Linearization of a warped $f(R)$ theory in the higher-order frame II: the equation of motion approach

TL;DR

This paper develops a gauge-independent method to linearize a five-dimensional $f(R)$ brane model directly in the higher-order frame using the equation of motion approach, avoiding conformal transformations.

Contribution

It provides a novel gauge-independent linearization technique for higher-order $f(R)$ theories directly in the equation of motion framework.

Findings

Derived all linear perturbation equations without fixing a gauge.

Established equivalence of master equations with previous quadratic action results.

Identified a constraint in the vector sector absent in previous approaches.

Abstract

Without using conformal transformation, a simple type of five-dimensional $f(R)-$brane model is linearized directly in its higher-order frame. In this paper, the linearization is conducted in the equation of motion approach. We first derive all the linear perturbation equations without specifying a gauge condition. Then by taking the curvature gauge we derive the master equations of the linear perturbations. We show that these equations are equivalent to those obtained in the quadratical action approach [Phys. Rev. D 95 (2017) 104060], except the vector sector, in which a constraint equation can be obtained in the equation of motion approach but absent in the quadratical action approach. Our work sets an example on how to linearize higher-order theories without using conformal transformation, and might be useful for studying more complicated theories.