Comparison of Pressure Profiles of Massive Relaxed Galaxy Clusters using Sunyaev-Zel'dovich and X-ray Data

We present Sunyaev-Zel'dovich (SZ) effect observations of a sample of 25 massive relaxed galaxy clusters observed with the Sunyaev-Zel'dovich Array (SZA), an 8-element interferometer that is part of the Combined Array for Research in Millimeter-wave Astronomy (CARMA). We perform an analysis of new SZA data and archival Chandra observations of this sample to investigate the integrated pressure -- a proxy for cluster mass -- determined from X-ray and SZ observations, two independent probes of the intra-cluster medium. This analysis makes use of a model for the intra-cluster medium introduced by Bulbul (2010) which can be applied simultaneously to SZ and X-ray data. With this model, we estimate the pressure profile for each cluster using a joint analysis of the SZ and X-ray data, and using the SZ data alone. We find that the integrated pressures measured from X-ray and SZ data are consistent. This conclusion is in agreement with recent results obtained using WMAP and Planck data, confirming that SZ and X-ray observations of massive clusters detect the same amount of thermal pressure from the intra-cluster medium. To test for possible biases introduced by our choice of model, we also fit the SZ data using the universal pressure profile proposed by Arnaud (2010), and find consistency between the two models out to r500 in the pressure profiles and integrated pressures.


Introduction
The Sunyaev-Zel'dovich (SZ) effect (Sunyaev & Zel'dovich, 1972) is a spectral distortion of the cosmic microwave background (CMB) caused by the scattering of CMB photons off the hot electrons of the intra-cluster medium (ICM). Over the past two decades, SZ observations with both single-dish and interferometric instruments have become routine (e.g., Birkinshaw et al., 1991;Carlstrom et al., 1996;Holzapfel et al., 1997;Carlstrom et al., 2002), and SZ surveys are now producing catalogs of newly-discovered clusters out to high redshift (Vanderlinde et al., 2010;Marriage et al., 2011;Williamson et al., 2011;Planck Collaboration et al., 2011a). SZ measurements are complementary to the X-ray measurements which have long been used to study clusters, but only in recent years have sufficiently large samples of objects been observed in the SZ to permit a rigorous comparison between these two techniques (e.g., Reese et al., 2002;Bonamente et al., 2006;LaRoque et al., 2006).
The SZ effect causes a perturbation ∆T of the CMB temperature T CM B given by ∆T where f (x) is the frequency dependence of the SZ effect (e.g., LaRoque et al., 2006); σ T is the Thomson cross-section; n e , T e and m e are the number density, temperature, and mass of the electrons, respectively; k is the Boltzmann constant; c the speed of light; and the integral is along the line of sight . At a given frequency, the amplitude of the effect depends linearly upon the Compton y parameter, which is defined implicitly in Equation 1. Note that the y parameter is proportional to the ICM pressure integrated along the line of sight. At frequencies below 218 GHz, the SZ effect causes a decrement in the CMB temperature in the direction of the cluster. The integral of y over the solid angle Ω subtended by the cluster, known as the (cylindrical) integrated Compton y parameter Y cyl = y dΩ, is expected to be a good proxy for cluster total mass since it traces the thermal energy content of the cluster gas. Alternatively, the Compton y parameter can be integrated spherically, where the volume V is a sphere centered on the cluster and D A is the angular diameter distance. X-ray data can also be used to constrain the density and temperature-and thus the pressure-of the ICM. Over the past decade, several groups have investigated the consistency between X-ray and SZ pressure measurements. Early measurements of the SZ signal from WMAP by, e.g., Lieu et al. (2006), Bielby & Shanks (2007), detected an SZ signal at a lower level than expected. Atrio- Barandela et al. (2008) showed that the isothermal beta model leads to an electron pressure profile that exceeds the measured values at large radii by a factor of few, and that the baryon profile is consistent with a model based on the Navarro et al. (1997) matter profile. Diego & Partridge (2010) also showed that contamination by compact radio sources may have led to underestimates of the SZ effect flux decrements in the WMAP data. More recent comparisons of Chandra X-ray data to stacked data from WMAP and Planck (Melin et al., 2011;Planck Collaboration et al., 2011b) have found consistency between SZ and X-ray measurements for large samples of clusters. Komatsu et al. (2011) also analyzed a sample of massive nearby clusters individually resolved by WMAP, again finding good agreement with X-ray predictions.
In this paper, we present Sunyaev-Zel'dovich Array (SZA) observations of the Allen et al. (2008) sample of massive relaxed galaxy clusters, together with archival Chandra X-ray observations that are available for all clusters in this sample. The sensitivity and resolution of our data permit us to measure the pressure profile and the integrated pressure out to r 500 -the radius within which the average cluster density is 500 times the critical density-for each cluster individually, without the need to resort to scaling relations between the X-ray luminosity and mass (as was done by Melin et al., 2011;Planck Collaboration et al., 2011b, for example). We use the Bulbul et al. (2010) model of the cluster pressure, density, and temperature. Since this model has a consistent parameterization for all thermodynamic quantities, it is especially well-suited for joint X-ray and SZ analysis. As a cross-check against model-dependent biases, we also fit the SZ data using the model of Arnaud et al. (2010) based on the numerical simulations of Nagai et al. (2007). We find consistency to within our measurement uncertainties both between the X-ray and SZ measurements, and between the Bulbul et al. (2010) and Arnaud et al. (2010) models.
The paper is structured as follows: Section 2 describes our observations and our sample, Section 3 presents our joint analysis technique, Section 4 describes our method of measuring the integrated Y sph (r 500 ) parameter (defined in Equation 2), Section 5 presents and discusses our results, and our conclusions are presented in Section 6.

