Mass and Hot Baryons in Massive Galaxy Clusters from Subaru Weak Lensing and AMiBA SZE Observations

1.4 MB·English

by  Keiichi Umetsu

DRAFTVERSIONJANUARY1,2009
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
MASSANDHOTBARYONSINMASSIVEGALAXYCLUSTERSFROMSUBARUWEAKLENSINGANDAMIBASZE
OBSERVATIONS1
KEIICHIUMETSU2,3,MARKBIRKINSHAW4,GUO-CHINLIU2,5,JIUN-HUEIPROTYWU6,3,ELINORMEDEZINSKI7,TOMBROADHURST7,
DORONLEMZE7,ADIZITRIN7,PAULT.P.HO2,8,CHIH-WEILOCUTUSHUANG6,3,PATRICKM.KOCH2,YU-WEILIAO6,3,KAI-YANG
LIN2,6,SANDORM.MOLNAR2,HIROAKINISHIOKA2,FU-CHENGWANG6,3,PABLOALTAMIRANO2,CHIA-HAOCHANG2,SHU-HAO
CHANG2,SU-WEICHANG2,MING-TANGCHEN2,CHIH-CHIANGHAN2,YAU-DEHUANG2,YUH-JINGHWANG2,HOMINJIANG2,
MICHAELKESTEVEN9,DEREKY.KUBO2,CHAO-TELI2,PIERREMARTIN-COCHER2,PETEROSHIRO2,PHILIPPERAFFIN2,TASHUN
WEI2,WARWICKWILSON9
DraftversionJanuary1,2009
9
ABSTRACT
0
0 Wepresentamultiwavelengthanalysisofasampleoffourhot(TX >8keV)X-raygalaxyclusters(A1689,
2 A2261, A2142, and A2390) using joint AMiBA Sunyaev-Zel’dovicheffect (SZE) and Subaru weak lensing
observations,combinedwith publishedX-raytemperatures,to examinethe distributionofmass andthe intr-
n
acluster medium (ICM) in massive cluster environments. Our observationsshow that A2261 is very similar
a
to A1689 in terms of lensing properties. Many tangential arcs are visible around A2261, with an effective
J
Einsteinradius 40′′(atz 1.5),whichwhencombinedwithourweaklensingmeasurementsimpliesamass
1 ∼ ∼
profilewellfittedbyanNFWmodelwithahighconcentrationc 10,similartoA1689andtoothermassive
clusters. TheclusterA2142showscomplexmasssubstructure,avnird∼displaysashallowerprofile(c 5),con-
] vir
∼
h sistentwithdetailedX-rayobservationswhichimplyrecentinteraction. TheAMiBAmapofA2142exhibits
p anSZEfeatureassociatedwithmasssubstructurelyingaheadofthesharpnorth-westedgeoftheX-raycore
- suggestingapressureincreaseintheICM.ForA2390weobtainhighlyellipticalmassandICMdistributions
o
atallradii,consistentwithotherX-rayandstronglensingwork. Ourclustergasfractionmeasurements,free
r
t from the hydrostatic equilibrium assumption, are overall in good agreement with published X-ray and SZE
s
a observations, with the sample-averaged gas fraction of fgas(<r200) =0.133 0.027, for our sample with
[ M =(1.2 0.1) 1015M h- 1. Whencomparedtothehcosmicbaryionfractio±n f =Ω /Ω constrainedby
h viri ± × ⊙ b b m
theWMAP5-yeardata,thisindicates f /f =0.78 0.16,i.e., (22 16)%ofthebaryonsaremissing
3 h gas,200i b ± ±
fromthehotphaseofclusters.
v
9 Subjectheadings:cosmology: observations,cosmic microwavebackground—galaxies: clusters: individual
6 (A1689,A2142,A2261,A2390)—gravitationallensing
9
0
. 1. INTRODUCTION inmergingclustersregardlessofthephysical/dynamicalstate
0
of the system (Cloweetal. 2006; Okabe&Umetsu 2008).
1 Clusters of galaxies, the largestvirialized systems known,
Thebulkofthebaryonsinclusters,ontheotherhand,reside
8 are key tracers of the matter distribution in the large scale
in the X-ray emitting intracluster medium (ICM), where the
0 structure of the Universe. In the standard picture of cosmic
X-ray surface brightnesstraces the gravitationalmass domi-
: structure formation, clusters are mostly composed of dark
v natedbyDM.Theremainingbaryonsareintheformoflumi-
matter(DM)asindicatedbyagreatdealofobservationalevi-
i nousgalaxiesandfaintintraclusterlight(Fukugitaetal.1998;
X dence,withtheaddedassumptionsthatDMisnonrelativistic
Gonzalezetal.2005). Sincerichclustersrepresenthighden-
(cold)andcollisionless,referredtoasCDM.Strongevidence
r
sity peaks in the primordial fluctuation field, their baryonic
a for substantial DM in clusters comes from multiwavelength
mass fraction and its redshift dependence can in principle
studies of interacting clusters (Markevitchetal. 2002), in
beusedtoconstrainthebackgroundcosmology(e.g.,Sasaki
whichweakgravitationallensingofbackgroundgalaxiesen-
1996; Allenetal. 2002, 2004, 2008). In particular, the gas
ablesustodirectlymapthedistributionofgravitatingmatter
mass to total mass ratio (the gas fraction) in clusters can
1BasedinpartondatacollectedattheSubaruTelescope,whichisoper- be used to place a lower limit on the cluster baryon frac-
atedbytheNationalAstronomicalSocietyofJapan tion,whichisexpectedtomatchthecosmicbaryonfraction,
2Institute ofAstronomyandAstrophysics, Academia Sinica, P.O.Box f Ω /Ω . However, non-gravitational processes associ-
b b m
23-141,Taipei10617,Taiwan ate≡dwithclusterformation,suchasradiativegascoolingand
3LeungcenterforCosmologyandParticleAstrophysics,NationalTaiwan AGN feedback, would break the self-similarities in cluster
University,Taipei10617,Taiwan
properties,whichcancausethe gasfractionto acquiresome
4Department ofPhysics, University ofBristol, Tyndall Avenue, Bristol
BS81TL,UK. massdependence(Bialeketal.2001;Kravtsovetal.2005).
5Department of Physics, Tamkang University, 251-37 Tamsui, Taipei The deep gravitational potential wells of massive clus-
County,Taiwan ters generate weak shape distortions of the images of back-
6DepartmentofPhysics,NationalTaiwanUniversity,Taipei10617,Tai-
ground sources due to differential deflection of light rays,
wan
7SchoolofPhysicsandAstronomy,TelAvivUniversity,TelAviv69978, resulting in a systematic distortion pattern around the cen-
Israel ters of massive clusters, known as weak gravitational lens-
8Harvard-Smithsonian CenterforAstrophysics,60GardenStreet,Cam- ing(e.g.,Umetsuetal.1999;Bartelmann&Schneider2001).
bridge,MA02138,USA
In the past decade, weak lensing has become a power-
9AustraliaTelescopeNationalFacility, P.O.Box76,EppingNSW1710,
ful, reliable measure to map the distribution of matter in
Australia
2 MassandHotBaryonsinMassiveGalaxyClusters
clusters, dominated by invisible DM, without requiring any forindividualclusters. In§5weexamineandcomparecluster
assumption about the physical/dynamical state of the sys- ellipticityandorientationprofilesonmassandICMstructure
tem (e.g., Cloweetal. 2006; Okabe&Umetsu 2008). Re- in the Subaru weak lensing and AMiBA SZE observations.
cently, cluster weak lensing has been used to examine the In§6wepresentourclustermodelsandmethodformeasur-
form of DM density profiles (e.g., Broadhurstetal. 2005b, ing cluster gas fraction profiles from joint weak-lensing and
2008;Mandelbaumetal.2008;Umetsu&Broadhurst2008), SZE observations, combined with published X-ray tempera-
aiming for an observational test of the equilibrium density ture measurements; we then derive cluster gas fraction pro-
profile of DM halos and the scaling relation between halo files, and constrain the sample-averaged gas fraction profile
mass and concentration, predicted by N-body simulations for our massive AMIBA-lensing clusters. Finally, a discus-
in the standard Lambda Cold Dark Matter (ΛCDM) model sionandsummaryaregivenin§7.
(Spergeletal.2007;Komatsuetal. 2008). Observationalre- Throughout this paper, we adopt a concordance
sultsshowthattheformoflensingprofilesinrelaxedclusters ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7, and
is consistent with a continuously steepening density profile h H /(100kms- 1Mpc- 1) = 0.7. Cluster properties are
0
≡
with increasing radius, well described by the general NFW determinedat the virial radiusr and radii(r ,r ,r ),
vir 200 500 2500
model(Navarroetal.1997),expectedforcollisionlessCDM correspondingtooverdensities(200,500,2500)relativetothe
halos. criticaldensityoftheuniverseattheclusterredshift.
