Reading this conference talk and the original paper by de Lapparent et al. (1986) I was rather surprised. John Huchra showed me his model during my first visit to Harvard, based on the First CfA Redshift Survey, where the filamentary distribution of galaxies was clearly seen. Our analysis, both qualitative and quantitative, indicated the filamentary character of the distribution. In Harvard I discussed the filaments using our latest data, as shown in Fig. 7.20, where it was clearly seen that filaments are very thin and are not cross-sections through larger shells.
The de Lapparent et al. (1986) paper and Geller (1987) talk had no quantitative tests. However, after the popular description of the structure by Geller & Huchra (1989) the term “walls” was widely used to describe the distribution of galaxies. When our “Nature” paper (Einasto et al., 1997a) was reported in “Physics Today”, the editors also used the term “walls” instead of “superclusters”. The editors sent the preliminary version of their report to me, and I explained that the term “superclusters” had already been in use for many years, starting from de Vaucouleurs (1953) and Abell (1958), and there is no need to replace it with “walls”, taking into account the filamentary character of the cosmic web. Walls have a completely different meaning as we see below.
New high-resolution simulations indicate that tenuous sheets really surround voids, but the density of these sheets is too low for galaxy formation — they consist of pre-galactic matter, see Einasto et al. (1986a) and the review by van de Weygaert (2002).
Filaments differ not only by their richness, but also by their topological behaviour. Most rich filaments form sections of the continuous web network, i.e. they are connected at both ends with the network. Examples are filaments which connect neighbouring superclusters. The filament found by Tago et al. (1986) is different. One of its tips joins the filament with the Local supercluster, but the other end just fades inside the Local supervoid, and has no connection with the Hercules supercluster on the other side of the void.
Several teams performed in last few years a more detailed study of the structure of filaments, including their objective definition. Aragon-Calvo et al. (2006); Aragón-Calvo et al. (2007b,a) used the Multiscale Morphology Filter (MMF) to automatically segment cosmic structure into basic components: clusters, filaments, and walls. The density field is calculated using the Delaunay Tessellation Field Estimator. Thereafter a series of morphology filters are applied that identify particular kinds of structure in the data. The method is referred to as the Multiscale Morphology Filter (MMF). To check the method it has been applied to a simple Voronoi model. Structural elements are defined as field if particles are located in the interior of Voronoi cells (these particles can be called void particles), as wall if particles lie within and around the Voronoi walls, as filament if particles are within and around the Voronoi edges, and as blobs if particles are within and around the Voronoi vertices. Aragón-Calvo et al. (2007b) applied the MMF technique to find the spin alignment of dark matter halos in filaments and walls.
Stoica et al. (2010) suggested using a marked point process to find filaments. This method does not use the density estimation step. Filaments are defined as cylinders filled with particles, and located between two clusters. Tempel et al. (2013) applied this filament search method to study the alignment of spiral and elliptical galaxies in filaments.
7.3.4 Walls
The term walls has been used in three different meanings: as walls in the cellular network, as a complex of clusters and superclusters between supervoids, and as rich superclusters with their neighbouring filaments.
In N-body and Voronoi models walls had the meaning of surfaces between cell interiors surrounded by cell edges. As shown by Einasto et al. (1986a) on the basis of a ΛCDM model, such walls exist only in unbiased models where all dark matter particles are present. The density of these walls is so low that here no galaxy formation takes place, thus in real galaxy samples there are no walls in this meaning.
In the early 1980’s we used the term ‘wall’ in another context, as a complex of rich clusters or superclusters which form a flattened disk between large low-density regions (supervoids). My own experience with the existence of such complexes goes back to the early 1980’s. When working in ESO as a visiting scientist in 19811 used the plotting facilities of ESO to make plots of galaxies and clusters in sequence at various coordinate systems to understand better the spatial distribution of these objects. One product of this work was the preparation of a movie together with Dick Miller, discussed elsewhere.
