  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.