The Yuan-Tseh Lee Array for Microwave Background
Anisotropy (Hoetal. 2008) is a platform-mounted interfer- 2. BASISOFCLUSTERSUNYAEV-ZEL’DOVICHEFFECTAND
WEAKLENSING
ometer array of up to 19 elements operating at 3mm wave-
length,specificallydesignedtostudythestructureofthecos- 2.1. Sunyaev-Zel’dovichEffect
micmicrowavebackground(CMB)radiation.Inthecourseof Webeginwithabriefsummaryofthebasicequationsofthe
early AMiBA operations we conducted Sunyaev-Zel’dovich thermalSZE. Our notationhere closely followsthe standard
effect (SZE) observations at 94GHz towards six massive notationofRephaeli(1995).
Abell clusters with the 7-element compact array (Wuetal.
The SZE is a spectral distortion of the CMB radiation re-
2008a). At 94GHz, the SZE signal is a temperature decre- sulting from the inverse Compton scattering of cool CMB
ment in the CMB sky, and is a measure of the thermal photonsbythehotICM.Thenon-relativisticformofthespec-
gas pressure in the ICM integrated along the line of sight tralchangewasobtainedbySunyaev-Zel’dovich(1972)from
(Birkinshaw1999;Rephaeli1995). Thereforeitisratherin- the Kompaneets equation in the non-relativistic limit. The
sensitivetotheclustercoreascomparedwiththeX-raydata,
change in the CMB intensity I due to the SZE is writ-
CMB
allowing us to trace the distribution of the ICM out to large tenintermsofitsspectralfunctiongandoftheintegralofthe
radii.
electron pressure along the line-of-sight as (Rephaeli 1995;
This paper presents a multiwavelength analysis of four Birkinshaw1999;Carlstrometal.2002):
nearby massive clusters in the AMiBA sample, A1689,
A2261, A2142, and A2390, for which high-quality deep ∆ISZE(ν)=Inormg[x(ν)]y(θ), (1)
Subaru images are available for accurate weak lensing mea-
where x(ν) is the dimensionless frequency, x
surements. This AMiBA lensing sample represents a sub- ≡
hν/(k T ) 1.66(ν/94GHz), with k being
set of the high-mass clusters that can be selected by their B CMB ≈ B
the Boltzmann constant and T = 2.725K be-
high (T >8keV) gas temperatures (Wuetal. 2008a). Our CMB
X ing the CMB temperature at the present-day epoch,
jointweaklensingandSZEobservations,combinedwithsup- I = (2h/c3)(k T /h)2 2.7 108Jysr- 1, and y(θ) is
porting X-ray information available in the published litera- norm B CMB ≃ ×
theComptonizationparameterdefinedas
ture,willallowustoconstraintheclustergasfractionswith-
out the assumption of hydrostatic equilibrium (Myersetal. +rmax k T σ +rmax ρ
1ie9s9.7O;uUrmceotmsupaentiaoln.2p0a0p5e)r,sccoommpplleemmeennttindgetXai-lrsayofbtahseedinsstturdu-- y=Z- rmax dlσthne(cid:18)mBece2(cid:19)= metch2Z- rmax dlµegmaspkBTgas,
(2)
ments,systemperformanceandverification,observationsand
where σ , m , c, and µ are the Thomson cross section, the
dataanalysis,andearlyscienceresultsfromAMiBA.Hoetal. th e e
electron mass, the speed of light, and the mass per electron
(2008)describethedesignconceptsandspecificationsofthe
in units of proton mass m , respectively; for a fully ionized
AMiBA telescope. Technical aspects of the instruments are H-Heplasma,µ =2/(1+Xp) 1.14,withX beingtheHydro-
describedin Chenetal. (2008) andKochetal. (2008a). De- e ≃
genprimordialabundancebymassfraction,X 0.76;r is
tails of the first SZE observationsand data analysis are pre- ≃ max
the cutoffradiusfor an isolated cluster (see §6.3). The SZE
sented in Wuetal. (2008a). Nishiokaetal. (2008) assess
spectralfunctiong(x)isexpressedas
the integrity of AMiBA data with several statistical tests.
Linetal. (2008) discuss the system performance and verifi- g(x)=g (x) 1+δ (x,T ) , (3)
NR SZE gas
cation. Liuetal. (2008) examine the levels of contamina- (cid:2) (cid:3)
where g (x) is the thermal spectral function in the non-
tionfromforegroundsourcesandtheprimaryCMBradiation. NR
relativisticlimit(Sunyaev&Zel’dovich1972),
Kochetal.(2008b)presentameasurementoftheHubblecon-
stant, H0, from AMiBA SZE and X-ray data. Huangetal. x4ex ex+1
(2008)discussclusterscalingrelationsbetweenAMiBASZE gNR(x)= (ex- 1)2(cid:18)xex- 1- 4(cid:19), (4)
andX-rayobservations.
Thepaperisorganizedasfollows.Webrieflysummarizein which is zero at the cross-over frequency x 3.83, or
0
§2thebasisofclusterSZEandweaklensing.In§3wepresent ν = 217GHz, and δ (x,T ) is the relativistic≃correction
0 SZE gas
aconcisesummaryoftheAMiBAtargetclustersandobserva- (Challinor&Lasenby1998;Itohetal. 1998). The fractional
tions. In§4wedescribeourweaklensinganalysisofSubaru intensity decrease due to the SZE with respect to the pri-
imagingdata,andderivelensingdistortionandmassprofiles mary CMB is maximized at ν 100GHz (see Figure 1 of
∼
Umetsuetal. 3
Zhangetal. 2002), which is well matched to the observing where ˆisthecomplexdifferentialoperator ˆ=(∂2- ∂2)/2+
D D 1 2
frequencyrange 86–102GHzof AMiBA. At the central fre- i∂ ∂ . The Green’s function for the two-dimensional Pois-
1 2
quencyν =94GHzofAMIBA,g(x) - 3.4. ForourhotX- sonequationis - 1(θ,θ′)=ln θ- θ′ /(2π),sothatequation
c
≃ △ | |
rayclusterswithT =8–10keV, therelativisticcorrectionto (11) can be solved to yield the following non-local relation
X
thethermalSZEis6–7%atν =94GHz. betweenκandγ (Kaiser&Squires1993):
c
2.2. ClusterWeakLensing 1
κ(θ)= d2θ′D∗(θ- θ′)γ(θ′) (12)
Weak gravitational lensing is responsible for the weak πZ
shape-distortion and magnification of the images of back- whereD(θ)isthecomplexkerneldefinedas
ground sources due to the gravitational field of intervening
foreground clusters of galaxies and large scale structures in θ2- θ2- 2iθ θ
theuniverse. Thedeformationoftheimagecanbedescribed D(θ)= 2 1 1 2. (13)
θ 4
bythe2 2Jacobianmatrix αβ(α,β=1,2)ofthelensmap- | |
× A
ping. TheJacobian isrealandsymmetric,sothatitcan Similarly,thespin-2shearfieldcanbeexpressedintermsof
αβ
A
bedecomposedas thelensingconvergenceas
αβ=(1- κ)δαβ- Γαβ, (5) 1
A γ(θ)= d2θ′D(θ- θ′)κ(θ′). (14)
Γαβ=(cid:18)+γγ21-γγ21(cid:19), (6) πZ
Notethataddinga constantmasssheetto κin equation(14)
where δαβ is Kronecker’s delta, Γαβ is the trace-free, sym- does not change the shear field γ(θ) which is observable in
metric shear matrix with γα being the components of spin- theweaklensinglimit,leadingtotheso-calledmass-sheetde-
2 complex gravitational shear γ := γ1+iγ2, describing the generacy(seeeq.[16])basedsolelyonshape-distortionmea-
anisotropicshapedistortion,andκisthelensingconvergence surements(e.g.,Bartelmann&Schneider2001;Umetsuetal.
