CHINESE JOURNAL OF PHYSICS
VOL. 48, NO. 2
APRIL 2010
Clustering Properties of the Luminous Red Galaxies (LRGs) with Different
Concentration Index
Xin-Fa Deng,∗ Bei Yang, Peng Jiang, Xiao-Xun Tang, and Cheng-Hong Luo
School of Science, Nanchang University, Jiangxi, China, 330031
(Received June 29, 2009)
From the Luminous Red Galaxy (LRG) sample of the Sloan Digital Sky Survey Data Release 6 (SDSS DR6), we construct a high-concentration LRG sample and a low-concentration
LRG sample and investigate the clustering properties of LRGs with different concentration
index using cluster analysis. Because the two LRG samples with different concentration
index have different number densities, we randomly extracted a subsample from the highconcentration LRG sample which has the same galaxy number and number density as the
low-concentration LRG sample, and again performed comparative studies of clustering properties between the high-concentration LRGs and low-concentration LRGs. Our result preferentially shows that the clustering properties of LRGs are not significantly correlated with
concentration index, which is not consistent with the result for Main galaxies.
PACS numbers: 98.65.Dx
I. INTRODUCTION
Many studies have shown that galaxies with different physical properties cluster differently [1–12]. For example, Brown et al. [1] showed that the galaxy correlation function
depends strongly on color, with red galaxies more strongly clustered than blue galaxies.
Deng et al. [11] also investigated clustering properties for Main galaxy samples with different g-r color, and found that the redder galaxies preferentially inhabit the dense groups
and clusters. Using the two-point correlation function of galaxies, many authors found that
the clustering amplitude increases with absolute magnitude [3–5, 13–15]. Galaxies with
different morphologies also show large variations of the galaxy correlation function, with
late-type galaxies having considerably weaker clustering than early-type galaxies [16–17].
Different types of galaxies trace the underlying large-scale structure of dark matter distribution in different ways. The dependence of clustering on galaxy properties provides tests
of cosmological models and galaxy formation theories. For example, the dependence of
clustering on galaxy luminosity obtained by many authors is consistent with hierarchical
models of galaxy formation, which predict that bright galaxies should be more strongly clustered than faint galaxies [18–19]. Undoubtedly, it is interesting to investigate the clustering
properties of galaxies with different physical parameters.
The unprecedented size of the Sloan Digital Sky Survey (SDSS) makes it possible to
quantify how the clustering signal depends on intrinsic galaxy properties, such as morphologies or the concentration index. The SDSS galaxy data contains two interesting samples:
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the Main galaxy sample [20] and the Luminous Red Galaxy (LRG) sample [21]. Main
galaxies mostly are located within the redshift interval 0.02 ≤ z ≤ 0.2 , while LRGs are
at higher redshift, intrinsically red, and luminous early-types. Many of the very luminous
early-type galaxies are massive. Clustering analyses of these galaxies show that they are
particularly interesting because they tend to reside in massive dark matter halos [22–25].
Zehavi et al. [26] investigated the intermediate-scale (0.3 to 40h−1 Mpc) clustering of 35,000
luminous early-type galaxies in the redshift region 0.16 ≤ z ≤ 0.44 from the Sloan Digital Sky Survey and found that LRGs are highly clustered and show stronger clustering
on smaller scales. Zehavi et al. [26] also noted the dependence of the clustering properties of LRGs on the luminosity: more luminous LRGs being yet more strongly clustered.
Interestingly, Zehavi et al. [26] showed that the LRG clustering strengths and mean separations are comparable to those of the poorest clusters or of rich groups. Eisenstein et
al. [27] also discovered the strong clustering of LRGs on a small-scale, and proposed that
the scale-dependent clustering of LRGs could serve as a test of the formation theories for
LRGs.
Clustering properties of different galaxy samples may be fairly different. To extract
common characters of clustering properties, exploring the same issue using different samples
is necessary. In the SDSS, a particularly interesting work is to perform comparative studies
of clustering properties between Main galaxies and LRGs. For example, Deng et al. [28]
found that the clustering properties of Luminous Red Galaxies is different from those of
Main galaxies: the structure of the Main galaxy sample is more filamentary, while that of
the LRG sample is a simple and central clustering.
Using the concentration index ci = R90 /R50 , Zehavi et al. [5] noted that highconcentration (early-types) galaxies have a steeper, higher amplitude correlation function than low-concentration (late-types) galaxies, as expected from previous studies of
morphology-dependent clustering. Deng et al. [10] performed comparative studies of the
clustering properties of Main galaxies between an early-type sample (high-concentration
galaxies) and a late-type sample (low-concentration galaxies). It was found that the distribution of late-type galaxies is more filamentary, which is in good agreement with the
conclusion obtained by Pandey & Bharadwaj [29]. Deng et al. [10] also noted that in the
late-type sample, the fractions of single, close double, and multiple galaxies are higher than
the ones of the early-type sample, but the fraction of grouped galaxies is lower. In this
study, we investigate the clustering properties of LRGs with different concentration index
and look for the difference between the Main galaxies and LRGs. Undoubtedly, this research will be beneficial to the further understanding of the dependence of LRG clustering
properties on galaxy properties and to the understanding of galaxy formation and evolution.