Observations
The SZA is an eight-element interferometer designed to detect and image the SZ effect from clusters at z> 0.1, and is part of the Combined Array for Research in Millimeterwave Astronomy (CARMA). The array is equipped with 30 and 90 GHz receivers; all SZA observations presented in this paper were taken at 30 GHz. At this frequency, the 3.5 m diameter SZA telescopes have a field-of-view (or primary beam) of 10. 7 FWHM. Interferometric data are proportional to the Fourier transform of the sky brightness. These visibility data, denoted V (u, v), are sampled at Fourier-plane coordinates (u, v) corresponding to the projected separation of pairs of telescopes (or baselines), as viewed by the source at the time of observation. At the time of the observations discussed in this work, the SZA antennas were arranged in a hybrid configuration, with six closely spaced telescopes and two "outriggers" located ∼50 m from the inner array. The inner six telescopes probe small (u, v) Fourier modes, sampling the angular scales where the SZ signal is largest for moderate-to high-redshift clusters (1 − 6 ). Baselines involving the outriggers are sensitive to angular scales down to ∼ 20 and are used to constrain the positions and fluxes of unresolved radio sources.
Of the 42 clusters in the Allen et al. (2008) sample of massive relaxed galaxy clusters, the SZA has observed the 31 objects above δ > −15 • at redshift z ≥ 0.09. The declination restriction is imposed by the latitude of the observatory in the Owens Valley, California (37 • 14 02 N, 118 • 16 56 W), while the exclusion of clusters at low redshift is due to the inability of an interferometer to constrain scales larger than that which the shortest antenna spacing can probe at the lowest frequency band. The largest angular wavelength measured by the SZA is 10.9 , which for massive low-redshift clusters is generally smaller than 2r 500 /D A . Of these 31 clusters observed with the SZA, Abell 2390 and Abell 611 were excluded from this analysis because they did not have available local background in their Chandra ACIS-S X-ray observations. Three additional clusters-3C295, ClJ1415.2+3612, and Abell 963-were discarded because of extended or otherwise difficult-to-remove radio source contamination, and one-RXJ0439.0+0521-because of a pointing error.
Our sample therefore consists of 25 clusters. The synthesized beam of the long (short) baseline data for this sample is approximately 15-30 (90-180 ), and the average rms noise in the maps is ∼ 0.25 − 0.30 mJy. In all cases, the Chandra data provide spatially resolved X-ray spectroscopy and sub-arcsecond imaging. A summary of the data is provided in Table 1.
Radio sources detected in the cluster fields are reported in Table 2. For each cluster field, we use the NRAO VLA Sky Survey (NVSS) and Faint Images of the Radio Sky at Twenty-centimeters (FIRST) 1.4 GHz catalogs as a reference for locating compact radio sources within 10 of the cluster center. Most radio sources in our observations have counterparts in the FIRST survey, which has an rms noise of 0.15 mJy at 1.4 GHz. Inverted spectrum sources that may be present at 30 GHz may not have counterparts at 1.4 GHz, but fortunately they comprise a small fraction of the source population (Muchovej et al., 2010).
For all 25 clusters in our sample we have available archival Chandra X-ray observations (Allen et al., 2008). Event files for all cluster observations and additional blank-sky composite event files used for background subtraction were reduced using CIAO 4.3.1 and CALDB 4.3. X-ray spectra are extracted in several annular regions for each cluster, centered at the peak of the X-ray emission. Emphasis is placed on the removal of periods of high background, and on the modeling of soft X-ray residuals that may be present after the subtraction of the blank-sky background. The method of analysis of the Chandra data and examples of the temperature and surface brightness profiles can be found in Bulbul et al. (2010) and Hasler et al. (2011). More details on the Chandra data for all clusters in this sample will be shown in a forthcoming paper, in which we will present the measurement of the gas mass fraction from the X-ray observations (Hasler et al. in prep.).
In Figure 1, we show the raw Chandra X-ray images (binned in the 0.7-7 keV energy band) for each of the 25 clusters, with contours obtained from the short baseline point source-removed SZA data overlaid.   3. Analysis of the SZA and Chandra data

