THE SLOAN GREAT WALL. MORPHOLOGY AND GALAXY CONTENT

We used the MAIN galaxy sample and the SDSS-LRG sample from the seventh data release of SDSS (Adelman-McCarthy et al. 2008; Abazajian et al. 2009). From these data we chose a subsample of the SGW region: 150° ⩽ R.A. ⩽ 220°, −4° ⩽ δ ⩽ 8°, within the distance limits 150 h⁻¹ Mpc ⩽D_com ⩽ 300 h⁻¹ Mpc. This region fully covers the SGW and its surroundings, including poor superclusters, which connect the SGW with other nearby rich superclusters. In this region there are 27,113 galaxies from the MAIN SDSS galaxy sample, and 3168 BRGs that have been selected from the SDSS database. These are nearby (z < 0.15) LRGs (cut I LRGs; Eisenstein et al. 2001).

We found groups of galaxies on the basis of the MAIN galaxy sample, as described in detail in Tago et al. (2010, hereafter T10) and Tago et al. (2008). We corrected the redshifts of galaxies for motion relative to the cosmic microwave background and computed the comoving distances (Martínez & Saar 2002) of galaxies using standard cosmological parameters: Hubble parameter H₀ = 100 h km s⁻¹ Mpc⁻¹, matter density Ω_m = 0.27, and dark energy density Ω_Λ = 0.73. Our SGW region sample contains about 4000 groups of galaxies.

In T10, groups of galaxies were defined by applying the Friends-of-Friends cluster analysis method introduced by Turner & Gott (1976), Zeldovich et al. (1982), and Huchra & Geller (1982) and modified by us (Tago et al. 2008). In this algorithm, galaxies are linked into systems using a variable linking length. By definition, a galaxy belongs to a group of galaxies if it has at least one group member galaxy closer than the linking length. To take into account selection effects when constructing a group catalog from a flux-limited sample, T10 calibrated the scaling of the linking length with distance, using nearby rich groups as a template. As a result, the maximum group sizes in the sky projection and the velocity dispersions of groups in our catalog are similar at all distances.⁵

Next, we calculated the luminosity density field of galaxies and found extended systems of galaxies (superclusters) in this density field. This procedure is described in more detail in Appendix A. The supercluster catalog is described in detail in Liivamägi et al. (2010).

To search for systems (high-density regions) in the distribution of galaxies and clusters of galaxies usually either the Friend-of-Friend algorithm or a density field approach is used. In the Friend-of-Friend method, galaxies or clusters of galaxies are collected together so that each object has at least one neighbor closer than the linking length (a fixed distance). For small linking lengths all objects are isolated. When the linking length increases, more objects are collected into systems, and as a result, the richness and lengths of these systems increase. At a certain linking length, percolation occurs and systems join rapidly to form a larger system that may penetrate the whole volume studied. Superclusters, as the largest still relatively isolated systems, are found using a linking length smaller than the percolation length (for details we refer to Zucca et al. 1993; Einasto et al. 1994, 2001). When using the density field to select systems, we search for connected systems with densities higher than a certain threshold. When we decrease the density threshold, then, as with increasing the linking length in the Friend-of-Friend method, more objects join the systems, and the richness and lengths of the systems increase. At a certain threshold the systems percolate the sample volume. Again, supercluster catalogs are compiled using a density threshold higher than the percolation threshold (Einasto et al. 2003a, 2003b, 2007a; Erdoğdu et al. 2004). Erdoğdu et al. (2004) used a varying density threshold to determine superclusters using the 2dF data; for details, refer to their study. While the Friend-of-Friend method tends to find elongated systems (Einasto et al. 1984), the density field approach does not have such a bias.

Comparisons of different catalogs of superclusters shows that rich superclusters can be identified in different ways, with similar parameters (e.g., maximum length). For example, the supercluster SCl 126 has been identified in Einasto et al. (2003a) using the SDSS data and the density field (supercluster N13 in the list), and has a length of 90 h⁻¹ Mpc; in Einasto et al. (2003b) this supercluster is determined using the 2dF data, (supercluster SCl 152 in the list) with a length of 112 h⁻¹ Mpc. In the Erdoğdu et al. (2004) list, this supercluster is SCNGP06.

In order to choose proper density levels to determine the SGW and the individual superclusters, which belong to the SGW, we compared the density field superclusters at a series of density levels (Figure 1). We analyzed the systems of galaxies at each level and examined the joining events when we move from higher density levels to lower ones. We will use the relative luminosity density D here and below (normalized to the mean density in the SDSS sample volume). At a very high-density level (D = 6.8), only the highest density part of the SGW is seen. This can be identified with the core region of the supercluster SCl 126 in the E01 catalog. We use this density level to define the core of this supercluster and refer to it as SCl 126c. As we decrease the density level, galaxies from the outskirts of SCl 126 join the SGW. At density level D = 4.9, the two richest superclusters in the SGW (SCl 126 and SCl 111) still form separate systems. At a lower density level, D = 4.7, these two superclusters join together to become one connected system, and several other superclusters also join this system. At a still lower density level (D = 4.0) these superclusters, which border the voids around the SGW, join the SGW. Several poor superclusters, which form a low-density extension of the SGW, join the SGW at this level, too. We show in Figure 1 the full SGW region and the surrounding superclusters at different density levels from D = 6.8 to D = 4.0. We used the information about the joining of superclusters to determine the SGW and the individual superclusters in the SGW as follows.

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First, we used the density level D = 4.9 to determine individual superclusters that belong to the SGW. In Figure 2, we show and mark all superclusters of galaxies in the SGW region. The data on these superclusters are presented in Table 1.

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The richest superclusters in the SGW are SCl 126 and SCl 111. The supercluster SCl 126 also includes the supercluster SCl 136 from the E01 catalog. We note that the supercluster SCl 111 in the E01 catalog actually corresponds to two superclusters in the SGW region, the first and second superclusters in Table 1. In what follows, we will call the first supercluster in this table SCl 111, as the second supercluster actually does not belong to the SGW (see Figure 2). We will use the notation SCl 126c for the core region of supercluster SCl 126, and the notation SCl 126o for the outskirts of this supercluster, defined by excluding the core SCl 126c from the whole supercluster.

We use the density level D = 4.7 to determine the whole SGW. This adds to the SGW also those poor superclusters which form the low-density extension of the SGW (superclusters 9, 8, and 13 in Table 1, which all belong to the SCl 91 in the E01 catalog). Thus, the highest density part of the SGW is the supercluster SCl 126 and the lowest density part is formed by poor superclusters, members of the supercluster SCl 91 in the E01 catalog. In addition, we combine those galaxies, groups, and poor superclusters in the volume defined at the start of this section, which do not belong to the SGW, into a comparison sample and call it the field sample (denoted as F). Our SGW sample contains 6138 galaxies and 817 groups, 351 of which host BRGs. The field sample contains 20,975 galaxies and 3020 groups; 1589 groups in the field have BRGs in them. In the SGW (field), 83% (68%) of all BRGs belong to groups.

In this paper, we use the g − r colors of galaxies and their absolute magnitudes in the r-band M_r. The absolute magnitudes of individual galaxies are computed using the KCORRECT algorithm (Blanton et al. 2003a; Blanton & Roweis 2007). We also accepted M_☉ = 4.53 (in the r photometric system). Evolution correction e has been calibrated according to Blanton et al. (2003b). All of our magnitudes and colors correspond to the rest frame at the redshift z = 0. We note that our colors are found using the rest-frame Petrosian magnitudes, not the model magnitudes as in Eisenstein et al. (2001), so there might be small differences in the colors.

As mentioned above, we will pay special attention to two populations of galaxies, the first ranked galaxies of groups and the BRGs (the galaxies that have the GALAXY_RED flag in the SDSS database). The BRGs are nearby (z < 0.15, cut I) LRGs (Eisenstein et al. 2001) and are similar to the LRGs at higher redshifts. Nearby BRGs do not form an approximately volume-limited population, but they are the brightest and the reddest galaxies in the SGW region, as shown in Figure 3.

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BRGs have been selected using model magnitudes, but in our analysis we will use rest-frame Petrosian magnitudes for all galaxies including BRGs, for concordance.

For a small subsample of galaxies (the BRGs and the first ranked galaxies in groups; Section 4), we also determined morphological types, using data from the SDSS visual database. We used color composite images and followed the classification used in the Galaxy Zoo project (Lintott et al. 2008)—elliptical galaxies are type 1, spiral galaxies are type 4, and those galaxies which show signs of merging or other disturbances are type 6. Types 2 and 3 denote clockwise and counterclockwise spiral galaxies, for which spiral arms are visible. Fainter galaxies, for which the morphological type is difficult to determine, are type 5. The use of color composite images may, in principle, introduce a bias with blue-looking galaxies tending to be classified as spirals and red-looking galaxies as ellipticals. We tried to avoid this by having several people classify the galaxies independently. A later comparison showed that in most cases these independent classifications coincided. Our sample of the BRGs and the first ranked galaxies consists of quite bright galaxies, so it was possible to determine their morphological type reliably.

