Carma observations of massive Planckdiscovered cluster candidates at
Carma observations of massive Planckdiscovered cluster candidates at associated with WISE overdensities: Breaking the sizeflux degeneracy
Abstract
We use a Bayesian software package to analyze CARMA8 data towards 19 unconfirmed Planck SZcluster candidates from RodríguezGonzálvez et al. (2015) that are associated with significant overdensities in WISE. We used two cluster parameterizations, one based on a (fixed shape) generalizedNFW pressure profile and another based on a gas density profile (with varying shape parameters) to obtain parameter estimates for the nine CARMA8 SZdetected clusters. We find our sample is comprised of massive, , relatively compact, systems. Results from the model show that our cluster candidates exhibit a heterogeneous set of brightnesstemperature profiles. Comparison of Planck and CARMA8 measurements showed good agreement in and an absence of obvious biases. We estimated the total cluster mass as a function of for one of the systems; at the preferred photometric redshift of 0.5, the derived mass, . Spectroscopic Keck/MOSFIRE data confirmed a galaxy member of one of our cluster candidates to be at . Applying a Planck prior in to the CARMA8 results reduces uncertainties for both parameters by a factor , relative to the independent Planck or CARMA8 measurements. We here demonstrate a powerful technique to find massive clusters at intermediate () redshifts using a crosscorrelation between Planck and WISE data, with highresolution followup with CARMA8. We also use the combined capabilities of Planck and CARMA8 to obtain a dramatic reduction by a factor of several, in parameter uncertainties.
1 Introduction
The Planck satellite (Tauber et al. 2010 and Planck Collaboration et al. 2011 I) is a third generation spacebased mission to study the Cosmic Microwave Background (CMB) and its foregrounds. It has mapped the entire sky at nine frequencies from 30 to 857 GHz, with an angular resolution of to 5′, respectively. Massive clusters have been detected in the Planck data via the SunyaevZel’dovich (SZ) effect (Sunyaev & Zel’dovich, 1972). Planck has published a cluster catalog containing 1227 entries, out of which 861 are confirmed associations with clusters. 178 of these were previously unknown clusters while a further 366 remain unconfirmed (Planck Collaboration et al. 2013 XXIX). The number of cluster candidates identified in the second data releases, PR2, has now reached 1653. Eventually, the cluster counts will be used to measure the cluster mass function and constrain cosmological parameters (Planck Collaboration et al., 2013 XX). However, using cluster counts to constrain cosmology relies, amongst other things, on understanding the completeness of the survey and measuring both the cluster masses and redshifts accurately (for a comprehensive review see, e.g., Voit 2005 and Allen, Evrard, & Mantz 2011). To do so, it is crucial to identify sources of bias and to minimize uncertainty in the translation from cluster observable to mass. Regarding cluster mass, since it is not a direct observable, the best massobservable relations need to be characterized in order to translate the Planck SZ signal into a cluster mass.
The accuracy of the Planck measurements of the integrated SZ effect at intermediate redshifts where, e.g., Xray data commonly reach out to, is limited by its resolution ( at SZrelevant frequencies) because the integrated SZ signal exhibits a wellknown degeneracy with the cluster angular extent (see e.g., Planck Collaboration et al. 2013 XXIX). Higher resolution SZ followup of Planckdetected clusters can help constrain the cluster size by measuring the spatial profile of the temperature decrement and identify sources of bias. Moreover, a recent comparison of the integrated SZ signal measured by the Arcminute MicroKelvin Imager (AMI; AMI Consortium: Zwart et al. 2008) on arcminute scales and by Planck showed that the Planck measurements were systematically higher by (Planck Collaboration et al. 2013e). This study, and its followup paper on 99 clusters (AMI and Planck Consortia: Perrott et al., 2014), together with another one by Muchovej et al. (2012) comparing CARMA8 and Planck data towards two systems, have demonstrated that cluster parameter uncertainties can be greatly reduced by combining both datasets.
Cluster  Union Name  RA  Dec  Short Baseline (02k)  Long Baseline (28 k) 

ID  
hh mm ss  deg min sec  
PSZ1G014.13+38.38  16 03 21.62  03 19 12.00  0.309  0.324  
P028  PSZ1G028.66+50.16  15 40 10.15  17 54 25.14  0.433  0.451 
P031    15 27 37.83  20 40 44.28  0.727  0.633 
P049    14 44 21.61  31 14 59.88  0.557  0.572 
P052    21 19 02.42  00 33 00.00  0.368  0.386 
P057  PSZ1G057.71+51.56  15 48 34.13  36 07 53.86  0.451  0.482 
PSZ1G086.93+53.18  15 13 53.36  52 46 41.56  0.622  0.599  
P090  PSZ1G090.82+44.13  16 03 43.65  59 11 59.61  0.389  0.427 
  14 55 13.99  58 51 42.44  0.653  0.660  
PSZ1G109.88+27.94  18 23 00.19  78 21 52.19  0.562  0.517  
P121  PSZ1G121.15+49.64  13 03 26.20  67 25 46.70  0.824  0.681 
P134  PSZ1G134.59+53.41  11 51 21.62  62 21 00.18  0.590  0.592 
P138  PSZ1G138.11+42.03  10 27 59.07  70 35 19.51  2.170  0.982 
PSZ1G171.01+39.44  08 51 05.10  48 30 18.14  0.422  0.469  
PSZ1G187.53+21.92  07 32 18.01  31 38 39.03  0.411  0.412  
PSZ1G190.68+66.46  11 06 04.09  33 33 45.23  0.450  0.356  
PSZ1G205.85+73.77  11 38 13.47  27 55 05.62  0.385  0.431  
P264    10 44 48.19  17 31 53.90  0.476  0.513 
  15 04 04.90  06 07 15.25  0.355  0.392 

