SUPERCLUSTER-VOID NETWORKJ. EinastoTartu Astrophysical Observatory, EE-2444, EstoniaAbstractWe study the distribution of rich clusters and superclusters of galaxies.We show that clusters located in rich superclusters (high-density regions)form a fairly regular network, and isolated clusters and clusters in poorsuperclusters (low-density regions) form a more uniform population.Quantitatively the regularity of the supercluster-void network can becharacterized by the cluster correlation function. We demonstrate thatclusters in low-density regions have almost zero correlation function on largescales, the correlation function of clusters in high-density regions has anoscillatory behaviour with period 130 h�1 Mpc.Simple toy models show that the correlation function has an oscillatorybehaviour in case when clusters are located in a regular grid.Numerical simulations with popular structure formation scenarios in-dicate that it is di�cult to form a population of high-density regions withproperties as the observed supercluster-void network. It is still open howto explain the presence of such a regularity.1. IntroductionOne outstanding problem of cosmology is the scale of transition to homo-geneity. On small scales galaxies and clusters of galaxies are clustered, onlarge scales the distribution is smooth. Quantitatively the clustering canbe characterized by the correlation function which has large values on smallscales, and is almost zero on large scales. On some intermediate scale thetransition from positive correlation to absence of correlation occurs. Thisscale can be considered as the scale of transition to homogeneity (Einastoand Gramann 1993).The correlation function of galaxies approaches zero at about 30h�1 Mpc, and of clusters of galaxies at about 70 h�1 Mpc. These scaleswere thought to be the scales of the transition to homogeneity. However,the distribution of galaxies indicates the presence of voids of diameter up to50 h�1 Mpc, voids in the distribution of clusters of galaxies are even largerwith diameters 100 h�1 Mpc and more. Thus it is not clear where the scaleof homogeneity begins.Additional indication that the homogeneity begins on rather largescales comes from the deep pencil-beam survey of redshifts of galaxies byBroadhurst et al. (1990). The distribution of galaxies is periodic with aperiod 128 h�1 Mpc. If this regularity is present in the tree-dimensionaldistribution of galaxies too then we must conclude that the transition scale1

to homogeneity is equal to or larger than the period found by Broadhurstet al.To investigate this problem deep three-dimensional data are needed.The best data available to study this problem is the redshift compilationof rich clusters of galaxies by Andernach, Tago and Stengler-Larrea (1995,1996). We have used this data to investigate the distribution of clustersof galaxies and to calculate the cluster correlation function, Sections 2 and3 below. In Sections 4 and 5 we compare the observed correlation func-tion with functions found for simple toy models and for popular models ofstructure formation.2. Distribution of superclustersUsing the revised and updated catalogue of data of Abell-ACO clustersof galaxies by Andernach, Tago and Stengler-Larrea (1996) Einasto et al.(1996) compiled a new list of clusters and a catalogue of superclusters. Thelist contains 1304 clusters of galaxies up to redshift z = 0:12 of richnessclasses 0 and above in both galactic hemispheres. Two thirds of clustershave measured redshifts, for the rest photometric distance estimates havebeen used. Practically all nearby clusters have measured redshifts, on thefar end of the list about 40% of clusters have measured redshifts. Thenew catalogue contains 220 superclusters with at least two member clus-ters (supercluster richness). 25 superclusters are very rich with at least 8members, approximately 25% of all clusters are members of these very richsuperclusters.The distribution of clusters in very rich superclusters in supergalacticcoordinates is shown in Figure 1. We see that the population of clustersin very rich superclusters forms a fairly regular network. Isolated clustersand clusters located in poor superclusters are much more uniformly spaced,see Einasto et al. (1994, 1996) for details. The regularity of the distribu-tion of clusters was noted also by Tully et al. (1992), they used the term\chessboard universe" to describe the regular location of clusters.3. Cluster correlation functionWe shall use the classical de�nition of the estimate of the two-point corre-lation function (Peebles 1980):�(r) = DD(r)RR(r) n2Rn2 � 1; (1)where DD(r) is the number of pairs of clusters of galaxies in the range ofdistances r�dr=2; dr is the bin size; RR(r) is the respective number of pairsin a Poisson sample of points; and n and nR are the mean number densitiesof particles in respective samples. The summation is over the whole volumeunder study, and it is assumed that the galaxy and Poisson samples haveidentical shape, volume and selection function.2

