11institutetext:  Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse 1, 85748 Garching, Germany
Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg,
Germany
IRAP, Universitt’e de Toulouse, CNRS, UPS, CNES, F-31028 Toulouse, France
Institute for Astro- and Particle Physics, University of Innsbruck,
Technikerstr. 25, 6020 Innsbruck, Austria
Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482
Potsdam, Germany
Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, the
Netherlands
Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121
Bonn, Germany

# The SRG/eROSITA All-Sky Survey

First catalog of superclusters in the western Galactic hemisphere ††thanks:
Tables 1, 2, A.1, and A.2 are only available in electronic form at the CDS via
anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

A. Liu e-mail<EMAIL_ADDRESS>E. Bulbul 11 M. Kluge 11 V. Ghirardini 11
X. Zhang 11 J. S. Sanders 11 E. Artis 11 Y. E. Bahar 11 F. Balzer 11 M.
Brüggen 22 N. Clerc 33 J. Comparat 11 C. Garrel 11 E. Gatuzz 11 S. Grandis 44
G. Lamer 55 A. Merloni 11 K. Migkas 6677 K. Nandra 11 P. Predehl 11 M. E.
Ramos-Ceja 11 T. H. Reiprich 77 R. Seppi 11 S. Zelmer 1111223344556677

Superclusters of galaxies mark the large-scale overdense regions in the
Universe. Superclusters provide an ideal environment to study structure
formation and to search for the emission of the intergalactic medium such as
cosmic filaments and WHIM. In this work, we present the largest-to-date
catalog of X-ray-selected superclusters identified in the first SRG/eROSITA
All-Sky Survey (eRASS1). By applying the Friends-of-Friends (FoF) method on
the galaxy clusters detected in eRASS1, we identified 1338 supercluster
systems in the western Galactic hemisphere up to redshift 0.8, including 818
cluster pairs and 520 rich superclusters with $\geq 3$ members. The most
massive and richest supercluster system is the Shapley supercluster at
redshift 0.05 with 45 members and a total mass of $2.58\pm 0.51\times
10^{16}M_{\odot}$. The most extensive system has a projected length of 127
Mpc. The sizes of the superclusters we identified in this work are comparable
to the structures found with galaxy survey data. We also found a good
association between the eRASS1 superclusters and the large-scale structures
formed by optical galaxies. We note that 3948 clusters, corresponding to
$45\%$ of the cluster sample, were identified as supercluster members. The
reliability of each supercluster was estimated by considering the
uncertainties in the redshifts of the galaxy clusters and the peculiar
velocities of clusters. Furthermore, 63% of the systems have a reliability
larger than 0.7. The eRASS1 supercluster catalog provided in this work
represents the most extensive sample of superclusters selected in the X-ray
band in terms of the unprecedented sample volume, sky coverage, redshift
range, the availability of X-ray properties, and the well-understood selection
function of the parent cluster sample, which enables direct comparisons with
numerical simulations. This legacy catalog will greatly advance our
understanding of superclusters and the cosmic large-scale structure.

###### Key Words.:

Galaxies: clusters: general – X-rays: galaxies: clusters – large-scale
structure of Universe

##  1 Introduction

In the hierarchical structure formation picture, galaxy clusters emerge from
the rarest and highest density peaks of initial fluctuations and become
gravitationally bound after going through several merging and accretion
processes (see Kravtsov & Borgani 2012, for a review). While clusters are
widely used to study cosmology and structure formation, their spatial
distribution also traces the cosmic overdense regions on a larger scale, such
as superclusters. As components of the cosmic web, clusters and superclusters
mark the density peaks in the large-scale structure. They are connected by
other elements, such as filaments, sheets, and walls, and are separated by
low-density regions, such as cosmic voids.

Superclusters are groups of galaxy clusters consisting of more than one member
cluster. Superclusters will not necessarily collapse in the future as they are
not gravitationally bound, and the physical connections between their member
clusters are relatively weak compared to virialized systems. Except for some
cases of merging clusters, in most of the supercluster systems, their members
have not yet interacted with other members and lie beyond the virial radius of
each other. These features make it difficult to define and identify
superclusters in observations quantitatively. Although the concept of a
supercluster was introduced about seventy years ago (de Vaucouleurs 1953), a
precise and widely accepted definition of superclusters is still absent. Tully
et al. (2014) and Einasto et al. (2019) proposed to define superclusters based
on their dynamic effect on the cosmic environment, calling them “basins of
attraction” and “cocoons;” Chon et al. (2015) suggested to call the
superclusters that will survive the cosmic expansion and eventually collapse
in the future “superstes-clusters,” to distinguish them from traditional
superclusters. In fact, some traditional superclusters, such as Laniakea
(Tully et al. 2014), are essentially compilations of several smaller
superclusters, and the whole system will disperse in the future (Chon et al.
2015).

Despite the difficulties in defining superclusters, there have been numerous
works aiming at finding superclusters based on optical galaxy surveys or
optical-selected cluster catalogs for decades (see, e.g., Abell 1961; Zucca et
al. 1993; Einasto et al. 1994, for some early attempts). Most of the works
where superclusters have been identified with galaxy catalogs used the method
of galaxy density field. They computed the density of galaxies across the sky
and searched for peaks of galaxy densities. For example, Einasto et al. (2007)
found 543 superclusters with the Two-degree-Field Galaxy Redshift Survey data
by selecting the peaks in the galaxy density field. Using the Sloan Digital
Sky Survey Data Release 7 galaxy catalog, Liivamägi et al. (2012) constructed
a set of supercluster catalogs by searching for regions with densities over a
selected threshold. Using a fixed density threshold and an adaptive local
density threshold, they found 982 and 1313 superclusters, respectively. On the
other hand, the works where superclusters were identified directly from
cluster catalogs prefer the Friends-of-Friends (FoF) method, mostly due to the
difficulties in precisely computing the density field of clusters as they are
much rarer than galaxies. For example, Sankhyayan et al. (2023) identified 662
superclusters in the redshift range [0.05, 0.42] by applying a modified FoF
method on the Wen-Han-Liu (WHL) SDSS cluster catalog (Wen et al. 2012). In
summary, the number of known optical superclusters up to now is on the order
of 103.

Compared to optical cluster surveys, X-ray cluster surveys have the advantage
in terms of sample purity, as the former often suffers from projection effects
(e.g., Costanzi et al. 2019; Myles et al. 2021). As clusters are more highly
biased than galaxies (see, e.g., Seppi et al. 2024), fewer of them are needed
to significantly trace the same underlying structure. Thanks to the
availability of large-area X-ray surveys, in the past decades, there have been
an increasing number of efforts in detecting superclusters directly using
X-ray cluster samples (Einasto et al. 2001; Chon et al. 2013; Adami et al.
2018; Böhringer & Chon 2021; Liu et al. 2022). The first X-ray flux-limited
supercluster sample was constructed by Chon et al. (2013), with the extended
ROSAT-ESO Flux-Limited X-ray (REFLEX II) galaxy cluster sample. They found 164
superclusters below redshift 0.4. In recent years, supercluster catalogs have
become important products of X-ray cluster surveys. In the XXL survey, Adami
et al. (2018) detected 35 superclusters with $N_{\rm cl}\geq 3$, and 39
cluster pairs, out to redshift 0.8. In the eROSITA Final Equatorial-Depth
Survey (eFEDS, Brunner et al. 2022), Liu et al. (2022) detected 84
superclusters including 19 rich systems with $N_{\rm cl}\geq 4$. Böhringer &
Chon (2021) identified eight superclusters in the local Universe ($z\leq
0.03$) from the Cosmic Large-Scale Structure in X-rays cluster survey,
compiled from the X-ray clusters in the ROSAT All-Sky Survey.