() Since the cluster selection criteria, as well as the data for the cluster extraction, are different to those for the PSZ catalog, not all the clusters in this work have an official Planck ID.

(a) Achieved rms noise in corresponding maps.
In this work, we have used the eight element CARMA interferometer, CARMA8 (see Muchovej et al. 2007 for further details) to undertake high spatial resolution followup observations at 31 GHz towards 19 unconfirmed Planck cluster candidates
This work is presented as a series of two articles. The first one, RodríguezGonzálvez et al. (2015), henceforth Paper 1, focused on the sample selection, data reduction, validation using ancillary data and photometricredshift estimation. This second paper is organized as follows. In Section 2 we describe the cluster parameterizations for the analysis of the CARMA8 data and present cluster parameter constraints for each model. In addition, we include Bayesian Evidence values between a model with a cluster signal and a model without a cluster signal to assess the quality of the detection and identify systems likely to be spurious. Planckderived cluster parameters and estimates of the amount of radiosource contamination to the Planck signal are given in Section 3. Improved constraints in the plane from the application of a Planck prior on to the CARMA8 results are provided in Section 3.3. In Section 4 we discuss the properties of the ensemble of cluster candidates, including their location, morphology and clustermass estimates and present spectroscopic confirmation for one of our targets. In this section we also compare the Planck and CARMA8 data and show how our results relate to similar studies. We note that, for homogeneity, since not all the cluster candidates in this work are included in the PSZ (Union catalog; Planck Collaboration 2013 XXIX), we assign a shorthand cluster ID to each system (see Table 1).
Throughout this work we use J2000 coordinates, as well as a CDM cosmology with , , , , , and . is taken as 70 km s Mpc.
2 Quantitative Analysis of CARMA8 Data
2.1 Parameter Estimation using Interferometric Data
In this work we have used McAdam, a Bayesian analysis package, for the quantitative analysis of the cluster parameters. This package has been used extensively to analyze cluster signals in interferometric data from AMI (see e.g., AMI Consortium: Schammel et al. 2012, AMI Consortium: RodríguezGonzálvez et al. 2012 & AMI Consortium: Shimwell et al. 2013 for real data and AMI Consortium: Olamaie et al. 2012 for simulated data) and once before on CARMA8 data (AMI Consortium: Shimwell et al., 2013b). McAdam was originally developed by Marshall, Hobson, & Slosar (2003) and later adapted by Feroz et al. (2009) to work on interferometric SZ data using an inference engine, MultiNest (Feroz & Hobson 2008 & Feroz, Hobson & Bridges 2008), that has been optimized to sample efficiently from complex, degenerate, multipeaked posterior distributions. McAdam allows for the cluster and radio source/s (where present) parameters to be be fitted simultaneously directly to the short baseline (SB; k) data in the presence of receiver noise and primary CMB anisotropies. The high resolution, long baseline (LB; k) data are used to constrain the flux and position of detected radio sources; these sourceparameter estimates are then set as priors in the analysis of the SB data (see Section 2.2.3). Our short integration times required all of the LB data to be used for the determination of radiosource priors and none of the LB data were included in the McAdam analysis of the SB data. Undertaking the analysis in the Fourier plane avoids the complications associated with going from the sampled visibility plane to the image plane. In McAdam, predicted visibilities at frequency and baseline vector , are generated and compared to the observed data through the likelihood function (see Feroz et al. 2009 for a detailed overview).
The observed SZ surface brightness towards the cluster electron reservoir can be expressed as
(1) 
where is the derivative of the black body function at – the temperature of the CMB radiation (Fixsen et al. 1996). The CMB brightness temperature from the SZ effect is given by
(2) 
Here, is the frequency ()dependent term of the SZ effect,
(3) 
where the term accounts for relativistic corrections (see Itoh et al. 1998), is the electron temperature, , is Planck’s constant and is the Boltzmann constant. To calculate the contribution of the cluster SZ signal to the (predicted) visibility data, the Comptonization parameter, , across the sky must be computed:
(4) 
Here, is the Thomson scattering crosssection, is the electron mass, , and are the electron density, temperature and pressure at radius respectively, is the speed of light and is the line element along the line of sight. The projected distance from the cluster center to the sky is denoted by , such that . The integral of over the solid angle subtended by the cluster is proportional to the volumeintegrated gas pressure, meaning this quantity correlates well with the mass of the cluster. For a spherical geometry this is given by
(5) 
When , Equation 5 can be solved analytically, as shown in AMI and Planck Consortia: Perrott et al. (2014), yielding the total integrated Compton parameter, , which is related to the SZ surface brightness integrated over the cluster’s extent on the sky through the angular diameter distance to the cluster () as .
Parameter  Prior 

