Advanced
LARGE SDSS QUASAR GROUPS AND THEIR STATISTICAL SIGNIFICANCE
LARGE SDSS QUASAR GROUPS AND THEIR STATISTICAL SIGNIFICANCE
Journal of The Korean Astronomical Society. 2015. Feb, 48(1): 75-82
Copyright © 2015, The Korean Astronomical Soiety
  • Received : January 31, 2015
  • Accepted : February 05, 2015
  • Published : February 28, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
CHANGBOM PARK
School of Physics, Korea Institute for Advanced Study, Heogiro 85, Seoul 130-722, Koreacbp@kias.re.kr,hmsong@kias.re.kr
HYUNMI SONG
School of Physics, Korea Institute for Advanced Study, Heogiro 85, Seoul 130-722, Koreacbp@kias.re.kr,hmsong@kias.re.kr
MARET EINASTO
Tartu Observatory, 61602 Toravere, Estonia;maret@to.ee
HEIDI LIETZEN
Universidad de La Laguna, Dept. Astrofísica, E-38206 La Laguna, Tenerife, Spain
PEKKA HEINÄMÄKI
Tuorla Observatory, University of Turku, Väisäläntie 20, Piikkiö, Finland;pekheina@utu.fi

Abstract
We use a volume-limited sample of quasars in the Sloan Digital Sky Survey (SDSS) DR7 quasar catalog to identify quasar groups and address their statistical significance. This quasar sample has a uniform selection function on the sky and nearly a maximum possible contiguous volume that can be drawn from the DR7 catalog. Quasar groups are identified by using the Friend-of-Friend algorithm with a set of fixed comoving linking lengths. We find that the richness distribution of the richest 100 quasar groups or the size distribution of the largest 100 groups are statistically equivalent with those of randomly-distributed points with the same number density and sky coverage when groups are identified with the linking length of 70 h −1 Mpc. It is shown that the large-scale structures like the huge Large Quasar Group (U1.27) reported by Clowes et al. (2013) can be found with high probability even if quasars have no physical clustering, and does not challenge the initially homogeneous cosmological models. Our results are statistically more reliable than those of Nadathur (2013), where the test was made only for the largest quasar group. It is shown that the linking length should be smaller than 50 h −1 Mpc in order for the quasar groups identified in the DR7 catalog not to be dominated by associations of quasars grouped by chance. We present 20 richest quasar groups identified with the linking length of 70 h −1 Mpc for further analyses.
Keywords
1. INTRODUCTION
The size and mass of large-scale structures 1 (hereafter LSS) in the universe have been often used as measures of amplitude of density fluctuations on large scales. The CfA Great Wall discovered by de Lapparent et al. (1986) in the CfA2 survey and the Sloan Great Wall in the SDSS survey ( Gott et al. 2005 ) cast doubt on the SCDM model ( Geller & Huchra 1989 ) and the ΛCDM model ( Sheth & Diaferio 2011 ), respectively. It has been argued that the universe did not have enough time for such large structures to form by the present epoch in these cosmological models, and that the gravitational instability theory for structure formation or the Cosmological Principle adopting large-scale homogeneity may not be valid (see also Clowes et al. 2012 ; Horvath et al. 2013 ).
On the other hand, a strong support for the Cosmological Principle comes from the smooth cosmic microwave background radiation, which shows anisotropies that agrees astonishingly well with those of the initially homogeneous isotropic ΛCDM model ( Clifton et al. 2012 ; Planck collaboration XV 2014 ). The fluctuation of the galaxy number density on large scales is also found to agree with the prediction of the ΛCDM model ( Scrimgeour et al. 2012 ).
It is expected that LSS of the universe contain a wealth of information on primordial density fluctuations and the history of their growth. This is because they represent density fluctuations on linear scales. It is then very important to rigorously study the physical properties and cosmological meaning of observed superstructures of the universe ( Park 1990 ; Park et al. 2012 ; Einasto et al. 2011 , 2014 ).
Recently, Clowes et al. (2013) reported a finding of a very large quasar group in the SDSS DR7 quasar catalog ( Schneider et al. 2010 ) with characteristic size of ~350 h −1 Mpc, longest dimension of ~870 h −1 Mpc, membership of 73 quasars, and mean redshift
PPT Slide
Lager Image
= 1.27. They claimed that the quasar group, named U1.27, is the largest ‘structure’ currently known in the universe. They argued that their finding might raise a question on validity of the large-scale homogeneity assumption, which is, in the form of Cosmological Principle, one of the fundamental assumptions of the standard cosmology. Their claim falls on the same kind of view applied to CfA and Sloan Great Walls.
