THE SHAPE, MULTIPLICITY, AND EVOLUTION OF SUPERCLUSTERS IN
CDM COSMOLOGY
James J. Wray,1 Neta A. Bahcall,1 Paul Bode,1 Carl Boettiger,1 and Philip F. Hopkins2
Received 2006 March 2; accepted 2006 August 22
ABSTRACT
We determine the shape, multiplicity, size, and radial structure of superclusters in the
CDM concordance cos-
mology from z ¼ 0 to z ¼ 2. Superclusters are defined as clusters of clusters in our large-scale cosmological simulation.
We find that superclusters are triaxial in shape;many have flattened since early times to become nearly two-dimensional
structures at present, with a small fraction of filamentary systems. The size and multiplicity functions are presented at
different redshifts. Supercluster sizes extend to scales of
100Y200 h1 Mpc. The supercluster multiplicity (richness)
increases linearly with supercluster size. The density profile in superclusters is approximately isothermal (
R2) and
steepens on larger scales. These results can be used as a new test of the current cosmology when compared with up-
coming observations of large-scale surveys.
Subject headinggs: cosmology: theory — galaxies: clusters: general — large-scale structure of universe
Online material: color figures
1. INTRODUCTION
Superclusters are the largest structures in the universe. Their
sizes span the range from a few times the radius of a typical gal-
axy cluster (
3Y4 h1 Mpc) to the lengths of the ‘‘Great Walls’’
of galaxies observed by the CfARedshift Survey (Geller &Huchra
1989) and the Sloan Digital Sky Survey (Gott et al. 2005). These
latter sizes of
102 h1 Mpc represent a significant fraction
(
10%) of the horizon scale.
Because they are so large relative to a typical survey volume,
superclusters have proven difficult to study, especially in a statis-
tical manner. Observationally, attempts to characterize the large-
scale structure using distributions of galaxies date back to Hubble
(1936), but these methods did not become powerful until the ad-
vent of galaxy-cluster and galaxy-redshift surveys. Abell (1958)
used the Palomar Sky Survey to construct a catalog of rich clusters,
withwhich hewas able to find evidence forwhat he called ‘‘second-
order clusters,’’ i.e., clusters of clusters, which are indeed super-
clusters. Subsequently, Gregory & Thompson (1978) studied
the supercluster containing the rich Coma Cluster, and Gregory
et al. (1981) studied the Perseus supercluster. Other observational
approaches to supercluster identification have been based, like
that of Abell (1958), on cluster catalogs (e.g., Oort 1983; Bahcall
& Soneira 1984; Bahcall 1988; Einasto et al. 1994; Kolokotronis
et al. 2002) or on analysis of smoothed density fields (Einasto
et al. 2003a, 2003b, 2006; Erdogdu et al. 2004). These have been
important in characterizing the low-redshift universe, but they
have not yet allowed the study of supercluster evolution from
early cosmic times (i.e., from redshifts of order unity). How do
superclusters form, and how do their shapes, multiplicities, sizes,
and structures evolve with time? The answers to these fundamen-
tal questions are not yet known.
Recently, there has been exciting progress on both the obser-
vational and computational fronts. The Sloan Digital Sky Survey
(SDSS; York et al. 2000; Stoughton et al. 2002) is collecting deep,
broadband imaging and spectroscopy of a large, contiguous por-
tion of the sky. SDSS has created the largest galaxy redshift
survey catalog to date, and much work is also underway to im-
prove photometric redshift techniques (e.g., Csabai et al. 2000;
Budava´ri et al. 2000), which would permit measurements of 3D
galaxy positions for the much larger SDSS imaging catalog. On
the theoretical side, a concordance cosmological model has gained
widespread acceptance (Bahcall et al. 1999; Bennett et al. 2003;
Spergel et al. 2003, 2006). Applying this model to cosmological
dynamics codes such as that described below, we can now re-
liably predict the properties and evolution of superclusters. The
time is therefore ideal for a computational analysis of large-scale
structure using new, powerful simulations, the results of which
can be used to determine the properties and evolution of super-
clusters. In addition, these results can be compared with obser-
vations, thus providing a new test of the current cosmological
model—complementary to that provided by CMB, large-scale
power spectra, and other studies—as well as shedding light on
structure formation and evolution.
We begin by describing, in x 2, the process by which clusters
and in turn superclusters were identified in the simulation output.
We provide quantitative descriptions of the multiplicity and size
distributions of the identified superclusters in xx 3.1 and 3.2. We
examine the radial structure of our superclusters in x 3.3 and the
dimensionality of superclusters in x 3.4. The evolution of the
above properties with redshift is presented in x 4. Results for
superclusters selected from only higher mass clusters are pre-
sented in x 5. Our conclusions are summarized in x 6.
2. SELECTION OF SUPERCLUSTERS
2.1. The Simulation
We use the efficient, parallel dark matter simulation code of
Bode & Ostriker (2003). A large-scale, high-resolution simula-
tion was evolved with a particle-mesh method to compute long-
range gravitational forces, and a tree code to treat high-density
regions.We use the cosmological parameters of the concordance
model: m ¼ 0:27, 
¼ 0:73, H0 ¼ 70 km s1 Mpc1, ns ¼
0:96, and
8 ¼ 0:84 (see Spergel et al. 2003; see also Spergel et al.
2006). A total of 12603 particles, each assigned themass 1:264 ;
1011 h1 M, is evolved in a periodic box 1500 h
1 Mpc on a
side, and positions are saved at the appropriate times in order
A
1 Princeton University Observatory, Princeton, NJ 08544; jwray@astro
.cornell.edu, neta@astro.princeton.edu, bode@astro.princeton.edu, cboettig@ astro
.princeton.edu.
