Chaos in anisotropic preinflationary universes

Abstract

We study the dynamics of anisotropic Bianchi type-IX models with matter and cosmological constant. The models can be thought as describing the role of anisotropy in the early stages of inflation, where the cosmological constant Λ plays the role of the vacuum energy of the inflaton field. The concurrence of the cosmological constant and anisotropy are sufficient to produce a chaotic dynamics in the gravitational degrees of freedom, connected to the presence of a critical point of saddle-center type in the phase space of the system. In the neighborhood of the saddle-center, the phase space presents the structure of cylinders emanating from unstable periodic orbits. The non-integrability of the system implies that the extension of the cylinders away from this neighborhood has a complicated structure arising from their transversal crossings, resulting in a chaotic dynamics. The invariant character of chaos is guaranteed by the topology of cylinders. The model also presents a strong asymptotic de Sitter attractor but the way out from the initial singularity to the inflationary phase is completely chaotic. For a large set of initial conditions, even with very small anisotropy, the gravitational degrees of freedom oscillate a long time in the neighborhood of the saddle-center before recollapsing or escaping to the de Sitter phase. These oscillations may provide a resonance mechanism for amplification of specific wavelengths of inhomogeneous fluctuations in the models. A geometrical interpretation is given for Wald's inequality in terms of invariant tori and their destruction by increasing values of the cosmological constant.

The Dynamics of Apparent Horizons in Robinson–Trautman Spacetimes

“…As a matter of fact, as an important piece of evidence of the accuracy of our numerical scheme, we have plotted in Fig. 2 the modulus of the residual equation (15) evaluated at the initial instant u = 0 for both initial data families (21) and (22) taking into account distinct values of the truncation orders N and M associated to the functions K(u, x) and V(u, x) (cf. (8) and (14)).…”

“…5. We started at u = 0 with the oblate spheroid initial data (21), and several plots are shown in subsequent instants until u f = 500 where a circle is formed. Indeed, this is a consequence from the fact that the asymptotic state is the Schwarzschild configuration An interesting application of our code is to follow the behavior of the apparent horizon mass that is basically the amount of mass enclosed by the apparent horizon.…”

Chaos in a Closed Friedmann-Robertson-Walker Universe: An Imaginary Approach

“…In this way we are able to relax the constraint on the vanishing-energy condition [10]. From now on, this constant will be absorbed by the Hamiltonian, i.e., H ≡ 2H − ρ r , and the tilde (˜) will be dropped to avoid cluttering of the equations.…”

Black Holes Collision in General Robinson-Trautman Spacetimes: Wave Forms and the Efficiency of the Gravitational Wave Extraction

“…3 the angular pattern is characterized by an antisymmetric structure of lobes, but in a distinct configuration. The final black hole is described by the stationary solution (6).…”

“…Some of these aspects were addressed in Refs. [5][6][7][8][9][10][11] in the context of axial symmetry after the development of a numerical code for integrating the field equations.…”