The full Fisher matrix for galaxy surveys

TL;DR

This paper derives a comprehensive Fisher matrix framework for galaxy surveys, incorporating multiple tracers, bias, and growth functions, extending previous models and providing exact calculations for isotropic, volume-limited surveys.

Contribution

The paper introduces a full Fisher matrix derivation for multi-tracer galaxy surveys, including cross-terms and functions of position, generalizing previous simplified models.

Findings

Derived the full Fisher matrix for multiple tracers.

Connected the new Fisher matrix to the classic Feldman-Kaiser-Peacock result.

Provided a method to compute the Fisher matrix for general survey geometries.

Abstract

Starting from the Fisher matrix for counts in cells, I derive the full Fisher matrix for surveys of multiple tracers of large-scale structure. The key assumption is that the inverse of the covariance of the galaxy counts is given by the naive matrix inverse of the covariance in a mixed position-space and Fourier-space basis. I then compute the Fisher matrix for the power spectrum in bins of the three-dimensional wavenumber k; the Fisher matrix for functions of position x (or redshift z) such as the linear bias of the tracers and/or the growth function; and the cross-terms of the Fisher matrix that expresses the correlations between estimations of the power spectrum and estimations of the bias. When the bias and growth function are fully specified, and the Fourier-space bins are large enough that the covariance between them can be neglected, the Fisher matrix for the power spectrum reduces to the widely used result that was first derived by Feldman, Kaiser and Peacock (1994). Assuming isotropy, an exact calculation of the Fisher matrix can be performed in the case of a constant-density, volume-limited survey. I then show how the exact Fisher matrix in the general case can be obtained in terms of a series of volume-limited surveys.