void Patrick Gramann Abstract Received th July accepted th September ters are equal in mean density while indistributions mo dels with neg pattern largescale structure of the Universe with its quasiregular Evolution analysis compare the mo dels withstructure observations Several They statistical include techniquesheight were the of the employed maxima to of thethe p ower sp ectra large on scale the and predicted smalldiscuss scale in p ower detail and the theand p osition impact CDM and mo dels of as mo del wellin parameters as their such double initial p owerlaw as p ower mo dels sp ectrumof We The mo dels and include dimensional HDM p cosmological ose we mo dels analyze which the dier evolution ofstructure structure consistent within with four these sets observations of the initial p owerby sp ectrum several authors which In lead this pap er to we prediction investigate prop erties of p erclusters maximum of the this mo del the p o or clusters structure the structure Smallscale uctuationsThe determine wavelength the of ne thetrum maximum of determines density the uctuations hassup scale erclusters a and of voids welldened can maximum b e repro duced only if the sp ec function is compared ters and sup erclusters In addition the cluster correlation scales the mo del with the HarrisonZeldovich sp ectrum on large structure on sup ercluster scales ulate astro-ph/9503037 8 Mar 1995 Your thesaurus co des are will b e inserted byAA hand manuscript later no Princeton University Observatory PeytonEurop Hall ean Princeton Southern NJ Observatory D Space Garching Telescope Europ eanTartu Co ordinating Astrophysical Facility Observatory D EE Estonia Garching Toravere University Observatory D Gottingen In mo dels with no p ower on large scales all sup erclus We conclude that the observed regular distribution of The b est agreement with observations was observed in distribution the a of of p ower of ne Frisch sup erclusters the Recently the the the of galaxies voids dened structure Veikko Saar distribution of masses of clusters distribution index correlation Universe p ower repro duce Jaan of n sp ectrum and and the observed Largescale Einasto function of well determine Ott Toomet voids has voids the Sup erclusterVoid on at cellular dened of resp ective small uctuations b een by galaxies clus Maret the clusters nature scales quasiregular h For this pur by p ointed observed rich Einasto Mp c In and and su of and mo d and out the the a at fairly regular intervals The characteristic sup erclustervoid lar tolani regular distribution pletely p ower on small scalesfunction the of clusters clusterdened is to o shallow In the HDM mo del no et al pattern like a threedimensional chessboard see of rich sup erclusters Key essentially given by initial conditions sic sup erclustervoid sup erclustervoid structure The results show that the ba sup ergalactic tic rectangular et al in Figures and by EETDA and Figure a of Lindner case sup erclusters imum ative p ervoids clusters Several INTRODUCTION emphasized by Tully largescale structure of the Universe metho ds numerical Wolfram We distribution Tully where rich clusters are plotted in sup ergalac words p ower Einasto et of also empty intervals recent of of al the et Broadhurst Network galaxies sup erclusters investigated is index al ab Sup erclusters Freudling and voids is insuciently developed in this equatorial sp ectrum cosmology theory the studies CDMmo dels co ordinates Bahcall of six network on sup erclusters regularity et al have in the vertical plane is very similar rich network is formed very early and is et large of et al plane and the sup erclusters and b hereafter found the is al this distribution lo oks An scales Klaus dynamical the have of largescale clusterdened z essential of a the cellular the no quasiregular galaxies galaxies h Bahcall separate J structure ASTROPHYSICS there welldened mass ASTRONOMY and The evolution Mp c Fricke prop erty distribution distribution voids are com EETDA Vet AND are EETDA can clustering distribution distribution scale sup ervoids voids The near at b e cellular of of the of the Mirt fairly Tully regu max seen su the the As of of

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 transition b etween large and

L is the disp ersion of mass density uctuations on small wavelengths in we use a sharp one

