Kneib et al. 2003), to measure the clusteringof dark matter around galaxies ensembles (e.g.,Natarajan et al. 1998; Hoekstra et al. 2004; Sheldon et al.2004; Mandelbaum et al. 2005), and to put con-straints on the matter density parameter Ωm

and the amplitude of the matter power spectrumσ8 (e.g., Hoekstra et al. 2006; Semboloni et al.2006; Schrabback et al. 2006; Hetterscheidt et al.2006; Jarvis et al. 2006). Most applications cur-rently only use the first order deformation in-duced by the mass distribution, but novel tech-niques are under development to take into ac-count second order deformations (also called flex-

Space Telescope, obtained at the Space Telescope ScienceInstitute, which is operated by AURA Inc, under NASAcontract NAS 5-26555; also based on data collected at: theSubaru Telescope, which is operated by the National As-tronomical Observatory of Japan; the European SouthernObservatory, Chile; Kitt Peak National Observatory, CerroTololo Inter-American Observatory, and the National Op-tical Astronomy Observatory, which are operated by theAssociation of Universities for Research in Astronomy, Inc.(AURA) under cooperative agreement with the NationalScience Foundation; the National Radio Astronomy Obser-vatory which is a facility of the National Science Founda-tion operated under cooperative agreement by AssociatedUniversities, Inc ; and the Canada-France-Hawaii Telescopeoperated by the National Research Council of Canada, theCentre National de la Recherche Scientifique de France andthe University of Hawaii.

ion; Goldberg & Bacon 2005; Bacon et al. 2006;Okura et al. 2006; Goldberg & Leonard 2006) andmay prove to be more efficient probes of compactstructures such as galaxies and groups of galaxies.Weak lensing measurements are particularly pow-erful when combined with the knowledge of thethree dimensional galaxy distribution. Sophisti-cated lensing ’tomography’ techniques that utilizeredshifts to analyze the 3D shear field are a sen-sitive probe of the growth of structure and theequation of state of dark energy (Jain & Taylor2003; Bernstein & Jain 2004; Bacon et al. 2005).Applied in many different ways, weak lensing tech-niques unravel the mass distribution of structuresand their evolution in the universe.

High quality measurements of weak shear de-pend on the accurate determination of the shapesand redshifts of distant, faint galaxies. The COS-MOS program has imaged the largest contiguousarea (1.64 degrees2) with the Hubble Space Tele-scope (HST) to date using the Advanced Camerafor Surveys (ACS) Wide Field Channel (WFC).The Full Width Half Maximum (FWHM) of thePoint Spread Function (PSF) of the ACS/WFCis 0.12′′ at the detector1, yielding a much bet-ter resolution of small galaxies than ground-basedsurveys, which are typically limited by the seeingto a PSF of FWHM ∼ 1′′. Shape measurementsalso benefit from ACS/WFC imaging compared toground-based observations because smaller correc-tions are required for the PSF and the shear mea-surements are less diluted by PSF smearing. Theimaging quality and unprecedented area of theCOSMOS ACS/WFC data combined with exten-sive follow-up observations at other wavelengths(Scoville et al. 2007; Koekemoer 2007) to provideaccurate photometric redshifts (Mobasher et al.2007), make COSMOS a unique data set for weaklensing studies.

To exploit the weak lensing potential of theCOSMOS ACS/WFC data, a carefully designedcatalog of resolved galaxies with shape measure-ments must be extracted from the imaging data.The challenges and requirements of such a cata-log are the following. First, the large survey sizemakes a robust automation of catalog generationessential. Second, the lensing sensitivity increases

1Before convolution with the detector pixels, the intrinsicwidth of the F814W PSF is 0.085′′

2

with the number density of resolved faint galax-ies. Thus, it is important to detect all galaxiesto faint magnitudes while taking care to minimizespurious detections that will add noise to weaklensing measurements. Third, the high spatial res-olution of the ACS/WFC allows for an excellentseparation of close pairs. Given accurate deblend-ing and a high number density of galaxies, onecan expect to measure shear statistically on sub-arcminute scales, where baryonic physics may be-gin to influence the dark matter distribution. Asthe COSMOS-ACS/WFC data set is likely to bethe only large space-based lensing survey until thelaunch of next-generation wide field space missionssuch as snap 2 or dune (Refregier et al. 2006), theknowledge acquired through the COSMOS datawill be unique and of crucial importance for thepreparation and design of these future missions.

This is the first paper of a series describing thegalaxy selection, the galaxy shape measurement,the lensing analysis, and the cosmological interpre-tation of COSMOS data. Details regarding PSFcorrections as well as tests for systematic effectsare presented in the second paper of this series(Rhodes et al. 2007). A three dimensional cosmicshear analysis is presented in the third paper ofthis series (Massey et al. 2007). Finally, a fourthpaper presents high resolution dark matter massmaps of the COSMOS field (Massey et al. 2007a).

In this paper, we describe our methods for con-structing a galaxy catalog from the ACS/WFCdata, to be used in subsequent weak lensing workwith COSMOS. Our goal is to produce a catalogof galaxies with photometric redshifts and PSF-corrected shape measurements, free of contami-nating stars, cosmic rays, diffraction spikes, andother artifacts. The paper is organized as follows.In §2 we present the data. In §3 we describe thepipeline that locates and measures the propertiesof all detected objects. In §4, we assess the qual-ity of the data and analyze the COSMOS redshiftdistribution. In §5 and §6 we present the PSFand CTE correction schemes, the shape and shearmeasurement methods, and our selection criteriafor the final lensing catalog. In §7 we extract theintrinsic shape noise of the galaxy sample as afunction of redshift and discuss the implicationsfor future weak lensing surveys. Where necessary,

2http://snap.lbl.gov

we assume a standard cosmological model withΩM = 0.3, ΩΛ = 0.7, H0 = 100h km s−1 Mpc−1

and h = 0.7.

2. The COSMOS ACS Data

The COSMOS HST ACS field (Scoville et al.2007; Koekemoer 2007) is a contiguous 1.64degrees2, centered at 10:00:28.6, +02:12:21.0(J2000). Between October 2003 and June 2005(HST cycles 12 and 13), the region was com-pletely tiled by 575 adjacent and slightly overlap-ping pointings of the ACS/WFC (see Figure 1).Images were taken through the wide F814W filter(“Broad I”). The camera has a 203′′×203′′ field ofview, covered by two 4096x2048 CCD chips with anative pixel scale of 0.05′′ (Ford et al. 2003). Themedian exposure depth across the field is 2028seconds (one HST orbit). At each pointing, four507 second exposures were taken, each ditheredby 0.25′′ in the x direction and 3.08′′ in the y di-rection from the previous position. This strategyensures that the 3′′ gap between the two chips iscovered by at least three exposures and facilitatesthe removal of cosmic rays. Pointings were takenwith two approximately 180 opposed orientationangles (PA V3= 100 ± 10 and 290 ± 10). In thispaper we use the “unrotated” images (as opposedto North up) to avoid rotating the original frameof the PSF. By keeping the images in the defaultunrotated detector frame, they can be stacked tomap out the observed PSF patterns. For sim-ilar reasons, we perform detection in individualACS/WFC tiles instead of on a larger mosaic(where the orientation of the PSF frame wouldbe unknown). Figure 2 shows a COSMOS/ACSpointing with the bright detections (F814W < 23)and the masking of stars, asteroid trails and imagedefects (see §3.5).

To build our catalog, we use version 1.3 of the“unrotated” ACS/WFC data which has been spe-cially reduced for lensing purposes (see Koekemoer2007, for technical details). Image registration,geometric distortion, sky subtraction, cosmic rayrejection and the final combination of the ditheredimages were performed by the MultiDrizzle al-gorithm (Koekemoer et al. 2002). As described in(Rhodes et al. 2007), the MultiDrizzle param-eters have been chosen for precise galaxy shapemeasurement in the co-added images. In par-

3

Fig. 1.— Date of observation for each of the sur-vey pointings. The PSF of the ACS/WFC varieson timescales that are much shorter than the pe-riod over which COSMOS was observed.

ticular, a finer pixel scale of 0.03′′/pix was usedfor the final co-added images (7000x7000 pixels),even though this implies more strongly correlatedpixel noise (see §3.8). Hereafter, when we refer topixels, we will assume a pixel scale of 0.03′′/pix.Pixelization acts as a convolution followed by are-sampling and, although current shear measure-ment methods can successfully correct for convolu-tion, the formalism to properly treat re-samplingis still under development for the next genera-tion of methods. Again following the recommen-dations of Rhodes et al. (2007), a Gaussian andisotropic multidrizzle convolution kernel was used,with scale=0.6 and pixfrac=0.8, small enoughto avoid smearing the object unnecessarily whilelarge enough to guarantee that the convolutiondominates the re-sampling. This process is thenproperly corrected by existing shear measurementmethods.

The ACS/WFC CCDs also suffer from imper-fect charge transfer efficiency (CTE) during read-out. As charges are transferred during the read-out process, a certain fraction are retained by

charge traps (created by cosmic ray hits) in thepixels. This causes flux to be trailed behind ob-jects as the traps gradually release their charge,spuriously elongating them in a coherent direc-tion that mimics a lensing signal. Since this ef-fect is produced by a fixed number of charge trapswithin the CCD substrate, it affects faint sources(with a larger fraction of their flux being trailed)more than bright ones. This is an insidious effectthat mimics an increasing shear signal as a func-tion of redshift, and prevents the traditional wayof dealing with the calibration of faint galaxies in alensing analysis by looking at bright stars. Ideallythis effect would be corrected for on a pixel-by-pixel basis in the raw images unfortunately ourcurrent physical understanding of this effect is in-sufficient and a more indepth analysis in still un-derway. The CTE effect can be quantified suffi-ciently well however that, in a first step, we canadopt a post-processing correction scheme basedon an object’s position, flux, and date of observa-tion. Further details regarding this model can befound in Rhodes et al. (2007).

3. The COSMOS ACS Galaxy Catalog

In this section, we discuss the construction ofthe COSMOS ACS/WFC source catalog. Thiscatalog is carefully cleaned of defects and arti-facts and is made publicly available through theInfrared Science Archive (IRSA) database1.

