[Paper Review] Fisher matrix for the angular power spectrum of multi-tracer galaxy surveys

L. Raul Abramo, João Vitor Dinarte Ferri+2 more|arXiv (Cornell University)|Apr 11, 2022

Galaxies: Formation, Evolution, PhenomenaPhysics and Astronomy76 references12 citations

TL;DR

This paper presents a semi-analytical method to compute the multi-tracer Fisher matrix for the angular power spectrum in both real and redshift space, without relying on the Limber approximation. By deriving exact analytical expressions for the inverse data covariance in the linear regime, it enables efficient forecasting of cosmological constraints from large-scale structure surveys with many tracers and thin redshift slices, overcoming numerical instabilities in high-dimensional covariance matrices.

ABSTRACT

Redshift evolution and peculiar velocities break the isotropy of cosmological surveys with respect to the directions parallel and transverse to the line of sight, limiting the accuracy of the Fourier representation to small areas and redshift ranges. In contrast to the Fourier space power spectrum, the full information about the two-point function of tracers of large-scale structure is encapsulated in the redshift-dependent angular power spectrum $C_\ell^{ij} (z_i,z_j)$ for the tracer species $i$ and $j$ at the redshift slices $z_i$ and $z_j$, expressed in harmonic space. In this paper we derive semi-analytical expressions for the multi-tracer Fisher matrix of angular power spectra, in real and in redshift space, which are exact in the linear regime of structure formation. Our expressions can be used to forecast the constraining power of galaxy surveys with many tracers and a large number of redshift slices, for which the derivation of the Fisher matrix from numerically evaluated covariance matrices may not be feasible or practical.

Motivation & Objective

• To develop a computationally efficient method for forecasting cosmological constraints from large-scale structure surveys with many tracers and thin redshift slices.
• To address the numerical challenges of inverting high-dimensional covariance matrices for angular power spectra in multi-tracer surveys.
• To provide an analytical framework for the inverse data covariance matrix in harmonic space, valid in the linear regime of structure formation.
• To enable accurate forecasting of ultra-large scale phenomena such as primordial non-Gaussianities and relativistic effects using angular power spectra.
• To offer a practical alternative to simulation-based covariance estimation for Fisher matrix forecasts in surveys with complex redshift and tracer configurations.

Proposed method

• Derives the data covariance matrix for angular power spectra in harmonic space, including shot noise and cross-correlations between multiple tracers.
• Inverts the data covariance matrix analytically in real space using exact expressions for radial integrals of spherical Bessel functions.
• Extends the formalism to redshift space by incorporating redshift-space distortions via the streaming velocity field, maintaining analytical control over radial mode coupling.
• Derives a semi-analytical expression for the inverse covariance (Fisher matrix) in redshift space, valid in the linear regime, without using the Limber approximation.
• Validates the method using a toy model with analytical solutions, confirming consistency with numerical results.
• Proposes that the Fisher matrix can be used both for forecasting and as a prior in MCMC likelihood analyses, especially in high-dimensional parameter spaces.

Experimental results

Research questions

• RQ1 How can the Fisher matrix for multi-tracer angular power spectra be computed efficiently in the linear regime without relying on numerical covariance inversion?
• RQ2 What is the analytical structure of the inverse data covariance matrix in harmonic space when redshift-space distortions are included?
• RQ3 How does the absence of the Limber approximation affect the radial mode coupling in the angular power spectrum covariance?
• RQ4 In what survey configurations does the semi-analytical Fisher matrix outperform simulation-based methods in terms of computational cost and stability?
• RQ5 Can the derived Fisher matrix be used reliably as a proxy for uncertainties in angular power spectrum measurements for MCMC likelihood exploration?

Key findings

• The paper derives an exact analytical expression for the inverse data covariance matrix in real space for multi-tracer angular power spectra, valid in the linear regime of structure formation.
• The method avoids the Limber approximation, enabling an exact semi-analytical treatment of radial mode coupling through the radial mixing matrix.
• In redshift space, the inverse covariance (Fisher matrix) is derived as a semi-analytical expression involving three double integrals of spherical Bessel functions, which can be evaluated numerically in ~50 CPU hours on a single core.
• The approach is particularly advantageous for surveys with many tracers and thin redshift slices, where numerical inversion of high-dimensional covariance matrices becomes unstable or infeasible.
• The Fisher matrix can be used directly in forecasting pipelines or as a mass matrix in Hamiltonian Monte Carlo samplers, improving sampling efficiency.
• The method is validated via a toy model, showing excellent agreement between analytical and numerical solutions, confirming the correctness of the derived expressions.