Correlation functions in 2D and 3D as descriptors of the cosmic web
aa r X i v : . [ a s t r o - ph . C O ] A p r Astronomy & Astrophysicsmanuscript no. corr2D c (cid:13)
ESO 2020April 8, 2020
On spatial and projected correlation functions
J. Einasto , , , G. Hütsi , and M. Einasto Tartu Observatory, 61602 Tõravere, Estonia Estonian Academy of Sciences, 10130 Tallinn, Estonia ICRANet, Piazza della Repubblica 10, 65122 Pescara, Italy National Institute of Chemical Physics and Biophysics, Tallinn 10143, EstoniaReceived ; accepted
ABSTRACT
Aims.
Our goal is to find the relation between the two-point correlation functions of the projected and spatial density fields of galaxies,and their influence to biasing, fractal and other geometrical properties of the cosmic web.
Methods.
Using spatial (3D) and projected (2D) density fields we calculate 3D and 2D correlation functions of galaxies, ξ ( r ), structurefunctions, g ( r ) = + ξ ( r ), and fractal functions, γ ( r ) = d log g ( r ) / d log r , for a biased Λ cold dark matter (CDM) simulation. Weanalyse how these functions describe biasing, fractal and other properties of the cosmic web. We compare the correlation functions ofspatial and projected density fields as descriptors of the cosmic web. Results.
Dominant elements of the cosmic web are clusters and filaments, separated by voids filling most of the volume. In individual2D sheets the positions of clusters and filament do not coincide. As a result, in projection clusters and filaments fill in 2D voids.This leads to the decrease of amplitudes of correlation functions (and power spectra) in projection. For this reason amplitudes of 2Dcorrelation functions are lower than amplitudes of 3D correlation functions, the di ff erence is the larger, the thicker are 2D samples. Conclusions.
Spatial correlation functions of galaxies contain valuable information on geometrical properties of the cosmic web, notavailable in angular correlation functions. 2D correlation functions do not contain information on voids in 3D density field, thus 3Dcorrelation functions cannot be calculated from 2D correlation functions.
Key words.
Cosmology: large-scale structure of Universe; Cosmology: dark matter; Cosmology: theory; Galaxies: clusters; Meth-ods: numerical
1. Introduction
The angular distribution of galaxies in the sky is close to a ran-dom one. Most galaxies belong to the field population, a frac-tion of galaxies is concentrated to clusters and superclusters.Until 1970’s the largest collection of data on the distributionof galaxies on sky was provided by the Lick catalog, reducedby Seldner et al. (1977) and Soneira & Peebles (1978). In late1970’s more redshift data were available, which allowed to studythe spatial distribution of galaxies. New data suggested that thespatial distribution of galaxies is more complex with galaxy andcluster chains and filaments surrounding large underdense re-gions — voids (Chincarini & Rood 1976; Gregory & Thompson1978; Jõeveer et al. 1978; Tully & Fisher 1978). Jõeveer et al.(1977) called this phenomenon as “cell structure”, presently as-tronomy community is using the term “cosmic web”, suggestedby Bond et al. (1996).The most commonly used statistic to study the general struc-ture of the cosmic web is the two-point correlation functionof galaxies. Early data allowed to find the angular correlationfunction directly from observational data. The spatial correla-tion function of galaxies can be calculated by numerical inver-sion of the particular integral equation. They are functions ofangular or spatial galaxy pair separations (distances) and de-scribe the excess probability of finding two galaxies separatedby this distance. Most analyses of the correlation function weremade using the assumption that the present density field is Gaus-sian and that power spectrum (and correlation function) charac-
Send o ff print requests to : J. Einasto, e-mail: [email protected] terises the density field completely and contains all the informa-tion needed to answer any statistical question about the densityfield (Tegmark et al. 1998). Actually the distribution of galax-ies in the cosmic web is more complex. Thus the presence of thecosmic web rises the question: How accurate are spatial and pro-jected correlation functions and transformations between them?The goal of this paper is to study the relationship between an-gular and spatial correlation functions. Observational data on po-sitions of galaxies are distorted by several e ff ects – random mo-tions of galaxies in clusters and the flow of galaxies and clusterstoward attractors (Kaiser 1987). To avoid complications causedby these e ff ects we shall study the relationship between pro-jected and spatial correlation functions using a simulated darkmatter (DM) model. According to presently accepted cosmolog-ical paradigm the growth of density fluctuations starts from tinyrandom perturbation at the very early stage of the evolution. Dur-ing evolution the phases of perturbations of di ff erent scales arepartly synchronised which leads to the formation of the filamen-tary cosmic web. The evolution and the present structure of theuniverse are well described by a Λ dominated cold dark matter( Λ CDM) model. The use of a dynamical model rather than an ac-tual observational data has one more advantage — in model weknow very well the distribution of all matter, including invisibleDM, which allows to determine the role of DM in quantitativedescription of the web.To find the relationship between projected and spatial corre-lation functions, as well as between matter and simulated galax-ies, we shall use a Λ CDM model. This model was calculated ina box of size 512 h − Mpc. We take advantage of the fact that
Article number, page 1 of 10 & Aproofs: manuscript no. corr2D for this model positions of all particles are known. This modelwas calculated using he following cosmological parameters: re-duced Hubble parameter h = .
73, matter density parameter Ω m = .
28, dark energy density parameter Ω Λ = .
