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arXiv:0809.5077v5 [astro-ph] 15 Jan 2009

Application of a Self-Similar Pressure Profile to Sunyaev-Zel’dovich Effect

Data from Galaxy Clusters

Tony Mroczkowski,1,2,3Max Bonamente,4,5John E. Carlstrom,6,7,8,9Thomas L. Culverhouse,6,7

Christopher Greer,6,7David Hawkins,10Ryan Hennessy,6,7Marshall Joy,5James W. Lamb,10

Erik M. Leitch,6,7Michael Loh,6,9Ben Maughan,11,12Daniel P. Marrone,6,8,13Amber Miller,1,14,15

Stephen Muchovej,2Daisuke Nagai,16,17Clem Pryke,6,7,8Matthew Sharp,6,9and David Woody10

ABSTRACT

We investigate the utility of a new, self-similar pressure profile for fitting

Sunyaev-Zel’dovich (SZ) effect observations of galaxy clusters. Current SZ imaging

instruments—such as the Sunyaev-Zel’dovich Array (SZA)—are capable of probing

clusters over a large range in physical scale. A model is therefore required that can ac-

curately describe a cluster’s pressure profile over a broad range of radii, from the core

1Columbia Astrophysics Laboratory, Columbia University, New York, NY 10027

2Department of Astronomy, Columbia University, New York, NY 10027

3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104

4Department of Physics, University of Alabama, Huntsville, AL 35899

5Department of Space Science, VP62, NASA Marshall Space Flight Center, Huntsville, AL 35812

6Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637

7Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637

8Enrico Fermi Institute, University of Chicago, Chicago, IL 60637

9Department of Physics, University of Chicago, Chicago, IL 60637

10Owens Valley Radio Observatory, California Institute of Technology, Big Pine, CA 93513

11Department of Physics, University of Bristol, Tyndall Ave, Bristol BS8 1TL, UK.

12Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138

13Jansky Postdoctoral Fellow, National Radio Astronomy Observatory

14Department of Physics, Columbia University, New York, NY 10027

15Alfred P. Sloan Fellow

16Department of Physics, Yale University, New Haven, CT 06520

17Yale Center for Astronomy & Astrophysics, Yale University, New Haven, CT 06520

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of the cluster out to a significant fraction of the virial radius. In the analysis presented

here, we fit a radial pressure profile derived from simulations and detailed X-ray anal-

ysis of relaxed clusters to SZA observations of three clusters with exceptionally high

quality X-ray data: A1835, A1914, and CL J1226.9+3332. From the joint analysis

of the SZ and X-ray data, we derive physical properties such as gas mass, total mass,

gas fraction and the intrinsic, integrated Compton y-parameter. We find that param-

eters derived from the joint fit to the SZ and X-ray data agree well with a detailed,

independent X-ray-only analysis of the same clusters. In particular, we find that, when

combined with X-ray imaging data, this new pressure profile yields an independent

electron radial temperature profile that is in good agreement with spectroscopic X-ray

measurements.

Subject headings: cosmology: observations — clusters: individual (Abell 1835, Abell

1914, CL J1226.9+3332) — Sunyaev-Zel’dovich Effect

1. Introduction

The expansion history of the universe and the growth of large-scale structure are two of the

most compelling topics in cosmology. As galaxy clusters are the largest collapsed objects in the

universe, taking a Hubble time to form, their abundance and evolution are critically sensitive to

the details of that expansion history. Cluster surveys can therefore provide fundamental clues to

the nature and abundance of dark matter and dark energy (see, e.g., White et al. 1993; Frenk et al.

1999; Haiman et al. 2001; Weller et al. 2002).

While clusters have been extensively studied using X-ray observations of the hot gas that

comprises the intracluster medium (ICM), radio measurements of the same gas via the Sunyaev-

Zel’dovich (SZ) effect (Sunyaev & Zel’dovich 1972) provide an independent and complementary

probe of the ICM (e.g., Carlstrom et al. 2002). The SZ effect arises from inverse Compton scat-

tering of cosmic microwave background (CMB) photons by the electrons in the ICM, imparting

a detectable spectral signature to the CMB that is independent of the redshift of the cluster. As

the SZ effect is a measure of the integrated line-of-sight electron density, weighted by tempera-

ture, i.e., the integrated pressure (see § 2.1), it probes different properties than the X-ray emission,

which is proportional to the square of the electron density. Cluster surveys exploiting the redshift

independence of the SZ effect are now being conducted by a variety of instruments, including the

SZA (Muchovej et al. 2007), the South Pole Telescope (Ruhl et al. 2004), the Atacama Cosmol-

ogy Telescope (Kosowsky 2003) and the Atacama Pathfinder Experiment (APEX) SZ instrument

(Dobbs et al. 2006).

