# The XMM Cluster Survey analysis of the SDSS DR8 redMaPPer Catalogue:
Implications for scatter, selection bias, and isotropy in cluster scaling
relations

P. A. Giles1 ID , A. K. Romer1 ID , R. Wilkinson1 ID , A. Bermeo1, D. J.
Turner1 ID , M. Hilton2,3 ID , E. W. Upsdell1 ID , P. J. Rooney1, S.
Bhargava1,4, L. Ebrahimpour5,6 ID , A. Farahi7 ID , R. G. Mann8, M.
Manolopoulou8, J. Mayers1, C. Vergara1, P. T. P. Viana5,6 ID , C. A. Collins9,
D. Hollowood10, T. Jeltema10, R. C. Nichol11, R. Noorali10, M.
Splettstoesser10, J. P. Stott12 ID
1Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH,
UK
2Astrophysics Research Centre, University of KwaZulu-Natal, Westville Campus,
Durban 4041, SA
3School of Mathematics, Statistics, and Computer Science, University of
KwaZulu-Natal, Westville Campus, Durban 4041, SA
4AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne
Paris Cité, F-91191 Gif-sur-Yvette, France
5Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, 4150-762 Porto, Portugal
6Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
7Department of Statistics and Data Sciences, The University of Texas at
Austin, Austin, TX 78712, USA
8Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
9Astrophysics Research Institute, Liverpool John Moores University, Liverpool
Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
10Santa Cruz Institute for Particle Physics, University of California, Santa
Cruz, 1156 High St, Santa Cruz, CA 95064, USA
11Institute of Cosmology and Gravitation, Dennis Sciama Building, Burnaby
Road, Portsmouth PO1 3FX, UK
12Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper we present the X-ray analysis of SDSS DR8 redMaPPer (SDSSRM)
clusters using data products from the XMM Cluster Survey (XCS). In total, 1189
SDSSRM clusters fall within the XMM-Newton footprint. This has yielded 456
confirmed detections accompanied by X-ray luminosity ($L_{X}$) measurements.
Of the detected clusters, 382 have an associated X-ray temperature measurement
($T_{X}$). This represents one of the largest samples of coherently derived
cluster $T_{X}$ values to date. Our analysis of the X-ray observable to
richness scaling relations has demonstrated that scatter in the
$T_{X}-\lambda$ relation is roughly a third of that in the $L_{X}-\lambda$
relation, and that the $L_{X}-\lambda$ scatter is intrinsic, i.e. will not be
significantly reduced with larger sample sizes. Our analysis of the scaling
relation between $L_{X}$ and $T_{X}$ has shown that the fits are sensitive to
the selection method of the sample, i.e. whether the sample is made up of
clusters detected “serendipitously” compared to those deliberately targeted by
XMM. These differences are also seen in the $L_{X}-\lambda$ relation and, to a
lesser extent, in the $T_{X}-\lambda$ relation. Exclusion of the emission from
the cluster core does not make a significant impact to the findings. A
combination of selection biases is a likely, but as yet unproven, reason for
these differences. Finally, we have also used our data to probe recent claims
of anisotropy in the $L_{X}-T_{X}$ relation across the sky. We find no
evidence of anistropy, but stress that this may be masked in our analysis by
the incomplete declination coverage of the SDSS DR8 sample.

###### keywords:

galaxies:clusters:general, X-rays:galaxies:clusters, X-rays:general

††pubyear: 2022††pagerange: The XMM Cluster Survey analysis of the SDSS DR8
redMaPPer Catalogue: Implications for scatter, selection bias, and isotropy in
cluster scaling relations–F

##  1 Introduction

Clusters of galaxies are the largest gravitationally bound objects in the
Universe, residing at the intersections of the dark matter filamentary
structure. Enabled by a new generation of imaging surveys, from across the
electromagnetic spectrum, clusters are expected to play an important role in
forthcoming attempts to measure cosmological parameters to percent level
accuracy (e.g. see Figure G2 in The LSST Dark Energy Science Collaboration et
al., 2018). Several different detection methods will be used to deliver
cluster samples of sufficient size, quality, and redshift grasp to meet the
requirements of Stage IV (and beyond) Dark Energy Experiments (Dodelson et
al., 2016). These methods include detections of spectral distortions to the
Cosmic Microwave Background (CMB), of extended X-ray emission, and of
projected overdensities (in the optical/near-IR band) of member galaxies.
Relevant, on-going, or soon to begin, experiments include the South Pole
Telescope, the Atacama Cosmology Telescope, the Simons Observatory
(CMB111pole.uchicago.edu, act.princeton.edu, simonsobservatory.org), the
eROSITA telescope (X-ray111mpe.mpg.de/eROSITA), the Dark Energy Survey (DES),
the Hyper Suprime-Cam Subaru Strategic Program, the Legacy Survey of Space and
Time and the EUCLID mission (optical/near-IR222darkenergysurvey.org,
hsc.mtk.nao.ac.jp, www.lsst.org, sci.esa.int/euclid).

Even after these new cluster samples become available, there will remain
significant challenges to overcome before unbiased cosmological parameters can
be reliably extracted. This has been illustrated by the surprising
inconsistency between parameter estimates derived from clusters compared to
those derived, using different techniques, from the same input data. For
example, there is a 2.4$\sigma$ tension with the DES Y1 galaxy clustering and
cosmic shear results (Abbott et al., 2019). Similar tension was found by the
Planck team when comparing their analysis of clusters with the CMB anisotropy
spectrum (Planck Collaboration et al., 2016). One way to address those
challenges is to exploit synergies between data sets collected at different
wavelengths (e.g. Wu et al., 2010; Grandis et al., 2021). The work presented
herein aims to provide X-ray support to the efforts of the DES and LSST-
DESC333lsstdesc.org collaborations to realise the potential of optical/near-IR
detected clusters for cosmological studies. For this, we use X-ray data
collected by the XMM-Newton telescope and analysed by the XMM Cluster Survey
team (Romer et al., 2001). We focus specifically on clusters identified using
the red-sequence Matched-filter Probabilistic Percolation technique (or
redMaPPer, Rykoff et al., 2014, 2016, hereafter RM). However, this work will
also be analogous to other cluster samples generated from optical/near-IR
surveys, e.g. those identified using the CAMIRA (Oguri, 2014) or WaZP (Aguena
et al., 2021) algorithms.

The extraction of cosmological parameters from RM samples relies on the use of
a Mass Observable Relation (MOR), i.e. a description of how the dark mater
halo mass scales with the detection observable. The latter is quantified in RM
samples by the so-called richness measure, which describes the number of
galaxies detected per cluster (see Section 2.1 for more information). The halo
mass is estimated from the weak lensing (WL) signal. However, the signal per
cluster is so small that it is necessary to bin the sample, by richness and
redshift, in order to measure the MOR (McClintock et al., 2019). The main
drawback of this binning, or “stacking” method is the loss of any information
about the intrinsic scatter of the observable with mass. As shown in Sahlén et
al. (2009), knowledge of the scatter and, its evolution with mass and
redshift, is needed for accurate parameter estimation. The benefit of X-ray
follow-up, such as that described herein, is that an X-ray observable to
richness scaling relation will preserve scatter information (e.g. Farahi et
al., 2019). This scatter information can then be used to infer scatter in the
stacked MOR, since X-ray observables have a tight correlation with mass, as
shown by observations (e.g., Lieu et al., 2016; Mantz et al., 2016) and
simulations (e.g., Le Brun et al., 2014; Farahi et al., 2018).

Another drawback of using WL to calibrate to the MOR for RM samples is that
the WL signal is diluted, at the RM determined cluster centroid, if there is
an offset between that and the dark matter halo centre of mass. The impact of
the offset needs to be modelled to mitigate the impact on derived cosmological
parameters and, for this, X-ray follow-up is essential. This is because the
X-ray surface brightness is a much better tracer of the underlying mass than
the projected galaxy density. This type of mis-centering correction using
X-ray data has been demonstrated in e.g., McClintock et al. (2019).

In summary, the work presented herein was motivated by the desire to support
RM cluster cosmology in two ways: estimating intrinsic scatter on the MOR and
determining a mis-centering model. The first step required to meet both goals
is to gather as much high quality X-ray data as possible, and in $\S$2 we
discuss the development of RM cluster samples with X-ray observations in the
XMM public archive. The X-ray analysis of these clusters is described in
$\S\ref{sec:specanalysis}$. We go on to present scaling relations between
X-ray and RM observables, and their associated scatter, in $\S$4. Analysis of
the samples for the purposes of mis-centering modelling is the subject of a
companion publication, Zhang et al. 2019. In §5 we explore the impact of
selection bias on scaling relation scatter measurements by splitting the
sample into clusters detected serendipitously and those specifically targeted
by XMM. We also investigate a recent claim of of anisotropy in scaling
relations across the sky. Conclusions are summarised in §6. Throughout this
paper we assume a cosmology of $\Omega_{M}$=0.3, $\Omega_{\Lambda}$=0.7 and
$H_{0}$=70 km s-1 Mpc-1.

##  2 Development of the SDSSRM-XCS cluster samples

In this section we describe the construction of the X-ray cluster samples used
throughout this work. The process starts with the parent SDSS optical cluster
catalog described in (Rykoff et al., 2014). A flowchart outlining the various
steps involved is shown in Figure 1.

Figure 1: A flowchart outlining the process used to generate a sample of RM
clusters with measured X-ray properties in the SDSS DR8 RM footprint.

###  2.1 The SDSS redMaPPer cluster catalogue

The red-sequence Matched-filter Probabilistic Percolation (again, denoted RM
throughout), cluster finding algorithm (Rykoff et al., 2014), is a powerful
tool for finding clusters from optical/near-IR photometric survey data and has
already been successfully applied to SDSS (Rykoff et al., 2014) and DES
(Rykoff et al., 2016). RM self-trains the red sequence model to any available
spectroscopic redshifts, and then calculates, in an iterative fashion,
photometric redshifts for each cluster identified. The richness estimated by
RM (hereafter, $\lambda_{\rm RM}$) of each cluster is calculated as the sum of
membership probabilities over all galaxies within a scale radius, Rλ, where
R${}_{\lambda}=1.0h^{-1}{\rm Mpc}(\lambda/100)^{0.2}$. The specific RM cluster
sample used throughout this work is based upon the 8th data release of the
Sloan Digital Sky Survey444https://www.sdss.org/ (or SDSS-DR8, Aihara et al.,
2011). The RM SDSS-DR8 catalog (Rykoff et al., 2014) contains a total of
396,047 clusters. The analysis was restricted to clusters with $\lambda_{\rm
RM}>$20, because numerical simulations show that, at this threshold, 99% of RM
clusters can be unambiguously mapped to an individual dark matter halo (Farahi
et al., 2016). Based upon this $\lambda_{\rm RM}$ cut, our initial sample
contained 66,028 clusters (we denote this as the ‘SDSSRM’ sample hereafter,
see Table 1).

###  2.2 The XCS image database and source catalogue

The results presented in this paper were derived using X-ray data from all
publicly available XMM observations555XMM database (as of September 2018) with
usable EPIC science data. The XMM observations were analysed as part of the
XMM Cluster Survey (Romer et al., 1999, hereafter XCS). The aim of XCS is to
catalogue and analyse all X-ray clusters detected during the XMM mission. This
includes both those that were the intended target of the respective
observation, and those that were detected serendipitously (e.g. in the
outskirts of an XMM observation targeting a quasar). The XCS reduction process
was fully described in Lloyd-Davies et al. (2011, hereafter LD11), but a brief
outline is as follows.

The data were processed using XMM-SAS version 14.0.0, and events lists
generated using the EPCHAIN and EMCHAIN tools. In order to exclude periods of
high background levels and particle contamination, we generated light curves
in 50s time bins in both the soft (0.1 – 1.0 keV) and hard (12 – 15 keV)
bands. An iterative 3$\sigma$ clipping process was performed on the light
curves; time bins falling outside this range were excluded.

Single camera (i.e. PN, MOS1 and MOS2) images, along with the corresponding
exposure maps, were then generated from the cleaned events files, spatially
binned with a pixel size of 4.35′′. The images and exposure maps were
extracted in the 0.5 – 2.0 keV band, which is typical for soft band X-ray
image analysis. Individual camera images were merged to create a single image
per observation, likewise the exposure maps. The mos cameras were scaled to
the PN during the merging by the use of energy conversion factors (ECFs)
derived using the xspec (Arnaud, 1996) package. The ECFs were calculated based
upon an absorbed power-law model.

