Characterizations of Perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity equipped with different gradient solitons

TL;DR

This paper explores perfect fluid spacetimes in $f( ext{R})$-gravity, analyzing various gradient solitons and establishing conditions for their properties, with implications for understanding dark matter in cosmology.

Contribution

It introduces new conditions and theorems for perfect fluid spacetimes with different gradient solitons in $f( ext{R})$-gravity, including existence proofs and potential equations.

Findings

Existence of $ ext{η}$-Ricci solitons demonstrated with examples

Conditions for expanding, steady, or shrinking solitons established

Theorems related to dark matter era derived

Abstract

The prime object of this article is to study the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when $\eta$-Ricci solitons, gradient $\eta$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons are its metrics. At first, the existence of the $\eta$-Ricci solitons is proved by a non-trivial example. We establish conditions for which the $\eta$-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying $f(\mathcal{R})$-gravity, when the potential vector field of $\eta$-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient $\eta$-Ricci solitons, gradient Einstein Solitons and gradient $m$-quasi Einstein solitons in $f(\mathcal{R})$-gravity, respectively. As a result, we establish some significant theorems about dark matter era.