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To maximize the utility of clusters as cosmological probes we must understand how to ac-

curately relate their observable properties to their total masses. The integrated SZ signal from a

cluster is proportional to the total thermal energy of the cluster, and is therefore a measure of the

underlying gravitational potential, and ultimately the dark matter content, within a cluster. The SZ

flux of a cluster thereby should provide a robust, low scatter proxy for the total cluster mass, Mtot

(see, for example, da Silva et al. 2004; Motl et al. 2005; Nagai 2006; Reid & Spergel 2006).

To date, cosmological studies combining SZA and X-ray data have relied almost exclusively

on the isothermal β-model (first used in Cavaliere & Fusco-Femiano 1976, 1978) to fit the SZ sig-

nal in the region interior to r2500(see Grego et al. 2000; Reese et al. 2002; LaRoque et al. 2006;

Bonamente et al. 2006, 2008, for applications of the isothermal β-model to SZ+X-ray data). Here

r2500is the radius within which the mean cluster density is a factor of 2500 over the critical density

of the universe at the cluster redshift. While the isothermal β-model recovers global properties of

clusters quite accurately in this regime (LaRoque et al. 2006), deep X-ray observations of nearby

clusters show that isothermality is a poor description of the cluster outskirts (r ∼ r500) (see e.g.

Piffaretti et al. 2005; Vikhlinin et al. 2005; Pratt et al. 2007, and references therein). An improved

model for the cluster gas, accurate to large radii, is therefore critical for the analysis and cosmo-

logical interpretation of SZ data obtained with new instruments that are capable of probing the

outer regions (r ∼ r500) of clusters. Such a model must be simple enough that it can be con-

strained by SZ data with limited angular resolution and sensitivity typical of data sets acquired by

SZ survey instruments optimized for detection, rather than imaging. While the β-model has the

virtue of simplicity, previous attempts to relax the assumption of isothermality typically required

high-significance, spatially-resolved X-ray spectroscopy; such data are seldom obtained in short

X-ray exposures of high-redshift clusters. This is particularly true for the cluster outskirts (see,

e.g., LaRoque et al. 2006). Attempts to move beyond the β-model have typically improved the

modeling of the gas density only within the core (r ? 0.15r500).

In this work, we investigate a new model for the cluster gas pressure by using it to fit SZ data

from theSZA and X-ray datafrom Chandra to threewell-studiedclusters: Abell 1835, Abell 1914,

and CL J1226.9+3332. The model was derived by Nagai et al. (2007) from simulations and from

detailed analysis of deep Chandra measurements of nearby relaxed clusters. The simplicity of

this model—and the fact that SZ data are inherently sensitive to the integrated electron pressure—

allow it to be used either in conjunction with X-ray imaging data, or fit to SZ data alone. The

outline of the paper is as follows: in § 2, we present the details of the model, and couple it with

an ICM density model that allows the inclusion of X-ray imaging data. In § 3, we apply this

method to three clusters, combining new data from the SZA with Chandra X-ray imaging data.

We demonstrate the utility of this model by applying it to SZ+X-ray data without relying on X-ray

spectroscopic information. Results from the joint SZ+X-ray analysis are then compared to results

from an X-ray-only analysis, including spectroscopic data, in § 4. We offer our conclusions in § 5.

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2. Cluster Gas Models

2.1. Sunyaev Zel’dovich Effect and X-ray Emission

ThethermalSZ effect isasmall(< 10−3)distortioninCMBintensitycausedby inverseComp-

tonscatteringofCMBphotonsbyenergeticelectronsinthehotintraclustergas(Sunyaev & Zel’dovich

1970, 1972). This spectral distortioncan be expressed, for dimensionlessfrequency x ≡ hν/kBTCMB,

where h is Planck’s constant, ν is frequency, kBis Boltzmann’s constant, and TCMBis the primary

CMB temperature, as the change ∆ISZrelative to the primary CMB intensity normalization I0,

∆ISZ

I0

=

kBσT

mec2

σT

mec2

?

g(x,Te)neTedℓ

(1)

=

?

g(x,Te)Pedℓ.

(2)

Here σTis the Thomson scattering cross-section of the electron, ℓ is the line of sight, and mec2

is an electron’s rest energy. The factor g(x,Te) encapsulates the frequency dependence of the SZ

effect intensity. For non-relativistic electrons,

g(x) =

x4ex

(ex− 1)2

?

xex+ 1

ex− 1− 4

?

.

(3)

We use the Itoh et al. (1998) relativistic corrections to Eq. 3, which are appropriate for fitting

thermal SZ observations, and discuss their impact on the fit ICM profiles in § 4.3. Note that we

have used the ideal gas law (Pe = nekBTe) to obtain Eq. 2 from Eq. 1; we use this to relate SZ

intensity directly to the ICM electron pressure.

