Noether symmetry approach in non-minimal derivative coupling gravity

TL;DR

This paper explores solutions in non-minimal derivative coupling gravity using Noether symmetry, revealing early universe behavior and new scalar field forms in both GR and NMDC models.

Contribution

It introduces a Noether symmetry approach to find exact solutions in NMDC gravity, highlighting early universe dynamics and scalar field behaviors.

Findings

Kinetic-dominant phase with a(t) ∝ t^{1/3} before inflation

New exponential forms of scalar fields in GR and NMDC models

Distinct scalar field solutions governed by exponential functions

Abstract

In this work, we examine solutions of the system of equations obtained by applying the Noether gauge symmetry (NGS) and its conserved quantity for the standard general relativity (GR) and the non-minimal derivative coupling (NMDC) cosmological model. We discover two salient features of the solutions. The first one is $a(t)\propto t^{1/3}$ for a kinetic-dominant phase which may emerge before inflationary period at very early time. The second one is a new form of scalar field $\phi(t)$ govern by the exponential cosmological solution for GR and NMDC $\phi_{\rm GR}(t)= \sqrt{c_{1}+c_{2}t+c_{3}e^{-\lambda t}} $ and $\phi_{\rm NMDC}(t)= \sqrt{c_{1}+c_{2}e^{-\lambda_{1}t}+c_{3}e^{-\lambda_{2}t}}$, respectively.