Probing cosmological isotropy with Planck SunyaevZeldovich galaxy clusters
Abstract
We probe the statistical isotropy hypothesis of the largescale structure with the second Planck SunyaevZeldovich (PSZ2) galaxy clusters data set. Our analysis adopts a statisticalgeometrical method which compares the 2point angular correlation function of objects in antipodal patches of the sky. Given possible observational biases, such as the presence of anisotropic sky cuts and the nonuniform exposure of Planck’s instrumentation, ensembles of Monte Carlo realisations are produced in order to assess the significance of our results. When these observational effects are properly taken into account, we find neither evidence for preferred directions in the sky nor signs of largeangle features in the galaxy clusters celestial distribution. The PSZ2 data set is, therefore, in good concordance with the fundamental hypothesis of largeangle isotropy of cosmic objects.
keywords:
Cosmology: Observations; largescale structure of Universe; galaxy clusters1 Introduction
The Cosmological Principle (CP) consists on a fundamental assumption in modern cosmology in which the Universe presents neither special directions, nor special positions, as discussed in Goodman (1995); Maartens (2011), for instance. In this sense, the success of FLRWbased models (especially CDM paradigm) in explaining the angular power spectrum of the Cosmic Microwave Background (CMB) temperature fluctuations (Ade et al., 2015a), the evolution and characterisation of the largescale structure of the Universe (LSS) (Aubourg et al., 2015), in addition to cosmological distances and ages (Alcaniz & Lima, 1999; Alcaniz et al., 2003; Simon et al., 2005; Stern et al., 2010; Moresco et al., 2012; Suzuki et al., 2012; Betoule et al., 2014), solidified the CP not only as a simplified mathematical hypothesis, but also as a valid physical assumption. Hence, it is crucial to test the CP with observational data for the sake of probing one of the underpinning assumptions of cosmology, besides obtaining a correct interpretation of the physical assumptions underlying the cosmic acceleration and structure formation of the Universe.
In the past years, some analyses reported a possible statistical isotropy violation in large scales, such as: i) CMB temperature fluctuations features (e.g., low multipole alignments, hemispherical power asymmetries, lack of angular correlations at large scales, and the presence of a nongaussian Cold Spot) (Eriksen et al., 2004; Bernui et al., 2007; Abramo et al., 2009; Akrami et al., 2014; Bernui et al., 2014; Ade et al., 2015d; Schwarz et al., 2016); ii) large velocity flows from analyses of kinematic SunyaevZeldovich effect in GCs (Kashlinsky et al., 2009, 2011; AtrioBarandela et al., 2015), although some controversy have been pointed out by Osborne et al. (2011), Ade et al. (2014) regarding the validity of these results
In the light of these puzzles, the validity of the CP must be put under scrutiny with other observational data in order to confirm or refute these results, given that further indications of violation in the statistical isotropy hypothesis would reinforce the demand for a complete reformulation of the concordance model of Cosmology. We can achieve it by directly studying the angular distribution of cosmic objects in antipodal patches of the celestial sphere. In this sense, the second release of Planck SunyaevZeldovich (PSZ2) catalogue (Ade et al., 2015b, c) provides an ideal observational sample for this purpose, since it comprises over 1600 galaxy clusters (GCs) in a wide sky coverage (), besides covering a deep redshift range (). Thus, it corresponds to the most complete allsky ensemble of GCs available at the present moment. Since these objects are excellent tracers of the largescale matter distribution of the Universe, they should follow the statistical isotropy which the standard scenario of structure formation is based upon. If any strong evidence for excessive correlations or anticorrelations in the GC distribution is detected, then we could state that the isotropy assumption could be invalid unless explained by limited sky coverage or by the systematics of the observational sample.
Therefore, the goal of this work is to probe the statistical isotropy assumption by means of a hemispherical comparison of the angular distribution of these PSZ2 catalogue. We also produce synthetic Monte Carlo realisations to account for the anisotropic signatures that could exist in this sample due to the removed area around the galactic plane, in addition to the sky exposure of Planck’s observational strategy. The paper is organised as follows: Section 2 is dedicated to the preparation of the observational sample while Section 3 discusses the methodology developed to carry out the isotropy analyses of the GCs distribution. In Section 4 we discuss the results of these analyses and the tests of statistical significance performed. The concluding remarks and perspectives of this work are presented in Section 5.
2 Data set
Our analyses are performed with the PSZ2 catalogue named Union, which has been obtained from the IRSA website

