Reconstruction method of $f(R)$ gravity for isotropic and anisotropic spacetimes

TL;DR

This paper extends the reconstruction method of $f(R)$ gravity to anisotropic Bianchi-I spacetimes, revealing that anisotropy acts as an independent degree of freedom, unlike in General Relativity, with applications to inflation and bounce models.

Contribution

The paper develops a novel reconstruction approach for $f(R)$ gravity in anisotropic spacetimes, highlighting the independent behavior of anisotropy and linking Ricci scalar definition to anisotropy evolution.

Findings

Reconstruction method applicable to anisotropic Bianchi-I spacetime.

Anisotropy acts as an independent degree of freedom in $f(R)$ gravity.

Application to inflationary and ekpyrotic scenarios.

Abstract

We present the reconstruction method of $f(R)$ gravity for the homogeneous and anisotropic Bianchi-I spacetime, which was previously formulated only for homogeneous and isotropic FLRW spacetime. We argue in this paper that for anisotropic spacetimes, the total anisotropy behaves as an independent metric degree of freedom on top of the average scale factor in $f(R)$ gravity. This is not like $GR$, where specifying the form of the average scale factor as a function of time also specify the total anisotropy as a function of time uniqely. We link this peculiar fact to an interesting intertwining between the definition of Ricci scalar for anisotropic metric and anisotropy evolution equation in $f(R)$ gravity. Consequently, specifying an anisotropic solution of $f(R)$ gravity implies specifying both the average scale factor and the total anisotropy as functions of time. The reconstruction method hence formulated is applied to two scenarios where anisotropy suppression is important, namely, a quasi de-Sitter expansion as required in inflation, and a power law contraction as required in ekpyrotic bounce models.