In the past, the correlation function was the most popular method for investigating
clustering properties. However by using the correlation function we cannot clearly understand the geometry of the distribution of galaxies. Cluster analysis [30] is a method which
has been widely applied to study the geometry of point samples and is more sensitive to the
geometry of the distribution of galaxies. By this method the galaxy sample can be separated
into isolated galaxies, close double, and multiple galaxies, galaxy groups or clusters, and
even superclusters. According to the results of Einasto et al. [30], superclusters consist of
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clusters and strings of galaxies. Galaxy strings also form bridges between superclusters and
join all superclusters to a single infinite network. In this study, we use cluster analysis [30]
to investigate the clustering properties of LRGs with different concentration index. Our
paper is organized as follows. In section II, we describe the data used. The cluster analysis
is discussed in section III. In section IV, we discuss the clustering properties of LRGs with
different concentration index. Our main results and conclusions are summarized in section
V.
In calculating the distance we use a cosmological model with a matter density Ω0 =
0.3, cosmological constant ΩΛ = 0.7 and Hubble’s constant H0 = 100h km·s−1 ·Mpc−1 with
h = 0.7.
II. DATA
The Sloan Digital Sky Survey (SDSS ) is one of the largest astronomical surveys to
date. Many of the survey properties were discussed in detail in the Early Data Release
paper [31]. Galaxy spectroscopic target selection can be implemented by two algorithms.
The Main galaxy sample [20] comprises galaxies brighter than rpetro < 17.77 (r-band apparent Petrosian magnitude). This sample has a median redshift of 0.10. The Luminous
Red Galaxy (LRG) algorithm [21] selects galaxies to rpetro < 19.5 that are likely to be
luminous early-types, based on the observed colors. In our work, the data was downloaded
from the Catalog Archive Server of the SDSS Data Release 6 [32] by the SDSS SQL Search
(http://www.sdss.org/dr6/).
Eisenstein et al. (2001) strongly advised the researcher that LRGs should be selected
at z > 0.15, and showed that the LRG sample appears to have approximately constant
passively evolved selection, physical size, and comoving number density out to z ' 0.4. From
this, the LRG sample can be called an approximately volume-limited one. Thus, we extract
all LRGs with the redshift 0.16 ≤ z ≤ 0.4 (with SDSS flag: Primtarget Galaxy Red, redshift
confidence level: zconf>0.95), and construct an approximately volume-limited sample which
contains 77148 LRGs. This sample is therefore the lower redshift regime of the LRGs and
does not extend into the z > 0.4 regime of just brighter objects.
R50 and R90 are the radii enclosing 50% and 90% of the Petrosian flux, respectively.
In this paper, we use the concentration index ci = R90 /R50 and divide the approximately
volume-limited LRG sample into two samples: the S1 sample (ci ≥ 2.86, containing 51174
galaxies) and the S2 sample (ci<2.86, containing 25974 galaxies). In the SDSS, it is nearly
impossible to classify SDSS galaxies into morphological classes through direct inspection of
the galaxy images, as in previous studies, due to the large number of galaxies. Whether
a galaxy is termed ‘early’ or ‘late’ is fairly subjective. Many authors developed different
methods or used different parameters, such as color, star formation rate indicators, and
concentration index as the morphology classification tool [33–38]. The concentration parameter is a good and simple morphological parameter. The Nakamura et al. [39] study
showed that ci = 2.86 separates galaxies at S0/a with a completeness of about 0.82 for both
late and early types. But we also note that when developing a selection criterion ci = 2.86,
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Nakamura et al. [39] only used nearby bright galaxies. It has been known for a long time
that concentration index is very sensitive to seeing, as shown in Blanton et al. [40]. At a
wide range of redshift as in this study, the concentration index may not be suitable for the
classification of galaxies.
III. CLUSTER ANALYSIS
The cluster analysis [30] used here is actually the friends-of-friends algorithm, by
which the galaxy sample can be separated into individual systems at a given neighbourhood
radius R. Starting from one galaxy of the sample, we search all galaxies within a sphere of
radius R around it, and call these close galaxies “friends”. These “friends” and the starting
galaxy are considered as belonging to the same system. Around the new neighbours, we
continue the above procedure using the rule “any friend of my friend is my friend”. When
no more new neighbours or ”friends” can be added, then the procedure stops and a system
is identified. Apparently, at small radii, most systems are some isolated single galaxies,
the rest being close double and multiple galaxies. At larger radii groups and clusters of
galaxies and even superclusters will be formed. Superclusters are the largest non-percolating
galaxy systems which contain clusters and groups of galaxies with their surrounding galaxy
filaments [41–44]. By selecting different neighbourhood radii, we can probe the structures
at different scales.
