2.1. Our sample of cores

For this study, we used the full core sample from the Density-based Core Catalogue (DCC, Paper I), a catalogue containing a total of 105 cores in 53 nearby rich superclusters, with redshifts between 0.02 and 0.15. The cores were selected from candidate structures that were initially identified using improved percolation techniques (based on the DBSCAN and FoF algorithms, e.g. Ester et al. Reference Ester, Kriegel, Sander and Xu1996; Berlind et al. Reference Berlind2006) applied to samples of galaxy systems present in the regions of rich superclusters of the Main SuperCluster Catalogue (MSCC, Chow-Martínez et al. Reference Chow-Martínez, Andernach, Caretta and Trejo-Alonso2014) based on Abell/ACO clusters (Abell Reference Abell1958; Abell, Corwin, & Olowin Reference Abell, Corwin and Olowin1989). The selection of cores was based on physically motivated density criteria proposed in the literature (see, for example, Dünner et al. Reference Dünner, Araya, Meza and Reisenegger2006; Chon, Böhringer, & Zaroubi Reference Chon, Böhringer and Zaroubi2015), defining them as structures with a high probability of future collapse and virialisation despite cosmic expansion.

In particular, in Paper I we defined cores as galaxy structures with masses $\mathcal{M}\geq 5\times 10^{14}h_{70}^{-1}\mathcal{M}_\odot$ , $\mathcal{R}\geq 7.86$ , and $\Delta_{\text{cr}}\geq 1.36$ , where

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

is the density ratio between the mean mass density of an overdense region, $\rho_{\text{ov}}$ , and the mean local background density, $\rho_{\text{b}}$ , and

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

is the density contrast, with $\rho_{\text{cr}}=3H^2(z)/8\pi G$ being the critical density of the Universe at redshift z. These parameters are used to assess whether a given structure is likely to remain gravitationally bound and eventually virialise in the future.

The DCC catalogue contains cores of some well-known superclusters of the Local Universe (such as Corona Borealis, Shapley, Ursa Major, Coma-Leo, Hercules, Böotes, among others), which have already been identified and studied in previous works (e.g. Kopylova & Kopylov Reference Kopylova and Kopylov1998; Bardelli et al. Reference Bardelli1994; Einasto et al. Reference Einasto2008). Additionally, new cores within other superclusters were identified, generating a systematically constructed and statistically significant sample for studying these structures.

For each MSCC supercluster in our sample, we selected galaxy samples with spectroscopic redshift from the Sloan Digital Sky Survey (SDSS DR13, Albareti et al. Reference Albareti2017), the 2dF Galaxy Redshift Survey (2dFGRS, Colless et al. Reference Colless2001), or the 6dF Galaxy Survey (6dFGS, Jones et al. Reference Jones2009), depending on the region of the sky where it was located. Galaxy systems were identified from these samples using iterative system identification algorithms based on the works of 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). Our full sample of systems consisted of a total of 3 337 groups and clusters, including about 527 Abell/ACO clusters, as well as many others that match systems from other published catalogues of galaxy systems.

Beyond the homogeneity in the identification and analysis algorithms (which also include adjustments to local densities), we also limited our samples of superclusters to the ones completely inside the area and restricted to the redshift limits of each survey, as captured in Figure 1 of paper I. SDSS-DR13 contains the Sloan Legacy Survey (DR7), which is complete and deep enough for our aims, despite receiving photometric and spectroscopic improvements through the subsequent DRs. SDSS-DR13 and 2dFGRS have similar depth, which guarantee the coverage to $z_{\textrm{lim}} \sim 0.15$ , while 6dFGS is shallower ( $z_{\textrm{lim}} \sim 0.08$ ) and was used only up to this redshift limit.

2.2. Core galaxy sample

The member galaxies of each DCC core were defined as all galaxies up to a distance of 3.5 $R_{\text{vir}}^i$ from the centroid of each i-th member system in the supercluster galaxy sample in 3D rectangular coordinates and corrected for the corresponding Finger-of-God effect (FoG, e.g. Coil Reference Coil2012). Following Santiago-Bautista et al. (Reference Santiago-Bautista, Caretta, Bravo-Alfaro, Pointecouteau and Andernach2020), we refer to this as the ‘supercluster box’. The transformations from equatorial to rectangular coordinates were performed in the form

(3) \begin{align}& X=D_{\text{c}}\cos{\delta}\cos{\alpha},\nonumber\\& Y=D_{\text{c}}\cos{\delta}\sin{\alpha},\nonumber\\& Z=D_{\text{c}}\sin{\delta},\end{align}

where $\alpha=\text{RA}$ , $\delta=\text{Dec}$ and

(4) \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

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

Within a distance of 3.5 $R_{\text{vir}}^i$ from the centre of each member system, we expect to include galaxies up to their turn-around zone (a region that encompasses galaxies on the zero-velocity surface of each system, as well as those that are decelerating while approaching this surface, and those that have begun moving towards collapse towards the system’s centre), along with galaxies in bridges between them and galaxies in the dispersed component of the cores.

3.1. Velocity distribution of galaxies

The line-of-sight velocity distributions of galaxies in the cores were studied from the spectroscopic redshifts $z_i \approx V_{r_i}/c$ of galaxies, where $V_{r_i}$ is the radial velocity of the i-th galaxy and c the speed of light. For each DCC core, the set of redshift values of its $N_g$ member galaxies was examined to explore whether it follows an underlying normal distribution. A preliminary analysis of skewness and kurtosis provided an initial qualitative assessment of the redshift distributions, revealing a variety of patterns: some closely resembling normal distributions, while others exhibited features such as flattened shapes, double peaks, or heavy tails. However, this initial exploration was intended only as a guide to identify possible deviations and to gain some insight into the underlying dynamics of the galaxies within the cores, as the subsequent analysis focused on quantifying these deviations using more robust statistical tests.

To formally assess the extent of these deviations, we applied a suite of statistical tests, including the Anderson–Darling (AD), Jarque–Bera (JB), and Lilliefors (L) tests. These tests evaluate whether the null hypothesis – that a sample of redshift values follows a normal distribution – can be rejected in favor of an alternative hypothesis suggesting significant deviation from normality. Each test returned a binary result, where $h_{\text{test}} = 1$ indicated rejection of the null hypothesis (when p-values $<0.05$ ), and $h_{\text{test}} = 0$ otherwise. The results of these tests, presented in Columns 2 and 3 of Table 1, provide a more rigorous and quantitative evaluation of the observed patterns, allowing us to distinguish between genuine deviations and random fluctuations from normality.

Table 1. Summary of the spatial, velocity, and dynamical properties of DCC cores.

By combining the results from the three statistical tests, we found that about 30% of the studied cores exhibit line-of-sight velocity distributions consistent with an underlying normal distribution within a significance level of 0.05, while the remaining 58% deviate from this behavior. Although the results for another 12% suggest that in some cores the radial velocity distributions are not far from normal, this alone is insufficient to conclude that they are dominantly relaxed structures. While a more advanced evolutionary state compared to their host superclusters is probable, the cores still remain highly substructured. Since the radial velocity distributions of galaxies in cores are mostly non-Gaussian, we infer that these galaxies have not yet reached a MaxwellianFootnote ^a 3D velocity distribution, which indicates that the cores are not dynamically relaxed, as expected. This lack of ‘relaxation’ in the velocity distributions is a key indicator that cores are still in the process of forming and evolving.

