[Paper Review] New leaves of the tree: percolation analysis for cosmic web with discrete points

Jiajun Zhang, Dalong Cheng+1 more|arXiv (Cornell University)|Aug 25, 2017

Galaxies: Formation, Evolution, Phenomena68 references5 citations

TL;DR

This paper introduces a novel continuum percolation analysis using discrete galaxy points via the Friends-of-Friends (FoF) algorithm to derive the S−bb relation, where S is the fractional mass of the largest connected group and bb is the linking length. It proposes the Probability Cloud Cluster Expansion Theory (PCCET) to link this relation to all-order correlation functions, demonstrating robustness against redshift distortion and observational incompleteness, while revealing limitations in Halo Abundance Matching (HAM) for preserving higher-order clustering beyond the two-point function.

ABSTRACT

Percolation analysis has long been used to quantify the connectivity of the cosmic web. Most of the previous work is based on density fields on grids. By smoothing into fields, we lose information about galaxy properties like shape or luminosity. Lack of mathematical model also limits our understanding of percolation analysis. In order to overcome these difficulties, we have studied percolation analysis based on discrete points. Using a Friends-of-Friends (FoF) algorithm, we generate the S-bb relation, between the fractional mass of the largest connected group (S) and the FoF linking length (bb). We propose a new model, the Probability Cloud Cluster Expansion Theory (PCCET) to relate the S-bb relation with correlation functions. We show that the S-bb relation reflects a combination of all orders of correlation functions. Using N-body simulation, we find that the S-bb relation is robust against redshift distortion and incompleteness in observation. From the Bolshoi simulation, with Halo Abundance Matching (HAM), we have generated a mock galaxy catalogue. Good matching of the projected two-point correlation function with observation is confirmed. However, comparing the mock catalogue with the latest galaxy catalogue from SDSS DR12, we have found significant differences in their S-bb relations. This indicates that the mock galaxy catalogue cannot accurately retain higher order correlation functions than the two-point correlation function, which reveals the limit of HAM method. As a new measurement, S-bb relation is applicable to a wide range of data types, fast to compute, robust against redshift distortion and incompleteness, and it contains information of all orders of correlation function.

Motivation & Objective

• To develop a percolation analysis method based on discrete galaxy points rather than smoothed density fields to preserve galaxy-specific properties like luminosity and shape.
• To overcome the limitations of traditional site percolation on grids, which lose point-level information and lack theoretical modeling.
• To propose a new theoretical framework, the Probability Cloud Cluster Expansion Theory (PCCET), to mathematically relate the S−bb relation to correlation functions of all orders.
• To test the robustness of the S−bb relation against observational effects such as redshift-space distortion and survey incompleteness.
• To evaluate the performance of Halo Abundance Matching (HAM) in reproducing the full clustering hierarchy by comparing mock and real galaxy catalogues.

Proposed method

• Apply the Friends-of-Friends (FoF) algorithm to discrete galaxy distributions to define connected groups based on a linking length bb.
• Define the S−bb relation as the fractional mass S of the largest connected group as a function of bb, serving as a new clustering diagnostic.
• Develop the Probability Cloud Cluster Expansion Theory (PCCET) to analytically relate the S−bb relation to the full set of correlation functions (up to all orders).
• Use the Bolshoi N-body simulation to generate a mock galaxy catalogue via Halo Abundance Matching (HAM), matching observed projected two-point correlation functions.
• Compare the S−bb relation in the HAM mock catalogue with that from the SDSS DR12 real galaxy catalogue to assess fidelity of higher-order clustering reproduction.
• Perform statistical analysis on both mock and real data to evaluate robustness under redshift-space distortion and survey incompleteness.

Experimental results

Research questions

• RQ1 Can the S−bb relation derived from discrete galaxy points serve as a robust and informative probe of cosmic web connectivity?
• RQ2 How does the S−bb relation relate to correlation functions of all orders, and can this relationship be theoretically modeled?
• RQ3 To what extent does the Halo Abundance Matching (HAM) method preserve higher-order clustering beyond the two-point function in mock galaxy catalogues?
• RQ4 How sensitive is the S−bb relation to observational systematics such as redshift-space distortion and survey incompleteness?
• RQ5 Why do mock catalogues generated via HAM fail to reproduce the S−bb relation observed in real data, despite matching the two-point function?

Key findings

• The S−bb relation is robust against redshift-space distortion and survey incompleteness, making it a reliable observational diagnostic.
• The S−bb relation reflects a combination of all orders of correlation functions, as formalized by the proposed Probability Cloud Cluster Expansion Theory (PCCET).
• Despite good agreement in the projected two-point correlation function, the HAM mock catalogue fails to reproduce the S−bb relation observed in SDSS DR12.
• This discrepancy indicates that HAM cannot accurately retain higher-order clustering information beyond the two-point function.
• The S−bb relation serves as a sensitive probe of the full clustering hierarchy and reveals limitations in current mock galaxy catalogue generation techniques.
• The continuum percolation approach using discrete points offers a fast, robust, and information-rich alternative to traditional grid-based percolation analysis.