Filaments of galaxies and Voronoi diagrams

The intersections between a spherical shell and the faces of Voronoi's polyhedrons are numerically evaluated. The nodes of these intersections are the points that share the same distances from three nuclei. The nodes are assumed to be the places where we will find clusters of galaxies. The cosmological environment is given by the coupling between the number of galaxies as function of the redshift and the LCDM cosmology.


Introduction
The first catalog of clusters of galaxies contained 2712 clusters Abell (1958). The updated version brought the number of rich clusters to 4073, each having at least 30 members Abell et al. (1989). The second catalog of galaxies contained 29418 galaxies and 9134 clusters and was organized in six books Zwicky et al. (1961Zwicky et al. ( , 1963Zwicky et al. ( , 1966aZwicky et al. ( ,b, 1965; Zwicky & Kowal (1968). The online version of the Zwicky catalog of galaxies contains 19369 galaxies Falco et al. (1999). Another line of research organized the observations of galaxies in slices; the first being the second CFA2 redshift Survey Huchra et al. (1999). Other catalogs made by slices or which can be organized in slices are: the 2dfGRS Norberg et al. (2002); the 6dF Galaxy Survey 6dFGS Jones et al. (2004)); and the SDSS DR12 with 208,478,448 galaxies Alam et al. (2015). All these slices present filaments of galaxies, which points toward a cellular structure for the 3D spatial distribution of galaxies. A model for the cellular structure of the local universe can be represented in Voronoi Diagrams Icke & van de Weygaert (1987);van de Weygaert & Icke (1989). In this model the galaxies can be inserted on the faces of irregular polyhedrons Zaninetti (1991aZaninetti ( , 1995. Recently, the analysis of filaments of galaxies with the Cosmic Web Reconstruction (CWR) has made enormous progress Chen et al. (2015aChen et al. ( ,b, 2016Chen et al. ( , 2017.
The rest of this paper is structured as follows. Section 2 introduces the adopted cosmological framework. Section 3 models the intersection between a sphere and the network of the Voronoi's faces. Finally, Section 4 reports on the theoretical filaments and clusters of galaxies at low and high values of redshift.

Adopted Cosmology
This section introduces two catalogs of galaxies (the ΛCDM and the pseudo-Euclidean cosmology), the statistics of the cosmic voids and the standard luminosity function (LF) for galaxies.

Luminosity distance
We review the existing knowledge on the luminosity distance, ( ; 0 , , Ω M , Ω Λ ), which in the ΛCDM cosmology can be expressed in terms of a Padé approximant. We should provide: the Hubble constant, 0 , as expressed in km s −1 Mpc −1 ; the velocity of light, , as expressed in km s −1 ; and the three numbers Ω M , Ω , and Ω Λ , (see Zaninetti (2016a) for more details). A numerical analysis of the distance modulus for the Union 2.1 compilation, see Suzuki et al. (2012), gives 0 = 69.81km s −1 Mpc −1 , Ω M = 0.239 and Ω Λ = 0.651. We now apply the minimax rational approximation, which is characterized by two parameters and , and we find a simplified expression for the luminosity distance, ,2,1 , when = 2 and = 1 The most simple model for the distance, , in the local universe is that of the pseudo-Euclidean cosmology: where we used 0 = 67.93km s −1 Mpc −1 , see Zaninetti (2016b). The differences between the two distances are the luminosity distance and the and the pseudo-Euclidean distance, which can be outlined in terms of a percentage difference: Δ. For example, for , Figure 1 reports the two distances. For ≤ 0.2, the percentage difference is lower than 10%. Therefore the boundary between low and high can be fixed at = 0.2.

Cosmic voids
A first catalog of cosmic voids can be found in Pan et al. (2012), where the effective radius of the voids, , has been derived The second catalog is that of radii up to redshift 0.12 ℎ −1 in (SDSS-DR7), see Varela et al. (2012), The third catalog is that of the Baryon Oscillation Spectroscopic Survey, see Mao et al. (2017) In the following, we will calibrate our code on the average value of the three previous values: = 29.2 ℎ −1 . Fig. 1: The distances that are adopted are: the luminosity distance, , in ΛCDM (full line); and the pseudo-Euclidean cosmology distance (dashed line).

