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We present a method for parametrizing linear cosmological perturbations oftheories of gravity, around homogeneous and isotropic backgrounds. The methodis sufficiently general and systematic that it can be applied to theories withany degrees of freedom (DoFs) and arbitrary gauge symmetries. In this paper, wefocus on scalar-tensor and vector-tensor theories, invariant under linearcoordinate transformations. In the case of scalar-tensor theories, we use ourframework to recover the simple parametrizations of linearized Horndeski and"Beyond Horndeski" theories, and also find higher-derivative corrections. Inthe case of vector-tensor theories, we first construct the most generalquadratic action for perturbations that leads to second-order equations ofmotion, which propagates two scalar DoFs. Then we specialize to the case inwhich the vector field is time-like (\`a la Einstein-Aether gravity), where thetheory only propagates one scalar DoF. As a result, we identify the completeforms of the quadratic actions for perturbations, and the number of freeparameters that need to be defined, to cosmologically characterize these twobroad classes of theories.

A general theory of linear cosmological perturbations: scalar-tensor and vector-tensor theories