Voronoi Tessellations form an attractive and versatile geometrical asymptotic
model for the foamlike cosmic distribution of matter and galaxies. In the
Voronoi model the vertices are identified with clusters of galaxies. For a
substantial range out to a scale in the order of the cellsize, their spatial
two-point correlation function is a power-law with a slope $\gamma \approx
1.95$. This study presents recent results showing that subsets of vertices
selected on the basis of their ``richness'', i.e. inflow volume, retain this
power-law correlation behaviour. Interestingly, they do so with a ``clustering
length'' $r_{\rm o}$ that is exactly linearly proportional to the average
inter-vertex distance in the sample, thus forming a realization of the
Szalay-Schramm prescription. For the geometry and structural patterns even more
significant is the finding tessellation vertices display a similar linear
increase for their correlation length $r_{\rm a}$, the coherence length at
which $\xi(r_{\rm a})=0$. Such patterns therefore exhibit positive correlations
out to distances considerably in excess of the cellsize. Most intriguing is the
implication of self-similar scaling, while these results may be regarded as the
presentation of a ``geometrical bias'' effect. A seemingly rigid structure
appears to represent a flexible and useful geometric model for exploring the
statistical and dynamical repercussions of the nontrivial cellular patterns in
Megaparsec scale cosmic structure.Comment: Contribution to ``Large Scale Structure in the X-ray Universe'',
  Workshop Santorini, Greece, September 1999, eds. M. Plionis and I.
  Georgantopoulos (Editions Frontieres). 5 pages of LaTex including 2+3
  postscript figures. Uses psfi

