Noether symmetries and duality transformations in cosmology

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.

12 pages; discussion improved; to appear in MPLA

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• Duality transformation and conformal equivalent scalar-tensor theories
• Integrability of geodesic motions in curved manifolds through non-local conserved charges
• Late-time-accelerated expansion arisen from gauge fields in an anisotropic background and a fruitful trick for Noether's approach
• A new symmetry of the spatially flat Einstein-Friedmann equations
• Raychaudhuri-based reconstruction of anisotropic Einstein-Maxwell equation in 1+3 covariant formalism of $f(R)$-gravity
• A description of classical and quantum cosmology for a single scalar field torsion gravity
• $O(d,d)$ symmetry in teleparallel dark energy