responsibleforthetrace-partoftheJacobianmatrix,describ- 1999). Ingeneral,theobservablequantityisnotthegravita-
ing the isotropic area distortion. In the weak lensing limit tionalshearγ butthereducedshear,
whereκ, γ 1,Γ inducesaquadrupoleanisotropyofthe
| |≪ αβ γ
backgroundimage, which can be observed from ellipticities g= (15)
ofbackgroundgalaxyimages. Thefluxmagnificationdueto 1- κ
gravitationallensingisgivenbytheinverseJacobiandetermi- inthesubcriticalregimewheredet >0(or1/g∗intheneg-
nant, ative parity regionwith det <0)A. We see that the reduced
1 1 A
µ= = , (7) sheargisinvariantunderthefollowingglobaltransformation:
det (1- κ)2- γ 2
A | | κ(θ) λκ(θ)+1- λ, γ(θ) λγ(θ) (16)
whereweassumesubcriticallensing,i.e.,det (θ)>0. → →
A
Thelensingconvergenceisexpressedasaline-of-sightpro- with an arbitrary scalar constant λ = 0 (Schneider&Seitz
jection of the matter density contrast δm =(ρm- ρ¯)/ρ¯out to 1995). 6
the source plane (s) weighted by certain combinationof co-
movingangulardiameterdistancesr(e.g.,Jainetal.2000), 3. AMIBASUNYAEV-ZEL’DOVICHEFFECTOBSERVATIONS
κ= 3H02Ωm χsdχ (χ,χ )δm dΣ Σ- 1, (8) 3.1. AMIBATelescope
2c2 Z G s a ≡Z m crit The AMiBA is a dual channel 86–102GHz (3-mm wave-
0
r(χ)r(χ - χ) length) interferometer array of up to 19-elements with dual
s
(χ,χs)= , (9) polarization capabilities sited at 3396m on Mauna-Loa,
G r(χs) Hawaii (latitude: +19.5◦, longitude: - 155.6◦) 10. AMiBA
where a is the cosmic scale factor, χ is the co-moving
isequippedwith4-laganalog,broadband(16GHzbandwidth
distance, Σ is the surface mass density of matter, Σ =
m m centeredat 94GHz)correlatorswhich outputa set of 4 real-
dχa(ρ - ρ¯),withrespecttothecosmicmeandensityρ¯,and
m numbercorrelationsignals(Chenetal.2008). Thesefourde-
RΣcritisthecriticalsurfacemassdensityforgravitationallens- grees of freedom (dof) correspond to two complex visibili-
ing,
ties in two frequency channels. The frequency of AMiBA
Σ = c2 Ds (10) operation was chosen to take advantage of the optimal fre-
crit 4πGD D quency window at 3mm, where the fractional decrement in
d ds
withD ,D ,andD beingthe(proper)angulardiameterdis- the SZE intensity relative to the primary CMB is close to
s d ds
tances from the observer to the source, from the observerto its maximum (see §2.1) and contamination by the Galac-
the deflecting lens, and from the lens to the source, respec- tic synchrotron emission, dust foregrounds, and the popula-
tively. Forafixedbackgroundcosmologyandalensredshift tion of cluster/background radio sources is minimized (see
z , Σ is a function of backgroundsource redshift z . For for detailed contamination analysis, Liuetal. 2008). This
d crit s
agivenmassdistributionΣ (θ),thelensingsignalispropor- makes AMiBA a unique CMB/SZE interferometer, and also
m
tionaltotheangulardiameterdistanceratio,D /D . complementsthe wavelength coverage of other existing and
ds s
In the present weak lensing study we aim to reconstruct planned CMB instruments: interferometers such as AMI
the dimensionless surface mass density κ from weak lens- at 15GHz (Kneissletal. 2001), CBI at 30GHz (Padinetal.
ing distortion and magnification data. To do this, we utilize 2001, 2002; Masonetal. 2003; Pearsonetal. 2003), SZA
the relation between the gradients of κ and γ (Kaiser 1995; at 30 and 90GHz (Mroczkowskietal. 2008), and VSA11 at
Crittendenetal.2002),
10http://amiba.asiaa.sinica.edu.tw/
κ(θ)=∂α∂βΓαβ(θ)=2ˆ∗γ(θ) (11) 11http://astro.uchicago.edu/sza/
△ D
4 MassandHotBaryonsinMassiveGalaxyClusters
34GHz(Watsonetal.2003);bolometerarrayssuchasACT,12 We note that AMiBA and SZA are the only SZE instru-
APEX-SZ13(Halversonetal.2008),andSPT.14 mentsmeasuringat3mm,butcomplimentaryintheirbaseline
In the initial operation of AMiBA, we used seven 0.6m coverage. With sensitivities of 20–30mJybeam- 1 typically
(0.58mtobeprecise)Cassegrainantennas(Kochetal.2006) achieved in 2-patch differencing observationsin 5–10 hours
co-mountedona6mhexapodplatforminahexagonalclose- ofneton-sourceintegration(Wuetal.2008a),wewouldex-
packedconfiguration(seeHoetal.2008). Ateachofthefre- pect > 5σdetectionsofSZEfluxes > 100–150mJyat3mm.
quencychannelscenteredatabout90and98GHz,thiscom- Finall∼y, our observing period (Apri∼l–August 2007) limited
pact configuration provides 21 simultaneous baselines with the range of right ascension (RA) of targets,15 since we re-
three baseline lengths of d =0.61, 1.05, and 1.21m, corre- stricted our science observations to nights (roughly 8pm to
sponding to angular multipoles l =2π√u2+v2(=2πd/λ) of 8am),wherewewouldexpecthighgainstabilitybecausethe
l 1194,2073,2394atν =94GHz. Thiscompact7-element ambient temperature varies slowly and little (Nishiokaetal.
c
ar≈ray is sensitive to multipole range 800 < l < 2600. With 2008). The SZE strong clusters in our AMiBA sample are
0.6-mantennas,theinstantaneousfield-of-∼view∼ofAMiBAis likely to have exceedingly deep potential wells, and indeed
about 23′ FWHM (Wuetal. 2008a), and its angular resolu- our AMiBA sample represents a class of hot X-ray clusters
tionrangesfrom2′ to6′ dependingontheconfigurationand with observed X-ray temperatures exceeding 8keV (see Ta-
weighting scheme. In the compact configuration, the angu- ble 1). We note that this may affect the generality of the
lar resolution of AMiBA is about 6′ FWHM using natural results presented in this study. A main-trail/lead differenc-
weighting(i.e., inversenoisevarianceweighting). Thepoint ing scheme has been used in our targeted cluster observa-
sourcesensitivityisestimatedtobe 63mJy(Linetal.2008) tionswhere the trail/lead(blank)field is subtractedfromthe
in 1hour of on-source integration i∼n 2-patch main-trail/lead main(cluster)field. Thisdifferencingschemesufficientlyre-
differencingobservations,wheretheoverallnoiselevelisin- moves contamination from ground spillover and electronic
creasedbyafactorof√2duetothedifferencing. DC offset in the correlator output (Wuetal. 2008a). A full
descriptionofAMiBAobservationsandanalysisoftheinitial
3.2. InitialTargetClusters sixtargetclusters,includingtheobservationstrategy,analysis
methodology,calibrations,andmap-making,canbefoundin
The AMiBA lensing sample, A1689, A2142, A2261,
Wuetal.(2008a,b).
A2390,isasubsetoftheAMiBAclustersample(seeWuetal.
2008a), composed of four massive clusters at relatively low 4. SUBARUWEAKLENSINGDATAANDANALYSIS
redshifts of 0.09 < z < 0.23 with the median redshiftof ¯z
0.2. The sample∼size∼is simply limited by the availability o≈f In this section we present a technical description of our
weaklensingdistortionanalysisoftheAMiBAlensingsam-
high quality Subaru weak lensing data. A1689 is a relaxed,
ple based on Subaru data. The present work on A1689 is
roundsystem,andisoneofthebeststudiedclustersforlens-
based on the same Subaru images as analyzed in our earlier
ingwork(e.g.,Broadhurstetal.2005b;Limousinetal.2007;
workof Broadhurstetal. (2005b) and Umetsu&Broadhurst
Umetsu&Broadhurst2008; Broadhurstetal. 2008). A2261
(2008), but our improved color selection of the red back-
isacompactclusterwitharegularX-raymorphology.A2142
groundhasincreasedthesamplesizeby 16%(§4.3). This
isa mergingcluster withtwosharpX-raysurfacebrightness ∼
work on A2142 is based on the same Subaru images as in
discontinuities in the cluster core (Markevitchetal. 2000;
Okabe&Umetsu (2008), but our inclusion of blue, as well
Okabe&Umetsu 2008). A2390 shows an elongated mor-
as red, galaxies has increased the sample size by a factor
phologybothinthe X-rayemissionandstrong-lensingmass
of 4 (cf. Table 6 of Okabe & Umetsu 2008), leading to a
distributions(Allenetal.2001;Frye&Broadhurst1998).Ta-
significant improvement of our lensing measurements. For
ble 1 summarizes the physical properties of the four target
A2261andA2390wepresentournewweaklensinganalysis
clustersinthismultiwavelengthstudy.
basedonSuprime-CamimagingdataretrievedfromtheSub-
In 2007, AMiBA with the seven small antennas (hence-
aruarchive,SMOKA.Thereaderonlyinterestedinthemain
forth AMiBA7) was in the science verification phase. For
resultmayskipdirectlyto§4.4.
our initial observations, we therefore selected those tar-
get clusters observable from Mauna Loa during the observ-
4.1. SubaruDataandPhotometry
ing period that were known to have strong SZEs at rel-
atively low redshifts (0.1 < z < 0.3) from previous experi- We analyze deep images of four high mass clusters in the
ments, such as OVRO obs∼erva∼tions at 30GHz (Masonetal. AMiBAsampletakenbythewide-fieldcameraSuprime-Cam
2001), BIMA/OVRO observations at 30GHz (Gregoetal. (34′ 27′; Miyazakietal. 2002) at the prime-focus of the
×
2001a; Reeseetal. 2002), VSA observations at 34GHz 8.3m Subaru telescope. The clusters were observed deeply
(Lancasteretal.2005),andSuZIEIIobservationsat145,221, in two optical passbands each with seeing in the co-added
and355GHz(Bensonetal.2004).Thetargetedredshiftrange images ranging from 0.55′′ to 0.88′′ (see Table 2). For
allows the target clusters to be resolved by the 6′ resolution each cluster we select an optimal combination of two filters
of AMiBA7, allowing us to derive useful measurements of that allows for an efficient separation of cluster/background
clusterSZEprofilesforourmultiwavelengthstudies. Atred- galaxies based on color-magnitude correlations (see Table
shiftsofz < 0.3(0.2),theangularresolutionofAMiBA7cor- 2). We use either Rc or i′ band for our weak lensing mea-
respondst∼o < 560kpch- 1 ( 400kpch- 1)inradius,whichis surements(describedin §4.2) for which the instrumentalre-
< 30–40%(∼20–30%)of∼thevirialradius1.5–2Mpch- 1 of sponse, sky background and seeing conspire to provide the
m∼assiveclust∼ers. TherequirementofbeingSZE strongisto best-quality images. The standard pipeline reduction soft-
ensure reliable SZE measurements at 3mm with AMiBA7. ware for Suprime-Cam (Yagietal. 2002) is used for flat-
fielding,instrumentaldistortioncorrection,differentialrefrac-
12http://www.hep.upenn.edu/act/act.html tion, sky subtraction and stacking. Photometric catalogsare
13http://bolo.berkeley.edu/apexsz
14http://pole.uchicago.edu 15TheelevationlimitofAMiBAis30◦.