Models for the thermodynamic quantities
We analyze the SZ and X-ray data using the Bulbul et al. (2010) model, which uses a consistent parameterization of the electron density, temperature and pressure, related through the ideal gas law at all radii, i.e., p e (r) = n e (r)kT e (r) for pressure p e , electron density n e , and temperature T e . All thermodynamic quantites depend on the gravitational potential, in which β describes the slope of the matter density at large radii and r s is a scale radius. The parameterization of the Bulbul et al. (2010) model does not allow the inner slope of the matter density to vary, which is fixed at r −1 as in the Navarro et al. (1997) model. The resolution of our SZ data can only effectively constrain the matter distribution on scales larger than the synthesized beam, which is of order 1 arcmin for these observations, and therefore we would not be able to place significant constraints on the inner slope. As explained in Bulbul et al. (2010), the potential is continuous at β = 2, the value of the Navarro et al. (1997) mass density model. The radial electron temperature profile is given by where τ cool (r) is the Vikhlinin et al. (2006) phenomenological core taper function, required to fit cool-core clusters, which is equal to one at large radii. The density is parameterized as n e (r) = n e0 φ(r) n τ −1 cool (r).
in such a way that the pressure distribution is not altered by the presence of the cool core. At large radii, where the effect of the cool core vanishes, the thermodynamic quantities are related by a simple polytropic equation of state. The electron pressure profile is therefore parameterized as and is independent of the presence of a cool core. The model therefore has five independent parameters for non-cool-core clusters: the scale radius r s ; the index β, the polytropic index n, and the normalization constants for the three thermodynamic quantities which satisfy n e0 kT 0 = P e0 . For cool-core clusters, the τ cool function τ cool (r) = α + (r/r cool ) γ 1 + (r/r cool ) γ adds three additional adjustable parameters. To test for model-dependent biases, we also use the Arnaud et al. (2010) model to fit the SZ data. This model describes the cluster pressure profile using an analytic function motivated by numerical simulations (Nagai et al., 2007) and X-ray observations of the REXCESS sample, The parameters p e,i and r p are left free in our fits to the SZ effect observations. The values (a, b, c) are the power law indices that describe the (intermediate, outer, inner) slopes of p e (r). We use the "universal" values (a, b, c) = (1.05, 5.49, 0.31) obtained by Arnaud et al. (2010) from a fit to X-ray observations of the REXCESS sample. Note that Arnaud et al. (2010) find different best-fit values for cool-core clusters. We choose to use the parameters fit to the entire sample because our sample was not selected based on the presence of a cool core, and in fact contains a few non cool-core clusters, namely 3C186, MS1137.5+6625 and CLJ1226.9+3332.