Most of these works used the FoF algorithm to identify superclusters, where
the linking length was determined based on a specific local overdensity ratio
at a specific redshift with respect to the sample average. It should be noted,
however, that the overdensity ratio $f$ can vary in different works (e.g.,
Chon et al. 2013; Böhringer & Chon 2021), and the detection of superclusters
is sensitive to the choice of $f$ (see, e.g., Chon et al. 2013). In the nearby
Universe, since the X-ray cluster surveys are almost complete at high fluxes,
one can also use the cluster X-ray luminosity function (XLF) to compute the
local overdensity with respect to the cosmic average (e.g., Böhringer & Chon
2021), which would ideally give the same results as using the sample average.
Other methods, such as the Voronoi tessellation, have also been used in the
identification of superclusters and are found to give similar results as FoF
(Adami et al. 2018). Recently, another novel technique based on the dendrogram
of galaxies was developed and applied to the identification of clustering
patterns of stars and galaxies (Liu et al. 2018; Yu & Hou 2022), which also
has promising prospects of application in supercluster detection.

Figure 1:  Mass and redshift distribution of the cluster sample used for
supercluster identification. Blue, yellow, and red histograms indicate
photometric, spectroscopic, and literature redshifts, respectively. The black
dashed line shows the redshift distribution of the overall eRASS1 cluster
sample.

Figure 2:  Cluster number density ($N/V$, left panel) and linking length ($l$,
right panel) plotted as functions of both redshift and exposure time, computed
using Eq. 5, with $f=10$.

A large and representative sample of superclusters will advance the studies on
both cosmology and astrophysics in many aspects. As the largest elements in
the cosmic web, superclusters retain the history of the formation and
evolution of the web. As “clusters of clusters”, they are ideal laboratories
to investigate the environmental effect on the evolution of galaxy clusters
and galaxies. Superclusters can also play essential roles in the search for
warm-hot intergalactic medium (WHIM) thanks to their association with cosmic
structures. Moreover, nonthermal physics such as magnetic fields and cosmic
rays can be studied in these dynamic systems. The amount of attempts to
directly use superclusters as cosmology probes is quite limited. This is owing
mostly to the difficulties in precisely identifying superclusters, such as
their centers, masses, and edges, and also to the sample volume: the total
number of X-ray superclusters is on the order of 102 (Einasto et al. 2001;
Chon et al. 2013; Adami et al. 2018; Liu et al. 2022). On the other hand, some
well-known supercluster systems have already been used to study the large-
scale structure, for example, mapping the structures in the nearby Universe
(e.g., Böhringer & Chon 2021; Böhringer et al. 2021), and detecting or
characterizing the inter-cluster filaments within supercluster systems (e.g.,
Ursino et al. 2015; Bulbul et al. 2016; Reiprich et al. 2021; Ghirardini et
al. 2021; Hoang et al. 2023).

A precondition of finding superclusters is large samples of clusters with high
sample purity and precise redshifts. These are available thanks to the recent
large-area surveys in X-ray, Sunyaev-Zeldovich (SZ), and optical bands. In
particular, the eROSITA (extended ROentgen Survey with an Imaging Telescope
Array, Predehl et al. 2021) X-ray telescope onboard Spectrum Roentgen Gamma
(SRG) will detect about $10^{5}$ clusters and groups during its lifetime
(Merloni et al. 2012; Liu et al. 2022; Bulbul et al. 2022), which will
substantially increase the sample size of X-ray clusters. In eFEDS, 84
supercluster systems are detected in the $\sim 140$ deg2 survey area (Liu et
al. 2022). Projected from the eFEDS results, the final eROSITA All-Sky Survey
(eRASS) is expected to detect thousands of superclusters in the western
Galactic hemisphere. In particular, thanks to the unprecedented sensitivity in
the soft X-ray band, eROSITA will detect a large number of galaxy groups,
which trace the large-scale structure even better than massive clusters owing
to the overwhelming advantage in numbers. In the eROSITA and eRASS era, we can
trace the large-scale structure with comparable power to the optical galaxy
surveys for the first time. The large sample of superclusters detected by
eROSITA will expand the study of superclusters from nearby Universe to higher
redshifts ($z\approx 1$), from single targets to representative populations,
and will greatly advance our understanding of superclusters and the related
cosmological and astrophysical topics.

In this work, we search for supercluster systems based on the galaxy cluster
catalog from the first eROSITA All-Sky Survey (eRASS1, Merloni et al. 2024;
Bulbul et al. 2024; Kluge et al. 2024). Leveraging this largest-ever X-ray
galaxy cluster sample, we aim to present the biggest X-ray supercluster
catalog to be used for further explorations of superclusters. The paper is
organized as follows. In Sect. 2, we introduce the eRASS1 galaxy cluster
sample used for supercluster detection. In Sect. 3, we describe the
identification of eRASS1 superclusters. In Sect. 4, we study the properties of
the eRASS1 superclusters and their members. Our conclusions are summarized in
Sect. 5. Throughout this paper, we adopt the concordance $\Lambda$CDM
cosmology with $\Omega_{\Lambda}=0.7$, $\Omega_{\mathrm{m}}=0.3$, and
$H_{0}=70$ km s-1 Mpc-1. However, we note that the exact choice of
cosmological parameters does not affect the results significantly. Quoted
error bars correspond to a 1$\sigma$ confidence level. To avoid confusion, we
refer to the supercluster systems with two member clusters as “cluster pairs”,
and the systems with more than two member clusters as “rich superclusters”,
unless noted otherwise.

##  2 Galaxy cluster sample

###  2.1 eRASS1 galaxy cluster catalog

The first eROSITA All-Sky Survey was completed on June 11, 2020. In the
western Galactic hemisphere111Defined as
($179.9442^{\circ}<l<359.9442^{\circ}$), nearly 9.3$\times 10^{5}$ X-ray
sources are detected in the 0.2–2.3 keV energy range where eROSITA is most
sensitive (Merloni et al. 2024). Among them, about $2.7\times 10^{4}$ are
classified as extended sources, namely, galaxy cluster candidates, according
to a simple cut on X-ray extent likelihood: $\mathcal{L}_{\rm ext}>3$ (Bulbul
et al. 2024).

Figure 3:  Distribution of the eRASS1 supercluster systems color-coded by
redshift. Each point represents a supercluster member. For clarity purposes,
only rich superclusters with $\geq 3$ member clusters are plotted.

