Probing Vainsthein-screening gravity with galaxy clusters using internal
  kinematics and strong and weak lensing
  We use high-precision combined strong/weak lensing and kinematics
measurements of the total mass profiles of the observed galaxy clusters
MACS~J1206.2-0847 and Abell~S1063, to constrain the relativistic sector of the
general DHOST dark energy theories, which exhibit a partial breaking of the so
called Vainsthein screening mechanism, on the linear level of scalar
fluctuations around a cosmological background. In particular, by using the
\textsc{MG-MAMMPOSSt} framework developed in Pizzuti et al. 2021, for the
kinematics analysis of member galaxies in clusters along with lensing mass
profile reconstructions, we provide new constraints on the coupling $Y_2$ which
governs the theory's relativistic contribution to the lensing potential. The
new bound from the combination of kinematics and lensing measurements of MACS
1206, $Y_2=-0.12^{+0.66}_{-0.67}$ at $2\sigma$, provides about a 2-fold
improvement on previous constraints. In the case of Abell~S1063 a $>2\sigma$
tension with the GR expectation arises. We discuss this in some detail, and we
investigate the possible sources of systematics which can explain the tension.
We further discuss why the combination of kinematics of member galaxies with
lensing is capable of providing much tighter bounds compared to kinematics or
lensing alone, and we explain how the number density profile of tracers, as
well as the choice of the velocity anisotropy profile affects the final
results.
