The structure of the Universe on 100 Mpc Scales
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THE STRUCTURE OF THE UNIVERSE ON 100MPC SCALES JAAN EINASTO Tartu Observatory,EE-61602,Estonia Observational evidence for the presence of a preferred scale around 100Mpc in the distribution of high-density regions in the Universe is summarised.Toy models with various degrees of regularity are analysed to understand better the observational data.Predictions of models which produce a feature on the initial power spectrum of matter are analysed and compared with new observational data on the CMB angular spectrum,cluster mass function,and galaxy and cluster power spectrum.1Introduction Until the mid-1970s the astronomical community accepted the paradigm that galax-ies are distributed in space more or less randomly and that about 5?10%of all galaxies are located in clusters which are also randomly distributed.A paradigm change occurred during a Symposium on Large-Scale Structure of the Universe in 1977in Tallinn.Several teams reported results of studies of the 3-dimensional dis-tribution of galaxies in superclusters,and demonstrated that superclusters consist of chains (?laments)of galaxies,groups and clusters,and that the space between galaxy systems is devoid of any visible galaxies (J?o eveer &Einasto 1978,Tarenghi et al.1978,Ti?t &Gregory 1978,Tully &Fisher 1978).The discovery of the ?lamentary character of superclusters and the presence of large empty voids made obsolete the early theories of structure formation –the hierarchical clustering model by Peebles (1980),and the whirl theory by Ozernoy (1978).The ?rst impression was that the winning theory was the pancake scenario by Zeldovich (1978).However,a more detailed analysis has shown that the original pancake scenario also has weak points –the structure forms too late and has no ?ne structure of faint galaxy ?laments in large voids (Zeldovich,Einasto &Shandarin 1982).Then the Cold Dark Matter model was suggested and generally accepted (Blumenthal et al 1984).Recently it was replaced by a CDM model dominated by a cosmological term (vacuum energy)–the LCDM model.However,this model
has also at least one weak point:in LCDM model superclusters are randomly distributed,whereas observational evidence suggests that rich superclusters and voids form a quasi-regular network of scale ～100?130h ?1Mpc (Einasto et al.1997a,1997d).Here we use the dimensionless Hubble constant h de?ned as H 0=100h km s ?1Mpc ?1.These data raise the question:Is the regularity of the distribution of superclusters a challenge for CDM models?My lecture tries to answer this question.First I shall summarise the observational evidence for the presence of the 130h ?1Mpc scale,thereafter I shall discuss attempts to explain the presence of this scale theoretically.
2Evidence for the 100–130Mpc Scale
The evidence for the presence of a scale in the distribution of superclusters has accumulated slowly.Zeldovich,Einasto &Shandarin (1982)noticed that voids
between superclusters have mean diameters about100h?1Mpc.Kopylov et al.
(1988)calculated the cluster correlation function and found a secondary peak at
～125h?1Mpc,which corresponds to superclusters on opposite sides of large https://www.sodocs.net/doc/5d10902255.html,ing a slightly di?erent method Mo et al.(1992)con?rmed the presence of
a feature in the cluster correlation function at～125h?1Mpc.
The most dramatic demonstration of the scale was made by Broadhurst,et al. (1990)who measured redshifts of distant galaxies in a pencil beam directed toward Northern and Southern galactic poles.Their data show that high-density regions in this direction alternate with low-density regions with surprising regularity of a period～128h?1Mpc.Finally,Einasto et al(1997a,1997b,1997d)calculated the cluster correlation function and power spectrum of a3-dimensional cluster sample and con?rmed the presence of a scale of～120h?1Mpc in the distribution of rich clusters of galaxies.
A similar phenomenon is observed in the distribution of Lyman-break galaxies (Broadhurst&Ja?e2000)at high redshift,z≈3.The correlation function of these distant galaxies has secondary peaks.A regularity on a similar scale has been found in the distribution of quasars by Roukema&Mamon(2000).All these data suggest that this scale is primordial and co-moves with the expansion;it can be used as a standard rod.
