2.1. Supercluster sample

We selected a sample of 53 rich superclusters from the Main SuperCluster Catalogue (MSCC, Chow-Martínez et al. Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014), an all-sky catalogue containing 601 superclusters, identified by a tunable Friends-of-Friends algorithm based on a 2012 version of the spectroscopic redshift compilation of Abell/ACO-clusters (see Andernach et al. Reference Andernach, Tago, Einasto, Einasto, Jaaniste, Fairall and Woudt2005, for a description), with redshifts in the interval $0.02\leq z\leq 0.15$ . The multiplicity $m_{\mathrm{sc}}$ of each MSCC-supercluster was defined as the number of its member clusters in the Abell/ACO catalogue (e.g., Abell Reference Abell1958; Abell, Corwin, & Olowin Reference Abell and Corwin1989).

The criteria to select the sample were: (i) choose only rich superclusters with $m_{\mathrm{sc}}\geq 5$ ; (ii) select all the well-sampled superclusters inside the Sloan Digital Sky Survey (SDSS-DR13, Albareti et al. Reference Albareti2017), as a relatively complete subsample; and (iii) include known and well-studied superclusters of the Local Universe in the southern sky with available redshifts in the 2dF Galaxy Redshift Survey (2dFGRS, Colless et al. Reference Colless2001) and the 6dF Galaxy Survey (6dFGS, Jones et al. Reference Jones2009). The first criterion allows better identification of internal structures in superclusters from samples of member galaxies and systems. The second criterion provides better statistics due to the relative homogeneity of SDSS, the currently largest, densest and most complete available galaxy redshift survey, containing 116 MSCC-superclusters of all $m_{\mathrm{sc}}$ . Finally, the third criterion offers the possibility of having information from independent sources and including the most studied superclusters in order to be able to compare our results with those of previous studies, and to obtain validation support to our analysis.

Thus, our sample contains 45 superclusters within the SDSS region, which correspond to those selected by Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020), along with another 8 in the southern sky with a relatively good number of galaxies in the 2dFGRS or 6dFGS regions. The selected sample includes several well-known superclusters such as Shapley, Pisces-Cetus, Ursa Major, Coma-Leo, Corona-Borealis, Hercules, Böotes, among others. The complete sample of 53 superclusters is presented in Table 1: column 1 shows the IDs of the superclusters in the MSCC; column 2 shows their proper names when they exist; columns 3 and 4 present, respectively, the mean RA and Dec (Equinox = J2000) of the supercluster positions taken from the MSSC; column 5 shows their mean redshifts $\bar{z}$ , and column 6 presents their multiplicities $m_{\mathrm{sc}}$ . The other columns of Table 1 will be described below.

2.2. Samples of galaxies and systems

From now on we reserve the term ‘systems’ to refer to first-order galaxy agglomerations such as groups and clusters, and the term ‘structures’ to refer to larger and second-order galaxy agglomerations (clusters of systems) such as filaments, cores, and superclusters. Catalogues of higher density of objects make it possible to better define the systems, favoring the detection of the poorest ones and increasing the number of systems available for the detection of possible larger structures formed by them. These catalogues also improve the possibility of studying the correlations between the properties of galaxy populations and their surroundings, as well as analyzing the connectivity between member systems and substructures within superclusters. So, in order to detect cores in MSCC-superclusters and study their properties, it is more convenient to apply the identification algorithms on a denser and more homogeneous galaxy sample to include those systems poorer and smaller than the Abell/ACO-clusters used by Chow-Martínez et al. (Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014) to compile the MSCC.

2.2.2. Southern galaxy sample

For superclusters located outside the SDSS region we compiled catalogues analogous to GalCat and SysCat (see the procedure below), but taking galaxies from the 2dFGRS or 6dFGS surveys (or just 2dF and 6dF for short), depending on where each of these superclusters had higher number density and greater coverage and homogeneity of galaxies. Both redshift surveys have also been quite useful and historically important for the study of the distribution of galaxies in the LSS.

The 2dF survey provides reliable redshifts for 221 414 galaxies brighter than an extinction-corrected nominal magnitude limit of $b_J = 19.45$ ( $r_F \sim 18.3$ for early-type or $r_F\sim 18.6$ for late-type galaxies, e.g., Cole et al. Reference Cole2005), covering an area of $\sim$ 1 500 deg $^2$ (only $\sim$ 4% of the sky) in regions of high Galactic latitude in both the northern and southern Galactic hemispheres and with median redshift of $z = 0.11$ . The four superclusters of our sample inside this database are all in the southern hemisphere.

The 6dF survey offers a catalogue of 125 071 galaxies, making near-complete samples with magnitude limits $K \leq 12.65$ , $H \leq 12.95$ , $J \leq 13.75$ , $r_F \leq 15.60$ , and $b_J \leq 16.75$ (e.g., Jones et al. Reference Jones2009), for almost half of the sky ( $\sim$ 17 000 deg $^2$ on the southern sky, $|b| > 10$ deg), with a median redshift of $z=0.053$ . Although 6dF covers a larger area of the sky ( $\sim$ 41%), it is shallower than 2dF and SDSS, allowing to cover only four MSCC-superclusters with $z\leq 0.08$ in our sample.

Table 1. Sample of MSCC-superclusters

¹ Taken from M. Einasto et al. (Reference Einasto, Einasto, Tago, Müller and Andernach2001), Chow-Martínez et al. (Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014), Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020), and references therein.

To build the supercluster boxes we first transform the redshift-angular coordinates of 2dF/6dF galaxies and Abell/ACO-clusters to rectangular coordinates. Thus, if $({\unicode{x03B1}},{\unicode{x03B4}})$ are the equatorial coordinates (RA and Dec) of a galaxy or system, its rectangular coordinates can be estimated in the form

(1) \begin{equation}\begin{split}& X=D_{\text{c}}\cos{{\unicode{x03B4}}}\cos{\unicode{x03B1}},\\& Y=D_{\text{c}}\cos{{\unicode{x03B4}}}\sin{{\unicode{x03B1}}},\\& Z=D_{\text{c}}\sin{{\unicode{x03B4}}},\end{split}\end{equation}

where,

(2) \begin{equation}D_{\text{c}}(z)=\frac{c}{H_0}\int_{0}^{z}\frac{dz'}{E(z')},\end{equation}

is the line-of-sight comoving distance (e.g., Hogg Reference Hogg2000) of the object defined by its redshift z, c is the speed of light, and

(3) \begin{equation}E(z)\equiv\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_{\Lambda}}.\end{equation}

Then, in a similar way as in Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020), we take all the 2dF/6dF galaxies located within the corresponding rectangular volume box that encloses the member Abell/ACO-clusters of each supercluster, including those found up to a distance of 20 $h_{70}^{-1}$ Mpc beyond the center of the farthest member cluster in each direction. For each supercluster box, we compile the southern GalCat catalogue containing the equatorial coordinates (FK5, Equinox = J2000.0), the redshift and the magnitudes of each galaxy. For 2dF galaxies we select the final $b_J$ magnitude corrected from extinction and the SuperCosmos R magnitude (e.g., Hambly et al. Reference Hambly2001), while for 6dF galaxies we select the recalibrated $b_J$ and $r_F$ magnitudes (e.g., Cole et al. Reference Cole2005). Photometric and spectroscopic data were taken directly from the 2dF and 6dF catalogues, both available in VizieR.Footnote a In each supercluster box, a system detection algorithm was applied to build the corresponding southern SysCat. The details of this algorithm are described in Section 4. The left panel of Fig. 4 below shows, for example, the raw supercluster box for the Shapley Supercluster (MSCC 389 and MSCC 401) extracted from 6dF.

Figure 1. Distribution of $M_r$ galaxy absolute magnitudes as a function of redshift. Included here are the three galaxy samples taken from the SDSS (blue circles), 2dF (red boxes) and 6dF (black diamonds) Surveys.

