X-Ray Properties of SPT-selected Galaxy Clusters at 0.2 < z < 1.5 Observed with XMM-Newton

SPT has detected 516 galaxy clusters via the SZE in the 2500 degree² SPT-SZ Survey at $0\lt z\lt 1.8$ with masses ${M}_{500}\geqslant 3\times {10}^{14}{M}_{\odot }$ (Bleem et al. 2015). The redshifts of many of these clusters have also been reported in Ruel et al. (2014) and Bayliss et al. (2016). XMM-Newton X-ray observations of 40 of these SZE-selected clusters have been performed through several programs (PIs: A. Andersson, B. Benson, J. Mohr, R. Suhada, E. Bulbul). An additional 33 clusters have been observed through various other small non-SPT programs. Five clusters have been excluded from this analysis, because one scattered below the detection threshold when better data were available (SPT-CL J2343−5521), and four observations are dominated by background flares (SPT-CL J0411−4819, SPT-CL J0013−4906, SPT-CL J0257−5732, SPT-CL J2136−6307).

We exclude clusters at $z\lt 0.20$ from the scaling relation analysis because their SZE mass estimates obtained via the ζ–${M}_{500}$ relation (Section 3.1) are impacted by the filtering adopted to remove signal from the primary CMB (see, e.g., Benson et al. 2013). From this sample of 68 clusters, 59 are at redshift z > 0.2 and have a total of 1000 or more filtered source counts in MOS observations and are therefore included in our final sample. The details of the XMM-Newton observations of these clusters are given in Table 1.

The final sample is shown in Figure 1 in redshift–mass space with an inset redshift histogram. This cluster sample is not a complete SZE signal-to-noise ratio selected cluster sample. It has a median mass and redshift of ${M}_{500}=4.77\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{med}}=0.45$, and five of the clusters lie at z > 1. Nevertheless, the sample we study here has median mass, median redshift, and fraction of z > 1 clusters similar to the SPT-SZ cosmology sample of de Haan et al. (2016), which has a median mass ${M}_{500}=4.57\times {10}^{14}{M}_{\odot }$ (with roughly 6% of the clusters at z > 1), although it has a lower median redshift, 0.45 versus 0.55.

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Our XMM-Newton data reduction is described in detail in Bulbul et al. (2012); here we summarize the main steps. The XMM-Newton EPIC-MOS data analysis is carried out with Science Analysis System (SAS) version 16.0.0 and the latest available calibration files from February 2017. The Extended Source Analysis (ESAS) tools are used to reduce the data and extract the final data products (Snowden et al. 2008). The event files are filtered from the periods with elevated backgrounds through light curve filtering. The good time interval files are produced and used to create cleaned event lists. The net exposure time after filtering the event lists for good time intervals is given in Table 1. There are three main detectors on board XMM-Newton: MOS1, MOS2, and PN. The back-illuminated PN observations can be more sensitive to proton flares than MOS observations.²⁰ As a result, the majority of the PN observation of some clusters in the sample is lost to background filtering. Additionally, Schellenberger et al. (2015) reports up to a 54% bias in temperature measurements between Chandra and PN temperature measurements in the soft 0.7–2 keV band, where the bulk of detected photon flux from the high-redshift clusters appears. To avoid creating potential biases in the X-ray observables, we only use MOS observations in this analysis. We examine the individual chips that may be affected by an anomalous background level and exclude them from further analysis (Kuntz & Snowden 2008).

The images are created in the 0.5–2 keV band from the filtered event files and used to detect point sources within the MOS field of view (FOV). The images are examined carefully for point sources missed by the CIAO algorithm wavdetect. An exposure map is created for each MOS detector and each pointing to account for chip gaps and mirror vignetting. The quiescent particle background (QBP) image is created from the filter-wheel closed data as described in Snowden et al. (2008). The images and exposure maps of MOS1 and MOS2 detectors are combined prior to the background subtraction. The CIAO tool wavdetect convolved with the XMM-Newton point-spread function (PSF) is used on the background-subtracted and exposure-corrected images to detect point sources within the MOS FOV. All these point sources are excluded from further analysis.

We extract spectra using the ESAS tool mos-spectra within a radius of ${R}_{500}$ for each cluster (see Section 3 for the details of the ${R}_{500}$ calculation). Redistribution matrix files (RMFs) and ancillary response files (ARFs) are created with rmfgen and arfgen, respectively. QPB is subtracted from the total spectra prior to the fitting. The spectral fitting of the source is done in the spectral fitting package XSPEC 12.9.0 (Arnaud 1996) with ATOMDB version 3.0.8 (Smith et al. 2001; Foster et al. 2012). The adopted solar abundances are from Lodders & Palme (2009). The Galactic column density is allowed to vary within 15% of the measured Kalberla et al. (2005) LAB value in our fits, following the approach described in McDonald et al. (2016). We use C-stat as a goodness-of-fit estimator in XSPEC.

Spectra are extracted from two apertures of $\lt {R}_{500}$ and 0.15 ${R}_{500}$–${R}_{500}$ (again, see Section 3 for discussion of ${R}_{500}$). The fits are performed in the 0.3–10 keV energy interval. The higher energy band 7–10 keV is used to constrain soft-proton contamination accurately. Soft-proton flares are largely removed by the light curve filtering. However, after the filtering, some residuals may remain in the data. These are modeled by including an extra power-law model component in the total model and the MOS diagonal response matrices provided in the SAS distribution (Snowden et al. 2008). The cluster emission is fit with an absorbed single temperature apec model with free metallicity and temperature. Constraining metallicity is challenging for low-count observations of some of our high-z clusters. In these cases, we fixed the metallicity at 0.3Z_⊙, the typical value at both low and high redshifts (Tozzi et al. 2003).

We also consider the X-ray foreground emission, including Galactic halo, local hot bubble, cosmic X-ray background due to unresolved extragalactic sources, and solar wind charge exchange. The ROSAT All-Sky Survey background spectra²¹ extracted beyond ${R}_{\mathrm{vir}}$ (discussed in Section 3) are used to model the soft X-ray background as described in Bulbul et al. (2012). The soft X-ray emission from the local hot bubble is modeled with a cool unabsorbed apec component with kT ≈ 0.1 keV and abundance of Z_⊙ at z = 0, while the Galactic halo is modeled with a warmer absorbed thermal component kT ≈ 0.25 keV and abundance of Z_⊙ at z = 0. The temperatures of the apec models are restricted, but the normalizations are allowed to vary in our fits. We model the cosmic X-ray background due to unresolved point sources using an absorbed power-law component with a spectral index of 1.4 (Hickox & Markevitch 2005) and normalization of ≈9 × 10⁻⁷ photons keV⁻¹ cm⁻² s⁻¹ at ≈1 keV (Moretti et al. 2003; Kuntz & Snowden 2008). The bright instrumental fluorescent lines Al–K (1.49 keV) and Si–K (1.74 keV) are not included in the MOS QBP files. Therefore, we model these instrumental lines by adding Gaussian models to our spectral fits to determine the best-fit energies and normalizations.

Because of scattering in the XMM-Newton mirrors, some of the flux that originates from one area of the sky is detected in a different area of the detector. This is not a major concern if the gradient in plasma temperature from core to outskirts is smooth; however, it may be important for clusters with a strong, cool core. Additionally, for high-redshift clusters, 0.15 ${R}_{500}$ (discussed in Section 3) is comparable to the PSF for XMM-Newton, so this PSF effect is crucial and must be accounted for when making spectral fits. This radial cross-talk or contamination effect is treated as an additional model component in XSPEC. The cross-talk ARFs for the contribution of X-rays originating from a region on the sky to another region on the detector are created using the SAS tool arfgen (Snowden et al. 2008). The cross-talk correction is applied to eliminate PSF effects for all clusters in our sample.

We derive the cluster mass ${M}_{500}$ based on the SZE signal-to-noise ratio ξ and redshift z as determined by SPT. The measured signal-to-noise ratio ξ is a biased observable subject to Gaussian noise that is extracted through a matched filter approach that employs a β model with three degrees of freedom: sky location (α, δ) and core radius θ_C. The mean value of the signal-to-noise ratio $\left\langle \xi \right\rangle$ is related to the underlying unbiased signal-to-noise ratio ζ as follows:

$$\begin{eqnarray}&&\left\langle \xi \right\rangle =\sqrt{{\zeta }^{2}+3},\end{eqnarray} \tag{ 2 }$$

for ζ > 2 (de Haan et al. 2016). The ζ–mass scaling relation is parameterized as follows:

$$\begin{eqnarray}&&\zeta ={A}_{\mathrm{SZ}}{\left(\displaystyle \frac{{M}_{500}}{4.3\times {10}^{14}{M}_{\odot }}\right)}^{{B}_{\mathrm{SZ}}}{\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{{C}_{\mathrm{SZ}}},\end{eqnarray} \tag{ 3 }$$

where the normalization is A_SZ, the mass trend parameter is B_SZ, the redshift trend parameter is C_SZ, and there is log-normal intrinsic scatter in the observables at a fixed mass of ${\sigma }_{\mathrm{ln}\zeta }$.

In this work, we marginalize over the parameters of the ζ–mass relation while fitting the parameters of the X-ray scaling relations that are investigated. This ensures that the final uncertainties in the X-ray observable–mass–redshift scaling relations include the systematic uncertainties associated with the imperfectly known SZE-based halo masses. In the interest of focusing on the X-ray scaling relations, we adopt priors on the parameters of the ζ–mass relation that correspond to the fully marginalized posterior distributions reported in de Haan et al. (2016; Table 2). This approach does not capture any covariances among the ζ–mass scaling relation parameters, but these are indeed small (see de Haan et al. 2016, Figure 5). The advantage is that the likelihood we must calculate in each iteration of the Markov chain involves our X-ray observables and the simple priors on the SZE ζ–mass relation parameters.

Table 2.  Gaussian Priors ${ \mathcal N }(\mu ,{\sigma }^{2})$ on the SZE Observable–Mass Relation Parameters and Uniform Priors ${ \mathcal U }(\min ,\,\max )$ on the X-Ray Observable–Mass Relation Parameters

Parameters	Priors
SZE $\zeta \mbox{--}{M}_{500}\mbox{--}z$ parameters
ASZ	${ \mathcal N }(4.842,{0.913}^{2})$
${B}_{\mathrm{SZ}}$	${ \mathcal N }(1.668,{0.083}^{2})$
${C}_{\mathrm{SZ}}$	${ \mathcal N }(0.550,{0.315}^{2})$
${\sigma }_{\mathrm{ln}\zeta }$	${ \mathcal N }(0.199,{0.069}^{2})$
X-ray ${ \mathcal X }\mbox{--}{M}_{500}\mbox{--}z$ parameters
${A}_{{T}_{{\rm{X}}}}$	${ \mathcal U }(0.1,20)$ keV
${A}_{{M}_{\mathrm{ICM}}}$	${ \mathcal U }({10}^{12},2\times {10}^{14})$ ${M}_{\odot }$
${A}_{{Y}_{{\rm{X}}}}$	${ \mathcal U }(5\times {10}^{12},2\times {10}^{15})$ keV ${M}_{\odot }$
${A}_{{L}_{{\rm{X}}}}$	${ \mathcal U }(2\times {10}^{43},1.2\times {10}^{45})$ ergs s−1
${B}_{{ \mathcal X }}$	${ \mathcal U }(0.1,3.5)$
${C}_{{ \mathcal X }}$	${ \mathcal U }(-4,4)$
${\sigma }_{\mathrm{ln}{ \mathcal X }}$	${ \mathcal U }(0.005,1.5)$
${\gamma }_{{ \mathcal X }}$	${ \mathcal U }(-4,4)$
${\delta }_{{ \mathcal X }}$	${ \mathcal U }(-4,4)$

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The priors we adopt on the SZE observable–mass relation are shown in Table 2, where ${ \mathcal N }(\mu ,{\sigma }^{2})$ corresponds to a Gaussian with mean μ and dispersion σ. These SZE ζ–mass parameter constraints emerge from a joint cosmology and mass calibration analysis that uses as input (1) the SPT cluster distribution in ξ and z (i.e., the number counts), (2) mass information from externally weak lensing calibrated ${Y}_{{\rm{X}}}$ measurements for 82 systems, and (3) external cosmological parameter constraints (for a more extensive discussion of SPT mass calibration, see, e.g., Bocquet et al. 2015; Chiu et al. 2018). For the baseline priors listed above, the external cosmological priors include a prior on the Hubble parameter (Riess et al. 2011) and a prior on the baryon density parameter from big bang nucleosynthesis (Cooke et al. 2014).

Although the mass calibration presented in de Haan et al. (2016) includes information from Chandra X-ray observations of 82 clusters, we stress that the mass information is dominated by the cluster distribution in ξ and redshift (i.e., the halo mass function information). That is, the ζ–mass–redshift relation used to calculate SPT-SZ masses does not simply follow the employed ${Y}_{{\rm{X}}} \mbox{-} \mathrm{mass}\mbox{--}\mathrm{redshift}$ relation, because it is a subdominant component of the mass information. Moreover, we adopt the resulting posteriors of the ζ–mass relation as the priors in this work, effectively marginalizing over the systematic uncertainties of all ingredients used in calibrating the cluster mass. Modeling these priors as independent Gaussian distributions is appropriate, given the lack of strong covariances in the joint parameter constraints presented in de Haan et al. (2016, see Figure 5). It is important to note that the correlated intrinsic scatter between the mass proxies of SZE and X-ray does not impact the mass calibration with the current sample size (de Haan et al. 2016; Dietrich et al. 2017), so we can use the existing ζ–mass–redshift relation with marginalized systematic uncertainties to investigate the X-ray observable-to-mass scaling relations.

To foreshadow an additional set of results that we present, we also adopt a separate set of priors derived from the second results column of Table 3 in de Haan et al. (2016), which also includes an external cosmological prior coming from BAO distance measurements (Anderson et al. 2014). This set of results is consistent with the baseline results, but has smaller uncertainties (because the cosmological uncertainties typically dominate the posterior distributions of the SZE ζ–mass parameters) and has a shift of ΔC_SZ = +0.3 that translates into a corresponding shift in the redshift trend parameters in the X-ray scaling relations.