We have calculated the errors of the correlation function from errorsof the number of pairs DD(r) (Mo, Jing and Borner 1992, hereafter MJB):�2DD(r) � DD(r) + b2NDD(r)2; (2)where N is the number of particles in the observed sample, and b is aconstant which depends on the high-order correlations. The �rst term ofequation (2) is the Poisson error, the second term is the cosmic error. Thevalue of the parameter b can be derived from the scatter of the correlationfunction for samples of clusters of N-body models calculated with di�erentinitial realization of the model, as well as from toy models of structure (seebelow). For pure Poisson model b = 0, in actual models it has value between0.8 and 1.4.We have calculated the correlation function for the whole sample of richclusters of galaxies, and for subsamples of clusters located in superclustersof various richness. The sample has a double conical shape, it is limitedwith distance 350 h�1 Mpc, and galactic latitude �8 degrees. We have cal-culated the Poisson distribution for this double conical volume taking intoaccount the selection function of clusters in distance and galactic latitude.To suppress random errors the correlation function has been smoothed witha Gaussian kernel of dispersion 15 h�1 Mpc. Results are shown in Figure 2.We have divided the whole sample of clusters into two populations, onein low-density regions (isolated clusters and clusters in poor superclusterswith a number of member clusters less than i), and clusters in high-densityregions (rich superclusters with at least i members). The threshold i wasvaried in broad limits between i = 2 and i = 8. Our results show that overthe whole threshold interval the population of clusters in low-density re-gions has an uniform correlation function on large scales. On the contrary,clusters in high-density regions have an oscillatory behaviour. Maxima andminima alternate with a period of � 130 h�1 Mpc. With increasing thresh-old i the amplitude of oscillations increases, and has at i = 8 the largestvalue. In Figure 2 we show the correlation function for clusters in low- andhigh-density regions for i = 6 and i = 8, samples ACO956 and ACO958,respectively. The 1� error corridor is considerably smaller than the am-plitude of oscillations. A small overall decrease of the correlation functionwith distance is due to inaccuracy of the selection function. The selectionfunction was calculated for the whole sample, the correlation function forclusters in low-density regions is rising, and in the mean the slope on largescales is zero.To investigate the in uence of estimated redshifts we have repeated allcalculations with subsamples of clusters with only measured redshifts, thetotal number of such clusters up to redshift z = 0:12 in our list is 869. A newselection function was derived, it decreases with distance much more rapidlyas for the whole sample. Again clusters were divided into two populations,3

and the correlation function was calculated separately for clusters in low-and high-density regions. Results for threshold multiplicity i = 6 and i = 8are given in lower panels of Figure 2, samples ACP956 and ACP958. Thecorrelation functions are very similar to ones calculated for the whole sampleand plotted in upper panels, only error corridors are larger as the numberof particles in samples is smaller. Again the amplitude of oscillations isthe largest for the subsample of clusters in high-density environment withi = 8.The oscillatory behaviour of the correlation function is rather surpris-ing. The presence of the �rst secondary maximum of the cluster correlationfunction was shown already by Kopylov et al. (1984, 1988), and con�rmedby Mo et al. (1992a, b), Einasto and Gramann (1993) and Fetisova et al.(1993). Further maxima were �rst detected by Saar et al. (1995). In thenext Section we discuss the meaning of the oscillatory behaviour of thecorrelation function.4. Model correlation functionsTo understand the unexpected oscillatory behaviour of the correlation func-tion of clusters of galaxies we calculate the correlation function for a num-ber of simple geometric toy models: random galaxies, random clusters,random superclusters, random voids, regular displacement of superclustersand voids. The correlation function is smoothed with a Gaussian window,this practically removes the Poisson noise. For each model we generateten random realizations, and calculate the error of the correlation functionsfrom the scatter of these realizations.As expected, the correlation function of the random galaxy and randomcluster model is close to zero over the whole distance interval. The randomsupercluster model has a large positive correlation on small scales and zeromean correlation on large scales, see Figure 3a.The random void model is known as the Voronoi tessellation (van deWeygaert & Icke 1989, van de Weygaert 1994 and references therein). Inthis model clusters of galaxies are located in corners of a structure which isformed by expanding volumes from randomly located void centres or seeds.This model has one free parameter, the number density of expansion centreswhich determines the scale of the model and the mean diameter of voids.We generated ten models of size 512 h�1 Mpc with 122 void centres, in thiscase the mean diameter of voids is 128 h�1 Mpc (van de Weygaert 1994).The mean number of clusters in a model is 827, about a factor of 2 less thanthe number of Abell clusters in a similar volume (Bahcall and Cen 1993).The correlation function for this model is shown in Figure 3b. It hasa high maximum on small scales, and a secondary maximum at about 170h�1 Mpc, corresponding to distance between superclusters across voids.This maximum is well seen in all individual models, the position of themaximum is also rather stable, only the amplitude and the shape of thecorrelation function on larger scales vary.4

