Chaos in the Einstein-Yang-Mills Equations

Abstract

Yang-Mills color fields evolve chaotically in an anisotropically expanding universe. The chaotic behaviour differs from that found in anisotropic Mixmaster universes. The universe isotropizes at late times, approaching the mean expansion rate of a radiation-dominated universe. However, small chaotic oscillations of the shear and color stresses continue indefinitely. An invariant, coordinate-independent characterisation of the chaos is provided by means of fractal basin boundaries. CfPA-97-TH-06Yang-Mills fields are central to quantum theories of elementary particles. They are of interest to dynamicists since they evolve chaotically in flat spacetime [1,2,3,4]. But how do Yang-Mills fields behave in the early universe? Does the chaos persist, or is it eradicated by the general relativistic effects of cosmological expansion? The earliest studies of the Einstein-Yang-Mills (EYM) system assumed isotropic cosmological expansion. This imposes special symmetries on the dynamics and the dynamical system is integrable: chaos cannot exist [5]. Recently, a Yang-Mills theory was formulated in an axisymmetric, spatially homogeneous universe [6]. The large number of degrees of freedom made an analysis of the full dynamics difficult, and the coordinate dependence of the standard chaotic indicators meant that the relativistic chaos could not be invariantly characterised.In this letter, we analyse general relativistic Yang-Mills chaos using invariant topological methods. We make the problem more tractable by an economical definition of variables, which reduces the dimension of the phase space substantially (from 8 − D to 5 − D). In the physical picture that emerges, the asymptotic evolution of the spatial volume of the universe imitates a radiation-dominated universe, while the shear diminishes chaotically.Previous studies of chaos in general relativity have focused on the non-axisymmetric Bianchi type VIII and IX (Mixmaster) universes, where the presence of anisotropic 3-curvature creates an infinite sequence of chaotic oscillations on approach to an initial Weyl curvature singularity at t = 0 [12,13,14]. This behaviour is intrinsically general relativistic. By contrast, the chaotic EYM cosmology that we study is different: chaos exists even when the metric is axisymmetric and the curvature is isotropic.We evolve the Yang-Mills fields in the simplest anisotropic metrics of Bianchi type I. The color degrees of freedom of the Yang-Mills gauge fields oscillate chaotically, while the expansion attenuates their overall energy. The EYM action iswhere F µν is the gauge-invariant field strength. The spacetime is described by the axisymmetric Bianchi I metric, with scale factors b(t) and c(t):

Chaotic symmetry breaking and dissipative two-field dynamics

“…1, which already shows a highly chaotic dynamics prior to symmetry breaking. Following the method of Box-Counting [10][11][12]16], around the initial condition (5.6) it is then considered a box in phase space (for the dimensionless variables) of size 10 −5 , inside which a large number of random points are taken (a total of 200.000 random points were used in each run). All initial conditions are then numerically evolved by using an eighthorder Runge-Kutta integration method and the fractal dimension is obtained by statistically studying the outcome of each initial condition at each run of the large set of points.…”

“…The method we apply in this work for quantifying chaos will then be particularly useful in our planned future applications of our model and it extensions to a cosmological context, in which case other methods may be ambiguous, like, for example, the determination of Lyapunov exponents, which does not give a coordinate invariant measure for chaos, as discussed in [11,12]. Also, other methods for studying chaotic systems, like for example by Poincaré sections, are not suitable in the case we are interested here, in which chaos is a transitory phenomenon as we will see.…”

“…We study chaos in our dynamical system of equations by means of the measure of the fractal dimension (or dimension information) [for a review and definitions, see e.g., [10]], which gives a topological measure of chaos for different space-time settings and it is a quantity invariant under coordinate transformations, providing then an unambiguous signal for chaos [11,12]. The method we apply in this work for quantifying chaos will then be particularly useful in our planned future applications of our model and it extensions to a cosmological context, in which case other methods may be ambiguous, like, for example, the determination of Lyapunov exponents, which does not give a coordinate invariant measure for chaos, as discussed in [11,12].…”

Fine-tuning solution for hybrid inflation in dissipative chaotic dynamics

“…We study chaos in our dynamical system of equations by means of the measure of the fractal dimension (or dimension information) [for a review and definitions, see e.g., Ref. [24]], which gives a topological measure of chaos for different space-time settings and it is a quantity invariant under coordinate transformations, providing then an unambiguous signal for chaos in cosmology and general relativity problems in general [25,26]. The method we apply in this work for quantifying chaos is then particularly useful in this cosmological pre-inflationary scenario context we are studying, in which case other methods may be ambiguous, like, for example, the determination of Lyapunov exponents, which does not give a coordinate invariant measure for chaos, as discussed in [25,26].…”

“…[24]], which gives a topological measure of chaos for different space-time settings and it is a quantity invariant under coordinate transformations, providing then an unambiguous signal for chaos in cosmology and general relativity problems in general [25,26]. The method we apply in this work for quantifying chaos is then particularly useful in this cosmological pre-inflationary scenario context we are studying, in which case other methods may be ambiguous, like, for example, the determination of Lyapunov exponents, which does not give a coordinate invariant measure for chaos, as discussed in [25,26]. Also, other methods for studying chaotic systems, like for example by Poincaré sections, are not suitable in the case we are interested here, in which case chaos is mostly a transitory phenomenon (it ends by the time the fields reach the potential minima, or when the inflaton enters the inflationary region).…”