Fisher matrix for multiple tracers: the information in the cross-spectra

TL;DR

This paper develops a comprehensive Fisher matrix framework for multiple tracers in cosmology, highlighting how cross-spectra can provide independent, cosmic-variance-free information that improves parameter constraints.

Contribution

It introduces general expressions for the multi-tracer Fisher matrix, including independent degrees of freedom in cross-spectra, and derives optimal survey strategies for multiple tracers.

Findings

Cross-spectra degrees of freedom are not limited by cosmic variance.

Uncertainties in cross-spectra decrease faster with tracer density than auto-spectra.

Provides formulas for optimal number of tracers in surveys.

Abstract

We derive general expressions for the multi-tracer Fisher matrix, both assuming that the cross-spectra are constrained by the auto-spectra, and also allowing for independent degrees of freedom in the cross-spectra. We show that, just like the ratios of power spectra, the independent degrees of freedom of the cross-spectra are also not constrained by cosmic variance. Moreover, whereas the uncertainties in the ratios of power spectra decrease with the number density of the tracers as $\sim 1/\sqrt{\bar{n}}$, the uncertainties in the independent degrees of freedom of the cross-spectra decrease even faster, as $\sim 1/\bar{n}$. We also derive simple expressions for the optimal number of tracers in a survey.