Using the merged images and exposure maps, we applied a bespoke wavdetect
(Freeman et al., 2002) based source detection routine, the XCS Automated
Pipeline Algorithm (xapa). Once the source detection stage was complete, xapa
proceeded to classify the resulting sources as either point-like or extended.
After removal of duplicates, a master source list (MSL) was generated. The MSL
used in this work contained a total of 326,294 X-ray sources, of which 35,575
were classified as extended detections.

###  2.3 Identifying SDSSRM clusters in the XCS footprint

The SDSSRM cluster sample (Sect 2.1) was compared to the footprint of the XCS
image archive (Sect 2.2). If a given RM centroid position fell within 15′ of
the aimpoint of one or more XMM observations, then that cluster was flagged as
having a preliminary XMM match. The matched list was then filtered based upon
the total exposure time, where the total exposure time is a combination of the
exposure times for each of the PN, MOS1 and MOS2 cameras, defined as
0.5$\times$PNexp+0.5$\times$(MOS1exp+MOS2exp). Only those clusters with a
total mean exposure (defined within a 5 pixel radius centered on the RM
position) of greater than 3ks, and a median exposure of greater than 1.5ks,
were retained in the match list. The median exposure limit excluded RM
clusters that had significant overlap with chip gaps or bad pixels. Next, an
additional exposure (mean and median) filter was carried out at a position
0.8Rλ away from the RM defined centre (in the direction away from the XMM
aimpoint). This was done to encapsulate the expected range of mis-centering
between RM and xapa centroids (see Zhang et al., 2019).

Based on these matching criteria, 1,246 SDSSRM clusters fall within the active
area of one or more XCS processed XMM observations. Hereafter, these 1,246
SDSSRM clusters are referred to as the ‘SDSSRM-XMM’ sample (see Table 1). We
then performed a visual inspection to remove clusters falling in observations
with abnormally high background levels (e.g. Figure 16(a)), and those that
were corrupted due to proximity to a very bright point source666Such sources
produce artifacts in the XMM images including readout trails and ghost images
of the telescope support structure. (e.g. Figure 16(b)). We removed 57
observations, therefore, after this filtering step, 1,189 clusters remained.
We denote this set as the ‘SDSSRM-XCS’ sample (see Table 1).

|
---|---

---

Figure 2: An example of a cluster in the SDSSRM-XMMext sample (Table 1). Top
left: SDSS optical image of the cluster XMMXCS J164020.2+464227.1 (SDSS
RMID=2), commonly referred to as Abell 2219, at a redshift of $z=0.23$ and
richness of $\lambda_{\rm RM}=$199\. The dashed yellow circle and the other
solid yellow shapes highlight the galaxies associated RMID=2. Galaxy members
of other nearby RM clusters nearby are circled in different colours (e.g.
green, turquoise and cream); Top right: XMM X-ray observation of the matched
XCS extended source. Green (red) regions highlight extended (point) XAPA
sources detected in the observation. Bottom middle: A zoom into the SDSS
optical image. The yellow circle (dashed), triangle, diamond, pentagon and
hexagon represent, respectively, the 1st, 2nd, 3rd, 4th and 5th most probable,
according to the RM algorithm, candidate for the central galaxy.

###  2.4 Cross matching the SDSSRM-XCS sample with XCS extended sources

Although all 1,189 SDSSRM-XCS clusters (Sect. 2.3) fall within the XCS defined
XMM footprint of SDSS, this doesn’t guarantee they are matched to an extended
xapa source. In this context a match was defined to mean that the respective
centroids were within $2$ $h^{-1}$ Mpc of each other, where the distance was
calculated assuming the RM cluster redshift. If more than one extended xapa
source met this criterion, we made the assumption that the closest (on the
sky) match was the correct association. By this definition, 782 – of the input
SDSSRM-XCS sample of 1,189 – were initially matched to an extended xapa source
(the remaining, 407 SDSSRM-XCS entries are discussed further in Sect. 2.4.2).

The 782 XCS extended sources matched to SDSS-RM clusters were then examined by
eye to exclude cases where the X-ray emission was unlikely to be physically
associated with the RM cluster in question. An example of a cluster that that
passed this test is shown in Figure 2 (one that did not is shown in Figure
17). The top left panel of Figure 2 shows, with yellow circles, all the
galaxies associated, by RM, with the cluster in question (other coloured
circles depict the galaxies associated with other RM clusters in the field).
In the bottom panel, the dashed circle highlights the position of the galaxy
defined by RM as the most likely central galaxy. The 2nd, 3rd, 4th and 5th
most likely candidates, are highlighted by the yellow triangle, diamond,
pentagon and hexagon respectively. The top right panel shows the XMM image of
the matched XCS extended source. Following the visual inspection process, only
456, of the 782 checked, clusters were retained. These 456 are referred to
hereafter as the ‘SDSSRM-XCSext’ sample, see Table 1 (the remaining 326
entries are discussed further in Sect. 2.4.2).

Table 1: Summary of the SDSSRM cluster sub-samples produced during the matching process described throughout Section 2 Sample | Brief description | # clusters | Relevant section
---|---|---|---
SDSSRM | SDSSRM DR8 clusters with a richness $\lambda_{\rm RM}>$20 | 66,028 |  $\S$ 2.1
SDSSRM-XMM |  | SDSSRM clusters fall within the active area of one or more
---
XCS processed XMM observations
1246 |  $\S$ 2.3
SDSSRM-XCS |  | As above, but after visual inspection to remove matches to
---
problematic XMM images
1189 |  $\S$ 2.4
SDSSRM-XCSext |  | SDSSRM-XCS clusters that are matched to an extended
---
XCS source
456 |  $\S$ 2.4
SDSSRM-XCSunm |  | SDSSRM-XCS clusters that are unmatched to an extended
---
XCS source
733 |  $\S$ 2.4.2

####  2.4.1 Accounting for incidences of redMaPPer mispercolations

The RM algorithm employs a process known a “percolation” that aims to assign
galaxies to the correct system when there are two or more RM clusters in close
proximity on the sky (Rykoff et al., 2014, $\S$ 9.3). However, sometimes this
process fails, with the result that RM assigns a low value of $\lambda_{\rm
RM}$ to a genuinely rich cluster when it is close (in projection) to a less
rich system, and vice versa. This RM failure mode is known as “mispercolation”
(see Hollowood et al., 2019). An example is shown in Figure 18. The yellow
circles in Figure 18 (a) highlight the galaxies associated with a
$\lambda_{\rm RM}=166$ RM cluster. From the distribution of the X-ray emission
of the system (Figure 18 (b)), it is clear that the large richness has been
incorrectly assigned to the low flux sub-halo of a nearby massive cluster
(incorrectly assigned a richness of $\lambda_{\rm RM}=20$).

During the visual inspection process that generated the SDSSRM-XCSext sample
(see Sect. 2.4), we identified three pairs of clusters affected by
mispercolation. In order to correct their $\lambda_{\rm RM}$ values, we
followed the method outlined in Hollowood et al. (2019), i.e. the originally
assigned $\lambda_{\rm RM}$ value for the main halo was manually switched with
that of the sub-halo. However, unlike Hollowood et al. (2019), we did not
remove the lower flux system from further analysis if $\lambda_{\rm
RM}\geq$20\. Table 5 provides properties of the clusters effected by
mispercolation. Of the 6 clusters effected by mispercolation, one is not
included in the final SDSSRM-XCSext sample as its richness has a value of
$\lambda_{\rm RM}<$20.

####  2.4.2 SDSSRM-XMM entries not associated with XCS extended sources

A total of 733 members of the SDSSRM-XCS sample are not included in the
SDSSRM-XCSext sample. This is because they have not been matched to an XCS
extended source. Of these, 407 are not close to any XCS extended source,
whereas the remaining 326 were close in projection, but were deemed, after the
eye-balling step, unlikely to be physically associated with it. Combined,
these 733 “unmatched” clusters are denoted as the SDSSRM-XCSunm sub-set. For
these clusters, we determine luminosity upper limits in Section 3.3 so that
they can be included in the scaling relation analysis presented in Section
4.3.

To better understand why certain clusters were not detected in their
respective XMM observation(s), we compared the distributions of their off-axis
distance, redshift and richness with those of the detected SDSSRM-${\rm
XCS}_{\rm ext}$ sample (see Figure 3). Here, we defined the off-axis distance
as the angular separation of the observation aimpoint to the RM defined
central galaxy: both the effective exposure time and the point spread function
(PSF) degrade significantly with off-axis distance. To emphasise the richness
difference between the two samples, the points are colour-coded by
$\lambda_{\rm RM}$. As expected, we find that the majority of SDSSRM-XCSunm
clusters fall at larger off-axis positions, higher redshifts, and lower
richnesses, than SDSSRM-XCSext clusters.

|
---|---
Figure 3: Distributions of off-axis distance, redshift ($z$) and richness
($\lambda_{\rm RM}$) for the SDSSRM-XCSext and SDSSRM-XCSunm subsets. The off-
axis distance is defined as the distance from the RM defined central galaxy to
the centre of the XMM observation. In each case, the points are colour-coded
by $\lambda_{\rm RM}$, given by the inset colour-bar.

###  2.5 False-Positive Rate

Figure 4: Distribution of the number of RMrd clusters matched to an extended
XCS source, repeated using 43 samples of random positions created to match the
size of the SDSS DR8 RM cluster sample (see 2.5).

In order to determine the false-positive rate of matches between the SDSS DR8
catalogue and the XCS MSL, we make use of the SDSSRM random
catalogue777http://risa.stanford.edu/redMaPPer/. Full details of the
construction of the random catalog can be found in Rykoff et al. (2014,
$\S$11). The random catalog is constructed such as to map the detectability of
clusters as a function of redshift and richness, taking into account the
large-scale structure that is already imprinted on the galaxy catalog. The
random catalogue contains $\approx$3$\times$106 clusters (which we denote as
RMrd), a factor $\sim$100 larger than the SDSS DR8 RM catalogue. We draw at
random from the RMrd clusters and create samples of equal size to the SDSS DR8
RM catalogue (i.e. 66,028 clusters), resulting in 43 separate catalogues of
RMrd clusters. For each separate catalogue, we first determined the number of
RMrd positions falling on an XMM observation using the method described in
Sect. 2.3 (i.e. a mean and median exposure cut of 3 ks and 1.5 ks
respectively). We note that the RMrd clusters do not contain a $R_{\lambda}$
estimate, therefore, we do not employ the additional exposure cut at a
position 0.8$R_{\lambda}$ away from the RM position (see Section 2.3). From
the 43 mock catalogues, we determined that, on average, 1548$\pm$33 RMrd
clusters fell inside the XMM footprint. Next, we matched the RMrd clusters to
XCS extended sources.

We defined a RMrd cluster to be associated with an extended source when the
centroid fell within the xapa detection region. Note that xapa provides
elliptical regions but, for this matching, we circularised the xapa region by
making the radius equal to the semi-major axis of the xapa source. Figure 4
shows the distribution of these associations for all 43 random catalogues.
Based upon a Gaussian fit to the distribution, we find we would, on average,
randomly match to an extended xapa source 22.8$\pm$5.0 times. We thus estimate
a contamination rate in the SDSSRM-XCS sample of $\simeq 1.5\%$ ($23/1548$).
We note that since we made the simplifying assumption of a RMrd match when
falling within a xapa (circularised) region, and no eyeballing performed, this
estimate is likely an upper limit.

##  3 X-ray Analysis of the SDSSRM-XCS sample

We used the XCS Post Processing Pipeline (XCS3P) to derive the X-ray
properties of the SDSSRM-XCSext clusters, i.e. their temperature ($T_{\rm X}$)
and luminosity ($L_{\rm X}$). XCS3P can be run in batch mode and applied to
hundreds of clusters at a time.

A detailed description of XCS3P can be found in LD11, but a brief overview is
as follows. Cluster spectra were extracted using the SAS tool evselect and fit
using xspec (Arnaud, 1996). The fits were performed in the 0.3-7.9 keV band
with an absorbed APEC model (Smith et al., 2001) using the $c$-statistic
(Cash, 1979). The APEC component accounts for the emission from a hot diffuse
gas enriched with various elements. Relative abundances of these elements are
defined as their ratio to Solar abundances ($Z_{\odot}$). The absorption due
to the interstellar medium was taken into account using a multiplicative Tbabs
model (Wilms et al., 2000) in the fit, with the value of the absorption
($n_{H}$) taken from HI4PI Collaboration et al. (2016) and frozen during the
fitting process. The abundance was fixed at 0.3 $Z_{\odot}$, a value typical
for X-ray clusters (Kravtsov & Borgani, 2012). The redshift was fixed to the
value as determined by RM. We note that redshift uncertainties are not taken
into account in the fit since the typical photometric redshift uncertainty for
SDSS RM clusters is very small ($\frac{\sigma_{z}}{1+z}\leq 0.04$ out to a
redshift of $z_{\lambda}$=0.6, see Fig. 9 in Rykoff et al., 2014). The APEC
temperature and normalisation were free to vary during the fitting process.
Temperature errors were estimated using the XSPEC ERROR command, and quoted
within 1-$\sigma$. Finally, luminosities (and associated 1-$\sigma$ errors)
were estimated from the best-fit spectra using the XSPEC LUMIN command (in
both the bolometric and 0.5–2.0 keV, rest-frame, bands).

