Noether Symmetries in $f(T,T_G)$ Cosmology

TL;DR

This paper investigates Noether symmetries in $f(T,T_G)$ gravity models, identifying symmetries for different functional forms to facilitate exact cosmological solutions, extending previous work with new models including exponential terms.

Contribution

It extends the analysis of Noether symmetries in $f(T,T_G)$ gravity to new functional forms, including exponential models, and demonstrates how to derive exact solutions using these symmetries.

Findings

Non-trivial Noether vectors are found for all studied models.

Symmetries enable the derivation of exact cosmological solutions.

Extension of previous models to include exponential $f(T,T_G)$ forms.

Abstract

All degrees of freedom related to the torsion scalar can be explored by analysing, the $f(T,T_G)$ gravity formalism where, $T$ is a torsion scalar and $T_G$ is the teleparallel counterpart of the Gauss-Bonnet topological invariant term. The well-known Noether symmetry approach is a useful tool for selecting models that are motivated at a fundamental level and determining the exact solution to a given Lagrangian, hence we explore Noether symmetry approach in $f(T,T_G)$ gravity formalism with three different forms of $f(T,T_G)$ and study how to establish nontrivial Noether vector form for each one of them. We extend the analysis made in \cite{capozziello2016noether} for the form $f(T,T_{G})=b_{0}T_{G}^{k}+t_{0}T^{m}$ and discussed the symmetry for this model with linear teleparallel equivalent of the Gauss-Bonnet term, followed by the study of two models containing exponential form of the teleparallel equivalent of the Gauss-Bonnet term. We have shown that all three cases will allow us to obtain non-trivial Noether vector which will play an important role to obtain the exact solutions for the cosmological equations.