Columns 7 and 8 of Table 1 show, respectively, the number $N_{g_{\text{box}}}$ of galaxies (GalCat) and the number $N_{\text{sys}}$ of detected galaxy systems (SysCat) within each supercluster box either from Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020) or the ones we detected from 2dF or 6dF; column 9 shows the survey from which the galaxy sample for each supercluster was drawn. In most cases, the number of systems identified within each supercluster box (in any of the SDSS, 2dF or 6dF regions) significantly exceeds the number of its member Abell/ACO-clusters (compare columns 6 and 8 of Table 1).

3.1. Distances and projection angle

For two galaxy systems observed with angular separation ${\unicode{x03B8}}_{jk}$ (in radians) and redshifts $z_j$ and $z_k$ , the physical distance $r_{jk}$ between them and their projected separation $R_{jk}$ in the sky-plane (both defined in units of $h_{70}^{-1}$ Mpc) are given, respectively, by

(4) \begin{equation}r_{jk}=\sqrt{D^2_{\text{c}}(z_j)+D^2_{\text{c}}(z_k)-2D_{\text{c}}(z_j)D_{\text{c}}(z_k)\cos{{\unicode{x03B8}}_{jk}}},\end{equation}

and,

(5) \begin{equation}R_{jk}\simeq{\unicode{x03B8}}_{jk}D_A(\bar{z})={\unicode{x03B8}}_{jk}\frac{D_{\text{c}}(\bar{z})}{1+\bar{z}},\end{equation}

where $D_A(z)$ is the angular diameter distance (e.g., Hogg Reference Hogg2000), and $\bar{z}$ the average redshift of the systems.

Furthermore, if $z_j\leq z_k$ , the tangent of the projection angle ${\unicode{x03C7}}$ between the separation vector along $r_{jk}$ and the sky-plane, at the midpoint between the pair of systems, is

(6) \begin{equation}\tan{{\unicode{x03C7}}}=\frac{z_k-z_j}{2z_j\tan{({\unicode{x03B8}}_{jk}/2)}},\end{equation}

with $0\leq{\unicode{x03C7}}\leq\pi/2$ (e.g., Sargent & Turner Reference Sargent and Turner1977). As can be seen below, this angle can be used to test the dynamical state of the pair of systems.

3.2 Binding tests

3.2.1. Pairwise gravitational binding

Consider two galaxy systems of masses $m_j$ and $m_k$ , with redshifts $z_j\leq z_k$ and separated by a distance $r_{jk}$ . Assuming linear orbits for the systems, i.e., with no rotations or discontinuities around the center of mass, one can use the Newtonian energy criterion, ${\unicode{x03C5}}_{jk}^2\leq 2G\mathcal{M}/r_{jk}$ , where ${\unicode{x03C5}}_{jk}$ is the relative velocity between systems, G the universal gravitational constant and $\mathcal{M}=m_j+m_k$ the pair total mass, to check the state of gravitational binding of the pair (e.g., Beers, Geller, & Huchra Reference Beers, Geller and Huchra1982). Thus, since $R_{jk}=r_{jk}\cos{{\unicode{x03C7}}}$ is the projected separation (in the sky-plane) between systems, and ${\unicode{x03C5}}_{r_{jk}}={\unicode{x03C5}}_{jk}\cos{{\unicode{x03C7}}}\approx H_0\,\ r_{jk}$ their relative radial velocity, then the pair binding criterion can be evaluated observationally in the form:

(7) \begin{equation}{\unicode{x03C5}}_{r_{jk}}^2R_{jk}\leq 2G\mathcal{M}\sin^2{{\unicode{x03C7}}}\cos{{\unicode{x03C7}}}.\end{equation}

3.2.2. Gravitationally bound structures

An alternative method can be tested to check the gravitational binding state of a ‘cluster of systems’ as a whole, if the individual masses of the member systems are known. A structure composed of N objects of masses $m_j$ , peculiar velocities ${\unicode{x03C5}}_j$ with respect to its center of mass, and relative separations $r_{jk}$ between them, will be gravitationally bound if $K\leq |W|$ , where K and W are the internal kinetic and potential energies, respectively, so that

(8) \begin{equation}\frac{1}{2}\sum_{j=1}^N{\unicode{x03C5}}_j^2m_j\leq \frac{1}{2}\sum_{j\neq k}^N \frac{Gm_jm_k}{r_{jk}}.\end{equation}

The above inequality can be expressed in the observationally more practical form (e.g., Schneider Reference Schneider2015):

(9) \begin{equation}\frac{1}{2}{\unicode{x03C3}}_{\text{sys}}^2\leq \frac{G\mathcal{M}}{r_G},\end{equation}

where ${\unicode{x03C3}}_{\text{sys}}^2\;:\!=\;\left\langle {\unicode{x03C5}}_i^2 \right\rangle$ is the squared velocity dispersion of systems inside the structure, $\mathcal{M}\;:\!=\;\sum m_k$ , and

(10) \begin{equation}r_G \;:\!=\; 2\mathcal{M}^2\left( \sum_{i\neq j} \frac{m_j m_k}{r_{jk}} \right)^{-1},\end{equation}

is the gravitational radius of the structure.

Since only the line-of-sight (radial) component of the velocity, ${\unicode{x03C5}}_{r_j}$ , can be estimated for each galaxy system, one can, to a first approximation, assume that ${\unicode{x03C3}}_{\text{sys}}^2= {\unicode{x03B2}}\left\langle {\unicode{x03C5}}_{r_j}^2 \right\rangle$ , with ${\unicode{x03B2}}\approx 3$ assuming a quasi-Maxwellian distribution of peculiar velocities of the systems inside the structures (see Bahcall Reference Bahcall1996) or ${\unicode{x03B2}}\approx 2.5$ assuming a weak anisotropy in the system velocity distributions (similar to that used for galaxy velocity distributions, e.g., Tully Reference Tully2015). Thus, the binding condition (9) can be evaluated in the form

(11) \begin{equation}{\unicode{x03B2}}{\unicode{x03C3}}_{{\unicode{x03C5}}_{\text{sys}}}^2 r_G \leq 2G\mathcal{M},\end{equation}

where, ${\unicode{x03C3}}_{{\unicode{x03C5}}_{\text{sys}}}^2=\left\langle {\unicode{x03C5}}_{r_j}^2 \right\rangle$ is the line-of-sight squared velocity dispersion of galaxy systems inside the structure.

3.3. Future virialized structures

Following the strategy adopted by Dünner et al. (Reference Dünner, Araya, Meza and Reisenegger2006), Chon et al. (Reference Chon, Böhringer and Zaroubi2015) and references therein, and thinking of each galaxy structure as an overdense region of mean mass density $\rho_{\text{ov}}$ , one can define the ratio

(12) \begin{equation}\mathcal{R}\equiv\frac{\rho_{\text{ov}}}{\rho_{\text{b}}},\end{equation}

between the overdensity and the local backgroundFootnote b mean mass density $\rho_{\text{b}}$ , as well as the density contrast

(13) \begin{equation}\Delta_{\text{cr}}\equiv\frac{\rho_{\text{ov}}}{\rho_{\text{cr}}}-1,\end{equation}

of the overdensity with respect to the critical density of the Universe, $\rho_{\text{cr}}=3H^2(z)/8\pi G$ , where $H(z)=H_0E(z)$ is the Hubble parameter, i.e., the Hubble constant at redshift z.

