https://pure.mpg.de/pubman/item/item_2533988_4/component/file_2533987/1712.03107.pdf

Dynamical systems applied to cosmology: dark energy and modified gravity

The Nobel Prize winning confirmation in 1998 of the accelerated expansion of our Universe put into sharp focus the need of a consistent theoretical model to explain the origin of this acceleration. As a result over the past two decades there has been a huge theoretical and observational effort into improving our understanding of the Universe. The cosmological equations describing the dynamics of a homogeneous and isotropic Universe are systems of ordinary differential equations, and one of the most elegant ways these can be investigated is by casting them into the form of dynamical systems. This allows the use of powerful analytical and numerical methods to gain a quantitative understanding of the cosmological dynamics derived by the models under study. In this review we apply these techniques to cosmology. We begin with a brief introduction to dynamical systems, fixed points, linear stability theory, Lyapunov stability, centre manifold theory and more advanced topics relating to the global structure of the solutions. Using this machinery we then analyse a large number of cosmological models and show how the stability conditions allow them to be tightly constrained and even ruled out on purely theoretical grounds. We are also able to identify those models which deserve further in depth investigation through comparison with observational data. This review is a comprehensive and detailed study of dynamical systems applications to cosmological models focusing on the late-time behaviour of our Universe, and in particular on its accelerated expansion. In self contained sections we present a large number of models ranging from canonical and non-canonical scalar fields, interacting models and non-scalar field models through to modified gravity scenarios. Selected models are discussed in detail and interpreted in the context of late-time cosmology.

This paper deals with the study of dynamical or phase space analysis of Bianchi I universe in Brans-Dicke gravity with chameleon scalar field. For this purpose, the matter contents are taken to be perfect fluid with magnetic field effects described by the non-linear Maxwell Lagrangian density. By taking some ansatz for the field potential and the interaction function in chameleon cosmology, we discuss three cases: Bianchi I universe with perfect fluid, FRW universe with magnetized perfect fluid and Bianchi I universe with magnetized perfect fluid. In all cases, we calculate fixed or critical points and discuss stability of the respective configuration for radiation as well as matter dominated eras. We also evaluate some cosmological parameters in each case for matter dominated era only and investigate their cosmological implications.

Magnetized chameleonic Brans-Dicke cosmology and phase space analysis

For the spatially open Friedmann–Robertson–Walker (FRW) Universe with stiff matter and radiation as non-interacting matter sources, the scale function coming from the integration of the Friedmann equation is expressed in terms of elliptic integrals. For a negative cosmological constant, we identify the allowed ranges for the model’s parameters. Within the quantum analysis, the Wheeler–De Witt (WDW) equation turns into a modified Morse equation whose solutions are Mathieu and Heun functions.

Mathieu and Heun Solutions to the Wheeler–De Witt Equation for Hyperbolic Universes

The $(k=-1)$-Robertson–Walker spacetime is under investigation. With the derived Hamilton operator, we are solving the Wheeler–De Witt Equation and its Schrodinger-like extension, for physically important forms of the effective potential. The closed form solutions, expressed in terms of Heun’s functions, allow us to comment on the occurrence of Universe from highly probable quantum states.

Quantum Analysis of k=-1 Robertson–Walker Universe

The dynamical behaviors of FRW Universe containing a posivive/negative potential scalar field in loop quantum cosmology scenario are discussed. The method of the phase-plane analysis is used to investigate the stability of the Universe. It is found that the stability properties in this situation are quite different from the classical cosmology case. For a positive potential scalar field coupled with a barotropic fluid, the cosmological autonomous system has five fixed points and one of them is stable if the adiabatic index $\gamma $ satisfies $0<\gamma <2$. This leads to the fact that the universe just have one bounce point instead of the singularity which lies in the quantum dominated area and it is caused by the quantum geometry effect. There are four fixed points if one considers a scalar field with a negative potential, but none of them is stable. Therefore, the universe has two kinds of bounce points, one is caused by the quantum geometry effect and the other is caused by the negative potential, the Universe may enter a classical re-collapse after the quantum bounce. This hints that the spatially flat FRW Universe containing a negative potential scalar field is cyclic.

Dynamical behaviors of FRW universe containing a positive/negative potential scalar field in loop quantum cosmology

The dynamical behaviors of FRW Universe containing a posivive/negative potential scalar field in loop quantum cosmology scenario are discussed. The method of the phase-plane analysis is used to investigate the stability of the Universe. It is found that the stability properties in this situation are quite different from the classical cosmology case. For a positive potential scalar field coupled with a barotropic fluid, the cosmological autonomous system has five fixed points and one of them is stable if the adiabatic index $\gamma$ satisfies $0<\gamma<2$. This leads to the fact that the universe just have one bounce point instead of the singularity which lies in the quantum dominated area and it is caused by the quantum geometry effect. There are four fixed points if one considers a scalar field with a negative potential, but none of them is stable. Therefore, the universe has two kinds of bounce points, one is caused by the quantum geometry effect and the other is caused by the negative potential, the Universe may enter a classical re-collapse after the quantum bounce. This hints that the spatially flat FRW Universe containing a negative potential scalar field is cyclic.

Kui Xiao

Papers

Dynamical behaviors of FRW universe containing a positive/negative potential scalar field in loop quantum cosmology

Dynamical behaviors of FRW Universe containing a positive/negative potential scalar field in loop quantum cosmology