On the evolution of inhomogeneous perturbations in the $\Lambda$CDM model and $f(R)$ modified gravity theories

TL;DR

This paper compares the evolution of inhomogeneous perturbations in the $\Lambda$CDM model and $f(R)$ modified gravity theories using a scalar-tensor formalism, revealing distinctive Yukawa-like solutions and model-independent radial profiles.

Contribution

It introduces a linear perturbation approach in the scalar-tensor formalism to analyze inhomogeneities in both models, highlighting differences in perturbation evolution and solutions.

Findings

Yukawa-like solutions distinguish $f(R)$ from $\Lambda$CDM.

Radial perturbation profiles are independent of specific $f(R)$ functions.

Results are applicable to any $f(R)$ model.

Abstract

We focus on weak inhomogeneous models of the Universe at low redshifts, described by the Lema\^itre-Tolman-Bondi (LTB) metric. The principal aim of this work is to compare the evolution of inhomogeneous perturbations in the $\Lambda$CDM cosmological model and $f(R)$ modified gravity theories, considering a flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric for the background. More specifically, we adopt the equivalent scalar-tensor formalism in the Jordan frame, in which the extra degree of freedom of the $f(R)$ function is converted into a non-minimally coupled scalar field. We investigate the evolution of local inhomogeneities in time and space separately, following a linear perturbation approach. Then, we obtain spherically symmetric solutions in both cosmological models. Our results allow us to distinguish between the presence of a cosmological constant and modified gravity scenarios, since a peculiar Yukawa-like solution for radial perturbations occurs in the Jordan frame. Furthermore, the radial profile of perturbations does not depend on a particular choice of the $f(R)$ function, hence our results are valid for any $f(R)$ model.