The eROSITA Final Equatorial-Depth Survey (eFEDS)

The CAMIRA algorithm (Oguri, 2014) first calculates the probability of each galaxy in the field being on the red sequence. Consequently, the galaxy clusters are defined as the overdensity of this type of galaxy population. At the same time, the selection method also returns the brightest cluster galaxy’s (BCG) position for each cluster. This cluster’s optical center is the position of the most luminous red-sequence galaxy close to the galaxy density peak. However, ultra-bright BCGs may not be correctly identified as BCGs because they are saturated in the HSC images, and the color estimates tend to be inaccurate. These reasons suggest that an additional confirmation of the BCG’s position is necessary.

To determine the BCGs of all the clusters, we used the optical/NIR data from SDSS (York et al., 2000), Pan-STARRS (Kaiser et al., 2002, 2010), 2MASS (Skrutskie et al., 2006), and WISE (Wright et al., 2010). Each survey has a different sensitivity and filter response. In particular, SDSS has an i-band saturation level of 14 mag for point sources, which is brighter than the 18 mag of the HSC. In fact, for nearly half of the clusters, the brightest galaxies exceeded the HSC’s saturation level and were re-identified in the following way. We searched for the brightest galaxy that locates within an $R_{500}$ radius from the cluster’s optical center, which is taken from the CAMIRA catalog. For the optical data, we used the $r$-band magnitudes to determine the brightest galaxy, whilst the $K_{s}$ band and $W_{1}$ values were used in the IR regime. The redshifts of the galaxies were either taken from the SDSS catalog or the NASA/IPAC Extragalactic Database (NED)²²2https://ned.ipac.caltech.edu/. The redshift constraints of the BCGs search is $\Delta z=|z_{\mathrm{galaxy}}-z_{\mathrm{cluster}}|=0.01\times(1+z_{\mathrm{cluster}})$ for the spectroscopic redshifts and $\Delta z=0.02\times({1+z_{\mathrm{cluster}}})$ for the photometric ones, respectively. Here, we considered uncertainties of redshift measurements and redshift tolerance of $3\sigma/c\sim 0.01$ for the typical velocity dispersion of rich clusters, $\sigma\sim 1000\leavevmode\nobreak\ {\rm km\,s^{-1}}$ (Fadda et al., 1996).

Two additional corrections have been made before the photometric data are used for the BCGs search. First, we corrected for the galactic extinction, using the Schlegel map (Schlegel et al., 1998), assuming an extinction law (Fitzpatrick, 1999) with $R_{V}$= 3.1. Second, appropriate $K$-correction methods are applied to obtain the magnitudes as in the rest frames of individual galaxies. We used the $K$-correction code version 2012 (Chilingarian et al., 2010; Chilingarian & Zolotukhin, 2012) for the SDSS, Pan-STARRS, and 2MASS data because this method performs well with bright sources and does not require the compulsory input of multiple filter bands. However, this method does not provide the coefficients for WISE filters; therefore, we applied the $K$-correction code from Blanton & Roweis (2007) on the WISE photometric data instead.

Finally, we assessed the BCG search result through a careful visual inspection. In the case that multiple BCG candidates are suggested for one cluster, we assigned the BCG to the elliptical galaxy with a more extended envelope. If the search from different surveys returns different BCGs, we gave more weight to the BCG results from the optical observations. There were two cases (HSC J091843+021231 and HSC J092041+024660), where we needed to check the information of the obvious BCG candidates manually. These were the seemingly best choices but were not selected by the BCG finding program. Indeed, these galaxies do not have photometric imprints in any of the aforementioned surveys. We found the redshift of these visible BCG candidates on NED and their brightness in GAMA (Wright et al., 2016). We thus confirmed that they are the BCGs of their host clusters because they also follow the spatial and redshift constraints.

