367 research outputs found
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
Multidimensional integrable vacuum cosmology with two curvatures
The vacuum cosmological model on the manifold describing the evolution of Einstein spaces of non-zero
curvatures is considered. For the Einstein equations are reduced to the
Abel (ordinary differential) equation and solved, when dim dim. The Kasner-like behaviour of the
solutions near the singularity is considered ( is synchronous
time). The exceptional ("Milne-type") solutions are obtained for arbitrary .
For these solutions are attractors for other ones, when . For dim and certain two-parametric
families of solutions are obtained from ones using "curvature-splitting"
trick. In the case , a family of non-singular
solutions with the topology is found.Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy).
Submitted to Classical and Quantum Gravit
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
Anisotropy and inflation in Bianchi I brane worlds
After a more general assumption on the influence of the bulk on the brane, we
extend some conclusions by Maartens et al. and Santos et al. on the asymptotic
behavior of Bianchi I brane worlds. As a consequence of the nonlocal
anisotropic stresses induced by the bulk, in most of our models, the brane does
not isotropize and the nonlocal energy does not vanish in the limit in which
the mean radius goes to infinity. We have also found the intriguing possibility
that the inflation due to the cosmological constant might be prevented by the
interaction with the bulk. We show that the problem for the mean radius can be
completely solved in our models, which include as particular cases those in the
references above.Comment: 10 pages, improved discussion on the likeliness of
non-isotropization, completed list of references, matches version to appear
in Class. Quantum Gra
(Non)Invariance of dynamical quantities for orbit equivalent flows
We study how dynamical quantities such as Lyapunov exponents, metric entropy,
topological pressure, recurrence rates, and dimension-like characteristics
change under a time reparameterization of a dynamical system. These quantities
are shown to either remain invariant, transform according to a multiplicative
factor or transform through a convoluted dependence that may take the form of
an integral over the initial local values. We discuss the significance of these
results for the apparent non-invariance of chaos in general relativity and
explore applications to the synchronization of equilibrium states and the
elimination of expansions
Theoretical Limits on the Equation-of-State Parameter of Phantom Cosmology
We investigate the restrictions on the equation-of-state parameter of phantom
cosmology, due to the minimum quantum gravitational requirements. We find that
for all the examined -parametrizations and for arbitrary phantom
potentials and spatial curvature, the phantom equation-of-state parameter is
not restricted at all. This is in radical contrast with the quintessence
paradigm, and makes phantom cosmology more robust and capable of constituting
the underlying mechanism for dark energy.Comment: 7 pages, 7 figure
On the resolution of cosmic coincidence problem and phantom crossing with triple interacting fluids
We here investigate a cosmological model in which three fluids interact with
each other involving certain coupling parameters and energy exchange rates. The
motivation of the problem stems from the puzzling `triple coincidence problem'
which naively asks why the cosmic energy densities of matter, radiation and
dark energy are almost of the same order of magnitude at the present time. In
our model, we determine the conditions under triple interacting fluids will
cross the phantom divide.Comment: 22 pages, 6 figures, to appear in Eur. Phys. J. C (2009
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 describing the evolution of 2 Einsteinian factor spaces,
and , 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
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
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
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
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