72, ampli-tude of the linear power spectrum σ = . Λ CDM modelwe calculate spatial and projected correlation functions of sim-ulated galaxies, ξ ( r ), structure functions, g ( r ) = + ξ ( r ), fractalfunctions, γ ( r ) = d log g ( r ) / d log r , and fractal dimension func-tions, D ( r ) = + γ ( r ). The comparison of these functions forvarious sets of data is the practical topic of the paper. Correla-tion function and its derivates allow to study also the biasing phe-nomenon and the fractal character of the distribution of galaxies.The paper is organised as follows. In the next section we de-scribe our simulation data, the methods to calculate correlationfunctions and their derivatives. In section 3, we compare spatialand projected correlation functions and their dependence on in-put parameters, on the biasing level of model samples, and onthe thickness of projected shells of the two-dimensional (2D)density field. In section 4 we compare properties of spatial andprojected density fields and the influence of di ff erences betweenthese fields to correlation functions. We also compare our resultswith others, and analyse properties of correlation functions bydetermining the bias parameters of model and real samples. Thelast section brings our conclusions.
2. Data and methods
In this section we describe the Λ CDM model and methods tocalculate 2D and 3D correlation functions and their derivates,structure functions and fractal dimension functions. We also de-scribe how to calculate bias parameters for 2D and 3D models.
Simulations of the evolution of the cosmic web were performedin a box of size L = h − Mpc, with resolution N grid =
512 and with N part = N particles. The initial density fluc-tuation spectrum was generated using the COSMICS code byBertschinger (1995), assuming concordance Λ CDM cosmolog-ical parameters (Bahcall et al. 1999): Ω m = . Ω Λ = . σ = .
84, and the dimensionless Hubble constant h = . Ω b = . ρ , using posi-tions of 27 nearest particles, including the particle itself. Den-sities were expressed in units of the mean density of the wholesimulation. In the study presented here we used particle-density-selected samples at the present epoch. Biased model samplescontain particles above a certain limit, ρ ≥ ρ , in units of themean density of the simulation. For the analysis we used particledensity limits as given in Table 1. Particle-density-selected sam-ples are referred to as biased model samples and are denoted asLCDM. i , where i denotes the particle-density limit ρ . The fullDM model includes all particles and corresponds to the particle-density limit ρ =
0, and therefore it is denoted as LCDM.00.The main data on biased model samples are given in Table 1. We
Table 1.
LCDM particle-density-limited 3D models.
Sample ρ b ( ρ ) r ξ (1) (2) (3) (4) (5)LCDM.00 0 1.000 4.86 0.729LCDM.01 1 1.285 6.75 1.203LCDM.02 2 1.447 7.83 1.530LCDM.05 5 1.677 9.21 2.061LCDM.10 10 1.849 10.14 2.507LCDM.20 20 2.031 11.20 3.021LCDM.50 50 2.342 12.56 3.725LCDM.100 100 2.626 14.9 5.038 Notes.
The columns are: (1) sample name; (2) particle-density limit ρ ; (3) bias parameter, calculated from 3D correlation functions of bi-ased models with particle-density limits ρ ; (4) correlation length r in h − Mpc; (5) correlation function amplitude ξ at r = . h − Mpc. also give the correlation length, r , and the amplitude of the cor-relation function at r = h − Mpc, ξ = ξ (6), found from spatialanalysis of the correlation function. To find correlation functions of LCDM samples conventionalmethod cannot be used, since the number of particles is too large.To find correlation functions of LCDM samples we used in thispaper the Szapudi et al. (2005) method. This method applies FastFourier Transform ( FFT) to calculate correlation functions andscales as O ( N log N ). As input the method uses density fields ongrids 1024 , 2048 , and 3072 of the L = h − Mpc model.Coordinates of all particles are known, thus it is easy to find den-sity fields with higher resolution. The GADGET-2 code allowsto follow internal structure of halos in general terms, but not thestructure of subhalos inside halos of characteristic scale of a few h − Mpc. Correlation functions were calculated up to the parti-cle separation L max = h − Mpc, and with 200, 400 and 600linear bins for 1024, 2048 and 3072 grids, respectively.In the following analysis we use the spatial correlation func-tion, ξ ( r ), and the pair correlation or structure function, g ( r ) = + ξ ( r ), to characterise the distribution of galaxies in space, fordetails see Martínez & Saar (2002). As usual, we use the separa-tion r , where the correlation function has unit value, ξ ( r ) = . r = r . Instead of theslope of the correlation function we calculate the log-log gradi-ent of the pair correlation function as a function of r , γ ( r ) = d log g ( r ) / d log r , (1)The γ ( r ) function characterises the e ff ective fractal dimension D ( r ) of samples at mean separation of galaxies equal to r (Martínez & Saar 2002), D ( r ) = + γ ( r ) . (2)We call the D ( r ) = + γ ( r ) function as the fractal dimensionfunction. The e ff ective fractal dimension of a random distribu-tion of galaxies is D =
3, and respectively γ =
0; in sheets D = γ = −
1; in a filamentary distribution D = γ = −
2; within clusters D = γ = − g ( r ) we use a linearfit in the distance interval plus and minus m steps from the par-ticular r value of g ( r ), presented as a table. The fit is found usingthe fit subroutine by Press et al. (1992), which gives the slopeand its error. This method cannot be applied to the first and last Article number, page 2 of 10inasto et al.: Projected correlation functions ξ LCDM.00LCDM.01LCDM.02LCDM.05LCDM.10LCDM.20LCDM.50LCDM.100 + ξ LCDM.00LCDM.01LCDM.02LCDM.05LCDM.10LCDM.20LCDM.50LCDM.100 D LCDM.00LCDM.01LCDM.02LCDM.05LCDM.10LCDM.20LCDM.50LCDM.100 ξ LCDM.00.0008LCDM.01.0008LCDM.02.0008LCDM.05.0008LCDM.10.0008LCDM.20.0008LCDM.50.0008LCDM.100.0008 + ξ LCDM.00.0008LCDM.01.0008LCDM.02.0008LCDM.05.0008LCDM.10.0008LCDM.20.0008LCDM.50.0008LCDM.100.0008 D LCDM.00.0008LCDM.01.0008LCDM.02.0008LCDM.05.0008LCDM.10.0008LCDM.20.0008LCDM.50.0008LCDM.100.0008 ξ LCDM.00.0001LCDM.01.0001LCDM.02.0001LCDM.05.0001LCDM.10.0001LCDM.20.0001LCDM.50.0001LCDM.100.0001 + ξ LCDM.00.0001LCDM.01.0001LCDM.02.0001LCDM.05.0001LCDM.10.0001LCDM.20.0001LCDM.50.0001LCDM.100.0001 D LCDM.00.0001LCDM.01.0001LCDM.02.0001LCDM.05.0001LCDM.10.0001LCDM.20.0001LCDM.50.0001LCDM.100.0001
Fig. 1.
