by Jaan Einasto
where α is the exponent at low luminosities (L/L*) 1, δ is the exponent at high luminosities (L/L*) 1, γ is a parameter that determines the speed of transition between the two power laws, and L* is the characteristic luminosity of the transition, similar to the characteristic luminosity of the Schechter function. The double power-law luminosity function was in general use in the 1970’s (Abell, 1977), however with a sharp transition from one to the other power law.
To check the correctness of our weighting procedure in the estimation of total luminosities we again shifted nearby groups to larger distances. Now we selected two subsamples of clusters at different true distances from the observer. The first subsample was chosen in the nearby region with distances 100 ≤ d < 200 h−1 Mpc, and the other sample in the distance interval 200 ≤ d < 300 h−1 Mpc. In both cases the number of visible galaxies in groups was chosen, Ngal ≥ 10. Next the clusters were shifted to progressively larger distances, galaxy apparent magnitudes were calculated, and galaxies inside the visibility window selected. The number of galaxies inside the visibility window for shifted clusters decreases; the mean number of galaxies in shifted clusters is shown in the upper panel of Fig. 7.12. We see that the mean number decreases almost linearly in the log N − d diagram. At the far side of our survey the mean number of remaining galaxies in clusters is between 1 and 2; some groups disappear, only the main galaxy remains.
The expected total luminosity of clusters, calculated on the basis of galaxies inside the visibility window, and using the procedure outlined above, is shown on the lower panel of Fig. 7.12. We see that the mean values of restored total luminosity of clusters are almost identical with the true luminosity at the initial distance. The restored luminosities of individual clusters have a scatter that increases with the distance.
A similar approach was used in the compilation of the group catalogues for the Sloan Survey galaxies. For the Data Release 5 of the Sloan Survey the group catalogue was published by Tago et al. (2008), and for the Data Release 8 by Tempel et al. (2012c). Analysis of the dependence of group properties on the mean distance from us shows that main properties (the mean size and expected total luminosity) do not depend on the distance, i.e. our group catalogues are homogeneous. Of course, the mean number of galaxies in groups decreases with distance, since only brighter group members remain in the observational window of apparent magnitudes.
Fig. 7.12 Upper panel: the mean number of galaxies in shifted clusters as a function of distance for the 2dF. Lower panel: restored mean total luminosities of shifted clusters. Solid line shows results for clusters located initially at distances 100 ≤ d < 200 h —1 Mpc, dashed line is for clusters of initial distance 200 ≤ d < 300 h —1 Mpc (Tempel et al., 2009).
7.2.3 Catalogues of superclusters
To investigate large-scale properties of the cosmic web more distant markers of the web are needed. The best markers of the web are rich clusters of galaxies. In earlier studies superclusters were defined using clusters of galaxies. George Abell (1958) in his catalogue found evidence for second-order clustering, and called these systems of clusters as clusters of clusters of galaxies. Soon the term “superclusters” was established, but as before, only systems consisting of several clusters were considered as superclusters. This definition does not take into account the property of the relatively rich nearby Virgo supercluster, where there is only one rich cluster surrounded by galaxy sheets and filaments.
In our first study of superclusters we compiled a list of ten probable superclusters (Jõeveer et al., 1978), using mostly Abell clusters as supercluster tracers. The Abell (1958) catalogue of rich clusters covers the Northern sky up to declination —27 degrees, and contains 2,712 clusters. When I visited George Abell in 1982, we discussed with him and his collaborator Harold Corwin the extension of the catalogue to the Southern sky. The main purpose of the catalogue is to find high- density knots in the cosmic web. Here the richness limit 30 galaxies in the cluster area is not so important, because less rich clusters also mark knots of the web. For this reason I suggested including all clusters found during the search into the catalogue, the poorer clusters being added as a Supplement to the main catalogue. So the Southern Survey adds 1,361 rich clusters to the main catalogue, bringing the total number of rich clusters to 4,076, and the Supplement has 1,174 Southern clusters which were not rich enough to be included into the main catalogue (Abell etal., 1989).
