The Y_SZ,Planck – Y_SZ,XMM scaling relation and its difference between cool-core and non-cool-core clusters

Our sample is extracted from an XMM-Newton bright cluster sample (XBCS) (Zhao 2015; Zhao et al. 2015) and the Planck PSZ2 catalog (Planck Collaboration et al. 2016b). For the XBCS, we select clusters with a flux-limited (f_X[0.1 – 2.4 keV] ≥ 1.0 × 10⁻¹¹ erg s⁻¹ cm⁻²) method from several cluster catalogs based on the ROSAT All-Sky Survey (RASS; De Grandi et al. 1999): HIghest X-ray FLUx Galaxy Cluster Sample (HIFLUGCS; Reiprich & Böhringer 2002), ROSAT-ESO Flux Limited X-ray catalog (REFLEX; Böhringer et al. 2004), Northern ROSAT Brightest Cluster Sample (NORAS; Böhringer et al. 2000), X-ray-bright Abell-type clusters (XBACs; Ebeling et al. 1996) and ROSAT Brightest Cluster Sample (BCS; Ebeling et al. 1998). Among the XBCS entries, 78 clusters are available in the PSZ2 catalog. The positions of the cluster centers identified by XMM-Newton and Planck exhibit some deviation. Clusters with conditions of ΔD > 4' or ΔD > 0.3 R₅₀₀ are excluded, where ΔD is the positional offset between two centers and the R₅₀₀ is the cluster radius where the mean density is 500 times the critical density of the universe at the cluster redshift. Our final sample consists of 70 clusters, covering the redshift from about 0.01 to 0.25. The mass within R₅₀₀ of these galaxy clusters ranges from 0.27 to 11.5 × 10¹⁴ M_⊙, while the R₅₀₀ ranges from 0.44 to 2.45 Mpc.

The PSZ2 catalog is constructed by blind detection over the full sky using three independent extraction algorithms: MMF1, MMF3 and PsW, with no prior positional information on known clusters. MMF1 and MMF3 are based on a matched-multi-frequency filter algorithm. PsW is a fast Bayesian multi-frequency algorithm. All three algorithms assume the generalized Navarro-Frenk-White (GNFW) pressure profile (Arnaud et al. 2010) as prior spatial characteristics for the cluster, given by

$$\begin{eqnarray}&&p(r)=\frac{{P}_{0}}{{({c}_{500}r/{R}_{500})}^{\gamma }{[1+{({c}_{500}r/{R}_{500})}^{\alpha }]}^{(\beta -\gamma )/\alpha }}\end{eqnarray} \tag{ 2 }$$

with the parameters (Planck Collaboration et al. 2014)

$$\begin{eqnarray*}&&[{P}_{0},{c}_{500},\gamma,\alpha,\beta ]=[8.40{h}_{70}^{-3/2},1.177,0.308,1.05,5.49],\end{eqnarray*}$$

where α, β and γ are the logarithmic slopes for the intermediate region (c₅₀₀r ∼ R₅₀₀), the outer region (c₅₀₀r ≫ R₅₀₀) and the core region (c₅₀₀r ≪ R₅₀₀), respectively, c₅₀₀ is the concentration parameter through which θ₅₀₀ (instead of radial coordinates, angular coordinates are more often used, as θ₅₀₀ = R₅₀₀/D_A) is related to the characteristic cluster scale θ_s (θ_s = θ₅₀₀/c₅₀₀), and P₀ is the normalization factor. θ_s and P₀ are free parameters in this profile.

For each detected source, each algorithm provides an estimated position, signal-to-noise ratio (S/N) value, a two-dimensional joint probability distribution for θ_s and the integrated Compton parameter within 5θ₅₀₀, Y_5R500 (see Planck Collaboration et al. 2016a, fig. 16).