It is possible to use physically motivated density criteria to decide which structures – on scales of superclusters or cores – will be able to survive the cosmic expansion and become virialized systems in the future. Assuming spherically symmetric overdense regions, the density criteria are based on theoretical estimates of the mean density that must be enclosed by its last – or ‘critical’ – shell, at a given cosmological epoch, so that it remains gravitationally bound to the overdensity in a future dominated by dark energy (e.g., Chiueh & He Reference Chiueh and He2002; Łokas & Hoffman Reference Łokas and Hoffman2002; Dünner et al. Reference Dünner, Araya, Meza and Reisenegger2006). Using the spherical collapse model in an $\Lambda$ CDM scenario, Chon et al. (Reference Chon, Böhringer and Zaroubi2015) estimated, for various cosmologies, threshold values for $\mathcal{R}$ and $\Delta_{\text{cr}}$ that characterize structures that are currently at turn-around (i.e., structures that have already decoupled from the Hubble flow and are at rest in the Eulerian frame of reference, so they are now beginning to collapse) compared to those that will collapse marginally. Likewise, Dünner et al. (Reference Dünner, Araya, Meza and Reisenegger2006) estimated very consistent values at the current time for $\mathcal{R}$ and $\Delta_{\text{cr}}$ within the last layer that will eventually stop its expansion. The threshold values of these two parameters are shown in Table 2.

Table 2. Current-epoch threshold values for density ratio ( $\mathcal{R}$ ) and density contrast ( $\Delta_{\text{cr}}$ ) characterizing structures at turn-around and those that are marginally collapsing, assuming a flat cosmology with $\Omega_{m,0}+\Omega_{\Lambda,0}=1$ , where $\Omega_{m,0}=0.3$ , $\Omega_{\Lambda,0}=0.7$ and $H_0=70$ km s $^{-1}$ Mpc $^{-1}$

Although the threshold values for $\mathcal{R}$ and $\Delta_{\text{cr}}$ parameters are valid only for spherical overdensities, they are approximately reasonable for other realistic structures: these density criteria have been successfully used to define bound structures in future extended N-body cosmological simulations (Dünner et al. Reference Dünner, Araya, Meza and Reisenegger2006); to build catalogues of zones – from the SDSS-DR7 region – that will become future virialized structures (FVSs, Luparello et al. Reference Luparello, Lares, Lambas and Padilla2011); to study whether currently known superclusters will survive cosmic expansion (the superstes-clusters, Chon et al. Reference Chon, Böhringer and Zaroubi2015); and to identify and study the properties of quasi-spherical superclusters also in SDSS-DR7 region (e.g., Heinämäki et al. Reference Heinämäki, Teerikorpi, Douspis, Nurmi and Einasto2022). As stated in Sankhyayan et al. (Reference Sankhyayan2023), a definition of ‘superclusters’ based on large gravitationally bound structures most often leads to the identification of the central regions of the superclusters identified by other criteria like overdensities in galaxy or light distribution and converging peculiar velocity fields regions. Thus, the cores we define here may not be confused with ordinary superclusters, but are a generalization of the bound or future virialized structures described above.

3.4. Percolation and FoF algorithm

The galaxy clustering, present in a hierarchical way at different scales, can be studied by percolation theory (e.g., Stauffer Reference Stauffer1979; Shandarin Reference Shandarin1983), introduced in cosmological studies by Zeldovich, Einasto, & Shandarin (Reference Zeldovich, Einasto and Shandarin1982), Melott et al. (Reference Melott1983) and Einasto et al. (Reference Einasto, Klypin, Saar and Shandarin1984). In this context, the set of coordinates of galaxies and/or systems constitutes the point distribution space where the clustering will be analyzed.

The well-known Friends-of-Friends (FoF) algorithm is a percolation technique that uses a single input parameter, the linking length $\varepsilon$ , as a distance criterion to link points and detect agglomerates in a data space: two points p and q are linked to each other (‘friends’) if the distance between them is $d_{pq}\leq\varepsilon$ ; also a third point s, such that $d_{ps}>\varepsilon$ , is linked to both p and q if $d_{qs}\leq\varepsilon$ (‘friend of friend’). The set of all points that are mutually friend constitute a ‘cluster’. We will use the term ‘cluster’, in quotes, to refer to any agglomeration of points in a generic data space and avoid confusion with conventional clusters of galaxies. If $\varepsilon$ is very small, the number of ‘clusters’ will be reduced to those with the highest density of points. As $\varepsilon$ gets large, the dense ‘clusters’ will begin to link to each other through their boundary points and sparse points in the data volume, so the number of detected ‘clusters’ will begin to decrease until, on some large scale, the entire sample percolates (e.g., Einasto et al. Reference Einasto, Klypin, Saar and Shandarin1984). The value $\varepsilon=\varepsilon_c$ for which the number of detected ‘clusters’ in the percolation process is maximized is called critical linking length (or critical percolation radius, e.g., Shandarin Reference Shandarin1983; Chow-Martínez et al. Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014).

The FoF technique is attractive, among other things, because it produces a single ‘cluster’ catalogue for each linking volume and does not assume any particular shape or geometry for the ‘clusters’ (e.g., Berlind et al. Reference Berlind2006). In astronomy, the FoF algorithm has often been used to detect galaxy clusters in redshift surveys (e.g., Huchra & Geller Reference Huchra and Geller1982; Berlind et al. Reference Berlind2006), superclusters or filamentary structures in samples of galaxies and clusters (e.g., Einasto et al. Reference Einasto, Klypin, Saar and Shandarin1984, Reference Einasto, Einasto, Tago, Müller and Andernach2001; Caretta et al. Reference Caretta, Maia, Kawasaki and Willmer2002; Chow-Martínez et al. Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014), and identify dark matter halos in N-body simulations (e.g., Davis et al. Reference Davis, Efstathiou, Frenk and White1985).

3.5. The DBSCAN algorithm

Density-based clustering is a family of unsupervised learning methods capable of identifying distinctive ‘clusters’ in a data space. The method is based on the idea that a ‘cluster’ is a region of high density of points, separated from other ‘clusters’ by neighbouring regions of low density of points, typically considered noise/outliers (e.g., Sander Reference Sander, Sammut and Webb2011; Kriegel et al. Reference Kriegel, Kröger, Sander and Zimek2011). In this work we use the density-based spatial clustering of applications with noise (DBSCAN, Ester et al. Reference Ester, Kriegel, Sander and Xu1996), one of the most popular and cited clustering algorithms in the scientific literature. This algorithm is advantageous because it does not require knowing, as an input parameter, the number k of ‘clusters’ to be detected (unlike k-means or k-medoid algorithms). Rather, it determines this parameter automatically from the data set ( $\mathcal{D}$ ), the radius of the neighbourhood around each point ( $\varepsilon$ ), and the minimum number of points in each ‘cluster’ ( $N_{\text{min}}$ ). Once the input parameters $(\mathcal{D},\varepsilon,N_{\text{min}})$ are established, DBSCAN searches for ‘clusters’ according to the following definitions:

As a result, DBSCAN allows to discover ‘clusters’ with arbitrary shapes (spherical, linear, elongated, etc., depending on the chosen $d_{pq}$ metric function), using few input parameters and with good efficiency in large databases (see Ester et al. Reference Ester, Kriegel, Sander and Xu1996), so that it can be successfully used for astronomical analysis. The DBSCAN technique is basically a percolation-based algorithm for linking points, supplemented with sufficient density criteria to define agglomerates and noise. A similar extended percolation method using the density field instead of points was used by Einasto et al. (Reference Einasto, Suhhonenko, Liivamägi and Einasto2018) to study the connectivity of over- and under-dense regions in the cosmic web.

4.1. Detection of galaxy systems in the southern sample

The detection of systems in superclusters (from the 2dF or 6dF regions) to compile the southern SysCat catalogue was performed by an automated algorithm based on those presented by Biviano et al. (Reference Biviano, Murante, Borgani, Diaferio, Dolag and Girardi2006) and Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020), executing the following steps in each box:

Figure 2. Percolation curves (PCs) obtained for three superclusters sampled in the 2dF or 6dF region, using galaxies as input data. The PCs show the variation in the number of first-order galaxy ‘clusters’, with $N_{\text{min}}=3$ galaxies, detected by the DBSCAN algorithm as the neighbourhood radius varies. The lines represent the smoothing-spline of data and their maxima are located at the critical neighbourhood radius, $\varepsilon_c$ .