h Mp c scale

m

A if k k

t

In most cases we use on large scales the Harrison

P k

Zeldovich sp ectrum with index n We assume see

n

A if k k

t

x that the maximum of the sp ectrum lies at wave

length h Mp c This wavelength is our scaling

The HDM mo del also b elongs to this series as the extreme

t

parameter ie the length scale of the mo dels is dened by

case having no p ower on small wavelengths We approxi

the maximum in the measured p ower sp ectrum

mate the sp ectrum with HarrisonZeldovich index n

Twodimensional mo dels were calculated using

on large scales and accept a sharp truncation at wavenum

cells and particles To investigate the inuence of spatial

b er k

t

resolution part of the calculations were carried twice using

In the third series of mo dels we vary the scale of the

cells and particles The spatial resolution of these

maximum of the power spectrum k Resp ec

t

mo dels is h Mp c and h Mp c resp ectively In

tive mo dels are designated as Mpk Mpk Mpk

dimensional mo dels we use cells and particles the

We use again a sharp transition b etween short and large

spatial resolution is h Mp c Our exp eriments with

wavenumbers These mo dels corresp ond to dierent

dimensional mo dels have shown that this resolution is

sizes of b oxes L since we assume that the scale of the

sucient to isolate clusters and sup erclusters of galaxies

maximum of the sp ectrum is xed

t

To investigate the inuence of dierent parameters of

The last series of mo dels is p erformed to investigate the

the sp ectrum in detail we have p erformed four series of

inuence of the height of the maximum of the power spec

simulations

trum The strong maximum case corresp onds to a sharp

In the rst series we varied the power on large scales

transition b etween the two p ower laws the weak max

The rst mo del has a simple p ower law mo del with density

imum case was taken according to a standard CDM sp ec

sp ectrum

trum We use the sp ectrum in the form given by Gramann

m

P k A

k

P k A

Here k k k L is the dimensionless wavenum

t

2

ak

b er is the wavelength k L is a normalizing

t t

logbk

wavenumber A is the amplitude m is the p ower index

In this formula the parameters are as follows a

The wavenumber k is taken equal to unity for the whole

h b h h is the Hubble constant

b ox length L This one p ower law mo del is designated as

is the density parameter in units of the critical closure

Mp

density and the wavenumber is dimensional k L

In the second mo del designated as Mpt we use a

The values of constants a and b corresp ond to cub e size

truncated p ower law

L expressed in h Mp c In calculations we have adopted

m

A if k k

t

the Hubble constant h

P k

If then the maximum of the sp ectrum is lo

if k k

t

cated at the wavelength h Mp c The maxi

t

mum of the sp ectrum is very shallow therefore we do

In this mo del k k k L is the truncation

t t t

dimensional simulations only using two values of the den

wavenumber and is the corresp onding wavelength

t

sity parameter mo dels CDM and CDM The p ower

The third mo del of the rst series is a double p ower

index in dimensional case diers by one unit from the

law mo del with a smo oth transition designated as Mp

dimensional case and for a shallow maximum it shifts

n

to a dierent wavelength thus the interpretation of

P k A

nm

dimensional mo dels in resp ect of the maximum wave

If n m then indices n and m characterize the sp ec length is dicult For the lowdensity mo del CDM we

trum at small and large k resp ectively at small wavenum assume the presence of a cosmological term where

nm

b ers and we get a simple p ower law with in For comparison with dimensional simu

dex n at large wavenumbers we can ignore in the de lations we calculate also a dimensional twopower law

nominator and have a simple p ower law with index m mo del Np using the sp ectrum

In the second series of mo dels we vary the power in The initial sp ectrum was generated in all mo del fam

dex on smal l scales m only The resp ective mo dels are ilies using identical particle distributions thus mo dels of

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig Sp ectra at the present ep o ch for all mo dels considered Observed mass sp ectrum is shown by dots Gramann and

Einasto Einasto Gramann and Saar The wavenumber k corresp onds to the size of the computational b ox The

observed galaxy sp ectrum is reduced to the mass sp ectrum It lies b elow the mo del sp ectra by a factor of due to overcorrection

for the bias factor in these pap ers

the same family corresp ond to the same realization of the resp onds to a disp ersion The disp ersion grows

structure In this case the dierences in the structure de almost linearly with the expansion factor a thus the rst

p end only on the parameter which is changed in the par ep o ch corresp onds approximately to a redshift z

ticular family From the density p ower sp ectrum we calculate also the

disp ersion on an h Mp c scale for the last ep o ch

On small scales initial uctuations were generated by

Sp ectra for all mo dels for the present ep o ch are plotted

a gaussian random pro cess In order to minimize the irreg

in Figure As noted ab ove the present ep o ch was iden

ularities caused by the small number of largescale mo des

tied in dimensional mo dels by the density uctuations

for large scales the initial amplitude of a mo de was xed

on the h Mp c scale in d mo dels by uctua

by the value of the sp ectrum according to the analytic

tions on the h Mp c scale Fluctuations on

expression while the sign of the amplitude was

h Mp c scale were calculated from the sp ectrum for

taken at random

the last ep o ch

We are also interested in the evolution of the structure

For this purp ose we identify four ep o chs characterized by

rms disp ersion of spatial density uctuations on the cell

Identication of galaxies

scale h Mp c Einasto et al a de

rived the disp ersion on the scale h Mp c for a sample The resulting distribution of particles in the mo del is in

in the Virgo sup ercluster After correction to the mat terpreted as the distribution of matter in the Universe

ter density uctuation they found a value for the which includes b oth luminous and dark matter In order

present ep o ch This disp ersion is characteristic for den to identify galaxies we assume that galaxy formation o c

sity smo othing on scales h Mp c If the density is curs only in regions where the density exceeds a certain

smo othed on h Mp c scale then the present ep o ch cor threshold Both observations Einasto et al a and

P Frisch et al Evolution of the Sup erclusterVoid Network

numerical simulations of galaxy formation Cen and Os clusters is chosen in such a way that the spatial density

triker suggest that the threshold density for galax of p o or clusters is four times higher than the density of

ies to form is close to the mean density of matter if Ab ell clusters The actual density of nearby Zwicky clus

smo othed on the scale of typical diameters of galaxy sys ters is times higher than the density of Ab ell clusters

tems R h Mp c In regions of lower density there but since not all of our mo dels contain such a high num