Next, we describe weak-lensing (WL) analyses for the CAMIRA clusters in the eFEDS field. We used the latest shape catalog and the weak-lensing mass calibration from the three-year HSC data (S19A; Li et al., 2022). The galaxy shapes are measured by the re-Gaussianization method (Hirata & Seljak, 2003) implemented in the HSC pipeline (see details in Mandelbaum et al., 2018a, b). The same shape catalog is used in the WL mass measurements for the eFEDS clusters (Chiu et al., 2022). We adopted the full-color and full-depth criteria for precise shape measurements and photometric redshift estimations.

The dimensional, reduced tangential shear, $\Delta\Sigma_{+}$, is computed by averaging the tangential component of a galaxy ellipticity $e_{+}=-(e_{1}\cos 2\varphi+e_{2}\sin 2\varphi)$ where $\varphi$ is the angle measured in sky coordinates from the RA direction to the line between the source galaxy and the lens. The formulation is specified by:

(e.g., Miyaoka et al., 2018; Medezinski et al., 2018a; Okabe et al., 2019; Miyatake et al., 2019; Murata et al., 2019; Umetsu et al., 2020; Okabe et al., 2021; Chiu et al., 2022). Here, the subscripts, $i$ and $k$, denote the $i$-th galaxy located in the $k-$th radial bin, and $z_{l}$ and $z_{s}$ are the cluster and source redshift, respectively. The inverse of the mean critical surface mass density, $\langle\Sigma_{{\rm cr}}(z_{l},z_{s})^{-1}\rangle$, is calculated by weighting the critical surface mass density, $\Sigma_{{\rm cr}}=c^{2}D_{s}/4\pi GD_{l}D_{ls}$, by the probability function of the photometric redshift, $P(z)$:

Here, $D_{l}$, $D_{s}$, and $D_{ls}$ are the angular diameter distances from the observer to the cluster, to the sources, and from the lens to the sources, respectively. The photometric redshift is estimated by the machine learning method (MLZ; Carrasco Kind & Brunner, 2014) calibrated with spectroscopic data (Nishizawa et al., 2020). The dimensional weighting function is expressed as

where $e_{\rm rms}$ and $\sigma_{e}$ are the root mean square of intrinsic ellipticity and the measurement error per component ($e_{\alpha}$; $\alpha=1$ or $2$), respectively. The shear responsivity, $\mathcal{R}$, and the calibration factor, $K$, are obtained by $\mathcal{R}=1-\sum_{ij}w_{i,j}e_{{\rm rms},i}^{2}/\sum_{ij}w_{i,j}$ and $K=\sum_{ij}m_{i}w_{i,j}/\sum_{ij}w_{i,j}$, with the multiplicative shear calibration factor $m$ (Mandelbaum et al., 2018a, b), respectively. We also conservatively subtracted an additional, negligible offset term for calibration. The radius position, $R_{k}$, is defined by the weighted harmonic mean (Okabe & Smith, 2016). We selected background galaxies behind each cluster using the photometric selection (p-cut) following Medezinski et al. (2018b):

where we allowed for a 2% contamination level with $p_{\rm cut}=0.98$.

In order to measure individual cluster masses, we used the NFW profile (Navarro et al., 1996, 1997). The three-dimensional mass density profile of the NFW profile is expressed as

where $r_{s}$ is the scale radius and $\rho_{s}$ is the central density parameter. The NFW model is also specified by the spherical mass, $M_{\Delta}=4\pi\Delta\rho_{\rm cr}r_{\Delta}^{3}/3$, and the halo concentration, $c_{\Delta}=r_{\Delta}/r_{s}$. Here, $r_{\Delta}$ is the overdensity radius. We treat $M_{\Delta}$ and $c_{\Delta}$ as free parameters and adopt $\Delta=500$. We computed the model of the dimensional, reduced tangential shear, $f_{\rm model}$, at the projected radius $R$ by integrating the mass density profile along the line of sight (Okabe et al., 2019; Umetsu, 2020);

where $\Sigma(R)$ is the local surface mass density at $R$, $\bar{\Sigma}(<R)$ is the average surface mass density within $R$, and $\mathcal{L}_{z}=\sum_{i}\langle\Sigma_{{\rm cr},i}^{-1}\rangle w_{i}/\sum_{i}w_{i}$. We chose the X-ray-defined centers and adopt an adaptive radial-bin choice (Okabe & Smith, 2016) for cluster mass estimation.

The log-likelihood of the weak-lensing analysis is expressed as

where $k$ and $m$ denote the $k-$th and $m-$th radial bins. We considered three components in the covariance matrix $C=C_{g}+C_{s}+C_{\rm LSS}$; the shape noise $C_{g}$, the errors of the source redshifts, $C_{s}$ and the uncorrelated large-scale structure (LSS), $C_{\rm LSS}$, of which the elements are correlated with each other (Schneider et al., 1998; Hoekstra, 2003). The details of calculations are described in Sect. 3 of Okabe & Smith (2016). We measured WL masses in 38 out of the 43 CAMIRA clusters that satisfy the full-color and full-depth conditions (Table 3). The signal-to-noise ratios, $S/N=(\Delta\Sigma_{+,k}C_{km}^{-1}\Delta\Sigma_{+,m})^{1/2}$, of individual clusters were small; $S/N<2$ for 3, $2\leq S/N<4$ for 20, and $4<S/N$ for 17 clusters, resulting in uncertainties in $M_{500}$. Here, for consistency with other X-ray observables, the center coordinates in the WL measurements were assumed to be equal to the X-ray centroids.

We also carried out the NFW model fitting with a free central position using two-dimensional shear pattern (Oguri et al., 2010; Okabe et al., 2011). The log-likelihood is expressed as

Here, the subscripts $\alpha$ and $\beta$ denote each shear component. We used the box size of $1\arcmin.5\times 1\arcmin.5$ for the shear pattern. We constrained the central positions for 23 clusters with good posterior distributions and compared them with X-ray centers, CAMIRA centers, and galaxy map peaks (Sect. 5.1).

The eFEDs data of seven telescope modules (TMs) were reduced in a standard manner by using the eROSITA Standard Analysis Software System (eSASS) version eSASSusers_201009 (Brunner et al., 2022). We extracted cleaned event files by applying a flag to reject bad events and selecting all valid patterns, namely single, double, triple, and quadruple events. The point sources were removed by referring to the main eFEDS X-ray source catalog³³3https://erosita.mpe.mpg.de/edr/eROSITAObservations/Catalogues/ (Brunner et al., 2022). We checked that the faint sources with low detection likelihood in the supplemental eFEDS catalog do not affect our X-ray analysis.

We extracted the X-ray image of each cluster from merged event files of seven TMs in the 0.5 – 2 keV band and corrected them for exposure and vignetting. The pixel size is 1″.5. The X-ray centroid was determined from the mean of the photon distribution within a circle of radius $R_{500}$ using the algorithm described in Ota et al. (2020); Starting with the BCG coordinates, we iterated the centroid search until its position converged within 1″. The X-ray peak within $R_{500}$ was measured using the 0.5 – 2 keV image smoothed with a $\sigma=3$ (pixels) Gaussian function. We note that we calculated the two-dimensional (2D) PSF image of the survey mode at each cluster coordinates using ermldet in the eSASS package and confirmed that it is almost symmetric and does not affect the present measurement.

