[Paper Review] First Release of Gauss-Legendre Sky Pixelization (GLESP) software package for CMB analysis

А. G. Doroshkevich, Pavel Naselsky+6 more|arXiv (Cornell University)|Jan 23, 2005

Astronomy and Astrophysical Research1 references3 citations

TL;DR

This paper introduces GLESP v1.0, a software package for cosmic microwave background (CMB) analysis based on Gauss-Legendre sky pixelization, enabling accurate spherical harmonic decomposition of CMB maps using orthogonal expansions via Legendre polynomial zeros. The key contribution is a robust, cross-platform toolkit for CMB map manipulation, harmonic analysis, and conversion between HEALPix and GLESP formats, with polarization support planned for future releases.

ABSTRACT

We report the release of the Gauss--Legendre Sky Pixelization (GLESP) software package version 1.0. In this report we present the main features and functions for processing and manipulation of sky signals. Features for CMB polarization is underway and to be incorporated in a future release. Interested readers can visit http://www.glesp.nbi.dk (www.glesp.nbi.dk) and register for receiving the package.

Motivation & Objective

• To provide a reliable, open-source software package for processing and analyzing CMB sky maps using the Gauss-Legendre Sky Pixelization (GLESP) scheme.
• To enable accurate spherical harmonic decomposition and synthesis of CMB temperature anisotropies via orthogonal expansion based on Legendre polynomial zeros.
• To support simulation of Gaussian CMB maps from given angular power spectra $C_\ell$ or $D_\ell$
• To facilitate interoperability between GLESP and HEALPix formats through bidirectional map conversion tools.
• To lay the foundation for future integration of CMB polarization analysis features in subsequent releases.

Proposed method

• The GLESP method uses zeros of associated Legendre polynomials to define pixel positions on the sphere, ensuring strict orthogonality in spherical harmonic expansion.
• The cl2map tool performs forward and inverse spherical harmonic transforms between pixelized maps and $a_{\ell m}$ coefficients.
• The rsalm tool applies beam convolution/deconvolution via $a'_{\ell m} = B_\ell a_{\ell m}$, with Tikhonov regularization for deconvolution.
• The f2fig tool converts GLESP FITS maps to GIF format with coordinate-based marker placement in multiple systems (equatorial, Galactic, spherical).
• The cmap and f2map tools enable bidirectional conversion between HEALPix and GLESP formats using spline interpolation or direct mapping.
• The ntot tool computes pixel count and resolution parameters for both GLESP and HEALPix schemes based on user-specified resolution.

Experimental results

Research questions

• RQ1 How can a sky pixelization scheme based on Gauss-Legendre quadrature points enable accurate and orthogonal spherical harmonic decomposition of CMB maps?
• RQ2 What are the essential computational tools required for end-to-end CMB data processing using the GLESP framework?
• RQ3 How can GLESP maps be effectively converted to and from HEALPix format while preserving map fidelity?
• RQ4 What are the performance characteristics and cross-platform compatibility of the GLESP software package?
• RQ5 What are the limitations of the current v1.0 release in handling CMB polarization, and how can they be addressed in future versions?

Key findings

• The GLESP software package v1.0 successfully implements a complete pipeline for CMB temperature map analysis, including harmonic decomposition, map synthesis, and beam convolution.
• The cl2map tool accurately generates Gaussian CMB maps from input $C_\ell$ or $D_\ell$ spectra by assigning independent Gaussian-distributed $a_{\ell m}$ coefficients with variance $\sqrt{C_\ell/2}$.
• The rsalm tool enables precise beam convolution and deconvolution using $B_\ell$ profiles or FWHM-based Gaussian beam models with Tikhonov regularization.
• The f2map and cmap tools provide reliable conversion between HEALPix and GLESP formats, supporting both spline interpolation and direct mapping.
• The package supports multiple coordinate systems for map visualization and position-based temperature queries, enhancing usability for researchers.
• The software is confirmed to operate on RedHat Linux, SGI Irix, and Solaris, ensuring broad cross-platform compatibility.