We use high-precision kinematic and lensing measurements of the total mass profile of the dynamically relaxed galaxy cluster MACS J1206.2-0847 at $z=0.44$ to estimate the value of the ratio $eta=Psi/Phi$ between the two scalar potentials in the linear perturbed Friedmann-Lemaitre-Robertson-Walker metric.[...] Complementary kinematic and lensing mass profiles were derived from exhaustive analyses using the data from the Cluster Lensing And Supernova survey with Hubble (CLASH) and the spectroscopic follow-up with the Very Large Telescope (CLASH-VLT). Whereas the kinematic mass profile tracks only the time-time part of the perturbed metric (i.e. only $Phi$), the lensing mass profile reflects the contribution of both time-time and space-space components (i.e. the sum $Phi+Psi$). We thus express $eta$ as a function of the mass profiles and perform our analysis over the radial range $0.5,Mpcle rle r_{200}=1.96,Mpc$. Using a spherical Navarro-Frenk-White mass profile, which well fits the data, we obtain $eta(r_{200})=1.01,_{-0.28}^{+0.31}$ at the 68% C.L. We discuss the effect of assuming different functional forms for mass profiles and of the orbit anisotropy in the kinematic reconstruction. Interpreting this result within the well-studied $f(R)$ modified gravity model, the constraint on $eta$ translates into an upper bound to the interaction length (inverse of the scalaron mass) smaller than 2 Mpc. This tight constraint on the $f(R)$ interaction range is however substantially relaxed when systematic uncertainties in the analysis are considered. Our analysis highlights the potential of this method to detect deviations from general relativity, while calling for the need of further high-quality data on the total mass distribution of clusters and improved control on systematic effects.
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