Content: Superclusters, Voids, Correlation function

Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology,  G.Petrov et al.


Superclusters are large conglomerations of clusters and groups of galaxies on a scale exceeding 100 million light years [1021 km or 30 megaparsecs (Mpc)]. Individual superclusters and "complexes" of superclusters interconnect to form the largest known structures in the universe: a sponge-like network of high-density regions , spaced apart by a labyrinth of voids (Fig. 1). The tendency in recent years is for ever-larger structures to be recognized, so that we may not have yet established the top of the hierarchy of clustering, that is, the largest inhomogeneities in the universe. The greatest strides have been made since the mid-1970s, as large numbers of galaxy redshifts have been obtained.

The first examinations of the large-scale distribution of galaxies were carried out in the 1930s independently by Edwin Hubble and Harlow Shapley. Hubble worked in numerous narrow selected fields. Shapley covered most of the sky using wide-angle photographs. He noted regions where the galaxy count was much higher than average and labeled these as "clouds" of galaxies. Many of his clouds are recognized today as superclusters.

Improved wide-angle photographic surveys, such as the classic National Geographic Society-Palomar Observatory Sky Survey, do reveal hundreds of thousands of galaxies, but give only a two-dimensional view of their distribution. To some extent, the third dimension (distance) can be gauged by the angular diameters of the galaxies. A nearby galaxy appears larger than a more distant one, although one may sometimes be misled by a small nearby galaxy mimicking the appearance of a larger distant galaxy. However, there would be little problem in deciding between nearby and distant clusters of galaxies. In this way, in the 1950s and 1960s, George Abell and Fritz Zwicky independently gauged the relative distances of clusters on the Palomar Sky Survey. Abell concentrated on rich clusters, whereas Zwicky's cluster boundaries lay far from the central condensations. Abell also noted that certain regions of the sky had a greater number of clusters than others. Thus, both suspected much larger entities. In a similar way, C.D. Shane, working from Lick photographs, described "superclusters" or (as he preferred to describe them, using Shapley's term) "clouds of galaxies."

The true recognition of superclusters required a three-dimensional view. When distances to relatively nearby galaxies were calibrated, Gerard de Vaucouleurs advocated the existence of a "supergalaxy," now recognized as our local supercluster.

At larger distances, the most effective way of obtaining galaxy distances is to measure redshifts. Due to the overall expansion of the universe, an observer in any galaxy would see its neighbors moving away from it. The velocity of recession increases with distance from the observer's galaxy according to Hubble's well known relation

V = H0 d ,

where V is the velocity of recession, d is the distance, and H0 is the Hubble constant (in the range 50-100 km s-1 Mpc-1 or 15-30 km s-1 per million light-years). The velocity can be measured by the Doppler shift of spectral features (redshift): either absorption or emission lines in the optical spectra, or 21-cm neutral hydrogen emission in the radio region. knowing the velocity, the distance can be inferred.

Strictly speaking the observed velocity is not entirely cosmological but should be expressed as

V0 = H0 d + Vs + Vp ,

where the velocity V0 is now corrected for our Sun's motion within the Galaxy and for the streaming motion of our galaxy, V0 is the streaming motion associated with the observed galaxy, and Vp is the observed galaxy's own individual motion over and above systematic streaming; Vp is significant (several hundred kilometers per second) in rich clusters. Because it is difficult to disentangle these velocities, and because of the uncertainty in the value of the Hubble constant, it is customary to plot data in "redshift space" (with dimensions shown as kilometers per second) rather than in conventional three-dimensional space. Gross structures are much the same in either space, but the peculiar velocities in rich clusters make the clusters appear stretched radially (the "Finger of God" effect) in redshift space (examples of this can be seen in Fig. 2).

In the past, obtaining the spectrum of a single galaxy called for a photographic exposure of some hours duration. The advent of electronic image intensifiers and the replacement of the photographic plate by charge-coupled devices (CCDs) and Reticon arrays has greatly accelerated the acquisition of redshifts. Around 1950 little more than 100 redshifts were known, by 1960, 1000 were known and twice that number had been collected by 10 years later. Yet, by 1980 the figure exceeded 10,000 and by 1990 it was around 40,000.