The X-ray emission from massive clusters arises predominantly as thermal bremsstrahlung

from electrons. The X-ray surface brightness produced by a cluster at redshift z is given by (e.g.,

Sarazin 1988)

1

4π(1 + z)4

where the integral is evaluated along the line of sight and the X-ray cooling function Λeeis propor-

tional to T1/2

a relatively weak dependence on Te. Separate spectroscopic observations of X-ray line emission

can be used to measure the gas temperature. Throughout this work, we use the Raymond-Smith

plasma emissivity code (Raymond & Smith 1977) with constant metallicity Z = 0.3Z⊙, yielding

µe= 1.17 and µ = 0.61 for the mean molecular weight of the electrons and gas, respectively, using

the elemental abundances provided by Anders & Grevesse (1989). The SZ and X-ray observables

∆ISZand SXare computed by evaluating the integrals in Eqs. 2 & 4 numerically.

SX=

?

n2

eΛee(Te,Z)dℓ,

(4)

e . The X-ray emission is therefore proportional to the square of the gas density, with

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2.2.Three-dimensional Models of the ICM Profiles

In this work, we adopt an analytic parameterization of the cluster radial pressure profile pro-

posed by Nagai et al. (2007) (hereafter N07),

Pe(r) =

Pe,i

(r/rp)c?

1 + (r/rp)a?(b−c)/a,

(5)

where Pe,iis a scalar normalization of the pressure profile, rpis a scale radius (typically rp ≈

r500/1.3), and theparameters (a,b,c)respectivelydescribe theslopes at intermediate(r ≈ rp), outer

(r > rp), and inner (r ≪ rp) radii. Note that Eq. 5 is a generalization of the analytic fitting for-

mula obtained in numerical simulations as a parameterization of the distribution of mass in a dark

matter halo (Navarro et al. 1997, NFW profile). This choice is reasonable because the gas pressure

distribution is primarily determined by the dark matter potential. The use of a parameterized pres-

sure profile is further motivated by the fact that self-similarity is best preserved for pressure, as

demonstrated by the low cluster-to-cluster scatter seen when using these parameters to fit numeri-

cal simulations. The NFW profile – in its pure form – has been applied by Atrio-Barandela et al.

(2008) to fit the electron pressure profiles of SZ observations of clusters, who demonstrated an

improvement over the application of the isothermal β-model to SZ cluster studies. In this work,

we adopt the fixed slopes (a,b,c) = (0.9,5.0,0.4), which N07 found to closely match the observed

profiles of the Chandra X-ray clusters and the results of numerical simulations in the outskirts of

a broad range of relaxed clusters.1

The density model used to fit the X-ray image data is a simplified version of the model em-

ployed by Vikhlinin et al. (2006) (hereafter V06) is

n2

e(r) = n2

e0

(r/rc)−α

?1 + (r/rc)2?3β−α/2

1

[1 + (r/rs)γ]ε/γ+

n2

e02

?1 + (r/rc2)2?3β2.

(6)

We refer to Eq. 6, used in the independent X-ray analysis to which we compare our SZ+X-ray

results, as the “V06 density model.” Since the cluster core contributes negligibly to the SZ signal

observedby theSZA, weexcludetheinner100kpcoftheclusterfrom theX-ray imagesused inthe

joint SZ+X-ray analysis. Recognizing α as the component introduced by Pratt & Arnaud (2002)

to fit the inner slope of a cuspy cluster density profile, and that the second β-model component (β2)

is present explicitly to fit the cluster core, we simplify the V06 density model to

ne(r) = ne0

?

1 + (r/rc)2?−3β/2[1 + (r/rs)γ]−0.5ε/γ,

(7)

1The original values published in N07 were (a,b,c) = (1.3,4.3,0.7). These have recently been updated, however,

and will be published in an erratum to N07.

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where rcis the core radius, and rsis the radius at which the density profile steepens with respect to

the traditional β-model. FollowingV06, we fix γ = 3, as this provides an adequate fit to all clusters

in the V06 sample. We refer to this model as the “Simplified Vikhlinin Model” (SVM). Note that

in the limit ε → 0, Eq. 7 reduces to the standard β-model. The SVM is thus a modification to the

β-model that has the additional freedom to extend from the core (r ? rc) to the outer regions of

cluster gas, spanning intermediate (r ? rs) to large radii (r ∼ r500).

With the electron pressure and densityin hand, we may also derivethe electron temperature of

the ICM using the ideal gas law, Te(r) = Pe(r)/kBne(r), where Peand neare given by Eqs. 5 and 7,

respectively. Note that Te(r) derived in this way is used in the analysis of X-ray surface brightness

(Eq. 4). Hereafter, we refer to this jointly-fit cluster gas model as the N07+SVM profile.

For comparison with previous work (e.g. Bonamente et al. 2008), we also employ the isother-

mal β-model for joint analysis of the SZ and X-ray data. In this model the density is given by

Eq. 7 with ε = 0 and Te(r) is a constant equal to the spectroscopically-measured temperature,

TX. The shape parameters of the isothermal β-model, rcand β, are jointly fit to the SZ and X-ray

data, while the X-ray surface brightness (Eq. 4) and SZ intensity profile (Eq. 2) normalizations are

independently determined from the X-ray and SZ data, respectively.

