by Jaan Einasto
Fig. 7.23 The Minkowski functional V3 (the Euler characteristic) for the observed superclusters according to list by Einasto et al. (2001) (Einasto et al., 2007d).
Figure 7.23 shows that at the mass fraction value of about 0.2, the value of V3 for the supercluster SCL9 begins to increase and reaches a maximum value at the mass fraction mf ≈ 0.7. Then the value of V3 begins to decrease. This indicates that the overall morphology of the supercluster SCL9 is clumpy; this supercluster consists of a large number of clumps or cores connected by relatively thin filaments, in which the density of galaxies is too low to contribute to the supercluster, starting at certain mass fraction values. The maximum value of the fourth Minkowski functional V3 shows that the supercluster SCL9 has the largest number of isolated clumps in it.
The second richest and largest supercluster among the observed superclusters in the Sloan region of sky is the supercluster SCL126, a member of the Sloan Great Wall. The V3 curve for the supercluster SCL126 shows several peaks at a high mass fraction, mf > 0.95. This indicates the presence of a very high density core region with several individual clumps in it — this is the main core region of the supercluster with several Abell clusters, which are also X-ray clusters.
Einasto et al. (2011c,d,e, 2012) investigated morphological properties of Sloan Digital Sky Survey rich superclusters, using the fourth Minkowski functional V3, the morphological signature (the curve in the shapefinders K1 − K2 plane) and the shape parameter (the ratio of the shapefinders K1/K2). The superclusters in our samples form three chains of superclusters; one of them is the Sloan Great Wall. Most superclusters have filament-like overall shapes. Superclusters can be divided into two sets: more elongated superclusters are more luminous, richer, have larger diameters and a more complex fine structure than less elongated superclusters. The fine structure of superclusters can be divided into four main morphological types: spiders, multispiders, filaments, and multibranching filaments.
The analysis demonstrates that almost all superclusters in our sample of rich superclusters are elongated; they have larger filamentarities K2 than planarities K1. The two most elongated superclusters are the richest and the most luminous. Almost all superclusters studied are elongated and have filamentarities that are larger than their planarities. More elongated superclusters are also more luminous, have larger diameters and contain a larger number of rich clusters. The values of the fourth Minkowski functional V3 show that they also have a more complicated inner morphology than less elongated superclusters.
7.3.6 Voids and supervoids
Already early studies of the distribution of galaxies and clusters showed that voids defined by objects of different type and luminosity have different sizes, see Figs. 5.5 and 5.4. The largest voids are determined by clusters and superclusters of galaxies, as seen from the distribution of near Abell clusters and superclusters in Fig. 7.24. Lindner et al. (1995) used the term “supervoids” for voids defined by superclusters of galaxies. As discussed above, supervoids are not empty; they are crossed by chains of galaxies and poor clusters or groups, as seen in Figs. 5.5 and 5.4.
Fig. 7.25 shows void diameters as a function of the size L of galaxy and Zwicky near cluster samples (Einasto et al., 1989). All galaxy samples are volume (absolute magnitude) limited; the limit corresponds to the apparent magnitude limit of the sample, m = 14.5, of the CfA first redshift survey, at the far side of the sample. The Figure shows an almost linear growth of void diameters with distance (limiting luminosity) in log–log scale.
Fig. 7.24 The distribution of Abell clusters in rectangular supergalactic coordinates in the interval X = — 75… 50 h−1 Mpc. The zone of avoidance is shown by dashed lines, outer contours of Local Voids by circles. Supercluster names are marked. H–R, H–C and A–C are for the Horologium–Reticulum, Hydra–Centaurus and Aquarius–Capricornus superclusters, respectively (Einasto et al., 1994b).
The right panel of the same Figure shows correlation lengths of the same samples as a function of the sample size (and the absolute magnitude limit). Here also in a log-log diagram the correlation length increases with sample size almost linearly. This is the result of biasing, with increasing luminosity limit the correlation length increases, as discussed above.
