[Paper Review] Rhombic cell analysis II. Application to the IRAS/PSCz Catalogue

T. Kiang, Yongfeng Wu+1 more|arXiv (Cornell University)|Dec 1, 2003

Stellar, planetary, and galactic studies4 citations

TL;DR

This study applies rhombic cell analysis to the IRAS/PSCz Catalogue to quantify large-scale structure morphology across samples of increasing luminosity. Despite inherent scatter due to discrete galaxy distributions, the flocking index and other numerical indices reveal that structure degrades significantly with increasing luminosity threshold, indicating that the universe's morphology shifts from island-like to lake-like configurations as only the most luminous galaxies are considered.

ABSTRACT

Rhombic cell analysis as outlined in the first paper of the present series is applied to samples of varying depths and liming luminosities of the IRAS/PSCz Catalogue. Numerical indices are introduced to summarize essential information. Because of the discrete nature of the anlaysis and of the space distribution of galaxies, the indices for a given sample must be regarded as each having an irreducible scatter. Despite the scatter, the mean indices show remarkable variations across the samples. The underlying factor for the variations is shown to be the limiting luminosity rather than the sampling depth. As samples of more and more luminous galaxies are considered over a range of some 2 magnitudes (a factor of some 50 in space density), the morpholgy of the filled and empty regions the galaxies define degrades steadily towards insignificance, and the degrading is faster for the filled than the empty region.

Motivation & Objective

• To investigate how large-scale structure morphology in galaxy distributions evolves with increasing luminosity limits using rhombic cell analysis.
• To quantify structural changes in filled (galaxy-rich) and empty (galaxy-poor) regions across different luminosity-limited samples.
• To assess whether observed variations in structural indices are due to luminosity effects or sampling depth by comparing with random selection tests.
• To introduce and validate numerical indices—particularly the flocking index η—for summarizing structural characteristics in discrete spatial distributions.
• To demonstrate that the irreducible scatter in indices arises from the discrete nature of galaxy distributions and cell placement, necessitating ensemble averaging.

Proposed method

• Construct a filled-out PSCz catalogue by combining real IRAS/PSCz galaxies with mock galaxies in galactic plane gaps, using flux assignment from the original catalogue.
• Convert redshifts to nominal distances using r = cz/100 (in Mpc), and compute luminosities via L = f × r², with log L = log f + 2 log r.
• Define luminosity-limited samples using a limiting luminosity L* derived from the flux limit (0.60 Jy) and distance r*, such that log L* = log 0.60 + 2 log r*.
• Apply rhombic cell analysis to 8 samples spanning r* from 75 to 250 Mpc and L* over 2.6 magnitudes, with each sample constrained by r < r* and L > L*.
• Introduce four numerical indices: η (flocking index), χ₁, χ₂, χ₂₁ to summarize n₁ and τ distributions of like and unlike neighboring cells.
• Compute mean and standard deviation of indices over 16 independent cell zero-offsets to account for positional sensitivity, separately for filled and empty cells.

Experimental results

Research questions

• RQ1 How does the morphology of galaxy distributions—specifically the clustering of like cells—change as the luminosity threshold increases across the IRAS/PSCz Catalogue?
• RQ2 Is the observed variation in structural indices primarily driven by limiting luminosity or sampling depth?
• RQ3 To what extent does random selection of galaxies from a sample destroy structural features, as measured by the flocking index?
• RQ4 How do the structural characteristics of filled (galaxy-rich) and empty (galaxy-poor) regions differ, and how do they evolve with luminosity?
• RQ5 Can numerical indices such as η, χ₁, χ₂, χ₂₁ reliably summarize the topological structure of galaxy distributions in large-scale surveys?

Key findings

• The flocking index η for filled cells decreases significantly with increasing luminosity threshold, indicating a steady degradation of clustering structure across samples.
• The mean flocking index η for the most luminous sample (S-250) is 0.56, compared to 2.63 for the least luminous (S-75), showing a strong luminosity-driven structural degradation.
• The degrading effect is stronger for filled regions than for empty regions, suggesting that the universe appears more like 'lakes' than 'islands' at high luminosity limits.
• A random selection of galaxies from the S-100 sample (S**-100) yields a mean η of -0.24, significantly lower than the original S-100 sample (η = 2.63), demonstrating that random selection inherently reduces structural coherence.
• The observed variation in indices is primarily due to limiting luminosity rather than sampling depth, as confirmed by comparison with depth-limited samples and random selection tests.
• The space number density ρ(L*) for the samples ranges from 10.7 to 0.13 galaxies per (100 Mpc)³, confirming a 75-fold decrease in space density across the luminosity range studied.