A Comparison and Joint Analysis of Sunyaev-Zel'dovich Effect Measurements from Planck and Bolocam for a set of 47 Massive Galaxy Clusters
Jack Sayers, Sunil R. Golwala, Adam B. Mantz, Julian Merten, Sandor M. Molnar, Michael Naka, Gregory Pailet, Elena Pierpaoli, Seth R. Siegel, Ben Wolman
DDraft version May 25, 2016
Preprint typeset using L A TEX style emulateapj v. 5/2/11
A COMPARISON AND JOINT ANALYSIS OF SUNYAEV-ZEL’DOVICH EFFECT MEASUREMENTS FROM
Planck
AND BOLOCAM FOR A SET OF 47 MASSIVE GALAXY CLUSTERS
Jack Sayers , Sunil R. Golwala , Adam B. Mantz , Julian Merten , Sandor M. Molnar , Michael Naka ,Gregory Pailet , Elena Pierpaoli , Seth R. Siegel , & Ben Wolman , Draft version May 25, 2016
ABSTRACTWe measure the SZ signal toward a set of 47 clusters with a median mass of 9 . × M (cid:12) and amedian redshift of 0.40 using data from Planck and the ground-based Bolocam receiver. When
PlanckXMM -like masses are used to set the scale radius θ s , we find consistency between the integratedSZ signal, Y , derived from Bolocam and Planck based on gNFW model fits using A10 shapeparameters, with an average ratio of 1 . ± .
030 (allowing for the (cid:39)
5% Bolocam flux calibrationuncertainty). We also perform a joint fit to the Bolocam and
Planck data using a modified A10 modelwith the outer logarithmic slope β allowed to vary, finding β = 6 . ± . ± .
76 (measurementerror followed by intrinsic scatter). In addition, we find that the value of β scales with mass andredshift according to β ∝ M . ± . × (1 + z ) − . ± . . This mass scaling is in good agreementwith recent simulations. We do not observe the strong trend of β with redshift seen in simulations,though we conclude that this is most likely due to our sample selection. Finally, we use Bolocammeasurements of Y to test the accuracy of the Planck completeness estimate. We find consistency,with the actual number of
Planck detections falling approximately 1 σ below the expectation fromBolocam. We translate this small difference into a constraint on the the effective mass bias for the Planck cluster cosmology results, with (1 − b ) = 0 . ± . Subject headings: galaxies: clusters: intracluster medium — astronomical databases: catalogs —cosmology: observations INTRODUCTION
The Sunyaev-Zel’dovich (SZ) effect has emerged as avaluable observational tool for studying galaxy clusters,particularly with the dramatic improvements in instru-mentation that have occurred over the past decade. Forexample, the South Pole Telescope (SPT, Bleem et al.2015), the Atacama Cosmology Telescope (ACT, Has-selfield et al. 2013), and
Planck (Planck Collaborationet al. 2015d) have delivered catalogs with a combinedtotal of more than 1000 SZ-detected clusters. Beyondthese large surveys, detailed studies of the gaseous intra-cluster medium (ICM) have been enabled by an addi-tional set of pointed SZ facilities with broad spectralcoverage and/or excellent angular resolution such as theMultiplexed SQUID/TES Array at Ninety GHz (MUS-TANG, Mason et al. 2010) the New IRAM KID Arrays(NIKA, Adam et al. 2016), and the MultiwavelengthSubmillimeter Inductance Camera (MUSIC, Sayers et al.2016).As the range of SZ instrumentation has become morediverse, the benefits of joint analyses using multipledatasets have increased. For example, a wide range of Division of Physics, Math, and Astronomy, California Insti-tute of Technology, Pasadena, CA 91125 Department of Physics, Stanford University, Stanford, CA94305 Department of Physics, University of Oxford, Oxford OX13RH, UK Institute of Astronomy and Astrophysics, Academia Sinica,Taipei 10617, Taiwan University of Southern California, Los Angeles, CA 90089 Department of Physics, McGill University, Montr´eal, QCH3A 2T8, Canada [email protected] studies have used data from two or more SZ receivers inorder to measure the spectral shape of the SZ signal (e.g.,Kitayama et al. 2004; Zemcov et al. 2010; Mauskopf et al.2012), mainly for the purpose of constraining the ICMvelocity via the kinetic SZ signal, but also to characterizerelativistic corrections to the classical SZ spectrum (e.g.,Sunyaev & Zeldovich 1980; Nozawa et al. 1998; Chlubaet al. 2012). Furthermore, recent analyses have begun toexploit the different angular sensitivities of the SZ facili-ties in order to obtain a more complete spatial picture ofthe cluster (e.g., Romero et al. 2015; Young et al. 2015;Rodr´ıguez-Gonz´alvez et al. 2015).In order for these joint analyses to be useful, the var-ious SZ instruments must provide measurements of theSZ signal that are consistent. Historically, this was of-ten not the case, likely due to large systematic errorsin the measurements (e.g., see the detailed discussionin Birkinshaw 1999). However, the situation has im-proved considerably with advances in modern SZ instru-mentation, and good