Y_5R500 and θ_s are strongly correlated, and we adopt θ₅₀₀, or equivalently θ_s, which is accurately derived from XMM-Newton observation (see Sect. 2.3) to break the Y_5R500 - θ_s degeneracy. Given the θ_s from X-ray, the expectation and standard deviation from the Y_5R500 conditional distribution are derived as the value of Y_5R500 and its uncertainty respectively. Uncertainties less than 0.05Y_5R500 would be assigned to the standard deviation of Y_5R500 in PSZ2, because they may be slightly underestimated by such Y_5R500 estimation (Planck Collaboration et al. 2016b). Finally, Y₅₀₀, denoted as Y_SZ,planck, is converted from Y_5R500 by Y_5R500 = 1.79 · Y₅₀₀ for the pressure profile mentioned above (Arnaud et al. 2010; Planck Collaboration et al. 2014).

The XBCS is built using a flux-limited method. We process the XMM-Newton data of the whole cluster sample in a complicated way. Here only a brief description of the XMM-Newton data process is presented, and more details can be referred to in Zhao et al. (2013, 2015). XMM-Newton pn/EPIC data acquired in extended full frame mode or full frame mode are analyzed with Science Analysis System (SAS) 12.6.0. We select events with FLAG = 0 and PATTERN≤4, in which contaminated time intervals are discarded. Then we correct vignetting effects and out-of-time events, remove prominent background flares and point sources, and subtract the particle background and the cosmic X-ray background. After that, the cluster region is divided into several rings centered on the X-ray emission peak, with the width of the rings depending on the net photon counts. The point spread function (PSF, pn: FWHM = 6''; XMM-Newton Users Handbook 2018) effect can be ignored because the minimum width of rings is set at 30''. By subtracting all the contributions from the outer regions, the de-projected spectrum of each ring is obtained (Chen et al. 2003, 2007; Jia et al. 2004, 2006).

XSPEC version 12.8.1 is used for spectral analysis. The de-projected temperature T_e, metallicity and normalizing constant norm at each ring are derived from the de-projected spectral fits with the thermal plasma emission model Mekal (Mewe et al. 1985) and Wabs model (Morrison & McCammon 1983). Fitting the simulated spectrum using T_e and abundance profiles in XSPEC, one can get the de-projected electron number density N_e at each ring.

Limited by the XMM-Newton field of view and the statistics of photons from clusters, the maximum observable radius of clusters, R_max, is usually smaller than R₅₀₀. In the case of R_max < R₅₀₀, T_e at r > R_max is set to the same value in the outermost ring. Linear interpolation is used to calculate T_e(r). For the fits of electron density profile N_e(r), the single β model and double β model are both adopted.

The single β model gives

$$\begin{eqnarray}&&{n}_{{\rm{e}}}(r)={n}_{0}{\left[1+{(\frac{r}{{r}_{{\rm{c}}}})}^{2}\right]}^{-\frac{3}{2}\beta },\end{eqnarray} \tag{ 3 }$$

where n₀ is the electron number density and r_c is the core radius.

The double β model is in the form of

$$\begin{eqnarray}&&{n}_{{\rm{e}}}(r)={n}_{01}{\left[1+{(\frac{r}{{r}_{{\rm{c1}}}})}^{2}\right]}^{-\frac{3}{2}{\beta }_{1}}+{n}_{02}{\left[1+{(\frac{r}{{r}_{{\rm{c2}}}})}^{2}\right]}^{-\frac{3}{2}{\beta }_{2}},\end{eqnarray} \tag{ 4 }$$

where n₀₁ and n₀₂ are the electron number density, and r_c1 and r_c2 are the core radius for the inner and outer components respectively (Chen et al. 2003).

For most clusters, the double β model fits significantly better than the single β model, however for some clusters, the improvements are negligible. As a result, 54 and 16 clusters are fitted with the double and single β model, respectively. Figure 1 shows a typical cluster profile. It clearly indicates that the double β model matches the electron number density data better than the single β model.

