Dynamical systems applied to cosmology: dark energy and modified gravity (1712.03107v3)

Abstract: 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.

• The paper applies dynamical systems techniques to reveal critical points and stability in cosmological models of dark energy and modified gravity.
• It employs methods like linear stability, Lyapunov analysis, and center manifold theory to convert complex equations into autonomous systems for deeper insights.
• The study presents numerical evidence of stable attractor solutions, highlighting models that naturally explain the universe's late-time acceleration.

Dynamical Systems Applied to Cosmology: Dark Energy and Modified Gravity

The academic paper under discussion performs a comprehensive analysis of dynamical systems techniques as applied to cosmological models, with a specific focus on dark energy and modified gravity. The paper is a detailed review touching upon the late-time cosmological behaviors of these models that are consistent with the observed universal acceleration. A key feature of the analysis is how these methods can be employed to either constrain or entirely rule out some cosmological models based purely on theoretical grounds.

Overview of Cosmological Dynamical Systems

The paper begins by introducing the framework of dynamical systems and how such frameworks can offer insights into the global dynamics of cosmological models. Theoretical tools discussed include linear stability theory, Lyapunov stability, and the center manifold theory among others, which are utilized to decipher the qualitative behavior of cosmological models at late times. By transforming cosmological equations into autonomous systems of differential equations, researchers can explore the stability, viability, and potential physical implications of different models.

Application to Various Cosmological Models

The authors apply these methods across a broad spectrum of cosmological models. Among these are:

• Canonical and Non-canonical Scalar Fields: The paper explores scenarios beyond a simple cosmological constant, including quintessence and phantom fields which provide dynamic forms of dark energy. The potential of each scalar field is cast into explicit forms (e.g., exponential or power-law potentials) and analyzed for fixed points that might indicate viable late-time behaviors.
• Interacting Dark Energy Models: The inclusion of interaction terms between dark energy and dark matter has been shown to enable scaling solutions which may address the cosmic coincidence problem—why the densities of dark energy and dark matter are of the same order today.
• Modified Gravity Theories: The paper also extends beyond General Relativity, considering scalar-tensor theories, $f(R)$ theories, and teleparallel models. Such models are analyzed to determine how modifications to gravity itself might provide explanations for cosmic acceleration.

Numerical Results and Stability Analysis

An impressive aspect of the paper is the thorough numerical analysis presented. For various models, critical points are identified, and their stability properties are explored, detailing when and why certain models may exhibit accelerated expansion late in the cosmic timeline. Notably, specific configurations of scalar fields or particular forms of the gravitational action often yield surprising results, such as stable attractor solutions that align well with observational data.

Theoretical Implications and Future Directions

The implications of such detailed stability analyses are substantial. By identifying stable fixed points, the authors spotlight which theoretical models can naturally lead to the observed accelerated expansion without additional tuning. Furthermore, the potential for future developments in cosmic acceleration theories is significant. With advancing observational capabilities, particularly in detecting high-redshift phenomena, the predictions of these models can be put to more stringent tests.

The review concludes that while some models can be ruled out due to their predicted instability or lack of consistency with observations, others offer promising explanations that merit further investigation. Continued exploration of these models, potentially incorporating new forms of matter or gravitational interactions, is recommended. The paper lays a robust methodological foundation for future work in theoretical cosmology, suggesting that innovative approaches to dynamical systems can significantly enhance our understanding of dark energy and the expanding universe.

In sum, the research scrutinizes a multitude of cosmological models through the lens of dynamical systems, offering a fresh pathway to investigate and potentially explain the enigmatic nature of dark energy and the accelerating universe.