Fisher matrix for the one-loop galaxy power spectrum: measuring expansion and growth rates without assuming a cosmological model

We introduce a methodology to extend the Fisher matrix forecasts to mildly non-linear scales without the need of selecting a cosmological model. We make use of standard non-linear perturbation theory for biased tracers complemented by counterterms, and assume that the cosmological distances can be measured accurately with standard candles. Instead of choosing a specific model, we parametrize the linear power spectrum and the growth rate in several $k$ and $z$ bins. We show that one can then obtain model-independent constraints of the expansion rate $E(z)=H(z)/H_0$ and the growth rate $f(k,z)$, besides the bias functions. We apply the technique to both Euclid and DESI public specifications in the range $0.6\le z \le 1.8$ and show that the gain in precision when going from $k_{\rm max} = 0.1$ to $0.2\,h$/Mpc is around two- to threefold, while it reaches four- to ninefold when extending to $k_{\rm max} = 0.3\,h$/Mpc. In absolute terms, with $k_{\rm max}=0.2\,h/$Mpc, one can reach high precision on $E(z)$ at each $z$-shell: 8-10% for DESI with $\Delta z=0.1$, 5-6% for Euclid with $\Delta z=0.2-0.3$. This improves to 1-2% if the growth rate $f$ is taken to be $k$-independent. The growth rate itself has in general much weaker constraints, unless assumed to be $k$-independent, in which case the gain is similar to the one for $E(z)$ and uncertainties around 5-15% can be reached at each $z$-bin. We also discuss how neglecting the non-linear corrections can have a large effect on the constraints even for $k_{\rm max}=0.1\,h/$Mpc, unless one has independent strong prior information on the non-linear parameters.