1. (1) The galaxies in the supercluster box were represented by the set of points with coordinates $\mathcal{D}=\lbrace({\unicode{x03B1}}_k, {\unicode{x03B4}}_k, 1\,000z_k), \,\ k=1,...,N_{g_{\text{box}}}\rbrace$ , compiled in its southern GalCat catalogue. The factor of 1 000 was applied to redshift z values to be comparable to the sky coordinate values ${\unicode{x03B1}}$ and ${\unicode{x03B4}}$ . The set of points in this form represents a pseudo-three-dimensional space where the – dimensionless- distances can be estimated with the metric

   (14) \begin{equation}d_{jk}=\sqrt{(\Delta{\unicode{x03B1}}_{jk}\cos{\bar{{\unicode{x03B4}}}})^2+(\Delta{\unicode{x03B4}}_{jk})^2+(1000\Delta z_{jk})^2}\end{equation}
   where $\Delta{\unicode{x03B1}}_{jk}={\unicode{x03B1}}_j-{\unicode{x03B1}}_k$ , $\Delta{\unicode{x03B4}}_{jk}={\unicode{x03B4}}_j-{\unicode{x03B4}}_k$ , $\Delta z_{jk}=z_j-z_k$ and $\bar{{\unicode{x03B4}}}$ is the mean declination between galaxies j and k.

2. (2) A DBSCAN-based algorithm, with input parameters $(\mathcal{D},\varepsilon,3)$ , was applied iteratively for a wide range of neighbourhood radii, $0\leq \varepsilon\leq 2.5$ , to analyze the percolation properties of galaxies and determine the – dimensionless – critical neighbourhood radius $\varepsilon_c$ .

   Fig. 2 shows an example of four percolation curves (PCs, i.e., number of detected ‘clusters’ vs. neighbourhood radius) obtained for three superclusters. The PCs were obtained with the data from the best box (in 2dF or 6dF) for each supercluster, except for one of them where the PCs were obtained with the data from both boxes (in 2dF and 6dF) for comparison. It can be seen that, in addition to its dependence on redshift, the radius $\varepsilon_c$ depends on factors such as the density and homogeneity of the survey in the region of a given supercluster. The critical neighbourhood radii obtained for the 8 superclusters in the 2dF and 6dF regions have mean and median values of $0.6$ and $0.5$ , respectively, similar to the dimensionless neighbourhood radius value obtained by Einasto et al. (Reference Einasto, Klypin, Saar and Shandarin1984) to detect galaxy clusters (and groups) by FoF.

3. (3) The radius $\varepsilon_c$ was established as the most appropriate to find ‘clusters’ with $N_{\text{min}}=3$ in the box. The projected centroid, $c_i=({\unicode{x03B1}}_i, {\unicode{x03B4}}_i)$ , and line-of-sight (radial) velocitiy, ${\unicode{x03C5}}_{r_i}$ , of each detected i-‘cluster’ was initially estimated as median values of its member galaxies. Not all the ‘clusters’ identified in this step are necessarily physical galaxy systems, but around their positions it is more likely to find them.

4. (4) All galaxies contained within an initial projected aperture $R_{a_i}=1$ $h_{70}^{-1}$ Mpc from each centroid $c_i$ and whose radial velocities were in the interval ${\unicode{x03C5}}_{r_i}\pm 3S_a$ , with $S_{a_i}=1\,000$ km s $^{-1}$ , were taken. That is, galaxy cylinders oriented along the line-of-sight in redshift space with radius $R_{a_i}$ and depth $6S_{a_i}$ were taken centered on $(c_i,{\unicode{x03C5}}_{r_i})$ .

5. (5) The cylinders with 5 or more galaxies were accepted as candidates for galaxy systems, while the others were rejected. For these candidates, line-of-sight velocities $V_{\text{LOS}_i}$ and velocity dispersions ${\unicode{x03C3}}_{{\unicode{x03C5}}_i}$ were estimated using Tukey’s biweight method (see Beers, Flynn, & Gebhardt Reference Beers, Flynn and Gebhardt1990), and centroids were recalculated.

6. (6) The virial mass of each candidate within the cylinder was determined as (e.g., Biviano et al. Reference Biviano, Murante, Borgani, Diaferio, Dolag and Girardi2006; Tully Reference Tully2015):

   (15) \begin{equation}\mathcal{M}_{\text{vir}_i}=\frac{{\unicode{x03B2}} \pi}{2G}{\unicode{x03C3}}_{{\unicode{x03C5}}_i}^2R_{\text{vp}_i},\end{equation}
   with ${\unicode{x03B2}}$ being, as above, an anisotropy parameter for the galaxy velocity distributions, and $R_{\text{vp}_i}$ , the projected mean radius, calculated in the form
   (16) \begin{equation}R_{\text{vp}_i}=\frac{N_i(N_i-1)}{\sum_{j<k} 1/R^{(i)}_{jk}},\end{equation}
   where $R^{(i)}_{jk}$ is the projected distance (in the sky-plane) between pairs of galaxies and $N_i$ the number of them within the i-th cylinder. Furthermore, assuming a spherical model for nonlinear collapse with virialization density $\rho_{\text{vir}}=18\pi^2[3H^2(z)/8\pi G]$ (e.g., Bryan & Norman Reference Bryan and Norman1998), the virial radius is then
   (17) \begin{equation}r_{\text{vir}_i}^3=\frac{3\mathcal{M}_{\text{vir}_i}}{4\pi\rho_{\text{vir}_i}}=\frac{{\unicode{x03B2}}{\unicode{x03C3}}_{{\unicode{x03C5}}_i}^2 R_{\text{vp}_i}}{18\pi H^2(z_i)}.\end{equation}

7. (7) For each candidate, the aperture $R_{a_i}$ was updated to the corresponding calculated $r_{\text{vir}_i}$ value, the median radial velocity $v_{r_i}$ to $V_{\text{LOS}_i}$ , and $S_a$ to ${\unicode{x03C3}}_{{\unicode{x03C5}}_i}$ , defining a new cylinder (including or excluding galaxies as the case may be). The process, from step (4) to (7), was repeated iteratively until finding the $r_{\text{vir}_i}\sim R_{a_i}$ convergence.

   Here, we assume an $r_{\text{vir}_i}\sim R_{a_i}$ convergence when

   (18) \begin{equation}\frac{|r_{\text{vir}_i}- R_{a_i}|}{R_{a_i}}\leq 0.05,\end{equation}
   that is, if the relative difference between $r_{\text{vir}_i}$ and $R_{a_i}$ was less than or equal to 5%.

8. (8) The candidates for which $r_{\text{vir}_i}$ converged before 20 iterations were accepted as real galaxy systems and their dynamical properties correspond to those estimated at the end of the last iteration, while those that did not converge were rejected. All the systems for which convergence $r_{\text{vir}_i}\sim R_{a_i}$ occurred achieved it well before 20 iterations.

Accepted systems become part of the corresponding southern SysCat catalogue of the respective supercluster, indicating for each member system its centroid $(\text{RA},\text{Dec},\bar{z})$ , position of its Brightest Cluster Galaxy (BCG), the number of member galaxies ( $N_g$ ), its line-of-sight velocity ( $V_{\text{LOS}}$ ), its radial galaxy velocity dispersion ( ${\unicode{x03C3}}_{{\unicode{x03C5}}}$ ), its virial mass ( $\mathcal{M}_{\text{vir}}$ ) and its projected ( $R_{\text{vp}}$ ) and virial ( $r_{\text{vir}}$ ) radii.

Fig. 3 shows the distribution of richness versus mass for our total sample of 3 337 SysCat systems (i.e., combining those identified in the SDSS, 2dF and 6dF regions). This distribution is very similar to the one obtained by Tempel et al. (Reference Tempel2014) for systems with $N_{\mathrm{gal}} \geq 5$ .