3.1. Detection Strategy

We use Version 2.4.3 of the SExtractor pho-tometry package (Bertin & Arnouts 1996) to ex-tract a source catalog of positions and variousphotometric parameters. In the construction ofthis catalog, our main concern is to pick out thesmall faint objects that contain most of the lens-ing signal. The detection strategy that we there-fore adopt is to configure SExtractor with verylow thresholds (even if this leads to more false de-tections in the catalog) and to control our sam-ple selection via subsequent “lensing cuts” (seeSect 6). We hope to thus reduce unknown selec-tion biases introduced by the SExtractor detectionalgorithm. When configured with low detectionthresholds however, SExtractor also inevitably a)

1http://irsa.ipac.caltech.edu/Missions/cosmos.html

4

Fig. 2.— COSMOS pointing acs I 095836+0141 unrot sci 12.fits with bright detections, masking and edgedefinition indicated. Adjacent images have a sizable overlap (shown here by the smaller magenta box) whichallows us to discard detections on the boundaries of each tile (defined by the larger blue box) without losingany objects in the final concatenated catalog. The automated masking of the diffraction spikes around brightstars (F814 < 23) is pictured here by the red polygons. The basic shape of the star masks is predefinedand then scaled with the magnitude of the star. The green rectangles correspond to the manual maskingof asteroid trails and various other image defects. Toward the right edge of this image, the dwarf galaxyL1-099 (Impey et al. 1996) is identified and flagged in a special category. Bright galaxies with F814 < 23are depicted by blue ellipses and bright stars with F814 < 23 by red circles.

5

overdeblends low surface brightness spirals andpatchy irregulars, b) deblends the outer featuresof bright galaxies, c) detects spurious objects inthe scattered light around bright objects and d)underdeblends close pairs.

The correct detection of close pairs enables lens-ing measurements on very small scales. However,overdeblending and spurious detections adds noiseto these measurements. In particular, false detec-tions around bright objects can have quite highsignal-to-noise (S/N) values and are not trivial toremove with lensing cuts ( see §6). The methodpresented here is a partial solution to b) c) andd). The overdeblending of low surface brightnessand patchy galaxies remains a difficult problem,however, especially for high resolution imaging,and calls for an improvement of existing detec-tion algorithms. With the advent of high resolu-tion multi-wavelength surveys, a possible solutionforward would be to incorporate color and mor-phological information into the detection process(e.g., Lupton et al. 2001).

While this overdeblending problem persists inour catalog, it affects less than 1% of the objects.This problem is furthermore mitigated by the cen-troiding process during the shape measurementstage. Indeed, objects for which the centroid algo-rithm fails to converge, which will often be the casefor overdeblended features, are discarded from thecatalog. To remedy the remaining problems b) c)and d), we adopt and improve the method (knownas the ‘Hot-Cold‘ method) employed in Rix et al.(2004). In this method, we run SExtractor twice,once with a configuration optimized for the detec-tion of only the brightest objects (“cold” step) andthen again with a configuration optimized for thefaint objects (“hot” step). This double extractionhelps improve the detection of close pairs. The twosamples are then merged together to form the finalcatalog and masks are created around the brightdetections minimizing the effects of c) and d).

For the “hot” and “cold” steps we vary fourmain parameters to optimize the detection: 1)detect threshold, the minimum signal-to-noise per pixel above the background level, 2)min area, the number of contiguous pixels ex-ceeding this threshold, 3) back size, the meshsize of the background map, 4) deblend nthresand deblend mincont, the parameters regulat-ing deblending. In both cases, the data are filtered

prior to detection by a 5 pixel (0.15′′) Gaussianfiltering kernel. Our choice of parameters for bothsteps is provided in Table 1.

The two-step method also allows one to adjustthe estimation of the background map accordingto the typical size of objects one expects to de-tect, improving detections with SExtractor. Thebackground map is constructed by computing anestimator for the local background on a grid ofmesh size back size. We adjust back size so asto capture the small scale variations of the back-ground noise while keeping it large enough not tobe affected by the presence of objects.

For each exposure, a weight map is producedby MultiDrizzle describing the combined noiseproperties of the read-out, the dark current andthe sky background (Koekemoer (2007)). Thesemaps describe the noise intensity at each pixel andare used to account for the spatial-dependent noisepattern in the co-added image with the SExtractorweight image option set to weight map.

Each ACS/WFC pointing consists of fourslightly offset, dithered exposures making cosmicray rejection more difficult and detection more un-reliable on the boundaries of each tile where thereare fewer than four input exposures. Because ad-jacent images overlap sufficiently, we can trim theedges of the images without actually removingdata (see Figure 2).

3.2. Bright Object Detection

In the first step we detect only the brightestand largest objects in the image, with 140 or morecontiguous pixels (corresponding to a diameter of0.4′′ for a circular object) rising more than 2.2sigma per pixel above the background level. Theback size parameter is set to 12′′, or 30 times thediameter of the smallest objects detected. The de-tection threshold and the deblending parametersdeblend nthres & deblend mincont are cal-ibrated heuristically on several images to separateclose pairs as much as possible without deblend-ing patchy, extended spiral galaxies. Because faintobjects are captured in a second run, we are freeto choose the value of the detection threshold dur-ing this step. We found that this flexibility greatlyhelped to calibrate the parameters that optimizethe deblending. Indeed, if detect thresholdis set to a low value (for example, to detect faint

6

Table 1

SExtractor configuration parameters

Parameter Bright objects Faint objects

DETECT MINAREA 140 18DETECT THRESH 2.2∗ 1.0DEBLEND NTHRESH 64 64DEBLEND MINCONT 0.04 0.065CLEAN PARAM 1.0 1.0BACK SIZE 400 100BACK FILTERSIZE 5 3BACKPHOTO TYPE local localBACKPHOTO THICK 200 200

∗Because of correlated noise (see § 3.8), the effective thresh-old levels are DETECT THRESH ∼ 1.25 for bright objects andDETECT THRESH ∼ 0.57 for faint objects.

object), close pairs will be detected as a single ob-ject and are difficult to deblend. Figure 3 illus-trates the improvements of this two-step methodcompared to a single-step method. During thisfirst step, all pixels associated with a detectionare recorded by SExtractor in an image called a“segmentation map”. These segmentation mapsare used at a later stage to merge the bright andthe faint catalogs (see §3.4). This first catalog ofbright objects is referred to as Ccold.

3.3. Faint Object Detection

In the second step, we configure SExtractor topick up the small, faint objects, taking care tochoose the detection parameters to be less con-servative than any subsequent lensing cuts (see§6). min area is set to 18 pixels (correspondingto a diameter of 1.2 times the FWHM of the PSF)and the detection threshold is one sigma abovethe background level. As objects detected at thisstep are smaller, the background estimation canbe improved by refining the mesh size of the back-ground map and setting back size to 100 pixels,or 20 times the diameter of the smallest objectsdetected. This second catalog is referred to here-after as Chot.

3.4. Merging the Two Samples

The final catalog is obtained by merging the de-tections from Ccold and Chot, keeping all objects inCcold and only the objects from Chot not detectedin Ccold. To determine which objects to discardfrom Chot, we use the segmentation maps createdduring the bright detection step. To begin with,

Fig. 3.— This figure illustrates the difficultyof correctly deblending close pairs while keepingpatchy spirals with strong star forming regions in-tact. Squares indicate detections from the faintstep and circles indicate detections from the brightstep. The top two panels show three objects thatare not detected or incorrectly deblended by thefaint step but that are picked by two step method.The bottom panel and the arrow towards the starin the upper left panel show that even with thismethod, a perfect configuration is still difficult toreach.

we enlarge the flagged areas in these segmentationmaps by approximately 20 pixels (0.6′′). We thendiscard all objects from Chot for which the cen-tral pixel lies within a flagged area of these maps.Thus, we remove duplicate detections and create amask around all bright objects, immediately clean-ing the catalog of a certain number of spurious

7

detections. By a visual inspection of the data,we estimate that this method solves about half ofthe deblending problems that we observe (exclud-ing the low surface brightness galaxies). The finalcatalog of raw SExtractor detections C1 contains1.8 million objects in total (see Table 2).

3.5. Cleaning the Catalog

Great care was taken to mask unreliable regionswithin images and to remove false detections fromthe catalog, especially those that can mimic a lens-ing signal. As illustrated in Figure 2, an auto-matic algorithm was developed to define polygo-nal shaped masks around stars with F814W < 19(the limit at which stars saturate in the COSMOSimages), with a size scaled by the magnitude of thestar. Objects near bright stars or saturated pix-els were masked to avoid shape biases due to anybackground gradient. All the images were then vi-sually inspected. In a few cases the automatic al-gorithm failed (very saturated stars for which thecentroid of the star is widely offset) and the stel-lar masks were corrected by hand. Other contam-inated regions of the images were also masked out,including reflection ghosts, asteroids, and satel-lite trails. Astronomical sources such as HII re-gions around bright galaxies, stellar clusters, andnearby dwarf galaxies, were also flagged and re-moved from the lensing catalog.

Objects with double entries in the catalog (fromthe overlap between adjacent images) are identi-fied and the counterpart with the highest SExtrac-tor flag (indicating a poor detection) is discarded,leaving a catalog of unique objects. However, theduplicated objects from overlapping regions are avaluable asset for consistency checks and are used,for example, to check the galaxy shape measure-ment error (see §7 and Figure 17).

The final clean catalog (C2) is free of spuriousor duplicate detections and contains 1.2 millionsources in total (see Table 2).

3.6. Star-Galaxy Classification

The correct identification of stars has two im-plications for the lensing analysis. First, brightstars are useful for PSF modelling and second,stars must be correctly identified in order to ap-ply our automatic masking algorithm of diffractionspikes. A robust star-galaxy classification is thus

necessary.

