arXiv:0810.0969v1 [astro-ph] 6 Oct 2008 · 2019. 5. 3. · arxiv:0810.0969v1 [astro-ph] 6 oct 2008...

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8DRAFT VERSIONFEBRUARY 25, 2019Preprint typeset using LATEX style emulateapj v. 10/10/03

MASS AND HOT BARYONS IN MASSIVE GALAXY CLUSTERS FROM SUBARU WEAK LENSING AND AMIBA SZEOBSERVATIONS1

KEIICHI UMETSU2,3, MARK BIRKINSHAW4, GUO-CHIN L IU2,5, JIUN-HUEI PROTY WU6,3, ELINOR MEDEZINSKI7, TOM BROADHURST7,DORON LEMZE7, ADI ZITRIN7, PAUL T. P. HO2, CHIH-WEI LOCUTUSHUANG6,3, PATRICK M. KOCH2, YU-WEI L IAO6,3, KAI -YANGL IN2,6, SANDOR M. M OLNAR2, HIROAKI NISHIOKA2, FU-CHENG WANG6,3, PABLO ALTAMIRANO 2, CHIA -HAO CHANG2, SHU-HAO

CHANG2, SU-WEI CHANG2, M ING-TANG CHEN2, CHIH-CHIANG HAN2, YAU-DE HUANG2, YUH-JING HWANG2, HOMIN JIANG2,M ICHAEL KESTEVEN8, DEREK Y. K UBO2, CHAO-TE L I2, PIERRE MARTIN-COCHER2, PETEROSHIRO2, PHILIPPE RAFFIN2, TASHUN

WEI2, WARWICK WILSON8

Draft version February 25, 2019

ABSTRACTWe present a multiwavelength analysis of a sample of four hot(TX > 8keV) X-ray galaxy clusters (A1689,A2261, A2142, and A2390) using joint AMiBA Sunyaev-Zel’dovich effect (SZE) and Subaru weak lensingobservations, combined with published X-ray temperatures, to examine the distribution of mass and the intra-cluster medium (ICM) in massive cluster environments. Our observations establish that A2261 is very similarto A1689 in terms of both weak and strong lensing properties.Many tangential arcs are visible around A2261,with an effective Einstein radius∼ 40′′ (at z∼ 1.5), which when combined with our weak lensing measure-ments implies a mass profile well fitted by an NFW model with a high concentrationcvir ∼ 10, similar to A1689and to other massive clusters. The cluster A2142 shows complex mass substructure, and displays a shallowerprofile (cvir ∼ 5), being well traced by the SZE in the AMiBA map, and consistent with detailed X-ray ob-servations which imply recent interaction. For A2390 we obtain highly elliptical mass and ICM distributionsat all radii, consistent with other X-ray and strong lensingwork. Our cluster gas fraction measurements, freefrom the hydrostatic equilibrium assumption, are overall in good agreement with published X-ray and SZEobservations, with the sample-averaged gas fraction of〈 fgas(< r200)〉 = 0.133± 0.027, for our sample with〈Mvir〉 = (1.2±0.1)×1015M⊙h−1. When compared to the cosmic baryon fractionfb = Ωb/Ωm constrained bythe WMAP 5-year data, this indicates〈 fgas,200〉/ fb = 0.78±0.16, i.e., (22±16)% of the baryons are missingfrom the hot phase of clusters.Subject headings:cosmology: observations, cosmic microwave background — galaxies: clusters: individual

(A1689, A2142, A2261, A2390) — gravitational lensing

1. INTRODUCTION

Clusters of galaxies, the largest virialized systems known,are key tracers of the matter distribution in the large scalestructure of the Universe. In the standard picture of cosmicstructure formation, clusters are mostly composed of darkmatter (DM) as indicated by a great deal of observational evi-dence, with the added assumptions that DM is non relativistic(cold) and collisionless, referred to as CDM. Strong evidencefor substantial DM in clusters comes from multiwavelengthstudies of interacting clusters (Markevitch et al. 2002), inwhich weak gravitational lensing of background galaxies en-ables us to directly map the distribution of gravitating matterin merging clusters regardless of the physical/dynamical stateof the system (Clowe et al. 2006; Okabe & Umetsu 2008).The bulk of the baryons in clusters, on the other hand, residein the X-ray emitting intracluster medium (ICM), where theX-ray surface brightness traces the gravitational mass domi-

1 Based in part on data collected at the Subaru Telescope, which is oper-ated by the National Astronomical Society of Japan

2 Institute of Astronomy and Astrophysics, Academia Sinica,P. O. Box23-141, Taipei 10617, Taiwan

3 Leung center for Cosmology and Particle Astrophysics, National TaiwanUniversity, Taipei 10617, Taiwan

4 University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.5 Tamkang University, 251-37 Tamsui, Taipei County, Taiwan6 Department of Physics, National Taiwan University, Taipei10617, Tai-

wan7 Tel Aviv University, Tel Aviv 69978, Israel8 Australia Telescope National Facility, P.O.Box 76, EppingNSW 1710,

Australia

nated by DM. The remaining baryons are in the form of lu-minous galaxies and faint intracluster light (Fukugita et al.1998; Gonzalez et al. 2005). Since rich clusters represent highdensity peaks in the primordial fluctuation field, their bary-onic mass fraction and its redshift dependence can in prin-ciple be used to constrain the background cosmology (e.g.,Sasaki 1996; Allen et al. 2002, 2004, 2008). In particular,the gas mass to total mass ratio (the gas fraction) in clusterscan be used to place a lower limit on the cluster baryon frac-tion, which is expected to match the cosmic baryon fraction,fb ≡ Ωb/Ωm. However, non-gravitational processes associ-ated with cluster formation, such as radiative gas cooling andAGN feedback, would break the self-similarities in clusterproperties, which can cause the gas fraction to acquire somemass dependence (Bialek et al. 2001; Kravtsov et al. 2005).

The deep gravitational potential wells of massive clustersgenerate weak shape distortions of the images of backgroundsources due to differential deflection of light rays, result-ing in a systematic distortion pattern around the centers ofmassive clusters, known as weak gravitational lensing (e.g.,Umetsu et al. 1999; Bartelmann & Schneider 2001). In thepast decade, weak lensing has become a powerful, reliablemeasure to map the distribution of matter in clusters, dom-inated by invisible DM, without requiring any assumptionabout the physical/dynamical state of the system (e.g., Cloweet al. 2006; Okabe & Umetsu 2008). Recently, cluster weaklensing has been used to examine the form of DM density pro-files (e.g., Broadhurst et al. 2005b, 2008; Mandelbaum et al.2008; Umetsu & Broadhurst 2008), aiming for an observa-

http://arxiv.org/abs/0810.0969v1

2 Mass and Hot Baryons in Massive Galaxy Clusters

tional test of the equilibrium density profile of DM halos andthe scaling relation between halo mass and concentration, pre-dicted byN-body simulations in the standard Lambda ColdDark Matter (ΛCDM) model (Spergel et al. 2007; Komatsuet al. 2008). Observational results show that the form of lens-ing profiles in relaxed clusters is consistent with a continu-ously steepening density profile with increasing radius, welldescribed by the general NFW model (Navarro et al. 1997),expected for collisionless CDM halos.

The Yuan-Tseh Lee Array for Microwave BackgroundAnisotropy (Ho et al. 2008) is a platform-mounted interfer-ometer array of up to 19 elements operating at 3mm wave-length, specifically designed to study the structure of the cos-mic microwave background (CMB) radiation. In course of theAMiBA experiment we have conducted Sunyaev-Zel’dovicheffect (SZE) observations at 94GHz towards six massiveAbell clusters with the 7-element compact array (Wu et al.2008a). At 94GHz, the SZE signal is a temperature decre-ment in the CMB sky, and is a measure of the thermal gaspressure in the intra-cluster medium integrated along the lineof sight (Birkinshaw 1999; Rephaeli 1995). Therefore it israther insensitive to the cluster core as compared with the X-ray data, allowing us to trace the distribution of the ICM outto large radii.

This paper presents a multiwavelength analysis of fournearby massive clusters in the AMiBA sample, A1689,A2261, A2142, and A2390, for which high-quality deep Sub-aru images are available for accurate weak lensing measure-ments. This AMiBA lensing sample represents a subset ofthe high-mass clusters that can be selected by their high(TX > 8keV) gas temperatures (Wu et al. 2008a). Our jointweak lensing and SZE observations, combined with support-ing X-ray information available in the published literature,will allow us to constrain the cluster gas fractions withoutthe assumption of hydrostatic equilibrium (Myers et al. 1997;Umetsu et al. 2005), complementing X-ray based studies.Our companion papers complement details of the instruments,system performance and verification, observations and dataanalysis, and early science results from AMiBA. Ho et al.(2008) describe the design concepts and specifications of theAMiBA telescope. Technical aspects of the instruments aredescribed in Chen et al. (2008) and Koch et al. (2008a). De-tails of the first SZE observations and data analysis are pre-sented in Wu et al. (2008a). Nishioka et al. (2008) assessthe integrity of AMiBA data with several statistical tests.Linet al. (2008) discuss the system performance and verification.Liu et al. (2008) examine the levels of contamination fromforeground sources and the primary CMB radiation. Kochet al. (2008b) present a measurement of the Hubble constantH0 from AMiBA SZE and X-ray data. Huang et al. (2008)discuss cluster scaling relations between AMiBA SZE and X-ray observations.

The paper is organized as follows. We briefly summarize in§2 the basis of cluster SZE and weak lensing. In §3 we presenta concise summary of the AMiBA target clusters and observa-tions. In §4 we describe our weak lensing analysis of Subaruimaging data, and derive lensing distortion and mass profilesfor individual clusters. In §5 we examine and compare clusterellipticity and orientation profiles on mass and ICM structurein the Subaru weak lensing and AMiBA SZE observations.In §6 we present our cluster models and method for measur-ing cluster gas fraction profiles from joint weak-lensing andSZE observations, combined with published X-ray tempera-ture measurements; we then derive cluster gas fraction pro-

files, and constrain the sample-averaged gas fraction profilefor our massive AMIBA-lensing clusters. Finally, summaryand discussions are given in §7.

Throughout this paper, we adopt a concordanceΛCDM cosmology with Ωm0 = 0.3, ΩΛ0 = 0.7, andh ≡ H0/(100kms−1Mpc−1) = 0.7. Cluster properties aredetermined at the virial radiusrvir and radii (r200, r500, r2500),corresponding to overdensities (200,500,2500) relative to thecritical density of the universe at the cluster redshift.

2. BASIS OF CLUSTER SUNYAEV-ZEL’DOVICH EFFECT ANDWEAK LENSING

2.1. Sunyaev-Zel’dovich Effect

We begin with a brief summary of the basic equations of thethermal SZE. Our notation here closely follows the standardnotation of Rephaeli (1995).

The SZE is a spectral distortion of the CMB radiation re-sulting from the inverse Compton scattering of cool CMBphotons by the hot ICM. The explicit form of the spectralchange was obtained by Sunyaev-Zel’dovich (1972) from theKompaneets equation in the non-relativistic limit. The changein the CMB intensityICMB due to the SZE is written in termsof its spectral functiong and of the integral of the electronpressure along the line-of-sight as (Rephaeli 1995; Birkin-shaw 1999; Carlstrom et al. 2002):

∆ISZE(ν) = Inormg[x(ν)]y(θ), (1)

where x(ν) is the dimensionless frequency,x ≡hν/(kBTCMB) ≈ 1.66(ν/94GHz), with kB beingthe Boltzmann constant andTCMB = 2.725K be-ing the CMB temperature at the present-day epoch,Inorm = (2h/c3) (kBTCMB/h)2 ≃ 2.7× 108JySr−1, andy(θ) isthe Comptonization parameter defined as

y =∫ +rmax

−rmax

dlσthne

(

kBTe

mec2

)

=σth

mec2

∫ +rmax

−rmax

dlρgas

µempkBTgas,

(2)whereσth, me, c, andµe are the Thomson cross section, theelectron mass, the speed of light, and the mass per electronin units of proton massmp, respectively; for a fully ionizedH-He plasma,µe = 2/(1+X)≃ 1.14, withX being the Hydro-gen primordial abundance by mass fraction,X ≃ 0.76; rmax isthe cutoff radius for an isolated cluster (see §6.3). The SZEspectral functiong(x) is expressed as

g(x) = gNR(x)[

1+ δSZE(x,Tgas)]

, (3)

where gNR(x) is the thermal spectral function in the non-relativistic limit (Sunyaev & Zel’dovich 1972),

gNR(x) =x4ex

(ex − 1)2

(

xex + 1ex − 1

− 4

)

, (4)

which is zero at the cross-over frequencyx0 ≃ 3.83, orν0 =217GHz, andδSZE(x,Tgas) is the relativistic correction (Challi-nor & Lasenby 1998; Itoh et al. 1998). The fractional intensitydecrease due to SZE with respect to the primary CMB is max-imized atν ∼ 100GHz (see Figure 1 of Zhang et al. 2002),which is well matched to the observing frequency range 86–102GHz of AMiBA. At the central frequencyνc = 94GHz ofAMIBA, g(x) ≃ −3.4. For our hot X-ray clusters withTX = 8–10keV, the relativistic correction to the thermal SZE is about6–7% atνc = 94GHz.