2.3. Parameter Estimation Using the Markov Chain Monte Carlo Method

Our models have five free parameters to describe the radial distribution of the gas density (see

Eq. 7) and two parameters for the electron pressure (see Eq. 5). Additional parameters such as

the cluster centroid, X-ray background level, as well as the positions, fluxes and spectral indices

of compact radio sources are also included where necessary. The Markov chain Monte Carlo

(MCMC) method is used to extract the model parameters from the SZ and X-ray data, as described

by Bonamente et al. (2004). In this section we provide a brief overview of this method, focusing

on the changes to accommodate the N07 pressure model.

The first step in fitting the SZ data is to compute the model image over a regular grid, sampled

at less than half the smallest scale the SZA can probe. This image is multiplied by the primary

beam of the SZA, transformed via FFT to Fourier space (where the data are naturally sampled by

an interferometer; see § 4.1), and interpolated to the Fourier-space coordinates of the SZ data. The

likelihood function for the SZ data is then computed directly in the Fourier plane, where the noise

properties of the interferometric data are well-characterized.

The first step of the MCMC method is the calculation of the joint likelihood L of the X-ray

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and SZ data with the model. The SZ likelihood is given by

ln(LSZ) =

?

i

?

−1

2

?

∆R2

i+ ∆I2

i

??

Wi,

(8)

where ∆Riand ∆Iiare the differences between model and data for the real and imaginary compo-

nents at each point i in the Fourier plane, and Wiis a measure of the Gaussian noise (1/σ2).

Since the X-ray counts, treated in image space, are distributed according to Poisson statistics,

the likelihood of the model fit is given by

ln(Limage) =

?

i

[Diln(Mi) − Mi− ln(Di!)],

(9)

where Miis the model prediction (including cluster and background components), and Diis the

number of counts detected in pixel i. The inner 100 kpc of the X-ray images—as well as any

detected X-ray point sources—are excluded from the fits by excluding these regions from the

calculation of ln(Limage) in Eq. 9.

The joint likelihood of the spatial and spectral models is given by L = LSZLXray. For

the N07+SVM fits, LXrayis simply Limage. Following Bonamente et al. (2004, 2006), the X-ray

likelihood for the β-model fits is LimageLXspec, as these must incorporate the likelihood LXspecof

the spectroscopic determination of TX. The likelihood is used to generate the Markov parameter

chains, and convergence of the chain to a stationary distribution is established using the Raftery-

Lewis and Geweke tests (Raftery & Lewis 1992; Gilks et al. 1996).

2.4. Calculation of Mgas, Mtot, and Yint

Withknowledgeofthethree-dimensionalgasprofiles, wecomputeglobalpropertiesofgalaxy

clusters as follows. The gas mass Mgas(r) enclosed within radius r is obtained by integrating the

gas density ρgas≡ µempne(r) over a spherical volume:

?r

0

Mgas(r) = 4πρgas(r′)r′2dr′.

(10)

The total mass, Mtot, can be obtained by solving the hydrostatic equilibrium equation as:

Mtot(r) = −

r2

Gρgas(r)

dP(r)

dr

,

(11)

where P = (µe/µ)Peis the total gas pressure. We then compute the gas mass fractions as fgas=

Mgas/Mtot.

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The line of sight Compton y-parameter, which characterizes the strength of Compton scatter-

ing by electrons, is defined

y ≡kBσT

mec2

We compute the volume-integrated Compton y-parameter, Y, from the pressure profile fit to the SZ

observationsfor both cylindricaland spherical volumes ofintegration. The cylindrically-integrated

quantity, Ycyl, is calculated within an angle θ on the sky, corresponding to physical radius R = θdA

at the redshift of the cluster,

?

neTedℓ.

(12)

Ycyl(R) ≡ 2πd2

A

?θ

0

y(θ) θ′dθ′= 2π

?R

0

y?r′?r′dr′=2πσT

mec2

?∞

−∞dℓ

?R

0

Pe?r′?r′dr′.

(13)

The last form makes explicit the infinite limits of integration in the line of sight direction, originat-

ing with the definition of y (Eqs. 1 and 2). The spherically-integrated quantity, Ysph, is obtained by

integrating the pressure profile within a radius r from the cluster center,

Ysph(r) =4πσT

mec2

?r

0

Pe(r′)r′2dr.

(14)

The parameter Ysph(r) is thus proportional to the thermal energy content of the ICM.

To compute the global cluster properties described above, one needs to define a radius out

to which all quantities will be calculated. Following LaRoque et al. (2006) and Bonamente et al.

