CLASH-VLT: Testing the Nature of Gravity with Galaxy Cluster Mass
  Profiles
  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 $0.5\,Mpc\le r\le
r_{200}=1.96\,Mpc$. 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.