Umetsu et al. 3

2.2. Cluster Weak Lensing

Weak gravitational lensing is responsible for the weakshape-distortion and magnification of the images of back-ground sources due to the gravitational field of interveningforeground clusters of galaxies and large scale structuresinthe universe. The deformation of the image can be describedby the 2×2 Jacobian matrixAαβ (α,β = 1,2) of the lens map-ping. The JacobianAαβ is real and symmetric, so that it canbe decomposed as

Aαβ = (1−κ)δαβ −Γαβ, (5)

Γαβ =

(

+γ1 γ2γ2 −γ1

)

, (6)

whereδαβ is Kronecker’s delta,Γαβ is the trace-free, sym-metric shear matrix withγα being the components of spin-2 complex gravitational shearγ := γ1 + iγ2, describing theanisotropic shape distortion, andκ is the lensing convergenceresponsible for the trace-part of the Jacobian matrix, describ-ing the isotropic area distortion. In the weak lensing limitwhereκ, |γ| ≪ 1,Γαβ induces a quadrupole anisotropy of thebackground image, which can be observed from ellipticitiesof background galaxy images. The flux magnification due togravitational lensing is given by the inverse Jacobian determi-nant,

µ =1

detA =1

(1−κ)2 − |γ|2 , (7)

where we assume subcritical lensing, i.e., detA(θ) > 0.The lensing convergence is expressed as a line-of-sight pro-

jection of the matter density contrast out to the source plane(S) weighted by certain combinationg of co-moving angulardiameter distances (e.g., Jain et al. 2000),

κ =3H2

0Ωm

2c2

∫ χs

0dχg(χ,χS)

δ

a≡∫

dΣmΣ−1crit (8)

where a is the cosmic scale factor,χ is the co-movingdistance;Σm is the surface mass density of matter,Σm =∫ χs

0 dχa(ρm − ρ), with respect to the cosmic mean densityρ,andΣcrit is the critical surface mass density for gravitationallensing,

Σcrit =c2

4πGDs

DdDds(9)

with Ds, Dd, andDds being the angular diameter distancesfrom the observer to the source, from the observer to the de-flecting lens, and from the lens to the source, respectively.Fora fixed background cosmology and a lens redshiftzd, Σcrit isa function of background source redshiftzs. For a given massdistributionΣ(θ), the lensing signal is proportional to the an-gular diameter distance ratio,Dds/Ds.

In the present weak lensing study we aim to reconstruct thedimensionless surface mass densityκ from weak lensing dis-tortion and magnification data. To do this, we utilize the rela-tion between the gradients ofκ andγ (Kaiser 1995; Critten-den et al. 2002),

κ(θ) = ∂α∂βΓαβ(θ) = 2D∗γ(θ) (10)

whereD is the complex differential operatorD = (∂21 −∂2

2)/2+i∂1∂2. The Green’s function for the two-dimensional Pois-son equation is−1(θ,θ′) = ln |θ −θ

′|/(2π), so that equation(10) can be solved to yield the following non-local relationbetweenκ andγ (Kaiser & Squires 1993):

κ(θ) =1π

∫

d2θ′ D∗(θ −θ′)γ(θ′) (11)

whereD(θ) is the complex kernel defined as

D(θ) =θ2

2 − θ21 − 2iθ1θ2

|θ|4 . (12)

Similarly, the spin-2 shear field can be expressed in terms ofthe lensing convergence as

γ(θ) =1π

∫

d2θ′ D(θ−θ′)κ(θ′). (13)

Note that adding a constant mass sheet toκ in equation (13)does not change the shear fieldγ(θ) which is observable in theweak lensing limit, leading to the so-calledmass-sheet degen-eracy based solely on shape-distortion measurements (e.g.,Bartelmann & Schneider 2001; Umetsu et al. 1999). In gen-eral, the observable quantity is not the gravitational shear γbut thereducedshear,

g =γ

1−κ(14)

in the subcritical regime where detA> 0 (or 1/g∗ in the neg-ative parity region with detA < 0). We see that the reducedshearg is invariant under the following global transformation:

κ(θ) → λκ(θ) + 1−λ, γ(θ) → λγ(θ) (15)

with an arbitrary scalar constantλ 6= 0 (Schneider & Seitz1995).

3. AMIBA SUNYAEV-ZEL’DOVICH EFFECT OBSERVATIONS

3.1. AMIBA Telescope

The AMiBA is a dual channel 86–102GHz (3-mm wave-length) interferometer array of up to 19-elements with dualpolarization capabilities sited at 3396m on Mauna-Loa,Hawaii (latitude:+19.5, longitude:−155.6) 9. AMiBA isequipped with 4-lag analog, broadband (16GHz bandwidthcentered at 94GHz) correlators which output a set of 4 real-number correlation signals (Chen et al. 2008). This fourdegrees-of-freedom (dof) correspond to two complex visibil-ities at two frequency channels. The frequency of AMiBAoperation is to take full advantage of the optimal frequencywindow at 3mm, where the fractional decrement in the SZEintensity relative to the primary CMB is close to its max-imum (see §2.1) and contamination by the Galactic syn-chrotron emission, dust foregrounds, and the population ofcluster/background radio sources is minimized (see for de-tailed contamination analysis, Liu et al. 2008). This makesAMiBA a unique CMB/SZE interferometer, and also comple-ments the wavelength coverage of other existing and plannedCMB instruments. interferometers such as CBI at 30GHz(Padin et al. 2001, 2002; Mason et al. 2003; Pearson et al.2003), SZA at 30 and 90GHz (Mroczkowski et al. 2008), andVSA10 at 34GHz (Watson et al. 2003); bolometer arrays suchas ACT,11 APEX-SZ12 (Halverson et al. 2008), and SPT.13

In the initial operation of AMiBA, we used seven 0.6m(0.58m to be precise) Cassegrain antennas (Koch et al. 2006)co-mounted on a 6m hexapod platform in a hexagonal close-packed configuration (see Ho et al. 2008). At each of the fre-quency channels centered at about 90 and 98GHz, this com-pact configuration provides 21 simultaneous baselines with

9 http://amiba.asiaa.sinica.edu.tw/10 http://astro.uchicago.edu/sza/11 http://www.hep.upenn.edu/act/act.html12 http://bolo.berkeley.edu/apexsz13 http://pole.uchicago.edu


three baseline lengths ofd = 0.61, 1.05, and 1.21m, corre-sponding to angular multipolesl = 2π

√u2 + v2(= 2πd/λ) of

l ≈ 1194,2073,2394 atνc = 94GHz. This compact 7-elementarray is sensitive to the multipole range of 800<∼ l <∼ 2600.With the 0.6m antenna, the instantaneous field-of-view ofAMiBA is about 23′ FWHM (Wu et al. 2008a), and its angularresolution ranges from 2′ to 6′ depending on the configurationand weighting scheme. In the compact configuration, the an-gular resolution of AMiBA is about 6′ FWHM usingnaturalweighting(i.e., inverse noise variance weighting). The pointsource sensitivity is estimated to be∼ 63mJy (Lin et al. 2008)in 1hour of on-source integration in 2-patch main-trail/leaddifferencing observations, where the overall noise level is in-creased by a factor of

√2 due to the differencing.

3.2. Initial Target Clusters

We observed with AMiBA7 six hot X-ray clusters, A1689,A1995, A2142, A2163, A2261, and A2390, at redshfits0.09 <∼ z <∼ 0.32 from April to August 2007 over a total ofabout 40 nights. The AMiBA lensing sample, A1689, A2142,A2261, A2390, is a subset of this AMiBA cluster sample,composed of four massive clusters at relatively low redshfitsof 0.09 <∼ z <∼ 0.23 with the median redshift ofz≈ 0.2. Thesample size is simply limited by the availability of high qual-ity Subaru weak lensing data. A1689 is a relaxed, round sys-tem, and is one of the best studied clusters for lensing work(e.g., Broadhurst et al. 2005b; Limousin et al. 2007; Umetsu& Broadhurst 2008; Broadhurst et al. 2008). A2261 is a com-pact cluster with a regular X-ray morphology. A2142 is amerging cluster with two sharp X-ray surface brightness dis-continuities in the cluster core (Markevitch et al. 2000; Okabe& Umetsu 2008). A2390 shows an elongated morphologyboth in the X-ray emission and strong-lensing mass distribu-tions (Allen et al. 2001; Frye & Broadhurst 1998). Table 1summarizes the physical properties of the four target clustersin this multiwavelength study.

In 2007, AMiBA7 was in the science verification phase.For our initial observations, we therefore selected those tar-get clusters which were observable from Mauna-Loa duringthe observing period and were known to be SZE bright atrelatively low redshifts (0.1 <∼ z <∼ 0.3) from previous experi-ments, such as OVRO observations at 30GHz (Mason et al.2001), BIMA/OVRO observations at 30GHz (Grego et al.2001a; Reese et al. 2002), VSA observations at 34GHz (Lan-caster et al. 2005), SuZIE II observations at 145, 221, and355GHz (Benson et al. 2004). The targeted redshift rangeallows that target clusters can be resolved in 6′ resolution ofAMiBA7, allowing us to derive useful measurements of clus-ter SZE profiles for our multiwavelength studies. At redshiftsof z <∼ 0.3 (0.2), the angular resolution of AMiBA7 corre-sponds to <∼ 560kpch−1 (∼ 400kpch−1) in radius, which is<∼ 30–40% (∼ 20–30%) of the virial radius 1.5–2Mpch−1 of

massive clusters. The requirement of being SZE strong is toensure reliable SZE measurements at 3mm with AMiBA7.We note that AMiBA and SZA are the the only SZE instru-ments measuring at 3mm, but complimentary in their base-line coverage. With sensitivities of 20–30mJybeam−1 typi-cally achieved in 2-patch differencing observations in 5–10hours of net on-source integration (Wu et al. 2008a), targetedSZE fluxes would be>∼ 100–150mJy at 3-mm wavelengthfor a >∼ 5σ detection. Finally, our observing period (April–August 2007) limited the range of right ascension (RA) of tar-

gets,14 since we restricted our science observations to night-time (roughly 8pm to 8am), where the variation of the ambi-ent temperature is slow and small (Nishioka et al. 2008), pro-viding high gain stability. These SZE bright clusters in ourAMiBA sample are likely to have exceedingly deep potentialwells, and indeed our AMiBA sample represents a class of hotX-ray clusters with observed X-ray temperatures exceeding8keV (see Table 1). We note that this may affect the gener-ality of the results presented in this study. A main-trail/leaddifferencing scheme has been used in our targeted cluster ob-servations where the trail/lead (blank) field is subtractedfromthe main (cluster) field. This differencing scheme sufficientlyremoves contamination from ground spillover and electronicDC offset in the correlator output (Wu et al. 2008a). A fulldescription of AMiBA observations and analysis of the initialsix target clusters, including the observation strategy, analysismethodology, calibrations, and map-making, can be found inWu et al. (2008a,b).

4. SUBARU WEAK LENSING DATA AND ANALYSIS

In this section we present a technical description of ourweak lensing distortion analysis on the AMiBA lensing sam-ple based on Subaru data. The present work on A1689 isbased on the same Subaru images as analyzed in our ear-lier work of Broadhurst et al. (2005b) and Umetsu & Broad-hurst (2008), but our improved color selection of the red back-ground has increased the sample size by∼ 16% (§4.3). Thiswork on A2142 is based on the same Subaru images as inOkabe & Umetsu (2008), but our inclusion of blue, as wellas red, galaxies has increased the sample size by a factorof 4 (cf. Table 6 of Okabe & Umetsu 2008), leading to asignificant improvement of our lensing measurements. ForA2261 and A2390 we present our new weak lensing analysisbased on Suprime-Cam imaging data retrieved from the Sub-aru archive, SMOKA. The reader only interested in the mainresult may skip directly to §4.5.

4.1. Subaru Data and Photometry

We analyze deep images of four high mass clusters in theAMiBA sample taken by the wide-field camera Suprime-Cam(34′ × 27′; Miyazaki et al. 2002) at the prime-focus of the8.3m Subaru telescope. The clusters were observed deeply intwo optical passbands each with seeing in the coadded imagesranging from 0.55′′ to 0.88′′ (see Table 2). For each clusterwe select an optimal combination of two filters that allows foran efficient separation of cluster/background galaxies basedon color-magnitude correlations (see Table??). We use eitherRc or i′ band for our weak lensing measurements (describedin §4.2) for which the instrumental response, sky backgroundand seeing conspire to provide the best-quality images. Thestandard pipeline reduction software for Suprime-Cam (Yagiet al. 2002) is used for flat-fielding, instrumental distortioncorrection, differential refraction, sky subtraction andstack-ing. Photometric catalogs are constructed from stacked andmatched images using SExtractor (Bertin & Arnouts 1996),and used for our color selection of background galaxies (see§4.3).

4.2. Weak Lensing Distortion Analysis

We use the IMCAT package developed by N. Kaiser15

to perform object detection, photometry and shape measure-ments, following the formalism outlined in Kaiser et al. (1995,

14 The elevation limit of AMiBA is 30.15 http://www.ifa.hawaii.edu/ kaiser/imcat

Umetsu et al. 5

KSB). Our analysis pipeline is implemented based on the pro-cedures described in Erben et al. (2001) and on verifica-tion tests with STEP1 data of mock ground-based observa-tions (Heymans et al. 2006). The same analysis pipeline hasbeen used in Umetsu & Broadhurst (2008), Okabe & Umetsu(2008), and Broadhurst et al. (2008).

