All-Sky Analysis of the General Relativistic Galaxy Power Spectrum

TL;DR

This paper develops a spherical Fourier analysis framework for the all-sky relativistic galaxy power spectrum, capturing large-scale effects and relativistic corrections for cosmological measurements.

Contribution

It introduces a spherical Fourier decomposition method that includes relativistic effects, connecting the spherical power spectrum with the standard 3D spectrum for all-sky galaxy surveys.

Findings

Spherical power spectrum recovers the 3D power spectrum at each wavenumber.

Relativistic effects like gravitational lensing are incorporated into the analysis.

Formulas for computing the relativistic spherical galaxy power spectrum are provided.

Abstract

We perform an all-sky analysis of the general relativistic galaxy power spectrum using the well-developed spherical Fourier decomposition. Spherical Fourier analysis expresses the observed galaxy fluctuation in terms of the spherical harmonics and spherical Bessel functions that are angular and radial eigenfunctions of the Helmholtz equation, providing a natural orthogonal basis for all-sky analysis of the large-scale mode measurements. Accounting for all the relativistic effects in galaxy clustering, we compute the spherical power spectrum and its covariance matrix and compare it to the standard three-dimensional power spectrum to establish a connection. The spherical power spectrum recovers the three-dimensional power spectrum at each wavenumber k with its angular dependence mu_k encoded in angular multipole l, and the contributions of the line-of-sight projection to galaxy clustering such as the gravitational lensing effect can be readily accommodated in the spherical Fourier analysis. A complete list of formulas for computing the relativistic spherical galaxy power spectrum is also presented.