van de Weygaert, Rien

English

arXiv

Weygaert, R. van de

University of Groningen Research Database

  
 University of Groningen
The Cosmic Foam and the Self-Similar Cluster Distribution
van de Weijgaert, Marinus
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The Cosmic Foam and the Self-Similar Cluster Distribution
Rien van de Weygaert
1Kapteyn Institute, University of Groningen, Groningen, the Netherlands
1 Foamlike Patterns in Cosmic Structure
Probing cosmic large scale structure on the basis of X-ray observations puts par-
ticular emphasis on the densest regions within the global matter distribution, the
rich clusters of galaxies. Understanding the relationship between the cluster dis-
tribution and the underlying matter distribution is therefore a key element in any
assessment of cosmic structure on the basis of samples of X-ray selected clusters.
By now, the foamlike arrangement of matter and galaxies on Megaparsec scales
has become a well-established feature of the cosmic matter distribution, consist-
ing of an assembly of anisotropic elements, filaments and walls of various sizes,
surrounding large underdense void regions and sprinkled with wholly or partially
virialized dense clumps of matter, varying in size from rich clusters of galaxies
down to small groups of a few galaxies. It is one of successes of gravitational
instability cosmic structure formation theories to find that this pattern of walls,
filaments and voids appears to be the generic outcome of these scenarios.
A major obstacle in quantifying this quintessential aspect of cosmic structure
is the lack of a systematic insight into the dynamical and statistical aspects of cel-
lular geometries as well as the absence of a readily available and well-established
mathematical machinery to evaluate and compare observations and simulations.
Stochastic geometry – the branch of mathematics concerned with nontrivial geo-
metrical concepts involving stochastic behaviour of one or more of their character-
istics – may be expected to contribute significantly to further such understanding.
2 Voronoi Tessellations
The canonical example of a stochastic geometrical model for a cellular division
of space is that of Voronoi tessellations. This space filling network of convex
polyhedra offers a surprisingly realistic and versatile representation of the char-
acteristics and features of the foamlike or cellular spatial arrangement of matter
in the Universe. In short, it is defined through a spatial distribution of nuclei[9].
Each nucleus corresponds to one Voronoi cell, which comprises that part of space
closer to this nucleus than to any of the other. The walls and edges forming
the polyhedrons’ surfaces are identified with the wall-like and filamentary su-
perclusters in the galaxy distribution, the vertices with and massive clusters of
galaxies, while the interior of the Voronoi cells corresponds to the large void re-
gions barren of galaxies. The morphology of artificial galaxy distributions set up
within this network bears a striking resemblance to that observed in the more
complicated circumstances of the real Cosmos or that of the artificial reality of
computer simulations of structure formation. Figure 1 shows a conglomerate of
Figure 1: A clump of Voronoi polyhedra from a tessellation of 1000 cells.
One cell is surface-shaded, the surrounding neighbouring cells are depicted
in wire-frame fashion. The corresponding vertices are indicated by solid
dots. Distribution of all Voronoi vertices within a cubic volume. Courtesy:
Jacco Dankers.
several neighbouring Voronoi cells, with one of the cells by shaded surface and a
wire-framed representation in the case of its neighbouring cells. The solid lines
are the edges (filaments) in the network, the dots (coloured red) are the vertices
of the Voronoi tessellation. In principle a Voronoi tessellation could be regarded
as a mere geometric toy model, a heuristic geometrical description of a substan-
tially more complicated reality. However, a detailed assessment of the role of
voids in structure formation provides a physical ground for them representing re-
ality, whether the observed or simulated one, in a more subtle way[5][4]. This
is certainly reinforced by their striking resemblance to the observed or simulated
foamlike appearance of the galaxy and matter distribution.
3 Voronoi Vertex Clustering
Even in the limited setting of Figure 1 it is evident that the vertex distribution is
not a random Poisson distribution. The full spatial distribution of Voronoi vertices
in the full cubic volume (righthans frame fig. 1) clearly involves a substantial
degree of clustering. This impression of strong clustering, on scales smaller than
or of the order of the cellsize λC, is most evidently confirmed by their two-point
correlation function ξ(r). Not only can we discern a clear positive signal but –
surprising at the time of its finding on the basis of similar computer experiments[8]
– out to a distance of at least r ≈ 1/4 λC the correlation function appears to be
an almost perfect power-law,
ξ(r) = (ro/r)
γ , (1)
with a slope γ ≈ 1.9 − 2.0. Its amplitude, traditionally expressed in terms of the
“clustering length” ro, at which ξ(ro) = 1 , has a value ro ≈ 0.29 λC. Beyond
this range, the power-law behaviour breaks down and following a gradual decline
the correlation function rapidly falls of to a zero value once distances are of the
order of the cellsize. However, rather than a characteristic geometric scale, ro is
more a measure for the “compactness” of the spatial clustering, set mainly by the
small-scale clustering. A more significant scale within the context of the geometry
of the spatial patterns in the density distribution is the “correlation length” ra,
the scale at which ξ(ra) = 0 . As a genuine scale of coherence, it is more relevant
to the morphology of the nontrivial spatial structures we seek to study. Beyond
ra the distribution of Voronoi vertices is practically uniform.
If we interpret the clustering length ro ≈ 20h
−1 Mpc, usually found for sam-
ples of rich clusters of galaxies[2], within the context of a Voronoi tessellation
it would imply a cellsize of λC ≈ 70h
−1 Mpc. Although the two-point cluster-
cluster correlation function reproduced by the Voronoi vertices fits very well to
the function yielded from the observations, the large cell size λc may be a com-
plication. It is surely well in excess of the 25h−1 − 35h−1 Mpc size of the voids in
the galaxy distribution. Moreover, also within the Voronoi concept itself it would
conflict with the clustering of objects dwelling in the walls and filaments of the
same tessellation framework. Clustering analysis of such configurations[9] reveals
that the two-point correlation function of galaxies confined to the walls – as well
as for those confined to the edges – also displays distinct power-law behaviour