agreement has been seen in mostrecent comparisons (e.g., Reese et al. 2012; Mauskopfet al. 2012; Rodr´ıguez-Gonz´alvez et al. 2015; Sayers et al.2016). Modest inconsistencies do still appear, althoughthey are often the result of assuming different spatialtemplates when performing the SZ analyses for separateinstruments (e.g., Benson et al. 2004; Planck Collabo-ration et al. 2013a; Perrott et al. 2015). In sum, thesystematics that plagued early SZ measurements appearto be largely absent from modern data. This fact, com-bined with the high degree of complementarity betweendifferent SZ facilities, has opened a promising future fordetailed cluster studies using multiple SZ datasets.In this work, we use SZ measurements from Planck a r X i v : . [ a s t r o - ph . C O ] M a y and the ground-based receiver Bolocam to study a setof 47 massive clusters. The manuscript is organized asfollows. In Section 2, the parametric model used to de-scribe the data is introduced, and in Section 3 the SZdata from Planck and Bolocam are detailed. Section 4compares the SZ signals measured by
Planck and Bolo-cam, and Section 5 presents the results from joint fits tothe two datasets. In Section 6, we use Bolocam SZ mea-surements to perform a test of the
Planck cluster surveycompleteness, and a summary of the manuscript is givenin Section 7. THE SZ EFFECT
The thermal SZ effect (Sunyaev & Zel’dovich 1972)describes the Compton scattering of CMB photons withhot electrons in the ICM according to∆ T ( ν ) = f ( ν, T e ) yy = (cid:90) k B σ T m e c p e dl where ∆ T ( ν ) is the observed surface brightness fluctua-tion in units of CMB temperature at the frequency ν , T e is the ICM electron temperature, f ( ν, T e ) describes thespectral dependence of the SZ signal including relativisticcorrections (e.g., Rephaeli 1995; Itoh et al. 1998; Nozawaet al. 1998; Itoh & Nozawa 2004; Chluba et al. 2012), y isthe SZ Compton parameter, k B is Boltzmann’s constant, σ T is the Thompson cross section, m e is the electronmass, c is the speed of light, p e is the ICM electron pres-sure, and dl is along the line of sight. In the absence ofrelativistic corrections, which are generally small and/orconstrained using a spectroscopic X-ray measurement ofthe value of T e , the SZ brightness gives a direct measureof the integrated ICM pressure. Therefore, SZ measure-ments are often used to constrain parametric models ofthe pressure, such as the generalized Navarro, Frenk, andWhite (gNFW, Navarro et al. 1997) model described inthe following section. The gNFW Model
Nagai et al. (2007) proposed the use of a gNFW modelto describe cluster pressure profiles according to P ( R ) = P ( R/R s ) γ (1 + ( R/R s ) α ) ( β − γ ) /α where P ( R ) is the pressure as a function of radius, P isthe normalization factor, R s is the scale radius, and α , β ,and γ control the logarithmic slope of the profile at R ∼ R s , R (cid:29) R s , and R (cid:28) R s . Often, the radial coordinatesare rescaled to angular coordinates denoted by θ and θ s , and R s is often recast in terms of a concentrationparameter, with C = R /R s = θ /θ s , and R denoting the radius where the average encloseddensity is 500 times the critical density of the universe.Therefore, for a given value of C , the values of R s and θ s are directly related to the cluster mass, M .Furthermore, the normalization is often given in termsthe SZ observable integrated within a specific radius, forexample Y = (cid:90) × θ y × πθdθ. Nagai et al. (2007) noted that, when P is scaled accord-ing to a factor that depends on the cluster’s mass andredshift and R s is recast in terms of C , that a single setof values for α , β , and γ provide an approximately uni-versal description of any cluster’s pressure profile. Sub-sequently, several groups have published different valuesfor these logarithmic slopes based on different samples,data, and analysis techniques (e.g., Arnaud et al. 2010;Plagge et al. 2010; Planck Collaboration et al. 2013b;Sayers et al. 2013; McDonald et al. 2014 and Mantz et al.2016), and the values given by Arnaud et al. (2010) arethe most widely used. The corresponding gNFW shapewith C = 1 . α = 1 . β = 5 .
49, and γ = 0 .
31 isoften referred to as the A10 model. DATA
Cluster Sample
This study focuses on a set of 47 clusters with pub-licly available data from Bolocam and Chandra . Datafor 45 of these clusters were published in Czakon et al.(2015), who named that sample the Bolocam X-ray SZ(BoXSZ) sample. Throughout this work, the slightly ex-panded set of 47 clusters is referred to as the BoXSZ + sample (see Table 1). Based on the Planck
MMF3 de-tection algorithm, 32 BoXSZ + clusters were detected by Planck , with 25 detected at a high enough significanceto be included in the
Planck cluster cosmology analysis(Planck Collaboration et al. 2015c,d).