g

are no galaxies b er of clusters we use this number appropriate for Zwicky

clusters

In addition we assume that in regions with density higher

than the mean density galaxies follow the matter distribu

tion Available observational data supp ort this assumption

Identication of superclusters

Vennik Hughes Under these assumptions all

test particles in these regions can b e considered as galax

Sup erclusters of galaxies were identied from the den

ies Dep ending on the size of the computational b ox and

sity after gaussian smo othing Sup erclusters were found as

the number of particles in a particular simulation every

highdensity regions in the density eld with a smo othing

particle corresp onds to a galaxy of certain mass including

length R This smo othing length was chosen considerably

b

the dark corona which contains most of the mass of the

larger than the smo othing length R h Mp c used

g

galaxy In dimensional mo dels with k there is one

in identifying clusters of galaxies

t

particle in a cell of size h Mp c and such a particle

The smo othing length R introduces another parameter to

b

corresp onds to a mass M a dwarf galaxy

our mo del To study the inuence of the smo othing length

In dimensional mo dels with k there is one particle

to the sup ercluster denition we test three values R

t

b

in a cell of size h Mp c which corresp onds to a mass

and h Mp c In Figure we plot equidensity con

M a massive giant galaxy

tours for all three smo othing lengths These calculations

The smo othing length R is an additional parameter to

show that the mean density contours the lowest contour

g

the mo del Einasto et al a have demonstrated that

level plotted are almost indep endent on the smo othing

the exact choice of this parameter has little inuence

parameter R provided R R In the following we

b b g

on the comparison b etween dierent mo dels In addi

use smo othing length R h Mp c to identify sup er

b

tion trial calculations with mo dels of lower resolution

clusters To avoid merging of nearby sup erclusters we use

R h Mp c have shown that prop erties of mo dels

a density threshold D higher than the threshold

g

t

with either resolution are virtually identical

to divide the matter into the clustered and nonclustered

comp onents

Another p ossibility to dene sup erclusters is select

Identication of clusters of galaxies

ing them from the cluster distribution using the cluster

analysis as used by EEDTA to investigate the largescale

Clusters of galaxies are identied as p eaks in the den

distribution of clusters and sup erclusters For the present

sity eld In calculating the density eld we use gaussian

analysis we prefer the use of the smo othed density eld

smo othing with disp ersion R h Mp c Cluster prop

g

In dimensional mo dels we have used b oth metho ds to

erties are determined by tting an ellipsoid to the density

dene sup erclusters clusters lo cated in rich sup erclusters

eld at the maximum The center of the ellipsoid gives

with at least rich clusters as members are plotted in

the p osition of the cluster and the volume ab ove a certain

Figure by lled circles

threshold density of the ellipsoid the cluster mass This

threshold density is chosen so that the cluster has dimen

sions comparable to real clusters of galaxies Using a trial

STATISTICAL TESTS

and error pro cedure we found that a density threshold in

units of the mean density D eciently separates

t

Mass distribution of clusters of galaxies

clusters of galaxies from eld galaxies If the threshold

In order to compare our cluster catalogue with observa

density is to o low then the resulting clusters are to o large

tions we have calculated the cluster mass function The

in volume and have an elongated shap e this threshold cor

observed cluster mass function found by Bahcall and Cen

resp onds already to a galaxy lament The p eak density

is given as the integrated function

in clusters of galaxies is several hundred in mean density

Z

units and the rise of the density near the p eak is very

1

rapid Thus the resulting cluster catalogue is not sensitive

N M nmdm

M

to the exact level of the threshold density

We distinguish b etween p o or and rich clusters of galax

where nmdm is the number of clusters in the mass inter

ies dep ending on the mass of the density enhancement

val m to m dm The mass function can b e represented

The transition density b etween p o or and rich clusters is

by the following Schechtertype analytic expression

chosen so that their spatial density agrees with those ob

M

served for Ab ell clusters of galaxies h Mp c



expM M N M a



Bahcall and Cen The lower mass limit of p o or M

P Frisch et al Evolution of the Sup erclusterVoid Network

1

Fig Equidensity contours of the mo del Mp with density smo othing at and h Mp c smo othing length from left

to right Note the approximate similarity of the lowest contour in all three cases

Table Parameters of the Mass Distribution Function



Mo del a M

h Mp c M

Mpk

Mpk

Mpk

Mpi

Mpi

Mpi

HDM

Mp

Mpt

Mp

Np

Fig The distribution of sup ercluster masses Dots note the

CDM

observed mass distribution calculated on the basis of the su

CDM

p ercluster catalogue by EETDA

To characterize dierent mo dels we also use the dis



Parameters a and M are found for all mo dels and are

tribution of masses of sup erclusters Results of this com

given in Table By comparing mo dels with observations

parison for mo dels Mp Mpt and Mp are given in Fig



we found parameters a and M in physical units

ure Sup ercluster masses are calculated as describ ed in

x using the density eld smo othed on an h Mp c

The cluster mass function for mo dels is plotted in Fig

scale Masses are expressed in units of the mean mass of

ure together with the observed function The Figure

sup erclusters for the mo del Mpt

shows that our mo dels repro duce the observed mass func

tion surprisingly well Only in the case of p o or clusters of A prop er estimate of the masses of real sup erclusters

galaxies the number of clusters in mo dels is smaller than is not obvious For this we need mass estimates of a large

given by observations In the following we shall use basi number of sup erclusters either from the velocity eld or by

cally rich clusters of galaxies For rich clusters there is no counting the number of galaxies in resp ective sup erclus

systematic deviation of mo dels from observations for the ters Presently there is to o little information available for a

ma jority of mo dels representative sample of sup erclusters As a rst approxi

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig Cluster mass distribution function Dots represent the observed function according to Bahcall and Cen lines

various mo dels see the Figure Note the similarity of mass distribution for mo dels with dierent largescale p ower upp er left

panel and with variable scale length of the maximum lower left panel

mation we can estimate the mass of sup erclusters from the Table Parameters of the Correlation Function

