Chaos and Quantum Chaos in Cosmological Models

Spatially homogeneous cosmological models reduce to Hamiltonian systems in a low dimensional Minkowskian space moving on the total energy shellH = 0. Close to the initial singularity some models (those of Bianchi type VIII and IX) can be reduced further, in a certain approximation, to a non-compact triangular billiard on a 2-dimensional space of constant negative curvature with a separately conserved positive kinetic energy. This type of billiard has long been known as a prototype chaotic dynamical system. These facts are reviewed here together with some recent results on the energy level statistics of the quantized billiard and with direct explicit semi-classical solutions of the Hamiltonian cosmological model to which the billiard is an approximation. In the case of Bianchi type IX models the latter solutions correspond to the special boundary conditions of a ‘no-boundary state’ as proposed by Hartle and Hawking and of a ‘wormhole’ state. INTRODUCTION The subject of chaos in cosmology is really an old one and predates the intense modern interest in chaotic dynamical systems. It started with the classical work of Belinsky et al 1969 [1] and of Misner 1969 [2] who found that the evolution of certain homogeneous cosmological models back into the past towards the ‘Big Bang’ shows an oscillatory behavior, some aspects of which in modern terminology one has to call chaotic. Useful references for these cosmological models are the books by Ryan 1972 [3] and Ryan and Shepley 1975 [4]. The chaotic aspects of their evolution have been examined in numerous papers starting with the review by Barrow 1982 [5]. Khalatnikov et al 1985 [6] give a detailed discussion. A dynamical systems approach was pioneered by Bogoyavlensky and Novikov 1973 [7] and elaborated by Ma and Wainwright 1989 [8]. Among the more recent works on this subject those of Pullin 1990 [9] and Rugh 1992 [10] may be mentioned. Most recently a Painlevé analysis was applied to this dynamical system by Cantopoulos et al 1993 [11] who found, surprisingly, that the system satisfies the Painlevé conditions which are necessary (but not sufficient) for a system to be integrable. I Early on, again before the explosion of the interest in chaos took place, Misner 1972 [12] in a seminal paper and his student Chitre 1972 [13] in his thesis introduced coordinates in which the most interesting homogeneous cosmologies, those of Bianchi types VIII and IX, sufficiently close to the singularity reduce to a special two-dimensional billiard on a homogeneous space of constant negative curvature. The study of this kind of billiards is an even more ancient subject in mathematics, dating back to the work of Hadamard 1898 [14], Artin 1924 [15] and Hopf 1937 [16]. An extensive review is due to Balazs and Voros 1986 [17]. Given a dynamical system of Hamiltonian form the temptation to quantize seems to be irresistable. Hence the field of quantum chaos, which is concerned with the quantization of classically chaotic systems with the aim of understanding features of chaos and quantum mechanics, hence also the field of quantum cosmology, which starts with the quantization of spatially homogeneous cosmological models with the aim to achieve an understanding of quantum features of cosmology. For the Bianchi type IX cosmology this was first done by Misner 1972 in [12]. However, the present interest in quantum cosmology was mainly triggered by the proposal of a specific initial condition, the ‘no-boundary-state’, by Hawking 1982 [18], and Hartle and Hawking 1983 [19]. Quantized versions of the homogeneous Bianchi type IX cosmological models were also investigated in this context, in semi-classical WKB-type approximations, e.g. in papers by Hawking and Luttrell 1984 [20], and Moss and Wright 1985 [21] who study consequences of the ‘no-boundary’condition and by Del Campo and Vilenkin 1989 [22], and Graham and Szépfalusy 1990 [23] who instead pose the boundary condition of an outgoing wave. From the point of view of quantum chaos, the Bianchi IX model has also been studied further in papers by Furusawa 1986 [24] and Berger 1989 [25] who study the dynamics of wave-packets and by Graham et al. 1991 [26] and Csordás et al. 1991 [27] who study level statistics in Misner’s billiard limit of the model. More recently further detailed studies of a class of billiards including the special one relevant to the Bianchi VIII and IX model have appeared in the quantum chaos literature (Bogomolny et al. 1992 [28, 29], Bolte et al. 1992 [30], Schmit 1991 [31] Eisele and Mayer 1993 [32]). Finally, as the most recent development, exact solutions of quantized Bianchi IX models were found independently by Moncrief and Ryan 1991 [33], Graham 1991 [34], and Bene and Graham 1993 [35]. It is now understood that the solution found in the first two references describe virtual quantum wormholes (D.Eath 1993 [36], Bene and Graham 1993 [35]), while a further exact solution by Bene and Graham 1993 [35] gives the ‘no-boundary-state’ of the model in a simple analytical form. The present paper has the purpose to review some of these results. In the next section we establish the necessary background on spatially homogeneous cosmological models. Then we proceed to consider Misner’s approximate asymptotic reduction of the Bianchi VIII and Bianchi IX models to 2-dimensional billiards on a space of constant negative curvature, and discuss some results following from this reduction. Next, the Bianchi VIII and IX models are quantized, giving rise to their Schrödinger equation, called Wheeler DeWitt equation in the present context. We then discuss some of the properties of the billiard approximation of the quantized model. Finally, we turn to exact solutions of the Wheeler DeWitt equation and their physical interpretation. SPATIALLY HOMOGENEOUS COSMOLOGICAL MODELS We consider the class of space-times such that through every point there passes a space-like 3-manifold which is left invariant by the actions of a three-dimensional Lie group (see e.g. Ryan and Shepley 1975 II [4]). Such space-times are called spatially homogeneous. The three-dimensional Lie groups have been classified by Bianchi into nine different types (‘Bianchi types’). They can be distinguished by the Lie algebra of the group-generators, called ‘Killing vectors’ in general relativity. It is convenient to use an invariant (non-coordinate) basis and its dual of basis 1-forms ω on the invariant 3-manifold. The space-time metric then takes the general form ds = −dt + gij(t)ωω (1) where dt is the time-like one-form orthogonal to the invariant 3-manifold and t is the standard cosmic time coordinate. The components gij(t) of the 3-metric then depend only on t. The basis 1-forms satisfy dω = 1 2 Cjkω j ∧ ω (2) (we recall the usual definition ω ∧ ω = ω ⊗ω − ω ⊗ω) where the Cjk are the structure constants of the homogeneity group. Of special interest here will be the Bianchi type VIII with − C23 = +C32 = +C31 = −C13 = +C12 = −C21 = 1 (3) and in particular the Bianchi type IX with Cjk = ǫijk. (4) In the latter case the homogeneity group is SO(3), the invariant 3-manifold is compact and topologically a 3-sphere. This case therefore includes the closed Friedmann-Robertson-Walker (FRW) space-time and all homogeneous anisotropic modifications thereof. The invariant 3-manifolds of Bianchi-type VIII are non-compact and describe open space-times. (The open FRW space-times are not of this particular Bianchi-type, however, see Ryan and Shepley 1975 [4]). We shall restrict the further discussion to diagonal models, where gij(t) can consistently be chosen diagonal, gij = 1 6π diag ( e+ √ 3β − , e √ 3β − , e+ ) (5) with three time-dependent parameters α, β+, β−. The special space-time metric (1) can now be inserted in the action functional of general relativity, in particular in its first-order (Hamiltonian) form first given by Arnowitt, Deser, and Misner 1962 [37]. We shall neglect the matter contribution in the action, which, in the anisotropic case, can be shown to become unimportant close to the initial singularity. The result is a reduced action of the form S = ∫ dλ (pν q̇ ν −N(λ)H(q, p)) (6) with