Essential Building Blocks of Dark Energy

TL;DR

The paper develops a systematic effective field theory (EFT) framework for cosmological perturbations in single-field dark energy and modified gravity by working in unitary gauge with an ADM-based Lagrangian. It identifies a minimal seven-operator basis that yields second-order equations of motion for linear perturbations and provides a dictionary to map any model into time-dependent EFT coefficients $f(t)$, $\Lambda(t)$, $c(t)$, and higher-order terms, clarifying the role of Horndeski/ generalized Galileon theories. The analysis shows that Horndeski theories are fully captured by six EFT operators at linear order (with $m_4^2=\tilde{m}_4^2$), while allowing $m_4^2\neq\tilde{m}_4^2$ extends beyond Horndeski, potentially altering high-k dispersion. The paper also derives the Newtonian-gauge perturbation equations and provides explicit expressions for the quasi-static gravitational coupling $G_{\rm eff}(k)$ and the gravitational slip $\gamma(k)$, establishing a direct path from EFT coefficients to observable signatures in large-scale structure and lensing.

Abstract

We propose a minimal description of single field dark energy/modified gravity within the effective field theory formalism for cosmological perturbations, which encompasses most existing models. We start from a generic Lagrangian given as an arbitrary function of the lapse and of the extrinsic and intrinsic curvature tensors of the time hypersurfaces in unitary gauge, i.e. choosing as time slicing the uniform scalar field hypersurfaces. Focusing on linear perturbations, we identify seven Lagrangian operators that lead to equations of motion containing at most two (space or time) derivatives, the background evolution being determined by the time dependent coefficients of only three of these operators. We then establish a dictionary that translates any existing or future model whose Lagrangian can be written in the above form into our parametrized framework. As an illustration, we study Horndeski's-or generalized Galileon-theories and show that they can be described, up to linear order, by only six of the seven operators mentioned above. This implies, remarkably, that the dynamics of linear perturbations can be more general than that of Horndeski while remaining second order. Finally, in order to make the link with observations, we provide the entire set of linear perturbation equations in Newtonian gauge, the effective Newton constant in the quasi-static approximation and the ratio of the two gravitational potentials, in terms of the time-dependent coefficients of our Lagrangian.