SExtractor produces a continuous stellarclassification index parameter ranging from 0 (ex-tended sources) to 1 (point sources). This indexhas two drawbacks: first the definition of the di-viding line is ambiguous and second, the neural-network classifier used by SExtractor is trainedwith ground-based images and is therefore onlyvalid for a sample of profiles similar to the orig-inal training set. With space-based images, thisindex becomes difficult to interpret, as illustratedin Figure 4 which depicts our star selection (de-scribed below) within the class star/mag autoplane.

We therefore test two alternative methods toclassify point sources and galaxies, one based onthe SExtractor parameter mu max (peak sur-face brightness above the background level) andthe other based on the half-light radius, Rhl(e.g., Peterson et al. 1979; Bardeau et al. 2005).Both methods are motivated by the fact that thelight distribution of a point source scales withmagnitude. Point sources therefore occupy awell-defined locus in a mu max/mag auto ora Rhl/mag auto plane. Figure 5 shows howwe can use this property to define stars (ID=2)and galaxies (ID=1) reliably up to a magnitudeof F814W ∼ 25. At fainter levels, the clas-sification begins to break down and the pointsources become indistinguishable from the smallgalaxies. We find that the two methods agreevery well, within 1% at magnitudes less thanF814W = 24 and within 2% at magnitudes lessthan F814W = 25. The small difference arisesmainly from a misclassification of objects by theRhl method because of the presence of a close pairthat distorts the estimation of Rhl. Overall, themu max method proved to be more robust and hasthe advantage of a tighter correlation of the stellarlocus and a clear break indicating the magnitudeat which the stars saturate (mag auto ∼ 19).Moreover, with this method, a surface brightnesscut at the faint end of the stellar sample is trivialto implement (at faint magnitudes, the catalog issurface brightness limited). The performance ofthis star galaxy separation scheme will be ana-lyzed in more detail in § 4.1.

Using the mu max method, we also define a setof objects that are more sharply peaked than thePSF, which is obviously non-physical. A visual

8

Table 2

Sumary of the construction of the ACS lensing catalog

Catalog of raw SExtractor detections: C1 Number Percent of C1

Total number of objects in C1 1.8 ×106

Number of Hot (faint) detections 1.6×106 88%Number of Cold (bright) detections 2.2×105 12%

Details of the cleaning process Number Percent of C1

Number of objects within the noisy border of a tile 3.2×105 17%Number of Hot detections with central pixel in Cold segmentation map 2.0×105 11%Number of objects within automatically defined star masks 2.4×104 1%Number of objects within manually defined masks 4.1×104 2%Number of objects detected more than once in adjacent tiles 6.6×104 4%

Catalog cleaned of image defects: C2 Number Percent of C2

Total number of objects in C2 1.2×106

Number of galaxies (ID = 1) 1.1×106 96 %Number of point sources (ID = 2) 2.8×104 2 %Number of fake detections (ID = 3) 1.7 ×104 2%

ACS galaxies from C2 with F814WAB < 26.5: C3 Number Percent of C3

Total number of objects in C3 7.0×105

Number of galaxies with a counterpart in the photometric catalog 6.0×105 85%Number of galaxies that have been matched but that are in ground based masks 8.3×104 12%Number of galaxies for which the redshift code did not converge 1.1×104 1.7%Total number of galaxies with accurate photometric redshifts, C3 5.0×105 71%

The final COSMOS ACS lensing catalog: C4 Number Galaxy number density

Total number of galaxies in final lensing catalog 3.9×105 66 arcmin2

Number of galaxies with accurate photometric redshifts 2.8×105 48 arcmin2

9

Fig. 4.— The SExtractor stellar index(class star) for our point source selection basedon the peak surface brightness of objects. Greypoints show the corresponding galaxy sample. Apoint source selection of the form class star >0.8 for example, will miss a certain number ofbright stars (F814W < 17) and will grossly mis-classify compact galaxies at 22 < F814W < 25.

inspection finds that these objects are mainly ar-tifacts, hot pixels, and residual cosmic rays. Weflag these spurious objects in our catalog (ID=3)and remove them for the lensing analysis.

Averaging over the COSMOS field, we find ∼ 15stars per pointing with 19 ≤ mag auto ≤ 23.This is an insufficient number to model the PSFin individual images using standard interpolationtechniques. However, it is a sufficient number toidentify the PSF pattern of each exposure givena finite set of recurring patterns (see § 5.1 andRhodes et al. (2007) for further details).

3.7. Photometric Redshifts

In addition to the ACS/WFC (F814W ) imag-ing, the COSMOS field has been imaged with Sub-aru Suprime-Cam (Bj ,Vj ,g+,r+,i+, z+,NB816),the Canada-French Hawaii Telescope (CFHT)(u∗, i∗) and the KPNO/CTIO (Ks). Details ofthe ground-based observations and the data re-duction are presented in Capak et al. (2007) andTaniguchi et al. (2007). Other observations weretaken in the UV with GALEX, in the X-ray with

Fig. 5.— Classification of point sources, galax-ies and artifacts within the mu max/ mag autoplane. Point sources follow the PSF and are de-limited by the solid region. Objects that are moresharply peaked than the PSF are contained in thedashed region and are considered to be artifacts.For clarity, only a 2% random selection of all ob-jects are included in this plot.

XMM-Newton and in the radio with VLA, CSOand IRAM. Yet more observations are underwayincluding intermediate and narrow-band imagingwith Subaru Suprime-Cam, deep Infrared imag-ing covering 1.0-2.2 microns (WIRCam/CFHT,WFCAM/UKIRT and ULBcam/UH2.2), and ob-servations with space-based facilities includingChandra and Spitzer Space Telescopes. This ex-tensive multi-wavelength data-set is a key com-ponent to COSMOS weak lensing measurementsbecause it allows us to accurately measure theCOSMOS redshift distribution, to separate fore-ground and background structures, and to removecontamination from intrinsic galaxy alignments(Heymans & Heavens 2003) and shear-ellipticitycorrelations (King 2005).

Photometric redshifts were determined bythe COSMOS photometric redshift code with aBayesian prior based on luminosity functions andallowing for internal extinction (Mobasher et al.2007). For each galaxy, the entire probabilitydistribution P (z), the most likely redshift anda confidence level for that redshift is calculated.The knowledge of the full P (z) allows us to apply

10

weight to galaxies in weak lensing measurementsaccording to the uncertainty in their measuredredshift. The accuracy of the photometric red-shifts are estimated based on extensive simula-tions and by comparison with a sample of 958galaxies with spectroscopic redshifts measuredby the ESO/VIMOS instrument as part as thezCOSMOS program (Lilly et al. 2007). Both theluminosity prior and the extinction correctionshave been shown to improve the accuracy of thephotometric redshifts when compared to the spec-troscopic sample. The accuracy of the currentphotometric redshifts down to F814W ∼ 22.5 is:

σ∆z/(1 + zs) = 0.031 (1)

with η = 1.0% of catastrophic errors, defined as∆z/(1+zs) > 0.15. This relation scales with mag-nitude in a similar fashion to Wolf et al. (2004).The accuracy of the photometric redshifts will con-tinue to improve as more data become available(in particular the deeper u∗, J , K, Spitzer andSubaru narrow band data).

The COSMOS optical and near infrared catalog(Capak et al. 2007) provides multi-band photom-etry for 89% of the COSMOS ACS/WFC galaxies.As demonstrated by Figure 6, the remaining 11%of the galaxies for which we lack multi-wavelengthinformation (and therefore photometric redshifts),are the small galaxies that cannot be detected bythe ground based imaging. In effect, although theSUBARU data is deeper for sources 1′′ in diam-eter, the ACS/WFC imaging will do a better jobat detecting anything smaller. Because we applya size cut to the the final lensing catalog how-ever (see § 6), many of these small galaxies will bediscarded from the final analysis. In total, afterremoving the galaxies with unreliable multi-bandphotometry (because they are masked out in theground-based data) as well as those for which thephotometric redshift code failed to converge, 76%of the galaxies in the COSMOS ACS/WFC lens-ing catalog (F814W < 26) have photometric red-shifts (See Figure 6). For further discussions onthe photometric redshifts and the COSMOS red-shift distribution, see § 4.3.

3.8. Noise Properties

The drizzling process introduces pattern-dependentcorrelations between neighboring pixels and can

20 22 24 26 28MAG_AUTO

103

104

105

106

Num

ber

coun

ts in

N d

egre

e−2 0

.5 m

ag−

1

ACS catalog C2

Match with ground based catalogReliable photoz

0 50 100 150 200ISOAREA_IMAGE (0.03" pixels)

103

104

105

Num

ber

ACS catalog C2

Match with ground basedcatalog

Fig. 6.— In the upper panel, the number countsof galaxies that have been correctly matched tothe ground based catalog (dotted line) are com-pared to the total number counts (solid line). Thedashed line indicates galaxies for which we con-sider the photometric redshift to be reliable. Thedifference between these two curves is primarilydue to larger masked areas in the ground-baseddata than in the ACS imaging. In the lowerpanel, the sizes of galaxies within the ACS catalog(solid line) are compared to the sizes of those thathave been matched with the ground based catalog(dashed line). The objects for which we do nothave multi-band photometry are small galaxiesthat are detected with ACS but not with groundbased imaging (seeing ∼ 1′′). The vertical dashedline shows the approximate size cut that we makein the final lensing catalog.

artificially reduce the noise levels in co-added im-ages. Noise and error estimates derived from driz-zled images will thus tend to underestimate thetrue noise levels of the image. One should in the-ory take into account the exact covariance matrixof the noise in order to derive error estimates for

11

drizzled images. For our purposes, however, asimple scaling of the noise level is each pixel bythe same constant factor is sufficient. The scal-ing factor that we adopt, FA, has been derivedfor MultiDrizzled images by Casertano et al.(2000). In principle FA is size dependent butconverges rapidly with increasing size toward anasymptotic value given by:

√

FA =

(s/p)(1 − s/(3p)) (s < p)(1 − p/(3s)) (p < s)

(2)

where p and s are respectively the pixfrac and thescale configuration parameters of MultiDrizzle.For the purpose of this paper, we assume FA tobe constant and equal to FA ∼ 0.316 (p = 0.8 ands = 0.6). By assuming a constant corrective factorregardless of size, we make less than a 10% erroron the noise estimation of the smallest objects de-tected.