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The influences of the center offsets ΔD between XMM-Newton observation and three Planck algorithm detections are considered. Because of the Planck blind detection, we cannot fix our X-ray cluster position to that provided by the Planck detection procedure and re-extract Y_SZ,planck. Instead, we correct the Y_SZ,XMM by changing its integral center. The cluster is assumed to be spherically symmetric, and Y_SZ,XMM within R₅₀₀ is given by

$$\begin{eqnarray}&&{Y}_{500}={D}_{{\rm{A}}}^{-2}\,\frac{{k}_{{\rm{B}}}\sigma T}{{m}_{{\rm{e}}}{c}^{2}}\,\displaystyle \underset{-{R}_{500}+\Delta D}{\overset{{R}_{500}+\Delta D}{\int }}\,\,\displaystyle \underset{-{R}_{y}}{\overset{{R}_{y}}{\int }}\,\,\displaystyle \underset{-{R}_{z}}{\overset{{R}_{z}}{\int }}{n}_{{\rm{e}}}{T}_{{\rm{e}}}dxdydz,\end{eqnarray} \tag{ 5 }$$

where ${R}_{y}=\sqrt{{R}_{500}^{2}-{x}^{2}}$ and ${R}_{z}=\sqrt{{R}_{500}^{2}-{x}^{2}-{y}^{2}}$.

We adopt the Monte-Carlo method to estimate the uncertainties of Y_SZ,XMM. For each cluster, we simulate T_e at each shell, and the parameters of the β model for the N_e profile 5000 times, following Gaussian distributions with their own uncertainties. Then the uncertainty of Y_SZ,XMM is obtained.

Emcee is the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) designed for Bayesian parameter estimation (Foreman-Mackey et al. 2013, the code can be downloaded at http://dan.iel.fm/emcee/current/). We employ emcee to fit the Y_SZ,CMB – Y_Sz,X–ray scaling relation in the linear form

$$\begin{eqnarray}&&Y=B\cdot X+A,\end{eqnarray} \tag{ 6 }$$

where A and B are estimated parameters, and X and Y denote the base-10 logarithm of Y_SZ,X–ray and Y_SZ,CMB (log₁₀ Y_SZ,X-ray, log₁₀ Y_SZ,CMB), respectively. The likelihood adopted in these fits is from equation (35) of Hogg et al. (2010), following Planck Collaboration et al. (2016b),

$$\begin{eqnarray}&&\mathrm{ln}L=-\frac{1}{2}\displaystyle \sum _{i=1}^{N}\left(\mathrm{ln}({\sigma }_{{\rm{i}}}^{2}+{\sigma }_{{\rm{int}}}^{2})+\displaystyle \sum _{i=1}^{N}\frac{{({Y}_{{\rm{i}}}-B\cdot {X}_{{\rm{i}}}-A)}^{2}}{{\sigma }_{{\rm{i}}}^{2}+{\sigma }_{{\rm{int}}}^{2}}\right),\end{eqnarray} \tag{ 7 }$$

where ${\sigma }_{{\rm{i}}}^{2}={\sigma }_{{Y}_{{\rm{i}}}}^{2}+{B}^{2}\cdot {\sigma }_{{X}_{{\rm{i}}}}^{2}$. N is the number of clusters, σ_int is the intrinsic scatter, and σ_Xi and σ_Yi are statistical errors in X_i and Y_i respectively. Three parameters, A, B and σ_int, are estimated in the fitting procedure. We also fix B = 1, and repeat the procedure above to obtain A and σ_{ins | B=1}. The ratio of Y_SZ,CMB/Y_SZ,X–ray> equals 10^A.

Y_SZ,planck and Y_SZ,XMM are all integrated within R₅₀₀. We construct five samples named MMF1, MMF3, PsW, MaxSN and NEAREST. Y_SZ,planck in the MMF1, MMF3 and PsW samples is given by the three corresponding Planck extraction algorithms, fixing θ_s in the (Y_5R500,θ_s) probability distribution plane at the X-ray θ_s. Four clusters are discarded from the PsW sample because the X-ray θ_s is beyond the scope of the PsW (Y_5R500,θ_s) plane. Collaboration et al. (2016b) demonstrated that the detection characteristics made by the three algorithms are consistent with each other by simulation. In order to construct a larger sample, we utilize them to build the MaxSN and NEAREST samples. In the MaxSN sample, the Y_SZ,planck of each cluster is assigned by the algorithm which gives the maximum S/N value, while in the NEAREST sample, the Y_SZ,planck of each cluster is set by the algorithm whose output position is closest to the X-ray center. With the accurately de-projected temperature and density distributions, we calculate Y_SZ,XMM correcting the impacts of the center offsets between XMM-Newton and the three Planck algorithms. The cluster properties are listed in Table 6. Differences between Y_SZ,XMM in the MaxSN and NEAREST samples are less than 2%, therefore we only present Y_SZ,XMM in the NEAREST sample in this table.