Object Detection

Objects are first detected as local peaks in the image byusing the IMCAT hierarchical peak-finding algorithmhfind-peakswhich for each object yields object parameters such asa peak position (x), an estimate of the object size (rg), the sig-nificance of the peak detection (ν). The local sky level and itsgradient are measured around each object from the mode ofpixel values on a circular annulus defined by inner and outerradii of 16× rg and 32× rg. In order to avoid contamina-tion in the background estimation by bright neighboring starsand/or foreground galaxies, all pixels within 3× rg of anotherobject are excluded from the mode calculation. Total fluxesand half-light radii (rh) are then measured on sky-subtractedimages using a circular aperture of radius 3× rg from the ob-ject center. Any pixels within 2.5× rg of another object areexcluded from the aperture. The aperture magnitude is thencalculated from the measured total flux and a zero-point mag-nitude. Any objects with positional differences between thepeak location and the weighted-centroid greater than|d| = 0.4pixels are excluded from the catalog.

Finally, bad objects such as spikes, saturated stars, andnoisy detections need to be removed from the weak lensinganalysis. We removed from our detection catalog extremelysmall or large objects withrg < 1 or rg > 10 pixels, objectswith low detection significance,ν < 7 (see Erben et al. 2001),objects with large raw ellipticities,|e| > 0.5 (see §4.2.0),noisy detections with unphysical negative fluxes, and objectscontaining more than 10 bad pixels,nbad> 10.

Weak Lensing Distortion Measurements

To obtain an estimate of the reduced shear,gα = γα/(1−κ)(α = 1,2), we measure using thegetshapesroutine in IMCATthe image ellipticityeα = Q11 − Q22,Q12/(Q11 + Q22) fromthe weighted quadrupole moments of the surface brightnessof individual galaxies defined in the above catalog,

Qαβ =∫

d2θW(θ)θαθβ I (θ) (α,β = 1,2) (16)

whereI (θ) is the surface brightness distribution of an object,W(θ) is a Gaussian window function matched to the size of theobject. In equation (16) the maximum radius of integration ischosen to beθmax = 4rg.

Firstly the PSF anisotropy needs to be corrected using thestar images as references:

e′α = eα − Pαβsm q∗β (17)

wherePsm is thesmear polarizabilitytensor being close to di-agonal, andq∗α = (P∗

sm)−1αβeβ∗ is the stellar anisotropy kernel.

We select bright, unsaturated foreground stars identified in abranch of the half-light radius vs. magnitude diagram to mea-sureq∗α. In order to obtain a smooth map ofq∗α which is usedin equation (17), we divided the co-added mosaic image (of∼ 10K×8K pixels) into rectangular blocks. The block lengthis based on the coherent scale of PSF anisotropy patterns, andis typically 2Kpixels. In this way the PSF anisotropy in indi-vidual blocks can be well described by fairly low-order poly-nomials. We then fitted theq∗ in each block independently

with second-order bi-polynomials,qα∗ (θ), in conjunction withiterative outlier rejection on each component of the residual:δe∗α = e∗α − (P∗

sm)αβq∗β(θ). The final stellar sample containstypically 500–1200 stars. Uncorrected ellipticity componentsof stellar objects have on average a mean offset of 1–2% witha few % of rms, or variation of PSF across the data field (see,e.g., Umetsu & Broadhurst 2008; Okabe & Umetsu 2008). Onthe other hand, the mean residual stellar ellipticityδe∗α aftercorrection is less than or about 10−4, with the standard erroron this measurement, a few×10−4. We show in Figure 1 thequadrupole PSF anisotropy fields as measured from stellar el-lipticities before and after the anisotropic PSF correction forour target clusters. Figure 2 shows the distributions of stel-lar ellipticity components before and after the PSF anisotropycorrection. In addition, we adopt a conservative magnitudelimit m< 25.5–26 ABmag, depending on the depth of thedata for each cluster, to avoid systematic errors in the shapemeasurement (see Umetsu & Broadhurst 2008). From the restof the object catalog, we select objects withrh > r∗h + σ(r∗h)pixels as a magnitude-selected weak lensing galaxy sample,wherer∗h is the median value of stellar half-light radiir∗h , cor-responding to half the median width of circularized PSF overthe data field, andσ(r∗h) is the rms dispersion ofr∗h .

Second, we need to correct image ellipticities for theisotropic smearing effect caused by atmospheric seeing andthe window function used for the shape measurements. Thepre-seeing reduced sheargα can be estimated from

gα = (P−1g )αβe′β (18)

with thepre-seeing shear polarizabilitytensorPgαβ defined as

Hoekstra et al. (1998),

Pgαβ = Psh

αβ −[

Psm(Psm∗)−1Psh∗]αβ

≈ Pshαβ − Psm

αβ

tr[Psh∗]tr[Psm∗]

(19)

with Psh being theshear polarizabilitytensor; In the secondequality we have used a trace approximation to the stellarshape tensors,Psh∗ andPsm∗. To apply equation (18) the quan-tity tr[Psh∗]/tr[Psm∗] must be known for each of the galaxieswith different sizescales. Following Hoekstra et al. (1998), werecompute the stellar shapesPsh∗ andPsm∗ in a range of filterscalesrg spanning that of the galaxy sizes (rg = [1,10]pixels).At each filter scalerg, the median〈tr[Psh∗]/tr[Psm∗]〉 over thestellar sample is calculated, and used in equation (19) as anestimate of tr[Psh∗]/tr[Psm∗]. Further, we adopt the scalar cor-rection scheme, namely

(Pg)αβ =12

tr[Pg]δαβ ≡ Psgδαβ (20)

(Erben et al. 2001; Hoekstra et al. 1998; Umetsu & Broad-hurst 2008). In order to suppress artificial effects due to thenoisyPs

g estimated for individual galaxies, we apply filteringto rawPs

g measurements. We compute for each object a me-dian value ofPs

g amongN-neighbors in the size and magni-tude plane to define object parameter space: firstly, for eachobject,N-neighbors with rawPs

g > 0 are identified in the size(rg) and magnitude plane; the median value ofPs

g is then usedas the smoothedPs

g for the object,〈Psg〉, and the varianceσ2

g ofg = g1 + ig2 is calculated using equation (18). The dispersionσg is used as an rms error of the shear estimate for individ-ual galaxies. We takeN = 30. Finally, we use the followingestimator for the reduced shear:gα = e′α/

⟨

Psg

⟩

.


4.3. Background Selection

It is crucial in the weak lensing analysis to make a secureselection of background galaxies in order to minimize con-tamination byunlensedcluster/foreground galaxies and henceto make an accurate determination of the cluster mass pro-file; otherwise dilution of the distortion signal arises fromthe inclusion of unlensed galaxies, particularly at small ra-dius where the cluster is relatively dense (Broadhurst et al.2005b; Medezinski et al. 2007). This dilution effect is sim-ply to reduce the strength of the lensing signal when aver-aged over a local ensemble of galaxies, in proportion to thefraction of unlensed galaxies whose orientations are randomlydistributed, thus diluting the lensing signal relative to the ref-erence background level derived from the background popu-lation (Medezinski et al. 2007).

To separate cluster members from the background andhence minimize the weak lensing dilution, we follow an ob-jective background selection method developed by Medezin-ski et al. (2007) and Umetsu & Broadhurst (2008). We selectred galaxies with colors redder than the color-magnitude se-quence of cluster E/S0 galaxies. The sequence forms a welldefined line in object color-magnitude space due to the rich-ness and relatively low redshifts of our clusters. These redgalaxies are expected to lie in the background by virtue ofk-corrections which are greater than for the red cluster se-quence galaxies; This has been convincingly demonstratedspectroscopically by Rines & Geller (2008). We also includeblue galaxies falling far from the cluster sequence to minimizecluster contamination.

Figure 3 shows for each cluster the mean distortion strengthaveraged over a wide radial range ofθ = [1′,18′] as a func-tion of color limit, done separately for the blue (left) and red(right) samples, where the color boundaries for the presentanalysis are indicated by vertical dashed lines for respectivecolor samples. Here we do not apply area weighting to en-hance the effect of dilution in the central region (see Umetsu& Broadhurst 2008). A sharp drop in the lensing signal isseen when the cluster red sequence starts to contribute signif-icantly, thereby reducing the mean lensing signal. Note thatthe background populations do not need to be complete inany sense but should simply be well defined and contain onlybackground. For A1689, the weak lensing signal in the bluesample is systematically lower than that of the red sample,so that blue galaxies in A1689 are excluded from the presentanalysis, as was done in Umetsu & Broadhurst (2008); on theother hand, our improved color selection for the red samplehas led to a∼ 16% increase of red galaxies. In the presentstudy we use for A2142 the same Subaru images as analyzedby Okabe & Umetsu (2008), but we have improved signifi-cantly our lensing measurements by including blue, as well asred, galaxies, where the sample size has been increased by afactor of 4.

An estimate of the background depth is required when con-verting the observed lensing signal into physical mass units,because the lensing signal depends on the source redshifts inproportion toDds/Ds. The mean depth is sufficient for ourpurposes as the variation of the lens distance ratio,Dds/Ds, isslow for our sample because the clusters are at relatively lowredshifts (zd ∼ 0.1−0.2) compared to the redshift range of thebackground galaxies. We estimate the mean depth〈Dds/Ds〉of the combined red+blue background galaxies by applyingour color-magnitude selection to Subaru multicolor photome-try of the HDF-N region (Capak et al. 2004) or the COSMOS

deep field (Capak et al. 2007), depending on the availabilityoffilters. The fractional uncertainty in the mean depth〈Dds/Ds〉for the red galaxies is typically∼ 3%, while it is about 5% forthe blue galaxies. It is useful to define the distance-equivalentsource redshiftzs,D (Medezinski et al. 2007; Umetsu & Broad-hurst 2008) defined as

⟨

Dds

Ds

⟩

zs

=Dds

Ds

∣

∣

∣

∣

zs=zs,D

. (21)

We find zs,D = 0.70+0.06−0.05,0.95+0.79

−0.30,0.98+0.24−0.16,1.00+0.25

−0.16 forA1689, A2142, A2261, and A2390, respectively. For thenearby cluster A2142 atz≃ 0.09, a precise knowledge of thesource redshift is not critical at all for lensing work. The meansurface number density (ng) of the combined blue+red sam-ple, the blue-to-red fraction of background galaxies (B/R), theestimated mean depth〈Dds/Ds〉, and the effective source red-shift zs,D are listed in Table 2.

4.4. Weak Lensing Map-Making

Weak lensing measurements of the gravitational shear fieldcan be used to reconstruct the underlying projected mass den-sity field. In the present study, we will use the dilution-free,color-selected background sample (§4.3) both for the 2D massreconstruction and the lens profile measurements16.

Firstly, we pixelize distortion data of background galax-ies into a regular grid of pixels using a Gaussianwg(θ) ∝exp[−θ2/θ2

f ] with θ f = FWHM/√

4ln2. Further we incorpo-rate in the pixelization a statistical weightug for an individualgalaxy, so that the smoothed estimate of the reduced shearfield at an angular positionθ is written as

gα(θ) =

∑

i wg(θ−θi)ug,igα,i∑

i wg(θ−θi)ug,i(22)

wheregα,i is the reduced shear estimate of theith galaxy at an-gular positionθi , andug,i is the statistical weight ofith galaxywhich is taken as the inverse variance,ug,i = 1/(σ2

g,i +α2), withσg,i being the rms error for the shear estimate ofith galaxy(see § 4.2.0) andα2 being the softening constant variance(Hamana et al. 2003). We chooseα = 0.4, which is a typicalvalue of the mean rmsσg over the background sample. Thecase withα = 0 corresponds to an inverse-variance weighting.On the other hand, the limitα≫ σg,i yields a uniform weight-ing. We have confirmed that our results are insensitive to thechoice ofα (i.e., inverse-variance or uniform weighting) withthe adopted smoothing parameters. The error variance for thesmoothed shearg = g1 + ig2 (22) is then given as

σ2g(θ) =

∑

i w2g,iu

2g,iσ

2g,i

(∑

i wg,iug,i)2 (23)

where wg.i = wg(θ − θi) and we have used〈gα,i gβ, j〉 =(1/2)σ2

g,iδKαβδ

Ki j with δK

αβ andδKi j being the Kronecker’s delta.

We then invert the pixelized reduced-shear field (22) to ob-tain the lensing convergence fieldκ(θ) using equation (11). Inthe map-making we assume the linear shearing in the weak-lensing limit, that is,gα = γα/(1− κ) ≈ γα. We adopt theKaiser & Squires inversion method (Kaiser & Squires 1993),

16 Okabe & Umetsu (2008) used the magnitude-selected galaxy samplein their map-making of nearby merging clusters to increase the backgroundsample size, while the dilution-free red background samplewas used in theirlensing mass measurements.

Umetsu et al. 7

which makes use of the 2D Green function in an infinitespace (§2.2). In the linear map-making process, the pix-elized shear field is weighted by the inverse of the variance(23). Note that this weighting scheme corresponds to us-ing only the diagonal part of the noise covariance matrix,N(θi ,θ j) = 〈∆g(θi)∆g(θ j )〉, which is only an approximationof the actual inverse noise weighting in the presence of pixel-to-pixel correlation due to non-local Gaussian smoothing.InTable 2 we list the rms noise level in the reconstructedκ(θ)field for our sample of target clusters. For all of the clusters,the smoothing scaleθ f is taken to beθ f = 1′ (θFWHM ≃1.665′),which is larger than the Einstein radius for our backgroundgalaxies. Hence our weak lensing approximation here is validin all clusters.