Umetsuetal. 5
constructedfromstackedandmatchedimagesusingSExtrac- where P is the smear polarizability tensor (which is close
sm
tor(Bertin&Arnouts1996),andusedforourcolorselection todiagonal),andq∗ =(P∗ )- 1 eβ isthestellaranisotropyker-
α sm αβ ∗
ofbackgroundgalaxies(see§4.3).
nel. Weselectbright,unsaturatedforegroundstarsidentified
inabranchofthehalf-lightradiusvs. magnitudediagramto
4.2. WeakLensingDistortionAnalysis measure q∗. In order to obtain a smooth map of q∗ which
α α
We use the IMCAT package developed by N. Kaiser16 is used in equation (18), we divided the co-added mosaic
to perform object detection, photometry and shape mea- image (of 10K 8K pixels) into rectangular blocks. The
surements, following the formalism outlined in Kaiseretal. blocklengt∼hisbas×edonthecoherentscaleofPSFanisotropy
(1995,KSB).Ouranalysispipelineisimplementedbasedon patterns, and is typically 2Kpixels. In this way the PSF
theproceduresdescribedinErbenetal.(2001)andonverifi- anisotropyinindividualblockscanbewelldescribedbyfairly
cationtestswithSTEP1dataofmockground-basedobserva- low-order polynomials. We then fitted the q∗ in each block
tions(Heymansetal. 2006). Thesame analysispipelinehas independently with second-order bi-polynomials, qα(θ), in
∗
beenusedinUmetsu&Broadhurst(2008),Okabe&Umetsu conjunction with iterative outlier rejection on each compo-
(2008),andBroadhurstetal.(2008). nentoftheresidual: δe∗ =e∗ - (P∗ )αβq∗(θ). Thefinalstel-
α α sm β
lar sample contains typically 500–1200 stars. Uncorrected
4.2.1. ObjectDetection
ellipticity components of stellar objects have on average a
Objects are first detected as local peaks in the image by mean offset (from a value of zero) of 1–2% with a few %
using the IMCAT hierarchical peak-findingalgorithm hfind- of rms, or variation of PSF across the data field (see, e.g.,
peakswhichforeachobjectyieldsobjectparameterssuchas Umetsu&Broadhurst2008;Okabe&Umetsu2008). Onthe
apeakposition(x),anestimateoftheobjectsize(r ),thesig- otherhand,themeanresidualstellarellipticityδe∗ aftercor-
g α
nificanceofthepeakdetection(ν). Thelocalskylevelandits rectionis less thanor about10- 4, with the standarderroron
gradientare measured around each object from the mode of this measurement, a few 10- 4. We show in Figure 1 the
pixelvaluesonacircularannulusdefinedbyinnerandouter quadrupolePSFanisotropy×fieldsasmeasuredfromstellarel-
radii of 16 rg and 32 rg. In order to avoid contamina- lipticitiesbeforeandafterthe anisotropicPSFcorrectionfor
× ×
tioninthebackgroundestimationbybrightneighboringstars our target clusters. Figure 2 shows the distributions of stel-
and/orforegroundgalaxies,allpixelswithin3 rgofanother larellipticitycomponentsbeforeandafterthePSFanisotropy
×
object are excludedfrom the mode calculation. Total fluxes correction. In addition, we adopt a conservative magnitude
andhalf-lightradii(rh) are then measuredon sky-subtracted limit m<25.5–26.0ABmag, dependingon the depth of the
imagesusingacircularapertureofradius3 rgfromtheob- data for each cluster, to avoid systematic errorsin the shape
×
ject center. Any pixels within 2.5 rg of another object are measurement(seeUmetsu&Broadhurst2008).Fromtherest
excludedfrom the aperture. The a×perturemagnitude is then of the object catalog, we select objects with r >r∗+σ(r∗)
h h h
calculatedfromthemeasuredtotalfluxandazero-pointmag- pixels as a magnitude-selectedweak lensing galaxy sample,
nitude. Any objects with positional differencesbetween the wherer∗isthemedianvalueofstellarhalf-lightradiir∗,cor-
peaklocationandtheweighted-centroidgreaterthan d =0.4 respondhingto half the median width of the circularizehdPSF
| |
pixelsareexcludedfromthecatalog. overthedatafield,andσ(r∗)isthermsdispersionofr∗.
h h
Finally, bad objects such as spikes, saturated stars, and Second, we need to correct image ellipticities for the
noisy detections need to be removed from the weak lensing isotropic smearing effect caused by atmospheric seeing and
analysis. We removedfrom our detection catalog extremely the windowfunction used for the shape measurements. The
small or large objects with rg <1 or rg >10 pixels, objects pre-seeingreducedsheargαcanbeestimatedfrom
withlowdetectionsignificance,ν<7(seeErbenetal.2001),
objects with large raw ellipticities, e > 0.5 (see §4.2.2), gα=(Pg- 1)αβe′β (19)
| |
noisydetectionswithunphysicalnegativefluxes,andobjects withthepre-seeingshearpolarizabilitytensorPg definedas
containingmorethan10badpixels,nbad>10. αβ
Hoekstraetal.(1998),
4.2.2. WeakLensingDistortionMeasurements tr[Psh∗]
Pg =Psh - Psm(Psm∗)- 1Psh∗ Psh - Psm (20)
Toobtainanestimateofthereducedshear,gα=γα/(1- κ) αβ αβ αβ ≈ αβ αβtr[Psm∗]
(cid:2) (cid:3)
(α=1,2),wemeasureusingthegetshapesroutineinIMCAT
the image ellipticity e = Q - Q ,Q /(Q +Q ) from with Psh being the shear polarizability tensor; In the sec-
α { 11 22 12} 11 22 ondequalitywehaveuseda trace approximationtothe stel-
the weighted quadrupole moments of the surface brightness
lar shape tensors, Psh∗ and Psm∗. To apply equation (19)
ofindividualgalaxiesdefinedintheabovecatalog,
the quantitytr[Psh∗]/tr[Psm∗] mustbe knownfor eachof the
galaxies with different sizescales. Following Hoekstraetal.
Q = d2θW(θ)θ θ I(θ) (α,β=1,2) (17)
αβ Z α β (1998), we recompute the stellar shapes Psh∗ and Psm∗ in
a range of filter scales r spanning that of the galaxy sizes
whereI(θ)isthesurfacebrightnessdistributionofanobject, g
(r = [1,10]pixels). At each filter scale r , the median
W(θ) is a Gaussian window function matched to the size of g g
tr[Psh∗]/tr[Psm∗] over the stellar sample is calculated, and
the object, and the object center is chosen as the coordinate h i
origin. Inequation(17)themaximumradiusofintegrationis usedinequation(20)asanestimateoftr[Psh∗]/tr[Psm∗]. Fur-
ther,weadoptascalarcorrectionscheme,namely
chosentobeθ =4r .
max g
Firstly the PSF anisotropyneedsto be correctedusing the 1
(P) = tr[P ]δ Psδ (21)
starimagesasreferences: g αβ 2 g αβ ≡ g αβ
e′ =e - Pαβq∗ (18) (Erbenetal. 2001; Hoekstraetal. 1998;
α α sm β
Umetsu&Broadhurst 2008). In order to suppress artifi-
16http://www.ifa.hawaii.edu/k˜aiser/imcat cial effects due to the noisy Ps estimated for individual
g
6 MassandHotBaryonsinMassiveGalaxyClusters
galaxies, we apply filtering to raw Ps measurements. We Anestimateofthebackgrounddepthisrequiredwhencon-
g
compute for each object a median value of Ps among vertingthe observedlensing signalinto physicalmass units,
g
N-neighbors in the size and magnitude plane to define the becausethelensingsignaldependsonthesourceredshiftsin
objectparameterspace: firstly, for each object, N-neighbors proportion to D /D . The mean depth is sufficient for our
ds s
withrawPs>0areidentifiedinthesize(r )andmagnitude purposes as the variation of the lens distance ratio, D /D ,
g g ds s
plane; the median value of Ps is then used as the smoothed is slow for our sample because the clusters are at relatively
g
cPagslcfourlattheedoubsijnecgt,eqhPugsait,ioannd(1t9h)e. vTahreiandcisepeσrg2sioofngσ=igs1u+siegd2aiss loofwthreedbsahcifktgsr(ozudn∼d0g.a1la- x0ie.2s.)coWmepeasretidmtaotethteheremdsehainftrdaenpgthe
g
anrmserroroftheshearestimateforindividualgalaxies. We Dds/Ds of the combinedred+blue backgroundgalaxiesby
h i
takeN =30. Finally,we usethe estimatorg =e′ / Ps for applyingourcolor-magnitudeselectiontoSubarumulticolor
α α g
thereducedshear. (cid:10) (cid:11) photometry of the HDF-N region (Capaketal. 2004) or the
COSMOS deep field (Capaketal. 2007), depending on the
4.3. BackgroundSelection availabilityof filters. Thefractionaluncertaintyin the mean
It is crucialin the weak lensing analysis to make a secure depth Dds/Ds fortheredgalaxiesistypically 3%,while
h i ∼
selection of background galaxies in order to minimize con- it is about 5% for the blue galaxies. It is useful to define
taminationbyunlensedcluster/foregroundgalaxiesandhence thedistance-equivalentsourceredshiftz (Medezinskietal.
