  The almost simultaneous detection of gravitational waves and a short
gamma-ray burst from a neutron star merger has put a tight constraint on the
difference between the speed of gravity and light. In the four-dimensional
scalar-tensor theory with second order equations of motion, the Horndeski
theory, this translates into a significant reduction of the viable parameter
space of the theory. Recently, extensions of Horndeski theory, which are free
from Ostrogradsky ghosts despite the presence of higher order derivatives in
the equations of motion, have been identified and classified exploiting the
degeneracy criterium. In these new theories, the fifth force mediated by the
scalar field must be suppressed in order to evade the stringent Solar System
constraints. We study the Vainshtein mechanism in the most general degenerate
higher order scalar-tensor theory in which light and gravity propagate at the
same speed. We find that the Vainshtein mechanism generally works outside a
matter source but it is broken inside matter, similarly to beyond Horndeski
theories. This leaves interesting possibilities to test these theories that are
compatible with gravitational wave observations using astrophysical objects.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory