Vainshtein regime in Scalar-Tensor gravity: constraints on DHOST theories

We study the screening mechanism in the most general scalar-tensor theories that leave gravitational waves unaffected and are thus compatible with recent LIGO/Virgo observations. Using the effective field theory of dark energy approach, we consider the general action for perturbations beyond linear order, focussing on the quasi-static limit. When restricting to the subclass of theories that satisfy the gravitational wave constraints, the fully nonlinear effective Lagrangian contains only three independent parameters. One of these, $\beta_1$, is uniquely present in degenerate higher-order theories. We compute the two gravitational potentials for a spherically symmetric matter source and we find that for $\beta_1 \ge 0$ they decrease as the inverse of the distance, as in standard gravity, while the case $\beta_1 < 0$ is ruled out. For $\beta_1 > 0$, the two potentials differ and their gravitational constants are not the same on the inside and outside of the body. Generically, the bound on anomalous light bending in the Solar System constrains $\beta_1 \lesssim 10^{-5}$. Standard gravity can be recovered outside the body by tuning the parameters of the model, in which case $\beta_1 \lesssim 10^{-2}$ from the Hulse-Taylor pulsar.