3.2. Projected distribution of galaxies

To study the spatial distribution of galaxies, we analyzed their projected positions in the plane of the sky to assess whether they exhibit a statistical tendency to follow the projected King profile:

(6) \begin{equation}\Sigma(r)=\Sigma_0\left[1+\left(\frac{r}{r_c} \right)^2 \right]^{-\gamma},\end{equation}

where the parameters $r_c$ , $\Sigma_0$ , and $\gamma$ are determined by fitting the model (6) to the projected distribution of member galaxies in each core.

The King profile was selected because, despite being a mass density profile, its projected form provides a good fit to the radial distribution of galaxies in regular clusters (e.g. Rood et al. Reference Rood, Page, Kintner and King1972; Adami et al. Reference Adami, Mazure, Katgert and Biviano1998). These clusters typically feature a dense core of galaxies surrounded by a sparse halo, where the galaxy number density decreases with distance. In contrast, other profiles, such as the Einasto and NFW profiles (e.g. Einasto Reference Einasto1965; Navarro, Frenk, & White Reference Navarro, Frenk and White1996), are better suited to describe the distribution of dark matter within clusters.

Since the MMCs represent the most significant gravitational potential wells in their respective host cores, we assume these clusters to be the main physical centres of gravitational attraction within the cores. Although in some cores (particularly those in Gaussian 1) a clearly dominant MMC is not present, we adopt the coordinates of the most massive cluster as a reference ‘centre’Footnote ^b for consistency across the sample. MMCs, even when not overwhelmingly dominant in mass, are generally rich systems located in high-density regions, suggesting a strong gravitational influence and a likely site of future collapse. Moreover, they often coincide with local peaks in the spatial distribution of galaxies.

Thus, taking the sky coordinates of each core’s MMC as its centre, we calculated the projected core-centric distances (in units of $h_{70}^{-1}$ Mpc) of its member galaxies as follows:

(7) \begin{equation}R_{i}\simeq \frac{\pi}{180}\frac{D_{\text{c}}(\bar{z})}{(1+\bar{z})} \left[(\alpha_{\text{MMC}}-\alpha_{i})^2\cos^2{\bar{\delta}}+(\delta_{\text{MMC}}-\delta_{i})^2\right]^{1/2},\end{equation}

where $\alpha_{\text{MMC}}$ and $\delta_{\text{MMC}}$ are the RA and Dec coordinates of the MMC (see columns 6 and 7 of Table 4 in Paper I), $\alpha_i$ and $\delta_i$ are the coordinates of each member galaxy, and $\bar{z}$ and $\bar{\delta}$ represent the mean redshift and declination of the core galaxies, respectively. As before, $D_{\text{c}}(\bar{z})$ is the comoving distance at redshift $\bar{z}$ , corresponding to the mean radial distance of each core.

We used a binning method to analyze the projected density distribution of galaxies as a function of the core-centric distance in the RA-Dec plane. This process involved counting only member galaxies in concentric circular annuli in 2D, centred on the MMC of the given core. Each ring was taken to have an area $A_r = 2\pi R \Delta R$ , where R is the mean radius of the ring and $\Delta R$ is its width. The width $\Delta R$ was kept constant for all radii, and all galaxies with core-centric distances $R_i$ between $R - \Delta R/2$ and $R + \Delta R/2$ were counted within the corresponding ring. The surface density of galaxies at a given radius R was then calculated as $\Sigma(R) = N_r/A_r$ , where $N_r$ is the number of galaxies in that ring.

For all cores, we adopted a fixed ring width of $\Delta R = 0.35$ $h_{70}^{-1}$ Mpc (a value that maximized the goodness-of-fit in most cores), while allowing the radius R to vary without overlap between bins. Using the Nonlinear Least Squares (NLS) method, we fitted the projected King profile (6) to the resulting set of $(R, \Sigma)$ pairs, determining the best-fit parameters $(\Sigma_0, r_c, \gamma)$ . Figure 2 provides an example of this procedure applied to the DCC 099 core in the Hercules Supercluster (MSCC 474). A similar analysis was performed for all other cores.

Figure 2. Top: 2D-density map of the projected distribution of galaxies in the DCC 099 core of the Hercules Supercluster (MSCC 474). Each black dot represents a member galaxy of the core. The space between the red dashed circles give a schematic representation of the rings (bins in 2D) on which galaxies were counted ( $\Delta R=0.35$ $h_{70}^{-1}$ Mpc). The width of the annuli plotted was chosen as 1 $h_{70}^{-1}$ Mpc to avoid saturation of circles in the graph. Bottom: $\Sigma$ vs. R plot for the DCC 099 core. The blue dots represent the surface number density of galaxies – in the ring – at distance R from the centre (the MMC) of the core. The solid red line is the projected King profile fitted to the data using the NLS method. The fit parameters for this case were $\Sigma_{0}=44.68$ gal/ $h_{70}^{-2}$ Mpc $^{2}$ , $r_c=2.56$ $h_{70}^{-1}$ Mpc, and $\gamma=1.78$ , with a goodness of fit $R_{\text{det}}^2=0.91$ .

Columns 6 to 9 of Table 1 present the set of parameters $(\Sigma_0,r_c,\gamma)$ obtained by fitting the King profile to the projected galaxy sample of each DCC core, along with the respective goodness of fit measured through the $\mathcal{R}^2_{\text{det}}$ statistic.Footnote ^c The profile fitting using the NLS method was not successful for approximately 33% of the DCC cores, resulting in unphysical parameters that were excluded from the analysis. The mean and median values of the corresponding distributions of parameters $(\Sigma_0,r_c,\gamma)$ in the sample of DCC cores, excluding outliers, are presented in Table 2. Notably, the characteristic radius $r_c$ of the cores, with a median value of $1.93$ $h_{70}^{-1}$ Mpc, is comparable to the virial radii of their MMCs (in contrast to the median values for galaxy clusters of about 0.25 $h_{70}^{-1}$ Mpc, e.g. Bahcall Reference Bahcall1996; Caretta et al. Reference Caretta2023). This may suggest that, although DCC cores are not yet regular galaxy structures, they have the potential to evolve into ‘core-halo supersystems’, with their central nuclei likely located close to their MMCs. It can be expected that these systems will serve as the formation regions for the future nuclei of marginally virialised (relaxed and regular) cores.