Gaussian centered at pointing centre,  
Gaussian centered at pointing centre,  
Uniform from 0.5 to 1.0  
Uniform from 0 to  
for  
& 0 outside this range  
for to  
& 0 outside this range with 
2.2 Models and Parameter Estimates
Analyses of Xray or SZ data of the intracluster medium (ICM) that aim to estimate cluster parameters are usually based on a parameterized cluster model. Cluster models necessarily assume a geometry for the SZ signal, typically spherical, and functional forms of two linearlyindependent thermodynamic cluster quantities such as electron temperature and density. These models commonly make assumptions such as, the cluster gas is in hydrostatic equilibrium or that the temperature or gas fraction throughout the cluster is constant. Consequently, the accuracy and validity of the results will depend on how well the chosen parameterization fits the data and on the effects of the model assumptions (see e.g., Plagge et al. 2010, Mroczkowski 2011 & AMI Consortium: RodríguezGonzálvez et al 2011 for studies exploring model effects in analyses of real data and AMI Consortium: Olamaie et al. 2012 and Olamaie, Hobson, & Grainge 2013 for similar work on simulated data). In this work we present cluster parameters calculated from two different models; one is based on a fixedprofileshape gNFW parameterization, for which typical marginalised parameter distributions for similar interferometric data from AMI have been shown in e.g., AMI and Planck Consortia: Perrott et al. (2014), and a second is based on the profile with variable shape parameters, where typical marginalised parameter distributions for comparable AMI data have been presented in AMI Consortium: RodríguezGonzálvez et al. (2012). Comparison of marginalised posteriors for CARMA and AMI data in AMI Consortium: Shimwell et al. (2013b) for the model showed the distributions to be very similar. The clusters presented here are at modest redshifts and are unlikely to be in hydrostatic equilibrium  adopting two models at least allows the dependency of the cluster parameters on the adopted model to be illustrated and a comparison with previous work to be undertaken.
Cluster model I: observational gNFW parameterization
For cluster model I, we have used a generalizedNFW (gNFW; Navarro, Frenk, & White 1996) pressure profile in the same fashion as in the analysis of Planck data (Planck Collaboration et al., 2011 VII) to facilitate comparison of cluster parameters. A gNFW pressure profile with a fixed set of parameters is believed to be a reasonable choice since (1) numerical simulations show low scatter amongst cluster pressure profiles, with the pressure being one of the cluster parameters that suffers least from the effects of nongravitational processes in the ICM out to the cluster outskirts and (2) the dark matter potential plays the dominant role in defining the distribution of the gas pressure, yielding a (pure) NFW form to the profile, which can be modified into a gNFW form to account for the effects of ICM processes (see e.g., Vikhlinin et al. 2005 and Nagai, Kravtsov, & Vikhlinin 2007). Using a fixed gNFW profile for cluster models has become regular practice (e.g., AtrioBarandela et al. (2008) for WMAP, Mroczkowski et al. (2009) for SZA, Czakon et al. (2014) for BOLOCAM and Plagge et al. (2010) for SPT data).
Assuming a spherical cluster geometry, the form of the gNFW pressure profile is the following:
(6) 
where is the normalization coefficient of the pressure profile and is the scale radius, typically expressed in terms of the concentration parameter . Parameters with a numerical subscript 500, like , refer to the value of that variable within —the radius at which the mean density is 500 times the critical density at the cluster redshift. The shape of the profile at intermediate regions (), around the cluster outskirts () and in the core regions () is governed by three parameters , , , respectively. Together with , they constitute the set of gNFW parameters. Two main sets of gNFW parameters have been derived from studies of Xray observations (inner cluster regions) and simulations (cluster outskirts) (Nagai, Kravtsov, & Vikhlinin 2007 and Arnaud et al. 2010). For ease of comparison with the results, as well as with SZinterferometer data e.g., from AMI in Planck Collaboration et al. (2013e), we have chosen to use the gNFW parameters derived by Arnaud et al.: .
Cluster ID  

″  ″  deg  ′  arcmin  
P014  
P086  
P097  
P109  
P170  
P187  
P190  
P205  
P351 
In our gNFW analysis, we characterize the cluster by the following set of sampling parameters (Table 2):
Here, are the displacement of the cluster decrement from the pointing centre, where the cluster right ascension is equal to the map center (provided in Table 1), is the ellipticity parameter, that is, the ratio of the semiminor and semimajor axes and is the position angle of the semimajor axis, measured N through E i.e. anticlockwise. We note that the projected cluster decrement is modeled as an ellipse and hence our model is not properly triaxial.
The priors used in this analysis are given in Table 2; they have been used previously for the blind detection of clusters in Planck data (Planck Collaboration et al., 2011 VII) and to characterize confirmed and candidate clusters in Planck Collaboration et al. (2013e). Cluster parameter estimates and the CARMA bestfit positions derived from model I are provided in Tables 3 and 4, respectively.
Cluster ID  RA  Dec 