3Power Spectrum and Correlation Function of Toy Models
The?uctuating density?eld can be described by the power spectrum and the cor-relation function.The power spectrum P(k)characterises the amplitude of density ?uctuations for various wavenumbers k=2π/l.It is determined by the power indices on large and small scales,n l,n s,by the transition scale l0(or wavenumber k0=2π/l0)and by the amplitudeσ8on scale8h?1Mpc.The correlation function ξ(r)characterises the clustering of galaxies or clusters in space;it is de?ned as the excess probability to?nd galaxies or clusters over a random distribution of points.
In the ideal case of data not being distorted by errors and being available over the whole space,the power spectrum and the correlation function form mutual pairs of Fourier transformations.In reality they complement each other.We are interested here in the study of the regularity of the structure.To understand bet-ter observational data we shall study?rst several toy models of di?erent geometry. These toy models are:random superclusters,Voronoi tessellation(randomly lo-cated void centers,and clusters located between voids as far from void centers as possible),and“regular rod models”(superclusters located randomly along regu-larly spaced parallel rods oriented along all3dimensions).Details of these models were discussed by Einasto et al.(1997c).In all three models clusters within super-clusters are generated in an identical way;they are distributed randomly according to an isothermal density law.Thus,on small scales correlation functions and power spectra of all three models are identical.Only on larger scales the models di?er. The random supercluster model has no built-in scale and no regularity in the dis-tribution of superclusters.The Voronoi tessellation model has a built-in scale–the mean size of voids,but no regularity in the distribution of superclusters as voids are distributed randomly.The regular rod model has a scale:the spacing of the
Figure 1.The distribution of clusters in Voronoi and regular rod models
(upper left and right panels,respectively),respective correlation functions (middle panels)and power spectra (lower panels).In the lower panels a wavenumber k =1corresponds to the size of the box in the upper panels.0100200300400500600700X 0
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grid of rods,and also a regularity:the rods form a regular lattice.In Figure 1we show the distribution of clusters in a sheet of the Voronoi and regular rod models (upper panels),the corresponding correlation functions (middle panels)and power
spectra (lower panels).
The analysis of these toy models allows us to make the following conclusions.On small scales (r <50h ?1Mpc)the correlation function and power spectrum characterise the distribution of clusters within superclusters,while on larger scales they describe the distribution of superclusters themselves.If superclusters are dis-tributed randomly,then on large scales (r >50h ?1Mpc)the correlation function is almost zero,ξ(r )=0,and the power spectrum P (k )is ?at.The correlation func-tion of the Voronoi model has one secondary minimum and maximum and is zero thereafter,and the power spectrum P (k )has a sharp maximum.The correlation function of the regular rod model is oscillating:it has regularly spaced minima and maxima;the power spectrum has a well-de?ned sharp maximum on a scale equal to the period of oscillation of the correlation function;it can be approximated by two power law.The position of the maximum of P (k )and the separation of maxima of ξ(r )correspond to the mean diameter of voids and to the separation of superclusters across voids,respectively.
For comparison we can say that all CDM-type models have a smooth power spectrum and a correlation function rather similar to those of the random super-cluster model.Here the con?ict between observations and model is well seen.
Figure 2.The distribution of Abell clusters (open circles)and X-ray clusters (squares)in super-galactic coordinates.Filled squares denote members of rich superclusters and open squares –isolated X-ray clusters and members of poor systems.In order to minimize the projection e?ects we plot only Abell clusters from very rich superclusters with at least 8members.
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4
Distribution of Superclusters 4.1Abell Clusters
We shall use Abell clusters of galaxies as indicators of the distribution of high-density regions in the
Universe.The distribution of Abell clusters in rich super-clusters is shown in Figure 2(open circles).We use clusters within a distance limit 350h ?1Mpc;the Northern galactic hemisphere is at right,the Southern one at left.
Power spectra of various galaxy and cluster samples are shown in the left panel of Figure 3;the correlation function of Abell clusters in rich superclusters is plotted in the right panel.We see that the correlation function of rich clusters is oscillating with a period 120h ?1Mpc.The mean power spectrum of clusters and deep galaxy samples has a sharp maximum on the same wavelength.These properties are similar to properties of the regular rod model.