number of rich clusters present in them These data can

Mo del r r

max

b e extracted from the recent catalogue of sup erclusters by

EETDA The distribution of sup ercluster masses again

Mpk

in units of the mean masses of sup erclusters is given in

Mpk

Figure

Mpk

Mpi

Mpi

Correlation function and power spectrum

Mpi

HDM

In Figure we plot the p ower sp ectra for all mo dels for

the present ep o ch In the panel for dimensional mo dels

Mp

we give also the observed p ower sp ectrum according to

Mpt

Gramann and Einasto and Einasto et al

Mp

other recent determinations by Peacock and West

Np

Fisher et al Park et al and Peacock and

CDM

Do dds give similar results

CDM

We have calculated the mass and cluster correlation

functions for all of our mo dels Clusters are divided into

two richness classes using mass limits as dened by the

has the value r was calculated from the mass mass distribution function The correlation length r de

correlation function and is given in Table ned as the distance for which the correlation function

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig The correlation function Dots are for the observed correlation function for rich clusters of galaxies Einasto et al

1

for the h Mp c sample in the Northern sup ergalactic hemisphere For mo dels the correlation function of p o or clusters of

galaxies is given

The cluster correlation function has a weak secondary secondary maximum is absent in the case of randomly lo

maximum at h Mp c as found for the Northern cated voids

sup ergalactic cluster sample by Einasto et al Most

These toy mo dels demonstrate that the secondary

cluster correlation functions of our simulated samples also

maximum is due to the presence of a quasiregular pattern

have a weak secondary maximum at r This value is

in the distribution of clusters The observed secondary

max

given in Table the correlation function for simulated

maximum in the cluster correlation function is not very

clusters for all mo del samples is given in Figure The

strong thus the regularity is not pronounced The mass

observed correlation function for Ab ell clusters is taken

correlation function has no secondary maximum excep

from Einasto et al

tions are only mo dels Mpi and HDM where b oth the

mass and cluster correlation functions have a secondary

maximum at the same lo cation

To understand b etter the geometrical origin of the

secondary maximum we constructed following Einasto

a series of toy mo dels The analysis of toy mo dels

Void diameter statistics

will b e published elsewhere here we summarize only the

main results of this study Toy mo dels have a builtin scale One kind of statistics we use in this pap er is the mean di

h Mp c either in the form of a rectangular grid ameter distribution of voids dened by dierent ob jects

or as a mean separation b etween voids A sharp secondary rich and p o or clusters of galaxies and single galaxies In

maximum is observed in all mo dels with a xed grid the the case of galaxies all test particles ab ove the threshold

strength of the maximum dep ends on the distribution of density D were considered as galaxies Rich and

t

clusters whether they are lo cated at grid corners edges p o or clusters were dened as describ ed ab ove Rich clus

or surfaces The maximum is shallower if the grid size is ters corresp ond to Ab ell clusters p o or clusters to nearby

changing within certain limits around the mean value The Zwicky clusters

P Frisch et al Evolution of the Sup erclusterVoid Network

Voids were found using the empty sphere metho d by Diameters of galaxydened voids were calculated for

Einasto et al the volume under study was divided a series of dimensional simulations for three dierent

into small cubic cells of size l Lk where L is the size ep o chs separately for high and lowdensity regions in

of the whole volume under study and k is a resolution side and outside of sup erclusters Results are given in

parameter Then for each cell center the distance to the Table We give also the scatter of diameters this num

nearest galaxy in the whole sample was found Cells where b er is not to b e confused with the statistical error of void