hh:mm:ss  dd:mm:ss  
P014  16:03:23.29  03:16:44.00 
P086  15:14:00.85  52:48:12.56 
P097  14:55:24.17  58:52:20.44 
P109  18:23:03.50  78:23:07.19 
P170  08:50:59.16  48:30:28.14 
P187  07:32:23.03  31:37:32.03 
P190  11:06:08.81  33:33:56.23 
P205  11:38:07.21  27:54:39.62 
P351  15:04:02.09  06:06:12.24 
Cluster model II: observational parameterization
For this cluster parameterization we fit for an elliptical cluster geometry, as we did for model I, and model the shape of the SZ temperature decrement with a like profile (Cavaliere & FuscoFemiano, 1978):
(7) 
where is the brightness temperature decrement at zero projected radius, while and —the power law index and the core radius—are the shape parameters that give the density profile a flat top at small and a logarithmic slope of at large . The sampling parameters for the cluster signal are:
with priors given in Table 5, which allow for the signal to be computed. Cluster parameter estimates derived from model II are provided in Table 6.
Parameter  Prior 

Gaussian centered at pointing centre,  
Gaussian centered at pointing centre,  
Uniform from 0.5 to 1.0  
Uniform from 0 to  
Uniform from to  
Uniform from to  
Uniform from to mK 
Cluster ID  

″  ″  deg  ″  microK  
P014  
P086  
P097  
P109  
P170  
P187  
P190  
P205  
P351 
It is important to note that, historically, in many SZ analyses the shape of the profile has been fixed to values obtained from fits to higher resolution Xray data. However, these Xray results primarily probe the inner regions of the cluster and, thus, can provide inadequate bestfit profile shape parameters for SZ data extending out to larger. A comparative analysis in Czakon et al. (2014) reveals systematic differences in cluster parameters derived from SZ data using a modelindependent method versus Xraydetermined cluster profiles. Several studies have now shown that fits to SZ data reaching and beyond preferentially yield larger values than Xray data, which tend to yield (see e.g., AMI Consortium: HurleyWalker et al. 2012 & Plagge et al. 2010). Using a suitable (and ) value for the aforementioned typical SZ data can yield results comparable to those of a gNFW profile. In our parameterization we allow the shape parameters, and , to be fit in McAdam, since they jointly govern the profile shape. While our data cannot constrain either of these variables independently, they can constrain their degeneracy.
Although a large fraction of clusters are welldescribed by the bestfit gNFW parameterizations, some are not, as can be seen from e.g., the spread in the gNFW parameter sets from fits to individual clusters in the REXCESS sample (Arnaud et al., 2010). In these cases, modelling the cluster using a fixed (inadequate) set of gNFW parameters will return biased, incorrect results, whereas using a model with varying and should provide more reliable results. This is shown in Figure 1 of AMI Consortium: RodríguezGonzálvez et al. (2012) where data from AMI for a relaxed and a disturbed cluster are analyzed with a parameterization and five gNFW parameterizations, four of which have gNFW sets of parameters drawn from the Arnaud et al. REXCESS sample, three from individual systems and one from the averaged (Universal) profile, and, lastly, one with the averageprofile values from an independent study by Nagai, Kravtsov, & Vikhlinin (2007). For both clusters, the Nagai parameterization lead to a larger degeneracy and larger parameter uncertainties than the Arnaud Universal parameterization. The mean and values obtained from using the , Universal (Arnaud) and Nagai gNFW profiles were consistent to within the 95% probability contours, but this was not the case for fits using the sets of gNFW parameters obtained from individual fits to REXCESS clusters, indicating that some clusters do not follow a single, averaged profile. Here, comparison of the Bayesian evidence values for beta and gNFWbased analyses showed that the data could not distinguish between them. Our CARMA data for this paper have a similar resolution to the AMI data but, typically, they have much poorer SNRs and similarly cannot determine which of the two profiles provides a better fit to the data. More recently, Sayers et al. (2013) have derived a new set of gNFW parameters from 45 massive galaxy clusters using Bolocam and Mantz et al. (2014) have further shown how the choice of model parameters can have a measurable effect on the estimated Yparameter.
All sets of gNFW parameters can lead to biases when applied to different sets of data. Given that there is no optimallyselected set of gNFW parameters to represent CARMA 31GHz data towards massive, mediumtohigh redshift clusters (z 0.5), we choose to base most of our analysis on the gNFW parameter set from the ’Universal’ profile derived by (Arnaud et al., 2010) as this facilitates comparison with the Planck analysis and parallel studies between Planck and AMI, an interferometer operating at 16 GHz with arcminute resolution.
Cluster Profiles Using Equation 7 for (the SZ temperature decrement), and the mean values for , and derived from model II fits to the CARMA8 data (Table 6), in Figure 1, left, we plot the radial brightness temperature profiles for our sample of CARMA8detected candidate clusters. We order them in the legend by decreasing CARMA8 , from Table 3, although in some cases the differences are small. We would expect clusters with the most negative values, the shallowest profiles and the largest to yield the largest values. While there is reasonable correspondence throughout our cluster sample, two clusters P351 and P187 are outliers in this relation. Computing for each cluster from 0 to its , determined from model I (Table 3) shows that P351 (P187) has the highest (fifth highest) volumeintegrated brightness temperature profile but only the fifth highest (second highest) .
In Figure 1, right, we plot the upper and lower limits of the brightness temperature profiles allowed by the profile uncertainties for three clusters, P014, P109 and P205, chosen to span a wide range of profile shapes. It can be seen that the cluster candidates display a range of brightness temperature profiles that can be differentiated despite the uncertainties. In Table 7, by computing the ratio of the integral of the brightness temperature profile within (a) the 100GHz Planck beam and (b) from Table 3, we quantify how concentrated each brightness temperature profile is. The profile concentration factors have a spread of a factor of but for six clusters they agree within a factor of . Furthermore, the derived ellipticities shown in Table 6 and 3, which can be constrained by the angular resolution of the CARMA8 data, show significant evidence of morphological irregularity suggesting that these clusters may be disturbed and heterogeneous systems.
Radiosource Model and Parameter Estimates
Radio sources are often strong contaminants of the SZ decrement and their contributions must be included in our cluster analysis. In this work, we jointly fit for the cluster, radio source and primary CMB signals in the SB data. The treatment of radio sources is the same for all cluster models. These sources are parameterized by four parameters,
where and are RA and Dec of the radio source, is the spectral index, derived from the low fractional CARMA8 bandwidth and is the 31GHz integrated source flux. We adopt the convention, where is flux and frequency.
The high resolution LB data were mapped in Difmap (Shepherd 1997) to check for the presence of radio sources. Radiopoint sources detected in the LB maps were modeled using the Difmap task Modelfit. The results from Modelfit were primarybeam corrected using a FWHM of 660 arcseconds by dividing them by the following factor:
(8) 
where is the distance of the source to the pointing centre and
(9) 
The primarybeam corrected values were used as priors in the analysis of the SB data (see Table 8). Modelfit values are given in Paper 1 and McADamderived values are provided in Table 11.
Cluster ID  