Because we adopt similar four-parameter scaling relations for both the SZE and X-ray observables, we denote the targeted X-ray scaling relation (e.g., Equation (10)) as ${r}_{{ \mathcal X }}=({A}_{{ \mathcal X }},{B}_{{ \mathcal X }},{C}_{{ \mathcal X }},{\sigma }_{\mathrm{ln}{ \mathcal X }})$ and the one used for estimating ${M}_{500}$ as ${r}_{\zeta }=({A}_{\mathrm{SZ}},{B}_{\mathrm{SZ}},{C}_{\mathrm{SZ}},{\sigma }_{\mathrm{ln}\zeta })$. The notation ${r}_{{ \mathcal X }}$ can be similarly extended to the five-parameter scaling relations for the X-ray observables, for which we define the functional forms in Section 4.1.

We stress that the cluster masses in our work include corrections for selection biases (e.g., Eddington bias, Malmquist bias), and therefore they reflect the unbiased distribution of cluster mass ${M}_{500}$ given the observable ξ and redshift z measured for each SZE-selected cluster.

We measure the temperature, metallicity, and luminosity by fitting the spectra extracted in the apertures of the core-included region (cin, $r\lt {R}_{500}$) and the core-excised region (cex, $0.15{R}_{500}\lt r\lt {R}_{500}$) with a single temperature thermal apec model. The best-fit core-included temperatures (${T}_{{\rm{X}},\mathrm{cin}}$), metallicity (${Z}_{{\rm{X}},\mathrm{cin}}$), and luminosities (${L}_{{\rm{X}},\mathrm{cin}}$) and core-excised temperatures (${T}_{{\rm{X}},\mathrm{cex}}$), metallicity (${Z}_{{\rm{X}},\mathrm{cex}}$), and luminosities (${L}_{{\rm{X}},\mathrm{cex}}$) are given in Table 3. In some clusters, the statistics of the observations are too poor to allow a determination of the global metallicity. In these cases, the metallicity is fixed to $0.3{Z}_{\odot }$ (Tozzi et al. 2003; McDonald et al. 2016). The metallicity constraints and their evolution with redshift in this sample are extensively discussed in McDonald et al. (2016).