at sub-cellular scales. The involved clustering length, however, is different from
that of the vertices in the same framework. For the wall galaxies it is but half
the value of that of the vertices, rw,o ≈ 0.14 λC[9],[10]. If rw,o is identified with
the galaxy-galaxy clustering length of rg,o ≈ 5h
−1 Mpc, this would yield a cellsize
of ≈ 35h−1 Mpc. The latter is suggestively similar to the size of the actually ob-
served voids, which may be a tantalizing hint for a profound relationship between
the clustering length ro and the typical cellular scale of the cosmic foam network.
Adressing this apparent inconsistency between vertex and wall clustering we
first observe that the vertex correlation function of eqn. (1) concerns the whole
sample of vertices, irrespective of any possible selection effects based on one or
more relevant physical aspects. In reality, it will be almost inevitable to invoke
some sort of biasing through the defining criteria of the catalogue of clusters. In-
terpreting the Voronoi model in its quality of asymptotic approximation to the
galaxy distribution, its vertices will automatically comprise a range of “masses”.
Neglecting the details of the temporal evolution, we may assign each Voronoi ver-
tex a “mass” equal to the total amount of matter that ultimately will flow towards
that vertex. Applying the “Voronoi streaming model”[9] as a reasonable descrip-
tion of the clustering process, it is reasonably straightforward if cumbersome to
calculate the “mass” or “richness” MV of each Voronoi vertex by pure geomet-
ric means[10]. It concerns the volume of a non-convex polyhedron centered on
the Voronoi vertex, with related Voronoi nuclei as one of the polyhedral vertices.
These nuclei are the ones that supply the Voronoi vertex with inflowing matter.
Figure 2: Voronoi tessellation scaling. Left insert: Two-point correlation
function ξ(r) Voronoi vertices. The clustering length ro (left) and the ratio
ra/ro (right) as a function of average vertex distance.
Evidently, the “mass” is larger for vertices on the surface of large Voronoi cells.
The vertex samples in our study consist of all Voronoi vertices with a richness
equal to higher than a limiting value, the “sample richness”MS[10]. Subsequently,
we determined the correlation function characteristics for each of the vertex sam-
ples. Assessing their behaviour as a function of the average distance λV between
the sample vertices, for samples encompassing from 10% up to 100% of the total
number of Voronoi vertices and thus for λV ≈ 0.5−1.5×λC, revealed a remarkable
and tantalizing scaling of the clustering phenomenon[10].
All subsamples of Voronoi vertices have a two-point correlation function that
out to a certain range retains a power-law behaviour almost exactly similar the
one of the full sample of vertcies. No significant difference in power law slope
between the various subsamples can be discerned, all have γ ∼ 1.8 − 2.0. On
the other hand, both clustering length ro and correlation length ra do display a
systematic dependence on sample richness. Both ro and ra increase proportionally
to the sample richness, and hence the average vertex distance λV in the samples !
The more massive the vertices are, the more strongly they cluster. In other words,
in the line of the random field “peak biasing” scheme[6], we here find a purely
geometrically based biasing scheme. Interesting in this respect is to observe that
the increase of rV,o is perfectly linearly proportional to the mutual vertex distance
λV in the sample (Fig. 2, left frame). This suggests that Voronoi vertex clustering
is a perfect realization of the clustering scaling description proposed by Szalay &
Schramm[7]. It also adheres to the increasing level of clustering that selections
of more massive clusters appear to display in large-scope N-body simulations[3]
although there are telling differences in detailed behaviour.
For our purpose, even more significant is the uncovered scaling of the cor-
relation length rV,a, similar in character to that of rV,o. The increase of ra also
turns out to be almost exactly linearly proportional to the average vertex distance.
The repercussions are manyfold. It means that the selected vertices still have a
strong positive correlation at scales where the poorer samples do not possess any
clustering. Moreover, samples with more massive clusters are expected to have a
clustering that extends out further than that of their poorer brethren. This may
be a significant observation in the light of the finding that samples of rich galaxy
clusters seem to have a positive correlation at scales of tens of Megaparsec, quite
in excess of scales with appreciable galaxy clustering. Finally, the implied con-
stant ratio between clustering and correlation length, ra/ro ≈ 1.86 (Fig. 2, right
frame), implies a perfect self-similar scaling of the Voronoi vertex distribution.
The complete correlation function ξs(r) of each selected subsample s, and not
just the part in the power-law range, is a self-similar mapping of an elementary
function ξel scaled by means of a characteristic lengthscale parameter Ls.
4 Bias and Cosmic Geometry: Conclusions
The above results form a tantalizing indication for the existence of self-similar
clustering behaviour in spatial patterns with a cellular or foamlike morphology. It
might hint at an intriguing and intimate relationship between the cosmic foamlike
geometry and a variety of aspects of the spatial distribution of galaxies and clus-
ters. One important implication is that with clusters residing at a subset of nodes
in the cosmic cellular framework, a configuration certainly reminiscent of the ob-
served reality, it would explain why the level of clustering of clusters of galaxies
becomes stronger as it concerns samples of more massive clusters. In addition,
it would succesfully reproduce positive clustering of clusters over scales substan-
tially exceeding the characteristic scale of voids and other elements of the cosmic
foam. At these Megaparsec scales there is a close kinship between the measured
galaxy-galaxy two-point correlation function and the foamlike morphology of the
galaxy distribution. In other words, the cosmic geometry apparently implies a ‘ge-
ometrical biasing” effect, qualitatively different from the more conventional “peak
biasing” picture[6].
References
[1] Bardeen, J.M., Bond, J.R., Kaiser, N., Szalay, A.S., 1986, ApJ 304, 15
[2] Bahcall, N.A., Soneira, R., 1983, ApJ 270, 20
[3] Colberg, J., 1998, Ph.D. thesis, Ludwig-Maximilian Univ. Mu¨nchen
[4] Dubinski, J., et al., 1993, ApJ 410, 458
[5] Icke, V. 1984, MNRAS 206, 1P
[6] Kaiser, N., 1984, ApJ , 284, L9
[7] Szalay, A., Schramm, D.N., 1985, Nature , 314, 718
[8] van de Weygaert, R., Icke, V., 1989, A&A 213, 1
[9] van de Weygaert, R., 1991, Ph.D. thesis, Leiden Univ.
[10] van de Weygaert, R., 1999, in preparation


