Dynamical systems applied to cosmology: Dark energy and modified gravity

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.

Disformally coupled quintessence

“…• If M −1 = 0 this system coincides with the standard coupled quintessence scenario, which was properly analysed in [7,18,20].…”

“…• Finally, if M −1 = 0 and α = 0 we are left with the standard (uncoupled) quintessence scenario, presented in [7,83].…”

“…IntroductionSince 1998 that we have been witnessing growing observational confirmation of the present accelerated expansion of the Universe [1][2][3][4]. This phenomenon can be attributed to an exotic fluid, the so-called Dark Energy (DE), which must amount to about 70% of the total content of the Universe, and whose effective negative pressure can successfully explain the observations (see [5][6][7] for recent reviews). Presently, the ΛCDM model is the most well-accepted cosmological model, consisting of a cosmological constant dark energy source, Λ, plus a dark matter component, needed in order to make formation of structure possible in the Universe [8].…”

Cosmological consequences of a scalar field with oscillating equation of state: A possible solution to the fine-tuning and coincidence problems

“…The terrible thing is, unlike the results of most dark energy models (see Ref. [10] for a review), here the critical points do not provide any quantitative information about the evolution of the system for λ 2 > 0.…”

“…3 in Ref. [10] for an intuition). It is believed that some dynamical dark energy models can alleviate, but not solve, this problem with the tracker property [11][12][13][14][15].…”