With greater numbers of redshifts available in the mid-1970s, Guido Chincarini pointed out that, even when clusters are avoided in a study, redshifts seem to favor certain values and to avoid others (for the region of sky involved). Thus superclustering was revealed in the third dimension, and three-dimensional mapping became possible. Much pioneering work was done in the region of the Coma cluster where a bridge to a neighboring cluster, Abell 1367, was discerned, with a void immediately in front of the structure.

In order to map completely the neighboring superclusters to our own "Virgo supercluster," one would need to work out to a redshift corresponding to several thousand kilometers per second. Within such a volume of space, there are hundreds of thousands of giant galaxies. Even with present technology, it is quite impossible to obtain all their redshifts (the number to be observed would be much greater because a vast number of background galaxies would have to be candidates). Thus, it is necessary to restrict observations to a "representative" sample. The most common approach is to observe all galaxies brighter than a selected limiting apparent luminosity (apparent magnitude). Such a choice is satisfactory for the capabilities of the telescopes involved, but nearer low-luminosity galaxies are included in the sample, whereas distant high-luminosity galaxies may be excluded. The data thin with distance, and low galactic latitudes (where light from distant objects is subject to extinction by matter in the MilKy Way) have to be avoided. Nevertheless, knowledge of the galaxy luminosity function (the relative numbers of galaxies versus luminosity) allows one to derive true number densities and other statistics. Difficulties could arise if the luminosity function varies with environment; such tendencies have been claimed in the literature. The alternative approach is to disregard a strict magnitude limit and to observe what appears to constitute a representative sample on the sky. This allows for initial mapping of superclusters (mainly because the intervening voids are almost completely empty), but cannot produce quantitative parameters. Whatever system of sampling is used, the outcome is plots revealing filamentary or sponge-like structures, reminiscent of aqueous media even though the data are in the form of discrete points.

Figure 1 gives an indication of neighboring superclusters. The nearest of these is the Hydra-Centaurus-Pavo supercluster. The names reflect the main constellations in which the structure is seen on the sky (although constellations are based on nearby stars in our galaxy, they conveniently represent general directions when looking far beyond those stars). Although the Hydra-Centaurus portion is seen on the sky on one side of the Milky Way and the Pavo portion on the other side, the agreement in redshift and general continuity of structure point to its being a single entity, with the main bulk lying in Centaurus or probably behind the foreground obscuration of the Milky Way. The Hydra condensation centers around the Hydra I cluster (redshift 3500 km s-1) and there is only a relatively weak bridge to the Centaurus concentration. The latter is dominated by the Centaurus cluster, which shows a composite structure in redshift space, with concentrations at both 3000 and 4500 km s-1. However, the weaker 4500 km s-1 concentration may be, to some extent, background galaxies because 4500 km s-1 is the dominant redshift for the bulk of the extended Centaurus superclustering. The same redshift (4500 km s-1) is picked up on the Pavo side, which contains a number of weaker clusters. All these clusters contain both elliptical and spiral galaxies, but the cores of both Hydra I and Centaurus have greater proportions of elliptical and SO galaxies. There seem to be some filamentary links between our own Virgo supercluster and the Centaurus concentration, almost as if our supercluster were something of an appendage. We are separated from the Hydra and Pavo concentrations by foam-like voids with only a sprinkling of galaxies around their perimeters.

The Coma supercluster, mentioned earlier, is separated from our own supercluster and Hydra-Centaurus by a number of voids. Its structure is centered on the Coma cluster, the nearest rich cluster of galaxies (composed almost entirely of elliptical and SO galaxies). From this central concentration, extensions run out more or less perpendicular to our line of sight. Although that running (west-wards) to the Abell 1367 cluster was the first discovered, present-day surveys (particularly the Harvard-Smithsonian "slices") show it extending both east, north, and southward, so much so that it has been referred to as the Great Wall, and its full extent is still being assessed.