In Figure 4 we show, for the four clusters, 2D maps of thelensing convergenceκ(θ) =Σ(θ)/Σcrit reconstructed from theSubaru distortion data (§4.4), each with the correspondinggravitational shear field overlaid. Here the resolution of theκ maps is 1′ in FWHM for all of the four clusters. The sidelength of the displayed region is 22′, corresponding roughly tothe instantaneous field-of-view of AMiBA (≃ 23′ in FWHM).In the absence of higher-order effects, weak lensing only in-duces curl-freeE-mode distortions, responsible for tangentialshear patterns, while theB-mode lensing signal is expected tovanish. For each case, a prominent mass peak is visible in thecluster center, around which the lensing distortion pattern isclearly tangential.

Also shown in Figure 4 are contours of the AMiBA fluxdensity due to the thermal SZE obtained by Wu et al. (2008a).The resolution of AMiBA7 is about 6′ in FWHM (§3). TheAMiBA map of A1689 reveals a bright and compact structurein the SZE, similar to the compact and round mass distribu-tion reconstructed from the Subaru distortion data. A2142shows an extended structure in the SZE elongated along thenorthwest direction, consistent with the direction of elonga-tion in the X-ray halo having a cometary appearance (Marke-vitch et al. 2000). In addition, it shows an excess SZE inthe northwest region of the cluster, associated with the masssubstructure seen in our lensingκ map (Figure 4). Okabe &Umetsu (2008) showed that this northwest substructure is alsoassociated with a slight excess luminosity of cluster sequencegalaxies, located ahead of a cold front at the northwest edgeof the central compact X-ray core. On the other hand, no X-ray counterpart to the northwest substructure was found in theX-ray images from Chandra and XMM-Newton observations(Okabe & Umetsu 2008). A2261 shows a filamentary massstructure with unknown redshift, extending to the west of thecluster core (Maughan et al. 2008), and likely backgroundstructures which coincide with redder galaxy concentrations.Our AMiBA and Subaru observations show a compact struc-ture both in mass and ICM. The elliptical mass distribution inA2390 agrees well with the shape seen by AMiBA in the ther-mal SZE, and is also consistent with other X-ray and stronglensing work. A quantitative comparison between the AMiBASZE and Subaru lensing maps will be given in §5.

4.5. Cluster Lensing Profiles

Lens Distortion

The spin-2 shape distortion of an object due to gravita-tional lensing is described by the complex reduced shear,g = g1 + ig2 (see equation [14]), which is coordinate depen-dent. For a given reference point on the sky, one can insteadform coordinate-independent quantities, the tangential distor-

tion g+ and the 45 rotated component, from linear combina-tions of the distortion coefficientsg1 andg2 as

g+ = −(g1cos2φ+ g2sin2φ), g× = −(g2cos2φ− g1sin2φ),(24)

whereφ is the position angle of an object with respect to thereference position, and the uncertainty in theg+ andg× mea-surement isσ+ = σ× = σg/

√2 ≡ σ in terms of the rms error

σg for the complex shear measurement. In practice, the ref-erence point is taken to be the cluster center, which is welldetermined from symmetry of the strong lensing pattern. Toimprove the statistical significance of the distortion measure-ment, we calculate the weighted average ofg+ andg×, and itsweighted error, as

〈g+(θm)〉=

∑

i ug,i g+,i∑

i ug,i, (25)

〈g×(θm)〉=

∑

i ug,i g×,i∑

i ug,i, (26)

σ+(θm) =σ×(θm) =

√

√

√

√

∑

i u2g,iσ

2i

(∑

i ug,i)2 , (27)

where the indexi runs over all of the objects located within themth annulus with a median radius ofθm, andug,i is the inversevariance weight forith object,ug,i = 1/(σ2

g,i + α2), softenedwith α = 0.4 (see §4.4).

Now we assess cluster lens-distortion profiles from thecolor-selected background galaxies (§4.3) for the four clus-ters, in order to examine the form of the underlying clustermass profile and to characterize cluster mass properties. Fig-ure 5 shows the azimuthally-averaged radial profiles of thetangential distortion,g+ (E mode), and the 45-rotated com-ponent,g× (B mode), where the cluster center is well deter-mined by the locations of the brightest cluster galaxies. Herethe presence ofB modes can be used to check for systematicerrors. For each of the clusters, the observedE-mode sig-nal is significant at the 12–16σ level out to the limit of ourdata (θ ∼ 20′). The significance level of theB-mode detec-tion is about 2.5σ for each cluster, which is about a factor of5 smaller thanE-mode.

The measuredg+ profiles are compared with two represen-tative cluster mass models, namely the NFW model and thesingular isothermal sphere (SIS) model (see Appendix A).The logarithmic gradientn≡ d lnρ(r)/d lnr of the NFW den-sity profile flattens continuously towards the center of mass,with a flatter central slopen = −1 and a steeper outer slope(n→ −3 whenr →∞) than a purely isothermal body (n = −2)interior to the inner characteristic radius,rs. A useful in-dex, the concentration, compares the virial radius,rvir , tors of the NFW profile,cvir = rvir/rs. We specify the NFWmodel with the halo virial massMvir and the concentrationcvir, and the SIS model with the 1D isothermal velocity dis-persion,σv. Table 3 lists the best-fitting parameters for thesemodels, together with the predicted Einstein radiusθE for afiducial source atzs = 1.5, corresponding roughly to medianredshifts of our blue background galaxies. For each cluster,the NFW profile provides a better fit to the data than the SISprofile. Both models provide statistically acceptable fits forA1689, A2261, and A2390. For our lowest-z cluster A2142,the curvature in theg+ profile is pronounced, and a SIS modelfor A2142 is strongly ruled out by the Subaru distortion dataalone, where the minimumχ2 is χ2

min = 39 with 8 dof.


Lens Convergence

Although the lensing distortion is directly observable, theeffect of substructure on the gravitational shear is non-local.Here we examine the lens convergence (κ) profiles using theshear-based 1D reconstruction method developed by Umetsu& Broadhurst (2008). See Appendix B.1 for details of thereconstruction method.

In Figure 6 we show for the four clusters model-independentκ profiles derived using the shear-based 1D re-construction method, together with the predictions from thebest-fit NFW models for theκ(θ) andg+(θ) data. The sub-structure contribution toκ(θ) is local, whereas the inver-sion from the observable distortion toκ involves a non-localprocess. Consequently the 1D inversion method requires aboundary condition specified in terms of the meanκ valuewithin an outer annular region (lying out to 18′–19′). Wedetermine this value for each cluster using the best-fit NFWmodel for theg+ profile (Table 3).

We find that the two sets of best-fit NFW parameters arein excellent agreement for all except A2261: For A2261, thebest-fit values ofcvir from theg+ andκ profiles are marginallyconsistent at 1− 2σ level. From Figures 4 and 6 we seethat the NFW fit to theg+ profile of A2261 is affected bythe presence of mass structures atθ ≃ 4′ and 10′, result-ing in a slightly shallower profile (cvir ≃ 6.4). The NFWfit to κ(θ) yields a steeper profile with a high concentra-tion, cvir ≃ 10.2, which implies a large Einstein angle ofθE ≃ 37′′ atzs = 1.5 (Table 3). This is in good agreement withour preliminary strong-lensing model based on the methodby Broadhurst et al. (2005a), in which aflexible light de-flection field is constrained by multiply-lensed images regis-tered in deep SubaruVRc and CFHT/WIRCamJHKs images(see Figure 7). This predicts an effective Einstein radius of∼ 40′′ at zs = 1.5. This motivates us to further improve thestatistical constraints on the NFW model by combining theouter lens convergence profile with the observed constrainton the inner Einstein radius. A joint fit of the NFW profileto theκ profile and the inner Einstein-radius constraint withθE = 40′′±4′′ (zs = 1.5) tightens the constraints on the NFWparameters (see for details, §5.4.2 of Umetsu & Broadhurst2008):Mvir = 1.25+0.17

−0.16×1015M⊙h−1 andcvir = 11.1+2.2−1.9 (Table

5). In the following analysis we will adopt this as our primarymass model of A2261.

For the strong-lensing cluster A1689, more detailed lens-ing constraints are available from joint observations withthe high-resolutionHubble Space Telescope(HST) Ad-vanced Camera for Surveys (ACS) and the wide-fieldSubaru/Suprime-Cam (Broadhurst et al. 2005b; Umetsu &Broadhurst 2008). In Umetsu & Broadhurst (2008) we com-bined all possible lensing measurements, namely, the ACSstrong-lensing profile of Broadhurst et al. (2005b) and Sub-aru weak-lensing distortion and magnification data, in a fulltwo-dimensional treatment, to achieve the maximum possi-ble lensing precision. Note, the combination of distortionandmagnification data breaks the mass-sheet degeneracy inher-ent in all reconstruction methods based on distortion infor-mation alone (Bartelmann et al. 1996). It was found that thejoint ACS and Subaru data, covering a wide range of radiusfrom 10 up to 2000kpch−1, are well approximated by a sin-gle NFW profile withMvir = (1.5±0.1+0.6

−0.3)×1015M⊙h−1 andcvir = 12.7± 1± 2.8 (statistical followed by systematic un-

certainty at 68% confidence),17 which properly reproducesthe Einstein radius tightly constrained by detailed strong-lens modeling (Broadhurst et al. 2005a; Halkola et al. 2006;Limousin et al. 2007):θE ≃ 52′′ at zs = 3.05 (or θE ≃ 45′′

at a fiducial source redshift ofzs = 1). With the improvedcolor selection for the red background sample (see §4.3), wehave redone a joint fit to the ACS and Subaru lensing ob-servations using the 2D method of Umetsu & Broadhurst(2008): The refined constraints on the NFW parameters areMvir ≃ 1.55× 1015M⊙h−1 andcvir ≃ 12.3, as listed in Table4. In the following, we will adopt this refined NFW profile asour primary mass model of A1689.

5. DISTRIBUTIONS OF MASS AND HOT BARYONS

Here we aim to compare the projected distribution of massand ICM in the clusters using our Subaru weak lensing andAMiBA SZE maps shown in §4.4. To make a quantitativecomparison, we first define the “cluster shapes” on weak lens-ing mass structure by introducing a spin-2 halo ellipticityehalo = ehalo

1 + iehalo2 , defined in terms of weighted quadrupole

shape momentsQhaloαβ (α,β = 1,2), as

ehaloα (θap) =

(

Qhalo11 − Qhalo

22

Qhalo11 + Qhalo

22

,2Qhalo

12

Qhalo11 + Qhalo

22

)

, (28)

Qhaloαβ (θap) =

∫

∆θ≤θap

d2θ∆θα∆θβ κ(θ), (29)

whereθap is the circular aperture radius, and∆θα is the angu-lar displacement vector from the cluster center. Similarly, thespin-2 halo ellipticity for the SZE is defined using the cleanedSZE decrement map−∆I (θ) ∝ y(θ) instead ofκ(θ) in equa-tion (28). The degree of halo ellipticity is quantified by the

modulus of the spin-2 ellipticity,|ehalo| =√

(ehalo1 )2 + (ehalo

2 )2,and the orientation of halo is defined by the position angleof the major axis,φhalo = arctan(ehalo

2 /ehalo1 )/2. In order to

avoid noisy shape measurements, we introduce a lower limitof κ(θ) > 0 and−∆I (θ) > 0 in equation (29). Practical shapemeasurements are done using pixelized lensing and SZE mapsshown in Figure 4. The images are sufficiently oversampled,so that the integral in equation (29) can be approximated bythe discrete sum. Note, a comparison in terms of the shapeparameters is optimal for the present case where a pair of theAMiBA and weak lensing images have different angular reso-lutions: θFWHM ≃ 6′ FWHM for AMiBA7, and θFWHM ≃ 1.7′

FWHM for Subaru weak lensing. When the aperture diam-eter is larger than the resolutionθFWHM, i.e.,θap> θFWHM/2,the halo shape parameters can be reasonably defined and mea-sured from the maps.

Now we measure as a function of aperture radiusθapthe cluster ellipticity and orientation profiles on mass andICM structure in the lensingκ and SZE decrement maps,respectively. For the Subaru weak lensing, the shapeparameters are measured atθap = [1,2,3, ...,11]× θFWHM

(1.7′ <∼ θap<∼ 18.3′); for the AMiBA SZE, θap = [1,2,3] ×

θFWHM (6′ <∼ θap<∼ 18′). The level of uncertainty in the halo

shape parameters is assessed by a Monte-Carlo error anal-ysis assuming Gaussian errors for weak lensing distortionand AMiBA visibility measurements (for the Gaussianity of

17 In Umetsu & Broadhurst (2008) cluster masses are expressed in units of1015M⊙ with h = 0.7. The systematic uncertainty inMvir is tightly correlatedwith that incvir through the Einstein radius constraint (θE = 45′′ atzs = 1) bythe ACS observations.