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge,
MA 02138; phopkins@cfa.harvard.edu.
907
The Astrophysical Journal, 652:907Y916, 2006 December 1
# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

to construct a light cone with vertex at one corner of the box.
Clusters are then identified in two different, partially over-
lapping volumes: a ‘‘low-redshift’’ sphere of radius 1500 h1 Mpc,
corresponding to a maximum redshift of z 0:57, and a ‘‘high-
redshift cone,’’ which includes the low-z sphere’s positive octant
(x > 0, y > 0, z > 0) and extends this octant out to z ¼ 3, a
distance of
4600 h1 Mpc. Using the cluster selection scheme
described below, a total of 1,442,616 clusters with Mvir  1:75 ;
1013 h1M are found in the low-z sphere, and 720,550 in the high-
z cone. Further details on this light cone simulation are given by
Hopkins et al. (2005, hereafter HBB).
2.2. Cluster Selection
Clusters are identified in the simulation using the friends-of-
friends (FOF) algorithm (see HBB; Davis et al. 1985). The FOF
results depend on the chosen linking length L. A common pa-
rameterization uses the linkage parameter b  L/rav, where rav is
the mean interparticle separation. This is useful in applications
where the mean density of objects, and in turn their mean sep-
aration, changes over time. Holding b constant imposes the same
minimum overdensity for clusters at all redshifts. We use the
value b ¼ 0:2 (see also HBB), corresponding to an overdensity
of
180 times the mean density. The center of each FOF cluster
is taken to be the position of the most bound particle. This cen-
ter is used to compute the virial radius Rvir, defined as the radius
enclosing the virial overdensity expected from spherical top-hat
collapse (which evolves with redshift in the
CDM cosmology).
In this analysis, we consider only clusters with virial mass thresh-
old (i.e., the total mass within the virial radius) Mvir  1:75 ;
1013 h1 M. This mass threshold is typical for a poor cluster of
galaxies. The mean mass of all clusters in the low-z sphere is

4:40 ; 1013 h1 M. In x 5 we discuss superclusters selected
from clusters with a higher mass threshold.
The ellipticities of the clusters identified at low and high z are
studied in detail by HBB. Relevant to our present work on su-
perclusters, they find alignments between nearby cluster pairs
out to separations of
100 h1 Mpc; the strength of alignment
increases with redshift and with decreasing cluster separation.
HBB explain these alignments as being due to clusters forming
and gathering material along ‘‘filamentary superstructures’’; they
provide evidence for overdensities along the separation vectors
between pairs of aligned clusters.
2.3. Supercluster Selection
As demonstrated by Abell (1958), Bahcall (1988), and refer-
ences therein, superclusters can be identified by their overden-
sity of clusters. We use the FOF method to identify superclusters
from the cluster sample defined above (xx 2.1Y2.2). The mean
density of clusters in the low-z sphere (we shall not discuss the
high-z cone until x 4) is nav ¼ 1:02 ; 104 (h1 Mpc)3, yielding
a mean intercluster separation of rav  n1/3av ¼ 21:4 h1 Mpc.
We therefore select linking lengths of L ¼ 3, 5, 7, and 10 h1Mpc,
corresponding to b  0:14, 0.23, 0.33, and 0.47, respectively.
All scales in this paper are in comoving coordinates. The differ-
ent linking lengths sample different portions of the supercluster
geometry. Small linking lengths identify only the dense cores of
superclusters, whereas larger linking lengths allow the FOF algo-
rithm to percolate to lower density filaments of clusters. Large
L-values therefore reach to the outer parts of superclusters, as well
as identify looser superclusters, and often connect structures that
appear discrete with smaller L.
We use Monte Carlo simulations to compare the significance
of the selected superclusters with that of random associations.
Clusters are placed at randomly chosen positions within a sphere
such that the total number density is the same as that for the
simulated clusters, and the FOF method is applied, as before,
with L ¼ 3, 5, 7, and 10 h1 Mpc. We find significantly fewer
clusters per supercluster, and significantly fewer superclusters
down to multiplicity as small as 3, for all four linking lengths
tested. This result implies that most of the superclusters are real,
even for L ¼ 10 h1 Mpc. The only exception is the case of
binary clusters (superclusters with only two members) selected
with L ¼ 10 h1 Mpc, of which we find only
25% more using
the simulated clusters than we do with the Monte Carlo sample.
The random rate of superclusters is typically <10% for N > 3
superclusters, decreasing to <1% for N > 6 superclusters. The
Monte Carlo results are compared with the
CDM multiplicity
functions below.
To help visualize the typical geometries of superclusters iden-
tified by the FOF method, we present supercluster maps of a typ-
ical region (a cube of side length 200 h1Mpc, concentricwith the
low-z sphere) for L ¼ 7 h1 Mpc (Figs. 1Y2). Each point is a
cluster, and a given symbol represents membership in the same
supercluster. The maps present a projection of the 3D distribution
onto the x-y plane in the first plot, and onto the x-z plane in the
second plot. Viewing the plots in turn corresponds to a 90 rota-
tion of the viewing angle about the x-axis. For clarity, we show
Fig. 1.—Clusters identified as supercluster members at redshift zero, for
L ¼ 7 h1 Mpc and minimum multiplicity 5. Clusters represented by similar
points are members of the same supercluster. See the text (x 2.3) for further
description.
Fig. 2.— Same parameters as Fig. 1, but a different projection.
WRAY ET AL.908 Vol. 652

only superclusters with at least five member clusters. Members of
a supercluster that extend outside the limits of our cubical region
are not shown, so some superclusters appear to have fewer mem-
bers than the imposed minimum. Comparing these maps to anal-
ogous figures generated with L ¼ 5 and 10 h1 Mpc reveals, as
expected, that increasing the linking length often causes neigh-
boring superclusters to be connected into one. This is especially
true when L increases from 7 to 10 h1 Mpc, resulting in many
large, complicated structures with central cores joining multiple
lower density filaments.
