Dark energy in scalar-vector-tensor theories

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-vectortensor 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.