Table 3.  Measurements of X-Ray Observables and Cluster Masses

	${R}_{500}$	${L}_{{\rm{X}},\mathrm{cin}}$	${L}_{{\rm{X}},\mathrm{cin},\mathrm{bol}}$	${T}_{{\rm{X}},\mathrm{cin}}$	${Z}_{{\rm{X}},\mathrm{cin}}$	${L}_{{\rm{X}},\mathrm{cex},\mathrm{bol}}$	${L}_{{\rm{X}},\mathrm{cex}}$	${T}_{{\rm{X}},\mathrm{cex}}$	${Z}_{{\rm{X}},\mathrm{cex}}$	${M}_{\mathrm{ICM}}$	${Y}_{{\rm{X}},\mathrm{cin}}$	${M}_{500}$
Cluster	(kpc)	(1044erg s−1)	(1044erg s−1)	(keV)	(Z⊙)	(1044erg s−1)	(1044erg s−1)	(keV)	(Z⊙)	(${10}^{13}{M}_{\odot }$)	(${10}^{14}{M}_{\odot }\mathrm{keV}$)	(${10}^{14}{M}_{\odot })$
SPT-CLJ0114-4123	1241	3.33 ± 0.28	11.60 ± 1.58	${5.62}_{-0.53}^{+0.47}$	0.3*	8.53 ± 0.34	2.56 ± 0.36	${5.01}_{-0.62}^{+0.86}$	0.3*	${8.02}_{-0.89}^{+0.90}$	4.50 ± 0.64	${5.86}_{-0.69}^{+0.85}$
SPT-CLJ0205-5829	759	4.64 ± 0.94	17.70 ± 2.64	${6.29}_{-1.13}^{+1.34}$	${0.31}_{-0.17}^{+0.19}$	13.70 ± 3.05	3.73 ± 1.18	${6.07}_{-0.65}^{+2.17}$	${0.30}_{-0.12}^{+0.29}$	${5.27}_{-0.61}^{+0.61}$	3.31 ± 0.75	${4.37}_{-0.55}^{+0.59}$
SPT-CLJ0217-5245	1110	1.40 ± 0.15	6.19 ± 0.58	${10.43}_{-1.64}^{+4.66}$	0.3*	5.35 ± 1.22	1.18 ± 0.23	${8.13}_{-1.95}^{+2.95}$	0.3*	${4.40}_{-0.40}^{+0.41}$	4.59 ± 1.45	${4.01}_{-0.61}^{+0.71}$
SPT-CLJ0225-4155	1144	3.57 ± 0.25	12.60 ± 1.07	${6.0}_{-0.33}^{+0.23}$	${0.21}_{-0.17}^{+0.27}$	9.04 ± 0.55	2.47 ± 0.11	${6.54}_{-0.41}^{+0.27}$	${0.19}_{-0.13}^{+0.27}$	${6.76}_{-0.80}^{+0.81}$	4.06 ± 0.52	${4.33}_{-0.64}^{+0.72}$
SPT-CLJ0230-6028	909	3.39 ± 0.63	11.10 ± 0.94	${4.81}_{-0.77}^{+0.70}$	0.3*	7.36 ± 0.43	2.24 ± 0.48	${4.86}_{-1.12}^{+1.20}$	0.3*	${4.83}_{-1.04}^{+1.11}$	2.32 ± 0.62	${3.43}_{-0.58}^{+0.61}$
SPT-CLJ0231-5403	921	1.50 ± 0.27	5.26 ± 0.90	${5.34}_{-1.09}^{+1.68}$	${0.50}_{-0.24}^{+0.21}$	4.43 ± 1.31	1.21 ± 0.22	${5.84}_{-1.51}^{+2.24}$	${0.49}_{-0.27}^{+0.37}$	${3.02}_{-0.48}^{+0.49}$	1.61 ± 0.49	${3.18}_{-0.62}^{+0.67}$
SPT-CLJ0232-4421	1507	7.24 ± 0.30	27.80 ± 1.24	${7.03}_{-0.41}^{+0.21}$	${0.35}_{-0.05}^{+0.06}$	14.90 ± 0.92	3.83 ± 0.2	${7.19}_{-0.50}^{+0.46}$	${0.31}_{-0.05}^{+0.05}$	${16.66}_{-0.81}^{+0.80}$	11.71 ± 0.76	${9.45}_{-1.10}^{+1.16}$
SPT-CLJ0233-5819	940	2.16 ± 0.31	7.40 ± 0.40	${5.12}_{-0.51}^{+0.50}$	${0.31}_{-0.10}^{+0.13}$	6.08 ± 0.30	1.79 ± 0.19	${5.04}_{-0.66}^{+0.63}$	${0.34}_{-0.07}^{+0.08}$	${4.41}_{-0.61}^{+0.65}$	2.26 ± 0.39	${3.70}_{-0.59}^{+0.61}$
SPT-CLJ0234-5831	1273	6.14 ± 0.46	20.00 ± 1.59	${4.67}_{-0.25}^{+0.34}$	${0.35}_{-0.05}^{+0.07}$	9.75 ± 1.26	2.86 ± 0.32	${5.21}_{-0.43}^{+0.55}$	${0.50}_{-0.17}^{+0.19}$	${6.72}_{-0.47}^{+0.47}$	3.13 ± 0.29	${6.70}_{-0.82}^{+0.84}$
SPT-CLJ0240-5946	1155	2.18 ± 0.22	9.18 ± 1.42	${8.60}_{-0.86}^{+1.17}$	${0.25}_{-0.17}^{+0.10}$	4.60 ± 0.48	1.15 ± 0.14	${7.65}_{-1.45}^{+1.89}$	0.3*	${4.30}_{-0.50}^{+0.48}$	3.69 ± 0.61	${4.85}_{-0.65}^{+0.70}$
SPT-CLJ0243-4833	1220	5.71 ± 0.55	21.50 ± 1.49	${6.26}_{-0.71}^{+0.41}$	${0.48}_{-0.13}^{+0.12}$	13.70 ± 0.71	3.47 ± 0.46	${6.85}_{-1.02}^{+0.70}$	${0.47}_{-0.17}^{+0.20}$	${9.46}_{-2.04}^{+1.99}$	5.92 ± 1.37	${6.47}_{-0.73}^{+0.90}$
SPT-CLJ0254-6051	1053	1.51 ± 0.22	5.03 ± 0.26	${5.13}_{-0.64}^{+1.12}$	${0.37}_{-0.16}^{+0.19}$	4.49 ± 0.62	1.36 ± 0.28	${5.12}_{-0.93}^{+1.18}$	${0.31}_{-0.11}^{+0.21}$	${4.49}_{-0.33}^{+0.33}$	2.30 ± 0.40	${3.86}_{-0.58}^{+0.67}$
SPT-CLJ0254-5857	1250	5.41 ± 0.21	21.70 ± 0.93	${7.62}_{-0.25}^{+0.25}$	${0.30}_{-0.04}^{+0.02}$	18.90 ± 0.54	4.73 ± 0.27	${7.60}_{-0.37}^{+0.29}$	${0.31}_{-0.06}^{+0.05}$	${10.95}_{-3.28}^{+3.73}$	8.34 ± 2.68	${6.52}_{-0.81}^{+0.81}$
SPT-CLJ0257-5732	981	0.35 ± 0.10	.97 ± 0.03	${3.48}_{-0.96}^{+1.31}$	0.3*	.86 ± 0.04	0.32 ± 0.08	${3.31}_{-0.91}^{+1.06}$	0.3*	${2.25}_{-0.40}^{+0.42}$	0.78 ± 0.29	${3.15}_{-0.69}^{+0.64}$
SPT-CLJ0304-4401	1274	3.40 ± 0.43	11.90 ± 0.95	${5.36}_{-0.33}^{+0.50}$	${0.42}_{-0.11}^{+0.11}$	9.70 ± 0.34	2.57 ± 0.19	${6.40}_{-0.87}^{+0.74}$	${0.46}_{-0.15}^{+0.16}$	${8.88}_{-0.86}^{+0.86}$	4.75 ± 0.59	${6.98}_{-0.77}^{+0.97}$
SPT-CLJ0317-5935	1022	2.34 ± 0.29	7.50 ± 0.37	${4.61}_{-0.59}^{+0.39}$	${0.29}_{-0.12}^{+0.10}$	5.76 ± 0.26	1.98 ± 0.2	${3.72}_{-0.48}^{+0.34}$	${0.28}_{-0.15}^{+0.14}$	${5.19}_{-0.67}^{+0.68}$	2.39 ± 0.40	${3.73}_{-0.61}^{+0.64}$
SPT-CLJ0330-5228	1193	8.22 ± 0.24	25.20 ± 0.84	${4.22}_{-0.06}^{+0.15}$	${0.13}_{-0.03}^{+0.03}$	25.00 ± 0.63	7.63 ± 0.24	${4.48}_{-0.10}^{+0.10}$	${0.10}_{-0.03}^{+0.02}$	${3.32}_{-0.37}^{+0.39}$	1.38 ± 0.17	${5.63}_{-0.66}^{+0.81}$
SPT-CLJ0343-5518	975	1.57 ± 0.51	4.81 ± 0.51	${4.09}_{-0.61}^{+0.90}$	0.3*	4.36 ± 0.49	1.32 ± 0.29	${4.87}_{-0.98}^{+0.91}$	${0.19}_{-0.19}^{+0.25}$	${3.07}_{-0.47}^{+0.49}$	1.25 ± 0.30	${3.52}_{-0.58}^{+0.65}$
SPT-CLJ0344-5452	827	2.02 ± 0.50	6.49 ± 0.22	${4.45}_{-0.60}^{+0.96}$	0.3*	4.52 ± 0.18	1.37 ± 0.32	${4.67}_{-0.73}^{+0.99}$	0.3*	${2.88}_{-0.43}^{+0.47}$	1.28 ± 0.30	${3.89}_{-0.54}^{+0.58}$
SPT-CLJ0354-5904	1063	1.60 ± 0.17	5.28 ± 0.26	${4.72}_{-0.49}^{+0.62}$	${0.54}_{-0.16}^{+0.18}$	4.66 ± 0.83	1.37 ± 0.25	${5.20}_{-0.77}^{+0.83}$	${0.40}_{-0.21}^{+0.21}$	${4.02}_{-0.35}^{+0.37}$	1.89 ± 0.28	${3.83}_{-0.59}^{+0.67}$
SPT-CLJ0403-5719	1008	2.68 ± 0.26	8.30 ± 0.78	${4.16}_{-0.29}^{+0.20}$	${0.43}_{-0.11}^{+0.10}$	4.96 ± 0.37	1.56 ± 0.18	${4.26}_{-0.30}^{+0.39}$	${0.80}_{-0.20}^{+0.27}$	${3.32}_{-0.37}^{+0.39}$	1.38 ± 0.17	${3.52}_{-0.59}^{+0.66}$
SPT-CLJ0406-5455	878	1.09 ± 0.18	4.36 ± 0.30	${7.23}_{-1.35}^{+2.14}$	${0.41}_{-0.22}^{+0.26}$	3.81 ± 0.20	0.95 ± 0.13	${7.26}_{-1.92}^{+2.89}$	${0.63}_{-0.21}^{+0.42}$	${2.57}_{-0.31}^{+0.32}$	1.86 ± 0.50	${3.28}_{-0.54}^{+0.63}$
SPT-CLJ0417-4748	1164	6.28 ± 0.36	23.60 ± 1.28	${6.17}_{-0.34}^{+0.48}$	${0.45}_{-0.08}^{+0.10}$	14.10 ± 2.55	3.6 ± 0.44	${6.78}_{-0.84}^{+1.49}$	${0.51}_{-0.13}^{+0.18}$	${6.26}_{-0.48}^{+0.48}$	3.85 ± 0.39	${6.22}_{-0.71}^{+0.85}$
SPT-CLJ0438-5419	1385	8.36 ± 0.37	34.80 ± 2.03	${8.09}_{-0.39}^{+0.48}$	${0.33}_{-0.04}^{+0.04}$	19.90 ± 1.78	5.09 ± 0.31	${7.06}_{-0.41}^{+0.61}$	${0.28}_{-0.09}^{+0.10}$	${12.53}_{-0.52}^{+0.52}$	10.13 ± 0.69	${8.68}_{-1.03}^{+1.03}$
SPT-CLJ0510-4519	1323	2.98 ± 0.14	10.40 ± 0.68	${5.93}_{-0.21}^{+0.28}$	${0.24}_{-0.05}^{+0.05}$	6.29 ± 0.43	1.81 ± 0.08	${5.87}_{-0.36}^{+0.37}$	${0.36}_{-0.09}^{+0.06}$	${7.06}_{-0.30}^{+0.30}$	4.18 ± 0.24	${5.73}_{-0.72}^{+0.85}$
SPT-CLJ0516-5430	1292	4.38 ± 0.22	17.40 ± 0.91	${7.64}_{-0.23}^{+0.23}$	${0.28}_{-0.03}^{+0.03}$	16.00 ± 1.04	4.38 ± 0.22	${7.55}_{-0.25}^{+0.25}$	${0.24}_{-0.04}^{+0.02}$	${9.64}_{-2.38}^{+2.51}$	7.37 ± 1.88	${5.96}_{-0.74}^{+0.78}$
SPT-CLJ0522-4818	1062	1.54 ± 0.18	5.67 ± 0.60	${6.26}_{-0.63}^{+1.02}$	${0.41}_{-0.13}^{+0.16}$	3.70 ± 0.49	0.96 ± 0.12	${6.90}_{-1.01}^{+1.49}$	${0.48}_{-0.23}^{+0.27}$	${2.73}_{-0.30}^{+0.31}$	1.71 ± 0.29	${3.37}_{-0.71}^{+0.75}$
SPT-CLJ0549-6205	1470	11.6 ± 0.48	48.80 ± 1.31	${8.60}_{-0.35}^{+0.42}$	${0.38}_{-0.05}^{+0.06}$	21.20 ± 1.27	4.96 ± 0.25	${8.97}_{-0.42}^{+1.27}$	${0.54}_{-0.13}^{+0.17}$	${11.60}_{-0.45}^{+0.46}$	9.97 ± 0.59	${9.66}_{-1.03}^{+1.31}$
SPT-CLJ0559-5249	1072	3.53 ± 0.46	13.60 ± 1.99	${6.64}_{-1.17}^{+1.17}$	${0.28}_{-0.15}^{+0.10}$	10.90 ± 1.34	2.87 ± 0.44	${6.59}_{-0.96}^{+1.40}$	${0.25}_{-0.19}^{+0.18}$	${7.09}_{-1.17}^{+1.14}$	4.71 ± 1.13	${5.03}_{-0.63}^{+0.69}$
SPT-CLJ0611-5938	992	1.27 ± 0.20	4.12 ± 0.77	${4.62}_{-0.78}^{+0.73}$	${0.33}_{-0.19}^{+0.22}$	3.00 ± 0.21	1.11 ± 0.21	${4.30}_{-0.76}^{+0.74}$	${0.45}_{-0.25}^{+0.30}$	${3.21}_{-0.87}^{+0.86}$	1.48 ± 0.40	${3.13}_{-0.67}^{+0.68}$
SPT-CLJ0615-5746	1098	15.9 ± 1.48	88.80 ± 6.94	${14.16}_{-1.32}^{+2.04}$	${0.65}_{-0.25}^{+0.22}$	56.90 ± 7.09	10.9 ± 1.58	${12.50}_{-1.99}^{+1.60}$	${0.36}_{-0.21}^{+0.26}$	${11.19}_{-1.05}^{+1.06}$	15.86 ± 2.40	${8.69}_{-0.99}^{+1.07}$
SPT-CLJ0637-4829	1258	1.01 ± 0.14	3.80 ± 0.75	${6.53}_{-1.38}^{+1.50}$	${0.21}_{-0.09}^{+0.35}$	2.75 ± 0.53	0.87 ± 0.11	${5.01}_{-0.95}^{+1.69}$	${0.24}_{-0.10}^{+0.40}$	6.29 ± 0.13	4.11 ± 0.94	${5.66}_{-0.68}^{+0.81}$
SPT-CLJ0638-5358	1459	6.43 ± 0.15	26.10 ± 0.99	${8.38}_{-0.29}^{+0.24}$	${0.32}_{-0.28}^{+0.36}$	13.10 ± 0.80	3.24 ± 0.14	${8.44}_{-0.48}^{+0.86}$	${0.33}_{-0.24}^{+0.39}$	9.77 ± 0.21	8.19 ± 0.32	${9.42}_{-1.09}^{+1.18}$
SPT-CLJ0658-5556	1664	13.3 ± 0.46	62.40 ± 3.48	${12.40}_{-0.54}^{+0.32}$	${0.28}_{-0.02}^{+0.03}$	45.00 ± 4.22	9.43 ± 0.51	${13.44}_{-0.32}^{+1.14}$	${0.29}_{-0.07}^{+0.07}$	${20.08}_{-1.04}^{+1.05}$	24.90 ± 1.56	${12.70}_{-1.38}^{+1.64}$
SPT-CLJ2011-5725	1067	2.12 ± 0.15	6.49 ± 0.14	${4.13}_{-0.13}^{+0.15}$	${0.39}_{-0.07}^{+0.08}$	4.44 ± 0.16	1.52 ± 0.1	${3.65}_{-0.21}^{+0.21}$	${0.56}_{-0.12}^{+0.14}$	${3.39}_{-0.34}^{+0.37}$	1.40 ± 0.15	${3.35}_{-0.64}^{+0.65}$
SPT-CLJ2017-6258	986	1.55 ± 0.21	4.55 ± 0.32	${3.65}_{-0.76}^{+0.61}$	${0.25}_{-0.08}^{+0.09}$	3.65 ± 0.31	1.29 ± 0.28	${3.39}_{-0.75}^{+0.81}$	0.3*	${7.78}_{-0.10}^{+0.09}$	4.56 ± 0.16	${4.03}_{-0.64}^{+0.68}$
SPT-CLJ2022-6323	1073	0.94 ± 0.14	3.90 ± 0.84	${8.45}_{-3.29}^{+4.91}$	0.3*	3.02 ± 0.78	0.83 ± 0.15	${5.73}_{-1.03}^{+2.28}$	0.3*	${3.45}_{-0.54}^{+0.55}$	2.91 ± 1.48	${3.80}_{-0.57}^{+0.67}$
SPT-CLJ2023-5535	1309	3.28 ± 0.26	14.80 ± 1.54	${10.93}_{-1.55}^{+2.00}$	${0.43}_{-0.25}^{+0.65}$	10.00 ± 0.99	2.47 ± 0.26	${8.31}_{-1.15}^{+1.69}$	${0.29}_{-0.06}^{+0.54}$	${8.43}_{-0.68}^{+0.71}$	9.22 ± 1.68	${6.49}_{-0.71}^{+0.81}$
SPT-CLJ2030-5638	1018	1.06 ± 0.17	2.99 ± 0.21	${3.46}_{-0.33}^{+0.39}$	0.3*	2.42 ± 0.14	0.81 ± 0.13	${3.88}_{-0.45}^{+0.45}$	0.3*	${2.62}_{-0.29}^{+0.27}$	0.91 ± 0.13	${3.35}_{-0.62}^{+0.64}$
SPT-CLJ2031-4037	1389	5.02 ± 0.24	20.50 ± 0.56	${8.14}_{-0.75}^{+1.22}$	${0.29}_{-0.09}^{+0.10}$	10.50 ± 0.41	2.8 ± 0.21	${6.67}_{-1.01}^{+0.89}$	${0.38}_{-0.16}^{+0.18}$	${7.79}_{-0.40}^{+0.41}$	6.34 ± 0.83	${7.95}_{-0.95}^{+0.99}$
SPT-CLJ2032-5627	1204	3.29 ± 0.12	10.80 ± 0.57	${4.99}_{-0.18}^{+0.19}$	${0.24}_{-0.05}^{+0.03}$	9.66 ± 0.42	2.92 ± 0.11	${5.07}_{-0.34}^{+0.28}$	${0.23}_{-0.03}^{+0.03}$	${5.93}_{-0.37}^{+0.36}$	2.95 ± 0.21	${4.77}_{-0.63}^{+0.71}$
SPT-CLJ2040-5725	803	3.68 ± 0.59	10.80 ± 0.88	${3.71}_{-0.26}^{+0.32}$	${0.23}_{-0.05}^{+0.10}$	7.18 ± 0.29	2.24 ± 0.39	${4.61}_{-0.52}^{+0.58}$	0.3*	${3.68}_{-0.34}^{+0.34}$	1.36 ± 0.10	${3.23}_{-0.51}^{+0.59}$
SPT-CLJ2040-4451	649	1.92 ± 0.57	5.83 ± 1.43	${3.75}_{-0.65}^{+0.85}$	0.3*	8.39 ± 1.79	2.92 ± 1.59	${4.78}_{-1.51}^{+1.50}$	${0.53}_{-0.27}^{+0.26}$	${3.34}_{-0.54}^{+0.55}$	1.17 ± 0.31	${3.31}_{-0.54}^{+0.53}$
SPT-CLJ2056-5459	889	1.91 ± 0.27	6.01 ± 0.25	${4.22}_{-0.45}^{+0.42}$	${0.52}_{-0.17}^{+0.22}$	5.07 ± 0.09	1.62 ± 0.16	${4.19}_{-0.65}^{+0.46}$	${0.63}_{-0.26}^{+0.29}$	${3.64}_{-0.30}^{+0.31}$	1.53 ± 0.20	${3.36}_{-0.57}^{+0.60}$
SPT-CLJ2106-5844	963	12.2 ± 0.85	55.80 ± 5.16	${9.43}_{-1.67}^{+0.70}$	0.3*	47.50 ± 3.48	10.5 ± 0.97	${9.19}_{-1.06}^{+0.88}$	0.3*	${11.73}_{-0.39}^{+0.38}$	11.05 ± 1.43	${7.14}_{-0.83}^{+0.86}$
SPT-CLJ2109-4626	737	1.81 ± 0.46	5.24 ± 0.66	${3.52}_{-0.36}^{+0.51}$	${0.51}_{-0.13}^{+0.31}$	3.50 ± 0.13	1.22 ± 0.18	${3.46}_{-0.53}^{+0.73}$	${0.85}_{-0.56}^{+0.55}$	${2.55}_{-0.44}^{+0.45}$	0.90 ± 0.19	${2.68}_{-0.56}^{+0.65}$
SPT-CLJ2124-6124	1113	1.01 ± 0.27	3.54 ± 0.69	${5.66}_{-1.05}^{+1.56}$	0.3*	4.71 ± 0.22	1.44 ± 0.17	${4.72}_{-0.65}^{+0.93}$	${0.24}_{-0.16}^{+0.25}$	${5.84}_{-0.79}^{+0.84}$	3.30 ± 0.89	${4.60}_{-0.63}^{+0.66}$
SPT-CLJ2130-6458	1151	1.88 ± 0.27	5.84 ± 0.64	${4.26}_{-0.42}^{+0.55}$	${0.26}_{-0.15}^{+0.18}$	3.83 ± 0.12	1.2 ± 0.13	${4.44}_{-0.52}^{+0.73}$	${0.30}_{-0.25}^{+0.14}$	${4.48}_{-0.85}^{+0.92}$	1.90 ± 0.43	${4.33}_{-0.58}^{+0.71}$
SPT-CLJ2131-4019	1232	6.00 ± 0.33	24.30 ± 1.98	${7.64}_{-0.59}^{+0.50}$	${0.37}_{-0.05}^{+0.09}$	14.10 ± 1.90	3.27 ± 0.34	${8.79}_{-2.15}^{+1.79}$	${0.43}_{-0.15}^{+0.18}$	${8.17}_{-0.68}^{+0.67}$	6.24 ± 0.68	${6.25}_{-0.72}^{+0.88}$
SPT-CLJ2136-6307	804	1.53 ± 0.60	3.57 ± 0.29	${2.58}_{-0.62}^{+1.03}$	0.3*	1.42 ± 0.39	3.54 ± 0.53	${2.04}_{-0.40}^{+0.79}$	0.3*	${4.23}_{-1.30}^{+1.41}$	1.09 ± 0.49	${3.24}_{-0.51}^{+0.60}$
SPT-CLJ2138-6008	1283	2.83 ± 0.14	10.50 ± 0.52	${6.59}_{-0.29}^{+0.42}$	${0.18}_{-0.08}^{+0.09}$	6.54 ± 0.16	1.9 ± 0.13	${5.43}_{-0.46}^{+0.66}$	${0.29}_{-0.15}^{+0.13}$	${6.25}_{-0.39}^{+0.41}$	4.11 ± 0.34	${6.10}_{-0.76}^{+0.79}$
SPT-CLJ2145-5644	1188	4.07 ± 0.38	15.40 ± 1.58	${6.36}_{-0.71}^{+0.57}$	${0.48}_{-0.14}^{+0.17}$	10.30 ± 1.21	2.85 ± 0.3	${5.82}_{-0.72}^{+0.71}$	${0.41}_{-0.11}^{+0.22}$	${8.64}_{-1.51}^{+1.54}$	5.49 ± 1.11	${5.82}_{-0.67}^{+0.82}$
SPT-CLJ2146-4633	921	2.94 ± 0.47	9.73 ± 0.26	${4.64}_{-0.24}^{+0.42}$	${0.63}_{-0.13}^{+0.07}$	9.04 ± 1.21	2.74 ± 0.65	${4.98}_{-0.43}^{+0.18}$	${0.83}_{-0.17}^{+0.09}$	${4.92}_{-1.65}^{+1.75}$	2.28 ± 0.80	${4.89}_{-0.65}^{+0.66}$
SPT-CLJ2200-6245	1067	1.08 ± 0.33	2.63 ± 0.30	${2.26}_{-0.38}^{+0.33}$	${0.32}_{-0.14}^{+0.07}$	1.99 ± 0.09	0.84 ± 0.24	${2.10}_{-0.31}^{+0.37}$	${0.30}_{-0.19}^{+0.55}$	${3.38}_{-0.85}^{+0.86}$	0.76 ± 0.22	${3.79}_{-0.57}^{+0.67}$
SPT-CLJ2248-4431	1633	16.8 ± 0.96	77.70 ± 6.91	${11.46}_{-0.63}^{+0.28}$	${0.26}_{-0.06}^{+0.02}$	44.00 ± 3.95	9.44 ± 0.48	${11.90}_{-0.69}^{+0.97}$	${0.22}_{-0.07}^{+0.07}$	${19.46}_{-0.89}^{+0.90}$	22.30 ± 1.30	${13.05}_{-1.44}^{+1.64}$
SPT-CLJ2332-5358	1137	2.24 ± 0.19	8.56 ± 0.99	${7.63}_{-0.97}^{+0.97}$	0.3*	5.75 ± 1.19	1.58 ± 0.25	${6.17}_{-0.84}^{+0.93}$	0.3*	${4.01}_{-0.57}^{+0.58}$	3.06 ± 0.58	${4.63}_{-0.60}^{+0.68}$
SPT-CLJ2337-5942	1112	8.2 ± 0.81	36.40 ± 2.42	${9.11}_{-1.01}^{+0.60}$	0.3*	24.40 ± 1.35	5.59 ± 0.76	${8.60}_{-1.33}^{+1.35}$	0.3*	${9.21}_{-1.05}^{+1.05}$	8.38 ± 1.21	${7.05}_{-0.81}^{+0.89}$
SPT-CLJ2341-5119	902	4.76 ± 0.39	19.40 ± 1.38	${7.47}_{-0.88}^{+0.71}$	${0.14}_{-0.08}^{+0.09}$	12.10 ± 0.31	3.48 ± 0.59	${5.34}_{-0.87}^{+1.51}$	0.3*	${5.26}_{-0.36}^{+0.37}$	3.93 ± 0.50	${4.94}_{-0.58}^{+0.68}$
SPT-CLJ2344-4243	1330	26.8 ± 0.47	145.00 ± 3.29	${14.89}_{-0.17}^{+0.32}$	${1.05}_{-0.02}^{+0.02}$	45.30 ± 2.24	9.09 ± 0.29	${12.23}_{-0.65}^{+0.31}$	${0.45}_{-0.04}^{+0.06}$	${14.83}_{-0.30}^{+0.31}$	22.08 ± 0.58	${9.60}_{-1.09}^{+1.20}$

