Mass Substructure in Abell 3128
AA ccepted for publication in A p J. Preprint typeset using L A TEX style emulateapj v. 5 / / MASS SUBSTRUCTURE IN ABELL 3128
J. M c C leary , I. dell ’A ntonio , & P. H uwe Department of Physics, Brown University, Box 1843, Providence, RI 02912, USA
Accepted for publication in ApJ.
ABSTRACTWe perform a detailed 2-dimensional weak gravitational lensing analysis of the nearby ( z = . ugrz imaging from the Dark Energy Camera (DECam). We have designed apipeline to remove instrumental artifacts from DECam images and stack multiple dithered observations withoutinducing a spurious ellipticity signal. We develop a new technique to characterize the spatial variation ofthe PSF which enables us to circularize the field to better than 0.5% and thereby extract the intrinsic galaxyellipticities. By fitting photometric redshifts to sources in the observation, we are able to select a sample ofbackground galaxies for weak lensing analysis free from low-redshift contaminants. Photometric redshiftsare also used to select a high-redshift galaxy subsample, with which we successfully isolate the signal froman interloping z = .
44 cluster. We estimate the total mass of Abell 3128 by fitting the tangential ellipticityof background galaxies with the weak lensing shear profile of an NFW halo, and also perform NFW fits tosubstructures detected in the 2-D mass maps of the cluster. This study yields one of the highest resolution massmaps of a low- z cluster to date, and is the first step in a larger e ff ort to characterize the redshift evolution ofmass substructures in clusters. Subject headings: galaxies: clusters: general – galaxies: clusters: individual (Abell 3128) – gravitationallensing: weak – techniques: image processing INTRODUCTION
Cosmological perturbation theory provides a framework inwhich cold dark matter organizes itself hierarchically, firstcollapsing into small structures which can overcome cosmo-logical expansion and then continuing to merge into increas-ingly large halos. Because small collapsed objects often sur-vive accretion onto a larger system to become sub-halos oftheir host, the hierarchical structure formation paradigm pre-dicts that dark matter halos should be rich in mass substruc-tures (Klypin et al. 2011; Gao et al. 2012).The amount of mass substructure that we observe shouldincrease with redshift, particularly for cluster-sized halos:mergers of galaxy- and group-size halos are more commonin the early Universe, and the resulting clusters have long dy-namical relaxation times (Gao et al. 2004). At higher red-shifts, then, we expect to observe an increasingly high fractionof the total cluster mass locked up in sub-regions of localizedmass enhancement. An observational study of cluster sub-structure and its cosmic evolution would probe the assemblyhistory of cluster-sized halos and test the CDM paradigm onsub-megaparsec scales. Characterizing substructure in clus-ters also has important implications for understanding the roleof the mass environment in the evolution of member galaxies.Correlating sub-halo locations and, e.g., star formation rateswould reveal the e ff ect of local mass environment (distinctfrom the larger-scale mass distribution) on galaxy properties.Studies of cluster substructure are already underway. Mostnotably, X-ray data have been used to obtain cluster massfunctions through the proxies of cluster gas emissivity andtemperature. However, these proxies are related to mass byscaling relations that rely on assumptions like the hydrostaticequilibrium of the intra-cluster medium. Accretion-inducedheating of cluster gas, as well as merger-induced bulk and tur-bulent motions, violate the assumption of hydrostatic equilib-rium at the substructure level, at which the cluster is dynami- Jacqueline [email protected] cally active. Moreover, the error induced by the assumption isgreater in the outskirts of higher redshift clusters, where merg-ers are more frequent and clusters are accreting more rapidly(Lau et al. 2013; Nelson et al. 2014). Hence, an accurate ob-servational study of cluster substructure requires an analysistechnique insensitive to the dynamical state of a cluster.Because of its freedom from assumptions about baryonicphysics, weak gravitational lensing (WL) has become a stan-dard tool for measuring mass concentrations in the Universe.Multiple observations show that individual dark matter sub-structures within a cluster are capable of producing their owndetectable weak lensing shear (Okabe & Umetsu 2008; Se-hgal et al. 2008; Huwe 2012). However, in the weak lens-ing regime, the distortion of background galaxies induced bythe intervening cluster is much smaller than the intrinsic un-certainty of galaxy shape measurement. To overcome thischallenge, obtaining the angular resolution needed to iden-tify cluster substructures, requires large numbers of resolvedbackground objects. We achieve this in our observations bytaking deep, wide-field images using the Dark Energy Cam-era (DECam) on the Blanco 4–m telescope at Cerro TololoInternational Observatory.In this paper, we describe the analysis pipeline we devel-oped to make weak lensing measurements on DECam dataand present the results of our first substructure study on anearby ( z = .
06) cluster, Abell 3128, which is one of thehighest resolution mass maps of a z < . z clusters (and their sub-structure) means that their distortions will be coherent over alarge swath of the sky. This greatly increases the number ofbackground galaxies from which to measure the WL signal, a r X i v : . [ a s t r o - ph . C O ] M a r McCleary et al.and makes substructure easier to detect at low redshifts. Fi-nally, Abell 3128 has a complex morphology that has beenwell studied at radio, optical and X-ray wavelengths (Rose etal. 2002; Werner et al. 2007), enabling the comparison of ourWL substructure analysis to other techniques.Our observations of Abell 3128 and reduction method aredescribed in § §
4, respectively. In § §
5. In § § WEAK GRAVITATIONAL LENSING
A thin gravitational lens deflects and distortsthe images ofbackground sources. It is customary to describe the mass dis-tribution of the lens in terms of its surface mass density Σ orits dimensionless convergence , κ = ΣΣ crit , (1)where the critical surface mass density Σ crit of the lens is de-fined as Σ crit = c π G D s D l D ls . (2)The quantities D s , D l and D ls are the angular diameter dis-tances to the background source (galaxy), the lensing cluster,and between the source (galaxy) and lensing cluster, respec-tively.Lenses with κ ≥ κ < κ ≥ κ produces the isotropic magnification of abackground source, and as such cannot be measured directlyfrom an image without some prior knowledge of the sourcesize. In the case of weak lensing, however, convergence canbe recovered from the shear γ of background source imagescaused by the tidal forces of the lens’ gravitational field. Inparticular, convergence is related to shear through the applica-tion of the inverse 2-D Laplacian to the 2-D gravitational po-tential of the lens, an integral transform first derived in Kaiser& Squires (1993) and Kaiser et al. (1995), with variants pub-lished by Fahlman et al. (1994) and Fischer & Tyson (1997).In the WL regime ( κ << , γ << e = − ( b / a )tilted at some position angle θ with respect to the image axes.Galaxy shapes are frequently decomposed into ellipticity mo-ments: e = e cos(2 θ ) is the projection of a galaxy’s ellipseonto the image x and y axes, and e = e sin(2 θ ) is its pro-jection onto the lines y = x and y = − x . With this in mind,the the shear induced by a lens can be written in terms of thegalaxies’ shapes as γ → e tan = − ( e cos(2 φ ) + e sin(2 φ )) . (3)In this so-called tangential ellipticity , φ is the angle from afiducial lens center to the galaxy measured counterclockwisefrom north. In other words, the factors e e φ ) and cos(2 φ ) rotate the galaxy’s e and e into the line tangent to the radial extending from thechosen lens center. In the absence of a gravitational lens,the average