Left panels:
Correlation functions, ξ ( r ), for LCDM models; central panels: pair correlation or structure functions, g ( r ) = + ξ ( r ); rightpanels: fractal dimension functions, D ( r ) = + γ ( r ). Upper panels are for 3D functions, middle panels are for mean 2D functions of sheets ofthickness T = h − Mpc, lower panels are for 2D function of the sheet of thickness T = h − Mpc. m steps of the table. For these r values the slope was calculatedusing just the previous and next value of g ( r ) in the Table.Correlation functions ξ ( r ), structure functions g ( r ) = + ξ ( r ),and fractal dimension functions D ( r ) = + γ ( r ) for 3D modelsare shown in upper panels of Fig. 1, for a set of biasing levels,expressed in particle density limits ρ , as given in Table 1. In the present work we use model data given in rectangularspatial coordinates. Projected correlation functions can be cal-culated using 2D density fields. First we calculated 2D densityfields on grid 2048 by integrating 3D fields, D ( x , y , z ), on grid2048 in z -direction: D ( x , y ) = Z z z D ( x , y , z ) dz . (3)The integration was made in n sequentially located sheets ofthe whole sample of size L = h − Mpc. The thickness ofsheets is T = L / n , with n = , , , . . . n = z -direction of thick-ness, T = L = h − Mpc, n = / = h − Mpc, and n = T = / = . h − Mpc. For each n we calculated corre-lation functions for all n sheets, and the mean correlation func-tion and its error was found for all sets with di ff erent n . In cal-culations for given n we used the mean density of the field forgiven particle density limit ρ and sheet thickness. Correlation functions were found using L max = h − Mpc for 87 logarith-mic distance bins. We label mean 2D sheets as LCDM. i , n , where i = ρ is the particle density limit used in selection of particlesto biased samples, and n is the number of sheets in z -direction,used to select particles for 2D samples.2D correlation functions ξ ( r ), structure functions g ( r ) = + ξ ( r ), and fractal dimension functions D ( r ) = + γ ( r ) areshown in Fig. 1: in middle panels for the mean of n = T = h − Mpc, and in lower panels for the one sheetof thickness T = h − Mpc.
The full LCDM model includes all particles, biased models in-clude particles with local density ρ values above a given thresh-old, ρ ≥ ρ . The model with all particles and with density thresh-old ρ = ρ ≥ ≤ r ≤ h − Mpc a plateau, sim-ilar to the plateau of power spectra around the wavenumber k ≈ . h Mpc − (Einasto et al. 2019). The present analysisshows that correlation functions of 2D samples have a similarplateau at the same location, r ≈ h − Mpc. We used the value of
Article number, page 3 of 10 & Aproofs: manuscript no. corr2D
Table 2.
Amplitudes of 2D correlators of LCDM models.
Sample ρ LCDM.01 1 0.0461 0.0889 0.1784 0.326 0.541 0.792 1.017 1.185LCDM.02 2 0.0579 0.1117 0.224 0.412 0.683 1.000 1.289 1.505LCDM.05 5 0.0764 0.1477 0.297 0.549 0.910 1.336 1.730 2.017LCDM.10 10 0.0917 0.1781 0.358 0.667 1.105 1.620 2.105 2.430LCDM.20 20 0.1092 0.212 0.430 0.810 1.337 1.961 2.558 2.933LCDM.50 50 0.1356 0.275 0.574 1.106 1.803 2.644 3.467 3.811LCDM.100 100 0.1652 0.354 0.745 1.456 2.333 3.430 4.493 3.939
Notes.
Table columns starting from the left: sample name, particle-density limit ρ , 2D correlation function amplitudes, ξ = ξ (6) for sampleparameter n = , , , , , , , n , the next row for sheet thickness T = / n h − Mpc.
Table 3.
Bias parameters of 2D LCDM models.
Sample ρ Notes.
