TY - JOUR
AU1 - Santucho,, V
AU2 - Luparello, H, E
AU3 - Lares,, M
AU4 - Lambas, D, G
AU5 - Ruiz, A, N
AU6 - Sgró, M, A
AB - ABSTRACT Studies of large-scale structures in the Universe, such as superstructures or cosmic voids, have been widely used to characterize the properties of the cosmic web through statistical analyses. On the other hand, the two-point correlation function of large-scale tracers such as galaxies or haloes provides a reliable statistical measure. However, this function applies to the spatial distribution of point-like objects, and therefore it is not appropriate for extended large structures that strongly depart from spherical symmetry. Here we present an analysis based on the standard correlation function formalism that can be applied to extended objects exhibiting arbitrary shapes. Following this approach, we compute the probability excess Ξ of having spheres sharing parts of cosmic structures with respect to a realization corresponding to a distribution of the same structures in random positions. For this aim, we identify superstructures defined as future virialized structures (FVSs) in semi-analytic galaxies in the MPDL2 MultiDark simulation. We have also identified cosmic voids to provide a joint study of their relative distribution with respect to the superstructures. Our analysis suggests that Ξ provides useful characterizations of the large-scale distribution, as suggested from an analysis of subsets of the simulation. Even when superstructure properties may exhibit negligible variations across the subsets, Ξ has the sensitivity to statistically distinguish sub-boxes that depart from the mean at larger scales. Thus, our methods can be applied in analysis of future surveys to provide characterizations of large-scale structure suitable to distinguish different theoretical scenarios. methods: data analysis, methods: statistical, large-scale structure of Universe, cosmology: observations 1 INTRODUCTION A network of interconnected filaments, walls, and dense clusters surrounded by large subdense void regions provides a useful description of the large-scale structure of the Universe (e.g. Einasto et al. 1997; Colless et al. 2001; Jaaniste, Einasto & Einasto 2004; Einasto 2006; Abazajian et al. 2009; Einasto et al. 2014). Galaxy surveys of large extension revealed this complex pattern (York et al. 2000; Colless et al. 2001), which is also supported by cosmological N-body simulations involving a cold dark matter (CDM)-type primordial power spectrum (Frisch et al. 1995; Bond, Kofman & Pogosyan 1996; Sheth et al. 2003; Shandarin, Sheth & Sahni 2004). In this scenario, the largest mass concentrations coincide with the intersections of walls and filaments conforming the locations of clusters and superclusters of galaxies. These massive objects have been studied and identified with a large variety of methods, such as linking Abell cluster positions (Zucca et al. 1993; Einasto et al. 1997), or directly from wide-area surveys of the large-scale galaxy distribution (Shectman et al. 1996; Colless et al. 2001; Stoughton et al. 2002). More recently, supercluster catalogues have been constructed by combining new methodologies of clustering analysis (Einasto et al. 1997; Einasto 2006; Liivamägi, Tempel & Saar 2012; Nadathur & Hotchkiss 2014; Chow-Martinez et al. 2014) providing a deeper insight into the nature, extension and distribution of the largest bound structures of the Universe. In the ΛCDM scenario, the present and future dynamics of the Universe is dominated by an accelerated expansion, hence an appropriate definition of large-scale structures can be obtained by the requirement of present overdense regions that will be bound and virialized systems in the future (Busha et al. 2005; Dünner et al. 2006; Araya-Melo et al. 2009; Luparello et al. 2011). Studies of the evolution of the superstructure network show that although some small systems may have undergone merger events with other superstructures during their past assembly history, this is not a dominant factor in the global evolution. Rather, the main transformation processes take place inside the systems (Einasto et al. 2019). Different attempts to provide useful characterizations of the large-scale structure have been performed either using the galaxy distribution (Tegmark et al. 2004; Scottez et al. 2016) as well as studying large-scale galaxy system properties such as superclusters, clusters, filaments and voids (Colberg et al. 2000; Brodwin et al. 2007; Chan, Hamaus & Desjacques 2014). Early studies included counts in cell moments (White 1979), the topological genus of smoothed density distributions (Gott, Melott & Dickinson 1986), Minkowsky functionals (Schmalzing, Kerscher & Buchert 1996), structure functions on minimal spanning trees (Colberg 2007), wavelet methods, etc. More recently, other approaches to extract high-order information have focused on non-linear transforms such as a logarithmic transformation method (Wang et al. 2011). It is worth considering that large-scale structure studies, mainly through detailed analysis of the two-point correlation function, have given important support to the settlement of the standard cosmological concordance model. High-order statistics has been used in present survey analyses (Verde et al. 2002; Baugh et al. 