A unifying description of dark energy

TL;DR

This work develops a unifying description of dark energy and modified gravity by formulating a general ADM-based action on uniform-scalar-field hypersurfaces that encompasses GR, quintessence, k-essence, F(R), Horndeski, beyond Horndeski, and Hořava-Lifshitz theories. The authors show that linear cosmological perturbations are fully characterized by five time-dependent functions $α_M$, $α_K$, $α_B$, $α_T$, $α_H$, and they derive the complete quadratic action for tensor, vector, and scalar modes, including the dark-energy perturbation equations in the Horndeski class. They connect this framework to the EFT of dark energy, discuss disformal transformations, and present practical pathways to confront models with observations through Newtonian-gauge perturbation equations, a fluid description, and quasi-static limits involving $G_{ m eff}$ and the gravitational slip $γ$. The approach reduces redundancies across models, clarifies degeneracies, and provides a straightforward, model-independent route to constrain broad classes of single-field dark-energy and modified-gravity theories with current and future data.

Abstract

We review and extend a novel approach that we introduced recently, to describe general dark energy or scalar-tensor models. Our approach relies on an ADM formulation based on the hypersurfaces where the underlying scalar field is uniform. The advantage of this approach is that it can describe in the same language and in a minimal way a vast number of existing models, such as quintessence models, $F(R)$ theories, scalar tensor theories, their Horndeski extensions and beyond. It also naturally includes Horava-Lifshitz theories. As summarized in this review, our approach provides a unified treatment of the linear cosmological perturbations about a FLRW universe, obtained by a systematic expansion of our general action up to quadratic order. This shows that the behaviour of these linear perturbations is generically characterized by five time-dependent functions. We derive the full equations of motion in the Newtonian gauge, and obtain in particular the equation of state for dark energy perturbations, in the Horndeski case, in terms of these functions. Our unifying description thus provides the simplest and most systematic way to confront theoretical models with current and future cosmological observations.