Eisenstein et al. (2001) warn that the sample of nearby LRGs may be contaminated by galaxies of lower luminosity not suitable for the use of nearby BRGs as a volume-limited sample. Nearby BRGs also may be of late type and/or, as Figure 3 shows, some of them may have blue g − r colors. Later we will compare the colors and morphological types of BRGs in color–magnitude diagrams to check for the probable presence of blue, late-type galaxies in our sample of BRGs.⁶

We divide the galaxies into red and blue populations using the color limit g − r ⩾ 0.7 (we call these galaxies red). We use volume-limited samples with an r-band magnitude limit of M_r = −19.25. We show in Figure 4 the distributions of the g − r colors of galaxies in the whole SGW and in the individual superclusters in the SGW. The numbers of galaxies in each sample are given in the caption. A comparison with the numbers of galaxies in the superclusters in Table 1 shows that all samples are almost volume-limited—the selection effects were taken into account correctly when we calculated the luminosity density field.

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In this paper, all the distributions (probability densities) were calculated within the R environment using the "density" command in the "stats" package (Ihaka & Gentleman 1996).⁷ The "density" function chooses the optimal bin width minimizing the MISE (mean integral standard error) of the estimate. This package does not provide the customary error limits (local error estimates), but the usual Poisson errors do not describe these either (we explained this in more detail in Einasto et al. 2008). We compare the density distributions using the full data (i.e., the integral distributions) and statistical tests such as the Kolmogorov–Smirnov test throughout the paper.

Figure 4 shows that in the whole SGW the fraction of red galaxies is larger than that in the field, which is evidence of large-scale morphological segregation (Einasto et al. 2007c). In the SGW (field), 65% (51%) of the galaxies are red. In addition, there are relatively more red galaxies in the core of the supercluster SCl 126 (SCl 126c) than in other superclusters (Table 3). This agrees with earlier findings by Einasto et al. (2007c, 2007d). We used the Kolmogorov–Smirnov test to estimate the statistical significance of the differences between the colors of galaxies in different superclusters. This test showed that the differences between the colors of galaxies in the SGW and in the field have a very high significance (the probability that the color distributions were drawn from the same sample is less than 10⁻⁵). The differences between the colors of galaxies in the core of the supercluster, SCl 126c, and in the outskirts of this supercluster, SCl 126o, and between the colors of galaxies in SCl 126c and the colors of galaxies in other superclusters also have a very high statistical significance. The differences between the colors of galaxies in other superclusters are marginal.

In the T10 catalog, the first ranked galaxy of a group is defined as the most luminous galaxy in the r band. We will use the same definition in the present paper; later we will compare the properties of those first rank galaxies that are BRGs with those that are not BRGs.

A three-dimensional interactive PDF graph that shows the spatial distribution of groups of galaxies in the SGW can be found at http://www.aai.ee/~maret/SGWmorph.html.

If we want to understand the formation, evolution, and present-day properties of cosmic structures then in addition to the usual statistics that characterize the clustering of galaxies (the two-point correlation function, Nth nearest neighbor, and others; see Martínez & Saar 2002 for a review about methods of characterizing galaxy distribution) we must use higher order statistics to describe structures such as superclusters, filaments, and clusters in which galaxies are embedded (Bond et al. 2010a). A number of methods are used to define and characterize cosmic structures (Pimbblet 2005; Stoica et al. 2010; van de Weygaert et al. 2009; Aragón-Calvo et al. 2010; Bond et al. 2010a, 2010b, and references therein). One possible approach is to determine cosmic structures (in our case, superclusters of galaxies) using, for example, a density field and then to study their morphology (Schmalzing & Buchert 1997; Sathyaprakash et al. 1998; Shandarin et al. 2004, and references therein).

Supercluster geometry (morphology) is defined by its outer (limiting) isodensity surface and its enclosed volume. When increasing the density level over the threshold overdensity D = 4.7 (Section 2.1), the isodensity surfaces move into the central parts of the supercluster. The morphology of the isodensity contours is (in the sense of global geometry) completely characterized by the four Minkowski functionals V₀–V₃ (we give the formulae in Appendix B).

Minkowski functionals, genus, and shapefinders have been used to study the morphology of the large-scale structure in the 2dF and SDSS surveys (Hikage et al. 2003; Park et al. 2005; Saar et al. 2007; James et al. 2007; Gott et al. 2008) and to characterize the morphology of superclusters (Sahni et al. 1998; Basilakos et al. 2001; Kolokotronis et al. 2002; Sheth et al. 2003; Basilakos 2003; Shandarin et al. 2004; Basilakos et al. 2006) in observations and simulations. These studies concern only the "outer" shapes of superclusters and do not treat their substructure. We expanded this approach in Einasto et al. (2007d), using the Minkowski functionals and shapefinders to analyze the full density distribution in superclusters, at all density levels.

The original luminosity density field, used to determine individual superclusters in the SGW and the full SGW complex, was calculated using all galaxies. To calculate the density field for the study of the morphology, we recalculated the density field for each individual structure studied. We used volume-limited galaxy samples and calculated the number density; this approach makes our results insensitive to selection corrections (although our SGW samples are already almost volume-limited; compare the numbers of galaxies in the full and volume-limited samples given in Table 1 and in the caption of Figure 4). To obtain the density field for estimating the Minkowski functionals, we used a kernel estimator with a B₃ box spline as the smoothing kernel, with a radius of 16 h⁻¹ Mpc (Saar et al. 2007; Einasto et al. 2007d). For the argument labeling the isodensity surfaces, we chose the (excluded) mass fraction m_f—the ratio of the mass in regions with densities lower than the density at the surface to the total mass of the supercluster. When this ratio runs from 0 to 1, the isosurfaces move from the outer limiting boundary into the center of the supercluster, i.e., the fraction m_f = 0 corresponds to the whole supercluster, and m_f = 1 to its highest density peak. This is the convention adopted in all papers devoted to the morphology of the large-scale galaxy distribution. The reason for this convention (the higher the density level, the higher the value of the mass fraction) is historical—the most popular argument for the genus and for the Minkowski functionals has been the volume fraction that grows with the density level; all other arguments are chosen to run in the same direction. We refer to Einasto et al. (2007d) for details of the calculations of the density field.

For a given surface, the four Minkowski functionals (from the first to the fourth, respectively) are proportional to the enclosed volume V, the area of the surface S, the integrated mean curvature C, and the integrated Gaussian curvature χ. The last of these, the fourth Minkowski functional, V₃, describes the topology of the surface; at high densities it gives the number of isolated clumps (balls) in the region and at low densities it gives the number of cavities (voids; see, e.g., Saar et al. 2007).

The fourth Minkowski functional, V₃, is well suited for describing the clumpiness of the galaxy distribution inside superclusters—the fine structure of superclusters. We calculate V₃ for galaxies of different populations in superclusters for a range of threshold densities starting with the lowest density used to determine superclusters up to the peak density in the supercluster core. Thus, we can see in detail how the morphology of superclusters is traced by galaxies of different types.

In addition to the fourth Minkowski functional V₃, we use the dimensionless shapefinders K₁ (planarity) and K₂ (filamentarity) (Sahni et al. 1998; Appendix B). In Einasto et al. (2007d), we showed that in the (K₁, K₂) shapefinder plane the morphology of superclusters is described by a curve (morphological signature) that is similar for all multibranching filaments. In this paper, we will use the morphological signature to characterize the morphology of superclusters and of galaxy populations in them.

There are two main effects that affect the reconstructed density field and its Minkowski functionals. The first is due to the fact that about 6%–7% of galaxy redshifts are missing because of fiber collisions (Zehavi et al. 2010). About 60% of these are approximately at the same distance as their close neighbors (Zehavi et al. 2002), but others are not. We found the collision groups in the SDSS, determined the galaxies that do not have redshifts because of fiber collisions, and ran Monte Carlo simulations, assigning to 60% of randomly selected non-redshift galaxies the redshift of their neighbors. The other 40% of non-redshift galaxies were omitted, assuming that their true redshift would disqualify them as members of the supercluster. This procedure should give upper limits for errors. We ran 1000 simulations and found that both the morphological signatures and V₃ did not change (at the 95% confidence level)—fiber collisions are too scarce to affect the estimates of the morphological characteristics.

