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

Abstract

Abstract 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) = E(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 ≤ z ≤ 1.8 and show that the gain in precision when going from k max = 0.1 to 0.2 h/Mpc is around two- to threefold, while it reaches four- to ninefold when extending to k max = 0.3 h/Mpc. In absolute terms, with k max = 0.2 h/Mpc, one can reach high precision on E(z) at each z-shell: 8–10% for DESI with Δz = 0.1, 5–6% for Euclid with Δz = 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 max = 0.1 h/Mpc, unless one has independent strong prior information on the non-linear parameters.