The scaling relations between Y_SZ,planck and Y_SZ,XMM are shown in Figure 2. The best-fitting parameters and the number of clusters for each sample are presented in Table 1. Firstly, we compare the MMF1, MMF3 and PsW samples, which are constructed by three independent detection algorithms. On the condition that the slope and normalization are free parameters, the Y_SZ,plank – Y_SZ,XMM relations in these three samples agree with each other. The intrinsic scatter in the MMF1 sample is relatively larger than that in the other algorithms. When we consider the relation with slope fixed to 1 (B = 1), the ratio of Y_SZ,Plank/Y_SZ,XMM for the MMF1 sample is significantly higher (∼ 4σ) than that of the MMF3 and PsW samples. This is due to the different background estimations and extraction strategies in the different algorithms. For the combined samples, MaxSN ad NEAREST, the Y_SZ,plank – Y_SZ,XMM relations between them are consistent. We regard the NEAREST sample as our reference sample, because the detection significance in each algorithm is different between the blind mode and the mode with a prior known cluster position, and the detection method which provides the position closest to the cluster's X-ray center is considered to be the most accurate.

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The NEAREST sample contains 70 clusters, in which 18, 18 and 34 detections are respectively made by algorithms MMF1, MMF3 and PsW, confirming that PsW produces the most accurate positions (Planck Collaboration et al. 2016b). The intercept and slope of the Y_SZ,plank – Y_SZ,XMM relations in this sample are A = –0.86 ± 0.30 and B = 0.83 ± 0.06 respectively. The intrinsic scatter is σ_ins = 0.14 ± 0.03. The ratio of Y_SZ,Plank/Y_SZ,XMM is 1.03±0.05 which is in excellent statistical agreement with unity. Our results indicate that the SZ signals detected by CMB and by X-ray observations are fully consistent.

There are two papers that study the Y_SZ,CMB – Y_Sz,X–ray scaling relation. Bonamente et al. (2012) present a sample of 25 massive relaxed galaxy clusters observed by the Sunyaev-Zel'dovich Array (SZA) and Chandra. They assume the ICM model which is introduced by Bulbul et al. (2010). This model can be applied simultaneously to SZ and X-ray data. Their ratio of Y_SZ,CMB/Y_SZ,X–ray is 1.06 ± 0.04, which is in good agreement with our results. DeMartino & Atrio-Barandela (2016) use a sample of 560 clusters whose properties are derived from Planck 2013 foreground cleaned nominal maps and ROSAT observations, to determine the SZ/X-ray scaling relations.

They calculate the angular size weighted Y_SZ, and obtain the relation ${\bar{Y}}_{{\rm{SZ}},Plank}=0.97{\bar{Y}}_{{\rm{SZ}},{\rm{X}}-{\rm{ray}}}$, which also agrees with ours.

The intrinsic scatter in our results σ_ins = 0.14 ± 0.03 is slightly larger than the prediction (∼ 10%). The extrapolation in both Planck and XMM-Newton may induce scatter or bias to our results. When determining Y_SZ,planck, Y₅₀₀ is obtained from Y_5R500. The shape of the GNFW pressure profile employed in the Planck analysis is fixed, which leaves a negligible impact on the scaling relation (Planck Collaboration et al. 2011c), but different shapes of pressure profile may have significantly different conversion factors from Y_5R500 to Y₅₀₀ (Sayers et al. 2016). To be more specific, each cluster has a unique pressure profile and a unique conversion factor, and converting Y₅₀₀ from Y_5R500 by a unified factor may induce scatter. In the extrapolation of cluster properties from X-rays, a flat temperature extending from ∼ 0.5R₅₀₀ to the cluster's outer region could overestimate Y_SZ,XMM.