Figure 3.Left:power spectra of galaxies and clusters of galaxies normalized to the amplitude of the 2-D APM galaxy power spectrum (references are given by Einasto et al.1999a).For clarity error bars are not indicated and spectra are shown as smooth curves rather than discrete data points.Bold lines show spectra for cluster data.Points with error bars show the spectrum of Abell clusters by Miller &Batuski (2000)adjusted to the galaxy spectrum amplitude by a relative bias factor b =3.2.Right:correlation function of Abell clusters located in superclusters with at least 8clusters (Einasto et al.1997b).
k [h Mpc ?1]103
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4.2APM and X-ray Clusters
The largest weight in the mean power spectrum plotted in Figure 3has the Abell cluster sample as it is the deepest and the largest in volume.However,Abell clusters were selected by visual inspection of Palomar survey plates and can be in?uenced by unknown selection e?ects.Thus an independent check of the power spectrum and correlation function is needed.We have used A(utomated)P(late)M(easuring)survey clusters and X-ray selected clusters and active galaxies for comparison.
In Figure 2we show both optically and X-ray selected clusters,demonstrating that rich superclusters are well seen in both cluster samples.The richest Abell
Figure 4.Correlation functions of clusters in rich superclusters:the left panel shows superclus-ters of richness N cl ≥3of X-ray selected clusters and active galaxies;the right panel shows superclusters of APM clusters of richness N cl ≥8.
r [h ?1 Mpc]?0.50.00.51.0
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1.0ξ(r )cluster superclusters are also rich in X-ray clusters.There exist di?erences in detail:not all Abell clusters are strong X-ray emitters,and some X-ray luminous clusters are not included in the Abell catalogue.Similarly,APM clusters do not coincide always with Abell clusters in the same volume.However,the comparison of spatial distributions of cluster samples shows that APM and X-ray selected clusters form rich superclusters very close to rich superclusters of Abell clusters.
Correlation functions of APM and X-ray selected clusters in rich superclusters are also rather similar to the correlation function of Abell clusters in rich super-clusters:compare Figures 3and 4.Again,there are di?erences in details:the correlation functions of APM and X-ray clusters have a secondary maximum at a separation of r ～190h ?1Mpc which is absent in the correlation function of the Abell cluster sample.This feature is due to the fact that the APM sample (and partly the X-ray selected sample)is dominated by two very rich superclusters surrounding an exceptionally large void between Horologium-Reticulum and Sculp-tor superclusters (Einasto et al.1997d),such that the maximum of the correlation function is shifted.On the other hand,the Abell sample contains a dozen very rich superclusters and di?erences in void diameters cancel each other out.Due to the smaller volume covered,the APM cluster sample is less suitable for large-scale studies.
The observational evidence can be summarised as follows:
?Rich superclusters and voids form a quasi-regular lattice;
?The mean diameter of cells (the mean separation of rich superclusters across voids)is 120?130h ?1Mpc.
5
Theoretical Implications 5.1Evolutionary Toy Models
To investigate the formation of a quasiregular cellular structure we have performed numerical simulations of the evolution of a toy model with two-power-law spectrum of index n l =1on large scales and n s =?2on small scales,and a sharp transition at
wavenumber k=0.05h Mpc?1(Frisch et al.1995).This power spectrum is rather similar to the observed power spectrum.The two-power-law model reproduces well the distribution of rich clusters of galaxies.Clusters form long?lamentary super-clusters and a quasiregular supercluster-void network.The correlation function and the power spectrum of the two-power-law model approximates well the observed functions.However,this model is phenomenological.Physically motivated models based on Dark Matter and in?ation properties are needed for comparison.
5.2Cold Dark Matter Models with a Bump
The standard high-density CDM model(?m=1)has a power spectrum very di?erent from observations.On small scales the COBE normalised CDM model has an amplitude of the spectrum which is much higher than the amplitude of the observed power spectrum of galaxies.As discussed above,this is di?cult to explain. LCDM models with cosmological term have better agreement with the observed power spectrum,but they lack the feature at the dominant observed scale.The Low-density Mixed Dark Matter model?ts better the observed power spectrum, but no feature is present either.
