Noether symmetries and duality transformations in cosmology

TL;DR

This paper explores the connection between Noether symmetries and duality transformations in cosmological models, revealing how symmetries relate to scale-factor duality and gravitational model equivalences.

Contribution

It demonstrates the link between Noether point symmetries and discrete duality symmetries in various cosmological theories, including Brans-Dicke and $f(R)$ gravity.

Findings

Noether symmetries imply the existence of reversal symmetries in suitable coordinates.

Scale-factor duality in the dilaton field relates to Noether symmetries of the field equations.

Duality transformations can relate different gravitational models in minisuperspace.

Abstract

We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian then there exists a coordinate system in which a reversal symmetry exists. Moreover as far as concerns the scale-factor duality symmetry of the dilaton field, we show that it is related to the existence of a Noether symmetry for the field equations, and the reversal symmetry in the normal coordinates of the symmetry vector becomes scale-factor duality symmetry in the original coordinates. In particular the same point symmetry as also the same reversal symmetry exists for the Brans-Dicke- scalar field with linear potential while now the discrete symmetry in the original coordinates of the system depends on the Brans-Dicke parameter and it is a scale-factor duality when $\omega_{BD}=1$. Furthermore, in the context of the O'Hanlon theory for $f\left( R\right) $-gravity, it is possible to show how a duality transformation in the minisuperspace can be used to relate different gravitational models.