Figure 3. Richness as a function of mass for the complete sample of 3 337 SysCat systems identified in the SDSS, 2dF, and 6dF regions.

4.1.1. FoG-effect correction

For each accepted system, a simple correction for the Finger-of-God effect (FoG, e.g., Coil Reference Coil2012) was performed by adjusting the position of its member galaxies in the final cylinder, so that their comoving distances were rescaled to stay within the calculated virial radius. Thus, if $D_{\text{c}_k}=D_{\text{c}}(z_k)$ is the initial comoving distance of the k-th member galaxy, at redshift $z_k$ , in the i-th system, its rescaled comoving distance is

(19) \begin{equation}D'_{\!\!\text{c}_k}=\frac{2r_{\text{vir}_i}}{{\unicode{x03B5}}} \left( D_{\text{c}_k}-D_{\text{c}}^{(i)} \right)+D_{\text{c}}^{(i)},\end{equation}

where $D_{\text{c}}^{(i)}=D_{\text{c}}(V_{\text{LOS}_i}/c)$ is the comoving distance to the centroid of the system, and ${\unicode{x03B5}}=\max_k{\left\lbrace D_{\text{c}_k} \right\rbrace}-\min_k{\left\lbrace D_{\text{c}_k} \right\rbrace}$ , with $k=1,...,N_i$ , is the distance (in the line of sight) between the nearest and the most distant galaxy in the system. The surface pressure term correction based on the concentration parameter was not applied here, however a virial approximation is sufficient for the objective of this work (e.g., Santiago-Bautista et al. Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020).

The FoG-effect can be seen in the three-dimensional distribution of galaxies obtained by transforming their coordinates from raw redshift-angular (provided by the source survey) to rectangular directly through equation (1), as shown in the left panel of Fig. 4. for the Shapley Supercluster. The regions of red points are galaxy systems which appear elongated in the line of sight due to the effect of the peculiar velocities of member galaxies (e.g., Coil Reference Coil2012). After the FoG-correction, re-estimating the rectangular coordinates only for member galaxies of systems using the rescaled comoving distances from equation (19) in equation (1), the systems and their spatial distribution within the supercluster can be clearly distinguished as shown in the right panel of Fig. 4.

4.1.2. System mass uncertainties

Although Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020) did not provide uncertainties for the dynamical masses of their systems, we estimate such uncertainties for all SysCat systems, including both those identified by them from the SDSS galaxy sample and those identified by us from the 2dF and 6dF galaxy samples. The mass uncertainties were estimated here through simple error propagation (i.e., $\Delta y = [\sum (\Delta x_i\partial y/\partial x_i)^2]^{1/2}$ ) such that, for each i-system

(20) \begin{equation}\Delta \mathcal{M}_{\mathrm{vir}_i}= \frac{{\unicode{x03B2}}\pi}{2G}{\unicode{x03C3}}_{{\unicode{x03C5}}_i}\sqrt{(2 R_{\mathrm{vp}_i}\Delta{\unicode{x03C3}}_{{\unicode{x03C5}}_i})^2+({\unicode{x03C3}}_{{\unicode{x03C5}}_i} \Delta R_{\mathrm{vp}_i})^2},\end{equation}

where $\Delta{\unicode{x03C3}}_{{\unicode{x03C5}}_i}$ represents the uncertainty in the system velocity dispersion, taken as ${\unicode{x03C3}}_{{\unicode{x03C5}}_i}/\sqrt{N_i}$ (e.g., Beers et al. Reference Beers, Flynn and Gebhardt1990), and

(21) \begin{equation}\Delta R_{\mathrm{vp}_i}\approx \frac{2R_{\mathrm{vp}_i}^2}{N_i(N_i-1)}\times \frac{\Delta{\unicode{x03B8}} (\bar{z}+1)}{D_{\mathrm{c}}(\bar{z})}\sqrt{\sum_{j<k} \left[{\unicode{x03B8}}^{(i)}_{jk}\right]^{-6}},\end{equation}

is the propagated uncertainty in the projected mean radius of the system (at average redshift $\bar{z}$ ) due to the astrometric precision $\Delta{\unicode{x03B8}}$ (in radians) of the galaxy angular position on the sky-plane (e.g., $\Delta{\unicode{x03B8}}\sim 0.1$ arsec for SDSS and $\Delta{\unicode{x03B8}}\sim 0.2$ arcsec for 2dF and 6dF). Here, ${\unicode{x03B8}}^{(i)}_{jk}$ represents the angular separation between pairs of observed member galaxies in the i-system.

The mass uncertainties with respect to redshift for the total sample of SysCat systems is shown in Fig. 5. Note that there is no significant trend of the relative uncertainties (to increase or decrease) with respect to redshift. It can be seen that the uncertainties obtained are typical for this type of systems when compared with other catalogues: our relative uncertainties have mean and median values of 0.367 and 0.351, respectively (see the marginal histogram in the vertical axis of Fig. 5), similar to those obtained, for example, for the GalWCat19 (Abdullah et al. Reference Abdullah, Wilson, Klypin, Old, Praton and Ali2020), which have mean and median values of 0.372 and 0.343, respectively.

Figure 4. The initial supercluster box for the Shapley Supercluster (MSCC 389 and MSCC 401): each dot (4 649 in total) represents an observed 6dF galaxy in 3D-rectangular coordinates. On both panels the red dots represent the members of galaxy systems. Left: galaxy positions before the FoG-effect correction. Right: galaxy positions after FoG-effect correction.

Figure 5. Distribution of relative mass uncertainties $\Delta \mathcal{M}_{\mathrm{vir}}/ \mathcal{M}_{\mathrm{vir}}$ as a function of redshift for SysCat systems. The marginal histograms in the horizontal and vertical axes show the total distribution of systems (SDSS+2dF+6dF) with respect to redshift and relative mass uncertainties, respectively.

We have also compared the masses obtained by our method with those from other sources in the literature such as Top70 (Caretta et al. 2023; Zúñiga et al. Reference Zúñiga, Caretta, González and Garca-Manzanárez2024), which comprise 70 nearby well sampled galaxy clusters; GalWCat19 (Abdullah et al. Reference Abdullah, Wilson, Klypin, Old, Praton and Ali2020), a catalogue with 1 800 groups up to $z=0.2$ from SDSS-DR13; and Tempel et al. (Reference Tempel2014), a flux-limited catalogue of groups from SDSS-DR10, all considering only galaxies with spectroscopic redshifts. The comparison (Fig. 6) shows consistent mass estimates between the four samples and methods.

Figure 6. Comparison of SysCat masses with system masses from other similar catalogues: GalWCat19 (Abdullah et al. Reference Abdullah, Wilson, Klypin, Old, Praton and Ali2020) masses are $\mathcal{M}_{100}$ calculated from a NFW profile, the ones closest to $\mathcal{M}_{\mathrm{vir}}$ ; Tempel et al. (Reference Tempel2014) are total masses estimated assuming a Herniquist density profile; and Top70 (Caretta et al. 2023; Zúñiga et al. Reference Zúñiga, Caretta, González and Garca-Manzanárez2024) are also virial masses, estimated in a similar way but with a different (and more complete) database. SysCat and Tempel groups tend to slightly overestimate the masses with respect to Top70, while GalWCat19 slightly underestimates them.

4.2. Detection of galaxy structures

The detection of structures inside superclusters of any redshift survey region (SDSS, 2dF or 6dF) was performed by an automated algorithm, similar to that of the detection of systems, but using the SysCat catalogues as input data. The procedure for each supercluster was:

1. (1) Using transformations from equation (1), the member systems in the corresponding box were represented by the set $\mathcal{D}=\left\lbrace (X_k,Y_k,Z_k),\,\ k=1,...,N_{\text{sys}} \right\rbrace$ of points, in rectangular coordinates, of their centroids compiled in SysCat. In this case, the set of points represents a three-dimensional space where the distances (in units of $h_{70}^{-1}$ Mpc) can be calculated with the standard Euclidean metric.