Umetsu et al. 9

AMiBA data, see Nishioka et al. 2008). For each cluster anddataset, we generate a set of 500 Monte Carlo simulations ofGaussian noise and propagate into final uncertainties in thespin-2 halo ellipticity,ehalo. Figure 8 displays, for the fourclusters, the resulting cluster ellipticity and orientation pro-files on mass and ICM structure as measured from the Subaruweak lensing and AMiBA SZE maps, shown separately forthe ellipticity modulus|ehalo| and the orientation, 2φhalo (twicethe position angle). Overall, a good agreement is found be-tween the shapes of mass and ICM structure up to large radii,in terms of both ellipticity and orientation. In particular, ourresults on A2142 and A2390 show that the mass and ICM dis-tributions trace each other well at all radii. At a large radiusof θap

>∼ 10′, the position angle of A2142 isφhalo ∼ 50. ForA2390, the position angle isφhalo∼ 30 at all radii.

6. CLUSTER GAS MASS FRACTION PROFILES

6.1. Method

In modeling the clusters, we consider two representativeanalytic models for describing the form of the cluster DMand ICM distributions, namely (1) the Komatsu & Seljak(2001, hereafter KS01) model of the universal gas den-sity/temperature profiles and (2) the isothermalβ model,where both are physically motivated under the hypothesisof hydrostatic equilibrium and polytropic equation-of-state,P ∝ ργ , with an additional assumption about the sphericalsymmetry of the system.

Joint AMiBA SZE and Subaru weak lensing observationsprobe cluster structures on angular scales up to∆θ ∼ 23′.18

At the median redshiftz ≃ 0.2 of our clusters, this maxi-mum angle covered by the data corresponds roughly tor200≈0.8rvir, exceptr500 ≈ 0.5rvir for A2142 atz = 0.09. In orderto better constrain the gas mass fraction in the outer partsof the clusters, we adopt a prior that the gas density pro-file ρgas(r) traces the underlying (total) mass density profile,ρtot(r). Such a relationship is expected at large radii, wherenon-gravitational processes such as radiative cooling andstarformation, have not had a major effect on the structure of theatmosphere so that the polytropic assumption remains valid(Lewis et al. 2000). Clearly this results in the gas mass frac-tion, ρgas(r)/ρtot(r), tending to a constant at large radius. Inthe context of the isothermalβ model, this simply means thatβ = 2/3.

In both models, for each cluster, the mass density profileρtot(r) is constrained solely by the Subaru weak lensing data(§4), the gas temperature profileTgas(r) is normalized by thespatially-averaged X-ray temperature (see Table 1), and theelectron pressure profilePe(r) = ne(r)kBTe(r) is normalized bythe AMiBA SZE data, wherene(r) is the electron number den-sity, andTe(r) = Tgas(r) is the electron temperature. The gasdensity is then given byρgas(r) = µempne(r).

6.2. Cluster Models

NFW-Consistent Model of Komatsu & Seljak 2001

The KS01 model describes the polytropic gas in hydro-static equilibrium with a gravitational potential described bytheuniversaldensity profile for collisionless CDM halos pro-posed by Navarro et al. (1996, hereafter NFW). See KS01,Komatsu & Seljak (2002, hereafter KS02), and Worrall &Birkinshaw (2006) for more detailed discussions. High mass

18 The FWHM of the primary beam patten of the AMiBA is about 23′,while the field-of-view of the Subaru/Suprime-Cam is about 34′

clusters with virial massesMvir>∼ 1015M⊙/h are so massive

that the virial temperature of the gas is too high for effi-cient cooling and hence the cluster potential simply reflectsthe dominant DM. This has been recently established by ourSubaru weak lensing study of several massive clusters (Broad-hurst et al. 2005b, 2008; Umetsu & Broadhurst 2008).

In this model, the gas mass profile traces the NFW profilein the outer region of the halo (rvir/2 <∼ r <∼ rvir ; See KS01),satisfying the adopted prior of the constant gas mass frac-tion ρgas(r)/ρtot(r) at large radii. This behavior is supportedby cosmological hydrodynamic simulations (e.g., Yoshikawaet al. 2000), and is recently found from the stacked SZE anal-ysis of the WMAP 3-year data (Atrio-Barandela et al. 2008).The shape of the gas distribution functions, as well as thepolytropic indexγgas, can be fully specified by the halo virialmass,Mvir , and the halo concentration,cvir = rvir/rs, of theNFW profile.

In the following, we use the form of the NFW profile to de-terminervir , r200, andr500. Table 6 summarizes for the fourclusters the NFW model parameters derived from our lensinganalysis (see §4.5). For each cluster we also list (r500, r200, rvir)deduced from the corresponding NFW density profile. Forcalculatingγgas and the normalization factorη(0) for a struc-ture constant (B in equation [16] of KS02), we follow the fit-ting formulae given by KS02, which are valid for a relevantrange of the halo concentration, 1< cvir < 25 (see Table 6).For our clusters,γgas is in the range of 1.15 to 1.20. Fol-lowing the prescription in KS01, we convert the X-ray clustertemperatureTX to the central gas temperatureTgas(0) of theKS01 model.

Isothermalβ Profile

The isothermalβ model provides an alternative consistentsolution of the hydrostatic equilibrium equation (Hattoriet al.1999), assuming the ICM is isothermal and its density pro-file follows ρgas(r) = ρgas(0)[1+ (r/rc)2]−3β/2 with the gas coreradiusrc. At large radii,r ≫ rc, where both of our SZE andweak lensing observations are sensitive, the total mass densityfollows ρtot(r) ∝ r−2. Thus we setβ = 2/3 to satisfy our as-sumption of constantρgas(r)/ρtot(r) at large radius. We adoptthe values ofrc andTX listed in Table 1, taken from X-ray ob-servations, and useTgas(r) = TX as the gas temperature for thismodel. At r ≫ rc, theρtot(r) profile can be approximated bythat of a singular isothermal sphere (SIS; see Appendix A.2)parametrized by the isothermal 1D velocity dispersionσv (seeTables 6), constrained by the Subaru distortion data (see §4.5).

Requiring hydrostatic balance gives an isothermal temper-atureTSIS equivalent toσv as

kBTSIS≡ µmpσ2v

23β

. (30)

For β = 2/3, kBTSIS = µmpσ2v , which can be compared with

the observedTX (Table 1). For our AMiBA-lensing clustersample, we found X-ray to SIS temperature ratiosTX/TSIS =0.82±0.03,1.65±0.15,0.94±0.05,1.28±0.15 for A1689,A2142, A2261, and A2390, respectively. For A2261 andA2390, the inferred temperature ratios are consistent withunity at 1–2σ. For the cold-front cluster A2142, the ob-served spatially-averaged X-ray temperature (cooling-flowcorrected; see Markevitch 1998) is significantly higher thanthe lensing-derived temperature. This temperature excessof∼ 4σ could be explained by the effects of merger boosts, asdiscussed in Okabe & Umetsu (2008). The temperature ratio


TX/TSIS for A1689, on the other hand, is significantly lowerthan unity. Recently, a similar level of discrepancy was alsofound in Lemze et al. (2008a), who performed a careful jointX-ray and lensing analysis of this cluster. A deprojected 3Dtemperature profile was obtained using a model-independentapproach to the Chandra X-ray emission measurements andthe projected mass profile obtained from the joint strong/weaklensing analysis of Broadhurst et al. (2005b). The projectedtemperature profile predicted from their joint analysis ex-ceeds the observed temperature by 30% at all radii, a level ofdiscrepancy suggested from hydrodynamical simulations thatdenser, more X-ray luminous small-scale structure can ob-served X-ray temperature measurements downward at aboutthe same level (Kawahara et al. 2007). By applying this+30%correction toTX, we haveTX/TSIS = 1.07±0.04 for A1689.

6.3. AMiBA SZE Data

We use our AMiBA data to constrain the remaining nor-malization parameter for theρgas(r) profile,ρgas(0). The cal-ibrated output of the AMiBA interferometer, after the lag-to-visibility transformation (Wu et al. 2008a), is the complexvis-ibility V(u) as a function of baseline vector in units of wave-length, u = d/λ, given as the Fourier transform of the skybrightness distribution∆I (θ) attenuated by the antenna pri-mary beam patternA(θ).

In targeted AMiBA observations at 94GHz, the sky signal∆I (θ) with respect to the background (i.e., atmosphere andthe mean CMB intensity) is dominated by the thermal SZEdue to hot electrons in the cluster,∆ISZE = Inormg(ν)y (seeeq. [1]). The Comptonization parametery is expressed asa line-of-sight integral of the thermal electron pressure (seeeq. [2]). In the line-of-sight projection of equation (2), thecutoff radiusrmax needs to be specified. We takermax≡ αrvirwith a dimensionless constantα which we set toα = 2. In thepresent study we found the line-of-sight projection in equa-tion (2) is insensitive to the choice ofα as long asα >∼ 1.

A useful measure of the thermal SZE is the integratedComptonization parameterY(θ),

Y(θ) = 2π∫ θ

0dθ′ θ′y(θ), (31)

which is proportional to the SZE flux, and is a measure ofthe thermal energy content in the ICM. The value ofY isless sensitive to the details of the model fitted than the centralComptonization parametery0 ≡ y(0), with the current config-uration of AMiBA. If the A(θ)y(θ) field has reflection sym-metry about the pointing center, then the imaginary part ofV(u) vanishes, and the sky signal is entirely contained in thereal visibility flux:

VRe(u) = 2πInormg(νc)∫ ∞

0dθ θA(θ)y(θ)J0(2πuθ)

≡2πI0

∫ ∞

0dθ θA(θ)

y(θ)y0

J0(2πuθ), (32)

whereI0 = Inormg(νc)y(0) is the central SZE intensity atνc =94GHz,J0(x) is the Bessel function of the first kind and or-der zero, andA(θ) is well approximated by a circular Gaus-sian with FWHM = 1.22(λ/D) ≃ 23′ at νc = 94GHz with anantenna diameter ofD = 60cm (Wu et al. 2008a). The ob-served imaginary flux can be used to check for the effectsof primary CMB and radio source contamination (Liu et al.2008). From our AMiBA data we derive in the Fourier do-main azimuthally-averaged visibility profiles〈V(u)〉 for indi-vidual clusters.

We constrain the normalizationI0 from χ2 fitting to the〈V(u)〉 profile (Liu et al. 2008). In order to convertI0 into thecentral Comptonization parameter, we take account of (i) therelativistic correctionδSZE(ν,Tgas) in the SZE spectral func-tion g(ν) (see eq. [3]) and (ii) the correction for contaminationby discrete radio point sources (Liu et al. 2008). The level ofcontamination inI0 from known discrete point sources hasbeen estimated to be about 6 to 35% in our four clusters (Liuet al. 2008). In all cases, a net positive contribution of thepoint sources was found in our 2-patch differencing AMiBAobservations (§3), indicating that there are more radio sourcestowards clusters than in the background (Liu et al. 2008).Thus ignoring the point source correction would systemati-cally bias the SZE flux estimates, leading to an underestimateof y0. The relativistic correction to the thermal SZE is about6–7% in ourTX range at 94GHz.

Table 7 summarizes, for our two models, the best-fittingparameter,y0, and theY-parameter interior to a cylinder ofradiusθ = 3′ that roughly matches to the AMiBA synthesizedbeam with 6′ FWHM. For each case, both of the cluster mod-els yield consistent values ofy0 andY(3′) within 1σ; in par-ticular, the inferred values ofY(3′) for the two models arein excellent agreement. Following the procedure in §6.1 weconverty0 into the central gas mass density,ρgas(0).

6.4. Gas Mass Fraction Profiles

We derive cumulative gas fraction profiles,

fgas(< r) =Mgas(< r)Mtot(< r)

, (33)

for our cluster sample using two sets of cluster models de-scribed in §6.2, whereMgas(< r) andMtot(< r) are the hot gasand total cluster masses contained within a spherical radiusr. In Table 8 we list for each of the clustersMgas and fgaswithin r500 andr200 (see also Table 6) calculated with the twomodels. Note that our total mass estimates do not require theassumption of hydrostatic balance, but are determined basedsolely on the weak lensing measurements. Gaussian errorpropagation was used to derive the errors onMgas(< r) andfgas(< r). We propagate errors on the individual cluster pa-rameters (Tables 1 and 6) by a Monte-Carlo method. ForA2142, the isothermal model increasingly overpredictsfgasat all radii r > r2500, exceeding the cosmic baryon fractionfb = Ωb/Ωm = 0.171±0.009 (Dunkley et al. 2008). For otherclusters in our sample, both cluster models yield consistentfgas andMgas measurements within the statistical uncertain-ties from the SZE and weak lensing data.

Our SZE/weak lensing-based measurements can be com-pared with other X-ray and SZE measurements. Gregoet al. (2001b) derived gas fractions for a sample of 18 clus-ters from 30GHz SZE observations with BIMA and OVROin combination with published X-ray temperatures. Theyfound fgas(< r500) = 0.140+0.041

−0.047h−170 and 0.053+0.139

−0.031h−170 (h70 =

h/0.7) for A1689 and A2261, respectively, in agreementwith our results. For A2142, thefgas and Mgas values in-ferred from the KS01 model is in good agreement withthose from the VSA SZE observations at 30GHz (Lancasteret al. 2005),Mgas(r500) = 6.1+1.7

−1.8×1013M⊙h−2 and fgas(r500) =0.123+0.080

−0.050h−170. From a detailed analysis of Chandra X-ray

data, Vikhlinin et al. (2006) obtainedfgas(< r500) = (0.141±0.009)h−3/2

72 (h72 = h/0.72) for A2390, in good agreement withour results.