The Cosmic Foam and the Self-Similar Cluster Distribution

Abstract: Voronoi Tessellations form an attractive and versatile geometrical asymptotic model for the foamlike cosmic distribution of matter and galaxies. In the Voronoi model the vertices are identified with clusters of galaxies. For a substantial range out to a scale in the order of the cellsize, their spatial two-point correlation function is a power-law with a slope $gamma approx 1.95$. This study presents recent results showing that subsets of vertices selected on the basis of their ``richness'', i.e. inflow volume, retain this power-law correlation behaviour. Interestingly, they do so with a ``clustering length'' $r_{rm o}$ that is exactly linearly proportional to the average inter-vertex distance in the sample, thus forming a realization of the Szalay-Schramm prescription. For the geometry and structural patterns even more significant is the finding tessellation vertices display a similar linear increase for their correlation length $r_{rm a}$, the coherence length at which $xi(r_{rm a})=0$. Such patterns therefore exhibit positive correlations out to distances considerably in excess of the cellsize. Most intriguing is the implication of self-similar scaling, while these results may be regarded as the presentation of a ``geometrical bias'' effect. A seemingly rigid structure appears to represent a flexible and useful geometric model for exploring the statistical and dynamical repercussions of the nontrivial cellular patterns in Megaparsec scale cosmic structure

