A Simplified Approach to General Scalar-Tensor Theories

TL;DR

The paper recasts Horndeski's general scalar-tensor theory for linear FRW perturbations within the EFT of Dark Energy, demonstrating that six time-dependent functions suffice to describe perturbations, with two fixed by the background evolution. It shows how explicit matching of Horndeski's Lagrangian to EFT coefficients yields a consistent mapping (including curvature) and identifies essential derivative-cancellation relations that guarantee second-order dynamics. In the quasistatic limit, the number of free functions reduces further to four (or even fewer if the background is fixed), enabling tighter, model-independent observational constraints via $G_{ m eff}(k,t)$ and the gravitational slip $\gamma(k,t)$. The results generalize prior work, cross-check existing analyses, and provide a streamlined parameter space to test Horndeski-inspired dark energy models with current and future surveys. The methodology also offers a pathway to apply similar EFT reductions to inflationary scenarios and higher-order perturbations.

Abstract

The most general covariant action describing gravity coupled to a scalar field with only second order equations of motion, Horndeski's theory (also known as "Generalized Galileons"), provides an all-encompassing model in which single scalar dark energy models may be constrained. However, the generality of the model makes it cumbersome to manipulate. In this paper, we demonstrate that when considering linear perturbations about a Friedmann-Robertson-Walker background, the theory is completely specified by only six functions of time, two of which are constrained by the background evolution. We utilise the ideas of the Effective Field Theory of Inflation/Dark Energy to explicitly construct these six functions of time in terms of the free functions appearing in Horndeski's theory. These results are used to investigate the behavior of the theory in the quasistatic approximation. We find that only four functions of time are required to completely specify the linear behavior of the theory in this limit, which can further be reduced if the background evolution is fixed. This presents a significantly reduced parameter space from the original presentation of Horndeski's theory, giving hope to the possibility of constraining the parameter space. This work provides a cross-check for previous work on linear perturbations in this theory, and also generalizes it to include spatial curvature.