P014  17.3 
P086  17.1 
P097  17.8 
P109  13.5 
P170  36.0 
P187  20.8 
P190  19.4 
P205  40.5 
P351  45.07 
Parameter  Prior 

,  Uniform between 
from the LBdetermined position  
Gaussian centered at bestfit Modelfit value  
with a of 20  
Gaussian centered at 0.6 with 
2.3 Quantifying the Significance of the CARMA8 SZ Detection or Lack thereof
Bayesian inference provides a quantitative way of ranking model fits to a dataset. Although the term model technically refers to a position in parameter space , here we refer to two model : a model class that allows for a cluster signal to be fit to the data, , and another, , that does not. The parameterization we have used for this analysis has been the gNFWbased model, Model I; for the case Model I was run as described in Section 2.2.1 and for the case it was run in the same fashion except for the prior on , which was set to 0, such that no SZ (cluster) signal is included in the model. Given the data , deciding whether or fit the data best can be done by computing the ratio:
(10) 
Here, is known as the Bayes Factor and is the prior ratio, that is, the probability ratio of the two model classes, which must be set before any information has been drawn from the data being analyzed. Here, we set the prior ratio to unity
(11) 
where is the dimensionality of the parameter space. represents an average of the likelihood over the prior and will therefore favour models with high likelihood values throughout the entirety of parameter space. This satisfies OccamÕs razor, which states that the models with compact parameter spaces will have larger evidence values than more complex models, unless the latter fit the data significantly better i.e., unnecessary complexity in a model will be penalized with a lower evidence value.
The derived Bayes factor is listed in Table 12 along with the corresponding classification of whether or not the cluster was considered to be detected. We find that all the SZ decrements considered to have high signaltonoise ratios in Paper I have Bayes Factors that indicate the presence of a cluster signature is strongly favoured. However, we do find some tension between the Paper I Modelfit and McAdam results for one of the candidate clusters, P014. In paper I this candidate cluster was catalogued as tentative (see Appendix B of Paper I for more details). The low SNR
3 Constraints from Planck
3.1 Cluster Parameters
We used the public Planck PR1 allsky maps to derive and values for our cluster candidates (Table 9). The values were derived using a multifrequency matched filter (Melin et al., 2006, 2012). The profile from Equation 4 is integrated over the cluster profile and then convolved with the Planck beam at the corresponding frequency; the matched filter leverages only the Planck high frequency instrument (HFI) data between 100857 GHz because it has been seen that the large beams at lower frequencies result in dilution of the temperature decrement due to the cluster. The beamintegrated, frequencydependent SZ signal is then fit with the scaled matched filter profile from Equation 2 to derive . The uncertainty in the derived is due to both the uncertainty in the cluster size (Planck Collaboration et al., 2013 XXIX), as well as the signal to noise of the temperature decrement in the Planck data. The large beam of Planck, FWHM at 100 GHz, makes it challenging to constrain the cluster size unless the clusters are at low redshift and thereby significantly extended. For this reason, Planck Collaboration et al. (2013 XXIX) provided the full range of contours which are consistent with the Planck data.
For the comparison here, there are two Planckderived estimates; was calculated using the cluster position and size () obtained from the higher resolution CARMA8 data, while was computed using the Planck data alone without using the CARMA8 size constraints. Similarly, is a measure of the angular size of the cluster using exclusively the Planck data; this value is weakly constrained and, thus, no clusterspecific errors are given for this parameter in Table 9. At this point, it is important to note that the quoted uncertainty for is an underestimate; the quoted error for this parameter is based on the spread in at the bestfit and is proportional to the signal to noise of the cluster in the Planck data i.e, without considering the error on which is very large. However, the uncertainty in is accurate since it propagates the true uncertainty in from the CARMA8 data into the estimation of this quantity from the Planck maps.
The uncertainty in from using the CARMA8 size measurement has gone down by on average, despite the fact that the does not include the uncertainty resulting from the unknown cluster size. If the true uncertainty in had been taken into account, the uncertainty would have gone down by more than an order of magnitude after application of the CARMA8 derived cluster size constraints. The mean ratio of to is 1.3 and, in fact, is only larger than its blind counterpart for two systems. Differences in the profile shapes account for being larger than for three systems, P014, P097 and P187, for which is smaller than measured by CARMA8.
Cluster ID  blind  blind  /  /  

