AA manuscript no ASTRONOMY will b e inserted by hand later AND Your thesaurus co des are ASTROPHYSICS Evolution of the Sup erclusterVoid Network Patrick Frisch Jaan Einasto Maret Einasto Wolfram Freudling Klaus J Fricke Mirt Gramann Veikko Saar Ott Toomet University Observatory D Gottingen Tartu Astrophysical Observatory EE Toravere Estonia Space Telescope Europ ean Co ordinating Facility D Garching Europ ean Southern Observatory D Garching Princeton University Observatory Peyton Hall Princeton NJ Received th July accepted th September Abstract Recently the observed cellular nature of the ative p ower index on large scales the mass distribution largescale structure of the Universe with its quasiregular function of clusters is to o shallow In the HDM mo del no pattern of sup erclusters and voids has b een p ointed out p ower on small scales the clusterdened voids are com by several authors In this pap er we investigate prop erties pletely empty CDMmo dels have no welldened max of the initial p ower sp ectrum which lead to prediction of imum of the sp ectrum and the cellular distribution of structure consistent with these observations For this pur sup erclusters and voids is insuciently developed in this p ose we analyze the evolution of structure within four sets case of and dimensional cosmological mo dels which dier We also investigated the dynamical evolution of the in their initial p ower sp ectrum The mo dels include HDM sup erclustervoid structure The results show that the ba and CDM mo dels as well as double p owerlaw mo dels We sic sup erclustervoid network is formed very early and is discuss in detail the impact of mo del parameters such as essentially given by initial conditions the large scale and small scale p ower and the p osition and height of the maxima of the p ower sp ectra on the predicted Key words cosmology theory galaxies clustering structure Several statistical techniques were employed to largescale structure of the Universe metho ds numerical compare the mo dels with observations They include the analysis of the distribution of voids dened by rich and p o or clusters of galaxies voids dened by galaxies clus INTRODUCTION ters and sup erclusters In addition the cluster correlation function is compared Several recent studies of the largescale distribution of We conclude that the observed regular distribution of clusters of galaxies have found a quasiregular cellular pattern of sup erclusters and clusterdened voids su sup erclusters and voids can b e repro duced only if the sp ec p ervoids Broadhurst et al Bahcall Tully trum of density uctuations has a welldened maximum et al Einasto et al b hereafter EETDA Vet The wavelength of the maximum determines the scale of astro-ph/9503037 8 Mar 1995 the structure Smallscale uctuations determine the ne tolani et al ab Sup erclusters separate sup ervoids structure of the Universe Largescale uctuations mo d at fairly regular intervals The characteristic scale of the sup erclustervoid network is h Mp c The regu ulate the ne structure and determine the quasiregular lar distribution of sup erclusters of galaxies can b e seen structure on sup ercluster scales in Figures and by EETDA and Figure a of Lindner The b est agreement with observations was observed in et al where rich clusters are plotted in sup ergalac the mo del with the HarrisonZeldovich sp ectrum on large scales a p ower index n on small scales and a tic rectangular co ordinates An essential prop erty of the maximum of the p ower sp ectrum at h Mp c In distribution is the regularity of the structure near the sup ergalactic equatorial plane z there are at fairly this mo del the distribution of masses of clusters and su regular intervals six rich sup erclusters The distribution p erclusters the correlation function of clusters and the of rich sup erclusters in the vertical plane is very similar void distribution repro duce well the resp ective observed distributions see Tully et al Bahcall and EETDA As In mo dels with no p ower on large scales all sup erclus emphasized by Tully et al this distribution lo oks like a threedimensional chessboard ters are equal in mean density while in mo dels with neg P Frisch et al Evolution of the Sup erclusterVoid Network Sup erclusters are made up of a net of galaxy laments ter asso ciated with galaxies as well as individual clusters and knots clusters of galaxies The same constituents and sup erclusters is given in x In x we discuss the sta are also found in sup ervoids which contain lamentary tistical tests to b e applied Next we analyze mo dels for galaxy systems The largescale distribution of galaxies in various mo del parameters We change separately in x the the