2004; Gaztañaga et al. 2005; Croton et al. 2007) since higher-order correlations can give useful information about the complex filamentary structure. Although straightforward calculation of high-order correlations is possible and has been computed in observations and in numerical simulations (see, for instance, Slepian & Eisenstein 2015, and references therein), beyond the three-point function estimations become challenging not only from the computational point of view, but also offering model interpretation in terms of covariances for a large number of data bins. However, it should be recalled that this statistic considers a distribution of point sources, not extended objects with arbitrary shapes where the centre definition cannot be properly addressed. When such structure sizes and mean separations are comparable, the two-point correlation function formalism cannot be used straightforwardly. In cases with spherical symmetry, as in the case of voids, this is overcome by comparing to a random distribution of points that take into account void sizes. It is worth noticing that although properties of voids have been used to test different cosmological models as well as modified gravity, overdense superstructures such as future virialized structures (FVSs, Luparello et al. 2011) have been less studied, mainly because of their complex spatial shape. Lares et al. (2017) studied the spatial correlations between voids and superstructures by defining a first modified version of the traditional two-point correlation function. They quantified the cross-correlation of voids relative to FVSs estimating the probability of finding a randomly placed sphere containing simultaneously void and FVS. That allowed them to measure the tendency of voids to be located at a given distance from a superstructure. In this framework, considering the random spheres as a suitable tool, we propose an alternative method, based on the formalism of the two-point correlation function, that can be applied to measure clustering on extended and non-symmetrical systems. Also, as the process includes a comparison with random data sets the results are easier to assess. Moreover, this allows us to avoid systematic and intrinsic bias associated with any particular sample of structures. This paper is organized as follows: in the next section, we describe the data and methods applied to the identification of FVSs and voids. Then, in Section 3 we introduce the definition of the structure correlation function, along with some tests intended to qualitatively interpret its behaviour. In Section 4 we compute the structure auto-correlation for FVSs, also exploring the effects of cosmic variance. In addition, we analyse the FVS–void structure cross-correlation. Finally, in Section 5 we present a summary and discussion of our results. 2 DATA The present work is based on the public semi-analytic catalogues MultiDark–Galaxies (Knebe et al. 2018), available at the CosmoSim (https://www.cosmosim.org) and Skies and Universes (https://www.skiesanduniverses.org) databases. In particular, we use the catalogue MDPL2-SAG,1 which was constructed populating the MDPL2 dark matter haloes with the semi-analytic model of galaxy formation and evolution (SAG, Cora et al. 2018). The MDPL2 simulation follows the evolution of 38403 particles in a cubic volume of side-length 1 h−1 Gpc, and forms part of the MultiDark suite of simulations (Riebe et al. 2013; Klypin et al. 2016). The adopted cosmology consists of a flat ΛCDM model with the Planck cosmological parameters: Ωm = 0.307, Ωb = 0.048, ΩΛ = 0.693, σ8 = 0.823, ns = 0.96 and a dimensionless Hubble parameter h = 0.678 (Planck Collaboration XIII 2016). The mass resolution is mp = 1.51 × 10|$^9\, h^{-1}{\rm \, M}_\odot$| per dark matter particle. Haloes and subhaloes have been identified with the rockstar halo finder (Behroozi, Wechsler & Conroy 2013a) and ConsistentTrees has been used to construct the merger trees (Behroozi, Wechsler & Conroy 2013b). Here we aim at presenting a useful statistic for large-scale structures, that hopefully could be applied to current or future observational galaxy catalogues. We have then selected a sample of semi-analytical galaxies brighter than −20.6 mag in the r band, so that the numerical density of the sample is roughly that of volume-limited samples of galaxies. 2.1 FVS catalogue As stated above, recent superclusters catalogues have been constructed mainly based on the smoothed density field approach (e.g. Einasto et al. 2007; Costa-Duarte, Sodré & Durret 2011; Liivamägi et al. 2012; Einasto et al. 2014). However, these systems are still at the beginning of the virialization process; thus the density threshold applied to delineate the superclusters has a certain degree of freedom. Taking this into account, Luparello et al. (2011) used a criterion motivated by dynamical considerations to calibrate this value and presented a catalogue of so-called future virialized structures. The procedure relies on combining the luminosity density field method (Einasto et al. 2007) with the theoretical criterion for the mass density threshold of Dünner et al. (2006). The method starts with a smooth luminosity density field obtained by convolving the spatial distribution of the galaxies with a kernel function weighted by galaxy luminosity. The