Another method to characterise void sizes is to use the void probability function VPF, introduced by White (1979). A number of authors have used this function. Einasto et al. (1991) derived the void probability function for a number of observational and model samples of various sizes, as used earlier to derive the distribution of void diameters (Einasto et al., 1989). The results show that both methods describe the void properties well. For the CfA2 sample the void probability function was found by Vogeley et al. (1994a). The VPF analysis of the CfA2 sample confirmed the principal results obtained from a similar analysis of the CfA1 sample by Einasto etal. (1991).
Fig. 7.25 The panel (a) shows the void diameters, Dm, as a function of the sample size, L, both expressed in units h−1 Mpc. Open circles mark diameters of galaxy samples, found with the beam method. Filled circles are for diameters found with empty sphere method; squares are forcluster samples. Panel (b) gives the correlation length Ro versus the sample size for the same galaxy and cluster samples (Einasto et al., 1989).
Lindner et al. (1995, 1996, 1997) investigated the structure of voids in more detail, in particular the distribution of faint galaxies in large voids defined by rich clusters and superclusters. He found the distribution of void diameters for galaxies of different absolute magnitude limits for various environments. The environment was defined using the luminosity density field, smoothed with a Gaussian kernel of radius 8 h−1 Mpc. The dependence of void sizes on the absolute magnitude limit is shown in Fig. 7.26. As expected, brighter galaxies form larger voids. In larger voids the central smoothed density is lower, and void galaxies are fainter.
In subsequent years there has been a lot of progress in the study of voids. In 2006 a special workshop on voids was organised in Aspen, and later in Amsterdam. At this workshop a program to compare void finder programs was initiated, and results were published by Colberg et al. (2008). One of the most effective void finders is the cosmic watershed method suggested by Platen et al. (2007).
In March 2012 Rien van de Weygaert visited Tartu Observatory; in November 2012 his previous collaborator Miguel Aragon-Calvo also visited us. Rien works in the Kapteyn Institute of the Groningen University. The Groningen team has concentrated their efforts on the detailed study of the cosmic web and voids. What impressed me in the work of the Groningen team was the study of the same phenomena we did many years earlier, but now the study was made much more methodically, using novel methods of analysis.
Fig. 7.26 The cumulative distributions of void diameters for various absolute magnitude limits in different environments. The environment is defined by the density field, smoothed with Gaussian dispersion 8 h−1 Mpc; high, medium and low density is defined using threshold density levels 2 and 1 in mean density units (Lindner et al., 1995).
Both Rien and Miguel had seminar talks in our Observatory. In his seminar talk (“The Alpha and Betti of the Universe”) Rien described his recent work in the study of the cosmic web. Together with his collaborators he initiated the void galaxy survey (van de Weygaert et al., 2011). Void galaxies are of great interest in understanding the evolution of galaxies, since in a low-density environment one can observe galaxies which have had little influence from interactions with other galaxies. Rien also described his results of the study of the void phenomenon using different techniques and galaxies (van de Weygaert & van Kampen, 1993; van de Weygaert & Platen, 2009; van de Weygaert et al., 2009, 2010; Aragon-Calvo et al., 2010b,c,a).
7.3.7 Cosmic web — cells and the cosmic foam
The largest natural units of the cosmic web are cells, defined by rich clusters and superclusters, as suggested already by Jõeveer et al. (1977); Jõeveer & Einasto (1978). The simplest characteristics of the cosmic web are the mean distances between superclusters, and the nearest neighb
our distribution. Both characterise the size of cells of the cosmic web.
Previous analysis has shown that the sizes of voids determined by superclusters of different richness are rather close. This result, and the absence of a randomly located population of rich clusters in voids, suggest that practically all rich clusters of galaxies are located in void walls, and the overall distribution of superclusters of different richness is rather similar. Einasto et al. (1997b) studied the distribution of superclusters of different richness in void walls. For that they calculated for each supercluster centre the distances to the centres of the three nearest superclusters, separately for poor, medium rich, and very rich superclusters (see Fig. 7.27).
