Future Attractors Research Articles

Scaling attractors for quintessence in a flat universe with a cosmological term

For evolution of a flat universe, we classify late time and future attractors with the scalingbehaviour of scalar field quintessence in the case of a potential which at definite values ofits parameters and initial data corresponds to exact scaling in the presence of acosmological constant.

A dynamical system approach to inhomogeneous dust solutions

We examine numerically and qualitatively the Lemaître–Tolman–Bondi (LTB) inhomogeneous dust solutions as a three-dimensional dynamical system characterized by six critical points. One of the coordinates of the phase space is an average density parameter, ⟨Ω⟩, which behaves as the ordinary Ω in Friedman–Lemaître–Robertson–Walker (FLRW) dust spacetimes. The other two coordinates, a shear parameter and a density contrast function, convey the effects of inhomogeneity. As long as shell crossing singularities are absent, this phase space is bounded or it can be trivially compactified. This space contains several invariant subspaces which define relevant particular cases, such as ‘parabolic’ evolution, FLRW dust and the Schwarzschild–Kruskal vacuum limit. We examine in detail the phase-space evolution of several dust configurations: a low-density void formation scenario, high-density re-collapsing universes with open, closed and wormhole topologies, a structure formation scenario with a black hole surrounded by an expanding background and the Schwarzschild–Kruskal vacuum case. Solution curves (except regular centers) start expanding from a past attractor (source) in the plane ⟨Ω⟩ = 1, associated with self-similar regime at an initial singularity. Depending on the initial conditions and specific configurations, the curves approach several saddle points as they evolve between this past attractor and two other possible future attractors: perpetually expanding curves terminate at a line of sinks at ⟨Ω⟩ = 0, while collapsing curves reach maximal expansion as ⟨Ω⟩ diverges and end up in a sink that coincides with the past attractor and is also associated with self-similar behavior.

Entropy and Syntropy: From Mechanical to Life Science

This work describes the qualities and implications, in the field of psychology, of two principles which can be observed in the physical and biological world: the principle of entropy and the principle of syntropy. The description of the qualities of entropic and syntropic phenomena can be found in the works of Luigi Fantappie (1901-1956), one of the major Italian mathematicians, who, while working on quantum mechanics and special relativity, discovered that all physical and chemical phenomena, which are determined by causes placed in the past, are governed by the principle of entropy, while all those phenomena which are attracted towards causes which are placed in the future (attractors), are governed by a principle which is symmetrical to entropy and which Fantappie named syntropy.

Late-time behaviour of the tilted Bianchi type VIh models

We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VIh using dynamical systems methods and numerical experimentation, with an emphasis on their future asymptotic evolution. We determine all of the equilibrium points of the type VIh state space (which correspond to exact self-similar solutions of the Einstein equations, some of which are new), and their stability is investigated. We find that there are vacuum plane-wave solutions that act as future attractors. In the parameter space, a ‘loophole’ is shown to exist in which there are no stable equilibrium points. We then show that a Hopf-bifurcation can occur resulting in a stable closed orbit (which we refer to as the Mussel attractor) corresponding to points both inside the loophole and points just outside the loophole; in the former case the closed curves act as late-time attractors while in the latter case these attracting curves will co-exist with attracting equilibrium points. In the special Bianchi type III case, centre manifold theory is required to determine the future attractors. Comprehensive numerical experiments are carried out to complement and confirm the analytical results presented. We note that the Bianchi type VIh case is of particular interest in that it contains many different subcases which exhibit many of the different possible future asymptotic behaviours of Bianchi cosmological models.

Equilibrium points of the tilted perfect fluid Bianchi VIh state space

We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $\gamma \in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $\gamma \in (1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having non-zero vorticity. In both cases the equilibrium points have a five dimensional stable manifold and may act as future attractors at least for the models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the self-similar metrics in terms of the state and group parameters. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group parameter $h$. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI$_h$.

Letter: On Tilted Perfect Fluid Bianchi Type VI0Self-Similar Models

We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{% Exact power law solutions of the Einstein equations}, 1985 Phys. Lett. \textbf{107}A 29-32] is the most general class of tilted self-similar models but the state parameter $\gamma $ lies in the interval $(\frac 65,\frac 32) $. The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI$_0$ models satisfying $n_\alpha ^\alpha =0$ in which it belongs. In addition the angle of tilt is asymptotically significant at late times suggesting that for the above subclasses of models the tilt is asymptotically extreme.