effect of the smoothing depends on the shape and size of the kernel function, an Epanechnikov kernel of radius r0 = 8 |$\, h^{-1}{\rm \, Mpc}$|⁠. The resulting density field is then mapped onto a grid with a resolution given by cubes of 1 |$\, h^{-1}{\rm \, Mpc}$| side. Then, a percolation method allows one to select the highest luminosity density groups of cells. These isolated overdensities are the basis of the large structures that will become virialized systems in the future. To this end, the luminosity density threshold adopted (DT = |$\rho _\mathrm{lum} / \bar{\rho }_\mathrm{lum}$| = 5.5) is calibrated, assuming a constant mass-to-luminosity relation and using the minimum mass overdensity necessary for a structure to remain bound in the future (Dünner et al. 2006). A lower limit for the total luminosity of a structure is also applied at Lstructure > 1012 L⊙, avoiding contamination from smaller systems. These criteria provide a suitable compromise of high completeness and low contamination. The described procedure was applied to the MDPL2-SAG catalogue, where we identify 3219 FVSs with volumes between 102 and ≃10|$^{4}(\, h^{-1}{\rm \, Mpc})^{3}$| and luminosities ranging from 1012 to ≃1014 (L⊙h−2). 2.2 Void catalogue We follow the void identification algorithm as described in Ruiz et al. (2015), which is a modified version of the algorithms presented in Padilla, Ceccarelli & Lambas (2005) and Ceccarelli et al. (2006). The method starts with an estimation of the density field performed with a Voronoi tessellation over the density tracers (SAG galaxies in this case). First, underdense regions are obtained by selecting all Voronoi cells below a threshold density. These cells are then used as centre candidates for larger underdense regions. The integrated density contrast in spherical regions around these cells, Δ(r), is then computed at increasing values of radius r. Then, void candidates are selected as the largest of each set of overlapping spheres that satisfy the condition Δ(Rvoid) < −0.9. Since this procedure can lead to spherical voids with edges that do not precisely fit with the surrounding structures, centres are displaced randomly so that the spheres are allowed to grow. The procedure of re-centring void locations provides systems whose borders are in better agreement with the local density field surroundings. We end up with a void catalogue comprising all the largest underdense non-overlapping spheres, with void radii Rvoid equal to those of re-centred spheres. Physical insight into the nature and evolution of void properties can be derived from void surroundings since the hierarchy of voids arises by mass assembly in the growing nearby structure network (Sheth & van de Weygaert 2004; Paranjape, Lam & Sheth 2012). In this scenario, some voids remain as underdense regions, while other voids evolve by collapsing into themselves together with the surrounding dense structures. Both possible behaviours, either void expansion or void collapse, are determined by the global surrounding density: large underdense regions with surrounding overdense shells are likely to be squeezed as the surrounding structures behave as a void-in-cloud system; voids in an environment more similar to the mean background density will expand and remain as underdense regions following the so-called void-in-void mode. Cumulative radial density profiles have proved to be a useful proxy to predict these two evolutionary modes (Ceccarelli et al. 2013; Paz et al. 2013; Ruiz et al. 2015) and can be used to perform a classification of voids: those with a smoothly rising but still underdense integrated radial profile resembling void-in-void systems, dubbed R-type voids; and those embedded in globally overdense regions, consistent with the void-in-cloud mode, defined as S-type (standing for shell-surrounded) voids. The final catalogue comprises 17 975 voids for the full MDPL2-SAG box, with radii in the range 7–30 |$\, h^{-1}{\rm \, Mpc}$|⁠. 3 METHOD Here, we describe a novel tool to characterize the clustering of non-symmetrical large-scale systems. We base our method on the two-point correlation function formalism, modified for application to non-symmetrical extended systems. As a first step, and following the idea of quantifying the different spatial distribution of superstructures from a randomly placed sample of the same systems, we use a reference randomly distributed sample by shuffling the actual position and orientation of the structures, with the restriction of no superposition. Thus, we end up with random catalogues that have the same number of structures and the same volume as the data. We repeat this procedure four times with different random seeds in order to generate different realizations of the random catalogues. We find that the results for the four random realizations are statistically indistinguishable. We apply this procedure for the generation of random catalogues of both cosmic voids and overdense structures in the following sections. In order to quantify the different clustering of the actual distribution and that derived from randomly shuffling the structure positions, we use the fraction of randomly placed spheres of a given diameter (a proxy for the scale of the correlation) that overlap with the structures in both data and