Another effect is the discreteness of galaxy catalogs (shot noise) that induces errors in the density estimates. To take this into account, first we have to define the statistical model for the spatial distribution of galaxies within a supercluster. One possibility is to use the Cox model, where we have a realization of a random field and a Poisson point process populates the supercluster by dropping galaxies there with an intensity proportional to the value of the realization in the neighborhood of a possible galaxy. We tested that model and found that it is not able to describe the structure of superclusters; we summarize the results in Appendix D.

After several erroneous approaches, we used the popular halo model ideology (see Cooray & Sheth 2002; Yang et al. 2007; van den Bosch et al. 2007, and references therein for details) to define a statistical model for superclusters. We assume that supercluster structure is defined by its dark matter halos and discreteness errors are caused by the random positions of galaxies within these halos. As the halo model assumes that the main galaxy of a halo lies at its center, the main galaxies of our groups and the isolated galaxies (the main galaxies of halos where other galaxies are too faint to see) remain fixed. For satellite galaxies, we use a smoothed bootstrap to simulate the distribution of satellites inside our halos—we select satellite galaxies by replacement, and add increments with the same distribution as the B₃ kernel we use for density estimation to their spatial positions. A smoothed bootstrap is known to be able to estimate pointwise errors of densities (Silverman & Young 1987; Davison & Hinkley 2009). As the B₃ spline (Equation (A3)) practically coincides with a Gaussian of the rms deviation σ = 0.6, we define the B₃ kernel width a = σ_r/0.6 for every halo (galaxy group) using the rms deviation σ_r of the radial positions of galaxies for that group. We generated mock superclusters with new galaxy distributions by smoothed bootstrap and re-calculated the density field and Minkowski functionals, repeating this procedure 1000 times. The figures in Section 3.4 where we compare the Minkowski functionals and shapefinders for galaxy populations of different superclusters in the SGW show the 95% confidence regions obtained by these simulations.

Next, we briefly describe how the fourth Minkowski functional V₃ and the shapefinders K₁ and K₂ describe the morphology of a system.

At small mass fractions, the isosurface includes the whole supercluster and the value of the fourth Minkowski functional V₃ = 1. As we move to higher mass fractions, the isosurface includes only the higher density parts of superclusters. Individual high-density regions in a supercluster, which in low values of mass fractions are joined together into one system, begin to separate from each other, and the value of the fourth Minkowski functional (V₃) increases. At a certain density contrast (mass fraction) V₃ has a maximum, showing the largest number of isolated clumps in a given supercluster. The fourth Minkowski functional V₃ also depends on the number of tunnels, which may appear in the supercluster topology as part of the galaxies do not contribute to the supercluster at intermediate density levels. At still higher mass fractions only the high-density peaks remain in the supercluster and the value of V₃ decreases again.

When we increase the mass fraction, the changes in the morphological signature accompany the changes of the fourth Minkowski functional. As the mass fraction increases, at first the planarity K₁ almost does not change, while the filamentarity K₂ increases—at higher density levels superclusters become more filament-like than the whole supercluster. Then the planarity also starts to decrease, and at a mass fraction of about m_f = 0.7 the characteristic morphology of a supercluster changes—we see the crossover from the outskirts of a supercluster to the core of a supercluster (Einasto et al. 2007d).

We calculate the Minkowski functionals and shapefinders for superclusters in the SGW, defined at four different density levels, to see the change of the morphology of the SGW with density (Figure 5). In the left panel of Figure 5, the individual mass fractions for higher density levels are rescaled and shifted so that they show the fraction of mass of systems at a high-density level relative to the total mass of the systems at the lowest density level, D = 4.0.

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At the highest density level (D = 6.8) only the core region of the supercluster SCl 126 contributes to the SGW. The left panel of Figure 5 shows that for this core, the value of the fourth Minkowski functional V₃ is small almost over the whole range of mass fractions; the largest value of V₃ is 5. The right panel of Figure 5 shows the corresponding morphological signature, with the starting planarity K₁ ≈ 0.1. As the mass fraction increases, the filamentarity K₂ increases and reaches a maximum value of about 0.72 at a mass fraction of m_f = 0.7, when the morphological signature changes. The shape of the morphological signature K₁–K₂ is consistent with a single filament (for details see Einasto et al. 2007d).

At the density level D = 4.9 new galaxies join the SGW and the SGW coincides with the whole supercluster SCl 126. The maximum value of the clumpiness V₃ = 12 and the values of the planarity K₁ and filamentarity K₂ are larger—the morphology of the supercluster resembles a multibranching filament (Einasto et al. 2007d), as shown by the morphological signature.

At the density level D = 4.7 the supercluster SCl 111 also joins the SGW (Figure 1, right panel). The maximum value of the fourth Minkowski functional is V₃ = 20. In the (K₁, K₂) plane, the morphological signature describes a planar system (Einasto et al. 2007d).

At a still lower density level, D = 4.0, the superclusters, which really do not belong to the SGW, join into one system with the SGW (Figure 1, right panel). The maximum value of the fourth Minkowski functional is V₃ = 33 and in the (K₁, K₂) plane the morphological signature becomes even more typical of a planar system, consisting of more than one individual supercluster. Interestingly, although the overall morphology changes strongly, we see that in all cases, the morphological signature changes at a mass fraction of m_f ≈ 0.7, a change of morphology at a crossover from the outskirts to the core regions (Einasto et al. 2008).

At high values of mass fractions, in addition to the core of the supercluster SCl 126, other high-density clumps also contribute to the full SGW, and they too add their contribution to the values of the fourth Minkowski functional V₃ and to K₁ and K₂ in Figure 5. Thus in Figure 5 the values of V₃ for a density level of, say, D = 4.9, at mass fractions that approximately correspond to the density level D = 6.8 (in Figure 5 this corresponds to a mass fraction of mf = 0.85) are larger than those from the density level D = 6.8 at the same mass fraction. In other words, the curves characterizing the morphology of the SGW at different density levels depend on all the systems encompassed by the density isosurfaces.

Next, we calculate the fourth Minkowski functional V₃ and the morphological signature K₁–K₂ for individual galaxy populations defined by their g − r color and for the BRGs, in the two richest superclusters in the SGW: the superclusters SCl 126 (separately for the core and for the whole supercluster) and SCl 111 (Figures 7–9). In Figures 7–9 we show the 95% confidence regions obtained by Monte Carlo simulations as explained in Section 3.1.

To help to understand the results of these calculations, in Figure 6 we show the sky distribution of galaxies in the superclusters SCl 126 and SCl 111. The distribution of all galaxies in the full superclusters is delineated with two-dimensional density contours. The location of groups with BRGs (we discuss the group membership of BRGs in the next section) is shown with different symbols—dots mark the members of both superclusters SCl 126 and SCl 111 at a high-density level D = 6.8, crosses mark members of other systems at this density level (these systems have not yet joined the superclusters SCl 126 and SCl 111), and ×'s mark groups with BRGs that do not belong to any supercluster at the density level D = 6.8 yet, but will be members of the superclusters SCl 126 and SCl 111 at the density level D = 4.9.

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In Figure 7, we present the results of the calculations of the Minkowski functional V₃ and the shapefinders for the core of the supercluster SCl 126c (the groups with BRGs in this system are marked with dots in Figure 6). Figure 7 shows that the clumpiness of red and blue galaxies and BRGs is rather similar over the whole range of mass fractions. The BRGs form some clumps at mass fractions of m_f = 0.3–0.4 and m_f = 0.5–0.7, which are not seen in the distribution of red and blue galaxies; their spatial distribution is less clumpy at these densities. The confidence regions show that these differences are statistically significant at least at the 95% confidence level.

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At a mass fraction of m_f = 0.8, the clumpiness of red galaxies is the largest, reaching a value of V₃ = 9. The differences between the clumpiness of red galaxies and other galaxy populations are statistically significant at least at the 95% confidence level. The clumpiness of blue galaxies and BRGs reaches the same value at a mass fraction of m_f = 0.80. At a higher mass fraction, m_f = 0.9, blue galaxies form some isolated clumps and the value of V₃ for blue galaxies is larger than that for red galaxies and BRGs (V₃ = 5 and V₃ ⩽ 3, correspondingly). At these highest mass fractions, the differences between the clumpiness of blue galaxies and all other galaxy populations are statistically significant at least at the 95% confidence level.

The morphological signatures of galaxies from different populations almost coincide at mass fractions of m_f < 0.7 (in the outskirts of the supercluster). The morphological signature for the core of the supercluster SCl 126c is characteristic of a filament (Einasto et al. 2007d). In this filament, the distribution of galaxies from different populations is rather homogeneous, showing similar clumpiness and morphological signatures.

