Matter and dynamics in closed cosmologies

2021

Journal of Cosmology and Astroparticle Physics, 2017

Dynamical system techniques are extremely useful to study cosmology. It turns out that in most of the cases, we deal with finite isolated fixed points corresponding to a given cosmological epoch. However, it is equally important to analyse the asymptotic behaviour of the universe. On this paper, we show how this can be carried out for 3-forms model. In fact, we show that there are fixed points at infinity mainly by introducing appropriate compactifications and defining a new time variable that washes away any potential divergence of the system. The richness of 3-form models allows us as well to identify normally hyperbolic non-isolated fixed points. We apply this analysis to three physically interesting situations: (i) a pre-inflationary era; (ii) an inflationary era; (iii) the late-time dark matter/dark energy epoch.

Inventiones Mathematicae, 1975

Illinois Journal of Mathematics - ILL J MATH, 2004

We study the omega-limit sets $\omega_X(x)$ in an isolating block $U$ of a singular-hyperbolic attractor for three-dimensional vector fields $X$. We prove that for every vector field $Y$ close to $X$ the set $ \{x\in U:\omega_Y(x)$ contains a singularity$\}$ is {\em residual} in $U$. This is used to prove the persistence of singular-hyperbolic attractors with only one singularity as chain-transitive Lyapunov stable sets. These results generalize well known properties of the geometric Lorenz attractor [GW] and the example in [MPu].

Journal of Mathematical Analysis and Applications, 1975

Advances in Theoretical and Mathematical Physics, 2009

This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the 'billiard attractor'. Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubble-normalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to 'cosmological billiards', as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding 'dual' representations of the asymptotic dynamics.

Physics Letters, 2015

In this paper we study some features of global dynamics for one Hamiltonian system arisen in cosmology which is formed by the minimally coupled field; this system was introduced by Maciejewski et al. in 2007. We establish that under some simple conditions imposed on parameters of this system all trajectories are unbounded in both of time directions. Further, we present other conditions for system parameters under which we localize the domain with unbounded dynamics; this domain is defined with help of bounds for values of the Hamiltonian level surface parameter. We describe the case when our system possesses periodic orbits which are found explicitly. In the rest of the cases we get some localization bounds for compact invariant sets.

Journal of Differential Equations, 1977

Given a semiflow on a metric space X (not necessarily locally compact), we relate notions of stability to the continuity of the orbital and limit set maps, K(x) and L(x), where K and L are considered as maps from X to 2x.

Since the 1960s, the big bang scenario has been generally accepted as the answer to the question of how did the universe come into being. At the formidable temperatures of the primeval universe, quantum mechanics and gravitation are inextricable, and the big bang model brought the program to reconcile these theories to the forefront of research in theoretical physics. Therefore the study of cosmological models over the past four decades was boosted by the realization of the importance of high energy physics and gravitation in the very early universe (inversely cosmology served as a testing ground for high energy particle physics), and by the observational revolution that occurred in the aftermath of space explorations. The contribution of particle physicists (many irreversibly converted to cosmology) revealed that events taking place as early as the ¯rst second after the big bang may determine processes that occur much later, when the universe is cold enough for structure to emerge....

Classical and Quantum Gravity, 2003

We investigate relativistic spherically symmetric static perfect fluid models in the framework of the theory of dynamical systems. The field equations are recast into a regular dynamical system on a 3dimensional compact state space, thereby avoiding the non-regularity problems associated with the Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space thus obtained is used to derive qualitative features and to prove theorems about mass-radius properties. The perfect fluids we discuss are described by barotropic equations of state that are asymptotically polytropic at low pressures and, for certain applications, asymptotically linear at high pressures. We employ dimensionless variables that are asymptotically homology invariant in the low pressure regime, and thus we generalize standard work on Newtonian polytropes to a relativistic setting and to a much larger class of equations of state. Our dynamical systems framework is particularly suited for numerical computations, as illustrated by several numerical examples, e.g., the ideal neutron gas and examples that involve phase transitions. We will see in Sec. 3 that, e.g., C 2 is sufficient, although this restriction can be weakened. Below we show how to handle even less restrictive situations like phase transitions. 7 However, note that ρ stands for the rest-mass density in the Newtonian case. 8 The region where m(r) < 0 can be analyzed with the same dynamical systems methods that are going to be used in the following. The treatment turns even out to be considerably simpler, cf. the case of Newtonian perfect fluids . If a solution to (3) satisfies 1 − 2Gm/(rc 2 ) > 0 initially at r = r 0 for initial data (m 0 , p 0 ), then this condition holds everywhere. This has been proved (for regular solutions), e.g., in [1]. Within the dynamical systems formulation this result can be established quite easily as we will see in Sec. 4. Solutions violating the condition 1 − 2Gm/(rc 2 ) > 0 could be treated with the dynamical systems methods presented in this paper as well. However, we refrain from a discussion of such solutions here.