Phase Space Analysis of Friedmann–Lemaitre–Robertson–Walker Cosmology

Abstract

Most of the physical phenomena can be described by non-linear dynamical systems. One can get exact solutions for some of the simplest non-linear dynamical systems but most of the non-linear dynamical systems have no exact solutions. Also, even if the system has exact solutions, it is too complicated to fully analyze the exact solutions because of the dependence of the solution behavior on the values of the system parameters, and the initial values of the variables. The dynamical system theory is providing us with powerful tools to analyze complicated non-linear dynamical systems qualitatively. One can use the dynamical system technique to attain all the different qualitative solutions for such a non-linear dynamical system like for example the standard cosmological models (Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) cosmological models). In this work, we present a complete dynamical study for a viscous Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) cosmological model in the presence of a cosmological constant. For the sake of clarification, our investigation is carried out in three different stages. In the first stage, we only consider ω (the equation of the state parameter) to be non-vanishing, and then Λ (the cosmological constant) is included while finally ξ (the bulk viscosity coefficient) is introduced. Viscous fluid FLRW cosmological models could be represented by twodimensional (planer) dynamical systems, where the system variables are Hubble’s parameter H and mass density ρ. Thus, in chapter 2, we explained in detail the essential dynamical tools for analyzing a planner dynamical system as well as the analysis of one-dimensional dynamical systems in order to form a solid understanding for important subjects like phase space, classification and stability of the fixed points, normal forms, and bifurcations. In chapter 3, essential dynamical tools are applied to the perfect fluid FLRW cosmological dynamical system. Also, in order to enhance our understanding of the