In some variants of the in?ation scenario primordial power spectra have a bump or peak.The model suggested by Lesgourgues,Polarski&Starobinsky(1998)has a break in amplitude and peak;the model by Chung et al.(1999)has a bump. Parameters of the model are the position and height of the bump(peak).Power spectra of the Starobinsky model are plotted in the upper left panel of Figure5. Spectra are based on an LCDM model with shape parameterΓ=?m h=0.2, and bump parameters p=1(no bump)and p=0.7.The upper right panel shows power spectra of an MDM model with cosmological parameters:baryon density?b=0.06,CDM density?c=0.29,HDM(neutrino)density?n=0.05, vacuum energy density?l=0.60,and Hubble parameter h=0.65.This set of parameters is rather close to the model suggested by Ostriker&Steinhardt(1995) and con?rmed by new data by Burles et al.(1999),Riess et al.(1998),Perlmutter et al.(1998),and Parodi et al.(2000)among others.One spectrum corresponds to the conventional MDM model,the other to the model with Chung peak of amplitude a=0.3.Calculations were made with the CMBFAST package by Seljak &Zaldarriaga(1996).
This Figure shows that a Starobinsky model with a bump is in good agreement with earlier observational data on the galaxy and cluster power spectrum but not with the new data by Miller&Batuski(2000).A mixed dark matter model with Chung peak of amplitude a=0.3is in good agreement with all observational data.
Finally we have checked the Chung model with other independent data.The lower left panel of Figure5shows the cluster mass function calculated using the Press&Schechter(1974)algorithm.These calculations show that the model with a bump produces more clusters of high mass.However within observational errors of the cluster mass functions of both models are in agreement with observations.A similar comparison with new CMB measurements is given in the lower right panel of Figure5.To improve the agreement with CMB data we have used a tilted MDM model with spectral index n l=0.90on large scales,keeping all other cosmological
Figure 5.Upper left:power spectra of an LCDM model with and without Starobinsky modi?ca-
tion.Upper right:power spectra of MDM models with and without Chung modi?cation.Lower left:cluster mass distributions for MDM models with and without Chung modi?cation.Observed cluster mass functions are shown according to Bahcall &Cen (1993)and Girardi et al.(1998).Lower right:angular power spectra of tilted MDM models with and without Chung modi?cation (amplitude parameter a =0.3),compared with BOOMERANG (de Bernardis et al.2000)and MAXIMA I (Hanani et al.2000)data.
k [h Mpc ?1]102
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l 02000400060008000l (2l +1)/4π [μK 2]parameters as in the model shown in the lower left panel.These calculations show that models with and without the Chung modi?cation are in satisfactory agreement with the CMB angular spectrum.Thus we conclude that this model represents well all available observational data.
However,there are several problems not solved yet.First,the observational question:How large is the region of the regular supercluster-void network?Then the question concerning the interpretation of models:How are spectral features (bump)and regularity related?We hope that this question can be answered with numerical simulations of LCDM models with and without a bump in the power spectrum,and such simulations are under way.Finally,the theoretical questions:Why is there a preferred scale and why does it have a value of ～130h ?1Mpc?What does the presence of a scale and a regularity tell us on the in?ation?
6Conclusions
?There exists good evidence on the presence of a scale of～130h?1Mpc:it measures the mean distance between rich superclusters across voids.?There are variants of the in?ation scenario suggesting the presence of a char-acteristic scale(bump)in the power spectrum.
?Models with a bump in the power spectrum are in agreement with other cos-mological data using sets of parameters within acceptable ranges.?Cellular large-scale structure may be the end of the fractal structure of the Universe,as suggested by Ru?ni,Song&Taraglio(1988).
We should note that models neither predict the position of the bump,nor the presence and extent of the regularity of the supercluster-void network.We should remember the maxim by Eddington:no experimental result can be taken seriously if not explained theoretically.Thus larger surveys are needed to clarify properties of the supercluster-void network.Also a further theoretical analysis are needed to ?nd the physical origin of the scale and the extent of the regularity.
Acknowledgments
I thank Maret Einasto,Mirt Gramann,Pekka Heinam¨a ki,Volker M¨u ller,Enn Saar, Aleksei Starobinsky,Ivan Suhhonenko and Erik Tago for fruitful collaboration and permission to use joint results in this talk.Heinz Andernach kindly improved the presentation.This study was supported by the Estonian Science Foundation grant 2625.
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