The next series of toy models was calculated with a built-in regularstructure (chessboard model). Clusters of galaxies were placed randomly insuperclusters located in corners of a rectangular net with step 130 h�1 Mpc.Additional clusters were put randomly along rods which join superclusters.Similar toy model was investigated earlier by Einasto (1992).The correlation function for this model is given in Figure 3c. We seepronounced oscillations with a period equal to the step of the net. Theerror corridor is very small in comparison to the amplitude of oscillations.This model shows that oscillations indicate the presence of regularity inthe distribution of clusters of galaxies. The overall shape of the correlationfunction is very similar to the observed correlation function of clusters inhigh-density environment, Figure 2b.We have investigated also the in uence of the selection function and thenon-trivial double-conical volume of the sample. Changes in the selectionfunction within reasonable limits change the overall slope of the correlationfunction on large scales. The positions of maxima and minima and theamplitude of oscillations are practically independent on the parameters ofthe selection function. The double conical shape of the sample has almostno in uence to the correlation function if the overall cubical sample and thesmaller double conical sample have similar distribution of clusters as in theregular net model. This is illustrated in Figure 3d, which shows correlationfunctions for our ten realizations of the regular net model calculated fordouble conical subsamples similar in form to observed samples. The generalform of the correlation function is the same, only the error corridor is larger,especially on large distances.5. Correlation function of clusters in N-body modelsWe have calculated several models of structure formation using the standardPM code with 1283 particles and 2563 cells. We have used also results ofmodel calculations in the Potsdam Astrophysical Institute, these modelswere calculated with 2563 particles and 5123 cells. Periodic initial conditionswere used in the computational volume of sidelength L = 768 h�1 Mpc forour models and L = 500 h�1 Mpc for Potsdam models. Three initialspectra were used, corresponding to standard CDM scenario with = 1and Hubble constant h = 0:5 (model CDM), broken scale invariant modelbased in double in ation scenario of structure formation (model BSI, fordetails cf. Gottl�ober, M�ucket and Starobinsky 1994), and a double power-law model with spectral index n = 1 on large scales, and index n = �1:5on small scales, and a transition at scale �t = 128 h�1 Mpc (see Frisch etal. 1995).Clusters of galaxies were selected using a friend-of-friends algorithmfor test particles representing the clustering of dark matter particles. Thenumber of particles in density enhancements determines the mass of clus-ters since the mass of each particle in the model is known from initial data.Clusters of galaxies corresponding to Abell clusters can be selected in two5

di�erent ways, either using the distribution of masses of Abell clusters orfrom the condition that the spatial density of clusters in model must coin-cide with the density of real clusters. We have used the second alternative,adopting for the spatial density the value N = 13:5� 10�6h�3Mpc3 (Bah-call, Cen 1993).To calculate the cosmic error we have divided the cubic computationalvolume into 8 subvolumes of sidelength L=2, the scatter of individual valuesof the cluster correlation function at various distances gives us the possibilityto derive the value of the parameter b in formula (2). This method givesresults in good agreement with other independent estimates from toy modelsconsidered above, as well as results obtained by MJB.The cluster correlation function for various models is shown in Fig-ure 4. We determined the correlation function for the whole cubical sampleas well as for double conical sample. Secondary maxima and minima arepresent in model cluster correlation functions but they are much weakerand less regular than in the case of real cluster samples. The reason for thisdiscrepancy is not yet clear.6. ConclusionsOur study of the distribution of clusters of galaxies has lead us to thefollowing main conclusions.� Clusters of galaxies can be divided into two populations: clusters lo-cated in high-density regions form a fairly regular three-dimensionalnetwork of superclusters and voids, clusters located in low-density re-gions are distributed more uniformly.� The correlation function of clusters of galaxies has an oscillatory be-haviour for clusters located in high-density superclusters and is prac-tically zero on large distances for clusters in low-density environment.� The period of oscillations, 130 h�1 Mpc, is equal to the scale of thesupercluster-void network, and the periodicity scale found by Broad-hurst et al.� Clusters of galaxies in popular models of structure formation are lo-cated less regularly than real clusters.We end up with several questions. Is the regularity observed in thedistribution of clusters of galaxies in high-density regions a global e�ect ordue to some disturbing e�ect in observations? If the e�ect is real, whatis its cause, i.e. why are high-density regions distributed regularly? Toanswer the �rst question other independent observational data are neededto �nd the distribution of high-density regions over very large scales. TheLas Campanas Redshift Survey seems to be presently the best one availableto answer this question. Analytic calculations show that oscillations of thecorrelation function take place only in case when the power spectrum hasa sharp transition between positive and negative spectral indexes. Whatcauses this transition? Clearly it is related to physics of dark matter in6