Planck
The 2015
Planck data release contains a range of prod-ucts related to the SZ signal toward clusters, and thisanalysis utilizes two of those products: 1) the R2.08 clus-ter catalog created with the MMF3 detection algorithm,which was the baseline catalog for the Planck cluster cos-mology analysis (Planck Collaboration et al. 2015c), and2) the R2.00 all-sky y -maps created based on the MILCAalgorithm (Planck Collaboration et al. 2015b), which, asdetailed below, show good consistency with the MMF3measurements for the clusters in the BoXSZ + sample.The MMF3 catalog provides a two-dimensional proba-bility density function (PDF) for each cluster as a func-tion of Y and θ s assuming an A10 profile. A con-straint on Y can therefore be obtained by marginal-izing over θ s , either with or without a prior. As an ex-ample of such a prior, the MMF3 catalog provides thevalues of M derived from the Planck data, based on ascaling relation calibrated using hydrostatic masses from
XMM , and these values of M provide a direct con-straint on θ s for an assumed value of C (see Figure 1).In addition to the MMF3 catalog, the value of Y can also be derived using the all-sky MILCA y -map byfitting an A10 model directly to the map according to thefollowing procedure. First, a prior on the value of θ s fromthe XMM -like measurements is used to set the angular http://irsa.ipac.caltech.edu/data/Planck/release_2/ancillary-data/bolocam/ http://irsa.ipac.caltech.edu/data/Planck/release_2/docs/ Because these masses and θ s values are calibrated basedon XMM measurements, they are referred to throughout thismanuscript as “
XMM -like”.
TABLE 1Cluster Sample
RA dec M θ ( CXO ) θ ( XMM ) Planck
BolocamCluster z hr deg 10 M (cid:12) arcmin arcmin SNR SNRAbell 2204 0.15 16:32:47 +05:34:32 10 . ± . . ± . . ± . − . ± . . ± . . ± . − . ± . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . − − . ± . . ± . − . ± . . ± . . ± . − − . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . − . ± . . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . − − . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . . ± . − − . ± . . ± . . ± . − − . ± . . ± . − − . ± . . ± . . ± . − − . ± . . ± . − − . ± . . ± . . ± . − − . ± . . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . − . ± . . ± . . ± . . ± . . ± . − − . ± . . ± . . ± . . ± . − − . ± . . ± . − . ± . . ± . . ± . . ± . . ± . Note . — From left to right the columns give: the cluster name, redshift,
Chandra
RA centroid,
Chandra deccentroid,
Chandra -derived mass,
Chandra -derived θ , XMM -like θ , Planck
MMF3 SNR, and Bolocam SNR. size of the model. Then, the three-dimensional model ofthe cluster is projected to a two-dimensional image withthe line-of-sight projection extending to a radial distanceof 5 × R . Next, the model is convolved with a 10 (cid:48) full-width half-maxima (FWHM) Gaussian profile to matchthe point spread function (PSF) of the MILCA y -map,and binned into square pixels with sides of 3 . (cid:48) . Tocompare to this candidate model, the full-sky HEALPix MILCA y -map data are rebinned into 100 (cid:48) × (cid:48) thumb-nails centered on each cluster with identical 3 . (cid:48) squarepixels (see Figure 1). Next, 1000 random noise mapsare generated from the sum of the inhomogeneous noisemap and the full-sky homogeneous noise spectrum underthe assumption that the noise is Gaussian. From thesenoise realizations, a variance per pixel is computed, andthe inverse of this variance is used as a weighting fac- http://healpix.jpl.nasa.gov tor when fitting the A10 model to the data. The fitsare performed using the generalized least squares routine MPFITFUN (Markwardt 2009), and the only free param-eter in the fits is the overall normalization of the A10model, Y .The homogeneous noise spectrum of the MILCA y -map is not white, and therefore the per-pixel variance ofthe random noise maps does not fully describe the data.As a result, the weighting factors used in the fits arein general sub-optimal. This causes the derived param-eter uncertainties from the fits to be larger than thosefrom an optimal fit, but it does not produce any biasin the parameter values. However, the parameter un-certainties will in general be mis-estimated using thisprocedure. Consequently, rather than estimating theseuncertainties using the per-pixel variance, they are de-termined using the 1000 noise realizations. Specifically,the best-fit model obtained from the data is added to Fig. 1.—
Examples of the SZ data used in this analysis for the cluster Abell 370. Left: Bolocam data (top) and 1 of the 1000 noiserealizations (bottom) smoothed to an effective FWHM of 1 . (cid:48) for visualization (the unsmoothed data are used for all analyses). Middle: Planck
MILCA y -map (top) and 1 of the 1000 noise realizations (bottom). Right, top: Planck
MMF3 PDF (blue contours separated by 0.1in probability) and the
Planck XMM -like constraint on the value of θ s (solid red, with each color separated by 0.1 in probability). Right,bottom: One-dimensional constraint on the value of Y based on the Planck
PDF and the
XMM prior on θ s . each of the 1000 noise realizations, all of which are thenfit using the same procedure as applied to the real data.For each of these fits, the value of θ s is varied accordingto its prior, thus fully including these uncertainties. Thespread of values obtained for a given parameter basedon these 1000 fits then provides the uncertainty on thatparameter.Based on the above fits, the value of Y obtainedfrom the MMF3 catalog is consistent with the value of Y obtained from the MILCA y -map, with a sample-mean ratio of 1 . ± .