Note. X-ray observables of the sample measured in core-included ($\mathrm{cin}$, $r\lt {R}_{500}$) and core-excised ($\mathrm{cex}$, $0.15{R}_{500}\lt r\lt {R}_{500}$) apertures. Parameters marked with * are fixed to the indicated values. From left to right are the cluster name, ${R}_{500}$, bolometric and soft-band luminosity, emission-weighted mean temperature, and metallicity, given first for the core-included and then for the core-excised measurements. Measured ICM masses ${M}_{\mathrm{ICM}}$, X-ray-derived integrated Compton-y ${Y}_{{\rm{X}},\mathrm{cin}}$, and halo mass ${M}_{500}$ determined from the SZE observations are then listed for each cluster.

Download table as:  ASCIITypeset images: 1 2

The X-ray surface brightness is extracted from background-subtracted, exposure-corrected images within 1.5 ${R}_{500}$ in the fitting environment Sherpa in CIAO (Freeman et al. 2001; Doe et al. 2007). We fit a two-dimensional β profile to determine the cluster centroids within software package Sherpa. This method also allows for precise measurements of X-ray centroids of the clusters in the sample. The X-ray surface brightness ${S}_{{\rm{X}}}$ (in units of erg s⁻¹ cm⁻² steradian⁻¹), produced by thermal bremsstrahlung and line emission, is expressed as

$$\begin{eqnarray}&&{S}_{{\rm{X}}}=\displaystyle \frac{1}{4\pi {\left(1+z\right)}^{4}}\int {n}_{{\rm{e}}}{n}_{{\rm{H}}}{{\rm{\Lambda }}}_{\mathrm{eH}}({T}_{{\rm{X}}},Z)\ {dl},\end{eqnarray} \tag{ 4 }$$

where ${{\rm{\Lambda }}}_{\mathrm{eH}}({T}_{{\rm{X}}},Z)$ is the band averaged emissivity, which is dependent on plasma temperature and metallicity, dl is the integral along the line of sight, and z is the cluster redshift. The electron and hydrogen number densities (n_e and n_H) have only weak dependence on plasma temperature and assumed abundance when derived from surface brightness in the 0.5–2 keV band (Mohr et al. 1999).

We fit the surface brightness profiles using an analytic density model (Bulbul et al. 2010, Bu10 hereafter):

$$\begin{eqnarray}&&{n}_{{\rm{e}}}(r)={n}_{{\rm{e}}0}{\left(\displaystyle \frac{1}{\left(\beta -2\right)}\displaystyle \frac{{\left(1+r/{r}_{{\rm{s}}}\right)}^{\beta -2}-1}{r/{r}_{{\rm{s}}}{\left(1+r/{r}_{{\rm{s}}}\right)}^{\beta -2}}\right)}^{n},\end{eqnarray} \tag{ 5 }$$

where ${n}_{{\rm{e}}0}$ is the normalization of the electron density profile, ${r}_{{\rm{s}}}$ is the scale radius, n is the slope of the density profile, and β is the slope of the dark matter potential. We assume that the dark matter halos of the SPT-selected sample follow the Navarro–Frenk–White (NFW) profile with a slope of $\beta =2$ (Navarro et al. 1997) and provide a good description of the electron density (Bu10; Bonamente et al. 2012). Application of the L'Hospital rule gives an electron density profile under the assumption of an NFW-like matter profile:

$$\begin{eqnarray}&&{n}_{{\rm{e}}}(r)={\tau }_{\mathrm{cool}}(r)\ {n}_{{\rm{e}}0}{\left(\displaystyle \frac{\mathrm{ln}(1+r/{r}_{{\rm{s}}})}{r/{r}_{{\rm{s}}}}\right)}^{n}.\end{eqnarray} \tag{ 6 }$$

The Bu10 density profile has been used for fitting both X-ray and SZE data (Landry et al. 2013; Romero et al. 2017). The core taper function ${\tau }_{\mathrm{cool}}(r)$ is used to fit the surface brightness profiles of cool-core clusters (Vikhlinin et al. 2006):

$$\begin{eqnarray}&&{\tau }_{\mathrm{cool}}(r)=\displaystyle \frac{\alpha +{\left(r/{r}_{\mathrm{cool}}\right)}^{\gamma }}{1+{\left(r/{r}_{\mathrm{cool}}\right)}^{\gamma }}.\end{eqnarray} \tag{ 7 }$$

For non-cool-core clusters, the parameter α is set to 1.

The Bu10 density model is projected along the line of sight and fit to the surface brightness profile obtained from background-subtracted and exposure-corrected X-ray images. The fitting is performed using a Markov chain Monte Carlo (MCMC) sampler within the emcee package in Python (Foreman-Mackey et al. 2013). The best-fit parameter values and their 1σ uncertainties for non-cool-core clusters (e.g., ${n}_{{\rm{e}}0}$, n, ${r}_{{\rm{s}}}$) and cool-core clusters (e.g., ${n}_{{\rm{e}}0}$, n, α, ${r}_{{\rm{s}}}$, and r_cool) are determined using a maximum-likelihood method. The surface brightness profile fit to the MOS observations of a non-cool-core cluster SPT-CL J0304−4401 and a cool-core cluster SPT-CL J2217−6509 are shown in Figure 2.

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To compute the ICM mass of a cluster within a given aperture of ${R}_{500}$, we use the enclosed ICM mass obtained by integrating the best-fit 3D ICM density profile:

$$\begin{eqnarray}&&{M}_{\mathrm{ICM}}=4\pi {\mu }_{{\rm{e}}}{m}_{{\rm{p}}}{\int }_{0}^{{R}_{500}}{n}_{{\rm{e}}}(r)\ {r}^{2}\ {dr},\end{eqnarray} \tag{ 8 }$$

where ${\mu }_{{\rm{e}}}$ is the mean molecular weight of the electrons, and m_p is the proton mass. The ICM mass measurements within ${R}_{500}$ for each cluster in the sample are given in Table 3. We use ${\mu }_{{\rm{e}}}=1.17$ when determining the cluster ICM mass. The integrated Compton-y parameter is the product of the ICM mass and temperature:

$$\begin{eqnarray}&&{Y}_{{\rm{X}}}={M}_{\mathrm{ICM}}\times {T}_{{\rm{X}}},\end{eqnarray} \tag{ 9 }$$

where ${T}_{{\rm{X}}}$ is the projected temperature measured within a 2D aperture either with or without the core, and ${M}_{\mathrm{ICM}}$ is integrated within a 3D sphere of radius ${R}_{500}$.

As described already in Section 3.1, there are remaining uncertainties in the SZE-based halo masses. This means that the extraction radius ${R}_{500}$ used above is not a single value for each cluster, but a distribution of values. To include these uncertainties, we marginalize over them when studying the X-ray scaling relations. As described in Section 4.3, this means that we evaluate the X-ray observable at a range of radii ${R}_{500}$ consistent with the SZE observable ξ and redshift z. Specifically, we use the best-fit density profile to calculate the ICM mass in each fit iteration. For ${L}_{{\rm{X}}}$ we extract the X-ray luminosity at a single radius—the baseline ${R}_{500}$—in this work, because we find the change in ${L}_{{\rm{X}}}$ due to the radial range in the surface brightness fit is negligible. For ${T}_{{\rm{X}}}$ we have in general too few photons to make spectral fits beyond the baseline ${R}_{500}$, so we adopt only a single radius for the temperature extraction. This means that for ${Y}_{{\rm{X}}}$ we are properly including the variation of the ${M}_{\mathrm{ICM}}$ component with ${R}_{500}$ but not the ${T}_{{\rm{X}}}$ component.

We use three functional forms to characterize the X-ray observable–mass–redshift scaling relations. In all cases, there are pivot masses and redshifts that should be chosen to be near the median values of the sample to reduce artificial covariances between the amplitude parameter and the mass and redshift trend parameters. For the X-ray observable ${ \mathcal X }$-to-mass scaling relations, the pivot mass and pivot redshift are ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{piv}}=0.45$, respectively.

The first form, similar to that used in Vikhlinin et al. (2009a) and many publications since, is defined as follows:

$$\begin{eqnarray}&&{ \mathcal X }={A}_{{ \mathcal X }}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{B}_{{ \mathcal X }}}{\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{{C}_{{ \mathcal X }}},\end{eqnarray} \tag{ 10 }$$

where the normalization and trend parameters in mass and redshift are ${A}_{{ \mathcal X }}$, ${B}_{{ \mathcal X }}$, and ${C}_{{ \mathcal X }}$, respectively, for the observable ${ \mathcal X }$. Note that the redshift trend in this formulation is expressed as a function of the Hubble parameter $H(z)={H}_{0}E(z)$, where ${E}^{2}{(z)={{\rm{\Omega }}}_{{\rm{M}}}(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}$ at late times in a flat ΛCDM universe. That is, in this parameterization, the redshift evolution of the X-ray observable–mass relation is attributed to an explicit cosmological dependence. In the case where the redshift evolution has a different cosmological dependence than adopted here (e.g., the evolution is non-self-similar), then assuming this form will lead to biases in cosmological analyses. We refer to Equation (10) as Form I hereafter.

The second form includes the expected self-similar evolution of the observable with redshift, which depends on the cosmologically dependent evolution of the critical density, while modeling departures of the observable from self-similar evolution with a function ${(1+z)}^{{\gamma }_{{ \mathcal X }}}$. With this form, we are adopting the view that the departures from self-similar evolution do not have a clearly understood cosmological dependence. Therefore, we model the departures with the cosmologically agnostic form ${(1+z)}^{{\gamma }_{{ \mathcal X }}}$ that has been adopted in many previous works (e.g., Lin et al. 2006). This form is defined as follows:

$$\begin{eqnarray}&&{ \mathcal X }={A}_{{ \mathcal X }}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{B}_{{ \mathcal X }}}{\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{{C}_{{ \mathcal X },\mathrm{SS}}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{{\gamma }_{{ \mathcal X }}},\end{eqnarray} \tag{ 11 }$$

where the normalization and mass trend are similarly characterized by the parameters ${A}_{{ \mathcal X }}$ and ${B}_{{ \mathcal X }}$, respectively. The redshift trend is modeled with ${C}_{{ \mathcal X },\mathrm{SS}}$ fixed to the self-similar expectation along with the factor ${(1+z)}^{{\gamma }_{{ \mathcal X }}}$ to describe the departure of the redshift trend from the self-similar expectation. For instance, ${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$ for the X-ray temperature–mass–redshift relation. In this way, the parameter ${\gamma }_{{ \mathcal X }}$ directly quantifies the deviation from the self-similar redshift trend. This form of the scaling relation is easily distinguishable, because it has a parameter ${\gamma }_{{ \mathcal X }}$ rather than ${C}_{{ \mathcal X }}$. We refer to Equation (11) as Form II hereafter.

The third form we adopt is much like Form II above, but it includes a cross term between cluster mass and redshift to characterize the possibility of having a redshift-dependent mass trend. Specifically, the third functional form is

$$\begin{eqnarray}&&{ \mathcal X }={A}_{{ \mathcal X }}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{B}_{{ \mathcal X }}^{{\prime} }}{\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{{C}_{{ \mathcal X },\mathrm{SS}}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{{\gamma }_{{ \mathcal X }}},\end{eqnarray} \tag{ 12 }$$

where the mass trend ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$ has a characteristic value of ${B}_{{ \mathcal X }}$ at the pivot redshift and an additional rate of variation ${\delta }_{{ \mathcal X }}$ with redshift. The normalization parameter ${A}_{{ \mathcal X }}$ and the redshift trend ${\gamma }_{{ \mathcal X }}$ are defined as in Form II. Specifically, the redshift trend is structured to capture the departures from the expected self-similar redshift evolution of the X-ray observable. It is worth mentioning that ${\delta }_{{ \mathcal X }}=0$ or statistically consistent with zero indicates there is no evidence for a redshift-dependent mass trend. We refer to Equation (12) as Form III hereafter.

For all three functional forms, we adopt log-normal intrinsic scatter in the observables at fixed mass, defined as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}{ \mathcal X }}\equiv {\sigma }_{\mathrm{ln}\left({ \mathcal X }| {M}_{500}\right)}.\end{eqnarray} \tag{ 13 }$$

In this way, each observable ${ \mathcal X }$-to-mass scaling relation is parameterized by either $({A}_{{ \mathcal X }},{B}_{{ \mathcal X }},{C}_{{ \mathcal X }},{\sigma }_{\mathrm{ln}{ \mathcal X }})$, $({A}_{{ \mathcal X }},{B}_{{ \mathcal X }},{C}_{{ \mathcal X },\mathrm{SS}},{\gamma }_{{ \mathcal X }},{\sigma }_{\mathrm{ln}{ \mathcal X }})$, or $({A}_{{ \mathcal X }},{B}_{{ \mathcal X }},{C}_{{ \mathcal X },\mathrm{SS}},{\gamma }_{{ \mathcal X }},{\sigma }_{\mathrm{ln}{ \mathcal X }},{\delta }_{{ \mathcal X }})$, and we denote these parameter sets by ${r}_{{ \mathcal X }}$ hereafter for simplicity. Note that the expected self-similar redshift evolution parameter ${C}_{{ \mathcal X },\mathrm{SS}}$ is fixed, so the first two parameterizations have four free parameters, and the last parameterization has five.