randomly placed structures. If there is a clustering signal in the distribution of superstructures, then these fractions would differ in such a way that can be quantified as a function of the scale with statistically significant estimators. The probability for a sphere of diameter d to be in contact with at least k structures, P(≥k, d), can be approximated by the fraction of random spheres of that diameter that are superimposed to k or more than k structures: $$\begin{eqnarray*} P(\ge k, d) = N(\ge k, d) / N(d), \end{eqnarray*}$$(1) where N(d) is the total number of random spheres. This quantity may differ for superstructures in the dataset (in our case, the simulation) and in the set with randomly placed superstructures. Then, we define a structure correlation function Ξ(≥k, d) of the order of k as the excess probability of finding a sphere of diameter d partially or totally overlapping at least k different structures w.r.t. an equivalent arrangement with random structures: $$\begin{eqnarray*} P_D(\ge k, d) = P_R(\ge k ,d) \, (1+\Xi (\ge k,d)). \end{eqnarray*}$$(2) Within this scheme, we define: D(≥k, d) as the number of random spheres that contain parts of at least k different structures, in the simulation box, and R(≥k, d) as the number of random spheres that contain parts of at least k different structures, in the random sample. We notice that when a structure contacts a random sphere in more than one location this intersection is considered a single overlap. Also, we fix the total number of spheres for each diameter bin N(d) to be the same in the simulation box and in the random sample. With these quantities, we estimate the ‘structure auto-correlation function’ of the order of 2 (k ≥ 2) as: $$\begin{eqnarray*} \Xi (\ge 2,d) = \frac{D(\ge 2, d)}{R(\ge 2, d)} -1, \end{eqnarray*}$$(3) and the ‘structure cross-correlation function’ as: $$\begin{eqnarray*} \Xi _\mathrm{VS}(\ge 2,d) = \frac{D_\mathrm{VS}(\ge 2, d)}{R_\mathrm{VS}(\ge 2, d)} -1, \end{eqnarray*}$$(4) where subscripts V and S represent a cosmic void and an overdense structure, respectively. Here we consider spheres in contact with at least one void and at least one FVS. Fig. 1 shows the scheme of our procedure, where the top panels sketch a simulated distribution of non-symmetrical extended objects (left) and its corresponding randomized sample, both in position and orientation (right). Contours of different colours are used to easily recognize the randomized structures. Three different sphere diameters are shown in the scheme and the numbers (+0 and +1) superposed to each of them indicate the contribution to the counts D(≥2, d) or R(≥2, d). It can be seen that spheres in contact with two or more structures contribute with one count and spheres that overlap with fewer than two structures do not contribute to the counts. The lower panel shows qualitatively the three different scales from the upper panels in which the auto-correlation gives positive, zero and negative values respectively. The three scales were arbitrarily chosen to illustrate possible behaviours of the correlation function computed by equation (3). Figure 1. Open in new tabDownload slide Simplified schematic representation of the procedure. Upper panels: shaded regions display a simulated distribution of non-symmetrical extended objects (left) and their corresponding randomized sample, both in position and orientation (right). The same structure in both panels is represented by the same contour colour. Some random spheres are displayed overlapping the structures and the numbers inside the spheres indicate the contribution to D(≥2, d) and R(≥2, d). Lower panel: The three spheres on the right qualitatively show the signs of Ξ(≥2, d) computed by equation (3), also shown to the left of the lower panel. These spheres correspond to the scales outlined in the top panels. Figure 1. Open in new tabDownload slide Simplified schematic representation of the procedure. Upper panels: shaded regions display a simulated distribution of non-symmetrical extended objects (left) and their corresponding randomized sample, both in position and orientation (right). The same structure in both panels is represented by the same contour colour. Some random spheres are displayed overlapping the structures and the numbers inside the spheres indicate the contribution to D(≥2, d) and R(≥2, d). Lower panel: The three spheres on the right qualitatively show the signs of Ξ(≥2, d) computed by equation (3), also shown to the left of the lower panel. These spheres correspond to the scales outlined in the top panels. 