First I used equatorial coordinates to plot galaxies and clusters. But soon I noticed that for relatively near objects it is better to use supergalactic coordinates, introduced by Gerard de Vaucouleurs. I plotted near rich clusters in two distance intervals, 37.5 ≤ r < 75 h−1 Mpc, and 75 ≤ r < 125 h−1 Mpc. To my surprise I found that in the first plot all rich clusters form a relatively narrow belt close to the supergalactic plane. In the plot of more distant clusters the distribution is uniform.
Then I remembered our first results of the distribution of galaxies and clusters by Jõeveer et al. (1977); Jõeveer & Einasto (1978), which showed the presence of a large void between the Virgo, the Coma and the Hercules superclusters. Thus I assumed that the belt of near clusters is a complex of clusters and superclusters which form one sidewall of this large void. However, this conclusion was only tentative. Since for Southern clusters I did not have data, the picture was not full.
At this time George Abell with his collaborator Harold Corwin had started to inspect the Southern sky to extend the Abell cluster catalogue to the South. So I wrote a letter to Harold asking if he can find in the Southern sky all nearby rich clusters, up to approximately redshift 12,500 km s−1. Soon I got the list of nearby rich clusters of galaxies. Thereafter I wrote letters to John Huchra and Massimo Tarenghi, who were engaged in redshift determinations of clusters of galaxies. John sent me redshifts from his redshift compilation, and Massimo was able to measure a number of new redshifts for these clusters.
To my joy the new Southern data confirmed my expectations — all close Southern clusters formed a continuation of the Northern part of the belt close to the supergalactic plane, and more distant clusters were evenly distributed. Time was pressing; it was spring 1982, and soon the next IAU General Assembly would took place in Athens, as well as a Symposium in Grete. So I prepared quickly short reports of our fresh results (Einasto & Miller, 1983; Einasto et al., 1983a). These papers contain Figures which show the location of clusters and superclusters, and show the presence of a disk, which forms a wall between two low-density regions, see Fig. 7.24. A figure with the distribution of galaxies and rich clusters in supergalactic z-coordinate was included into the review paper by Zeldovich et al. (1982) (Fig. 7.21). The main conclusions of the study were formulated as follows (Einasto & Miller, 1983; Einasto et al., 1983a):
Fig. 7.21 Distribution of galaxies and rich clusters of galaxies in supergalactic z coordinate. Solid and dotted lines are for galaxy distributions uncorrected and corrected for incompleteness, respectively; the dashed line is for rich cluster distribution (Zeldovich et al., 1982).
(1) The distribution of rich clusters suggests the presence of two local cells, the Northern Local Cell and the Southern Local Cell, each about 100 h−1 Mpc in diameter;
(2) The Virgo, Coma, Perseus–Pisces, Lynx–Ursa Major, Hydra-Centaurus and Pavo–Corona Australes superclusters form a disk between the Northern and the Southern Local Cells; its diameter is about 125 h−1 Mpc, and thickness about 25 h−1 Mpc;
(3) Superclusters located at distances 75…125 h−1 Mpc (Hercules, Ursa Majoris–Leo, and several Southern superclusters) form side-walls of Local Cells. The Hercules and Ursa Majoris–Leo superclusters form a wall between the Northern Local Cell and the Bootes Cell; the Perseus–Pisces supercluster is located in a wall between the Northern Local Cell and a cell beyond that supercluster, seen in Figs. 5.7, 5.8 and 5.9.
The concentration of nearby Abell clusters close t
o the Supergalactic plane was confirmed by Brent Tully & Fisher (1987) and Peter Shaver (1991), see the discussion by Jim Peebles (2012). Brent also discovered independently local voids, the Northern Local Void is often called Tully’s Void. Our earlier papers on local voids and the concentration of Abell clusters between these voids were published only in proceedings of IAU (Einasto & Miller, 1983; Einasto et al., 1983a). Evidently this was not enough to be noticed by the community.
Our new study confirmed our earlier results that the principal structural elements in cell walls are galaxy and cluster chains. Within superclusters chains are rich and consist mainly of clusters. In cell interiors there are several chains, consisting of galaxies and poor clusters. Galaxy chains connect neighbouring superclusters into a single lattice, and there are no large sheets of galaxies, uniformly filled with galaxies. Small sheets surround some clusters (Virgo).