this distance has a maximum are lo cated in centers of diameters

voids and the resp ective distance corresp onds to the void

radius The void search algorithm nds any maxima in

the distance eld We want to study welldened voids

THE INFLUENCE OF THE POSITION OF

ie voids surrounded from all sides by galaxies To avoid

THE MAXIMUM OF THE SPECTRUM

illdened voids we exclude from the void sample all voids

with center co ordinates close to sample b oundaries

Mo dels Mpk through Mpk can b e interpreted in two

Most voids are elongated and our program identies

dierent ways One interpretation is that the physical size

several close voids with almost identical void radius Lind

of these three mo dels is the same say h Mp c and

ner et al studied the eect of overlapping voids

the maximum of the sp ectrum has three dierent val

t

and demonstrated that this phenomenon has little eect

ues h Mp c and h Mp c The second

on mean void diameter statistics Thus we can ignore this

interpretation is that h Mp c but the size of the

t

eect

b ox L changes from h Mp c to h Mp c

A technical parameter in void search is the resolution

In this Section we use the former interpretation in order

parameter k Lindner et al studied the inuence of

to investigate whether prop erties of the largescale struc

this parameter and showed that statistically the mean void

ture can b e changed simply by tuning the length scale of

diameter is indep endent of the value of this parameter

the initial p ower sp ectrum

however the higher this parameter the smaller are random

In upp er panels of Figure we give the distribution of

uctuations of results For this reason we have used high

particles representing galaxies in lower panels the distri

values k and k in and dimensional cases

bution of the smo othed at R h Mp c smo othing

resp ectively

b

length density eld We see clearly the concentration of

The size of voids dened by clusters dep ends critically

simulated galaxies and sup erclusters into circular struc

on the number density of clusters In the dimensional

tures which are similar to structures observed by Vettolani

case we use the number density of rich clusters Ab ell and

et al a b in a deep dimensional survey of

ACO clusters of all richness classes found by Bahcall and

faint galaxy redshifts The scale of these structures in the

Cen h Mp c the density of Zwicky

mo dels varies in mo del Mpk the circular structures have

clusters is times higher

a diameter ab out half of the computational b ox in mo del

Mean diameters of voids for a Poisson distribution of

Mpk the structures are ab out two times smaller and in

particles are for simulated galaxies in our dimensional

mo del Mpk they are smaller by another factor of two

simulations h Mp c for simulated Ab ell and Zwicky

clusters h Mp c and h Mp c resp ectively Ac

Now we consider the cluster correlation function The

tual mean diameters of voids are larger thus in the

secondary p eak of the correlation function characterizes

dimensional case the void statistics is meaningful

the mean distance b etween neighboring sup erclusters In

In our dimensional calculations the mean diameter

Figure for these mo dels the correlation function is plot

of galaxydened voids for a Poisson distribution is

ted under the assumption that the size of the computa

h Mp c that of rich clusterdened voids ab out

tional b ox for all mo dels is h Mp c We see that in

h Mp c approximately equal to the mean size of voids

mo del Mpk the secondary p eak of the correlation func

in the observed cluster distribution EEDTA Thus the

tion is lo cated at h Mp c In mo del Mpk we

number density of clusters in dimensional simulations is

see a maximum at h Mp c in mo del Mpk at

to o low to apply the void statistics For this reason we

distance h Mp c A lo ok at Figure conrms that

give in Table mean diameters of clusterdened voids

these maxima corresp ond to sup ercluster distances across

only for dimensional mo dels

voids

The observed mean distance b etween sup erclusters

Table Mean diameters of cluster dened voids

across voids is h Mp c regardless of the p osition

of the region on the sky In other words this scalelength

Mo del D D

r ich poor

is an invariant observational parameter This means that

the p osition of the maximum in the sp ectrum is not a

Np

free parameter Because it denes the scale of the cellu

CDM

lar structure of the largescale distribution of clusters and

CDM galaxies it is xed by the observations

P Frisch et al Evolution of the Sup erclusterVoid Network

1

Fig The distribution of simulated galaxies and the density eld smo othed with disp ersion h Mp c in mo dels Mpk

Mpk and Mpk

INFLUENCE OF SMALLSCALE POWER During the evolution of the HDM mo del also small

scale uctuations form and at the present ep o ch the slop e

of the sp ectrum in the HDM mo del on small scales is

To investigate the inuence of the change of the amplitude

m close to the slop e in mo del Mpi see

of smallscale uctuations we use mo dels Mpi Mpi

Figure and further discussion of the evolution in x

Mpi and HDM In these mo dels the initial amplitude of

The structure of these mo dels is however very dierent

smallscale p erturbations with resp ect to mediumscaled

In mo del Mpi middle panel of Figure galaxy la

p erturbations decreases when we move to larger negative

ments are lo cated over the whole space including sup er

p ower index and is absent in the HDM mo del

voids while in the HDM model there are no galaxies in

The basic parameter which varies in this series of mo d

supervoids In other words small scale p ower which forms

els is the number of massive clusters relative to the number

in the HDM mo del during the evolution is generated by

of lowmass clusters This change can b e characterized by

structures lo cated in highdensity regions lowdensity re

the slop e of the mass function see Table and Figure

gions remain empty

Mo del Mpi agrees b est with observations In other mo d

els the slop e of the mass function is to o small ie there

This dierence in the evolution can b e expressed also

are to o many very rich clusters in the mo del The HDM

by the statistics of galaxydened voids Mean sizes of

mo del deviates most from the observations This dierence

galaxydened voids are given in Table separately for

in mean masses of clusters can b e seen in the distribution

high and lowdensity regions within sup erclusters and

of particles shown in Figure

sup ervoids This table demonstrates that in the HDM

mo del galaxydened voids in lowdensity regions have the Another asp ect of smallscale uctuations is the pres

same size as clusterdened voids in other mo dels This re ence of galaxies in clusterdened voids sup ervoids If

sult is also exp ected visual insp ection of the distribution initially there is no p ower on small scales then in sup er

of simulated galaxies shows a complete absence of galax voids no galaxies form at all Our HDM mo del corresp onds

ies and clusters of galaxies in sup ervoids In highdensity just to this case

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig The distribution of simulated galaxies and the density eld in mo dels Mpi Mpi and HDM

Table Mean diameters of galaxy dened voids

compare sp ectra at the present ep o ch also the evolution

ary history of the structure is imp ortant

Mo del D D F

low hig h low

Mp

INFLUENCE OF LARGESCALE POWER

Mp

Mp

To investigate the inuence on the largescale p ower to

Mpt

the formation of the structure we use mo dels Mp Mp

Mpt

and Mpt These mo dels dier by the strength of the

Mpt

largescale p ower In mo del Mp the sp ectrum on large

scales has the same index as on small scales in this mo del

Mp

long wavelength p erturbations have the largest amplitude

Mp

In mo del Mp the largescale p ower sp ectrum has the

Mp

HarrisonZeldovich index n in mo del Mpt the am

HDM

plitude of the sp ectrum on large scales is zero

HDM

The parameter most inuenced by the change in the

HDM

large scale p ower is the mass of sup erclusters Sup er

clusters were dened using a smo othing length R

b

h Mp c and a density threshold D x The

t

regions galaxydened voids are much smaller than in all

distribution of sup ercluster masses is shown in Figure

other mo dels

for all three mo dels considered Masses are expressed in

This analysis shows that lamentary structure of units of mean masses of sup erclusters for mo del Mpt We

galaxies in lowdensity regions develops only if there is see that the distribution of sup ercluster masses is rather

some p ower on small scales initial ly It is not sucient to dierent The truncated mo del Mpt has sup erclusters of

