Persistent Homology and Non-Gaussianity

TL;DR

This paper introduces topological persistence diagrams as a new statistical tool in Topological Data Analysis for analyzing CMB temperature maps, demonstrating higher sensitivity to local non-Gaussianity than previous methods.

Contribution

It presents the application of persistence diagrams to CMB data, showing improved sensitivity to primordial non-Gaussianity over traditional topological statistics.

Findings

Persistence diagrams are more sensitive to local non-Gaussianity than genus and Betti number curves.

They can constrain $NL^{ m loc}$ to 35.8 at 68% confidence, better than Betti numbers.

Expected to provide competitive constraints with Minkowski Functionals on observational data.

Abstract

In this paper, we introduce the topological persistence diagram as a statistic for Cosmic Microwave Background (CMB) temperature anisotropy maps. A central concept in `Topological Data Analysis' (TDA), the idea of persistence is to represent a data set by a family of topological spaces. One then examines how long topological features `persist' as the family of spaces is traversed. We compute persistence diagrams for simulated CMB temperature anisotropy maps featuring various levels of primordial non-Gaussianity of local type. Postponing the analysis of observational effects, we show that persistence diagrams are more sensitive to local non-Gaussianity than previous topological statistics including the genus and Betti number curves, and can constrain $\Delta f_{NL}^{\rm loc}= 35.8$ at the 68\% confidence level on the simulation set, compared to $\Delta f_{NL}^{\rm loc}= 60.6$ for the Betti number curves. Given the resolution of our simulations, we expect applying persistence diagrams to observational data will give constraints competitive with those of the Minkowski Functionals. This is the first in a series of papers where we plan to apply TDA to different shapes of non-Gaussianity in the CMB and Large Scale Structure.