direction of the Coma and Hercules sup erclusters was p osition of the maximum in x smallscale p ower index studied by Lindner et al In front of these sup er in x largescale p ower index and in x the strength of clusters there is a large lowdensity region without any the maximum In this analysis of mo dels we use dierent rich clusters of galaxies the Northern Lo cal sup ervoid statistics sizes of voids dened by dierent ob jects the This lowdensity region embo dies numerous galaxy la mass distribution of clusters and sup erclusters the corre ments The basic dierence b etween the constituents of lation function In x we present results of the study of sup erclusters and sup ervoids lies in their richness in su the evolution of the structure in dierent mo dels The dis p erclusters systems of galaxies are much richer and denser cussion of our results is given in x Finally we summarize than in sup ervoids principal results of the study In this pap er h denotes the Hubble constant in units These observations raise several imp ortant questions of kms Mp c What causes the formation of a regular network of sup er clusters and sup ervoids Why do b oth sup erclusters and sup ervoids consist of lamentary structures of dierent MODELS richness A large b o dy of numerical simulations of the formation Model simulations of the structure in the Universe exists in the literature We use the particlemesh co de by Gramann to sim Mo dels with CDM p erturbation sp ectra have b een used ulate the evolution of the distribution of mass Most of the Efstathiou et al West et al Little and Wein investigations are done on twodimensional simulations b erg as well as simple p owerlaw mo dels Efstathiou Two dimensional mo delling is not only less demanding in et al Weinberg and Gunn Beacom et al using computer resources it also facilitates the easy in Melott Shandarin and combined CDM and HDM terpretation and graphical representation of results How mo dels Davis Summers and Schleger Holtzman and ever some asp ects of the structure can b e checked only Primack Klypin et al The formation of a by threedimensional mo delling we therefore also use a regular network of sup erclusters and sup ervoids has b een limited number of threedimensional mo dels Principal pa addressed so far only by Einasto and Gramann rameters of mo dels are given in Table They considered the problem however only qualitatively A quantitative study of the inuence of p erturbations of Table Data on mo dels used various scales using a broad set of initial conditions is still lacking Mo del D n m k L t The goal of this pap er is to mo del two observed h Mp c features of the large scale structure First we fo cus on the question what kind of initial conditions are required Mpk to repro duce the quasiregular distribution of sup erclus Mpk ters and sup ervoids Then we study which conditions are Mpk needed to explain the lamentary structure of galaxies in Mpi sup erclusters and sup ervoids Since initial conditions are Mpi given by the p ower sp ectrum our task is to nd a suitable Mpi initial p ower sp ectrum Direct determinations of the p ower sp ectrum on the scales of interest have to o large random HDM errors these scales are close to the size of the largest Mp volume for which data are available Thus we use an in Mpt direct metho d to investigate the b ehavior of the Universe Mp on large scales by mo delling the evolution of the struc ture and by comparing results with observed prop erties Np using various statistics For that purp ose we carried out CDM Nb o dy simulation to simulate the distribution of matter CDM Since the aim is the study of the inuence of largescale mo des of density p erturbations we consider a computa tional b ox which is considerably larger than the scale of Designations in the table are as follows D is the di the maximum of the p ower sp ectrum mension of the simulation is the density parameter n and m are eective p ower indices of the sp ectrum on large The pap er is organized as follows A description of the and short wavelengths resp ectively index indicates that mo del simulations and rules to identify the clustered mat P Frisch et al Evolution of the Sup erclusterVoid Network the sp ectral index corresp onds to dimensional case the designated as Mpi Mpi Mpi the last number cor index in dimensional case used in d calculations is resp onds to the absolute value of the sp ectral index in n n and m m k is the wavenumber cor dimensional case actual simulations have b een p erformed t resp onding to the maximum of the sp ectrum wavenumber in D and the index is resp ectively lower in absolute k corresp onds to the size of the computational b ox value Instead of a smo oth