3.1 Tests with simple geometrical models In order to address the performance of the Ξ(≥2, d) correlation we have constructed five different models of structures in the same grid of the simulation data. The models follow simple distribution prescriptions and geometrical shapes, so that the behaviour of the correlation function can be assessed. We acknowledge the restriction of |$30 \, h^{-1}{\rm \, Mpc}$| as the minimum distance between the structure centres. This constraint is applied to all models in order to avoid overlapping of structures. We adopted two structure shapes: one is a set of cells consistent with a sphere of radius |$8 \, h^{-1}{\rm \, Mpc}$| (adding up to 2106 |$\, h^{-1}{\rm \, Mpc}^3$|⁠), and the other, a prism with 26, 9, 9 |$\, h^{-1}{\rm \, Mpc}$| a side. The volumes of these structures are approximately the mean volume of the FVSs identified in the MDPL2-SAG simulation. One of the structure models consists of spheres (S-model) and the remaining models of prisms (P-models). The total volume and number of structures (set as 8673 objects) is the same for all five models. The spheres in the S-model are distributed so that they show a clustering signal in the two-point correlation function of their centres, as shown in the upper inset panel of Fig. 2. In the construction of the P-models, we randomly chose a fraction μ of the S-model centres to place a pair of nearby and parallel prisms, and a fraction of 1–2μ to place one isolated prism. The prisms are oriented along a randomly chosen axis. We have adopted P-models with μ values of 0, 0.05, 0.1, and 1/3, and where the pairs of prisms have a constant separation of |$2 \, h^{-1}{\rm \, Mpc}$|⁠. We notice that in the cases where μ ≠ 0 the prism centres have a very different exclusion restriction. We acknowledge that the original exclusion restriction is modified since pairs of prisms are arranged so that their separation is significantly lower than the |$30 \, h^{-1}{\rm \, Mpc}$| of the original prescription. These new distributions of prisms also affect the original two-point correlation function. We stress the fact that in all cases we impose the condition of no overlapping of structures, consistent with the definition of FVSs. The results are shown in Fig. 2. The effect of FVS shapes can be seen by comparing the S-model (red dashed line) and the P-model with μ = 0 (blue dashed line), where the largest difference occurs at small scales due to the effects of the elongated prism shapes. The larger extension of the prism-shaped structures in one of the axes with respect to spherically shaped structures makes it more probable that a random sphere will contact two or more different prisms. For the P-model with μ = 0, the negative correlation values of ΞP(≥2, d) correspond to the imposed exclusion distance between centres. At increasing separations beyond this exclusion scale, the effects of the original standard two-point correlation of the centres can also be distinguished in ΞP(≥2, d) as positive values at large scales. Beyond |${\gtrapprox}100 \, h^{-1}{\rm \, Mpc}$| both the standard and the Ξ(≥2, d) correlations tend to zero. Figure 2. Open in new tabDownload slide Structure auto-correlation function in toy models. The red dashed line corresponds to the S-model auto-correlation ΞS(≥2, d). Solid and dashed cyan lines show the prism auto-correlation ΞP(≥2, d) for different μ values, labelled in the figure. For a better comparison, the lower inset displays an enlarged region of the structure auto-correlation functions. The upper inset shows the two-point correlation function of the centres used to place the structures in the S-model and P-model with μ = 0. Shaded regions correspond to jackknife uncertainties. Figure 2. Open in new tabDownload slide Structure auto-correlation function in toy models. The red dashed line corresponds to the S-model auto-correlation ΞS(≥2, d). Solid and dashed cyan lines show the prism auto-correlation ΞP(≥2, d) for different μ values, labelled in the figure. For a better comparison, the lower inset displays an enlarged region of the structure auto-correlation functions. The upper inset shows the two-point correlation function of the centres used to place the structures in the S-model and P-model with μ = 0. Shaded regions correspond to jackknife uncertainties. The reader can also appreciate the effects of the introduction of a significant clumping at small scales in the different P-models corresponding to μ values of 0.05, 0.1, and 1/3. At small separations, the large positive values of ΞP(≥2, d) correspond to an excess of close pairs of prisms. We notice that even a small fraction of prism pairs, μ = 0.05, gives a positive correlation at small scales, showing negative values at intermediate separations due to the imposed exclusion condition. We notice that at larger scales, the positive correlation of the original distribution of centres provides positive values of ΞP(≥2, d) as well. For higher values of μ (i.e. 0.1, 1/3), this positive signal is absent due to the fact that there are practically no traces of the original spatial distribution. 4 ANALYSIS AND RESULTS Using equation (3), we compute the FVS structure auto-correlation function (ΞFVS(≥2, d)) in the MDPL2-SAG catalogue and show it in Fig. 3(a). To that end, we generate random spheres and estimate the correlation from the fractions of spheres that contain structures. For smaller scales, where the variance is large due to the structure, we use a larger number of spheres. The choice of the distribution of the sphere radii was made so that their total volume is equivalent to 100 times the simulation box volume. As can be seen in the plots, due to the intrinsic dispersion of ΞFVS(≥2, d) at small scales the jackknife uncertainty estimates are significantly higher than at larger scales. Histograms in the inset (panel b) represent the fraction of the number of spheres that are in contact with at least two structures relative to the total number of spheres, as a function of sphere diameter. The solid lines correspond