(2006), we compute global properties of clusters enclosed within the overdensity radius r∆, within

which the average density ?ρ? of the cluster is a specified fraction ∆ of the critical density, via

4

3πρc(z)∆r3

∆= Mtot(r∆),

(15)

where ρc(z) is the critical density at cluster redshift z, and ∆ ≡ ?ρ?/ρc(z). In this work, we

evaluate cluster properties at density contrasts of ∆ = 2500 and ∆ = 500. The overdensity radius

r2500has been used in previous OVRO and BIMA interferometric SZ studies (e.g. LaRoque et al.

2006; Bonamente et al. 2008) as well as in many X-ray cluster studies (e.g. Vikhlinin et al. 2006;

Allen et al. 2007), while r500is a radius reachable with SZA and deep Chandra X-ray data.

3. Data

3.1. Cluster Sample

For this work, we selected three massive clusters that are well studied at X-ray wavelengths,

and span a range of redshifts (z = 0.17–0.89) and morphologies, on which to test the joint analysis

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of the Chandra X-ray and SZA data. We assume a ΛCDM cosmology with ΩM= 0.3, ΩΛ= 0.7,

and h = 0.7 throughout our analysis.

Located at z = 0.25, Abell 1835 (A1835) is an intermediate-redshift, relaxed cluster, as

evidenced by its circular morphology in the X-ray images and its cooling core (Peterson et al.

2001). To demonstrate the applicability of our technique for high redshift clusters, we analyzed

CL J1226.9+3332 (CL1226), an apparently relaxed cluster at z = 0.89 (Maughan et al. 2004,

2007). To assess how this method performs on somewhat disturbed clusters, we also analyzed

Abell 1914 (A1914), an intermediate redshift (z = 0.17) cluster with a hot subclump near the

core. When the subclump is not excluded from the X-ray analysis, Maughan et al. (2008, hereafter

M08) find a large X-ray centroid shift in the density profile, which they use as an indicator of an

unrelaxed dynamical state.

In the following sections, we discuss the instruments, data reduction, and analysis of the SZ

and X-ray data. Details of these observations, including the X-ray fitting regions, the unflagged,

on-source integration times, and the pointing centers used for the SZ observations, are presented

in Tables 1 and 2. We also review an independent, detailed X-ray-only analysis, with which we

compare the results of our joint SZ+X-ray analyses.

3.2. Sunyaev-Zel’dovich Array Observations

The Sunyaev-Zel’dovich Array is an interferometric array comprising eight 3.5-meter tele-

scopes, and is located at the Owens Valley Radio Observatory. For the observations presented

here, the instrument was configured to operate in an 8-GHz-wide band covering 27–35 GHz using

the 26–36 GHz receivers (hereafter referred to as the “30-GHz” band) or covering 90–98 GHz us-

ing the 80–115 GHz receivers (the “90-GHz” band). See Muchovej et al. (2007) and Marrone et al.

(in preparation) respectively for more details about commissioning observations performed with

the 30-GHz and the 90-GHz SZA instruments. Details of the observations presented here, includ-

ing the sensitivity and effective resolution (the synthesized beam) of the long and short baselines,

are given in Table 1.

An interferometer directly measures the amplitude and phase of Fourier modes of the sky

intensity, with sensitivity to a range of angular scales on the sky given by ∼ λ/B, where B is the

projected separation of any pair of telescopes, i.e., a baseline. The field of view of the SZA is given

by the primary beam of a single telescope, approximately 10.7′at the center of the 30-GHz band.

At 30 GHz, optimaldetection of thearcminute-scale bulk SZ signal from clusters requires the short

baselines of a close-packed array configuration; six of the SZA telescopes were arranged in this

configuration for the observations presented here, yielding 15 baselines with sensitivity to ∼ 1–5′

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scales. The two outer antennas, identical to the inner six, providean additional 13 long baselines in

this configuration, with sensitivity to small-scale structure, allowing simultaneous measurement of

compactradiosourcesunresolvedbythelongbaselines(angularsize? 20′′)whichcouldotherwise

contaminate the SZ signal. Observations at 90 GHz with the SZA probe scales at three times the

resolution of the 30-GHz observations for the same array configuration. The short baselines of the

90-GHz observations thereby bridge the gap between long and short baseline coverage at 30-GHz.

SZA data are processed in a complete pipeline for the reduction and calibration of interfero-

metric data, developed within the SZA collaboration. Absolute flux calibrations are derived from

observations of Mars, scaled to the predictions of Rudy (1987). Data from each observation are

bandpass-calibrated using a bright, unresolved, flat-spectrum radio source, observed at the start or

end of an observation. The data are regularly phase-calibrated using radio sources near the targeted

field; these calibrators are also used to track small variations in the antenna gains. Data are flagged

for corruption due to bad weather, sources of radio interference and other instrumental effects that

could impact data quality. A more detailed account of the SZA data reduction pipeline is presented

in Muchovej et al. (2007).