To measure the gas temperature and bolometric luminosity, we extracted the spectra from a circular region of a radius $R_{500}$ centered on the X-ray centroid. Here, it is reasonable to assume that the cluster center is represented by the X-ray centroid (Sect. 5.1, Fig. 3). The TM1,2,3,4,6/TM5,7 spectra in the 0.3 – 10 keV/1 – 10 keV band were simultaneously fit using XSPEC 12.1.1. For TM5 and TM7, the energies below 1 keV were excluded due to the light-leak contamination (Predehl et al., 2021). The spectral model consists of cluster emission and background components. For the cluster component, we assumed the APEC thin-thermal plasma model (Smith et al., 2001; Foster et al., 2012), with the Galactic absorption model tbabs (Wilms et al., 2000). The redshift and metal abundance were fixed at the optical value (Table 2) and 0.3 solar, respectively. The hydrogen column density $N_{\rm H}$ in the tbabs model was fixed at a value taken from Willingale et al. (2013). For the background components, the Galactic emission and cosmic X-ray background were determined by fitting an annulus with inner and outer radii of 2.5 and 4 Mpc from the cluster. The instrumental background was estimated based on the filter wheel closed (FWC) data⁴⁴4The FWC spectral model version 1.0 (https://erosita.mpe.mpg.de/edr/eROSITAObservations/EDRFWC/). Table 3 lists the resultant gas temperature, $kT,$ and bolometric luminosity, $L_{\rm X}$. We note that for three clusters (HSC J084548+020640, HSC J084656+01383, and HSC J090754+005732), we fixed $kT$ at a value expected from the $N-T$ relation (Oguri et al., 2018) and the richness due to the large statistical uncertainty.

We calculated the spatial separation from the cluster’s X-ray centroid to the BCG’s position and report it as the BCG-and-X-ray centroid offset, $D_{\mathrm{XC}}$. In addition, we estimate the distance between the cluster’s X-ray peak to its BCG $D_{\mathrm{XP}}$. Figure 2 demonstrates the distributions of $D_{\rm XC}$ and $D_{\rm XP}$ for 43 clusters in our sample. The median values of these offsets in the unit of kpc and $R_{500}$ are shown in Table 4.

Next, we compare the projected distance between 2D weak-lensing mass and three kinds of cluster centers, namely X-ray centroid, galaxy map peak, and the CAMIRA coordinates for 23 clusters. We estimated the statistical errors of the probability distributions by the bootstrap resampling method. As shown in Fig. 3, all the histograms are well described by double Gaussian distributions:

where $p_{i}(r)=(r/\sigma_{i}^{2})\exp\left[-r^{2}/2\sigma_{i}^{2}\right]$ and $f_{\rm cen}$ is the fraction of the central component (Oguri et al., 2010, 2018). The first standard deviation, $\sigma_{1}$, for the X-ray centroid is smaller than those of the CAMIRA and galaxy ones, suggesting that the X-ray centroids are closer to the WL mass centers. Although the values of $\sigma_{1}$ for the CAMIRA centers are the largest and the histogram is slightly skewed from the double Gaussian distributions, the amplitude of the CAMIRA centers at $r\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}60$ kpc is comparable to that of the X-ray centers. Although the value of $\sigma_{1}$ for the galaxy peak is the second smallest among the first components of the three centers, the amplitude at $r\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}100$ kpc is $\sim 60-70$ % of the others. The second standard deviation, $\sigma_{2}$, of the galaxy peak is the smallest among the three centers.

In this subsection, we use three different methods to classify the morphology of galaxy clusters: i) BCG-X-ray offset, ii) concentration parameter, and iii) galaxy peak-finding method.

First, the BCG-X-ray offset: We divided the clusters into two groups following the criteria used in Sanderson et al. (2009), namely, “relaxed” clusters with a small peak offset ($D_{\rm XP}<0.02R_{500}$) and “disturbed” clusters with a large offset ($D_{\rm XP}>0.02R_{500}$). There is only one cluster that posseses a small offset of $D_{\mathrm{XP}}\leq 0.02R_{500}$, which corresponds to a very small fraction of relaxed clusters $2.3\pm 2.2\,(^{+0.0}_{-2.3})$% or ¡ 5%. Here, the statistical and systematic errors are calculated via the bootstrap resampling method and by varying the smoothing scales between 2 and 4 pixels, respectively.

