Cosmology with Persistent Homology: a Fisher Forecast (2403.13985v2)

Abstract: Persistent homology naturally addresses the multi-scale topological characteristics of the large-scale structure as a distribution of clusters, loops, and voids. We apply this tool to the dark matter halo catalogs from the Quijote simulations, and build a summary statistic for comparison with the joint power spectrum and bispectrum statistic regarding their information content on cosmological parameters and primordial non-Gaussianity. Through a Fisher analysis, we find that constraints from persistent homology are tighter for 8 out of the 10 parameters by margins of 13-50%. The complementarity of the two statistics breaks parameter degeneracies, allowing for a further gain in constraining power when combined. We run a series of consistency checks to consolidate our results, and conclude that our findings motivate incorporating persistent homology into inference pipelines for cosmological survey data.

• The paper demonstrates that persistent homology yields tighter constraints (by 13-50%) on key cosmological parameters compared to conventional statistics.
• It employs innovative persistence diagrams vectorized into histograms to analyze dark matter halo data from Quijote simulations.
• Combining persistent homology with power spectrum methods breaks parameter degeneracies, enhancing overall inference accuracy.

Cosmology with Persistent Homology: A Fisher Forecast

The paper "Cosmology with Persistent Homology: a Fisher Forecast" explores the application of persistent homology—a tool from computational topology—for extracting cosmological information. It evaluates how well this method can constrain cosmological parameters and primordial non-Gaussianity when compared to more conventional methods like the power spectrum and bispectrum analysis.

Overview

Persistent homology provides a mechanism to quantify and analyze the topology of data across multiple scales, capturing features such as clusters, loops, and voids within the cosmic web. This paper applies persistent homology to dark matter halo catalogs derived from the Quijote simulations, comparing its efficacy to traditional joint power spectrum and bispectrum statistics.

Methodology

The core of this inquiry lies in developing summary statistics from persistence diagrams, which represent the birth and death of topological features detected in the halo distributions. Through a Fisher analysis that comprehensively compares these persistence-based statistics against conventional correlation function methods, the research identifies their respective capacities to constrain cosmological parameters and detect signatures of primordial non-Gaussianity.

1. Data and Filtration: The authors utilize the Quijote halo catalogs with various cosmological parameters, applying a novel $\alpha$-DTM$\ell$-filtration that incorporates local point density measurements to delay the addition of simplices for outliers.
2. Statistics from Persistence Diagrams: Persistent homology yields persistence diagrams which are vectorized into histograms of the number of features born and died at different scales. These histograms serve as the data vector in a Fisher information context.
3. Comparison Metrics: By constructing summary statistics over multiple nearest-neighbor parameters, the team assesses how much additional information persistent homology can recover, beyond traditional two- and three-point correlation functions.

Results and Discussion

The paper reports significant findings regarding the effectiveness of persistent homology compared to conventional statistics:

• Persistent homology achieves tighter constraints for 8 out of the 10 parameters, improving precision by 13-50%, particularly in relation to $\sigma_8$, total neutrino mass, and dark energy equation of state parameters.
• The combination of statistics from both persistent homology and power spectrum methodologies leads to substantial improvements in parameter constraints, demonstrating the complementary nature of the information captured by these methods.
• The analysis reveals distinct degeneracy directions between parameters when using persistent homology, allowing it to break degeneracies better when combined with traditional metrics.

The paper also tackles the challenge of analyzing intricate non-linear structures inherent in the large-scale universe, which often evade traditional statistical methods. The persistent homology approach provides a promising avenue for cosmological analysis because of its inherent multi-scale sensitivity to topological features.

Implications and Future Prospects

The research underscores the potential role of persistent homology in future cosmological surveys, where extracting maximum information from observational data is paramount. It suggests several pathways for further inquiry:

• Simulation-Driven Optimization: As computational resources improve, fully leveraging persistent homology for cosmological inference may be achieved through simulation-driven optimizations, potentially combined with machine learning techniques.
• Extensions to Real Data: Beyond simulations, the methods proposed could handle realistic galaxy surveys, taking into account observational constraints like redshift errors and sample biases.
• Integration into Broader Inference Pipelines: Persistent homology could be embedded within broader machine-learning frameworks aimed at improving efficiency in cosmological parameter inference.

This paper represents a significant stride in applying topological data analysis within cosmology, illuminating how persistent homology can enrich our understanding of cosmic structures while providing robust, complementary insights to prevailing analytical methods.