There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Dynamics of dark energy

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution

We present new determinations of the cosmic expansion history from red-envelope galaxies. We have obtained for this purpose high-quality spectra with the Keck-LRIS spectrograph of red-envelope galaxies in 24 galaxy clusters in the redshift range 0.2 < z < 1.0. We complement these Keck spectra with high-quality, publicly available archival spectra from the SPICES and VVDS surveys. We improve over our previous expansion history measurements in Simon et al. (2005) by providing two new determinations of the expansion history: H(z) = 97±62 km sec^(−1) Mpc^(−1) at z ≃ 0.5 and H(z) = 90±40 km sec^(−1) Mpc^(−1) at z ≃ 0.9. We discuss the uncertainty in the expansion history determination that arises from uncertainties in the synthetic stellar-population models. We then use these new measurements in concert with cosmic-microwave-background (CMB) measurements to constrain cosmological parameters, with a special emphasis on dark-energy parameters and constraints to the curvature. In particular, we demonstrate the usefulness of direct H(z) measurements by constraining the dark-energy equation of state parameterized by w_0 and w_a and allowing for arbitrary curvature. Further, we also constrain, using only CMB and H(z) data, the number of relativistic degrees of freedom to be 4±0.5 and their total mass to be < 0.2 eV, both at 1σ.

/pdf/cosmic-chronometers-constraining-the-equation-of-state-of-1c35umok8o.pdf

Cosmic chronometers: constraining the equation of state of dark energy. I: H(z) measurements

We investigate evolutional paths of an extended quintessence with a nonminimally coupled phantom scalar field $\ensuremath{\psi}$ to the Ricci curvature. The dynamical system methods are used to investigate typical regimes of dynamics at the late time. We demonstrate that there are two generic types of evolutional scenarios which approach the attractor (a focus or a node type critical point) in the phase space: the quasioscillatory and monotonic trajectories approach the attractor which represents the Friedmann-Robertson-Walker model with the cosmological constant. We demonstrate that the dynamical system admits an invariant two-dimensional submanifold and discuss that which cosmological scenario is realized depends on the behavior of the system on the phase plane $(\ensuremath{\psi},{\ensuremath{\psi}}^{\ensuremath{'}})$. We formulate simple conditions on the value of the coupling constant $\ensuremath{\xi}$ for which trajectories tend to the focus in the phase plane and hence damping oscillations around the mysterious value $w=\ensuremath{-}1$. We describe this condition in terms of slow-roll parameters calculated at the critical point. We discover that the generic trajectories in the focus-attractor scenario come from the unstable node. We also investigate the exact form of the parametrization of the equation of state parameter $w(z)$ (directly determined from dynamics) which assumes a different form for both scenarios.

Extended quintessence with nonminimally coupled phantom scalar field

In the paper we investigate observational constraints on coupling to gravity constant parameter {xi} using distant supernovae SNIa data, baryon oscillation peak (BOP), the cosmic microwave background radiation (CMBR) shift parameter, and the H(z) data set. We estimate the value of this parameter to constrain the extended quintessence models with nonminimally coupled to gravity phantom scalar field. The combined analysis of observational data favors a value of {xi} which lies in close neighborhood of the conformal coupling. While our estimations are model dependent they give rise to an indirect bound on the equivalence principle.

Coupling constant constraints in a nonminimally coupled phantom cosmology

Dissipative or conservative cosmology with dark energy

In this publication we investigate dynamics of a flat FRW cosmological model with a non-minimally coupled scalar field with the coupling term ξRψ2 in the scalar field action. The quadratic potential function V(ψ) ∝ ψ2 is assumed. All the evolutional paths are visualized and classified in the phase plane, at which the parameter of non-minimal coupling ξ plays the role of a control parameter. The fragility of global dynamics with respect to changes of the coupling constant is studied in details. We find that the future big rip singularity appearing in the phantom scalar field cosmological models can be avoided due to non-minimal coupling constant effects. We have shown the existence of a finite scale factor singular point (future or past) where the Hubble function as well as its first cosmological time derivative diverge.

Non-minimally coupled scalar field cosmology on the phase plane

The dynamics of the Brans-Dicke theory with a quadratic scalar field potential function and barotropic matter is investigated. The dynamical system methods are used to reveal complexity of dynamical evolution in homogeneous and isotropic cosmological models. The structure of phase space crucially depends on the parameter of the theory ωBD as well as barotropic matter index wm. In our analysis these parameters are treated as bifurcation parameters. We found sets of values of these parameters which lead to generic evolutional scenarios. We show that in isotropic and homogeneous models in the Brans-Dicke theory with a quadratic potential function the de Sitter state appears naturally. Stability conditions of this state are fully investigated. It is shown that these models can explain accelerated expansion of the Universe without the assumption of the substantial form of dark matter and dark energy. The Poincare construction of compactified phase space with a circle at infinity is used to show that phase space trajectories in a physical region can be equipped with a structure of a vector field on nontrivial topological closed space. For ωBD < −3/2 we show new types of early and late time evolution leading from the anti-de Sitter to the de Sitter state through an asymmetric bounce. In the theory without a ghost we find bouncing solutions and the coexistence of the bounces and the singularity. Following the Peixoto theorem some conclusions about structural stability are drawn.

https://iopscience.iop.org/article/10.1088/1475-7516/2013/12/016/pdf

Orest Hrycyna

Papers

Extended quintessence with nonminimally coupled phantom scalar field

Coupling constant constraints in a nonminimally coupled phantom cosmology

Dissipative or conservative cosmology with dark energy

Non-minimally coupled scalar field cosmology on the phase plane

Dynamical complexity of the Brans-Dicke cosmology