The columns are from the left: sample name, particle-density limit ρ , bias parameters for n = , , , , , , , n , the next row for sheet thickness T = / n h − Mpc. the correlation function at r = h − Mpc to calculate amplitudesof correlation functions, ξ = ξ (6), and to find bias parametersof 2D LCDM models, defined as follows: b ( T , ρ ) = p ξ D ( T , ρ ) /ξ D (0) . (4)Here ξ D ( T , ρ ) is the value of 2D correlation function for thick-ness T at r = h − Mpc, ξ = ξ (6), and particle selection limit ρ . Table 2 gives amplitudes of 2D correlation functions for ourset of biased models with varying ρ , and a set of 2D modelthicknesses T . In Eq. (4) ξ D ( ρ =
0) is the value of 3D correla-tion function of the full DM model LCDM.00 at r = h − Mpc.In this way 2D correlation functions define the bias parameter,reduced to the density field of all matter. 2D bias parameters aregiven in Table 3. Correlation amplitudes and bias parameters de-pend on two parameters, the thickness of sheets, T = / n ,and the particle density limit of biased LCDM samples, ρ . Am-plitudes of 2D correlation functions for full unbiased samplesLCDM.00 are printed in italics, the amplitude of the 3D correla-tion function of the model LCDM.00 in boldface.
3. Comparison of spatial and projected correlationfunctions
In this section we compare spatial correlation functions and theirderivates with respective projected functions. We focus the anal-ysis to study the influence of the thickness of 2D sheets to the be-haviour of correlation functions and their derivates. These prop-erties describe the multifractal nature of the cosmic web.
LCDM model samples are based on all particles of the simula-tion and contain detailed information on the distribution of mat-ter in regions of di ff erent density. Top panels of Fig. 1 show cor-relation functions, ξ ( r ), structure functions, g ( r ) = + ξ ( r ), andfractal dimension functions, D ( r ) = + γ ( r ) for 3D samples.Middle panels give the same functions for the 2D samples ofthickness T = / n = h − Mpc, and lower panels for the 2Dsamples of thickness 512 h − Mpc. Amplitudes ξ of correlationfunctions are given in Table 1 for the 3D model, and in Table 2for 2D models.The first impression from the Figure and Tables is that am-plitudes of correlation functions of all 2D models continuouslyincrease with the increase of the particle density threshold ρ ofmodels, and decrease with the increase of the thickness of 2Dmodels.The second important impression is that 2D correlation andstructure functions have amplitudes much less than amplitudesof respective 3D functions, see Table 2. We show in Fig. 2 2Dcorrelation functions of LCDM models for various thickness of2D sheets, from the maximum thickness T = h − Mpc (num-ber of sheets n = n = T = . h − Mpc. Left panel of Fig. 2 plots 2D corre-lation functions for the whole DM sample with particle densityselection limit ρ =
0; next panels are for particle selection limits ρ = ρ =
10. Bold dashed black lines are for 3D correla-tion functions of the 3D samples with the same particle densitylimit ρ . These 3D correlation functions are practically identicalto 2D correlation functions calculated as the mean of n = Article number, page 4 of 10inasto et al.: Projected correlation functions ξ LCDM.00.0001LCDM.00.0002LCDM.00.0004LCDM.00.0008LCDM.00.0016LCDM.00.0032LCDM.00.0064LCDM.00.2048LCDM.00 1 10 100r [Mpc/h]0.0010.010.11101001000 ξ LCDM.05.0001LCDM.05.0002LCDM.05.0004LCDM.05.0008LCDM.05.0016LCDM.05.0032LCDM.05.0064LCDM.05.2048LCDM.05 1 10 100r [Mpc/h]0.0010.010.11101001000 ξ LCDM.10.0001LCDM.10.0002LCDM.10.0004LCDM.10.0008LCDM.10.0016LCDM.10.0032LCDM.10.0064LCDM.10.2048LCDM.10
Fig. 2.
2D correlation functions for LCDM models for di ff erent thickness of 2D samples. Left panel is for LCDM model with all particles, ρ = ρ =
5, and right panel for models with particle density limit ρ =
10. Lines of various colourmark 2D samples of di ff erent thickness. Thick dashed lines show 3D correlation functions for the same ρ limits. ξ LCDM.00LCDM.00.0004LCDM.01.0004LCDM.02.0004LCDM.05.0004LCDM.10.0004LCDM.20.0004 1 10 100r [Mpc/h]0.0010.010.11101001000 ξ LCDM.00LCDM.00.0008LCDM.01.0008LCDM.02.0008LCDM.05.0008LCDM.10.0008LCDM.20.0008 1 10 100r [Mpc/h]0.0010.010.11101001000 ξ LCDM.00LCDM.00.0016LCDM.01.0016LCDM.02.0016LCDM.05.0016LCDM.10.0016LCDM.20.0016
Fig. 3.