Optical identification of the cluster candidates is performed with eROMaPPer,
a highly parallelized version of the red-sequence-based cluster finder tool
redMaPPer (Rykoff et al. 2014, 2016). The public photometric data of the DESI
Legacy Imaging Surveys DR9 and DR10 (Dey et al. 2019) are used for the
confirmation of clusters and the computation of photometric redshifts
$z_{\lambda}$. Where possible, we also derive the spectroscopic redshifts of
the eRASS1 clusters using publicly available galaxy spectroscopic redshifts.
The final choice of cluster redshift BEST_Z is made as follows. If the cluster
has at least three spectroscopic members, we determine its BEST_Z as $z_{\rm
spec}$, which is the mean of the spectroscopic redshifts of the members.
Otherwise, if the optical central galaxy has a spectroscopic redshift $z_{\rm
spec,cg}$, we determine BEST_Z as $z_{\rm spec,cg}$. When neither of the above
conditions is met, we adopt the photometric redshift $z_{\lambda}$ as the
BEST_Z. Finally, for a few hundred clusters without spectroscopic or
photometric redshifts, we adopt the literature redshifts $z_{\rm lit}$ by
matching with public cluster catalogs using a matching radius of 2′. In
summary, the eRASS1 galaxy cluster catalog contains 12247 optically confirmed
clusters up to redshift 1.32. The details of the optical follow-up methodology
are presented in Kluge et al. (2024).

To obtain the X-ray properties of the eRASS1 clusters, a multi-band X-ray
imaging analysis is performed for each cluster, using the tool MBProj2D
(Sanders et al. 2018). By forward-fitting cluster’s X-ray images in seven
bands from 0.3 keV to 7 keV, MBProj2D provides the best-fit cluster physical
model. Products for each cluster include the azimuthally-averaged electron
density profile described in the form of the model in Vikhlinin et al. (2006),
a single global temperature under the isothermal assumption, and other derived
quantities such as flux, luminosity, count rate, gas mass, etc. In particular,
the above quantities are given as a function of radius and in multiple energy
ranges (see, e.g., Liu et al. 2023, for a recent application of MBProj2D).
Masses within $R_{500}$222$R_{500}$ is the radius within which the average
matter density is 500 times the critical density at cluster’s redshift of the
clusters are computed based on the scaling relation between X-ray count rate,
redshift, and mass, after calibrated with weak lensing shear signal (see more
details in Grandis et al. 2024; Ghirardini et al. 2024). The masses of eRASS1
clusters span a range $5\times 10^{12}M_{\odot}<M_{500}<2\times
10^{15}M_{\odot}$.

A contamination probability estimator, $P_{\rm cont}$, is also computed for
each eRASS1 cluster. The calculation of $P_{\rm cont}$ is based on a mixture
model that takes into account the cluster’s redshift, X-ray count rate, and
optical richness. It gives the probability of a cluster to be a contaminant.
According to the results of $P_{\rm cont}$ for each cluster, we estimate that
around 1700 of the 12247 eRASS1 clusters are spurious. This is computed by
$\sum P_{\rm cont}$. Thus the purity of the eRASS1 cluster sample,
$\sum(1-P_{\rm cont})/N_{\rm cl}$, is about $86\%$. Most of the contaminants
in the catalog are AGN, misclassified as extended sources due to the sizable
PSF of eROSITA. More details about the eRASS1 cluster catalog are provided in
Bulbul et al. (2024) and Kluge et al. (2024).

Figure 4:  Multiplicity of the eRASS1 superclusters. Cluster pairs and
superclusters (with $\geq 3$ member clusters) are plotted in yellow and blue,
respectively.  Figure 5:  Average separation and total length of the
superclusters. Both quantities are given in projected co-moving distance.
Cluster pairs and rich superclusters are plotted in yellow and blue,
respectively.

###  2.2 Sample selection

We apply several additional selections on the primary eRASS1 cluster catalog
published in Bulbul et al. (2024) to obtain a subsample of clusters with
higher purity and more reliable redshifts, which is suitable for supercluster
detection. Firstly, about $1900$ clusters with $P_{\rm cont}$ larger than 0.3
are ignored to enhance the purity of the sample. Secondly, since the detection
of superclusters is sensitive to cluster redshifts, we exclude the clusters
with unreliable redshifts. These include about $900$ cases when the errorbar
in redshift is large ($\delta z/(1+z)>0.02$) or when the photometric redshift
exceeds the limiting redshift at cluster’s sky position (see Bulbul et al.
2024; Kluge et al. 2024, for more details). Thirdly, about 300 clusters are
found to be duplicates based on the optical data. Namely, more than 70% of
their members are also identified as members of another cluster, which has a
higher extent likelihood (see Kluge et al. 2024, and the optical follow-up
catalog, for more information). These clusters are also excluded from the
sample. Fourthly, we exclude about $200$ clusters with less than 5 X-ray
photons within $R_{500}$ to further reduce the contamination level. Finally, a
few nearby clusters below redshift 0.005, such as Virgo, are not considered.
With the above selections, we obtain a subsample of 8862 clusters in the
redshift range of [0.0056, 1.32], with a purity of 96.4% estimated from the
$P_{\rm cont}$ of the remaining clusters in the subsample.

In Fig. 1, we present the mass $M_{500}$ and redshift distribution of the
cluster sample. Most high redshift clusters do not pass the selection criteria
due to the larger error bar in their photometric redshifts. Among the 8862
clusters in the sample, 2758 have spectroscopic redshifts as BEST_Z, 5885 and
219 have photometric and literature redshifts respectively. The median
redshift of the sample is 0.28, slightly lower than the overall eRASS1 cluster
sample, where $z_{\rm median}\approx 0.31$. The loss of high redshift clusters
will limit our supercluster identification to below $z\approx 1$.

##  3 Supercluster identification

Figure 6:  General properties of the supercluster systems identified in
eRASS1. Left panel: redshift distribution of eRASS1 superclusters. Cluster
pairs and superclusters (with $\geq 3$ member clusters) are plotted in yellow
and blue, respectively. Right panel: supercluster total mass versus redshift.
Color code denotes the number of member clusters. For clarity, only rich
superclusters with $\geq 3$ members are shown in this plot.

Figure 7:  Examples of supercluster systems detected in eRASS1. Upper panels:
the most distant supercluster system identified in eRASS1: 1eRASS-SC
J0530-4138, a cluster pair at redshift 0.802. The red crosses mark the
position of the two member clusters: 1eRASS J052957.4-413822 at $z=0.811\pm
0.007$ and 1eRASS J053040.8-413904 at $z=0.793\pm 0.012$. Lower panels: a rich
supercluster 1eRASS-SC J0529-2226 detected at redshift 0.17, with the four
member clusters marked as red crosses. The left panels show the eROSITA X-ray
exposure-corrected images in the 0.2–2.3 keV band, after smoothing with a
Gaussian of $\sigma=12\arcsec$. The right panels show the optical images from
the Legacy Survey, with the X-ray emission overlaid as white contours.  Figure
8:  Reliability of the eRASS1 superclusters accounting for cluster redshift
uncertainties. In the left panel, each data point represents a supercluster
system color-coded by its multiplicity. Shown in the right panel is the
histogram of reliability for all the systems (black), superclusters with $\geq
3$ members (blue), and cluster pairs (yellow).