tangential ellipticity (cid:104) e tan (cid:105) should vanish whenconsidered over many background galaxies with no intrinsicalignment. Hence, the (cid:104) e tan (cid:105) measured at a point in the imageis an unbiased estimator of the WL shear. By measuring the asystematic deviation from zero average ellipticity with a sam-ple of galaxies widely separated from each other in redshiftspace, we may reconstruct a cluster’s convergence. If in ad-dition the redshifts of the cluster and background galaxies areknown, the cluster’s surface mass density may be recovered.For a comprehensive treatment, see reviews by Bartelmann &Schneider (2001) and Wittman (2002a).In this study, we identify and characterize mass peaks us-ing the aperture mass statistic M ap first introduced by Schnei-der (1996). Measured some angular distance θ away from thecluster center, M ap is given by the convolution of the conver-gence κ with an aperture mass filter U ( | θ − θ | ): M ap ( θ ) = n (cid:90) d θκ ( θ ) U ( | θ − θ | ) . (4)The aperture mass filter U ( | θ − θ ]) smooths the convergenceover some characteristic aperture θ . By design, the M ap isa local measurement involving only the shear from galaxieswithin an angle θ of the center at position θ ; the filter is “com-pensated” so that its first order moment vanishes on scaleslarger than the aperture size.If the aperture mass filter U ( | θ − θ ]) in Equation 4 is trans-formed as Q ( | θ − θ | ) = θ (cid:90) θ θ (cid:48) U ( | θ − θ (cid:48) | ) d θ (cid:48) − U ( θ ) , (5)we can replace κ in the aperture mass statistic with the tan-gential ellipticity of background galaxies e tan . For a discretedataset of background sources, the aperture mass thus has theform M ap ( θ ) = n (cid:88) i e tan i ( θ ) Q ( | θ − θ | ) , (6)where the sum is taken over all galaxies in the observation(Schneider 1996). We apply Equation 6 to our observationsto build WL convergence maps.A variety of aperture mass filters exist which can be usedin Equation6 exist, but the one best suited to our search forsubstructure was introduced by Schirmer et al. (2004) as partof the GaBoDS survery. The Schirmer filter was originallydesigned to pick out shear signal from clusters embedded inlarge-scale structure, and is given by Q ( x ) = + e a − bx + e dx − c ) tanh( x / x c ) π R S ( x / x c ) , (7)where R S is the Schirmer filter size and x = r / R S is a scaleddistance between the cluster center and the point in consid-eration (Hetterscheidt et al. 2005). To optimize the filter fordetection of NFW shear profiles, the parameters in Equation7 are tuned to a = , b = , c = , d =
50 and x c = . R S ) in the aper-ture mass statistic, we can discern both the main cluster sig-nal and its substructures while simultaneously characterizingtheir respective scales: noting that the Schirmer filter weightspeak sharply at a value of x c R S , the structures identified haveass Substructure in Abell 3128 3size ∼ . R S . Since the Schirmer filter is not monotoni-cally decreasing, it is di ffi cult to assign it with a Gaussian-type FWHM. Instead, a smoothing length can be obtained bycomputing the radius which encompasses 50% of the filter’sweight. For the form of the Schirmer filter given above, thisradius is 0 . R S , equivalent to a Gaussian with standard de-viation σ = π/ e c = e cos(2 φ ) − e sin(2 φ ) . (8)Since gravitational lensing creates no B-mode signal, the re-placement of e tan with e c is frequently used as a statisticalcontrol. OBSERVATIONS
The DECam imager consists of 62 2048 × andcaptures 3 square degrees (2.2 square degrees wide field) at0.265” / pixel resolution (DePoy et al. 2008; Flaugher et al.2012). The camera’s wide field of view allows us to imagethe entire virial region of even a low-redshift cluster in a sin-gle pointing, making it e ffi cient for our study of cluster sub-structure.Observations of A3128 were made over eleven days from8th-24th November 2012 in the ugrz Y filter set by Dara Nor-man and the DECam science verification team as part of thatinstrument’s science verification program. To ensure sky cov-erage in the gaps between science CCDs, the telescope wasdithered in a “center + rectangle” pattern. The dithers arelarge enough to overlap adjacent CCDs by several hundredpixels, providing more uniform depth at the chip edges and al-lowing construction of a catalog covering the entire 1 . ◦ × . ◦ uniformly. The exposure time of each pointing varied by fil-ter: 720 s in u , 600 seconds in g , 300 seconds in r and 240seconds each in Y and z . The final exposure times across thefield were 10,800 seconds in u , 3600 seconds in g , 5400 sec-onds in r , 2630 seconds in Y and 2160 seconds in z , with atleast two complete dithers in each band. The mean seeing in r was 0.94”, and after calibration of source number counts vs.magnitude against the Subaru-COSMOS catalog (Taniguchiet al. 2007), the observations have a 50% completeness depthof m = .
97 in u , m = .
76 in g , m = .
62 in r and m = .
85 in z . We note that the Y and z filter profiles overlapconsiderably, such that the narrower Y essentially just coversthe longer-wavelength portion of the z filter. Imaging in thesetwo filters is redundant for the purposes of our analysis, and sowe make no use of the Y band data beyond making a stackedimage.The CTIO + DECam system’s sensitivity is greatest in r band, and this filter also optimizes the balance between highbackground galaxy luminosity and reasonably low sky noise(both of which increase with increasing wavelength). Follow-ing the successful observing strategy of the Deep Lens Sur-vey (Wittman et al. 2002b), we observed A3128 in r when theseeing FWHM reached < .
0” and in ugz
Y otherwise. Ac-cordingly, the r -band imaging has uniformly good resolutionas well as a greater depth than the imaging in other bands. Note that as of the time the A3128 data was collected, one of the CCDsat the southern edge of the array (N30) was non-functional.
Shear measurements are thus made exclusively in the r band,while other filters are used to provide color information forphotometric redshifts (see § ANALYSIS
Image Processing
Reconstructing a two-dimensional mass maps from galaxyshapes is an involved procedure. The intrinsic galaxy elliptic-ities are ∼
30 times larger than the distortions we are tryingto measure, and a priori it is impossible to disentangle WL-induced shear from the shape of a single galaxy. Moreover,anisotropy in the PSF field shears incoming light from galax-ies and obscures our weak lensing signal. For a camera aslarge as DECam, which reaches the edge of the focal plane ofits telescope, this e ff ect is substantial. The number and size ofDECam’s CCDs also makes removing instrumental artifactsfrom observations a technical challenge. In the following sec-tion, we list the image processing steps undertaken to over-come these di ffi culties and measure the mass substructure ofA3128. Although a community reduction pipeline now exists,we developed our own independent image processing pipelineas part of the science verification program for DECam. CCD-Level Reduction
The CCD-level image reduction applied to each exposurein the dataset includes the standard complement of overscansubtraction and trimming, bias subtraction, and dome flatfield correction. These tasks were accomplished with theMSCRED package in IRAF . We apply to CCD images anempirically-determined correction for the crosstalk that oc-curs as neighboring amplifiers are read out in parallel. A “treering” pattern of concentric circles of light and dark pixels ap-pears in all DECam object and flat field exposures; these arenot an artifact of gain variations on the chips, but actuallyrepresent the physical shifting of charge between pixel wells.Tree rings in object exposures are successfully camouflagedby the flat fielding step, although flat fielding away the treerings in object exposures is tantamount to turning an astro-metric error into a photometric one. Given the tiny amplitudeof the tree rings ( ∼ .
2% of pixel flux value), the error intro-duced is dwarfed by the m ≥ .