Approximately 55% of the DCC cores could be fitted by the King profile achieving a goodness of fit $\mathcal{R}^2_{\text{det}} > 0.9$ , while only $\sim$ 8% of them were fitted with a goodness of fit $\mathcal{R}^2_{\text{det}} < 0.8$ . Thus, most of the cores in the sample exhibit a projected distribution of galaxies that can be adequately described by a King profile, suggesting an evolutionary tendency towards dynamical states statistically consistent with this density model. However, the observed density distributions still display ‘humps’ as a function of radius, indicating the presence of substructures within the cores. These humps can affect the values of the fit parameters, which, in principle, assuming that the distribution of galaxies follows the distribution of total matter (dark and baryonic) and vice versa in these structures, should be related to the equilibrium state of the cores. For instance, and as anticipated, the $\gamma$ values obtained for the distributions of core galaxies remain far from $\gamma = 1$ , which is the expected terminal value for a projected galaxy distribution in dynamical equilibrium (e.g. Sarazin Reference Sarazin1986; Adami et al. Reference Adami, Mazure, Katgert and Biviano1998).

4.1. Minkowski functionals and shapefinders

A comprehensive morphological study of bodies in $n$ dimensions requires both topological and geometrical descriptors to characterise their connectivity, content, and shape (e.g. Mecke, Buchert, & Wagner Reference Mecke, Buchert and Wagner1994). Minkowski functionals (MFs) constitute a family of $n + 1$ morphological descriptors for extended bodies, grounded in well-established principles of integral geometry (e.g. Wiegand, Buchert, & Ostermann Reference Wiegand, Buchert and Ostermann2014). The morphological properties of an $n$ -dimensional body are determined by the $(n - 1)$ -dimensional hypersurface that encloses it (e.g. Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003). Thus, the morphology of a closed two-dimensional surface embedded in three-dimensional Euclidean space is comprehensively characterized by four MFs (e.g. Einasto et al. Reference Einasto2007a; Bag et al. Reference Bag2019):

1. 1. The volume enclosed by the surface: $V$ ,

2. 2. The surface area: $S$ ,

3. 3. The integrated mean curvature of the surface:

   (8) \begin{equation} C = \frac{1}{2} \oint \left( \frac{1}{R_1} + \frac{1}{R_2} \right) dS, \end{equation}

4. 4. The integrated Gaussian curvature (or Euler characteristic) of the surface:

   (9) \begin{equation} \chi = \frac{1}{2\pi} \oint \frac{1}{R_1 R_2} dS, \end{equation}

Table 2. Mean (with standard deviation) and median values for the $\Sigma_0$ , $r_c$ and $\gamma$ parameters (excluding outliersFootnote ^d ) of the King profile fit to the projected distributions of galaxies in the sample of DCC cores. Median values are given as asymmetric ranges with $\Delta\text{Q}_1=\mathrm{Median}-\text{Q}_1$ and $\Delta\text{Q}_3=\text{Q}_3-\mathrm{Median}$ as lower and upper indices, where $\text{Q}_1$ and $\text{Q}_3$ are the 25th and 75th percentiles, respectively.

where in Equations (8) and (9), ${R_1}$ and ${R_2}$ denote the two local principal radii of curvature at any point on the surface. Furthermore, the Euler characteristic can be expressed in terms of the genus, which quantifies the number of topological handles that the surface exhibits and provides a measure of the connectivity of the structure, distinguishing between isolated underdense regions (voids) and interconnected features (Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003). Essentially, it describes the number of holes in a closed surface or three-dimensional object and can be defined as follows:

(10) \begin{equation}\mathcal{G} = 1 - \frac{\chi}{2}.\end{equation}

Both $\chi $ and ${\mathcal G}$ serve as measures of the surface topology (multiply connected surfaces have ${\mathcal G} \gt 0$ , while those that are simply connected have ${\mathcal G} = 0$ , e.g. Sahni, Sathyaprakash, & Shandarin Reference Sahni, Sathyaprakash and Shandarin1998). Thus, while the genus provides insight into the connectivity of a surface, the other three MFs are sensitive to local surface deformations, effectively characterizing the geometry and shape of the bodies (Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003).

To characterise the shape and determine the characteristic dimensions of a structure, we employ ‘shapefinders’ (Sahni et al. Reference Sahni, Sathyaprakash and Shandarin1998), which are defined from the MFs as follows (e.g. Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003; Bag et al. Reference Bag2019):

1. 1. Thickness:

   (11) \begin{equation}\mathcal{T}=3V/S,\end{equation}

2. 2. Breadth:

   (12) \begin{equation}\mathcal{B}=S/C,\end{equation}

3. 3. Length:

   (13) \begin{equation}\mathcal{L}=\frac{C}{4\pi(1+|\mathcal{G}|)},\end{equation}

4. 4. Planarity:

   (14) \begin{equation}\mathcal{P}=\frac{\mathcal{B}-\mathcal{T}}{\mathcal{B}+\mathcal{T}},\end{equation}

5. 5. Filamentarity:

   (15) \begin{equation}\mathcal{F}=\frac{\mathcal{L}-\mathcal{B}}{\mathcal{L}+\mathcal{B}},\end{equation}

This set of five shapefinders includes three quantities with dimensions of length ( ${\mathcal T}$ , ${\mathcal B}$ , and ${\mathcal L}$ ) and two dimensionless ratios ( ${\mathcal P}$ and ${\mathcal F}$ ), providing a robust framework for quantifying the morphology of structures.

The shapefinders ${\mathcal T}$ , ${\mathcal B}$ , and ${\mathcal L}$ estimate the three principal physical extensions of a three-dimensional structure. These shapefinders are spherically normalized, ensuring that $V = (4\pi /3){\mathcal T}{\mathcal B}{\mathcal L}$ , where $V$ is the volume enclosed by the structure. Generally, for a convex surface, the relation ${\mathcal T} \le {\mathcal B} \le {\mathcal L}$ holds; if not, the smallest dimension is designated as ${\mathcal T}$ and the largest as ${\mathcal L}$ to maintain order (Bag et al. Reference Bag2019). In cases where $C \lt 0$ , it is possible to redefine $C \to |C|$ to ensure that both ${\mathcal B}$ and ${\mathcal L}$ remain positive. Oblate ellipsoids (pancake-like shapes) are characterized by ${\mathcal T} \lt {\mathcal B} \approx {\mathcal L}$ , while prolate ellipsoids (filamentary structures) are described by ${\mathcal T} \approx {\mathcal B} \lt {\mathcal L}$ (Einasto et al. Reference Einasto2007a).

On the other hand, the shapefinders $\mathcal{P}$ and $\mathcal{F}$ are dimensionless quantities that allow us to determine the shape of an object. For example, in some works (e.g. Sahni et al. Reference Sahni, Sathyaprakash and Shandarin1998; Einasto et al. Reference Einasto2007a; Bag et al. Reference Bag2019) these shapefinders have been characterized so that:

• • For spheres: $\mathcal{T}=\mathcal{B}=\mathcal{L}$ , that is, $\mathcal{P}=\mathcal{F}=0$ ,

• • For ideal filaments $\mathcal{P}\approx 0$ , $\mathcal{F}\approx 1$ ,

• • For real filaments: $\mathcal{F}\gg\mathcal{P}$ ,

• • For ideal pancakes: $\mathcal{P}\approx 1$ , $\mathcal{F}\approx 0$ ,

• • For planar objects (sheets or pancakes): $\mathcal{P}\gg\mathcal{F}$ ,

• • For ideal ribbons: $\mathcal{P}\approx\mathcal{F}\approx 1$ ,

• • For ribbon-like objects: $\mathcal{P}/\mathcal{F}\approx 1$ .