to the simulation box, and the dashed lines to the control sample. As can be seen (panel a), ΞFVS(≥2, d) exhibits a positive correlation at scales less than 70 |$\, h^{-1}{\rm \, Mpc}$|⁠. Beyond this scale, the FVS spatial distribution shows an anti-correlation up to scales of roughly 150 |$\, h^{-1}{\rm \, Mpc}$|⁠, which smoothly approaches zero at large scales, consistent with a random distribution. The strong positive correlation signal at small separations can be understood in terms of the arrangement of FVSs within the large-scale structure network as, for instance, large-scale filaments. Figure 3. Open in new tabDownload slide FVS structure auto-correlation function ΞFVS(≥2, d) in the MDPL2-SAG catalogue. (a) ΞFVS(≥2, d) for MDPL2-SAG. The inset (b) shows the distribution of the number of spheres D(≥2, d) (solid line) and R(≥2, d) (dashed line) involved in the computation of ΞFVS(≥2, d), as exemplified in equation (1). (c) Contribution of the ΞFVS(2, d) (thin line) and the ΞFVS(≥3, d) structure correlation functions (thick line) to the total structure auto-correlation (light-grey line). In both panels, shaded regions correspond to jackknife uncertainties. (d) The number of spheres involved in the determination of the ΞFVS(2, d) and ΞFVS(≥3, d) structure correlations in diameter bins for the simulation box. Figure 3. Open in new tabDownload slide FVS structure auto-correlation function ΞFVS(≥2, d) in the MDPL2-SAG catalogue. (a) ΞFVS(≥2, d) for MDPL2-SAG. The inset (b) shows the distribution of the number of spheres D(≥2, d) (solid line) and R(≥2, d) (dashed line) involved in the computation of ΞFVS(≥2, d), as exemplified in equation (1). (c) Contribution of the ΞFVS(2, d) (thin line) and the ΞFVS(≥3, d) structure correlation functions (thick line) to the total structure auto-correlation (light-grey line). In both panels, shaded regions correspond to jackknife uncertainties. (d) The number of spheres involved in the determination of the ΞFVS(2, d) and ΞFVS(≥3, d) structure correlations in diameter bins for the simulation box. According to our definition, any configuration of two or more structures in contact with a randomly placed sphere makes the same contribution to our correlation measure, regardless of the common volume fraction or of the number of contributing structures. In order to deepen our understanding of the FVS structure auto-correlation function, we have considered separately the number of structures contained in the random spheres to compute ΞFVS(k, d). Fig. 3(c) shows the FVS auto-correlation function considering k = 2 and k ≥ 3 separately. The grey solid line represents the correlation of FVSs computed in Section 3, and the thick solid line measures the contribution of spheres containing only two different FVSs |$(\Xi _\mathrm{FVS}(2, d))$|⁠. Similarly, the thin solid line represents ΞFVS(≥3, d), corresponding to the contribution of three or more FVSs. We acknowledge that the ΞFVS(2, d) and ΞFVS(≥3, d) terms dominate the signal of the full structure correlation at different scales. Although the mean number of structures contained in the spheres scales with diameter, the contribution of pairs in sufficiently large spheres will be negligible since it is increasingly difficult to find regions containing only two structures. As can be seen in Fig. 3(c), at scales beyond ≃100 |$\, h^{-1}{\rm \, Mpc}$| the the main contribution is due to the ΞFVS(≥3, d) term. To further understand these results, we show in Fig. 3(d) the histograms corresponding to the fraction of spheres (with respect to the total number of spheres with the same diameter range) containing k = 2 (thin solid line) and k ≥ 3 (thick solid line) structures where the k ≥ 3 contribution clearly dominates beyond |$100 \, h^{-1}{\rm \, Mpc}$|⁠. 4.1 Subsamples and cosmic variance With the aim of exploring the effects of cosmic variance on the structure correlations we define eight non-overlapping sub-boxes of 500 |$\, h^{-1}{\rm \, Mpc}$| sides of the MDPL2-SAG catalogue. For each sub-box we compute the FVS structure auto-correlation function. The results obtained are shown in the upper panel of Fig. 4. The lower panel shows the difference of the structure correlation of each sub-box with respect to the full box in units of the mean correlation variance of each sub-box. The grey region corresponds to one standard deviation from the full box correlation. As can be noticed, there is one distinct sub-box in which the structure correlation function shows the highest correlation with respect to the other sub-boxes although we notice that its significance is only at the 2σ level. To analyse this feature in further detail, and in particular its possible dependence on differences in characteristics of FVSs across the sub-boxes, we have analysed the distribution function of several intrinsic properties of FVSs in the MDPL2-SAG sub-boxes. The upper panel of Fig. 5 presents the probability density of FVS luminosities (left-hand panel), FVS volumes (middle panel) and FVS multiplicity, i.e. the number of SAG galaxies within the limiting magnitude (right-hand panel) in the MDPL2-SAG sub-boxes. These FVS properties are remarkably similar for the different sub-boxes, indicating that the observed clustering differences cannot be attributed to differences in FVS intrinsic