Hojman Symmetry in $f(T)$ Theory

TL;DR

This paper explores Hojman symmetry within $f(T)$ gravity, deriving exact cosmological solutions and revealing restrictions on the functional form of $f(T)$, offering an alternative to Noether symmetry approaches.

Contribution

It demonstrates the existence of Hojman symmetry in $f(T)$ gravity and derives exact solutions, highlighting differences from Noether symmetry methods.

Findings

Hojman symmetry exists in $f(T)$ theory.

Exact cosmological solutions are obtained.

Functional form of $f(T)$ is restricted to power-law or hypergeometric types.

Abstract

Today, $f(T)$ theory has been one of the popular modified gravity theories to explain the accelerated expansion of the universe without invoking dark energy. In this work, we consider the so-called Hojman symmetry in $f(T)$ theory. Unlike Noether conservation theorem, the symmetry vectors and the corresponding conserved quantities in Hojman conservation theorem can be obtained by using directly the equations of motion, rather than Lagrangian or Hamiltonian. We find that Hojman symmetry can exist in $f(T)$ theory, and the corresponding exact cosmological solutions are obtained. We find that the functional form of $f(T)$ is restricted to be the power-law or hypergeometric type, while the universe experiences a power-law or hyperbolic expansion. These results are different from the ones obtained by using Noether symmetry in $f(T)$ theory. Therefore, it is reasonable to find exact cosmological solutions via Hojman symmetry.