,

,

.
The first two queries eliminate 180 of 1654 objects, which provide a sample purity of 85%, whereas the last one reduces the sample to 1066 GCs. While the latter is not the main source of uncertainty, we note that the exclusion of the GCs with no redshift determination mildly increases the average signaltonoise () of their detection, hence corresponding to the most reliable sources of the catalogue
3 Methodology: The SigmaMap
Here we describe the angulardistribution estimator which leads to quantify deviations from statistical isotropy in a given set of cosmic events with known positions on the celestial sphere (Bernui et al., 2007, 2008). Our primary purpose is to illustrate the procedure for defining the discrete function on the celestial sphere in order to generate its associated map, called sigmamap. Then, one compares the sigmamap obtained from the data catalogue with a large set of sigmamaps generated from statistically isotropic simulated ensembles in order to assess a measure of (possible) deviation from statistical isotropy in the data set in analysis.
Let be a spherical cap region on the celestial sphere, of degrees of radius, centred at the th pixel, , where are the angular coordinates of the center of the th pixel. Both, the number of spherical caps and the coordinates of their center are defined using the HEALPix pixelisation scheme (Górski et al., 2005). The spherical caps are such that their union completely covers the celestial sphere . We assume galactic coordinates throughout our analyses. Let be the catalogue of cosmic objects located in the th spherical cap . The 2point angular correlation function (2PACF) of these objects (Padmanabhan, 1993), denoted as , is the difference between the normalised frequency distribution and that expected from the number of pairsofobjects with angular distances in the interval , where and is the bin width. The expected distribution is the average of normalized frequency distributions obtained from a large number of simulated realisations of isotropically distributed objects in , containing the same number of objects as in the data set in analysis. This 2PACF estimator is nothing else than the well known , termed natural estimator in the literature (Bernui & Teixeira, 1999; Bernui, 2005). A positive (negative) value of indicates that objects with angular separation are correlated (anticorrelated), while zero indicates no correlation.
Let us define now the scalar function , for , which assigns to the cap, centred at , a real positive number . We define a measure of the angular correlations in the cap as
(1) 
To obtain a quantitative measure of the angular correlation signatures of the GC’s sky map, we choose and cover the celestial sphere with hemispheres, then calculate the set of values using eq. (1). Patching together the set in the celestial sphere according to a coloured scale, we obtain a sigmamap. We quantify the angular correlation signatures of a given sigmamap from a data set by calculating its angular power spectrum. Similar power spectra are calculated, for comparison, with isotropically distributed samples of cosmic objects.
Since the sigmamap assigns a real number value to each pixel in the celestial sphere, that is, , it is possible to expand it in spherical harmonics: , where the set of values , defined by , is the angular power spectrum of the sigmamap. Because we are interested in the largescale angular correlations, we shall concentrate on , i.e., scales larger than in the sky. Therefore, the smallscale clustering of data points that could arise due to possible visual effects from angular projections, or even nonlinearities in the structure formation scenario, is not an issue in our analyses since we are only interested in the largeangle statistics of the sample.
4 Statistical significance tests
4.1 Monte Carlo isotropic simulations
As previously mentioned, the statistical significance of our analyses is assessed by comparison of the sigmamaps of Monte Carlo (MCs) realisations with the sigmamap computed from the real data.
We produce an ensemble of 500 statistically isotropic data sets, hereafter termed MCs iso, with the same available area and approximately the same number of data points as the real data. This is carried out according to the following steps

Let be a uniform distribution of real random numbers: , where (the subindex I stands for Isotropy). Given a specific pixel , we generate a real number . If , then we assign to this pixel a GC, otherwise, it remains empty
^{6} . 
We repeat such operation for all pixels of the map (), thus creating a simulated MCs iso map with GCs in the full sky. By repeating this methodology 500 times, we obtain our set of MCs iso maps.

The final step of this procedure consists in applying the Planck’s foreground mask to this set of simulated maps, thereby resulting in a map with . An example of such MC iso map, produced under these procedures, is displayed in the left panel of Figure 3
^{7} .
4.2 Monte Carlo anisotropic simulations
In this section we detail the construction of the second set of simulated maps, i.e., an ensemble of 500 realisations which take into account the nonuniform sky exposure (NUSE) function of the Planck satellite, henceforth MCs aniso. For this purpose, we consider the information given by the Planck collaboration
In order to produce a MC aniso ensemble, we establish the following steps:

Let and be two uniform distributions of real random numbers in the interval . Given a pixel , we generate two real numbers and and define the number as: (the subindex A stands for anisotropy). If or , we assign a GC to this pixel , otherwise, it remains empty. Note that the first condition imposes higher probabilities to assign a GC toward the ecliptic poles (the most intense patches in the map in Figure 2), while the second one ensures that, outside such ecliptic regions (thus ), the isotropy condition predominates.