03 photometric uncertainty ofthe images. For more details regarding this and other CCDartifacts peculiar to DECam, see Plazas et al. (2014).To mitigate the > (cid:48)(cid:48) pointing errors in DECam sci-ence verification data, objects in the observation are matchedagainst a list of reference celestial coordinates in the USNO-B catalog. We fit a linear relation, which may include a zeropoint shift, scale change, and axis rotation, between the ob-served positions and the reference coordinates on both coor-dinate axes. The fit is used to update the image world coordi-nate system so that it is registered to the reference coordinatesystem defined by USNO-B. PSF Correction
Because the DECam CCD array is so large, the point spreadfunction (PSF) has significant and spatially varying contribu-tions both from the curvature of the focal plane and the Blanco4-m optics. Such anisotropies in the point spread function caninduce spurious shear signal, and so the accuracy of our massmaps relies on extremely precise characterization of the DE-Cam PSF. Distortions in the PSF field of an image can be IRAF is distributed by the National Optical Astronomy Observatory
McCleary et al.traced by systematic variations in the shapes of its stars sincethese are point sources and should appear perfectly round inan isotropic PSF field. Consequently, the first step in circular-izing the PSF is the identification of stars in the observation.Next, polynomials are fit to the spatial variation of the follow-ing combinations of second-order intensity moments: I xx − I yy , I xx + I yy and I xy . Finally, the intensity moment fits are used toderive a PSF circularization kernel, which is convolved withimage pixels in the stacking stage (cf. 4.1.3) as in Bernstein& Jarvis (2002).The default procedure is to go through these steps for ev-ery CCD of every exposure in the dataset, and for most ap-plications, this would yield a su ffi ciently circular PSF. How-ever, the DECam PSF field is severely under-sampled by un-saturated stars in any single exposure; the CCD chips them-selves are large, and A3128 is at high galactic latitude. Conse-quently, applying the standard PSF circularization techniqueto the DECam A3128 data only lowers the mean stellar ellip-ticity from 5% to 1%, which is still high enough to a ff ect theWL shear signal.To improve the PSF modelling, we combine stellar cata-logs from sequential exposures on the same CCD chip anduse these “super-catalogs” of stars to fit 2-D polynomials tothe spatial variation of intensity moments. High-order poly-nomial terms (fourth- and fifth-order) of the intensity momentfits capture e ff ects like focal plane curvature and tend to bestable over the course of contiguous dithers. Because thelower order terms in the intensity moment polynomials cap-ture time-dependent e ff ects like seeing or telescope drift, theyare usually best fit using individual CCD exposure catalogsrather than super-catalogs.The particular grouping of exposures used to build super-catalogs is also determined empirically for each CCD. Stellarellipticity is minimized when the higher-order terms of the I xx − I yy polynomials are fit by super-catalogs assembled froma single dither of five exposures. Meanwhile, the higher-orderterms of the I xy and I xx + I yy polynomials should be fit usingsuper-catalogs assembled from as many contiguous exposuresas possible.For each of the 61 functional CCDs, we determine empir-ically both the degree of the polynomial fits and which of itsterms should be obtained through super-catalogs. We piecetogether the final forms of fits to I xx − I yy , I xx + I yy and I xy from whichever combination of terms ultimately yields thelowest stellar ellipticities. This procedure allows for the cir-cularization of stellar PSFs to better than 0.5% (see Figure 1).We verified that the magnitude of stars selected does not af-fect the polynomial fitting by examining the residuals of thePSF fits to stars. These show no discernible trend in the rangeof magnitudes considered (16 < m r < ff ectiveness of this PSF circularization scheme may bequantified by constructing two-point shear correlation func-tions, defined as C i j = (cid:104) e i ( r ) × e j ( r + θ ) (cid:105) , (9)where e i is the i th ellipticity component of an object at posi-tion r, and brackets denote an average over all pairs within aseparation θ . A third correlation function, C = (cid:104) e ( r ) × e ( r + θ ) + e ( r ) × e ( r + θ ) (cid:105) , (10)should be zero and is frequently used to test for systematic er-ror in PSF correction schemes (Massey et al. 2005). Star-starauto-correlation functions and star-galaxy cross-correlation functions are shown in Figure 2. The star-star auto-correlationfunctions C and C (top left panel) show a small signal onsmall scales that we attribute to some over-correction of theDECam PSF in the areas near stars. The star-galaxy corre-lation functions (bottom panels) show the same correlationon small scales, with a magnitude several times smaller thanthe galaxy-galaxy auto-correlation signal (which traces WLshear). The negligible value of the C in both star-star andstar-galaxy pairs confirms that the PSF circularization schemeintroduces no major systematic error to shape measurement.It should be noted that even after circularization, the mea-sured ellipticity moments do not yet represent the true shapesof the galaxies. Both atmospheric seeing and the circulariza-tion of the image PSF make galaxies appear more round thanthey really are, and e ff ectively dilutes the WL shear signal.We correct for this “smearing” at the catalog level as detailedin § Stacking
Once we have corrected for CCD artifacts and, in the caseof the r -band exposures, obtained polynomial fits to the PSF,we proceed to stack CCD exposures into a single image. Ourprocedure for stacking closely follows the one used in the theDeep Lens Survey; see Wittman et al. 2006 for full technicaldetails. The steps undertaken to produce our stacked imagesare summarized here.1. Source Detection & Characterization . – For eachCCD image in the dataset, Source Extractor (Bertin& Arnouts 1996) is used to make a catalog of high S / N objects. Source Extractor also generates the skybackground-subtracted images that will be the final in-puts to the final stack image. Subtracting the skybackground at this stage eliminates the need to matchsky levels at the stacking stage. At this stage, theELLIPTO program (Bernstein & Jarvis 2002) is usedto measure the so-called “adaptive” second-order mo-ments of objects in the CCD image. Adaptive mo-ments are centrally weighted by an elliptical Gaussian,and their measurement is equivalent to finding the best-fit elliptical Gaussian for each object. Unlike SEx-tractor’s intensity-weighted moments, which are com-puted within some limiting isophote, adaptive momentsdo not depend on magnitude. This property makesadaptive moments more advantageous for galaxy shapemeasurement and the identification of stars (Wittman etal. 2006).2. Star Identification . – As discussed in § r band stellarcatalogs are individually inspected and manually ad-justed as needed. The stars used for r band PSF fitsare highlighted in the size-magnitude diagram of Fig-ure 3. There and elsewhere in the paper, size is de-fined as the sum of the second-order intensity momentsass Substructure in Abell 3128 5
00 x (pix)
00 x (pix)
Figure 1.
Whisker plot showing spatial variation of the PSF across the Abell 3128 image. Each stick represents a star that was used to circularize the PSF,with length proportional to the magnitude of its measured ellipticity, and orientation equal to its position angle. Blank regions correspond to CCD chip edges,large cluster galaxies or saturated stars.
Left:
Stars in a Abell 3128 stack made without circularization correction. The mean segment size corresponds to stellarellipticities ∼ e ≤ . Right:
Stack made with the multi-chip circularization correction described below. Mean ellipticityhas been reduced to 0.005.
Figure 2.
Correlation functions between tangential ellipticity components for objects in the A3128 observation. Star-star auto-correlation functions are plottedat top left, and star-galaxy cross-correlation functions are contrasted with galaxy-galaxy auto-correlation functions in the other three panels. All ellipticitycomponents are defined with respect to the image axes. I xx + I yy with an additional factor ρ to correct for non-Gaussianity (Bernstein & Jarvis 2002)3. Master Catalogs . – The DLS survey found that the MS-CRED astrometric calibration of images is not goodenough to stack them directly; small shifts betweenoverlapping exposures would lead to spurious stretch-ing of galaxy shapes (Wittman et al. 2006). To pre-cisely define the astrometry of the final stack image, we match all the catalogs in equatorial coordinates toproduce a master catalog. Every object that was ob-served in at least three exposures (within the toleranceof 1 (cid:48)(cid:48) .
8) has its mean right ascension, declination andmagnitude recorded. Subsequently, the master catalogpositions are used to define a coordinate system for thestack (a simple tangent plane projection with no opticaldistortion) and then transformed to pixel coordinates in McCleary et al.the final stack image.4.
Pixel Coordinate Transformations . – For every CCD ineach exposure, matches between the Source Extractorcatalog and the master catalog are used to define a trans-formation from CCD pixel coordinates to final stackimage. For the ugz
Y data not subject to WL analy-sis, less stringent astrometry requirements allow for theDLS default of a third-order polynomial. In the case of r band exposures, the coordinate transformations mustbe defined by a fourth-order polynomial. Lower or-der polynomials underfit the variation at the edges ofthe CCD, resulting in a slight elongation of galaxies inthe final stack image and ultimately leading to bandsof spurious shear signal in the WL convergence maps.However, many CCDs from the edge of the exposure( ∼ PSF Circularization . – Once the pixel transformationpolynomials are determined, we generate PSF circular-ization profiles for every CCD image in the dataset. Bydefault, the adaptive moment combinations I xx − I yy , I xx + I yy and I xy are fit automatically with 4th orderpolynomials. The exceptions are PSF profiles for the r -band imaging, which are prepared in advance usingthe multi-stellar catalog procedure laid out in 4.1.2. Figure 3.