###  3.1 Updates to XCS3P since LD11

Improvements have been made to XCS3P since LD11 was published, and these are
described in the sub-sections below.

####  3.1.1 Spectral extraction region

In LD11, the spectral extraction region was based on the xapa (see Section
2.2) characterized detection region i.e. an elliptical aperture defined using
the lengths of the xapa determined major and minor axes. The extraction region
has since been updated to be within a circular overdensity radius
($r_{\Delta}$). Overdensity radii are defined as the radius at which the
density is $\Delta$ times the critical density of the Universe at the cluster
redshift. We used two radii common in the X-ray cluster literature i.e.
$r_{\rm 500c}$ and $r_{\rm 2500c}$, where the radii were estimated using the
relation given in Arnaud et al. (2005):

$\displaystyle E(z)r_{\Delta}$ $\displaystyle=B_{\delta}\left(\frac{T_{\rm
X}}{5~{}{\rm keV}}\right)^{\beta},$ (1)

where $E(z)$=$\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}}$. In the case of
$r_{500c}$, $B_{\delta}$=1104 kpc and $\beta$=0.57. The process is iterative
because we do not know a priori what $T_{\rm X}$ is; an initial temperature
was calculated within the xapa defined elliptical source region which is then
used to estimate $r_{\rm 500c}$ (using Equ. 1). A new $T_{\rm X}$ value was
then measured from a spectrum extracted from a circular region with $r_{\rm
500c}$ radius. The new $T_{\rm X}$ was then used to define a new $r_{\rm
500c}$ value. The process was repeated until $r_{\rm 500c}$ converged (the
ratio of the new to old $0.9>r_{\rm 500c,new}/r_{\rm 500c,old}<1.1$). We
employed the condition that at least three iterations were performed,
regardless of the convergence. To account for the background in the spectral
analysis, we made use of a local background annulus centered on the cluster,
with an inner and outer radii of 1.05$r_{\rm 500c}$ and 1.5$r_{\rm 500c}$
respectively (see blue edged outer annulus in Fig 6). It is also beneficial to
compute core excluded properties for analysis (e.g. the use of core-excluded
luminosities reduces the scatter in the luminosity-mass relation, see Mantz et
al., 2018). Therefore, we repeat the process described above, but exclude the
inner 0.15$r_{\rm 500c}$ region (as used in many studies in the literature
e.g. Pratt et al., 2009; Maughan et al., 2012; Lovisari et al., 2020).

In the $r_{\rm 2500c}$ case, Equation 1 was used, with $B_{\delta}$=491 kpc
and $\beta$=0.56. The local background was taken into account using an annulus
centered on the cluster with an inner and outer radius of 2$r_{\rm 2500c}$ and
3$r_{\rm 2500c}$ respectively. In all other respects, the derivation of
$T_{\rm X,2500}$ values followed that used for the $T_{\rm X,500}$ values.

####  3.1.2 Selection of spectra

Figure 5: Comparison of the measured temperature when only including spectra
in the simultaneous fit that pass quality controls ($T_{X,filt}$, Sect.
3.1.2), and the temperature when using all available spectra in the
simultaneous fit ($T_{X,all}$, Sect. 3.1.2). The top plot shows the
distribution of $T_{X,all}$-$T_{X,filt}$ against $T_{X,filt}$, and the bottom
plot simply $T_{X,all}$ against $T_{X,filt}$. Grey squares highlight clusters
failing to converge during the iteration process (see Sect. 3.1.1) when
determining $T_{X,all}$. The remaining clusters are colour-coded by the ratio
of the number of spectra when including all available spectra to the number
passing the quality controls ($\frac{ns_{\rm all}}{ns_{\rm filt}}$). The black
dashed line represents the 1:1 relation.

In the LD11 version of XCS3P, all available spectra were used in a
simultaneous xspec fit ($ns_{\rm all}$). This included spectra derived from
each of the three (PN, MOS1 and MOS2) XMM cameras and, where available,
multiple XMM observations (up to 25 per cluster in some cases). However, we
have subsequently discovered that using all available spectra, irrespective of
data quality, can increase the measured temperature. This is demonstrated in
Figure 5 which compares the temperature estimated using all available spectra
($T_{X,all}$) to those determined by filtering out spectra that did not
($T_{X,filt}$), individually, produce a fitted temperature (complete with
$1\sigma$ upper and lower limit values) in the range 0.08 $<$ $T_{X}$ $<$ 20
keV. The number of available spectra after filtering is defined as $ns_{\rm
filt}$. In Figure 5 (top-plot), we show the distribution of the difference in
temperature ($T_{X,all}$-$T_{X,filt}$) against $T_{X,filt}$, with Figure 5
(bottom-plot) showing $T_{X,all}$ against $T_{X,filt}$. Each point represents
a cluster and is colour coded by the ratio of the number of spectra used when
determining $T_{X,all}$ and $T_{X,filt}$ (defined as $\frac{ns_{\rm
all}}{ns_{\rm filt}}$). Grey squares indicate clusters that do not fulfil the
criteria of a converged $r_{\rm 500c}$ temperature for the $T_{X,all}$
analysis. Therefore, by using a filtered sample, we were able to extract more
$T_{X}$ values. Moreover, where $T_{X,all}$ and $T_{X,filt}$ differ, the
former are systematically higher. This suggests that there is residual
background flaring in low signal to noise observations, because the particle
background has a hard spectrum. For these reasons, XCS3P now only uses
filtered spectra sets during the simultaneous fitting.

####  3.1.3 Measurement of luminosity uncertainties

When estimating the luminosity in xspec, the absorption component ($n_{H}$)
must be set to zero in order to represent conditions at the cluster (i.e.
unabsorbed). However, the luminosity uncertainties will be in error if they
are also determined while $n_{H}$ is set to zero, since the uncertainties are
determined from the spectral fit to the absorbed data. This error was present
in the LD11 version of XCS3P, and has now been corrected. In the latest
version of XCS3P, the uncertainties are determined using an initial luminosity
($L_{ini}$) calculation, before $n_{H}$ has been set to zero. Then, $n_{H}$ is
set to zero and the luminosity extracted ($L_{0}$). The uncertainties are then
scaled by the ratio of $L_{0}$ to $L_{ini}$.

Figure 6: Updated approach to source masking in XCS3P compared to LD11. In
each image the green ellipse represents the cluster extraction region, the
blue edged annulus represents the background region and the red small circles
are excluded point sources. The nearby extended source is excluded using the
red hashed ellipse. In the LD11 analysis, Figure (a), the exclusion region was
too large. This has been corrected in the current version of XCS3P (Figure b,
see Section 3.1.4).

####  3.1.4 Exclusion of extended sources

The method used in LD11 to exclude nearby extended sources (NES) sometimes
overestimated the area to ‘drill out’ around the NES, because the exclusion
area was scaled by number of NES counts. Figure 6 (a) highlights the region
used to exclude a NES in the LD11 analysis (red hashed ellipse). In this case,
the excluded region overlaps with the source extraction region (green circle),
removing a fraction of the source flux. Therefore, the scaling factor used in
LD11 has been deprecated, see Figure 6 (b).

###  3.2 Luminosity estimates when $T_{X}$ is fixed

Not all 456 clusters in the SDSSRM-XCSext sample yielded a reliable
temperature measurement. However, it was still possible to estimate a
luminosity value for them from the extracted spectra using an adapted version
of the iterative procedure outlined in Sect. 3. In this adaptation, the
temperature was fixed in the spectral fit. Initially, spectra were extracted
within the xapa defined region, and an XSPEC fit was performed with the
$T_{X}$ in the model fixed at 3 keV. This produced an initial luminosity value
which was fed into the luminosity-temperature relation presented in Sect. 4.1
(with parameters given in Table 3) to derive a more appropriate $T_{X}$ value.
An $r_{\rm 500c}$ was estimated using Equation 1 using this $T_{X}$ value and
a new spectrum was extracted and fit. The process was repeated until the
change in the $r_{\rm 500c}$ radius was less than within 10%. Luminosities
estimated in this way are denoted $L_{\rm Fixed~{}Tx,52}^{r500}$. To test the
validity of this method, we applied it to all clusters in the SDSSRM-XCSext
sample, of which 351 have a measured $L_{X}$ from the spectral extraction
method described above (throughout Sect. 3). In Figure 7, these luminosities
(estimated using a fixed $T_{X}$) are compared to the luminosities estimated
for the 382 clusters in the SDSSRM-XCS${}_{T_{X}}$ sample (see Sect 3.4, i.e.,
clusters where the luminosities were estimated from the spectral analysis with
$T_{X}$ free). The 1:1 relation is highlighted by the solid black line. This
comparison shows there is a good agreement between the two luminosity
estimates.

Figure 7: Comparison of luminosity determined using a fixed temperature
(L${}_{\rm Fixed~{}Tx,52}^{r500}$) estimated from the luminosity temperature
relation (see Sect. 3.2 for details), to that determined from spectra
extracted within $r_{500}$ (see Sect. 3). The solid black line indicates the
1:1 relation.

###  3.3 Upper limit estimates in the absence of an XCS detection

There are 733 SDSSRM-XCS clusters that have no corresponding confirmed match
to an XCS extended source (the SDSSRM-XCSunm sample, see Table 1). For these
systems, we calculated upper limit luminosities in the following way. First,
we assumed each RM cluster has a temperature of 3 keV and calculated $r_{\rm
500c}$ using Equation 1 (note we only estimated upper limits within $r_{\rm
500c}$). We used a fixed temperature of 3 keV for the upper limit analysis to
avoid bias coming from the correlation between the richness and luminosity (as
would happen if one were to estimate the temperature from $\lambda_{\rm RM}$).
We then measured a 3$\sigma$ upper limit on the count-rate within those
apertures, using the SAS tool eregionanalyse, which implements the method of
Kraft et al. (1991). Point and extended sources are masked out from the
analysis. The background region had radii with inner and outer values of
1.05$r_{\rm 500c}$ & 1.5$r_{\rm 500c}$ respectively.

In order to convert the count-rate upper limit into a luminosity upper limit,
we used an energy conversion factor (ECF). First, an Auxiliary Response File
(ARF) and Redistribution Matrix File (RMF) was produced at the position of the
RM cluster assuming the relevant overdensity radius. The ARF and RMF were then
used to generate a fake spectrum in xspec using the fakeit tool. The process
requires the use of a model with which to produce the fake spectrum, for which
we assumed a Tbabs$\times$APEC model (the same one used to estimate cluster
properties as in Sect. 3). We assumed an $nH$ calculated at the RM position,
the redshift as determined by RM, and the abundance fixed at 0.3 Z⊙. The
temperature was assumed to be 3 keV. An arbitrarily high exposure time, of 100
ks, was used to generate the spectrum. The ECF was then calculated as the
ratio between the count-rate and the measured flux from the fake spectrum.
Using this ECF, the count-rate upper limit is converted to a flux, and finally
converted to a luminosity upper limit. Using this method, we measure upper
limit luminosities for 599 of the SDSSRM-XCSunm sample (representing
$\approx$80% of the input sample)888The remaining clusters could not have an
upper limit measured since they fall on or near a chip gap, or the region was
masked due to the presence of a point source or un-associated extended source.

###  3.4 Introducing the various SDSSRM-XCS sub-samples

In Table 2 we overview the various sub-samples of SDSSRM clusters that have
been analysed in this work. The cluster sample that we use most (e.g.,
Sections 4.1, 4.2, 5.1) is known as ‘SDSSRM-XCS${}_{T_{X},vol}$’. It contains
151 clusters that have accurate temperature estimates, defined as having an
average percentage temperature error of $T_{X,err}<$25%, and falling in the
redshift range corresponding to the SDSSRM volume-limited sample (as estimated
in Rykoff et al., 2014, 0.1$\leq$ z $\leq$0.35). The SDSSRM-
XCS${}_{T_{X},vol}$ sample is a subset of the SDSSRM-XCS${}_{T_{X}}$ sample,
which contains 382 clusters with $T_{X,err}<$100% and no redshift limits
imposed (see Sect. 4.3).