2. (2) A DBSCAN-based algorithm, with input parameters $(\mathcal{D},\varepsilon,2)$ , was applied for a range of neighbourhood radii, $0\leq\varepsilon\leq 25$ $h_{70}^{-1}$ Mpc, to analyze the percolation properties of galaxy systems and determine the critical radius $\varepsilon_c$ .

   Fig. 7 shows an example of PCs obtained for seven of the superclusters of Table 1 sampled in some of the surveys (SDSS, 2dF or 6dF). Again, the curves reach their maximum, i.e., greatest number of detected ‘clusters’, when $\varepsilon=\varepsilon_c$ . The value of $\varepsilon_c$ obtained for each supercluster is shown in column 9 of Table 3. The variation of $\varepsilon_c$ with respect to the mean redshift $\bar{z}$ of the supercluster, i.e., the so-called percolation function (PF, e.g., Chow-Martínez et al. Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014), can be seen in Fig. 8. The critical radius that maximizes the detected amount of ‘clusters’ of galaxy systems in superclusters is, as expected, an increasing function of their mean redshift. For the total sample, $\varepsilon_c$ has mean and median values of $10.4$ $h_{70}^{-1}$ Mpc and $9.6$ $h_{70}^{-1}$ Mpc, respectively.

3. (3) The radius $\varepsilon_c$ was established as the most appropriate to find galaxy structures with $N_{\text{min}}=2$ in the supercluster box.

   The number $N_{\text{str}}$ of structures detected by the algorithm within each supecluster box is shown in column 10 of Table 3.

Figure 7. Percolation curves (PCs) obtained for seven of the superclusters in the sample, using galaxy systems as input data. The PCs show the variation in the number of second-order galaxy ‘clusters’, i.e., ‘clusters’ of galaxy systems with $N_{\text{min}}=2$ , detected by the DBSCAN algorithm as the neighbourhood radius varies. The lines represent the smoothing-spline of data and their maxima are located at the critical neighbourhood radius, $\varepsilon_c$ .

A total of 155 structures were identified in the full sample of superclusters. While in principle each SysCat-object represents a virialized – physical – galaxy system, the structures detected at this stage are only distance-linked ‘clusters’ of systems, but there is no guarantee that they are gravitationally bound structures or that they have enough density to survive the cosmic expansion. Several of the detected structures correspond to filaments, chains of galaxy systems that generally connected with each other if the neighbourhood radius $\varepsilon$ is allowed to grow as reported by Einasto et al. (Reference Einasto, Klypin, Saar and Shandarin1984). In what follows we will focus on selecting only those structures that meet the necessary criteria to be considered cores of superclusters.

4.2.1. Extensive mass of structures

The multiplicity m of each detected structure was defined as the number of systems (linked by DBSCAN) that compose it, and its extensive mass, i.e., the sumFootnote c of the dynamical masses of its constituent parts, can be estimated in the form

(22) \begin{equation}\mathcal{M}_{\text{ext}}=\sum_{i=1}^{m} \mathcal{M}_{\text{vir}_k},\end{equation}

where $M_{\text{vir}_k}$ is the virial mass of each system in the structure. Assuming that all systems inside the structure are viralized, one might expect the total mass of a structure to be only slightly greater than its extensive mass, that is $\mathcal{M}_{\text{tot}}\geq \mathcal{M}_{\text{ext}}$ . The difference in mass would be given by the matter (probably gas, galaxies or dark matter) that resides between systems or as a dispersed component, i.e., not contained in galaxy systems.

In particular, the mass $\mathcal{M}^{\text{sc}}_{\text{ext}}$ of the superclusters was estimated from the above equation, but using $m=N_{\text{sys}}$ . The value of $\mathcal{M}^{\text{sc}}_{\text{ext}}$ (and its respective propagated uncertainty) for each supercluster in the sample is shown in column 5 of Table 3. For comparison, the masses of some well-known supercluster are available in the literature: the total mass of the Coma-Leo (MSCC 295) and Hercules (MSCC 474) superclusters are estimated in the ranges of $(168-189)\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ and $(67-260)\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ , respectively (e.g., Böhringer & Chon Reference Böhringer and Chon2021); the mass of the Shapley Supercluster (MSCC 389 and MSCC 401) is estimated in the range of (142–285) $\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ (e.g., Ragone et al. Reference Ragone2006; Sheth & Diaferio Reference Sheth and Diaferio2011); and the mass of the Corona-Borealis Supercluster (MSCC 463) lie in the range of (67–420) $\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ (e.g., Small et al. Reference Small, Ma, Sargent and Hamilton1998; Einasto et al. Reference Einasto2021). As can be seen, our estimates using $\mathcal{M}^{\text{sc}}_{\mathrm{ext}}$ are consistent with the mass ranges determined by other authors.

Table 3. General properties of the MSCC-superclusters in the sample

4.2.2. Volume estimations

The volume of any structure can be approximated by the volume enclosed by the surface enclosing its member galaxies (represented as points in FoG-corrected rectangular coordinates from equation (1)). Such enclosing surfaces can be built by triangulating boundary points using alpha-shape based algorithms (e.g., Edelsbrunner & Mücke Reference Edelsbrunner and Mücke1994) that allow polyhedral surfaces to be built around sets of points in three-dimensional space.

In particular, in this work a MATLAB alpha-shape based algorithm (see MATLAB 2023) was used, which allows the generation of a single polyhedral surface around a set of points (x, y, z) in three-dimensional space and returns an estimate of the volume enclosed by it. In such algorithm the fit of polyhedral surfaces can be adjusted (tightened/loosened) by means of a ‘shrink factor’ $s_{\text{f}}$ ranging from 0, generating the convex hull, to 1, generating the compact boundary. Fig. 9 shows the polyhedral surface fit, with $s_{\text{f}}=1$ , made for the Shapley Supercluster from the galaxies contained in its – FoG-corrected and purified – supercluster box. The volumes $V_{\text{sc}}$ of the sampled superclusters are shown in column 6 of Table 3. Such values were estimated by fitting polyhedral surfaces, with $s_{\text{f}}=1$ , to the set of member galaxies in each refined supercluster box.

Figure 8. Percolation function (PF) obtained for the supercluster sample: critical neighbourhood radius for detecting galaxy structures increases as a function of supercluster redshift. The solid black line represents the best fit (goodness, $\mathcal{R}^2_{\text{det}}=0.34$ ) to the data: a power law $az^{b}+c$ with coefficients $a=685.20$ , $b=2.26$ , and $c=6.58$ . In general, the critical radius used to detect galaxy structures increases with mean redshift due to incompleteness and thus the loss of galaxy density.

Figure 9. Polyhedral surface (compact boundary, with shrink factor $s_{\text{f}}=1$ ) fitted to the – purified and FoG-corrected – sample of member galaxies of Shapley Supercluster (MSCC 389 and MSCC 401). The volume enclosed by the compact boundary is $V_{\text{sc}}=191.1$ $h_{70}^{-3}$ Mpc $^3$ , smaller than that which would enclosed by the convex hull (e.g., with $s_{\text{f}}=0$ ) since the latter include voids of considerable size.

5.1. Purifying the galaxy sample in superclusters

To identify cores, we first refined the samples of supercluster ‘member’ galaxies from both southern and SDSS samples as follows:

1. (1) Inside each supercluster box, a FoF algorithm was applied to the set of $N_{\text{sys}}$ systems identified there to determine the smallest radius $\varepsilon_{\text{sc}}$ linking them all. For this, each (SysCat) system was visualized as a single point, its centroid ( $c_i$ ) in rectangular coordinates.

2. (2) For each galaxy ( $\text{g}_*$ ) in the corresponding initial FoG-corrected supercluster box ( $\mathcal{B}_{\text{sc}}$ ), its Euclidean distance $d(\text{g}_*,c_i)$ to each galaxy system was calculated. Galaxies were provisionally labelled as ‘linked’ to the nearest galaxy system.

