$f(R)$-gravity from Killing Tensors

TL;DR

This paper identifies new integrable $f(R)$ gravity models in FLRW spacetime using Killing tensors and conservation laws, leading to simplified equations and insights into cosmological fluid evolution, including a CPL-like dark energy model.

Contribution

The paper introduces three new integrable $f(R)$ models using Killing tensors and contact symmetries, advancing the understanding of $f(R)$ gravity in cosmology.

Findings

Three new integrable $f(R)$ models found.

Reduction of field equations to two first-order ODEs.

One model exhibits CPL-like dark energy behavior.

Abstract

We consider $f\left(R\right) $-gravity in a Friedmann-Lema\^itre-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of $f\left(R\right) $ and the field equations to be invariant under Lie-B\"acklund transformations which are linear in the momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether's Theorem. We find three new integrable $f\left(R\right) $ models, for which with the application of the conservation laws we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. Where we find that for the one integrable model the cosmological fluid has an equation of state parameter, in which in the latter there is a linear behavior in terms of the scale factor which describes the Chevallier, Polarski and Linder (CPL) parametric dark energy model.