Size-magnitude diagram of the object catalog generated from thefinal stacked r image. At this stage, the catalog is filtered for objects withSource Extractor or ELLIPTO error flags, but no other cuts are applied. Thestellar locus is the stripe of objects with size ∼ . < m r <
21; redstars mark objects used in our multi-catalog PSF circularization procedure.Objects in the “second stellar locus” with sizes around 2 pixels and m r < m ∼ Photometric Calibration . – Before CCD exposures arestacked, their photometry is calibrated to ensure consis-tent object magnitudes everywhere in the final stack im-age. This is made more di ffi cult since, due to the curva-ture of the focal plane, pixels at the edges of the DECamCCD subtend more sky area than pixels at the center ofthe array. To correct for focal plane distortion, catalogmagnitudes are multiplied by the Jacobian of the pixelcoordinate transformations computed in step 4 and thengathered into a master photometric catalog. For each CCD exposure, we then derive a relative photometrico ff set by matching its catalog to the master photomet-ric catalog and computing the 3 σ clipped mean of themagnitude di ff erences of the matching objects.Finally, to produce the stacked image, we implement the DeepLens Survey’s DLSCOMBINE algorithm. For each pixelin the output image, DLSCOMBINE loops over contribut-ing pixels in the input images and applies the relevant badpixel masks, PSF circularization kernels, coordinate transfor-mations and photometric o ff sets. A 3 σ clipping is appliedbefore the mean pixel value is returned. A 3-color compositeimage made from the z , r and g stacked images is shown inFigure 4. Source Selection
We produce a source catalog from the final stacked r imageusing Source Extractor (Bertin & Arnouts 1996). The SourceExtractor detection significance and deblending thresholds aredeliberately set to low values so that the faint backgroundgalaxies on which we perform WL analysis will be recog-nized. To clean out the accompanying multitude of junkSource Extractor detections, we perform a number of cutsto the initial object catalog. All detections with high SourceExtractor and ELLIPTO error flag values may easily be fil-tered out, but ridding the catalog of pixel noise “detections”presents a special problem. Because their few counts are con-tained within a small isophotal area, SExtractor frequently as-signs them reasonable magnitudes, and measurement of theiradaptive moments produces no error flags. To filter out suchpixel noise from the object catalog, we removed detectionswith ELLIPTO-determined fluxes of less than 100 counts.The cuts on error flags decreased the initial object catalog of1.4 million objects by about 22%, and the flux cut decreasedit by a further 43%.The object catalog is then subject to the the size and magni-tude cuts typical of weak lensing studies, which must removestars, low-redshift galaxies and noisy sources while maintain-ing a large sample. The criteria for inclusion in the final sam-ple were isophotal r magnitudes between 17.2 and 24.6, andobject size between 6.0 and 200 pixel , where size was de-fined in 4.1.2. The atypically generous upper limit of the sizecut reflects the fact that A3128 is at very low redshift, and its“background” contains many large galaxies. Requiring thatobjects be detected in all four bandpasses de facto constitutesan additional catalog cut, eliminating 30,000 objects from thefinal catalog. At this point, the filtering has reduced the cata-log down to 200,000 sources in total (25 sources arcmin − ).Any galaxies in the foreground of A3128 will not besheared by the cluster, and so their presence in the final sourcecatalog dilutes the convergence measured on the lens. Tofilter out low redshift contaminants, we derived photometricredshifts using BPZ (Ben´ıtez 2000) with the standard HDFprior. The ugrz magnitudes of cataloged objects were submit-ted to the program, although since the u-band observations arerather shallower than the grz observations, the photometricredshifts are essentially three-point fits. We used the defaultCWWSB template set (E, Sbc, Scd, Irr, SB3, and SB2) withno modifications, and allowed three levels of interpolation be-tween neighboring templates. The range to be considered wasrestricted to 0 . < z BPZ < .
0. To evaluate BPZ results onlow-redshift galaxies, we identified 10 galaxies in the obser-vations with spectroscopic redshifts z ∼ .
06 and used BPZto determine their photometric redshifts. This test calls at-ass Substructure in Abell 3128 7
Figure 4.
Sequence of zrg composite images showing progressively higher magnifications of the Abell 3128 field. Top: The full DECam 1.5 ◦ x 1.5 ◦ field of viewof the cluster and surrounding region. Left: Close-up view of the central 32’ x 30’ of Abell 3128. Right: 3’ x 3’ image showing the background cluster ACT-CLJ0330-5227, which hosts SUMSS J033057-52281, a radio source at z = Figure 5.
Redshift distribution of the final catalog, filtered for low-redshiftgalaxies. The majority of our sources are at z ∼ .
30, with secondary peaksat z ∼ .
46 and z ∼ . tention to the uncertainty in BPZ results, as all 10 galaxieswere assigned redshifts between 0.07 and 0.13. This range iscomparable to the per-galaxy rms error found by other studies (e.g. Sehgal et al. 2008). While most galaxies that are trulyin the foreground of A3128 are large enough to be eliminatedby the size cuts described above, we nonetheless took the low-redshift uncertainty into account when developing our redshiftselection criterion of z BPZ > .
19 at greater than 95% prob-ability. About 169,000 objects (21 sources arcmin − ) remainafter this latest cut is made.One last step remains before we can make convergencemaps for Abell 3128. Even after circularization, atmosphericseeing and the circularization procedure itself still smearout the measured adaptive moments, making galaxies appearrounder than they really are and diluting the WL shear signal.As a correction, each galaxy’s ellipticity is divided by a factor R which relates the size of the galaxy to the mean stellar PSFsize (Bernstein & Jarvis 2002). Galaxies must have a mea-sured R > . − . Convergence Mapping and Quantified Detection ofSubstructure
The catalog finally contains the true shapes of confirmedgalaxies behind A3128, and may be used to reconstruct the McCleary et al.
Figure 6.
An aperture mass peak in five maps of consecutively larger filterradii.
Top:
Detection significance of aperture mass in units of σ . Bottom:
Signal-to-noise of the aperture mass. In this example adapted from our A3128observation, the signal peaks in both significance and signal-to-noise at aSchirmer filter radius of 6500 pixels. cluster’s projected mass. To extract the aperture mass sig-nal of A3128 from the tangential ellipticities of backgroundsources, this study relies on software developed by co-authorHuwe as part of his thesis work (Huwe 2013, Huwe & Del-lAntonio 2014, in preparation). Our particular implementa-tion of this software is presented here.Our software first bins the A3128 image into blocks 200pixels on a side to reduce computation time. Using the adap-tive moments of galaxies in the catalog, the software then re-turns the filtered aperture mass statistic (Equations 6 and 7)within each 200-pixel block. To quantify our mass recon-struction of the A3128 observation, we construct a WL signal-to-noise map as follows. Random noise maps are generatedby recalculating the aperture mass statistic (with the sameSchirmer filter and pixel block sizes) on a catalog of shuf-fled galaxy positions and moments. The random maps willinitially be dominated by non-Gaussian shape measurementerror, so the randomization process is repeated 100 times. As-suming that the errors in the aperture mass reconstruction willthen be Gaussian, the variance of each image block in the ran-dom maps represents the 1 σ noise level of the M ap statistic.Dividing the M ap signal map by the variance of the randommaps, pixel block by pixel block, yields an estimate for thesignal-to-noise.Because WL distortion is tangential to the direction to thecenter of mass, an image should have no systematic B-mode(curl-like) distortion in the shapes of background galaxy. Thelensing signal should thus vanish when e tan , the E-mode (curl-free) component of shear, is replaced with the B-mode com-ponent e c defined in Equation 8. Any significant WL peaksobtained when e tan is replaced with e c in Equation 6 wouldnot come from the cluster, but instead indicate some system-atic error in the analysis. Since most systematics are expectedto add equal power to E- and B-modes (Jarvis et al. 2003), wegenerate B-mode signal-to-noise maps to control for bias inour analysis.Both E- and B-mode signal-to-noise maps treat the errorsin the convergence field as Gaussian, but this assumption maynot be warranted. To further quantify confidence in the resultsof our mass reconstruction, we calculate the detection signif-icance of features in the signal-to-noise map with the follow-ing algorithm. Using the same filtered aperture mass statisticas before (Equation 6), the software creates a signal file andthen iterates through some large number of random noise re-constructions which are stored in memory. At every 200-pixelblock of the observation, the software tallies how many noise reconstructions had a greater magnitude of WL signal thanthe signal file. This number should be close to 0 for blocksnear the cluster center, but in massless regions of the observa-tion will be roughly 50% of the total number of random iter-ations. When inserted into an inverse cumulative distributionfunction, this number is converted into a Gaussian-type con-fidence σ which quantifies the significance of the shear signalin that pixel block. The maximum attainable σ will depend onthe number of noise iterations; our software generates 100000random maps which corresponds to a maximum confidence of4 . σ .In addition to a magnitude given by its σ value, the soft-ware assigns to each significance map pixel the sign of thecorresponding pixel block in the original signal map. Hence,regions in the significance maps with negative values corre-spond to statistically significant underdensities in the massdistribution, while positive σ means an area of mass enhance-ment compared to the mean.To search for mass concentrations, the significance mapsare thresholded above + . σ , and potential substructure peaksare identified by inspection. For each group of contiguous im-age blocks with significance greater than + . σ , we followthe feature through a range of Schirmer filter scales. Rela-tively small Schirmer filter radii x c do not encompass all theshear signal from the feature, whose significance will sub-sequently be suppressed. As the filter radius increases, thesignificance of the detection increases before peaking at someSchirmer filter size which is then the characteristic scale ofthat substructure. Further filter expansions eventually lead tothe merging of the substructure signal into the overall clustersignal. An example of this increase and decrease in signifi-cance is shown in Figure 6. NFW Shear Profile Fitting