We briefly introduce the likelihood and fitting framework below and refer the reader to previous publications for more details (Liu et al. 2015; Chiu et al. 2016c). This likelihood is designed to obtain the parameters of the targeted X-ray observable–mass–redshift scaling relation (e.g., ${r}_{{ \mathcal X }}$) for a given sample that is selected using another observable (e.g., the SPT signal-to-noise ratio ξ), for which the observable–mass–redshift relation is already known (e.g., Equation (3) used in this work). Specifically, the ith term in the likelihood ${{ \mathcal L }}_{i}$ contains the probability of obtaining the X-ray observable ${{ \mathcal X }}_{i}$ for the ith cluster at redshift z_i with SZE signal-to-noise rato ${\xi }_{i}$, given the scaling relations ${r}_{{ \mathcal X }}$ and ${r}_{\xi }$:

$$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{i}({r}_{{ \mathcal X }},{r}_{\zeta }) & = & P({{ \mathcal X }}_{i}| {\xi }_{i},{z}_{i},{r}_{{ \mathcal X }},{r}_{\zeta })\\ & = & \displaystyle \frac{\displaystyle \int {{dM}}_{500}P({{ \mathcal X }}_{i},{\xi }_{i}| {z}_{i},{r}_{{ \mathcal X }},{r}_{\zeta })\ n({M}_{500},{z}_{i})}{\displaystyle \int {{dM}}_{500}P({\xi }_{i}| {z}_{i},{r}_{{ \mathcal X }},{r}_{\zeta })\ n({M}_{500},{z}_{i})},\end{array}\end{eqnarray} \tag{ 14 }$$

where $n({M}_{500},{z}_{i})$ is the mass function whose inclusion allows the Eddington bias correction to be included when determining the mass corresponding to the SZE observable ξ at redshift z. The integrals are over the relevant range of the mass M₅₀₀ used in the mass function. The Tinker et al. (2008) mass function is used with fixed cosmological parameters in calculations of $n({M}_{500},{z}_{i})$, although given the mass range of the SPT sample the use of a mass function determined from hydrodynamical simulations would make no difference (Bocquet et al. 2016).

We ignore correlated scatter between the X-ray observable ${ \mathcal X }$ and SZE observable ξ in our analysis. This will not impact our results, because in previous studies of the SPT-SZ sample no evidence of correlated scatter between the X-ray ${Y}_{{\rm{X}}}$, X-ray-based ${M}_{\mathrm{ICM}}$, and the SZE signal-to-noise ratio has emerged (de Haan et al. 2016; Dietrich et al. 2017). In future analyses with much larger X-ray samples, we plan to explore again the evidence for correlated scatter in the X-ray and SZE properties of the clusters. As discussed in Liu et al. (2015), in this limit of no correlated X-ray and SZE scatter, there are no selection effects to be accounted for in the X-ray scaling relation.

Based on Bayes's theorem, the best-fit scaling relation parameters ${r}_{{ \mathcal X }}$ and ${r}_{{ \mathcal X }}$ are obtained by maximizing the probability

$$\begin{eqnarray}&&P({r}_{{ \mathcal X }},{r}_{\zeta })\propto { \mathcal L }({r}_{{ \mathcal X }},{r}_{\zeta }){ \mathcal P }({r}_{{ \mathcal X }},{r}_{\zeta }),\end{eqnarray} \tag{ 15 }$$

where ${ \mathcal P }({r}_{{ \mathcal X }},{r}_{\zeta })$ is the prior on ${r}_{{ \mathcal X }}$ and ${r}_{\zeta }$ (see Table 2), and the likelihood ${ \mathcal L }({r}_{{ \mathcal X }},{r}_{\zeta })$ is evaluated using Equation (14) as follows:

$$\begin{eqnarray}&&{ \mathcal L }({r}_{{ \mathcal X }},{r}_{\zeta })=\displaystyle \prod _{i=1}^{{N}_{\mathrm{cl}}}\,{{ \mathcal L }}_{i}({r}_{{ \mathcal X }},{r}_{\zeta }),\end{eqnarray} \tag{ 16 }$$

where i runs over the ${N}_{\mathrm{cl}}$ clusters. We use the Python package emcee to explore the parameter space. The intrinsic scatter and measurement uncertainties of ${\xi }_{i}$ for each cluster are taken into account while evaluating Equation (16). We have verified that this likelihood recovers unbiased scaling relation parameters by testing it against large mocks (>1300 clusters). Moreover, it has been further optimized in the goal of obtaining the parameters of scaling relations in a high dimensional space (Chiu et al. 2016c).

We note that in each iteration of the chain, we use the current value of ${R}_{500}$ for each cluster to recalculate the ${M}_{\mathrm{ICM}}$ (see Section 3.2). For the temperature ${T}_{{\rm{X}}}$ and the luminosity ${L}_{{\rm{X}}}$, we extract only once at the ${R}_{500}$ appropriate for the model $\zeta \mbox{--}{M}_{500}\mbox{--}z$ parameter values in our priors (see Table 2), because the impact of adjusting ${R}_{500}$ at each iteration is small.

As discussed in Section 3.1, we marginalize over the parameters of the ζ–${M}_{500}$–z relation while fitting the X-ray observable ${ \mathcal X }$–${M}_{500}$–z relations (i.e., ${r}_{\zeta }$ and ${r}_{{ \mathcal X }}$, respectively). Specifically, we adopt informative priors on ${r}_{\zeta }$, which have been obtained in a joint cosmology and mass calibration analysis described in de Haan et al. (2016). Our baseline priors on ${r}_{\zeta }$ are listed in Table 2 and correspond to the posterior distributions for each parameter reported in the first results column of de Haan et al. (2016, Table 3). We explore a second set of priors on ${r}_{\zeta }$ corresponding to the posterior parameter distributions from the second results column of de Haan et al. (2016, Table 3), and we report those results in Table 5.

Our approach allows us to effectively marginalize over the remaining uncertainties in our ${M}_{500}$ estimates. In each iteration of the chain, each cluster has a different halo mass ${M}_{500}$ and associated radius ${R}_{500}$. The X-ray observables ${M}_{\mathrm{ICM}}$ and ${Y}_{{\rm{X}}}$ defined in Section 3.2 are then extracted at this radius ${R}_{500}$ and used to determine the likelihood for this iteration. Final uncertainties on the X-ray observable scaling relation parameters ${r}_{{ \mathcal X }}$ therefore include not only those due to measurement uncertainties but also due to the (largely systematic) uncertainties in the underlying halo masses.

In the fitting, we apply the uniform priors listed in Table 2 on ${r}_{{ \mathcal X }}$ during the likelihood maximization. With this approach, we evaluate the scaling relation Forms I, II, and III. In Table 2 we present the parameter in the first column and the form of the prior in column two. In this table, ${ \mathcal N }$ denotes a normal or Gaussian distribution, and ${ \mathcal U }$ represents a uniform or flat distribution between the two values presented.

In a final step, we report the parameters of the Form III relation also while fixing the parameters of ${r}_{\zeta }$ to the best-fit values derived in de Haan et al. (2016; i.e., the central values listed in Table 2). Through the comparison of the results when marginalizing over the posterior distributions with those when fixing the ${r}_{\zeta }$ parameters, we can gauge the impact of the remaining systematic uncertainties on the SZE-based halo masses.

We note that the de Haan et al. (2016) priors we adopt when estimating cluster halo masses are derived using the cluster mass function information (distribution of clusters in signal-to-noise ratio ξ and z) together with a sample of 82 ${Y}_{{\rm{X}}}$ measurements that have been calibrated to mass first through hydrostatic masses (Vikhlinin et al. 2009a) and later through weak lensing (Hoekstra et al. 2015). We note that the mass information from the ${Y}_{{\rm{X}}}$ measurements is subdominant in comparison to that from the mass function information (see prior and posterior distributions on ${r}_{\xi }$ parameters in Figure 5 of de Haan et al. 2016).

Moreover, the follow-up studies using weak lensing masses of 32 SPT-SZ clusters (Dietrich et al. 2017) and using dynamical masses from 110 SPT-SZ clusters (Capasso et al. 2019) have provided independent mass calibration of the $\zeta \mbox{--}{M}_{500}\mbox{--}z$ relation and cross-checks of cluster masses, and they are all in excellent agreement with the cluster masses in de Haan et al. (2016), as we adopt for our study. Ongoing work with the Dark Energy Survey weak lensing will further improve our knowledge of the $\zeta \mbox{--}{M}_{500}\mbox{--}z$ relation, allowing even more accurate cluster halo mass estimates in the future (e.g., Stern et al. 2018).

Before cluster mass measurements were available, the emission-weighted ICM temperature was viewed as the most robust mass proxy available and was therefore employed in early studies of cluster scaling relations (Smith et al. 1979; Mohr & Evrard 1997; Mushotzky & Scharf 1997; Mohr et al. 1999). Early attempts to constrain the ${T}_{{\rm{X}}}$–mass relation using hydrostatic masses were carried out first for low-temperature clusters using spatially resolved spectroscopy from ROSAT (David et al. 1993) and then later for clusters with a broad range of temperatures using the ASCA observatory (Finoguenov et al. 2001).

By combining the virial condition (${GM}/R\sim T$) and the definition of the virial radius (${R}_{500}\sim {[{M}_{500}/{\rho }_{\mathrm{crit}}]}^{1/3}$), one can show that the self-similar expectation for the ${T}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relation is

$$\begin{eqnarray}&&{T}_{{\rm{X}}}\propto {M}_{500}^{2/3}\ E{\left(z\right)}^{2/3}.\end{eqnarray} \tag{ 17 }$$

As noted in Section 4.2, we examine the scaling relations with and without the core region, and for three different scaling relation functional forms.

5.1.1. Parameter Constraints

We present the parameters associated with the ${T}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ and ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relations in Table 4. The marginalized posteriors of the parameters and joint parameter confidence regions using the core-excised and core-included observables are contained in Figures 5 and 6. Here we provide the best-fit ${T}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ and ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relations for the Form II scaling relation. For the core-included X-ray emission-weighted mean temperature,

$$\begin{eqnarray}\begin{array}{rcl}{T}_{{\rm{X}},\mathrm{cin}} & = & {6.41}_{-0.66}^{+0.64}\,\mathrm{keV}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{0.80}_{-0.12}^{+0.09}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{2}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.36}_{-0.26}^{+0.27}},\end{array}\end{eqnarray} \tag{ 18 }$$

with the intrinsic scatter of ${0.18}_{-0.04}^{+0.05}$. For the core-excised X-ray temperature ${T}_{{\rm{X}},\mathrm{cex}}$, the best-fit relation is

$$\begin{eqnarray}\begin{array}{rcl}{T}_{{\rm{X}},\mathrm{cex}} & = & {6.09}_{-0.51}^{+0.76}\,\mathrm{keV}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{0.80}_{-0.08}^{+0.11}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{2}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.33}_{-0.28}^{+0.23}},\end{array}\end{eqnarray} \tag{ 19 }$$

with intrinsic scatter of ${0.13}_{-0.04}^{+0.05}$. As for all relations, the mass and redshift pivots are ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{piv}}=0.45$.

Table 4.  Best-fit Parameters for the Various X-Ray Observable–Halo Mass–Redshift Scaling Relations

