The Raychaudhuri Equation in inhomogeneous FLRW space-time : A $f(R)$-gravity model

TL;DR

This paper explores the Raychaudhuri equation within inhomogeneous FLRW spacetime in the context of $f(R)$ gravity, deriving explicit solutions, analyzing geodesic behavior, and discussing quantum aspects to understand singularity conditions.

Contribution

It formulates the Raychaudhuri equation in $f(R)$ gravity for inhomogeneous FLRW spacetime and investigates geodesic convergence and singularity avoidance.

Findings

Raychaudhuri equation expressed as a harmonic oscillator with variable frequency.

Conditions identified for geodesic convergence and singularity avoidance.

Explicit solutions for the expansion scalar in inhomogeneous $f(R)$ gravity.

Abstract

In general description of the Raychaudhuri equation it is found that this first order non-linear differential equation can be written as a second order linear differential equation in the form of Harmonic Oscillator with varying frequency. Further, the integrability of the Raychaudhuri equation has been studied and also the expansion scalar is obtained in an explicit form. Subsequently, $f(R)$ gravity theory has been studied in the background of inhomogeneous FLRW spacetime with an aim to formulate the Raychaudhuri Equation. A congruence of time-like geodesics has been investigated using the Raychaudhuri Equation to examine whether the geodesics converge or not and some possible conditions are determined to avoid singularity. Finally, a brief quantum description has been presented.