Recalling that aperture mass maps return only the relativemass enhancements in an observation, we have written soft-ware which fits A3128 and its substructures with axisymmet-ric NFW weak lensing shear profiles to constrain their physi-cal masses. Di ff erent algorithms are employed to fit the obser-vation with single and multiple NFW halos, but in both casesthe software first obtains the scaling factor Σ cr (Equation 2)for each galaxy in the catalog. Requisite angular diameterdistances for Σ cr are computed from the galaxies’ BPZ red-shifts. The software then varies the M of an NFW halo andcomputes the corresponding r under a Planck
XVI cosmol-ogy. The halo’s concentration c is obtained by inserting its M into the empirical relation of Bhattacharya et al. (2013)for their full cluster sample. The r and c parameters be-come the r s and δ c which characterize an NFW halo’s massdistribution and hence its shear profile.To fit a single NFW mass to the primary WL peak ofA3128, as M is varied the software follows the prescriptionof Wright & Brainerd (2000) to compute the halo’s reducedshear at the location of every background galaxy. We find thebest-fit M by using the parabolic extrapolation method ofPress et al. (2007) to minimize χ residuals between the NFWhalo’s shear profile and the e tan measured on the image.Simultaneous fitting of NFW masses to multiple substruc-tures requires a di ff erent approach: the (tensor) reduced shearfrom multiple NFW peaks does not add linearly, so the pre-scription of Wright & Brainerd (2000) is not directly applica-ble. Instead, we use the fact that background galaxies expe-rience an individual displacement (cid:126)β from each NFW halo. Inass Substructure in Abell 3128 9the weak lensing approximation, the displacements from mul-tiple NFW haloes add linearly: (cid:126)β tot = (cid:80) (cid:126)β i . The total shearand convergence at every point in the image can then be builtfrom derivatives of the Jacobian ( ∂(cid:126)β tot /∂(cid:126)θ ) using the formu-lae of Golse & Kneib (2002). Presupposing the locations oftheir centers have been established, we vary the M of NFWhaloes centered on each substructure, obtain the correspond-ing reduced shear at the location of every background galaxy,and minimize the profiles’ χ residuals against the galaxies’tangential ellipticities.We emphasize that neither the single-peak nor multiple-peak mass estimates in this work result from fitting 1-DNFW shear profiles to azimuthally averaged galaxy elliptic-ities, though such an approach is common in the literature.Rather, our NFW fitting method uses the full positional in-formation of every galaxy in the catalog. Tangential shearprofiles shown below (Figure 13) are for illustrative purposesonly. We also note that in all mass estimates, NFW shear pro-files are centered on the aperture mass peak’s highest signal-to-noise pixel in convergence maps. However, due to our bin-ning scheme (cf. § (cid:48)(cid:48) ) on the observation. The resultant am-biguity in the identified center of a WL peak could bias massestimates through a mis-computation of the galaxy ellipticity;we investigate this potential centroid bias in § RESULTS
Detection of Primary Cluster Aperture Mass
Applying the procedure laid out in § σ > .
42 at all Schirmer filters larger than6000 pixels. To identify the aperture size which best charac-terizes the cluster, we constructed signal-to-noise maps usingSchirmer filters up through R S = S / N of 8.4at two distinct locations with an aperture size of R S = ∼ . R S , the primary A3128 signal spans roughly 4.4’ onthe image.From the location of the highest signal pixels in the S / N map of Figure 7(b), we might assign the primary weak lens-ing peak of Abell 3128 to coordinates α = h m s . , δ = − ◦ (cid:48) (cid:48)(cid:48) . However, the presence of a massive high-redshift cluster (visible in Figure 4) only 6’ from the A3128X-ray center confounds the location of its center of mass. In-stead, we defer this question to § σ aperture masses appear in Figure 7: the S / N ∼ σ = .
42. These clumps are better characterized at smallSchirmer filter sizes, as described in in § e c (Equation 8) controls for any systematic error in the galaxycatalog ( § R S = S / N maps of Figure 8 display a negative signal nearthe cluster center. Given that systematic errors should addequal power to E- and B-modes, it is expected to see the neg-ative signal in both the B-mode S / N map of Figure 7(c) andthe E- and B-mode S / N maps of Figure 8. Table 1
Centroids from Previous Studies α δ
ID (J2000.0) (J2000.0) ReferenceA3128 Optical Center 3 h m s − ◦ (cid:48) (cid:48)(cid:48) Rose et al. (2002)A3128 X-ray Center 3 h m s − ◦ (cid:48) (cid:48)(cid:48) Werner et al. (2007)X-ray “SW Peak” 3 h m s − ◦ (cid:48) (cid:48)(cid:48) Werner et al. (2007)ACT-CL J0330-5227 3 h m s − ◦ (cid:48) (cid:48)(cid:48) Menanteau et al. (2010)
Equation 8 shows that negative WL signal is caused by anet tendency of objects in a region to be aligned radially out-wards. Furthermore, the DECam PSF is known to have astrong radial component (cf. the “pincushion” of Figure 1).Hence, this negative signal is attributable to an undercorrec-tion of the PSF “pincushion” near the bright galaxies of thecluster, likely due to gaps in stellar coverage. This interpre-tation is supported by the weakening of the e ff ect at largeSchirmer filter radii, where the inclusion of more galaxies atrandom orientations washes out any localized residual corre-lations in the PSF.We constrain the magnitude of this systematic e ff ect as fol-lows. We amalgamate the star and galaxy convergence mapsand tally up WL signal enclosed in a circle centered on thecluster. This value is then compared to the equivalent signal inthe convergence map made from galaxies alone. We find thatthe combined stars / galaxies convergence map has 5% morepower than the galaxies-only signal map, likely because thesystematic undercorrection of the PSF subtracted away someof the cluster’s original convergence signal. Since WL shearand mass scale linearly, we expect that the NFW fits presentedin Table 3 underestimate the true mass by ∼ High-Redshift Background Cluster
Visible in the last panel of Figure 4 is an interloping high-redshift cluster of galaxies, complete with the blue arc that isthe hallmark of strong gravitational lensing. Multiple studieshave confirmed that this background “cluster behind a clus-ter,” identified in the literature as ACT-CL J0330-5227, lies at z = .
44 and is the source of the northeastern lobe of X-rayemission in A3128 (Werner et al. 2007). All cataloged galax-ies with z > .
44 will bear lensing signal from both clusters,and so the higher-redshift cluster must be precisely character-ized to prevent confusion with substructure in A3128.To disambiguate the signals of A3128 and ACT-CL J0330-5227, we prepared two subsamples using BPZ redshiftsand probabilities: an intermediate-redshift sample containinggalaxies at 0 . < z < . z > . z = Figure 7.
Reconstructed weak lensing convergence maps for Abell 3128, all with 52” / pixel resolution on the observation. Left:
Significance map made using aSchirmer aperture of radius 10000 pixels ( = . σ . Center:
Signal-to-noise map and44’ Schirmer aperture, which yielded the maximal cluster S / N of 8.4 at two separate locations. Right:
B mode S / N map for 10000 pixel aperture radius (primarypeak filter size). Figure 8.
Reconstructed aperture mass maps with 52” resolution made from the catalog of stars used to circularize the DECam PSF .
Left:
Signal-to-noise mapmade with a Schirmer aperture of radius 10000 pixels ( = Right:
B mode S / N map for 10000 pixel aperture size. Note thatboth maps have significant areas of negative WL signal, indicating a net tendency of background galaxies to be alined radially outwards.) .
31, and the high-redshift catalog contains 52,847 galaxies ata mean redshift of z = .
68. The intermediate-redshift galax-ies experience WL distortion only from A3128, since they liebehind A3128 but in front of ACT-CL J0330-5227; the high-redshift galaxies are behind both clusters and experience dis-tortion from both.In Figure 9, significance maps made with the two redshiftsubsamples are compared to the the full galaxy sample map.The X-ray position of cluster ACT-CL J0330-5227 and the“southwest peak” of A3128 X-ray emission are marked withblack stars. Using the high-redshift galaxy subsample (leftpanel), we detect at high confidence the WL signal from ACT-CL J0330-5227 and both the A1 and A2 substructures ofA3128. In the reconstruction with the intermediate-redshiftgalaxy subsample (center panel), the WL signal of A3128 isstill distinct but the high-redshift cluster has dropped out ofview. A reconstruction made with the full galaxy sample isshown in the right panel of Figure 9; the inclusion of a largenumber of galaxies at z < .