arcmin  arcmin  arcmin  
P014  13.2 6.4  4.23  1.61  0.98  
P086  6.9 2.5  4.23  1.18  1.14  
P097  3.8 0.8  0.92  1.22  0.29  
P109  5.4 1.2  0.92  0.91  0.30  
P170  11.7 4.1  4.75  1.65  1.76  
P187  11.5 4.1  3.35  1.10  0.82  
P190  7.9 4.7  4.75  1.33  1.36  
P205  10.3 4.9  3.35  0.97  0.68  
P351  14.2 9.7  6.75  1.60  1.27 
Cluster  Flux  Flux  Flux  143GHz Planck  Radio Source 

ID  densities  densities  densities  SZ decrement  Contamination 
at 1.4 GHz  at 100 GHz  at 143 GHz  inside Beam  to Planck SZ  
P014  131.20  6.10  4.70  77.3  6.1 
P028  120.70  5.60  4.40  90.2  4.9 
P031  14.10  0.60  0.50  57.6  0.9 
P049  62.60  2.90  2.30  55.7  4.1 
P052  293.00  13.60  10.50  67.6  15.5 
P057  126.80  6.00  4.60  96.5  4.8 
P090  4.30  0.20  0.20  42.0  0.5 
P097  5.90  0.20  0.20  38.4  0.5 
P109  54.90  2.60  2.00  108.4  1.8 
P121  31.30  1.50  1.10  79.9  1.4 
P134  15.80  0.70  0.60  54.9  1.1 
P138  37.00  1.70  1.30  66.8  1.9 
P170  18.40  0.90  0.70  141.1  0.5 
P187  71.10  3.40  2.60  70.4  3.7 
P190  18.90  0.80  0.70  48.6  1.4 
P205  15.80  0.70  0.60  82.1  0.7 
P264  8.40  0.40  0.30  61.1  0.5 
P351  67.10  3.10  2.40  101.7  2.4 
3.2 Estimation of Radiosource Contamination in the Planck 143GHz Data
In order to assess if there are any clusterspecific offsets in the Planck values, we estimate the percentage of radiosource contamination to the Planck SZ decrement at 143 GHz—an important Planck frequency band for cluster identification—from the 1.4GHz NVSS catalog of radio sources. Spectral indices between 1.4 and 31 GHz were calculated in Table 3 in Paper 1 for sources detected in both our CARMA8 LB data and in NVSS, giving a mean value of of 0.72. We use this value for to predict the sourceflux densities at 100 and 143 GHz of all NVSS sources within 5′of the CARMA8 pointing center, following the same relation as we did earlier, .
The accuracy of the derived 100 and 143GHz source fluxes is uncertain. Firstly, there is source variability due to the fact the NVSS and CARMA8 data were not taken simultaneously, which could affect the 1.431 GHz spectral index. Secondly, we assume the spectral index between 1.4 and 31 GHz is the same as for 1.4 to 143 GHz, which need not be true. Thirdly, we deduce from a small number of sources, all of which must be bright in the LB data and apply this to lowerflux sources found in the deeper NVSS data, for which might be different. However, previous work shows that this value for is not unreasonable. Comparison of 31GHz data with 1.4GHz data on field sources has been previously done by Muchovej et al. (2010) and Mason et al. (2009). For the former, the 1.4to31 GHz spectralindex distribution peaked at 0.7 while, for the latter, it had a mean value of 0.7. The Muchovej et al. study also investigated the spectral index distribution between 5 and 31 GHz and located its peak at . Radio source properties in cluster fields have been characterized in e.g., Coble et al. (2007) tend to have a steep spectrum. In particular, the 1.4to31 GHz spectral index for the Coble et al. study had a mean value of 0.72. Sayers et al. (2013) explored the 1.4to31 GHz radio source spectral properties towards 45 massive cluster systems and obtained a median value for of 0.89, which they showed was consistent with the 30to140 GHz spectral indices. The radio source population used to estimate the contamination to the Planck 143 GHz signal is likely to be a combination of field and clusterbound radio sources due to the size of the Planck beam and the fact that some of the candidates might be spurious Planck detections. Overall, given the differences in the source selection and in frequency, and the agreement with other studies, our choice for of seems to be a reasonable one.
In Table 10 we list the sum of all the predicted radiosourceflux densities at 100 and 143 GHz of all the NVSSdetected sources within 5′ of our pointing centre. This yields an approximate measure of the radiosource contamination in the Planck beam at these frequencies. The mean of the sum of all integrated sourceflux densities at 1.4 GHz is 61.0 mJy (standard deviation, s.d. 71.6); at 100 GHz it is 2.8 mJy (s.d.) and at 143 GHz it is 2.2 mJy (s.d.=2.6). The SZ decrement towards each cluster candidate within the 143 GHz Planck beam is given in Table 10, together with the (expected) percentage of radiosource contamination to the Planck cluster signal at this frequency, which on average amounts to . The mean percentage contamination to the Planck SZ decrement would drop to if we used the Sayers et al. (2013) and would increase to if we used a flatter of . Thus, we expect the flux density from unresolved radio sources towards our cluster candidates to be an insignificant contribution to the Planck SZ flux although individual clusters may have radio source contamination at the % level.
Source ID  Cluster ID  /deg  /deg  /Jy  