Note that in this context, the word ‘ideal’ refers to a theoretical or simplified representation of the structure, as commonly used in the literature.

4.2. Morphometry of cores

The MFs are typically defined for extended bodies with well-established (smooth and differentiable) boundary surfaces. However, they can also be applied to galaxy distributions by constructing an extended object from the point set of galaxy coordinates. This is achieved by defining a limiting surface that encloses the member galaxies of a structure, thus allowing its morphology to be characterized by the four MFs (V, S, C, $\chi$ ) that describe the enclosing surface (e.g. Einasto et al. Reference Einasto2007a, Reference Einasto2008). Given the global and additive nature of MFs, their applicability can be extended to surfaces with singular edges and corners (Mecke et al. Reference Mecke, Buchert and Wagner1994). This versatility makes them suitable for analyzing irregular or discontinuous structures, providing valuable insights into the geometry and topology of cores. In this work, we employ two methods to generate enclosing surfaces around the core member galaxies, adjusting the definition of the MFs (and shapefinders) in each case to match the specific characteristics of the generated surfaces.

4.2.1. Main method: Polyhedral surfaces

Polyhedral surfaces that envelop the DCC cores were constructed by triangulating boundary points from the three-dimensional distribution of their member galaxies. The triangulation was performed using the 3D alpha-shape algorithm (e.g. Edelsbrunner & Mücke Reference Edelsbrunner and Mücke1994), implemented through the MATLAB boundary function. This function identifies and returns the set of boundary points of a distribution for a given compactness level, modulated by the shrink factor $s_{\text{f}}$ , and subsequently allows the generation of a polyhedral envelope, ranging from the convex hull, for $s_{\text{f}}=0$ , to a tightly fitted compact boundary, for $s_{\text{f}}=1$ (see MATLAB 2023). The left panel of Figure 3 illustrates an example of the polyhedral surface fitted to member galaxies in the main core (DCC 099) of the Hercules Supercluster (MSCC 474).

Figure 3. Left: Polyhedral fit of the optimal surface (with shrink factor, $s_{\text{f}}=1$ ) enclosing the distribution of sampled member galaxies in DCC 099, the main core (A2147-A2151-A2152A-A2153A) of the Hercules Supercluster (MSCC 474). Right: The alternative ellipsoidal fit for the same galaxy distribution. The lighter-colored regions on the ellipsoid correspond to automatically generated cut planes in the upper XY and right YZ sections, enhancing the 3D visualisation of the surface. The polyhedral and ellipsoidal surfaces shown are best-fit models of the main overdense body of thecore, not a strict envelope of all its member galaxies. Due to the complex and unrelaxed nature of the structures, some galaxies, particularly those in the more diffuse or peripheral regions, may fall outside these fitted surfaces.

The obtained triangulated polyhedral surface defines a single well-defined boundary around the member galaxies. Morphological properties of these surfaces, including their geometric and topological characteristics, are determined by the MFs adapted for triangulated surfaces (e.g. Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003; Bag et al. Reference Bag2019):

• • The total surface area is computed as

  (16) \begin{equation} S = \sum_{k=1}^{N_T} S_k, \end{equation}
  where $S_k$ is the area of the k-th triangle, and $N_T$ is the total number of triangles forming the surface.

• • The volume enclosed by the surface is the sum of contributions from $N_T$ tetrahedra:

  (17) \begin{equation} V = \sum_{k=1}^{N_T} V_k, \quad V_k = \frac{1}{3} S_k (\boldsymbol{n}_k \cdot \boldsymbol{P}_k), \end{equation}
  where $V_k$ is the volume of the k-th tetrahedron with base $S_k$ in the k-th triangle, with normal vector $\boldsymbol{n}_k$ and centroid vector position $\boldsymbol{P}_k$ , and apex in an arbitrary origin. Note that the sums of S and V are maximum for $s_{\text{f}}=0$ and minimum for $s_{\text{f}}=1$ .

• • The integrated mean curvature is calculated as

  (18) \begin{equation} C = \frac{1}{2} \sum_{j,k} l_{jk} \phi_{jk} \epsilon, \end{equation}
  where $l_{jk}$ is the length of the common edge between adjacent triangles j and k, $\phi_{jk}$ is the angle between their normals, and $\epsilon = 1$ for convex and $\epsilon = -1$ for concave edges (Sheth et al. Reference Sheth, Sahni, Shandarin and Sathyaprakash2003).

• • The Euler Characteristic and Genus are defined as

  (19) \begin{equation} \chi = N_T - N_E + N_V, \quad \mathcal{G} = 1 - \chi / 2, \end{equation}
  where $N_E$ and $N_V$ are the number of edges and vertices, respectively.

This approach ensures a detailed and versatile morphological analysis of cores, enabling the characterisation of both their geometry and topology.

4.2.2. Alternative method: Ellipsoidal fitting

To validate the polyhedral surface method described above, continuous ellipsoidal surfaces, whose MFs can be calculated analytically, were fitted to the same galaxy samples for each core. The algorithm proposed by Petrov (Reference Petrov2015) was used to estimate the semi-major axes (a, b, c) of the ellipsoid that best represents the 3D galaxy distribution in rectangular coordinates. The fitting process was performed iteratively by extracting boundary points using the boundary function for different values of the shrink factor $(0\leq s_{\text{f}}\leq 1)$ . For each $s_{\text{f}}$ , a new set of boundary points was obtained and used to fit an ellipsoidal surface. The optimal ellipsoid was then selected based on the maximum goodness of fit, measured by the Chi-square statistic.

Table 3. Minkowski functional and shapefinders for DCC cores (ellipsoidal fits). In this fit all cores have $\chi=2$ , i.e., $\mathcal{G}=0$ .

The right panel of Figure 3 illustrates an example of the best ellipsoidal surface fit for the DCC 099 core. The parameters of the best fits for each core are presented in columns 2–5 of Table 3. The parametric equation for an ellipsoid with semi-major axes a, b, and c is given by:

(20) \begin{equation}\boldsymbol{r}(\theta, \phi) = a (\! \sin{\theta}\cos{\phi})\boldsymbol{i} + b (\! \sin{\theta}\sin{\phi})\boldsymbol{j} + c (\! \cos{\theta})\boldsymbol{k},\end{equation}

where $0 \leq \phi \leq 2\pi$ and $0 \leq \theta \leq \pi$ .