To use theAbell catalogue for the study of the cosmic web, distances to clusters are needed. For most nearby rich clusters redshifts were found by special observing programs. However, the number of clusters with known distances was still too low. Thus Erik Tago in collaboration with Heinz Andernach started to collect redshifts of galaxies belonging to Abell clusters, using all available galaxy redshift sources.
In our early studies of the distribution of galaxies and clusters (Jõeveer & Einasto, 1978; Jõeveer et al., 1978; Einasto et al., 1980a) we noticed another problem in the definition of superclusters — superclusters are connected by filaments of galaxies, groups and clusters, forming a continuous cosmic web. Thus it was not evident where a particular supercluster ends and the another one begins.
Following Zeldovich’s ideas we expected that the evolution of density perturbations starts from pancaking to form flat sheets (Zeldovich, 1970, 1978). According to this scenario filaments form by the flow of pre-galactic matter towards the crossing of sheets, and clusters by the flow of matter towards the corners of the cellular network. Thus we considered the possibility of identifying one cell wall with surrounding cluster chains as a supercluster (Einasto et al., 1980a). When more redshift data arrived we noticed that cell walls do not form continuous surfaces (Zeldovich et al., 1982; Einasto et al., 1983b; Tago et al., 1984), thus cell walls cannot be used to define superclusters. We concluded that it is reasonable to consider superclusters as high-density regions of the cosmic web, using in their identification both clusters and galaxies.
Fig. 7.13 Left: the length of the largest supercluster, Dmax in h−1 Mpc, as a function of the neighborhood radius r in dimensionless Poisson units. Solid line shows all clusters, dotted line shows Abell clusters of richness R = 0, and dashed line shows clusters of richness R ≥ 1. Heavy solid line shows the Rosat Bright X-ray Survey sample. Right: multiplicity functions of systems of Abell clusters as a function of the neighbourhood radius R. The solid line shows the fraction of isolated clusters, the short-dashed line shows the fraction of clusters in superclusters of multiplicity 2 to 31, and the dashed line shows the fraction of clusters in merged superclusters with at least 32 member clusters (Einasto et al., 2001).
In the early 1990’s the Tago and Andernach collection of Abell cluster redshifts was complete enough to use it to compile new catalogues of superclusters. Our first supercluster catalogue based on all near Abell clusters was published by Maret Einasto et al. (1994b). Here again the formal definition of a supercluster was essential. Maret constructed a number of supercluster catalogues for varying neighbourhood radii, calculated sizes of all superclusters, and found the length of the largest supercluster, see the left panel of Fig. 7.13. Further she calculated for various neighbourhood radii multiplicity functions of systems of various richness, separately for isolated clusters, for superclusters of multiplicity 2… 32, and for merged superclusters with multiplicity at least 32 clusters, see the right panel of Fig. 7.13. If we understand superclusters as the largest non-percolating high- density regions in the cosmic web, then it is reasonable to use for their selection the neighbourhood radius just below the radius where superclusters start to merge. Using data shown in Fig. 7.13 Maret chose R = 24 h−1 Mpc to select superclusters of Abell clusters.
The redshift collection by Tago and Andernach was improving, so it was possible to improve also the supercluster catalogues (Einasto et al., 1997b, 2001). The last catalogue contains superclusters up to redshift z = 0.13 and contains in addition to Abell clusters also X-ray clusters, using the Rosat Bright X-ray Survey sample. In this redshift inte
rval the mean spatial density of Abell clusters is practically distance independent, thus the supercluster sample is volume limited.
In the early 2000’s data on deep redshift surveys became available, which covered large areas of the sky. First of such surveys was the Las Campanas Redshift Survey; the Two-degree Field (2dF) Survey and the Sloan Digital Sky Survey (SDSS) followed. We used all these surveys to compile catalogues of groups of galaxies and superclusters.
In earlier supercluster catalogues we used the FoF method, applied to clusters. Now data on individual galaxies were known, and a more effective method was needed to make use of all the data. We found that for supercluster searches the luminosity density field of galaxies can be used. But there is a problem here: galaxies have random velocities in groups and clusters, thus, in order to useredshifts as distance indicators, peculiar velocities must be eliminated. To do this, catalogues of groups of galaxies are needed, which contain information on peculiar velocities.
