Noether symmetry approach in the cosmological alpha-attractors

TL;DR

This paper uses Noether symmetry to analyze cosmological alpha-attractors, enabling exact solutions and revealing how the Hubble parameter's evolution depends on initial parameters, with potential for de Sitter behavior.

Contribution

It introduces a method to exactly solve the dynamics of cosmological alpha-attractors using Noether symmetry in the Einstein frame.

Findings

The dynamical system can be fully integrated with Noether symmetry.

The potential can be exactly derived from the symmetry.

The Hubble parameter can evolve to a constant, indicating de Sitter behavior.

Abstract

In cosmological framework, Noether symmetry technique has revealed a useful tool in order to examine exact solutions. In this work, we first introduce the Jordan-frame Lagrangian and apply the conformal transformation in order to obtain the Lagrangian equivalent to Einstein-frame form. We then analyse the dynamics of the field in the cosmological alpha-attractors using the Noether sysmetry approach by focusing on the single field scenario in the Einstein-frame form. We show that with a Noether symmetry the coresponding dynamical system can be completely integrated and the potential exhibited by the symmetry can be exactly obtained. With the proper choice of parameters, the behavior of the scale factor displays an exponential (de Sitter) behavior at the present epoch. Moreover, we discover that the Hubble parameters strongly depends on the initial values of parameters exhibited by the Noether symmetry. Interestingly, it can retardedly evolve and becomes a constant in the present epoch in all cases.