A Random Walk Model for Spatial Galaxy Distribution

Vladimir V Uchaikin,Elena V Kozhemyakina,Ilya I Kozhemyakin,Vladimir A Litvinov

doi:10.3390/math9010098

Vladimir V Uchaikin, Elena V Kozhemyakina + Show 2 more

Open Access

https://doi.org/10.3390/math9010098

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Abstract

A new statistical model of spatial distribution of observed galaxies is described. Statistical correlations are involved by means of Markov chain ensembles, whose parameters are extracted from the observable power spectrum by adopting of the Uchaikin–Zolotarev ansatz. Markov chain trajectories with the Lévy–Feldheim distributed step lengths form the set of nodes imitating the positions of galaxy. The model plausibly reproduces the two-point correlation functions, cell-count data and some other important properties. It can effectively be used in the post-processing of astronomical data for cosmological studies.

Highlights

IntroductionThe presence of a boundary region between the fractal and homogeneous parts of the mesofractal can play a decisive role in establishing the horizon of a representative (fair) sample
In Mandelbrot’s model, the entire set of random points correlated according to the same law as galaxies was generated by the nodes of a single random trajectory of an infinite Markov chain
The distances between successive nodes are distributed according to a power law, consistent with the characteristic of decreasing correlations in the observed samples

Summary

Introduction

The presence of a boundary region between the fractal and homogeneous parts of the mesofractal can play a decisive role in establishing the horizon of a representative (fair) sample To investigate this problem in the frame of the basic cosmological principle, stating the equality of all points in space when choosing the origin [1], Mandelbrot offered interpreting the left hand side of Equation (1) as the expected number of Markov chain nodes with the transition probability. (r0 /r )α , r > r0 , 1, r < r0 , α > 0 This law sufficiently reproducing inverse-power correlations observed in spatial galaxy distribution, turned out to be rather awkward when computing multiple convolutions [2].

Power Spectrum and Correlation Function

Estimation of UZ Spectrum Parameters

The OZ-Equation Solving

Cell Counts

Separation of Cluster and Background

Conclusions