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig The distribution of simulated galaxies and the density eld in mo dels Mp Mpt and Mp

approximately equal masses the disp ersion of masses ex tary structure is similar in all mo dels What diers is the

pressed in units of mean masses of sup erclusters in mo del strength of laments A close insp ection of plots shows

Mpt is only In mo del Mp some sup erclusters have that some laments which are lo cated in the Mpt mo del

masses exceeding mean masses by a factor of six In mo del in highdensity regions are in mo dels Mp and Mp sit

Mp largest sup erclusters have masses over ten times the uated in lowdensity regions This dierence is caused by

mean mass a change in the p ower of largescale mo des This demon

strates that large wavelength p erturbations mo dulate the

The observed mass distribution in sup erclusters using

strength of galaxy systems formed by smallscale p ertur

the number of rich clusters as an indicator of the sup er

bations

cluster mass is shown in Figure by lled circles Here

we use the data from EEDTA Although the metho ds to

determine sup ercluster masses see x were dierent

the general trend of the mass distribution is similar to the

Void statistics shows that voids dened by galaxies

trend we see in the mo dels The mo del Mp agrees b est

have almost identical size for all three mo dels see Ta

with observations In mo del Mpt there are to o few su

ble This result demonstrates that galaxydened voids

p erclusters with large mass in mo del Mp there are to o

are insensitive to the largescale p erturbations Similarly

many

a lo ok to Figure shows that largescale mo des do not

inuence the mass distribution of clusters

The distribution of simulated galaxies and density con

tours for all three mo dels is shown in Figure Visual in

sp ection of the distribution suggests that sup erclusters in

the mo del Mpt have almost equal masses In the mo del

To summarize our analysis we can say that most quan

Mp the disp ersion is larger and in the mo del Mp the

titative tests are insensitive to changes in the largescale

largest

mo des of density p erturbations The mass of sup erclusters

Since we have used identical random numbers to gen is the only parameter which dep ends primarily on these

erate the initial sp ectrum on small scales the lamen mo des

P Frisch et al Evolution of the Sup erclusterVoid Network

Fig The distribution of simulated galaxies upp er panels and clusters of galaxies lower panels in mo dels Np CDM and

1

CDM A h Mp c thick sheet of dimensional simulations has b een plotted Clusters lo cated in rich sup erclusters with at

least four member clusters are plotted as lled circles isolated clusters and clusters in p o or sup erclusters as dots

INFLUENCE OF THE HEIGHT OF THE distance is much larger than in the real Universe In mo del

MAXIMUM OF THE SPECTRUM CDM sup erclusters are distributed more regularly and

the number of rich sup erclusters is comparable with the

number of real rich sup erclusters However there is no

To study the inuence of the height of the maximum of

preferred scale of the sup erclustervoid network Only in

the sp ectrum we use two CDM mo dels and a double p ower

mo del Np we see a regular pattern with constant scale

law mo del with a sharp transition of the p ower index eqn

This visual impression is conrmed by the cluster cor

relation functions which has no real minimum and sec

The distribution of simulated galaxies in b oth CDM

ondary maximum in mo del CDM a weak minimum with

mo dels and the Np mo del is shown in Figure Slices

out a secondary maximum in mo del CDM and a well de

h Mp c thick are plotted The presence of high and

ned minimum and shallow secondary maximum in mo del

lowdensity regions is well seen in all three mo dels Since

Np

a rather thick sheet is shown and the resolution of the

calculations is four times lower than in the dimensional

The cluster correlation function is the basic quanti

mo dels the distribution of particles is much smo other

tative statistics which is sensitive to dierences in the

than those plotted in Figure

largescale distribution of sup erclusters observed in mo d

In lower panels of Figure we show the distribution els CDM CDM and Np The mean diameter of cluster

of rich clusters of galaxies in these three mo dels Clusters dened voids in mo del CDM is considerably smaller than

lo cated in rich sup erclusters are plotted as lled circles in the other two mo dels and is practically equal to the di

The distribution of galaxies and rich clusters of galaxies ameter of voids in a Poisson sample with the same number

is rather irregular in mo del CDM In this mo del there density of ob jects This test also demonstrates the dier

are only a few very rich sup erclusters as measured by the ence b etween the mo del CDM on the one side and mo d

number of rich clusters in sup erclusters and their mutual els CDM and Np on the other Other quantitative tests