023 for the 32 BoXSZ + clusterscontained in the MMF3 catalog (see Figure 2). Further,the uncertainty on Y is also consistent between thetwo, with a sample-mean ratio of 0 . ± . There-fore, on average, Y values obtained from fits to theMILCA y -maps are equivalent to Y values obtainedfrom the MMF3 catalog. Bolocam
This analysis makes use of the publicly available fil-tered Bolocam maps, which contain an image of the clus-ter that has been high-pass filtered according to a two-dimensional transfer function included with the data.Analogous to the MILCA y -maps, 1000 noise realizationsof the Bolocam maps are provided. The A10 model fits An identical fitting procedure was also applied to the
Planck
NILC y -maps. While the value of Y is consistent between theNILC y -maps and the MMF3 catalog with a sample-mean ratio of0 . ± . y -mapsare systematically lower with a sample-mean ratio of 0 . ± . y -maps. As a result, the NILC y -maps are not considered in this analysis. are performed using the same procedure applied to theMILCA y -maps, with the following differences: 1) theBolocam data have a 58 (cid:48)(cid:48) FWHM PSF, 2) the modelmust be convolved with the transfer function of the spa-tial high-pass filter, and 3) the transfer function of themean signal level of the map is equal to 0, and so anadditional nuisance parameter is included in the fits todescribe the mean signal. COMPARISON OF SZ MEASUREMENTS
The Bolocam fit results from Section 3.3 can be directlycompared to the
Planck -derived results from Section 3.2,which were based on identical A10 model shapes and
XMM -like priors on the value of θ s , along with a nearlyidentical fitting procedure. The weighted mean ratiobetween the Bolocam and
Planck values of Y ob-tained from these fits is 1 . ± . Planck
SZ measure-ments to ground-based data. For example, Planck Col- One subtlety is that the frequency-dependent relativistic cor-rections to the SZ signal were not included in any of the fits, andthis could potentially bias the values of Y derived from Planck compared to the values derived from Bolocam. However, this biasshould be minimal for two main reasons. First, the most sensitive
Planck
SZ channel is centered on 143 GHz, which is nearly identicalto the Bolocam observing band centered on 140 GHz. Second, at140 GHz the typical relativistic corrections for the BoXSZ + clus-ters are (cid:46) Y values. Fig. 2.—
Left: the value of Y obtained from A10 fits to the Planck
MILCA y -maps compared to the value of Y recovered from Planck using the MMF3 algorithm. On average, the two results are consistent. Right: the value of Y obtained from A10 fits to theBolocam data compared to the value of Y recovered from the Planck
MILCA y -maps. Given the 5% flux calibration uncertainty onthe Bolocam data, the two results are consistent on average. In both plots, clusters above the Planck cluster cosmology cut (MMF3 SNR >
6) are shown in black, while MMF3 detections below the cut are shown in red. Both plots contain all 32 BoXSZ + clusters detected by Planck using the MMF3 algorithm. laboration et al. (2013a) fit A10 models to a set of 11clusters using
XMM priors on θ s and SZ data from theArcminute Microkelvin Imager (AMI). They found anaverage ratio of 0 . ± .
05 between the values of Y derived from AMI and Planck , indicating good agree-ment. A later comparison by Perrott et al. (2015), us-ing AMI observations of 99 clusters, found systemati-cally lower values of Y from AMI relative to Planck .However, the value of θ s was allowed to float in the fitsperformed in their analysis, and therefore some or all ofthe difference in Y values may be a result of usingdifferent pressure profile shapes when fitting AMI and Planck . More recently, Rodr´ıguez-Gonz´alvez et al. (2015)compared SZ measurements from
Planck and the Com-bined Array for Research in Millimeter-wave Astronomy(CARMA-8) for a set of 19 clusters. Like Perrott et al.(2015), they floated the value of θ s in their fits, although,unlike Perrott et al. (2015), they obtained consistency,with a CARMA-8/ Planck ratio of 1 . ± . JOINT FITS TO
Planck
AND BOLOCAM ANDCOMPARISONS TO PREVIOUS PRESSURE PROFILERESULTS
Motivated by the good agreement between
Planck andBolocam in measuring the value of Y based on iden-tical A10 profile shapes, the data can be combined tojointly constrain a more general gNFW shape. Specifi-cally, given that Bolocam and Planck are most sensitiveto the gNFW shape at large radii, the value of the outerlogarithmic slope β is allowed to vary in these fits whilethe other parameters are fixed to the A10 values. In orderto apply these fits to the largest sample possible, namelythe full set of 47 BoXSZ + clusters, an external prior onthe value of θ s is required due to the fact that XMM -likepriors only exists for 32 BoXSZ + clusters. This θ s prioris obtained from previously published values of M de-rived using data from Chandra , mainly from Sayers et al.(2013) based on the analysis methods detailed in Mantzet al. (2010). Two BoXSZ + clusters are absent fromSayers et al. (2013), and so the Chandra -derived M Recall from Section 2.1 that M uniquely determines θ s fora given C . of Abell 1689 is obtained from Mantz et al. (2010) andthe Chandra -derived M of Abell 2744 is obtained fromEhlert et al. (2015).One subtlety is that the Chandra -derived values of M are systematically larger than the XMM -like val-ues. In particular, the
XMM -like M values are knownto be biased low by (cid:39)
30% compared to lensing masses(von der Linden et al. 2014; Planck Collaboration et al.2015c), while the
Chandra M values described aboveare (cid:39)
10% higher than these same lensing masses (Mantzet al. 2014; Applegate et al. 2016). As a result, the
XMM -like values of θ s are smaller than the Chandra values of θ s ,with an average ratio of 1.16 for the 32 BoXSZ + clustersin the MMF3 catalog. Since θ s sets the angular scaleof the gNFW profile, this is equivalent to a change inthe value of C . However, since β is allowed to varyin these fits, and β and C are highly degenerate overthe angular scales probed by Planck and Bolocam, thisdiffering choice of θ s values does not significantly impactthe derived profile shape in the radial range where Bolo-cam and Planck are sensitive, though the specific valueof β derived from these fits does depend on the choice of θ s (i.e., of C ).To better understand the results of these jointly con-strained gNFW models, linear fits of P and β were per-formed versus M and z using LINMIXERR (Kelly 2007),with the results shown in Figure 3. Only modest cor-relations exist and the strongest trend is found in β ver-sus M . These fits find a cluster-to-cluster scatter of (cid:39)
30% for the value of P and (cid:39)
15% for the value of β .If the linear fits versus mass are evaluated at the medianvalue for the BoXSZ + sample, M = 9 . × M (cid:12) ,then the results are P /P = 9 . ± . ± .