3. (3) All galaxies within a radius $\varepsilon_{\text{sc}}$ around the centroid of the system to which they were ‘linked’ were taken as a ‘member’ of the respective supercluster. The final, purified, supercluster box ( $\mathcal{G}_{\text{sc}}$ ) only contains the galaxies selected as ‘members’. This set of ‘member’ galaxies can be represented, for each supercluster, in the form

   (23) \begin{equation}\mathcal{G}_{\text{sc}}=\left\lbrace \text{g}_* \in \mathcal{B}_{\text{sc}} | \min_i\left\lbrace d(\text{g}_*,c_i)\right\rbrace \leq \varepsilon_{\text{sc}}, i=1,...,N_{\text{sys}} \right\rbrace.\end{equation}

Galaxies not selected by this criterion were considered as ‘field’ galaxies, not belonging to superclusters. There is no univocal criterion to define the membership of a galaxy to a supercluster, so here the criterion was to include all galaxies that are members of systems as well as those that are between them, forming galaxy bridges or being part of the disperse component of the supercluster up to a distance that we consider reasonable, e.g., $\varepsilon_{\text{sc}}$ from each identified galaxy system.

The radius $\varepsilon_{\text{sc}}$ and the number $N_{g_{\text{sc}}}$ of ‘member’ galaxies found for each supercluster are shown respectively in columns 10 and 11 of Table 1.

5.2. Definition of cores

Rich superclusters have central regions of high density of galaxies (and systems) that are absent in poor superclusters (e.g., Einasto et al. Reference Einasto2007b, Reference Einasto2008). Such regions, commonly called cores, may be observationally defined as collections of rich (and poor) clusters, groups and individual galaxies, being identified by their high galaxy density contrasts ( ${\unicode{x03B4}}>10$ , according to Einasto et al. Reference Einasto2007b, Reference Einasto2021). Given these characteristics, it could be said that cores are “internal structures of superclusters that have begun to materialize as recognizable entities” (e.g., Araya-Melo et al. Reference Araya-Melo, Reisenegger, Meza, van de Weygaert, Dünner and Quintana2009), like ‘compact superclusters’ (e.g., Pearson et al. Reference Pearson, Batiste and Batuski2014). The most important characteristic of these entities is probably that they are the largest structures in the Universe that are expected to survive cosmic expansion, reaching, at least marginally, dynamical equilibrium.

In this work only rich MSCC-superclusters (with $m\geq 5$ member Abell/ACO-clusters) have been selected, so the presence of cores is expected in them. To identify the cores we think of them as massive and gravitationally bound galaxy structures, within superclusters, comprised of two or more galaxy systems (groups or clusters), with high probability of future collapse and virialization. The cores of superclusters involve high local densities in regions of small volume (e.g., Marini et al. Reference Marini2004). Based on the above, we choose as core candidates those structures that, at the present-epoch, meet the following conditions:

1. i. Have extensive masses $\mathcal{M}_{\mathrm{ext}}\geq 5\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ .

2. ii. Are composed by gravitationally bound galaxy systems.

3. iii. Have a density ratio $\mathcal{R}\geq 7.86$ and an overdensity $\Delta_{\mathrm{cr}} \geq 1.36$ .

The first condition allows the massive systems to be used as markers (or ‘seed’ points, e.g., Chow-Martínez Reference Chow-Martínez2019) of cores; rich galaxy clusters, for example, are powerful indicators of local density in supercluster environments (Einasto et al. Reference Einasto2008). The minimum mass $5\times 10^{14}h_{70}^{-1}\mathcal{M}_{\odot}$ (e.g., $\sim 10^{15}h_{70}^{-1}\mathcal{M}_{\odot}$ chosen by Araya-Melo et al. (Reference Araya-Melo, Reisenegger, Meza, van de Weygaert, Dünner and Quintana2009) to select bound superclusters in cosmological simulations) is of the order of magnitude of the richest clusters, so any mass above that represents a considerable galaxy agglomeration. Thus, searching for systems with large virial masses increases the probability of finding nucleation regions, i.e., zones of higher concentration of matter in the form of clusters, groups, individual galaxies and hot gas.

The spherical density criterion described in Section 3.3 forms a key ingredient in our definition of cores and for their selection process from the identified structures. Conditions (ii) and (iii) above, together, guarantee, at least theoretically, that galaxy structures once gravitationally bound will begin to collapse to virialize in the future according to the threshold values presented in Table 2. In fact, condition (iii) could guarantee, by itself, that the structures under these density conditions will remain bound despite the expansion. However, an alternative method is used here to analyze the current gravitational binding state of the structures, which is why the second condition is taken independently.

Figure 10. Schematic of a galaxy structure identified by DBSCAN with a critical neighbourhood radius $\varepsilon_c$ . Its initial member systems are enclosed by the red dashed line, and $d_f$ is the distance between the MMC and the Farthest Galaxy System (FGS) from it inside the structure. The systems outside the dashed line (if they exist) are called Surrounding Galaxy Systems (SGS). If any SGS is found in the spherical shell of radii $d_f$ and $d_f+1.5\varepsilon_{c}$ , centered on the MCC, it is provisionally taken as a member of the structure. Whether or not a member system remains in the structure will depend on the gravitational binding criteria.

Figure 11. Flow chart of the semi-automatic CorSel-Algorithm.

5.3. Core selection

Using the three conditions stated above to define cores, these were selected from among the structures identified within each supercluster in the sample. The selection process was developed through a semi-automatic analysis as follows:

1. (1) The extensive mass of each structure is calculated by equation (22) and used as an approximation for its total mass. Only structures with $\mathcal{M}_{\mathrm{ext}}\geq 5\times 10^{14}$ $h_{70}^{-1}\mathcal{M}_{\odot}$ are accepted as core candidates.

2. (2) For each structure, its MMC, i.e., the member galaxy system with the largest virial mass, is identified and the distance $d_f$ from this to the farthest constituent system of the initial structure is determined. Then, all the systems in the surroundings of the structure (if any in the supercluster box), within the spherical shell between $d_f$ and $d_f+1.5\varepsilon_{c}$ , centered on the MMC centroid, are provisionally included as members of the – enlarged – structure (see Fig. 10). Here, $\varepsilon_{c}$ is the corresponding critical neighbourhood radius used to identify the initial structure (see column 7 of Table 3). In this stage, a new $\mathcal{M}_{\mathrm{ext}}$ for the enlarged structure is calculated if it is the case.

3. (3) The state of gravitational binding between pairs of member systems inside the enlarged structure is analyzed using the criterion established in equation (7): a member is classified as directly-bound ( $\mathcal{D}_b$ -member) if it meets the criterion paired with the MMC or as indirectly-bound ( $\mathcal{I}_b$ -member) if it does not meet the criterion with the MMC, but is linked to a third member that is directly-bound to both separately (like a ‘gravitational FoF’). These two sets of members form, together with the MMC, the pairwise bound region of the structure. Members of the enlarged structure that are neither directly- nor indirectly-bound to the MMC are classified as possibly-bound ( $\mathcal{P}_b$ -member). This can happen because these $\mathcal{P}_b$ -members can still be bound to the larger mass of the whole structure, even if not bound to a specific neighbour.

4. (4) In a complementary way, the gravitational binding state of the structure as a whole is analyzed using the criterion established in equation (11). For this, the line-of-sight velocity dispersion ${\unicode{x03C3}}_{{\unicode{x03C5}}_{\mathrm{sys}}}$ of systems inside the structure is estimated from their mean velocities using Tukey’s biweight robust method (e.g., Beers et al. Reference Beers, Flynn and Gebhardt1990), which is best suited for the small amount of member radial velocity data, and ${\unicode{x03B2}}=2.5$ , taken assuming a weak anisotropy in the velocity distribution of member systems. Furthermore, in order to estimate the gravitational radius $r_G$ , the virial mass $\mathcal{M}_{\mathrm{vir}_k}$ of each member system of the structure is used in equation (10).