In the 1990’s more data on Abell clusters became available, and Einasto et al. (1994b, 1997b) compiled catalogues of superclusters. Maret found that very rich superclusters are concentrated to a Dominant Supercluster Plane which is situated at a right angle with respect to the plane of the Local Supercluster and adjacent nearby superclusters. Einasto et al. (1997a) found that rich clusters of galaxies, located in rich superclusters, form a quasi-regular lattice. Part of this lattice coincides with the Dominant Supercluster Plane, see Fig. 7.29 below.
de Lapparent et al. (1986) applied the term “Great Wall” to designate the Coma supercluster and its extension towards the Hercules supercluster, a rich filament of galaxies, as seen in our absolute magnitude limited distribution of galaxies across the Coma and Hercules superclusters, shown in Fig. 7.7 (Einasto et al., 1999a). Praton et al. (1997) investigated the “wall” phenomenon in detail using numerical simulations. The authors showed that structures perpendicular to the line of sight are enhanced in redshift space, due to the Finger-of-God effect and distortions caused by infall. The effect is enhanced in apparent-magnitude-limited samples of galaxies near the maximum of the sensitivity of the sample. The authors call this the bull’s-eye effect.
The Sloan Great Wall (Vogeley et al., 2004) is a complex of several very rich superclusters. Its structure has been recently investigated by Einasto et al. (2010). In front and beyond the Sloan Great Wall there are large under-dense regions, as shown by Liivamägi et al. (2012). The richest supercluster in this complex is SCL126 from the list by Einasto et al. (2001). In redshift-slices of apparent-magnitude-limited samples the Sloan Great Wall is enhanced due to the bull’s-eye effect. If corrected for incompleteness, the region remains a very rich one (see Fig. 7.14), but not so extreme as in apparent-magnitude-limited slices.
7.3.5 Superclusters
The largest elements of the cosmic web are superclusters of galaxies. Their properties can be divided into physical and general properties, similarly to properties of smaller elements of the web. Physical properties characterise the internal structure and physical nature of supercluster populations. The most important general properties of superclusters are their luminosity and richness.
Einasto et al. (2006) used two independent parameters to quantitatively characterise the richness of superclusters: the multiplicity and the total luminosity. These parameters were calculated for the 2dF and SDSS Data Release 4 superclusters, and for comparison superclusters found for the Millennium Simulation galaxy samples by Croton et al. (2006). The colour systems of our various samples are different: r in SDSS and Mill.A8, bj in 2dFGRS and g in Mill.F8.
To characterise the multiplicity of superclusters we used density field (DF) clusters, defined as high-density peaks of the density field, smoothed on a scale of 8 h−1 Mpc. We defined the multiplicity of a supercluster by the number of DF- clusters in it. The spatial density of DF-clusters in our observational samples is 62 per million cubic h−1 Mpc, about twice the spatial density of Abell clusters, 25 per million cubic h−1 Mpc (Einasto et al., 1997b). Thus the expected multiplicity of superclusters is about two times higher than the multiplicity of Abell superclusters of Einasto et al. (2001).
The other integral parameter of a supercluster is its total luminosity, determined by summing luminosities of all galaxies and groups of galaxies inside the threshold isodensity contour, which was used in the definition of superclusters. We are interested in the fraction of rich and very rich superclusters relative to poor superclusters. To avoid complications due to the use of different color systems and mean luminosities, we defined relative luminosities as the luminosity in terms of the mean luminosity of poor superclusters, i.e. superclusters that contain only one DF-cluster or Abell-cluster, and hence are classified as richness class 1.
Fig. 7.22 shows the relative luminosity functions and multiplicity functions for the observational and model samples. The most striking feature of the figure is the demonstration of the presence of numerous very luminous superclusters in observational samples, and the absence of such systems in simulated samples. This difference between real and simulated supercluster richness is well seen using both richness criteria, the multiplicity and luminosity functions.
Fig. 7.22 A comparison of relative luminosity functions and multiplicity functions of observational and model supercluster samples in the left and right panels, respectively. In the left panel we show relative luminosity functions for observational samples SDSS DR4, 2dF, and the combined sample Obs; in the right panel we use the combined observational sample Obs and the Abell supercluster sample (here multiplicity is defined by the number of Abell clusters; isolated Abell clusters are considered as richness class 1 superclusters) (Einasto et al., 2006).