1  P014  240.831 0.001  3.282 0.001  0.0081 0.0012  0.5 0.4 
2  P014  240.875 0.001  344.155 0.001  0.0089 0.0022  0.6 0.4 
1  P109  275.718 0.002  78.384 0.002  0.0018 0.0004  0.6 0.5 
1  P170  132.813 0.002  48.619 0.002  0.0046 0.0007  0.5 0.5 
1  P187  113.084 0.001  31.688 0.001  0.0037 0.0004  0.5 0.5 
1  P351  226.077 0.002  5.914 0.002  0.0028 0.0005  0.6 0.5 
3.3 Improved Constraints on and from the Use of a Planck Prior on in the Analysis of CARMA8 Data
Due to their higher resolution (a factor of ), the CARMA8 data are better suited than the Planck data to constrain . On the other hand, the large Planck beam (FWHM at 100 GHz) allows the sampling parameter for our clusters (all of which have ) to be measured directly, which is not the case for the CARMA8 data due to its finite sampling of the uv plane and the missing zerospacing information (a feature of all interferometers). We have exploited this complementarity of the Planck and CARMA8 data to reduce uncertainties in and . In order to do this, we filtered out the parameter chains (henceforth chains) for the analysis of the CARMA8 data (model I) that had values of outside the range allowed by the Planck results (Table 9). We refer to the results from the remaining set of chains as the joint results (Table 13). In Figure 5 we plot the 2D marginalized distributions for and for the CARMA8 data alone (black contours) and for the joint results (magenta contours). Similar approaches comparing Planck data with higher resolution SZ data have been undertaken by Planck Collaboration et al. (2013e) (with AMI), Muchovej et al. (2012) (with CARMA), Sayers et al. (2013) (with BOLOCAM) and AMI and Planck Consortia: Perrott et al. (2014) (with AMI). Clearly, the introduction of cluster size constraints from high resolution interferometry data provides a powerful way to shrink the uncertainties in phase space.
Cluster  Bayes Factor  Degree of detection 

P014  3.4e+01  D 
P014b  4.2e01  ND 
P028  4.6e01  ND 
P031  5.0e02  NC 
P049  2.2e01  ND 
P052  9.5e01  ND 
P057  6.1e01  ND 
P086  3.5e+03  D 
P090  1.2e+00  ND 
P097  7.9e+02  D 
P109  8.4e+02  D 
P121  4.9e01  ND 
P134  3.0e02  NC 
P134b  3.0e04  NC 
P170  7.2e+08  D 
P187  8.0e+08  D 
P190  5.6e+18  D 
P205  1.3e+09  D 
P264  2.3e+00  ND 
P351  2.7e+01  D 
4 Discussion
4.1 Use of Priors
When undertaking a Bayesian analysis, it is important not only to check that the priors on individual parameters are sufficiently wide, such that the distributions are not being truncated, but also that the effective prior is not biasing the cluster parameter results. Here the term effective prior refers to the prior that is being placed on a model parameter while taking into account the combined effect from all the priors given to the set of sampling parameters. What may seem to be inconspicuous priors on individual parameters can occasionally jointly reshape the high dimensional parameter space in unphysical ways; this was noticed in e.g., AMI Consortium: Zwart et al. (2011). Biases from effective priors should be investigated by undertaking the analysis without data i.e., by setting the likelihood function to a constant value. Such studies for the models used in this work have been presented in AMI Consortium: RodríguezGonzálvez et al. (2012), AMI Consortium: Olamaie et al. (2012) and Olamaie, Hobson, & Grainge (2013) and have determined that the combination of all the model priors does not bias the results.
Cluster ID  