The mean density of galaxies is ρ = N/V (N is the number of galaxies contained in
the volume V ). The Poisson radius (radius of the sphere with unit population) is R0 =
(3/4πρ)1/3 . To compare samples with a different number density we express all distances
in dimensionless radii r = R/R0 . The Poisson radii (comoving distance) are 22.38 Mpc for
the S1 sample, and 28.05 Mpc for the S2 sample.
IV. COMPARATIVE STUDIES OF CLUSTERING PROPERTIES OF LRGS WITH
DIFFERENT CONCENTRATION INDEX
In order to study how the geometry of the systems changes with increasing neighbourhood radius, we calculate the maximum lengths of the systems. The maximal length
of a system is defined as the maximum distance between members of this system. Figure 1
shows the galaxy number Nmax of the richest system and the maximal length of the largest
system as a function of the dimensionless radius r for two LRG samples with different
concentration index. Like Deng et al. [28], we define L0 = V 1/3 (the edge length of the
cube) as the rough estimate of the edge length of the sample volume V , and express the
maximal length of the largest system as the dimensionless length dmax = Dmax /L0 . The
edge length L0 is 1339.10 Mpc for the two LRG samples. As seen from Figure 1, on the
average, richer and larger systems can be more easily formed in the S1 sample (containing
high-concentration LRGs), which disagrees with the result of the Main galaxies obtained
by Deng et al. [10]. Deng et al. [10] found that richer and larger systems can be more
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FIG. 1: Clustering properties for the S1 sample (solid lines) and the S2 sample (dashed lines): (a)
the galaxy number Nmax of the richest system, (b) the maximal length Dmax /L0 of the largest
system, as a function of the dimensionless radius r.
easily formed in the late-type sample (containing low-concentration galaxies). But we also
note that the number density of the S1 sample is two times larger than the one of the S2
sample. In a sample with larger number density, richer systems can be more easily formed.
Although many authors used dimensionless radii to express distances, we still suspect that
this replacement can not completely correct above bias. Maybe this difference of clustering
properties between the two LRG samples with different concentration index is partially due
to this drawback of cluster analysis.
With increasing neighbourhood radius, various systems merge into strings and later
into a string network. At a certain critical radius rc , called the percolation radius, the
largest string system reaches the opposite sidewalls of the sample. In this study, the percolation radius rc is defined as the radius at which the maximal length Dmax of the largest
system approximates to the edge length L0 : rc ' 1.24 for the S1 sample, and rc ' 1.38 for
the S2 sample. As indicated by Einasto et al. [30], the percolation radius depends on two
factors: the degree of clustering and the degree of concentration of the strings. It is evident
that the more filamentary the structure is, the easier it is to reach percolation. The percolation radius rc is a good indicator of the filamentary structure. Pandey & Bharadwaj [29,
45–46] studied how the filaments depend on galaxy properties. For example, Pandey &
Bharadwaj [29] showed that the degree of filamentarity exhibits a luminosity dependence,
with the brighter galaxies having a more concentrated and less filamentary distribution
as compared to the faint ones, and that the blue galaxies and the spirals have a higher
filamentarity as compared to the red galaxies and the ellipticals, respectively, at large filling factors. Deng et al. [10] found that the structure of the late-type sample (containing
low-concentration galaxies) is more filamentary. But in this study, we found that highconcentration LRGs have a more filamentary distribution, which is not consistent with the
result of the Main galaxies [10]. We also note that the distribution of LRGs is less filamen-
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FIG. 2: Histograms of the multiplicity functions for the S1 sample (solid lines) and the S2 sample
(dashed lines) at different radii: (a) at r = 0.7, (b) at r = 0.8, (c) at r = 0.9. The error bars for the
S2 sample are 1σ Poissonian errors. Error bars for the S1 sample are omitted for clarity.
tary than the one of Main galaxies, which is in agreement with the conclusion obtained by
Deng et al. [28].
In order to describe the distribution of systems having different sizes, we analyze the
multiplicity functions: the fraction of galaxies in systems of membership from n to n + dn,
which depends on the dimensionless radii r. We divide the interval from 1 to N (the total
number of galaxies) into 7 subintervals: n=1; 2 ≤ n < 5; 5 ≤ n < 20, 20 ≤ n < 50,
50 ≤ n < 100, 100 ≤ n < 200, and n ≥ 200, and then construct histograms of the
multiplicity functions at different radii (r = 0.7, r = 0.8, r = 0.9). In each histogram,
systems which contain one galaxy are in the first bin, systems which contain from 2 to 4
galaxies are in the second bin, systems with 5 to 19 galaxies in the third bin and so on.