To identify superclusters in eRASS1, we adopt the similar FoF method as Liu et
al. (2022). For a random cluster in the sample, the algorithm searches for
neighboring clusters (namely, “friends”) closer than a specific distance,
called the “linking length” $l$. Then, for each of the “friends”, the
algorithm continues to search for “friends of friends” until no new
neighboring clusters are found. The co-moving distances between clusters are
computed using their redshifts and X-ray centroids (RA_XFIT, DEC_XFIT)
obtained from the fit of MBProj2D. The 3D comoving distance between two
clusters, assuming a flat universe with $\Omega_{k}=0$, can be computed as:

$D=\sqrt{D_{\rm C,1}^{2}+D_{\rm C,2}^{2}-2D_{\rm C,1}D_{\rm C,2}\cdot{\rm
cos}\theta},$ (1)

where $D_{\rm C,1}$ and $D_{\rm C,2}$ are the line-of-sight co-moving
distances of the two clusters at (R.A.=$\alpha_{1}$, Dec.=$\delta_{1}$,
$z=z_{1}$) and (R.A.=$\alpha_{2}$, Dec.=$\delta_{2}$, $z=z_{2}$), $\theta$ is
the angular separation of the clusters. The line-of-sight co-moving distance
of a cluster at redshift $z$ can be written as:

$D_{\rm C}(z)=D_{\rm H}\cdot\int_{0}^{z}\frac{{\rm
d}z^{\prime}}{E(z^{\prime})},$ (2)

where $D_{\rm H}\equiv c/H_{0}$ is the Hubble distance, and the function
$E(z)$ is defined as:

$E(z)\equiv\sqrt{\Omega_{M}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}}\ \
(\Omega_{k}=0).$ (3)

The angular separation of the two clusters $\theta$ can be computed as:

${\rm cos}\theta={\rm sin}\delta_{1}{\rm sin}\delta_{2}+{\rm
cos}\delta_{1}{\rm cos}\delta_{2}{\rm cos}(\alpha_{1}-\alpha_{2}).$ (4)

The linking length, in co-moving 3D distance, is computed as a function of the
average cluster number density $n$ and the desired overdensity ratio $f$:
$l\equiv(n\times f)^{-1/3}$. Here, $n^{-1/3}$ is simply the average distance
between two neighboring clusters assuming that the clusters are uniformly
distributed in a 3D space without structures such as superclusters. Thus
multiplying $n$ with the overdensity ratio $f$ implies that the cluster
density in a supercluster is $f$ times that of the space average. In some
other works, a factor of $4\pi/3$ is included in the computation of local
cluster density, by assuming spherical collapse, so that the linking length
becomes $l=(4\pi nf/3)^{-1/3}$ (e.g., Zucca et al. 1993). However, we note
that most supercluster systems, especially cluster pairs, do not have regular
shapes. Estimating the volume of a low-multiplicity supercluster with a sphere
of radius $l$ will likely overestimate the volume and underestimate the
density. We therefore use the more general definition of $l$ without the
assumption of spherical collapse: $(n\times f)^{-1/3}$.

The selection of eRASS1 clusters is a strong function of X-ray count rate,
which confines the detection of high-redshift clusters to the high-luminosity
regime, known as the Malmquist bias, a common selection bias for flux-limited
surveys. Additionally, surface brightness patterns also have non-negligible
effects on the selection of X-ray clusters: high redshift clusters with bright
cores are more likely to be misidentified as point sources because of their
smaller angular sizes and the large PSF of X-ray telescopes (e.g., Bulbul et
al. 2022). The combination of these selection effects leads to a decrease in
the number density of eRASS1 clusters, and thus a rapid increase in linking
length, at high redshifts. Additionally, as the survey depth of eRASS1 is not
uniform, where the ecliptic pole areas have longer exposure than the
equatorial areas, the distribution of eRASS1 clusters generally shows the same
pattern of nonuniformity as the eRASS1 exposure map. The magnitude of this
nonuniformity is much lower compared to the evolution of cluster number
density with redshift. However, for some studies on superclusters, for
example, the comparison between supercluster members and isolated clusters,
one would prefer a more uniform selection after accounting for the
nonuniformity of the parent cluster sample. Therefore, in the computation of
cluster number density, we consider not only the dependency on redshift but
also the influence of exposure time. This is different than what Liu et al.
(2022) have done on eFEDS clusters, where cluster density is computed only as
a function of redshift because the survey depth of eFEDS is nearly uniform.
Our linking length is then defined as:

$l(z,t)=\left(\frac{N(z,t)}{V(z,t,A(t))}\cdot f\right)^{-1/3},$ (5)

where $N$ and $V$ are the number of clusters and the corresponding survey
volume in the redshift range [$z-\delta z,z+\delta z$] and exposure range
[$t-\delta t,t+\delta t$]. The thickness of the shell, $2\delta z$, is fixed
at 0.01. $\delta t$ is adjusted to make sure the corresponding survey area is
larger than 100 deg2. $A(t)$ is the eRASS1 depth curve (Bulbul et al. 2024).
$f$ is the overdensity ratio, and we adopt $f=10$, which is a common choice in
many previous works (e.g., Chon et al. 2013; Adami et al. 2018; Liu et al.
2022).

Since our cluster density decreases rapidly at high redshifts, we empirically
set an upper limit on linking length: $l<=50$ Mpc, to avoid spurious
detections at high redshifts. Namely, we set 50 Mpc as the maximum distance
between two clusters that are believed to be connected with each other. We
note that the choice of 50 Mpc as the upper limit of linking length is
relatively conservative compared to previous works. This will help reduce
false detections and systems with very large distances between members. As a
comparison, the linking length can be as large as 70–80 Mpc in XXL (Adami et
al. 2018) and eFEDS (Liu et al. 2022). We also note that the thickness of the
volume shell for computing the local cluster density in Eq. 5,
$2\delta_{z}=0.01$, ranges between 27 Mpc (at $z=0.8$) and 42 Mpc (at
$z=0.05$), both are smaller than the upper limit of linking length. We plot in
Fig. 2 the cluster density and linking length as functions of redshift and
exposure time. As expected, the linking length shows a much stronger
dependence on redshift than the exposure time. In summary, our selection of
superclusters includes the cluster sample selection described in Sect. 2.2,
and the linking length plotted in the right panel of Fig. 2. These selection
procedures need to be accounted for in statistical studies of superclusters
and in the comparison with numerical simulations.

In the identification of superclusters, we use the best redshifts of the
eRASS1 clusters (BEST_Z column in the eRASS1 cluster catalog) to compute the
distance between clusters. However, the 3D distance between clusters is
sensitive to the precision of cluster redshifts. Therefore, for each
supercluster system detected using BEST_Z, we must check how robust the
detection is over the clusters’ redshift uncertainties. In addition to that,
the peculiar velocities of clusters can also contribute to their redshifts and
are also an important source of uncertainties in the distances. For clusters
with spectroscopic redshifts, the distance uncertainties might be dominated by
peculiar motions. The peculiar velocities of clusters are not well-
constrained, except for the very nearby Universe. We, therefore, adopt a
typical value of $v_{\rm pec}=cz_{\rm pec}=400$ km/s (see, e.g., Dolag &
Sunyaev 2013) and add this component to the uncertainties of redshift in
quadrature with the original redshift uncertainties. We then perform the
following simulations to estimate the reliability of each supercluster over
the total uncertainties of the clusters’ redshifts. In each simulation, we
randomly vary the redshift of each cluster in the sample over its total
uncertainty. The same FoF method is then employed to identify superclusters in
each simulated cluster sample. The simulations are performed 1000 times. We
note that directly comparing the superclusters in each simulation by simply
matching the central coordinates is not straightforward, and might bring
misleading results. For instance, the addition or subtraction of a single
member in the outskirts of a rich supercluster can cause a significant shift
in its central position, even if the dominant part of the superclusters is
robustly detected in both cases. Therefore, instead of estimating the
detection rate of the superclusters in the simulations, we compute for each
cluster the frequency of being a supercluster member. Then, for each
supercluster detected using BEST_Z, we define its reliability as the average
frequency of its member clusters.

