Covariant Formulation of the Invariant Measure for the Mixmaster Dynamics

We provide a Hamiltonian analysis of the Mixmaster Universe dynamics showing the covariant nature of its chaotic behavior with respect to any choice of time variable. We construct the appropriate invariant measure for the system (which relies on the existence of an ``energy-like'' constant of motion) without fixing the time gauge, i.e. the corresponding lapse function. The key point in our analysis consists of introducing generic Misner-Chitr\'e-like variables containing an arbitrary function, whose specification allows one to set up the same dynamical scheme in any time gauge.Comment: 11 pages, 1 figur

Billiard Representation for Multidimensional Cosmology with Intersecting p-branes near the Singularity

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N-1)-dimensional Lobachevsky space, N = n+l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N-2)-dimensional sphere by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in ``truncated'' D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard.Comment: 27 pages Latex, 3 figs., submit. to Class. Quantum Gra

Multidimensional integrable vacuum cosmology with two curvatures

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 =$dim $M_1, N_2 =$ dim$M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found.Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravit

Non-minimally coupled scalar field cosmology on the phase plane

In this publication we investigate dynamics of a flat FRW cosmological model with a non-minimally coupled scalar field with the coupling term $\xi R \psi^{2}$ in the scalar field action. The quadratic potential function $V(\psi)\propto \psi^{2}$ is assumed. All the evolutional paths are visualized and classified in the phase plane, at which the parameter of non-minimal coupling $\xi$ 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 diverges.Comment: revtex4, 20 pages, 12 figs; (v2) title changed, analysis of critical points at infinity added, accepted to JCA

Geometry of dynamics, Lyapunov exponents and phase transitions

The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.Comment: REVTeX, 10 pages, 5 PostScript figures, published versio

Integration of D-dimensional 2-factor spaces cosmological models by reducing to the generalized Emden-Fowler equation

The D-dimensional cosmological model on the manifold $M = R \times M_{1} \times M_{2}$ describing the evolution of 2 Einsteinian factor spaces, $M_1$ and $M_2$, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of the perfect fluid components are both equal to 2, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin within discrete-group analysis. Using the integrable classes of this equation one generates the integrable cosmological models. The corresponding metrics are presented. The method is demonstrated for the special model with Ricci-flat spaces $M_1,M_2$ and the 2-component perfect fluid source.Comment: LaTeX file, no figure

Toda chains with type A_m Lie algebra for multidimensional m-component perfect fluid cosmology

We consider a D-dimensional cosmological model describing an evolution of Ricci-flat factor spaces, M_1,...M_n (n > 2), in the presence of an m-component perfect fluid source (n > m > 1). We find characteristic vectors, related to the matter constants in the barotropic equations of state for fluid components of all factor spaces. We show that, in the case where we can interpret these vectors as the root vectors of a Lie algebra of Cartan type A_m=sl(m+1,C), the model reduces to the classical open m-body Toda chain. Using an elegant technique by Anderson (J. Math. Phys. 37 (1996) 1349) for solving this system, we integrate the Einstein equations for the model and present the metric in a Kasner-like form.Comment: LaTeX, 2 ps figure

Stabilization of the Yang-Mills chaos in non-Abelian Born-Infeld theory

We investigate dynamics of the homogeneous time-dependent SU(2) Yang-Mills fields governed by the non-Abelian Born-Infeld lagrangian which arises in superstring theory as a result of summation of all orders in the string slope parameter $\alpha'$. It is shown that generically the Born-Infeld dynamics is less chaotic than that in the ordinary Yang-Mills theory, and at high enough field strength the Yang-Mills chaos is stabilized. More generally, a smothering effect of the string non-locality on behavior of classical fields is conjectured.Comment: 7 pages, 5 figure

Observational Constraints on the Generalized Chaplygin Gas

In this paper we study a quintessence cosmological model in which the dark energy component is considered to be the Generalized Chaplygin Gas and the curvature of the three-geometry is taken into account. Two parameters characterize this sort of fluid, the $\nu$ and the $\alpha$ parameters. We use different astronomical data for restricting these parameters. It is shown that the constraint $\nu \lesssim \alpha$ agrees enough well with the astronomical observations.Comment: Accepted by IJMPD; 18 pages; 10 Figure

Cosmological zoo -- accelerating models with dark energy

ecent observations of type Ia supernovae indicate that the Universe is in an accelerating phase of expansion. The fundamental quest in theoretical cosmology is to identify the origin of this phenomenon. In principle there are two possibilities: 1) the presence of matter which violates the strong energy condition (a substantial form of dark energy), 2) modified Friedmann equations (Cardassian models -- a non-substantial form of dark matter). We classify all these models in terms of 2-dimensional dynamical systems of the Newtonian type. We search for generic properties of the models. It is achieved with the help of Peixoto's theorem for dynamical system on the Poincar{\'e} sphere. We find that the notion of structural stability can be useful to distinguish the generic cases of evolutional paths with acceleration. We find that, while the $\Lambda$CDM models and phantom models are typical accelerating models, the cosmological models with bouncing phase are non-generic in the space of all planar dynamical systems. We derive the universal shape of potential function which gives rise to presently accelerating models. Our results show explicitly the advantages of using a potential function (instead of the equation of state) to probe the origin of the present acceleration. We argue that simplicity and genericity are the best guide in understanding our Universe and its acceleration.Comment: RevTeX4, 23 pages, 10 figure