Fisher matrix for the one-loop galaxy power spectrum: measuring expansion and growth rates without assuming a cosmological model
Luca Amendola, Massimo Pietroni, Miguel Quartin
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 Δ z = 0.1, 5–6% for Euclid with Δ 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 max = 0.1 h /Mpc, unless one has independent strong prior information on the non-linear parameters.