A cosmological exclusion plot: Towards model-independent constraints on modified gravity from current and future growth rate data

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)