We derive an effective equation and action for comoving curvature perturbations and gravitational waves (GWs) in terms of a time, momentum and polarization dependent effective speed, encoding the effects of the interaction among metric perturbations or with other fields, such as dark energy and dark matter. The structure of the effective actions and equations is the same for scalar and tensor perturbations, and the effective actions can be written as the Klein-Gordon action in terms of an appropriately defined effective metric, dependent on the effective speed. The effective action reproduces, and generalizes to higher order in perturbations, results obtained for GWs in the effective field theory of inflation and dark energy, or for curvature perturbations in systems with multiple scalar fields, encoding in the effective speed the effects of both entropy and anisotropy. The effective approach can also be applied to the solutions of theories with field equations different from the Einstein equations, by defining an appropriate effective energy-momentum tensor. As an example, we show that for a minimally coupled scalar field in general relativity, the effective speeds of curvature perturbations and gravitational waves are frequency and polarization dependent, due to their coupling in the action beyond the quadratic order.

The first version of this paper was submitted to JCAP on February 7th 2023, and the first report was received on January 14th 2024. This is a revised version based on the first referee report