As implemented by SExtractor, the formulasfor the flux and magnitude uncertainties (for bothauto and the iso quantities) are given by:

flux err =√

Aσ2 + F/g (3)

mag err =2.5

ln 10

flux err

F(4)

where A is the area (in pixels) over which the fluxF (in ADU) is summed and g is the detector gain.To correct the magnitude and flux errors reportedby SExtractor for the correlated noise, we replaceσ, the standard deviation of the noise (in ADU)estimated by SExtractor, by σ/

√FA within the

above equations. The significance of a COSMOSdetection, after this correction is applied, is de-fined as s/n = flux auto/fluxerr auto (seeFigure 7).

4. Quality Assessment of the ACS Catalog

4.1. Galaxy and Stellar Counts

The same mu max parameter used to classifystars and galaxies can also be used as an indica-tion of the background level and the depth of thedata. For each image i we calculate the mode mi

of the mu max parameter. We then divide the mi

into two bins according to the angle of the tele-scope with the Sun at the moment of the pointing.

Fig. 7.— The significance of COSMOS de-tections defined as flux auto/fluxerr autowhere fluxerr auto has been corrected for cor-related noise.

The histogram of mi (Figure 8) for these two binsreveals that the depth of the data is bimodal de-pending on whether the angle of the sun is less orgreater than a critical angle of 70 deg. This is vi-sually evident when we inspect the density map ofthe very faint objects (Figure 8). 96 pointings outof 575 have a Sun angle less than the critical valuemaking them slightly shallower than the average.

The number counts serve as a check of the ap-proximate photometric calibration and the depthof the data. Figure 9 shows the counts for galaxiesand stars compared to the reported HDF F814Wcounts (Williams et al. 1996). The magnitudes aregiven in the AB-system. Stars have been sub-tracted from the galaxy counts up to F814W =25. At fainter magnitudes their contribution isnegligible. We plot raw number counts only, i.e.we do not correct for incompleteness at the faintend. To facilitate comparisons with other surveys,we fit the galaxy counts between F814W = 20and F814W = 26, to an exponential of the formN = B × 10A×mag where N has units of numberdegree−2 0.5 mag−1. For the deeper set of images(sun angle > 70 degrees), we find A = 0.332 andlog10(B) = −3.543. The raw galaxy and stellarnumber counts are provided in Table 3.

We also fit the stellar counts to models as shownin Figure 9. The star count predictions have beendone using the Besancon model of the Galaxy

12

22.6 22.8 23.0 23.2 23.4Mode of the mu_max parameter

0

50

100

150

200

250

300

Sun angle > 70 deg

Sun angle < 70 deg

Fig. 8.— The top panel shows the histogram of mi

(mode of the mu max parameter for each image)demonstrating that images with a sun angle lessthan 70 degrees are not quite as deep as imageswith a sun angle greater than 70 degrees. Thebottom panel shows the density of faint objects(27 < F814W < 28) within the COSMOS field.The red circles indicate the pointings for whichthe angle with the sun is less than 70 degrees.

(Robin et al. 2003, 2004) and are described in de-tail in (Robin et al. 2007). By extrapolating thestellar number counts, we estimate that the galaxycatalog has less than a 3% contamination fromstars at magnitudes greater than 25. The fit tostellar models is excellent between F814W = 20

and F814W = 25, and at magnitudes less than19, we visually inspect the catalog to check thatthe star selection is correct to within 0.5%. Thissmall error arises mainly from false detections bySExtractor of the diffraction spikes of bright, sat-urated stars.

20 22 24 26 28MAG_AUTO

102

103

104

105

106

Num

ber

coun

ts in

N d

egre

e−2 0

.5 m

ag−

1

Stellar counts

HDF Total mag

Sun angle > 70 degSun angle < 70 deg

20 21 22 23 24 25 26MAG_AUTO

100

1000

N d

egre

e−2 0

.2 m

ag−

1

Stellar models− Stellar counts

Fig. 9.— The top panel shows the galaxy and stel-lar number counts as compared to the HDF. Thedashed curve corresponds to images with a sun-angle of less than 70 degrees and the solid curvecorresponds to images with a sun-angle greaterthan 70 degrees. Poisson error bars are also in-dicated but are very small. The bottom panelshows the point source selection for the catalogcompared to stellar models computed from evolu-tionary tracks and constrained by local Hipparcosdata.

4.2. Completeness

The probability that a galaxy enters our cat-alog will depend on its size and surface bright-ness profile. To quantify the completeness anddetection limits of our SExtractor configuration,

13

Table 3

Cosmos F814W galaxy and stellar number counts

F814W Galaxy density Stellar density

log10(n) deg−2 0.5 mag−1 log10(n) deg−2 0.5 mag−1

20.25 3.138 ± 1.503 2.807 ± 1.29720.75 3.323 ± 1.600 2.865 ± 1.32921.25 3.514 ± 1.691 2.930 ± 1.36121.75 3.686 ± 1.777 2.936 ± 1.36522.25 3.853 ± 1.860 2.981 ± 1.38522.75 4.022 ± 1.945 3.003 ± 1.40123.25 4.180 ± 2.023 3.043 ± 1.41423.75 4.352 ± 2.110 3.081 ± 1.43724.25 4.523 ± 2.196 3.143 ± 1.46324.75 4.682 ± 2.275 3.206 ± 1.49325.25 4.814 ± 2.340 · · ·

25.75 4.956 ± 2.412 · · ·

.

Note.—Galaxy counts are derived for the 479 images with a sunanglegreater than 70 degrees. Magnitudes are the SExtractor mag auto

we insert fake objects with a Gaussian profile ofvarying FWHM and total magnitude into emptyregions of an ACS image and test how well theseobjects can be recovered with our pipeline. Eachartificial source was considered to be correctly de-tected if its centroid was within 10 pixels and themag auto parameter was within 0.5 mag of theinput value. From this analysis, we determineour completeness as a function of magnitude andFWHM and the results are shown in Figure 10.The completeness is about 90% for objects witha FWHM of 0.2′′ at F814W=26.6. These valuesshould only be used as a rough estimate however,as we do not actually model galaxies, but use asimple Gaussian profile for artificial objects.

4.3. The COSMOS Redshift Distribution

The estimation of the redshifts of galaxies isthe major astrophysical uncertainty inherent toweak lensing methods. To first approximation,cosmic shear and tomography are mainly sensi-tive to the median redshift of the sources whilegalaxy-galaxy lensing benefits greatly from theknowledge of precise spectroscopic redshifts forthe foreground lenses (Kleinheinrich et al. 2005).With the depth and area coverage of COSMOS,we surpass the current capability for completespectroscopic follow-up. For most forthcomingweak lensing surveys, this will also be the case,hence the importance of the photometric red-shift technique to measure redshifts for a major-

ity of the galaxies, to and beyond today’s spec-troscopic limits (Ilbert et al. 2006). COSMOSpresents a unique advantage, in terms of cur-rent weak lensing surveys, of a prodigious multi-wavelength follow-up combined with the plannedmeasurement of ∼ 50000 spectroscopic redshiftsby the ongoing zCOSMOS program (Lilly et al.2007). Upon completion, this data set will pro-vide the COSMOS lensing catalog with accuratephotometric redshifts and will be vital for refin-ing and improving the photometric technique inpreparation for forthcoming weak lensing surveys.We present here a first analysis of the COSMOSredshift distribution. A more detailed study of thesystematic trends in the photometric redshifts andof their effects on the redshift distribution is be-yond the scope of this paper and will be addressedelsewhere when more data becomes available. Forthe purposes of this paper, we adopt the mag-nitude dependent parametrization of the redshiftdistribution, common to many other weak lensingstudies as given by Baugh & Efstathiou (1993),

n(z, mag) ∝ z2 exp[− zzo(mag)

1.5]

z0(mag) = zm(mag)1.412

(5)

where zm is the median redshift of the survey asa function of magnitude. We calculate zm for theCOSMOS ACS data by bins of ∆F814W = 0.25for 20 < F814W < 24 and derive the best linearfit to zm, given by:

14

Fig. 10.— Completeness of the COSMOS F814Wcatalog as a function of total magnitude andFWHM determined by inserting fake objects intoan ACS image. The thick contours show the per-centage of fake objects recovered by SExtractor.The thin contours are the lines of constant sur-face brightness, in units of mag arcsec−2, assum-ing a Gaussian profile. The grey points representa random sample of objects from the COSMOScatalog plotted as a function of mag auto andfwhm image. The dashed horizontal line indi-cates the size of the ACS PSF. Note that the sim-ulations only consider objects with Gaussian pro-files whereas in reality, the COSMOS objects ex-hibit a wide variety of profiles.

zm = (0.18 ± 0.01)× F814W − (3.3 ± 0.2) (6)

The majority of the galaxies for which we haveno redshift estimate at F814W < 24 are thosein masked regions and their exclusion from thisderivation do not affect this results. At F814W <24, our redshift incompleteness is less than 4% andthe dominant source of error is the photometricredshift uncertainty expressed previously in Equa-tion 1. We note that a significant number of galax-ies fainter then this limit do have photometric red-shifts. The fact that these galaxies represent a sta-tistically incomplete sample, does not matter forsome applications. For example, these additionalgalaxies are used in the cosmic shear measurementby Massey et al. (2007).

In Figure 11, we compare the median redshift

20 21 22 23 24 25 26F814WAB

0.5

1.0

1.5

2.0

diffe

rent

ial m

edia

n re

dshi

ft

COSMOS

CFHTLS

UDF

HDF

ssa22

Fig. 11.— Median redshift of COSMOS comparedto various photometric redshift surveys. For clar-ity, error bars are only shown for the COSMOS,CFHTLS and UDF surveys

of COSMOS to the UDF survey (Coe et al. 2006),the CFHTLS survey (Ilbert et al. 2006), the H-HDF-N survey (Capak et al. 2004) and SSA22(Hu et al. 2004, Capak et al. 2004). Table 4is a summary of the data and the methods em-ployed by these different photometric redshift sur-veys. All photometric redshifts have been com-puted with a Bayesian prior based on luminosityfunctions. The agreement that we see between thevarious surveys at z < 1 is quite remarkable. Atz > 1 however, the scatter in Figure 11 indicatesthe limits of current photometric techniques. Fur-ther simulations are clearly necessary in order tounderstand the biases introduced in the redshiftdistribution at z > 1. Although we reserve a fulldiscussion for a future paper, we can already high-light some of the issues at hand.

