Dark energy FRW cosmology - dynamical system reconstruction

We develop a simple method of dark energy reconstruction using a geometrical form of the luminosity-distance relation. In this method the FRW dynamical system with dark energy is reconstructed instead of the equation of state parameter. We give several examples which illustrate the usefulness of our method in fitting the redshift transition from the decelerating to accelerating phase as the value of the Hubble function at the transition.Comment: Talk presented at Spanish Relativity Meeting 2007, Puerto de la Cruz, Tenerife, Spain, 10-14 September 200

Dynamical System Approach to Cosmological Models with a Varying Speed of Light

Methods of dynamical systems have been used to study homogeneous and isotropic cosmological models with a varying speed of light (VSL). We propose two methods of reduction of dynamics to the form of planar Hamiltonian dynamical systems for models with a time dependent equation of state. The solutions are analyzed on two-dimensional phase space in the variables $(x, \dot{x})$ where $x$ is a function of a scale factor $a$. Then we show how the horizon problem may be solved on some evolutional paths. It is shown that the models with negative curvature overcome the horizon and flatness problems. The presented method of reduction can be adopted to the analysis of dynamics of the universe with the general form of the equation of state $p=\gamma(a)\epsilon$. This is demonstrated using as an example the dynamics of VSL models filled with a non-interacting fluid. We demonstrate a new type of evolution near the initial singularity caused by a varying speed of light. The singularity-free oscillating universes are also admitted for positive cosmological constant. We consider a quantum VSL FRW closed model with radiation and show that the highest tunnelling rate occurs for a constant velocity of light if $c(a) \propto a^n$ and $-1 < n \le 0$. It is also proved that the considered class of models is structurally unstable for the case of $n < 0$.Comment: 18 pages, 5 figures, RevTeX4; final version to appear in PR

Multidimensional cosmological models with hydrodynamical energy-momentum tensor : part II : analysis of dynamical systems at infinity

The dynamics of the full class of multidimensional cosmological models with topology FRW $\times$ T$^{D}$, (where T$^{D}$ is a D-dimensional torus) near the singularity is investigated. Phase portraits show possible evolutions of FRW $\times$ T$^{D}$ models with hydrodynamical energy-momentum tensor. The problem of stability of solutions with a "crack-of-doom" singularity is also discussed

Multidimensional cosmological models with hydrodynamical energy-momentum tensor : part I : analysis of dynamical systems in finite domains

The dynamics of the full class of multidimensional cosmological models with topology FRW $\times$ T$^{D}$, where T$^{D}$ i a D-dimensional torus is investigated. Phase portraits show possible evolutions of FRW $\times$ T$^{D}$ models with a hydrodynamical energy-momentum tensor. Typical solutions for late times are studied. The stability of solutions, with dynamical reduction and inflation as dynamical effects of extra dimensions, is also discussed

Non-integrability of density perturbations in the FRW universe

We investigate the evolution equation of linear density perturbations in the Friedmann-Robertson-Walker universe with matter, radiation and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no "closed form" solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic's algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated.Comment: 13 pages. The latest version with added references, and a relevant new paragraph in section I

Can the initial singularity be detected by cosmological tests?

In the present paper we raise the question whether initial cosmological singularity can be proved from the cosmological tests. The classical general relativity predict the existence of singularity in the past if only some energy conditions are satisfied. On the other hand the latest quantum gravity applications to cosmology suggest of possibility of avoiding the singularity and replace it with the bounce. The distant type Ia supernovae data are used to constraints on bouncing evolutional scenario where square of the Hubble function $H^2$ is given by formulae $H^2=H^2_0[\Omega_{m,0}(1+z)^{m}-\Omega_{n,0}(1+z)^{n}]$, where $\Omega_{m,0}, \Omega_{n,0}>0$ are density parameters and $n>m>0$. We show that the on the base of the SNIa data standard bouncing models can be ruled out on the $4\sigma$ confidence level. If we add the cosmological constant to the standard bouncing model then we obtain as the best-fit that the parameter $\Omega_{n,0}$ is equal zero which means that the SNIa data do not support the bouncing term in the model. The bounce term is statistically insignificant the present epoch. We also demonstrate that BBN offer the possibility of obtaining stringent constraints of the extra term $\Omega_{n,0}$. The other observational test methods like CMB and the age of oldest objects in the Universe are used. We also use the Akaike informative criterion to select a model according to the goodness of fit and we conclude that this term should be ruled out by Occam's razor, which makes that the big bang is favored rather then bouncing scenario.Comment: 30 pages, 7 figures improved versio

Testing and selection of cosmological models with (1+z)6(1+z)^6 corrections

In the paper we check whether the contribution of $(-)(1+z)^6$ type in the Friedmann equation can be tested. We consider some astronomical tests to constrain the density parameters in such models. We describe different interpretations of such an additional term: geometric effects of Loop Quantum Cosmology, effects of braneworld cosmological models, non-standard cosmological models in metric-affine gravity, and models with spinning fluid. Kinematical (or geometrical) tests based on null geodesics are insufficient to separate individual matter components when they behave like perfect fluid and scale in the same way. Still, it is possible to measure their overall effect. We use recent measurements of the coordinate distances from the Fanaroff-Riley type IIb (FRIIb) radio galaxy (RG) data, supernovae type Ia (SNIa) data, baryon oscillation peak and cosmic microwave background radiation (CMBR) observations to obtain stronger bounds for the contribution of the type considered. We demonstrate that, while $\rho^2$ corrections are very small, they can be tested by astronomical observations -- at least in principle. Bayesian criteria of model selection (the Bayesian factor, AIC, and BIC) are used to check if additional parameters are detectable in the present epoch. As it turns out, the $\Lambda$CDM model is favoured over the bouncing model driven by loop quantum effects. Or, in other words, the bounds obtained from cosmography are very weak, and from the point of view of the present data this model is indistinguishable from the $\Lambda$CDM one.Comment: 19 pages, 1 figure. Version 2 generally revised and accepted for publicatio

The Mixmaster Universe in Five Dimensions

We consider a five dimensional vacuum cosmology with Bianchi type-IX spatial geometry and an extra non-compact coordinate. Finding a new class of solutions, we examine and rule out the possibility of deterministic chaos. We interpret this result within the context of induced matter theory.Comment: 13 page

Extended Quintessence with non-minimally coupled phantom scalar field

We investigate evolutional paths of an extended quintessence with a non-minimally coupled phantom scalar field $\psi$ 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 quasi-oscillatory and monotonic trajectories approach to the attractor which represents the FRW model with the cosmological constant. We demonstrate that dynamical system admits invariant two-dimensional submanifold and discussion that which cosmological scenario is realized depends on behavior of the system on the phase plane $(\psi, \psi')$. We formulate simple conditions on the value of coupling constant $\xi$ for which trajectories tend to the focus in the phase plane and hence damping oscillations around the mysterious value $w=-1$. 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. It is also investigated 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.Comment: revtex4, 15 pages, 9 figures; (v2) published versio

Topological quantum numbers and curvature -- examples and applications

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss-Bonnet theorem and the Poincar\'{e}-Hopf theorem.Comment: LaTeX2e, 26 pages, 4 figure