arcmin  arcmin  
P014  
P086  
P097  
P109  
P170  
P187  
P190  
P205  
P351 
4.2 Characterization of the Cluster Candidates
Cluster position and Morphology
The mean separation (and standard deviation, s.d.) of the CARMA8 centroids from Model I and the Planck position is (0.5); see Tables 3 and 6 for offsets from the CARMA8 SZ decrement to the Planck position. This offset is comparable to the offsets between Planck and Xray cluster centroids found for the ESZ (Planck Collaboration et al. 2011 VII) and the PSZ (Planck Collaboration et al. 2013 XXIX), which were typically and , respectively. The cluster candidate with the largest separation, , is P014. The highresolution CARMA8 data allow for the reduction of positional uncertainties in the Planck catalog for candidate clusters from a few arcminutes to within . This is crucial, amongst other things, for the efficient followup of these candidate systems at other wavelengths. P351 has the largest positional uncertainties for both parameterizations (), indicative of a poorer fit of the models to the data, since the noise in the CARMA8 data is one of the smallest of the sample. Interestingly, this cluster stands out in the parameterization for having the shallowest profile (Figure 1), and in the gNFW parameterization for having the largest . Overall, the positional uncertainties from the shapefitting model I, tends to be larger than that from the radial profile based model II, typically by a factor of 1.2 and reaching a factor of 2.8. The different parameter degeneracies resulting from each analysis is likely to be the dominant cause for this. As shown in Figure 1 of AMI Consortium: RodríguezGonzálvez et al. (2012), in the plane, the 2D marginalized distribution for the cluster size is significantly narrower for the parameterization (model II) than for the gNFW parameterization (model I).
On average, cluster candidates with CARMA8 detections have with an s.d. of and , with s.d. of 0.9 (see Table 3). The largest cluster has , (P351) and the smallest , (P170). In Paper 1 we estimated the photometric redshifts for our cluster candidates with a CARMA8 SZ detection and found that, on average, they appeared to be at ; in Section 4.3.1 we report on the spectroscopic confirmation of P097 at . The relatively small values for would support the notion that our systems are at intermediate redshifts (). In comparison, for the MCXC catalog of Xrayidentified clusters (Piffaretti et al. 2011), whose mean redshift is 0.18, the mean Xrayderived is a factor of 2 larger. In Figure 2 we plot the average within a series of redshift ranges starting from for all MCXC clusters (in grey) and for only the more massive, , clusters (in blue), which should be more representative of the cluster candidates analyzed here (see Figure 3) and mark the average CARMA8derived for our clusters with an orange line. This plot suggests the values for our clusters are most comparable with the values for MCXC clusters at .
The resolution of the CARMA8 data, together with the often poor signaltonoise ratios and complications in the analysis, e.g., regarding the presence radio sources towards some systems, makes getting accurate measurements of the ellipticity challenging, with typical uncertainties in of 0.2 (Table 3). The mean and standard deviation of for our sample is . The values are therefore consistent with unity, to within the uncertainties. Nevertheless, the use of a spherical model is physically motivated and allows the propagation of realistic sources of uncertainty. Moreover, comparison of models with spherical and elliptical geometries for similar data from AMI is presented in AMI Consortium: HurleyWalker et al. (2012) which show the Bayesian evidences are too alike for model comparison, indicating that the addition of complexity to the model by introducing an ellipticity parameter is not significantly penalised. In AMI and Planck Consortia: Perrott et al. (2014) modelling of AMI cluster data with and ellipsoidal GNFW profile instead of a spherical profile had a negligible effect on the constraints in . Our CARMA data with higher noise levels and generally more benign source environments should show even smaller effects.
values close to 1 would be expected for relaxed systems, whose projected signal is close to spheroidal, unless the main merger axis is along the line of sight. On the other hand, disturbed clusters should have . Some evidence for a correlation of cluster ellipticity and dynamical state has been found in simulations, e.g, Krause et al. (2012) and data, e.g., Kolokotronis et al. (2001) (Xray), Plionis (2002) (Xray and optical) and AMI Consortium: RodríguezGonzálvez et al. (2012) (SZ), although this correlation has a large scatter. Hence, from the derived fits to the data, we conclude that our sample is likely to be mostly comprised by large, dynamically active systems, unlikely to have fully virialized which is not surprising given the intermediate redshifts of the sample.
4.3 ClusterMass Estimate
To estimate the total cluster mass within , we use the Olamaie, Hobson, & Grainge (2013) cluster parameterization, which samples directly from
Spectroscopic Redshift Determination for P097
We have measured the spectroscopic redshift of a likely galaxy member of P097 through Keck/MOSFIRE Yband spectroscopy. We deem it a likely member, given that it is situated close to the peak of the CARMA SZ decrement (within the contour) and close to a group of tightly clustered galaxies, as shown in Figure 4, of similar colours. We detect clear evidence for H, [NII], [SII] at redshifts of 0.565, with the strongest lines being present in a Sloandetected galaxy located at RA,DEC of (14:55:25.3,+58:52:33.86; J2000). We also find evidence of velocity structure in this galaxy with the H being double peaked with the line components separated by 150 km/s. These data will be published in a separate paper. We calculated for this cluster, as we did for P190 in the previous section, setting the redshift prior to a delta function at and obtained , supporting the notion that our sample of clusters are some of the most massive clusters at . Further followup of this sample in the Xrays and through weak lensing measurements with Euclid will help constrain the mass of these clusters more strongly.
4.9  6.5  

6.6  4.0  
5.2  3.8  