2D correlation functions for LCDM models as functions of the particle selection limit, ρ ; for di ff erent thickness of 2D samples. Leftpanel is for 2D correlation functions of thickness 128 h − Mpc, middle panel for thickness 64 h − Mpc, right panel for thickness 32 h − Mpc,corresponding to numbers of sheets, n = , ,
16, respectively. Lines of various colour show 2D functions found for di ff erent particle density limit ρ . 3D correlation functions of the full LCDM model with particle density limit ρ = The main lesson from this Figure is: 2D correlation func-tions have always lower amplitudes than respective 3D correla-tion functions, the di ff erence is the larger the greater is the thick-ness of 2D samples.In Fig. 3 we show 2D correlation functions of LCDM mod-els for di ff erent biasing levels, expressed in particle density limit, ρ . Left panel is for the 2D samples with the number of sheets n = T = h − Mpc; middle and right panelsare for 2D correlation functions with n = n =
16, corre-sponding to thicknesses T = , h − Mpc, respectively. Withdi ff erent colours 2D samples of various particle density limit, ρ ,are shown. The limit ρ =
10 corresponds approximately to L ∗ galaxies, as shown by Einasto et al. (2019). Bold dashed blacklines are for the 3D correlation function of the full DM sampleLCDM.00 with particle density limit ρ = ff ers fromthe shape of 3D correlation functions — 2D correlation functionsare much shallower, and have lower amplitudes. For this reason2D correlation functions cross with 3D correlation function ofmatter at di ff erent distance r and particle density limit ρ . All 2Dcorrelation functions for the thickest sheet with T = h − Mpchave amplitudes lower than the amplitude of the 3D correlationfunction for matter. 2D correlation functions of samples of thick-ness 128, 64, and 32 h − Mpc, shown in Fig. 3, cross 2D func-tions with 3D correlation function of matter at decreasing dis-tances r and decreasing particle density limit ρ with decreasingthe thickness of 2D sheets. We can compare correlation functionsat distance r = h − Mpc, used in the determination of ampli-tudes. 2D correlation functions of various particle density limit ρ have at some sheet thickness level amplitudes equal to theamplitude of 3D correlation function of all matter, ξ ≈ . b m ≈
1. In this way2D correlation functions can mimic 3D DM correlation func- tion, depending on the thickness of sheets used in calculation of2D correlation functions.On large distance correlation functions are slightly negativedue to normalisation (Davis & Peebles 1983; Peebles 1980). Theradius r z where correlation function becomes negative is equal to r z ≈ h − Mpc for all LCDM samples.
Fractal properties of density fields can be studied by correlationfunctions, especially by the log-log gradient of the pair corre-lation function as a function, γ ( r ) = d log( g ( r )) / d log( r ), andthe fractal dimension function, D ( r ) = + γ ( r ). Right panels ofFig. 1 show fractal dimension functions, calculated using the 3Dcorrelation function of the LCDM model, and using 2D corre-lation functions for samples of thickness 64 and 512 h − Mpc.As discussed by Einasto et al. (2020) and references therein, thefractal dimension function describes the geometry of the cosmicweb and its fractal character at various distances (pair separa-tions) of objects. At small distances up to r ≈ h − Mpc thefractal dimension function is determined by the distribution ofparticles within DM halos, at larger distances by the distributionof matter in the whole cosmic web.Fig. 1 shows that 2D samples have properties which are sim-ilar to properties of 3D samples, but there are also large andimportant di ff erences. The similarity is in the general shape offractal dimension functions which describe the geometry at dif-ferent scales — at small scales the structure of DM halos, and atlarger scales the structure of the web in general. Both 2D and 3Dfractal dimension function have minima at r ≈ h − Mpc. Allfractal dimension functions approach the value D = ff erences are more important. Thedepth of the minimum near r ≈ h − Mpc is di ff erent: in 2D Article number, page 5 of 10 & Aproofs: manuscript no. corr2D
Fig. 4.
2D luminosity density fields of biased LCDM.05 models with particle density threshold ρ =
5, Top panels show the 2D density fieldsof thickness T = h − Mpc at z = , ,
300 locations. Bottom panels are from left to right of thickness T = , , h − Mpc. Weshow central sections of 2D density fields of size 75 × h − Mpc at identical z locations . Colour scale is logarithmical, the code is identical in allpanels. samples the depth is much smaller, and is located at a smaller dis-tance r . At very small separations inside halos fractal dimensionof 2D samples of di ff erent ρ limits have a scatter larger than inthe 3D case. At smallest distances 3D correlation functions havealmost constant γ (0 . ≈ − .
5, which corresponds to fractal di-mension D (0 . ≈ .
5. In comparison, 2D samples of thickness T = h − Mpc have at these small distances γ (0 . ≈ − . D (0 . ≈ . T = h − Mpc has at this distance re-gion γ (0 . ≈ − . D (0 . ≈ .
6. This di ff erence between slopesof 3D and 2D correlation functions is expected since the slopeof the 2D correlation function on small and medium scales is re-lated to the slope of 3D correlation function on these scales asfollows: γ D = + γ D . On large scales both slopes correspond toa random distribution. Note also, that the minimum of the frac-tal dimension function near r ≈ h − Mpc is for 2D functionsmuch shallower, and the position of the minimum is shifted to-wards smaller distance.For particle density limit ρ ≤
10, which corresponds togalaxies of luminosity, L ≤ L ∗ , the fractal dimension function isalmost flat for the 2D sample of thickness 512 h − Mpc, and witha modest minimum for the 2D sample of thickness 64 h − Mpc.In other words, 2D correlation functions of normal galaxies havean almost constant slope. In this way our study of 2D and 3D cor- relation functions confirms earlier studies of the 2D correlationfunctions.