ARTS repository - University of Groningen

   University of GroningenThe Cosmic Foam and the Self-Similar Cluster DistributionWeygaert, R. van dePublished in:ArXivIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Weygaert, R. V. D. (1999). The Cosmic Foam and the Self-Similar Cluster Distribution. ArXiv.https://arxiv.org/abs/astro-ph/9912279CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022arXiv:astro-ph/9912279 v1   14 Dec 1999The Cosmic Foam and the Self-Similar Cluster DistributionRien van de Weygaert1Kapteyn Institute, University of Groningen, Groningen, the Netherlands1 Foamlike Patterns in Cosmic StructureProbing cosmic large scale structure on the basis of X-ray observations puts par-ticular emphasis on the densest regions within the global matter distribution, therich clusters of galaxies. Understanding the relationship between the cluster dis-tribution and the underlying matter distribution is therefore a key element in anyassessment of cosmic structure on the basis of samples of X-ray selected clusters.By now, the foamlike arrangement of matter and galaxies on Megaparsec scaleshas become a well-established feature of the cosmic matter distribution, consist-ing of an assembly of anisotropic elements, filaments and walls of various sizes,surrounding large underdense void regions and sprinkled with wholly or partiallyvirialized dense clumps of matter, varying in size from rich clusters of galaxiesdown to small groups of a few galaxies. It is one of successes of gravitationalinstability cosmic structure formation theories to find that this pattern of walls,filaments and voids appears to be the generic outcome of these scenarios.A major obstacle in quantifying this quintessential aspect of cosmic structureis the lack of a systematic insight into the dynamical and statistical aspects of cel-lular geometries as well as the absence of a readily available and well-establishedmathematical machinery to evaluate and compare observations and simulations.Stochastic geometry – the branch of mathematics concerned with nontrivial geo-metrical concepts involving stochastic behaviour of one or more of their character-istics – may be expected to contribute significantly to further such understanding.2 Voronoi TessellationsThe canonical example of a stochastic geometrical model for a cellular divisionof space is that of Voronoi tessellations. This space filling network of convexpolyhedra offers a surprisingly realistic and versatile representation of the char-acteristics and features of the foamlike or cellular spatial arrangement of matterin the Universe. In short, it is defined through a spatial distribution of nuclei[9].Each nucleus corresponds to one Voronoi cell, which comprises that part of spacecloser to this nucleus than to any of the other. The walls and edges formingthe polyhedrons’ surfaces are identified with the wall-like and filamentary su-perclusters in the galaxy distribution, the vertices with and massive clusters ofgalaxies, while the interior of the Voronoi cells corresponds to the large void re-gions barren of galaxies. The morphology of artificial galaxy distributions set upwithin this network bears a striking resemblance to that observed in the morecomplicated circumstances of the real Cosmos or that of the artificial reality ofcomputer simulations of structure formation. Figure 1 shows a conglomerate ofFigure 1: A clump of Voronoi polyhedra from a tessellation of 1000 cells.One cell is surface-shaded, the surrounding neighbouring cells are depictedin wire-frame fashion. The corresponding vertices are indicated by soliddots. Distribution of all Voronoi vertices within a cubic volume. Courtesy:Jacco Dankers.several neighbouring Voronoi cells, with one of the cells by shaded surface and awire-framed representation in the case of its neighbouring cells. The solid linesare the edges (filaments) in the network, the dots (coloured red) are the verticesof the Voronoi tessellation. In principle a Voronoi tessellation could be regardedas a mere geometric toy model, a heuristic geometrical description of a substan-tially more complicated reality. However, a detailed assessment of the role ofvoids in structure formation provides a physical ground for them representing re-ality, whether the observed or simulated one, in a more subtle way[5][4]. Thisis certainly reinforced by their striking resemblance to the observed or simulatedfoamlike appearance of the galaxy and matter distribution.3 Voronoi Vertex ClusteringEven in the limited setting of Figure 1 it is evident that the vertex distribution isnot a random Poisson distribution. The full spatial distribution of Voronoi verticesin the full cubic volume (righthans frame fig. 1) clearly involves a substantialdegree of clustering. This impression of strong clustering, on scales smaller thanor of the order of the cellsize λC, is most evidently confirmed by their two-pointcorrelation function ξ(r). Not only can we discern a clear positive signal but –surprising at the time of its finding on the basis of similar computer experiments[8]– out to a distance of at least r ≈ 1/4 λC the correlation function appears to bean almost perfect power-law,ξ(r) = (ro/r)γ , (1)with a slope γ ≈ 1.9 − 2.0. Its amplitude, traditionally expressed in terms of the“clustering length” ro, at which ξ(ro) = 1 , has a value ro ≈ 0.29 λC. Beyondthis range, the power-law behaviour breaks down and following a gradual declinethe correlation function rapidly falls of to a zero value once distances are of theorder of the cellsize. However, rather than a characteristic geometric scale, ro ismore a measure for the “compactness” of the spatial clustering, set mainly by thesmall-scale clustering. A more significant scale within the context of the geometryof the spatial patterns in the density distribution is the “correlation length” ra,the scale at which ξ(ra) = 0 . As a genuine scale of coherence, it is more relevantto the morphology of the nontrivial spatial structures we seek to study. Beyondra the distribution of Voronoi vertices is practically uniform.If we interpret the clustering length ro ≈ 20h−1 Mpc, usually found for sam-ples of rich clusters of galaxies[2], within the context of a Voronoi tessellationit would imply a cellsize of λC ≈ 70h−1 Mpc. Although the two-point cluster-cluster correlation function reproduced by the Voronoi vertices fits very well tothe function yielded from the observations, the large cell size λc may be a com-plication. It is surely well in excess of the 25h−1 − 35h−1 Mpc size of the voids inthe galaxy distribution. Moreover, also within the Voronoi concept itself it wouldconflict with the clustering of objects dwelling in the walls and filaments of thesame tessellation framework. Clustering analysis of such configurations[9] revealsthat the two-point correlation function of galaxies confined to the walls – as wellas for those confined to the edges – also displays distinct power-law behaviourat sub-cellular scales. The involved clustering length, however, is different fromthat of the vertices in the same framework. For the wall galaxies it is but halfthe value of that of the vertices, rw,o ≈ 0.14 λC[9],[10]. If rw,o is identified withthe galaxy-galaxy clustering length of rg,o ≈ 5h−1 Mpc, this would yield a cellsizeof ≈ 35h−1 Mpc. The latter is suggestively similar to the size of the actually ob-served voids, which may be a tantalizing hint for a profound relationship betweenthe clustering length ro and the typical cellular scale of the cosmic foam network.Adressing this apparent inconsistency between vertex and wall clustering wefirst observe that the vertex correlation function of eqn. (1) concerns the wholesample of vertices, irrespective of any possible selection effects based on one ormore relevant physical aspects. In reality, it will be almost inevitable to invokesome sort of biasing through the defining criteria of the catalogue of clusters. In-terpreting the Voronoi model in its quality of asymptotic approximation to thegalaxy distribution, its vertices will automatically comprise a range of “masses”.Neglecting the details of the temporal evolution, we may assign each Voronoi ver-tex a “mass” equal to the total amount of matter that ultimately will flow towardsthat vertex. Applying the “Voronoi streaming model”[9] as a reasonable descrip-tion of the clustering process, it is reasonably straightforward if cumbersome tocalculate the “mass” or “richness” MV of each Voronoi vertex by pure geomet-ric means[10]. It concerns the volume of a non-convex polyhedron centered onthe Voronoi vertex, with related Voronoi nuclei as one of the polyhedral vertices.These nuclei are the ones that supply the Voronoi vertex with inflowing matter.Figure 2: Voronoi tessellation scaling. Left insert: Two-point correlationfunction ξ(r) Voronoi vertices. The clustering length ro (left) and the ratiora/ro (right) as a function of average vertex distance.Evidently, the “mass” is larger for vertices on the surface of large Voronoi cells.The vertex samples in our study consist of all Voronoi vertices with a richnessequal to higher than a limiting value, the “sample richness” MS[10]. Subsequently,we determined the correlation function characteristics for each of the vertex sam-ples. Assessing their behaviour as a function of the average distance λV betweenthe sample vertices, for samples encompassing from 10% up to 100% of the totalnumber of Voronoi vertices and thus for λV ≈ 0.5−1.5×λC, revealed a remarkableand tantalizing scaling of the clustering phenomenon[10].All subsamples of Voronoi vertices have a two-point correlation function thatout to a certain range retains a power-law behaviour almost exactly similar theone of the full sample of vertcies. No significant difference in power law slopebetween the various subsamples can be discerned, all have γ ∼ 1.8 − 2.0. Onthe other hand, both clustering length ro and correlation length ra do display asystematic dependence on sample richness. Both ro and ra increase proportionallyto the sample richness, and hence the average vertex distance λV in the samples !The more massive the vertices are, the more strongly they cluster. In other words,in the line of the random field “peak biasing” scheme[6], we here find a purelygeometrically based biasing scheme. Interesting in this respect is to observe thatthe increase of rV,o is perfectly linearly proportional to the mutual vertex distanceλV in the sample (Fig. 2, left frame). This suggests that Voronoi vertex clusteringis a perfect realization of the clustering scaling description proposed by Szalay &Schramm[7]. It also adheres to the increasing level of clustering that selectionsof more massive clusters appear to display in large-scope N-body simulations[3]although there are telling differences in detailed behaviour.For our purpose, even more significant is the uncovered scaling of the cor-relation length rV,a, similar in character to that of rV,o. The increase of ra alsoturns out to be almost exactly linearly proportional to the average vertex distance.The repercussions are manyfold. It means that the selected vertices still have astrong positive correlation at scales where the poorer samples do not possess anyclustering. Moreover, samples with more massive clusters are expected to have aclustering that extends out further than that of their poorer brethren. This maybe a significant observation in the light of the finding that samples of rich galaxyclusters seem to have a positive correlation at scales of tens of Megaparsec, quitein excess of scales with appreciable galaxy clustering. Finally, the implied con-stant ratio between clustering and correlation length, ra/ro ≈ 1.86 (Fig. 2, rightframe), implies a perfect self-similar scaling of the Voronoi vertex distribution.The complete correlation function ξs(r) of each selected subsample s, and notjust the part in the power-law range, is a self-similar mapping of an elementaryfunction ξel scaled by means of a characteristic lengthscale parameter Ls.4 Bias and Cosmic Geometry: ConclusionsThe above results form a tantalizing indication for the existence of self-similarclustering behaviour in spatial patterns with a cellular or foamlike morphology. Itmight hint at an intriguing and intimate relationship between the cosmic foamlikegeometry and a variety of aspects of the spatial distribution of galaxies and clus-ters. One important implication is that with clusters residing at a subset of nodesin the cosmic cellular framework, a configuration certainly reminiscent of the ob-served reality, it would explain why the level of clustering of clusters of galaxiesbecomes stronger as it concerns samples of more massive clusters. In addition,it would succesfully reproduce positive clustering of clusters over scales substan-tially exceeding the characteristic scale of voids and other elements of the cosmicfoam. At these Megaparsec scales there is a close kinship between the measuredgalaxy-galaxy two-point correlation function and the foamlike morphology of thegalaxy distribution. In other words, the cosmic geometry apparently implies a ‘ge-ometrical biasing” effect, qualitatively different from the more conventional “peakbiasing” picture[6].References[1] Bardeen, J.M., Bond, J.R., Kaiser, N., Szalay, A.S., 1986, ApJ 304, 15[2] Bahcall, N.A., Soneira, R., 1983, ApJ 270, 20[3] Colberg, J., 1998, Ph.D. thesis, Ludwig-Maximilian Univ. München[4] Dubinski, J., et al., 1993, ApJ 410, 458[5] Icke, V. 1984, MNRAS 206, 1P[6] Kaiser, N., 1984, ApJ , 284, L9[7] Szalay, A., Schramm, D.N., 1985, Nature , 314, 718[8] van de Weygaert, R., Icke, V., 1989, A&A 213, 1[9] van de Weygaert, R., 1991, Ph.D. thesis, Leiden Univ.[10] van de Weygaert, R., 1999, in preparation