However, Nadathur (2013) pointed out that the statistical significance of the huge quasar group has not been properly studied in Clowes et al. (2013) , and that Clowes et al.’s claim on the cosmological implication of existence of the large quasar group needs to be examined quantitatively. Nadathur showed that the algorithm used to identify the quasar groups frequently finds such large-size structures even in homogeneous simulations of a Poisson point process with the same number density as the quasar catalog. They concluded that Clowes et al.’s interpretation of U1.27 as a physical ‘structure’ is misleading.
Even though it is cosmologically interesting to search for associations of quasars 2 ( Webster 1982 ; Clowes & Campusano 1991 ; Komberg et al. 1996 ; Williger et al. 2002 ; Clowes et al. 2012 , 2013 ; Einasto et al. 2014 ), one needs to be careful in interpreting the quasar groups identified with specific criteria. This is because the size and richness of LSS depend sensitively on how they are identified. Evolution of the objects used to trace LSS and evolution of LSS themselves can also complicate the interpretation of identified LSS. In particular, the quasar phenomenon has short lifetime and astrophysics of quasar activity can affect the results ( Komberg et al. 1996 ; Wold et al. 2000 ; Miller et al. 2004 ; Sochting et al. 2004 ; Coldwell & Lambas 2006 ; Hutchings et al. 2009 ; Lietzen et al. 2009 , 2011 ; Bradshaw et al. 2011 ; Portinari et al. 2012 ; Trainor & Steidel 2012 ; krumpe et al. 2013 ; Fanidakis et al. 2013 ; Shen et al. 2013 ; DiPompeo et al. 2014 ; Karhunen et al. 2014 ; Song et al. 2015 ). Quasar samples usually have very low spatial number density, which can introduce shot noise effects in group identification. This situation demands the use of well-defined samples and rigorous statistical tests.
In this paper we identify quasar groups using a volume-limited sample drawn from the SDSS DR7 quasar catalog that has the maximum contiguous angular size. We present the statistical significance of not only the largest quasar group, but also the quasar groups within a large range of richness. We also address a question of whether or not the huge quasar group found by Clowes et al. is a statistically-significant object in terms of multiplicity and size. We find the critical linking length (LL hereafter) for quasar group identification which can result in a catalog dominated by physically clustered groups.
2. SDSS QUASAR CATALOG
We use the fifth edition of the SDSS quasar catalog from SDSS DR7, which is a compilation of quasars probed in the SDSS-I and SDSS-II quasar survey ( Schneider et al. 2010 ). The catalog contains 105783 spectroscopically confirmed quasars in the sky of area covering approximately 9380deg 2 . They are brighter than Mi = −22.0 (corrected for the Galactic extinction, and K -corrected to z = 2 in a cosmology with H 0 = 70km/s/Mpc, Ω m = 0.3, and Ω Λ = 0.7) and have at least one broad emission line with full width at half maximum larger than 1000km/s or interesting/complex absorption features. The quasars show a wide redshift distribution from 0.065 to 5.46, but the majority of quasars (76%) is concentrated below redshift 2. They have the apparent i magnitude of 14.86 < i < 22.36, where the lower limit comes from the maximum brightness limit of the target selection on quasar candidates to avoid saturation and cross-talk in the spectra. The catalog does not include several classes of active galactic nuclei such as Type 2 quasars, Seyfert galaxies, and BL Lacertae objects.
Even though the catalog contains a large number of quasars, not all the quasars are suitable for statistical analyses for several reasons. It is mainly because the quasar target selection for the follow-up spectroscopy was not done uniformly in its early version (see Richards et al. 2006 ; Shen et al. 2007 , for more details). Shen et al. (2011) provided a parameter, in the 10 th column in their catalog, named as ‘uniform flag’ to take the uniformity problem into account. The catalog of Schneider et al. also has its own uniform flag in the 35th column. The uniform flag is assigned based on the achievement of the target selection algorithm. But Shen et al. made more careful evaluation of the selection effects by finding out quasars that are uniformly selected as high- z candidates but later identified as low- z quasars. Shen et al. has one more class than Schneider et al. as follows: 0 = not in the uniform sample; 1 = uniformly selected and with the galactic extinction-corrected i magnitude less than 19.1 at z < 2.9 or 20.2 at z > 2.9; 2 = uniformly selected by the QSO_HiZ branch in the algorithm and with measured spectroscopic redshift z < 2.9 and i > 19.1. By selecting those whose Shen et al. (2011) ’s uniform flag is 1, we construct a uniform sample of 59679 quasars with Galactic extinction corrected i magnitude less than 19.1. It rejects about a half of the quasars in the original catalog, mostly at z < 2.9.