Method of analysis
As in previous work with the SZA (e.g., Mroczkowski et al., 2009;Hasler et al., 2011), we relate the point-source subtracted interferometric SZ visibilities to the unitless integrated Compton y by introducing Y (u, v), defined as Here g(x) corrects for the frequency dependence of the SZ flux, and I 0 = 2(k B T CM B ) 3 /(hc) 2 is the primary CMB intensity. The SZ models and compact radio sources are fit directly and simultaneously in Fourier space, where the statistical properties of the model fits are better understood and the noise is Gaussian. This is done simply by building up the sky brightness image, Fourier transforming it, and computing the likelihood of the model. The X-ray data consist of spectroscopic temperature measurements taken in clustercentric annuli, and an X-ray image in units of surface brightness (counts s −1 cm −2 sr −1 ). The X-ray surface brightness S x varies with the line of sight integral of the electron density and temperature distributions as where is the line of sight through the cluster, n e is the electron density, T e is the electron temperature, A is the metallicity, and Λ ee (T e , A) is the X-ray cooling function (in units of counts cm 3 s −1 ) as a function of electron temperature and metallicity. Each cluster was divided in a number of annuli according to the total number of photons detected, and for each annular region the temperature and abundance were free parameters. The surface brightness is only marginally sensitive to the choice of outer limit of integration in Equation 10: we find that the masses vary by less than 1% when the outer limit ranges between 2 and 5 Mpc. We therefore choose a limit of 2 Mpc, which corresponds to approximately the virial radius for clusters in this mass range. We use the Mazzotta We first estimate the pressure profile of the ICM by jointly fitting the SZ and Xray data with the Bulbul et al. (2010) model. Both datasets are used simultaneously to constrain all three thermodynamic quantities, with the global shape parameters β, n and r s (and the cool-core parameters when applicable) linked among the profiles. Both datasets contribute to the determination of the shape of the pressure profile, with SZ observations contributing primarily at the largest radii where the sensitivity of Chandra to the diffuse cluster emission is limited. Instead of linking the normalization of the pressure profile (P e0 ) to the product of the normalizations of the density and temperature (n e0 and T 0 ), we let the normalizations be free, and check a posteriori that P e0 = n e0 × kT 0 in accordance with the ideal gas law. The normalization of the pressure is determined by the SZ data, and the normalizations of temperature and density by the X-ray data.
This method results in the measurement of the shape of the pressure profile, p e (r)/P e0 , and two normalizations determined independently by each of the two datasets. The two normalizations are left free to vary because in principle systematic uncertainties in the two datasets could lead to different values, and we do not want to assume an a priori agreement between them. The fit uses a Markov chain Monte Carlo method (Bonamente et al., 2004), and computes the angular diameter distance assuming a Ω Λ = 0.73, Ω M = 0.27 and h = 0.73 cosmology.
To obtain a measurement of the integrated pressure that depends only on the SZ data, we also perform another fit in which we fix the shape parameters of the Bulbul et al. Measurements of the ICM pressure using SZ and X-ray data are subject to different sources of systematic uncertainty that could affect the calculation of the Y sph parameter (Hasler et al., 2011). Systematic errors that integrate down with sample size include cluster asphericity, the effect of X-ray background, and the presence of kinetic SZ effect; these errors are included in the calculation of the ratio between the various measurements of Y sph (r 500 ), and of the weighted averages and χ 2 min values in Sections 4.1 and 4.2, following the prescriptions of Hasler et al. (2011).

