  Most cosmological constraints on modified gravity are obtained assuming that
the cosmic evolution was standard $\Lambda$CDM in the past and that the present
matter density and power spectrum normalization are the same as in a
$\Lambda$CDM model. Here we examine how the constraints change when these
assumptions are lifted. We focus in particular on the parameter $Y$ (also
called $G_{\mathrm{eff}}$) that quantifies the deviation from the Poisson
equation. This parameter can be estimated by comparing with the
model-independent growth rate quantity $f\sigma_{8}(z)$ obtained through
redshift distortions. We reduce the model dependency in evaluating $Y$ by
marginalizing over $\sigma_{8}$ and over the initial conditions, and by
absorbing the degenerate parameter $\Omega_{m,0}$ into $Y$. We use all
currently available values of $f\sigma_{8}(z)$. We find that the combination
$\hat{Y}=Y\Omega_{m,0}$, assumed constant in the observed redshift range, can
be constrained only very weakly by current data, $\hat{Y}=0.28_{-0.23}^{+0.35}$
at 68\% c.l. We also forecast the precision of a future estimation of $\hat{Y}$
in a Euclid-like redshift survey. We find that the future constraints will
reduce substantially the uncertainty, $\hat{Y}=0.30_{-0.09}^{+0.08}$ , at 68\%
c.l., but the relative error on $\hat{Y}$ around the fiducial remains quite
high, of the order of 30\%. The main reason for these weak constraints is that
$\hat{Y}$ is strongly degenerate with the initial conditions, so that large or
small values of $\hat{Y}$ are compensated by choosing non-standard initial
values of the derivative of the matter density contrast.
  Finally, we produce a forecast of a cosmological exclusion plot on the Yukawa
strength and range parameters, which complements similar plots on laboratory
scales but explores scales and epochs reachable only with large-scale galaxy
surveys. (abridged)

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology