Effective action approach to cosmological perturbations in dark energy and modified gravity

TL;DR

This paper develops an effective action formalism to parameterize linear perturbations in dark energy and modified gravity models without requiring a specified Lagrangian, relying only on field content and quadratic perturbations. By imposing isotropy and using an Eulerian perturbation scheme, the authors reduce the perturbation description to a finite set of background-dependent functions and provide explicit expressions for the perturbed energy-momentum tensor and conservation equations in two generic cases: no extra fields and scalar fields. They demonstrate decoupling conditions that substantially simplify the perturbation dynamics, derive generalized entropy and anisotropic-stress descriptions, and extend the framework to derivative-based modified gravity theories such as $F(R)$ and Gauss–Bonnet models. The approach offers a physically transparent, parameter-efficient route to compare theories with observations and to connect perturbation phenomenology to underlying field content.

Abstract

In light of upcoming observations modelling perturbations in dark energy and modified gravity models has become an important topic of research. We develop an effective action to construct the components of the perturbed dark energy momentum tensor which appears in the perturbed generalized gravitational field equations, δG_{μν} = 8πGδT_{μν} + δU_{μν} for linearized perturbations. Our method does not require knowledge of the Lagrangian density of the dark sector to be provided, only its field content. The method is based on the fact that it is only necessary to specify the perturbed Lagrangian to quadratic order and couples this with the assumption of global statistical isotropy of spatial sections to show that the model can be specified completely in terms of a finite number of background dependent functions. We present our formalism in a coordinate independent fashion and provide explicit formulae for the perturbed conservation equation and the components of δU_{μν} for two explicit generic examples: (i) the dark sector does not contain extra fields, L = L(g_{μν}) and (ii) the dark sector contains a scalar field and its first derivative L = L(g_{μν}, φ, \nabla_μφ). We discuss how the formalism can be applied to modified gravity models containing derivatives of the metric, curvature tensors, higher derivatives of the scalar fields and vector fields.

Figures (2)

• Figure 1: Schematic view of the Eulerian and Lagrangian coordinate systems. The Lagrangian system can be said to be comoving, and the Eulerian system as being fixed. The Lagrangian system retains the density of a field, whereas the Eulerian system does not. This is schematically depicted by the "grid square" becoming deformed in the Lagrangian system on the left, to accommodate the movement of "particles" upon evolution in time. The grid square in the Eulerian system has remained fixed, meaning that the number of particles in a given square changes upon evolution. In cosmology we are perturbing against a fixed background: the FRW background, however calculations are often easier to perform in a comoving system. This means that physical relevance is taken from equations perturbed according to a Eulerian scheme.
• Figure 2: Schematic view of the foliation and evolution, with three example world-lines drawn on, each piercing two 3D surfaces; $u^{\mu}$ is a time-like vector satisfying $u^{\mu}u_{\mu} = -1$. A quantity $X$ on a surface with spacetime location $(t, x^i)$ can be transformed into a quantity on the same surface but at a different location by transforming the coordinate on the surface, $x^i \rightarrow x^i+\xi^i$. This is a diffeomorphism which drags one world-line into another. If the time coordinate is transformed $t \rightarrow t+u^0$ then the quantity is evaluated on a different 3d surface, but on the same world-line. Thus, if we were to have a transformation $x^{\mu} \rightarrow x^{\mu} + \xi^{\mu} + \chi u^{\mu}$, where $\chi$ is an arbitrary scalar field, the time-like part of $\xi^{\mu}$ is redundant. Hence, we are free to set $\xi^{\mu}u_{\mu}=0$, fixing the time-like part of the diffeomorphism field to be zero. So, we should have the interpretation that $\xi^{\mu}$ moves between world-lines and $u^{\mu}$ moves along world-lines.