Furthermore, it is interesting to compare our results with thedetailed joint X-ray and lensing analysis of A1689 by Lemze

Umetsu et al. 11

et al. (2008a), who derived deprojected profiles ofρtot(r),ρgas(r), andTgas(r) assuming hydrostatic equilibrium, using amodel-independent approach to the Chandra X-ray emissionprofile and the projected lensing mass profile of Broadhurstet al. (2005b). A steep 3D mass profile was obtained by thisapproach, with the inferred concentration ofcvir = 12.2+0.9

−1.0,consistent with the detailed lensing analysis of Broadhurstet al. (2005b) and Umetsu & Broadhurst (2008), whereas theobserved X-ray temperature profile falls short of the derivedprofile at all radii by a constant factor of∼ 30% (see Figure 15of Lemze et al. 2008a). With the pressure profile of Lemze etal. we findy0 = (4.7±0.3)×10−4, which is in agreement withour KS01 prediction,y0 = (4.2±1.0)×10−4 (Table 7). The in-tegrated Comptonization parameter predicted by the Lemze etal. model isY(3′) = (3.0±0.1)×10−10, which roughly agreeswith the AMiBA measurement ofY(3′) = (2.5±0.6)×10−10.Alternatively, adopting the observed temperature profile inLemze et al. model reduces the predicted SZE signal bya factor of∼ 30%, yieldingy0 ≃ 3.3× 10−4 and Y(3′) ≃2.1× 10−10, again in agreement with the AMiBA measure-ments. Therefore, more accurate SZE measurements are re-quired to further test and verify this detailed cluster model.

We now use our data to find the average gas fraction pro-file over our sample of four hot X-ray clusters. The weightedaverage ofMvir in our AMiBA-lensing sample is〈Mvir〉 =(1.19± 0.08)× 1015M⊙h−1, with a weighted-mean concen-tration of 〈cvir〉 = 8.9± 0.6. The weighted average of thecluster virial radius is〈rvir〉 ≃ 1.95Mpch−1. At each radiuswe compute the sample-averaged gas fraction,〈 fgas(< r)〉,weighted by the inverse square of the statistical 1σ uncer-tainty. In Figure 9 we show for the two models the resulting〈 fgas〉 profiles as a function of radius in units ofrvir , alongwith the published results for other X-ray and SZE obser-vations. Here the uncertainties (cross-hatched) represent thestandard error (1σ) of the weighted mean at each radius point,including both the statistical measurement uncertaintiesandcluster-to-cluster variance. Note A2142 has been excludedfor the isothermal case (see above). The averaged〈 fgas〉 pro-files derived for the isothermal and KS01 models are consis-tent within 1σ at all radii, and lie below the cosmic baryonfraction fb = 0.1713± 0.009 constrained by the WMAP 5-year data (Dunkley et al. 2008). Atr = 〈r200〉 ≃ 0.79〈rvir〉, theKS01 model gives

〈 fgas,200〉 = 0.133±0.020±0.018 (34)

where the first error is statistical, and the second is thestandard error due to cluster-to-cluster variance. This ismarginally consistent with〈 fgas,200〉 = 0.109±0.013 obtainedfrom the averaged SZE profile of a sample of 193 X-ray clus-ters (TX >3keV) using the WMAP 3-year data (Afshordi et al.2007). Atr = 〈r500〉 ≃ 0.53〈rvir〉, we have

〈 fgas,500〉 = 0.126±0.019±0.016 (35)

for the KS01 model, in good agreement with the ChandraX-ray measurements for a subset of sixTX > 5keV clustersin Vikhlinin et al. (2006). Atr = 〈r2500〉 ≃ 0.25〈rvir〉, whichis close to the resolution limit of AMiBA7, we have for theKS01 model

〈 fgas,2500〉 = 0.105±0.015±0.012, (36)

again marginally consistent with the Chandra gas fractionmeasurements in 26 X-ray luminous clusters withTX > 5keV.

7. DISCUSSION AND CONCLUSIONS

We have obtained secure, model-independent profiles ofthe lens distortion and projected mass (Figures 5 and 6) byusing the shape distortion measurements from high-qualitySubaru imaging, for our AMiBA lensing sample of four high-mass clusters. We utilized weak lensing dilution in deep Sub-aru color images to define color-magnitude boundaries forblue/red galaxy samples, where a reliable weak lensing sig-nal can be derived, free of unlensed cluster members (Figure3). Cluster contamination otherwise preferentially dilutes theinner lensing signal leading to spuriously shallower profiles.With the observed lensing profiles we have examined clustermass-density profiles dominated by DM. For all of the clus-ters in our sample, the lensing profiles are well described bythe NFW profile predicted for collisionless CDM halos.

A qualitative comparison between our weak lensing andSZE data, on scalesr >∼ 3′ limited by the current AMiBA res-olution, shows a good correlation between the distributionofmass (weak lensing) and hot baryons (SZE) in massive clus-ter environments (§4.4), as physically expected for high massclusters with deep gravitational potential wells (§4.4). Wehave also examined and compared, for the first time, the clus-ter ellipticity and orientation profiles on mass and ICM struc-ture in the Subaru weak lensing and AMiBA SZE observa-tions, respectively. For all of the four clusters, the mass andICM distribution shapes are in good agreement at all relevantradii in terms of both ellipticity and orientation (Figure 8). Inthe context of the CDM model, the mass density dominatedby collisionless DM is expected to have a more irregular andelliptical distribution than the ICM density due to inherent tri-axiality of CDM halos. We however do not see such a ten-dency in our lensing and SZE datasets, likely limited by thecurrent uncertainty and resolution in the AMiBA SZE mea-surements.

We have obtained cluster gas fraction profiles (Figure 9)for the AMiBA-lensing sample (TX > 8keV) based on jointAMiBA SZE and Subaru weak lensing observations (§6.4).Our cluster gas fraction measurements are overall in goodagreement with previously-published values. Atr = 〈r200〉 ≃0.79〈rvir〉, corresponding roughly to the maximum avail-able radius in our joint SZE/weak lensing data, the sample-averaged gas fraction is〈 fgas,200〉 = 0.133± 0.027 for theNFW-consistent KS01 model, representing the global gasfraction averaged over our high-mass cluster sample with amean virial mass of〈Mvir〉 = (1.2±0.1)×1015M⊙h−1. Whencompared to the cosmic baryon fractionfb, this indicates〈 fgas,200〉/ fb = 0.78± 0.16, i.e., a fraction of (22± 16)% ismissing from the hot phase in our cluster sample (cf. Af-shordi et al. 2007; Crain et al. 2007). This missing clusterbaryon fraction is partially made up by observed stellar andcold gas fractions of∼ several % in ourTX range (Gonzalezet al. 2005). Crain et al. (2007)

Halo triaxiality may affect our projected total and gas massmeasurements based on the assumption of spherical symme-try, producing an orientation bias. A degree of triaxialitydepends on the collisional nature, and is inevitable for col-lisionless gravitationally collapsed structures. Discussion ofthe likely effect of triaxiality on the measurements of lensingproperties has been examined analytically (Oguri et al. 2005;Sereno 2007; Corless & King 2007), and in numerical investi-gations (Jing & Suto 2002; Hennawi et al. 2007). The effect oftriaxiality will be lesser for the collisional ICM, which followsthe gravitational potential being smoother and rounder than


the total mass density distribution. For an unbiased measure-ment of the gas mass fractions, a large, homogeneous sampleof clusters would be needed to beat down the orientation bias.

Possible biases in X-ray spectroscopic temperature mea-surements (Mazzotta et al. 2004; Kawahara et al. 2007) mayalso affect our gas fraction measurements based on the over-all normalization by the observed X-ray temperature. Thiswould need to be seriously taken into account in the futureinvestigation with a larger cluster sample of higher statisticalprecision.

Our joint analysis of high quality Subaru weak lensing andAMiBA SZE observations allows for a detailed study of in-dividual clusters. The merging cluster A2142 shows complexmass substructure (Okabe & Umetsu 2008), and displays ashallower profile withcvir ∼ 5, consistent with detailed X-rayobservations which imply recent interaction. Due to its low-z and lowcvir, the curvature in the lensing profiles is highlypronounced, so that a SIS profile for A2142 is strongly ruledout by the Subaru distortion data alone. For this cluster, ourAMiBA SZE map shows an extended structure in the ICMdistribution, elongated along the northwest direction. This di-rection of elongation in the SZE halo is in good agreementwith that in the X-ray halo having a cometary appearance seenby the Chandra X-ray observations (Markevitch et al. 2000;Okabe & Umetsu 2008). In addition, an extended excess SZEcan be seen in the northwest region of the cluster. A jointweak-lensing, optical-photometric, and X-ray analysis ofOk-abe & Umetsu (2008) revealed northwest mass substructure inthis SZE excess region, located ahead of the northwest coldfront seen in Chandra X-ray emission. The northwest masssubstructure is also seen in our weak lensing mass map (Fig-ure 4) based on the much improved color selection for thebackground sample. A slight excess luminosity of cluster se-quence galaxies associated with the northwest substructure isalso found in Okabe & Umetsu (2008), while no X-ray coun-terpart to the northwest substructure is seen in the Chandraand XMM-Newton images (Okabe & Umetsu 2008). Thisclearly demonstrates the potential of SZE observations as apowerful tool for measuring the distribution of ICM in clus-ter outskirts where the X-ray emission measure (∝ n2

e) is lesssensitive. This also demonstrates the potential and reliabilityof AMiBA, and the power of multiwavelength cluster analysisfor probing the distribution of mass and baryons in clusters.A further detailed multiwavelength analysis of A2142 will beof great importance for further understanding of the clustermerger dynamics and associated physical processes of the in-tracluster gas.

For A2390 we obtain a highly elliptical mass distribution atall radii from both weak and strong lensing (Frye & Broad-hurst 1998). The elliptical mass distribution agrees wellwith the shape seen by AMiBA in the thermal SZE (Fig-ures 4 and 8). Our joint lensing, SZE, and X-ray modelingleads to a relatively high gas mass fraction for this cluster,fgas,500∼ 0.153 for the NFW-consistent case, which is in goodagreement with careful X-ray work by Vikhlinin et al. (2006),fgas,500 = (0.141±0.009)h−3/2

72 .We have refined for A1689 the statistical constraints on

the NFW mass model of Umetsu & Broadhurst (2008), withour improved color selection for the red background sam-ple, where all possible lensing measurements are combinedto achieve the maximum possible lensing precision,Mvir =(1.55+0.13

−0.12) × 1015M⊙h−1 and cvir = 12.3+0.9−0.8 (quoted are sta-

tistical errors at 68% confidence level), confirming again the

high concentration found by detailed lensing work (Broad-hurst et al. 2005b; Halkola et al. 2006; Limousin et al. 2007;Umetsu & Broadhurst 2008). The AMiBA SZE measure-ments at 94GHz support the compact structure in the ICMdistribution for this cluster (Figure 4). Recently, good con-sistency was found between high-quality multiwavelengthdatasets available for this cluster (Lemze et al. 2008a,b).Lemze et al. (2008a) performed a joint analysis of ChandraX-ray, ACS strong lensing, and Subaru weak lensing mea-surements, and derived an improved mass profile in a model-independent way. Their NFW fit to the derived mass profileyields a virial mass ofMvir = (1.58±0.15)×1015M⊙h−1 anda high concentration ofcvir = 12.2+0.9

−1.0, both of which are inexcellent agreement with our full lensing constraints. Morerecently, Lemze et al. (2008b) further extended their multi-wavelength analysis to combine their X-ray/lensing measure-ments with two dynamical datasets from VLT/VIRMOS spec-troscopy and Subaru/Suprime-Cam imaging. Their joint lens-ing, X-ray, and dynamical analysis provides a tight constrainton the cluster virial mass: 1.5 < Mvir/(1015M⊙h−1) < 1.65.Their purely dynamical analysis constrains the concentrationparameter to becvir > 9.8 for A1689, in agreement with ourindependent lensing analysis and the joint X-ray/lensing anal-ysis of Lemze et al. (2008a). We remark that NFW fits tothe Subaru outer profiles alone give consistent but somewhathigher concentrations,cvir ∼ 15 (Table 3; see also Umetsu &Broadhurst 2008 and Broadhurst et al. 2008). This slight dis-crepancy can be explained by that the mass density slope atlarge radii (θ >∼ 5′) for A1689 is slightly steeper than for theNFW profile where the asymptotic decline tends toρNFW(r)∝r−3 (Broadhurst et al. 2005b; Medezinski et al. 2008; Lemzeet al. 2008a; Umetsu & Broadhurst 2008; Lemze et al. 2008b).For accurate measurements of the outermost lensing profile,awider optical/near-infrared wavelength coverage is requiredto improve the contamination-free selection of backgroundgalaxies including blue background galaxies behind this richcluster.

