Dynamical System Perspective of Cosmological Models Minimally Coupled with Scalar Field

S Surendra Singh,Chingtham Sonia

doi:10.1155/2020/1805350

Abstract

The stability criteria for the dynamical system of a homogeneous and isotropic cosmological model are investigated with the interaction of a scalar field in the presence of a perfect fluid. In this paper, we depict the dynamical system perspective to study qualitatively the scalar field cosmology under two special cases, with and without potential. In the absence of potential, we get a two-dimensional dynamical system, and we study the analytical as well as geometrical behavior. For the dynamical system with potential, we analyze different potential forms: simple exponential potential form (Vϕ=Voe−λϕ), double exponential potential form Vϕ=Voexp−Aexp2αϕ, and inverse power law potential form (Vϕ=Voϕ−α). We generate an autonomous system of ordinary differential equations (ASODE) for each case by introducing new dimensionless variables and obtain respective fixed points. We also analyze the type, nature, and stability of the fixed points and how their behavior reflects towards the cosmological scenarios. Throughout the whole work, the investigation of this model has shown us the deep connection between these theories and cosmic acceleration phenomena. The phase plots of the system at different conditions and different values of γ have been analyzed in detail, and their geometrical interpretations have been studied. The perturbation plots of the dynamical system have been analyzed with emphasis on our analytical findings. We have evaluated the total energy density (Ωϕ) at the fixed points and also found out the suitable range of γ and λ for a stable model.

Highlights

The dynamical system approach, nowadays, has become one of the most suitable and viable ways for qualitative specification of various cosmic features, specially for studying the possible asymptotic states at early as well as late times of the evolving Universe
In this work, a dynamical system perspective of scalar field FRW cosmology without potential as well as with potential
In the absence of potential, we found a two-dimensional dynamical system and four critical points P, Q, R, and S

Summary

Introduction

The dynamical system approach, nowadays, has become one of the most suitable and viable ways for qualitative specification of various cosmic features, specially for studying the possible asymptotic states at early as well as late times of the evolving Universe. It is quite an accepted fact that inflation is one such theory of exponential expansion of the early Universe which corresponds to energy scales ∼1016 GeV [6,7,8,9,10,11,12,13]. This large scale deviation in the energy scales have drawn many researchers towards these two phenomena and how the difference has come up. Scalar field cosmology has paved its grand way to study dark energy which is taken as the main entity for the cosmic expansion with acceleration at late time.

Dynamical System Analysis

Conclusion