44 dilutes the signal of ACT-CL J0330-5227 and makes it harder to discern. The fact thatthe background cluster does appear or disappear with such se-lections bolsters our confidence in the galaxies’ photometricredshifts, and by extension the angular diameter distances re-quired to fit NFW profiles to aperture masses.Based on the absence of high- z cluster signal in Figure 9(b),the intermediate-redshift galaxy subsample may be used to authentically identify the A3128 barycenter. From the loca-tion of the highest σ pixel in large-aperture significance maps,we report the primary weak lensing peak (and presumablybarycenter) of the cluster at α = h m s . , δ = − ◦ (cid:48) (cid:48)(cid:48) .This WL center is o ff set by 4.7 arcminutes from the opticalcenter of the galaxy distribution, but by 6.24 arcminutes fromthe published X-ray center. Instead, the WL potential centercoincides more closely with the “southwest peak” of X-rayemission. Having established a location for ACT-CL J0330-5227 and the A3128 barycenter in our WL reconstructions,we may proceed in identification of substructures in A3128with more assurance. High Significance Substructure
Our significance maximization procedure yields two sub-structures within A3128 proper, which are visible in the left-hand panel of Figure 10. The peak to the cluster’s southwest(A2) saturates our significance maps at σ = .
42 at apertureswith Schirmer filters larger than R S = R S = . σ structure in all significance maps made with apertures6000 pixels and larger. However, the two substructures canbe still distinguished in S / N maps up through an aperture sizeof 9000 pixels. Both aperture masses achieve their maximalass Substructure in Abell 3128 11 Figure 9.
Significance maps marked with the published positions of the Abell 3128 X-ray “southwest peak” (bottom right star) and the z = .
44 cluster ACT-CLJ0330-5227 (top left star). Pixels in all maps span 52” on the observation. The two extended high- σ regions apparent in all three panels are the principalsubstructures of A3128; they are discussed as A1 and A2 in § Left:
Close-up of map made with background galaxy redshift restricted to z ≥ .
44 and aSchirmer aperture of 4000 pixels. The high-redshift cluster is plainly visible.
Center:
Map made with galaxies at redshifts between 0 . ≤ z ≤ . Right:
Map made withthe full background galaxy sample and a 4000 pixel aperture. signal-to-noise at R S = S / N = .
6, and the northeast peak with S / N = .
6. Recall-ing that the Schirmer filter’s weight peaks at 0 . R S , the cor-responding angular scale of the substructures is 3.5 arcmin-utes. The northeast peak (A1) is spatially coincident with thebrightest cluster galaxy, and both peaks are near the opticalcenter of the cluster. Note that both of these features are spa-tially coincident with small-scale features in the “southwestpeak” of X-ray emission (cf. Tables 1 and 2), and are dis-tinct from the northeastern lobe of X-ray emission which isthe high-redshift cluster ACT-CL J0330-5227.Within the same range of Schirmer apertures, we detect athigh significance two substructures adjacent to the primaryA3128 peak, circled in white in Figure 11 and with coordi-nates listed in Table 2. At R S = . σ while its neighbor-ing feature (G2) achieves 4 . σ with a 5000 pixel aperture.At their respective characteristic aperture sizes, the featuresachieve S / N of 5.1 and 4.3. By R S = Table 2
Substructures Identified in Convergence Maps α δ
ID (J2000.0) (J2000.0)A1 / “Northeastern Peak” 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) A2 / “Southwestern Peak” 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) B1 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) B2 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) B3 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) G1 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) G2 3 h m s . − ◦ (cid:48) (cid:48)(cid:48) The three high- σ features towards the bottom left of Fig-ure 10(a) match up to regions of mass enhancement in thesignal-to-noise image of Figure 10(c). The leftmost peak (B1) is just at the edge of the detector, but peak B2 is less than anarcminute from the rich group ACO S 366 ( z ∼ . ff ects and a genuine DM enhancement associated withACO S 366. Peak B1 achieves its maximum σ = .
42 with aSchirmer filter size of 5000 pixels, and peak B2 does the sameat R S = S / N ∼ . − . α = h . m , δ = − ◦ (cid:48) ). While there is acorresponding bridge of galaxies in the DECam image, thereare only moderate enhancements in the significance map. Wefind no significant WL features at the location of A3125 itself. Mass Estimates
Having constrained the locations of the A3128 primaryaperture mass and its substructures with WL convergencemaps, we proceed to parametrically fit them with NFWmasses following the procedure in § Single NFW fit to Primary A3128 Aperture Mass
Centering a single NFW peak at the A3128 barycenteryields a best-fit mass of M = . ± . × M (cid:12) . The cor-responding r for this mass is 2.1 Mpc, which at the distanceof A3128 spans about 7000 pixels on the observation. Uncer-tainty is estimated through a jackknife approach wherein werandomly resample 50% of the galaxy catalog and recomputethe best-fit mass. The variance of 2000 realizations is taken asa measure of the NFW fitting procedure’s internal consistencyand becomes the error bar on the best-fit mass. Simultaneous NFW fits to Abell 3128 and ACT-CLJ0330-5227
NFW shear profiles were fit to the A3128 barycenterand ACT-CL J0330-5227 simultaneously, using both the fullgalaxy sample and the high-redshift subsample ( z gal > . § Figure 10.
High resolution convergence maps made with with 52” / pixel resolution on the A3128 observation. Top Left:
Significance map made with a Schirmerfilter of R S = = (cid:48) ), in which the two substructures attain 4.42 σ detection significance. Several peaks to the bottom left (B1, B2 and B3) alsoattain high significance. Top Right:
Magnified view of significance map at left, highlighting central cluster.
Bottom left:
E-mode signal-to-noise map made witha Schirmer filter of R S = S / N > . Bottom Right:
B mode S / N map made with R S = G1 G2
Figure 11.
WL significance map made with a Schirmer aperture of R S = = (cid:48) ). At this filter size size, the two substructures in A3128 have nearlymerged, but other structures surrounding the cluster are now visible. Inset:
Magnified view of the WL significance map, overlaid on a zrg composite image ofA3128. The two high significance peaks circled in white are likely associated with galaxy groups recently accreted onto the cluster.
50% random resampling of the catalog. All fits were subjectto 2000 resamplings except the full galaxy sample / Centroid 1fits, which were resampled 4000 times.In fits made with the full galaxy sample, the masses of thetwo clusters sum to about 2 . × M (cid:12) , but the allocationof this mass between A3128 and ACT-CL J0330-5227 variessignificantly ( (cid:38) σ error bars) depending on the A3128 cen-ter chosen. When the A3128 shear profile is centered at thebarycenter, the high- z cluster is assigned a lower mass thanwhen the A3128 profile is centered at the southwest substruc- ture peak. A similar picture emerges in the 2-peak fits withthe high-redshift galaxy sample ( z gal > . . × M (cid:12) ) regardless of A3128 centroid chosen, butthe distribution of this mass between the two clusters variessignificantly. We note that compared to the fits with the fullgalaxy sample, the variance of the A3128 and ACT-CL J0330-5227 masses in Table 3 are slightly higher – an expected by-product of the smaller sample size.Table 3 also contains the results of a two-peak fit madeass Substructure in Abell 3128 13with the intermediate-redshift subsample of galaxies (0 . < z gal < .
4) of § (cid:38) . σ level. Because of the lowsource density of the intermediate-redshift subsample (7-10arcmin − ), and moreover because those galaxies have an unfa-vorable D S / D LS , the underestimate likely reflects a WL signaltoo weak to beat down the random shape noise of backgroundgalaxies. Table 3
NFW Masses for High-Significance Aperture MassesA3128 Centroid 1 A3128 Centroid 2Galaxy Sample A3128 Mass a High- z Mass A3128 Mass b High- z Mass(10 M (cid:12) ) (10 M (cid:12) ) (10 M (cid:12) ) (10 M (cid:12) )Full z gal sample 11.5 ± ± ± ± z gal > .