The photometric redshift technique relies on de-tecting and measuring the strength of broad spec-tral features. These same features are used bycolor selection techniques to select objects at spe-cific redshifts. The key features are the 4000Abreak, the Lyman break at 912A, Lyman absorp-tion at 1216A, and coronal line absorption be-tween 1500-2500A (see Adelburger et. al. 2005).Photometric redshifts are very robust if one ormore of these features are detectable in the avail-able data. However, at faint magnitudes the pho-

15

Table 4

Present date photometric redshift surveys

Survey Area Imaging data Calibration spectra Technique

COSMOS 1.67 deg2 Bj , Vj , g+, r+, i+, z+, NB816, u∗, i∗, Ks 958 COSMOS/BPZ

UDF 11.97 arcmin2 B, V, i′, z′, Ja Hb 76 BPZCFHTLS 3.2 deg2 u∗, g′, r′, i′, z′, J, K 2867 Le Phare

H-HDF-N 0.2 deg2 Uj , Bj , Vj , Rc, Ic, Z+, HK′ 2149 BPZ

SSA22 0.2 deg2 U∗

j , Bj , Vj , Rc, Ic, z+, J, H, K, HK′ 452 BPZ

a5.76 arcsec2

b160 arcmin2

tometric errors and detection limits are often toolarge to constrain these features.

For example, problems arise for COSMOS be-tween 1.5 < z < 3.2 where the measurable fea-tures, the 4000A break, and the coronal line ab-sorption features, are difficult to detect. Indeed,at these redshifts, the 4000A break is well into theIR where it is difficult to obtain deep data. Atypical object at z ≃ 2 will be ∼ 1.4 magnitudesfainter at I than K. This means objects fainterthan F814W > 23.5 are not constrained by thepresent K-band data. At the other end of thespectrum the coronal absorption feature has a typ-ical strength of ∼ 0.15 magnitudes, which requiresa 25σ detection in u∗ to accurately differentiatefrom a similar break in z < 0.5 galaxies. Withthe present u∗ data, this corresponds to objectsbrighter than F814W < 24.5 for typical galaxies.

In conclusion, the high redshift tail of the COS-MOS redshift distribution will be more accuratelydetermined with the forthcoming NIR data andthe future deep zCOSMOS spectroscopy whichspecifically targets the 1.5 < z < 3 region and willallow proper calibration down to F814W ∼ 24. Amore detailed analysis of the photometric redshiftswill be conducted once the new data is available.

5. PSF Correction and Shear Measure-

ment

In this section, we measure the shapes of galax-ies and correct them for the convolution with thetelescope’s PSF and for other instrumental effects.For each galaxy, we construct an unbiased localestimator of the shear and derive the associatedmeasurement error.

5.1. PSF Modelling

The ACS/WFC PSF is not as stable as onemight naively hope from a space-based camera.As shown in Rhodes et al. (2007), gradual changesto both the size and the ellipticity pattern of thePSF due to telescope “breathing”, causes the PSFto change considerably on timescales of weeks.The long period of time over which the COSMOSfield was observed forces us to take account ofthese variations (see Figure 1). Although otherstrategies have been demonstrated successfully forobservations conducted on a shorter time span,it would be inappropriate for us to assume, likeLombardi et al. (2005) or Jee et al. (2005), thatthe PSF is constant or even, like Heymans et al.(2005), that the focus is piecewise constant.

Fortunately, most of the PSF variation can beascribed to a single physical parameter. Thermalexpansions and contractions of HST alter the dis-tance between the primary and secondary mirrors.As the “effective focus” deviates from nominal, thePSF becomes larger and more elliptical, with thedirection of elongation depending upon the po-sition above or below nominal focus (c.f. Krist2005). The thermal load on HST is constantlychanging, in a complicated way, as it passes inand out of the shadow of the Earth and is rotatedto different pointings.

As described in Rhodes et al. (2007), we havemodified version 6.3 of the TinyTim ray-tracingprogram (Krist and Hook 2004) to create a gridof model PSF images, at varying focus offsets. Bycomparing the ellipticity of ∼ 20 stars in each im-age to these, we can determine the image’s effec-tive focus. Tests of this algorithm on ACS/WFCimages of dense stellar fields confirm that the best-

16

Fig. 12.— Adopted PSF model across the sur-vey. The colors correspond to deviations in theapparent focus of the telescope away from nom-inal (µm). These are caused by expansion andcontraction of the HST due to thermal variations.Note that the focus values are clustered and notrandomly distributed. See Rhodes et al. 2007 formore details about the PSF pattern at particularfocus positions.

fit effective focus can be repeatably determinedfrom a random sample of ten stars brighter thanF814W = 23 with an rms error less than 1µm.The effective focus of the COSMOS images areshown in Figure 12. An alternative correctionscheme based on PSF models constructed fromdense stellar fields has also been suggested bySchrabback et al. (2006).

Once images have been grouped by their effec-tive focus position, we can combine the few starsin each image into one large catalog. We interpo-late the PSF model parameters using a polynomialfit (of order 3× 2× 2 in each CCD separately), inthe usual weak lensing fashion (c.f. Massey et al.2002). See Rhodes et al. (2007) for more detailsconcerning the PSF modelling scheme.

5.2. Galaxy Shape Measurement

We use the shape measurement method de-veloped for space-based imaging by Rhodes et al.(2000, hereafter RRG). The RRG method hasbeen optimized for space-based images withsmall PSFs and has previously been used onweak lensing analyses of WFPC2 and STISdata (Rhodes et al. 2001; Refregier et al. 2002;Rhodes et al. 2004). In a manner similar tothe common “KSB” method (Kaiser et al. 1995),RRG measures the second and fourth orderGaussian-weighted moments of each galaxy:

Iij =

∑

wIxixj∑

wI, (7)

Iijkl =

∑

wIxixjxkxl∑

wI. (8)

The sum is over all pixels, w is the size of theGaussian weight function, I is the pixel inten-sity, and the xi coordinates are measured in pixels.The Gaussian weight function is necessary to sup-press divergent sky noise contributions in the mea-surement of the quadripole moments. The RRGmethod is well-suited to the small, diffraction-limited PSF obtained from space, because it de-creases the noise on the shear estimators by cor-recting each moment for the PSF linearly, and onlydividing them to form an ellipticity at the last pos-sible moment.

After the moments have been corrected for thePSF, an ellipticity ε = (e1, e2) and size measure,d, are calculated for each galaxy:

e1 =Ixx − Iyy

Ixx + Iyy, (9)

e2 =2Ixy

Ixx + Iyy, (10)

d =

√

(Ixx + Iyy)

2. (11)

Note that the d parameter is a measure ofgalaxy size but that its value will depend on thechoice of the width of the Gaussian weight func-tion, w.

17

5.3. Shear Measurement

The estimator ε = (e1, e2) is not yet a shearestimator, because it does not respond linearly tochanges in shear. It must first be normalized bya shear susceptibility factor (also known as the“shear polarizability”),

γ = γ1, γ2 =ε

G, (12)

where the shear susceptibility factor, G, is mea-sured from moments of the global distribution ofε and other, higher order shape parameters (seeequation 28 in Rhodes et al. 2000). The RRG for-malism does not allow for G to be calculated forany individual galaxy. However, G can be cal-culated for an ensemble of galaxies by averagingover a population’s shape moments. Previous in-carnations of RRG that have been used to measurecosmic shear (Rhodes et al. 2001; Refregier et al.2002; Rhodes et al. 2004), have made use of a sin-gle value of G for the entire survey. Adoptingthis approach for the COSMOS data would yielda value of G = 1.13. However, the Shear TEst-ing Program (Massey et al. 2007) showed that Gcan vary significantly as a function of object flux,whether this be due to evolution in galaxy mor-phologies as a function of redshift, or noise in thewings of faint galaxies that simply impedes themeasurement of their radial profiles and higher or-der moments. An increase in shear susceptibilitywith object S/N has also been seen in KSB-typeanalyses (Massey et al. 2004a).

Our tests have confirmed that a constant valuewould be insufficiently precise for a survey thesize of COSMOS, and would particularly affect thekind of 3D analysis for which COSMOS is so well-suited. We therefore calculate G from the COS-MOS data in bins of S/N (see Figure 13). Sim-ulated images of the COSMOS data are createdusing the shapelets-based method of Massey et al.(2004b) (see §5.4). The variations of G as a func-tion of S/N are apparent in the COSMOS dataare well reproduced by the simulated data. ForCOSMOS galaxies, we find that variations in G asa function of S/N are well-fit by

G = 1.125 + 0.04 arctan

(

S/N − 17

4

)

. (13)

Adopting the above model, we derive G for each

galaxy as a function of S/N .

Fig. 13.— Interpolation of the shear susceptibilityfactor G. The solid circles show G, calculated inbins of S/N , for the COSMOS data. The opencircles show the same for the simulated COSMOSimages. The solid line shows the shear suscepti-bility model adopted for the data.

5.4. Calibration via Simulated Images

Using the shapelets-based method of Massey et al.(2004b), we have created simulated images withthe same depth, noise properties, PSF, and galaxymorphology distribution as the real COSMOSdata. A known shear signal was applied to theimages, which we have then attempted to mea-sure using the same pipeline as the data. Thisexercise is similar to the Shear TEsting Program(STEP; Heymans et al. 2006; Massey et al. 2006)but tailored exclusively to COSMOS.

