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

TL;DR

This paper develops a general, coordinate-independent effective action framework for modeling linear cosmological perturbations in dark energy and modified gravity, using only the background field content and quadratic Lagrangian perturbations.

Contribution

It introduces a novel formalism that constructs perturbed energy-momentum components without requiring the explicit Lagrangian density, applicable to a wide class of dark energy and modified gravity models.

Findings

Provides explicit formulas for perturbed conservation equations.

Demonstrates the method with models containing scalar fields and no extra fields.

Applicable to models with derivatives of metrics, curvature, and vector fields.

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, {\delta}G_{\mu\nu} = 8{\pi}G{\delta}T_{\mu\nu} + {\delta}U_{\mu\nu} 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 {\delta}U_{\mu\nu} for two explicit generic examples: (i) the dark sector does not contain extra fields, L = L(g_{\mu\nu}) and (ii) the dark sector contains a scalar field and its first derivative L = L(g_{\mu\nu}, {\phi}, \nabla_{\mu}{\phi}). 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.