In Figure 8, we show the fourth Minkowski functional V₃ and the morphological signature K₁–K₂ for the whole supercluster SCl 126. This figure reveals several differences in comparison with the core of the supercluster. At mass fractions of 0.15 < m_f < 0.25 BRGs form some isolated clumps (for BRGs, V₃ = 3) while V₃ for red galaxies has a small minimum (V₃ = 0) suggesting that at a low-density level in the outskirts of the supercluster there are differences in how the substructure of the supercluster is delineated by BRGs and red galaxies. The value of the fourth Minkowski functional V₃ for red galaxies (V₃ = 0) suggests that the population of red galaxies may have tunnels through them at these mass fractions (see Appendix B), but the differences between the clumpiness of red galaxies and BRGs at this mass fraction interval are not statistically significant at the 95% confidence level, as the confidence regions in Figure 8 show.

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At mass fractions of m_f ≈ 0.3 the clumpiness of red galaxies and BRGs increases, while the clumpiness of blue galaxies decreases and even becomes negative. This means that at intermediate densities (up to m_f ≈ 0.6), red galaxies and BRGs form isolated clumps, while blue galaxies are located in lower density systems which do not contribute to the supercluster at these mass fractions. Some of these isolated clumps belong to a separate system marked with crosses in Figure 6 (left panel). Negative values of V₃ for blue galaxies at mass fractions of m_f ≈ 0.45 suggest that these systems have tunnels through them. These systems belong to the outskirts of the supercluster SCl 126. At this mass fraction interval, the differences between the clumpiness of red galaxies and BRGs and that of blue galaxies are statistically significant at least at the 95% confidence level.

At higher mass fractions, m_f ⩾ 0.5, the clumpiness of blue galaxies increases. The clumpiness of red and blue galaxies reaches a maximum value, V₃ = 13, at mass fractions of m_f = 0.7–0.8, where the morphological signature shows a change in the morphology of the supercluster. At still higher mass fractions, we do not see differences between the fourth Minkowski functionals of galaxy populations that are statistically significant at the 95% confidence level.

For most of the mass fraction interval, the clumpiness of red galaxies is larger than the clumpiness of blue galaxies. The differences between the morphological signatures of red and blue galaxies are statistically significant for the high values of mass fractions at the 95% confidence level.

Figure 9 shows the fourth Minkowski functional V₃ and the morphological signature K₁–K₂ for another rich supercluster in the SGW, the supercluster SCl 111. The curves of V₃ differ from those for the supercluster SCl 126. At mass fractions of 0.15 < m_f < 0.35 the V₃ curves suggest that red galaxy and BRG populations have tunnels through them. At these mass fractions some galaxies already do not contribute to the supercluster (these are galaxies that do not belong to any supercluster at higher density levels in Figure 6) so that tunnels can form. At the same time individual clumps are connected so that tunnels form between them (otherwise red galaxies and BRGs would form isolated clumps instead of tunnels). At mass fractions of m_f > 0.35 the value of V₃ for red galaxies and BRGs increases showing that the number of isolated clumps formed by these galaxies grows. The V₃ curves for red galaxies and BRGs reach a maximum value (V₃ = 9–10) at a mass fraction of m_f ≈ 0.6 and then decrease. Such a clumpiness curve is characteristic of a poor supercluster, a system dominated by a few high-density clumps with rich clusters in it (a spider; see Einasto et al. 2007d). Figure 6 (right panel) shows low-density regions (groups that do not belong to any supercluster at densities of D = 6.8) and high-density regions that already form systems at densities of D = 6.8 in this supercluster.

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The value of the fourth Minkowski functional V₃ for blue galaxies in the supercluster SCl 111 V₃ = 1 up to mass fractions of m_f < 0.3 shows that at these mass fractions blue galaxies form one system without isolated clumps or tunnels through them. In other words, in the outskirts of this supercluster blue galaxies are distributed more homogeneously than red galaxies or BRGs. In the mass fraction interval 0.35 < m_f < 0.45 the value of V₃ for blue galaxies decreases—some blue galaxies do not contribute to superclusters anymore and tunnels form. At a mass fraction of m_f > 0.45 the value of V₃ increases (the number of isolated clumps increases) and reaches the maximum (V₃ = 9) at the mass fraction m_f = 0.8. After that, the clumpiness of blue galaxies decreases.

In Figure 9, we see that for almost the whole mass fraction interval the differences between the values of the fourth Minkowski functional V₃ for galaxies from different populations are statistically significant at least at the 95% confidence level.

The morphological signature of supercluster SCl 111 also differs from that of supercluster SCl 126. The values of the planarity K₁ at small mass fractions are larger than those for the SCl 126, and the maximum values of the filamentarity K₂ are lower (K₂ ⩽ 0.7, while for the supercluster SCl 126 the filamentarity K₂ ⩽ 0.9). Interestingly, for certain mass fractions (0.45 < m_f < 0.65) the planarity of BRGs is larger than that of red and blue galaxies. This may be due to the large clumpiness of BRGs in the same mass fraction interval (Figure 9, left panel).

Figure 9 shows that the differences between morphological signatures of red galaxies, BRGs, and blue galaxies are statistically significant at the 95% confidence level at a wide range of mass fractions.

We have seen that the morphology of the SGW varies from a simple filament to a multibranching filament and then, as we lower the level of the density, becomes characteristic of a very clumpy, planar system. The changes in morphology suggest, in addition to the analysis of the properties of individual superclusters (their richness and size), that a proper density level to delineate individual superclusters is D = 4.9. A lower density level, D = 4.7, can be used to select the whole SGW.

The overall morphology of the galaxy systems in the SGW region changes strongly when we change the density level, but in all cases, the morphological signature changes at a mass fraction of m_f ≈ 0.7, at a crossover from the outskirts to the core regions of superclusters.

The morphology of the core of supercluster SCl 126 is characteristic of a simple filament with small clumpiness. Differences between the clumpiness and morphological signatures of individual galaxy populations are small, telling us that all galaxy populations delineate the core of the supercluster in the same way. The morphology of the whole supercluster SCl 126 can be described as characteristic of a multibranching filament, where the clumpiness of red galaxies and BRGs is larger than the clumpiness of blue galaxies, especially in the outskirts of the supercluster. The supercluster SCl 111 resembles a multispider (we described the morphological templates in more detail in Einasto et al. 2007d). Here again, the clumpiness of red galaxies and BRGs is larger than the clumpiness of blue galaxies. In the outskirts of both superclusters, blue galaxies are distributed more homogeneously than red galaxies and BRGs. In the outskirts of superclusters, galaxy populations have tunnels through them.

The morphology of superclusters can be characterized by the ratio of the shapefinders, K₁/K₂ (the shape parameter). This approach was used, for example, by Basilakos (2003) to study the shapes of superclusters—a large value of this ratio indicates planar systems. Our calculations show that the values of the shape parameter are as follows: for the core of supercluster SCl 126, SCl 126c, K₁/K₂ ≈ 0.33 and for the full supercluster SCl 126, K₁/K₂ ≈ 0.28; we get the same value for all galaxy populations. In supercluster SCl 111, K₁/K₂ = 0.47 for the full supercluster and for red galaxies, while for BRGs K₁/K₂ = 0.52 and for blue galaxies K₁/K₂ = 0.61. This is another difference between the superclusters in the SGW.

We assessed the statistical significance of the differences between the fourth Minkowski functional V₃ and morphological signatures of different galaxy populations in superclusters using Monte Carlo simulations and showed that in at least at some mass fraction intervals (depending on the populations) the differences between the galaxy populations are statistically significant at least at the 95% confidence level.

Figure 10 shows the sky distribution of the BRGs and galaxy groups in superclusters SCl 126 (left panel) and SCl 111 (right panel). Here we plotted groups with symbols of sizes proportional to their richness. In this figure, we mark those groups that host up to four BRGs with small dots and groups with at least five BRGs with stars. A comparison with Figure 6 shows that in the supercluster SCl 126 groups with a large number of BRGs are located both in the core and in the outskirts. In supercluster SCl 111, assemblies of groups with a large number of BRGs already form systems at a density level of D = 6.8, while the poor groups and groups with a few BRGs in them do not belong to any supercluster at this density level.

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The richest group in the core of supercluster SCl 126, which corresponds to the Abell cluster A1750 located at R.A. ≈ 2027 and decl. ≈ −19 (Figure 10, left panel), hosts 17 BRGs. In the outskirts of supercluster SCl 126 the number of BRGs is the largest in a group that corresponds to the Abell cluster A1809, located at R.A. ≈ 2083 and decl. ≈ 51; this group hosts 12 BRGs. Isolated clumps seen in the V₃ curve at mass fractions of about m_f ≈ 0.2 (Figure 8) are probably at least partly due to this group. In supercluster SCl 111, the richest group corresponds to the Abell cluster A1516, located at R.A. ≈ 1844 and decl. ≈ 37, and hosts 15 BRGs. The locations of the groups with the largest numbers of BRGs in Figure 10 are marked with the largest stars.