The accuracy of the X-ray peak position depends on the statistical quality of the X-ray observations, which broadly varies among clusters. In fact, based on the XMM-Newton observations of the CAMIRA clusters, Ota et al. (2020) reported that the clusters with low photon statistics tend to show larger centroid and peak offsets. We thus calculated the standard error of the peak offset $\delta D_{\rm XP}$ based on a comparison of X-ray images of each cluster with different smoothing scales ($\sigma=2,3,4$ pixels). As a result, $\delta D_{\rm XP}$ ranges from 0% to 55 % (the mean is 5%), which is smaller than that of the XMM-Newton study (Ota et al., 2020).

At high redshifts, however, $0.02R_{500}$ is comparable to the accuracy of the attitude determination (Predehl et al., 2021). With regard to how much the results would change based on the threshold, when we set the threshold value to $0.05R_{500}$, the percentage of relaxed clusters was $9$%. This example suggests that the systematic uncertainty of the relaxed fraction is not negligible. Accordingly, we assigned the typical positional accuracy of 4″.7 (Brunner et al., 2022) to the systematic error of $D_{\rm XP}$. This increases the number (or fraction) of relaxed clusters that fall in $D_{\mathrm{XP}}\leq 0.02R_{500}$ to 7 (or 16%). Consequently, we estimate the fraction of relaxed clusters as $2(<16)$% from the BCG-X-ray offset (Table 4).

We should note that this modest fraction of relaxed clusters is potentially impacted by other effects. First, the selection of the BCG itself faces undeniable obstacles. The lack of redshift information and the use of photometric redshifts with large errors could lead to a poor choice of BCG. In some cases, the true BCG may not be detected due to being overbright for the optical instruments.

Second, there is the concentration parameter: if we refer to the concentration parameter of the X-ray surface brightness, $C_{{\rm SB},R_{500}}$, and apply $C_{{\rm SB},R_{500}}>0.37$ to identify relaxed clusters (Lovisari et al., 2017), the fraction of relaxed clusters is estimated as 39% for 33 optical clusters with X-ray counterparts in the eFEDS catalog within $R_{500}$ from the X-ray centroid and $\Delta z<0.02$. Here, $C_{{\rm SB},R_{500}}$ is defined as $C_{{\rm SB},R_{500}}=S_{B}(<0.1R_{500})/S_{B}(<R_{500})$ (Maughan et al., 2012) and we quoted the value measured by Ghirardini et al. (2022), based on the X-ray morphological study of the eFEDS clusters. Because all ten unmatched clusters are disturbed clusters, according to the BCG-X-ray offset measurements, we consider the above estimate to be an upper limit on the relaxed fraction.

Third, the galaxy peak-finding method: Since our optically selected sample covers a wide range of morphologies, we expect a large percentage of merging clusters. These complex systems could be classified using the peak-finding method (Okabe et al., 2019). Thus, we also checked the fraction of merging clusters by finding peaks of member-galaxy distribution (Ramos-Ceja et al., 2022). The threshold corresponds to the peak height of the richness $N=15$ at each redshift. In the eFEDS field, the fraction of clusters with single over multiple peaks is 27/73% for the high-richness CAMIRA clusters, while the fraction of single over multiple peaks is 83/17% for the eFEDS clusters.

To be conservative, we quote the results from three kinds of measurements and estimate the relaxed fraction of our sample to be $2(<39)$%.

To derive the temperature- and mass-observable relations of the high-richness clusters, we fit the data to the power-law model (Eq. 10) via the Bayesian regression method (Kelly, 2007):

The quantities $a$, $b$, and the intrinsic scatter are treated as free parameters. According to the self-similar model and Hubble’s law, $E(z)=[\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$ describes the redshift evolution of the scaling relation. The luminosity of the cluster in the hydrostatic equilibrium follows $E(z)^{-1}L\propto T^{2}$ and $E(z)^{-1}L\propto[E(z)M]^{4/3}$. In Fig. 4, we corrected the redshift evolution by applying the self-similar model and plotted $E(z)^{-1}L$ against $T$ or $E(z)M$ since no clear consensus has been reached on the evolution of the scaling relations (Giodini et al., 2013). Table 5 lists the best-fit parameters for the $N-T$ and $L-T$ relations of 43 clusters and the $N-M$ and $L-M$ relations for 38 clusters with the weak-lensing measurements (Sect. 3.2). The correlation coefficient is 0.62 – 0.70 for the four relations.