The largest sub-sample of SDSSRM clusters with measured luminosities is known
as SDSSRM-XCS${}_{L_{X}}$ and contains 456 clusters (no $z$ limits imposed).
In this case, the $L_{X}$ values were estimated with a fixed (not fitted)
$T_{X}$ parameter (see Sect. 3.2). A subset of SDSSRM-XCS${}_{L_{X}}$ clusters
in the 0.1$\leq$ z $\leq$0.35 range has 178 entries and is known as SDSSRM-
XCS${}_{L_{X},vol}$.

Finally, the SDSSRM-XCS${}_{L_{X}}$ sample is supplemented with upper limit
luminosities determined for the SDSSRM-XCSunm sample. These luminosities are
added to the SDSSRM-XCS${}_{L_{X}}$ sample to create a sample of 1055
clusters, which we denote as the SDSSRM-XCS${}_{L_{X}+upper}$ subset (no $z$
limits). A subset of SDSSRM-XCS${}_{L_{X}+upper}$ clusters in the 0.1$\leq$ z
$\leq$0.35 range has 222 entries and is known as SDSSRM-
XCS${}_{L_{X}+upper,vol}$. This subset is used in the analyses presented in
Section 4.3.

In Appendix D, an outline of the sample table is provided (see Table 6),
giving details of the column headers and a description of each column. The
first entry of the table is provided for reference.

####  3.4.1 Comparison to the literature

Sample size

|
---|---
(a) | (b)
Figure 8: (a) Temperature distribution of the SDSSRM-XCS${}_{T_{X}}$ (grey
histogram) and SDSSRM-XCS${}_{T_{X},vol}$ (lightblue histogram) samples.
Distributions from Hollowood et al. (2019, brown line), Lovisari et al. (2020,
green line), Migkas et al. (2020, yellow line), Molham et al. (2020, cyan
line) and Liu et al. (2021, red line) are highlighted for comparison. Note
that Hollowood et al. (2019) and Migkas et al. (2020) temperatures are scaled
from Chandra to XMM using Rykoff et al. (2016) and Schellenberger et al.
(2015) respectively, and the Liu et al. (2021) temperatures are scaled by the
offset between eROSITA and XMM found in Turner et al. (2021). (b) Comparison
of the measured temperatures in the SDSSRM-XCS${}_{T_{X},vol}$ sample to
Hollowood et al. (2019, brown squares), Lovisari et al. (2020, green downward
triangles), Migkas et al. (2020, yellow stars) and Molham et al. (2020, cyan
diamonds), with 43, 20, 20 and 15 clusters in common respectively. In most
cases $T_{X}^{r500}$ is compared, however, in the case of Migkas et al. (2020)
we compare our $T_{X}^{(0.15-1)r_{500}}$ to their $T_{X}^{(0.2-0.5)r_{500}}$.
The 1:1 relation is given by the black solid line.

We have delivered one of the largest cluster samples with coherently measured
$T_{X}$ values to date. The only equivalent sample is the XCS First Data
release (Mehrtens et al., 2012, XCSDR1). XCSDR1 included 401 clusters with
measured temperatures distributed across the entire extragalactic sky (i.e.
extending beyond the SDSSDR8 footprint). In Figure 8(a), we show the number of
SDSSRM-XCS${}_{T_{X}}$ clusters per $T_{X}$ bin (the subset of 157 in the
SDSSRM-XCS${}_{T_{X},vol}$ subsample is highlighted in blue). The $T_{X}$
value distributions from a non-comprehensive list of other recently published
samples are over-plotted as spline curves, as described below:

  * •

The cyan curve shows the 57 XMM derived $T_{X}$ values for SDSS DR8 RM
clusters (spanning a redshift range of 0.14$<z<$0.59) in the Molham et al.
(2020, hereafter Mol20) sample. This sample is a subset of the X-CLASS catalog
(Clerc et al., 2012).

  * •

The red curve shows the 95 eROSITA derived $T_{X}$ values X-ray selected
clusters (spanning a redshift range 0.049$<z<$0.708) in the Liu et al. (2021,
hereafter L21)999Liu+21 sample. This sample is a subset of 542 clusters
extracted from the 140 deg2 contiguous eROSITA Final Equatorial-Depth Survey
(eFEDS). We note that the eFEDS $T_{X}$ values were derived from spectra
extracted from a circular $r<$ 500 kpc region, and that they have been scaled
by a factor of 1.25 to account for the measured $T_{X}$ offset between eROSITA
and XMM (Turner et al., 2021).

  * •

The brown curve shows the 97 Chandra derived $T_{X}$ values for SDSSRM
clusters ($0.1\leq z\leq 0.35$) in the Hollowood et al. (2019, hereafter
H19)101010Hollowood+ 19 sample sample. For the purposes of illustration, the
Chandra $T_{X}$ values are scaled to XMM using the calibration found in Rykoff
et al. (2016).

  * •

The green curve shows the 120 XMM derived $T_{X}$ values for Planck clusters
(spanning a redshift range of 0.059$<z<$0.546) in the Lovisari et al. (2020,
hereafter L20)111111Lovisari+ 20 sample. This sample is a subset of the Planck
Early Sunyaev-Zeldovich (Planck Collaboration et al., 2011) cluster catalog.

  * •

The yellow curve shows the 313 XMM and Chandra derived $T_{X}$ values for
X-ray selected clusters (spanning a redshift range 0.004$<z<$0.447, with 70%
at $z<$0.1) in the Migkas et al. (2020, hereafter Mig20) sample121212Migkas+
20 sample. This sample is a subset of the Meta-Catalogue of X-ray detected
Clusters of galaxies (MCXC Piffaretti et al., 2011). We note that the Chandra
values in Mig20 are scaled to XMM using Schellenberger et al. (2015, as used
in Mig20), and that all 313 $T_{X}$ values were derived from spectra extracted
from a (0.2-0.5)$r_{500}$ region.

Temperature estimates

To demonstrate the reliability of the $T_{X}$ values estimated in this work,
we have compared our values (using the SDSSRM-XCS${}_{T_{X},vol}$ sample) to
those for clusters in common with the H19, L20, Mig20 and Mol20 samples
mentioned above. There are 43, 20, 20 and 15 examples respectively. For the
purposes of this comparison, a Chandra-to-XMM scaling as been applied to the
H19 and Mig20 $T_{X}$ values as described above. Figure 8(b) plots the
comparison of the temperature for these three literature samples. The black
line shows the 1:1 relation, highlighting both that the various $T_{X}$
measurements are broadly consistent, and that the XCS values generally have
smaller errors.

Table 2: Summary of the SDSSRM cluster sub-samples used for scaling relation analysis in this work. Sample | Brief description | # clusters | Relevant sections
---|---|---|---
SDSSRM-XCS${}_{T_{X}}$ |  | SDSSRM-XCSext clusters with a measured temperature value
---
(with $T_{X,err}<$100%)
382 |  $\S$3.4
SDSSRM-XCS${}_{T_{X},vol}$ | As above, but limited to systems with 0.1$<$z$<$0.35 and $T_{X,err}<$25% | 151 |  $\S$3.4
SDSSRM-XCS${}_{L_{X}}$ |  | SDSSRM-XCSext clusters where the luminosity was measured
---
assuming a fixed temperature
456 |  $\S$3.2
SDSSRM-XCS${}_{L_{X},vol}$ | As above, but limited to systems with 0.1$<$z$<$0.35 | 178 |  $\S$3.2
SDSSRM-XCS${}_{L_{X}+upper}$ |  | SDSSRM-XCS${}_{L_{X}}$ sample, supplemented with upper limit luminosities
---
1055 |  $\S$3.2, 3.3
SDSSRM-XCS${}_{L_{X}+upper,vol}$ | As above, but limited to systems with 0.1$<$z$<$0.35 | 222 |  $\S$3.2, 3.3

##  4 Scaling relations derived from the SDSSRM-XCS samples

In this section we present the scaling relations derived from some of the
SDSSRM-XCS samples described in Section 3.4 and Table 2. In Sections 4.1 and
4.2, we focus on the sample with the most robustly measured X-ray properties
and that is restricted to the RM volume limited redshift range i.e., the
SDSSRM-XCS${}_{T_{X},vol}$ cluster sample (see Sect. 3.4). In Section 4.3 we
present fits to samples with less conservative cuts, to explore the relative
importance of sample size over measurement accuracy. Fits to the scaling
relations were performed in log space using the R package LInear Regression in
Astronomy (lira131313lira is available as an R package from
https://cran.r-project.org/web/packages/lira/index.html), fully described in
Sereno (2016). For these analyses we used core-included temperatures and soft
band luminosities within $r_{\rm 500c}$ unless otherwise stated. Temperature
and luminosities estimated in this way are denoted $T_{X}^{r500}$ and
$L_{X,52}^{r500}$ respectively.

Table 3: Best-fit parameters of the cluster scaling relations (see $\S$4.1 and $\S$4.2 for details). For each relation, parameters are given for the SDSSRM-XCS${}_{T_{X},vol}$ ($T_{X,err}<25\%$ and 0.1$\leq$z$\leq$0.35) cluster sample. Best-fit parameters are given for the $L_{X}-T_{X}$, $L_{X}-\lambda_{\rm RM}$ and $T_{X}-\lambda_{\rm RM}$ relations, given by equations 2, 3 and 4 respectively. Relation | Fit | Normalisation | Slope | Scatter | Figure
---|---|---|---|---|---
(sample) |  |  |  |  |
$L^{r500}_{X,52}-T^{r500}_{X}$ |  | $A_{LT}$ | $B_{LT}$ | $\sigma_{LT}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | LIRA | 0.97$\pm$0.06 | 2.63$\pm$0.12 | 0.68$\pm$0.04 | 9
| LINMIX | 0.98$\pm$0.06 | 2.63$\pm$0.12 | 0.69$\pm$0.03 | –
SDSSRM-XCS${}_{T_{X}}$ | LIRA | 0.94$\pm$0.04 | 2.49$\pm$0.08 | 0.64$\pm$0.03 |  11(a)
$L^{r500}_{X,bol}-T^{r500}_{X}$ |  | $A_{LbT}$ | $B_{LbT}$ | $\sigma_{LbT}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | LIRA | 3.05$\pm$0.18 | 3.07$\pm$0.12 | 0.68$\pm$0.04 | –
$L^{r500}_{X,52}-\lambda_{\rm RM}$ |  | $A_{L\lambda}$ | $B_{L\lambda}$ | $\sigma_{L\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | LIRA | 0.98$\pm$0.09 | 1.61$\pm$0.14 | 1.07$\pm$0.06 |  10(a)
| LINMIX | 0.98$\pm$0.09 | 1.62$\pm$0.14 | 1.08$\pm$0.06 | –
SDSSRM-XCS${}_{L_{X+upper},vol}$ | LIRA | 1.08$\pm$0.10 | 1.84$\pm$0.12 | 1.09$\pm$0.06 |  11(b)
$T^{r500}_{X}-\lambda_{\rm RM}$ |  | $A_{T\lambda}$ | $B_{T\lambda}$ | $\sigma_{T\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | LIRA | 1.01$\pm$0.03 | 0.59$\pm$0.04 | 0.33$\pm$0.02 |  10(b)
| LINMIX | 1.01$\pm$0.03 | 0.59$\pm$0.05 | 0.33$\pm$0.01 | –

###  4.1 The Luminosity-Temperature relation derived from the SDSSRM-
XCS${}_{T_{X},vol}$ sample

Figure 9: Luminosity-Temperature relation of the SDSSRM-XCS${}_{T_{X},vol}$
subset (blue circles). The best-fit to the data (see Sect. 4.1) is represented
by the black solid line, and the lightblue shaded region represents the 68%
confidence interval. The grey bands represent the 1, 2, and 3$\sigma$
intrinsic scatter.