The four MFs for an ellipsoidal body are expressed as follows (e.g. Lipschutz Reference Lipschutz1969; Sahni et al. Reference Sahni, Sathyaprakash and Shandarin1998):

(21) \begin{equation}V = \frac{4}{3}\pi abc,\end{equation}

(22) \begin{equation}S = \oint \sqrt{EG - F^2} \, d\theta \, d\phi,\end{equation}

(23) \begin{equation}C = \frac{1}{2} \oint \left[\frac{EN + GL - 2FM}{EG - F^2}\right] dS,\end{equation}

(24) \begin{equation}\chi = \frac{1}{2\pi} \oint \left[\frac{LN - M^2}{EG - F^2}\right] dS,\end{equation}

where:

\begin{align*} E &= \boldsymbol{r}_\theta \cdot \boldsymbol{r}_\theta, & F &= \boldsymbol{r}_\theta \cdot \boldsymbol{r}_\phi, & G &= \boldsymbol{r}_\phi \cdot \boldsymbol{r}_\phi, \\ L &= \boldsymbol{r}_{\theta\theta} \cdot \boldsymbol{n}, & M &= \boldsymbol{r}_{\theta\phi} \cdot \boldsymbol{n}, & N &= \boldsymbol{r}_{\phi\phi} \cdot \boldsymbol{n},\end{align*}

$\boldsymbol{n}$ is the unit normal vector to the surface at any point, defined as $\boldsymbol{n} = \boldsymbol{r}_\theta \times \boldsymbol{r}_\phi / |\boldsymbol{r}_\theta \times \boldsymbol{r}_\phi|$ . The differential area is $dS = \sqrt{EG - F^2} \, d\theta \, d\phi$ , and $\boldsymbol{r}_\theta = \partial \boldsymbol{r}/\partial \theta$ , $\boldsymbol{r}_\phi = \partial \boldsymbol{r}/\partial \phi$ , $\boldsymbol{r}_{\theta\theta} = \partial^2 \boldsymbol{r}/\partial \theta^2$ , $\boldsymbol{r}_{\phi\phi} = \partial^2 \boldsymbol{r}/\partial \phi^2$ , $\boldsymbol{r}_{\theta\phi} = \partial^2 \boldsymbol{r}/\partial \theta \partial \phi$ are the first- and second-order partial derivatives of $\boldsymbol{r}$ with respect to $\theta$ and $\phi$ .

This approach provides an analytical framework for validating the polyhedral surfaces, enabling a comparative morphological analysis of the cores.

Table 4. Minkowski functional and shapefinders for DCC cores (polyhedral fits).

4.2.3. Dimensions and global topology of cores

Once the MFs have been determined using both the polyhedral and ellipsoidal surface methods, the shapefinders can be directly computed using the definitions in Equations (11)–(15). The resulting values of V, S, C, $\chi$ , $\mathcal{G}$ , and the shapefinders $\mathcal{T}$ , $\mathcal{B}$ , $\mathcal{L}$ , $\mathcal{P}$ , and $\mathcal{F}$ for each DCC core, based on polyhedral (with $s_{\text{f}}=1$ ) and ellipsoidal surface fits, are summarized in Tables 3 and 4, respectively.

The shapefinders $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ , which have units of length, are particularly useful for estimating the dimensions of the cores. The smallest shapefinder, $\mathcal{T}$ , represents the thickness of the cores; the intermediate shapefinder, $\mathcal{B}$ , is analogous to the breadth (width); and the largest shapefinder, $\mathcal{L}$ , characterizes the length. It should be noted that $\mathcal{L}$ is not the actual physical length of the structure, but rather a morphological measure related to the integrated curvature of its surface (Einasto et al. Reference Einasto2007a). Thus, $\mathcal{L}$ can become significantly large for irregularly shaped or curved surfaces, without necessarily corresponding directly to a linear extension, as in elongated structures. The mean and median values of these dimensions are presented in Table 5. As shown in the table, the extension in any dimension of the DCC cores (as measured by $\mathcal{T}$ , $\mathcal{B}$ , and  $\mathcal{L}$ ) does not exceed $10 \, h_{70}^{-1}\,\mathrm{Mpc}$ for either surface method. These results are consistent with those of Einasto et al. (Reference Einasto2007a), who reported similar scales for the densest central regions of superclusters constrained to high-mass fraction areas, which coincide with their cores.

Although the polyhedral and ellipsoidal surface fits share the same boundary-point selection process, the mean (and median) values of $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ obtained with these two methods differ significantly. These differences stem from the intrinsic nature of each approach. The polyhedral fit with $s_{\textrm{f}}=1$ yields a more compact representation of the surface enclosing the member galaxies of the cores, capturing (via triangulation) finer substructures and morphological details that are typical of dynamically evolving regions. Consequently, this compactness produces smaller values of $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ , since, by construction, surface area and volume are minimal at $s_{\textrm{f}}=1$ (and increase toward $s_{\textrm{f}}=0$ ). In contrast, the ellipsoidal fit enforces a smoother, global shape that encloses most of the core member galaxies within an ellipsoidal boundary, which systematically returns larger values for the shapefinders.

This dual strategy (compact polyhedral fits for structural detail and ellipsoidal fits for global extent) provides a robust morphological characterisation of the cores: the results of the polyhedral fit can be interpreted as a lower bound on the dimensions of these structures, while those of the ellipsoidal fits serve as an upper bound. The ellipsoidal method is included for comparison because it is a conservative technique widely used in the literature. Furthermore, exploratory tests (not included in the manuscript for brevity) employing intermediate shrink factors (e.g. $0 \le s_{\textrm{f}} < 1$ ) in polyhedral fits confirm that the qualitative classification, in particular the dominance of filamentary morphologies, remains consistent and does not critically depend on the specific choice of $s_{\textrm{f}}$ .

Figure 4 shows scatterplots of the three characteristic dimensions ( $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ from Table 4) of the DCC cores versus their extensive mass ( $\mathcal{M}_{\text{ext}}^c$ , defined as the sum of the virial masses of the member galaxy systems of a core, and presented in Table 5 of Paper I). Statistical correlations were evaluated using a Pearson test with a 95% confidence level. The correlation coefficients for $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ with mass were $0.70$ , $0.64$ , and $0.40$ , respectively. These correlations are statistically significant with a significance level of $\alpha_s=0.05$ . These results suggest that the dimensions of the cores are positively correlated with their mass: more massive structures tend to exhibit greater thickness, breadth, and length. This finding is consistent with previous studies by Shandarin et al. (Reference Shandarin, Sheth and Sahni2004), which reported similar trends in superclusters.

Figure 4. Length, breadth, and thickness ( $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ shapefinders) versus extensive mass $\mathcal{M}_{\text{ext}}^c$ for DCC cores. The three solid blue lines correspond to the best linear fit in each case. The slope m and vertical intercept b of each fit are displayed in the upper-left inset of the corresponding panel. The units of each axis must be understood in terms of $h_{70}^{-1}$ .