Although the Virgo, Centaurus, and Coma superclusters dominate the northern galactic hemisphere, the other side of the obscuring band of the Milky Way is dominated by the Perseus-Pisces supercluster (which has been mapped extensively by the radio redshifts of Martha P. Haynes and Riccardo Giovanelli). Particularly interesting in this supercluster is a filamentary central condensation that is well marked by elliptical galaxies. It is some 4000 km s-1 long in redshift space and runs perpendicular to the line of sight. Toward one end lies the Perseus cluster. Voids again intervene between this supercluster and ours their peripheral galaxies provide tenuous interconnections. Toward the south the supercluster continues and connects to another heavy wall-like structure, in the Sculptor region, that runs at an angle to our line of sight.

It has been remarked that these surrounding structures give a sort of "tree ring," appearance to the distribution within distances from our galaxy that correspond to redshifts of several thousand kilometers per second. More redshifts still are needed to define the nature and extent of such larger patterns. What is relevant is that, whenever a volume of space is sampled, there always seems to be structure with a dimension comparable to that of the volume surveyed. This has led to considerations of fractal structures (identical forms repeated on ever-increasing scales) occurring in the universe. If this is correct, although one would gain a geometrical interpretation to the nature of the structures, it would make a physical explanation extremely difficult.

Beyond redshifts of several thousand kilometers per second, a number of further superclusters have been mapped tentatively. The volume is incompletely sampled, although it is unlikely that very conspicuous superclusters would have been overlooked. For example, in the north there is a pair of superclusters in Hercules (at redshifts around 10,000 km s-1), whereas in the south Shapley's cloud of galaxies in Horologium is resolved into two superclusters (at redshifts of 12,000 and 18,000 km s-1) seen along a common line of sight (see Fig. 2).

An alternative approach for reaching out to larger distances is to assume that Abell's clusters, which mark peak number densities, flag the high points of superclusters. Thus, distant superclusters can be recognized as groupings (in redshift space) of Abell clusters, and examinations of even larger volumes of space can be carried out. On this basis, ever larger conglomerations (to scales of 30,000 km s-1) have been claimed. The work recently has been extended to the southern skies.

The future holds exciting prospects. Just as human eyes scanned the galaxies on the wide-angle photographs, the finest-quality photographs of the U.K. Schmidt telescope and the new Palomar sky surveys are now being scrutinized by machine and millions of galaxies already have been detected and cataloged. From these sky densities comes evidence of possible larger superclusters. In parallel, the development of fiber-optic spectrographs that can be used to observe many galaxies simultaneously may lead to the mass determination of redshifts. We can look forward to finding even more remarkable structures in the superclustering of galaxies.


Definition of the problem
In extragalactic astronomy, the basic unit is a galaxy with a characteristic length of 20 kpc that typically moves within a group or clusters with diameter 50 kpc to 5 Mpc which in turn is located in a supercluster with diameter 50 Mpc or larger. Superclusters and voids are often discussed together because they are identified and studied on a common data base and it is likely that they share a common origin and represent two complementary effects of related evolutionary processes.

The structure of superclusters, the material entities that make up the contiguous shell, was reviewed by Oort, 1983, and the voids by Rood, 1988. Voids were not immediately recognized in the surface distribution of galaxies because they are superimposed by background and especially foreground galaxies.

Voids were recognized only after Doppler velocities were measured for statistically homogeneous samples of galaxies in selected solid angles of the sky, which provided direct information on the three-dimensional distribution of galaxies.

In extragalactic astronomy, the three-dimensional region that constitutes a void is transparent and empty or nearly empty of galaxies. It is often useful to think of a void as for a discrete entity - a region containing significantly fewer galaxies than predicted by the appropriate Poisson distribution. The occurrence of physical groupings of galaxies evidently requires the presence

<>of voids defined in this way. There are two ways in which to study an individual void observationally:
 a) the void can be probed with telescopic sensors in order to detect galaxies within it

b) the structure and content of the contiguous shell of superclusters surrounding the void can be studied.

Early surveys of galaxies over large solid angles of the sky indicated a superclustering of galaxies, i.e. predominant occurrence of galaxies, groups, and clusters within larger structures now called superclusters (Lundmark, Holmberg, Shapley, Oort, de Vaucoleurs, Shane & Wirtanen, Abell).

