http://arxiv.org/abs/1904.10514
The Cosmic Microwave Background Radiation (CMBR) represents the first light to travel during the early stages of the universe’s development. This sphere of relic radiation gives the strongest evidence for the Big Bang theory to date, and refined analysis of its angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. Satellites collect CMBR data over a sphere using a Hierarchical Equal Area isoLatitude Pixelation (HEALPix) grid. While this grid gives a quasiuniform discretization of a sphere, it is not well suited for doing fast \emph{and} accurate spherical harmonic analysis — a central component to computing and analyzing the angular power spectrum of the massive CMBR data sets. In this paper, we present a new method that overcomes these issues through a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky’s fast spherical harmonic transform (Slevinsky, 2017). The method has a quasi-optimal computational complexity of $\mathcal{O}(N\log^2 N)$ with an initial set-up cost of $\mathcal{O}(N^{3/2}\log N)$, where $N$ represents the number of points in the HEALPix grid. Additionally, we provide the first analysis of the method used in the current HEALPix software for computing the spherical harmonic coefficients. Numerical results illustrating the effectiveness of the new technique over the current method are also included.
K. Drake and G. Wright
Thu, 25 Apr 19
46/58
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