44 13.8 ± ± ± ± . < z gal < . ± < c ± < a Centered at cluster barycenter: α = h m s . , δ = − ◦ (cid:48) (cid:48)(cid:48) b Centered at SW substructure peak: α = h m s . , δ = − ◦ (cid:48) (cid:48)(cid:48) c Fits with this galaxy sample consistently assigned to the high-redshift cluster mini-mum mass in the allowed range, down to 1 × M (cid:12) The χ landscape shown in Figure 12(a) indicates that theA3128 mass is well constrained by the full galaxy samplefits, but that they only place marginal constraints on the high-redshift cluster mass. Eliminating all galaxies in front of thehigh-redshift cluster leads to slightly better constraints on themass of ACT-CL J0330-5227, as evidenced by the tighter χ ellipse about the best-fit masses in Figure 12(b). Coupled withthe fact that the high- z cluster mass experiences significantlygreater variance in all fits, Figure 12 suggest that A3128 dom-inates the WL signal embedded in the background galaxy cat-alogs. NFW Fits to Substructures
We also attempted to fit masses to the two high-significance, small-aperture substructures identified withinthe A3128 peak (A1 and A2 in Table 2). The lowest χ fitassigned a mass of 1 . × M (cid:12) to the northeastern substruc-ture and the minimum boundary value mass of 0 . × M (cid:12) to the southwestern. These results were replicated in a simul-taneous 3 NFW shear profile fit which also included ACT-CLJ0330-5227. Although the combined masses of the two sub-structures matches the best-fit A3128 mass in Table 3, theirallocation appears inconsistent with the larger size and higher S / N of the southwestern substructure and the fact that it satu-rates significance maps sooner than the northeastern substruc-ture. The shunting of the A3128 mass to the northeast aper-ture mass likely reflects its proximity to the cluster barycenter(400 pixels away) compared to the southwest aperture mass(over 1000 pixels away). Figure 12(c) supports an interpre-tation where the total mass of the cluster is constrained to1 . × M (cid:12) by this set of fits, but that the two substructurescannot be resolved with individual NFW profiles.Finally, NFW masses were fit to the two infalling groupsidentified in Figure 11 (G1 and G2 in Table 2) in a simul-taneous 4 NFW shear profile fit with A3128 and ACT-CLJ0330-5227. To guarantee the distinctness of G1 and G2 fromthe shear profiles of the two clusters, A3128 and ACT-CLJ0330-5227 masses were fixed to their respective full sam-ple / Centroid 1 values of 1 . × M (cid:12) and 1 . × M (cid:12) .The fits returned masses of 2 . ± . × M (cid:12) for G1, and2 . ± . × M (cid:12) for G2, where the error bars come from the variance of 2000 jackknife resamplings, as before. Global Tests of NFW Fitting Procedure
In Figure 13, the best-fit A3128 single-peak tangential shearprofile is compared with the azimuthally averaged galaxy el-lipticity signal. We again emphasize that mass estimates inthis work do not result from fitting 1-D NFW profiles tobinned galaxy ellipticities; the shear profiles shown below arefor comparison only. The wide area of the A3128 observationand large number of background galaxies allows for a fine ra-dial binning and detailed inspection of galaxy ellipticity sig-nal. The negative value in the first radial bin is a manifestationof the PSF undercorrection near the cluster center discussedin § R − R c , the galaxy ellipticity signal shouldapproach zero; however Figure 13 shows a noticeable down-ward trend in galaxy ellipticity. As in § ff ect of raising the masses by 10%.Uncertainties on cluster masses in Table 3 presume that ourmeasurements obey Gaussian statistics. This assumption maybe tested using the distribution of cluster masses returned bythe jackknife procedure. The left panel of Figure 14 showsa 2-D histogram of cluster masses returned by 4000 randomresamplings of the full galaxy catalog. The right panel of Fig-ure 14 shows the distribution of allowed A3128 NFW massesreturned in the resampling, i.e., the left panel collapsed alongthe y-direction of ACT-CL J0330-5227 mass. Starting at thethe median mass ( ∼ . × M (cid:12) ), we sum 34.1%, 47.7% and49.9% of the returned masses on either side of the distribution.The equivalent 68% confidence interval is (7 . , . × M (cid:12) and the 95% confidence interval is (5 . , . × M (cid:12) . Whensumming over 99% of the returned mass range, the bound onthe low-mass end is 4 . × M (cid:12) . However, the upper bound-ary value of the sampled mass range is reached before we canfind an equivalent upper bound. The 68% confidence intervalis symmetric about the median (i.e., 1 . + . − . × M (cid:12) ), and isalso roughly equivalent to the 1 σ variance of Table 3. How-ever, the 95% and 99% confidence intervals are not equiva-lent to 2 and 3 σ , nor are they symmetric about the median.This skewness to high masses suggests that the distribution4 McCleary et al. . . . . . . . A3128 Mass ( M (cid:12) ) . . . . . . . AC T - C L J0330 - M a ss ( M (cid:12) ) . . . . . . . A3128 Mass ( M (cid:12) ) . . . . . . . AC T - C L J0330 - M a ss ( M (cid:12) ) . . . . . . . Southwestern Aperture Mass ( M (cid:12) ) . . . . . . . N o r t h e a s t e r n A p e r t u r e M a ss ( M (cid:12) ) . . . . . . . . . . . Figure 12. χ landscapes for fits of NFW shear profiles to the WL peaks identified in the reconstructions. For ease of viewing, the values have been rescaled toeach distribution’s respective ( χ − χ min ) × Left:
Residuals from parametric mass fits to ACT-CL J0330-5227 and A3128 (centered at its barycenter) madewith the full galaxy sample.
Center:
Residuals from parametric mass fits to ACT-CL J0330-5227 and A3128 (centered at its barycenter) with z gal > . Right:
Residuals from fits to the two A3128 substructures with the mass of ACT-CL J0330-5227 fixed to 1 . × M (cid:12) . R-R c (arcmin) − . − . − . − . . . . . . . R e d u c e d s h e a r (cid:15) tan (cid:15) g NFW
Figure 13.
Tangential shear profile for the 1-peak NFW fit to the A3128barycenter (solid red line), overplotted on the azimuthually averaged tangen-tial ellipticity signal of background galaxies (solid blue line). The dashed lineis the B-mode signal of galaxy ellipticity. Error bars on the galaxy ellipticitysignal are the value of reduced shear in a radial bin ( ∼ (cid:104) e tan (cid:105) / √
2) divided by √ N galaxies in that bin. The best-fit mass of 1 . × M (cid:12) was used for thetheoretical curve. of allowed masses departs from Gaussianity at the endpoints,though it is Gaussian near the best-fit value.The finite resolution of our significance maps leads to to anuncertainty in the coordinates of WL peak centroids. Any re-sulting mis-centering of NFW profiles might bias the reportedmasses. This potential systematic was probed through a slewof mass fits in which the identified centers of A3128 and ACT-CL J0330-5227 were individually shifted north, south, east,and west by 200 pixels (the size of the centroid uncertainty).The mass of ACT-CL J0330-522 varied an average of 7.3%from its Table 3 value, while the mass of A3128 only variedby 2.7%. Since the variations are smaller than the error barsreported in Table 3, the uncertainty of centroid coordinates isprobably not a dominant source of error in our results. DISCUSSION
With increasing R S , the smaller aperture mass peaks de-tected in the periphery of A3128 such as ACO S 366 (B1 in Figure 10) display the gradual increase and decrease ofsignificance and S / N expected from § σ = .
42 at allSchirmer radii considered. In addition, the two A3128 sub-structures merge into the primary cluster signal at R S = S / N maps throughmuch larger kernel sizes. These results suggest that the pri-mary cluster and its substructures are detected at much higherconfidence than 4 . σ . They also underscore a fundamentallimitation of the significance maps, which are only as goodas the number of random noise iterations performed. Un-fortunately, the gain in significance is a slow function of thenumber of random maps. For example, to achieve σ = . × iterations, and σ = .
33 requires 1 × iterations or 100 times more computation time. Given unlim-ited computational resources, the two substructures could beteased apart from the primary cluster signal at R S > σ of detection confi-dence.We note that the departure from Gaussianity in Figure 14does not invalidate our use of significance maps to evaluateWL signal. The “ σ ” of a significance map is a Gaussian-equivalent confidence, representing a signal pixel’s distancefrom the mean, and is rooted in an exact pixel probability dis-tribution (see § σ of our significance maps – as well as our S / N values, whichassume a Gaussian noise distribution – would in fact onlybe pseudo-Gaussian. However, the relative pixel-to-pixel en-hancements in significance maps would still be genuine: a 2 σ peak is still 95.4% less likely than the mean value of WLsignal, a 3 σ peak is still 99.7% less likely than the mean valueof WL signal, etc. For this reason, the significance maps andthe distribution of allowed NFW masses are complementary:while jackknife resampling makes no assumptions about thedistribution of WL signal pixels, it must be remade for eachaperture mass as it contains no spatial information. On theother hand, the significance maps (whether strictly Gaussianor not) do reveal the 2-D distribution of projected mass.While ACT-CL J0330-5227, A3128, and the A3128 sub-structures are all unambiguously detected in WL reconstruc-tions at very high significance, their parametric mass fits aresubject to some degeneracy. Both in simultaneous NFW fitsto the two clusters and in simultaneous fits to the A3128 sub-ass Substructure in Abell 3128 15 Figure 14.
