What ξ\xi? Cosmological constraints on the non-minimal coupling constant

In dynamical system describing evolution of universe with the flat Friedmann-Robertson-Walker symmetry filled with barotropic dust matter and non-minimally coupled scalar field with a constant potential function an invariant manifold of the de Sitter state is used to obtain exact solutions of the reduced dynamics. Using observational data coming from distant supernovae type Ia, the Hubble function $H(z)$ measurements and information coming from the Alcock-Paczy$\'n$ski test we find cosmological constraints on the non-minimal coupling constant $\xi$ between the scalar curvature and the scalar field. For all investigated models we can exclude negative values of this parameter at the $68\%$ confidence level. We obtain constraints on the non-minimal coupling constant consistent with condition for conformal coupling of the scalar field in higher dimensional theories of gravity.Comment: (v2) 11 pages, 2 figs. published versio

Cosmological dynamics with non-minimally coupled scalar field and a constant potential function

Dynamical systems methods are used to investigate global behavior of the spatially flat Friedmann-Robertson-Walker cosmological model in gravitational theory with a non-minimally coupled scalar field and a constant potential function. We show that the system can be reduced to an autonomous three-dimensional dynamical system and additionally is equipped with an invariant manifold corresponding to an accelerated expansion of the universe. Using this invariant manifold we find an exact solution of the reduced dynamics. We investigate all solutions for all admissible initial conditions using theory of dynamical systems to obtain a classification of all evolutional paths. The right-hand sides of the dynamical system depend crucially on the value of the non-minimal coupling constant therefore we study bifurcation values of this parameter under which the structure of the phase space changes qualitatively. We found a special bifurcation value of the non-minimal coupling constant which is distinguished by dynamics of the model and may suggest some additional symmetry in matter sector of the theory.Comment: 39 pages, 8 multiple figs; v2. 41 pages, 8 multiple figs. published versio

Paths of Friedmann-Robertson-Walker brane models

Dynamics of brane-world models of dark energy is reviewed. We demonstrate that simple dark energy brane models can be represented as 2-dimensional dynamical systems of a Newtonian type. Hence a fictitious particle moving in a potential well characterizes the model. We investigate the dynamics of the brane models using methods of dynamical systems. The simple brane-world models can be successfully unified within a single scheme -- an ensemble of brane dark energy models. We characterize generic models of this ensemble as well as exceptional ones using the notion of structural stability (instability). Then due to the Peixoto theorem we can characterize the class of generic brane models. We show that global dynamics of the generic brane models of dark energy is topologically equivalent to the concordance $\Lambda$CDM model. We also demonstrate that the bouncing models or models in which acceleration of the universe is only transient phenomenon are non-generic (or exceptional cases) in the ensemble. We argue that the adequate brane model of dark energy should be a generic case in the ensemble of FRW dynamical systems on the plane.Comment: revtex4, 14 pages, 11 figures; (v2) title changed, published versio

Dynamical complexity of the Brans-Dicke cosmology

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 $\omega_{\textrm{BD}}$ as well as barotropic matter index $w_{m}$. 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 $\omega_{\textrm{BD}}<-3/2$ 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.Comment: 34 pages, 14 figs; (v2) 36 pages, 16 figs, refs. added, JCAP (in press

Route to Lambda in conformally coupled phantom cosmology

In this letter we investigate acceleration in the flat cosmological model with a conformally coupled phantom field and we show that acceleration is its generic feature. We reduce the dynamics of the model to a 3-dimensional dynamical system and analyze it on a invariant 2-dimensional submanifold. Then the concordance FRW model with the cosmological constant $\Lambda$ is a global attractor situated on a 2-dimensional invariant space. We also study the behaviour near this attractor, which can be approximated by the dynamics of the linearized part of the system. We demonstrate that trajectories of the conformally coupled phantom scalar field with a simple quadratic potential crosses the cosmological constant barrier infinitely many times in the phase space. The universal behaviour of the scalar field and its potential is also calculated. We conclude that the phantom scalar field conformally coupled to gravity gives a natural dynamical mechanism of concentration of the equation of state coefficient around the magical value $w_{\text{eff}}=-1$. We demonstrate route to Lambda through the infinite times crossing the $w_{\text{eff}}=-1$ phantom divide.Comment: revtex4, 7 pages, 3 figures; (v2) refs. added, typos corrected; (v3) published versio