Another important point to note is that our supercluster identification,
similar to all the other supercluster catalogs in both optical and X-ray
bands, is clearly affected by the selection of the parent eRASS1 cluster
sample and the detection algorithm. In principle, there is no universal
division between supercluster members and isolated clusters. Obviously, an
isolated cluster identified in a shallower survey can become a supercluster
member in a deeper survey, when its faint neighbors are detected. Therefore,
one has to specify the selection of the cluster sample and the detection
criteria associated with a supercluster sample. Similarly, the comparison
between supercluster members and isolated clusters also requires that the two
classes of clusters have consistent selection functions, despite that the
division of the two classes might differ in different surveys.

With the 8862 eRASS1 clusters and the linking length defined in Eq. 5, we
identify 1338 supercluster systems in eRASS1, including 818 cluster pairs and
520 rich superclusters with $>=3$ members. 3948 clusters are identified as
members of these supercluster systems. In addition to this primary
supercluster catalog obtained with $f=10$, we also employ the same FoF method
to search for supercluster systems with a lower overdensity ratio $f=3$ and a
higher overdensity ratio $f=50$. As a result, we detect 1270 and 929
superclusters corresponding to $f=3$ and $f=50$, respectively. In both cases,
the numbers of superclusters are lower than that of $f=10$. This result is
consistent with Chon et al. (2013) where the authors show in their Figure 2
that the number of detected superclusters reaches its maximum value between
$f=5$ and $f=10$. We also note that 4867 and 2205 clusters are identified as
supercluster members for $f=3$ and $f=50$, consistent with the trend that the
lower the overdensity ratio, the more clusters are linked as superclusters. We
base our following analysis on the primary supercluster catalog with $f=10$.

Table 1:  General properties of superclusters identified in eRASS1.

Name | Redshift | RA | Dec | Multiplicity | Total mass | Reliability | $d_{\rm mem}$ | $L$
---|---|---|---|---|---|---|---|---
[-] | [-] | [deg] | [deg] | [-] | [$10^{14}M_{\odot}$] | [-] | [Mpc] | [Mpc]
… |  |  |  |  |  |  |  |
1eRASS-SC J1208-8349 | 0.169 | 182.0510 | -83.8305 | 3 | 27.52$\pm$5.88 | 0.78 | 10.95 | 26.00
1eRASS-SC J1338-0413 | 0.169 | 204.6015 | -4.2213 | 2 | 24.65$\pm$5.24 | 0.66 | 11.16 | 22.31
1eRASS-SC J0527-4654 | 0.170 | 81.7702 | -46.9022 | 2 | 5.60$\pm$1.35 | 0.76 | 3.66 | 7.32
1eRASS-SC J0442-5600 | 0.170 | 70.6324 | -56.0091 | 2 | 3.24$\pm$0.89 | 0.64 | 7.23 | 14.46
1eRASS-SC J0349-3207 | 0.170 | 57.4379 | -32.1185 | 11 | 80.64$\pm$16.23 | 0.95 | 22.14 | 74.55
… |  |  |  |  |  |  |  |

333Column 1: supercluster name. Column 2: supercluster redshift, computed from
the average of its member clusters. Columns 3 and 4: central coordinate of
supercluster. Column 5: multiplicity, defined as the number of member
clusters. Column 6: the total mass of superclusters, scaled from the masses of
the member clusters, see the text for more details. Column 7: reliability.
Column 8: average distance of supercluster members to the center. Column 9:
supercluster length, defined as the largest distance between its members. The
table is sorted by supercluster redshifts. Only part of the table is presented
here. The full table is available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/.

Table 2:  Properties of member clusters of the eRASS1 superclusters.

SC name | Cluster name | Redshift | RA | Dec | $M_{500}$ | $L_{500}$ | $P$
---|---|---|---|---|---|---|---
[-] | [-] | [-] | [deg] | [deg] |  [$10^{14}M_{\odot}$] |  [$10^{43}$ erg s-1] | [-]
… |  |  |  |  |  |  |
1eRASS-SC J1208-8349 | 1eRASS J121629.5-845559 |  0.1712$\pm$0.0050 | 184.1226 | -84.9345 | $1.25\pm 0.43$ | $2.11\pm 0.94$ | 0.66
1eRASS-SC J1208-8349 | 1eRASS J121825.5-824718 |  0.1682$\pm$0.0044 | 184.6240 | -82.7890 | $2.06\pm 0.25$ | $4.02\pm 0.62$ | 0.83
1eRASS-SC J1208-8349 | 1eRASS J114936.9-834601 |  0.1666$\pm$0.0043 | 177.4063 | -83.7681 | $4.04\pm 0.34$ | $10.28\pm 0.86$ | 0.84
1eRASS-SC J1338-0413 | 1eRASS J134048.2-045446 |  0.1661$\pm$0.0004 | 205.2020 | -4.9161 | $1.55\pm 0.34$ | $2.93\pm 0.61$ | 0.65
1eRASS-SC J1338-0413 | 1eRASS J133600.2-033133 |  0.1726$\pm$0.0004 | 204.0011 | -3.5265 | $5.04\pm 0.39$ | $16.31\pm 1.47$ | 0.67
1eRASS-SC J0527-4654 | 1eRASS J052823.3-464208 |  0.1673$\pm$0.0004 | 82.0974 | -46.7015 | $1.02\pm 0.15$ | $1.55\pm 0.32$ | 0.77
1eRASS-SC J0527-4654 | 1eRASS J052548.3-470619 |  0.1719$\pm$0.0043 | 81.4431 | -47.1028 | $0.48\pm 0.15$ | $0.60\pm 0.20$ | 0.75
… |  |  |  |  |  |  |

444Column 1: supercluster name. Column 2: member cluster name. Column 3:
cluster redshift. Columns 4 and 5: cluster coordinate. Column 6: $M_{500}$.
Column 7: $L_{500}$. Column 8: membership probability. The table is sorted by
supercluster redshifts. Only part of the table is presented here. The full
table is available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

Figure 9:  Comparison of the large-scale structures traced by galaxies (gray
dots) and X-ray superclusters identified in this work. The redshifts and
positions of the galaxies are from a spectroscopic galaxy compilation of
published catalogs (see Kluge et al. 2024). Each yellow triangle represents a
supercluster member. Plotted is the slice $-3^{\circ}<$ Dec. $<3^{\circ}$.