3. SUPERCLUSTER PROPERTIES
3.1. Multiplicity Function
The multiplicity of a supercluster is the number of clusters it
contains, N. In Figure 3 we present, for various linking lengths,
the integrated supercluster multiplicity function—the number den-
sity (per cubic comoving h1Mpc) of superclusters withN or more
members—as a function of N. Also shown for comparison are the
corresponding results from random catalogs, as described in the
previous section.
Longer linking lengths yield more superclusters of all multi-
plicities, as expected. Shorter linking lengths yield curves that
rise more steeply toward low multiplicities, implying that they
preferentially find small superclusters with low multiplicity. Con-
versely, longer linking lengths find a higher percentage of larger,
richer superclusters. The shorter linking lengths naturally trace
high-density regions (e.g., dense cores of superclusters), while
larger linking lengths characterize lower density regions, such
as the outer parts of superclusters or diffuse superclusters. Small
L can also divide structures that are identified with larger link-
ing lengths. The fact that we see superclusters with
40 or more
clusters may seem surprising, but the total number density of
such structures is only 108 per (h1 Mpc)3, and it is based on
a very low mass threshold of cluster members, Mvir  1:75 ;
1013 h1 M. Such clusters are numerous, and any comparison
with observations should of course account for this threshold.
For superclusters selected using higher mass clusters, see x 4.1.
We repeated the above analysis after selecting superclusters in
redshift-space coordinates; the resulting multiplicity functions
are nearly identical to the real-space functions at low multiplic-
ities, and are shifted to slightly higher N (by up to 20%) at the
highest multiplicities.
3.2. Size Function
The supercluster size function—i.e., the number of super-
clusters above a given size—is presented in Figure 4 for different
linking lengths. The size of a supercluster is defined as the max-
imum distance of any member cluster from the supercluster
center of mass. As with the multiplicity function, the size func-
tion rises more steeply for shorter linking lengths. This implies
that the longer linking lengths find a greater ratio of large su-
perclusters to small superclusters, analogous to the result for the
multiplicity function. Similar results are found using redshift-
space coordinates; for the largest linking length used, the sizes
are slightly (P5%) larger.
We find that the largest supercluster radii are
80 h1 Mpc,
corresponding to diameters of
160 h1 Mpc. This is consistent
with the largest structures seen in observations (e.g., Bahcall &
Soneira 1984; Geller & Huchra 1989; Gott et al. 2005), and it is
also the largest distance at which HBB detect cluster alignment
in the simulation.
3.3. Radial Structure
The average multiplicity of superclusters is plotted as a func-
tion of supercluster size for different linking lengths in Figure 5.
For N ¼ 2 or 3—objects more aptly called binary or triple
clusters than superclusters—the maximum radius is set mainly
by the separation(s) of the two or three clusters; thus the average
multiplicity changes slowly over this range of separations. Be-
yond this regime, the multiplicity and size of superclusters ap-
pear to be related nearly linearly. This result suggests that the
spherically averaged density profile in superclusters is roughly
Fig. 3.—Integrated supercluster multiplicity functions for linking lengths
L ¼ 3, 5, 7, and 10 h1 Mpc, all at z  0. Curves show superclusters identified in
our randomMonte Carlo simulations to test the physical reality of the superclusters.
[See the electronic edition of the Journal for a color version of this figure.]
Fig. 4.—Integrated size functions for linking lengths L ¼ 3, 5, 7, and 10 h1
Mpc at z  0. The size measurement along the horizontal axis is, for a given
supercluster, the distance of themember clustermost distant from the supercluster
center of mass. All are comoving scales. [See the electronic edition of the Journal
for a color version of this figure.]
SUPERCLUSTER SHAPE AND EVOLUTION 909No. 2, 2006

isothermal; i.e., n(r) / r2, where the cluster number density n(r)
satisfies
N ¼
Z Rmax
0
n(r)4r 2dr: ð1Þ
This profile yields multiplicity N / Rmax for n(r) / r2.
We can investigate the spherically averaged density profile
more directly by examining the distribution of clusters within in-
dividual superclusters and determining the average number den-
sity profile for the entire sample. To do so, we superpose the
centers of mass (calculated using only the mass in clusters) of all
superclusters, and scale all superclusters to the same size by
dividing all cluster coordinates (relative to the center of mass) for
a given supercluster by its Rmax. We add up the total number of
clusters inside radial bins, divided by the total number of clusters
in superclusters for the given linking length. We have included
only superclusters with five or more members; because the multi-
plicity function drops off rapidly, most of the superclusters in-
cluded have only five or six members. This fact affects the
combined profile in two ways. First, only in a few cases will one
of the five members happen to lie very near the supercluster’s
center of mass; thus it is only at R /Rmaxk 0:2 that we begin
to find a significant fraction of the clusters. Second, one out of
roughly five clusters in each supercluster lies at R /Rmax ¼ 1.
Thus, for the smaller linking lengths, the integrated number frac-
tion approaches a limit of
0.8; for the larger linking lengths,
which find richer superclusters, the limiting fraction is slightly
higher. The fraction of clusters within R /Rmax increases linearly
with radius for the range 0:3PR /RmaxP0:7. This linear relation-
ship indeed implies n(R)
R2 for the supercluster density pro-
file in this range. The profile steepens on larger scales.