In Figure 11, we compare the number of BRGs in groups in supercluster SCl 126 (in the core and in the outskirts) and in supercluster SCL 111. This figure shows that most groups with BRGs host only a few BRGs (the mean number of BRGs in groups is 2 in both superclusters SCl 126 and SCl 111). Groups with five or more BRGs are also richer than groups with a smaller number of BRGs, with at least 20 member galaxies. Although the richest group in the core of supercluster SCl 126c hosts more BRGs than the richest group in the outskirts of this supercluster, the overall membership of BRGs in groups is statistically similar in both of these regions, and also in supercluster SCl 111. Table 2 shows that approximately 40% of BRGs are first ranked galaxies in groups.

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There are also BRGs that do not belong to any group (isolated BRGs). In the core of supercluster SCl 126, SCl 126c, 11% of all BRGs do not belong to groups, while in the outskirts of this supercluster this fraction is 18%. In supercluster SCl 111, 20% of all BRGs do not belong to groups. In this respect, there is a clear difference between superclusters.

Tempel et al. (2009) showed by analyzing the luminosity functions of galaxies in the 2dFGRS groups that most of the bright isolated galaxies should be the first ranked galaxies of groups, where the fainter members are not observed since they fall outside of the observational window of the flux-limited sample. Thus, we may suppose that there are no truly isolated BRGs, but they are the first ranked galaxies of fainter groups.

Figure 10 shows that there are also many groups without BRGs (empty circles without a dot or a star). In supercluster SCl 111, many of these groups are located loosely between the three concentrations of this supercluster (Figure 10, right panel). We present a summary of the properties of groups with and without BRGs in superclusters in Table 2. First, this table shows that groups with BRGs are richer and have larger velocity dispersions and higher luminosities than groups without BRGs. Second, groups with BRGs in the core of the SCL 126, SCl 126c, are richer and have larger velocity dispersions and higher luminosities than groups with BRGs in the outskirts of this supercluster or groups in supercluster SCl 111. The Kolmogorov–Smirnov test shows that the probability that the distributions of the properties of groups with BRGs in SCl 126c and SCl 126o are drawn from the same distribution is 0.04, and 0.01 for groups in SCl 126c and SCl 111. This shows another difference between these systems.

Next, we divide BRGs into two populations: those that are the first ranked galaxies in groups and those that are not (we refer to these as satellite BRGs). We compare the colors, luminosities, and morphological types of these two sets of BRGs and those first ranked galaxies that are not BRGs in the two richest superclusters in the SGW, SCl 126 and SCl 111.

We summarize the colors and morphologies of the galaxies of the two superclusters in Table 3. This table shows that most BRGs are red, although a small fraction of BRGs have blue colors. The fraction of red galaxies among these first ranked galaxies that are not BRGs is smaller. We note that in the core of supercluster SCl 126 the fraction of red galaxies among those first ranked galaxies that are not BRGs is larger and the fraction of spirals (and red spirals) among them is smaller than in the outskirts of this supercluster or in supercluster SCl 111.

A large fraction of red galaxies in our sample are spirals. The images of some BRGs in the SGW are shown in Figure 12. Galaxies with visible spiral arms are located in groups of different richness, and more than half of them do not belong to any group. We see also that a considerable number of BRGs and first ranked galaxies may be merging or have traces of past merging events.

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The scatter plots of the colors of ellipticals and spirals among the first ranked and satellite BRGs are shown in Figure 13 (see also Table 3). Both show that for the red elliptical first ranked galaxies the color scatter is small (about 0.03), while for the red spirals it is larger (0.04–0.06). Some satellite BRGs have blue colors. Figure 13 shows that most BRGs with blue colors are spirals, but there are also two elliptical galaxies among them.

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Figures 14 and 15 show the colors and luminosities of BRGs and those first ranked galaxies that are not BRGs versus the number of galaxies in the groups where they reside. We see that the first ranked galaxies with bluer colors, fainter first ranked galaxies, and fainter BRGs reside in poor groups. In almost all groups with N_gal > 10 the first ranked galaxy is a BRG. The Kolmogorov–Smirnov test shows that the probabilities that the colors and luminosities of BRGs and the first ranked galaxies in systems studied are drawn from the same distributions are very small.

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A comparison of the richness of groups where elliptical or spiral BRGs reside shows that almost all the spiral first ranked BGRs belong to poor groups with less than 13 member galaxies. At the same time, spiral satellite BRGs can be found in groups of any richness.

Our results are consistent with those of Berlind et al. (2006) who showed that the first ranked galaxies in more luminous groups are more luminous than the first ranked galaxies in less luminous groups (see also Yang et al. 2008; Tempel et al. 2009, and references therein).

Next, we calculated the peculiar velocities for the red (g − r > 0.7) elliptical and spiral BRGs in three systems studied, for three sets of galaxies (the first ranked BRGs, satellite BRGs, and those first ranked galaxies that are not BRGs; Table 3).

Our calculations show that in both superclusters SCl 126 and SCl 111 about half of the first ranked galaxies have large peculiar velocities. The largest peculiar velocities of the first ranked galaxies are even of the order of 1000 km s⁻¹.

In supercluster SCl 126c, the peculiar velocities of elliptical BRGs are larger than those of spiral galaxies (both in the case of the first ranked galaxies and satellite BRGs). The statistical significance of differences of peculiar velocities between the elliptical and spiral first ranked galaxies is low, according to the Kolmogorov–Smirnov test. The differences between the peculiar velocities of elliptical and spiral satellite galaxies are statistically significant at the 95% confidence level.

In the superclusters SCl 126o and SCl 111, elliptical BRGs have smaller peculiar velocities than spiral BRGs, but those elliptical first ranked galaxies that do not belong to BRGs have larger peculiar velocities than spiral galaxies. The peculiar velocities of BRGs in these two superclusters are smaller than in supercluster SCl 126c. According to the Kolmogorov–Smirnov test, these differences are statistically significant at the 92% level in both superclusters.

Now we compare the distribution of groups with elliptical and spiral BRGs as their first ranked galaxies in the sky (Figures 16 and 17). These figures show that in both superclusters the groups with elliptical first ranked BRGs fill the supercluster volume in a much more uniform manner than the groups with spiral first ranked BRGs. This is unexpected, since groups with an elliptical BRG as the first ranked galaxy are richer, and, as we showed, groups in the core of supercluster SCl 126 are richer than in other systems studied.

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To quantify the distribution of groups in superclusters (the substructure of superclusters) and to calculate the projected density contours, we have used the R package mclust (Fraley & Raftery 2006). This package builds normal mixture models for the data, using multidimensional Gaussian densities with varying shape, orientation, and volume. It finds the optimal number of concentrations, the corresponding classification (membership of each concentration), and the uncertainty of the classification. The uncertainty of the classification is defined using the probabilities of each galaxy belonging to a particular component and is calculated as 1 minus the highest probability that a galaxy belongs to a component. The uncertainty of the classification can be used as a statistical estimate of how well groups are assigned to the components.

To measure how well components are determined and to find the best model for a given data set, mclust uses the Bayes Information Criterion (BIC) which is based on the maximized loglikelihood for the model, the number of variables, and the number of mixture components. The model with the smallest BIC value among all models calculated by mclust is considered the best. For details we refer to Fraley & Raftery (2006).

We present the results in Figures 16 and 17. In these figures we plot components of the best model according to BIC, determined by mclust in the distribution of groups with elliptical and spiral BRGs as the first ranked galaxies with different symbols; large symbols (in the case of triangles and circles—large filled symbols) correspond to those groups for which the uncertainty of the classification is larger than 0.05. Groups with spiral BRGs as the first ranked galaxies in supercluster SCl 111 all have an uncertainty of classification smaller than 0.05.

Figures 16 and 17 show that groups with elliptical and spiral BRGs as the first ranked galaxies populate superclusters in a different manner. In the supercluster SCl 126, groups with elliptical BRGs as the first ranked galaxies form three concentrations in the supercluster, and groups with a spiral BRG as the first ranked galaxy form two concentrations. The mean uncertainty of the classification calculated by mclust for groups with elliptical first ranked BRGs is 2.0 × 10⁻³ and for the groups with spiral first ranked BRGs it is 2.0 × 10⁻⁴. Very small uncertainties show very high probabilities that groups belong to a particular component. Groups with the largest uncertainties of classification are located where components overlap.