Our Subaru observations have established that A2261 isvery similar to A1689 in terms of both weak and stronglensing properties: Our preliminary strong lens modeling re-veals many tangential arcs and multiply-lensed images aroundA2261, with an effective Einstein radiusθE ∼ 40′′ at z ∼1.5 (Figure 7), which when combined with our weak lens-ing measurements implies a mass profile well fitted by anNFW profile with a concentrationcvir ∼ 10, similar to A1689(Umetsu & Broadhurst 2008), and considerably higher thantheoretically expected for the standardΛCDM model, wherecvir ∼ 5 is predicted for the most massive relaxed clusters withMvir

>∼ 1015M⊙ (Bullock et al. 2001; Neto et al. 2007; Duffyet al. 2008).

Such a high concentration is also seen in other several well-studied massive clusters from careful lensing work (Gavazziet al. 2003; Kneib et al. 2003; Broadhurst et al. 2005b;Limousin et al. 2007; Lemze et al. 2008a; Broadhurst et al.2008). A chance projection of structure along the line-of-sightand the orientation bias due to halo triaxiality may potentiallyaffect the projected lensing measurements, and hence thelensing-based concentration measurement (e.g., Oguri et al.2005). A statistical bias in favor of prolate structure pointedto the observer is unavoidable at some level, as this orientationboosts the projected surface mass density and hence the lens-ing signal. In the context of theΛCDM model, this leads toan increase of∼ 18% in the mean value of the lensing-basedconcentrations (Hennawi et al. 2007). A larger bias of∼ 30

Umetsu et al. 13

up to 50% is expected for CDM halos selected by the pres-ence of large gravitational arcs (Hennawi et al. 2007; Oguri& Blandford 2008). Our cluster sample is identified by theirbeing X-ray/SZE bright, with the added requirement of theavailability of high-quality multi-band Subaru/Suprime-Camimaging (see §3.2). Hence, it is unlikely that the four clustersare all particularly triaxial with long axes pointing to theob-server. Indeed, in the context ofΛCDM, the highly ellipticalmass distribution of A2390 would suggest that its major axisis likely parallel to the sky plane, and that, its actual concen-tration is higher than the projected measurement ofcvir ≃ 7.

The ongoing upgrade of AMiBA to 13-elements with 1.2mantennas will improve the spatial resolution and dynamicrange, and the 13-element AMiBA (AMiBA13) will be sen-sitive to structures on scales down to 2′, matching to the an-gular scales probed by ground-based weak lensing observa-tions. For our initial target clusters, joint constraints withAMiBA7 and AMiBA13 data will complement the baselinecoverage, which will further improve our multiwavelengthanalysis of the relation between mass and hot baryons in theclusters. A joint analysis of complementary high-resolutionlensing, SZE, and X-ray observations will be of great inter-est to address the issue of halo triaxiality and to further im-prove the constraints on the cluster density profiles (Sereno2007). This upgrade will also make the instrument faster by

a factor of∼ 60 in singled pointed observations. Our con-straints can be further improved in the near future by observ-ing a larger sample with AMiBA13. A detailed comparisonbetween X-ray based and SZE/weak lensing-based gas frac-tion measurements will enable us to test the degree of clumpi-ness (

√

n2e/ne) and of hydrostatic balance in hot cluster gas.

The high angular resolution (2′) of AMiBA13 combined withdynamically-improved imaging capabilities will allow fordi-rect tests of the gas pressure profile in deep single pointedobservations (Molnar et al. 2008).

We are grateful to N. Okabe, M. Takada, and Y. Rephaelifor valuable discussions. We thank the Ministry of Education,the National Science Council, and the Academia Sinica fortheir support of this project. We thank the Smithsonian Astro-physical Observatory for hosting the AMiBA project staff atthe SMA Hilo Base Facility. We thank the NOAA for locatingthe AMiBA project on their site on Mauna Loa. We thank theHawaiian people for allowing astronomers to work on theirmountains in order to study the Universe. We thank all themembers of the AMiBA team for their hard work. The workis partially supported by the National Science Council of Tai-wan under the grant NSC95-2112-M-001-074-MY2. Supportfrom the STFC for MB is also acknowledged.

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Umetsu et al. 15

TABLE 1. TARGET CLUSTERS ANDAM IBA/X- RAY PROPERTIES

Cluster z 1arcmina AMiBA7 b X-rayc

SZE flux Image FWHM TX θc Refs(kpch−1) (mJy) (arcmin) (keV) (arcmin)

A1689 0.183 129.6 −168±28 5.7 9.66+0.22−0.20 0.44±0.01 3

A2142 0.091 71.4 −316±23 9.0 9.7±1.0 3.14±0.22 1, 4, 5A2261 0.224 151.8 −90±17 5.8 8.82+0.37

−0.32 0.26±0.02 3A2390 0.228 153.3 −158±24 8.0 10.1±1.1 0.47±0.05 2

REFERENCES. — [1] Markevitch et al. (1998); [2] Boehringer et al. (1998); [3] Reese et al. (2002); [4] Sanderson et al. (2003); [5] Lancaster et al. (2005).

NOTE. — Uncertainties are 68% confidence.aPhysical scale in kpch−1 units corresponding to 1′ at the cluster redshift.bSZE properties from AMiBA7 at 94GHz: cluster peak SZE flux (mJy) and angular size (′) in FWHM measured from the cleaned image (Wu et al. 2008a).cPublished X-ray properties: X-ray temperature (keV), X-ray core radius (kpch−1), and references. For A2142,TX andθc are taken from Ref. [1], and Refs. [4,5],

respectively. For A2390 a 10% error is assumed for (TX ,β), for which no error estimate was presented in the original reference.

TABLE 2. SUBARU WEAK LENSING DATA AND BACKGROUND GALAXY SAMPLE

Cluster Filters Seeinga ngb B/Rc 〈Dds/Ds〉d zs,D

e σκ

f

(arcsec) (arcmin−2)

A1689 Vi′ 0.88 8.8 0 0.70±0.02 0.70+0.06−0.05 0.029

A2142 g′Rc 0.55 30.4 2.1 0.88±0.04 0.95+0.79−0.30 0.021

A2261 VRc 0.65 13.8 1.5 0.72±0.04 0.98+0.24−0.16 0.032

A2390 VRc 0.70 20.7 2.1 0.72±0.04 1.00+0.25−0.16 0.026

aSeeing FWHM in the final co-added image in the redder band.bSurface number density of blue+red galaxies.cFraction of blue to red galaxies in the blue+red background sample.dDistance ratio averaged over the redshift distribution of the blue+red sample.eEffective source redshift (see eq. [21]) corresponding to the mean depth〈Dds/Ds〉.fRMS noise level in the reconstructedκ map.

TABLE 3. SUMMARY OF BEST-FIT MASS MODELS FROMSUBARU DISTORTION DATA

Cluster Tangential reduced shear,g+ Lensing convergence,κSIS NFW NFW

σv χ2/dof θE Mvir cvir χ2/dof θE Mvir cvir χ2/dof θE(kms−1) (′′) (1015M⊙/h) (′′) (1015M⊙/h) (′′)

A1689 1403±41 11/9 47±3 1.09+0.18−0.16 15.6+4.8

−3.3 7.3/8 47+15−14 1.05+0.18

−0.15 15.8+14.2−8.0 5.3/8 46+26

−31A2261 1276±43 8.7/8 37±3 1.35+0.26

−0.22 6.4+1.9−1.4 7.7/7 20+16

−11 1.26+0.20−0.17 10.2+7.1

−3.5 9.8/8 37+25−19

A2142 970±27 39/8 25±1 1.07+0.22−0.16 5.6+0.9

−0.8 2.1/7 1.2+2.9−0.9 1.06+0.19

−0.16 4.9+1.2−1.0 20/10 0.5+2.3

−0.4A2390 1139±38 3.8/8 30±2 0.90+0.15

−0.14 6.9+2.3−1.5 3.8/7 15+13

−8 0.92+0.15−0.12 7.3+6.9

−2.9 8.1/8 17+26−14

NOTE. — A flat prior of cvir ≤ 30 is assumed for the halo concentration of the NFW model. TheEinstein radiusθE is calculated for a background source atzs = 1.5, correspondingroughly to the mean depth of blue+red background galaxies.

TABLE 4. REFINED NFW MODEL OF A1689

Parameter Value

Mvir (1015M⊙ h−1) ..... 1.55+0.13−0.12

cvir ..... 12.3+0.9−0.8

χ2/dof ..... 352/421Einstein radius,θpred

E (′′) ..... 50+6.5−6.0

NOTE. — Improved statistical constraints on the NFW parameters of A1689. This table lists the best-fit NFW parameters (Mvir ,cvir) for A1689 obtained from a joint fit to ACSstrong lensing and Subaru weak lensing distortion+magnification data, presented in Umetsu & Broadhurst (2008), but with our improved color selection of the red background samplefor Subaru distortion measurements. The errors represent 68% statistical confidence intervals estimated from∆χ2 = χ2 −χ2

min = 1. The Einstein radiusθpredE listed is predicted for a

background source atzs = 1.5.


TABLE 5. REFINED NFW MODEL OF A2261

Parameter Value

Mvir (1015M⊙ h−1) ..... 1.25+0.17−0.16

cvir ..... 11.1+2.2−1.9

χ2/dof ..... 9.85/9Einstein radius,θpred

E (′′) ..... 40±11

NOTE. — Improved statistical constraints on the NFW parameters of A2261. This table lists the best-fit NFW parameters (Mvir ,cvir) for A2261 obtained from combined Subarulensing convergence measurements (θ > 1′) and the observed Einstein constraint,θobs

E = 40′′ ±4′′ atzs ≃ 1.5. The errors represent 68% statistical confidence intervals estimated from

∆χ2 = χ2 −χ2min = 1. The Einstein radiusθpred

E listed is predicted for a background source atzs = 1.5.

TABLE 6. CLUSTER MASS MODELS FOR GAS MASS FRACTION MEASUREMENTS

Cluster NFW model SIS model r500 r200 rvirMvir cvir σv

(1015M⊙h−1) (kms−1) (Mpch−1) (Mpch−1) (Mpch−1)

A1689 1.55+0.13−0.12 12.3+0.9

−0.8 1403±41 1.16±0.06 1.70±0.08 2.13±0.10A2142 1.07+0.22

−0.16 5.6+0.9−0.8 970±27 0.99±0.06 1.51±0.09 1.98±0.12

A2261 1.25+0.17−0.16 11.1+2.2

−1.9 1276±43 1.06±0.04 1.56±0.06 1.94±0.08A2390 0.90+0.15

−0.14 6.9+2.3−1.5 1139±38 0.92±0.05 1.38±0.07 1.73±0.09

TABLE 7. AM IBA VISIBILITY ANALYSIS

Cluster KS01 isothermalβ(= 2/3)y0 Y(3′) y0 Y(3′)

(10−4) (10−10) (10−4) (10−10)

A1689 4.15±1.00 2.5+0.6−0.6 4.31±1.10 2.6+0.6

−0.6A2142 2.29±0.28 3.5+0.5

−0.5 2.00±0.25 4.0+0.5−0.5

A2261 3.00±0.84 1.5+0.5−0.4 4.25±1.22 1.6+0.5

−0.4A2390 2.87±0.61 1.9+0.6

−0.5 3.40±0.72 2.1+0.8−0.5

NOTE. — The effect of radio point source contamination in the thermal SZE is corrected for (see Liu et al. 2008). The relativistic correction to the SZE is also taken into account.

TABLE 8. CLUSTER GAS PROPERTIES DERIVED FROM THEAM IBA/SUBARU DATA

Cluster KS01 + NFW isothermalβ(= 2/3) + SISMgas,500 Mgas,200 fgas,500 fgas,200 Mgas,500 Mgas,200 fgas,500 fgas,200

(1013M⊙h−2) (1013M⊙h−2)

A1689 8.8+2.3−2.2 11.5+3.0

−3.0 0.115+0.029−0.029 0.119+0.031

−0.030 7.8+2.0−1.8 11.8+3.0

−2.7 0.108+0.026−0.025 0.111+0.027

−0.026A2142 7.2+1.5

−1.3 11.2+2.6−2.2 0.169+0.046

−0.034 0.183+0.049−0.037 – – – –

A2261 6.3+1.9−2.1 8.4+2.7

−2.8 0.103+0.036−0.033 0.108+0.040

−0.035 5.4+1.5−1.5 8.1+2.3

−2.3 0.103+0.031−0.030 0.105+0.032

−0.030A2390 6.1+2.4

−1.8 8.8+4.0−2.7 0.153+0.075

−0.049 0.164+0.084−0.053 5.6+1.5

−1.5 8.8+2.4−2.4 0.145+0.041

−0.041 0.151+0.042−0.043

NOTE. — The derived gas fractionsfgas scale with the Hubble parameter asfgas ∝ h−1 (h = 0.7 adopted here). Confidence intervals are quoted at the 1σ (68%) level. Here weexclude the results from the isothermal model for A2142 which overpredicts at all relevant radiir > r2500 fgascompared with the cosmic baryon fraction,fb = Ωb/Ωm = 0.1713±0.009.

Umetsu et al. 17

A2142A1689

A2261 A2390

FIG. 1.— The quadrupole PSF anisotropy field for individual clusters as measured from stellar ellipticities before and after the PSF anisotropy correction. Foreach cluster field, the left panel shows the raw ellipticity field of stellar objects, and the right panel shows the residual ellipticity field after the PSF anisotropycorrection. The orientation of the sticks indicates the position angle of the major axis of stellar ellipticity, whereas the length is proportional to the modulus ofstellar ellipticity. A stick with the length of 5% ellipticity is indicated in the top right of the right panel.