The average mass of clusters in superclusters, M ¼ 4:8 ;
1013 h1 M for the L ¼ 7 h1 Mpc superclusters, is some-
what larger than the average mass of clusters in the total sample,
M ¼ 4:4 ; 1013 h1 M, indicating that superclusters contain
slightly more massive clusters, on average, than the mean. The
L ¼ 7 h1 Mpc superclusters withN  5 members have an even
higher average cluster mass of M ¼ 5:5 ; 1013 h1 M, imply-
ing that richer superclusters contain more massive clusters than
poorer superclusters.
3.4. Supercluster Shapes
Wemeasure the shape, and in particular the dimensionality, of
a given supercluster by fitting a 3D ellipsoid to the distribution of
its member clusters. We use the code developed by HBB to fit
ellipsoids to the dark matter particles comprising their clusters.
The code constructs, for a given supercluster, the 3 ; 3 matrix of
second moments of cluster positions relative to the supercluster
center of mass; i.e.,
Ii j ¼
X
xi xjm; ð2Þ
where xi and xj are two of the position coordinates for a given
cluster, andm is the mass of the cluster; the sum is over all mem-
ber clusters. The eigenvalues of this matrix are simply the three
axis lengths of the best-fit ellipsoid for the supercluster, times a
known constant. Following the notation used by HBB, we shall
denote these lengths a1, a2, and a3 for the primary, secondary,
and tertiary axes, respectively; i.e., a1  a2  a3.We restrict our
analysis to superclusters with N  5; for poorer superclusters,
the ‘‘best-fit ellipsoid’’ would not be very meaningful.
We have already examined size distributions for superclusters
(x 3.2), so we now focus on the axis ratios, which allow us to
probe the shape and dimensionality of superclusters. Figure 6
displays the distribution of primary axis ratios a2/a1 for super-
clusters selected with L ¼ 5, 7, and 10 h1 Mpc. Figure 7 shows
the distribution of secondary axis ratios a3/a2, and Figure 8 shows
a3/a1. The mean and peak axis ratios from each curve in these
figures are listed in Table 1.
The secondary axis ratios are generally smaller than the pri-
mary axis ratios. We find that a primary axis ratio of a2/a1  0:6
is typical for superclusters selected using a broad range of link-
ing lengths; this corresponds to a primary ellipticity of 1  1
a2/a1  0:4. The secondary axis ratio is more strongly dependent
Fig. 5.—Multiplicity vs. size of superclusters, z  0.Rmax is defined as in Fig. 4.
Linking lengths L are quoted in h1 Mpc. [See the electronic edition of the Journal
for a color version of this figure.]
Fig. 6.—Distributions of supercluster primary axis ratios for various linking
lengths, at z  0. Linking lengths L are quoted in h1 Mpc. [See the electronic
edition of the Journal for a color version of this figure.]
WRAY ET AL.910 Vol. 652

on the linking length used to find the superclusters. For L ¼
5 h1 Mpc, we find a peak a3/a2 of
0.4, corresponding to a
secondary ellipticity of 2  1 a3/a2  0:6; for L ¼ 10 h1 Mpc,
we find a peak of a3/a2  0:2, or 2  0:8. The peak a3/a1 ratios
range from0.13 forL ¼ 10 h1Mpc to0.23 for L ¼ 5 h1Mpc.
These measurements allow us to construct a picture of the
typical shape and dimensionality of superclusters. The inter-
pretation of different ellipticities can be described as follows:
1. A combination of high primary and low secondary axis
ratios is typical of two-dimensional pancake-like structures (two
dimensions are large, and one is small).
2. A combination of low primary and high secondary axis
ratios is typical of one-dimensional filamentary structures (one
dimension is large, and two are small).
3. A combination of primary and secondary axis ratios that
are both significantly less than unity is typical of structures most
appropriately described as triaxial (each dimension is a different
size).
4. If both axis ratios are close to unity, then the distribution is
nearly spherical.
The results we find suggest that superclusters are typically tri-
axial structures nearing 2D, pancake-like structures at z  0.
They are neither spherical nor purely filamentary in nature. As
the linking length is increased, the resulting superclusters be-
come more two-dimensional; the cores of superclusters are thus
more triaxial, spreading out to more pancake-like structures on
larger scales.
We plot the bivariate distribution of a2/a1 and a3/a2 for L ¼
7 h1Mpc in Figure 9 (the distributions for L ¼ 10 and 5 h1Mpc
are similar). Applying the interpretation rules listed above, 1D
structures reside in the upper left corner of Figure 9, 2D in the
lower right corner, and 3D (spherical) in the upper right. Points
in the middle or toward the lower left corner represent triaxial
structures.
Figure 9 shows that a large majority of the superclusters are
more nearly two-dimensional than one-dimensional. No nearly
spherical superclusters are found with this linking length, but
there is a smaller population of filamentary structures. Similar dis-
tributions are found for L ¼ 10 and 5 h1 Mpc, although these
show more flattened structures (2D) as L increases, and more tri-
axial structures as L decreases. This is the same trend depicted in
Figure 7 as a shift toward lower secondary axis ratio with increas-
ing linking length. The shorter linking lengths also yield signifi-
cantly higher fraction of filamentary structures, i.e., superclusters
with low primary axis ratio and a secondary axis ratio approach-
ing unity.
This trend suggests that most superclusters at z  0 are triaxial
structures with one dimension considerably smaller than the
other two. This geometry is apparent in Figure 9, and becomes
more significant when a larger FOF linking length is used. Since
the superclusters are not completely flat, smaller linking lengths
may probe only the dense core regions, which are smaller in size
than the supercluster thickness. Thus, while the thickness sets an
upper bound to the tertiary axis length identified with any linking
length, the higher L-values find longer primary and secondary
axes by probing farther out into the pancake. This causes larger
linking lengths to find a smaller mean secondary axis ratio, and
reveal the underlying 2D nature of the large-scale superclusters.