For every data set we analyzed in detail three best models found by mclust. For groups with elliptical BRGs as the first ranked galaxies in supercluster SCl 126, the two best models have three components and the third best model has four components. The best model with two components for these groups is, according to BIC, very unlikely, and in this case mclust collects groups indicated with circles and triangles in Figure 16 into one component. These components differ from those determined using groups with spiral BRGs as the first ranked galaxies in this supercluster (Figure 16).

In supercluster SCl 111, the groups with an elliptical BRG as the first ranked galaxy form two barely separated concentrations in the supercluster, and groups with spiral first ranked BRGs from three well-defined concentrations (Figure 17). The mean uncertainty of the mclust classification for the groups with elliptical first ranked BRGs is 9.4 × 10⁻⁹; for groups with spiral first ranked BRGs it is 7.5 × 10⁻⁷.

The best model with three components for groups with elliptical BRGs as the first ranked galaxies in supercluster SCl 111 is, according to BIC, much less likely. In this model, groups from the component shown by triangles in Figure 17 are divided between two components, the leftmost groups forming a separate component. Thus, components differ from those determined in the distribution of groups with spiral BRGs as the first ranked galaxies in this supercluster.

If the components have a regular shape (ellipsoids or circles) as in supercluster SCl 111 (Figure 17, the component with mean coordinates R.A. ≈ 1844 and decl.S ≈ 367), the scatter of the sky coordinates can be used to estimate the compactness/looseness of the sky distribution of groups in components. The rms scatter of coordinates in the case of groups with elliptical first ranked BRGs is σ ≈ 39, for groups with spiral first ranked BRGs σ ≈ 08 which shows a more compact distribution of groups with spiral first ranked galaxies in this component. However, in supercluster SCl 126 the sky distribution of groups is such that the scatter of coordinates of all components is large (Figure 16) so we cannot use this estimate. In this supercluster, we compared the fraction of groups with spiral and elliptical first ranked BRGs in the core and the outskirts of the supercluster. A full 47% of groups with a spiral first ranked BRG lie in the core region, while among groups with an elliptical BRG as the first ranked galaxy only 39% lie in the core region.

The number of groups with an elliptical first ranked galaxy in both superclusters is larger than the number of groups with a spiral first ranked galaxy. Thus, we can resample groups with an elliptical first ranked galaxy by randomly diluting the sample of groups so that in the final sample the number of groups is equal to the number of groups with a spiral first ranked galaxy, and then run mclust again to search for components among the resampled set of groups. However, a word of caution is needed—if mclust will detect a different number of components among the randomly resampled groups compared to the original sample, the sky coordinates of these components depend on this particular realization of the resampling. We run the resampling procedure for both superclusters 10,000 times. Our calculations show that in supercluster SCl 126, in about 27% of all cases mclust detected two components in the distribution of groups with an elliptical first ranked galaxy, as in the case of the groups with a spiral first ranked galaxy, but the sky coordinates of these components varied strongly, covering the whole right ascension and declination interval of supercluster SCl 126. The same is true for supercluster SCl 111, where mclust detected three components in 14% of cases in resampled groups with an elliptical first ranked galaxy. This shows that when groups with an elliptical first ranked galaxy are resampled to the same number of objects as groups with a spiral first ranked galaxy there is no clear difference in the number of components between samples. To better understand the differences between the distribution of groups with elliptical and spiral first ranked galaxies in superclusters, a study of a larger sample of superclusters is needed.

We note that Berlind et al. (2006) showed that groups with first ranked galaxies with bluer colors are more strongly clustered than groups with first ranked galaxies with redder colors. In contrast, Wang et al. (2008) showed that SDSS groups with red central galaxies are more strongly clustered than groups of the same mass but with blue central galaxies. As we show, colors and morphological types of galaxies are not well correlated. Therefore, we plan to study a larger sample of groups with first ranked galaxies of different morphological types and colors in a larger set of superclusters to search for possible differences in their distribution.

The comparison of the group membership and properties (colors, luminosities, and morphological types), as well as the peculiar velocities and the sky distribution of BRGs and those first ranked galaxies that are not BRGs in the two SGW superclusters revealed a number of differences between these superclusters.

In all three systems studied (the core and the outskirts of supercluster SCl 126 and supercluster SCl 111) rich groups with at least 20 member galaxies host at least 5 BRGs and the richest group in the core of supercluster SCl 126 hosts 17 BRGs. The fraction of those BRGs that do not belong to any group in the core of supercluster SCl 126 (SCl 126c) is about two times smaller than in other systems studied.

In the core of supercluster SCl 126 (SCl 126c), groups with BRGs are richer and have larger velocity dispersions than groups with BRGs in the outskirts of this supercluster and in supercluster SCl 111.

In the core of supercluster SCl 126 (SCl 126c) the fraction of red galaxies among those first ranked galaxies that are not BRGs is larger and the fraction of spirals among them is smaller than in the outskirts of this supercluster and in supercluster SCl 111.

About 1/3 to 1/2 of all BRGs are spirals. The scatter of the g − r colors of elliptical BRGs is about two times smaller than that of spiral BRGs, showing that elliptical BRGs form a much more homogeneous population than spiral BRGs.

Peculiar velocities of elliptical BRGs in the core of supercluster SCl 126 (SCl 126c) are larger than those of spiral BRGs, while in the outskirts of this supercluster, as well as in supercluster SCl 111, the situation is opposite. Since both elliptical and spiral satellite BRGs lie in groups of all levels of richness, this difference is not caused by the richness of the host group (this could be the case if spiral satellites were residing preferentially in poor groups). In the core of supercluster SCl 126 (SCl 126c) the peculiar velocities of galaxies are larger than in superclusters SCl 126o and SCl 111, suggesting that groups in the core of supercluster SCl 126 are dynamically more active than those in the other two systems studied.

In the two superclusters, groups with an elliptical first ranked BRG are located more uniformly than groups with a spiral first ranked BRG.

We note that, as we saw in Section 3.3, although the outskirts of supercluster SCl 126o and supercluster SCl 111 are determined by a lower density level than the core of supercluster SCl 126c, they contain high-density regions. Thus, the differences between the systems are not entirely due to the differences in density.

We recently analyzed the morphology of individual superclusters, applying Minkowski functionals and shapefinders, and using the full density distribution in superclusters, at all density levels (Einasto et al. 2007d, 2008). In these papers, we used data from the 2dFGRS to delineate the richest superclusters and to study their morphology, as well as the morphology of individual galaxy populations.

Our studies also included supercluster SCl 126, the richest supercluster in the SGW. A comparison of the results shows several differences and similarities. Due to sample limits a part of supercluster SCl 126 was not covered by the 2dFGRS. This is seen in Figure 10 (left panel) which shows an extended filament at decl. δ > 2° not visible in the 2dFGRS data. Due to this, the clumpiness of galaxies in this supercluster (V₃ ⩽ 10) in Einasto et al. (2008) was smaller than that found here; there are also small differences between the morphological signatures of this supercluster found here and in Einasto et al. (2008). Both studies showed that the differences between the clumpiness of red and blue galaxies are larger in the outskirts of the supercluster and smaller in the core of the supercluster. This shows that the scale-dependent bias between the 2dFGRS and SDSS due to the r-band selection of the SDSS (Cole et al. 2007) does not (strongly) affect the values of the Minkowski functionals.

However, even in the present study, using the data from the SDSS DR7, we cannot be sure that the SGW is fully covered by the observations. At low declinations, the SGW extends to the DR7 limit, and it may be possible that the SGW continues at lower declinations. Future observations are needed to uncover the total extent of the SGW.

Our present study showed that the peculiar morphology of supercluster SCl 126 is especially strongly expressed in the core region of the supercluster. The morphology of another supercluster in the SGW, supercluster SCl 111, resembles that of simulated superclusters studied in Einasto et al. (2007d). This agrees with our comparison of the morphology of individual richest superclusters (Einasto et al. 2007d) in observations and in simulations that showed that the morphology of the simulated superclusters differs from the morphology of supercluster SCl126.

The morphology of the full supercluster SCl 126 can be described as a multibranching filament while the core of this supercluster resembles a very rich filament with only a small number of clumps at high-density levels. This makes the core of supercluster SCl 126 an extreme case of a cosmic filament. Supercluster SCl 111, by contrast, consists of several high-density clumps connected by lower density filaments. Such typical filaments between rich clusters have lengths up to about 25 h⁻¹ Mpc (Bond et al. 2010a, 2010b; Choi et al. 2010), although filaments with a length of about 40 h⁻¹ Mpc have also been detected (Pimbblet et al. 2004). Even in the case of intracluster filaments comparison with model filaments shows discrepancies: while Stoica et al. (2010) show that model filaments tend to be shorter than observed filaments and they do not form an extended network, Bond et al. (2010b) find that filaments from observations and models are in good agreement. Thus, even in the case of typical intracluster filaments future studies are needed to further compare the properties of observed and model filaments.