In the SZA cluster observations presented here, the A1835 field contains three detectable

compact sources at 31 GHz: a 2.8 ± 0.3 mJy central source, a 1.1 ± 0.4 mJy source about one

arcminute from the cluster center, and a 0.8 ± 0.4 mJy source 5.5 arcminutes from center. The po-

sitions of these sources are in good agreement with those from the NVSS (which only contains the

central source) and the FIRST surveys. The SZA observation of CL1226 contains one detectable

compact source, identified in both FIRST and NVSS, with flux at 31 GHz of 3.9 ± 0.2 mJy. This

source is 4.3 arcminutes from the cluster center (see also Muchovej et al. 2007) and therefore lies

outside the field of view of the SZA 90 GHz observations. Two compact sources, with positions

constrained by NVSS and FIRST, were detected at 31 GHz in the A1914 field. The fluxes of these

sources are 1.8 ± 0.4 mJy and 1.2 ± 0.3 mJy. The stronger was detected in both the NVSS and the

FIRST surveys, while the weaker was only detected in the FIRST survey. Table 3 summarizes the

properties of the compact radio sources detected in the SZA observations.

When fitting compact sources detected in the SZ observations, we calculated the spectral

indices from the measured flux at 31 GHz and 1.4 GHz, where the latter was constrained by either

the NVSS (Condon et al. 1998) or FIRST (White et al. 1997) survey, respectively. The source

fluxes and approximate coordinates are first identified using the interferometric imaging package

Difmap (Shepherd 1997). We first refine the source positions by fitting a model in the MCMC

routine, and then fix these positions to their best-fit values when fitting the cluster SZ model. We

leave the source flux a free parameter, so that the cluster SZ flux and any compact sources are fit

simultaneously.

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3.3.X-ray Observations

All X-ray imaging data used in this analysis were obtained with the Chandra ACIS-I detector,

which provides spatially resolved X-ray spectroscopy and imaging with an angular resolution of

∼ 0.5′′and energy resolution of ∼ 100–200 eV. Table 2 summarizes the X-ray observations of

individual clusters.

For the X-ray data used in the joint SZ+X-ray analysis, images were limited to 0.7–7 keV in

order to exclude the data most strongly affected by background and by calibration uncertainties.

The X-ray images—which primarily constrain the ICM density profiles—were binned in 1.97′′

pixels; this binningsets thelimitingangularresolutionof our processed X-ray data, as theChandra

point response function in the center of the X-ray image is smaller than our adopted pixel size. The

X-ray background was measured for each cluster exposure, using source-free, peripheral regions

of the adjacent detector ACIS-I chips. Additional details of the Chandra X-ray data analysis are

presented in Bonamente et al. (2004, 2006).

In § 4.3 & 4.4 we compare the results of our joint SZ+X-ray analysis to the results of inde-

pendent X-ray analyses. For A1914 and CL1226 we use the data and analysis described in detail in

M08and Maughan et al. (2007)(hereafter M07), respectively. ForA1835, theACIS-I observations

used became public after the M08 sample was published, and we therefore present its analysis here

for the first time. The observation of A1835 was calibrated and analyzed using the same methods

and routines described in M08, which we now briefly review.

In the X-ray-only analyses, blank-sky fields are used to estimate the background for both the

imaging and spectral analysis. The imaging analysis (primarily used to obtain the gas emissivity

profile) is performed in the 0.7–2 keV energy band to maximizesignal to noise. Similar to the joint

SZ+X-ray analyses, these images were also binned into 1.97′′pixels.

For the spectral analysis, spectra extracted from a region of interest were fit in the 0.6–9 keV

band with an absorbed, redshifted Astrophysical Plasma Emission Code (APEC) (Smith et al.

2001) model. This absorption was fixed at the Galactic value. This spectral analysis was used

to derive both the global temperature TX, determined within the annulus r ∈ [0.15,1.0]r500, and

the radial temperature profile fits of the V06 temperature profile, given by

T3D(r) = T0

?(r/rcool)acool+ Tmin/T0

(r/rcool)acool+ 1

??

(r/rt)−a

(1 + (r/rt)b)c/b

?

.

(16)

We refer the reader to V06 for details, but note that this temperature profile is the combination of a

cool core component (first set of square brackets, where the core temperature declines to Tmin) and

a decline at large radii (second set of square brackets, where temperature falls at r ? rt).

An important consideration when using a blank-sky background method is that the count rate

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at soft energies can be significantly different in the blank-sky fields than in the target field, due

to differences between the level of the soft Galactic foreground emission in the target field and

that in the blank-sky field. This was accounted for in the imaging analysis by normalizing the

background image to the count rate in the target image in regions far from the cluster center. In

the spectral analysis, this was modeled by an additional thermal component that was fit to a soft

residual spectrum (the difference between spectra extracted in source free regions of the target and

background datasets; see Vikhlinin et al. (2005)). The exception to this was the XMM-Newton data

used in addition to the Chandra data for CL1226. As discussed in M07, a local background was

found to be more reliable for the spectral analysis in this case, thus requiring no correction for the

soft Galactic foreground.