The key element in our scheme is the restoration of the expected total luminosity of superclusters as accurately as possible. This goal can be achieved using the weights for galaxies and groups in the calculation of the density field; for details see the previous section.
The final step in the selection of superclusters is the proper choice of the threshold density to separate high and low-density galaxy systems. We compiled supercluster catalogues in a wide range of threshold densities. For each threshold density we found a list of superclusters, calculated the number of superclusters, and the maximal diameter of the largest system. At a low threshold density the largest systems span the whole observational volume. At a very high threshold density only the densest parts of the density field are considered as superclusters. The optimal choice is in the medium range of threshold density, which yields about 100-150 h−1 Mpc for the size of largest superclusters, as known from the study of nearby rich superclusters.
The first supercluster catalogue based on the luminosity density field was compiled from the Las Campanas Redshift Survey. This survey covered six 1.5 × 80 degree slices: three slices located in the Northern Galactic cap, and three slices in the Southern Galactic cap. The slices are so thin that only 2-dimensional luminosity density fields can be found. We calculated high-resolution and low- resolution density fields using Gaussian filters with smoothing scale 0.8 h−1 Mpc and 10 h−1 Mpc, respectively (Einasto et al., 2003a). A high-resolution field was used to find density field clusters, which are some equivalent of Abell clusters. A low-resolution field was used to find superclusters. The density threshold to define superclusters was chosen so that the largest superclusters do not form percolating systems.
Next we used the 2dF Galaxy Redshift Survey data to compile catalogues of superclusters for the Northern and Southern regions of the 2dFGRS. Altogether 543 superclusters at redshifts 0.009 ≤ z ≤ 0.2 were found (Einasto et al., 2007b). We analysed methods of compiling supercluster catalogues using complete flux-limited galaxy catalogues. Results of the Millennium Simulation (Springel et al., 2005) were used to investigate possible selection effects and errors. We found that the most effective method is the density field method using smoothing with an Epanech- nikov kernel of radius 8 h−1 Mpc. Superclusters were defined as galaxy systems larger than groups and clusters which have a certain minimal overdensity of the smoothed luminosity density field, but are still non-percolating. Superclusters form intermediate-scale galaxy systems between groups and filaments, and the whole cosmic web. Similar procedures were applied in the preparation of the supercluster catalogue for the SDSS Data Release 4 (Einasto et al., 2006), and a preliminary catalogue for SDSS Data Release 7.
The group catalogue for SDSS DR8 was compiled by Tempel et al. (2012c), and the supercluster catalogue for SDSS DR7 by Juhan Liivamägi et al. (2012). Here a number of improvements in the calculation of the density field and the supercluster search were made. In the calculation of the density field instead of the Epanechnikov kernel the B3 kernel was used (see Martinez & Saar (2002)). The SDSS density field of the main galaxy sample was calculated with the kernel size 8 h−1 Mpc, the field for the Luminous Red Giant (LRG) sample with the kernel of size 16 h−1 Mpc. Superclusters were found for a series of density thresholds to find the best way to define superclusters. It was found that it is not possible to use a fixed threshold for the whole sample, since superclusters in different regions of space have different mean densities.
To see the problem we plot in Fig. 7.14 the high-resolution luminosity density field for a spherical layer at a distance 240 h−1 Mpc from us. The thickness of the shell is 10 h−1 Mpc. In the plotting we used SDSS coordinates η, λ. Because we use a spherical shell at a fixed distance, distance dependent selection effects are excluded. We see that in the lower part of the Figure a huge complex of several superclusters is located — this complex is called the Sloan Great Wall; actually it consists of several very rich superclusters. Here the overall luminosity density is higher, and galaxy and cluster chains joining superclusters to the cosmic web have much higher luminosities than in other regions.
As a solution of the problem Juhan prepared two sets of supercluster catalogues, one set with a number of fixed threshold densities, and the second set with adaptive local threshold densities chosen individually for each supercluster. The idea is to follow the growth of individual superclusters from a compact volume around its centre, by lowering the density level and observing the supercluster mergers. By defining a supercluster as the volume within the density contour until the first merger, we can break the large-scale structure into a collection of compact components. Every component (supercluster) then has its own limiting density level Da.