Brans-Dicke theory and the emergence of \Lambda CDM model

The dynamics of the Brans-Dicke theory with a scalar field potential function is investigated. We show that the system with a barotropic matter content can be reduced to an autonomous three-dimensional dynamical system. For an arbitrary potential function we found the values of the Brans-Dicke parameter for which a global attractor in the phase space representing de Sitter state exists. Using linearized solutions in the vicinity of this critical point we show that the evolution of the Universe mimics the $\Lambda$CDM model. From the recent Planck satellite data, we obtain constraints on the variability of the effective gravitational coupling constant as well as the lower limit of the mass of the Brans-Dicke scalar field at the de Sitter state.Comment: 15 pages; (v2) 12 pages, refs. added, PhysRevD (in press

The non-minimal coupling constant and the primordial de Sitter state

Dynamical systems methods are used to investigate dynamics of a flat Friedmann-Robertson-Walker cosmological model with the non-minimally coupled scalar field and a potential function. Performed analysis distinguishes the value of non-minimal coupling constant parameter $\xi=\frac{3}{16}$, which is the conformal coupling in five dimensional theory of gravity. It is shown that for a monomial potential functions at infinite values of the scalar field there exist generic de Sitter and Einstein-de Sitter states. The de Sitter state is unstable with respect to expansion of the Universe for potential functions which do not change faster than linearly. This leads to a generic cosmological evolution without the initial singularity.Comment: 22 pages; 6 multiple figures; (v2) published versio

Dynamics and cosmological constraints on Brans-Dicke cosmology

We investigate observational constraints on the Brans-Dicke cosmological model using observational data coming from distant supernovae type Ia, the Hubble function $H(z)$ measurements, information coming from the Alcock-Paczy{\'n}ski test, and baryon acoustic oscillations. Our analysis is based on the modified Friedmann function resulting form dynamical investigations of Brans-Dicke cosmology in the vicinity of a de Sitter state. The qualitative theory of dynamical systems enables us to obtain three different behaviors in the vicinity of this state. We find for a linear approach to the de Sitter state $\omega_{\textrm{BD}}=-0.8606^{+0.8281}_{-0.1341}$, for an oscillatory approach to the de Sitter state $\omega_{\textrm{BD}}=-1.1103^{+0.1872}_{-0.1729}$, and for the transient de Sitter state represented by a saddle-type critical point $\omega_{\textrm{BD}}=-2.3837^{+0.4588}_{-4.5459}$. We obtain the mass of the Brans-Dicke scalar field at the present epoch as $m_{\phi}\sim H_{0}$. The Bayesian methods of model comparison are used to discriminate between obtained models. We show that observational data point toward vales of the $\omega_{\textrm{BD}}$ parameter close to the value suggested by the low-energy limit of the bosonic string theory.Comment: 13 pages, 5 figs; (v2) new analysis, refs. added, PhysRevD (in press

Regular and chaotic dynamics in scalar field cosmology

This dissertation is devoted to investigations of dynamics of Friedmann-Robertson-Walker cosmological models with a non-minimally coupled scalar field and a barotropic matter content. Using dynamical systems methods two types of evolution: the complex behaviour and the regular expansion of the universe were studied in details. Different manifestations of the coexistence of chaotic and regular behaviour in the conformally coupled scalar field cosmology were demonstrated. Also, the fragility of cosmological evolution with respect to the value of non-minimal coupling parameter was presented. The full characteristic of the dynamical evolution of a non-minimally coupled scalar field cosmological models was performed. It was shown that the inclusion of both types of matter opens an enormous dynamical complexity of possible evolutional paths of the universe, which are not present in standard cosmological models or in the models filled with a scalar field only. The generic evolutional path, which does not depend on the form of the scalar field potential function, was found. It leads to new and natural possibility of unified description of the cosmological evolution. Within one framework of a non-minimally coupled scalar field cosmology all the major epochs in the history of the universe emerge as critical points of the corresponding dynamical system (a finite scale factor singularity, an inflation (slow-roll and fast-roll), a radiation era, a barotropic matter domination era and finally the present accelerated expansion epoch)

Constraints on oscillating dark energy models

The oscillating scenario of route to Lambda was recently proposed by us arXiv:0704.1651 as an alternative to a cosmological constant in a explanation of the current accelerating universe. In this scenario phantom scalar field conformally coupled to gravity drives the accelerating phase of the universe. In our model $\Lambda$CDM appears as a global attractor in the phase space. In this paper we investigate observational constraints on this scenario from recent measurements of distant supernovae type Ia, CMB R shift, BAO and $H(z)$ observational data. The Bayesian methods of model selection are used in comparison the model with concordance $\Lambda$CDM one as well as with model with dynamical dark energy parametrised by linear form. We conclude that $\Lambda$CDM is favoured over FRW model with dynamical oscillating dark energy. Our analysis also demonstrate that FRW model with oscillating dark energy is favoured over FRW model with decaying dark energy parametrised in linear way.Comment: revtex4, 7 pages, 3 figures; (v2) 12 pages, 11 figures; Phys. Lett. B (in press