Performing this operation for all pixels of the map (), we obtain a simulated MC aniso map with GCs in the full sky. We repeat this operation 500 times in order to obtain a set of 500 MCs aniso maps.

We apply the Planck’s foreground mask to this set, resulting in a simulated map with . An example of a simulated MC aniso map produced according to this procedure is displayed, for the sake of illustration, in the right panel of Figure 3.
Hence, we are able to test the statistical isotropy of the PSZ2 data, considering that a possible source of clustering arises due to this anisotropic sky exposure toward the ecliptic poles, which could be manifested as strong correlations and anticorrelations in the sigmamap analysis, when compared to the typical fluctuations of isotropic distributions, as shown in Bernui et al. (2008); Ukwatta & Wózniak (2016). This is done by comparing the PSZ2 sigmamap power spectrum with both MCiso and MCaniso ensembles, given that this power spectrum realises the angular features of these data sets. By means of this analysis, we can test whether systematic effects, like the NUSE and the foreground mask, could lead to false anisotropic signals, or whether there is indeed an intrinsic anisotropy in the data.
5 Data analysis and results
The left panel of Figure 4 depicts, for illustrative reasons, the 2PACF (see Eq. (1)) obtained in two different hemispheres along the starting from the hemisphere centre. The hemispheres and , as indicated in this panel as the blue and red curves, respectively, correspond to those where the maximal and minimal sigma values (noting that the sigma denotes the sum of the square of this over all these ) were attained, respectively.
It is noticeable that the angular correlations fluctuates much more in the former case, hence indicating larger angular correlations (and anticorrelations) in the GCs encompassed in this region. When approximating the sigmamap results obtained through the whole celestial sphere as a dipole, we obtain a direction whose maximal value is located towards the northwestern patch of the sky, so, the minimal value points towards the southeast. Even though this direction resembles the maximal CMB power asymmetry localisation, we are unable to ascribe this signal to any of these features in a statistically significant manner. Thus, we provide no support for a significant preferred direction in the PSZ2 map which could be associated with the aforementioned CMB features or the nearby velocity flow.
Furthermore, we show the power spectrum of the sigmamap, namely , as black crosses in the right panel of Figure 4. The red curve denotes the mean sigmamap power spectra of 500 MCs aniso, while the blue curve represents the mean spectra from the idealistic isotropic realisations. In all these cases, the central values of these curves are given by their arithmetic average and the error bars correspond to the mean absolute deviation (MAD) of these simulated data sets. We adopt the MAD instead of the standard deviation (STD) because the ’s of these 500 MCs are very skewed to the right, i.e., they present a longtailed distribution, so that the MAD gives a more robust estimation of the uncertainty around their mean value than the STD. We note that the PSZ2 sigmamap behaviour is much closer to the average of 500 MCs aniso than the isotropic realisations, yet the multipole moments , , and , corresponding to the angular scales , and , respectively, are slightly in tension with them. Such mild discrepancies could be ascribed to some other features in the data besides those we are addressing, as possible impurities and contamination that had not been eliminated in our quality tests.
Nevertheless, we point out that, since these MCs aniso give the upper limit of the possible amount of anisotropy in each angular scale because of these selection effects, and since the apparent anisotropy of the GCs distribution matches this prediction in most of the angular scales, there is no significant suggestion of anisotropy in the PSZ2 data apart from mild anisotropic features. Accordingly, such signals can be accounted by the asymmetric sky cut and the anisotropic sky exposure. Thus, we can conclude that the GC distribution of the PSZ2 sample is in good concordance with the statistical isotropy hypothesis underlying the standard model of Cosmology.
6 Conclusions
In this work, the cosmological isotropy of the largescale structure of the Universe has been probed with the largest allsky catalogue of GCs available in the literature, i.e., the second release of Planck Sunyaev Zeldovich sources (PSZ2) (Ade et al., 2015b, c). This has been accomplished by mapping the angular distribution of GCs through the celestial sphere using a geometricalstatistical test, named sigmamap. Given the nature of this method, any departure from the statistical isotropy of the data would be revealed by discrepancies of its power spectrum when compared to isotropic data sets. Since we had performed this test in large angular scales, such discrepancy would suggest that the statistical isotropy assumption may not hold, thus the concordance model of Cosmology would need to be completely reformulated.
