A late time accelerated FRW model with scalar and vector fields via Noether symmetry

Babak Vakili

doi:10.1016/j.physletb.2014.10.028

Abstract

We study the evolution of a three-dimensional minisuperspace cosmological model by the Noether symmetry approach. The phase space variables turn out to correspond to the scale factor of a flat Friedmann–Robertson–Walker (FRW) model, a scalar field with potential function V(ϕ) with which the gravity part of the action is minimally coupled and a vector field of its kinetic energy is coupled with the scalar field by a coupling function f(ϕ). Then, the Noether symmetry of such a cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generator of the desired symmetry. We explicitly calculate the form of the coupling function between the scalar and the vector fields and also the scalar field potential function for which such symmetry exists. Finally, by means of the corresponding Noether current, we integrate the equations of motion and obtain exact solutions for the scale factor, scalar and vector fields. It is shown that the resulting cosmology is an accelerated expansion universe for which its expansion is due to the presence of the vector field in the early times, while the scalar field is responsible of its late time expansion.

Highlights

Symmetries have always played a central role in conceptual discussion of the classical and quantum physics
We have studied a scalar-vector field model of cosmology in a Noether symmetry point of view, in such a way that in its action, in addition to a minimally coupling between the scalar field and gravity, there is a coupling between the scalar and the kinetic energy of the vector field
We have considered a flat FRW metric and set up the phase space by taking the scale factor a, scalar field φ and the vector field A as the independent dynamical variables

Summary

Introduction

Symmetries have always played a central role in conceptual discussion of the classical and quantum physics. From a more general point of view, it can be shown that all such conservation laws are particular cases of the so-called Noether theorem, according to which for every one-parameter group of coordinate transformation on the configuration space of a system, which preserves the Lagrangian function, there exists a first integral of motion [1] In mathematical language this means that if the vector field X is the generator of the above diffeomorphism, the Lie derivative of the Lagrangian function along it should vanish: LX L = 0 [2]. In cosmology, when the models are expressed in terms of the minisuperspace variables usually the scale factors, matter fields and their conjugate momenta play the role of dynamical variables In these models it can be shown that the evolution of the system can be obtained from an action of the form (2) [4]. Since the existence of a symmetry results in a constant of motion, we can integrate the field equations which would lead to the expansion law of the universe

The model

Noether symmetry

Cosmological dynamics

Summary