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

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.

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