A Fast and Accurate Algorithm for Spherical Harmonic Analysis of the Cosmic Microwave Background Radiation. (arXiv:1904.10514v1 [math.NA])

A Fast and Accurate Algorithm for Spherical Harmonic Analysis of the Cosmic Microwave Background Radiation. (arXiv:1904.10514v1 [math.NA])
<a href="http://arxiv.org/find/math/1/au:+Drake_K/0/1/0/all/0/1">Kathryn P. Drake</a>, <a href="http://arxiv.org/find/math/1/au:+Wright_G/0/1/0/all/0/1">Grady B. Wright</a>

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 $mathcal{O}(Nlog^2
N)$ 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.

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 $mathcal{O}(Nlog^2
N)$ 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.

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