University of Groningen

   University of GroningenThe Cosmic Foam and the Self-Similar Cluster DistributionWeygaert, R. van dePublished in:Default journalIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Weygaert, R. V. D. (1999). The Cosmic Foam and the Self-Similar Cluster Distribution. Default journal.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019arXiv:astro-ph/9912279 v1   14 Dec 1999The Cosmic Foam and the Self-Similar Cluster DistributionRien van de Weygaert1Kapteyn Institute, University of Groningen, Groningen, the Netherlands1 Foamlike Patterns in Cosmic StructureProbing cosmic large scale structure on the basis of X-ray observations puts par-ticular emphasis on the densest regions within the global matter distribution, therich clusters of galaxies. Understanding the relationship between the cluster dis-tribution and the underlying matter distribution is therefore a key element in anyassessment of cosmic structure on the basis of samples of X-ray selected clusters.By now, the foamlike arrangement of matter and galaxies on Megaparsec scaleshas become a well-established feature of the cosmic matter distribution, consist-ing of an assembly of anisotropic elements, filaments and walls of various sizes,surrounding large underdense void regions and sprinkled with wholly or partiallyvirialized dense clumps of matter, varying in size from rich clusters of galaxiesdown to small groups of a few galaxies. It is one of successes of gravitationalinstability cosmic structure formation theories to find that this pattern of walls,filaments and voids appears to be the generic outcome of these scenarios.A major obstacle in quantifying this quintessential aspect of cosmic structureis the lack of a systematic insight into the dynamical and statistical aspects of cel-lular geometries as well as the absence of a readily available and well-establishedmathematical machinery to evaluate and compare observations and simulations.Stochastic geometry – the branch of mathematics concerned with nontrivial geo-metrical concepts involving stochastic behaviour of one or more of their character-istics – may be expected to contribute significantly to further such understanding.2 Voronoi TessellationsThe canonical example of a stochastic geometrical model for a cellular divisionof space is that of Voronoi tessellations. This space filling network of convexpolyhedra offers a surprisingly realistic and versatile representation of the char-acteristics and features of the foamlike or cellular spatial arrangement of matterin the Universe. In short, it is defined through a spatial distribution of nuclei[9].Each nucleus corresponds to one Voronoi cell, which comprises that part of spacecloser to this nucleus than to any of the other. The walls and edges formingthe polyhedrons’ surfaces are identified with the wall-like and filamentary su-perclusters in the galaxy distribution, the vertices with and massive clusters ofgalaxies, while the interior of the Voronoi cells corresponds to the large void re-gions barren of galaxies. The morphology of artificial galaxy distributions set upwithin this network bears a striking resemblance to that observed in the morecomplicated circumstances of the real Cosmos or that of the artificial reality ofcomputer simulations of structure formation. Figure 1 shows a conglomerate ofFigure 1: A clump of Voronoi polyhedra from a tessellation of 1000 cells.One cell is surface-shaded, the surrounding neighbouring cells are depictedin wire-frame fashion. The corresponding vertices are indicated by soliddots. Distribution of all Voronoi vertices within a cubic volume. Courtesy:Jacco Dankers.several neighbouring Voronoi cells, with one of the cells by shaded surface and awire-framed representation in the case of its neighbouring cells. The solid linesare the edges (filaments) in the network, the dots (coloured red) are the verticesof the Voronoi tessellation. In principle a Voronoi tessellation could be regardedas a mere geometric toy model, a heuristic geometrical description of a substan-tially more complicated reality. However, a detailed assessment of the role ofvoids in structure formation provides a physical ground for them representing re-ality, whether the observed or simulated one, in a more subtle way[5][4]. Thisis certainly reinforced by their striking resemblance to the observed or simulatedfoamlike appearance of the galaxy and matter distribution.3 Voronoi Vertex ClusteringEven in the limited setting of Figure 1 it is evident that the vertex distribution isnot a random Poisson distribution. The full spatial distribution of Voronoi verticesin the full cubic volume (righthans frame fig. 1) clearly involves a substantialdegree of clustering. This impression of strong clustering, on scales smaller thanor of the order of the cellsize λC, is most evidently confirmed by their two-pointcorrelation function ξ(r). Not only can we discern a clear positive signal but –surprising at the time of its finding on the basis of similar computer experiments[8]– out to a distance of at least r ≈ 1/4 λC the correlation function appears to bean almost perfect power-law,ξ(r) = (ro/r)γ , (1)with a slope γ ≈ 1.9 − 2.0. Its amplitude, traditionally expressed in terms of the“clustering length” ro, at which ξ(ro) = 1 , has a value ro ≈ 0.29 λC. Beyondthis range, the power-law behaviour breaks down and following a gradual declinethe correlation function rapidly falls of to a zero value once distances are of theorder of the cellsize. However, rather than a characteristic geometric scale, ro ismore a measure for the “compactness” of the spatial clustering, set mainly by thesmall-scale clustering. A more significant scale within the context of the geometryof the spatial patterns in the density distribution is the “correlation length” ra,the scale at which ξ(ra) = 0 . As a genuine scale of coherence, it is more relevantto the morphology of the nontrivial spatial structures we seek to study. Beyondra the distribution of Voronoi vertices is practically uniform.If we interpret the clustering length ro ≈ 20h−1 Mpc, usually found for sam-ples of rich clusters of galaxies[2], within the context of a Voronoi tessellationit would imply a cellsize of λC ≈ 70h−1 Mpc. Although the two-point cluster-cluster correlation function reproduced by the Voronoi vertices fits very well tothe function yielded from the observations, the large cell size λc may be a com-plication. It is surely well in excess of the 25h−1 − 35h−1 Mpc size of the voids inthe galaxy distribution. Moreover, also within the Voronoi concept itself it wouldconflict with the clustering of objects dwelling in the walls and filaments of thesame tessellation framework. Clustering analysis of such configurations[9] revealsthat the two-point correlation function of galaxies confined to the walls – as wellas for those confined to the edges – also displays distinct power-law behaviourat sub-cellular scales. The involved clustering length, however, is different fromthat of the vertices in the same framework. For the wall galaxies it is but halfthe value of that of the vertices, rw,o ≈ 0.14 λC[9],[10]. If rw,o is identified withthe galaxy-galaxy clustering length of rg,o ≈ 5h−1 Mpc, this would yield a cellsizeof ≈ 35h−1 Mpc. The latter is suggestively similar to the size of the actually ob-served voids, which may be a tantalizing hint for a profound relationship betweenthe clustering length ro and the typical cellular scale of the cosmic foam network.Adressing this apparent inconsistency between vertex and wall clustering wefirst observe that the vertex correlation function of eqn. (1) concerns the wholesample of vertices, irrespective of any possible selection effects based on one ormore relevant physical aspects. In reality, it will be almost inevitable to invokesome sort of biasing through the defining criteria of the catalogue of clusters. In-terpreting the Voronoi model in its quality of asymptotic approximation to thegalaxy distribution, its vertices will automatically comprise a range of “masses”.Neglecting the details of the temporal evolution, we may assign each Voronoi ver-tex a “mass” equal to the total amount of matter that ultimately will flow towardsthat vertex. Applying the “Voronoi streaming model”[9] as a reasonable descrip-tion of the clustering process, it is reasonably straightforward if cumbersome tocalculate the “mass” or “richness” MV of each Voronoi vertex by pure geomet-ric means[10]. It concerns the volume of a non-convex polyhedron centered onthe Voronoi vertex, with related Voronoi nuclei as one of the polyhedral vertices.These nuclei are the ones that supply the Voronoi vertex with inflowing matter.Figure 2: Voronoi tessellation scaling. Left insert: Two-point correlationfunction ξ(r) Voronoi vertices. The clustering length ro (left) and the ratiora/ro (right) as a function of average vertex distance.Evidently, the “mass” is larger for vertices on the surface of large Voronoi cells.The vertex samples in our study consist of all Voronoi vertices with a richnessequal to higher than a limiting value, the “sample richness”MS[10]. Subsequently,we determined the correlation function characteristics for each of the vertex sam-ples. Assessing their behaviour as a function of the average distance λV betweenthe sample vertices, for samples encompassing from 10% up to 100% of the totalnumber of Voronoi vertices and thus for λV ≈ 0.5−1.5×λC, revealed a remarkableand tantalizing scaling of the clustering phenomenon[10].All subsamples of Voronoi vertices have a two-point correlation function thatout to a certain range retains a power-law behaviour almost exactly similar theone of the full sample of vertcies. No significant difference in power law slopebetween the various subsamples can be discerned, all have γ ∼ 1.8 − 2.0. Onthe other hand, both clustering length ro and correlation length ra do display asystematic dependence on sample richness. Both ro and ra increase proportionallyto the sample richness, and hence the average vertex distance λV in the samples !The more massive the vertices are, the more strongly they cluster. In other words,in the line of the random field “peak biasing” scheme[6], we here find a purelygeometrically based biasing scheme. Interesting in this respect is to observe thatthe increase of rV,o is perfectly linearly proportional to the mutual vertex distanceλV in the sample (Fig. 2, left frame). This suggests that Voronoi vertex clusteringis a perfect realization of the clustering scaling description proposed by Szalay &Schramm[7]. It also adheres to the increasing level of clustering that selectionsof more massive clusters appear to display in large-scope N-body simulations[3]although there are telling differences in detailed behaviour.For our purpose, even more significant is the uncovered scaling of the cor-relation length rV,a, similar in character to that of rV,o. The increase of ra alsoturns out to be almost exactly linearly proportional to the average vertex distance.The repercussions are manyfold. It means that the selected vertices still have astrong positive correlation at scales where the poorer samples do not possess anyclustering. Moreover, samples with more massive clusters are expected to have aclustering that extends out further than that of their poorer brethren. This maybe a significant observation in the light of the finding that samples of rich galaxyclusters seem to have a positive correlation at scales of tens of Megaparsec, quitein excess of scales with appreciable galaxy clustering. Finally, the implied con-stant ratio between clustering and correlation length, ra/ro ≈ 1.86 (Fig. 2, rightframe), implies a perfect self-similar scaling of the Voronoi vertex distribution.The complete correlation function ξs(r) of each selected subsample s, and notjust the part in the power-law range, is a self-similar mapping of an elementaryfunction ξel scaled by means of a characteristic lengthscale parameter Ls.4 Bias and Cosmic Geometry: ConclusionsThe above results form a tantalizing indication for the existence of self-similarclustering behaviour in spatial patterns with a cellular or foamlike morphology. Itmight hint at an intriguing and intimate relationship between the cosmic foamlikegeometry and a variety of aspects of the spatial distribution of galaxies and clus-ters. One important implication is that with clusters residing at a subset of nodesin the cosmic cellular framework, a configuration certainly reminiscent of the ob-served reality, it would explain why the level of clustering of clusters of galaxiesbecomes stronger as it concerns samples of more massive clusters. In addition,it would succesfully reproduce positive clustering of clusters over scales substan-tially exceeding the characteristic scale of voids and other elements of the cosmicfoam. At these Megaparsec scales there is a close kinship between the measuredgalaxy-galaxy two-point correlation function and the foamlike morphology of thegalaxy distribution. In other words, the cosmic geometry apparently implies a ‘ge-ometrical biasing” effect, qualitatively different from the more conventional “peakbiasing” picture[6].References[1] Bardeen, J.M., Bond, J.R., Kaiser, N., Szalay, A.S., 1986, ApJ 304, 15[2] Bahcall, N.A., Soneira, R., 1983, ApJ 270, 20[3] Colberg, J., 1998, Ph.D. thesis, Ludwig-Maximilian Univ. Mu¨nchen[4] Dubinski, J., et al., 1993, ApJ 410, 458[5] Icke, V. 1984, MNRAS 206, 1P[6] Kaiser, N., 1984, ApJ , 284, L9[7] Szalay, A., Schramm, D.N., 1985, Nature , 314, 718[8] van de Weygaert, R., Icke, V., 1989, A&A 213, 1[9] van de Weygaert, R., 1991, Ph.D. thesis, Leiden Univ.[10] van de Weygaert, R., 1999, in preparation

