The cosmic web, a complex network of galaxies and matter, provides insights into cosmological models. Key components include galaxy clusters, walls, filaments, and voids. Superclusters, coherent regions spanning tens to hundreds of megaparsecs, are the largest known structures, containing multiple galaxy clusters and groups. Their formation and evolution remain an open question, necessitating detailed studies using large, statistically significant catalogs. Current definitions of superclusters vary; they can be defined as gravitationally bound regions, unbound overdense regions, or regions of converging peculiar velocity flows. While gravitationally bound definitions focus on high density regions overcoming cosmic expansion, velocity-based definitions consider regions where galactic peculiar velocities converge, acting as attractors. Observationally identifying these regions, particularly for distant galaxies, is challenging due to limited peculiar velocity data for distant objects; simulations are useful in this respect. The Friends-of-Friends (FoF) algorithm is commonly used to identify superclusters, although other methods such as density threshold cuts on the luminosity density field and the watershed method also exist. Previous studies have shown that supercluster size, mass, and luminosity evolve with redshift, with size decreasing as luminosity and mass increase, impacting galaxy evolution within them. The ΛCDM model suggests that structure growth slows down below z ≈ 0.5 due to dark energy; however, highly overdense structures continue to grow. This paper aims to identify and characterize superclusters across a wide redshift and sky area, comparing observational findings with cosmological simulations to improve our understanding of these massive cosmic structures.

Existing supercluster catalogs have limitations in redshift coverage, sky area, or the completeness of data. Chow-Martínez et al. (2014) used a tunable FoF algorithm on Abell/ACO clusters, reaching redshifts up to z ≤ 0.15, but covering a vast sky area. Liivamägi et al. (2012) utilized the luminosity density field from SDSS DR7, spanning 0.02 ≤ z ≤ 0.5 but with less sky coverage compared to the present study. Several studies focused on specific superclusters like Saraswati (Bagchi et al., 2017), Sloan Great Wall (Gott et al., 2005), and Corona Borealis (Einasto et al., 2021c), offering detailed analyses of their properties but lacking the statistical power of a large catalog. Existing work highlighted the evolution of supercluster size, mass, and luminosity with redshift, and the effect of supercluster environment on galaxies and clusters; however, the need for a comprehensive study using a large catalog spanning a significant redshift range remained unaddressed.

This study uses the Wen-Han-Liu (WHL) cluster catalog from the SDSS-III DR12, which includes both spectroscopic and photometric redshifts for groups and clusters in the redshift range 0.05 ≤ z ≤ 0.8. Focusing on the redshift range 0.05 ≤ z ≤ 0.42, 85,686 groups and clusters were selected for supercluster identification. The WHL catalog provides cluster richness (R L * ,500 ), which is used to estimate the halo mass (M 500c ) using an established empirical relation. To account for mass beyond the virial radius, the bound halo mass (M halo ) is approximated as M halo ≈ M 5.6c ∼ 2.2 × M 200c. A modified Friends-of-Friends (mFoF) algorithm is employed to identify superclusters, considering the non-uniform distribution of WHL clusters with redshift. Delaunay triangulation is used for efficient distance calculations. Weighted linking lengths (l i j ) are determined using radial selection weights (w r,i ) that account for the redshift-dependent number density of clusters. The linking length (l o ) is chosen to maximize the number of superclusters while maintaining a minimum number of clusters (N min = 2) per supercluster. Superclusters are defined as candidate superclusters with at least 10 member clusters. The mass of a supercluster is calculated by summing the bound halo masses of its constituent clusters. Supercluster size is determined as the maximum distance between any two member clusters. The supercluster's position is calculated as the virial mass-weighted average position of its member clusters. Density contrast (δ) is estimated using the volume of the supercluster's convex hull, calculated using the Qhull algorithm. The Horizon Run 4 (HR4) N-body simulation is utilized to create a mock WHL cluster catalog, matching the sky distribution, redshift distribution, and mass distribution of the observed WHL clusters. The same mFoF algorithm is applied to both the observed and mock clusters for comparison. The peculiar velocities of member clusters in the mock superclusters are analyzed to determine their dynamical state relative to the supercluster center of mass.

The mFoF algorithm identified 662 superclusters in the redshift range 0.05 ≤ z ≤ 0.42, making it one of the largest supercluster catalogs to date at this redshift. Around 12% of the WHL clusters reside within a supercluster, and ~28% of superclusters contain at least one Abell cluster. The median mass and size of the superclusters are approximately 6 × 10¹⁵ M⊙ and 65 Mpc, respectively. The most massive supercluster, discovered at z ~ 0.25 and named the Einasto Supercluster, has a mass of ~2.57 × 10¹⁶ M⊙ and a size of ~111 Mpc. Several previously known superclusters, including Saraswati and the Sloan Great Wall, were rediscovered. A power-law relationship between supercluster density contrast and size is observed, with a slope of α ~ -2. A comparison of the properties of observed and simulated superclusters shows good agreement except for some differences in the mass distribution, likely due to the presence of fewer massive clusters in the simulation. The analysis of peculiar velocities in the mock superclusters shows that ~90% of member clusters are gravitationally influenced by the supercluster. The supercluster environment shows a slight positive bias in the mass of its member clusters; there's a slightly higher probability of finding more massive clusters within superclusters compared to the field. This effect was confirmed using a Kolmogorov-Smirnov test.

The findings address the need for a large, statistically robust supercluster catalog to investigate their properties and impact on cosmic structure formation. The power-law correlation between density contrast and size merits further investigation into its relation to supercluster morphology and evolution. The observed differences in the mass distribution between the observation and simulations highlight the importance of accurate modeling of cluster mass functions in simulations. The influence of the supercluster environment on cluster growth, while weak, suggests that superclusters are not passive structures but actively affect their surroundings. The study's large catalog combined with the dynamical information from the simulations allows a detailed analysis of the large-scale distribution of matter in the Universe and its evolution. The extensive catalog provides a valuable resource for future studies targeting individual superclusters, their detailed composition, and their role in the larger structure of the Universe.

This study presents a substantial advance in the understanding of superclusters by constructing the largest catalog of superclusters in the redshift range 0.05 ≤ z ≤ 0.42, using data from the WHL cluster catalog. The catalog and its associated analysis provide valuable information for future research on the evolution of the cosmic web and the properties of its largest structures. Future work should focus on detailed morphological analyses to explain the observed relationship between density contrast and size, further refined simulations to improve mass function modeling, and targeted multi-wavelength observations of the identified superclusters.

The study's reliance on the WHL cluster catalog introduces potential biases related to cluster detection efficiency, and mass estimations that are based on optical richness introduce uncertainties. The mass estimations of superclusters are limited to the sum of member cluster masses, neglecting the mass of inter-cluster matter. The incompleteness of low-mass clusters at higher redshifts might affect the identification of superclusters, although this was mitigated by adjusting the linking length. The use of convex hull to estimate supercluster volume introduces a degree of approximation in density contrast estimations. The selection effects in the SDSS survey need further investigation and analysis.