s,D
to make an accurate determination of the cluster mass pro- 2007;Umetsu&Broadhurst2008)definedas
file; otherwise dilution of the distortion signal arises from D D
the inclusion of unlensed galaxies, particularly at small ra- ds = ds . (22)
(cid:28) D (cid:29) D (cid:12)
dius where the cluster is relatively dense (Broadhurstetal. s zs s (cid:12)zs=zs,D
2005b; Medezinskietal. 2007). This dilution effect is sim- We find zs,D = 0.70-+00..0056,0.95-+00..37(cid:12)(cid:12)09,0.98-+00..1264,1.00+- 00..2156 for
ply to reduce the strength of the lensing signal when aver- A1689, A2142, A2261, and A2390, respectively. For the
aged over a local ensemble of galaxies, in proportionto the nearbyclusterA2142atz 0.09,apreciseknowledgeofthe
≃
fractionofunlensedgalaxieswhoseorientationsarerandomly sourceredshiftisnotcriticalatallforlensingwork.Themean
distributed,thusdilutingthelensingsignalrelativetotheref- surface number density (ng) of the combinedblue+red sam-
erencebackgroundlevelderivedfromthebackgroundpopu- ple,theblue-to-redfractionofbackgroundgalaxies(B/R),the
lationMedezinskietal.(2007). estimatedmeandepth Dds/Ds ,andtheeffectivesourcered-
h i
To separate cluster members from the background and shiftzs,D arelistedinTable2.
hence minimize the weak lensing dilution, we follow
4.4. WeakLensingMap-Making
an objective background selection method developed by
Medezinskietal. (2007) and Umetsu&Broadhurst (2008). Weaklensingmeasurementsofthegravitationalshearfield
We select red galaxies with colors redder than the color- canbeusedtoreconstructtheunderlyingprojectedmassden-
magnitudesequence of cluster E/S0 galaxies. The sequence sity field. Inthe presentstudy, we willuse the dilution-free,
forms a well defined line in object color-magnitude space color-selectedbackgroundsample(§4.3)bothforthe2Dmass
due to the richness and relatively low redshifts of our clus- reconstructionandthelensprofilemeasurements17.
ters. Theseredgalaxiesareexpectedtolieinthebackground Firstly, we pixelize distortion data of background galax-
by virtue of k-correctionswhich are greater than for the red ies into a regular grid of pixels using a Gaussian wg(θ)
∝
clustersequencegalaxies;Thishasbeenconvincinglydemon- exp[- θ2/θ2]withθ =FWHM/√4ln2. Furtherwe incorpo-
f f
strated spectroscopicallybyRines&Geller (2008). We also rateinthepixelizationastatisticalweightu foranindividual
g
includebluegalaxiesfallingfarfromtheclustersequenceto galaxy, so that the smoothed estimate of the reduced shear
minimizeclustercontamination. fieldatanangularpositionθiswrittenas
Figure3showsforeachclusterthemeandistortionstrength w (θ- θ)u g
averaged over a wide radial range of θ =[1′,18′] as a func- g¯α(θ)= Pi gw (θ- iθ)gu,i α,i (23)
tionofcolorlimit, doneseparatelyfortheblue(left)andred i g i g,i
(right) samples, where the color boundaries for the present where gα,i is the reducedPshear estimate of the ith galaxy at
angular position θ, and u is the statistical weight of ith
analysis are indicated by vertical dashed lines for respec- i g,i
tive color samples. Here we do not apply area weighting galaxytakenastheinversevariance,ug,i=1/(σg2,i+α2),with
to enhance the effect of dilution in the central region (see σg,i being the rms error for the shear estimate of ith galaxy
Umetsu&Broadhurst2008). Asharpdropinthelensingsig- (see § 4.2.2) and α2 being the softening constant variance
nalisseenwhentheclusterredsequencestartstocontribute (Hamanaetal.2003). We chooseα=0.4,whichisatypical
significantly,therebyreducingthemeanlensingsignal. Note value of the mean rms σ¯ overthe backgroundsample. The
g
thatthebackgroundpopulationsdonotneedtobecompletein casewithα=0correspondstoaninverse-varianceweighting.
anysensebutshouldsimplybewelldefinedandcontainonly Ontheotherhand,thelimitα σ yieldsauniformweight-
g,i
≫
background. ForA1689,theweaklensingsignalin theblue ing. Wehaveconfirmedthatourresultsareinsensitivetothe
sample is systematically lower than that of the red sample, choiceofα(i.e.,inverse-varianceoruniformweighting)with
sothatbluegalaxiesinA1689areexcludedfromthepresent theadoptedsmoothingparameters.Theerrorvarianceforthe
analysis,aswasdoneinUmetsu&Broadhurst(2008);onthe smoothedshearg¯=g¯ +ig¯ (23)isthengivenas
1 2
other hand, our improved color selection for the red sample w2 u2 σ2
shtausdyledwetousae∼fo1r6A%21i4n2crtehaesesaomfereSdubgaarlauxiimesa.geInsatsheanparleyszeendt σg¯2(θ)= Piiwg,gi,iug,gi,ig,2i (24)
by Okabe&Umetsu (2008), but we have improved signifi- (cid:0)P (cid:1)
cantlyourlensingmeasurementsbyincludingblue,aswellas 17 Okabe&Umetsu(2008)used themagnitude-selected galaxy sample
intheirmap-makingofnearbymergingclusterstoincreasethebackground
red,galaxies,wherethe samplesize hasbeenincreasedbya
samplesize,whilethedilution-freeredbackgroundsamplewasusedintheir
factorof4. lensingmassmeasurements.
Umetsuetal. 7
where w = w (θ - θ) and we have used g g = tivecomparisonbetweentheAMiBASZEandSubarulensing
g.i g i α,i β,j
h i
(1/2)σ2 δK δK withδK andδK beingtheKronecker’sdelta. mapswillbegivenin§5.
g,i αβ ij αβ ij
Wetheninvertthepixelizedreduced-shearfield(23)toob-
tainthelensingconvergencefieldκ(θ)usingequation(12).In 4.5. ClusterLensingProfiles
themap-makingwe assume linearshearin theweak-lensing 4.5.1. LensDistortion
limit, that is, gα = γα/(1- κ) γα. We adopt the Kaiser The spin-2 shape distortion of an object due to gravita-
≈
& Squiresinversionmethod(Kaiser&Squires1993), which tional lensing is described by the complex reduced shear,
makesuseofthe2DGreenfunctioninaninfinitespace(§2.2). g=g +ig (see equation [15]), which is coordinate depen-
1 2
Inthelinearmap-makingprocess,thepixelizedshearfieldis dent. Foragivenreferencepointonthesky,onecaninstead
weighted by the inverse of the variance (24). Note that this formcoordinate-independentquantities,thetangentialdistor-
weightingschemecorrespondstousingonlythediagonalpart tiong+ andthe45◦ rotatedcomponent,fromlinearcombina-
of the noise covariance matrix, N(θi,θj)=h∆g(θi)∆g(θj)i, tionsofthedistortioncoefficientsg1andg2as
which is only an approximation of the actual inverse noise
weightinginthepresenceofpixel-to-pixelcorrelationdueto g+=- (g1cos2φ+g2sin2φ), g×=- (g2cos2φ- g1sin2φ),
(25)
non-local Gaussian smoothing. In Table 2 we list the rms
noise level in the reconstructed κ(θ) field for our sample of whereφisthepositionangleofanobjectwithrespecttothe
targetclusters. Foralloftheclusters,thesmoothingscaleθ referenceposition,andtheuncertaintyintheg+ andg× mea-
f
tihsetaEkeinnstteoinberaθdfiu=s1f′o(rθoFWurHMba≃ckg1r.6o6u5n′d),gwalhaixcihesi.slHaregnecrethoaunr sσugrefomretnhteiscoσm+p=leσx×sh=eσarg/m√e2as≡ureσmienntte.rmInsporfacthtiecer,mthseerrreof-r
weaklensingapproximationhereisvalidinallclusters. erence point is taken to be the cluster center, which is well
InFigure4weshow,forthefourclusters, 2Dmapsofthe determinedbythe locationsof the brightestclustergalaxies.