The simulated COSMOS images are each 4′×4′,and contain ∼ 500 galaxies after applying thesame catalog cuts that were applied to the realdata (see §6). Simulated galaxy morphologiesare based on those observed in the Hubble DeepFields (Williams et al. 1996, 1998), parametrizedas shapelets and randomly rotated/flipped beforebeing sheared. Different input galaxies were usedin each simulated image, as if they were pointing todifferent patches of the sky, to keep them indepen-dent. To simplify later analysis, all of the galaxieswithin an image were sheared by the same amount.A total of 41 images were made, with shears ap-plied in integer steps from −10% to +10% in theγ1 component (while γ2 was fixed at zero) andsimilarly for the γ2 component. The images werethen convolved with a model ACS PSF. Again tosimplify the analysis, this was a constant PSF ob-tained from TinyTim. Its (e1, e2) ellipticity is(−0.21%,−2.07%) ± (0.14, 0.10). No stars wereincluded in the simulated images; a separate starfield was created, from which the PSF moments

18

could be measured. Noise was added to all of theseimages, to the same depth as the COSMOS ob-servations, and with a similar (but isotropic) cor-relation between adjacent pixels to mimic the ef-fects of MultiDrizzle and unresolved backgroundsources.

Fig. 14.— Calibration of the RRG shear mea-surement method from simulated COSMOS im-ages containing a known input shear. The mea-sured shear on the y-axis includes the shear cali-bration factor C. Squares show measurements ofγ1, diamonds show measurements of γ2. The solidline is a linear fit to deviations from the ideal caseof γmeasured = γinput for all points. The dashedline is a quadratic fit demonstrating that the cur-vature terms are negligible.

The recovered shear measurement from thesimulated data is presented in Figure 14. Wefind that, in order to correctly measure the in-put shear on COSMOS-like images, the RRGmethod requires an overall calibration factor ofC = (0.86+0.07

−0.05)−1, so that

γ = C × ε

G. (14)

The necessity for such calibration factors haslong been known in the field (e.g. Bacon et al.2001; Erben et al. 2001), and is in accord with re-sults from STEP1. STEP2 and Highet al. (2007)suggest that this may be intrinsic with KSB-related methods and, furthermore, that the cali-

Linear STEP fit〈c〉 ( 2.1 ± 40.0)× 10−4

c1 ( 5.6 ± 28.5)× 10−4

c2 (-1.3 ± 28.0)× 10−4

〈m〉 (-0.3 ± 9.4) × 10−2

m1 (-12.2 ± 6.5) × 10−2

m2 ( 11.6 ± 6.3) × 10−2

Quadratic STEP fit〈c〉 (-23.3 ± 121.8)× 10−4

〈m〉 ( 20.1 ± 51.5)× 10−2

〈q〉 -2.84 ± 4.59

Table 5: — Calibration of the RRG shear mea-surement method on simulated COSMOS imagescontaining a known shear, described using STEPparameters. Figures are supplied after the appli-cation of the shear calibration factor.

bration can vary for the two components of shear.For this reason, we fit each component separately,and use final calibration factors of C1 = (0.80)−1

for γ1 and C2 = (0.92)−1 for γ2. After this re-calibration, Table 5 shows STEP-like estimates ofthe additive bias 〈c〉 and the multiplicative bias〈m〉 obtained by fitting deviations of the recov-ered shear from the input shear. Both of theseare consistent with ideal shear recovery, althoughthe error on these estimates will be propagatedthrough subsequent analyses.

For the kind of 3D shear analysis for whichCOSMOS is so well-suited, the simultaneous cal-ibration of shears from an entire population ofgalaxies is insufficient. Since our companion pa-per, Massey et al. (2007), is concerned with thegrowth of the shear signal as a function of red-shift, it is crucial that the shear calibration beequally precise for both distant and relativelynearby galaxies. Given that more distant galax-ies are fainter and smaller and that the details ofthe shear measurement depend upon a fixed PSFsize, pixel size, and noise level, this requirementis not trivial. We have therefore split the sim-ulated galaxy catalog in half by magnitude (atF814W = 25.4) and by size (at d = 5.0 pix-els), and repeated the analysis. We find that ourshear calibration 〈m〉 is robust for galaxies of dif-

19

ferent fluxes within 1% and of different sizes within4%. Redshifts were not available for the simulatedgalaxies, so a direct split in redshift was not pos-sible. Although this effect is clearly small in theregime of our current measurements, it will be sig-nificant in future weak lensing surveys where theerror budget will be dominated by systematic un-certainties.

5.5. Error on the Shear Estimator

For each galaxy, the error on the measuredshear is estimated using the same method as im-plemented in the Photo pipeline (Lupton et al.2001) to analyze data from the Sloan Digital SkySurvey. For each object, we assume that the opti-mal moments are the same as the moments corre-sponding to a best fit Gaussian. This formulationallows us to determine the covariance matrix ofthe moments in the usual way of non-linear leastsquares. To be precise, we model the image witha 2D elliptical Gaussian model

M(x) =f

2π|Q|1/2×

exp[−1

2(x − µ)T Q−1(x − µ)]. (15)

This model has six parameters: the flux, f , thetwo centroids, µ, and the three moments whichform the elements of the symmetric matrix Q.These parameters are noted pl. Next, we derivethe χ2 in the usual way

χ2 ≡ 1

σ2

∑

ij

[M(xij) − Iij ]2, (16)

where σ is the sky noise level. We can now com-pute the 6 × 6 Fisher matrix which is the matrixof the second derivatives of the χ2

Fkl ≡ 1

2

∂2χ2

∂pk∂pl

=1

σ2

∑

ij

∂M(xij)

∂pk

∂M(xij)

∂pl

− 1

σ2

∑

ij

[M(xij) − Iij ]∂2M(xij)

∂pk∂pl.

As is customary, we drop the second term which isproportional to the residuals. This term is usuallyvery small compared to the first. The covariance

matrix of the Gaussian parameters are then theinverse of the Fisher matrix. The 3 × 3 block ofthe covariance matrix corresponding to the sec-ond moments can then be extracted. Because thewhole 6 × 6 matrix was inverted, this correctlymarginalizes over centroid errors and other modelparameter degeneracies. In this way, it differs fromformulas which assume a constant (non-adaptive)weighting function and perfect centroiding.

This Fisher matrix does not depend explic-itly on the data; it only depends on the bestfit parameters and can be computed analytically.It is simply a function of four numbers: thethree second order moments defined in Equation 7and the signal-to-noise ratio (f/σ), which can beparametrized by the magnitude error (for exam-ple, magerr auto). Since the flux computedwith SExtractor is not exactly equal to f (whichwould be the best fit Gaussian amplitude) we al-low for a single calibration factor and multiply thecovariance matrix by this. We calibrate this factorwith image simulations and verify that the errorsare correctly predicted.

Since the ellipticity components are computedfrom the moments, the variances of the ellipticitycomponents can be computed by linearly propa-gating the covariance matrix of the moments. Fi-nally, the two ellipticity components can be shownto be uncorrelated with each other.

6. Final Galaxy Selection

6.1. Lensing Cuts

Estimations of the gravitational shear will beimproved by averaging only those galaxies withprecise shape measurements on the condition thatno ellipticity selection bias is introduced by the“lensing cuts”. We apply strict cuts to the C2 cat-alog that are designed to extract a sample of re-solved galaxies with reliable shape measurements.The resulting catalog is referred to as C4. Our“lensing cuts” are summarized in Table 6 and arebased on the four following parameters:

1. The estimated significance of each galaxy de-tection, where the significance is defined ass/n = flux auto/fluxerr auto,

2. The first order moments, Ixx and Iyy,

3. The total ellipticity, e =√

e21 + e2

2,

20

4. The galaxy size as defined by the RRG dparameter (see §5.2).

The final size cut is designed to select galaxieswith well resolved shapes. Indeed, PSF correc-tions become increasingly significant as the size ofa galaxy approaches that of the PSF and the in-trinsic shape of a galaxy becomes more difficult tomeasure. In COSMOS images, the typical size (asdefined by d) of a star is about d⋆ = 2.2 pixels(0.066′′). Our size cut is thus equivalent to select-ing galaxies with dg > 1.6 × d⋆. Note that in thissection, as well as in all following sections, d hasnot been corrected for the PSF.

The ellipticity cut at e < 2 may be surprisinggiven that, by definition, ellipticities are restrictedto e ≤ 1. In reality, however, because of noise,it is possible to measure an ellipticity of e > 1.Selecting galaxies with e < 1 could introduce anunwanted ellipticity bias, but, because we are onlyinterested in ensemble averages, an acceptable so-lution is to cut out only a small number of largeoutliers (e > 2).

Note that further cuts may be required forsome applications (for example, to isolate onlythose objects with well-measured photometric red-shifts). In addition to these cuts, a galaxy-by-galaxy weighting scheme may also be used to min-imize the impact of shape measurement noise.

6.2. Effective Galaxy Number Density

Once the “lensing cuts” have been applied,the C4 catalog contains only those galaxies whichare useful for lensing analyses with the COSMOSdata. Predictions for future weak lensing surveysbased on COSMOS results must consider the “ef-fective number” of galaxies that are actually usefulfor lensing purposes (and not the number of rawdetections). With these considerations in mind,we define the “effective galaxy number density”,Ng(z), as the total number of galaxies within C4,per unit area, and with redshifts below a given red-shift, z. The equivalent quantities as a function ofgalaxy magnitude and size are Ng(m) and Ng(d).For each galaxy, the SExtractor mag auto pa-rameter is used to estimate the magnitude and theRRG d parameter is used as an estimate of thesize. We also consider the derivatives of Ng(z),Ng(m), and Ng(d) so that:

Ng(z) =

∫ z

0

ng(z′)dz′, (17)

Ng(m) =

∫ m

20

ng(m′)dm′, (18)

Ng(d) =

∫ ∞

d

ng(d′)dd′. (19)

The total number density of galaxies in C4 isnoted as Ng and is significantly lower than thenumber density of detected galaxies. In total, thefinal lensing catalog C4 contains 3.9×105 galaxieswith accurate shape measurements and 2.8×105

galaxies with both shape and photometric red-shift measurements. Table 2 shows a summaryof the different steps leading to this catalog. Thesurveyed area of COSMOS is 1.64 deg2 leadingto an overall number density of Ng ∼66 galaxiesper arcmin2 for the first sample and ∼50 for thesecond. These numbers can be contrasted to amore sparse 15-25 galaxies per arcmin2 typicallyresolved with deep, ground-based surveys. In Fig-ures 15 and 16 we show the effective densities de-fined above as well as their corresponding deriva-tives. From these figures, we can draw the follow-ing conclusions:

• Over 60% of the COSMOS source galaxiesare at redshifts higher than z = 1. TheCOSMOS weak lensing data is therefore apowerful probe of the dark matter distribu-tion from z ∼ 1 to the present day.