On the other hand, a standard interpretation of the genus in the cosmology literature defines it as (e.g. Shandarin et al. Reference Shandarin, Sheth and Sahni2004):

(25) \begin{equation} \mathcal{G} = (\text{number of holes}) - (\text{number of isolated regions}) + 1,\end{equation}

where ‘holes’ are considered complex mathematical objects that, in three-dimensional structures, typically manifest as tunnels crossing the structure from one side to the other, as in a toroidal shape. The term ‘isolated regions’ refers to the number of disconnected parts of the fitted surface that define the structure’s boundaries. Thus, Equation (26) can be rewritten as:

(26) \begin{equation} \mathcal{G} = N_{\text{tunn}} - N_{\text{is,surf}} + 1,\end{equation}

where $N_{\text{tunn}}$ and $N_{\text{is,surf}}$ represent the number of tunnels and isolated surfaces, respectively (e.g. Bag et al. Reference Bag2019).

Table 5. Mean (with standard deviation) and median values for the $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ shapefinders estimated from polyhedral and ellipsoidal surface fits to the DCC cores. Median values are given as asymmetric ranges with $\Delta\text{Q}_1=\mathrm{Median}-\text{Q}_1$ and $\Delta\text{Q}_3=\text{Q}_3-\mathrm{Median}$ as lower and upper indices, where $\text{Q}_1$ and $\text{Q}_3$ are the 25th and 75th percentiles, respectively.

Visual inspections using basic rotation in 3D and rendering tools available in standard scientific software (e.g. MATLAB and Python libraries) revealed that the DCC cores exhibit significant substructure patterns, frequently containing tunnels and isolated systems. The top panel of Figure 5 shows the distribution of $\mathcal{G}$ values obtained from the fitted polyhedral surfaces of the DCC cores, as presented in Table 4. The wide range of $\mathcal{G}$ values (from $-1$ to 8) reflects the diversity of complex three-dimensional shapes exhibited by these structures. Interpreting the genus as a structural feature of such surfaces is not unique and becomes non-trivial when the regions have highly intricate geometries (e.g. Shandarin et al. Reference Shandarin, Sheth and Sahni2004).

Figure 5. Top: Distribution of (genus) $\mathcal{G}$ values for DCC cores. Bottom: Distribution of $\mathcal{G}$ values versus the extensive mass $\mathcal{M}_{\text{ext}}^c$ of DCC cores.

For this analysis, the genus obtained from ellipsoidal surface fits was not considered, since ellipsoids are topologically homeomorphic to spheres, which have genus $\mathcal{G}=0$ . As a result, a meaningful topological study based on this type of surface fit is not feasible.

Various non-integer values of $\mathcal{G}$ were obtained because the genus was computed using the formula $\mathcal{G}=1-\chi/2$ , where $\chi$ is the Euler characteristic. Since $\chi$ can take odd values depending on the discretisation and resolution of the surface representation, $\mathcal{G}$ is not always an integer. However, genus is fundamentally a topological invariant that describes the number of handles or holes in a surface, and it is conventionally defined for integer values. Non-integer values are difficult to interpret in this context, as they do not correspond to well-defined topological classes. Therefore, we restrict our analysis to integer values as defined in Equation (26). Under this criterion, about 10% of the DCC cores are topologically isomorphic to a sphere (i.e. $\mathcal{G}=0$ ), $\sim$ 18% to a toroid (i.e. $\mathcal{G}=1$ ), and $\sim$ 10% to a pretzel (i.e. $\mathcal{G}=2$ ). Higher genus values indicate increasingly complex topologies. Structures with negative genus (about 11% of the cores) can be interpreted as coarser configurations with multiple isolated member systems.

Additionally, the bottom panel of Figure 5 shows the distribution of all genus values (both integer and non-integer) versus the extensive mass of cores, though no significant statistical correlation can be inferred from this relationship. This suggests that the topological complexity of cores is not directly governed by their mass, possibly due to the influence of other structural and dynamical factors.

4.2.4. The spectrum of shapes of the cores

In addition to MFs and shapefinders, Tables 3 and 4 present the morphological parameter $\mathcal{P}/\mathcal{F}$ , which enables the statistical analysis of the shapes of the cores. The shapefinder formalism allows for classifying cores into two basic morphological types:

1. 1. Pancakes (oblate structures), with $\mathcal{P}/\mathcal{F}>1$ , and

2. 2. Filaments (prolate structures), with $0\leq\mathcal{P}/\mathcal{F}\leq1$ .

Here, ribbon-like objects were excluded, as defining the range $\mathcal{P}/\mathcal{F}\approx 1$ for this morphology is somewhat arbitrary (e.g. Costa-Duarte et al. Reference Costa-Duarte, Sodré and Durret2011). The distribution of the morphological parameter $\mathcal{P}/\mathcal{F}$ , referred to as the shape spectrum, for the DCC cores is shown in Figure 6. Outliers (15 data points from polyhedral fits and 9 from ellipsoidal fits) were excluded from the analysis. The polyhedral surface fits reveal that about 75% of the cores can be classified as filaments, while the remaining $\sim$ 25% can be identified as pancakes. The ellipsoidal fits corroborate these findings, indicating that approximately 62% of the cores have filamentary morphologies, while $\sim$ 38% exhibit pancake-like shapes. The shape spectrum obtained from both methods confirms a statistical tendency of DCC cores towards filamentary morphologies.

Figure 6. The ‘shape spectrum’ of DCC cores. Distribution of the shape statistic $\mathcal{P}/\mathcal{F}$ in the sample of cores excluding outliers and using two surface fit (polyhedral and ellipsoidal) methods. The structures are classified as filaments if $0\leq \mathcal{P}/\mathcal{F}\leq 1$ or as pancakes if $\mathcal{P}/\mathcal{F}>1$ .

Despite the differences in the values of the characteristic dimensions (see Table 5) obtained from polyhedral and ellipsoidal surface fitting, the analyses of planarity and filamentarity remain consistent for both methods. Since these quantities are defined as combined ratios of $\mathcal{T}$ , $\mathcal{B}$ , and $\mathcal{L}$ (see Equations 14 and 15), their stability suggests that both fitting approaches preserve the overall structural characteristics of the cores (see Section 4.2.4). This reinforces the robustness of our morphological classification, regardless of the specific method used to estimate absolute dimensions. Therefore, while the polyhedral fit provides a more precise morphological description, the ellipsoidal fit remains a useful comparative reference within the framework of large-scale structure studies.

Since filaments are the most common features in rich superclusters (e.g. Einasto et al. Reference Einasto2007a; Cautun et al. Reference Cautun, van deWeygaert, Jones and Frenk2014), it is reasonable to expect that cores, being internal substructures of superclusters, tend towards these morphologies. This is consistent with the hierarchical growth these structures experience, as cores are primarily ‘nourished’ through the filaments to which they are connected. Filaments act as massive channels for transporting matter in the Universe (e.g. Libeskind et al. Reference Libeskind2018, and references therein), primarily along directions of maximum anisotropic gravitational attraction (e.g. Zeldovich Reference Zeldovich1970), in this case towards cores, which are the densest regions within superclusters.

On the other hand, pancake-like cores are less common in rich superclusters but represent another class of structures that can evolve into compact, virialised objects. Pancake cores are more prevalent in lumpy, less filamentary superclusters and are often isolated or disconnected from other parts of their host superclusters.