##  4 Properties of the eRASS1 superclusters

We present in this section the properties of the primary eRASS1 supercluster
catalog. The spatial distribution of the rich superclusters is presented in
Fig. 3. Although we have accounted for the dependency of cluster density on
exposure time, there is still a slight overdensity of identified supercluster
systems around the South Ecliptic Pole, where eRASS1 is deeper than other
regions. Among the 8862 clusters in the sample, 3948 ($45\%$) are identified
as supercluster members. This fraction is only slightly lower than the result
in Böhringer & Chon (2021), where $51\%$ of the clusters are found to be
supercluster members. The difference is likely due to the fact that Böhringer
& Chon (2021) adopt a much lower overdensity ratio ($f=2$). The distribution
of multiplicity, defined as the number of member clusters in a supercluster,
is shown in Fig. 4. The richest system we identify, 1eRASS-SC J1307-3016, also
known as the Shapley supercluster, has 45 members with a median redshift of
0.050. For each supercluster system, we compute the average distance of its
members to the center of the system ($d_{\rm mem}$, where the center is
defined as the algebraic mean coordinate of the members), and the total length
($L$), which is defined as the maximum distance between its members. Both
quantities are computed in the projected 2D distance. The distribution of the
average separation and length are shown in Fig. 5. $d_{\rm mem}$ can be used
to quantify how compact is a supercluster system. For rich superclusters, the
distribution of $d_{\rm mem}$ reaches a peak value at around $15$ Mpc, and
extends to 40 Mpc (see the left panel of Fig. 5). The median $d_{\rm mem}$ of
rich superclusters is 12.9 Mpc. On the other hand, most of the cluster pairs
have a low $d_{\rm mem}$ smaller than $15$ Mpc, with a small fraction
extending to 25 Mpc, due to the upper limit of 50 Mpc on linking length. The
median $d_{\rm mem}$ of cluster pairs is 8.4 Mpc, much lower than rich
superclusters. A similar trend can be found in the distribution of total
length $L$ (see the right panel of Fig. 5). The $L$ of rich superclusters
peaks at around $30$ Mpc, with a median value of 33.5 Mpc, while the median
$L$ of cluster pairs is 16.9 Mpc. The most extensive system we detect in
eRASS1, 1eRASS-SC J1140-1939 at redshift 0.303, consisting of 10 member
clusters, has a projected comoving length of 127 Mpc. As a comparison, the
superclusters identified with SDSS DR7 data (Liivamägi et al. 2012) have
average and maximum diameters of 22 Mpc and 120 Mpc, similar to the sizes of
the eRASS1 X-ray superclusters.

We also estimate the total cluster mass of the supercluster systems by simply
summing up the virial masses $M_{\rm 200}$ of its member clusters, where we
adopt the approximation $M_{\rm 200}\approx 1.46\times M_{500}$ by assuming a
Navarro-Ferenk-White (NFW) profile (Navarro et al. 1997) with concentration
$c\equiv r_{200}/r_{\rm s}=4$ (Reiprich et al. 2013). The total cluster mass
is then converted to the total supercluster mass by adopting the relation
found by Chon et al. (2014) with cosmological $N$-body simulations: $M_{\rm
tot,Cl}=0.39\pm 0.077\times M_{\rm tot,SC}$. The most massive eRASS1
supercluster is the Shapley supercluster, 1eRASS-SC J1307-3016, with a total
mass of $2.58\pm 0.51\times 10^{16}M_{\odot}$ and a length of $L=111$ Mpc. The
mass estimation of the Shapley supercluster is available in several previous
works. For example, Reisenegger et al. (2000) reported a total mass of
$1.9\times 10^{16}M_{\odot}$ within $12$ Mpc, using a caustic method of
galaxies, and assuming a spherical collapse model. Ragone et al. (2006)
measured a total mass of $2.3\times 10^{16}M_{\odot}$, using 122 galaxy
systems in a area of $12\times 15$ deg2, with the masses outside the galaxy
systems corrected. More recently, Chon et al. (2014) found a total mass of
$1.91\times 10^{16}M_{\odot}$ within $17.7$ Mpc. We note that, given the
difference in radius, data, and methods, our result is in broad agreement with
the values reported by the previous works. A dedicated analysis on the Shapley
supercluster with eROSITA will be performed in another work (Sanders et al.,
in prep.).

Figure 10:  Comparison of the filaments detected with SDSS galaxies (Malavasi
et al. 2020) (curves) and the eRASS1 X-ray superclusters. The supercluster
members are plotted with dots. Color codes denote redshifts for both the
filaments and supercluster members. To reduce overlapping between optical
filaments, only the region ($125^{\circ}<$ R.A. $<210^{\circ}$, $3^{\circ}<$
Dec. $<30^{\circ}$) and redshift range $0.15<z<0.25$ are plotted in the
figure.

The redshift distribution of the supercluster systems is shown in Fig. 6. The
supercluster systems at high redshifts are dominated by cluster pairs, owing
to the decrease in the number density of clusters in the sample. Also shown in
Fig. 6 in the right panel is the mass-redshift relation of rich superclusters
with $\geq 3$ members, where the masses of the systems span a range of
[$6\times 10^{13}$–2$\times$ 10${}^{16}M_{\odot}$]. The most distant system,
1eRASS-SC J0530-4138, is a cluster pair at redshift $0.802$, consisting of two
members, 1eRASS J052957.4-413822 and 1eRASS J053040.8-413904. We show in Fig.
7 the optical image of this cluster pair from the Legacy Imaging Survey. The
photometric redshifts of the two members are $0.811\pm 0.007$ and $0.793\pm
0.012$, and the masses ($M_{500}$) are $3.6_{-0.8}^{+1.0}\times
10^{14}M_{\odot}$ and $4.2_{-0.6}^{+0.8}\times 10^{14}M_{\odot}$. Due to the
large error bar in redshifts, we are not able to precisely constrain the 3D
distance between the two members. Therefore, this system has relatively low
reliability, $P=0.50$. The projected distance is about $3.7$ Mpc, implying
that the two members will probably merge as a massive cluster. Although this
cluster pair is identified for the first time in this work, the two member
clusters are already detected in Sunyaev–Zeldovich (SZ) surveys by the Atacama
Cosmology Telescope (ACT, Hilton et al. 2021) and the South Pole Telescope
(SPT, Bocquet et al. 2019). The reported photometric redshifts of 1eRASS
J052957.4-413822 and 1eRASS J053040.8-413904 are $0.793\pm 0.010$ and
$0.795\pm 0.010$ in ACT, and $0.775\pm 0.050$ and $0.775\pm 0.048$ in SPT.
Therefore, our redshifts are consistent with both SZ surveys within $2\sigma$,
indicating that the detection of this cluster pair is reliable. Also shown in
Fig. 7 is another example of a rich eRASS1 supercluster: 1eRASS-SC J0529-2226
consisting of four members at an average redshift of 0.17.

The reliabilities of the supercluster systems are computed using the method
described in Sect. 3, and are presented in Fig. 8. Among the 1338 systems, 841
($63\%$) have reliability larger than 0.7. For rich superclusters with $\geq
3$ members and cluster pairs, this fraction is $87\%$ and $48\%$, indicating
that the identification of rich superclusters is generally more reliable than
cluster pairs. On average, low-redshift systems have higher reliability than
high-redshift ones, as can be seen from the left panel of Fig. 8. This is
probably due to the fact that the fraction of spectroscopic redshift is larger
at low redshifts (see Fig. 1), which is about $10\times$ more precise than
photometric redshift. Dedicated spectroscopic follow-up of the eROSITA
clusters is ongoing or planned with SDSS-V (Kollmeier et al. 2017; Almeida et
al. 2023) and 4MOST (Finoguenov et al. 2019). It is expected that the
reliability of the superclusters will be further improved when more
spectroscopic redshifts are available.

The properties of the eRASS1 superclusters, including the average redshift,
coordinate, multiplicity, total mass, average member separation $d_{\rm mem}$,
and total length $L$, are present in Table 1. Since most of the superclusters
do not have a regular shape, the average redshift and coordinate alone cannot
locate the supercluster precisely. We, therefore, provide in Table 2 the
properties of the member clusters for all the superclusters. In addition to
the primary supercluster catalog obtained with $f=10$, we also provide in the
appendix a supplementary supercluster catalog with the information of members,
corresponding to $f=50$.

