Effective field theory of modified gravity with two scalar fields: dark energy and dark matter

TL;DR

The work extends the EFT of cosmological perturbations to a two-field dark sector, introducing a second scalar $\phi$ (dark matter) with kinetic energy $X$ alongside a Horndeski-like dark-energy sector. By employing a unitary gauge and a 3+1 ADM decomposition, the authors derive second-order perturbation equations and establish explicit no-ghost and Laplacian-stability conditions, including cross-derivative constraints to avoid higher-order time-space terms. They specialize to Horndeski dark energy with k-essence dark matter, obtaining scalar and tensor propagation speeds and the quasi-static, sub-horizon effective gravitational coupling $G_{\rm eff}$, including Brans–Dicke-like limits. The framework reproduces the perfect-fluid dark-matter limit and provides a systematic way to construct viable two-field models compatible with cosmological observations. Overall, the paper sets a robust, model-independent foundation to study the interplay between dark energy and dark matter in modified gravity theories at linear order and on sub-horizon scales.

Abstract

We present a framework for discussing the cosmology of dark energy and dark matter based on two scalar degrees of freedom. An effective field theory of cosmological perturbations is employed. A unitary gauge choice renders the dark energy field into the gravitational sector, for which we adopt a generic Lagrangian depending on three-dimensional geometrical scalar quantities arising in the ADM decomposition. We add to this dark-energy associated gravitational sector a scalar field $φ$ and its kinetic energy $X$ as dark matter variables. Compared to the single-field case, we find that there are additional conditions to obey in order to keep the equations of motion for linear cosmological perturbations at second order. For such a second-order multi-field theory we derive conditions under which ghosts and Laplacian instabilities of the scalar and tensor perturbations are absent. We apply our general results to models with dark energy emerging in the framework of the Horndeski theory and dark matter described by a k-essence Lagrangian $P(φ,X)$. We derive the effective coupling between such an imperfect-fluid dark matter and the gravitational sector under the quasi-static approximation on sub-horizon scales. By considering the purely kinetic Lagrangian $P(X)$ as a particular case, the formalism is verified to reproduce the gravitational coupling of a perfect-fluid dark matter.