The M07/M08 X-ray analysis methods exploit the full V06 density and temperature models

(Eqs. 6 & 16, respectively) to fit the emissivity and temperature profiles derived for each cluster,

and the results of these fits are used to derive the total hydrostatic mass profiles of each system.

Uncertainties for the independent, X-ray-only analysis method are derived using a Monte Carlo

randomization process. These fits involved typically ∼1000 realizations of the temperature and

surface brightness profiles, fit to data randomized according to the measured noise. We refer the

reader to M07 and M08, where this fitting procedure is described in detail.

4. Results

4.1. SZ Cluster Visibility Fits

InterferometricSZdataareintheformofvisibilitiesVν(u,v)(see, forexample,Thompson et al.

2001), which for single, targeted cluster observations with the SZA can be expressed in the small

angle approximation as

? ?

Here u and v (in number of wavelengths) are the Fourier conjugates of the direction cosines x and y

(relativetotheobservingdirection), Aν(x,y)istheangularpowersensitivitypattern ofeach antenna

at frequency ν, and Iν(x,y) is the intensity pattern of the sky (also at ν). Eq. 17 is recognizable as a

2-D Fourier transform, so the visibilities give the flux for the Fourier mode for the corresponding

u,v-coordinate.

Vν(u,v) =

Aν(x,y)Iν(x,y)e−i2π(ux+vy)dxdy.

(17)

By combining Eqs. 1 and 17, we can remove the frequency dependence from the measured

cluster visibilities, just as we have related SZ intensity to the frequency-independent Compton

y-parameter. We define the frequency-independent cluster visibilities Y(u,v) as

Vν(u,v) ≡ g(x)I0Y(u,v),

(18)

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where I0(in units of flux per solid angle) is

I0=2(kBTCMB)3

(hc)2

.

(19)

Additionally, we rescale Y(u,v) by the square of the angular diameter distance, d2

remove the redshift dependence from the cluster SZ signal. Note that, while we use the non-

relativistic g(x) (Eq. 3) to compute Y(u,v) (Eq. 18) , we only use this for display purposes. The

effects of assuming the classical SZ frequency dependence are discussed in § 4.3.

Figure 1 shows the maximum-likelihood fits of the N07 profile and the isothermal β-model to

each cluster’s visibility data, from which we have subtracted the detected radio sources (Table 3).

We also removed the frequency dependence of the SZ effect by rescaling the cluster visibilities to

Y(u,v) (Eq. 18). For the purposes of plotting, this rescaling is useful when binning the SZ signal

across 8 GHz of bandwidth as well as when plotting the 30-GHz and 90-GHz SZA data taken on

CL1226.

A, in order to

As indicated in Fig. 1, both the isothermal β-model and N07 model (which was fit jointly

with the SVM) fit the available data equally well. However, as

sky), the isothermal β-model extrapolates to a much larger value of Y(u,v). This corresponds to

the much larger values of Ycylthat are computed at large radii using fit β-models (see §4.4).

The middle panel of Fig. 1 shows the combined 30+90 GHz observations of CL1226, which

has a smaller angular extent than A1835 or A1914 due to its distance (compare, for example, the

values of r500for each cluster, listed in Table 5). The dynamic range and u,v-space coverage pro-

vided by the 30-GHz SZA observations (black points) on CL1226 were insufficient for constrain-

ing the radial pressure profile of the cluster. The short baselines of the 90-GHz SZA observations

(middle three points) help to provide more complete u,v-space coverage, as discussed in §3.2.

√u2+ v2→ 0 (large scales on the

4.2. X-ray Surface Brightness Fits

The X-ray surface brightness (Eq. 4), ignoring the data within a 100 kpc radius, was modeled

separately with both the isothermal β-model, using the spectroscopically-determined, global TX

(measured within r ∈ [0.15,1.0]r500), and the SVM, using the temperature derived from the N07

pressure profile fit to the SZ data. Figure 2 shows the maximum-likelihood fits to the surface

brightness of each cluster for both the SVM and isothermal β-model. For plotting purposes, the X-

ray data are radially-averaged around the cluster centroid, which is determined in the joint SZ+X-

ray analysis by fitting the two-dimensional X-ray imaging data with the spherically-symmetric

SVM and isothermal β-model profiles.

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4.3.ICM Profiles

Figure 3 shows the three-dimensional ICM radial profiles derived from the joint analysis of

SZA + Chandra X-ray surface brightness data for A1835, CL1226 and A1914 (from left to right).

From top to bottom, we show the electron pressure, the gas density and the derived electron tem-

perature profiles, each as a function of cluster radius. In all panels, we compare the ICM profiles

derived from the N07+SVM model to the results of a traditional isothermal β-model analysis, indi-

cated by solid and dot-dashed lines, respectively. The hatched regions indicate the 68% confidence

interval for each derived parameter.

