Large-scale Modeling of the Observed Power Spectrum Multipoles

TL;DR

This work addresses the challenge of accurately modeling the observed power spectrum multipoles on the largest scales for wide-area galaxy surveys by developing a framework based on the discrete spherical Fourier-Bessel (dSFB) basis. It provides an exact mapping from the dSFB power spectrum to the conventional power spectrum multipoles (PSM), enabling non-perturbative treatment of wide-angle effects, redshift evolution, window convolution, and integral-constraint corrections. The approach leverages the separability of angular and radial modes in the dSFB basis, supports efficient precomputation of geometry-only mapping matrices, and is validated against large ensembles of lognormal mocks with realistic survey windows. The results demonstrate the significant impact of WA and IC on even high-order multipoles, the need for multipole-specific treatment of redshift evolution, and the practicality of using this framework for robustly constraining local primordial non-Gaussianity and related relativistic effects in all-sky, Stage-IV surveys.

Abstract

Current and upcoming large-scale structure surveys are pushing toward increasingly wide angular coverage, where wide-angle effects (arising from the varying line of sight across the curved sky) become critical for accurate modeling of the three-dimensional galaxy power spectrum. At the same time, these survey's broader redshift reach makes the effects of redshift evolution (beyond the effective-redshift approximation) non-negligible on large radial scales. Additional observational effects such as the survey window function and integral constraints also become significant on these large scales, necessitating a careful theoretical treatment to robustly constrain local primordial non-Gaussianities and relativistic effects. In this work, we present a consistent and accurate theoretical framework for modeling the commonly used power spectrum multipoles (PSM) on large scales using the discrete spherical Fourier-Bessel (dSFB) basis. This basis ensures numerical stability and allows an exact separation between angular and radial modes. Using the dSFB basis, we study the impact of wide-angle effects and redshift evolution on the PSM, and incorporate the effects of window function convolution and integral constraints. We validate our PSM modeling using lognormal mocks under radial integral constraints with realistic survey geometries, demonstrating the readiness of our framework for application to all-sky galaxy surveys.

Figures (16)

• Figure 1: The dSFB-to-gSFB mapping function $\mathcal{V}_{n\ell}^{a}(k)$ with $\ell=15$ and $n=2$ at a redshift bin $z=0.2-0.5$ as defined in Eq. \ref{['eq:d-to-g']} for different radial orders $a$ in the gSFB basis. The dashed line indicates the value of $k_{n\ell}$ for the dSFB mode considered. We see that the peak of the mapping function gets shifted away from the dSFB wavenumber $k_{n\ell}$ when the gSFB radial function takes a different order than the angular multipole of the dSFB mode ($a\neq\ell$).
• Figure 2: The dSFB-to-cSFB mapping function $\mathcal{V}_{n\ell}(k)$ as defined in Eq. \ref{['eq:Vlnk']} for different angular $\ell$ modes and radial $n$ modes at a redshift bin $z=0.2$ to $0.5$. The dashed lines indicate the value of $k_{n\ell}$ for each dSFB mode. The mapping functions peak at each of the corresponding Fourier $k_{n\ell}$ values of the dSFB mode.
• Figure 3: The diagonal and off-diagonal contribution of the dSFB power spectrum to the power spectrum monopole and quadrupole at a redshift bin $z=0.2$ to $0.5$. We show the relative deviation of the approximated PSM (including all off-diagonal contributions up to a certain $\Delta n=|n_1-n_2|$) from the true PSM. The grey band indicates the region within $1\%$ of the true PSM. The PS quadrupole has significantly more contributions from the off-diagonal SFB PS than the monopole.
• Figure 4: The power spectrum dipole, octupole, and hexadecapole obtained by summing all off-diagonal components of the dSFB PS up to a certain $\Delta n$ at a redshift bin $z=0.2-0.5$. The black dashed lines indicate the correct PSM calculated with all components of the dSFB PS. We plot the absolute PSM instead of the relative error as in the case of \ref{['fig:off_diag_contribute_02']}, since these multipoles either cross or approach zero, making the relative error a less effective indicator of accuracy.
• Figure 5: Comparison between the exact PSM modeled through the dSFB basis and the plane-parallel (p.p.) approximations for PS monopole, quadrupole, and hexadecapole for a uniform spherical shell of $z=0.2$ to $0.5$. The grey band in the bottom subplot indicates the region within $1\%$ of the true PSM. Here we fix the linear bias $b_1$, growth factor $D$ and growth rate $f$ to be constant across the redshift bin for both p.p. and projected SFB calculations such that we only illustrate the impact of WA effects without redshift evolution. WA effects become more prominent for higher multipoles.