  The scalar-vector-tensor theories with second-order equations of motion can
accommodate both Horndeski and generalized Proca theories as specific cases. In
the presence of a perfect fluid, we study the cosmology in such a most general
scheme of scalar-vector-tensor theories with parity invariance by paying
particular attention to the application to dark energy. We obtain a closed-form
expression of the background equations of motion by using coefficients
appearing in the second-order action of scalar perturbations. We also derive
conditions for the absence of ghost and Laplacian instabilities of tensor and
vector perturbations and show that the existence of matter does not
substantially modify the stabilities of dynamical degrees of freedom in the
small-scale limit. On the other hand, the sound speed of scalar perturbations
is affected by the presence of matter. Employing the quasi-static approximation
for scalar perturbations deep inside the sound horizon, we derive analytic
expressions of Newtonian and weak lensing gravitational potentials as well as
two scalar perturbations arising from the scalar and vector fields. We apply
our general framework to dark energy theories with the tensor propagation speed
equivalent to the speed of light and show that the observables associated with
the growth of matter perturbations and weak lensing potentials are generally
affected by intrinsic vector modes and by interactions between scalar and
vector fields.

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