Similarly, for a given L, the largest, highest-multiplicity super-
clusters (N > 10) are on average flatter (a3/a2  0:24 for L ¼
10 h1 Mpc) than those with N ¼ 5Y10 (a3/a2  0:35). This
flatness is still apparent in projection: for superclusters selected
(from Mvir  1:75 ; 1013 h1 M clusters) in redshift space,
the projected axis ratios have a broad distribution with a mean
a2/a1 ¼ 0:4  0:005 and a peak a2/a1  0:35, for all linking
lengths L ¼ 5, 7, and 10 h1 Mpc.
The triaxiality of superclusters can also be quantified using
the measurement commonly employed for geometrical descrip-
tions of elliptical galaxies, first introduced explicitly by Statler
(1994):
T  1 (a2=a1)
2
1 (a3=a1)2
: ð3Þ
Fig. 7.—Distributions of supercluster secondary axis ratios for various link-
ing lengths, at z  0. Linking lengths L are quoted in h1 Mpc. [See the electronic
edition of the Journal for a color version of this figure.]
Fig. 8.—Distributions of supercluster third axis ratios a3/a1 for various
linking lengths, at z  0. Linking lengths L are quoted in h1 Mpc. [See the elec-
tronic edition of the Journal for a color version of this figure.]
SUPERCLUSTER SHAPE AND EVOLUTION 911No. 2, 2006

With a1  a2  a3, T will take on values between 0 and 1 (unless
a1 ¼ a2 ¼ a3, in which case T is undefined). Two-dimensional
pancake structures approach T ! 0, while filaments approach
T ! 1 (note, however, that the increase is not linear with axis
ratio). Intermediate values of T represent triaxial systems, with
triaxiality increasing with T. By this measure our low-z super-
clusters are triaxial, ranging from a mean of T  0:65 for L ¼
10 h1 Mpc to T  0:69 for L ¼ 5 h1 Mpc.
For all linking lengths we see a tail in the distribution that de-
scribes structures that are very nearly one-dimensional. Thus,
some mass concentrations are filamentary at z  0, especially in
the relatively dense regions probed by L ¼ 5 h1 Mpc. Indeed,
wewould expect structures that have collapsed along two dimen-
sions to be of higher average density than those that have col-
lapsed along only one dimension. This result agrees with HBB’s
identification of filamentary density enhancements in the large-
scale structure; here we add the prediction of more diffuse flattened
structures surrounding and possibly connecting the filaments.
4. SUPERCLUSTER EVOLUTION
We examine the time evolution of the supercluster properties
studied above. We study superclusters in two redshift slices at
0:8 z 1:2 and 1:7 z 2:4, denoted z  1 and z  2,
respectively.
Superclusters are identified by the FOFmethod as before. How-
ever, since the cluster abundance decreases at higher redshift, and
their mean separation thus increases, we compare superclusters
found at different redshifts notwith the same linking length L, but
instead with the same linkage parameter b  L/rav, where rav is
the mean comoving intercluster separation. As discussed in x 2.2,
this approach sets a minimum overdensity for detection of su-
perclusters, which is constant with redshift.
This use of a constant linkage parameter accounts for the re-
duced density of clusters at high redshift. However, because
clusters are clustered, the density of superclusters depends on the
relative rates of cluster formation in regions of high density ver-
sus low density. Hierarchical supercluster formation, in which
clusters gravitate toward each other over time, also contributes to
a reduced supercluster density at early times.
We indeed find a lower density of superclusters in the high-z
cone. The total number of superclusters decreases by a factor of a
few from z  0 to z  1, and by over an order of magnitude from
z  1 to z  2. In order to construct a sufficiently large sample,
we consider broad ranges in redshift, especially for the z  2
slice. In addition, we omit the b-value corresponding to the L ¼
3 h1 Mpc (z  0) case from our subsequent analysis, as it re-
sults in too few superclusters at high redshift.
As stated in x 2.3, our z  0 linking lengths of L ¼ 5, 7, and
10 h1 Mpc correspond to b ¼ 0:23, 0.33, and 0.47, respec-
tively. We use these latter linkage parameters b to refer to the
linking lengths used in our subsequent analysis. A given value of
the b-parameter can be converted to an actual comoving linking
length at a given redshift simply by multiplying by the mean
intercluster separation at that redshift. The mean cluster densities
in our z  1 and z  2 shells are nav ¼ 3:53 ; 105 and 2:57 ;
106 (h1 Mpc)3, corresponding to mean cluster separations of
rav ¼ 30:5 and 73.0 h1 Mpc, respectively. Therefore, b ¼ 0:23,
0.33, and 0.47 correspond to L ¼ 7, 10, and 14.3 h1 Mpc at
z  1, and 16.8, 24, and 34.3 h1 Mpc at z  2.
4.1. Multiplicity Function
The evolution of the multiplicity function is presented in Fig-
ure 10. We present the integrated multiplicity function for all
three redshifts, using two different values of b. The number den-
sity of superclusters drops noticeably with increasing redshift.
Monte Carlo simulations analogous to those described in x 2.3
were also run at z  1 and 2, with results similar to those ob-
tained at low z: the multiplicity functions are physically signif-
icant for all multiplicities and linking lengths, except for binary
clusters in the b ¼ 0:47 case, where only
30% more binary
clusters are found over the Monte Carlo results.