   University of GroningenThe Cosmic Foam and the Self-Similar Cluster DistributionWeygaert, R. van dePublished in:ArXivIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Weygaert, R. V. D. (1999). The Cosmic Foam and the Self-Similar Cluster Distribution. ArXiv.https://arxiv.org/abs/astro-ph/9912279CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-10-2023arXiv:astro-ph/9912279 v1   14 Dec 1999The Cosmic Foam and the Self-Similar Cluster DistributionRien van de Weygaert1Kapteyn Institute, University of Groningen, Groningen, the Netherlands1 Foamlike Patterns in Cosmic StructureProbing cosmic large scale structure on the basis of X-ray observations puts par-ticular emphasis on the densest regions within the global matter distribution, therich clusters of galaxies. Understanding the relationship between the cluster dis-tribution and the underlying matter distribution is therefore a key element in anyassessment of cosmic structure on the basis of samples of X-ray selected clusters.By now, the foamlike arrangement of matter and galaxies on Megaparsec scaleshas become a well-established feature of the cosmic matter distribution, consist-ing of an assembly of anisotropic elements, filaments and walls of various sizes,surrounding large underdense void regions and sprinkled with wholly or partiallyvirialized dense clumps of matter, varying in size from rich clusters of galaxiesdown to small groups of a few galaxies. It is one of successes of gravitationalinstability cosmic structure formation theories to find that this pattern of walls,filaments and voids appears to be the generic outcome of these scenarios.A major obstacle in quantifying this quintessential aspect of cosmic structureis the lack of a systematic insight into the dynamical and statistical aspects of cel-lular geometries as well as the absence of a readily available and well-establishedmathematical machinery to evaluate and compare observations and simulations.Stochastic geometry – the branch of mathematics concerned with nontrivial geo-metrical concepts involving stochastic behaviour of one or more of their character-istics – may be expected to contribute significantly to further such understanding.2 Voronoi TessellationsThe canonical example of a stochastic geometrical model for a cellular divisionof space is that of Voronoi tessellations. This space filling network of convexpolyhedra offers a surprisingly realistic and versatile representation of the char-acteristics and features of the foamlike or cellular spatial arrangement of matterin the Universe. In short, it is defined through a spatial distribution of nuclei[9].Each nucleus corresponds to one Voronoi cell, which comprises that part of spacecloser to this nucleus than to any of the other. The walls and edges formingthe polyhedrons’ surfaces are identified with the wall-like and filamentary su-perclusters in the galaxy distribution, the vertices with and massive clusters ofgalaxies, while the interior of the Voronoi cells corresponds to the large void re-gions barren of galaxies. The morphology of artificial galaxy distributions set upwithin this network bears a striking resemblance to that observed in the morecomplicated circumstances of the real Cosmos or that of the artificial reality ofcomputer simulations of structure formation. Figure 1 shows a conglomerate ofFigure 1: A clump of Voronoi polyhedra from a tessellation of 1000 cells.One cell is surface-shaded, the surrounding neighbouring cells are depictedin wire-frame fashion. The corresponding vertices are indicated by soliddots. Distribution of all Voronoi vertices within a cubic volume. Courtesy:Jacco Dankers.several neighbouring Voronoi cells, with one of the cells by shaded surface and awire-framed representation in the case of its neighbouring cells. The solid linesare the edges (filaments) in the network, the dots (coloured red) are the verticesof the Voronoi tessellation. In principle a Voronoi tessellation could be regardedas a mere geometric toy model, a heuristic geometrical description of a substan-tially more complicated reality. However, a detailed assessment of the role ofvoids in structure formation provides a physical ground for them representing re-ality, whether the observed or simulated one, in a more subtle way[5][4]. Thisis certainly reinforced by their striking resemblance to the observed or simulatedfoamlike appearance of the galaxy and matter distribution.3 Voronoi Vertex ClusteringEven in the limited setting of Figure 1 it is evident that the vertex distribution isnot a random Poisson distribution. The full spatial distribution of Voronoi verticesin the full cubic volume (righthans frame fig. 1) clearly involves a substantialdegree of clustering. This impression of strong clustering, on scales smaller thanor of the order of the cellsize λC, is most evidently confirmed by their two-pointcorrelation function ξ(r). Not only can we discern a clear positive signal but –surprising at the time of its finding on the basis of similar computer experiments[8]– out to a distance of at least r ≈ 1/4 λC the correlation function appears to bean almost perfect power-law,ξ(r) = (ro/r)γ , (1)with a slope γ ≈ 1.9 − 2.0. Its amplitude, traditionally expressed in terms of the“clustering length” ro, at which ξ(ro) = 1 , has a value ro ≈ 0.29 λC. Beyondthis range, the power-law behaviour breaks down and following a gradual declinethe correlation function rapidly falls of to a zero value once distances are of theorder of the cellsize. However, rather than a characteristic geometric scale, ro ismore a measure for the “compactness” of the spatial clustering, set mainly by thesmall-scale clustering. A more significant scale within the context of the geometryof the spatial patterns in the density distribution is the “correlation length” ra,the scale at which ξ(ra) = 0 . As a genuine scale of coherence, it is more relevantto the morphology of the nontrivial spatial structures we seek to study. Beyondra the distribution of Voronoi vertices is practically uniform.If we interpret the clustering length ro ≈ 20h−1 Mpc, usually found for sam-ples of rich clusters of galaxies[2], within the context of a Voronoi tessellationit would imply a cellsize of λC ≈ 70h−1 Mpc. Although the two-point cluster-cluster correlation function reproduced by the Voronoi vertices fits very well tothe function yielded from the observations, the large cell size λc may be a com-plication. It is surely well in excess of the 25h−1 − 35h−1 Mpc size of the voids inthe galaxy distribution. Moreover, also within the Voronoi concept itself it wouldconflict with the clustering of objects dwelling in the walls and filaments of thesame tessellation framework. Clustering analysis of such configurations[9] revealsthat the two-point correlation function of galaxies confined to the walls – as wellas for those confined to the edges – also displays distinct power-law behaviourat sub-cellular scales. The involved clustering length, however, is different fromthat of the vertices in the same framework. For the wall galaxies it is but halfthe value of that of the vertices, rw,o ≈ 0.14 λC[9],[10]. If rw,o is identified withthe galaxy-galaxy clustering length of rg,o ≈ 5h−1 Mpc, this would yield a cellsizeof ≈ 35h−1 Mpc. The latter is suggestively similar to the size of the actually ob-served voids, which may be a tantalizing hint for a profound relationship betweenthe clustering length ro and the typical cellular scale of the cosmic foam network.Adressing this apparent inconsistency between vertex and wall clustering wefirst observe that the vertex correlation function of eqn. (1) concerns the wholesample of vertices, irrespective of any possible selection effects based on one ormore relevant physical aspects. In reality, it will be almost inevitable to invokesome sort of biasing through the defining criteria of the catalogue of clusters. In-terpreting the Voronoi model in its quality of asymptotic approximation to thegalaxy distribution, its vertices will automatically comprise a range of “masses”.Neglecting the details of the temporal evolution, we may assign each Voronoi ver-tex a “mass” equal to the total amount of matter that ultimately will flow towardsthat vertex. Applying the “Voronoi streaming model”[9] as a reasonable descrip-tion of the clustering process, it is reasonably straightforward if cumbersome tocalculate the “mass” or “richness” MV of each Voronoi vertex by pure geomet-ric means[10]. It concerns the volume of a non-convex polyhedron centered onthe Voronoi vertex, with related Voronoi nuclei as one of the polyhedral vertices.These nuclei are the ones that supply the Voronoi vertex with inflowing matter.Figure 2: Voronoi tessellation scaling. Left insert: Two-point correlationfunction ξ(r) Voronoi vertices. The clustering length ro (left) and the ratiora/ro (right) as a function of average vertex distance.Evidently, the “mass” is larger for vertices on the surface of large Voronoi cells.The vertex samples in our study consist of all Voronoi vertices with a richnessequal to higher than a limiting value, the “sample richness” MS[10]. Subsequently,we determined the correlation function characteristics for each of the vertex sam-ples. Assessing their behaviour as a function of the average distance λV betweenthe sample vertices, for samples encompassing from 10% up to 100% of the totalnumber of Voronoi vertices and thus for λV ≈ 0.5−1.5×λC, revealed a remarkableand tantalizing scaling of the clustering phenomenon[10].All subsamples of Voronoi vertices have a two-point correlation function thatout to a certain range retains a power-law behaviour almost exactly similar theone of the full sample of vertcies. No significant difference in power law slopebetween the various subsamples can be discerned, all have γ ∼ 1.8 − 2.0. Onthe other hand, both clustering length ro and correlation length ra do display asystematic dependence on sample richness. Both ro and ra increase proportionallyto the sample richness, and hence the average vertex distance λV in the samples !The more massive the vertices are, the more strongly they cluster. In other words,in the line of the random field “peak biasing” scheme[6], we here find a purelygeometrically based biasing scheme. Interesting in this respect is to observe thatthe increase of rV,o is perfectly linearly proportional to the mutual vertex distanceλV in the sample (Fig. 2, left frame). This suggests that Voronoi vertex clusteringis a perfect realization of the clustering scaling description proposed by Szalay &Schramm[7]. It also adheres to the increasing level of clustering that selectionsof more massive clusters appear to display in large-scope N-body simulations[3]although there are telling differences in detailed behaviour.For our purpose, even more significant is the uncovered scaling of the cor-relation length rV,a, similar in character to that of rV,o. The increase of ra alsoturns out to be almost exactly linearly proportional to the average vertex distance.The repercussions are manyfold. It means that the selected vertices still have astrong positive correlation at scales where the poorer samples do not possess anyclustering. Moreover, samples with more massive clusters are expected to have aclustering that extends out further than that of their poorer brethren. This maybe a significant observation in the light of the finding that samples of rich galaxyclusters seem to have a positive correlation at scales of tens of Megaparsec, quitein excess of scales with appreciable galaxy clustering. Finally, the implied con-stant ratio between clustering and correlation length, ra/ro ≈ 1.86 (Fig. 2, rightframe), implies a perfect self-similar scaling of the Voronoi vertex distribution.The complete correlation function ξs(r) of each selected subsample s, and notjust the part in the power-law range, is a self-similar mapping of an elementaryfunction ξel scaled by means of a characteristic lengthscale parameter Ls.4 Bias and Cosmic Geometry: ConclusionsThe above results form a tantalizing indication for the existence of self-similarclustering behaviour in spatial patterns with a cellular or foamlike morphology. Itmight hint at an intriguing and intimate relationship between the cosmic foamlikegeometry and a variety of aspects of the spatial distribution of galaxies and clus-ters. One important implication is that with clusters residing at a subset of nodesin the cosmic cellular framework, a configuration certainly reminiscent of the ob-served reality, it would explain why the level of clustering of clusters of galaxiesbecomes stronger as it concerns samples of more massive clusters. In addition,it would succesfully reproduce positive clustering of clusters over scales substan-tially exceeding the characteristic scale of voids and other elements of the cosmicfoam. At these Megaparsec scales there is a close kinship between the measuredgalaxy-galaxy two-point correlation function and the foamlike morphology of thegalaxy distribution. In other words, the cosmic geometry apparently implies a ‘ge-ometrical biasing” effect, qualitatively different from the more conventional “peakbiasing” picture[6].References[1] Bardeen, J.M., Bond, J.R., Kaiser, N., Szalay, A.S., 1986, ApJ 304, 15[2] Bahcall, N.A., Soneira, R., 1983, ApJ 270, 20[3] Colberg, J., 1998, Ph.D. thesis, Ludwig-Maximilian Univ. München[4] Dubinski, J., et al., 1993, ApJ 410, 458[5] Icke, V. 1984, MNRAS 206, 1P[6] Kaiser, N., 1984, ApJ , 284, L9[7] Szalay, A., Schramm, D.N., 1985, Nature , 314, 718[8] van de Weygaert, R., Icke, V., 1989, A&A 213, 1[9] van de Weygaert, R., 1991, Ph.D. thesis, Leiden Univ.[10] van de Weygaert, R., 1999, in preparation