The predominantly anisotropic morphologies (flattened and elongated) of cores are indicative of their current quasi-linear dynamical stage, as predicted by the Zeldovich model (e.g. Araya-Melo et al. Reference Araya-Melo, Reisenegger, Meza, van deWeygaert, Dünner and Quintana2009). As cores evolve, their morphologies change, leading to various intermediate shapes over time. According to the Zeldovich formalism, which defines a morphological sequence linked to stages of an anisotropic gravitational collapse (see, e.g. Cautun et al. Reference Cautun, van deWeygaert, Jones and Frenk2014), cores that are mostly filaments are the largest bound structures closest to becoming virialised objects, as they only need to collapse along a single axis (the longest of the three). In contrast, pancake-shaped cores still need to collapse along two axes before reaching virialisation. However, since they are already bound structures with significant overdensities, their collapse and future virialisation are inevitable.

The evolution of cores rarely occurs in isolation within the LSS. As previously noted, most cores are dense filaments or located at filament intersections within superclusters, constantly accreting matter from their surroundings. The boxplot in Figure 7 shows that filamentary cores ( $0\leq\mathcal{P}/\mathcal{F}\leq 1$ ) are relatively more massive than pancakes ( $\mathcal{P}/\mathcal{F}>1$ ). The mass distribution of filaments extends towards higher values, while pancake masses are distributed within a narrower, lower-mass range. Pancake-like cores are often surrounded by less dense regions, as evidenced by density contrast analyses and connectivity studies between cores and their host superclusters. As core richness increases with mass, these results align with Costa-Duarte et al. (Reference Costa-Duarte, Sodré and Durret2011), who found that filaments in superclusters tend to be richer than pancakes on average.

Figure 7. Boxplots of extensive mass $\mathcal{M}_{\text{ext}}^c$ for the two morphological classifications of DCC cores, filaments ( $0\leq \mathcal{P}/\mathcal{F}\leq 1$ ) and pancakes ( $\mathcal{P}/\mathcal{F}>1$ ), based on the polyhedral (poly-fit) and ellipsoidal (ell-fit) fits surfaces. Filaments have a relatively greater mass than pancakes.

Finally, we estimate the fraction of cores with approximately spherical shapes (i.e. $\mathcal{P} \approx \mathcal{F} \approx 0$ ). Allowing a tolerance of up to 0.05 in both $\mathcal{P}$ (planarity) and $\mathcal{F}$ (filamentarity), only $\sim$ 8% of the DCC cores are approximately spherical based on ellipsoidal fits. None of these cores fall within this range when using polyhedral surface fits. Decreasing the tolerance further reduces the fraction of cores with comparable axes ( $a\approx b\approx c$ ). The virtual absence of spherical objects underscores the relative youth of the cores, which may reflect the fact that the primordial density field did not contain spherical overdense regions (e.g. Bardeen et al. Reference Bardeen, Bond, Kaiser and Szalay1986; Kravstov & Borgani Reference Kravstov and Borgani2012), and that the early stages of contraction and gravitational collapse occur in strongly flattened and elongated geometries (e.g. Zeldovich Reference Zeldovich1970; Araya-Melo et al. Reference Araya-Melo, Reisenegger, Meza, van deWeygaert, Dünner and Quintana2009; Cautun et al. Reference Cautun, van deWeygaert, Jones and Frenk2014).

5.1. The $H_Z$ -entropy estimator

To quantitatively characterise the evolutionary state of galaxy systems within an entropy-increasing framework, a continuous entropy estimator has been introduced (see Zúñiga et al. Reference Zúñiga, Caretta and Andernach2024b), defined as:

(27) \begin{equation}H_Z\equiv\ln{\left( \frac{\mathcal{M}_{\text{vir}}}{\frac{4}{3}\pi R_{\text{vir}}^3} \frac{1}{\rho_0} \right)} +\frac{\beta}{2}\ln{\left( \frac{\beta}{2} \frac{\sigma_v^2}{\sigma_{v_0}^2} \right)},\end{equation}

where $\mathcal{M}_{\text{vir}}$ , $R_{\text{vir}}$ , and $\sigma_v$ denote the virial mass, virial radius, and line-of-sight galaxy velocity dispersion of a system, respectively. The parameter $\beta$ accounts for the velocity anisotropy of galaxies ( $\beta=3$ for Maxwellian distributions, e.g. Tully Reference Tully2015). The constants $\rho_0=(10^{14}\mathcal{M}_\odot/\text{Mpc}^3)h_{70}^2$ and $\sigma_{v_0}= 1 \,\ \text{km s}^{-1}$ are fiducial reference values for density and velocity dispersion used to render the arguments of the logarithms dimensionless. The $H_Z$ estimator is a physically motivated, dimensionless quantity that traces the net change in specific entropy ( $\Delta s$ ) experienced by a system from its formation to the present epoch. Since it is defined in terms of (optical) observational parameters, it can be readily computed once a sufficient number of member galaxies are identified.

The $H_Z$ estimator was first tested on a sample of 70 well-sampled galaxy clusters in the Local Universe (the Top70 cluster sample,Footnote ^e Caretta et al. Reference Caretta2023). The test results suggest a strong correlation between $H_Z$ and the dynamical state of clusters (Zúñiga et al. Reference Zúñiga, Caretta and Andernach2024b). Specifically, $H_Z$ correlates with the level of gravitational assembly, yielding lower values for dynamically younger, substructured clusters and higher values for more relaxed, evolved systems. Furthermore, $H_Z$ exhibits a robust correlation with other continuous dynamical indicators derived from both optical and X-ray observations.

5.2. Entropy of cores

The use of the $H_Z$ estimator can be extended to the study of structures larger than clusters due to self-similarity. Gravity-dominated systems are expected to follow a comparable evolutionary path on a global scale (e.g. simulation results indicate that supercluster-scale structures tend to evolve into very rich cluster-like systems, Araya-Melo et al. Reference Araya-Melo, Reisenegger, Meza, van deWeygaert, Dünner and Quintana2009). Based on its successful application to characterise the dynamical state of galaxy clusters (Zúñiga et al. Reference Zúñiga, Caretta and Andernach2024b), we propose here to extend the use of the $H_Z$ estimator to larger structures, such as superclusters and their cores. To this end, we generalise the $H_Z$ estimator as follows:

(28) \begin{equation}H_Z=\ln{\left( \frac{\bar{\rho}}{\rho_0} \right)} + \frac{\beta}{2}\ln{\left( \frac{\beta}{2} \frac{\sigma_v^2}{\sigma_{v_0}^2} \right)},\end{equation}

where $\bar{\rho}$ and $\sigma_v$ represent the average mass density (e.g. $\bar{\rho}=\mathcal{M}_{\text{ext}}/V$ ) and the line-of-sight galaxy velocity dispersion of the structure under study, respectively. This generalisation avoids using virial parameters when estimating entropy for non-relaxed systems.

Using the average densities and galaxy velocity dispersions estimated for the cores (see Table 5 of Paper I and Table 1) and superclusters (see Table 3 of Paper I and Table 6), we computed the $H_Z$ -entropy for each of these structures. The $H_Z$ values obtained for the complete MSCC-supercluster sample are listed in column 6 of Table 6, while the $H_Z$ values for DCC cores are shown in column 12 of Table 1.

