Document Type

Article

Publication Date

9-1-2020

Abstract

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a central component to computing and analyzing the angular power spectrum of the massive CMB data sets. The method uses a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky's fast spherical harmonic transform. For a HEALPix grid with N pixels (points), the computational complexity of the method is O(N log2 N), with an initial set-up cost of O(N log3/2 N). This compares favorably with O(N3/2) runtime complexity of the current methods available in the HEALPix software when multiple maps need to be analyzed at the same time. Using numerical experiments, we demonstrate that the new method also appears to provide better accuracy over the entire angular power spectrum of synthetic data when compared to the current methods, with a convergence rate at least two times higher.

Copyright Statement

This is an author-produced, peer-reviewed version of this article. © 2020, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International license. The final, definitive version of this document can be found online at Journal of Computational Physics, https://doi.org/10.1016/j.jcp.2020.109544

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