Abstract: Voronoi Tessellations form an attractive and versatile geometrical asymptotic model for the foamlike cosmic distribution of matter and galaxies. In the Voronoi model the vertices are identified with clusters of galaxies. For a substantial range out to a scale in the order of the cellsize, their spatial two-point correlation function is a power-law with a slope $gamma approx 1.95$. This study presents recent results showing that subsets of vertices selected on the basis of their ``richness'', i.e. inflow volume, retain this power-law correlation behaviour. Interestingly, they do so with a ``clustering length'' $r_{rm o}$ that is exactly linearly proportional to the average inter-vertex distance in the sample, thus forming a realization of the Szalay-Schramm prescription. For the geometry and structural patterns even more significant is the finding tessellation vertices display a similar linear increase for their correlation length $r_{rm a}$, the coherence length at which $xi(r_{rm a})=0$. Such patterns therefore exhibit positive correlations out to distances considerably in excess of the cellsize. Most intriguing is the implication of self-similar scaling, while these results may be regarded as the presentation of a ``geometrical bias'' effect. A seemingly rigid structure appears to represent a flexible and useful geometric model for exploring the statistical and dynamical repercussions of the nontrivial cellular patterns in Megaparsec scale cosmic structure.

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The Cosmic Foam and the Self-Similar Cluster Distribution

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