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.

Figures (35)

• Figure 1: Classification of cosmological models analysed with dynamical systems techniques.
• Figure 2: Phase space portrait of the dynamical system (\ref{['043']})--(\ref{['044']}). The yellow/shaded area denotes the region of the phase space where the universe is accelerating.
• Figure 3: Evolution of the relative energy density of dark matter ($\Omega_{\rm m}$), radiation ($\Omega_{\rm r}$) and dark energy ($\Omega_\Lambda$), together with the effective EoS parameter ($w_{\rm eff}$) in the $\Lambda$CDM model. The vertical dashed line indicates the present cosmological time.
• Figure 4: Global phase space of the spatially curved $\Lambda$CDM model in the single fluid case Goliath:1998na. The global space has been obtained by patching the $k<0$ (rectangular part) and the $k>0$ phase spaces (triangular parts) along the $k=0$ invariant submanifold that connects the points $F$ and $dS$. Note that the different sections of this plot represent different phase spaces defined by different variables. For this reason the yellow area representing the accelerating regime is not delimited by a smooth boundary.
• Figure 5: Phase space with $\lambda=1$ and $w=0$. The only attractor is Point $C$ which represents an accelerating solution. For values $\lambda^2>2$ Point $C$ would lie outside the acceleration region (yellow/shaded) and would not be an inflationary solution. The red/dashed line highlights the heteroclinic orbit connecting Point $O$ to Point $C$. The yellow/shaded region denotes the part of the phase space where the universe is accelerating ($w_{\rm eff}<-1/3$).

Theorems & Definitions (14)

• Definition 1: Critical point or fixed point or equilibrium point
• Definition 2: Stable fixed point
• Definition 3: Asymptotically stable fixed point
• Definition 4: Hyperbolic point
• Definition 5: Lyapunov function
• Theorem 1: Lyapunov stability
• Definition 6: Centre Manifold
• Theorem 2: Existence
• Theorem 3: Stability
• Theorem 4: Approximation