4. Discussion
Here we discuss di ff erences of 2D and 3D density fields, whichlead to di ff erences in respective correlation functions. Thereafterwe discuss quantitative di ff erences of 2D and 3D correlationfunctions, as descriptors of the cosmic web. Assuming Poisson character for the line-of-sight distributionof galaxies methods to calculate angular and spatial corre-lation functions were elaborated by Limber (1953, 1954),Totsuji & Kihara (1969), Peebles (1973) and Groth & Peebles(1977). For summaries of these classical methods see Peebles(1980) and Martínez & Saar (2002).The cosmic web is very rich in details. Spatial correlationfunction and its derivates (structure function and fractal dimen-sion function) characterise structural properties of the cosmicweb only in a general and global way. To understand what prop-erties of the web can be studied by correlation functions let ushave a look on the geometry of the cosmic web, as given by 2Dand 3D data.
Article number, page 6 of 10inasto et al.: Projected correlation functions
In Fig. 4 we show slices of 2D density fields of the modelLCDM.05 using particle density limit ρ =
5. Top panelsof Fig. 4 present 2D density fields of thin slices of thickness T = h − Mpc in x , y coordinates at various z locations. Thethickness of the 2D density field in the bottom left panel is T = h − Mpc, i.e. the whole cube of our LCDM.05 model. Inthe middle panel the thickness is T = h − Mpc, in the rightpanel T = h − Mpc. Galaxies can form in regions of localdensity ρ ≥
3; particle density limit ρ = M r − h = − . M r > − .
5, are excluded. Incalculation of 2D density fields we used resolution 2048 × × h − Mpc sectionsof fields.The 2D density field in top panels is so thin that its mor-phological properties are approximately similar to propertiesof the 3D density field of the same model, see Fig. 14 ofEinasto et al. (2019). However, as seen from Tables 2 and 3, am-plitudes of correlation functions and bias parameters of stacked2D LCDM.05.0064 model sheets of thickness 8 h − Mpc di ff erslightly from amplitudes and bias parameters of the 2D modelLCDM.05.2048, which is approximately equivalent to the 3Dmodel LCDM.05, compare Tables 1 and 2.Dominant elements of the cosmic web are clusters and fila-ments, seen in all panels of Fig. 4, surrounded by zero densityvoids, occupying most of the volume of the density field. Thecharacteristic scale of halos is a few h − Mpc. Clusters and fil-aments in sheets at di ff erent z locations are located in various x , y -positions, compare di ff erent top panels of Fig. 4. Correla-tion functions are sensitive to particle / galaxy separations , not locations . For this reason statistical properties of 2D correlationfunctions of thin sheets at various z -locations are similar, andclose to statistical properties of 3D correlation functions. Thisstatistical similarity is realised in the mean correlation functionof n = T = / n = . h − Mpc,and shown in Fig. 2 together with the 3D correlation function ofthe same particle density limit ρ .Another important elements of the cosmic web are regions ofzero volume density — voids. The filling factors of high-densityregions of models LCDM.05 and LCDM.10 are 0.10 and 0.06,respectively, the rest of the volume has zero density. This meansthat 3D density fields, corresponding to galaxies, as well as thinslices of the 2D density field are dominated by zero density cells.In thick 2D sheets clusters and filaments at various z are pro-jected to the 2D x , y plane at di ff erent positions, and in this wayfill in voids in the 2D density field. This is clearly seen in bot-tom panels of Fig. 4. The thicker are 2D density fields, the lessthey contain zero density cells. We conclude that the essentialdi ff erence between 2D and 3D (and thin 2D) density fields is thealmost absence of visible zero density regions in thick 2D fields.Power spectra and correlation functions of our LCDM mod-els were calculated using density fields. The power spectrum isdefined as: P ( k ) = h| δ k | i , (5)where k is the wavenumber, δ = ρ − ρ is density in mean density units. The density field used tofind power spectra or correlation functions can be divided intofour main regions: zero-density regions with ρ = δ = − ≤ ρ ≤ | δ | ≤
1, medium densityregions with 2 ≤ ρ ≤
10, and high-density regions with ρ > ff erent density regions. In full DM model allbasic regions are present. The density contrast in low-densityregions fluctuates between 0 and 1 and has a mean contrast about0.5. In density fields of biased LCDM models all previous low-density cells, which had previously | δ | ≈ .
