Exact Cosmological Solutions of $f(R)$ Theories via Hojman Symmetry

TL;DR

This paper demonstrates how Hojman symmetry can be used to find new exact cosmological solutions in $f(R)$ gravity theories, revealing features not accessible through Noether symmetry.

Contribution

The work applies Hojman symmetry to $f(R)$ theories in both metric and Palatini formalisms, discovering novel solutions previously unreported in literature.

Findings

New exact solutions in $f(R)$ gravity via Hojman symmetry

Hojman symmetry reveals features different from Noether symmetry

Supports Hojman symmetry as a useful tool in cosmology

Abstract

Nowadays, $f(R)$ theory has been one of the leading modified gravity theories to explain the current accelerated expansion of the universe, without invoking dark energy. It is of interest to find the exact cosmological solutions of $f(R)$ theories. Besides other methods, symmetry has been proved as a powerful tool to find exact solutions. On the other hand, symmetry might hint the deep physical structure of a theory, and hence considering symmetry is also well motivated. As is well known, Noether symmetry has been extensively used in physics. Recently, the so-called Hojman symmetry was also considered in the literature. Hojman symmetry directly deals with the equations of motion, rather than Lagrangian or Hamiltonian, unlike Noether symmetry. In this work, we consider Hojman symmetry in $f(R)$ theories in both the metric and Palatini formalisms, and find the corresponding exact cosmological solutions of $f(R)$ theories via Hojman symmetry. There exist some new solutions significantly different from the ones obtained by using Noether symmetry in $f(R)$ theories. To our knowledge, they also have not been found previously in the literature. This work confirms that Hojman symmetry can bring new features to cosmology and gravity theories.