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On Levy-Walk Model for Correlations in Spatial Galaxy Distribution Physics & Astronomy International Journal Mini Review Open Access On levy-walk model for correlations in spatial galaxy distribution Abstract Volume 3 Issue 2 - 2019 The model of stochastic fractal is briefly described, basic results regarding properties VV Uchaikin of spatial structures with long-range correlations of power-law type are reviewed, and Ulyanovsk State University, 42 Lev Tolstoy Str., Ulyanovsk some applications to galaxy distribution in the Universe are discussed. 432700, Russia Correspondence: VV Uchaikin, Ulyanovsk State University, 42 Lev Tolstoy Str., Ulyanovsk 432700, Russia, Email Received: March 14, 2019 | Published: March 29, 2019 Introduction distribution of gravitational force magnitude (Holtsmark distribution). Section IV presents the distribution of number of points inside the One of important problems complicating the extraction of spherical cells in the stochastic fractal model. In Section V the question information from cosmological surveys, is necessity to supplement about the global mass density of the fractal Universe is investigated observational data subtracted of foreground or for some (i.e. with the empirical distribution density over masses adopted. In Section technical) reasons removed from the survey. This particularly VI the ergodic problem in observational cosmology is discussed. affects the extraction of information from the largest observable scales. Maximum-likelihood estimators are not quite applicable for Power spectrum reconstructing the full-sky because of non-Caussian character of The important statistical characteristic of the large-scale structure observed matter distribution. To solve this problem, many authors of the Universe is the two-point spatial galaxy correlation function,1 modify implementations of such estimators which are robust to the which determines the joint probability leakage of contaminants from within masked regions. The trouble arises from the need to satisfy the cosmological principle of an 2 δPn=1 + ξ( r) δδ V12 V arbitrary choice of the reference frame and long-distant correlations to find two galaxies in volumes δV and δV , placed at the distance in spatial galaxy distribution observed. We are going to discuss here 1 2 one а such approach and to show some results of its application. The r =xx21 − from each other ( is the mean galaxy number per unite discussion about homogeneity (or inhomogeneity) of the large-scale volume, i.e. the concentration). According to the assumption about visible matter distribution in the Universe is attracting the attention of homogeneity and isotropy of matter distribution in the Universe the both the astronomers and theoreticians of cosmology.1–4 In recent years mean galaxy number density n should not depend on coordinates and the more complete and deeper galaxy surveys become available,5,6 ξ(r) depends only on distance between the points chosen. Instead of however, this does not eliminate the question about fractality of the ξ(r) it is often used its Fourier transformation Pk( ) called power large-scale structure. The fractal concept, introduced by Mandelbrot,7 spectrum1 applied to the galaxy distribution means the presence of the overdence −13ik.r regions and voids extending to all scales, at least to those could be Pk( )= V∫ξ(re) d r, (2) reached by now. Mathematically, it formulates as 3 −ikr 3 NR( )∝ RD (1) ξπ(r) =( V/8 ) ∫ Pre( ) dr. (3) for the ensemle averaged number of galaxies inside a sphere Figure 1 shows the power spectrum obtained from from the of radius centered at any galaxy, is the fractal dimension. Lick Observatory catalog of Shane and Wirtanen,8,9 where is Such a relation naturally originates from the model of self-similar expressed in units of “waves per box”, physical wavelengths are ( hierarchical clustering.3 The aim of this work is not to prove or −1 λ = 260h Mpc/ k . There exists several approximations of power disapprove the fractality of the matter distribution in the Universe but only to deduce some conclusions about statistical properties of the spectrum. The most simple of them is structures with the long-range correlations of power type. The paper γ P( k) = Ak (4) is organized as follows: Section I deals with the power spectrum being the Fourier transformation of two-point correlation function found in with being the amplitude of the spectrum and γ ∈−( 3, 0) in the galaxy distribution. From approximation of the real galaxy power order to allow for the convergence of the integral of Pk( ) at large spectrum it follows that galaxy distribution can be simulated within wavelength. The value for the spectral index corresponds the frame of random walk model (Section II). Section III concerns to the scale-free Harrison–Zel’dovich spectrum10 that describes the the cosmological monopole and dipole which are the characteristics fluctuations generated in the framework of the canonical inflationary of large-scale homogeneity and isotropy. Here is also present the scenario.11–13 Substituting (4) into (3), we get Submit Manuscript | http://medcraveonline.com Phys Astron Int J. 2019;3(2):82‒88. 