Fig. 7.27 The distribution of distances between centres of superclusters. Upper panel shows the distributions for poor and medium rich superclusters, lower panel the distributions for very rich superclusters. Curves correspond to the first (line with short dashes), second (line with long dashes) and third (solid line) neighbour (in the lower panel the last two lines are interchanged) (Einasto et al., 1997b).
On the upper panel of Fig. 7.27 these distributions are given for poor and medium rich superclusters. We see firstly, that these distances are small, and secondly, that these distributions are smooth and do not show the presence of any preferred distance between superclusters. That would be seen as peaks in the distance distribution.
Fig. 7.28 The cumulative distribution of void diameters. The solid line shows diameters of voids determined by all clusters (CL), the long-dashed line shows diameters of voids defined by clusters that belong to superclusters (SCL), and the dotted line shows diameter of voids defined by centres of superclusters (SCLC) (Einasto et al., 1994b).
The distributions of distances between very rich superclusters (Fig. 7.27 lower panel) are different. None of these distributions is as smooth as in the upper panel. The most important feature in this Figure is the presence of a peak in the distribution of distances of the second and the third neighbour in the interval 110 < DNN2;3 < 150 h−1 Mpc — over 75% of very rich superclusters have a second or third neighbour at this distance interval. The median distances to the second and third neighbours are, correspondingly, DNN2 = 115 and DNN3 = 142 h−1 Mpc.
Einasto et al. (1994b) analysed the distribution of void diameters as defined by Abell clusters of galaxies. Distributions were found for three cases: voids defined by all clusters, voids defined by clusters located in superclusters, and voids defined by supercluster centres, see Fig. 7.28. The median diameter of voids defined by clusters in superclusters is 120 h−1 Mpc, and that of centres of superclusters is 145 h−1 Mpc. These values characterise the scale of cells of the Universe and the mean sizes of voids between superclusters.
van de Weygaert (2002) characterized the cosmic web as a “cosmic foam” — a tenuous space-filling frothy network permeating the interior of the Universe. Sheth & van de Weygaert (2004) investigated the hierarchy of voids in more detail.
Miguel in his seminar talk in Tartu described his algorithm to simulate the evolution of the cosmic web, the Multum In Parvo (MIP) constrained ensemble simulation, which allows a very high resolution model with minimal computer resources (Aragon-Calvo, 2012). He used these high-resolution simulations to investigate the internal structure of voids. His simulations confirmed that the cosmic web has properties of a cellular distribution. He compared the cosmic cells with other cellular systems in Nature on very different scales, from molecular to cosmic scales. He found that all cellular systems have similar properties, depending on the number of neighbouring cells. If the number of neighbouring cells is small, then during the evolution the cell shrinks and disappears (Sheth & van de Weygaert, 2004). If the number of neighbours is large, then the cell expands. The most stable configuration is a cell with 8 neighbouring cells. Such cells have the structure of a honeycomb.
Miguel’s high-resolution simulations also showed in more detail the hierarchical nature of the cellular and void structure. Within a large cell (void) there are sub-cells (sub-voids), within sub-cells there are sub-sub-cells (sub-sub-voids) etc. The highest level in this hierarchy have super-cells (super-voids), surrounded by superclusters, which are well visible both in simulations and in Nature. The next level is also seen both in Nature and in simulations. Lower levels in this hierarchy are located inside sub-cells (sub-voids). Here the density of void surfaces is so low that in most cases no galaxy formation can take place here. These substructures are seen only in high-resolution simulations of dark matter. But their existence is highly probable taking into account the similarity of the cellular structure in other natural phenomena of different scale. Thus the cosmic web has properties of a “cosmic foam”, as already suggested by Rien ten years ago.
When we introduced the term “cellular structure of the Universe” in the late 1970’s, we did not guess that this term could have such a deep physical meaning.