lensing convergence κ(θ) = Σ (θ)/Σ reconstructed from To improvethe statistical significance of the distortion mea-
m crit
the Subaru distortion data (§4.4), each with the correspond- surement, we calculate the weighted average of g+ and g×,
ing gravitationalshear field overlaid. Here the resolution of anditsweightederror,as
Tthheeκsifideeldlenisg∼th1o.f6t6h5e′diinspFlWayHedMrefgoironalliso2f2t′h,ecfoorurerscplounsdteirnsg. g+(θm) = iug,ig+,i, (26)
h i P u
roughlytotheinstantaneousfield-of-viewofAMiBA ( 23′ i g,i
inFWHM).Intheabsenceofhigher-ordereffects,weak≃lens- g (θ ) = Piug,ig×,i, (27)
ingonlyinducescurl-freeE-modedistortions,responsiblefor h × m i P u
i g,i
tangentialshear patterns, while the B-modelensing signalis
P
expectedtovanish. Foreachcase,a prominentmasspeakis u2 σ2
visibleintheclustercenter,aroundwhichthelensingdistor- σ+(θm)=σ×(θm)=vuPi ug,i i2, (28)
tionpatternisclearlytangential. u i g,i
t
Also shown in Figure 4 are contours of the AMiBA flux (cid:0)P (cid:1)
wheretheindexirunsoveralloftheobjectslocatedwithinthe
densityduetothethermalSZEobtainedbyWuetal.(2008a).
mthannuluswithamedianradiusofθ ,andu istheinverse
The resolution of AMiBA7 is about6′ in FWHM (§3). The m g,i
variance weight for ith object, u = 1/(σ2 +α2), softened
AMiBA map of A1689 reveals a bright and compact struc- g,i g,i
withα=0.4(see§4.4).
ture in the SZE, similar to the compact and round mass
Now we assess cluster lens-distortion profiles from the
distribution reconstructed from the Subaru distortion data.
color-selected background galaxies (§4.3) for the four clus-
A2142 shows an extended structure in the SZE elongated
ters, in order to examine the form of the underlying cluster
along the northwest-southeast direction, consistent with the
mass profile and to characterize cluster mass properties. In
direction of elongation of the X-ray halo, with its general
theweaklensinglimit(κ, γ 1),theazimuthallyaveraged
cometary appearance (Markevitchetal. 2000). In addition, | |≪
A2142 shows a slight excess in SZE signals located 10′ tangentialdistortionprofilehg+(θ)i(eq. [26])isrelatedtothe
∼ projectedmassdensityprofile(e.g.,Bartelmann&Schneider
northwest of the cluster center, associated with mass sub-
2001)as
structure seen in our lensing κ map (Figure 4); This slight
excess SZE appears extended for a couple of synthesized g+(θ) γ+(θ) =κ¯(<θ)- κ(θ) , (29)
h i≃h i h i
beams, although the per-beam significance level is marginal where denotes the azimuthal average, and κ¯(< θ) is
(2- 3σ).Okabe&Umetsu(2008)showedthatthisnorthwest h···i
the mean convergence within a circular aperture of radius
mcluassstersusbesqturuenctcueregailsaxailesos,alyssinogciate5d′wahitehadaosflitghhetneoxrctehswseosft θ defined as κ¯(< θ) = (πθ2)- 1 |θ′|≤θd2θ′κ(θ′). Note that
∼ equation (29) holds for an arbitRrary mass distribution. With
edge of the central X-ray gas core. On the other hand, no
the assumption of quasi-circular symmetry in the projected
X-raycounterparttothenorthwestsubstructurewasfoundin
massdistribution,onecanexpressthetangentialdistortionas
theX-rayimagesfromChandraandXMM-Newtonobserva-
tions (Okabe&Umetsu 2008). A2261 shows a filamentary g+(θ) [κ¯(<θ)- κ(θ) ]/[1- κ(θ) ]inthenon-linearbut
mass structurewith unknownredshift, extendingto the west shub-criiti≃cal(det (θ)h>0)iregimeh. i
A
Figure5showstheazimuthally-averagedradialprofilesof
of the cluster core (Maughanetal. 2008), and likely back-
groundstructureswhichcoincidewithreddergalaxyconcen- the tangentialdistortion, g+ (E mode), andthe 45◦-rotated
h i
component, g (Bmode).HerethepresenceofBmodescan
trations(see§4.5.2fordetails). OurAMiBAandSubaruob- ×
h i
beusedto checkforsystematicerrors. Foreachoftheclus-
servations show a compactstructure both in mass and ICM.
TheellipticalmassdistributioninA2390agreeswellwiththe ters,theobservedE-modesignalissignificantatthe12–16σ
shape seen by AMiBA in the thermal SZE, and is also con- leveloutto the limit of our data (θ 20′). The significance
∼
sistentwithotherX-rayandstronglensingwork. Aquantita- levelof the B-modedetectionis about2.5σ foreach cluster,
whichisaboutafactorof5smallerthanE-mode.
8 MassandHotBaryonsinMassiveGalaxyClusters
Themeasuredg+ profilesarecomparedwithtworepresen- In Figure 6 we show, for the four clusters, model-
tative cluster mass models, namely the NFW model and the independentκ profiles derived using the shear-based 1D re-
singularisothermalsphere(SIS)model.Firstly,theNFWuni- constructionmethod,togetherwithpredictionsfromthebest-
versaldensityprofilehasatwo-parameterfunctionalformas fitNFWmodelsfortheκ(θ)andg+(θ)data. Thesubstructure
contributionto κ(θ) is local, whereas the inversionfrom the
ρ
ρ (r)= s , (30) observabledistortiontoκinvolvesanon-localprocess. Con-
NFW (r/rs)(1+r/rs)2 sequentlythe1Dinversionmethodrequiresaboundarycon-
ditionspecifiedintermsofthemeanκvaluewithinanouter
whereρ isacharacteristicinnerdensity,andr isacharacter-
s s annularregion(lyingoutto18′–19′).Wedeterminethisvalue
isticinnerradius.Thelogarithmicgradientn dlnρ(r)/dlnr
oftheNFWdensityprofileflattenscontinuo≡uslytowardsthe foreachclusterusingthebest-fitNFWmodelfortheg+ pro-
centerofmass,withaflattercentralslopen=- 1andasteeper file(Table3).
outer slope (n - 3 when r ) than a purely isother- We find that the two sets of best-fit NFW parameters are
mal structure (n→= - 2). A use→ful∞index, the concentration, inexcellentagreementforallexceptA2261: ForA2261,the
compares the virial radius, r , to r of the NFW profile, best-fit values of cvir from the g+ and κ profiles are are in
vir s poorer agreement. From Figures 4 and 6 we see that the
c =r /r . WespecifytheNFWmodelwiththehalovirial
mvairss Mvir sand the concentration c instead of ρ and r .18 NFW fit to the g+ profile of A2261 is affected by the pres-
vir vir s s enceofmassstructuresatouterradii, θ 4′ and10′, result-
We employ the radial dependence of the NFW lensing pro- ≃
ing in a slightly shallower profile (c 6.4) than in the κ
files,κNFW(θ)andγ+,NFW(θ),givenbyBartelmann(1996)and analysis. It turns out that these masvsirs≃tructures are associ-
Wright&Brainerd (2000). Next, the SIS density profile is
atedwithgalaxyoverdensitieswhosemeancolorsareredder
givenby thantheclustersequenceforA2261atz=0.224,∆(V- R )
c
ρSIS(r)= 2πσGv2r2, (31) (bVe-phRycs)i-ca(Vlly- uRnca)sAs2o26c1ia∼ted+0b.a6c,kagnrdouhnedncoebtjhecetys.areThliekeNlyFW≡to
fit to κ(θ) yields a steeper profile with a high concentration,
where σv is the one-dimensionalisothermal velocity disper- c 10.2,whichimpliesalargeEinsteinangleofθ 37′′
vir E
sionoftheSIShalo. ThelensingprofilesfortheSISmodel, ≃ ≃
atz =1.5(Table3). Thisisingoodagreementwithourpre-
s
obtainedbyprojectionsofthethree-dimensionalmassdistri-
liminary strong-lensing model (Zitrin et al. in preparation)
bution,arefoundtobe
based on the method by Broadhurstetal. (2005a), in which
thedeflectionfieldisconstructedbasedonthesmoothedclus-
θ
κSIS(θ)=γ+,SIS(θ)= 2Eθ, (32) ter light distribution to predict the appearance and positions
ofcounterimagesofbackgroundgalaxies. Thismodelisre-
where θ is the Einstein radius defined by θ fined as new multiply-lensed images are identified in deep
E E
4π(σv/c)2Dds/Ds. ≡ SubaruVRcandCFHT/WIRCamJHKs images,andincorpo-
Table 3 lists the best-fitting parameters for these models, ratedtoimprovetheclustermassmodel. Figure7showsthe
together with the predicted Einstein radius θ for a fiducial tangentialcriticalcurvepredictedforabackgroundsourceat
E
sourceat zs =1.5, correspondingroughlyto the medianred- zs 1.5,overlaidontheSubaruV+Rcpseudo-colorimagein
shifts of our blue background galaxies. For a quantitative the∼central6.7′ 6.7′regionofA2261.Thepredictedcritical
×
comparison of the models, we introduce as a measure of curve is a nearly circular Einstein ring, characterized by an
the goodness-of-fitthe significance probabilityQ(ν/2,χ2/2) effective radius of θE 40′′ (see Oguri&Blandford 2008).