<>From independent counts of galaxies to faint magnitudes in a large number of small survey fields distributed over the sky, Hubble concluded that groups and clusters of galaxies are aggregations drawn from the general field, which is everywhere and in all directions approximately uniform. <>

Early studies of the surface distribution of galaxies featured three different models:
 a) superclustering of galaxies

b) super-large rich clusters

c) group and clusters drawn from a uniform general field.

To discriminate between these models, we need to know the three-dimensional distribution of galaxies in large homogeneous samples.

<>Chincarini & Rood, 1970 pointed out  in  the constellation Hercules the redshifts segregate into a small number of groups. They hypothesized that galaxies occur in groups and that the apparent field of galaxies in two dimensional distribution is the result of a superposition of such groups - model (a). <> 

Peebles, 1975
showed the Universe conforms to model (a), the superclustering of galaxies, based on correlation functions derived from the surface distribution of galaxies over large solid angle of the sky. <> 

Studying the correlation function from large sets of data, Peebles et al., 1980, found that galaxies and systems of galaxies are correlated on all scales up to the noise limit of their data, between approx 10 and 100 Mpc. The results are
consistent with (a) and (b) but not with (c). <> 

, 1983 pointed out a large void in Hercules from the analysis of data from 1976 - 1981. In fact “VOID” is a natural contraction of “region devoid of galaxies”. <> 

et al., 1985 failed to detect a nonclustered homogeneous background of galaxies from the three-dimensional distribution of galaxies with mp < 14.5 covering a large solid angle in the sky. Their result is consistent with the conclusions of Soneira & Peebles, 1977. <> 

There are many works to confirm the presence of the voids in theoretical aspect. As early as 1970, Zeldovich et al., 1977, developed theoretical models showing the "cell-like" large-scale structure formed by nonlinear dissipative collapse of gaseous density perturbations, forming "pancakes", followed by condensation into galaxies.

et al, 1978 found that the observed large-scale three-dimensional distribution of galaxies exhibits cellular structure in accordance with Zeldovich et al. models. <> 

Many cosmologists in Western Europe and USA assumed that galaxies formed initially by Jeans instabilities from random density perturbations that evolved into a hierarchy of larger structures that could be tracked by N-body computer experiments.

, 1981, and Ostriker & Cowie, 1986, suggested that cosmic explosions with an energy release 10^61 ergs would create blast waves that might trigger or amplify the formation of galaxies, and that might also produce large-scale structure by tending to create voids with outward moving galaxies on their contiguous shells. <> 

Johnson S.B. et. al., 1994
take CCD images of the Bootes void. Several dozen galaxies have been found within this void. However a 4,000 km/s diameter volume displaced to the southwest of the original void center remains completely empty raising the possibility that the true void is at a lower declination and is smaller than originally defined. There is a deficit of galaxies over roughly a four magnitude range centered on M* at the distance of the void center. <> 

N.A., 1995 represent the SDSS. The Sloan Digital Sky Survey (SDSS) will provide a complete imaging and spectroscopic survey of the high-latitude northern sky. The 2D survey will image the sky in five colors and will contain nearly 5x10^7 galaxies to approx 23m. The spectroscopic survey will obtain spectra of the brightest 10^6 galaxies, 10^5 quasars, and 10^3 rich clusters of galaxies (to approx 18.3-19.3 m, respectively). The survey will identify a complete sample of several thousand rich clusters of galaxies, both in 2D and 3D - the largest automated sample yet available.
The extensive cluster sample can be used to determine critical clustering properties such as the luminosity-function, velocity-function, and mass-function of clusters of galaxies (a critical test for cosmological models), detailed cluster dynamics and Omega_dyn, the cluster correlation function and its
dependence on richness, cluster evolution, superclustering and voids to the largest scales yet observed, the motions of clusters and their large-scale peculiar velocity field, as well as detailed correlations between X-ray and optical properties of clusters, the density-morphology relation, and cluster-quasar associations. The large redshift survey, reaching to a depth of >600,h^-1 Mpc, will accurately map the largest scales yet observed, determine the power-spectrum and correlation function on these large scales for different type galaxies, and study the clustering of quasars to high redshifts (z > 4). The implications of the survey for cosmological models, the dark matter, and Omega are also discussed. <> 