Table 6. General properties of the MSCC superclusters in the sample.

To compare evolutionary states of galaxy systems and structures at different scales, we took the $H_Z$ -entropy values for the Top70 cluster sample from Table 2 of Zúñiga et al. (Reference Zúñiga, Caretta and Andernach2024b). These clusters, among the best-sampled galaxy systems in the nearby Universe, belong to MSCC superclusters and cover a broad range from poor to rich systems (with ICM temperatures from 1 to 12 keV).

The top panel of Figure 8 displays boxplots of $H_Z$ -entropy distributions for clusters, cores, and superclusters. It is evident that clusters exhibit higher entropies (median value of $15.4$ ), consistent with systems near virial equilibrium. In contrast, superclusters show the lowest entropies (median value of $11.1$ ), indicating their less advanced evolutionary state among the studied galaxy structures. Meanwhile, cores present intermediate entropy values (median value of $12.9$ ), supporting the hypothesis that these regions are dynamically more evolved than their rich host superclusters.

Figure 8. Distributions of $H_Z$ -entropy (top panel) and probability ${\mathcal{P}}_{\mathrm{relax}}$ (bottom panel) values for clusters (the Top70 sample), cores (from DCC), and superclusters (from MSCC).

5.3. Relaxation probability of cores

To complement the entropy-based approach, we evaluate the relaxation probability parameter, $\mathcal{P}_{\text{relax}}$ , as defined in Zúñiga et al. (Reference Zúñiga, Caretta and Andernach2024b). This parameter provides an independent method for assessing the dynamical state of galaxy structures based on the distribution of their member galaxies in phase space. virialised cores are expected to exhibit spatial and velocity distributions similar to those of relaxed galaxy clusters. These distributions are typically modeled using profiles derived from equilibrium assumptions, such as the King profile for radial distributions, uniform azimuthal distributions, and Gaussian velocity distributions (e.g. Saslaw & Hamilton Reference Saslaw and Hamilton1984; Sarazin Reference Sarazin1986; Adami et al. Reference Adami, Mazure, Katgert and Biviano1998; Sampaio & Ribeiro Reference Sampaio and Ribeiro2014).

The $\mathcal{P}_{\text{relax}}$ parameter is determined by comparing the observed probability density functions (PDFs) of galaxies within a core to equilibrium models. This approach mirrors that used in Zúñiga et al. (Reference Zúñiga, Caretta and Andernach2024b) for galaxy clusters. Briefly, the raw observational coordinates of member galaxies, namely the triples $(\text{RA}, \text{Dec}, z)$ of right ascension, declination, and redshift, are distributed within a solid angle that can be approximated by a cylinder with a circular base in the plane of the sky and depth along the line of sight. Within this cylinder, each galaxy’s position can be expressed as $(r, \theta, z)$ , where r represents its projected distance from the core centroid, $\theta$ its azimuthal angle relative to the local north direction in the projected sky distribution, and z its redshift, serving as a proxy for radial velocity. Assuming statistical independence,Footnote ^f the galaxy distribution in each core can be described by an empirical joint PDF, $\bar{f}_{r\theta z} = \bar{f}_r(r)\bar{f}_\theta(\theta)\bar{f}_z(z)$ . Here, $\bar{f}_r(r)$ , $\bar{f}_\theta(\theta)$ , and $\bar{f}_z(z)$ represent the observed PDFs for the radial-r, azimuthal- $\theta$ , and redshift-z variables, respectively.

To estimate these observed PDFs, we apply a smoothing technique using a kernel density estimator over normalized galaxy counts within bins of width $\Delta r = 0.35\, h_{70}^{-1}$ Mpc, $\Delta \theta = 12^\circ$ , and $c\Delta z = 250$ km s $^{-1}$ for the r, $\theta$ , and z variables, respectively. These bin widths were chosen from a range of test values to optimise the fit of the distribution functions during histogram smoothing. A standard Gaussian smoothing kernel is applied using the same bin widths over the intervals $[0, R_h]$ , $[0, 360^\circ]$ , and $[z_{\mathrm{min}}, z_{\mathrm{max}}]$ for $\bar{f}_r$ , $\bar{f}_\theta$ , and $\bar{f}_z$ , respectively. Here, $z_{\mathrm{min}}$ and $z_{\mathrm{max}}$ are the minimum and maximum redshifts of the galaxies in the structure, and

(29) \begin{equation}R_h\equiv \frac{2N(N-1)}{\sum_{i\neq j}R_{ij}^{-1}},\end{equation}

represents the harmonic radius of the structure calculated from the projected distances $R_{ij}$ (in Mpc) between its N sampled galaxies (see column 10 of Table 1). Since $\bar{f}_r$ , $\bar{f}_\theta$ , and $\bar{f}_z$ depend on the dynamical state of the structure, they describe the current distributions of its member galaxies.

To define the equivalent relaxed PDFs for each variable in each core (or supercluster), we adopt a simple reference equilibrium model. virialised cores are assumed to exhibit spherical galaxy distributions with a homogeneous core-halo configuration and isotropic velocities, lacking net angular momentum. For simplicity, we use the King-type radial, continuous uniform, and normal distributions ${f}_r^\mathrm{eq}(r)$ , ${f}_\theta^\mathrm{eq}(\theta)$ , and ${f}_z^\mathrm{eq}(z)$ from Zúñiga et al. (Reference Zúñiga, Caretta and Andernach2024b) as equilibrium models for the r, $\theta$ , and z variables, respectively. The similarity between observed and relaxed PDFs is quantified using the Hellinger distance ( $0 \leq H(\bar{f}_k, f_k^\mathrm{eq}) \leq 1$ ; Hellinger Reference Hellinger1909), where lower Hellinger distances indicate distributions closer to equilibrium, corresponding to higher relaxation probabilities since $\mathcal{P}_{\mathrm{relax}} \equiv 1 - H$ .

The $\mathcal{P}_{\mathrm{relax}}$ values computed for DCC cores are shown in column 13 of Table 1. Additionally, the bottom panel of Figure 8 illustrates the $\mathcal{P}_{\mathrm{relax}}$ distributions for the Top70 clusters, DCC cores, and MSCC superclusters. For superclusters, these values were estimated similarly (see column 7 of Table 6), while the Top70 cluster values are from Zúñiga et al. (Reference Zúñiga, Caretta and Andernach2024b). Consistent with the $H_Z$ -entropy analysis, the relaxation probability results also support the hypothesis that cores, with an average $\left\langle \mathcal{P}_{\mathrm{relax}} \right\rangle = 0.64$ , represent galaxy structures in an intermediate evolutionary state. Clusters, being the most relaxed structures, exhibit $\left\langle \mathcal{P}_{\mathrm{relax}} \right\rangle = 0.83$ , while superclusters, the least relaxed, have $\left\langle \mathcal{P}_{\mathrm{relax}} \right\rangle = 0.56$ .