The $L^{r500}_{X,52}-T^{r500}_{X}$ relation is shown in Figure 9. The SDSSRM-
XCS${}_{T_{X},vol}$ data points are shown as blue circles. A power law
relation between $L^{r500}_{X,52}$ and $T^{r500}_{x}$ is fit to the data,
which we express as

$\displaystyle{\rm
log}\left(\frac{L^{r500}_{X,52}}{E(z)^{\gamma_{LT}}L_{0}}\right)$
$\displaystyle={\rm log}(A_{LT})+B_{LT}{\rm
log}\left(\frac{T^{r500}_{X}}{T_{0}}\right)\pm\sigma_{L|T},$ (2)

where $A_{LT}$ denotes the normalisation, $B_{LT}$ the slope, $\gamma_{LT}$
the evolution with redshift and $\sigma_{L|T}$ the intrinsic scatter. Note
that the intrinsic scatter is given in natural log space and can be
interpreted as the fractional scatter. We assumed $T_{0}=4$ keV,
$L_{0}=0.8\times 10^{44}$ erg s-1 (roughly the median values for the SDSSRM-
XCS${}_{T_{X},vol}$ sample) and a self-similar evolution of the relation where
$\gamma_{LT}=1$. The fit to the SDSSRM-XCS${}_{T_{X},vol}$ sample is
highlighted by the blue solid line in Figure 9, with the lightblue shaded
region representing the 68% uncertainty. The grey bands represent the 1, 2 and
3$\sigma$ intrinsic scatter. This scaling relation was used to estimate
luminosities when the $T_{X}$ was fixed, rather than fitted (see Sect. 3.2).
The best-fit lira parameters of the $L^{r500}_{X,52}-T^{r500}_{X}$ relations
are given in Table 3. For comparison, we performed a fit using the linmix
routine (Kelly, 2007), with best-fit parameters also given in Table 3. Many
literature studies using X-ray luminosities determine relations using the
bolometric luminosity. Therefore, we also fitted the bolometric luminosity -
temperature ($L^{r500}_{X,bol}-T^{r500}_{X}$) relation, with the best-fit
parameters from given in Table 3.

###  4.2 The X-ray observable-Richness relations derived from the SDSSRM-
XCS${}_{T_{X},vol}$ sample

|
---|---
(a) | (b)
Figure 10: Luminosity-richness (a) and temperature-richness (b) relation of
the SDSSRM-XCS${}_{T_{X},vol}$. For each relation, the best-fit to the data
(see Sect. 4.2) is represented by the blue solid line, and the light blue
shaded region represents the 68% confidence interval of the mean logarithmic
relation. The grey bands represent the 1, 2, and 3$\sigma$ intrinsic scatter.

The $L_{X,52}^{r500}-\lambda_{\rm RM}$ and $T_{X}^{r500}-\lambda_{\rm RM}$
relations for the SDSSRM-XCS${}_{T_{X},vol}$ (steel blue circles) are shown in
Figures 10(a) and (b) respectively. We fit for the
$L_{X,52}^{r500}-\lambda_{\rm RM}$ and $T_{X}^{r500}-\lambda_{\rm RM}$
relations, again, which we express as:

$\displaystyle{\rm
log}\left(\frac{L_{X,52}^{r500}}{E(z)^{\gamma_{L\lambda}}L_{0}}\right)$
$\displaystyle={\rm log}(A_{L\lambda})+B_{L\lambda}{\rm
log}\left(\frac{\lambda_{\rm RM}}{\lambda_{0}}\right)\pm\sigma_{L|\lambda},$
(3)

$\displaystyle{\rm log}\left(\frac{T_{X}^{r500}}{T_{0}}\right)$
$\displaystyle={\rm log}(A_{T\lambda})+B_{T\lambda}{\rm
log}\left(\frac{\lambda_{\rm RM}}{\lambda_{0}}\right)\pm\sigma_{T|\lambda},$
(4)

where $A_{L\lambda}$ and $A_{T\lambda}$ denote the normalisations,
$B_{L\lambda}$ and $B_{T\lambda}$ represent the slopes and
$\sigma_{T|\lambda}$ and $\sigma_{L|\lambda}$ denote the intrinsic scatters
(once again the values are given in natural log space). We assumed
$L_{0}=0.8\times 10^{44}$ erg s-1 in equation 3 and $T_{0}=4$ keV in equation
4, and in both relations assumed $\lambda_{0}=60$ (again, all roughly
corresponding to the median values for the SDSSRM-XCS${}_{T_{X},vol}$ sample).
Self-similar evolution for each relation is assumed such that
$\gamma_{L\lambda}=1$ in equation 3. We note that the $E(z)$ correction
cancels out in the $T_{X}-\lambda_{\rm RM}$ relation (hence the absence of the
$E(z)$ parameter in Equ. 4). The best-fit lira parameters for each relation
are given in Table 3 (again, linmix parameters are also provided for
comparison) and the best-fit relations are given by the blue solid lines in
Figures 10(a) and (b), with the 68% uncertainty given by the light blue shaded
region. The grey bands represent the 1, 2 and 3$\sigma$ intrinsic scatter. A
comparison of these results to those the literature are presented in Section
4.4.

In summary, we find that the measured scatter of the $L_{X}-\lambda_{\rm RM}$
relation is roughly three times that of the $T_{X}-\lambda_{\rm RM}$. This is
not due to measurement error (indeed the percentage errors on the $L_{X}$
values are much smaller than those on the $T_{X}$ values) but likely because
non gravitational physics impacts the luminosity to a much greater extent than
it does the temperature. Even expanding the sample of $L_{X}$ values by a
large factor (as will be possible with the eROSITA All Sky Survey Predehl et
al. 2021) will not bring the scatter down below that shown in Figure 10(a).

###  4.3 Scaling relations with all available X-ray data

The SDSSRM-XCS${}_{T_{X},vol}$ sample only contains a fraction of the X-ray
information available for SDSSRM-XCS clusters. In this section we investigate
whether there is a benefit to including additional clusters with less precise
individual measurements.

To explore the impact of $T_{X}$ measurement errors on the derived
$L_{X}-T_{X}$ relation (see Sect. 4.1), we have added all 382 clusters with a
measured $T_{X}$ value in the SDSS-XCS${}_{T_{X}}$ sample. The results are
shown in Figure 11(a) and best-fit parameters given in Table 3. It is clear
that there is no significant change in the fitted relation when less accurate
$T_{X}$ values are included. There is some marginal benefit to including more
clusters in the fit (e.g. the scatter drops a little, although not
significantly).

To explore the impact of $L_{X}$ measurement errors, and, to some extent,
sample incompleteness, on the derived luminosity-richness relation, we make
use of luminosities estimated with a fixed temperature (see Sect. 3.2) for all
456 clusters in the SDSSRM-XCSext sample, combined with luminosity upper
limits (Section 3.3) where available. The results are shown in Figure 11(b)
and best-fit parameters given in Table 3. It is clear that when less accurate
$L_{X}$ values, and upper limits, are included that the measured scatter goes
up a little, but does not change significantly. However, there are perceptible
changes to the slope and normalisation, which are likely a result of a
combination of the change in $L_{X}$ measurement method, and in the selection
function.

In summary, it is probably worthwhile including all available $T_{X}$ values
when assessing $T_{X}-\lambda$ scatter for cosmological studies, i.e. the
fitted parameters are robust to both measurement errors and selection effects.
However, one should exercise more caution when using $L_{X}-\lambda$
relations. The impact of selection on the $L_{X}-\lambda$ relation will be
explored in Upsdell et al. (prep), which explores completeness and
contamination in the low $\lambda$ regime using an XCS analysis of contiguous
XMM survey regions (totalling $\sim$57 deg2) that overlap with the DES Year 3
data release (Abbott et al., 2018).

|
---|---
(a) | (b)
Figure 11: (a) Luminosity-temperature relation of the SDSSRM-XCS${}_{T_{X}}$
sample (blue circles). The best-fit to the data is represented by the black
solid line, and the lightblue shaded region represents the 68% uncertainty.
(b) Luminosity-richness relation of the SDSSRM-XCS${}_{L_{X}+upper,vol}$
subset. Clusters are given by the blue points and luminosity upper limits for
SDSSRM-XCS without an XCS detection given by the black circles (and downward
arrows). The black solid line represents a fit to the data (including upper
limits) with the lightblue shaded region highlighting the 68% uncertainty. In
each plot, the grey bands represent the 1, 2, and 3$\sigma$ intrinsic scatter.

###  4.4 Comparison to the literature

|
---|---
(a) | (b)
|
(c) | (d)
Figure 12: Comparison of the SDSSRM-XCS${}_{T_{X},vol}$ temperature-richness
relation to previously published relations. In each case, the black solid line
represents a fit to the SDSSRM-XCS${}_{T_{X},vol}$ data (as given in Table 3)
and the light-blue shaded region the corresponding 68% uncertainty. Comparison
to the fit provided in Rozo & Rykoff (2014, green dashed line and light-green
shaded region highlighting the 68% uncertainty) using the R14ACCEPT sample
(note the relation has been scaled to XMM temperatures using Rykoff et al.,
2016) is shown in (a). Comparison to the data given in Hollowood et al. (2019)
is shown plot (b), scaled to XMM temperatures using Rykoff et al. (2016), with
the LIRA fit to the data given by the brown dash line and the 68% uncertainty
given by the brown shaded region. Comparison to the data given in Oguri et al.
(2018) is shown in plot (c), with the LIRA fit to the data given by the purple
dash line and 68% uncertainty given by the light-purple shaded region. Note,
the richness is estimated from CAMIRA, denoted $N_{\rm mem(CAMIRA)}$.
Comparison to the data given in Farahi et al. (2019, dark-red triangles) is
shown in plot (d), with the LIRA fit given by the dark-red dashed line and the
68% uncertainty given by the red shaded region (note the agreement between the
fits obscures much of the comparison, and in this case the $T_{\rm X}$ values
were measured in a $r_{2500}$ apertures).

Figure 19(b) and (d) demonstrate that $T_{X}$ estimates values are insensitive
to the details of the measurement process, be that the extraction aperture, or
the inclusion of the cluster core. Furthermore, the comparisons in Figure 8(b)
show consistency of our measured $T_{X}$ to those in the literature.
Therefore, we can have confidence that comparisons of the $T_{X}$-$\lambda$
scaling relations presented in Section 4.2 with those available in the
literature will be meaningful. We do not make similar comparisons to relation
involving $L_{X}$ since $L_{X}$ estimates can vary significantly for a given
cluster depending on the adopted methodology, see Figures 19(a) and (c).
Furthermore, relations involving $L_{X}$ are more dependent on sample
selection than $T_{X}$.

We first compare to the $T_{X}$-$\lambda_{\rm RM}$ scaling relations presented
in Rozo & Rykoff (2014, hereafter RR14). These are based on SDSSDR8 RM
clusters ($0.1<z<0.3$), and so the $\lambda$ values are consistent with those
used herein. Two samples are presented in RR14, one contains 25 XMM derived
$T_{X}$ values taken from the first XCS data release (Mehrtens et al., 2012),
hereafter the RR14XCS sample. The other contains 54 Chandra derived $T_{X}$
values, hereafter the RR14ACCEPT sample. These 54 are a subsample of the 329
clusters in the ACCEPT database (Cavagnolo et al., 2009). The input data
vectors used in the RR14 are not available, therefore the comparison here is
limited to the fitted relations (taken from Table 2 of that paper). It is
important to note that, for the ACCEPT sample, the RR14 fit was scaled to
account for the offset in Chandra and XMM temperature measurements (using
Rykoff et al., 2016). As expected (given that the $T_{X}$ methodology was very
similar to that used herein), there is excellent agreement in the case of the
RR14XCS sample. The fit to the RR14ACCEPT sample is also consistent with that
to the SDSSRM-XCS${}_{T_{X},vol}$ sample. However, as can be seen in Figure
12(a), the slope is steeper (although at $<$3$\sigma$). There is consistency
in normalisation at the pivot point ($\lambda_{0}$=60). This contrary to that
found in RR14, who found a $\approx$40% difference between their fits to
RR14ACCEPT and to R14XCS because RR14 did not carry out any Chandra to XMM
$T_{X}$ scaling.

In Figure 12(b) we compare our $T_{X}$-$\lambda$ scaling relation to that
derived from the H19 sample of 97 SDSSRM clusters (brown curve/points in
Figure 8). In this case, the input data vector was available, so we were able
to perform a new fit following the approach in Section 4.2, i.e with
$\lambda_{0}$=60 and $T_{0}$=4 keV, to maximise uniformity in the method. The
comparison of the data and fits are given in Figure 12(c, with the appropriate
Chandra and XMM $T_{X}$ scaling applied). The H19 data are given by the brown
squares, with the LIRA fit given by the brown dashed line (and brown shaded
region highlighting the 68% uncertainty). We obtain fit parameters of the
normalisation and slope of $A_{T\lambda,H19}=1.16\pm 0.04$ and
$B_{T\lambda,H19}=0.50\pm 0.05$ respectively. There is a small (14%) offset in
normalisation at the pivot point ($\lambda_{0}$=60) significant at the
2.9$\sigma$ level. While not significant, we assess the impact of the choice
of Chandra-to-XMM temperature scaling on the above comparison. Therefore, we
rescaled the H19 temperatures to XMM using the scaling found in Schellenberger
et al. (2015) and re-fit the H19 $T_{X}$-$\lambda$ relation. We obtain fit
parameters of $A_{T\lambda,H19}=1.23\pm 0.03$ and $B_{T\lambda,H19}=0.44\pm
0.05$. The offset offset in normalisation increases to $\approx$20%,
significant at the 5.1$\sigma$ level. This highlights the potential difficulty
of combining Chandra and XMM data. However, we note, one cannot exclude the
effects of differences in selection between the two archival samples.