Scaling relation	${A}_{{ \mathcal X }}$	${B}_{{ \mathcal X }}$	${C}_{{ \mathcal X }}$	${\sigma }_{\mathrm{ln}{ \mathcal X }}$	${\gamma }_{{ \mathcal X }}$	${\delta }_{{ \mathcal X }}$
${T}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${6.36}_{-0.64}^{+0.70}$	0.80 ± 0.10	0.33 ± 0.27	0.18 ± 0.04	⋯	⋯
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${6.41}_{-0.66}^{+0.64}$	${0.79}_{-0.12}^{+0.08}$	$\tfrac{2}{3}$	${0.18}_{-0.04}^{+0.05}$	$-{0.36}_{-0.26}^{+0.27}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}(\tfrac{1+z}{1+{z}_{\mathrm{piv}}})$	${6.48}_{-0.69}^{+0.58}$	${0.79}_{-0.10}^{+0.09}$	$\tfrac{2}{3}$	${0.18}_{-0.04}^{+0.04}$	$-{0.22}_{-0.35}^{+0.29}$	${0.81}_{-0.46}^{+0.56}$
III with fixed SZE parameters	6.41 ± 0.22	${0.78}_{-0.09}^{+0.08}$	$\tfrac{2}{3}$	${0.16}_{-0.03}^{+0.04}$	$-{0.20}_{-0.25}^{+0.23}$	${0.77}_{-0.47}^{+0.57}$
${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${6.17}_{-0.63}^{+0.71}$	${0.83}_{-0.10}^{+0.09}$	${0.28}_{-0.23}^{+0.28}$	${0.13}_{-0.05}^{+0.05}$	⋯	⋯
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${6.09}_{-0.51}^{+0.76}$	${0.80}_{-0.08}^{+0.11}$	$\tfrac{2}{3}$	${0.13}_{-0.05}^{+0.04}$	$-{0.33}_{-0.28}^{+0.23}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${6.31}_{-0.69}^{+0.57}$	${0.81}_{-0.08}^{+0.09}$	$\tfrac{2}{3}$	${0.13}_{-0.04}^{+0.05}$	$-{0.30}_{-0.28}^{+0.27}$	${0.35}_{-0.41}^{+0.53}$
III with fixed SZE parameters	${6.17}_{-0.17}^{+0.20}$	${0.79}_{-0.06}^{+0.10}$	$\tfrac{2}{3}$	${0.12}_{-0.03}^{+0.04}$	$-{0.29}_{-0.25}^{+0.19}$	${0.38}_{-0.43}^{+0.41}$
${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=0$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${6.80}_{-0.87}^{+1.09}$	${1.260}_{-0.11}^{+0.10}$	${0.17}_{-0.29}^{+0.28}$	${0.12}_{-0.08}^{+0.04}$	⋯	⋯
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{0}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${7.37}_{-1.35}^{+0.76}$	${1.26}_{-0.09}^{+0.12}$	0	${0.10}_{-0.07}^{+0.04}$	${0.18}_{-0.31}^{+0.30}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${7.09}_{-1.11}^{+0.91}$	${1.26}_{-0.09}^{+0.11}$	0	${0.10}_{-0.07}^{+0.05}$	${0.16}_{-0.31}^{+0.33}$	${0.16}_{-0.44}^{+0.47}$
III with fixed SZE parameters	${7.02}_{-0.27}^{+0.21}$	${1.26}_{-0.07}^{+0.09}$	0	0.07 ± 0.05	${0.20}_{-0.22}^{+0.20}$	${0.26}_{-0.51}^{+0.42}$
${Y}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{5}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	4.70 ± 1.1	${2.00}_{-0.14}^{+0.19}$	${0.44}_{-0.54}^{+0.46}$	${0.15}_{-0.12}^{+0.05}$	⋯	⋯
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	4.60 ± 1.1	${1.99}_{-0.15}^{+0.17}$	$\tfrac{2}{3}$	${0.16}_{-0.12}^{+0.05}$	$-{0.21}_{-0.45}^{+0.50}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${4.52}_{-0.91}^{+1.23}$	${2.00}_{-0.17}^{+0.16}$	$\tfrac{2}{3}$	${0.16}_{-0.10}^{+0.07}$	$-{0.28}_{-0.40}^{+0.56}$	${0.77}_{-0.53}^{+0.74}$
III with fixed SZE parameters	${4.57}_{-0.21}^{+0.25}$	${1.98}_{-0.10}^{+0.16}$	$\tfrac{2}{3}$	${0.07}_{-0.05}^{+0.09}$	$-{0.09}_{-0.32}^{+0.34}$	${1.01}_{-0.71}^{+0.61}$
${Y}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{5}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	4.31 ± 0.96	${2.01}_{-0.13}^{+0.20}$	0.44 ± 0.49	${0.16}_{-0.11}^{+0.04}$	⋯	⋯
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${4.50}_{-1.1}^{+1.0}$	${2.02}_{-0.17}^{+0.16}$	$\tfrac{2}{3}$	${0.11}_{-0.08}^{+0.07}$	$-{0.17}_{-0.50}^{+0.47}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${4.54}_{-0.98}^{+1.09}$	${2.01}_{-0.14}^{+0.18}$	$\tfrac{2}{3}$	${0.13}_{-0.08}^{+0.07}$	$-{0.20}_{-0.47}^{+0.52}$	${0.55}_{-0.56}^{+0.78}$
III with fixed SZE parameters	${4.40}_{-0.22}^{+0.23}$	${2.04}_{-0.15}^{+0.10}$	$\tfrac{2}{3}$	${0.04}_{-0.03}^{+0.08}$	$-{0.14}_{-0.32}^{+0.33}$	${0.67}_{-0.74}^{+0.66}$
${L}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=2$
I: ${ \mathcal X }(z)\propto E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${4.20}_{-0.92}^{+0.91}$	${1.93}_{-0.20}^{+0.16}$	${1.72}_{-0.46}^{+0.53}$	${0.25}_{-0.10}^{+0.10}$	⋯	⋯
II: ${ \mathcal X }(z)\propto E{\left(z\right)}^{2}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${4.12}_{-0.94}^{+0.91}$	${1.89}_{-0.13}^{+0.23}$	2	${0.27}_{-0.12}^{+0.08}$	$-{0.20}_{-0.49}^{+0.51}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${4.39}_{-0.99}^{+0.82}$	${1.93}_{-0.18}^{+0.19}$	2	${0.28}_{-0.11}^{+0.07}$	$-{0.13}_{-0.46}^{+0.63}$	${0.71}_{-0.72}^{+0.89}$
III with fixed SZE parameters	${3.96}_{-0.24}^{+0.22}$	${1.95}_{-0.18}^{+0.14}$	2	${0.24}_{-0.06}^{+0.08}$	$-{0.02}_{-0.48}^{+0.32}$	${0.84}_{-0.80}^{+0.81}$
${L}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=2$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${2.84}_{-0.53}^{+0.60}$	${1.60}_{-0.13}^{+0.17}$	${1.86}_{-0.43}^{+0.47}$	${0.27}_{-0.10}^{+0.07}$	⋯	—
II: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{2}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${2.84}_{-0.50}^{+0.53}$	${1.60}_{-0.15}^{+0.16}$	2	${0.27}_{-0.11}^{+0.07}$	$-{0.10}_{-0.42}^{+0.47}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${2.89}_{-0.51}^{+0.55}$	${1.56}_{-0.16}^{+0.18}$	2	${0.28}_{-0.08}^{+0.07}$	${0.10}_{-0.60}^{+0.35}$	${0.30}_{-0.62}^{+0.86}$
III with fixed SZE parameters	${2.66}_{-0.11}^{+0.17}$	${1.60}_{-0.16}^{+0.14}$	2	${0.26}_{-0.05}^{+0.05}$	$-{0.01}_{-0.42}^{+0.33}$	${0.60}_{-0.75}^{+0.79}$
${L}_{{\rm{X}},\mathrm{cin},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{4}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{7}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${15.4}_{-3.3}^{+2.8}$	${2.15}_{-0.19}^{+0.24}$	${1.90}_{-0.53}^{+0.55}$	${0.29}_{-0.13}^{+0.09}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{\tfrac{7}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${14.8}_{-2.7}^{+3.5}$	${2.19}_{-0.17}^{+0.21}$	$\tfrac{7}{3}$	${0.29}_{-0.13}^{+0.08}$	$-{0.14}_{-0.57}^{+0.62}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${13.8}_{-3.9}^{+3.2}$	${2.12}_{-0.18}^{+0.23}$	$\tfrac{7}{3}$	${0.31}_{-0.12}^{+0.08}$	$-{0.26}_{-0.60}^{+0.58}$	${1.53}_{-1.11}^{+0.31}$
III with fixed SZE parameters	${14.94}_{-1.01}^{+0.65}$	${2.24}_{-0.15}^{+0.13}$	$\tfrac{7}{3}$	${0.22}_{-0.10}^{+0.08}$	$-{0.17}_{-0.33}^{+0.43}$	${1.67}_{-0.97}^{+0.28}$
${L}_{{\rm{X}},\mathrm{cex},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{4}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{7}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	${10.2}_{-2.1}^{+2.6}$	${1.89}_{-0.18}^{+0.17}$	${2.01}_{-0.44}^{+0.53}$	${0.29}_{-0.12}^{+0.07}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{\tfrac{7}{3}}{\left(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	10.7 ± 2.3	${1.88}_{-0.17}^{+0.19}$	$\tfrac{7}{3}$	${0.27}_{-0.13}^{+0.07}$	$-{0.26}_{-0.43}^{+0.53}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${10.4}_{-2.2}^{+2.4}$	${1.86}_{-0.16}^{+0.21}$	$\tfrac{7}{3}$	${0.28}_{-0.09}^{+0.07}$	${0.02}_{-0.58}^{+0.48}$	${0.76}_{-0.71}^{+0.76}$
III with fixed SZE parameters	${9.93}_{-0.49}^{+0.58}$	${1.90}_{-0.18}^{+0.13}$	$\tfrac{7}{3}$	${0.25}_{-0.06}^{+0.07}$	$-{0.18}_{-0.32}^{+0.48}$	${0.80}_{-0.57}^{+0.93}$

Note. The first column contains the scaling relation identifier. The next six columns show best-fit parameters and associated fully marginalized 1σ uncertainties of the scaling relation normalization ${A}_{{ \mathcal X }}$, mass trend ${B}_{{ \mathcal X }}$, E(z) redshift trend ${C}_{{ \mathcal X }}$, log-normal intrinsic scatter ${\sigma }_{\mathrm{ln}{ \mathcal X }}$, departure from self-similarity in redshift trend ${(1+z)}^{{\gamma }_{{ \mathcal X }}}$, and redshift dependence ${\delta }_{{ \mathcal X }}$ of the mass trend.

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The mass trend parameters of the ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relations using Forms I, II, and III (see Equations (10)–(12)) are ${0.83}_{-0.10}^{+0.09}$, ${0.80}_{-0.08}^{+0.11}$, and ${0.81}_{-0.08}^{+0.09}$, respectively, showing consistency at better than the 1σ confidence level. All derived mass trends are consistent with the self-similar expectation at the $\approx 1.6\sigma$ level. Note that there is neither a significant redshift dependence in the mass trend (${\delta }_{{T}_{{\rm{X}},\mathrm{cex}}}={0.35}_{-0.41}^{+0.53}$) nor a strong deviation (${\gamma }_{{T}_{{\rm{X}},\mathrm{cex}}}=-{0.33}_{-0.28}^{+0.23}$) from a self-similar redshift trend. In addition, as can be seen in Figure 5, there is a mild degeneracy between the slope and redshift evolution, such that the mass trend can be pushed back closer to the self-similar expectation with a stronger deviation of the redshift evolution from self-similarity. Fixing the SZE ζ–mass relation does not change the best-fit parameters but reduces the uncertainty on the normalizations ${A}_{{T}_{{\rm{X}}}}$ by a factor of between two and three (see Table 4). This indicates that only the normalizations of the ${T}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relations are dominated by the systematic uncertainties in the SZE-based halo masses.

This behavior is clearly visible in Figure 5, where we show the joint confidence contours and fully marginalized posterior distributions for each of the ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relation variables for all three forms of the relation. In the figure, the self-similar parameter expectations are marked with dashed red lines. The preference for the mass trend to deviate by ≈2σ and the redshift trend to deviate by ≈1σ can be seen both in the joint parameter constraint panels and the fully marginalized single-parameter distributions. Evidence for parameter covariance is clearest in the parameter pair ${C}_{{T}_{{\rm{X}},\mathrm{cex}}}$ and ${\delta }_{{T}_{{\rm{X}},\mathrm{cex}}}$. We include the four $\zeta \mbox{--}{M}_{500}\mbox{--}z$ relation parameters to show the strong positive correlations among the corresponding parameters A_SZ and ${A}_{{T}_{{\rm{X}},\mathrm{cex}}}$, B_SZ and ${B}_{{T}_{{\rm{X}},\mathrm{cex}}}$, and C_SZ and ${C}_{{T}_{{\rm{X}},\mathrm{cex}}}$. Also, a strong negative correlation among the scatter parameters ${\sigma }_{\mathrm{ln}\zeta }$ and ${\sigma }_{\mathrm{ln}{T}_{{\rm{X}},\mathrm{cex}}}$ is indicated by the same parameters. This is as expected and follows from the importance of the SZE-based masses in the ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relation and the fact that the quadrature sum of the scatter in $\zeta \mbox{--}{M}_{500}\mbox{--}z$ and ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ is constrained by the measurement-error-corrected scatter in the data about the best-fit ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relation. In all other cases that follow, we exclude the $\zeta \mbox{--}{M}_{500}\mbox{--}z$ parameters from the plots to conserve space, but the correlations persist, as expected. Finally, this plot makes clear (black lines) that the improvement from fixing the $\zeta \mbox{--}{M}_{500}\mbox{--}z$ scaling relation parameters at their best-fit values is a dramatic decrease in the uncertainties of ${A}_{{T}_{{\rm{X}},\mathrm{cex}}}$ but only has a modest impact on the other parameters.

For the core-included ${T}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ relation, the mass and redshift trends as well as the normalization are consistent with those for the core-excluded case within the quoted 1σ uncertainties. The discernible difference in the two relations comes from intrinsic scatter. The log-normal intrinsic scatter for the core-excised observable at fixed mass ${\sigma }_{\mathrm{ln}{T}_{{\rm{X}},\mathrm{cex}}}$ is ≈0.12, approximately a factor of 1.5 smaller than in the case of the core-included observable. Figure 6 contains the joint and single parameter constraints for this relation.

5.1.2. Comparison to Previous Results

We show the mass and redshift trends in the ${T}_{{\rm{X}}}$ scaling relations in the top panels of Figures 3 and 4, respectively. In both figures, the core-excised measurements are on the left and core-included on the right. In the case of the mass trends, all X-ray observable measurements are corrected to the pivot redshift ${z}_{\mathrm{piv}}=0.45$ using the best-fit redshift trend from the Form II scaling relation, and in the case of the redshift trends, all are corrected to the pivot mass ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ using the best-fit mass trend from Form II. Also shown are the self-similar expectations (red dashed line), and the gray region corresponds to the 1σ allowed region for the relation. One of the outlier clusters in the sample is SPT-CLJ0217-5245, whose temperature is high compared to the expected temperature from the luminosity scaling relations (see Section 5.4). We note that the noise-dominated spectrum of this cluster makes it challenging to determine its temperature from the shape of the continuum bremsstrahlung emission.

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Our analysis shows steeper mass trends than those measured before. Vikhlinin et al. (2009a) found a self-similar slope (${B}_{{T}_{x}}=0.65\pm 0.03$) in the ${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relation for X-ray-selected clusters observed with Chandra in the redshift range $0.02\lt z\lt 0.9$ and with hydrostatic mass measurements in the range ${10}^{14}{M}_{\odot }\lesssim {M}_{500}\lesssim {10}^{15}{M}_{\odot }$, while a similar mass slope of ${B}_{{T}_{X}}=0.67\pm 0.07$ was reported in Arnaud et al. (2005) covering the XMM-Newton observations of low-redshift clusters at $z\lesssim 0.15$ with a hydrostatic mass range of $9\times {10}^{13}{M}_{\odot }\lesssim {M}_{500}\lesssim 8.4\times {10}^{14}{M}_{\odot }$. Our result for the mass slope is $1.6\sigma$ and $1.1\sigma$ away, respectively, from these results. In Mahdavi et al. (2013), the mass slopes of ${0.51}_{-0.16}^{+0.42}$ and ${0.70}_{-0.08}^{+0.11}$ were derived using weak lensing and hydrostatic masses, respectively, for a sample of 50 galaxy clusters at $z\lesssim 0.5$ with a mass range similar to those we study here; no tension with our results is seen. A recent result based on the 100 brightest clusters selected in the XXL survey (Lieu et al. 2016) gave slopes of $\approx {0.56}_{-0.10}^{+0.12}$, and this slope became ≈0.60 ± 0.05 if combined with the lower mass groups. Given their preference for a shallower than self-similar relation, these results are in tension at 1.5σ and 2.0σ with ours. In Mantz et al. (2016), a mass trend of ≈0.66 ± 0.05 was reported for X-ray-selected clusters with redshift range $0.07\lt z\lt 1.06$ and mass range $3\times {10}^{14}{M}_{\odot }\leqslant {M}_{500}\lesssim 2\times {10}^{15}{M}_{\odot }$. This result is in 1.4σ tension with ours.