5, have now zerodensity and | δ | =
1. This leads to the increase of the amplitudeof power spectra and correlation functions. When we considerdensity fields of increasing particle density limit, then with theincrease of the density limit an increasing fraction of mediumdensity cells, which had previously densities in the interval 2 ≤ ρ ≤
10, also change to zero-density cells with | δ | =
1, whichleads to further increase of the amplitude of power spectra andcorrelation functions.To summarise the comparison of 2D and 3D density fields wecan say, that the fraction of zero density regions of 2D fields de-creases due to projection e ff ect with the increase of the thicknessof the 2D field. For this reason amplitudes of power spectra andcorrelation functions of biased models are always higher thanamplitudes of power spectra and correlation functions of unbi-ased full DM models . For the same reason amplitudes of powerspectra and correlation functions of galaxies are always higherthan amplitudes of power spectra and correlation functions ofmatter . The first conclusion is known as the biasing phenomenon(Kaiser 1984), but the second conclusion is often ignored.Analytical description of the relation between 2D and 3Dtwo-point correlators is given in the Appendix. Norberg et al. (2001) calculated correlation functions of galax-ies of the 2dF redshift survey to analyse clustering properties ofgalaxies of various luminosity. Authors used observational datain spherical coordinates, and calculated the angular correlationfunction, w p ( r p ), by integrating over the measured ξ ( r p , π ), us-ing the equation w p ( r p ) = Z r max r min ξ ( r p , π ) d π, (6)where π and r p are pair separations parallel and perpendicularto the line of sight, and r min and r max are minimal and maxi-mal distances of galaxies in samples. Projected correlation func-tions were found with integrating upper limit π = h − Mpc.Thereafter projected correlation functions were transformed tospatial correlation functions using equation similar to Eq. (A.7.Galaxies were selected in conical shell of various thickness from ∼ h − Mpc for faintest galaxies ( M b − h ≈ −
18) to ∼ h − Mpc for most luminous subsamples. Norberg et al.(2001) found that the luminosity dependence of the relative biasis well fitted with the relation b / b ∗ = . + . L / L ∗ .Tegmark et al. (2004) calculated 3D power spectra of galax-ies from the SDSS survey. Power spectra were calculated forgalaxies in absolute magnitude bins of size 1 mag, the thick-ness of shells varies from ∼ h − Mpc for the nearest shell to ∼ h − Mpc for the shell for brightest galaxies, see Table 1of Tegmark et al. (2004). The luminosity dependence of relativebias parameter is rather similar to the Norberg et al. (2001) fit.Zehavi et al. (2005, 2011) investigated projected correlationfunctions of SDSS galaxies of di ff erent luminosity. Authors ap-plied standard practice and computed projected correlation func-tions using Eq. (6), and real-space correlation functions usingEq. (A.7 by Davis & Peebles (1983). Samples of various abso-lute magnitude bins are located in spherical shells of thickness,similar to thicknesses of shells in the Tegmark et al. (2004) anal-ysis. The luminosity dependence of the relative bias parameter Article number, page 7 of 10 & Aproofs: manuscript no. corr2D is similar to the dependence found by Norberg et al. (2001) andTegmark et al. (2004).In papers cited above authors assumed that density fluctua-tions at present epoch can be modeled as a homogeneous andisotropic random field.
Comparison with earlier analyses shows that the luminosity de-pendence of the relative correlation functions and power spectraare in good mutual agreement. More di ffi cult is the determina-tion of the absolute levels of amplitudes of correlation functionin respect to matter. As seen from left panel of Fig. 2, the ampli-tude of the 2D correlation function of the full DM model dependscritically on the thickness of sheets to derive 2D density field forthe determination of correlation function. At small separations r ≈ h − Mpc the di ff erence in the amplitude of 2D correlationfunctions of the real 3D distribution and projected 2D distribu-tion for thickness T = h − Mpc over hundred times, at r ≈ ff erence is smaller, but islarge anyway. In other words, the determination of the absolutebias level in respect to matter is a very di ffi cult task when using2D analyses.For this reason it is preferable to use 3D analysis. Thepresent work is not a replacement of earlier analyses of obser-vational data. Our goal was to show where the di ffi culties lie.The present analysis of correlation functions of simulated uni-verse suggest that the bias parameter of L ∗ galaxies is approx-imately b ∗ = . ± .
15, in agreement with the analysis ofpower spectra of biased Λ CDM model by Einasto et al. (2019).Klypin et al. (2016) used several MultiDark simulations of boxsizes 250 − h − Mpc with 3840 particles to investigate DMhalo concentrations and profiles. They found that the bias factorof power spectra of halos to power spectra of mass is 1.95, seeFig. 2 of Klypin et al. (2016).It should be noted, that the bias parameter b of galaxies tomatter is simply related to the fraction of matter in the clus-tered component of the cosmic web, F c , associated with galaxies(Einasto et al. 1994, 1999): b = / F c . (7)Einasto et al. (1999) and Einasto et al. (2019) showed that thisrelation is rather accurate at high fraction F c levels, 1 ≥ F c ≥ .
7. The limit F c = . F c less rapidly than suggested by Eq. (7).
5. Conclusions
We investigated spatial and projected correlation functions usinga biased Λ CDM model. Biased model samples contain particlesabove a certain limit, ρ ≥ ρ = , , , , , , , T = L / n , where L = h − Mpc is the length ofthe Λ CDM model, and n = , , , . . . ξ = ξ ( r = b ( T , ρ ) = p ξ D ( T , ρ ) /ξ D (0). Our mainconclusions are as follows. 1. Dominant elements of the cosmic web are clusters and fila-ments, separated by voids filling most of the volume. In indi-vidual 2D sheets clusters and filaments are located at di ff er-ent positions. As a result in projection clusters and filamentsfill in 2D voids, which leads to the decrease of amplitudesof correlation functions (and power spectra). For this reasonamplitudes of 2D correlation functions are lower than ampli-tudes of 3D correlation functions, the di ff erence is the larger,the thicker are 2D samples.2. Biasing of samples and thickening of 2D sheets influenceamplitudes of correlation functions in di ff erent directions.For this reason at certain biasing (luminosity) level and thick-ness of samples galaxy correlation functions can imitate cor-relation functions of matter.3. 2D correlation functions are flatter than 3D correlation func-tions. Contrast in fractal dimension between small and largeseparations (halos and web) is in 2D correlation functionsmuch smaller than in 3D correlation functions.4. 3D correlation functions cannot be calculated from 2D cor-relation functions, because 2D correlation functions do notcontain information on voids in 3D density field. Acknowledgements.
