[Paper Review] Multifractal analysis and lacunarity spectrum of the galaxies of the ninth Sloan Digital Sky Survey (SDSS) data release

C. A. Chacón-Cardona, Rigoberto A. Casas-Miranda|arXiv (Cornell University)|Dec 19, 2012

Complex Systems and Time Series Analysis2 references3 citations

TL;DR

This study applies multifractal analysis and lacunarity spectrum computation to a magnitude-limited sample of 164,168 galaxies from the SDSS DR9 survey, covering 120° < α < 240° and 0° < δ < 60°, to investigate large-scale clustering. Using the sliding-window technique on the generalised correlation integral, it finds persistent multifractal behaviour up to 180 Mpc/h without a transition to homogeneity, with distinct dimensional growth in low- and high-density regions and oscillatory lacunarity indicating recurring voids and clusters.

ABSTRACT

In this work, we develop a statistical analysis of the large-scale clustering of matter in the Universe from the fractal point of view using galaxies from the Ninth Sloan Digital Sky Survey (SDSS) Data Release (DR9). From the total set of galaxies, a magnitude-limited sample of galaxies with redshifts in the range 0 &lt; z &lt; 0.15 was created. The sample covers the largest completely connected area of the celestial sphere within the catalogue, with limits in right ascension from 120 to 240 degrees and declination from 0 to 60 degrees, which is a region that includes the largest galactic samples that have been studied from the fractal viewpoint to date. The sample contains 164,168 galaxies. Using the sliding-window technique, the multifractal dimension spectrum and its dependence on radial distance are determined. This generalisation of the concept of fractal dimension is used to analyse large-scale clustering of matter in complex systems. Likewise, the lacunarity spectrum, which is a quantity that complements the characterisation of a fractal set by quantifying how the set fills the space in which it is embedded, is determined. Using these statistical tools, we find that the clustering of galaxies exhibits fractal behaviour that depends on the radial distance for all calculated quantities. A transition to homogeneity is not observed in the calculation of the fractal dimension of galaxies; instead, the galaxies exhibit a multifractal behaviour whose dimensional spectrum does not exceed the physical spatial dimension for radial distances up to 180 Mpc/h from each centre within the sample. Our results and their implications are discussed in the context of the formation of large-scale structures in the Universe.

Motivation & Objective

• To investigate the large-scale clustering of galaxies in the local Universe using multifractal geometry and lacunarity analysis.
• To test whether the standard cosmological principle—homogeneity and isotropy on large scales—is supported by observed galaxy distributions.
• To determine if a transition to homogeneity occurs at scales below 180 Mpc/h, as suggested by some prior studies.
• To characterise the structural complexity of galaxy clustering through multifractal dimension spectra and lacunarity, avoiding assumptions of homogeneity.
• To assess the consistency of observed galaxy distributions with models that assume spatial homogeneity, using data-driven statistical methods.

Proposed method

• Constructed a magnitude-limited galaxy sample from SDSS DR9 with redshifts 0 < z < 0.15, covering 120° < α < 240° and 0° < δ < 60°, resulting in 164,168 galaxies.
• Applied the sliding-window technique to compute the generalised correlation integral across radial distances up to ~600 Mpc/h, where space-time curvature effects are negligible.
• Used the generalised correlation integral to derive the multifractal dimension spectrum for varying structure parameters q, distinguishing low-density (q < 1) and high-density (q ≥ 1) regions.
• Calculated the lacunarity spectrum based on the pre-factor F of the correlation integral to quantify spatial heterogeneity and detect voids and clusters.
• Compared results with those from the Millennium N-body simulation to validate methodology and interpret findings in the context of cosmological structure formation.
• Avoided cosmological assumptions by analyzing data directly without assuming the FRW metric, aiming to reduce bias in homogeneity detection.

Experimental results

Research questions

• RQ1 Does the galaxy distribution in the SDSS DR9 sample exhibit a transition to homogeneity at radial distances below 180 Mpc/h?
• RQ2 How does the multifractal dimension spectrum vary with radial distance and structure parameter q across different density regimes?
• RQ3 To what extent does the lacunarity spectrum reflect the presence of voids and clustered regions in the large-scale structure?
• RQ4 Is the observed galaxy clustering consistent with a multifractal model, or does it show signs of approaching monofractality at large scales?
• RQ5 Can the lacunarity spectrum detect structural transitions that are not evident in the multifractal dimension alone?

Key findings

• No transition to homogeneity was observed in the galaxy clustering up to 180 Mpc/h; instead, multifractal behaviour persisted across all radial distances studied.
• For low-density regions (q < 1), the fractal dimension showed a clear tendency toward homogeneity beyond 50 Mpc/h, with dimensional growth followed by a decline toward the physical dimension.
• For high-density regions (q ≥ 1), the fractal dimension reached high values near cluster centres and did not asymptotically approach the physical dimension, indicating persistent multifractal structure at large scales.
• The multifractal dimension spectrum exhibited two distinct regions: a high-dimensional growth phase below 30 Mpc/h and a smooth decay with fractal dimension below the physical dimension beyond 30 Mpc/h.
• The lacunarity spectrum showed oscillatory behaviour with alternating high and low lacunarity values, indicating recurring voids and dense clusters, and did not monotonically decrease, contradicting expectations for homogeneity.
• Lacunarity values increased again after 50 Mpc/h, reinforcing the absence of a homogeneity transition and supporting the conclusion that the sample remains multifractal across the entire scale range.