The redshift evolution of the multiplicity function depends on
the linking length of the superclusters. The vertical distances be-
tween the b ¼ 0:47 curves in Figure 10 increase with increasing
multiplicity, implying that the abundance of the richest (highest-N )
superclusters found with this linking length drops off especially
swiftly with increasing z. However, for b ¼ 0:23 the vertical dis-
tances decrease with increasing N, to the extent that the abun-
dance of the richest superclusters found with this linking length
is in fact roughly the same at z  1 as it is today. We interpret
TABLE 1
Supercluster Axis Ratios at z  0, Cluster Masses Mvir  1:75 ; 1013 h1 M
L Mean a2/a1 Peak a2/a1 Mean a3/a2 Peak a3/a2 Mean a3/a1 Peak a3/a1
5........................................... 0.60  0.009 0.64 0.50  0.007 0.41 0.29  0.004 0.23
7........................................... 0.57  0.004 0.62 0.43  0.003 0.31 0.23  0.001 0.19
10......................................... 0.54  0.002 0.64 0.33  0.001 0.20 0.16  0.001 0.13
Notes.—Peaks and mean values of the distributions shown in Figs. 6Y8. Statistical errors are given for the mean values.
Fig. 9.—Bivariate distribution of primary and secondary axis ratios for low-z
superclusters found with linking length L ¼ 7 h1 Mpc. See the text (x 3.4) for
interpretation.
WRAY ET AL.912 Vol. 652

these trends as further evidence that small linking lengths (e.g.,
b ¼ 0:23) select the densest regions of superclusters. Whereas
the larger linking lengths do not find many rich superclusters
at high redshift because of the lack of clusters, the smaller link-
ing lengths find the same dense supercluster cores even at much
earlier times. These dense cores are likely the sites of the largest
overdensities present at early times, and the first places where
observable structures form through gravitational collapse. Thus
smaller linking lengths relative to the mean intercluster separation
are needed to locate the highest overdensities in the observable
universe. The lack of rich superclusters at high z is also evidence
that these structures form largely through gravitational accretion
of clusters that are initially too distant to be identified as super-
cluster members.
Supercluster size remains highly correlated with multiplicity
at high z, but with a lower N-Rmax normalization than at low z.
The size function does not evolve significantly with redshift for
the larger superclusters (see Fig. 11), while, as expected, the num-
ber of cluster members decreases.
4.2. Dimensionality
Using the method described in x 3.4, we fit ellipsoids to the
superclusters identified at z  1 and z  2. Because we only
perform these fits on superclusters with N  5, which are rare at
high redshift, the statistical uncertainties are larger. For example,
only 68 superclusters are identified at z  2 for b ¼ 0:23.
We use the axis ratios of the best-fit ellipsoids to characterize
the evolution of supercluster dimensionality. Tables 2 and 3
present themean primary and secondary axis ratios, respectively,
for b ¼ 0:23, 0.33, and 0.47 at z  0, 1, and 2. We also construct
the axis ratio distributions (similar to Figs. 6Y7) as a function of z
for b ¼ 0:33 and 0.47 (there are too few b ¼ 0:23 superclusters
at high redshift to create meaningful distributions). The results
for b ¼ 0:47 are shown in Figures 12Y13. The number of high-
redshift superclusters is too low to construct bivariate distribu-
tions as in Figure 9, but we can consider how the changes in
mean ratios correspond to translations of these distributions.
The strongest evolution observed is the increase in the sec-
ondary axis ratio with increasing redshift, for b ¼ 0:47 (Fig. 13).
Most of the change occurs between z  0 and z  1, with only a
slight additional increase between z  1 and z  2 (see Table 3).
Typical secondary axis ratios increase from a3/a2  0:25 at low
z to a3/a2  0:45 at high z. Thus the flattened structures ob-
served with b ¼ 0:47 at z  0 (see x 3.4) appear less flattened at
high redshifts, indicating their gravitational collapse over time.
Very little evolution is evident in the secondary axis ratio for
superclusters identified with b ¼ 0:33, but the population of
high-density filaments with a3/a2  1 evolves between z  1
and z  0, as there are almost none of these filamentary struc-
tures at high redshift. The z ¼ 0 distributions for b ¼ 0:33 are
shown in Figures 6Y7; at higher redshifts the distributions are
similar to the corresponding curves in Figures 12Y13.
The primary axis ratios are slightly smaller at higher redshifts
for both b ¼ 0:33 and 0.47. Typical values decrease from a2/a1 
0:6 at z  0 to a2/a1  0:5 at z  1Y2. In a plot similar to Figure 9,
Fig. 11.—Evolution of the integrated supercluster size function (comoving
scales) with redshift. Linking lengths L are quoted in h1 Mpc. [See the electronic
edition of the Journal for a color version of this figure.]
TABLE 2
Evolution of the Primary Axis Ratio a2/a1, Mvir  1:75 ; 1013 h1 M
z b = 0.23 b = 0.33 b = 0.47
0......................................... 0.60  0.009 0.57  0.004 0.54  0.002
1......................................... 0.44  0.03 0.47  0.01 0.48  0.01
2......................................... 0.52  0.06 0.50  0.03 0.50  0.02
Notes.—Mean values are given as a function of the linking-length parameter b.
Statistical errors are given. Note that b ¼ 0:23 corresponds to Lz¼0 ¼ 5 h1 Mpc,
b ¼ 0:33 to L0 ¼ 7 h1 Mpc, and b ¼ 0:47 to L0 ¼ 10 h1 Mpc.
TABLE 3
Evolution of the Secondary Axis Ratio a3/a2, Mvir  1:75 ; 1013 h1 M
z b = 0.23 b = 0.33 b = 0.47
0......................................... 0.50  0.007 0.43  0.003 0.33  0.001
1......................................... 0.36  0.02 0.40  0.01 0.44  0.01
2......................................... 0.41  0.05 0.41  0.03 0.46  0.01
Notes.—Mean values are given as a function of the linking-length parameter b.