the very early stages of the evolution of the Universe, and thus containsinformation on processes in these early stages.Acknowledgements This study was made in collaboration with as-tronomers from Tartu and G�ottingen Observatory and Potsdam Astrophys-ical Institute. I thank my colleagues M. Einasto, P. Frisch, S. Gottl�ober, V.M�uller, V. Saar, E. Tago and R. van de Weygaert for fruitful collaborationand for the permission to use our results prior to publication.ReferencesAndernach, H., Tago, E., and Stengler-Larrea, E., 1995, Astroph. Lett &Comm. 31, 27Andernach, H., Tago, E., and Stengler-Larrea, E., 1996, (in preparation)Bahcall, N. A., Cen, R. Y., 1993, ApJ, 407, L49Batuski, D.J., Bahcall, N.A., Olowin, R.P. & Burns, J.O. 1989. ApJ, 341,599Einasto, J., Gramann, M. 1993, ApJ, 407, 443Einasto, M., 1992, MNRAS, 258, 571Einasto, M., Einasto, J., Tago, E., Dalton, G., Andernach, H., 1994, MN-RAS, 269, 301Einasto, M., Tago, E., Jaaniste, J., Einasto, J., Andernach, H., 1996 (inpreparation)Fetisova, T. S., Kuznetsov, D. Y., Lipovetsky, V. A., Starobinsky, A. A. &Olowin, R. P. 1993, Pis'ma v Astr. Zh. 19, 508 (engl. transl. in Astron.Lett. 19, 198)Frisch, P., Einasto, J., Einasto, M., Freudling, W., Fricke, K.J., Gramann,M., Saar, V., Toomet, O., 1995, AA, 296, 611Gottl�ober, S., M�ucket, J.P., Starobinsky, A.A., 1994, ApJ, 434, 417Kopylov, A. I., Kuznetsov D. Y., Fetisova T. S. & Shvarzman V. F. 1984,Astr. Tsirk. 1347, 1Kopylov, A. I., Kuznetsov D. Y., Fetisova T. S. & Shvarzman V. F. 1988, inLarge Scale Structure of the Universe, eds. J. Audouze, M.{C. Pelletan,A. Szalay (Kluwer), 129Mo H.J., Deng Z.G., Xia X.Y., Schiller P., & B�orner G. 1992a: AA, 257, 1Mo, H.J., Jing, Y.P., B�orner, G., 1992, ApJ 392, 452Mo H.J., Xia X.Y., Deng Z.G., B�orner G., & Fang, L.Z. 1992b: AA, 256,L23Peebles, P.J.E., 1980, The Large-Scale Structure of the Universe, PrincetonUniv. PressSaar, V., Tago, E., Einasto, J., Einasto, M., Andernach, H., 1995, SISSApreprint Astro-ph-950553Tully, R. B., Scaramella, R., Vettolani, G., Zamorani, G. 1992, ApJ, 388, 9van de Weygaert, R. 1994, AA, 283, 361van de Weygaert, R., Icke, V. 1989, AA, 213, 17

Figure 1. Distribution of clusters in high-density regions in supergalacticcoordinates. Only clusters in superclusters with at least 8 members areplotted. In lower panels clusters in the northern and southern galactichemispheres are plotted separately. The supergalactic Y = 0 plane coincidesalmost exactly with the galactic equatorial plane.8

Figure 2. The correlation function of clusters of galaxies. In upper panelsthe function is determined for the whole sample of 1304 clusters, in lowerpanels only 869 clusters with measured redshifts were used. The error corri-dor of the correlation function for clusters in low-density regions is markedwith long-dashed curves, and the error corridor for clusters in high-densityregions with short-dashed lines. In left panels the transition multiplicitybetween high- and low-density regions was put to i = 6, in right panels toi = 8 (see text for explanation). 9

Figure 3. The correlation functions for ten realizations of various toy mod-els of point distribution: random superclusters (upper left panel), Voronoimodel (upper right panel), chessboard model (in lover left panel the wholecubic samples, and lover right panel double conical subsamples).10

Figure 4. The correlation functions for N-body models of structure forma-tion: CDM, BSI, and double power model 2p. Solid lines show the correla-tion function for all clusters in cubic samples, short- and long-dashed curvesshow the correlation function for subsamples of clusters in high-density re-gions: in superclusters with at least 8 and 4 members, respectively. Inlover right panel the dashed curves show the correlation function for clus-ters in high-density regions (transition multiplicity i = 8) in double conicalsubsamples oriented towards x, y, and z-axes.11