Using the Noether symmetry approach to probe the nature of dark energy

Abstract

We propose to use a model-independent criterion based on first integrals of motion, due to Noether symmetries of the equations of motion, in order to classify the dark energy models in the context of scalar field (quintessence or phantom) FLRW cosmologies. In general, the Noether symmetries play an important role in physics because they can be used to simplify a given system of differential equations as well as to determine the integrability of the system. The Noether symmetries are computed for nine distinct accelerating cosmological scenarios that contain a homogeneous scalar field associated with different types of potentials. We verify that all the scalar field potentials, presented here, admit the trivial first integral namely energy conservation, as they should. We also find that the exponential potential inspired from scalar field cosmology, as well as some types of hyperbolic potentials, include extra Noether symmetries. This feature suggests that these potentials should be preferred along the hierarchy of scalar field potentials. Finally, using the latter potentials, in the framework of either quintessence or phantom scalar field cosmologies that contain also a non-relativistic matter(dark matter) component, we find that the main cosmological functions, such as the scale factor of the universe, the scalar field, the Hubble expansion rate and the metric of the FRLW space-time, are computed analytically. Interestingly, under specific circumstances the predictions of the exponential and hyperbolic scalar field models are equivalent to those of the $\Lambda$CDM model, as far as the global dynamics and the evolution of the scalar field are concerned. The present analysis suggests that our technique appears to be very competitive to other independent tests used to probe the functional form of a given potential and thus the associated nature of dark energy.Comment: Accepted for publication in Physical Review D (13 pages

Noether symmetry approach in Gauss–Bonnet cosmology

“…As discussed in [52], their existence allows to select physically observable universes and, specifically, to select analytical models related to observations. In particular, it is possible to classify dark energy behaviors related to Noether symmetries and then discriminate between physical and unphysical solutions (see [53,[67][68][69]). These topics will be the argument of a forthcoming, detailed study.…”

General scalar–tensor cosmology: analytical solutions via noether symmetry

“…Finally, expanding the action up to second order, we found the condition for stability of the de Sitter solution and showed it to be an attractor of the system for 3H d S + q d S > 0 in all five cases. It deserves mentioning at this point that symmetries are not only a mathematical tool to solve dynamical systems but also bring about the opportunity to physically select an observable universe and "particularly" to single out analytical models related to observation [96] which would provide a tool to classify dark energy models related to Noether symmetry; see [97][98][99].…”

Comment on “A study of phantom scalar field cosmology using Lie and Noether symmetries”

“…We would like to remark that the application of the complete Noether's theorem in scalar field cosmology can be found in 6 and the application of Lie point symmetries in, 7 hence the results of 1 are not new in the literature.…”