P Frisch et al Evolution of the Sup erclusterVoid Network

cluster mass function mass correlation function show no

signicant dierences b etween these three mo dels

The absence of a clear pattern in the distribution of

sup erclusters is a generic prop erty of all mo dels with a

shallow maximum as it is the case with CDM mo dels

The mo del CDM has more p ower on large scales but the

maximum of the sp ectrum is not well dened thus the

distribution of rich clusters is organized not so well as in

mo dels with a sharp maximum mo del Np

EVOLUTION OF THE SUPERCLUSTER

VOID NETWORK

To investigate the evolution of the structure we have

stored the distribution of mo del particles at four ep o chs

corresp onding to density disp ersions and on

the h Mp c scale In order to illustrate the evolution

of the sp ectrum we plot in Figure sp ectra of these

ep o chs for two mo dels Mp and HDM The evolution of

the sp ectrum for the rest of the mo dels is very similar

to the evolution of the mo del Mp We see that sp ectra

at dierent ep o chs of the mo del Mp run almost parallel

to each other ie the evolution is basically linear In the

HDM mo del the growth of the sp ectrum on small scales is

much more rapid here a p owerlaw sp ectrum with index

m forms this corresp onds to index m in the

threedimensional case

The distribution of simulated galaxies in mo del Mp

for three ep o chs corresp onding to density disp ersion

Fig The evolution of sp ectra of mo dels Mp and HDM

and is shown in Figure We plot in this Figure also

The solid curve denotes initial sp ectrum dotted and dashed

the density eld smo othed on R h Mp c scale for

b

lines sp ectra at time steps and as measured by the

all three ep o chs

density disp ersion

Mean void diameters for simulated galaxies for these

ep o chs are given for several mo dels in Table Void di

This comparison shows that clusters of galaxies are still

ameters were found separately for voids in sup erclusters

in the pro cess of b eing formed while their richness grows

and in sup ervoid regions

steadily

Our results show that in all mo dels voids b etween l

The comparison of the smo othed with disp ersion

aments in sup ercluster regions are smaller than in low

h Mp c density eld shows that all sup erclusters ob

density regions The diameter of voids increases with time

served in the present ep o ch can b e followed already in the

This increase is due to two dierent pro cesses First dur

ep o ch only the density contrast is much smaller All

ing the evolution smaller voids join to form larger ones

highdensity regions can b e easily identied at all ep o chs

compare the distribution of particles at ep o chs and

During the evolution the density contrast increases but

in Figure and second the dimensions of walls b etween

dimensions of sup erclusters are rather stable We see only

voids shrink The dierence of void diameters in high and

a very small contraction of laments and expansion of

lowdensity regions increases with time in other words

voids

voids in lowdensity regions expand faster than in high

density regions

The particle distribution in Figure shows that at

DISCUSSION

the ep o ch clusters of galaxies start to form This

ep o ch can b e identied with the b eginning of the inten

Inuence of the maximum of the spectrum

sive galaxy formation Most of the matter in this ep o ch

A quasiregular network of sup erclusters is p ossible only if is still distributed rather smo othly In the second ep o ch

there is a welldened maximum of the p ower sp ectrum the clusters are b etter seen but only in the third ep o ch

Our mo dels Mpk and Np with a sharp transition of which corresp onds to the present time we see a well de

the p ower index probably exaggerate the regularity of the veloped network of clusters laments and sup erclusters

P Frisch et al Evolution of the Sup erclusterVoid Network

1

Fig The evolution of the distribution of simulated galaxies and smo othed density eld smo othing length h Mp c The

lowest density contour corresp onds to mean density level

structure cf the distribution of clusters in rich sup erclus see Kaiser and Peacock for the discussion of this

ters in these mo dels Figures and and in observations problem

Figure by EETDA On the other hand if the maximum

One p ossibility to derive the scale of the maximum of

of the sp ectrum is very shallow as in CDMmo dels then

the p ower sp ectrum is to use data on the correlation func

the distribution of sup erclusters is rather irregular The

tion of rich clusters of galaxies on large scales The scale of

b est agreement with observations has probably the mo del

the secondary maximum is related to the scale of the max

with a smo oth but rapid transition b etween p ositive and

imum of the sp ectrum However a direct relation b etween

negative sp ectral index approximated by formula

these scales is valid only in analytic mo dels Einasto and

Gramann Our simulations and the study of various

The comparison of mo dels with dierent scale of the

toy mo dels has shown that in realistic situations this re

maximum wavelength demonstrates clearly that the mu

lationship is only approximate In many simulations the

tual distance b etween sup erclusters and the scale of the

secondary maximum of the cluster correlation function is

sup erclustervoid network is determined by the scale of

very shallow and the scale of the secondary maximum

the maximum of the sp ectrum

do es not coincide exactly with the scale of the maximum

in the p ower sp ectrum

An imp ortant problem is the determination of the p o

sition of the maximum of the sp ectrum from observations We b elieve that a combined approach which takes

Direct determinations of the sp ectrum on large scales have into account all available direct and indirect data is

large random errors since the size of observational samples probably the b est in the determination of the scale

t

is close to the scale of interest In principle deep one and Best available estimates for the scale of the maximum

twodimensional samples can give b etter results However come from the deep p encilb eam survey by Broadhurst

not all one or twodimensional samples cross sup ervoids et al from the deep twodimensional survey by

through central regions thus in most cases only little in Vettolani et al ab from the distribution of sup er

formation on the actual p ower sp ectrum can b e extracted clusters and voids by EEDTA from the secondary maxi