As shown in the top panels of Fig. 3, the pressure profiles derived from the N07 model and

the isothermal β-model show good agreement within r2500, but deviate by ∼3–5-σ in the cluster

outskirts. This is a consequence of the fact that clusters exhibit a significant decline in tempera-

ture beyond r2500, as determined from spectroscopic X-ray observations (see the bottom panel of

Fig. 3). The pressure profile derived from the isothermal β-model analysis is therefore biased sys-

tematically high beyond r2500. In contrast, the N07 model, which fits the pressure directly, is free

to capture the true shape of the pressure profile well beyond the radius at which the assumption of

isothermality becomes invalid.

For all three clusters there is little evidence for the second component of the electron density

allowed by the SVM; the density is fit equally well by either the SVM or a single-component β-

model, as illustrated by the center row of panels in Fig. 3 (see also Table 2). For all three clusters,

the fits of the SVM agree to within 1–2% of the full V06 density profile fits (not shown) outside

the core; discrepancies at this level are easily attributed to differences in the fitting of the X-ray

background, and to the differences between the APEC and Raymond-Smith emissivity models

used respectively in the M07/M08 X-ray analysis and the joint SZ+X-ray analyses.

In the bottom panels, the electron temperature profiles inferred from the N07+SVM pro-

files are compared to temperature profiles derived from deep spectroscopic Chandra X-ray ob-

servations (and XMM-Newton in the case of CL1226; see M07). The comparison shows that the

radial electron temperature profiles derived from the N07+SVM profiles are in good agreement

with independent X-ray measurements for the relaxed clusters A1835 and CL1226, which exhibit

radially decreasing temperature profiles in the cluster outskirts (see also Markevitch et al. 1998;

Vikhlinin et al. 2005). The disturbed cluster A1914, however, shows less overall agreement be-

tween the derived N07+SVM radial temperature profile and the M08 fit of the V06 temperature

profile. Since the N07 pressure profile fit to the SZA observation of A1914 agrees with that derived

from M08 within their respective 68% confidence intervals, the temperature discrepancy is likely

due to deviations from the spherical symmetry implicitly assumed in this analysis. Additionally,

scales greater than about six arcminutes are beyond the radial extents probed by the SZA; it is

unsurprising the agreement becomes poorer at radii larger than this.

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We note that we fit the SZ data using relativistic corrections to the SZ frequency depen-

dence provided by Itoh et al. (1998). These corrections are appropriate for the thermal SZ effect

at the SZA observing frequencies of 30 and 90 GHz. Compared to fits assuming the classical SZ

frequency dependence, a pressure profile fit using the relativisticcorrections has both a higher nor-

malization and larger upper error bars. This is noticeable when including higher frequency data,

where the relativistic correction is larger (∼ 5% at 90 GHz versus ∼ 3% at 30 GHz for cluster tem-

peratures ∼ 8 keV). This increase in the pressure fit is due to the diminished magnitude of the SZ

effect when usingthese corrections (for frequencies below the nullin theSZ spectrum, ? 218 GHz;

see e.g. Itoh et al. (1998)). The pressure profile therefore must adjust to fit the observed SZ flux.

The larger upper error bars on the fit pressure profile arise from a more subtle effect. Since the

temperature is derived from the simultaneously fit pressure and density profiles, and the SZ effect

diminishes as electrons become more relativistic (i.e. hotter), the upper error bar of the pressure

fit must increase to fit the same noise in the observation (compared to the non-relativistic case).

The lower error bar is less affected, as lower electron temperatures require smaller relativistic

corrections. The resulting asymmetric error bars can be seen in the derived temperature profile of

CL1226, which relied on 90 GHz data, in Figure 3.

4.4. Measurements of Y, Mgas, Mtot, and fgas

InTables4and5, wereportglobalpropertiesofindividualclustersderivedfromtheN07+SVM

model fits to the SZ+X-ray data. We calculate all quantities at overdensity radii r2500and r500, and

compare them to results from both the isothermal β-model analysis of the same data, as well as to

the X-ray-only analysis.

The N07 pressure profile has just two free parameters—Pe,iand rp—which exhibit a degener-

acy. Figure 4 shows this degeneracy in fits of the N07 profile to the SZA observations of A1835.

Similar to the rc− β degeneracy of the β-model (see Grego et al. 2001, for example), these two

quantities are not individually constrained by our SZ observations, but they are tightly correlated

and the preferred region in the Pe,i− rpplane encloses approximately constant Ycyl. As a result,

the 68% confidence region for Ycylis more tightly constrained than the large variation in Pe,ior

rpmight individually indicate. Figure 4 also shows that the inclusion of X-ray data has only a

marginal effect on the value of Ycylderived from the SZ fit. This is as expected, due to the weak

dependence of the X-ray surface brightness on temperature (see § 2.1) and the fact that the N07

profile is not linked to the SVM density profile’s shape. This indicates that X-ray data are not

necessary to constrain Ycyl, although they do limit the range of accepted radial pressure profiles.

At both r2500and r500, the measurements of Ycylderived from the joint N07+SVM and the