Our analysis has found no statistically significant indication for anomalous anisotropy in the PSZ2 catalogue, as well as no links with the CMB features or large velocity flows, that have been previously suggested by other authors. As a matter of fact, we have found that the power spectra of the sigmamap computation presents good concordance between the real data and the MC realisations in most of the angular scales probed. Nevertheless, we point out that this agreement happens only if the incomplete sky coverage, and especially the anisotropic sky exposure, are properly incorporated in these simulated data sets. When the latter is neglected, we have noticed that the sigmamap power spectra are substantially lower than the real data’s. Since the NUSE provides an upper limit of the anisotropy of the GC sky maps introduced by the Planck’s scanning strategy, such observational feature seems to be the main cause of the power excess we have found in PSZ2 largeangle correlations.
We have concluded that the GC angular distribution, which comprises the most massive objects composing the largescale structure of the Universe, indeed agrees with the cosmic isotropy assumption in large angular scales. Yet, we stress that it is crucial to repeat these tests, or even propose new ones, given the prospect of future GC survey such as eROSITA (Merloni et al., 2012), which is expected to provide an allsky sample of , thus enormously enhancing the precision of the test carried out in this work. Furthermore, the advent of much larger galaxy or SN data sets by future surveys, such as LSST (Abell et al., 2009) and SKA (Maartens et al., 2015; Schwarz et al., 2015), may provide assessments of the concordance model and the CP with unprecedented precision in the next decade.
acknowledgments
We thank Mariachiara Rossetti for useful discussions about the PSZ2 catalogue. CAPB Jr. acknowledges CAPES for the financial support. AB and ISF acknowledge the Science without Borders Program of CAPES and CNPq, for a PVE project (88881.064966/201401) and a PDE fellowship (234529/201408), respectively. JSA acknowledges financial support from CNPq, FAPERJ and INEspaço. We also acknowledge the HEALPix package for the derivation of many of the results presented in this work. This research has made use of the NASA / IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Planck SunyaevZeldovich catalogue used here is based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.
Footnotes
 pagerange: Probing cosmological isotropy with Planck SunyaevZeldovich galaxy clusters–Probing cosmological isotropy with Planck SunyaevZeldovich galaxy clusters
 The possibility of anisotropy in the cosmological expansion was also investigated with Type Ia Supernovae luminosity distance data (Schwarz & Weinhorst, 2007; Antoniou & Perivolaropoulos, 2010; Cai & Tuo, 2012; Turnbull et al., 2012; Kalus et al., 2013; Feindt et al., 2013; Jiménez et al., 2015; Appleby et al., 2015; Bengaly et al., 2015; Javanmardi et al., 2015). No significant result supporting an anisotropic expansion has been detected so far.
 http://irsa.ipac.caltech.edu
 Note also that there is a subset of the PSZ2 catalogue, named cosmology, which corresponds to the highestquality GC detections (, with a sample purity increased to 95% in ) data set constructed by Planck team in order to constrain cosmological parameters such as and (Ade et al., 2015c). We do not focus on this subsample since our goal is to test the isotropy assumption with the largest data set possible in terms of the number of GCs and sky coverage.
 As a matter of fact, we sample in the sky, since this number corresponds to the expected when no foreground mask is considered. In all these cases, the celestial sphere is divided in equal area pixels, thus corresponding to a grid (Górski et al., 2005).
 In order to produce GCs in the full sky each pixel should have % of probability to be occupied because .
 Note that the angular distribution of the GCs in these simulated maps has been extensively tested to confirm their statistical isotropy feature (for details of such tests, see Bernui et al. (2004))
 http://pla.esac.esa.int/pla/#maps
 The NUSE map provides the average number of observations for each sky pixel () of the four lowest frequency channels of Planck’s instrumentation, namely 100, 143, 217, and 353 GHz, according to , where is the maximal number of observations for each channel. We did not consider the maps for the two highest frequencies (545 and 857 GHz) because they present quite lower values compared to the lower frequency channels, which could bias this normalisation procedure adopted to obtain the final NUSE map. However, we stress that the NUSE should be regarded as an approximation for possible directional weights adopted in Planck’s detection algorithm of GCs, as such procedure is beyond the scope of this work.
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