Joint SZ and X-ray fit using the Bulbul et al. (2010) model
The integrated pressure, which we quantify in terms of the Compton y parameter, is expected to be a good proxy for total cluster mass. Since the SZA measures the integrated flux within Fourier modes on the sky, our SZ data relate most directly to the integrated Compton-y parameter Y cyl . However, it is conventional in X-ray analyses to report spherically integrated quantities. We therefore quantify the integrated pressure using the spherically-integrated Compton y parameter Y sph out to r 500 . The overdensity radius r 500 is given by with ∆ = 500, where ρ c (z) is the critical density of the universe at the cluster redshift. The total cluster mass is calculated under the assumption of hydrostatic equilibrium; for the Bulbul et al. (2010) model, the total mass is given by where the matter density normalization is given by ρ i = (kT 0 (n+1)(β−1))/(4πGµm p r 2 s ); µ is the mean molecular weight, and m p is the proton mass.
Using the method of analysis discussed in Section 3.2, we first compare Y sph normalized using n e0 and T 0 constrained by the X-ray data with Y sph normalized using P e0 constrained by the SZ data. This comparison is summarized in Table 3. The normalizations are in good agreement: the weighted average of the ratio between the measurements using the SZ and X-ray normalization is 1.06 ± 0.04. This indicates that systematic uncertainties do not produce a large overall offset between the two observables.
Below, we refer to Y sph as the measurement obtained from the joint fit using the Xray normalization. We adopt this value since the joint profile makes use of all information available from both the X-ray and SZ observations including the effect of the cool core, and since both normalizations are in agreement.

SZ-only fit using the Bulbul et al. (2010) average pressure profile
We also fit only the SZA data to the Bulbul et al. (2010) average pressure profile, which consists of the pressure profile of Equation 6 with P e0 and r s as free parameters and the two shape parameters fixed at n = 3.5 and β = 2.0. We use this model to compute Y as described above, which we refer to as Y sph,SZ,B10 . The value of r 500 used in computing Y sph,SZ,B10 is determined from the joint fit. These results are shown in Table 4, and are plotted against the joint fit Y sph in Figure 2. We find that the weighted mean of the ratio between the measurements is given by Y sph,SZ,B10 /Y sph = 0.90 ± 0.05, where the uncertainty is the standard deviation of the weighted mean. A linear fit of the two measurements to a y = x model results in a χ 2 min = 35.3 for 25 degrees of freedom, and we measure a scatter of 16%.

Comparison between the Bulbul et al. (2010) and Arnaud et al. (2010) pressure profiles applied to the SZ data
The SZA data were also fit to the Arnaud et al. (2010) model using the same value of r 500 as above. The best-fit parameters are shown in Table 5. We compare the results from the Bulbul et al. (2010)  A fit of the two measurements to a y = x model assuming the values are independent results in a χ 2 min =5.6 for 25 degrees of freedom, consistent with the presence of negligible scatter between the two measurements. The low value of χ 2 min is likely due to correlated errors, since the two measurements make use of the same data. Figure 4 shows the average Arnaud et al. (2010) and Bulbul et al. (2010) pressure profiles for our sample. The two parameterizations result in fits that are consistent at all radii within r 500 . The consistency between the pressure profiles and the integrated Y (r 500 ) values measured from the two models indicate that the choice of parameterization for the gas pressure does not introduce a significant bias in the calculation of the integrated pressure within r 500 .

