Invariant Measure for the Mixmaster Dynamics

Abstract: We provide a Hamiltonian analysis of the Mixmaster Universe dynamics showing the covariant nature of its chaotic behavior with respect to any choice of time variable. We construct the appropriate invariant measure for the system (which relies on the existence of an “energy-like” constant of motion) without fixing the time gauge, i.e. the corresponding lapse function. The key point in our analysis consists of introducing generic Misner-Chitré-like variables containing an arbitrary function, whose specification allows one to set up the same dynamical scheme in any time gauge. PACS number(s): 04.20.Jb, 98.80.Dr The original Belinski-Khalatnikov-Lifshitz (BKL) analysis of the asymptotic dynamics of Bianchi type VIII and IX cosmological models [1] (the so-called Mixmaster universes [2]) showed the existence of a chaotic behavior in their approach to the initial cosmological singularity. Probably the most suitable description of this BKL oscillatory regime, especially in view of its chaotic nature, is via the Hamiltonian formulation of the Mixmaster Universe dynamics expressed in Misner-Chitré-like variables [3],[16]. The advantage of this approach, apart from the immediate interpretation of the system evolution as a chaotic scattering process, consists of replacing the discrete BKL map by a geodesic flow in a space of continuous variables. In this approach a key result is obtaining an invariant measure for the Mixmaster evolution [4, 5] (see also [6, 7]). In spite of the achievements made in describing the Mixmaster stochasticity, however, because of its relativistic nature, the dynamics can be viewed from different reference frames, thus leaving open the question of the covariance of the observed chaos. Interest in these covariance aspects has increased in recent years in view of the contradictory and often dubious results that have emerged on this topic. The confusion which arises regarding the effect of a change of the time variable in this problem depends on some special properties of the Mixmaster model when represented as a dynamical system, in particular the vanishing of the Hamiltonian and its non-positive definite kinetic terms (a typical feature of a gravitational system). These special features prevent the direct application of the most common criteria provided by the theory of dynamical systems for characterizing chaotic behavior (for a review, see [8]). Although a whole line of research opened up, following this problem of covariant characterization for the chaos in the Mixmaster model [9]–[13], the first widely accepted indications in favor of covariance were derived by a fractal formalism in [14] (see also [15]). Indeed the requirement of a covariant description of the Mixmaster chaoticity when viewed in terms of continuous dynamical variables, due to the discrete nature of the fractal approach, leaves this subtle question open and prevents a general consensus from being reached. Here we show how the derivation of an invariant measure for the Mixmaster model (performed in [5, 7] within the framework of the statistical mechanics) can be extended to a generic time gauge (more directly than in previous approaches relying on fractal methods [14]) provided that suitable Misner-Chitré-like variables are chosen. More precisely, after a standard Arnowitt-Deser-Misner (ADM) reduction of the variational problem when written in terms of a generic time variable, we show how asymptotically close to the cosmological singularity, the Mixmaster dynamics can be modeled by