In Figure 1 we plot the quasars in the Schneider et al. (2010) ’s catalog in black. Only the northern hemisphere of the SDSS survey regions is adopted and the SDSS survey coordinates η and λ are used for the plot as the survey boundaries are more simply defined in this coordinate system. The red points in the figure are quasars marked as those of the uniform subsample by Shen et al.
PPT Slide
Lager Image
Distribution of all the quasars in the original Schneider et al. (2010)’s catalog (black or red points) and the quasars in the Shen et al. (2011)’s uniform sample (red points) in the SDSS survey coordinates. A contiguous region defined by the green line is the region under analysis in our study.
Since the sky distribution of the uniform quasar sample is not perfectly contiguous, we define a new boundary for our sample by excluding isolated patches and trimming the jagged boundaries to make one clean contiguous region. The green line in Figure 1 defines the region under our study. The area of the region is 1.754 steradian. This is nearly the maximum possible contiguous sample that can be drawn from the DR7 catalog.
Figure 2 shows the uniform-sample quasars located within our mask in the i -band absolute magnitude and redshift plane. In Figure 3 it can be seen that the comoving space number density of quasars is roughly uniform between z = 0.8 and 1.8, but is significantly higher between z = 0.1 and 0.8.
PPT Slide
Lager Image
Distribution of the uniform-sample quasars in the redshift versus i-band absolute magnitude plane. The blue line is a low-redshift absolute magnitude cut used in our study making the sample nearly constant in comoving number density between z = 0.1 and 1.8.
PPT Slide
Lager Image
Comoving number density of quasars. The black line is for the whole uniform sample, and the red line is when the low-redshift absolute magnitude cut (Equation (1)) is applied. The blue dashed line is the mean density between z = 0.1 and 1.8 after the cut is applied.
Song et al. (2015) has compared the distribution of the SDSS quasars at 0.4 ≤ z ≤ 0.7 with the smooth density field of the galaxies in the SDSS DR12 BOSS survey and found that the probability of finding a quasar increases monotonically as the local galaxy number density increases with no significant redshift dependence. This implies that LSS found in the distribution of quasars should be similar to those in the galaxy distribution. Then a quasar sample having a constant comoving number density should be as useful as a volumelimited galaxy sample in finding LSS.
In order to make a volume-limited quasar sample with uniform density we apply an absolute magnitude cut line given by
PPT Slide
Lager Image
in the range of 0.1 < z < 0.8. The cut is shown as a blue line in Figure 2 . The resulting subsample has a nearly uniform number density of 1.31 × 10 −6 ( h −1 Mpc) −3 in a wide range of 0.1 < z < 1.8 as shown by the red histogram in Figure 3 . The corresponding mean quasar separation is 91.5 h −1 Mpc. By varying the absolute magnitude cut one can effectively correct a quasar sample for the luminosity evolution. The use of uniform comoving density sample makes us sample the same kind of cosmological objects over a wide redshift range.
Our final sample contains 32276 quasars with i < 19.1 in the redshift range 0.1 ≤ z ≤ 1.8 covering 5758deg 2 . On the other hand, Clowes et al. (2013) and Nadathur (2013) did not use the sample made to meet the precondition of uniformity for statistical works. They just applied an i -band apparent magnitude cut of i ≤ 19.1 to Schneider et al. (2010) ’s sample, where i here is not corrected for the Galactic extinction. This made their sample affected by non-uniform quasar target selection which Shen et al. (2011) corrected, and the structures identified are affected by the Galactic extinction. Our sample is from the homogeneous sample and has a true flux limit. It is also much larger in the sky area (5758 deg 2 versus 3725 deg 2 ) and in redshift range (0.1 ≤ z ≤ 1.8 versus 1.0 ≤ z ≤ 1.8). It makes our sample contain 1.72 times more quasars than that used by Clowes et al. (2013) , Nadathur (2013) , and Einasto et al. (2014) .
3. IDENTIFICATION OF QUASAR GROUPS
To identify groups of quasars we apply the Friendof-Friend (FoF) algorithm to the ‘uniform’ constant-number density sample described in the previous section. We first use LL= 70 h −1 Mpc for a comparison with Clowes et al. (2013) ’s results. This corresponds to the threshold overdensity of δ = (
PPT Slide
Lager Image
/ LL ) 3 − 1 = (91.5/70) 3 −1 = 1.23, where
PPT Slide
Lager Image
is the mean quasar separation of the sample. We find 3233 quasar groups having 3 or more members. Figure 4 shows the positions of top ten richest groups on the sky (upper panel) and in the x-y plane of Equatorial coordinate frame (bottom panel). The second richest one labelled with ‘2’ has 70 member quasars and the maximum extent of 703 h −1 Mpc. The maximum extent is defined as the comoving distance between the most distant two members in a group. This group corresponds to huge Large Quasar Group, U1.27, reported by Clowes et al. The group identified by us has 3 new members and does not have 6 members compared to Clowes et al.’s list. We discover a rich quasar group, marked with ‘1’ in Figure 4 , that contains 95 members and has the maximum extent of 712 h −1 Mpc. This group was not found by previous studies because its center is located at z = 0.366, which is outside the redshift limit of 1.0 < z < 1.8 adopted by Clowes et al., Nadathur (2013) , and Einasto et al. (2014) . This group is much richer and has a size larger than U1.27. A catalog of the twenty richest quasar groups is given in Table 1 for future analyses.