We also calculate the Y_SZ,XMM whose N_e(r) is fitted with only the single β model. The resulting ratio is Y_SZ,Planck/Y_SZ,XMM = 0.89 ± 0.05, deviating nearly 3σ from our previous result. Many studies argue that the isothermal β model is inadequate to fit ICM and may overestimate the SZ signal (Lieu et al. 2006; Bielby & Shanks 2007; Hallman et al. 2007; Atrio-Barandela et al. 2008; Mroczkowski et al. 2009; Allison et al. 2011). Assuming two components in the ICM when fitting the electron distribution, the double β model works well within R₅₀₀ (Chen et al. 2007).

We construct a subsample including 55 clusters, which are overlapping clusters between the HIFLUGCS and ours. In this subsample, we refer to data in the NEAREST sample to investigate the cool core influences on the scaling relations. We adopt two methods to distinguish CCCs from NCCCs using X-ray data. The first method follows the definition in Zhao et al. (2013) (hereafter Z13): clusters with a central cooling time ${t}_{{\rm{c}}}\lt 7.7{h}_{70}^{-1/2}$ (Rafferty et al. 2006) and a temperature drop larger than 30% from the peak are classified as CCCs. This divides the sample into 28 NCCCs and 27 CCCs. The second method follows the definition in Chen et al. (2007) (hereafter C07): clusters with significant classical mass deposition rate ≥ 0.01 M_⊙ yr⁻¹ are classified as CCCs. Instead of calculating the mass deposition rate by ourselves, we directly use their classification which divides the sample into 29 NCCCs and 26 CCCs.

Figure 3 displays the CCCs' and NCCCs' scaling relations between Y_SZ,planck and Y_SZ,XMM. The best-fit parameters for each subsample are presented in Table 2.

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In the Z13 classification criteria, intrinsic scatter in the Y_SZ,plank – Y_SZ,XMM scaling relation of CCCs (∼ 0.11) is slightly smaller than that of NCCCs (∼ 0.20), and the Y_SZ,Plank/Y_SZ,XMM ratio of CCCs tends to be less than that of NCCCs. Due to the relatively large uncertainties, we observe weak evidence for the discrepancies between CCCs and NCCCs. Under the C07 criteria, disagreements between CCCs and NCCCs become more significant, especially for the intrinsic scatter which is ∼ 0.04 and ∼ 0.28 for CCCs and NCCCs, respectively. These results are not only obtained in the NEAREST sample, they remain the same in other samples, which are shown in Table 3.

To validate our results, we use Y_SZ,planck taken from three papers, Y₅₀₀ in Planck Collaboration et al. (2011c), Y_z in the PSZ1 catalog (Planck Collaboration et al. 2014) and Y_blind in the PSZ2 catalog (Planck Collaboration et al. 2016b), to discuss the cool-core influences on the Y_SZ,plank – Y_SZ,XMM scaling relations. Y₅₀₀ in Planck Collaboration et al. (2011c) is obtained by algorithm re-extraction from Planck maps at the X-ray position and with the X-ray size. Y_z in PSZ1 is calculated using redshift information. Y_blind in PSZ2 is the blind detection which has high average bias because of the overestimated size. Our Y_SZ,plank – Y_SZ,XMM is derived from Y_blind restricting by our X-ray size. Under C07 cool-core criteria, CCCs and NCCCs show clear discrepancies in the SZ and X-ray measurements no matter which Y_SZ,planck we used. The results are listed in Table 4 and displayed in Figure 4.

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We also examine the Y_SZ,Planck – Y_X scaling relation. Compared with Y_SZ,XMM, which requires accurate temperature and electron number density distribution, Y_X, which is equal to the mean temperature multiplied by the gas mass, is much easier to obtain. Therefore the Y_SZ,Planck – Y_X scaling relation is more widely used in comparing SZ and X-ray data. Here we define ${Y}_{{\rm{X}}}={T}_{{\rm{X}}}\cdot {M}_{{\rm{gas}}}\cdot ({D}_{{\rm{A}}}^{-2}({\sigma }_{{\rm{T}}}/{m}_{{\rm{e}}}{c}^{2})/({\mu }_{{\rm{e}}}{m}_{{\rm{p}}}))$, where T_X is the volume average temperature determined within the region [0.2,0.5]R₅₀₀, M_gas is the gas mass within R₅₀₀, $4\pi {m}_{{\rm{p}}}\displaystyle {\int }_{0}^{Rv}{n}_{{\rm{e}}}(r){r}^{2}dr$, with m_p the proton mass and μ_e the mean molecular weight of the electrons, and the factor ${D}_{{\rm{A}}}^{-2}({\sigma }_{{\rm{T}}}/{m}_{{\rm{e}}}{c}^{2})/({\mu }_{{\rm{e}}}{m}_{{\rm{p}}})$ is used to convert the unit from Mpc² to arcmin².