When comparing models with observations we used the sample Mill.F8 by Einasto et al. (2007b), which is formed using similar selection criteria as observational samples. The most luminous simulated superclusters of the Mill.F8 sample have a relative luminosity of about 15 in terms of the mean luminosity of richness class 1 superclusters. The most luminous superclusters of real samples have a relative luminosity of about 100, i.e. they are about 6 times more luminous. The richest model superclusters of the sample Mill.F8 have a multiplicity of 10, whereas the richest real superclusters have DF cluster multiplicities over 70. The number of Abell clusters in the richest Abell supercluster is 34 (Einasto et al., 2001). The differences between real and simulated samples are observed not only in the region of most luminous superclusters: over the whole richness scale the number of DF- clusters in simulated samples is smaller than in samples of real superclusters, and the relative luminosity function lies lower.
The absence of very rich superclusters in simulated samples can be explained either as a real deviation of ΛCDM models from reality or by too small volume of the model. This problem has been studied by many astronomers; the most recent and perhaps most important contribution came from the analysis by Park et al. (2012).
Changbom Park with collaborators used the largest so far simulation of a ΛCDM universe, the so-called Horizon Run 2 (HR2) (Kim et al., 2011). This simulation was made for a box of side length 10 Gpc, it contains 60003 particles and allows to extract dark matter subhalos of the minimum mass 5.2 × 1012M. The mean subhalo separation is 12.5 h−1 Mpc, equal to that of the SDSS volume limited galaxy sample used by the authors. Park et al. (2012) made 200 SDSS- like surveys of simulated galaxies for the present epoch on the basis of HR2, and analysed the properties of these mock survey samples in exactly the same way as the observational data. Out of the 200 mock samples, 137 samples contain rich supercluster complexes similar to or richer than the Sloan Great Wall (SGW). The authors conclude that the SGW type structures can be easily found in surveys like the SDSS in the ΛCDM universe. This analysis shows that the absence of very rich systems of galaxies in previous simulations was the result of a too small size of the simulation.
To study the internal structure of superclusters a number of methods have been used; an overview of such methods is given by Martίnez & Saar (2002). In particular, Minkowski functionals can be used to describe overall structural prope
rties of superclusters; they describe the volume, the surface, the mean curvature, and the integrated Gaussian curvature of the system. Einasto et al. (2007d) derived these parameters for the richest systems of the 2dF supercluster catalogue by Einasto et al. (2007a,b). Sahni et al. (1998) and Shandarin et al. (2004) introduced shapefinders, a set of combinations of Minkowski functionals: H1 (thickness), H2 (width), and H3 (length). These quantities have dimensions of length, and are normalized to give Hi = R for a sphere of radius R. Additionally Sahni et al. (1998) defined their combinations — shapefinders K1 (planarity) and K2 (filamentarity). Einasto et al. (2007d) found that the information about the shapes of superclusters can be best described by their morphological signature, i.e. the path in the shapefinder K1 − K2 plane for varying mass fraction, calculated for various threshold densities to derive the supercluster.
The results of these calculations of Minkowski functionals and shapefinders are presented in Fig. 7.23. For morphological study we used volume-limited galaxy samples; this makes our results insensitive to selection corrections. As the argument labeling the isodensity surfaces, we chose the mass fraction mf — the ratio of the mass in regions with density 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 mf =0 corresponds to the whole supercluster, and mf = 1 to its highest density peak. At small mass fractions the isodensity surface includes the whole supercluster. As we move to higher mass fractions, the iso-density surfaces include only higher density parts of superclusters, and their volumes and areas get smaller. At very high mass fractions only the highest density clumps in superclusters give their contribution to the supercluster. Individual high density regions in a supercluster, which at low mass fraction are joined together into one system, begin to separate from each other, and the value of the fourth MF (V3) increases. At a certain density contrast (mass fraction) V3 has a maximum showing the largest number of isolated clumps in a given supercluster at the spatial resolution determined by the smoothing kernel. At still higher density contrasts only the highest density peaks contribute to the supercluster.
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