In Figure 12(c) we compare our $T_{X}$-$\lambda$ scaling relation to that
based on the CAMIRA analysis of Hyper Suprime-Cam (HSC) observations Oguri et
al. (2018). The CAMIRA algorithm is similar to RM, in that it identifies
clusters using the red-sequence, but the estimated richness values will
differ. The $T_{X}$-$\lambda$ scaling relation analysis based on 50 CAMIRA
clusters is presented in Oguri et al. (2018), where the input $T_{X}$ values
were derived from XMM observations. For these 50 clusters, 34 $T_{X}$ values
were taken from Giles et al. (2016) and 16 $T_{X}$ values taken from Clerc et
al. (2014). Again, we were able to refit the input data using the approach in
Section 4.2, as they were kindly made available to us via priv. comm. by the
authors. We obtain fit parameters of the normalisation and slope of
$A_{T\lambda,O18}=1.04\pm 0.10$ and $B_{T\lambda,O18}=0.56\pm 0.11$
respectively. Figure 12(d) compares the SDSSRM-XCS${}_{T_{X},vol}$
$T_{X}-\lambda_{\rm RM}$ relation and the fit to the CAMIRA data (given by the
purple diamonds with the best-fit relation given by the purple dashed line and
light purple shaded region the 68% uncertainty). We note that richness is
defined as $N_{\rm mem}$ in Oguri et al. (2018), but we keep the notation of
$\lambda_{RM}$ in Figure 12(c) for clarity in the comparisons. As seen in
Figure 12(c), the two relation are fully consistent, albeit with the caveat
that $\lambda_{RM}\neq N_{mem}$.

Finally, we compare to the $T_{X}$-$\lambda_{\rm RM}$ scaling relations
presented in Farahi et al. (2019, hereafter F19). The relations are
constrained using RM clusters detected within 1500 deg2 of the DES (using the
1st year of DES observations Drlica-Wagner et al., 2018). DES RM clusters were
matched to XMM detected clusters using the same processes outlined in this
work, resulting in a sample of 110 clusters used for the $T_{X}$-$\lambda_{\rm
RM}$ scaling analysis. The clusters fall within 0.2$<$z$<$0.7 and do not
contain a temperature error cut (unlike in the SDSSRM-XCS${}_{T_{X},vol}$
sample). Furthermore, the temperatures are determined within r2500. The input
data vector was obtained, and the $T_{X}^{2500}$-$\lambda_{\rm RM}$ relation
fit following Section 4.2 i.e., with $\lambda_{0}=$60 and $T_{0}=$4 keV. The
comparison of the data and fits are given in Figure 12(d). The F19 data are
given by the dark-red triangles, with the LIRA fit given by the dark-red
dashed line (and red shaded region the 68% uncertainty). The SDSSRM-
XCS${}_{T_{X},vol}$ relation is mostly obscured by the F19 fit, because the
results are so consistent.

In summary, the results presented here (and in Sect. 3.4.1) are consistent
with those in the literature and based on the largest compilation of $T_{X}$
and $\lambda_{\rm RM}$ data to date. Furthermore, the extremely consistent
comparison between this work and the results in F19 (Fig 12(d)), highlights
that our sample can be combined with clusters from the DES for further
analysis (see further discussion in Sect. 5.2).

##  5 Discussion

As mentioned above (see Sect. 3.4), we have compiled one of the largest
samples of consistently derived $T_{X}$ values to date. This allows us to
explore factors that might influence measured (as opposed to intrinsic)
scaling relations. For example, in Sect. 5.1, we explore the impact of
selection on the relations, specifically the difference between targeted and
serendipitous detections. And, in Sect. 5.2, we investigate the recent claims
of an anisotropy across the sky in the measured $L_{X}-T_{X}$ relation (Migkas
et al., 2020).

###  5.1 The dependence of scaling relations on detection type (targeted or
serendipitous)

Table 4: Best-fit parameters of the cluster scaling relations for the SDSSRM-XCS${}_{T_{X},vol}$ cluster sample, split between the targeted and serendipitous sub-samples (as defined in Sect. 5.1). Best-fit parameters are given for the $L_{X}-T_{X}$, $L_{X}-\lambda_{\rm RM}$ and $T_{X}-\lambda_{\rm RM}$ relations, given by equations 2, 3 and 4 respectively. Relation | Fit | Normalisation | Slope | Scatter | Figure
---|---|---|---|---|---
(sample) |  |  |  |  |
$L^{r500}_{X,52}-T^{r500}_{X}$ |  | $A_{LT}$ | $B_{LT}$ | $\sigma_{LT}$ |
SDSSRM-XCS${}_{T_{X},vol}$ |  |  |  |  |
Targets | LIRA | 1.04$\pm$0.09 | 2.63$\pm$0.20 | 0.74$\pm$0.06 |  13(a)
Serendipitous | LIRA | 0.66$\pm$0.09 | 2.00$\pm$0.22 | 0.52$\pm$0.06 |  13(a)
$L^{r500}_{X,52}-\lambda_{\rm RM}$ |  | $A_{L\lambda}$ | $B_{L\lambda}$ | $\sigma_{L\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ |  |  |  |  |
Targets | LIRA | 1.42$\pm$0.16 | 1.13$\pm$0.19 | 1.06$\pm$0.07 |  14(a)
Serendipitous | LIRA | 0.48$\pm$0.09 | 1.23$\pm$0.27 | 0.79$\pm$0.08 |  14(a)
$T^{r500}_{X}-\lambda_{\rm RM}$ |  | $A_{T\lambda}$ | $B_{T\lambda}$ | $\sigma_{T\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ |  |  |  |  |
Targets | LIRA | 1.14$\pm$0.03 | 0.45$\pm$0.05 | 0.27$\pm$0.02 |  14(b)
Serendipitous | LIRA | 0.81$\pm$0.06 | 0.50$\pm$0.12 | 0.34$\pm$0.04 |  14(b)
---
(a)
(b)
Figure 13: Luminosity-temperature relation of the SDSSRM-XCS${}_{T_{X},vol}$ sample, split between targeted (red circles) and serendipitous (blue triangles) clusters. The fiducial core-included $r_{500}$ relations are shown in Plot (a, $L_{X,52}^{r500}-T_{X}^{r500}$), and the core-excluded relation in Plot (b, $L_{X,52}^{(0.15-1)r500}-T_{X}^{(0.15-1)r500}$). The red and blue lines represent a fit to the targeted and serendipitous clusters respectively (fit using equation 2). The shaded regions around the respective lines represents the 68% confidence interval of the mean logarithmic relation. |
---|---
(a) | (b)
|
(c) | (d)
Figure 14: X-ray observable-richness relations of the SDSSRM-
XCS${}_{T_{X},vol}$ sample, split between targeted (red circles) and
serendipitous (blue triangles) clusters. Figure (a) plots the
$L_{X,52}^{r500}-\lambda_{\rm RM}$ relation and Figure (b) plots the
$T_{X}^{r500}-\lambda_{\rm RM}$ relation, using the fiducial core included
properties. Figure (c) plots the $L_{X,52}^{(0.15-1)r500}-\lambda_{\rm RM}$
relation and Figure (d) plots the $T_{X}^{(0.15-1)r500}-\lambda_{\rm RM}$
relation, using core excluded properties. In each case, the red line and blue
line represent a fit to the targeted and serendipitous clusters respectively.
Shaded regions around each fit highlights the 1$\sigma$ uncertainty.

We have separated the SDSSRM-XCS${}_{T_{X},vol}$ clusters into those that were
the target of their respective XMM-Newton and those that were detected
“serendipitously”. The classification was done based upon a visual inspection
of the X-ray images and information from the XMM-Newton Science Archive
(namely the target name and target type). Of the 151 SDSSRM-
XCS${}_{T_{X},vol}$ clusters, 98 were flagged as being XMM-Newton targets, and
53 as serendipitous detections. We then revisited the scaling relations
presented in Table 3. The results are presented in Table 4, plotted in Figures
13(a), 14(a) and 14(b). In all cases, the measured normalisation of the
targeted sub-sample is higher than that of the serendipitous sub-sample
(ranging between 2.9 to 5.1-$\sigma$). This remains true even when the
emission from the cluster cores is excluded, see Table 7 and Figures 13(b),
14(c) and 14(d). While the measured slope of the
$L_{X,52}^{r500}-T_{X}^{r500}$ differs, it is only significant at the
$\approx$2$\sigma$ level. There is very little change in the richness scaling
relations.

The current data are not sufficient to draw a firm conclusion as to the cause
of these differences. However, they are unlikely to be due to a systematic in
the XCS analysis methods, i.e. whereby biases in measured $L_{X}$ or $T_{X}$
values are correlated with location on the detector: LD11 studied the effect
of measuring temperatures for the same clusters that were detected at a high
off-axis position and then re-observed at the on-axis aimpoint. LD11 found a
1:1 relationship between the measured temperatures, finding no systematic
offset (see Fig 18 in LD11).

Instead, we suggest the cause is due to incompleteness in the sub-samples.
There is a dearth of X-ray bright objects in the serendipitous sub-sample
because these clusters are intrinsically very rare and so have a low projected
sky density: a small area serendipitous survey is unlikely to come across them
by accident. In the targeted sample, many XMM (and Chandra) targets were
historically drawn from samples detected by the ROSAT All-Sky Survey (which
had a relatively bright flux limit) and followed-up clusters with a high
luminosity. Figure A1 in Mantz et al. 2010 demonstrates how biases
(specifically a luminosity limit) can significantly flatten the measured slope
of a scaling relation. In addition, both sub-samples are incomplete at the low
flux end due to biases in selection. It is possible to model the impact of
incompleteness (as was done in Mantz et al. 2010) but is beyond the scope of
this work. The true normalisation and slope of the $L_{X}-\lambda$ relation
should be uncovered by the X-ray selected samples from the eRASS project, but
in the mean time it would be prudent to use only $T_{X}-\lambda$ relations for
cosmological studies (as these are the least impacted by the sub-sample
choice, see Figure 14,(b) & (d)).

###  5.2 Investigating $L_{X}-T_{X}$ isotropy with the SDSS-XCSext sample

Recently, Mig20 made a claim relating to a possible anisotropy across the sky
in the luminosity-temperature relation (Migkas et al., 2020). This claim, if
true, would add additional systematics and uncertainty when using cluster
number counts as a cosmological probe. The Mig20 result was based on 313
clusters with measured $T_{X}$ values (the yellow curve in Figure 8(a)). These
313 are made up of a compilation of both XMM and re-scaled Chandra $T_{X}$
values. So we felt it was worthwhile to re-explore the Mig20 result using the
larger (382) SDSSRM-XCS${}_{T_{X}}$ sample of clusters, with $T_{X}$ values
drawn only from one telescope. Additional motivation comes from the results
presented in Section 5.1, the difference in normalisation seen in Figure 13 is
larger than that presented in Mig20.

To demonstrate the robustness of our technique, we first repeated the analysis
presented in Mig20, using the same input data vectors. In brief, the method is
as follows: The sky is binned into regions over the full range of galactic
longitude ($l$) and latitude ($b$), using a bin width of $\Delta l$=1∘ and
$\Delta b$=1∘ (creating 65,160 bins on the sky). At the centre of each bin, a
cone with a radius $\theta$ is used to find a subset of all clusters within an
angular separation of $\theta_{d}$ from the coordinates of the bin. Using this
subset, the ‘local’ $L_{X}-T_{X}$ relation is fit using LIRA, following the
same method as described in Sect. 4.1. Following Mig20, a statistical
weighting is applied to each cluster in the subset by increasing the size of
the uncertainties by a factor

$\displaystyle cos(\theta_{d}/\theta\times 90^{\circ}).$ (5)

At each position on the sky, the local $L_{X}-T_{X}$ normalisation, $A$, is
divided by the normalisation of the all sky $L_{X}-T_{X}$ relation ($A_{\rm
all}$), with sky maps plotted based upon $A$/$A_{\rm all}$. In Figure 20, we
replicate the results presented in Mig20 Figure 8 for the $\theta$=60∘ cone
(thus confirming that our method is robust). This test also shows that the
dipole feature is present irrespective of the linear regression fitting method
used. Whereas we used LIRA, the Mig20 analysis used a fitting method
equivalent to the BCES Y|X fitting method (Akritas & Bershady, 1996).