98 and β = 6 . ± . ± .
76 (where the first value representsmeasurement uncertainty and the second indicates in-trinsic cluster-to-cluster scatter). Compared to the A10model, with P /P = 8 .
40 and β = 5 .
49, both of thesevalues are slightly larger and indicate a higher pressure In determining P , relativistic corrections are applied basedon spectroscopic Chandra measurements from Sayers et al. (2013)Mantz et al. (2010), and Babyk et al. (2012), using on an effectiveobserving frequency of 140 GHz.
Fig. 3.—
Best-fit parameters of the joint gNFW fit to Bolocam and
Planck . The normalization P (top row) and outer logarithmic slope β (bottom row) were allowed to float, while C , α , and β were fixed to the A10 values using a prior on θ s from Chandra . From left toright the plots indicate cluster mass and redshift, with 68% confidence intervals of linear fits from
LINMIXERR overlaid in orange. At themedian mass of the BoXSZ + sample, the linear fits versus M provide values of P /P = 9 . ± . ± .
98 and β = 6 . ± . ± . in the cluster center with a steeper fall-off at large radius.However, in interpreting these results, it is important tonote that, while β provides one metric for understandingthe pressure profile at large radius, it does not uniquelydescribe a single shape due to the strong degeneraciesbetween the gNFW parameters. A more robust met-ric is the ratio between the integrated SZ signal at R and at 5 × R , with Arnaud et al. (2010) obtaining Y / Y = 0 .
56. This result can be compared to thevalue of Y / Y = 0 . ± . ± .
10 obtained fromour joint Bolocam/
Planck fits to the BoXSZ + clusters(see Table 2).As mentioned in Section 2.1, a range of other analysesbeyond Arnaud et al. (2010) have also constrained gNFWprofiles in large samples of clusters. In particular, sev-eral groups have examined these profiles at large radiususing either simulations or SZ observations. For exam-ple, recent simulations from both Kay et al. (2012) andBattaglia et al. (2012) note a trend of increasing β withredshift, and both Battaglia et al. (2012) and Le Brun While Y / Y is a more robust metric than β for com-paring outer profile shapes, the general convention in the literaturehas been to report gNFW fit parameters directly. Therefore, thecomparisons presented in this section generally include both values. et al. (2015) find increasing values of β with increasingmass. Specifically, evaluating the Le Brun et al. (2015)fits at the median mass of the BoXSZ + sample yields β = 4 .
63 and Y / Y = 0 .
63, the latter indicating anouter profile shape that is consistent with our joint Bolo-cam/
Planck fit. Battaglia et al. (2012) used a param-eterization allowing P , β , and C to vary with massand redshift according to functional forms described by,for example β ∝ M b M (1 + z ) b z . Evaluating their “AGN Feedback ∆ = 500” fit at the me-dian mass and redshift of the BoXSZ + sample results ina value of β = 5 .
75 and Y / Y = 0 .
63, both in rel-atively good agreement with our joint Bolocam/
Planck fits.Given the good agreement of our results with Battagliaet al. (2012), we also fit an identical functional formto the joint Bolocam/
Planck constraints on β , finding b M = 0 . ± .
026 and b z = − . ± .
09. These can becompared to the values of b M = 0 .
048 and b z = 0 .