5. (5) If the structure contains a pairwise bound region and $\mathcal{P}_b$ -members, then one of the following two cases could occur:

   1. (a) If the structure is bound as a whole (or globally bound), then the $\mathcal{P}_b$ -members are considered as bound members and the entire enlarged structure is accepted.

   2. (b) Instead, if the structure is not globally bound, the farthest $\mathcal{P}_b$ -member from the MMC is removed and step (4) is performed again. The process is repeated until a globally bound structure is obtained, stripped of unbound members, or until only the pairwise bound region remains.

   On the other hand, cases may occur in which there are no pairwise bound systems within a structure (i.e., there is no a pairwise bound region), but it is globally bound. These type of structures are also accepted.

   In any case, the structures accepted after this step continue to be considered core-candidates of the corresponding supercluster, while the others are discarded.

6. (6) The member galaxies of each core-candidate are defined as all galaxies up to a distance of $3.5R_{\mathrm{vir}_k}$ from the centroid of each k-th member system in the corresponding FoG-corrected supercluster box. At the chosen distance we expect to select the galaxies of member systems up to about their turn-around zone, those present in bridges between them, as well as galaxies that can be considered as the disperse component of the core-candidate. The photometric and spectroscopic data of core galaxies are extracted from the respective GalCat catalogue.

7. (7) The mean densities of each core-candidate and its corresponding host supercluster are estimated, respectively, in the form

   (24) \begin{equation}\rho_{\text{c}}=\frac{\mathcal{M}_{\text{ext}}^c}{V_{\text{c}}}, \,\ \text{and} \,\ \rho_{\text{sc}}=\frac{\mathcal{M}^{\text{sc}}_{\text{ext}}}{V_{\text{sc}}},\end{equation}
   where $\mathcal{M}_{\text{ext}}^c$ and $\mathcal{M}^{\text{sc}}_{\text{ext}}$ are the extensive masses of the core-candidate and the supercluster, and $V_{\text{c}}$ and $V_{\text{sc}}$ are the respective volumes estimated from the spatial distribution of their member galaxies using the alpha-shape based method described in Section 4.2.2, with $s_{\text{f}}=0.5$ for core-candidates and $s_{\text{f}}=1$ for superclusters. The reason for this choice of shrink factors is explained below.

The masses $\mathcal{M}^{\text{sc}}_{\text{ext}}$ and volumes $V_{\text{sc}}$ used here for superclusters are those compiled in columns 5 and 6 of Table 3, while the respective (mass and galaxy) densities calculated from these values are shown in columns 7 and 8 of the same table. The estimated gravitational radius, extensive mass, volume and density values for the core-candidates are presented in Table 5 only for those that were finally selected as cores (see Section 6).

Now, assuming $\rho_{\text{sc}}$ to be the local background density of each core-candidate in its respective supercluster, its density ratio is calculated from equation (12) in the form

(25) \begin{equation}\mathcal{R}=\frac{\rho_{\text{c}}}{\rho_{\text{sc}}},\end{equation}

and its density contrast with respect to the critical density from equation (13) as

(26) \begin{equation}\Delta_{\text{cr}}=\frac{8\pi G\rho_{\text{c}}}{3H^2(z)}-1\end{equation}

where z is the mean redshift of the host supercluster using the cosmology parameters described in Section 1. Finally, only candidates that meet criteria $\mathcal{R}\geq 7.86$ and $\Delta_{\text{cr}} \geq 1.36$ (see Table 2) are considered cores, and the other structures were rejected.

Figure 12. Top: The three cores identified for the Shapley Supercluster (MSCC 389 and MSCC 401). The red points enclosed by yellow polyhedral surfaces correspond to the member galaxies of the systems of the respective cores. The labels in each core inform about the main Abell/ACO-clusters that comprise them. Bottom: the main core of the Shapley Supercluster with 448 member 6dF-galaxies. The red dots represent the member galaxies of systems (Abell/ACO-clusters in this case), while the blue dots represent galaxies in bridges and disperse component; CM and GC are, respectively, the center of mass and the geometric center (centroid) of the core; the black ‘+’ symbols represent the centroids of the member clusters from the Abell/ACO catalogue.

We took the values $s_{\text{f}}=0.5$ and $s_{\text{f}}=1$ , respectively, for the polyhedral fit shrink factors of the core-candidates and superclusters, because with these values we have the worst case scenario for the density ratio (contrast) of a core-candidate, possibly underestimating its density and overestimating the density of its local environment (the host supercluster). If a candidate meets the criteria under these conditions, they will meet the criteria under any other volume setting. A summary of the core selection process (or CorSel Algorithm) is shown in Fig. 11.

The top panel of Fig. 12 shows an example of the cores identified in the Shapley Supercluster labelled with the respective main Abell/ACO-clusters that constitute them. The polyhedral surfaces fitted (with $s_{\text{f}}=0.5$ ) to the member galaxies of the core, and used to determine the volumes of these, are also shown. The bottom panel of Fig. 12 shows in greater detail the distribution of member galaxies in the so called main core (i.e., A3556-A3558-A3562 and other clusters) of Shapley Supercluster and the Abell/ACO-clusters that match its member systems. The three cores identified for the Shapley Supercluster also correspond to the regions with high surface density of galaxies in the RA-Dec plane, as can be seen in Fig. 13. Note that there is a forth surface overdensity in this figure, north of the main core, corresponding to the region of A1736 cluster: this region do not correspond to a core because the overdensity is caused by a projection effect (A1736 is composed of two rich systems aligned towards the LOS, e.g., Caretta et al. 2023). These cases can be identified in all superclusters that have cores in our sample.

Figure 13. 2D surface density map of the RA-Dec distribution of galaxies in the Shapley Supercluster (MSCC 389 and MSCC 401). Three of the regions with the highest density of galaxies correspond to the cores identified for the Shapley Supercluster: DCC 066 ( $\mathrm{RA}=194.211$ , $\mathrm{Dec}=-30.698$ ), DCC 067 ( $\mathrm{RA}=202.604$ , $\mathrm{Dec}=-31.075$ ), and DCC 068 ( $\mathrm{RA}=207.434$ , $\mathrm{Dec}=-32.631$ ). The density peaks correspond to the MMCs of each core.

Figure 14. Top: The histogram shows the number of superclusters as function of the number of cores they contain. Bottom: Box plot showing the relationship between the extensive mass of superclusters ( $\mathcal{M}_{\text{ext}}^{\text{sc}}$ in units of $10^{14}h^{-1}_{70}\mathcal{M}_{\odot}$ ) and the number of cores they contain; the boxes represent the interquartile range with the red line indicating the median mass, the whiskers add 1.5 times the interquartile range, and notches display the confidence interval (with a significance level of 0.05) around the median.

The number of cores ( $N_{\text{crs}}$ ) found within each supercluster box, following the steps and criteria described above, is shown in column 11 of Table 3. From the histogram in the top panel of Fig. 14 it can be noted that about 83% of the MSCC rich superclusters in the sample have at least one core, about 19% have two cores, and about 36% have three or more cores. Additionally, the boxplot in the bottom panel of Fig. 14 shows a clear correlation between the number of cores and the extensive mass (e.g, $\mathcal{M}_{\text{ext}}^{\text{sc}}$ in Table 3) of their host superclusters. This correlation is expected given that the more massive a supercluster is, the greater the probability of forming regions (internal structures) with sufficient density to bind gravitationally and break away from the cosmic expansion. The sampled MSCC-superclusters with masses below $\sim$ $10^{15}h_{70}^{-1} \mathcal{M}_{\odot}$ tend to have fewer than three cores, while those with masses above that value tend to have three or more.