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

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Most cosmological constraints on modified gravity are obtained assuming that the cosmic evolution was standard ΛCDM in the past and that the present matter density and power spectrum normalization are the same as in a ΛCDM model. Here we examine how the constraints change when these assumptions are lifted. We focus in particular on the parameter Y (also called Geff) that quantifies the deviation from the Poisson equation. This parameter can be estimated by comparing with the model-independent growth rate quantity fσ₈(z) obtained through redshift distortions. We reduce the model dependency in evaluating Y by marginalizing over σ₈ and over the initial conditions, and by absorbing the degenerate parameter Ωₘ,₀ into Y. We use all currently available values of fσ₈(z). We find that the combination Y=YΩₘ,₀, assumed constant in the observed redshift range, can be constrained only very weakly by current data, Y=0.28₋₀.₂₃⁺⁰.³⁵ at 68% c.l. We also forecast the precision of a future estimation of Y in a Euclid-like redshift survey. We find that the future constraints will reduce substantially the uncertainty, Y=0.30₋₀.₀₉⁺⁰.⁰⁸, at 68% c.l., but the relative error on Y around the fiducial remains quite high, of the order of 30%. The main reason for these weak constraints is that Y is strongly degenerate with the initial conditions, so that large or small values of 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)