∼
to find by chance a value of χ2 as poor as the observed Thismotivatesustofurtherimprovethestatisticalconstraints
value for a given number of dof, ν (see §15.2 in Pressetal. ontheNFWmodelbycombiningtheouterlensconvergence
1992).19 We findwith ourbest-fitNFW modelsQ-valuesof profilewith theobservedconstraintonthe innerEinsteinra-
Q 0.50,0.95,0.36,and0.80,andwithourbest-fitSISmod- dius. A joint fit of the NFW profile to the κ profile and the
els≃Q 0.28,5.0 10- 6,0.37, and 0.87, for A1689, A2142, inner Einstein-radius constraint with θE =40′′ 4′′ (zs = 1)
A2261≃, and A239×0, respectively. Both models provide sta- tightenstheconstraintsontheNFWparameters(±see§5.4.2of
toiustricloalwlyesatc-zcecplutasbtelerAfit2s1f4o2r,Ath1e6c8u9r,vAat2u2r6e1i,natnhdeAg+23p9ro0fi.leFoisr Uanmdectsvuir =&1B1.r1o-+a12d..92h;uTrshti2s0m08o)d:elMyviierl=ds1.a2n5-+E00i..n1167st×ei1n0r1a5dMiu⊙sho- 1f
pronounced,andaSISmodelforA2142isstronglyruledout θ =(40 11)′′ atz =1.5. Inthefollowinganalysiswewill
E s
bytheSubarudistortiondataalone,wheretheminimumχ2is adoptthis±asourprimarymassmodelofA2261.
χ2 =39with8dof. For the strong-lensing cluster A1689, more detailed
min
lensing constraints are available from joint observa-
4.5.2. LensConvergence tions with the high-resolution Hubble Space Telescope
(HST) Advanced Camera for Surveys (ACS) and the
Although the lensing distortion is directly observable, the
wide-field Subaru/Suprime-Cam (Broadhurstetal. 2005b;
effect of substructure on the gravitational shear is non-
Umetsu&Broadhurst 2008). In Umetsu&Broadhurst
local. Hereweexaminethelensconvergence(κ)profilesus-
(2008) we combined all possible lensing measurements,
ing the shear-based 1D reconstructionmethod developedby
namely, the ACS strong-lensing profile of Broadhurstetal.
Umetsu&Broadhurst(2008). See AppendixA.1 for details
(2005b) and Subaru weak-lensing distortion and magnifica-
ofthereconstructionmethod.
tiondata, ina fulltwo-dimensionaltreatment,to achievethe
maximumpossiblelensingprecision. Note, the combination
18Weassumetheclusterredshiftzdisequaltotheclustervirialredshift. ofdistortionandmagnificationdatabreaksthemass-sheetde-
19 NotethataQvaluegreaterthan0.1indicatesasatisfactoryagreement
between the data and the model; if Q∼>0.001, the fit maybe acceptable, generacy (see eq. [16]) inherent in all reconstruction meth-
e.g. inacasethatthemeasurementerrorsaremoderatelyunderestimated;if ods based on distortion informationalone (Bartelmannetal.
Q∼<0.001,themodelmaybecalledintoquestion.
Umetsuetal. 9
1996). ItwasfoundthatthejointACSandSubarudata,cov- [1,2,3,...,11] θ (1.7′ < θ < 18.3′); for the AMiBA
FWHM ap
ering a wide range of radii from 10 up to 2000kpch- 1, are SZE,θ =[1,2×,3] θ (6∼′ < θ∼< 18′).Thelevelofun-
ap FWHM ap
wellapproximatedbyasingleNFWprofilewithMvir=(1.5 certaintyinthehalo×shapeparam∼eters∼isassessedbyaMonte-
0.1+- 00..63)×1015M⊙h- 1andcvir=12.7±1±2.8(statisticalfo±l- CarloerroranalysisassumingGaussianerrorsforweaklens-
lowed by systematic uncertaintyat 68%confidence).20 This ing distortion and AMiBA visibility measurements (for the
properlyreproducestheEinsteinradius,whichistightlycon- Gaussianity of AMiBA data, see Nishiokaetal. 2008). For
strained by detailed strong-lens modeling (Broadhurstetal. eachclusteranddataset,wegenerateasetof500MonteCarlo
2005a; Halkolaetal. 2006; Limousinetal. 2007): θ 52′′ simulationsofGaussiannoiseandpropagateintofinaluncer-
E
at z = 3.05 (or θ 45′′ at a fiducial source redsh≃ift of taintiesinthespin-2haloellipticity,ehalo. Figure8displays,
s E
≃
z =1). With the improvedcolor selection for the red back- forthefourclusters,theresultingclusterellipticityandorien-
s
ground sample (see §4.3), we have redone a joint fit to the tation profileson massand ICM structureas measuredfrom
ACS and Subaru lensing observations using the 2D method theSubaruweaklensingandAMiBASZEmaps,shownsep-
of Umetsu&Broadhurst (2008): The refined constraints on arately for the ellipticity modulus ehalo and the orientation,
cthe =N1F2W.3+p0a.9r,amyieetledrisngaraenMEivnirs=tei1n.5ra5d+- 00iu..11s32×of15001+56M.5⊙ahrc- s1ecanadt 2isφfhoaluon(dtwbiectewteheenpthoesistihoanpeasngolfem).aOs|svaenrad|llI,CaMgosotrducatgurreeeumpetnot
vir - 0.8 - 6.0
z =1.5. In the following, we will adopt this refined NFW largeradii,intermsofbothellipticityandorientation.Inpar-
s
profileasourprimarymassmodelofA1689. ticular, our resultson A2142andA2390show that the mass
and pressure distributions trace each other well at all radii.
5. DISTRIBUTIONSOFMASSANDHOTBARYONS Atalargeradiusofθ > 10′,thepositionangleofA2142is
ap
Hereweaimtocomparetheprojecteddistributionofmass φhalo 50◦. For A2390∼, the position angle is φhalo 30◦ at
∼ ∼
and ICM in the clusters using our Subaru weak lensing and allradii.
AMiBA SZE maps. To make a quantitativecomparison, we
first define the “cluster shapes” on weak lensing mass struc- 6. CLUSTERGASMASSFRACTIONPROFILES
turebyintroducingaspin-2haloellipticityehalo=ehalo+iehalo,
1 2 6.1. Method
definedintermsofweightedquadrupoleshapemomentsQhalo
αβ In modeling the clusters, we consider two representative
(α,β=1,2),as
analyticmodelsfor describingthe cluster DM and ICM dis-
Qhalo- Qhalo 2Qhalo tributions,namely(1)theKomatsu&Seljak(2001,hereafter
ehαalo(θap)=(cid:18)Q1h1alo+Q2h2alo,Qhalo+12Qhalo(cid:19), (33) KS01)modeloftheuniversalgasdensity/temperatureprofiles
11 22 11 22 and(2)theisothermalβmodel,wherebotharephysicallymo-
Qhalo(θ )= d2θ∆θ ∆θ κ(θ), (34) tivated under the hypothesis of hydrostatic equilibrium and
αβ ap Z∆θ≤θap α β psuomlyptrtoiopnicabeqouuatttihoens-opfh-esrtiactael,sPym∝mρeγt,rywoifththaensyadstdeimtio.nal as-
whereθapisthecircularapertureradius,and∆θαistheangu- Joint AMiBA SZE and Subaru weak lensing observations
lardisplacementvectorfromtheclustercenter. Similarly,the probe cluster structures on angular scales up to ∆θ 23′.21
spin-2haloellipticityfortheSZEisdefinedusingthecleaned At the median redshift ¯z 0.2 of our clusters, thi∼s maxi-
SZEdecrementmap- ∆I(θ) y(θ)insteadofκ(θ) inequa- mumanglecoveredbythe≃datacorrespondsroughlytor
tion (34). The degree of halo∝ellipticity is quantified by the 0.8r , exceptr 0.5r for A2142 at z=0.09. In o20r0d≈er
vir 500 vir
≈
modulusof the spin-2ellipticity, ehalo = (ehalo)2+(ehalo)2, to better constrain the gas mass fraction in the outer parts
| | q 1 2 of the clusters, we adopt a prior that the gas density pro-
and the orientation of halo is defined by the position angle
file ρ (r) traces the underlying(total) mass density profile,
of the major axis, φhalo = arctan(ehalo/ehalo)/2. In order to gas
2 1 ρ (r). Such a relationship is expected at large radii, where
avoidnoisyshapemeasurements,weintroducealowerlimit tot
non-gravitationalprocesses,suchasradiativecoolingandstar
ofκ(θ)>0and- ∆I(θ)>0inequation(34). Practicalshape
formation,havenothadamajoreffectonthestructureofthe
measurementsaredoneusingpixelizedlensingandSZEmaps
atmosphere so that the polytropic assumption remains valid
shownin Figure 4. The imagesare sufficientlyoversampled
(Lewisetal. 2000). Clearlythisresultsinthegasmassfrac-
thattheintegralinequation(34)canbeapproximatedbythe
tion, ρ (r)/ρ (r), tending to a constant at large radius. In
discretesum.Note,acomparisonintermsoftheshapeparam- gas tot
thecontextoftheisothermalβ model,thissimplymeansthat
etersisoptimalforthepresentcasewherethepairedAMiBA
β=2/3.
and weak lensing images have differentangular resolutions:
In both models, for each cluster, the mass density profile
θ 6′FWHMforAMiBA7,andθ 1.7′FWHM
foFrWSHuMb≃aruweaklensing.WhentheapertFuWreHdMia≃meterislarger ρtot(r) is constrainedsolely bythe Subaruweaklensing data
(§4), the gastemperatureprofileT (r) isnormalizedbythe
thantheresolutionθ ,i.e.,θ >θ /2,thehaloshape gas
FWHM ap FWHM spatially-averaged X-ray temperature (see Table 1), and the
parameterscanbereasonablydefinedandmeasuredfromthe
electronpressureprofileP(r)=n (r)k T(r)isnormalizedby
maps. e e B e
theAMiBASZEdata,wheren (r)istheelectronnumberden-
Now we measure as a function of aperture radius θ e
ap sity, and T(r)=T (r) is the electron temperature. The gas
the cluster ellipticity and orientation profiles for projected e gas
densityisthengivenbyρ (r)=µ m n (r).
mass and ICM pressure as represented by the lensing κ gas e p e
and SZE decrement maps, respectively. For the Subaru
weak lensing, the shape parameters are measured at θ = 6.2. ClusterModels
ap
6.2.1. NFW-ConsistentModelofKomatsu&Seljak2001
20 InUmetsu&Broadhurst(2008)clustermassesareexpressedinunits
of1015M⊙ withh=0.7. ThesystematicuncertaintyinMvir istightlycor-
related with that incvir through the Einstein radius constraint bythe ACS 21 TheFWHMoftheprimarybeampatten oftheAMiBAisabout23′,
observations. whilethefield-of-viewoftheSubaru/Suprime-Camisabout34′
10 MassandHotBaryonsinMassiveGalaxyClusters
The KS01 model describes the polytropic gas in hydro- corrected; see Markevitch 1998) is significantly higher than
static equilibrium with a gravitational potential described the lensing-derivedtemperature. This temperatureexcess of
by the universal density profile for collisionless CDM 4σ could be explainedby the effectsof mergerboosts, as
∼
halos proposed by Navarroetal. (1996, hereafter NFW). discussedinOkabe&Umetsu(2008). Thetemperatureratio
See KS01, Komatsu&Seljak (2002, hereafter KS02), and T /T for A1689, on the other hand, is significantly lower
X SIS
Worrall&Birkinshaw (2006) for more detailed discussions. thanunity. Recently, a similar levelof discrepancywas also
High mass clusters with virial masses M > 1015M /h are foundinLemzeetal.(2008a),whoperformedacarefuljoint
vir ⊙
so massive that the virial temperature of the∼gas is too high X-rayandlensinganalysisofthiscluster. A deprojected3D
forefficientcoolingandhencetheclusterpotentialsimplyre- temperatureprofilewas obtainedusinga model-independent
flects the dominant DM. This has been recently established approach to the Chandra X-ray emission measurements and
byourSubaruweaklensingstudyofseveralmassiveclusters theprojectedmassprofileobtainedfromthejointstrong/weak
(Broadhurstetal.2005b,2008;Umetsu&Broadhurst2008). lensing analysis of Broadhurstetal. (2005b). The projected
In this model, the gas mass profile traces the NFW pro- temperature profile predicted from their joint analysis ex-
file in the outer region of the halo (r /2 < r < r ; see ceedstheobservedtemperatureby30%atallradii,alevelof
vir vir
KS01), satisfying the adopted prior of the∼co∼nstant gas discrepancysuggestedfromhydrodynamicalsimulationsthat
mass fraction ρ (r)/ρ (r) at large radii. This behav- find that denser, more X-ray luminous small-scale structure
gas tot
ior is supported by cosmological hydrodynamic simula- canbiasX-raytemperaturemeasurementsdownwardatabout
tions (e.g., Yoshikawaetal. 2000), and is recently found thesamelevel(Kawaharaetal.2007). Ifweacceptthis+30%
from the stacked SZE analysis of the WMAP 3-year data correctionforT ,theratioT /T 1.07 0.04forA1689,
X X SIS
→ ±
(Atrio-Barandelaetal. 2008). The shape ofthe gasdistribu- consistentwithβ=2/3.
tion functions, as well as the polytropic index γ , can be
gas
6.3. AMiBASZEData
fullyspecifiedbythehalovirialmass,M ,andthehalocon-
vir
centration,c =r /r ,oftheNFWprofile. We use our AMiBA data to constrain the remaining nor-
vir vir s
In the following, we use the form of the NFW profile to malizationparameterfortheρ (r)profile,ρ (0). Thecal-
gas gas
determine r , r , r , and r . Table 4 summarizes the ibrated output of the AMiBA interferometer, after the lag-
vir 200 500 2500
NFWmodelparametersderivedfromourlensinganalysisfor to-visibilitytransformation(Wuetal.2008a),isthecomplex
the fourclusters(see §4.5). For eachcluster we also list the visibility V(u) as a function of baseline vector in units of
corresponding (r ,r ,r ,r ). For calculating γ and wavelength, u=d/λ, given as the Fourier transform of the
2500 500 200 vir gas
the normalization factor η(0) for a structure constant (B in sky brightness distribution ∆I(θ) attenuated by the antenna
equation[16]ofKS02),wefollowthefittingformulaegiven primarybeampatternA(θ).
byKS02,whicharevalidforhaloconcentration,1<c <25 IntargetedAMiBA observationsat94GHz,theskysignal
vir
(see Table 4). For our clusters, γ is in the range of 1.15 ∆I(θ) with respect to the background (i.e., atmosphere and
gas
to 1.20. Followingthe prescriptionin KS01, we convertthe the mean CMB intensity) is dominated by the thermal SZE
X-ray cluster temperature T to the central gas temperature due to hot electrons in the cluster, ∆I = I g(ν)y (see
X SZE norm
T (0)oftheKS01model. eq. [1]). The Comptonization parameter y is expressed as
gas
a line-of-sight integral of the thermal electron pressure (see
6.2.2. IsothermalβProfile
eq. [2]). In the line-of-sight projection of equation (2), the
The isothermalβ modelprovidesan alternativeconsistent cutoffradiusr needstobespecified.Wetaker α r
max max r vir
≡
solutionofthehydrostaticequilibriumequation(Hattorietal. with a dimensionless constant α which we set to α = 2.
r r
1999), assuming the ICM is isothermal and its density pro- In the present study we found the line-of-sight projection
filefollowsρgas(r)=ρgas(0)[1+(r/rc)2]- 3β/2withthegascore in equation (2) is insensitive to the choice of αr as long as
radiusrc. Atlargeradii, r rc, wherebothofourSZEand αr > 1.
weaklensingobservationsa≫resensitive,thetotalmassdensity A∼useful measure of the thermal SZE is the integrated
followsρ (r) r- 2. Thuswe set β =2/3 to satisfy our as- ComptonizationparameterY(θ),
tot
∝
sumptionofconstantρgas(r)/ρtot(r)atlargeradius. Weadopt θ
the values of r and T listed in Table 1, taken from X-ray Y(θ)=2π dθ′θ′y(θ′), (36)
c X
Z
observations,and use T (r)=T as the gastemperaturefor 0
gas X
thismodel. Atr r ,theρ (r)profilecanbeapproximated which is proportional to the SZE flux, and is a measure of
c tot
by that of a SIS≫(see §4.5.1) parametrizedby the isothermal the thermal energy content in the ICM. The value of Y is
1D velocity dispersion σ (see Table 4), constrained by the lesssensitivetothedetailsofthemodelfittedthanthecentral
v
Subarudistortiondata(see§4.5). Comptonizationparametery0 y(0),withthecurrentconfig-
Requiringhydrostaticbalancegivesanisothermaltemper- uration of AMiBA. If the A(θ≡)y(θ) field has reflection sym-
atureT ,equivalenttoσ ,as metry about the pointing center, then the imaginary part of
SIS v
V(u)vanishes,andtheskysignalisentirelycontainedinthe
2
k T µm σ2 . (35) realvisibilityflux. IftheA(θ)y(θ)fieldisfurtherazimuthally
B SIS≡ p v3β symmetric,therealvisibilityfluxisexpressedbytheHankel
For β =2/3, k T =µm σ2, which can be compared with transformoforderzeroas
B SIS p v
the observed T (Table 1). For our AMiBA-lensing cluster ∞
X VRe(u)=2πI g(ν ) dθθA(θ)y(θ)J (2πuθ)
sample, we foundX-rayto SIS temperatureratiosT /T = norm c 0
X SIS Z
0
0.82 0.03,1.65 0.15,0.94 0.05,1.28 0.15forA1689,
± ± ± ± ∞ y(θ)
A2142, A2261, and A2390, respectively. For A2261 and 2πI dθθA(θ) J (2πuθ), (37)
0 0
A2390, the inferred temperature ratios are consistent with ≡ Z0 y0
unity at 1–2σ. For the merging cluster A2142, the ob- where I = I g(ν )y is the central SZE intensity at ν =
0 norm c 0 c
served spatially-averaged X-ray temperature (cooling-flow 94GHz, J (x) is the Bessel function of the first kind and or-
0