[Paper Review] Investigation of clustering of galaxies, clusters and superclusters by the method of correlation Gamma-function

А. В. Тихонов, D. I. Makarov+1 more|arXiv (Cornell University)|Jun 15, 2001

Advanced Scientific Research Methods3 citations

TL;DR

This study uses the correlation Gamma-function (conditional density) to analyze clustering in galaxies, clusters, and superclusters across scales from 50 kpc to 250 Mpc. It finds a pronounced break in clustering at ~30 Mpc, with a steep power-law drop (index 0.9–1.5) on small scales and a transition to near-homogeneous distribution (slope ≈ 0) on large scales, indicating that large-scale structure is more complex than a single fractal dimension can describe.

ABSTRACT

Using the apparatus of correlation Gamma-function (``conditional density''), we have analyzed spatial clustering of objects from several different samples of galaxies, clusters and superclusters. On small scales the distribution of objects obeys a power law drop of density with a power index of 0.9--1.5. On a scale of ~30 Mpc for independent samples of bright galaxies and clusters of galaxies we have detected a pronounced break in the slope of the Gamma-function with a power index decreasing to ~0.3. The clustering is much less pronounced in the region from 40 to 100 Mpc, and there is reason to suppose that the distribution of objects changes to homogeneous on scales larger than 100 Mpc. This is indicated by the slope of the Gamma-function close to 0 for a sample of rich clusters of galaxies up to 250 Mpc. The slope of the Gamma-function prior to the break, which characterizes the degree of clustering of matter, changes essentially and in a complex manner when passing to brighter (massive) objects. This suggests that the large-scale structure of the visible Universe even on small scales is considerably more complex than the fractal distribution described by one dimension (monofractal).

Motivation & Objective

• To investigate the spatial clustering of galaxies, clusters, and superclusters using the correlation Gamma-function as a statistical tool.
• To determine whether the large-scale structure of the universe exhibits fractal-like behavior or transitions to homogeneity on large scales.
• To assess how clustering properties vary with object brightness and mass, particularly comparing galaxies, clusters, and superclusters.
• To evaluate the influence of sample selection effects and boundary conditions on Gamma-function results.
• To test the hypothesis that the universe transitions from fractal clustering to homogeneity at scales larger than ~100 Mpc.

Proposed method

• Applies the Gamma-function (conditional density) to quantify spatial clustering, which measures the expected number of objects within a sphere of radius R around a given object.
• Uses multiple astronomical samples: CfA2+SSRS2 galaxies, APM clusters, Abell clusters, and Einasto superclusters, each with different depth and completeness.
• Employs a power-law fit to the Gamma-function to extract the slope (power index) γ, which characterizes the rate of density decrease with scale.
• Analyzes the behavior of the Gamma-function across scales from 50 kpc to 250 Mpc, identifying breaks and transitions in clustering strength.
• Compares results across different object types (galaxies, clusters, superclusters) to assess how clustering depends on luminosity or mass.
• Considers the impact of selection effects, boundary conditions, and sample incompleteness on Gamma-function shape and interpretation.

Experimental results

Research questions

• RQ1 Does the correlation Gamma-function reveal a break in clustering at intermediate scales, and if so, at what scale?
• RQ2 How does the power-law slope of the Gamma-function vary with object type (galaxies, clusters, superclusters), and what does this imply about large-scale structure?
• RQ3 Is there evidence for a transition to homogeneity in the distribution of matter at large scales, and at what scale does this occur?
• RQ4 To what extent do selection effects and sample boundaries influence the shape of the Gamma-function?
• RQ5 Can the observed clustering behavior be explained by a single fractal dimension, or is the structure more complex than monofractal models suggest?

Key findings

• On small scales (~50 kpc to 30 Mpc), the Gamma-function exhibits a power-law drop with a slope (power index) ranging from 0.9 to 1.5, consistent with fractal clustering.
• A pronounced break in the Gamma-function occurs at approximately 30 Mpc for bright galaxies and clusters, where the slope decreases to ~0.3, indicating a change in clustering dynamics.
• Between 40 and 100 Mpc, clustering is significantly weaker, with the Gamma-function slope remaining low, suggesting a transition toward homogeneity.
• For rich clusters and superclusters, the Gamma-function approaches a slope close to zero at scales up to 250 Mpc, indicating a strong tendency toward homogeneity on large scales.
• The slope prior to the break (γ₁) varies systematically across samples, indicating that clustering strength depends on object brightness or mass, and cannot be described by a single fractal dimension.
• The observed behavior of the Gamma-function—especially the break and subsequent leveling off—supports the idea that large-scale structure is more complex than a simple monofractal model, with multiple physical mechanisms likely at play.