As described in Akino et al. (2022), a multi-variate analysis is needed to properly correct the selection bias and the dilution effect and incorporate the weak-lensing mass calibration. Thus, to correct for the selection bias due to the richness cut of $N>40$, we simultaneously fit two kinds of relations, $N-T$ and $L-T$ or $N-M$ and $L-M$, by the hierarchical Bayesian regression method (HiBRECS; Akino et al., 2022) (Fig. 4). Because the slopes of $N-T$ and $N-M$ were not well constrained due to the large data scatter, we fixed them at 1.05 and 0.70, respectively, that were deduced from the best-fit mass-richness relation (Okabe et al., 2019). We also assumed the intrinsic scatter of the weak-lensing mass to be $\ln{\sigma_{M}}=-1.54$ (Umetsu et al., 2020). Table 5 summarizes the best-fit scaling relations obtained from three types of fitting codes.

From Table 5, we find that the fitting results with the Kelly and 1D HiBRECS codes agree well. The Kelly code gave a marginally shallower $N-T$ slope of $0.61\pm 0.21,$ as compared to the expectation from $M\propto T^{3/2}$ and $M\propto N^{1.4}$(Okabe et al., 2019). In the present analysis, the impact of selection bias correction was small; the 1D and 2D HiBRECS analyses gave consistent results within the error bars. Therefore, in Sect. 6, we quote the above results based on the 2D HiBRECS code, which can properly handle the bias correction and the mass calibration.

As mentioned in Sect. 5, only $2(<16)$% clusters out of our cluster sample possess a small X-ray peak offset $D_{\mathrm{XC}}\leq 0.02R_{500}$. Compared with the previous XMM-Newton measurements of 17 optical clusters with $N>20$, $29\pm 11(\pm 13)$% (Ota et al., 2020), the two results agree within the errors.

Rossetti et al. (2016) investigated the dynamical states of 132 galaxy clusters of a Sunyaev-Zel’dovich (SZ) selected a sample at a median redshift of $z=0.16$ to find that $52\pm 4$% of the sample has $D_{\mathrm{XP}}\leq$ $0.02R_{500}$. The SZ selection method tends to pick out dynamically-active systems, which leads to a relatively lower percentage of small peak shift. On the other hand, the relaxed ratio is higher in the X-ray selected sample, at $\approx$ 74%, indicating that clusters collected in an X-ray flux-limited survey are subject to the cool-core bias. This accounts for a large fraction of X-ray bright, hence, dynamically relaxed clusters with a small peak shift. In addition, Migkas et al. (2021) found that approximately 44% of the X-ray flux-limited eeHIFLUGCS sample have a small peak shift of $D_{XP}<0.02R_{500}$. Therefore, our sample’s relaxed fraction is considerably small compared to the above SZ and X-ray samples. We note that Ghirardini et al. (2022) reported that the eFEDS-selected cluster sample is not biased toward cool-core clusters, but that it does contain a similar fraction of cool-cores as SZ surveys.

As shown in the upper-left panel of Fig. 2, we assessed the distribution of the centroid offset by the double Gaussian model (Eq. 9) since there is a tail toward large $D_{\rm XC}$. The fitting yields $\sigma_{1}=28\pm 2$ kpc, $\sigma_{2}=95\pm 6$ kpc, and $f_{\rm cen}=0.46\pm 0.04$. In comparison with the positional offset between the CAMIRA and XXL clusters (Oguri et al., 2018), the present sample has a marginally smaller fraction of well-centered clusters and a lower tail component.