P Frisch et al Evolution of the Sup erclusterVoid Network

lowdensity regions sup ervoids In that study we inves mum of the correlation function of rich clusters of galaxies

tigated the Northern Lo cal Void but plots of galaxies in by Einasto and Gramann and from the study of

other regions of the sky demonstrate that this is a common the second derivative of the correlation function by Mo

prop erty of sup ervoids et al a b Based on these determinations we

adopted h Mp c The formal error of this pa

t

Our study shows that the presence of some initial

rameter as estimated from the scatter of individual de

p ower on small and intermediate scales in the mo dels is

terminations is rather small h Mp c The actual

crucial for the development of ne structure in sup ervoids

error due to cosmic scatter and p ossible systematic errors

Mo dels with some p ower on small scales can b e called

is much larger A new determination of this parameter is

CDMtype The ne structure is absent in all HDMtype

an imp ortant tasks of the observational cosmology

mo dels where there is no p ower on small scales initially

The formation of some p ower on small scales in later

ep o chs do es not change the situation our calculations

Inuence of largescale density perturbations

show clearly that ne structure forms only in highdensity

regions of HDM mo dels

Simple p owerlaw mo dels are essentially scalefree Prop

Another manifestation of the dierence b etween CDM

erties of these mo dels have b een studied by Efstathiou

and HDMtype mo dels is the void diameter statistics

et al Weinberg and Gunn and others Our

Galaxydened voids in CDMtype mo dels are consid

double p owerlaw mo dels give us the means to investigate

erably smaller than clusterdened voids In HDMtype

the inuence of largescale p erturbations to scaling prop

mo dels galaxydened voids are identical in diameter to

erties of galaxies

clusterdened voids in the real Universe and in most of

Data discussed in earlier Sections demonstrate that

our simulations In other words there is no void hierarchy

most quantitative statistics are insensitive to largescale

in HDMtype mo dels

p erturbations The mass correlation function mass distri

bution of clusters and the distribution of galaxydened

This dierence b etween CDMtype and HDMtype

voids is essentially determined by medium and smallscale

mo dels may b e called in terms of dierent structure for

p erturbations and are practically not inuenced by large

mation concepts topdown and b ottomup In the

scale p erturbations These statistics dene directly or in

classical topdown scenario rst ob jects to form are sup er

directly a certain scale correlation length mean void di

clusters of galaxies which during the evolution fragment

ameter etc and this scale is practically the same for all

into galaxies and small systems of galaxies In the clas

mo dels provided smallscale p erturbations as measured

sical b ottomup scenario the formation starts from small

by the p ower sp ectrum on medium and small scales are

units which merge during the evolution to form systems of

identical

larger scale In particular sup erclusters form in the second

The insensitivity of scaling prop erties to largescale

scenario only gradually in later ep o chs and their growth

density p erturbations do es not mean that these p erturba

should continue indenitely

tions have no inuence at all Our analysis has demon

Our calculations show that either picture is to o simpli

strated that largescale mo des determine the geomet

ed The lo cation of sup erclusters is given already by the

ric pattern of the sup erclustervoid network sup ercluster

initial conditions the largescale p erturbations determine

mass distribution and the strength of laments in dier

the lo cation of sup erclusters and sup ervoids Furthermore

ent environment These asp ects of the largescale distri

clusters and laments of galaxies are inuenced by the

bution are more dicult to express in quantitative terms

largescale environment in sup ervoids these systems re

The geometric pattern of sup erclusters and voids can b e

main p o or in sup erclusters they are richer from the very

measured by the sizes of voids and by the cluster cor

b eginning of the structure formation A similar conclu

relation function on large scales The mass distribution

sion has b een drawn also by Kofman et al on the

function of sup erclusters can b e estimated in mo dels di

basis of twodimensional adhesion mo del of the structure

rectly by counting the mass in all highdensity regions

evolution

and indirectly by the number of rich clusters in sup erclus

On the other hand the formation of ne structure in

ters In the real world only this indirect approach can b e

the topdown scenario is not sucient to pro duce la

applied presently to determine the mass of sup erclusters

ments in sup ervoids The presence of smallscale structure

The sup ercluster mass distribution test gives preference to

in the Universe was rst demonstrated using the multiplic

mo dels with a HarrisonZeldovich sp ectral index on large

ity function test Zeldovich Einasto and Shandarin

scales

Einasto et al The HDMmo del lacks this prop erty

there are only systems of large multiplicity The present

study demonstrates that this dierence is more fundamen

Fine structure of supervoids

tal It is not sucient to pro duce the lamentary struc

The basic result of the study by Lindner et al was ture in sup erclusters only the HDMtype mo dels can in

the establishing of the ne structure and void hierarchy in fact achieve this in later evolutionary stages but such

P Frisch et al Evolution of the Sup erclusterVoid Network

structure must also b e formed in sup ervoids where the Acknowledgements This study started during a visit of JE

HDMtype mo dels fail completely and ME to Europ ean Southern Observatory and contin

Recently interest in the HDMmo del has b een revived ued during visits to GottingenUniversity Observatory JE

Beaky et al Blanchard et al A number of and ME thank the sta of these Observatories for supp ort

mo dels were presented and it has b een suggested that and very stimulating atmosphere allowing to establish

these mo dels can predict the observed structure Accord this collab oration Fruitful discussions with Prof N Bah

ing to our results these suggestions obviously ignore the call and Drs E Saar and D Weinberg are acknowledged

presence of lamentary systems in lowdensity regions This study was supp orted by Estonian Science Founda

which cannot b e accounted for in HDMmo dels tion grant and International Science Foundation grant

To summarize the comparison of mo dels with observa LDC PF was partly funded by the Studienstiftung

tions shows that conventional HDM and CDM mo dels are des Deutschen Volkes

excluded by the combined evidence presented here The

b est agreement with observations was found for a mo del

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