A1689 A2142

A2390A2261

FIG. 2.— Stellar ellipticity distributions before and after the PSF anisotropy correction for individual clusters. For each cluster field, the left panel shows theraw ellipticity components (e∗1 ,e

∗2 ) of stellar objects, and the right panel shows the residual ellipticity components (δe∗1 ,δe∗2 ) after the PSF anisotropy correction.


FIG. 3.— Top panels: Mean shape distortions (g+,g×) averaged over the entire cluster region (1′ < θ < 18′) for the four clusters done separately for theblue and red samples, in order to establish the boundaries ofthe color distribution free of cluster members.Bottom panels: Respective numbers of galaxies as afunction of color-limit in the red (right) and the blue (blue) samples.

Umetsu et al. 19

FIG. 4.— Mass maps of the central 22′ ×22′ of four AMiBA/Subaru clusters reconstructed from Subaru weak lensing data, with the gravitational shear fieldof background galaxies overlaid; 10% ellipticity is indicated top right, and the resolution characterized by GaussianFWHM is shown bottom right. Also overlaidare contours of the SZE flux densities at 94GHz, observed withthe 7-element AMiBA, given in units of 1σ reconstruction error. The resolution of AMiBA, givenin Gaussian FWHM, is 6′ For all of the four clusters, the distribution of the SZE signal is well correlated with the projected mass distribution,indicating that thehot gas in the clusters traces well the underlying gravitational potential dominated by unseen dark matter. The dark blue regions in the mass map of A2142 areoutside of the Subaru observations.

(a) A1689 (red) (d) A2390 (blue+red)(b) A2261 (blue+red) (c) A2142 (blue+red)

FIG. 5.— Azimuthally-averaged radial profiles of the tangential reduced shearg+ (upper panels) for the four clusters based on the combined red and bluebackground samples. The solid and dashed curves show the best-fitting NFW and SIS profiles for each cluster. Shown below isthe 45 rotated (×) component,g×.


FIG. 6.— Model-independent radial profiles of the lensing convergenceκ(θ) = Σm(θ)/Σcrit for the four clusters derived from a variant of the non-linearaperture-mass densitometry. For each cluster, the best-fitting NFW model for theκ profile is shown with a solid line. The dashed curve shows the best-fittingNFW model for theg+ profile in Figure 5.

Umetsu et al. 21

FIG. 7.— SubaruV + Rc pseudo-color image of the central 6.7′ × 6.7′ (2K× 2K pixels) region of the cluster A2261 atzd = 0.226. Overlaid is the tangentialcritical curve for a background source atzs ∼ 1.5 based on strong lensing modeling of multiply-lensed images and tangential arcs registered in deep SubaruVRcand CFHT/WIRCamJHKS images. The effective radius of the critical curve defines the Einstein radius,θE ≈ 40′ atzs ∼ 1.5.


FIG. 8.— Cluster ellipticity and orientation profiles on mass and ICM structure as a function of aperture radiusθap, measured from the Subaru weak lensingand AMiBA SZE maps shown in Figure 4. For each cluster, the toppanel shows the halo ellipticity profile|ehalo|(θap), and the bottom panel shows the orientationprofile 2φ(θap), whereφhalo represents the position angle of the major axis as measured from weighted quadrupole shape moments.

Umetsu et al. 23

FIG. 9.— Gas mass fraction profiles〈 fgas(< r)〉 = 〈Mgas(< r)/Mtot(< r)〉 averaged over the sample of four hot (TX > 8keV) clusters (A1689, A2142, A2261,A2390) obtained from joint AMiBA SZE and Subaru weak lensingobservations, shown for the NFW-consistent Komatsu & Seljak 2001 model (black) and theisothermalβ model withβ = 2/3 (blue), along with published results (square, triangle, andcircle) from other X-ray and SZE observations. The isothermal resultsexclude the cluster A2142 (see §6.4). For each model, thecross-hatchedregion represents 1σ uncertainties for the weighted mean at each radius point, includingboth the statistical measurement uncertainties and cluster-to-cluster variance. The black horizontal bar shows the constraints on the cosmic baryon fraction fromthe WMAP 5-year data.


APPENDIX

LENS MODELS

The NFW Lens Model

The NFW universal density profile has a two-parameter functional form as

ρNFW(r) =ρs

(r/rs)(1+ r/rs)2(A1)

whereρs is a characteristic inner density, andrs is a characteristic inner radius. The virial properties arerelated In stead of usingρs andrs, we introduce for an NFW halo the virial massMvir and the concentration parameter,cvir ≡ rvir/rs, defined as the ratioof the virial radiusrvir to the scale radius. The virial mass and virial radius are related through the following equation:

Mvir =4π3ρ(zvir)∆virr

3vir , (A2)

where∆vir is the mean overdensity with respect to the mean cosmic density ρ(zvir) at the virialization epochzvir , predicted bythe dissipationless spherical tophat collapse model (Peeebles 1980; Eke, Cole, & Frenk 1996). We assume the cluster redshift zdis equal to the cluster virial redshiftzvir . We use the following fitting formula in a flat 3-space with cosmological constant (seeOguri, Taruya, & Suto 2001):

∆vir = 18π2(1+ 0.4093ω0.9052vir ), (A3)

whereωvir ≡ 1/Ωm(zvir) − 1.The inner densityρs can be then expressed in terms of other virial properties of the NFW halo:

ρs = ρ(zvir)∆vir

3c3

vir

ln(1+ cvir) − cvir/(1+ cvir). (A4)

Hence, for a given cosmological model and a halo virial redshift, we can specify the NFW model with the halo virial massMvirand the halo concentration parametercvir.

For an NFW profile, it is useful to decompose the convergenceκ(θ) and the averaged convergenceκ(θ) as

κNFW(x) =b2

f (x), (A5)

κNFW(x) =bx2

g(x), (A6)

whereb = 4ρsrs/Σcrit(zd,zs) is the dimensionless scaling convergence,x= θ/(rs/Dd) is the dimensionless angular radius, andf (x)andg(x) are dimensionless functions. We have analytic expressions for f (x) andg(x) as (Bartelmann 1996; Wright & Brainerd2000):

f (x) =

11−x2

(

−1+ 2√1−x2

arctanh√

1−x1+x

)

(x< 1),13 (x = 1),

1x2−1

(

+1− 2√x2−1

arctan√

x−1x+1

)

(x> 1),

(A7)

g(x)= ln(x

2

)

+

2√1−x2

arctanh√

1−x1+x (x< 1),

1 (x = 1),2√x2−1

arctan√

x−1x+1 (x> 1).

(A8)

The tangential shearγ+,NFW(θ) is then evaluated by

γ+,NFW(θ) = κNFW(θ) −κNFW(θ). (A9)

For a given source redshiftzs, the Einstein radius is then readily calculated by 1 =κNFW(θ); or more explicitly, using equation(A8) we have

θ2E = bθ2

sg(θE/θs), (A10)

whereθs ≡ rs/Dd = rvir/(cvirDd) is the angular size of the NFW scale radius. This equation for θE can be solved numerically, forexample, by the Newton-Raphson method.

The SIS Lens Model

The singular isothermal sphere (SIS) density profile is given by

ρSIS(r) =σ2

v

2πGr2, (A11)

Umetsu et al. 25

whereσv is the one-dimensional isothermal velocity dispersion of the singular isothermal halo. The lensing convergence for aSIS lens is simply given by

κSIS(θ) =θE

2θ(A12)

with the Einstein radiusθE = 4π(σv/c)2Dds/Ds. Then, the averaged convergence interior to radiusθ is

κSIS(θ) =θE

θ= 2κSIS(θ). (A13)

The tangential shear is then evaluated by

γ+,SIS(θ) = κSIS(θ) −κSIS(θ) =θE

2θ. (A14)

ONE-DIMENSIONAL MASS RECONSTRUCTION FROM DISTORTION DATA

Following the method developed by Umetsu & Broadhurst (2008), we derive an expression for the discrete convergence profileusing a non-linear extension of the weak lensing aperture densitometry technique.

Non-Linear Aperture Mass Densitometry

For a shear-based estimation of the cluster mass profile we use a variant of the weak lensing aperture densitometry, or so-calledtheζ-statistic (Fahlman et al. 1994; Clowe et al. 2000) of the form:

ζc(θ)≡2∫ θinn

θ

d lnθ′γ+(θ′)

+2

1− (θinn/θout)2

∫ θout

θinn

d lnθ′γ+(θ′)

= κ(θ) − κ(θinn < ϑ < θout), (B1)

whereκ(θ) is the azimuthal average of the convergence fieldκ(θ) at radiusθ, κ(θ) is the average convergence interior to radiusθ, θinn and θout are the inner and outer radii of the annular background region in which the mean background contribution,κb ≡ κ(θinn <ϑ< θout), is defined; theγ+(θ) = κ(θ)−κ(θ) is an azimuthal average of the tangential component of the gravitationalshear at radiusθ (Fahlman et al. 1994), which is observable in the weak lensing limit: γ+(θ) ≈ 〈g+(θ)〉. This cumulative massestimator subtracts from the mean convergenceκ(θ) a constantκbg for all aperturesθ in the measurements, thus removing anyDC component in the control regionθ = [θinn,θout]. Note that theκb is a non-observable free parameter. This degrees of freedomcan be used to fix the outer boundary condition, and hence to derive a convergence profileκ(θ).

In the non-linear regime, theγ+(θ) is not a direct observable. Therefore, non-linear corrections need to be taken into accountin the mass reconstruction process (Umetsu & Broadhurst 2008). In the subcritical regime (i.e., outside the critical curves), theγ+(θ) can be expressed in terms of the the averaged tangential reduced shear as〈g+(θ)〉 ≈ γ+(θ)/[1 − κ(θ)] assuming a quasicircular symmetry in the projected mass distribution (Broadhurst et al. 2005b; Umetsu et al. 2007). This non-linear equation (B1)for ζc(θ) can be solved by an iterative procedure: Since the weak lensing limit (κ, |γ|, |g| ≪ 1) holds in the background regionθinn ≤ θ ≤ θmax, we have the following iterative equation forζc(θ):

ζ(k+1)c (θ)≈2

∫ θinn

θ

d lnθ′〈g+(θ)〉[1 −κ(k)(θ)]

+2

1− (θinn/θout)2

∫ θout

θinn

d lnθ′〈g+(θ′)〉, (B2)

whereζ(k+1)c represents the aperture densitometry in the (k+ 1)th step of the iteration (k = 0,1,2, ...,Niter); theκ(k+1) is calculated

from ζ(k+1)c using equation (B10). This iteration is preformed by starting with κ(0) = 0 for all radial bins, and repeated until

convergence is reached at all radial bins. For a fractional tolerance of 1×10−5, this iteration procedure converges withinNiter ∼ 10iterations. We compute errors forζc andκ with the linear approximation.

Discretized Estimator for the Lensing Convergence

In the continuous limit, the averaged convergenceκ(θ) and the convergenceκ(θ) are related by

κ(θ) =2θ2

∫ θ

0d lnθ′θ′2κ(θ′), (B3)

κ(θ) =1

2θ2

d(θ2κ)d lnθ

. (B4)

For a given set of annular radiiθm (m= 1,2, ...,N), discretized estimators can be written in the following way:

κm≡ κ(θm) =2θ2

m

m−1∑

l=1

∆ lnθl θ2l κ(θl ), (B5)

κl ≡κ(θl ) = αl2κl+1 −αl

1κl (l = 1,2, ...,N − 1), (B6)


where

αl1 =

12∆ lnθl

(

θl

θl

)2

, αl2 =

12∆ lnθl

(

θl+1

θl

)2

, (B7)

with ∆ lnθl ≡ (θl+1 −θl )/θl andθl being the area-weighted center of thel th annulus defined byθl andθl+1; in the continuous limit,we have

θl ≡2∫ θl+1

θl

dθ′θ′2/(θ2l+1 − θ2

l )

=23θ2

l + θ2l+1 + θlθl+1

θl + θl+1. (B8)

The technique of the aperture densitometry allows us to measure the azimuthally averaged convergenceκ(θ) up to an additiveconstantκb, corresponding to the mean convergence in the outer background annulus with inner and outer radii ofθinn andθout,respectively:

κ(θ) = ζc(θ) + κb. (B9)

Substituting equation (B9) into equation (B6) yields the desired expression as

κ(θl ) = αl2ζc(θl+1) −αl

1ζc(θl ) + (αl2 −αl

1)κb. (B10)

Finally, the error covariance matrix ofκl is expressed as

Ckl ≡ 〈δκkδκl 〉=αk2α

l2C

ζk+1,l+1 +αk

1αl1C

ζk,l

−αk1α

l2C

ζk,l+1 −αk

2αl1C

ζk+1,l , (B11)

whereCζkl ≡ 〈δζkδζl 〉 is the bin-to-bin error covariance matrix of the aperture densitometry measurements which is calculated by

propagating the rms errorsσ+(θl ) for the tangential shear measurement.

arXiv:0810.0969v1 [astro-ph] 6 Oct 2008 · 2019. 5. 3. · arxiv:0810.0969v1 [astro-ph] 6 oct 2008...

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Transcript of arXiv:0810.0969v1 [astro-ph] 6 Oct 2008 · 2019. 5. 3. · arxiv:0810.0969v1 [astro-ph] 6 oct 2008...