Fig. 7.14 The luminosity density field of the SDSS in a spherical shell of 10 h−1 Mpc thickness at a distance of 240 h−1 Mpc. To enhance the faint filaments in voids between the superclusters, the density scale is logarithmic, in units of the mean luminosity density for the whole DR7. The rich complex in the lower area of the picture is part of the Sloan Great Wall; it consists of two very rich superclusters, SCL 111 and SCL 126 in the list by Maret Einasto et al. (2001) (Suhhonenko et al., 2011).
Figure 7.15 gives an example of how supercluster diameters and luminosities change during mergers when lowering the density level and, while still growing, remain relatively stable in between. Supercluster SCl 24 in this Figure is a part of the Sloan Great Wall. At densities D < 4.7 it actually includes all of the SGW superclusters (Einasto et al., 2010).
The Liivamagi et al. (2012) supercluster catalogues contain 982 and 1313 superclusters in the fixed and adaptive threshold catalogues of the main galaxy sample, respectively. The LRG supercluster catalogues contain 3761 and 2701 superclusters in the fixed and adaptive catalogues. Analysis of supercluster catalogues shows that the main supercluster properties do not depend on the distance from the observer, i.e. selection effects have been corrected properly. The authors find that superclusters are well-defined systems, and the properties of the superclusters of the main and LRG samples are similar. The Millennium simulation galaxy catalogue provides similar superclusters to those observed.
Fig. 7.15 An example of the dependence of the supercluster diameter and the total luminosity on the density level D. Vertical thin dashed lines show splitting/merger events, and the thick dashed line shows the adaptive density threshold. The minimum diameter limit 16 h−1 Mpc is also shown. The lines begin at the level where the object separates from the larger structure (Liivamagi et al., 2012).
7.3 Elements of the cosmic web
7.3.1 Galaxies in different environments
Already in the early stages of the study of galactic populations I found that there exists a close correlation between physical and dynamical properties of populations: colour and mass–luminosity relation (Einasto & Kaasik, 1973), and a relation between dynamic properties and ages of galaxies (Einasto, 1973, 1974b). These relations suggested the presence of fundamental relations between various properties of galaxies. About ten years earlier we der
ived the fundamental plane of stars (Tiit & Einasto, 1964), and made preparations to apply this method to galaxies by compiling a bibliography of galaxies (Brosche et al., 1974). However, in the early 1970’s we had so many programs running simultaneously that this study was postponed. A few years later the problem was investigated in much more detail by Faber & Jackson (1976). Sandra used her own determinations of velocity dispersions in elliptical galaxies, and found the Faber–Jackson fundamental plane for elliptical galaxies.
In the further study of galaxies and their environment we found that giant galaxies are surrounded by dark matter halos and dwarf satellite galaxies, and that satellite galaxies are morphologically segregated (Einasto et al., 1974c, 1975b). Dark halos form the largest and the most massive populations of the central giant galaxies. Satellite galaxies move inside the halo of the maternal galaxy. As noted earlier, to avoid confusion (a dwarf galaxy inside a giant galaxy), we called such systems hypergalaxies (Einasto et al., 1974a). Hypergalaxies are actually small groups of galaxies with one concentration center. In terms of DM simulations hypergalaxies are equivalent to halos, and dwarf satellite galaxies are equivalent to subhalos. Einasto et al. (1974a) assumed that hypergalaxies are the primary sites of galaxy formation, similar to stars which form in stellar associations and star- forming regions (Ambartsumian, 1958). Further Einasto et al. (1974a) assumed that large multicomponent groups and clusters are formed by merging of hypergalaxies.
The morphological segregation of satellite galaxies is actually an early hint to the presence of the density–morphology relation, discussed in detail by Dressler (1980). Einasto & Einasto (1987) showed that the density–morphology relation is valid not only in clusters, but also in the environment of clusters up to distance ∼10−15 h−1 Mpc.