Fisher matrix for the angular power spectrum of multi-tracer galaxy surveys

TL;DR

This paper derives semi-analytical Fisher matrix expressions for multi-tracer angular power spectra in galaxy surveys, enabling efficient forecasts of survey constraining power in the linear regime.

Contribution

It provides exact, semi-analytical formulas for the Fisher matrix of angular power spectra, improving forecasting for multi-tracer galaxy surveys with many redshift slices.

Findings

Exact Fisher matrix expressions derived for real and redshift space.

Facilitates forecasting for surveys with many tracers and slices.

Applicable in the linear regime of structure formation.

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.