Y_SZ,Planck – Y_X relations, with C07 and Z13 criteria, are shown in Figure 5. We find similar results in the Y_SZ,Planck – Y_X relation as in the Y_SZ,plank – Y_SZ,XMM relation, which indicate that SZ and X-ray observations of CCCs and NCCCs are inconsistent, although discrepancies in the Y-ratio between CCCs and NCCCs in the Y_SZ,Planck – Y_X relation are smaller than those in the Y_SZ,plank – Y_SZ,XMM relation. Intrinsic scatters of CCCs and NCCCs still significantly disagree with each other. The results are listed in Table 5. We emphasize that Y_SZ,Planck/Y_X = 0.92 ± 0.05 is completely consistent with the prediction in X-ray, 0.924±0.004 (Arnaud et al. 2010).

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Our sample is an intersection of the X-ray sample with flux limit, and the Planck sample with S/N cut. The selection effects of the Malmquist bias (Stanek et al. 2006) and Eddington bias (Maughan 2007) may cause the results to deviate due to scatters in these scaling relations around the limit/cut. To quantify these effects on scaling relations, complicated computations are required to generate a large mock cluster sample from the assumed mass function, to mimic the observed sample with the same selection criteria (Vikhlinin et al. 2009a; Planck Collaboration et al. 2011a,c; Rozo et al. 2012; Czakon et al. 2015; De Martino & Atrio-Barandela 2016). For the Y-ratio, the correction is negligible according to Planck Collaboration et al. (2011c); Rozo et al. (2012); Czakon et al. (2015); De Martino & Atrio-Barandela (2016). The bias should be fairly small for very luminous objects (Planck Collaboration et al. 2011b; Rozo et al. 2012; Planck Collaboration et al. 2016b). As the galaxy clusters in our sample are very bright clusters with strong SZ detections, we believe the bias of the Eddington effect and Malmquist effect is fairly small in our Y_SZ,plank – Y_SZ,XMM scaling relation. The discrepancies between CCCs and NCCCs are due to other reasons. However, we should also bear in mind that our Y_SZ,plank – Y_SZ,XMM scaling relation is derived from the most luminous clusters. Applications to dimmer clusters with this scaling relation should be considered carefully.

Most CCCs are relaxed systems while NCCCs are undergoing more disturbing processes, like merging. Therefore, the intrinsic scatter of CCCs is smaller than that of NCCCs. The ratio of Y_SZ,Plank/Y_SZ,XMM in CCCs (NCCCs) has a trend to be smaller (larger) than unity, which implies that the outskirt pressure profiles of CCCs and NCCCs could have substantial differences, instead of following a universal profile.

Because of the different dynamical states of CCCs and NCCCs, it is natural to believe that the Y_SZ,CMB – Y_Sz,X–ray scaling relation of CCCs and NCCCs could have discrepancies, but previous measurements show little difference between them (Planck Collaboration et al. 2013b; Rozo et al. 2012; DeMartino & Atrio-Barandela 2016). This contradiction may be mainly due to our high quality X-ray data. We processed the XMM-Newton data in detail, and no scaling relation was adopted during the data analysis. Another reason may be due to the cool-core classification criteria. In our results, the CCC and NCCC discrepancies are more significant with the C07 definition, therefore the mass deposition rate may be much closer to the physical nature of CCCs and NCCCs than the central gas density, core entropy excess and central cooling time, which previous works apply to distinguish CCCs from NCCCs.