Flaglets for studying the large-scale structure of the Universe

TL;DR

This paper introduces the Fourier-Laguerre transform and flaglets, novel tools for analyzing 3D spherical data in cosmology, enabling precise, localized, scale-dependent analysis of galaxy surveys to study the Universe's large-scale structure.

Contribution

It presents the construction and application of Fourier-Laguerre transforms and flaglets for analyzing galaxy survey data in cosmology, with exact reconstruction capabilities.

Findings

Exact Fourier-Laguerre and flaglet transforms achieved.

Potential application demonstrated for void detection in galaxy surveys.

Enables scale-dependent, localized analysis of large-scale structure.

Abstract

Pressing questions in cosmology such as the nature of dark matter and dark energy can be addressed using large galaxy surveys, which measure the positions, properties and redshifts of galaxies in order to map the large-scale structure of the Universe. We review the Fourier-Laguerre transform, a novel transform in 3D spherical coordinates which is based on spherical harmonics combined with damped Laguerre polynomials and appropriate for analysing galaxy surveys. We also recall the construction of flaglets, 3D wavelets obtained through a tiling of the Fourier-Laguerre space, which can be used to extract scale-dependent, spatially localised features on the ball. We exploit a sampling theorem to obtain exact Fourier-Laguerre and flaglet transforms, such that band-limited signals can analysed and reconstructed at floating point accuracy on a finite number of voxels on the ball. We present a potential application of the flaglet transform for finding voids in galaxy surveys and studying the large-scale structure of the Universe.