[Paper Review] Is the Universe a fractal? Results from the SSRS2

A. Cappi, Christophe Benoıst+2 more|arXiv (Cornell University)|Apr 8, 1998

Scientific Research and Discoveries3 references3 citations

TL;DR

This paper investigates whether the large-scale structure of the universe exhibits fractal properties using data from the Southern Sky Redshift Survey 2 (SSRS2). It confirms that the galaxy distribution appears fractal at small to intermediate scales due to luminosity-dependent correlation amplitude and incomplete sampling of large-scale power, but finds no contradiction with cosmological homogeneity at ~100 h⁻¹ Mpc, concluding that the apparent fractal behavior is an artifact of survey depth and luminosity selection rather than true self-similarity across all scales.

ABSTRACT

We perform a fractal analysis of the Southern Sky Redshift Survey 2, following the methods prescribed by Pietronero and collaborators, to check their claim that the galaxy distribution is fractal at all scales, and we discuss and explain the reasons of some controversial points, through tests on both galaxy samples and simulations. We confirm that the amplitude of the two-point correlation function does not depend on the sample depth, but increases with luminosity. We find that there is no contradiction between the results of standard and non-standard statistical analysis; moreover, such results are consistent with theoretical predictions derived from standard CDM models of galaxy formation, and with the hypothesis of homogeneity at large scale ($\sim 100$ \h). However, for our SSRS2 volume-limited subsamples we show that the first zero-point of the autocorrelation function $ξ(s)$ increases linearly with the sample depth, and that its value is comparable to the radius of the maximum sphere which can be completely included in the sampled volume; this implies that the true zero-crossing point of $ξ(s)$ has not been reached. We conclude that the apparent fractal behavior is due to a combination of a luminosity-dependent correlation amplitude and the recovering of power at larger scales in deeper samples.

Motivation & Objective

• To test Pietronero et al.'s claim that the galaxy distribution is fractal at all scales using the SSRS2 redshift survey.
• To resolve controversies in interpreting two-point correlation functions and autocorrelation functions in large-scale structure analysis.
• To assess whether the observed fractal-like behavior is consistent with standard cosmological models and the hypothesis of large-scale homogeneity.
• To investigate how sample depth and luminosity affect the apparent correlation amplitude and zero-crossing point of the correlation function.

Proposed method

• Performs a fractal analysis on volume-limited subsamples of the SSRS2 galaxy redshift survey using methods developed by Pietronero and collaborators.
• Analyzes the two-point correlation function ξ(s) and its zero-crossing point to assess scale-invariant clustering.
• Compares results from real galaxy samples with those from N-body simulations to test consistency with standard CDM models.
• Evaluates the dependence of the correlation amplitude on sample depth and luminosity to isolate systematic effects.
• Uses the radius of the largest sphere fully contained within the survey volume as a reference to interpret the zero-crossing point of ξ(s).
• Applies statistical tests to reconcile discrepancies between standard and non-standard statistical approaches in large-scale structure analysis.

Experimental results

Research questions

• RQ1 Does the galaxy distribution in the SSRS2 survey exhibit fractal behavior across all scales, as claimed by Pietronero et al.?
• RQ2 How does the zero-crossing point of the autocorrelation function ξ(s) depend on survey depth and sample luminosity?
• RQ3 To what extent do luminosity-dependent correlation amplitudes and incomplete sampling of large-scale power mimic fractal behavior?
• RQ4 Are the observed results consistent with standard CDM models of galaxy formation and the cosmological principle of large-scale homogeneity?
• RQ5 Can the apparent fractal structure be explained by systematic effects rather than intrinsic scale-invariant clustering?

Key findings

• The amplitude of the two-point correlation function does not depend on sample depth but increases with luminosity, indicating a luminosity-dependent clustering strength.
• The first zero-crossing point of the autocorrelation function ξ(s) increases linearly with sample depth and is comparable to the radius of the largest sphere that fits within the survey volume, suggesting the true zero-crossing has not yet been reached.
• There is no contradiction between standard and non-standard statistical analyses of the data, and both are consistent with theoretical predictions from standard CDM models.
• The observed fractal-like behavior is not evidence of true self-similarity across all scales but results from the combination of luminosity-dependent correlation amplitude and the recovery of power at larger scales in deeper samples.
• The hypothesis of homogeneity at scales of ~100 h⁻¹ Mpc remains consistent with the data, as the zero-crossing point has not yet been reached in the current survey depth.
• The apparent fractal structure is therefore an artifact of survey geometry and luminosity selection, not a fundamental property of the large-scale universe.