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Fisher matrix for multiple tracers: all you can learn from large-scale structure without assuming a model

The galaxy power spectrum is one of the central quantities in cosmology. It contains information about the primordial inflationary process, the matter clustering, the baryon-photon interaction, the effects of gravity, the galaxy-matter bias, the cosmic expansion, the peculiar velocity field, etc.. Most of this information is however difficult to extract without assuming a specific cosmological model, for instance $Λ$CDM and standard gravity. In this paper we explore instead how much information can be obtained that is independent of the cosmological model, both at background and linear perturbation level. We determine the full set of model-independent statistics that can be constructed by combining two redshift bins and two distinct tracers. We focus in particular on the statistics $r(k,z_1,z_2)$, defined as the ratio of $fσ_8(z)$ at two redshift shells, and we show how to estimate it with a Fisher matrix approach. Finally, we forecast the constraints on $r$ that can be achieved by future galaxy surveys, and compare it with the standard single-tracer result. We find that $r$ can be measured with a precision from 3 to 11%, depending on the survey. Using two tracers, we find improvements in the constraints up to a factor of two.