Persistent homology of the cosmic web. I: Hierarchical topology in $\Lambda$CDM cosmologies

TL;DR

This paper applies multiscale topological data analysis to $\ ext{Lambda}$CDM simulations, revealing how the cosmic web's topology evolves and links to structure formation, with persistence diagrams providing detailed insights beyond traditional metrics.

Contribution

It establishes the first connection between persistence diagrams and cosmic structure formation, demonstrating their effectiveness in capturing hierarchical topology evolution.

Findings

Persistence diagrams reveal key transitions in cosmic structure formation.

Apexes in Betti number curves relate to the collapse and merging of cosmic structures.

Persistence diagrams outperform traditional topology metrics in describing cosmic web evolution.

Abstract

Using a set of $\Lambda$CDM simulations of cosmic structure formation, we study the evolving connectivity and changing topological structure of the cosmic web using state-of-the-art tools of multiscale topological data analysis (TDA). We follow the development of the cosmic web topology in terms of the evolution of Betti number curves and feature persistence diagrams of the three (topological) classes of structural features: matter concentrations, filaments and tunnels, and voids. The Betti curves specify the prominence of features as a function of density level, and their evolution with cosmic epoch reflects the changing network connections between these structural features. The persistence diagrams quantify the longevity and stability of topological features. In this study we establish, for the first time, the link between persistence diagrams, the features they show, and the gravitationally driven cosmic structure formation process. By following the diagrams' development over cosmic time, the link between the multiscale topology of the cosmic web and the hierarchical buildup of cosmic structure is established. The sharp apexes in the diagrams are intimately related to key transitions in the structure formation process. The apex in the matter concentration diagrams coincides with the density level at which, typically, they detach from the Hubble expansion and begin to collapse. At that level many individual islands merge to form the network of the cosmic web and a large number of filaments and tunnels emerge to establish its connecting bridges. The location trends of the apex possess a self-similar character that can be related to the cosmic web's hierarchical buildup. We find that persistence diagrams provide a significantly higher and more profound level of information on the structure formation process than more global summary statistics like Euler characteristic or Betti numbers.