In Fig. 2, the multiplicity functions are shown for high-concentration LRGs (the S1
sample) and low-concentration LRGs (the S2 sample). The (1σ) error bars are Poissonian
errors. As seen from Fig. 2, the fractions of single, close double, and multiple galaxies
in the S2 sample (containing low-concentration LRGs) are higher than the ones of the S1
sample (containing high-concentration LRGs), but the fraction of grouped galaxies in the
S2 sample is lower. This is consistent with the result for the Main galaxies. Deng et al. [10]
found that in the late-type sample (containing low-concentration galaxies), the fractions of
single, close double, and multiple galaxies are higher than the ones of the early-type sample
(containing high-concentration galaxies), but the fraction of grouped galaxies is lower. It
is noteworthy that in the study of Deng et al. [10] the number density of the late-type
sample (containing low-concentration galaxies) is much larger than the one of the earlytype sample (containing high-concentration galaxies), while in this study the S2 sample
(containing low-concentration LRGs) has a lower number density.
We must be cautious about the difference of clustering properties produced by the
statistical method. For example, in a sample with a larger number density, richer systems
can be more easily formed by cluster analysis. As Deng et al. [10] did, we believe that the
use of dimensionless radii cannot completely correct such a drawback. Thus, we randomly
extract a subsample from the S1 sample, which has the same galaxy number and number
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FIG. 3: Clustering properties for the high-concentration LRG subsample (solid lines) and the S2
sample (dashed lines): (a) the galaxy number Nmax of the richest system, (b) the maximal length
Dmax /L0 of the largest system as a function of the dimensionless radius r.
density as the S2 sample, and again perform the above analyses. Figure 3 shows the galaxy
number Nmax of the richest system and the maximal length of the largest system as a
function of the dimensionless radius r for the high-concentration LRG subsample and the
S2 sample. The percolation radius rc of the high-concentration LRG subsample is rc '
1.31. We find that the difference of clustering properties between the high-concentration
LRG subsample and the S2 sample is much smaller the one between the S1 sample and
the S2 sample. This shows that above difference of clustering properties between highconcentration LRGs and low-concentration LRGs is mainly due to the drawback of the
statistical method, not a physical effect. We preferentially conclude that there is not a
significant difference of clustering properties between the high-concentration LRGs and lowconcentration LRGs. Deng et al. [47] compared distributions of some physical properties
of member galaxies of LRG groups with the ones of isolated LRGs, and found that the
correlation between luminosity and environment still exists in the LRG sample, but colors,
morphologies, and the structural parameter of LRGs are nearly independent of the local
density. This is consistent with our conclusion. But for the Main galaxies of SDSS, Deng
et al. [10] found that the clustering properties of galaxies are significantly correlated with
the concentration index or morphologies.
V. SUMMARY
For a long time, it has been known that galaxies with different properties cluster differently. In this study, we use the Luminous Red Galaxy (LRG) sample of the Sloan Digital
Sky Survey Data Release 6, and investigate clustering properties of LRGs with different
concentration index, using cluster analysis. We note that two LRG samples with different
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FIG. 4: Histograms of multiplicity functions for the high-concentration LRG subsample (solid lines)
and the S2 sample (dashed lines) at different radii: (a) at r = 0.7, (b) at r = 0.8, (c) at r = 0.9.
The error bars for the S2 sample are 1σ Poissonian errors. Error bars for the high-concentration
LRG subsample are omitted for clarity.
concentration index have a different number density; randomly extracted a subsample from
the high-concentration LRG sample, which has the same galaxy number and number density as the low-concentration LRG sample; and again performed comparative studies of the
clustering properties betweenthe high-concentration LRGs and low-concentration LRGs.
These analyses preferentially indicate that the clustering properties of LRGs are not significantly correlated with concentration index, which is not consistent with the results for
Main galaxies. Our studies also show that, when comparing clustering properties of samples
with a different number density, we must be cautious about the difference of the clustering
properties produced by the statistical method.
Acknowledgements
We thank the anonymous referee for many useful comments and suggestions. Our
study was supported by the National Natural Science Foundation of China (NSFC, Grant
10863002) and also supported by the Program for Innovative Research Team of Nanchang
University.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department
of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.
The SDSSWeb site is http://www.sdss.org.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural
History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge,
Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University,
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the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics
and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST),
Los Alamos National Laboratory, the Max Planck Institute for Astronomy (MPIA), the
Max Planck Institute for Astrophysics (MPA), New Mexico State University, Ohio State
University, University of Pittsburgh, University of Portsmouth, Princeton University, the
US Naval Observatory, and the University of Washington.
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