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 $H(z)/H_0$ and the growth rate $f(k,z)$, besides the bias
functions. We apply the technique to both Euclid and DESI public specifications
in the redshift range $0.6-1.8$ and show that the precision on $H(z)$ from
increasing the cut-off scale improves abruptly for $k_{\rm max} > 0.17\,h$/Mpc
and reaches subpercent values for $k_{\rm max} \approx 0.3\,h$/Mpc. Overall,
the gain in precision when going from $k_{\rm max} = 0.1\,h$/Mpc to $k_{\rm
max} = 0.3\,h$/Mpc is around one order of magnitude. The growth rate has in
general much weaker constraints, unless is assumed to be $k$-independent. In
such case, the gain is similar to the one for $H(z)$ and one can reach
uncertainties around 5--10\% 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.