615 ob-tained by Battaglia et al. (2012), although some cautionis required because the values of C were not varied inour fits as they were by Battaglia et al. (2012). These re- TABLE 2gNFW Outer Profile Shapes
Analysis Data Type β Y / Y this work SZ observations 6.13 0.66Le Brun et al. (2015) simulations 4.63 0.63Ramos-Ceja et al. (2015) SZ power spectrum 6.35 0.69Sayers et al. (2013) SZ observations 3.67 0.28Planck Collaboration et al. (2013b) SZ/X-ray observations 4.13 0.48Battaglia et al. (2012) simulations 5.75 0.63Plagge et al. (2010) SZ observations 5.5 0.53Arnaud et al. (2010) X-ray observations/simulations 5.49 0.56Nagai et al. (2007) X-ray observations/simulations 5.0 0.52 Note . — Measurements of the outer pressure profile shape in large samples of clusters.The columns show the reference to the analysis, the type of data used in the analysis, thevalue of β , and the value of Y / Y . In the case of Le Brun et al. (2015) their “medianAGN 8.0” fits were used, and were scaled to the median mass of the BoXSZ + sample usingtheir fitting formulae. In the case of Battaglia et al. (2012), their “AGN Feedback ∆ = 500”fits were used, and were scaled to the median mass and redshift of the BoXSZ + sample usingtheir fitting formulae. Uncertainties are not available for most analyses, and so they havebeen omitted. Fig. 4.—
The ensemble-average best-fit gNFW profile to the jointBolocam/
Planck data for the BoXSZ + sample of clusters (black).Profiles for the 47 individual BoXSZ + clusters are shown as thingray lines. For comparison, the ensemble-average profiles fromother published gNFW fits to large samples of clusters are shownin red (Arnaud et al. 2010), green (Planck Collaboration et al.2013b), and blue (Sayers et al. 2013). The profiles extend over theapproximate radial range probed by the data in each analysis. Theensemble-average profiles show good agreement at R (cid:46) . R ,but noticeably diverge at larger radii. sults indicate that the trend in mass seen in the Battagliaet al. (2012) simulations is reproduced in our fits, but thetrend in redshift is not.The lack of a redshift trend could be a result of theBoXSZ + sample selection, which is biased toward relaxedcool-core systems at low- z and toward disturbed mergingsystems at high- z (see Sayers et al. 2013). For example,13 BoXSZ + clusters are defined as relaxed based on theSPA criteria of Mantz et al. (2015), and these clustersproduce a value of β = 6 . ± .
37. In contrast, 10BoXSZ + clusters are defined as merging based on eitherfailing the Symmetry/Alignment criteria or containing Mantz et al. (2015) use SPA to stand for symmetry, peakiness,and alignment, and relaxed clusters must pass a threshold in allthree criteria. Some known merging clusters pass the peakiness a radio relic/halo based on the analysis of Feretti et al.(2012) and Cassano et al. (2013), and these clusters pro-duce a value of β = 5 . ± .
61. Therefore, an excess ofcool-core clusters at low- z (which have larger values of β on average), and an excess of merging clusters at high- z (which have smaller values of β on average), will artifi-cially introduce a trend of decreasing β with redshift forthe BoXSZ + sample.Other groups have used SZ observations to constraingNFW profile shapes at large radii. For example, Plaggeet al. (2010) fit SZ data from a set of 15 clusters, find-ing β = 5 . Y / Y = 0 .
53. More recently,Planck Collaboration et al. (2013b) used
Planck obser-vations of a larger cluster sample to constrain β = 4 . Y / Y = 0 .
48 (see Table 2 and Figure 4). Bothof these analyses indicate a shallower outer profile thanour joint Bolocam/
Planck fits, although some of this dif-ference may be a result of sample selection. Specifi-cally, the Plagge et al. (2010) sample contains clusterswith a median redshift of 0.28 and a median mass of M ∼ × M (cid:12) , and the Planck Collaboration et al.(2013b) sample contains clusters with a median redshiftof 0.15 and a median mass of M = 6 . × M (cid:12) .If the parameterization of Battaglia et al. (2012) is usedto rescale their gNFW fits to the median mass and red-shift of the BoXSZ + sample, then the resulting valueof Y / Y from both the Plagge et al. (2010) andthe Planck Collaboration et al. (2013b) fits is equal to0.56, closer to our value of 0 . ± . ± .
10. The causeof the remaining difference is unclear, although it couldbe related to the mass estimates used in these analyses.In particular, Planck Collaboration et al. (2013b) used
XMM -derived masses to set the value of θ , and, asnoted above, these masses are known to be biased low, re-sulting in a different profile shape, and thus Y / Y ratio, for a given set of gNFW parameters.In another recent work, Sayers et al. (2013) obtained,from a joint fit to Bolocam observations of all the clus-ters in the BoXSZ sample, β = 3 .
67 and Y / Y = 0 .
28, with an overall profile that noticeably divergesfrom our joint Bolocam/
Planck fit at large radius. This test, and so therefore merging clusters were partially selected basedon failing the Symmetry and Alignment portions of the test.