Lindner U., 1996
investigate the distribution of normal (faint) galaxies and blue compact galaxies (BCGs) in voids by analyzing their distribution as a function of distance from the void centers and by employing nearest neighbour statistics between objects of various subsamples. They find that galaxies in voids defined by brighter galaxies tend to be concentrated close to the walls of voids in a hierarchical manner, similar to the behavior of brighter galaxies. The behavior of BCGs is in this respect similar to the one found for normal dwarf galaxies. <> 

B.S. and Mellot A.L., 1996 compare the properties of voids in real space to their properties in redshift space. Both the void probability function (VPF) and the underdense probability function (RE) are enhanced in redshift space. With their algorithm that detects individual voids, voids are ellipses whose enclosed density of galaxies falls below a threshold density. When voids are identified using this algorithm, the mean void size and the maximum void size both increase in going from real space to redshift space. <> 

El-ad H. et al., 1996
present a new void search algorithm for automated detection of voids in three-dimensional redshift surveys. Based on a model in which the main features of the large-scale structure (LSS) of the universe are voids and walls, they classify the galaxies into wall galaxies and field galaxies, and  define voids as continuous volumes that are devoid of any wall galaxies. Field galaxies are allowed within the voids. The algorithm makes no assumptions regarding the shapes of the voids, and the only constraint that is imposed is that the voids are always thicker than a preset limit, thus eliminating connections between adjacent voids through small breaches in the walls. <> 

El-ad H. and Piran T., 1997
present the VOID FINDER algorithm, a novel tool for objectively quantifying voids in the galaxy distribution. The algorithm first classifies galaxies as either wall galaxies or field galaxies. Then, it identifies voids in the wall-galaxy distribution. Voids are defined as continuous volumes that do not contain any wall galaxies. The voids must be thicker than an adjustable limit, which is refined in successive iterations. In this way, we identify the same regions that would be recognized as voids by the eye. The voids have a scale of at least 40,h^-1 Mpc and an average -0.9 underdensity. Faint galaxies do not fill the voids, but they do populate them more than bright ones. <> 

Kuhn B. et al., 1977
present the results of a search for intrinsically faint galaxies towards three regions with known voids and the Hercules supercluster. They point out that no clear indication of a void-population was found. <> 

C.C. et al., 1997 analyse the spatial distribution of our sample of ELGs in comparison with the distribution of normal galaxies. Overall both distributions trace the same structures. Nevertheless they also found a few ELGs in voids, from which at least 8 lie in the very well defined nearby voids. The isolated galaxies seem to form fainter structures that divide the larger voids into smaller ones. From their estimations of the expected number of void galaxies they did not find an underlying homogenous void population. <> 

El-ad H. and Piran T., 2000
present a comparison between the voids in two nearly all-sky redshift surveys: the Optical Redshift Survey (ORS) and the IRAS 1.2-Jy survey. While the galaxies in these surveys are selected differently and their populations are known to be biased relative to each other, the two void distributions are similar. <> 

N.A. and Geller M.J., 2000 analyze the optical properties of approx 300 galaxies within and around three prominent voids of the Center for Astrophysics Redshift Survey determining CCD morphologies and H_alpha equivalent widths from their imaging and spectroscopic survey. Both the morphological mix and the H_alpha line width distribution of galaxies at modest underdensities, 0.5<n/n   _d1le1, are indistinguishable from the control sample at modest overdensities, 1<n/n_d1le2. Both density regions contain a similar fraction of galaxies with early-type (E and S0) morphologies and with absorption-line spectra (approx35%). In the voids, where the luminous galaxies presumably formed more recently, there should be more gas and dust present for active star formation triggered by nearby companions. <> 

Y. and Piran T., 2001 introduce a simple model for the formation of voids. In this model the underdensity of galaxies in voids is the product of two factors. The first arises from a gravitational expansion of the negative density perturbation. The second is due to biasing: galaxies are less likely to form in an underdense region. <> 