Our analysis of substructures of superclusters as delineated by different galaxy populations revealed several differences and similarities. In the core of supercluster SCl 126 (SCl 126c) the morphology of different galaxy populations is similar. In the outskirts of this supercluster, the clumpiness of different galaxy populations is rather different. Here BRGs form some isolated clumps not seen in the clumpiness curves of other galaxy populations. The clumpiness of blue galaxies is small; blue galaxy populations may have tunnels through them. The differences between the clumpiness of red and blue galaxies is the largest at intermediate mass fractions. In this respect, our present results are similar to those obtained about this supercluster using the 2dFGRS data (Einasto et al. 2008). The differences between the clumpiness of red and blue galaxy populations in supercluster SCl 111 are even larger than in supercluster SCl 126.

Interestingly, in another rich supercluster studied in Einasto et al. (2008), the Sculptor supercluster, the clumpiness of blue galaxies was larger than the clumpiness of red galaxies. In Einasto et al. (2007d), we showed that the clumpiness of observed superclusters for bright and faint galaxies has a much larger scatter than that of simulated superclusters. This shows that superclusters that we have studied so far show a large diversity in how different galaxy populations determine their substructure. Explaining large variations in the distribution of different galaxy populations between superclusters is a challenge for galaxy formation models.

We also calculated the shape parameter K₁/K₂ (the ratio of shapefinders) for different galaxy populations in superclusters SCl 126 and SCl 111 and showed that the shape parameters of different galaxy populations in supercluster SCl 126 are similar. In supercluster SCl 111, the shape parameter for red galaxies has a smaller value than that for blue galaxies. A larger value of the shape parameter indicates more planar systems. This calculation shows another difference between these superclusters and suggests that the large-scale segregation of red and blue galaxy populations in supercluster SCl 111 is larger than in supercluster SCl 126. We note that Basilakos (2003) used the shape parameter to characterize observed superclusters and to compare observed and simulated superclusters.

The large clumpiness of the red subsystem and the small clumpiness of the blue subsystem show that blue galaxies are distributed more homogeneously than red galaxies, especially in the outskirts of superclusters. This is evidence of a large-scale morphological segregation of galaxies in superclusters, which was already discovered in the first studies of superclusters and later confirmed (Giovanelli et al. 1986; Kaldare et al. 2003; Wolf et al. 2005; Haines et al. 2006, among others) and quantified using the correlation function of galaxies at large scales (Einasto 1991; Guzzo et al. 1991) (the "2-halo" contribution in halo models; see Zehavi et al. 2005; Blake et al. 2008).

Earlier studies of the Minkowski functionals of the whole SDSS survey region (Gott et al. 2008, and references therein) showed the SGW as a very strong density enhancement in the overall distribution of galaxies, which causes a "meatball" shift of the genus of SDSS. Gott et al. (2008) analyzed the topology of mock SDSS samples from the Millennium simulations and concluded that such systems are not found in the simulations. Similar conclusions were reached by Einasto et al. (2006).

This all shows that the simulation of very rich superclusters is complicated and present-day simulations do not yet explain all the features of observed superclusters. Future N-body simulations for very large volumes and with more power at large scales are needed to model structures like the SGW more accurately than the present simulations have done.

The formation and evolution history of superclusters is a complex subject. The best way to follow their real evolution in time is to look for superclusters in deep surveys. In this respect, an especially promising project is the ALHAMBRA Deep survey (Moles et al. 2008) which will provide us with data about (possible) galaxy systems at very high redshifts, which can be searched for and analyzed using morphological methods. Comparing superclusters at different redshifts will clarify many questions about their evolution. A good template for such a comparison is the recently confirmed distant (z = 0.55) supercluster (Tanaka et al. 2009).

The finding that most BRGs lie in groups is consistent with the strong small-scale clustering of the LRGs (Zehavi et al. 2005; Eisenstein et al. 2005a; Blake et al. 2008) that has been interpreted in the framework of the halo occupation model, where more massive halos host a larger number of LRGs (van den Bosch et al. 2008; Zheng et al. 2009; Tinker et al. 2010; Watson et al. 2010). van den Bosch et al. (2008) also showed that more massive halos host more massive satellites, and satellites at smaller halocentric radii and in more massive halos are redder. Zheng et al. (2009) showed that, according to the halo model, most LRGs are the first ranked galaxies in groups, while in our sample about 60% of BRGs are not the first ranked galaxies. This difference may be due to the use of a nearby sample in our study.

The LRGs, selected so that they form almost a volume-limited sample of galaxies up to redshifts of about z = 0.5, are mainly used as tracers of the large-scale structure in the universe at large distances. Our results show that in supercluster SCl 126, rich groups with BRGs are located both in the high-density core and in the outskirts. In supercluster SCl 111, rich groups with a larger number of BRGs can be found in high-density regions, while in lower density regions there are poor groups with a small number of BRGs. Thus, the brightest red galaxies trace the large-scale structure in a somewhat different manner than other galaxies; these differences are larger in lower density regions of superclusters.

We plan to study a larger sample of BRGs and superclusters to further understand the large-scale distribution of BRGs which is important for several purposes. These galaxies (and their more distant cousins, the LRGs) have been mainly used to study baryonic acoustic oscillations (Eisenstein et al. 2005b; Hütsi 2006; Martínez et al. 2009; Kazin et al. 2010). Many observational projects to better measure these oscillations have been proposed and accepted. As an example, a new photometric galaxy redshift survey called Physics of the Accelerating Universe (PAU) will measure positions and redshifts for over 14 million luminous red galaxies over 8000 deg² in the sky, in the redshift range 0.1 < z < 0.9 with a precision that is needed to measure baryon acoustic oscillations (Benítez et al. 2009).

Recently, Ho et al. (2009) investigated the distribution of luminous red galaxies in 47 distant X-ray clusters (0.2 < z < 0.6) and showed that the brightest LRGs are concentrated in the cluster centers, while a significant fraction of LRGs lies farther away from the centers. This agrees with our results about the large peculiar velocities of BRGs.

We found that a large fraction of BRGs are spirals. Images of a few BRGs are shown in Figure 12; images of all BRGs in the SGW are displayed on the Web page http://www.aai.ee/~maret/SGWmorph.html, which shows that our morphological classification of these bright galaxies is reliable. A recent analysis of colors and morphological types of galaxies in the Galaxy Zoo project (Bamford et al. 2009; Masters et al. 2010) showed that, in fact, many spiral galaxies are red, and, moreover, all massive galaxies are red independent of their morphology. Our results are in agreement with these much larger data. Skibba et al. (2009) showed that red spiral galaxies can be found in moderate dense environments and they are often satellite galaxies in the outskirts of halos. Our results show, however, that spiral red galaxies may be the first ranked galaxies in groups. The scatter of colors of red spiral galaxies is large which could be a hint that their colors have changed recently in a group environment while their morphology has not changed yet.

In Einasto et al. (2008) we showed, using the data from the 2dFGRS, that in the supercluster SCl 126 red galaxies with spectra typical of late-type galaxies are located in poor groups at intermediate densities. In the present study, we see that red spiral galaxies also reside in rich groups in high-density cores of superclusters, often being the first ranked galaxies in groups. This means that the processes that change the colors of spirals to red must occur in both high- and low-density regions. The same was concluded by Masters et al. (2010). This, evidently, is a challenge for galaxy evolution scenarios (Kauffmann 1996); we plan to explore this by studying the detailed properties of individual SGW groups.

According to cold dark matter models, cluster (group) halos form hierarchically via the merging of smaller mass halos (Loeb 2008). If at present galaxy groups were virialized (as shown, e.g., by Araya-Melo et al. 2009b) and the galaxies in groups follow the gravitational potential, then we should expect that the first ranked galaxies of groups would lie at the centers of groups and have small peculiar velocities (regardless of the group richness or its velocity dispersion; Ostriker & Tremaine 1975; Merritt 1984; Malumuth 1992). However, our results show that a significant fraction of first ranked galaxies in groups have large peculiar velocities. Our results are consistent with those by Coziol et al. (2009) based on the data about rich clusters of galaxies but do not fit well into the picture where the brightest galaxies in groups are considered the central galaxies (Yang et al. 2005). The large peculiar velocities of the first ranked galaxies in groups indicate that all galaxy groups are not yet virialized. Tovmassian & Plionis (2009) arrived at the same conclusion on the basis of the analysis of poor groups, using, in particular, our group catalog based on the SDSS DR5 (Tago et al. 2008). Niemi et al. (2007) showed that a significant fraction of nearby groups of galaxies are not gravitationally bound systems. This is important because the masses of observed groups are often estimated assuming that the groups are bound.