We then apply the same method to the SDSSRM-XCS${}_{T_{X}}$ sample. Note we
use core excluded properties for this, in line with Mig20, who used
(0.2-0.5)$r_{500}$ values. The ratio of $A$/$A_{\rm all}$ over the sky is then
determined where there are $>$30 clusters in the bin. Figures 15(a) and (b)
displays the sky distribution of $A$/$A_{\rm all}$, assuming cones of
$\theta$=60∘ and $\theta$=75∘ respectively. The $\theta$=60∘ cone was chosen
as the dipole feature found in Mig20 is the most prominent at this scale. The
$\theta$=75∘ cone was chosen to increase the sky coverage. Based upon the
distribution of $A$/$A_{\rm all}$ (Fig. 15), we do not observe the anisotropy
feature found in Mig20 using either cone search radius. However, it is not
possible to rule out the existence of an anisotropy feature because there is a
region in the Southern sky where we are not able to measure $A$/$A_{\rm all}$
because SDSS is a northern survey. The strong edge features around the empty
area correspond to local regions where all the clusters have an angular
separation of $\geq$55∘. Assuming Equation 5, and $\theta_{d}$>55∘, the
uncertainties on the measured cluster properties are divided by $\leq$0.13.
The resulting local $L_{X}-T_{X}$ relation thus becomes unconstrained.

In summary, while our sample size is larger than the one presented in Mig20,
and we have replicated the results using Mig20 data, further data is required
due to the SDSS sky coverage. For this, the sample used here will be combined
with RM clusters detected from the DES Y3 Gold catalogue (Sevilla-Noarbe et
al., 2021) to improve the sky coverage. This technique (of combining SDSS-RM
and DES-RM clusters) has been successfully applied in Wetzell et al. (2021) to
measure the correlations between velocity dispersion, $\lambda_{\rm RM}$,
$T_{\rm X}$ and $L_{\rm X}$ for RM clusters. The results shown in Figure 12(d)
also give us confidence that SDSS-RM and SDSS-DES cluster samples can be
jointly analysed.

|
---|---
(a) | (b)
Figure 15: Ratio of the local best fit
$L^{(0.15-1)r_{500}}_{X,52}-T^{(0.15-1)r_{500}}_{X}$ normalisation ($A$) to
the all sky normalisation ($A_{\rm all}$), as a function of sky position. Maps
are created assuming at each sky position, clusters within a cone of
$\theta$=60∘ (a) and $\theta$=75∘ (b) are used for the local relation. The
black line highlights the Galactic plane.

##  6 Summary

In this paper we detail the X-ray analysis of SDSS DR8 redMaPPer (SDSSRM)
clusters using data products from the XMM Cluster Survey (XCS). In summary:

  * •

In total, 1189 SDSSRM clusters fall within the cleaned XMM-Newton footprint.
This has yielded 456 confirmed detections accompanied by X-ray luminosity
($L_{X}$) measurements. Using an updated version of the XCS Post Processing
Pipeline (XCS3P), we have extracted 382 X-ray temperature measurements
($T_{X}$) from these 459 clusters. This represents one of the largest samples
of coherently derived cluster $T_{X}$ values to date. We have also shown that
the reliability of derived $T_{X}$ values improves when low quality spectra
are removed from joint fits.

  * •

We find that the SDSSRM clusters in the XMM footprint that were not detected
are primarily lower richness systems (75% at $\lambda<30$). It was possible to
estimate $L_{X}$ upper limits for most 599 (of 733) of these non-detections.

  * •

Our analysis of the X-ray observable to richness scaling relations has
demonstrated that scatter in the $T_{X}-\lambda$ relation is roughly a third
of that in the $L_{X}-\lambda$ relation, and that the $L_{X}-\lambda$ scatter
is intrinsic, i.e. will not be significantly reduced with larger sample sizes.

  * •

Our analysis of the scaling relation between $L_{X}$ and $T_{X}$ has shown
that the fits are sensitive to the selection method of the sample, i.e.
whether the sample is made up of clusters detected “serendipitously” compared
to those deliberately targeted by XMM. These differences are also seen in the
$L_{X}-\lambda$ relation and, to a lesser extent, in the $T_{X}-\lambda$
relation. Exclusion of the emission from the cluster core does not make a
significant impact to the findings. A combination of selection biases is a
likely, but as yet unproven, reason for these differences.

  * •

We have used our data to probe recent claims of anisotropy in the
$L_{X}-T_{X}$ relation across the sky (Migkas et al., 2020). We find no
evidence of anistropy, but stress that this may be masked in our analysis by
the incomplete declination coverage of the SDSS DR8 sample.

The methods outlined in this work have further been employed in the analysis
of large cluster samples, such as those constructed from the RM analysis of
the Dark Energy Survey data e.g. (Zhang et al., 2019; Farahi et al., 2019).
Although optically selected samples are free from X-ray selection biases, when
matching to available X-ray data, future archival studies should consider only
the use of serendipitously detected X-ray clusters to avoid observer biases.
Furthermore, future use of the XMM Cluster Survey data will be of critical
importance for upcoming cluster samples such as those constructed from the
Legacy Survey of Space and Time undertaken by the Vera C. Rubin Observatory,
of which currently $\approx$450 deg2 of the LSST sky has been covered by XMM.

## Data availability

The data underlying this will be made available once the paper has been
refereed.

## Acknowledgements

PG, RW, KR and SB recognises support from the Science and Technology
Facilities Council. MH acknowledges financial support from the National
Research Foundation. We thank M. Sereno for useful discussions on the use of
the LIRA fitting package. We thank M. Oguri for providing the CAMIRA data used
in Figure 12(c).

## References

  * Abbott et al. (2018) Abbott T. M. C., et al., 2018, ApJS, 239, 18
  * Abbott et al. (2019) Abbott T. M. C., et al., 2019, Phys. Rev. D, 100, 023541
  * Aguena et al. (2021) Aguena M., et al., 2021, MNRAS, 502, 4435
  * Aihara et al. (2011) Aihara H., et al., 2011, ApJS, 193, 29
  * Akritas & Bershady (1996) Akritas M. G., Bershady M. A., 1996, ApJ, 470, 706
  * Arnaud (1996) Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society of the Pacific Conference Series Vol. 101, Astronomical Data Analysis Software and Systems V. p. 17
  * Arnaud et al. (2005) Arnaud M., Pointecouteau E., Pratt G. W., 2005, A&A, 441, 893
  * Cash (1979) Cash W., 1979, ApJ, 228, 939
  * Cavagnolo et al. (2009) Cavagnolo K. W., Donahue M., Voit G. M., Sun M., 2009, ApJS, 182, 12
  * Clerc et al. (2012) Clerc N., Pierre M., Pacaud F., Sadibekova T., 2012, MNRAS, 423, 3545
  * Clerc et al. (2014) Clerc N., et al., 2014, MNRAS, 444, 2723
  * Dodelson et al. (2016) Dodelson S., Heitmann K., Hirata C., Honscheid K., Roodman A., Seljak U., Slosar A., Trodden M., 2016, arXiv e-prints, p. arXiv:1604.07626
  * Drlica-Wagner et al. (2018) Drlica-Wagner A., et al., 2018, ApJS, 235, 33
  * Farahi et al. (2016) Farahi A., Evrard A. E., Rozo E., Rykoff E. S., Wechsler R. H., 2016, MNRAS, 460, 3900
  * Farahi et al. (2018) Farahi A., Evrard A. E., McCarthy I., Barnes D. J., Kay S. T., 2018, MNRAS, 478, 2618
  * Farahi et al. (2019) Farahi A., et al., 2019, MNRAS, 490, 3341
  * Freeman et al. (2002) Freeman P. E., Kashyap V., Rosner R., Lamb D. Q., 2002, ApJS, 138, 185
  * Giles et al. (2016) Giles P. A., et al., 2016, A&A, 592, A3
  * Grandis et al. (2021) Grandis S., et al., 2021, MNRAS, 504, 1253
  * HI4PI Collaboration et al. (2016) HI4PI Collaboration et al., 2016, A&A, 594, A116
  * Hollowood et al. (2019) Hollowood D. L., et al., 2019, ApJS, 244, 22
  * Kelly (2007) Kelly B. C., 2007, ApJ, 665, 1489
  * Kraft et al. (1991) Kraft R. P., Burrows D. N., Nousek J. A., 1991, ApJ, 374, 344
  * Kravtsov & Borgani (2012) Kravtsov A. V., Borgani S., 2012, ARA&A, 50, 353
  * Le Brun et al. (2014) Le Brun A. M. C., McCarthy I. G., Schaye J., Ponman T. J., 2014, MNRAS, 441, 1270
  * Lieu et al. (2016) Lieu M., et al., 2016, A&A, 592, A4
  * Liu et al. (2021) Liu A., et al., 2021, arXiv e-prints, p. arXiv:2106.14518
  * Lloyd-Davies et al. (2011) Lloyd-Davies E. J., et al., 2011, MNRAS, 418, 14
  * Lovisari et al. (2020) Lovisari L., et al., 2020, ApJ, 892, 102
  * Mantz et al. (2010) Mantz A., Allen S. W., Ebeling H., Rapetti D., Drlica-Wagner A., 2010, MNRAS, 406, 1773
  * Mantz et al. (2016) Mantz A. B., et al., 2016, MNRAS, 463, 3582
  * Mantz et al. (2018) Mantz A. B., Allen S. W., Morris R. G., von der Linden A., 2018, MNRAS, 473, 3072
  * Maughan et al. (2012) Maughan B. J., Giles P. A., Randall S. W., Jones C., Forman W. R., 2012, MNRAS, 421, 1583
  * McClintock et al. (2019) McClintock T., et al., 2019, MNRAS, 482, 1352
  * Mehrtens et al. (2012) Mehrtens N., et al., 2012, MNRAS, 423, 1024
  * Migkas et al. (2020) Migkas K., Schellenberger G., Reiprich T. H., Pacaud F., Ramos-Ceja M. E., Lovisari L., 2020, A&A, 636, A15
  * Molham et al. (2020) Molham M., et al., 2020, MNRAS, 494, 161
  * Oguri (2014) Oguri M., 2014, MNRAS, 444, 147
  * Oguri et al. (2018) Oguri M., et al., 2018, PASJ, 70, S20
  * Piffaretti et al. (2011) Piffaretti R., Arnaud M., Pratt G. W., Pointecouteau E., Melin J. B., 2011, A&A, 534, A109
  * Planck Collaboration et al. (2011) Planck Collaboration et al., 2011, A&A, 536, A8
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A24
  * Pratt et al. (2009) Pratt G. W., Croston J. H., Arnaud M., Böhringer H., 2009, A&A, 498, 361
  * Predehl et al. (2021) Predehl P., et al., 2021, A&A, 647, A1
  * Romer et al. (1999) Romer A. K., Viana P. T. P., Liddle A. R., Mann R. G., 1999, ArXiv Astrophysics e-prints,
  * Romer et al. (2001) Romer A. K., Viana P. T. P., Liddle A. R., Mann R. G., 2001, ApJ, 547, 594
  * Rozo & Rykoff (2014) Rozo E., Rykoff E. S., 2014, ApJ, 783, 80
  * Rykoff et al. (2014) Rykoff E. S., et al., 2014, ApJ, 785, 104
  * Rykoff et al. (2016) Rykoff E. S., et al., 2016, ApJS, 224, 1
  * Sahlén et al. (2009) Sahlén M., et al., 2009, MNRAS, 397, 577
  * Schellenberger et al. (2015) Schellenberger G., Reiprich T. H., Lovisari L., Nevalainen J., David L., 2015, A&A, 575, A30
  * Sereno (2016) Sereno M., 2016, MNRAS, 455, 2149
  * Sevilla-Noarbe et al. (2021) Sevilla-Noarbe I., et al., 2021, ApJS, 254, 24
  * Smith et al. (2001) Smith R. K., Brickhouse N. S., Liedahl D. A., Raymond J. C., 2001, ApJ, 556, L91
  * The LSST Dark Energy Science Collaboration et al. (2018) The LSST Dark Energy Science Collaboration et al., 2018, arXiv e-prints, p. arXiv:1809.01669
  * Turner et al. (2021) Turner D. J., et al., 2021, arXiv e-prints, p. arXiv:2109.11807
  * Upsdell et al. (prep) Upsdell E., et al., in prep
  * Wetzell et al. (2021) Wetzell V., et al., 2021, arXiv e-prints, p. arXiv:2107.07631
  * Wilms et al. (2000) Wilms J., Allen A., McCray R., 2000, ApJ, 542, 914
  * Wu et al. (2010) Wu H.-Y., Rozo E., Wechsler R. H., 2010, ApJ, 713, 1207
  * Zhang et al. (2019) Zhang Y., et al., 2019, MNRAS, 487, 2578

##  Appendix A Examples of problematic XMM observations

Here we show examples of SDSSRM clusters that were removed from the SDSSRM-XMM
sample due to high levels of background, Fig 16(a), and strong point source
contamination, Fig 16(b). See Section 2.3 for further details.

|
---|---
(a) | (b)
Figure 16: Examples of problematic XMM observations found during the visual
inspection process described in Sect 2.3. (a) Soft-band field-of-view image of
the XMM observation ObsiD=0556213801. The observation was rejected from
further analysis because it is effected by periods of high background rates;
(b) XMM image in the 0.5-2.0 keV band of the region surrounding SDSSRM cluster
RMID=42060 (centroid indicated with the dashed yellow circle). The cluster
falls nearby to a bright point source that has created artifacts in the image
(characterised by the “spokes”) and was subsequently dropped from the cluster
sample.