The LF for galaxies
We now review the actual knowledge on the Schechter function, see Schechter (1976) which provides a useful standard for the LF of galaxies where sets the slope for low values of luminosity, , * is the characteristic luminosity and Φ * is the normalisation. The equivalent distribution in absolute magnitude is where * is the characteristic magnitude as derived from the data. The scaling with ℎ is * − 5 log 10 ℎ and Φ * ℎ 3 [ −3 ]. According to formula (1.104) in Padmanabhan (1996) or formula (1.117) in Padmanabhan (2002) , the joint distribution in redshift, z, and flux, f, for galaxies in the pseudo-Euclidean cosmology, is where Ω, and represent the differential of the solid angle, the redshift and the flux, respectively, and Φ is the Schechter LF. The critical value of , , is The number of galaxies in and as given by formula (10) has a maximum at = − , where which can be re-expressed as where ⊙ is the reference magnitude of the sun at the considered bandpass. On replacing the flux with the apparent magnitude Figure 2 reports the number of observed galaxies of the 2MRS catalog for a given apparent magnitude and for the theoretical curve. Table 1 reports the parameters that were adopted in this model. 1346734 ⊙ 2 are organized in frequencies versus heliocentric redshift, (empty circles); and, the error bar is given by the square root of the frequency. The maximum frequency of observed galaxies is at = 0.017. The full line is the theoretical curve generated by Ω ( ), see equation (10), and the maximum is at = 0.0133. The parameters are given in Table 1 The number of galaxies, ( , , ) comprised between a minimum value of flux, , and maximum value of flux , can be computed through the following integral .
The indefinite integral exists in terms of the Whittaker function , ( ), see Abramowitz & Stegun (1965) The number of all of the galaxies for 2MRS as function of the redshift is visible in Figure 3. More details can be found in Zaninetti (2014).

Voronoi Diagrams
We now review the existing knowledge on the Voronoi diagrams. The faces of the Voronoi Polyhedra share the same property; i.e., they are equally distant from two nuclei or seeds. The intersection between a plane and the faces produces diagrams that are similar to the edges displacement in 2D Voronoi diagrams. From the point of view of the observations, it is very useful to study the intersection between a slice which crosses the center of the box and the faces of irregular polyhedrons where the galaxies presumably reside. According to the nomenclature reported in Okabe et al. (2000), this cut is classified as (2, 3). The parameters that will be used in the following are the kind of nuclei, which can be Poissonian or not Poissonian, the number of nuclei, the of the box in , and the number of , for example 1400, that are used to build the diagrams, see Zaninetti (1991b). The parameters adopted in the simulation of the pseudo-Euclidean cosmology are reported in Table2.
According to the nomenclature reported in Okabe et al. (2000), this cut is classified as (2, 3) and Figure 4 reports a typical example. This Figure also reports the spherical nodes that, in the absence of an official definition, can be defined as the locus of intersection between the lines of (2, 3). The spherical nodes are equally distant from three or four nuclei. The cross-sectional area of Fig. 3: All the galaxies for 2MRS are organised in frequencies versus redshift, (empty circles); the error bar is given by the square root of the frequency. The maximum frequency of the observed galaxies is at = 0.017. The full line is the theoretical curve generated by the integral equation (16), which has maximum is at = 0.0237. The parameters are given in Table 1. (2, 3); cut on the X-Y plane, black points, the parameters are given in Table 2. The spherical nodes are marked by great green points.   Table 2. the VP can also be visualised through a spherical cut that is characterised by a constant value of the distance to the center of the box, which in this case is expressed in units. This intersection is not present in the Voronoi literature and therefore can be classified as a "new" topic. It may be called , (2, 3), where the index , stand for Poissonian and sphere respectively, see Figure 5. More details can be found in Zaninetti (2006).

Astrophysical Applications
This section reports the theoretical spatial display for clusters at low redshift, simulation of 2MRS, and at high redshift, simulation of redMaPPer.  Table 2.

The 2MRS catalog
The photometric maximum in the number of galaxies as a function of , see formula (16), has a maximum as a function of the redshift. Figure 6 reports an Hammer-Aitoff projection of the galaxies of 2MRS, as well the same number of , (2, 3). Figure 7 reports a cut of a given thickness, Δ, of 2MRS and a number of (2, 3) chosen to scale as the number of galaxies.

The redMaPPer catalog
The number of clusters of the redMaPPer catalog as a function of the redshift has a maximum at ≈ 0.346, see Figure 8. Figure 9 reports the galaxies and clusters of the redMaPPer catalog. Figure 10 reports the same number of , (2, 3) and the spherical nodes with the same parameters as in Table2.

Conclusions
Adopted Cosmology The local universe can be considered Euclidean up to ≈ 0.2, see Figure 1. At higher values of redshift, we need a a luminosity distance Fig. 7: A cut of the 3D spatial distribution of 2MRS in the = 0 plane when Δ = 4 Mpc, the squared box has a side of 614 : we have 1384 galaxies (green filled circles). The same number of (2, 3) (red squares) with radial scaling as the number of real galaxies with the same parameters as in Table 2.   9: Cut of the 3D spatial distribution of redMaPPer catalog at 0.1 < < 0.11: galaxies (black points) and clusters (red circles).
Clusters of galaxies The geometrical nature for the clusters of galaxies is introduced in the framework of the intersection generated by a sphere with the faces of the Voronoi diagrams. At the same time, the exact number of galaxies and clusters as function of the redshift is regulated by photometric rules; i.e., a maximum in the number of galaxies, see Figure 2 for the 2MRS catalog. This maximum is also visible for the number of clusters as a function of the redshift, see Figure 8 for the redMaPPer catalog. A theoretical result for the spatial distribution of galaxies and clusters for the redMaPPer catalog is reported in Figure 10.