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The future Euclid space satellite mission will offer an invaluableopportunity to constrain modifications to general relativity at cosmic scales.We focus on modified gravity models characterised, at linear scales, by ascale-independent growth of perturbations while featuring different testabletypes of derivative screening mechanisms at smaller nonlinear scales. Weconsider 3 specific models, namely Jordan-Brans-Dicke (JBD), the normal branchof Dvali-Gabadadze-Porrati (nDGP) gravity and $k$-mouflage (KM) gravity. Weprovide forecasts from spectroscopic and photometric primary probes by Euclidon the cosmological parameters and the extra parameters of the models,respectively, $\omega_{\rm BD}$, $\Omega_{\rm rc}$ and $\epsilon_{2,0}$, whichquantify the deviations from general relativity. This analysis will improve ourknowledge of the cosmology of these modified gravity models. The forecastsanalysis employs the Fisher matrix method applied to weak lensing (WL);photometric galaxy clustering (GC$_{ph}$); spectroscopic galaxy clustering(GC$_{sp}$) and the cross-correlation (XC) between GC$_{ph}$ and WL. For theEuclid survey specifications we define three scenarios, characterised bydifferent cuts in $\ell$ and $k$, to assess the constraining power of nonlinearscales. For each model we consider two fiducial values for the correspondingmodel parameter. In an optimistic setting at 68.3\% confidence interval, withEuclid alone we find the following percentage relative errors: for$\log_{10}{\omega_{\rm BD}}$, with a fiducial value of $\omega_{\rm BD}=800$,35% using GC$_{sp}$ alone, 3.6% using GC$_{ph}$+WL+XC and 3.3% usingGC$_{ph}$+WL+XC+GC$_{sp}$; for $\log_{10}{\Omega_{\rm rc}}$, with a fiducialvalue of $\Omega_{\rm rc}=0.25$, we find respectively 90%, 20% and 17%;finally, for $\epsilon_{2,0}=-0.04$ respectively 5%, 0.15% and 0.14%.(abridged)

Euclid: Constraining linearly scale-independent modifications of gravity with the spectroscopic and photometric primary probes