Fig. 5.—
The ratio of Y measured from A10 fits to Bolocamusing a Chandra prior on θ s and Y measured from the Planck
MMF3 algorithm using the A10 model with an
XMM -like prior on θ s . The Chandra value of θ s is larger by an average fraction of1.16, resulting in systematically larger values of Y measured byBolocam. The Y ratio is shown as a function of M , with the68% confidence interval of linear fits from LINMIXERR overlaid inorange. This linear fit provides a mapping from the Bolocam mea-surements of Y to the Planck measurements of Y , allowingfor a test of the Planck completeness using the Bolocam data. is particularly surprising because the cluster samples arenearly identical, and the only significant difference is theinclusion of
Planck data in our current analysis. Becausethe Bolocam observations were made from the groundat a single observing frequency, they have have reducedsensitivity to large angular scales as a result of both at-mospheric fluctuations and primary CMB anisotropies.In contrast,
Planck is able to remove CMB anisotropiesvia its multiple observing channels, and it is not subjectto atmospheric fluctuations. Therefore, the
Planck dataare likely to provide more robust constraints on large an-gular scales. Though efforts were made in Sayers et al.(2013) to account for the atmospheric and CMB noise,they may be the primary cause of the shallower outerprofile found in that work.Beyond these SZ observations of large samples of in-dividual clusters, Ramos-Ceja et al. (2015) used mea-surements of the SZ power spectrum on small angu-lar scales from the South Pole Telescope (SPT, Re-ichardt et al. 2012) to constrain the average pressureprofile shape. They found that the A10 model needsto be adjusted to have an outer slope β = 6 . ± . Y / Y = 0 . ± .
03) in order to match the SPTmeasurements. Further, if this value of β is adopted, thentheir analysis implies little or no evolution in its value asa function of redshift. These results are consistent withour findings. TEST OF THE
Planck
CLUSTER COMPLETENESSESTIMATE
An accurate characterization of the completeness ofthe
Planck cluster survey is required for cosmologicalanalyses, and the discrepancy between the Planck clus-ter and CMB power spectrum cosmological results moti-vates special attention to such a characterization (PlanckCollaboration et al. 2015c). The details of how the com-pleteness is estimated are given in Planck Collaborationet al. (2015d) and summarized below. First, a set of clus- ters based on spherical profiles obtained from simulatedclusters (Le Brun et al. 2014; McCarthy et al. 2014) areinserted into both real and simulated
Planck maps. TheMMF3 algorithm is then applied to these maps, and theprobability of detecting a cluster above a given SNR isdetermined as a function of Y and θ s based on a bruteforce Monte-Carlo, which has been publicly released aspart of the MMF3 catalog. Ideally, the accuracy of thecompleteness function would be tested using a catalog ofreal clusters with known positions, θ s , and Y . In theabsence of such a catalog, Planck Collaboration et al.(2015d) undertook a somewhat less demanding test us-ing the MCXC (Piffaretti et al. 2011) and SPT (Bleemet al. 2015) cluster catalogs, which contain cluster posi-tions and θ s values, but not Y values.The BoXSZ + sample enables a better approximationof the ideal test of the Planck completeness because it haspositions, θ s , and Y estimates for each cluster. Specifi-cally, the positions and θ s values are obtained from Chan-dra , the latter rescaled by a factor of 1.16 to account forthe average difference between the
Chandra and
XMM values. This rescaling is required because
XMM -derived θ s values were used to calibrate the Planck completeness.Although it would be better to use the
XMM θ s valuesfor all of the BoXSZ + clusters, they only exist for theclusters detected by the MMF3 algorithm, significantlylimiting the value of such a test. In order to obtain Y estimates from Bolocam, the following procedure is ap-plied. First, the Bolocam value of Y for each BoXSZ + cluster is generated from A10 model fits to the Bolo-cam data using the Chandra value of θ s . Next, for the32 BoXSZ + clusters in the MMF3 catalog, the Planck value of Y is derived from the MMF3 PDF using the XMM -like value of θ s in order to mimic the computationof Y values used in the Planck completeness estimate.The ratio of the Bolocam and
Planck Y values is thenfit as a function of M using LINMIXERR (see Figure 5).The results of this linear fit, including the (cid:39)
25% intrin-sic scatter, are then used to rescale the Bolocam Y measurements for all of the BoXSZ + clusters. By fittingversus M , this ensures that the mass dependence ofthe profile shape found in Section 5 is fully included inthe conversion from Bolocam to Planck measurementsof Y . As part of this rescaling, an additional 5% un-certainty is added to account for the Bolocam flux cali-bration uncertainty, although the overall error budget isdominated by the intrinsic scatter in the linear fit.The Chandra and Bolocam values of θ s and Y ,rescaled to mimic the XMM and
Planck values as de-scribed in the previous paragraph, are then inserted intothe
Planck
SNR = 6 completeness estimate to determinea detection probability for each BoXSZ + cluster (see Fig-ure 6). One subtlety is that the noise in the Planck mapsis not uniform over the full sky, and it is therefore nec-essary to account for this variation when calculating thedetection probability for each BoXSZ + cluster. Specif-ically, this variation is accounted for by comparing thenoise RMS within the MILCA y -map thumbnail centeredon each cluster to the average noise RMS within the re-gion of sky satisfying the cuts used for the Planck clusteranalysis. In general, the local noise is within 5% of theaverage, and the most extreme local noise deviation is12%.The left panel of Figure 6 shows the probability for ev-
Fig. 6.—
Left: The probability for each BoXSZ + cluster to be detected with SNR > Planck using the MMF3 algorithm in ascendingorder of detection probability. The probability is computed using the Bolocam measurement of Y , rescaled according to the linearfit shown in Figure 5. Black diamonds denote the clusters actually detected by Planck and red triangles denote non-detections. Thevertical bars represent the range of probabilities predicted from the Bolocam measurement of Y , with most of the uncertainty dueto the intrinsic scatter in the linear model used to rescale the Bolocam measurements. Right: 68% (dark orange) and 95% (light orange)confidence regions for the total cumulative number of Planck
MMF3 clusters with SNR > Planck detections is given as a solid black line, and is consistent with, but somewhat low,compared to the predictions based on Bolocam. ery BoXSZ + cluster to be detected by the Planck
MMF3algorithm with a SNR >
6. There are no obvious out-liers, with
Planck detecting all of the clusters with a prob-ability of ∼ ∼
0. To provide a quantitative test, a simulation wasperformed based on the estimated detection probabili-ties. For each run of the simulation, a random value wasdrawn for each BoXSZ + cluster based on the detectionprobability distribution for that cluster, and the total cu-mulative number of detections was computed. The sim-ulation was repeated 10000 times, and the resulting 68%and 95% confidence regions on the cumulative detectionsare plotted in the right panel of Figure 6. The averagenumber of detections in the simulations is 27.6, and 16%of the simulations result in fewer than the actual numberof clusters detected by Planck , which is 25.This result provides a more extensive validation of the
Planck completeness estimate, although
Planck does de-tect slightly fewer clusters than expected given the Bolo-cam Y measurements. Such a shortfall could partiallyexplain the tension seen between the CMB-derived andcluster-derived cosmological constraints (Planck Collab-oration et al. 2015c,a). For example, Planck Collabora-tion et al. (2015c) quantifies the level of tension in termsof a cluster mass bias, with a value of (1 − b ) = 0 . − b ) (cid:39) . . Planck de-tections from the BoXSZ + sample can be recast as aneffective mass bias. In order for the average numberof predicted detections to equal the actual number of25, the Bolocam Y measurements would need to belower by a factor of 0 . ± .
11. Based on the Y / M scaling relation derived in Planck Collaboration et al.(2014), this corresponds to an effective mass bias factor of(1 − b ) = 0 . ± .
06. This effective bias is multiplicativewith the true mass bias, and would bring the
Planck clus- ter results into better agreement with the
Planck
CMBresults. SUMMARY
We fit A10 models to the
Planck
MILCA y -maps usingan XMM -like prior on the value of θ s , obtaining Y values consistent with those determined from the Planck
MMF3 algorithm using the same θ s prior. We also de-rived Y from ground-based Bolocam observations,finding a Bolocam/ Planck Y ratio of of 1 . ± . Planck measureconsistent SZ signals. Our results are in good agree-ment with previous comparisons between
Planck andthe ground-based AMI and CARMA-8 receivers, whichyielded similar consistencyWe also performed joint fits to the Bolocam and
Planck data, using a gNFW model with the outer logarithmicslope β allowed to vary with the other shape parame-ters fixed to the A10 values. These fits produce aver-age values of β = 6 . ± . ± .
76 and Y / Y = 0 . ± . ± .
10, which are in good agreement withrecent simulations for clusters matching the masses andredshifts of the BoXSZ + sample. Compared to simu-lations, our data are also consistent with the trend ofincreasing β with increasing cluster mass, but they donot reproduce the relatively strong trend of increasing β with increasing redshift, likely due to selection effectsin the BoXSZ + sample. Previous SZ measurements of β and Y / Y indicate lower values than our results,although some or all of this difference may be due toa combination of different median masses and redshiftswithin those samples, different mass measurements usedto set the cluster radial scale, and/or measurement noise.Using Bolocam measurements of Y and Chandra measurements of θ s , both rescaled to account for system-atic differences relative to Planck measurements of Y and XMM measurements of θ s , we compute the detectionprobability for each BoXSZ + cluster using the publiclyavailable Planck completeness estimate. We estimatethat
Planck should detect an average of 27.6 BoXSZ + (cid:39) σ of the actual numberof Planck detections, which is 25. Our results thereforeprovide a further validation of the
Planck completenessestimate. Taking the small discrepancy at face value,however, may suggest that
Planck detects fewer clustersthan expected. Translated to an effective mass bias, thisdiscrepancy yields (1 − b ) = 0 . ± .
06. This effectivemass bias is multiplicative with the true mass bias of(1 − b ) (cid:39) . Planck cluster-derived and CMB-derived cosmologi-cal parameters that has not been explained by the lensingmeasurements (Planck Collaboration et al. 2015a,c). ACKNOWLEDGMENTS
We acknowledge the assistance of: Kathy Deniston,who provided effective administrative support at Cal-tech; James Bartlett and Jean-Baptiste Melin, whoprovided useful discussions; JS was supported by aNASA/ADAP award; MN, GP, and BW were supportedby the Caltech Summer Research Connection program.SRS was supported by a NASA Earth and Space ScienceFellowship and a generous donation from the Gordon andBetty Moore Foundation.
Facilities:
Caltech Submillimeter Observatory,
Planck , Chandra ..