PPT Slide
Lager Image
(upper panel) Ten richest quasar groups detected with the linking length of 70h−1 Mpc. The second richest one in magenta on the bottom right labelled with 2 corresponds to the huge Large Quasar Group, U1.27, of Clowes et al. (2013). (bottom panel) Same as the upper plot but in the Cartesian coordinates defined in the Equatorial coordinate system. Grey points are all the quasars in our sample.
The twenty richest SDSS DR7 quasar groups identified with the linking length of 70h−1Mpc or overdensity threshold of 1.23.
PPT Slide
Lager Image
The full catalog with information on members of groups can be downloaded at http://astro.kias.re.kr/Quasar/DR7_ 20groups. a Coordinates of the geometric center of groups. The mean comoving distance is in units of h−1Mpc. b The number of member quasars. c The maximum extent of groups in h−1Mpc.
To check if our group finding is consistent with previous works we perform the FoF group finding to a sample from the original Schneider et al. (2010) ’s catalog that is limited only by i < 19.1 not corrected for Galactic extinction and a redshift range of 1.0 < z < 1.8. In this case the richest group is found to have 73 members that are exactly the same as the member quasars of U1.27 listed in Table 1 of Clowes et al. This proves that the difference in our quasar catalog is not due to the group finding algorithm but to the sample definition. We also compared our quasar groups with those identified by Einasto et al. (2014 , see their Table A.3) who also used Schneider et al.’s catalog constrained by i < 19.1, and found a very good agreement at z > 1.0. We choose our sample instead of that used by previous works because the observed sample must be uniformly defined for statistical hypothesis tests against random catalogs.
The observed quasar distribution has random perturbations along the line-of-sight due to the random redshift error which is typically about 0.004 in the case of SDSS DR7 catalog ( Schneider et al. 2010 ). Therefore, it is necessary to examine if the quasar group finding is significantly affected by this error. We will discuss it in the next section.
4. COMPARISON WITH RANDOM DISTRIBUTIONS
We generate random point sets which have the same number density and the same angular coverage on the sky as our quasar sample. The random catalogs also have the mean point separation of 91.5 h −1 Mpc throughout the entire redshift range from z = 0.1 to 1.8. We make 10000 random mock samples. For each mock sample we find groups of points in the same way as for the quasar sample. We then measure richness and size of the groups in quasar sample and random mock samples. Richness is the number of member quasars/points belonging to a given group. Size is the maximum co-moving extent of the group.
In the upper panel of Figure 5 we plot the cumulative distribution function (CDF) of richness. It is the number of groups whose richness is greater than or equal to Ng . The circles are from the observed quasar sample. Richness of the group corresponding to U1.27 group is 70, and is marked with an arrow. The solid line and error bars are the mean and standard deviations obtained from 10000 random catalogs. Similarly, the bottom panel shows the number of observed quasar groups whose size is greater than or equal to L, and the results from 10000 mock catalogs. We find that U1.27 is 703 h −1 Mpc long (marked with an arrow) and is only the fourth largest one. It can be seen that the richness and size distributions of the observed quasars are statistically indistinguishable with those of random catalogs.
PPT Slide
Lager Image
Cumulative distribution functions of richness (upper panel) and maximum extent (bottom panel) of the groups identified in the observed SDSS quasar catalog (red circles) and in random point catalogs (solid line with error bars). A set of 10000 random point catalogs is used for the comparison. The group corresponding to the huge Large Quasar Group, U1.27, reported by Clowes et al. (2013) is marked with downward arrows.