We showed that in the core of the richest supercluster in the SGW, the supercluster SCl 126, the peculiar velocities of elliptical BRGs are large. One possible reason for that may be that (rich) groups there have formed recently via the merging of poorer groups consisting mostly of red elliptical galaxies. In Figure 12, the left panel shows the image of the first ranked galaxy of the richest group in the core of supercluster SCl 126 which demonstrates clear signs of merging (the two brightest galaxies in this group have a common red halo). This group is the Abell cluster A1750, where available optical and X-ray data show evidence of merging (Donnelly et al. 2001; Belsole et al. 2004; Hwang & Lee 2009), supporting our assumption. In addition, Tinker et al. (2010) in their analysis of the distribution of distant red galaxies in the halo model framework mention that distant red galaxies must have formed their stars before they became satellites in halos. This is possible if these galaxies were members of poorer groups before the groups merged into a richer one. The scatter of colors of elliptical BRGs in the groups in the core of the supercluster SCl 126 is small. This suggests that even if the rich groups in the SCl 126c formed recently, their galaxies obtained their colors when residing in poor groups, before the merger of the groups.

In the core of supercluster SCL 126, the scatter of colors of spiral BRGs is larger and the peculiar velocities of them are smaller than those of elliptical BRGs. A detailed study of groups in different superclusters of the SGW is needed to understand these differences.

In the outskirts of supercluster SCl 126 and in supercluster SCl 111, spiral BRGs have larger peculiar velocities than elliptical BRGs.

Evolution of galaxy groups depends on the environment, being faster in the high-density cores of superclusters (see, e.g., Tempel et al. 2009). This may partly explain the differences between the properties of different superclusters in the SGW, where the highest global densities are found in supercluster SCl 126.

One unexpected result of our study is that in the richest superclusters in the SGW, SCl 126 and SCl 111, groups with an elliptical first ranked BRGs are located more uniformly than groups with a spiral first ranked BRG. Groups with an elliptical BRG as the first ranked galaxy are richer, and rich groups should be concentrated in the denser parts of the supercluster. We plan to study this difference in more detail, using a larger sample of superclusters.

Of course, our results depend on how good our group catalog is. As we mentioned, in order to take into account selection effects, we carefully rescaled the linking parameter with distance. As a result, the maximum projected sizes in the sky and velocity dispersions of our groups in our catalog are similar at all distances (Tago et al. 2008; T10). In the group catalog the group richness decreases rapidly starting from D = 300 h⁻¹ Mpc, due to the use of a flux-limited galaxy sample. We used the same distance limit, to minimize selection effects, which still might affect our results. A good agreement with other results (e.g., Coziol et al. 2009) suggests that our choice of parameters to select groups has been reliable.

To calculate a luminosity density field, we must convert the spatial positions of galaxies, $\mathbf {r}_i$, and their luminosities, L_i, into spatial (luminosity) densities. The standard approach is to use kernel densities (Silverman 1986):

$$\begin{equation} \rho (\mathbf {r}) = \sum _i K\left(\mathbf {r} - \mathbf {r}_i; a\right) L_i, \end{equation} \tag{ A1 }$$

where the sum is over all galaxies and $K\left(\mathbf {r}; a\right)$ is a kernel function of a width a. Good kernels for calculating densities on a spatial grid are generated by box splines B_J. Box splines are local and they are interpolating on a grid:

$$\begin{equation} \sum _i B_J \left(x-i \right) = 1, \end{equation} \tag{ A2 }$$

for any x and a small number of indices that give non-zero values for B_J(x). We use the popular B₃ spline function

$$\begin{eqnarray} B_3(x) &=& \frac{1}{12} (|x-2|^3 - 4|x-1|^3 + 6|x|^3\nonumber\\ && - 4|x+1|^3 + |x+2|^3). \end{eqnarray} \tag{ A3 }$$

We define the (one-dimensional) B₃ box spline kernel K⁽¹⁾_B of the width a as

$$\begin{equation} K_B^{(1)}(x;a,\delta) = B_3(x/a)(\delta / a), \end{equation} \tag{ A4 }$$

where δ is the grid step. This kernel differs from zero only in the interval x ∈ [ − 2a, 2a]; it is close to a Gaussian with σ = 1 in the region x ∈ [ − a, a], so its effective width is 2a (see, e.g., Saar 2009). The kernel preserves the interpolation property exactly for all values of a and δ, where the ratio a/δ is an integer. (This kernel can also be used if this ratio is not an integer, and a ≫ δ; the kernel sums to 1 in this case, too, with a very small error). This means that if we apply this kernel to N points on a one-dimensional grid, the sum of the densities over the grid is exactly N.

The three-dimensional kernel K⁽³⁾_B is given by the direct product of three one-dimensional kernels:

$$\begin{equation} K_B^{(3)}(\mathbf {r};a,\delta) \equiv K_3^{(1)}(x;a,\delta) K_3^{(1)}(y;a,\delta) K_3^{(1)}(z;a,\delta), \end{equation} \tag{ A5 }$$

where $\mathbf {r} \equiv \lbrace x,y,z\rbrace$. Although this is a direct product, it is isotropic to a good degree (Saar 2009).

To use galaxy data, we first applied the k+e-correction to each galaxy (Section 2.2). Also, we had to consider the luminosities of the galaxies that lie outside the observational window of the survey. Assuming that every galaxy is a visible member of a density enhancement (a group or cluster), we estimated the amount of unobserved luminosity and weighed each galaxy accordingly:

$$\begin{equation} L_{w} = W_L(d)\; L_{{\rm obs}}. \end{equation} \tag{ A6 }$$

Here W_L(d) is a distance-dependent weight:

$$\begin{equation} W_L(d) = \frac{\int _0^\infty L\;F(L)dL}{\int _{L_1(d)}^{L_2(d)} L\;F(L) dL}, \end{equation} \tag{ A7 }$$

where F(L) is the luminosity function and L₁(d) and L₂(d) are the luminosity window limits at a distance d.

The luminosity weights for the groups of the SDSS DR7 in the region of the SGW are plotted as a function of the distance from the observer in Figure 18. We see that the mean weight is slightly higher than unity (about 1.4) within the sample limits, 150 ⩾ D_com ⩽ 300 h⁻¹ Mpc. At small distances very bright member galaxies of groups are missing; they lie outside the observational window. When the distance is larger, the weights increase due to the absence of faint galaxies.

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Using the rms velocity σ_v, translated into distance, and the rms projected radius σ_r from the group catalog (T10), we suppress the cluster finger redshift distortions. We divide the radial distances between the group galaxies and the group center by the ratio of the rms sizes of the group finger:

$$\begin{equation} d_{{\rm gal},f} = d_{{\rm group}} + (d_{{\rm gal},i} - d_{{\rm group}})\; \sigma _r / \sigma _v. \end{equation} \tag{ A8 }$$

This removes the smudging effect the fingers have on the density field.

The densities were calculated on a Cartesian grid based on the SDSS η, λ coordinate system, as it allowed the most efficient fit of the galaxy sample cone into a brick. The grid coordinates were found as follows:

$$\begin{eqnarray} x&=&{-}d_{{\rm gal}} \sin \lambda,\nonumber \\ y&=&d_{{\rm gal}} \cos \lambda \cos \eta,\nonumber\\ z&=&d_{{\rm gal}} \cos \lambda \sin \eta. \end{eqnarray} \tag{ A9 }$$

We used a 1 h⁻¹ Mpc step grid and chose the kernel width a = 8 h⁻¹ Mpc. As we noted above, this means that the kernel differs from zero within the radius 16 h⁻¹ Mpc, but significantly so only inside the 8 h⁻¹ Mpc radius.

Before extracting superclusters we apply the DR7 mask constructed by P. Arnalte-Mur (Martínez et al. 2009) to the density field and convert densities into units of mean density. The mean density is defined as the average over all pixel values inside the mask. The mask is designed to follow the edges of the survey and the galaxy distribution inside the mask is assumed to be homogeneous.

We create a set of density contours by choosing a relative density threshold D and define connected volumes above this threshold as superclusters. We assemble galaxy superclusters by collecting all the galaxies that are located in the supercluster volume.

Different thresholds yield different supercluster catalogs. A supercluster catalog combines both the density field and galaxy information. We find the supercluster location, volume, diameter, total luminosity, and the numbers of galaxies and groups. For a supercluster ID we use the coordinates of a "marker galaxy" that we choose to be a bright galaxy nearest to the highest density peak in the supercluster volume (often this is the first ranked galaxy in the most luminous group of a supercluster).