Statistical errors are given.
Fig. 10.—Evolution of the integrated supercluster multiplicity function with
redshift. Linking lengths L are quoted in h1 Mpc. [See the electronic edition of
the Journal for a color version of this figure.]
SUPERCLUSTER SHAPE AND EVOLUTION 913No. 2, 2006

this would correspond to a leftward shift in the distribution, i.e., a
shift toward greater triaxiality at high redshift, and a shift to a 2D
geometry as the systems approach z  0. Quantitatively, our high-z
superclusters have mean triaxiality T  0:79 for both linking
lengths, significantly higher than the mean T for any linking length
at low z.
We conclude that superclusters had highly triaxial structure at
early times, and that between a cosmological look-back time of

8 Gyr (corresponding to z  1 for the cosmology described in
x 2.1) and the present day, many collapsed along their smallest
dimensions to form the more nearly pancake-like structures ob-
served in the simulation at z  0. In the same span of time, some
structures in regions of highest density also collapsed along a
second dimension to form nearly one-dimensional filaments.
5. SUPERCLUSTERS SELECTED
USING HIGH-MASS CLUSTERS
How do the supercluster properties change if superclusters
are selected using higher mass clusters as their seeds? While the
abundance of high-mass clusters is significantly lower, and the
statistical uncertainties larger, the massive clusters are more easily
detected in observational surveys. We repeat the analysis described
above for superclusters selected using the rarer, higher mass clus-
ters with Mvir  1 ; 1014 h1 M. This threshold corresponds to
typical ‘‘rich clusters’’ (e.g., Abell 1958; Bahcall 1988). There
are 9:68 ; 104 clusters above this mass threshold at z  0, and
4:88 ; 103 clusters at z  1. The cluster abundance is 6:8 ;
106 (h1Mpc)3 at z  0, decreasing to 0:8 ; 106 (h1 Mpc)3
at z  1. The intercluster mean separation is 52.7 h1 Mpc at
z  0, increasing to 107.9 h1 Mpc at z  1.
Superclusters are selected from these clusters at z  0 and
z  1 using the linkage parameters b ¼ 0:33 and 0.47; these cor-
respond to L ¼ 17:4 and 24.8 h1 Mpc, respectively, at z  0,
and L ¼ 35:6 and 50.7 h1 Mpc at z  1. The large linking
lengths L reflect the large mean intercluster separation of these
massive clusters. We also include, for comparison, the smaller
linking length of L ¼ 10 h1 Mpc at z  0. There are fewer su-
perclusters identified, and a smaller number of cluster members
in each, as compared to those selected using the low-mass clus-
ters. The overall structure of the superclusters is not significantly
changed, as described below.
The multiplicity function of the superclusters is presented in
Figure 14 for both z  0 and 1. The results show, as expected,
a similar shape to the multiplicity function of superclusters se-
lected using the more numerous low-mass clusters (Figs. 3, 10,
and 11), but with a greatly reduced amplitude: the number of
Fig. 12.—Evolution of supercluster primary axis ratios with redshift. All
curves are generated from superclusters selected with linking-length parameter
b ¼ 0:47. The points are connected for clarity. Linking lengths L are quoted in
h1 Mpc. [See the electronic edition of the Journal for a color version of this
figure.]
Fig. 13.—Evolution of supercluster secondary axis ratios with redshift. All
curves are generated from superclusters selected with linking-length parameter
b ¼ 0:47. Linking lengths L are quoted in h1 Mpc. [See the electronic edition of
the Journal for a color version of this figure.]
Fig. 14.—Integrated supercluster multiplicity functions for superclusters se-
lected using only high-mass clusters, Mvir  1014 h1 M. Linking lengths L are
quoted in h1 Mpc. [See the electronic edition of the Journal for a color version of
this figure.]
WRAY ET AL.914 Vol. 652

superclusters at a given multiplicity is lower, and—equivalently—
the supercluster multiplicity (richness) is smaller for a given super-
cluster abundance. This is partly due to the much smaller number
of massive clusters. The evolution of the multiplicity function is
stronger than that of superclusters selected with low-mass clus-
ters (see Fig. 10). Superclusters with linkage parameters of b 
0:2Y0:33 contain up to
10Y15 rich clusters at z  0. The super-
cluster multiplicity decreases sharply at higher redshifts.
The supercluster size function is presented in Figure 15. The
maximum size of z  0 superclusters ranges from
40 h1 Mpc
(i.e., twice the plotted Rmax radius) for L ¼ 10 h1 Mpc, to

90 h1Mpc for L ¼ 17:4 h1Mpc (b ¼ 0:33), to
200 h1Mpc
forL ¼ 24:8 h1Mpc (b ¼ 0:47). For a given linkage parameter b,
the maximum supercluster size is somewhat larger than that of
superclusters selected with low-mass clusters, since the linking-
length values L are larger. For a given value of b, no significant
evolution is observed in the size of the largest superclusters, re-
flecting the existence of these extended structures at early times.
The same result is observed for superclusters selected with the
lower mass threshold clusters. The number of cluster members
within these large structures decreases with redshift, as seen above,
due to the lowered cluster abundance at high z. The supercluster
multiplicity versus size relation is similar—but reduced in am-
plitude, as expected—to the superclusters selected with the lower
mass threshold (Fig. 5). A nearly linear relation between multi-
plicity and size is observed for all but the binary clusters.
The ellipticity distribution shows that superclusters exhibit a
triaxial geometry on average, but with significantly larger scatter
than the more numerous superclusters selected with lower mass
clusters (Figs. 12Y13). The mean primary and secondary axis
ratios are each
0.4Y0.5; for example, for b ¼ 0:33 at z  0, the
mean values are a2/a1 ¼ 0:46  0:01 and a3/a2 ¼ 0:40  0:01.
