Topological Analysis of High-Resolution CMB Maps

TL;DR

This paper introduces new numerical tools for topological analysis of high-resolution all-sky CMB maps, including neighbor relationships, extrema search, and efficient simulation of gradient and curvature fields.

Contribution

It develops and releases the HEALFAST software package with algorithms for topological analysis and fast simulation of derivative fields on the sphere.

Findings

Neighbor relationship routines for HEALPIX tessellation.

Linear-scaling extrema search algorithm.

Efficient on-the-fly simulation of gradient and curvature fields.

Abstract

We report the development of numerical tools for the topological analysis of sub--degree resolution, all--sky maps. Software to be released in the HEALFAST (V0.9) package defines neighbour relationships for the HEALPIX tessellation of the sphere. We apply this routine to a fast extrema search which scales strictly linearly in the number of pixels, $N_{p}$. We also present a highly efficient algorithm for simulating the gradient vector and curvature tensor fields ``on--the--fly'' with the temperature map, needing only of order $N_{p} \log_2 N_{p}$ more operations.