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.

http://arxiv.org/icons/sfx.gif