These ratios do not change significantly either with linking length
or with redshift to z  1. The distribution of ellipticities around
this mean is, however, quite broad. The supercluster shapes are
not significantly different from those selected using the lower
mass clusters, except for the stronger flattening observed in the lat-
ter at z  0, where the mean triaxial shape has evolved slightly
toward a flattened two-dimensional shape (with a2/a1
0:6 and
a3/a2
0:3Y0:4). The superclusters selected with high-mass
clusters are triaxial, with somewhat higher triaxiality at z  0
than those selected using lower mass clusters; i.e., the high-mass
superclusters are approaching a slightly more filamentary nature,
with mean axis ratios of
0.4 in both a2/a1 and a3/a2 (see Fig. 9),
and a mean triaxiality of T ¼ 0:8Y0:85.
6. CONCLUSIONS
We have investigated the multiplicities, sizes, radial struc-
tures, and dimensionalities of superclusters from z  0 to z  2
in
CDM cosmology. The results improve our understanding of
the properties (especially shapes) and evolution of superclusters,
and can also be used as predictions for testing the cosmology
with future observations.
At low redshift, we predict a total number density of
4 ;
106, 2 ; 106, and 3 ; 107 (h1 Mpc)3 superclusters with at
least five member clusters with mass Mvir  1:75 ; 1013 h1 M
for superclusters selected with linking length L ¼ 10, 7, and
5 h1Mpc, respectively. Themaximum size of these superclusters
ranges from
150 h1 Mpc for L ¼ 10 h1 Mpc to
30 h1 Mpc
for L ¼ 5 h1 Mpc. The abundance of superclusters decreases
rapidly with increasing redshift. There are many more super-
clusters at low z than at high z largely because there aremanymore
clusters that have formed by the present time and that have grav-
itated to form superclusters. The richest superclusters are almost
entirely missing at high redshifts; these structures form late in
cosmic time. However, the densest ‘‘core’’ regions of super-
clusters, traced well by FOF linking lengths
25% of the mean
intercluster separation, are present at z  1 almost as abundantly
as they are today. These are likely the sites of the highest overden-
sities in the early universe.
The spherically averaged density profiles of superclusters are
well fit by an isothermal profile, n(r)
r2, over a broad range
in radius. The profile is shallower at small radii and is steeper at
large radii. A nearly linear relation exists between supercluster
size and multiplicity, arising from this density profile.
The clusters residing in superclusters are more massive on av-
erage than unclustered clusters, and clusters residing in rich super-
clusters are more massive than those in poorer superclusters.
We find that superclusters are triaxial in shape, especially at
early times, where the mean triaxiality of our sample is T  0:79.
Over time, many collapse along a single dimension to approach
two-dimensional shapes on the largest scales (i.e., those sampled
by the FOF linking length L ¼ 10 h1 Mpc at z  0, using low-
mass clusters with Mvir  1:75 ; 1013 h1 M); their cores re-
main triaxial. Quantitatively, the mean triaxiality for long linking
lengths decreases to T  0:65 at z  0, whereas for smaller L
(i.e., in the cores) it decreases to T  0:69. Themean primary axis
ratio of superclusters at z  0 is
0.6, corresponding to a primary
ellipticity 1  0:4. Typical secondary axis ratios at z  0 are
a3/a2  0:4 and a3/a1  0:23 for L ¼ 5 h1 Mpc, and a3/a2 
0:2 and a3/a1  0:13 for L ¼ 10 h1 Mpc. Gravitational collapse
increases ellipticity, as the potential gradient is larger along the
minor rather than the major axis (Lin et al. 1965); collapse thus
leads to lower dimensionality (seeYoshisato et al. 2006 for amore
recent treatment). A nonnegligible population of dense, filamen-
tary superstructures is also present at z  0, although not at higher
redshift.
Superclusters selected using higher mass clusters, Mvir 
1014 h1 M, show consistent results, with a reduced multiplicity
function, as expected. These superclusters are triaxial in shape,
with somewhat greater mean triaxiality at z  0 (T  0:8Y0:85)
than superclusters with lower mass clusters.
Fig. 15.—Integrated supercluster size functions for superclusters selected
using only high-mass clusters,Mvir  1014 h1 M. Linking lengths L are quoted
in h1 Mpc. [See the electronic edition of the Journal for a color version of this
figure.]
SUPERCLUSTER SHAPE AND EVOLUTION 915No. 2, 2006

These properties, derived from a large-scale, high-resolution
cosmological simulation, provide direct information on the size,
content, shape, and evolution of superclusters, as well as testable
predictions of the currently favored
CDM model with which
future and currently accumulating observations can be compared.
Once SDSS is complete, the best statistical analyses to date of
large-scale structure using observations will be possible, espe-
cially if accurate and well-behaved photo-z algorithms are devel-
oped. There are other exciting observing programs on the horizon.
The Pan-STARRS Project (Kaiser et al. 2002) is expected to see
first light in 2006, and will carry out multiband imaging surveys to
identify galaxy clusters, as well as gravitational weak lensing ob-
servations to map the matter distribution in the universe. The Large
Synoptic Survey Telescope (Tyson 2002), which is expected to see
first light in 2012, will provide improved weak-lensing maps that
will probe the dark matter distribution to the largest scales. The
analysis presented here will aid in the comparison of these major
observational data with the predictions of the concordance cos-
mological model.
This research was supported in part by NSF grant AST 04-
07305. Computations were performed on the National Science
Foundation Terascale Computing System at the Pittsburgh Super-
computingCenter; additional computational facilities at Princeton
were provided by NSF grant AST 02-16105.
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