characteristics. We have also studied the luminosity distribution of the galaxies within each sub-box and the results are in complete agreement among the sub-boxes. Besides these analysis, we have also analysed whether differences in the structure correlation function could be directly related to variance in the galaxy auto-correlation function. Hence, we have analysed the standard two-point galaxy–galaxy correlation function in the different sub-boxes. The lower panel in Fig. 5 shows the standard two-point correlation function of galaxies for the eight MDPL2-SAG sub-boxes. It can be seen that all sub-boxes show very similar correlation functions at scale <40 |$\, h^{-1}{\rm \, Mpc}$| although we notice that beyond this scale the same box exhibiting the higher FVS structure correlation also departs from the trend of the remaining sub-boxes. This can be seen in the inset of the lower panel of Fig. 5, where the above-mentioned sub-box is represented by a dashed line. However, we stress the fact that, in both FVS structure and standard correlation functions, all determinations are within cosmic variance uncertainties. We have also checked if the superstructure correlation excess of the sub-box with the highest correlation is also present in both ΞFVS(2, d) and ΞFVS(≥3, d), finding a consistent excess in both cases. Thus, to conclude from this analysis, we find that the larger excess of superstructure correlation of this sub-box relies on very large-scale correlations rather than on intrinsic FVS properties. Figure 4. Open in new tabDownload slide FVS structure auto-correlation function for MDPL2-SAG sub-boxes. The dashed line corresponds to the box with the particular behaviour mentioned in Section 4.1. The lower panel shows the difference of the structure correlation of each sub-box with respect to the full box in units of the mean correlation variance of each sub-box. The shaded region represents one standard deviation from the full box correlation. The box highlighted in the upper panel is shown with a dashed line. Figure 4. Open in new tabDownload slide FVS structure auto-correlation function for MDPL2-SAG sub-boxes. The dashed line corresponds to the box with the particular behaviour mentioned in Section 4.1. The lower panel shows the difference of the structure correlation of each sub-box with respect to the full box in units of the mean correlation variance of each sub-box. The shaded region represents one standard deviation from the full box correlation. The box highlighted in the upper panel is shown with a dashed line. Figure 5. Open in new tabDownload slide Top: Probability density of FVS luminosity (left), volume (middle) and galaxy number density (right) distributions for FVSs in MDPL2-SAG sub-boxes. Bottom: Two-point correlation function of galaxies for MDPL2-SAG sub-boxes. The inset shows the two-point correlation function beyond 40 |$\, h^{-1}{\rm \, Mpc}$| in linear scales. The dashed line corresponds to the sub-box mentioned in Section 4.1 and the grey region shows its jackknife uncertainty. Figure 5. Open in new tabDownload slide Top: Probability density of FVS luminosity (left), volume (middle) and galaxy number density (right) distributions for FVSs in MDPL2-SAG sub-boxes. Bottom: Two-point correlation function of galaxies for MDPL2-SAG sub-boxes. The inset shows the two-point correlation function beyond 40 |$\, h^{-1}{\rm \, Mpc}$| in linear scales. The dashed line corresponds to the sub-box mentioned in Section 4.1 and the grey region shows its jackknife uncertainty. To further explore the possible origin of the structure correlation excess in the sub-box with the highest correlation, we imposed a lower threshold density for FVS identification (see Section 2) and studied the resulting FVS properties in each MDPL2-SAG sub-box. This analysis shows that, particularly in this sub-box, the currently identified FVSs were part of a larger superstructure. We argue that this fact can explain the actual clustering excess with respect to the other boxes. Thus, this test shows that the FVS structure auto-correlation function contains information on the properties of the underlying density field and large-scale structure. 4.2 FVS–void structure cross-correlation In previous sections we have studied the correlated positions of superstructures by means of an appropriate statistics. Since the cosmic web comprises underdense regions as well as superstructures, it is useful to analyse their joint distribution. In this subsection we compute the structure cross-correlation function between FVSs and voids, ΞSV(≥2, d), by applying equation (4) for FVSs (S) and voids (V). We consider the spheres that are in contact with at least one void and at least one FVS. The left-hand panel of Fig. 6 shows the FVS–void structure cross-correlation function for the total MDPL2-SAG catalogue. The upper line corresponds to ΞSV(≥2, d) between FVSs and S-type voids, and the bottom line to ΞSV(≥2, d) between FVSs and R-type voids. We can distinguish two opposite behaviours: S-type voids are correlated with FVSs while R-type voids tend to avoid FVS proximity, consistently with previous studies (Lares et al. 2017). As stated in Lares et al. (2017), this is somewhat related to the selection of voids according to their environment. Moreover, they found that this