To estimate the probability of finding groups like the observed quasar groups in random catalogs we perform a χ 2 test. We define
PPT Slide
Lager Image
as a measure of the deviation of a particular CDF ϕ from the average CDF
PPT Slide
Lager Image
of random catalogs. Here σi is the standard deviation of the CDFs of random catalogs in the i -th bin, and the summation is over the range where the average CDF of random catalogs is between 3 and 100. The choice ensures that a large part of CDF is compared. The histograms in Figure 6 are the distributions of χ 2 of 10000 random catalogs and the arrows correspond to the observed SDSS quasar catalog generated with LL= 70 h −1 Mpc. The probability of having χ 2 greater than
PPT Slide
Lager Image
in random catalogs is 58.1% and 92.4% for richness and size distributions, respectively. Therefore, it can be concluded that the quasar groups identified with LL= 70 h −1 Mpc are consistent with those of random point distributions.
PPT Slide
Lager Image
Distributions of χ2 defined as in Equation (2) for random point samples. The upper panel is for richness of groups identified with the linking length of 70 h−1Mpc, and the bottom panel is for maximum extent of groups. The χ2 for the observed sample are marked with arrows.
On the other hand, we find that the probability of finding a random group whose richness is equal to or greater than 70 is 92.3%. Similarly, the probability of finding a random group with the maximum extent equal to or greater than 703 h −1 Mpc is 100%. Therefore, both richness and size of the objects like U1.27 are of no surprise even in random distributions of points with no physical clustering when LL is as large as 70 h −1 Mpc.
Since quasars do have intrinsic spatial clustering, it is expected that these distribution functions are statistically different from those of random catalogs for some smaller LL or for higher density thresholds. Figure 7 shows the CDFs of richness and size of quasar groups when LL is taken to be 60, 50, and 40 h −1 Mpc. Also shown are the average CDFs (solid lines with error bars) of random point groups identified with the corresponding LL. It is clear that the CDFs of observed group richness and size become more different from those of random catalogs as LL decreases.
PPT Slide
Lager Image
Cumulative distribution functions of richness (upper panel) and maximum extent (bottom panel) when the linking length for group identification is taken to be 60, 50, and 40 h−1Mpc. Circles are from observation, and solid lines with error bars are from 1000 random catalogs.
It will be then interesting to search for the critical LL for which the observed CDF becomes significantly different from those of random points. We calculate the probability that the CDF of random point groups happens to show a deviation from the mean CDF as large as or greater than the difference of the observed CDF from the mean. This is estimated by counting the cases with
PPT Slide
Lager Image
. A high probability means that the observed groups are statistically consistent with random ones. Figure 8 shows the probability as a function of LL used to find groups. For both richness and size it can be seen that the observed groups are consistent with those of random catalogs when LL is chosen to be larger than about 50 h −1 Mpc. Clowes et al. chose LL= 70 h −1 Mpc and it is now clear that the choice makes the group catalog dominated by randomly connected groups. Figures 7 and 8 indicate that the groups identified with LL< 50 h −1 Mpc are likely to be physically-clustered genuine groups of quasars.
PPT Slide
Lager Image
Probability of finding random catalogs whose distribution function deviates from their mean more than or equal to that of the observed sample. The deviation is measured by χ2 defined in Equation (2). Probability is measured for richness (solid line) and size (dashed line) distribution functions as a function linking length.
As mentioned in the previous section the random redshift error of order of 0.004 seems problematic in finding physical structures as the error corresponds to ~ 12 h −1 Mpc. We use the redshifts calculated by Hewett & Wild (2010, hereafter HW10) whose redshift errors are much smaller and redo the group findings. According to the error estimates for the redshifts of HW10, the RMS redshift error for quasars at 0.1 < z < 1.8 is found to be only 0.00024, more than 10 times smaller than the nominal value for the SDSS DR7.
It will be a good consistency check to compare two group catalogs derived from the redshifts from Schneider et al. (2010) and HW10. We find that the new quasar group catalog is very close to the previous one obtained using Schneider et al. (2010) ’s redshifts. All major groups are detected again and only a few members are affected. As a result, the richness and size distributions change only slightly. It is reassuring that the quasar groups and the statistics are robust against the redshift error of Schneider et al. (2010) . When LL = 50 h −1 Mpc, we still find a very good agreement of both distribution functions. Therefore, all of our results are essentially unchanged by using more accurate quasar redshifts. We stick to Schneider et al. (2010) ’s redshifts in our paper since there is practically no change in the conclusions and it is easier to compare our groups directly to those of previous works. The groups identified in the redshift space are also affected by the peculiar velocities of quasars. Influence of peculiar velocity on group identification is to some extent unavoidable. But this is relatively small because the RMS pairwise peculiar velocity of galaxies is only about 300 km/s.
5. SUMMARY AND CONCLUSIONS
We make a wide contiguous volume-limited sample of quasars statistically uniformly selected from the SDSS DR7 quasar catalog and identify quasar groups using the FoF algorithm. The statistical properties of the observed quasar groups are compared with those of random point sets. We make the following findings.