Pasini et al. (2022) showed that BCGs with radio-loud active galactic nucleus (AGN) are more likely to lie close to the cluster center than radio-quiet BCGs and that the relations between the AGN and the intracluster medium (ICM) hold regardless of the dynamical state of the cluster (see also Pasini et al., 2020).

In what follows, we discuss the $L-T$ and $L-M$ relations based on a comparison with previous observations and theoretical models. The $L-T$ relation shows a large intrinsic scatter of $\sigma_{L|T}=0.63,$ as noted by the previous X-ray studies (e.g., Ota et al., 2006; Pratt et al., 2009). The observed $L-T$ slope of $2.08\pm 0.46$ agrees with $2.2\pm 0.6\,(\pm 0.2)$ derived from the XMM-Newton observations of the CAMIRA clusters (Ota et al., 2020). In contrast, a steeper slope of $\sim 3$ has been reported by many X-ray observations in the past (Giodini et al., 2013). From the analysis of 265 eFEDS clusters, Bahar et al. (2022) obtained the best-fit $L_{\rm bol}-T$ slope of $3.01^{+0.13}_{-0.12}$, suggesting a strong deviation from the self-similar model, $L\propto T^{2}$. Because of the small fraction of relaxed clusters, we consider the present measurement is less affected by the cool-core emission.

The scatter of $L-M$ relation is comparably large, $\sigma_{L|M}=0.65$, which confirms previous reports (e.g., Pratt et al., 2009). The $L-M$ slope of $1.52\pm 0.34$ is consistent with $1.51\pm 0.09$ derived for 232 clusters at $z=0.05-1.46$ based on a compilation of 14 published X-ray data sets (Reichert et al., 2011). The best-fit $L-M$ relation of the present optical sample also agrees with the result on 25 shear-selected clusters in the eFEDS field (Ramos-Ceja et al., 2022) and the luminosity-mass-and-redshift relation of the eFEDS cluster sample with the HSC weak-lensing mass calibration (Chiu et al., 2022) within the measurement errors though the fitting functions are not exactly the same.

Finally, we discuss the interpretation of our results by referring to the baseline scaling relations. The self-similar model (Kaiser, 1986) predicts simple relations between X-ray properties of ICM and mass in the absence of baryonic physics such as AGN feedback and radiative cooling. Moreover, these baseline relations were derived without considering the fact that less massive clusters tend to be more concentrated and have higher characteristic densities in the hierarchical structure formation in a CDM universe (Navarro et al., 1997). On the other hand, Fujita & Aung (2019) constructed the new baseline luminosity-temperature and mass relations by considering the mass-concentration relationship. This is again the case when additional physics, such as feedback and radiative cooling, do not work. They showed that the baseline relations should be shallower than the conventional self-similar model and follow $L\propto T^{1.6-1.8}$ and $L\propto M^{1.1-1.2}$. They also suggested that the $L-T$ relation of high-mass clusters should be close to the baseline model because the feedback from stars and AGN is less effective.

To carry out a comparison with their model, we derived the scaling relations without $E(z)$-correction in the same manner as in Fujita & Aung (2019). From rows 5-8 of Table 5, the fitted slopes of $L-T$ and $L-M$ relations agree with the baseline models within the $1\sigma$ errors. Here, the baseline relations show little difference between merging and non-merging clusters (Fig. 3 of Fujita & Aung, 2019), while disturbed clusters dominate our sample. The nature of the fundamental plane can explain the above trend because the $L-T$ relation is close to the edge-on view of the fundamental plane and the clusters do not substantially deviate from the thin plane, even during a merging process (Fujita et al., 2018). Therefore, the small $L-T$ slope observed in the high-richness massive clusters agrees with the prediction of the revised baseline model by Fujita & Aung (2019). We plan to extend the analysis using the eROSITA and HSC surveys to test the shallow scaling relations.