7.3.8 Regularity of the cosmic web
Already our first pictures of the wedges shown in Fig. 5.5 and 5.4 show that there is some regularity in the distribution of rich superclusters. The relatively rich Perseus–Pisces supercluster is located at a distance of about 50 h−1 Mpc. The Local supercluster is fairly poor; the closest rich supercluster is the Coma supercluster almost behind the Local one. The distance between the Coma and the Perseus–Pisces superclusters is about 120 h−1 Mpc. Behind the Perseus–Pisces supercluster there is a relatively empty region; the next region containing several rich clusters is located at a distance about 150 h−1 Mpc, i.e. the distance between the Perseus–Pisces supercluster and the system of superclusters behind is about 100 h−1 Mpc.
The Bootes void found by Kirshner et al. (1981) has a diameter of about 100 h−1 Mpc. It is surrounded by several rich superclusters, one of which is the Hercules supercluster. In our terminology “cellular structure” we had in mind just large low-density regions surrounded by rich superclusters.
Maret Einasto compiled catalogues of superclusters formed by Abell clusters (Einasto et al., 1994b, 1997b, 2001). She noticed that in the nearest neighbour distribution of superclusters there are peaks: the first neighbour is located at a distance of about 60 h−1 Mpc, this is usually on the same side of the wall between two adjacent supervoids. But the second and third nearest neighbours are located, as a rule, across the supervoid, and have a mean distance of about 100 h−1 Mpc.
This issue became topical after the discovery by Broadhurst et al. (1990) that the distribution of high-density regions of galaxies may be quasi-regular or periodic. Broadhurst et al. found such a regularity in the direction of the North and South Galactic poles.
Thus we started to investigate the distribution of rich clusters and superclusters in more detail. Our study confirmed that high-density regions marked by rich superclusters form a quasi-regular lattice (Einasto et al., 1997b,a), see Fig. 7.29.
Initially we believed that this quasi-regularity is due to the presence of a bump on the corresponding scale in the power spectrum of density fluctuations. To check this possibility I made a series of numerical simulations in Fermilab during my visit in 2000. My results were negative.
Then I started to think of other reasons for such quasi-regularity. I discussed the problem with Enn. It was clear that the present density field is the sum of density perturbations of various scales which have various amplitudes and phases. The power spectrum characterises only the amplitudes of density waves of different scales. But phases of density perturbations also play an important role in the formation of the cosmic web. So far the phase information has been almost ignored by the astronomical community. In the next Chapter I shall describe our efforts to understand the evolution of the cosmic web, taking into account the phase information of density perturbations.
Fig. 7.29 The distribution of 319 clusters in 25 very rich superclusters with at least 8 members (including 58 clusters with photometric distance estimates) illustrates the network in the cluster distribution in supergalactic coordinates. In the lower panels clusters in
the northern and southern Galactic hemispheres are plotted separately. The supergalactic Y = 0 plane coincides almost exactly with the galactic equatorial plane, i.e. with the zone of avoidance due to galactic absorption. The grid with step- size 120 h−1 Mpc corresponds approximately to distances between high-density regions across voids. In the two upper panels and in the lower right panel several superclusters overlap due to projection but are actually well-separated in space (Einasto et al., 1997a).
7.3.9 Baryonic acoustic oscillations
The early Universe consisted of a hot plasma of electrons, baryons and photons, and dark matter. The plasma was so hot that electrons, baryons and photons were tightly coupled. Overdensities of matter (baryons and dark matter) attract matter gravitationally towards the center of the overdensity, while the heat of photon–matter interactions creates outward pressure. Competing forces of gravity and pressure create oscillations, similar to sound waves in air. In this way the pressure forms a spherical sound wave of baryons and photons around each overdense region. These sound waves move outwards from the overdensity. The dark matter does not interact with the hot plasma, and stays at the center of the sound wave at the origin of the overdensity.