• About 18 galaxies per arcmin−2 (73%) arediscarded when we select only those galax-ies with accurate photometric redshifts. Theprimary cause of this loss are the larger ar-eas masked out in the ground-based data ascompared to the ACS data. Future space-based weak lensing surveys could recover theremaining 27% by using space-based, multi-wavelength imaging to derive photometricredshifts.

• The effective number density rises verysteeply with decreasing galaxy size; over50% of our total number of sources haved < 5 pixels (0.15′′). The current size cutfor the COSMOS data is d = 3.6 pixels andthe size of the ACS PSF is d⋆ = 2.2 pix-els (0.066′′). By pushing this size barrier

21

Table 6

Lensing cuts applied to C2

Parameter Galaxies retained in C4

Ixx and Iyy Finitea Ixx and Iyy

RRG size parameterb d d > 3.6 pixelsSignificance S/N > 4.5Total ellipticityc e e < 2

aIndicating that the RRG code converged

bUncorrected for the PSF

cCorrected for the PSF

to even smaller values, future surveys couldvery quickly obtain much higher effectivedensities.

Finally, the cuts that we have applied to theCOSMOS data (in particular the size cut) arestricter than would be necessary without signifi-cant CTE effects that degrade the shape measure-ments of the faintest galaxies. Next generationspace-based missions designed to avoid the prob-lems encountered and identified in COSMOS, aswell as future implementations of the COSMOScatalog, will undoubtedly achieve higher effectivenumber densities.

7. Intrinsic Shape Noise: a Fundamental

Limit to Weak Lensing Measurements

Under the assumption of weak gravitationallensing, a source galaxy with intrinsic shape εint

and observed ellipticity εobs is related to the grav-itational lensing induced shear γ according to:

εobs = εint + γ. (20)

Throughout this paper, the gravitational shearis noted as γ whereas γ represents our estimatorof γ. The above relationship indicates that galax-ies would be ideal tracers of the distortions causedby gravitational lensing if the intrinsic shape εint

of each source galaxy was known a priory. Aquick glance at an ACS image however, revealsthat galaxies display a very wide variety of shapeswhich unfortunately prevents the extraction ofγ for any single galaxy. Lensing measurementsthus exhibit an intrinsic limitation, encoded in thewidth of the ellipticity distribution of the galaxypopulation, noted here as σint, and often referred

to as the “intrinsic shape noise”. Because theshape noise (of order σint ∼ 0.26) is significantlylarger than weak shear (typically γ ∼ 0.01 for cos-mic shear), γ must be estimated by averaging overa large number of galaxies. In this case equation20 simplifies to:

< εobs >=< γ > . (21)

The uncertainty in the shear estimator, σγ ,arises from a combination of unavoidable intrinsicshape noise, σ2

int =< ε2int > and the measurement

error of galaxy shapes σ2meas:

σ2γ = σ2

int + σ2meas. (22)

In the following analysis, σγ will be referred toas the shape noise and σint will be called the in-trinsic shape noise. The former includes the shapemeasurement error, σmeas, and hence will vary ac-cording to the data-set as well as the shape mea-surement method that is employed. Note that theuncertainty contributions from, photon noise, PSFcorrection, CTE calibration, and the shape mea-surement method are all included in our definitionof σ2

meas. The weak lensing distortions averagedover the whole COSMOS field are small, and rep-resent a negligible perturbation to equation 22.

Instead of the simple arithmetic mean of equa-tion 21, many lensing practitioners adopt in someform or another, an optimized weighting scheme inorder to estimate γ which often incorporates boththe measurement error and the shape noise (fore.g., Bernstein & Jarvis 2002). A constant value(of order 0.3) is often assumed for σint . However,it would not be surprising that the same processesthat shape galaxy formation also lead to a vari-ation of σint as a function of magnitude, galaxy

22

Fig. 15.— In the upper panel we show the ef-fective number density of galaxies as a function ofF814W magnitude. Only resolved galaxies withprecise shape measurement are included in thesecounts. In the lower panel, we show the effec-tive number density of galaxies as a function ofredshift. The effective density still evolves sharplyafter z > 1 demonstrating that the COSMOS lens-ing data is a powerful probe of structures at z < 1.

type, or redshift. Furthermore, as weak lensingsurveys increase in both scale and depth, it of in-tense interest to obtain accurate estimates of theintrinsic shape noise floors that these surveys mustconfront. For these reasons, we undertake a mea-surement of the shape noise, σγ , as well as theintrinsic shape noise, σint, as a function of magni-tude, size and redshift. A more detailed analysisof the intrinsic shape noise as a function of galaxymorphology will be the subject of a future paper.

First, we estimate the shape noise σγ directlyfrom the COSMOS data as a function of size andF814W magnitude. To derive σγ we consider:

Fig. 16.— In the upper panel we show the ef-fective number density of galaxies as a functionof size. The dotted vertical line indicates thesize cut that we make in order to extract galax-ies with precise shape measurements. This sizecut is such that dg > 1.6 × d⋆. Ng rises verysharply as a function of decreasing d demonstrat-ing that small galaxies make up the majority ofour lensing sources. The lower panel shows Ng forCOSMOS galaxies with accurate photometric red-shifts and for magnitude cuts of F814W < 25 andF814W < 24.5.

• The mean variance of both shear compo-nents (including correction factors), σγ =(σγ1 + σγ2)/2

• Galaxies with well measured shapes (see § 6)

Second, we derive an empirical estimation of theshape measurement error, σmeas, using a sampleof 27000 galaxies that belong to overlapping re-gions of adjacent pointings. Each of these galax-ies provide us with two independent shape mea-

23

Fig. 17.— The observed scatter in the shear as afunction of F814W magnitude and size. The scat-ter is a combination of intrinsic shape noise, σint,and a shape measurement error, σmeas. The shapemeasurement error is determined by a theoreticalmodel and tested using galaxies from overlappingregions. Shaded regions indicate the RMS width ofthe measurement error distribution. The intrinsicshape noise appears to increase slightly as a func-tion of magnitude but is independent of size. Atfaint magnitudes and small galaxy sizes, the shearscatter increases rapidly due to large measurementerrors. The dashed vertical line indicates the sizeand magnitude cut that we apply in order to con-struct Figure 18.

surements. Using these overlaps, we find that theshape measurement error is a function of both sizeand magnitude and increases beyond σmeas = 0.1for mag auto > 24.5 and d < 7. Third, we com-pare this empirical determination of σmeas to thetheoretical one derived in § 5.5. We find that thetheoretical model of § 5.5 does remarkably well inpredicting σmeas as a function of size and magni-tude. Thus confident in the validity of this model,we adopt it for subsequent derivations. Finally,using equation 22 and the shear measurement er-ror σmeas, we extract the intrinsic shape noise ofour galaxy sample as a function of size, magnitudeand redshift. The results are shown in Figures 17and 18.

As can be seen in Figure 17, large measure-ment errors lead to an increase of σγ at small sizesand faint magnitudes. The intrinsic shape noisehowever, appears to change little with either sizeof magnitude and remains constant at a value ofσint ∼ 0.26. The slight apparent increase of theintrinsic shape noise at fainter magnitudes is prob-ably due to the simplified measurement error esti-mator that we are using (indeed, the overlaps in-dicate slightly higher errors). From this analysis,we can draw the following conclusions:

• The intrinsic shape noise varies little fromz = 0 to z = 3. Deep space based weak lens-ing surveys will therefore confront equivalentintrinsic shape noise floors as their shallowercounterparts.

• Measurement errors lead to an increase inthe shape noise as a function of size andmagnitude. A joint improvement in bothimaging quality as well as shape measure-ment methodology will lead to shape noisesthat are closer to the intrinsic floor of 0.26.

• We have yet to explore if the intrinsic shapenoise varies as a function of galaxy type. Ifso, shear measurements could be improvedby incorporating a galaxy-type discriminantinto the weighting scheme. This will be thefocus of a future paper.

8. Conclusion

We have carefully constructed a weak lens-ing catalog from 575 ACS/HST tiles, the largest

24

Fig. 18.— Intrinsic shape noise as a functionof photometric redshift. Galaxies have first beenselected to have measurement errors less than 0.1(d > 7 and mag auto < 24.5) so that the scatterin the observed shear closely matches the intrinsicshape noise. A linear fit to the shape noise as afunction of redshift reveals a flat distribution witha mean of σint = 0.26.

space-based survey to date. We have establishedthe quality of this catalog and analyzed the COS-MOS redshift distribution, showing broad agree-ment with other photometric redshift surveys outto z ∼ 1. The photometric redshifts are currentlylimited by the lack of deep K-band data butwill rapidly improve as this data soon becomesavailable. Shapes have been measured for over3.9× 105 galaxies and corrected for distortions in-duced by PSF and CTE effects. Simulations havebeen used in order to calibrate our shear measure-ment method and a STEP-like analysis has beenperformed, demonstrating our ability accuratelymeasure shear with negligible additive and multi-plicative bias. The effective number density of theCOSMOS weak lensing catalog is 66 galaxies perarcmin−2 (48 when we consider only those withaccurate photometric redshifts). A large fractionof these galaxies are at z > 1 making COSMOS apowerful probe of the dark matter distribution andits evolution from z = 1 to the present day. TheCOSMOS survey is also of foremost importancefor the preparation and design of future wide fieldspace-based lensing missions. Regarding the de-sign of such missions, our main conclusions fromworking with the COSMOS data are the following:

1. Understanding and correcting for the timevarying PSF and calibrating CTE effectswere two of the most difficult challenges en-countered with the COSMOS data. Reduc-ing these two systematic effects should bea key specification in the design of next-generation telescopes and instruments.

