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High-precision cosmology requires the analysis of large-scale surveys in 3D spherical coordinates, i.e. spherical Fourier-Bessel decomposition. Current methods are insufficient for future data-sets from wide-field cosmology surveys. The aim of this paper is to present a public code for fast spherical Fourier-Bessel decomposition that can be applied to cosmological data or 3D data in spherical coordinates in other scientific fields. We present an equivalent formulation of the spherical Fourier-Bessel decomposition that separates radial and tangential calculations. We propose the use of the existing pixelisation scheme HEALPix for a rapid calculation of the tangential modes. 3DEX (3D EXpansions) is a public code for fast spherical Fourier-Bessel decomposition of 3D all-sky surveys that takes advantage of HEALPix for the calculation of tangential modes. We perform tests on very large simulations and we compare the precision and computation time of our method with an optimised implementation of the spherical Fourier-Bessel original formulation. For surveys with millions of galaxies, computation time is reduced by a factor 4-12 depending on the desired scales and accuracy. The formulation is also suitable for pre-calculations and external storage of the spherical harmonics, which allows for additional speed improvements. The 3DEX code can accommodate data with masked regions of missing data. 3DEX can also be used in other disciplines, where 3D data are to be analysed in spherical coordinates. The code and documentation can be downloaded at http://ixkael.com/blog/3dex.
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The spherical Fourier-Bessel (SFB) decomposition is a natural choice for the radial/angular separation that allows optimal extraction of cosmological information from large volume galaxy surveys. In this paper we develop a SFB power spectrum estimator that allows the measurement of the largest angular and radial modes with the next generation of galaxy surveys. The code measures the pseudo-SFB power spectrum, and takes into account mask, selection function, pixel window, and shot noise. We show that the local average effect is significant only in the largest-scale mode, and we provide an analytical covariance matrix. By imposing boundary conditions at the minimum and maximum radius encompassing the survey volume, the estimator does not suffer from the numerical instabilities that have proven challenging in the past. The estimator is demonstrated on simplified Roman-like, SPHEREx-like, and Euclid-like mask and selection functions. For intuition and validation, we also explore the SFB power spectrum in the Limber approximation. We release the associated public code written in Julia.
Cosmological random fields are often analysed in spherical Fourier-Bessel basis. Compared to the Cartesian Fourier basis this has an advantage of properly taking into account some of the relevant physical processes (redshift-space distortions, redshift evolution). The observations usually come in redshift slices and have a partial sky coverage. These masking effects strongly correlate Fourier-Bessel modes that are meant for a perfect spherical geometry and result in a lot of redundant measurements. This work proposes a new Fourier basis that is better suited for measurements in redshift shells and results in fewer Fourier modes, with the radial modes strictly uncorrelated on large scales and the angular modes with significantly reduced redundancy. I argue that the spherical Fourier analysis of cosmological fields should always use these new modes instead of the historically established Fourier-Bessel eigenfunctions. The new angular modes on the other hand have number of practical advantages and disadvantages and whether or not to adopt them for a particular analysis should be made on a case by case basis.
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We present an efficient and accurate algorithm for principal component analysis (PCA) of a large set of two dimensional images, and, for each image, the set of its uniform rotations in the plane and its reflection. The algorithm starts by expanding each image, originally given on a Cartesian grid, in the Fourier-Bessel basis for the disk. Because the images are bandlimited in the Fourier domain, we use a sampling criterion to truncate the Fourier-Bessel expansion such that the maximum amount of information is preserved without the effect of aliasing. The constructed covariance matrix is invariant to rotation and reflection and has a special block diagonal structure. PCA is efficiently done for each block separately. This Fourier-Bessel based PCA detects more meaningful eigenimages and has improved denoising capability compared to traditional PCA for a finite number of noisy images.
Future precision cosmology from large-scale structure experiments including the Dark Energy Spectroscopic Instrument (DESI) and Euclid will probe wider and deeper cosmic volumes than those covered by previous surveys. The Cartesian power spectrum analysis of anisotropic galaxy clustering based on the Fourier plane wave basis makes a number of assumptions, including the local plane-parallel approximation, that will no longer be valid on very large scales and may degrade cosmological constraints. We propose an approach that utilises a hybrid basis: on the largest scales, clustering statistics are decomposed into spherical Fourier modes which respect the natural geometry of both survey observations and physical effects along the line of sight, such as redshift-space distortions, the Alcock--Paczynsky and light-cone effects; on smaller scales with far more clustering modes, we retain the computational benefit of the power spectrum analysis aided by fast Fourier transforms. This approach is particularly suited to the likelihood analysis of local primordial non-Gaussianity $f_textrm{NL}$ through the scale-dependent halo bias, and we demonstrate its applicability with $N$-body simulations. We also release our public code Harmonia (https://github.com/MikeSWang/Harmonia) for galaxy clustering likelihood inference in spherical Fourier or hybrid-basis analyses.
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