Signature of $f\left(R\right)$ gravity via Lema\^itre-Tolman-Bondi inhomogeneous perturbations

TL;DR

This paper compares inhomogeneous cosmological models in the local Universe using LTB and $f(R)$ gravity, revealing distinct radial profiles of inhomogeneities that could help differentiate between dark energy and modified gravity explanations for cosmic acceleration.

Contribution

It introduces an analytical and numerical approach to compare LTB and $f(R)$ gravity models, highlighting the unique Yukawa-like radial profiles in $f(R)$ theories.

Findings

Radial inhomogeneities follow a power-law in $ ext{Lambda}$LTB models.

$f(R)$ gravity induces Yukawa-like radial profiles in inhomogeneities.

Time-dependent perturbations do not diverge, ensuring physical viability.

Abstract

We analyze inhomogeneous cosmological models in the local Universe, described by the Lema\^itre-Tolman-Bondi (LTB) metric and developed using linear perturbation theory on a homogeneous and isotropic Universe background. Focusing on the different evolution of spherical symmetric inhomogeneities, we compare the $\Lambda$LTB model, in which the cosmological constant $\Lambda$ is included in the LTB formalism, with inhomogeneous cosmological models based on $f\left(R\right)$ modified gravity theories viewed in the Jordan frame. We solve the system of field equations for both inhomogeneous cosmological models adopting the method of separation of variables: we integrate analytically the radial profiles of local perturbations, while their time evolution requires a numerical approach. The main result of the analysis concerns the different radial profiles of local inhomogeneities due to the presence of a non-minimally coupled scalar field in the Jordan frame of $f\left(R\right)$ gravity. While radial perturbations follow a power-law in the $\Lambda$LTB model, Yukawa-like contributions appear in the $f\left(R\right)$ theory. Interestingly, this latter peculiar behavior of radial profile is not affected by the choice of the $f\left(R\right)$ functional form. The numerical solution of time-dependent perturbations exhibits a non-diverging profile. This work suggests that investigations about local inhomogeneities in the late Universe may allow us to discriminate if the present cosmic acceleration is caused by a cosmological constant term or a modified gravity effect.