We consider the finite interactions of the generalized Proca theory including the sixth-order Lagrangian and derive the full linear perturbation equations of motion on the flat Friedmann-Lemaître-Robertson-Walker background in the presence of a matter perfect fluid. By construction, the propagating degrees of freedom (besides the matter perfect fluid) are two transverse vector perturbations, one longitudinal scalar, and two tensor polarizations. The Lagrangians associated with intrinsic vector modes neither affect the background equations of motion nor the second-order action of tensor perturbations, but they do give rise to nontrivial modifications to the no-ghost condition of vector perturbations and to the propagation speeds of vector and scalar perturbations. We derive the effective gravitational coupling $G_{\mathrm{eff}}$ with matter density perturbations under a quasistatic approximation on scales deep inside the sound horizon. We find that the existence of intrinsic vector modes allows a possibility for reducing $G_{\mathrm{eff}}$. In fact, within the parameter space, $G_{\mathrm{eff}}$ can be even smaller than the Newton gravitational constant $G$ at the late cosmological epoch, with a peculiar phantom dark energy equation of state (without ghosts). The modifications to the slip parameter $\eta$ and the evolution of the growth rate $f{\sigma }_8$ are discussed as well. Thus, dark energy models in the framework of generalized Proca theories can be observationally distinguished from the $\mathrm{\Lambda }\mathrm{CDM}$ model according to both cosmic growth and expansion history. Furthermore, we study the evolution of vector perturbations and show that outside the vector sound horizon the perturbations are nearly frozen and start to decay with oscillations after the horizon entry.

We consider higher-order, scalar-tensor theories which appear degenerate when restricted to the unitary gauge but are not degenerate in an arbitrary gauge. We dub them U-degenerate theories. We provide a full classification of theories that are either DHOST or U-degenerate and that are quadratic in second derivatives of the scalar field and discuss its extension to cubic and higher-order theories. Working with a simple example of U-degenerate theory, we find that, for configurations in which the scalar field gradient is time-like, the apparent extra mode in such a theory can be understood as a generalized instantaneous, or “shadowy” mode, which does not propagate. Appropriate boundary conditions, required by the elliptic nature of part of the equations of motion, lead to the elimination of the apparent instability associated with this extra mode.

For a massive vector field with derivative self-interactions, the breaking of the gauge invariance allows the propagation of a longitudinal mode in addition to the two transverse modes. We consider generalized Proca theories with second-order equations of motion in a curved space-time and study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric background. We show explicitly that cubic-order self-interactions lead to the suppression of the longitudinal mode through the Vainshtein mechanism. Provided that the dimensionless coupling of the interaction is not negligible, this screening mechanism is sufficiently efficient to give rise to tiny corrections to gravitational potentials consistent with solar-system tests of gravity. We also study the quartic interactions with the presence of nonminimal derivative coupling with the Ricci scalar and find the existence of solutions where the longitudinal mode completely vanishes. Finally, we discuss the case in which the effect of the quartic interactions dominates over the cubic one and show that local gravity constraints can be satisfied under a mild bound on the parameters of the theory.