  • 1. The observed SDSS DR7 quasar groups identi- fied with LL= 70h−1Mpc are statistically consistent, in terms of size and richness, with those of random points clustered by chance. The size or richness distribution functions of those quasar groups can be found in random catalogs with 92.4% and 58.1% probabilities, respectively.
  • 2. The probabilities of finding groups larger or richer than the large quasar group U1.27, reported byClowes et al. (2013), in a random point catalog are 100% and 92%, respectively.
  • 3. The quasar groups are statistically dominated by physically clustered quasars when LL for group identi- fication is less than 50h−1Mpc.
We emphasize that the observed LSS should not be used to draw cosmological conclusions without making elaborate statistical tests. Size and richness of cosmic structures depend sensitively on how those structures are identified. For example, one can always find structures as large as the survey size if the threshold density level used to identify them is lowered close to the percolation density. Even in a random point distribution without any physical clustering an infinitely long structure can be identified when LL is increased to 0.864 times the mean particle separation, which is the critical value for percolation. It is conceivable that one would claim discovery of groups of galaxies or quasars even larger than the Sloan Great Wall in existing or future redshift surveys. The lesson of the past is that one will again need to make statistical tests before drawing conclusions on cosmological implication of those objects.
Our results suggest that the large quasar groups identified with LL > 50 h −1 Mpc can be considered as associations of smaller physically-clustered quasar systems that are connected by a very large linking length. The smaller quasar systems are similar to the supercluster complexes in the local universe, as was suggested in Einasto et al. (2014) . Hutsemékers et al. (2014) showed that quasar spin axes in U1.27 seem to be aligned parallel to this structure. However, from their Figure 6 one can see that alignment is better if this quasar group is divided into smaller systems, in agreement with our interpretation of quasar systems.
There have also been several studies trying to de- fine and measure the homogeneity scale of the universe ( Yadav et al. 2010 ; Marinoni et al. 2012 ; Scrimgeour et al. 2012 ). However, definition of the homogeneity scale has been either arbitary or dependent on observational samples. We point out that, in the current popular cosmological models that adopt the Cosmological Principle, the notion of the critical scale of homogeneity does not make sense. Homogeneity is only achieved asymptotically in these models as the scale of observation increases, and there is no characteristic scale intrinsic to the universe above which the universe can be suddenly regarded homogeneous (see also Nadathur 2013 ). Instead of looking for arbitrarily-defined homogeneity scales or naively examining the size of the largest structures the observed LSS should be statistically compared with the structures identified in the same way in the mock survey samples obtained from simulated initiallyhomogeneous universes. In our forthcoming paper we will use mock quasars formed in simulated universes to identify large groups of quasars and statistically compare their properties with those of observed quasars. We will examine if the large structures can be explained in the framework of the standard hierarchical structure formation model in the given Hubble time.
We use ‘large-scale structure’ as a generic term for structures larger than galaxy clusters. In this terminology the large-scale structures of the universe are classified into high-density and low-density large-scale structures. The high-density large-scale structures are classified into categories of superclusters, filaments, and walls depending on their internal density and morphology. The low-density large-scale structures are voids, and classified into bubbles, and tunnels depending on their morphology. In this scheme all the names that have been used to describe ‘large’ structures in the universe are unified.
In this paper we call the large loose groupings of quasars ‘quasar groups.’ This terminology can give a misleading idea that they are compact and physically interacting internally as in galaxy groups. Einasto et al. (2014) used ‘quasar systems’ to avoid the confusion. The name ‘quasar association’ may also be an appropriate name for these loose quasar groups since the name ‘stellar association’ has been already used in astronomy for loose groups of stars that are not gravitationally bound.
Acknowledgements
The authors thank the anonymous referee for helpful comments and reviewing our paper quickly, and also thank Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation Linux Cluster System). M. Einasto was supported by the ETAG project IUT26-2, and by the European Structural Funds grant for the Centre of Excellence “Dark Matter in (Astro)particle Physics and Cosmology” TK120. H. Lietzen acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under the 2011 Severo Ochoa Program MINECO SEV-2011-0187. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. 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 SDSS Web Site ishttp://www.sdss.org.
References
Bradshaw E. J. , Almaini O. , Hartley W. G. , Chuter R. W. , Simpson C. , Conselice C. J. , Dunlop J. S. , McLure R. J. , Cirasuolo M. 2011 Environments of Active Galactic Nuclei at z < 1.5 in the UKIDSS Ultra-Deep Survey MNRAS 415 2626 -    DOI : 10.1111/j.1365-2966.2011.18888.x
Clifton T. , Clarkson C. , Bull P. 2012 Isotropic Blackbody Cosmic Microwave Background Radiation as Evidence for a Homogeneous Universe PRL 109 051303 -    DOI : 10.1103/PhysRevLett.109.051303
Clowes R. G. , Campusano L. E. 1991 A 100-200 MPC Group of Quasars MNRAS 249 218 -    DOI : 10.1093/mnras/249.2.218
Clowes R. G. , Campusano L. E. , Graham M. J. , Sochting I. K. 2012 Two Close Large Quasar Groups of Size ~ 350 Mpc at z ~ 1.2 MNRAS 419 556 -    DOI : 10.1111/j.1365-2966.2011.19719.x
Clowes R. G. , Harris K. A. , Raghunathan S. 2013 A Structure in the Early Universe at z ~ 1.3 that Exceeds the Homogeneity Scale of the R-W Concordance Cosmology MNRAS 429 2910 -    DOI : 10.1093/mnras/sts497
Coldwell G. V. , Lambas D. G. 2006 Properties of Galaxies in Sloan Digital Sky Survey Quasar Environments at z < 0.2 MNRAS 371 786 -    DOI : 10.1111/j.1365-2966.2006.10712.x
de Lapparent V. , Geller M. J. , Huchra J. P. 1986 A Slice of the Universe ApJ 302 L1 - 1
Einasto M. , Liivamägi L. J. , Tempel E. , Saar E. , Tago E. , Einasto P. , Enkvist I. , Einasto J. , Martínez V. J. , Heinämäki P. , Nurmi P. 2011 The Sloan Great Wall. Morphology and Galaxy Content ApJ 736 51 -    DOI : 10.1088/0004-637X/736/1/51
Einasto M. , Tago E. , Lietzen H. , Park C. , Heinämäki P. , Saar E. , Song H. , Liivamägi L. J. , Einasto J. 2014 Tracing a High Redshift Cosmic Web with Quasar Systems A&A 568 46 -
Geller M. J. , Huchra J. P. 1989 Mapping the Universe Science 246 897 -    DOI : 10.1126/science.246.4932.897
DiPompeo M. A. , Myers A. D. , Hickox R. C. , Geach J. E. , Hainline K. N. 2014 The Angular Clustering of Infrared-Selected Obscured and Unobscured Quasars MNRAS 442 3443 -    DOI : 10.1093/mnras/stu1115
Fanidakis N. , Maccio A. V. , Baugh C. M. , Lacey C. G. , Frenk C. S. 2013 The Most Luminous Quasars Do Not Live in the Most Massive Dark Matter Haloes at Any Redshift MNRAS 436 315 -    DOI : 10.1093/mnras/stt1567
Gott J. R. , Jurić M. , Schlegel D. 2005 A Map of the Universe ApJ 624 463 -    DOI : 10.1086/428890
Hewett P. C. , Wild V. 2010 Improved Redshifts for SDSS Quasar Spectra MNRAS 405 2302 -
Horvath I. , Hakkila J. , Bagoly Z. 2013 The Largest Structure of the Universe, Defined by Gamma-Ray Bursts arXiv1311.1104
Hutchings J. B. , Scholz P. , Bianchi L. 2009 Environments of QSOs at Redshift 0.9-1.3 AJ 137 3533 -    DOI : 10.1088/0004-6256/137/3/3533
Hutsemékers D. , Braibant L. , Pelgrims V. , Sluse D. 2014 Alignment of Quasar Polarizations with Large-Scale Structures A&A 572 18 -
Karhunen K. , Kotilainen J. K. , Falomo R. , Bettoni D. 2014 Low-Redshift Quasars in the SDSS Stripe 82. The Local Environments MNRAS 441 1802 -    DOI : 10.1093/mnras/stu688
Komberg B. V. , Kravtsov A. V. , Lukash V. N. 1996 The Search for and Investigation of Large Quasar Groups MNRAS 282 713 -    DOI : 10.1093/mnras/282.3.713
Krumpe M. , Miyaji T. , Coil A. L. 2013 Clustering Measurements of broad-Line AGNs: Review and Future arXiv1308.5976
Lietzen H. , Heinämäki P. , Nurmi P. , Liivamagi L. J. , Saar E. , Tago E. , Takalo L. O. , Einasto M. 2011 Large-Scale Environments of z < 0.4 Active Galaxies A&A 535 21 -