In the upper left panel of Figure 7, we further compare our results of ${T}_{{\rm{X}},\mathrm{cex}}$ to the simulated clusters at z = 0.1 from the C-Eagle cosmological hydrodynamical simulations (Barnes et al. 2017), together with the nearby clusters from Pratt et al. (2009) and the clusters at z ≲ 0.5 from Vikhlinin et al. (2009a). In addition, we overplot the best-fit relation from Mantz et al. (2016) as a blue dashed line and the self-similar prediction with the normalization anchored to our best-fit value (gray dashed line).

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It is important to note that there exist nonnegligible systematic differences among these studies, especially in the estimation of cluster masses. In Pratt et al. (2009), the cluster mass is estimated using the ${Y}_{{\rm{X}},\mathrm{cex}}$-${M}_{500}$ relation derived from hydrostatic mass estimates in nearby relaxed clusters. For the sake of consistency, we take the ${Y}_{{\rm{X}}}$-inferred masses from Vikhlinin et al. (2009a). We scale up the cluster masses from Pratt et al. (2009) and Vikhlinin et al. (2009a) by a factor of 1.12 to account for the offsets between hydrostatic masses and our masses (Bocquet et al. 2015). For Mantz et al. (2016), the cluster masses are calibrated using weak lensing, for which we do not expect significant systematic offsets with our results (de Haan et al. 2016; Dietrich et al. 2017; Schrabback et al. 2018). For the simulated clusters in Barnes et al. (2017), we directly use the true halo masses.

To make the figure, we rescale each reported ${T}_{{\rm{X}},\mathrm{cex}}$ from the literature studies to the pivotal redshift ${z}_{\mathrm{piv}}$ by multiplying by ${\left(E({z}_{\mathrm{piv}})/E(z)\right)}^{\tfrac{2}{3}}$, because we observe that the core-excised temperature is evolving as predicted by the self-similar evolution. As seen in Figure 7, our results are consistent in terms of normalization and mass trends with the simulations (Barnes et al. 2017) and other observed clusters over the common mass range, except that we observe a shift in normalization in comparison with Mantz et al. (2016).

In summary, the previous results show mass trends that are in agreement with the self-similar prediction (i.e., the value of 2/3), while the fully marginalized posterior of our mass trend parameter is steeper than self-similar at ≈1.6σ and in tension with these previous results at a similar or lower level. One difference between our work and these others is that we simultaneously fit the mass and redshift trends of the scaling relation, exploiting the fact that our SZE-selected sample is approximately mass selected out to redshift ≈1.4. Most of these previous analyses have assumed self-similar redshift evolution, because their data sets tend to cover very different mass ranges at low and high redshift, introducing strong degeneracies in the mass and redshift trend parameters. Thus, our sample provides the first direct constraint on the deviation from self-similarity for massive clusters out to z ≈ 1.4 that accounts for both mass and redshift trends. Only the analysis of larger samples with improved halo mass estimates will allow us to definitively determine departures from self-similarity in the mass trends of the ${T}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relations.

The ${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relation and its redshift trends have important implications for ICM mass fractions and baryon fraction within clusters, because a majority of the baryons reside within the ICM (Lin et al. 2003; Chiu et al. 2016b). The expression for the self-similar scaling of the ${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ relation is

$$\begin{eqnarray}&&{M}_{\mathrm{ICM}}\propto {M}_{500}.\end{eqnarray} \tag{ 20 }$$

That is, in the simplest universe with no feedback or radiative processes, the ICM mass fraction would be expected to be identical in halos of all masses and at all redshifts.

5.2.1. Parameter Constraints

We present the best-fit parameters of the ${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ relations using the scaling relation Forms I, II, and III (Equations (10)–(12)) in Table 4, and the marginalized posteriors of the single and joint parameter constraints are presented in Figure 8. We do not present core-excised values for the ICM mass, because the central core region contains only a negligible portion of the ICM. The best-fit ${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relation using Form II is

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{ICM}} & = & {7.37}_{-1.35}^{+0.76}\times {10}^{13}{M}_{\odot }{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{1.26}_{-0.09}^{+0.12}}\\ & & \times {\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{{0.18}_{-0.31}^{+0.30}},\end{array}\end{eqnarray} \tag{ 21 }$$

with intrinsic scatter of ${0.10}_{-0.07}^{+0.04}$. As before, the mass and redshift pivots are ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{piv}}=0.45$.

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We find that the mass trend parameter is ${B}_{{M}_{\mathrm{ICM}}}={1.26}_{-0.09}^{+0.12}$, which is steeper than the self-similar scaling at the ≈2.9σ level. Although the uncertainties are large, there are no significant redshift trends observed. The data provide no evidence for a redshift-dependent mass slope, given that ${\delta }_{{M}_{\mathrm{ICM}}}$ of ${0.16}_{-0.44}^{+0.47}$. The normalization ${A}_{{M}_{\mathrm{ICM}}}$ of ${7.37}_{-1.35}^{+0.76}\times {10}^{13}{M}_{\odot }$ suggests an ICM mass fraction of ≈(16.0 ± 2)% at the pivot mass and redshift. A consistent picture is suggested by all three functional forms. Furthermore, fixing the SZE parameters ${r}_{\zeta }$ does not shift the best-fit parameters, but reduces the uncertainty of the normalization by a factor of four and the uncertainties on the mass and redshift trends by a factor of two.

5.2.2. Comparison to Previous Results

The X-ray estimated integrated pressure ${Y}_{{\rm{X}}}$ is of interest because of relatively low intrinsic scatter, its connection to the SZE observable, and its relative insensitivity to the influence of feedback from AGN and star formation (Kravtsov et al. 2006; Nagai et al. 2007; Bonamente et al. 2008; Vikhlinin et al. 2009a; Andersson et al. 2011; Benson et al. 2013). The self-similar expectation of the ${Y}_{{\rm{X}}}$ to mass scaling relations is

$$\begin{eqnarray}&&{Y}_{{\rm{X}}}\propto {M}_{500}^{5/3}\ E{\left(z\right)}^{2/3},\end{eqnarray} \tag{ 22 }$$

which results from ${Y}_{{\rm{X}}}$ being the product of ${M}_{\mathrm{ICM}}$ and ${T}_{{\rm{X}}}$ together with the dependence of the ${T}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relation on the evolution of the critical density.

5.3.1. Parameter Constraints

Similar to previous sections, we present the ${Y}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relation derived using both the core-included ${Y}_{{\rm{X}},\mathrm{cin}}$ and core-excised ${Y}_{{\rm{X}},\mathrm{cex}}$ observables for the scaling relations Forms I, II, and III (Equations (10)–(12), respectively). The best-fit parameters and uncertainties of the scaling relations are listed in Table 4 for both ${Y}_{{\rm{X}}}$ observables, and the marginalized posteriors of the single and joint parameters are presented in Figure 9.

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The best-fit ${Y}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relation using functional Form II in the core-included case is

$$\begin{eqnarray}\begin{array}{rcl}{Y}_{{\rm{X}},\mathrm{cin}} & = & {4.6}_{-1.1}^{+1.1}\times {10}^{14}\,{M}_{\odot }\mathrm{keV}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{1.99}_{-0.15}^{+0.17}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{2}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.21}_{-0.45}^{+0.50}},\end{array}\end{eqnarray} \tag{ 23 }$$

with intrinsic scatter of ${0.16}_{-0.12}^{+0.05}$. For the core-excised observable ${Y}_{{\rm{X}},\mathrm{cex}}$, the best-fit relation is

$$\begin{eqnarray}\begin{array}{rcl}{Y}_{{\rm{X}},\mathrm{cex}} & = & {4.50}_{-1.10}^{+1.00}\times {10}^{14}\,{M}_{\odot }\mathrm{keV}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{2.02}_{-0.17}^{+0.16}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{2}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.17}_{-0.50}^{+0.47}},\end{array}\end{eqnarray} \tag{ 24 }$$

with intrinsic scatter ${0.11}_{-0.08}^{+0.07}$. As for all other cases, the mass and redshift pivots are ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{piv}}=0.45$.

For ${Y}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relations, we observe the mass trend ${B}_{{Y}_{{\rm{X}}}}$ that is in tension with the self-similar prediction at the ≈2σ level for the core-included and core-excised X-ray observables. On the other hand, the redshift trends for all three functional forms are consistent with self-similarity within the quoted 1σ uncertainty. There is no evidence for a redshift-dependent mass trend. Fixing the SZE parameters ${r}_{\zeta }$ leads to no major parameter shifts, but does reduce the parameter uncertainties on the normalization by a factor of about four and, interestingly, leads to a reduction in the estimate of the intrinsic scatter. With intrinsic scatter at the ≈10% level as with the ${M}_{\mathrm{ICM}}$, the ${Y}_{{\rm{X}}}$ observable with or without core excision offers an outstanding single cluster mass proxy.

5.3.2. Comparison to Previous Results

We extract the X-ray luminosity obtained from the core-included aperture of $\lt {R}_{500}$ in the 0.5–2 keV band (i.e., the soft-band luminosity ${L}_{{\rm{X}}}$) and the 0.01:100 keV band (i.e., the bolometric luminosity ${L}_{{\rm{X}},\mathrm{bol}}$) to study the ${L}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relations. In previous studies, the ${L}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relations have tended to exhibit larger scatter if cluster cores are included in the analysis (Pratt et al. 2009), due to the complex cool-core phenomenon that affects the central regions of clusters. Indeed, it was argued long ago that the primary driver of the ${L}_{{\rm{X}}}$ − ${T}_{{\rm{X}}}$ relation scatter was this cool-core phenomenon (Fabian et al. 1994), and with the availability of cluster samples extending to high redshift, it was shown to be true out to $z\approx 0.8$ (O'Hara et al. 2006). Therefore, we also additionally extract the X-ray luminosities obtained from the core-excised aperture of (0.15–1)${R}_{500}$ in both soft and bolometric bands. As a result, we derive four ${L}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relations: (1) core-included and soft-band luminosity to mass ${L}_{{\rm{X}},\mathrm{cin}}$–${M}_{500}$, (2) core-included and bolometric luminosity to mass ${L}_{{\rm{X}},\mathrm{cin},\mathrm{bol}}$–${M}_{500}$, (3) core-excised and soft-band luminosity to mass ${L}_{{\rm{X}},\mathrm{cex}}$–${M}_{500}$, and (4) core-excised and bolometric luminosity to mass ${L}_{{\rm{X}},\mathrm{cex},\mathrm{bol}}$–${M}_{500}$ scaling relations. The self-similar expectation of the ${L}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ scaling relation is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}}} & \propto & {M}_{500}\ E{\left(z\right)}^{2}.\\ {L}_{{\rm{X}},\mathrm{bol}} & \propto & {M}_{500}^{4/3}\ E{\left(z\right)}^{7/3}\end{array}\end{eqnarray} \tag{ 25 }$$

for the soft-band and bolometric luminosities, respectively, where for the soft band we have assumed that the emissivity is independent of temperature (see discussion in Mohr et al. 1999).

5.4.1. Parameter Constraints

The resulting best-fit scaling relation parameters and uncertainties are listed in Table 4, and the marginalized posteriors of the single and joint parameter constraints for the core-included and core-excised observables appear in Figure 10 (0.5–2.0 keV) and Figure 11 (bolometric).

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For the core-included, soft-band 0.5–2.0 keV X-ray luminosity ${L}_{{\rm{X}},\mathrm{cin}}$, the best-fit relation is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}},\mathrm{cin}} & = & {4.12}_{-0.94}^{+0.91}\times {10}^{44}\mathrm{erg}/{\rm{s}}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{1.89}_{-0.13}^{+0.23}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{2}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.20}_{-0.49}^{+0.51}},\end{array}\end{eqnarray} \tag{ 26 }$$

with intrinsic scatter of ${0.27}_{-0.12}^{+0.08}$. For the ${L}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ relation, the best-fit relation is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}},\mathrm{cex}} & = & {2.84}_{-0.50}^{+0.53}\times {10}^{44}\mathrm{erg}/{\rm{s}}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{1.60}_{-0.15}^{+0.16}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{2}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.10}_{-0.42}^{+0.47}},\end{array}\end{eqnarray} \tag{ 27 }$$

with intrinsic scatter of ${0.27}_{-0.11}^{+0.07}$. As before, the mass and redshift pivots are ${M}_{\mathrm{piv}}=6.35\times {10}^{14}{M}_{\odot }$ and ${z}_{\mathrm{piv}}=0.45$.

The soft-band, core-excised ${L}_{{\rm{X}}}\mbox{--}{M}_{500}\mbox{--}z$ relation shows a mass trend that is $\approx 4\sigma$ higher than the self-similar trend (${B}_{{L}_{{\rm{X}}}}=1$), while the core-included relation is steeper and exhibits a tension of ≈6.8σ with the self-similar behavior. The redshift trends for both core-included and core-excised luminosities are consistent with the self-similar trend of ${C}_{{L}_{{\rm{X}}}}=2$. There is no evidence for a redshift-dependent mass slope in either soft-band ${L}_{{\rm{X}}}$ measurement. Fixing the SZE parameters ${r}_{\zeta }$ (the black curves in Figure 10) does not change the overall picture except that the uncertainties of the normalization are reduced by about a factor of four.

The characteristic luminosities at the pivot mass and redshift for core-included clusters are ≈45% higher than the core-excised luminosities, a difference of ≈1σ. Interestingly, the scatter of the two relations is similar at ≈27%.