Discussion
The agreement we find between SZ and X-ray measurements of the Y sph (r 500 ) parameter is consistent with a simple scenario in which the SZ decrement and the X-ray emission from massive relaxed clusters originate from the same highly-ionized thermal plasma, with only small contributions from other possible sources of emission. This result is in agreement with earlier ∼ 30 GHz SZ studies using the Owens Valley Radio Observatory (OVRO) and the Berkeley Illinois Maryland Array (BIMA) millimeter arrays, in which the same value of the gas mass fraction was measured using SZ and X-ray data  Our results also support the findings by Melin et al. (2011) andPlanck Collaboration et al. (2011b) of an overall agreement between the two measurements of the thermal pressure.
We find scatter between the SZA and Chandra Y sph estimates at a level of 16%. A possible source of systematic error that could give rise to this scatter, and that is particularly relevant to our measurements out to r 500 , is elongation of the cluster along the line of sight. We use spherically symmetric models in the analysis; an intrinsically prolate cluster (elongated along the line of sight), when fit to a spherical model, will have its X-ray surface brightness -and therefore the corresponding Y sph parameterunderestimated with respect to the corresponding SZ measurement (e.g., Cooray, 2000;De Filippis et al., 2005;Ameglio et al., 2007). This is due to the quadratic dependence of the X-ray surface brightness profile on the density, as opposed to the linear dependence of the SZ effect. Our sample has just three clusters with a statistically significant deviation from the Y sph = Y sph,SZ line, but in the direction of Y sph /Y sph,SZ > 1, and therefore consistent with oblateness (compression along the line of sight) rather than prolateness. The fact that the Allen et al. (2008) sample of relaxed clusters is X-ray selected may lead to including preferentially oblate clusters as their surface brightness will be boosted. An alternative interpretation for the presence of scatter between the SZA and Chandra estimates of Y is that some of these clusters are disturbed and have undergone a recent merger, as is almost certainly the case for RXJ1347.5-1145 (Mason et al., 2010;Johnson et al., 2011). A merger would result in clumping of the gas, and therefore an overstimate of the gas mass and Y from X-ray measurements, as suggested by Simionescu et al. (2011) to explain the observations of the Persues cluster. Clumping would not affect the SZ observations, because of the linear dependence of the signal on density.
The fit of the SZ data to the universal pressure profile of Arnaud et al. (2010), and to the average pressure profile based on the Bulbul et al. (2010) parameterization of the pressure, are statistically acceptable for all clusters, with a similar χ 2 for the two models. The agreement between Y sph at r 500 using the two models indicates that the integrated pressure is not highly sensitive to (reasonable) choices of parameterization.
We have adopted throughout our analysis the value of r 500 determined from the joint SZ and X-ray observations. In the absence of X-ray information, one may instead adopt a fiducial value of the gas mass fraction f gas to determine r 500 (e.g., Joy et al., 2001;Bonamente et al., 2008;Mroczkowski, 2011), or other means based on SZ-mass scaling relations. The additional assumptions required to estimate r 500 from SZ data only will likely contribute additional scatter to the Y sph − Y sph,SZ relation, when r 500 used to measure Y sph,SZ is estimated directly from the SZ data.

Conclusions
We have presented the joint analysis of SZA and Chandra observations of the Allen et al. (2008) sample of massive and relaxed galaxy clusters. We have collected sensitive SZ data for all clusters at declination ≥ −15 • with no significant contamination from foreground or intrinsic radio sources, for a total of 25 clusters in the redshift range 0.09 ≤ z ≤ 1.06. We also used the X-ray imaging and spectroscopic Chandra data that are available for all clusters, and performed a cluster-by-cluster comparison of the integrated pressure. The Y sph value estimated from the joint SZ and X-ray data, and from the SZ data alone, agree within a few percent at r 500 , indicating that the SZ and X-ray signal from massive relaxed clusters is consistent with a common thermal origin. We therefore confirm the findings of Melin et al. (2011) andPlanck Collaboration et al. (2011b), and find no evidence for the presence of significant sources of systematic uncertainty in the measurements of the ICM pressure from SZ and X-ray observations of massive relaxed clusters. We also determined an average pressure profile based on the Bulbul et al. (2010) model, with shape parameters (n = 3.5 and β = 2.0) determined by a joint fit to Chandra X-ray data and our SZA observations of the Allen et al. (2008) sample of massive relaxed clusters. We have shown that measurements of the radial profile of the pressure out to r 500 , and of Y sph,SZ at r 500 , agree between the Arnaud et al. (2010) and the Bulbul et al. (2010) average pressure profiles out to r 500 . Our conclusions indicate that both models are adequate for describing cluster radial pressure profiles and measuring the integrated thermal energy content in relaxed clusters.

Acknowledgments
The operation of the SZA is supported by NSF through grant AST-0604982 and AST-0838187. Partial support is also provided from grant PHY-0114422 at the University of Chicago, and by NSF grants AST-0507545 and AST-05-07161 to Columbia University. CARMA operations are supported by the NSF under a cooperative agreement, and by the CARMA partner universities. Support for TM was provided by NASA through