##  Appendix B Example of a cluster excluded from the SDSSRM-XCS sample after
visual inspection

Here we show an example of SDSSRM-XCS clusters that were initially matched to
an extended XCS source, but after visual inspection (see Sect. 2.4), the X-ray
emission was found not to be associated with the RM cluster. In Figure 17, the
SDSSRM-XCS cluster has been matched to an extended source where the X-ray
emission comes from an outflow from a low redshift galaxy. The extended XCS
source was deemed un-associated with the SDSSRM cluster in question.

|
---|---
(a) | (b)
Figure 17: An example of SDSS-RM cluster that is not physically associated
with nearest extended X-ray source. (a) SDSS optical image of the cluster SDSS
RMID=55078, $z=0.39$, $\lambda=26$. The yellow dashed circle shows the
position of the RM defined central galaxy (remaining circles/shapes as defined
in Fig 2); (b) Corresponding XMM X-ray observation. Green (red) outlines
highlight xapa extended (point) sources. In this case the extended emission is
associated with an outflow from a nearby galaxy

##  Appendix C Clusters effected by mispercolation

In Section 2.4.1, we identified three pairs of clusters effected by
mispercolation. In Figure 18, an example of a mispercolated cluster is shown,
and Table 5 highlights the three pairs of clusters effected by mispercolation
and detail manual adjustments made to their properties.

|
---|---
(a) | (b)
Figure 18: Example of a mispercolated cluster (as described in Sect. 2.4.1). Each image measures 36${}^{\prime}\times$36′ on a side. (a) SDSS optical image with the yellow circles highlighting galaxies associated with the lower flux cluster ($\lambda_{\rm RM}=166$). Pink circles highlight galaxies RM associated with the higher flux cluster ($\lambda_{\rm RM}=20$). Other RM clusters nearby are highlighted by their respective colours (the colourbar inlay gives the redshift and richness of the highlighted clusters); (b) Corresponding XMM image. Table 5: Clusters effected by mispercolation. RMID | $\lambda_{\rm RM}$ | z | XCS match | swap $\lambda_{\rm RM}$ | Notes
---|---|---|---|---|---
9 | 151 | 0.32 | XMMXCS J100213.9+203222.7 | 15 | Dropped from sample
12 | 15 | 0.32 | XMMXCS J100227.5+203102.1 | 151 | Retained, $\lambda_{\rm RM}$ swapped with RMID 9
21 | 39 | 0.30 | XMMXCS J092021.2+303014.5 | 129 | Retained, $\lambda_{\rm RM}$ swapped with RMID 23
23 | 129 | 0.29 | XMMXCS J092052.5+302803.5 | 39 | Retained, $\lambda_{\rm RM}$ swapped with RMID 21
34 | 166 | 0.30 | XMMXCS J231148.8+034046.7 | 20 | Retained, $\lambda_{\rm RM}$ swapped with RMID 41
41 | 20 | 0.30 | XMMXCS J231132.6+033759.9 | 166 | Retained, $\lambda_{\rm RM}$ swapped with RMID 34

##  Appendix D Information available in the SDSSRM-XCS online database

Table 6 provides the header of the table along with a description of each
column. The first entry of the table is provided for reference.

Table 6: Properties of the SDSSRM-XCS sample. The headers for the online table
and the column descriptions are outlined here, along with an example from the
first entry in the table.

RMID | XCSID |  XMM ObsID | R.A.RM | Dec.RM | R.A.XCS | Dec.XCS
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
EXPeff | $z_{RM}$ | Rλ | $\lambda_{RM}$ | $r_{\rm 500c}$ | $T_{X}^{r500}$ | $L_{X,52}^{r500}$
(8) | (9) | (10) | (11) | (12) | (13) | (14)
$L_{X,bol}^{r500}$ | $L_{X52,fixedT_{X}}^{r500}$ | $L_{X,52,upper}^{r500}$ | SDSSRM-XCSext |  |  |
(15) | (16) | (17) | (18) |  |  |
2 | XMMXCS J164020.2+464227.1 | 0605000501 | 250.08 | 46.71 | 250.08 | 46.71
21 | 0.233 | 1.15 | 199 | 1505 | 10.4${}^{+0.2}_{-0.2}$ | 9.8${}^{+0.4}_{-0.4}$
50.2${}^{+0.6}_{-0.6}$ | 9.9${}^{+0.04}_{-0.04}$ | – | Y |  |  |

(1): redMaPPer member ID; (2): XCS Name; (3): XMM observation ID,
corresponding to the ObsID with the largest effective exposure time; (4)
redMaPPer R.A. (deg); (5): redMaPPer Declination (deg); (6) R.A. of matched
extended XCS source (deg); (7): Declination of matched extended XCS source
(deg); (8): Effective exposure time at position of redMaPPer cluster (ks);
(9): redMaPPer photometric redshift; (10): Scale-radius $R_{\lambda}$, where
$R_{\lambda}=1.0(\lambda/100)^{0.2}h^{-1}$ (Mpc); (11): redMaPPer richness;
(12): Overdensity radius $r_{\rm 500c}$ (kpc); (13): X-ray temperature within
$r_{\rm 500c}$ (keV); (14) X-ray luminosity in the 0.5-2.0 keV band (rest-
frame) within $r_{\rm 500c}$ (1044 erg s-1); (15) Bolometric X-ray luminosity
within $r_{\rm 500c}$ (1044 erg s-1); (16) X-ray luminosity in the 0.5-2.0 keV
band (rest-frame) within $r_{\rm 500c}$ assuming a fixed $T_{X}$ from the
$L_{X}-T_{X}$ relation; (17) Luminosity upper limit; (18) SDSSRM cluster
matched to an XCS extended source (Y/N).

##  Appendix E Additional scaling relation fits

Table 7: Best-fit parameters of the scaling relations studied in this work when considering core excluded and $r_{2500}$ cluster properties. In each case, parameters are given for the SDSSRM-XCS${}_{T_{X},vol}$ cluster sample, and the targeted and serendipitous sub-samples (as defined in Sect. 5.1). Best-fit parameters are given for the $L_{X}-T_{X}$, $T_{X}-\lambda_{\rm RM}$, and $L_{X}-\lambda_{\rm RM}$ relations, given by equations 2, 3 and 4 respectively (see Sects. 4.1 and 4.2). Relation | Normalisation | Slope | Scatter | Figure
---|---|---|---|---
(sample) |  |  |  |
Core-excluded relations
$L_{X,52}^{(0.15-1)r500}-T_{X}^{(0.15-1)r500}$ | $A_{LT}$ | $B_{LT}$ | $\sigma_{LT}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 0.74$\pm$0.03 | 2.46$\pm$0.10 | 0.51$\pm$0.04 | –
Targets | 0.73$\pm$0.05 | 2.58$\pm$0.16 | 0.53$\pm$0.04 |  13(b)
Serendipitous | 0.54$\pm$0.07 | 1.84$\pm$0.21 | 0.43$\pm$0.06 |  13(b)
$L_{X,52}^{(0.15-1)r500}-\lambda_{\rm RM}$ | $A_{L\lambda}$ | $B_{L\lambda}$ | $\sigma_{L\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 0.79$\pm$0.06 | 1.49$\pm$0.12 | 0.88$\pm$0.06 | –
Targets | 1.06$\pm$0.10 | 1.13$\pm$0.16 | 0.88$\pm$0.07 |  14(c)
Serendipitous | 0.42$\pm$0.07 | 1.15$\pm$0.25 | 0.66$\pm$0.08 |  14(c)
$T_{X}^{(0.15-1)r500}-\lambda_{\rm RM}$ | $A_{T\lambda}$ | $B_{T\lambda}$ | $\sigma_{T\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 1.04$\pm$0.03 | 0.58$\pm$0.05 | 0.32$\pm$0.02 | –
Targets | 1.17$\pm$0.04 | 0.43$\pm$0.05 | 0.26$\pm$0.02 |  14(d)
Serendipitous | 0.80$\pm$0.07 | 0.46$\pm$0.13 | 0.34$\pm$0.04 |  14(d)
r2500 relations
$L_{X,52}^{r2500}-T_{X}^{r2500}$ | $A_{LT}$ | $B_{LT}$ | $\sigma_{LT}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 0.57$\pm$0.04 | 2.89$\pm$0.13 | 0.71$\pm$0.05 | –
Targets | 0.68$\pm$0.06 | 2.69$\pm$0.19 | 0.71$\pm$0.06 | –
Serendipitous | 0.44$\pm$0.07 | 2.56$\pm$0.33 | 0.62$\pm$0.08 | –
$L_{X,52}^{r2500}-\lambda_{\rm RM}$ | $A_{L\lambda}$ | $B_{L\lambda}$ | $\sigma_{L\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 0.57$\pm$0.06 | 1.69$\pm$0.15 | 1.14$\pm$0.07 | –
Targets | 0.88$\pm$0.11 | 1.15$\pm$0.20 | 1.13$\pm$0.09 | –
Serendipitous | 0.43$\pm$0.07 | 1.60$\pm$0.24 | 0.66$\pm$0.03 | –
$T_{X}^{r2500}-\lambda_{\rm RM}$ | $A_{T\lambda}$ | $B_{T\lambda}$ | $\sigma_{T\lambda}$ |
SDSSRM-XCS${}_{T_{X},vol}$ | 1.01$\pm$0.03 | 0.59$\pm$0.04 | 0.30$\pm$0.02 |  12(d)
Targets | 1.10$\pm$0.04 | 0.49$\pm$0.05 | 0.29$\pm$0.02 | –
Serendipitous | 0.85$\pm$0.06 | 0.42$\pm$0.11 | 0.27$\pm$0.04 | –
|
---|---
(a) | (b)
|
(c) | (d)
Figure 19: Comparison of cluster properties measured for our fiducial analysis
(i.e. core-included, $r_{500}$). Plots (a) and (b) compare the core excluded
(0.15-1.0$r_{500}$) luminosities and temperatures respectively. Plots (c) and
(d) compare $r_{2500}$ luminosities and temperatures respectively. In each
plot, the 1:1 relation is highlighted by the solid black line.

In Table 7, we present results of our cluster scaling relation analyses using
other apertures: core-excluded (0.15-1)$r_{500}$, and $r_{2500}$. The X-ray
properties derived from these apertures are compared to those from our
fiducial $r_{\rm 500c}$ analysis in Figure 19.

##  Appendix F Replicating the observed $L_{X}-T_{X}$ anisotropy

In Section 5.2, we show the results of our investigation into the possible
anisotropic behaviour of the $L_{X}-T_{X}$ relation using the SDSS-
XCS${}_{T_{X}}$ cluster sample. While we conclude that the SDSS-
XCS${}_{T_{X}}$ sample does not have the required sky coverage to probe such
effects, here, we show that the method (adopted from Mig20) indeed replicates
the results shown in Mig20. Cluster data was obtained from Mig20 and using the
replicated method (see Sect. 5.2), the results shown in Figure 20.

Figure 20: Ratio of the local best fit $L_{X}-T_{X}$ normalisation ($A$) to
the all sky normalisation ($A_{\rm all}$), as a function of sky position. The
data used to create the map was taken from Mig20 (Migkas et al., 2020), hence
replicating the results presented therein (see Mig20 Fig 8). The map is
generated using cones of $\theta$=60∘ at each sky position. The black solid
line represents the Galactic plane.