2. Because 1) small galaxies are not as read-ily detected from the ground and 2) becauselarger areas are masked out in the ground-based data than in the ACS imaging, we lose27% of our source sample when we selectonly those with accurate photometric red-shifts.

3. The effective number density of galaxies is avery sensitive function of survey depth andresolution. The capability to resolve and ac-curately measure the shapes of very small,faint galaxies will be key in obtaining num-ber densities of over 66 galaxies arcmin−2.

4. Finally, we have derived the intrinsic shapenoise of the galaxy sample and demonstratedthat it remains fairly constant as a func-tion of size, magnitude and redshift. Weaklensing measurements with deep space-basedimaging are therefore on par with more shal-low imaging in terms of the intrinsic shapenoise floors that they must overcome.

The COSMOS weak lensing data describedin this paper has already been used to mea-sure cosmological parameters and to demonstratethe feasiblity of the “tomography” technique(Massey et al. 2007). Striking weak lensing massmaps of the COSMOS field have been made thatreveal tantalizing evidence of a complex interplaybetween the baryon and the dark matter distri-bution (Massey et al. 2007a). Yet more lensinganalyses are underway including a galaxy-galaxylensing and a group-galaxy lensing study that willundoubtedly lead to a better understanding of therelationship between baryonic and dark matterstructures and of its evolution over cosmic time.

25

Acknowledgments

The HST COSMOS Treasury program was sup-ported through NASA grant HST-GO-09822. Wewish to thank our referee for useful comments andKevin Bundy for carefully reading the manuscript.We also thank Tony Roman, Denise Taylor, andDavid Soderblom for their assistance in planningand scheduling of the extensive COSMOS obser-vations. We gratefully acknowledge the contribu-tions of the entire COSMOS collaboration consist-ing of more than 70 scientists. More informationon the COSMOS survey is availableat http://www.astro.caltech.edu/$\sim$cosmos.It is a pleasure to acknowledge the excellent ser-vices provided by the NASA IPAC/IRSA staff(Anastasia Laity, Anastasia Alexov, Bruce Berri-man and John Good) in providing online archiveand server capabilities for the COSMOS data-sets. The COSMOS Science meeting in May 2005was supported in part by the NSF through grantOISE-0456439. CH is supported by a CITA Na-tional fellowship and, along with LVW, acknowl-edges support from NSERC and CIAR. In France,the COSMOS project is supported by CNES andthe Programme National de Cosmologie. JPKacknowledges support from CNRS.

REFERENCES

Bacon, D. J., Goldberg, D. M., Rowe, B. T. P., &Taylor, A. N. 2006, MNRAS, 365, 414

Bacon, D. J., Refregier, A., Clowe, D., & Ellis,R. S. 2001, MNRAS, 325, 1065

Bacon, D. J. et al. 2005, MNRAS, 363, 723

Bardeau, S., Kneib, J.-P., Czoske, O., Soucail, G.,Smail, I., Ebeling, H., & Smith, G. P. 2005,A&A, 434, 433

Bartelmann, M. & Schneider, P. 2001, Phys. Rep.,340, 291

Baugh, C. M. & Efstathiou, G. 1993, MNRAS,265, 145

Bernstein, G. & Jain, B. 2004, ApJ, 600, 17

Bernstein, G. M. & Jarvis, M. 2002, AJ, 123, 583

Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393

Capak, P. et al. 2004, AJ, 127, 180

Capak, P. L. et al. 2007, ApJS,

Casertano, S. et al. 2000, AJ, 120, 2747

Coe, D., Benıtez, N., Sanchez, S. F., Jee, M.,Bouwens, R., & Ford, H. 2006, AJ, 132, 926

Erben, T., Van Waerbeke, L., Bertin, E., Mellier,Y., & Schneider, P. 2001, A&A, 366, 717

Ford, H. C. et al. 2003, in Future EUV/UV andVisible Space Astrophysics Missions and Instru-mentation. Edited by J. Chris Blades, OswaldH. W. Siegmund. Proceedings of the SPIE, Vol-ume 4854, pp. 81-94 (2003)., 81–94

Goldberg, D. M. & Bacon, D. J. 2005, ApJ, 619,741

Goldberg, D. M. & Leonard, A. 2006, ArXiv As-trophysics e-prints

Hetterscheidt, M., Simon, P., Schirmer, M., Hilde-brandt, H., Schrabback, T., Erben, T., &Schneider, P. 2006, ArXiv Astrophysics e-prints

Heymans, C. & Heavens, A. 2003, MNRAS, 339,711

Heymans, C. et al. 2006, MNRAS, 368, 1323

Heymans, C. et al. 2005, MNRAS, 361, 160

High, F. W. 2007, in preparation

Hoekstra, H. et al. 2006, ApJ, 647, 116

Hoekstra, H., Yee, H. K. C., & Gladders, M. D.2004, ApJ, 606, 67

Ilbert, O. et al. 2006, ArXiv Astrophysics e-prints

Impey, C. D., Sprayberry, D., Irwin, M. J., &Bothun, G. D. 1996, ApJS, 105, 209

Jain, B. & Taylor, A. 2003, Physical Review Let-ters, 91, 141302

Jarvis, M., Jain, B., Bernstein, G., & Dolney, D.2006, ApJ, 644, 71

Jee, M. J., White, R. L., Benıtez, N., Ford, H. C.,Blakeslee, J. P., Rosati, P., Demarco, R., &Illingworth, G. D. 2005, ApJ, 618, 46

Kaiser, N., Squires, G., & Broadhurst, T. 1995,ApJ, 449, 460

26

King, L. J. 2005, A&A, 441, 47

Kleinheinrich, M. et al. 2005, A&A, 439, 513

Kneib, J.-P., Hudelot, P., Ellis, R. S., Treu, T.,Smith, G. P., Marshall, P., Czoske, O., Smail,I., & Natarajan, P. 2003, ApJ, 598, 804

Koekemoer, A. M. 2007, ApJS

Koekemoer, A. M., Fruchter, A. S., Hook, R. N.,& Hack, W. 2002, in The 2002 HST Calibra-tion Workshop : Hubble after the Installationof the ACS and the NICMOS Cooling Sys-tem, Proceedings of a Workshop held at theSpace Telescope Science Institute, Baltimore,Maryland, October 17 and 18, 2002. Editedby Santiago Arribas, Anton Koekemoer, andBrad Whitmore. Baltimore, MD: Space Tele-scope Science Institute, 2002., p.337, ed. S. Ar-ribas, A. Koekemoer, & B. Whitmore, 337–+

Krist, J. 2005, Instrument Science Rep. ACS 2003-06 (Baltimore: STScI)

Lilly, S. J. et al. 2007, ApJS, 999, 999

Lombardi, M. et al. 2005, ApJ, 623, 42

Lupton, R., Gunn, J. E., Ivezic, Z., Knapp, G. R.,& Kent, S. 2001, in ASP Conf. Ser. 238: Astro-nomical Data Analysis Software and SystemsX, ed. F. R. Harnden, Jr., F. A. Primini, &H. E. Payne, 269–+

Mandelbaum, R., Seljak, U., Kauffmann, G., Hi-rata, C. M., & Brinkmann, J. 2005, ArXiv As-trophysics e-prints

Massey, R., Bacon, D., Refregier, A., & Ellis, R.2002, in ASP Conf. Ser. 283: A New Era inCosmology, ed. N. Metcalfe & T. Shanks, 193–+

Massey, R. et al. 2006, ArXiv Astrophysics e-prints

Massey, R., Refregier, A., Conselice, C. J., David,J., & Bacon, J. 2004a, MNRAS, 348, 214

—. 2004b, MNRAS, 348, 214

Massey, R. et al. 2007, ApJS,

Massey, R., Rhodes, J., Ellis, R., Scoville, N.,Leauthaud, A., Finoguenov, A., Capak, P., Ba-con, D., Aussel, H., Kneib, J.-P., Koekemoer,A., McCracken, H., Mobasher, B., Pires, S., Re-fregier, A., Sasaki, S., Starck, J.-L., Taniguchi,Y., Taylor, A., & Taylor, J. 2007a, Nature, 445

Mobasher, B. et al. 2007, ApJS

Natarajan, P., Kneib, J.-P., Smail, I., & Ellis, R. S.1998, ApJ, 499, 600

Okura, Y., Umetsu, K., & Futamase, T. 2006,ArXiv Astrophysics e-prints

Peterson, B. et al. 1979, ApJ

Refregier, A., Rhodes, J., & Groth, E. J. 2002,ApJ, 572, L131

Refregier, A. et al. 2006, in High-Power Laser Ab-lation VI. Edited by Phipps, Claude R.. Pro-ceedings of the SPIE, Volume 6265, pp. (2006).

Rhodes, J., Refregier, A., Collins, N. R., Gardner,J. P., Groth, E. J., & Hill, R. S. 2004, ApJ, 605,29

Rhodes, J., Refregier, A., & Groth, E. J. 2000,ApJ, 536, 79

—. 2001, ApJ, 552, L85

Rhodes, J. D. et al. 2007, ApJS,

Rix, H.-W. et al. 2004, ApJS, 152, 163

Robin, A. et al. 2007, ApJS

Robin, A. C., Reyle, C., Derriere, S., & Picaud, S.2003, A&A, 409, 523

Schrabback, T. et al. 2006, ArXiv Astrophysics e-prints

Scoville, N. Z. et al. 2007, ApJS

Semboloni, E. et al. 2006, A&A, 452, 51

Sheldon, E. S. et al. 2004, AJ, 127, 2544

Taniguchi, Y. et al. 2007, ApJS, 999-999

Tyson, J. A., Wenk, R. A., & Valdes, F. 1990,ApJ, 349, L1

Williams, R. et al. 1998, Bulletin of the AmericanAstronomical Society, 30, 1366

27

Williams, R. E. et al. 1996, AJ, 112, 1335

Wolf, C. et al. 2004, A&A, 421, 913

This 2-column preprint was prepared with the AAS LATEXmacros v5.2.

28