We compare our eRASS1 supercluster catalog with the known superclusters
published in the literature. In many of the published supercluster catalogs,
the location of superclusters is given as an average coordinate of the
members. As superclusters are not bounded systems with irregular shapes, and
they often span an area of several square degrees, it is almost infeasible to
directly match the superclusters with the central coordinates. A practical way
is to match the member clusters.

We first compare the eRASS1 supercluster catalog with eFEDS. As eFEDS is
almost $10$ times deeper than eRASS1, only a small fraction (63 out of 542) of
eFEDS clusters are detected in eRASS1. In the eFEDS survey, Liu et al. (2022)
detected 19 rich superclusters with $\geq 4$ members. In this work, using the
eRASS1 cluster sample, we detect 8 superclusters (including cluster pairs) in
the eFEDS footprint. Most of them are already found in the eFEDS survey,
despite that we report lower multiplicities for some systems in this work than
the values in Liu et al. (2022) due to the much lower source density in eRASS1
compared to eFEDS. For example, eFEDS-SC3 ($z=0.196,N=10$) is identified as
1eRASS-SC J0859+0306 ($z=0.197,N=3$) in this work; eFEDS-SC5 ($z=0.269,N=7$)
is identified as 1eRASS-SC J0841-0036 ($z=0.267,N=2$) in this work; eFEDS-SC6
($z=0.281,N=4$) is identified as 1eRASS-SC J0921+0221 ($z=0.283,N=2$) in this
work; eFEDS-SC12 ($z=0.358,N=4$) is identified as 1eRASS-SC J0935+0051
($z=0.359,N=2$) in this work. The only exception is 1eRASS-SC J0932-0110, a
cluster pair at $z=0.238$. The two members of this system both have
counterparts in the eFEDS cluster sample. However, one of its members, 1eRASS
J093024.6-020635 (eFEDS J093025.7-020507), has a lower redshift in the eFEDS
cluster catalog: 0.220$\pm 0.006$, while the redshift measured in the eRASS1
cluster catalog is 0.241$\pm 0.006$. Both results are photometric redshifts.
Therefore, it is not identified as a cluster pair in eFEDS. We note that the
optical confirmation of eFEDS clusters was performed with the multi-component
matched filter cluster confirmation tool (MCMF, Klein et al. 2022) and the
data from HSC-SSP, Legacy Survey DR8, and unWISE. In eRASS1, we use the
eROMaPPer tool on Legacy Survey DR9 and DR10 (Kluge et al. 2024). The slight
difference in the redshift of this cluster is probably because we use
different optical survey data and methods for cluster confirmation in eFEDS
and eRASS1.

35 rich supercluster systems and 39 cluster pairs are detected using the
XXL365 cluster catalog (Adami et al. 2018). In eRASS1, we only detect 11 XXL
clusters (Bulbul et al. 2024) due to the relatively shallow depth of eRASS1
compared to XXL, thus consequently limiting the detection rate of XXL
superclusters. Only one cluster pair is identified in eRASS1: Id17 in the XXL
cluster pair catalog at redshift 0.378. In eRASS1, we identify this cluster
pair as a rich system: 1eRASS-SC J2321-5326, with four member clusters at a
median redshift of 0.369. Except for the two members reported in Adami et al.
(2018), XLSSC513 and XLSSC525, we find two additional members: 1eRASS
J232541.0-531638 at $z=0.370$ (XLSSC547) and 1eRASS J232708.1-513733 at
$z=0.364$.

We also make a general comparison with large-area optical spectroscopic
surveys. In Fig. 9, we plot a slice of cosmic volume at $z<0.2$ and
$-3^{\circ}<$ Dec. $<3^{\circ}$. A few more examples for different slices are
shown in Fig. 11. We find a broad association in the large-scale structures
formed by galaxies and traced by the X-ray superclusters we identified in this
work. We further compare the eRASS1 superclusters with the cosmic filaments
detected with SDSS galaxies (Malavasi et al. 2020) in Fig. 10. To reduce
overlapping between optical filaments, only a small part of the sky
($125^{\circ}<$R.A. $<210^{\circ}$, $3^{\circ}<$ Dec. $<30^{\circ}$) and
redshift range $0.15<z<0.25$ are shown in Fig. 10. As a result, most of the
eRASS1 superclusters are associated with the filaments. Most of the
supercluster members are connected by galaxy filaments.

##  5 Conclusions

We present in this work the first catalog of superclusters in the western
Galactic hemisphere detected from the eRASS1 survey. We base our supercluster
detection on the optically-confirmed galaxy clusters detected in eRASS1
(Bulbul et al. 2024). By applying several additional filters on the primary
eRASS1 cluster catalog, with the aim of obtaining a purer subsample of
clusters with more reliable redshifts, we select a subsample of 8862 eRASS1
clusters. Based on a contamination estimator $P_{\rm cont}$ computed using the
cluster’s X-ray count rate, redshift, and optical richness, this cluster
subsample has a high purity of 96.4%. A Friends-of-Friends method is employed
to identify supercluster systems, where the linking length as a function of
redshift and exposure time is computed to make sure that the local cluster
density in superclusters is at least 10 times that of the average value of the
sample. The false detection rate at high redshifts where the cluster density
is too low is controlled by conservatively setting an upper limit of 50 Mpc on
the linking length.

With the above data and method, we identify 1338 supercluster systems up to
redshift 0.8, including 818 cluster pairs and 520 rich superclusters with
$\geq$ 3 members. Among the 8862 selected clusters, 3948 clusters (about 45%
of the sample) are members of these supercluster systems. The most massive and
richest system, 1eRASS-SC J1307-3016, also known as the Shapley supercluster,
consists of 45 members at an average redshift 0.050, and has a total mass of
$2.58\pm 0.51\times 10^{16}M_{\odot}$. The most extensive system, 1eRASS-SC
J1140-1939 at redshift 0.303, has a total length of 127 Mpc. The sizes of the
superclusters we identify in this work are comparable to the structures found
with galaxy survey data. A good association is found between the eRASS1
superclusters and the large-scale structures formed by optical galaxies. We
compute the reliability of each supercluster by accounting for the
uncertainties in cluster redshifts and the peculiar velocities of the
clusters. Thanks to the high accuracy of eRASS1 clusters’ redshifts, 63% of
the supercluster systems have a reliability larger than 0.7. This will be
further improved when more spectroscopic redshifts of the eRASS1 clusters are
available from SDSS-V (Kollmeier et al. 2017; Almeida et al. 2023) and 4MOST
(Finoguenov et al. 2019).

The eRASS1 supercluster catalog presented in this work represents the most
extensive sample of superclusters detected in the X-ray band in terms of
sample volume, sky coverage, redshift range, and the availability of X-ray
properties. Another advantage of the eRASS1 supercluster catalog is that the
superclusters are identified on the basis of a cluster sample with a well-
understood selection function, thus making it convenient to compare with large
numerical simulations. This legacy catalog will greatly advance our
understanding of the evolution of the cosmic large-scale structure.

