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Hierarchical Clustering in $Λ$CDM Cosmologies via Persistence Energy

Michael Etienne Van Huffel, Leonardo Aldo Alejandro Barberi, Tobias Sagis

TL;DR

Problem: Understanding the evolving cosmic web under $\Lambda$CDM requires tools that capture topology across scales. Approach: The paper combines DTFE-based density fields, cubical filtrations, and LITE-embedded persistence diagrams to compute Persistence Energy over redshift. Contributions: It demonstrates that Persistence Energy tracks hierarchical clustering, showing energy decreases and rightward birth shifts for $H_0$, $H_1$, and $H_2$ diagrams; provides an open-source implementation. Significance: The method offers a robust, scalable, topology-driven lens on cosmic structure formation with potential as a diagnostic for cosmological models.

Abstract

In this research, we investigate the structural evolution of the cosmic web, employing advanced methodologies from Topological Data Analysis. Our approach involves leveraging LITE, an innovative method from recent literature that embeds persistence diagrams into elements of vector spaces. Utilizing this methodology, we analyze three quintessential cosmic structures: clusters, filaments, and voids. A central discovery is the correlation between \textit{Persistence Energy} and redshift values, linking persistent homology with cosmic evolution and providing insights into the dynamics of cosmic structures.

Hierarchical Clustering in $Λ$CDM Cosmologies via Persistence Energy

TL;DR

Problem: Understanding the evolving cosmic web under CDM requires tools that capture topology across scales. Approach: The paper combines DTFE-based density fields, cubical filtrations, and LITE-embedded persistence diagrams to compute Persistence Energy over redshift. Contributions: It demonstrates that Persistence Energy tracks hierarchical clustering, showing energy decreases and rightward birth shifts for , , and diagrams; provides an open-source implementation. Significance: The method offers a robust, scalable, topology-driven lens on cosmic structure formation with potential as a diagnostic for cosmological models.

Abstract

In this research, we investigate the structural evolution of the cosmic web, employing advanced methodologies from Topological Data Analysis. Our approach involves leveraging LITE, an innovative method from recent literature that embeds persistence diagrams into elements of vector spaces. Utilizing this methodology, we analyze three quintessential cosmic structures: clusters, filaments, and voids. A central discovery is the correlation between \textit{Persistence Energy} and redshift values, linking persistent homology with cosmic evolution and providing insights into the dynamics of cosmic structures.
Paper Structure (16 sections, 1 equation, 9 figures)

This paper contains 16 sections, 1 equation, 9 figures.

Table of Contents

  1. Introduction
  2. The Cosmic Web
  3. Data Collection and Sources
  4. Related Work and Contribution
  5. Background
  6. Methodology
  7. DTFE
  8. Filtration of subhalo galaxies positions
  9. Embedding of Persistence Diagrams
  10. Lattice Integrated Topological Embedding - (LITE)
  11. Persistence Energy
  12. Results
  13. Conclusions and further work
  14. Conclusions
  15. Further work
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Overarching methodology presented in this paper.
  • Figure 2: DTFE log-density estimation at various redshifts.
  • Figure 3: DPDs of $H_{1}$ at different redshifts. Heatmap scale is in logarithm.
  • Figure 4: Persistence energy variation with redshifts and Gabor parameters for $H_1$ diagram.
  • Figure 5: log-Persistence energy for the first three homology groups, using the identity function as a transform.
  • ...and 4 more figures