82 © 2019 Uchaikin. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: On levy-walk model for correlations in spatial galaxy distribution ©2019 Uchaikin 83 3 23Γ+(γ32) ( γπ +) −+( γ ) g( x) =+− pq( x) ∫g ( xx') pd( x ') x ', (8) ξπ(r)= AV /2 sinr . ( ) γ +22 so that ξ (r) = ( AV/ c3) g( xx) , r= c . The positive function g ( x) can be interpreted as a density of collisions of some particle starting its moving at the origin and 3 performing the first collision in dx' with probability pd( xx'') where it stops with the probability 1 − q or performs the next jump 3 into dx'' with the probability qp( x′′− x ′ ), d x ′′ and so forth. So in this way of interpretation the points of collisions are positions of galaxies considered to be the point-like objects. The random walk model applied to simulation of galaxy distribution was firstly proposed by B.Mandelbrot.7 In his version the transition probability has the form (− ?) of the pure power law px()α r . Figure 1 Power spectrum P(k) (dots with error bars). The solid curve is The integral approximate formula for , c = 0.018, and q = 0.99. The 3 dashed curve corresponds to the law . ∫=−g( xdx) 1/( 1 q) , < (9) q 1, Thus, the detected power law shape for the correlation function gives the mean number of all collisions of the particle including −α the final one, and the function itself can be expressed by its ξα(rr) ~ , ~ 1.8, Neumann’s series expansion turns into a constant logarithmic slope of the power spectrum with j−1 g ( x) = ∑∞ q pj( x) , (10) (− 1) j=1 spectral index γα=3 − at least at scales r≤ 10 h Mpc [1,14] : −+3 α where P( k) = Ak . pp( xx) ≡ ( ) (11) Here the exponent a is equivalent to the fractal dimension D . As 1 one can see from Figure 1, the formula is in agreement with observed and results in the region 3≤≤k 30, but it may be improved in the region 3 p( x) =∫− p( xx''') pd( x) x (12) of small values of k which affects the ξ (r) at large distances. jj+1 Obviously, there are many appropriate representations of it, but one of them leads us to a very useful and significant analogy: is the multiple convolution of the distribution density px(). Here it should be noted that the probability density (7) with α ∈ (0, 2] belongs − α to the set of symmetrical three-dimensional stable distributions.16 e ()ck Pk( ) = A , (5) One of them, withα = 2, is the famous Gauss’ law, but the others are − α 1e−q ()ck utilized by physicists rarely enough, although in probability theory they play the same role in the problem of summation of independent where c > 0 and 01≤≤q can be chosen in appropriate way (see random values with infinite variance. Generally, the densities are not Figure 1). The point is that the inverse Fourier transformation of expressed in terms of elementary functions and must be calculated expression (5) leads to the integral equation numerically. Two properties of the laws are very important for the problem. First, the stable density with α < 2 has an inverse power tail 3 ξξ(r) = AVp( r// c33) +( q c) ∫−( x x') p( x '/ c) d x ', (6) at large distances, 1 πα −−α 3 where pj( x) Γ+(2α ) sinr , 3 2 α 2π 1 −−ik.3x k p( xk) =∫= e dk, k . (7) and second, multiple convolutions of this density are expressed in π 3 8 terms of original density with rescaled argument: The equation obtained is one form of the Ornstein-Zernike α equation17 and can be used to clear up the relationship between 1 −−ik. x jk 3− 3/α − α p( x) =∫= e dkx j pj( 1/ ). statistical mechanics and the concepts under consideration. On the j 8π 3 other hand, this equation leads us directly to the random walks model as a tool for construction of random point distribution with given Now we consider an infinite set of independent trajectories of correlations. such kind starting from different random points of birth distributed by Poisson uniform law with the mean density n . Using the generating Random walk model 0 functional technique, we obtained in our work,18 that in this case Let us consider the integral equation Citation: Uchaikin VV. On levy-walk model for correlations in spatial galaxy distribution. Phys Astron Int J. 2019;3(2):82‒88. DOI: 10.15406/paij.2019.03.00162 Copyright: On levy-walk model for correlations in spatial galaxy distribution ©2019 Uchaikin 84 nc3 gr( x) = ξ ( ) , (13) vr= Ht( ) , forming the background for observations of peculiar 2q motions of galaxies.
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