Schmidt J.D. et al., 2001
show the sizes and shapes of voids in a galaxy survey depend not only on the physics of structure formation, but also on the sampling density of the survey and on the algorithm used to define voids. Using an N-body simulation with a tauCDM power spectrum, they study the properties of voids in samples with different number densities of galaxies, in both redshift space and real space. When voids are defined as totally empty regions of space, their characteristic volume is strongly dependent on sampling density; when they are defined as regions whose density is 0.2 times the mean galaxy density, the dependence is less strong. <> 

G., A. Kniazev and J. Fried, 2004 presented photometry and morphology of faint galaxies in the direction of 1600+18 in Hercules void. Coordinates of ca. 1200 faint galaxies in a field of one square degree centered at 1600+18 (1950) (Hercules void), and m(B),diameters, position angles and morphological classification are presented. The distribution of the magnitudes of the galaxies in this direction is compared with “Log Normal and Gauss ones and with similar results from SDDS studies of galaxies. Major axes luminosity profiles are analysed. Some candidates for primeval galaxies - Low surface brightness galaxies were detected in the direction of the void.

The superclustering of galaxies (model “a”) and the presence of voids are now accepted as fact, a paradigm among workers on large-scale structure.<> <> 
Amongst the famous known voids are Coma and Hercules, which are practically devoid of matter, whereas Bootes and Perseus-Pisces are not empty (approx 10 emission line galaxies in Boo, furthermore evidence for obscuring matter).
  <>The structure of superclusters, the material entities that make up the contiguous shell, as pointed by Oort, allows us to study the vicinity of the voids itself. Particularly interesting are correlations of spectral properties of galaxies and Hubble morphological type with local density of galaxies.


The testing of theories of structure formation using observations of the large-scale structure of the distribution of galaxies requires a statistical approach. Theoretical studies of the problem of structure formation generally consist of performing numerical N-body simulations on powerful computers. Such simulations show how galaxies would form and cluster according to some well-defined assumptions about the form of primordial density fluctuations, the nature of any dark matter and the parameters of an underlying cosmological model, such as the density parameter and Hubble constant. The simulated Universe is then compared with observations, and this requires a statistical approach: the idea is to derive a number (a `statistic') which encapsulates the nature of the spatial distribution in some objective way. If the model matches the observations, the statistic should have the same numerical value for both model and reality. It should always be borne in mind, however, that no single statistic can measure every possible aspect of a complicated thing like the distribution of galaxies in space. So a model may well pass some statistical tests, but fail on others which might be more sensitive to particular aspects of the spatial pattern. Statistics therefore can be used to reject models, but not to prove that a particular model is correct.

One of the simplest (and most commonly used) statistical methods appropriate for the analysis of galaxy clustering observations is the correlation function or, more accurately, the two-point correlation function. This measures the statistical tendency for galaxies to occur in pairs rather than individually. The correlation function, usually denoted by xi (r), measures the number of pairs of galaxies found at a separation r compared with how many such pairs would be found if galaxies were distributed at random throughout space. More formally, the probability of finding two galaxies in small volumes dV1 and dV2 separated by a distance r is defined to be be

dP = n2 (1 + xi (r)) dV1 dV2

where n is the average density of galaxies per unit volume. A positive value of xi (r) thus indicates that there are more pairs of galaxies with a separation r than would occur at random; galaxies are then said to be clustered on the scale r. A negative value indicates that galaxies tend to avoid each other; they are then said to be anticlustered. A completely random distribution, usually called a Poisson distribution, has xi (r) = 0 for all values of r.

Estimates of the correlation function of galaxies indicate that xi (r) is a power-law function of r:

xi (r) approx (r/r0)-1.8

where the constant r0 is usually called the correlation length. The value of r0 depends slightly on the type of galaxy chosen, but is around 5 Mpc for bright galaxies. This behaviour indicates that these galaxies are highly clustered on scales of up to several tens of millions of light years in a roughly fractal pattern. On larger scales, however, xi (r) becomes negative, indicating the presence of large voids (see large-scale structure). The correlation function xi (r) is mathematically related to the power spectrum P(k) by a Fourier transformation; the function P(k) is also used as a descriptor of clustering on large scales.

Compiled by G.T.Petrov, 2004