In a forthcoming paper (Liu et al. in prep.), we will utilize the eRASS1
supercluster catalog to investigate the environmental effects on the evolution
of galaxy clusters. This can be done by simply comparing the X-ray properties
of supercluster members and isolated clusters after accounting for selection
effects (see, e.g., Manolopoulou et al. 2021). As these two classes of
clusters naturally have different clustering magnitudes, such a comparison can
also be used to test the dark matter halo formation theories, such as the halo
assembly bias (Gao et al. 2005; Gao & White 2007).

A task for X-ray astronomy in the next decade is to find the “missing
baryons”, which account for 30%–40% of the total baryonic mass in the Universe
(e.g., Bregman 2007; Shull et al. 2012). These “missing baryons” are probably
residing in the vast space between the nodes of the cosmic web, namely,
galaxies and clusters, in the form of a low-density and X-ray emitting gas,
with a temperature of $\sim 10^{6}$ K (e.g., Fang 2018; Nicastro et al. 2018).
They can be detected either from their emission in the soft X-ray band (e.g.,
Eckert et al. 2015; Bulbul et al. 2016; Veronica et al. 2023) or through their
absorption on bright background sources (Nicastro et al. 2022; Štofanová et
al. 2023). Detecting and characterizing the intergalactic medium (IGM) and
circumgalactic medium (CGM) are important science goals for X-ray missions in
the near future, such as the Hot Universe Baryon Surveyor (HUBS, Bregman et
al. 2023) and Athena (Nandra et al. 2013). Superclusters are the most probable
reservoirs of the missing baryons. Therefore, the eRASS1 X-ray supercluster
catalog will provide promising targets for future X-ray missions. Limited by
the survey depth of eROSITA, the IGM emission in most of the individual
supercluster systems is expected to be fainter than the detection limit of
eRASS. Thus we need to apply image and spectrum stacking techniques to search
for possible X-ray emissions from these cosmic filaments. The results of this
work are presented in a separate paper (Zhang et al. 2024). Dedicated
multiwavelength analysis of individual systems, for example, the Shapley
supercluster, will also be performed, to investigate the merging processes in
member clusters (Sanders et al. in prep., di Gennaro et al. in prep.).

###### Acknowledgements.

We greatly thank the anonymous referee for his/her constructive comments that
helped improve the paper. This work is based on data from eROSITA, the soft
X-ray instrument aboard SRG, a joint Russian-German science mission supported
by the Russian Space Agency (Roskosmos), in the interests of the Russian
Academy of Sciences represented by its Space Research Institute (IKI), and the
Deutsches Zentrum für Luft- und Raumfahrt (DLR). The SRG spacecraft was built
by Lavochkin Association (NPOL) and its subcontractors and is operated by NPOL
with support from the Max Planck Institute for Extraterrestrial Physics (MPE).
The development and construction of the eROSITA X-ray instrument was led by
MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP
(FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA.
The eROSITA data shown here were processed using the eSASS/NRTA software
system developed by the German eROSITA consortium.
A. Liu, E. Bulbul, V. Ghirardini, C. Garrel, S. Zelmer, and X. Zhang
acknowledge financial support from the European Research Council (ERC)
Consolidator Grant under the European Union’s Horizon 2020 research and
innovation program (grant agreement CoG DarkQuest No 101002585). M.B.
acknowledges support from the DFG under Germany’s Excellence Strategy - EXC
2121 “Quantum Universe” – 390833306.
This work made use of SciPy (Jones et al. 2001), Matplotlib, a Python library
for publication-quality graphics (Hunter 2007), Astropy, a community-developed
core Python package for Astronomy (Astropy Collaboration et al. 2013), NumPy
(Van Der Walt et al. 2011).

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##  Appendix A Supplementary catalog of superclusters identified with $f=50$

Table 3:  General properties of superclusters identified in eRASS1 with
overdensity ratio $f=50$.

Name | Redshift | RA | Dec | Multiplicity | Total mass | Reliability | $d_{\rm mem}$ | $L$
---|---|---|---|---|---|---|---|---
[-] | [-] | [deg] | [deg] | [-] | [$10^{14}M_{\odot}$] | [-] | [Mpc] | [Mpc]
… |  |  |  |  |  |  |  |
1eRASS-SC J0538-4823 | 0.196 | 84.6960 | -48.3927 | 2 | 6.87$\pm$1.77 | 0.45 | 1.57 | 3.13
1eRASS-SC J1242+2731 | 0.196 | 190.7033 | 27.5222 | 3 | 38.61$\pm$8.07 | 0.89 | 4.51 | 8.93
1eRASS-SC J0332-0711 | 0.196 | 53.0321 | -7.1875 | 2 | 15.67$\pm$4.89 | 0.70 | 6.62 | 13.25
1eRASS-SC J0329-2056 | 0.196 | 52.2812 | -20.9459 | 2 | 33.81$\pm$7.10 | 0.39 | 6.51 | 13.01
1eRASS-SC J1818-7208 | 0.196 | 274.5161 | -72.1427 | 2 | 25.11$\pm$5.51 | 0.61 | 6.89 | 13.79
… |  |  |  |  |  |  |  |

555The meanings of columns are the same as Table 1. Only part of the table is
presented here. The full table is available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/.

Table 4:  Properties of member clusters of the eRASS1 superclusters identified
with overdensity ratio $f=50$.

SC name | Cluster name | Redshift | RA | Dec | $M_{500}$ | $L_{500}$ | $P$
---|---|---|---|---|---|---|---
[-] | [-] | [-] | [deg] | [deg] |  [$10^{14}M_{\odot}$] |  [$10^{43}$ erg s-1] | [-]
… |  |  |  |  |  |  |
1eRASS-SC J0538-4823 | 1eRASS J053835.4-481700 |  0.1935$\pm$0.0047 | 84.6509 | -48.2846 | $0.68\pm 0.18$ | $1.11\pm 0.34$ | 0.45
1eRASS-SC J0538-4823 | 1eRASS J053857.4-482954 |  0.1978$\pm$0.0052 | 84.7412 | -48.5007 | $1.15\pm 0.24$ | $2.19\pm 0.55$ | 0.45
1eRASS-SC J1242+2731 | 1eRASS J124325.7+271700 |  0.1924$\pm$0.0022 | 190.8563 | 27.2810 | $5.83\pm 0.57$ | $20.96\pm 2.19$ | 0.81
1eRASS-SC J1242+2731 | 1eRASS J124356.3+274155 |  0.1958$\pm$0.0013 | 190.9820 | 27.6989 | $2.19\pm 0.31$ | $5.04\pm 0.94$ | 0.99
1eRASS-SC J1242+2731 | 1eRASS J124104.0+273521 |  0.1990$\pm$0.0008 | 190.2715 | 27.5866 | $2.29\pm 0.30$ | $5.77\pm 0.98$ | 0.87
… |  |  |  |  |  |  |

666The meanings of columns are the same as Table 2. Only part of the table is
presented here. The full table is available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-
bin/qcat?J/A+A/.

##  Appendix B A few more examples of comparison between eRASS1 superclusters
and optical LSS

Figure 11:  Same as Fig. 9, but for different slices. Top left: $45^{\circ}<$
R.A. $<100^{\circ}$, $-55^{\circ}<$ Dec. $<-45^{\circ}$. Top right:
$60^{\circ}<$ R.A. $<95^{\circ}$, $-22^{\circ}<$ Dec. $<-15^{\circ}$. Bottom:
$180^{\circ}<$ R.A. $<220^{\circ}$, $-35^{\circ}<$ Dec. $<-25^{\circ}$.