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

Abstract

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 ∼ 1/√n̅, the uncertainties in the independent degrees of freedom of the cross-spectra decrease even faster, as ∼ 1/n̅. We also derive simple expressions for the optimal number of tracers in a survey.