Scaling solution, radion stabilization, and initial condition for brane-world cosmology

We propose a new, self-consistent and dynamical scenario which gives rise to well-defined initial conditions for five-dimensional brane-world cosmologies with radion stabilization. At high energies, the five-dimensional effective theory is assumed to have a scale invariance so that it admits an expanding scaling solution as a future attractor. The system automatically approaches the scaling solution and, hence, the initial condition for the subsequent low-energy brane cosmology is set by the scaling solution. At low energies, the scale invariance is broken and a radion stabilization mechanism drives the dynamics of the brane-world system. We present an exact, analytic scaling solution for a class of scale-invariant effective theories of five-dimensional brane-world models which includes the five-dimensional reduction of the Horava-Witten theory, and provide convincing evidence that the scaling solution is a future attractor.

Asymptotic dynamics of the exceptional Bianchi cosmologies

In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI−1/9. This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous G2 cosmologies.Using the orthonormal frame formalism and Hubble-normalized variables, we show that the exceptional Bianchi cosmologies differ from the non-exceptional Bianchi cosmologies of type VIh in two significant ways. Firstly, the models exhibit an oscillatory approach to the initial singularity and hence are not asymptotically self-similar. Secondly, at late times, although the models are asymptotically self-similar, the future attractor for the vacuum-dominated models is the so-called Robinson–Trautman SH model instead of the vacuum SH plane wave models.

The Weyl tensor in spatially homogeneous cosmological models

We study the evolution of the Weyl curvature invariant in all spatially homogeneous universe models containing a non-tilted γ-law perfect fluid. We investigate all the Bianchi and Thurston type universe models and calculate the asymptotic evolution of Weyl curvature invariant for generic solutions to the Einstein field equations. The influence of compact topology on Bianchi types with hyperbolic space sections is also considered. Special emphasis is placed on the late-time behaviour where several interesting properties of the Weyl curvature invariant occur. The late-time behaviour is classified into five distinctive categories. It is found that for a large class of models, the generic late-time behaviour of the Weyl curvature invariant is to dominate the Ricci invariant at late times. This behaviour occurs in universe models which have future attractors that are plane-wave spacetimes, for which all scalar curvature invariants vanish. The overall behaviour of the Weyl curvature invariant is discussed in relation to the proposal that some function of the Weyl tensor or its invariants should play the role of a gravitational ‘entropy’ for cosmological evolution. In particular, it is found that for all ever-expanding models the measure of gravitational entropy proposed by Grøn and Hervik increases at late times.

Closed cosmologies with a perfect fluid and a scalar field

Closed, spatially homogeneous cosmological models with a perfect fluid and a scalar field with exponential potential are investigated, using dynamical systems methods. First, we consider the closed Friedmann-Robertson-Walker models, discussing the global dynamics in detail. Next, we investigate Kantowski-Sachs models, for which the future and past attractors are determined. The global asymptotic behaviour of both the Friedmann-Robertson-Walker and the Kantowski-Sachs models is that they either expand from an initial singularity, reach a maximum expansion and thereafter recollapse to a final singularity (for all values of the potential parameter kappa), or else they expand forever towards a flat power-law inflationary solution (when kappa^2<2). As an illustration of the intermediate dynamical behaviour of the Kantowski-Sachs models, we examine the cases of no barotropic fluid, and of a massless scalar field in detail. We also briefly discuss Bianchi type IX models.

Self-similar sphericallysymmetric cosmological models with a perfect fluid and a scalar field

Self-similar, spherically symmetric cosmological models with a perfectfluid and a scalar field with an exponential potential are investigated.New variables are defined which lead to a compact state space, anddynamical systems methods are utilized to analyse the models. Due tothe existence of monotonic functions, global dynamical results can bededuced. In particular, all of the future and past attractors forthese models are obtained and the global results are discussed. Theessential physical results are that initially expanding models alwaysevolve away from a massless scalar field model with an initialsingularity and, depending on the parameters of the models, eitherrecollapse to a second singularity or expand forever towards a flatpower-law inflationary model. The special cases in which there is nobarotropic fluid and in which the scalar field is massless areconsidered in more detail in order to illustrate the asymptoticresults. Some phase portraits are presented and the intermediatedynamics and hence the physical properties of the models arediscussed.

Scaling solutions inRobertson-Walker spacetimes

We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p = (-1) and a non-interacting scalar field with an exponential potential V() = V0e-. We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where 1 ( < (2/3),2 < 3 and > (2/3),2 < 2). Another is the zero-curvature scaling solution, first identified by Wetterich, where the energy density of the scalar field is proportional to that of matter with 3/2 ( < (2/3),2 > 3). We find that this matter scaling solution is unstable to curvature perturbations for > (2/3). The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with 2/2 ( > (2/3),2 > 2). We find that solutions with 0 are never late-time attractors.