Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity

TL;DR

This paper provides a minimal, non-redundant framework to describe linear structure formation in general Horndeski scalar-tensor theories with a single universal scalar. By certifying five time-dependent functions—the background history $H(t)$ and four dark-energy properties ${\alpha_i(t)}$—plus a fixed today matter density $\Omega_{m0}$, it unifies the treatment of perturbations across diverse models and delivers a direct translation to EFT. The authors derive the complete set of linear perturbation equations in Newtonian gauge, discuss stability, and identify observable quantities such as the effective Newton constants and the slip parameter, highlighting the braiding and Compton scales that govern regime transitions. They also map this parameterization to specific models (e.g., $f(R)$, k-essence, Galileons) and discuss observational constraints from large-scale structure, as well as local tests and screening effects. The result is a practically implementable tool that can test a broad class of dark-energy and modified-gravity theories with a single numerical framework, clarifying what can and cannot be inferred about the underlying acceleration mechanism from linear cosmological data.

Abstract

We present a turnkey solution, ready for implementation in numerical codes, for the study of linear structure formation in general scalar-tensor models involving a single universally coupled scalar field. We show that the totality of cosmological information on the gravitational sector can be compressed - without any redundancy - into five independent and arbitrary functions of time only and one constant. These describe physical properties of the universe: the observable background expansion history, fractional matter density today, and four functions of time describing the properties of the dark energy. We show that two of those dark-energy property functions control the existence of anisotropic stress, the other two - dark-energy clustering, both of which are can be scale-dependent. All these properties can in principle be measured, but no information on the underlying theory of acceleration beyond this can be obtained. We present a translation between popular models of late-time acceleration (e.g. perfect fluids, f (R), kinetic gravity braiding, galileons), as well as the effective field theory framework, and our formulation. In this way, implementing this formulation numerically would give a single tool which could consistently test the majority of models of late-time acceleration heretofore proposed.