behaviour also has dynamical implications when considered in the context of the large-scale structure, manifested in the infall of voids that are close to FVSs independently of void type. Given the results obtained for the cosmic variance of the structure auto-correlation function of FVSs ΞFVS(≥2, d) from Section 4.1, we have also explored ΞSV(≥2, d). The results are shown in the right-hand panel of Fig. 6, where it can be seen that the sub-box with the highest auto-correlation also shows a slight excess of the structure cross-correlation amplitude between FVSs and S-type voids. However, for all sub-boxes the structure cross-correlations are within cosmic variance uncertainties. Thus, we envisage that cross-correlations between superstructures and voids may also provide useful statistical characterizations of the spatial distribution in the next generation of large-scale surveys. Figure 6. Open in new tabDownload slide FVS–void structure cross-correlation for the total MDPL2-SAG catalogue. The left-hand panel shows FVS–void structure cross-correlations for FVSs and S- and R-type void samples. The right-hand panels (b and c) show the same correlation as in (a) but for the eight simulation sub-boxes in order to visualize the cosmic variance. Shaded regions in both panels correspond to jackknife uncertainties within each sub-box. Figure 6. Open in new tabDownload slide FVS–void structure cross-correlation for the total MDPL2-SAG catalogue. The left-hand panel shows FVS–void structure cross-correlations for FVSs and S- and R-type void samples. The right-hand panels (b and c) show the same correlation as in (a) but for the eight simulation sub-boxes in order to visualize the cosmic variance. Shaded regions in both panels correspond to jackknife uncertainties within each sub-box. 5 CONCLUSIONS In this paper we have addressed the computation of spatial correlations of large-scale cosmic structures with arbitrary shape. These structures are determinant in the distribution of galaxies at large scales and, being highly asymmetrical, standard N-point correlation function methods are not suitable for statistical studies. Our method is based on the probability excess Ξ(k, d) of finding spheres of diameter d overlapping with cosmic structures of a given species with respect to a randomized distribution of the same structures. We have applied this procedure to superstructures and voids identified in the MDPL2-SAG catalogue and we have tested its performance in deriving useful characterizations of the large-scale distribution properties. We have analysed cosmic variance effects in subsets of the MDPL2-SAG catalogue on Ξ(≥2, d) and on the standard galaxy correlation function statistics. We find that both Ξ(≥2, d) and the standard correlation function are suitable to detect departures from the correlation of the full box due to cosmic variance. Thus, the structure correlation can provide a useful complementary characterization of the large-scale mass distribution in future large galaxy surveys. We stress the fact that, even when the superstructure properties do not strongly vary among the different sub-boxes, the structure correlation function and the standard correlation function of haloes are still sensitive to the cosmic variance. Regarding the FVS–void structure cross-correlation, our results are consistent with those presented in Lares et al. (2017), where FVSs and R-type (S-type) voids have negative (positive) correlations. This provides further support to the ability of our methods to produce useful characterizations of the large-scale structure. We argue that, since FVSs emerge as features of the cosmic web such as knots in the filamentary structure of the supercluster–void network, their formation and evolution are intertwined, thus allowing the use of superstructures as an alternative proxy for the statistical description of the large-scale network of the distribution of galaxies. ACKNOWLEDGEMENTS This work was partially supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and the Secretaría de Ciencia y Tecnología, Universidad Nacional de Córdoba, Argentina. Plots were made using python software and postprocessed with inkscape. This research has made use of NASA’s Astrophysics Data System. The CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (https://www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in Europe (PRACE, https://www.prace-ri.eu) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, https://www.lrz.de). Footnotes 1 " doi:10.17876/cosmosim/mdpl2/007 REFERENCES Abazajian K. N. et al. ., 2009 , ApJS , 182 , 543 10.1088/0067-0049/182/2/543 Crossref Search ADS Crossref Araya-Melo P. A. , Reisenegger A., Meza A., van de Weygaert R., Dünner R. o., Quintana H., 2009 , MNRAS , 399 , 97 10.1111/j.1365-2966.2009.15292.x Crossref Search ADS Crossref Baugh C. M. et al. ., 2004 , MNRAS , 351 , L44 10.1111/j.1365-2966.2004.07962.x Crossref Search ADS Crossref Behroozi P. 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TI - Spatial correlations of extended cosmological structures
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/staa732
DA - 2020-05-21
UR - https://www.deepdyve.com/lp/oxford-university-press/spatial-correlations-of-extended-cosmological-structures-7ATCAV01PQ
SP - 3227
VL - 494
IS - 3
DP - DeepDyve
ER -