Euclid preparation. Constraining parameterised models of modifications of gravity with the spectroscopic and photometric primary probes

The Euclid mission has the potential to understand the fundamental physical nature of late-time cosmic acceleration and, as such, of deviations from the standard cosmological model, LCDM. In this paper, we focus on model-independent methods to modify the evolution of scalar perturbations at linear scales. We consider two approaches: the first is based on the two phenomenological modified gravity (PMG) parameters, $μ_{\rm mg}$ and $Σ_{\rm mg}$, which are phenomenologically connected to the clustering of matter and weak lensing, respectively; and the second is the effective field theory (EFT) of dark energy and modified gravity, which we use to parameterise the braiding function, $α_{\rm B}$, which defines the mixing between the metric and the dark energy field. We discuss the predictions from spectroscopic and photometric primary probes by Euclid on the cosmological parameters and a given set of additional parameters featuring the PMG and EFT models. We use the Fisher matrix method applied to spectroscopic galaxy clustering (GCsp), weak lensing (WL), photometric galaxy clustering (GCph), and cross-correlation (XC) between GCph and WL. For the modelling of photometric predictions on nonlinear scales, we use the halo model to cover two limits for the screening mechanism: the unscreened (US) case, for which the screening mechanism is not present; and the super-screened (SS) case, which assumes strong screening. We also assume scale cuts to account for our uncertainties in the modelling of nonlinear perturbation evolution. We choose a time-dependent form for $\{μ_{\rm mg},Σ_{\rm mg}\}$, with two fiducial sets of values for the corresponding model parameters at the present time, $\{\barμ_0,\barΣ_0\}$, and two forms for $α_{\rm B}$, with one fiducial set of values for each of the model parameters, $α_{\rm B,0}$ and $\{α_{\rm B,0},m\}$. (Abridged)