Lietzen H. , Heinämäki P. , Nurmi P. , Tago E. , Saar E. , Liivamagi J. , Tempel E. , Einasto M. , Einasto J. , Gramann M. , Takalo L. O. 2009 Environments of Nearby Quasars in Sloan Digital Sky Survey A&A 501 145 -
Marinoni C. , Bel J. , Buzzi A. 2012 The Scale of Cosmic Isotropy JCAP 10 036 -
Miller L. , Croom S. M. , Boyle B. J. , Loaring N. S. , Smith R. J. , Shanks T. , Outram P. 2004 200-Mpc-Sized Structure in the 2dF QSO Redshift Survey MNRAS 355 385 -    DOI : 10.1111/j.1365-2966.2004.08330.x
Nadathur S. 2013 Seeing Patterns in Noise: GigaparsecScale ‘Structures’ that Do Not Violate Homogeneity MNRAS 434 398 -    DOI : 10.1093/mnras/stt1028
Park C. 1990 Large N-Body Simulations of a Universe Dominated by Cold Dark Matter MNRAS 242 59P -    DOI : 10.1093/mnras/242.1.59P
Park C. , Choi Y.-Y. , Kim J. , Gott J. R. , Kim S. S. , Kim K.-S. 2012 The Challenge of the Largest Structures in the Universe to Cosmology ApJ 759 L7 -    DOI : 10.1088/2041-8205/759/1/L7
Ade P. A. R. , Aghanim N. 2014 Planck 2013 Results. XV. CMB Power Spectra and Likelihood A&A 571 15 -
Portinari L. , Kotilainen J. , Falomo R. , Decarli R. 2012 On the Cosmological Evolution of the Black Hole-Host Galaxy Relation in Quasars MNRAS 420 732 -    DOI : 10.1111/j.1365-2966.2011.20086.x
Richards G. T. , Strauss M. A. , Fan X. 2006 The Sloan Digital Sky Survey Quasar Survey: Quasar Luminosity Function from Data Release 3 AJ 131 2766 -    DOI : 10.1086/503559
Schneider D. P. , Richards G. T. , Hall P. B. 2010 The Sloan Digital Sky Survey Quasar Catalog. V. Seventh Data Release AJ 139 2360 -    DOI : 10.1088/0004-6256/139/6/2360
Scrimgeour M. I. , Davis T. , Blake C. 2012 The WiggleZ Dark Energy Survey: the Transition to Large Scale Cosmic Homogeneity MNRAS 425 116 -    DOI : 10.1111/j.1365-2966.2012.21402.x
Shen Y. , Richards G. T. , Strauss M. A. 2011 A Catalog of Quasar Properties from Sloan Digital Sky Survey Data Release 7 ApJS 194 45 -    DOI : 10.1088/0067-0049/194/2/45
Shen Y. , McBride C. K. , White M. 2013 Cross Correlation of SDSS DR7 Quasars and DR10 BOSS Galaxies: The Weak Luminosity Dependence of Quasar Clustering at z ~ 0.5 ApJ 778 98 -    DOI : 10.1088/0004-637X/778/2/98
Shen Y. , Strauss M. A. , Oguri M. 2007 Clustering of High-Redshift (z ≥ 2.9) Quasars from the Sloan Digital Sky Survey AJ 133 2222 -    DOI : 10.1086/513517
Sheth R. K. , Diaferio A. 2011 How Unusual Are the Shapley Supercluster and the Sloan Great Wall? MNRAS 417 2938 -    DOI : 10.1111/j.1365-2966.2011.19453.x
Sochting I. K. , Clowes R. G. , Campusano L. E. 2004 Relation of Radio-Quiet Quasars to Galaxy Clusters at z < 0.3 MNRAS 347 1241 -    DOI : 10.1111/j.1365-2966.2004.07306.x
Song H. , Park C. , Lietzen H. , Einasto M. 2015 in preparation
Trainor R. F. , Steidel C. C. 2012 The Halo Masses and Galaxy Environments of Hyperluminous QSOs at z ~ 2.7 in the Keck Baryonic Structure Survey ApJ 752 39 -    DOI : 10.1088/0004-637X/752/1/39
Webster A. 1982 The Clustering of Quasars from an Objective-Prism Survey MNRAS 199 683 -    DOI : 10.1093/mnras/199.3.683
Williger G. M. , Campusano L. E. , Clowes R. G. , Graham M. J. 2002 Large-Scale Structure at z = 1.2 Outlined by Mg II Absorbers ApJ 578 708 -    DOI : 10.1086/342614
Wold M. , Lacy M. , Lilje P. B. , Serjeant S. 2000 Clustering of Galaxies around Radio Quasars at 0.5 ≤ z ≤ 0.8 MNRAS 316 267 -    DOI : 10.1046/j.1365-8711.2000.03473.x
Yadav J. K. , Bagla J. S. , Khandai N. 2010 Fractal Dimension as a Measure of the Scale of Homogeneity MNRAS 405 2009 -