Left:
Right:
Distribution of most probable NFW masses for A3128 with an unconstrained ACT-CLJ0330-5227 mass. This plot is equivalent to projecting the 2-D histogram at left along its y-axis of ACT-CL J0330-5227 masses. The spike at the end of thex-axis is caused by projecting all masses greater than 1 . × into the highest bin. structures, the sum of the masses is more tightly constrainedthan their respective magnitudes. This is evident from allthree χ plots of Figure 12, in which the best-fit ellipses tiltabout a line of constant mass. Table 3 also attests to the de-generacy of mass fits: with both the full and high-redshiftgalaxy samples, the aggregate mass of ACT-CL J0330-5227and A3128 is conserved, but their respective magnitudes varyat the (cid:46) σ level depending on the chosen A3128 NFW pro-file centroid. The significant dependence of cluster mass onA3128 centroid signals that the exact assignment of mass inour two-peak fits is driven by sources near the center of thecluster, and that galaxies further away only bear the overallsignal. As it happens, the source density in our observation isdepressed near the cluster center – a consequence of studyinglow redshift clusters whose member galaxies subtend wideangles. This explains the degeneracy in our parametric massfits, and particularly our inability to resolve the A3128 sub-structures with NFW shear profiles.Figure 13 shows a downward trend in galaxy ellipticity atdistances past 50-60 arcminutes from the cluster center, whichwe attributed above to an under-correction of the “pincush-ion” in the DECam PSF. A residual radial gradient in the PSFwas also invoked explain the non-zero correlation at smallscales visible in Figure 2, and the negative galaxy signal inFigure 8. To avoid underestimating the WL signal in the ob-servation, the NFW mass fits were supplied exclusively withgalaxies within 58 arcminutes of the cluster center. In fu-ture work, we will attempt to improve our PSF circulariza-tion scheme, perhaps adopting the method of Miyatake et al.(2013) to reject exposures with unsatisfactory PSF correction.Aside from highlighting the shortcomings of our PSF cor-rection scheme, Figure 13 has another interesting feature. Thelarge number of galaxies in our observation allows for a finebinning of the galaxy ellipticity signal which, in turn, allowsus to appraise the success of an NFW profile in describing it.Ignoring the inner 5 arcminutes, it is clear that the galaxy el-lipticity distribution tends to zero faster than predicted by anNFW fit: galaxy ellipticities are already consistent with zeroby 35 arcminutes, while the NFW fit does not settle to zeroon the scale of the image. Cosmological simulations such ashave been published by Diemer & Kravtsov (2014) report asimilar conclusion, that the density profiles of cluster-size ha-los fall o ff more quickly than predicted by either the NFW or Einasto mass models. In future work, we will attempt to fitthe galaxy ellipticity distributions with the latest generationof mass distributions. CONCLUSIONS
The significance maps and the distribution of most probablemasses in Figure 14 are completely independent ways of con-firming the WL signal from Abell 3128. Significance mapsemploy the aperture mass statistic of Equation 6 to sum upbackground galaxies’ tangential ellipticity and return the con-vergence signal at each point in the observation. Figure 14results from the parametric fitting of an NFW halo shear pro-file to the tangential ellipticities of background galaxies. Witheither method, the weak lensing signal from Abell 3128 is de-tected at high significance. In particular, the probability dis-tribution of A3128 mass obtained by randomly resampling ofthe full galaxy catalog skews to high masses, from which itfollows that low values of A3128 mass are more strongly dis-favored than higher masses. Our confidence in the WL de-tection of A3128 is thus higher than might be indicated bysignificance maps alone.Given a su ffi cient density of background galaxies (and withthe caveat that the number of degrees of freedom decreaseswith added NFW peaks), the tools we have developed allowfor the fitting of NFW masses to an arbitrary number of sub-structures. Our average source density of 20 galaxies per ar-cminute was enough to constrain the mass of Abell 3128 to(1 . ± . × M (cid:12) through both the simultaneous NFWfits with the high-redshift cluster ACT-CL J0330-5227 and thesubstructure mass fits. However, the background galaxy den-sity drops to ∼
13 arcmin − near the cluster center becauseof the large apparent size of its member galaxies. Combinedwith the unfavorable distance ratio of background galaxiesin our observations, these issues prevent the fitting of NFWmasses to the two central substructures. Moving forward,we expect that all WL studies of low-redshift galaxy clus-ters will be similarly a ff ected, and this places a lower limiton the source density required to resolve their substructureswith NFW shear profiles.Since they can be mistaken for mass substructures, inter-loping high-redshift clusters like ACT-CL J0330-5227 pose aproblem for systematic studies of low-redshift clusters. Evenwhen they are not fully resolved in WL convergence maps,6 McCleary et al. Figure 15.
Abell 3128 raw convergence map for a 10000 pixel Schimer aperture size, colorized and superimposed on zrg composite image. The bright clumpon the top left is Abell 3128; the fainter clumps on the bottom left are from infalling groups at z = . − .
08. The dispersed arc of signal along the bottom isfrom Abell 3125, a cluster disturbed by a recent passage near Abell 3128 (Werner et al. high-redshift background clusters can have knock-on e ff ectson NFW mass estimates because their presence may blur thelocation of the lower-redshift cluster’s barycenter. The testsperformed in § z ≥ .
44. The success of the redshift tests sug-gests a natural means of authenticating potential mass sub-structures: a localized WL signal enhancement that appearsonly when distant galaxies are used – like ACT-CL J0330-5227 in Figure 9 – is almost certainly an interloping high-redshift cluster.Abell 3128 is one of the lowest redshift clusters to havebeen studied with weak gravitational lensing in such detail.The advent of wide-angle cameras such as DECam makes thesystematic studies of low-redshift clusters possible. In partic-ular, the work on A3128 presented here is the pilot for our WLstudy of the mass distributions of a complete, volume-limitedsample of massive galaxy clusters between 0 . < z < . ff orts in per-forming all observations made during the DECam scienceverification phase, including those of Abell 3128 used inthis study. The authors also thank the anonymous refereefor suggesting many useful improvements to the analysis.This project used data obtained with the Dark Energy Cam-ass Substructure in Abell 3128 17era (DECam), which was constructed by the Dark EnergySurvey (DES) collaborating institutions: Argonne NationalLab, University of California Santa Cruz, University of Cam-bridge, Centro de Investigaciones Energeticas, Medioambi-entales y Tecnologicas-Madrid, University of Chicago, Uni-versity College London, DES-Brazil consortium, Universityof Edinburgh, ETH-Zurich, University of Illinois at Urbana-Champaign, Institut de Ciencies de l’Espai, Institut de Fisicad’Altes Energies, Lawrence Berkeley National Lab, Ludwig-Maximilians Universitat, University of Michigan, NationalOptical Astronomy Observatory, University of Nottingham,Ohio State University, University of Pennsylvania, Universityof Portsmouth, SLAC National Lab, Stanford University, Uni-versity of Sussex, and Texas A&M University. Funding forDES, including DECam, has been provided by the U.S. De-partment of Energy, National Science Foundation, Ministryof Education and Science (Spain), Science and TechnologyFacilities Council (UK), Higher Education Funding Council(England), National Center for Supercomputing Applications,Kavli Institute for Cosmological Physics, Financiadora de Es-tudos e Projetos, Fundao Carlos Chagas Filho de Amparo aPesquisa, Conselho Nacional de Desenvolvimento Cientfico eTecnolgico and the Ministrio da Cincia e Tecnologia (Brazil),the German Research Foundation-sponsored cluster of excel-lence “Origin and Structure of the Universe” and the DES col-laborating institutions. REFERENCESAndernach, H., Plionis, M., L´opez-Cruz, O., Tago, E., & Basilakos, S. 2005,Nearby Large-Scale Structures and the Zone of Avoidance, 329, 289Bartelmann, M., & Schneider, P. 2001, Phys. Rep., 340, 291Ben´ıtez, N. 2000, ApJ, 536, 571Bernstein, G. M., & Jarvis, M. 2002, AJ, 123, 583Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393Bhattacharya, S., Habib, S., Heitmann, K., & Vikhlinin, A. 2013, ApJ, 766,32Clowe, D., De Lucia, G., & King, L. 2004, MNRAS, 350, 1038Coe, D., Ben´ıtez, N., Broadhurst, T., & Moustakas, L. A. 2010, ApJ, 723,1678DePoy, D. L., Abbott, T., Annis, J., et al. 2008, Proc. SPIE, 7014,Diemer, B., & Kravtsov, A. V. 2014, arXiv:1407.4730 Du ffff