Fisher matrix for multiple tracers: model independent constraints on the redshift distortion parameter

We show how to obtain constraints on $\beta =f\mathrm{/}b$, the ratio of the matter growth rate and the bias that quantifies the linear redshift-space distortions, that are independent of the cosmological model, using multiple tracers of large-scale structure. For a single tracer the uncertainties on $\beta$ are constrained by the uncertainties in the amplitude and shape of the power spectrum, which is limited by cosmic variance. However, for two or more tracers this limit does not apply, since taking the ratio of power spectra cosmic variance cancels out, and in the linear (Kaiser) approximation one measures directly the quantity $(1+{\beta }_1{\mu }^2{)}^2\mathrm{/}(1+{\beta }_2{\mu }^2{)}^2$, where $\mu$ is the angle of a given mode with the line of sight. We provide analytic formulae for the Fisher matrix for one and two tracers, and quantify the signal-to-noise ratio needed to make effective use of the multiple-tracer technique. We also forecast the errors on $\beta$ for a survey like Euclid.