Our special thanks are to Enn Saar for many stimulatingdiscussions. This work was supported by institutional research funding IUT26-2and IUT40-2 of the Estonian Ministry of Education and Research, and by theEstonian Research Council grant PRG803. We acknowledge the support by theCentre of Excellence “Dark side of the Universe” (TK133) financed by the Eu-ropean Union through the European Regional Development Fund.
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Article number, page 8 of 10inasto et al.: Projected correlation functions
Appendix A: Two-point function of the projected field
Here we briefly describe the relation between 2D and 3D two-point correlators. In the following we assume su ffi ciently smallsurvey volume, such that evolutionary and lightcone e ff ects can be neglected. We assume the validity of c osmological principle, i.e. ,statistical homogeneity and isotropy of the cosmic density field. Additionally, we do not include redshift-space distortions, since wefocus only on real-space two-point functions.The density contrast is defined as usual δ ( r ) ≡ n ( r )¯ n − , (A.1)where n ( r ) is the comoving number density of tracer objects at spatial location r and ¯ n is the corresponding average density.In the following we consider plane-parallel and spherical projections of the 3D field separately. AppendixA.1: Plane-parallelgeometry
Assuming appropriately normalized projection / selection function w ( r ), namely Z d r w ( r ) ≡ , (A.2)the projected 2D overdensity field can be expressed as δ ( r ⊥ ) = Z d r k w ( r k ) δ ( r k , r ⊥ ) . (A.3)The two-point correlation function of the projected density field can now be obtained. According to the cosmological principlewe can always choose one point to be at the origin, i.e. , r ⊥ , = , and for the other we assume r ⊥ , = R with modulus R ≡ | R | . Thusone can write ξ ( R ) = h δ ( ) δ ( R ) i = Z d r w ( r ) Z d r w ( r ) h δ ( r , ) δ ( r , R ) i == Z d r w ( r ) Z d r w ( r ) ξ (cid:16) p R + ( r − r ) (cid:17) (A.4)In case of uniform selection within range [0 , L ], i.e. w ( r ) = / L , the above result can be expressed as ξ ( R ) = Z d x Z d x ξ (cid:16) p R + L ( x − x ) (cid:17) . (A.5)In Eq. (A.4) the ξ part of the integrand is peaked around r = r . If the selection function varies smoothly in comparison, it canbe pulled out of the 2nd integral (this is the essence of the Limber approximation, e.g. Peebles (1980)), giving ξ ( R ) ≃ Z ∞ d r w ( r ) Z ∞−∞ d x ξ (cid:16) √ R + x (cid:17) = Z ∞ d r w ( r ) Z ∞ d x ξ (cid:16) √ R + x (cid:17) , (A.6)where we have introduced new dummy variables r → r and r − r → x . In case of uniform selection within range [0 , L ] the aboveresult can be expressed as ξ ( R ) ≃ L Z ∞ d x ξ (cid:16) √ R + x (cid:17) = L Z ∞ R d r r √ r − R ξ ( r ) (A.7) AppendixA.2: Sphericalgeometry
The above results can be extended for the case of spherical geometry. Here the results have the simplest form once the radialselection / projection function is normalized as Z d r r w ( r ) ≡ . (A.8)In that case the analog of Eq. (A.4) reads ξ ( Θ ) = Z d r r w ( r ) Z d r r w ( r ) ξ (cid:18) q r + r − r r cos Θ (cid:19) , (A.9)where cos Θ ≡ ˆr · ˆr with ˆr , ˆr the radial unit vectors marking the two points. Article number, page 9 of 10 & Aproofs: manuscript no. corr2D
The case with uniform selection in the range [ D − L / , D + L / i.e. analogue of Eq. (A.5), can be given as ξ ( R ) = Z D + L / D − L / d r r Z D + L / D − L / d r r ξ (cid:18) q r + r − r r cos ( Θ = R / D ) (cid:19) . (A.10)In case of slowly varying selection and with small-angle approximation the Eq. (A.9) can be recast as a follows (this is a standardLimber’s formula) ξ ( Θ ) ≃ Z ∞ d r r w ( r ) Z ∞−∞ d x ξ (cid:16) √ x + r Θ (cid:17) , (A.11)where new dummy variables r → r and r − r → x were introduced. Appendix B: Projected two-point function
Very often, in order to avoid complications caused by the redshift-space distortions, the spatial two-point correlation function whichis evaluated as a 2D function of radial and transverse separations, is integrated along the radial direction, resulting in the followingquantity w ( r ⊥ ) = Z ∞−∞ ξ ( r k , r ⊥ ) d r k = Z ∞ r ⊥ d r r q r − r ⊥ ξ ( r ) . (B.1)The above has a form of the Abel integral equation, which is often inverted to recover the real-space correlation function. However,in the presence of measurement errors this inversion is not a well posed problem, and thus needs additional assumptions for regu-larisation purposes (often one assumes specific functional form for ξ , e.g. simple power-law). In addition, instead of using a fullinformation, which would demand somewhat more complicated modelling in order to capture also signal stored in the redshift-spacedistortions, the line-of-sight integration results in a generic signal loss.It is instructive to note that the above result is practically identical to Eq. (A.7). This is only so under the validity of the Limberapproximation, i.e. , in that case the projection and correlator calculation operations e ff ectively commute. In general, there is a cleardi ff erence between the projected correlation function and correlation function of the projected field.erence between the projected correlation function and correlation function of the projected field.