Similarly, for the bolometric luminosities, the best-fit ${L}_{{\rm{X}},\mathrm{cin},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ and ${L}_{{\rm{X}},\mathrm{cex},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ relations are

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}},\mathrm{cin}} & = & {14.8}_{-2.7}^{+3.5}\times {10}^{44}\mathrm{erg}/{\rm{s}}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{2.19}_{-0.17}^{+0.21}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{7}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.14}_{-0.57}^{+0.62}},\end{array}\end{eqnarray} \tag{ 28 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{L}_{{\rm{X}},\mathrm{cex}} & = & {10.7}_{-2.3}^{+2.3}\times {10}^{44}\mathrm{erg}/{\rm{s}}{\left(\displaystyle \frac{{M}_{500}}{{M}_{\mathrm{piv}}}\right)}^{{1.88}_{-0.17}^{+0.19}}\\ & & \times {\left(\displaystyle \frac{E(z)}{E({z}_{\mathrm{piv}})}\right)}^{\tfrac{7}{3}}{\left(\displaystyle \frac{1+z}{1+{z}_{\mathrm{piv}}}\right)}^{-{0.26}_{-0.43}^{+0.53}},\end{array}\end{eqnarray} \tag{ 29 }$$

with intrinsic scatter of ${0.29}_{-0.13}^{+0.08}$ and ${0.27}_{-0.13}^{+0.07}$, respectively. The same pivot mass and redshift as before are used.

As expected, the bolometric luminosity relations have steeper mass trends than those of the soft-band luminosities. Similar to the soft band, the bolometric luminosity-to-mass scaling relations have mass trends that are steeper than self-similar (${B}_{{L}_{{\rm{X}}}}=\tfrac{4}{3}$) with a significance of ≈3.2σ and ≈5.1σ for the core-excised and core-included luminosities, respectively. The redshift trends of the scaling relations are all consistent with the self-similar trend ${C}_{{L}_{{\rm{X}}}}=\tfrac{7}{3}$, and there is a preference for a redshift-dependent mass trend in the core-included luminosity scaling relation. Fixing the SZE parameters ${r}_{\zeta }$ does not result in significant differences except by decreasing the uncertainties of the normalization by a factor of three to five, and it also slightly reduces the scatter in the core-included relation. Both relations exhibit intrinsic scatter at around the 27% level, which is comparable to that in the soft band.

5.4.2. Comparison to Previous Results

Currently, the redshift trend parameter on the SZE $\zeta \mbox{--}{M}_{500}\mbox{--}z$ relation is the least well constrained, and this leads to additional uncertainty in understanding the X-ray observable–mass relations. In de Haan et al. (2016, the second column of Table 3), an analysis within the context of a flat ΛCDM model was undertaken where additional external cosmological priors from BAO were added. This helped reduce the cosmological parameter space consistent with the SPT cluster sample distribution in ξ and redshift, tightening up $\zeta \mbox{--}{M}_{500}\mbox{--}z$ parameter uncertainties. In addition, the redshift evolution parameter was shifted upward from ${C}_{\zeta }=0.55\pm 0.3$ to ${C}_{\zeta }=0.80\pm 0.15$. The combination of the shift and the reduction in uncertainties have motivated us to present the scaling relations derived using the X-ray observables together with these SZE-based halo masses. Table 5 contains the results of these relations. We recommend that those particularly interested in obtaining precise redshift trends in the scaling relations should use these results.

Table 5.  Best-fit Scaling Relation Parameters When Priors on the SZE $\zeta \mbox{--}{M}_{500}\mbox{--}z$ Relation Parameters Are Taken from de Haan et al. ( 2016, Table 3, column 2)

Scaling relation	${A}_{{ \mathcal X }}$	${B}_{{ \mathcal X }}$	${C}_{{ \mathcal X }}$	${\sigma }_{\mathrm{ln}{ \mathcal X }}$	${\gamma }_{{ \mathcal X }}$	${\delta }_{{ \mathcal X }}$
${T}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{{C}_{{ \mathcal X }}}$	6.50 ± 0.66	${0.78}_{-0.09}^{+0.10}$	${0.44}_{-0.26}^{+0.27}$	${0.18}_{-0.05}^{+0.04}$	⋯	⋯
II: ${ \mathcal X }{\left(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${6.51}_{-0.70}^{+0.58}$	${0.81}_{-0.10}^{+0.09}$	$\tfrac{2}{3}$	${0.18}_{-0.04}^{+0.04}$	$-{0.21}_{-0.22}^{+0.24}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${6.42}_{-0.65}^{+0.67}$	${0.77}_{-0.09}^{+0.10}$	$\tfrac{2}{3}$	${0.17}_{-0.03}^{+0.05}$	$-{0.14}_{-0.26}^{+0.33}$	${0.60}_{-0.47}^{+0.57}$
III with fixed SZE parameters	${6.44}_{-0.23}^{+0.24}$	${0.79}_{-0.09}^{+0.08}$	$\tfrac{2}{3}$	${0.17}_{-0.03}^{+0.04}$	$-{0.05}_{-0.27}^{+0.25}$	${0.81}_{-0.53}^{+0.49}$
${T}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${6.26}_{-0.68}^{+0.57}$	${0.80}_{-0.08}^{+0.10}$	${0.43}_{-0.22}^{+0.23}$	${0.13}_{-0.06}^{+0.04}$	⋯	⋯
II: ${ \mathcal X }{\left(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${6.25}_{-0.75}^{+0.49}$	${0.81}_{-0.09}^{+0.08}$	$\tfrac{2}{3}$	${0.14}_{-0.06}^{+0.04}$	$-{0.20}_{-0.22}^{+0.21}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${6.18}_{-0.62}^{+0.63}$	${0.81}_{-0.10}^{+0.10}$	$\tfrac{2}{3}$	${0.14}_{-0.05}^{+0.04}$	$-{0.17}_{-0.23}^{+0.28}$	${0.35}_{-0.45}^{+0.41}$
III with fixed SZE parameters	${6.14}_{-0.17}^{+0.22}$	${0.81}_{-0.08}^{+0.08}$	$\tfrac{2}{3}$	${0.13}_{-0.04}^{+0.04}$	-0.13 ± 0.23	${0.45}_{-0.48}^{+0.41}$
${M}_{\mathrm{ICM}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=0$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${6.60}_{-1.0}^{+1.1}$	${1.27}_{-0.11}^{+0.08}$	${0.43}_{-0.23}^{+0.24}$	${0.11}_{-0.08}^{+0.04}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{0}{(1+z)}^{{\gamma }_{{ \mathcal X }}}$	${7.14}_{-1.01}^{+1.00}$	${1.26}_{-0.10}^{+0.09}$	0	${0.11}_{-0.08}^{+0.04}$	${0.39}_{-0.24}^{+0.25}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${6.89}_{-0.91}^{+1.15}$	${1.27}_{-0.10}^{+0.10}$	0	${0.11}_{-0.08}^{+0.03}$	${0.40}_{-0.24}^{+0.23}$	${0.19}_{-0.54}^{+0.40}$
III with fixed SZE parameters	${6.93}_{-0.19}^{+0.28}$	${1.27}_{-0.08}^{+0.08}$	0	${0.08}_{-0.06}^{+0.04}$	${0.41}_{-0.22}^{+0.21}$	${0.11}_{-0.47}^{+0.44}$
${Y}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{5}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${4.5}_{-1.1}^{+1.0}$	${2.00}_{-0.14}^{+0.16}$	${0.80}_{-0.35}^{+0.42}$	${0.15}_{-0.09}^{+0.08}$	⋯	⋯
II: ${ \mathcal X }{\left(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${4.7}_{-1.0}^{+1.1}$	${2.01}_{-0.14}^{+0.16}$	$\tfrac{2}{3}$	${0.15}_{-0.11}^{+0.07}$	${0.15}_{-0.34}^{+0.39}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	4.9 ± 1.1	${2.03}_{-0.15}^{+0.13}$	$\tfrac{2}{3}$	${0.17}_{-0.10}^{+0.06}$	${0.32}_{-0.41}^{+0.39}$	${1.16}_{-0.77}^{+0.44}$
III with fixed SZE parameters	${4.59}_{-0.27}^{+0.22}$	1.97 ± 0.11	$\tfrac{2}{3}$	${0.07}_{-0.06}^{+0.07}$	${0.23}_{-0.33}^{+0.35}$	${0.73}_{-0.55}^{+0.77}$
${Y}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{5}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{2}{3}$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${4.25}_{-1.02}^{+0.96}$	${1.99}_{-0.16}^{+0.14}$	${0.77}_{-0.42}^{+0.37}$	${0.13}_{-0.10}^{+0.05}$	⋯	⋯
II: ${ \mathcal X }{\left(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E{\left(z\right)}^{\tfrac{2}{3}}(1+z\right)}^{{\gamma }_{{ \mathcal X }}}$	${4.35}_{-1.10}^{+0.90}$	${2.00}_{-0.15}^{+0.16}$	$\tfrac{2}{3}$	${0.05}_{-0.03}^{+0.12}$	${0.09}_{-0.29}^{+0.44}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${4.70}_{-1.2}^{+1.1}$	${1.99}_{-0.14}^{+0.18}$	$\tfrac{2}{3}$	${0.13}_{-0.10}^{+0.05}$	${0.35}_{-0.41}^{+0.38}$	${0.47}_{-0.65}^{+0.63}$
III with fixed SZE parameters	${4.31}_{-0.18}^{+0.30}$	1.98 ± 0.12	$\tfrac{2}{3}$	${0.04}_{-0.03}^{+0.09}$	${0.27}_{-0.35}^{+0.33}$	${0.72}_{-0.69}^{+0.57}$
${L}_{{\rm{X}},\mathrm{cin}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=2$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${4.13}_{-0.98}^{+0.87}$	${1.93}_{-0.18}^{+0.15}$	${2.01}_{-0.37}^{+0.44}$	${0.28}_{-0.12}^{+0.07}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{2}{(1+z)}^{{\gamma }_{{ \mathcal X }}}$	${4.15}_{-0.81}^{+1.10}$	${1.91}_{-0.15}^{+0.18}$	2	${0.25}_{-0.13}^{+0.08}$	${0.20}_{-0.43}^{+0.41}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${4.33}_{-0.89}^{+1.11}$	${1.85}_{-0.16}^{+0.21}$	2	${0.28}_{-0.11}^{+0.08}$	${0.21}_{-0.44}^{+0.53}$	${0.82}_{-0.94}^{+0.61}$
III with fixed SZE parameters	${3.86}_{-0.20}^{+0.22}$	1.94 ± 0.15	2	${0.23}_{-0.07}^{+0.08}$	${0.33}_{-0.37}^{+0.42}$	${0.79}_{-0.65}^{+0.83}$
${L}_{{\rm{X}},\mathrm{cex}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=1$	${C}_{{ \mathcal X },\mathrm{SS}}=2$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${2.83}_{-0.52}^{+0.53}$	${1.57}_{-0.14}^{+0.17}$	${2.17}_{-0.43}^{+0.34}$	${0.26}_{-0.09}^{+0.07}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{2}{(1+z)}^{{\gamma }_{{ \mathcal X }}}$	${2.82}_{-0.49}^{+0.61}$	${1.63}_{-0.16}^{+0.13}$	2	${0.26}_{-0.09}^{+0.08}$	${0.20}_{-0.34}^{+0.41}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${2.77}_{-0.51}^{+0.55}$	${1.57}_{-0.16}^{+0.18}$	2	${0.28}_{-0.07}^{+0.07}$	${0.24}_{-0.32}^{+0.51}$	${0.47}_{-0.57}^{+0.93}$
III with fixed SZE parameters	${2.67}_{-0.15}^{+0.16}$	1.58 ± 0.15	2	${0.26}_{-0.05}^{+0.06}$	0.25 ± 0.38	${0.75}_{-0.74}^{+0.76}$
${L}_{{\rm{X}},\mathrm{cin},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{4}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{7}{3}$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${13.8}_{-3.2}^{+3.6}$	${2.21}_{-0.20}^{+0.15}$	${2.28}_{-0.41}^{+0.46}$	${0.29}_{-0.12}^{+0.09}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{\tfrac{7}{3}}{(1+z)}^{{\gamma }_{{ \mathcal X }}}$	14.3 ± 3.2	2.18 ± 0.19	$\tfrac{7}{3}$	${0.26}_{-0.13}^{+0.09}$	${0.03}_{-0.37}^{+0.54}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${14.8}_{-2.9}^{+3.3}$	${2.16}_{-0.16}^{+0.19}$	$\tfrac{7}{3}$	${0.28}_{-0.11}^{+0.09}$	${0.19}_{-0.50}^{+0.44}$	${1.14}_{-0.76}^{+0.69}$
III with fixed SZE parameters	${14.55}_{-0.70}^{+0.98}$	2.23 ± 0.15	$\tfrac{7}{3}$	${0.19}_{-0.10}^{+0.08}$	${0.35}_{-0.42}^{+0.39}$	${1.34}_{-0.74}^{+0.49}$
${L}_{{\rm{X}},\mathrm{cex},\mathrm{bol}}\mbox{--}{M}_{500}\mbox{--}z$ Relation		${B}_{{ \mathcal X },\mathrm{SS}}=\tfrac{4}{3}$	${C}_{{ \mathcal X },\mathrm{SS}}=\tfrac{7}{3}$
I: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{{C}_{{ \mathcal X }}}$	${10.6}_{-2.2}^{+2.5}$	${1.87}_{-0.16}^{+0.19}$	${2.31}_{-0.35}^{+0.45}$	${0.26}_{-0.13}^{+0.08}$	⋯	⋯
II: ${ \mathcal X }{(M,z)\propto {M}_{500}^{{B}_{{ \mathcal X }}}E(z)}^{\tfrac{7}{3}}{(1+z)}^{{\gamma }_{{ \mathcal X }}}$	${10.5}_{-2.5}^{+2.1}$	${1.87}_{-0.14}^{+0.20}$	$\tfrac{7}{3}$	${0.27}_{-0.14}^{+0.07}$	${0.10}_{-0.35}^{+0.47}$	⋯
III: as II with ${B}_{{ \mathcal X }}^{{\prime} }={B}_{{ \mathcal X }}+{\delta }_{{ \mathcal X }}\mathrm{ln}\left(\tfrac{1+z}{1+{z}_{\mathrm{piv}}}\right)$	${9.9}_{-2.1}^{+2.2}$	${1.80}_{-0.22}^{+0.17}$	$\tfrac{7}{3}$	${0.30}_{-0.09}^{+0.07}$	${0.23}_{-0.49}^{+0.45}$	${0.73}_{-0.70}^{+0.78}$
III with fixed SZE parameters	${10.06}_{-0.61}^{+0.54}$	${1.86}_{-0.16}^{+0.15}$	$\tfrac{7}{3}$	${0.25}_{-0.07}^{+0.05}$	${0.36}_{-0.49}^{+0.32}$	${0.88}_{-0.57}^{+0.76}$

Note. The table layout is the same as in Table 4.

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