Bianchi models are posited by the BKL picture to be essential building blocks towards an understanding of generic cosmological singularities. We study the behaviour of spatially homogeneous anisotropic vacuum spacetimes of Bianchi type VIII and IX, as they approach the big bang singularity. It is known since 2001 that generic Bianchi IX spacetimes converge towards the so-called Mixmaster attractor as time goes towards the singularity. We extend this result to the case of Bianchi VIII vacuum. The BKL picture suggests that particle horizons should form, i.e. spatially separate regions should causally decouple. We prove that this decoupling indeed occurs, for Lebesgue almost every Bianchi VIII and IX vacuum spacetime.
The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative Painlevé test; this proves the inexistence of any vacuum solution other than the three known ones.
In the loop quantum cosmology effective dynamics for the vacuum Bianchi type I and type IX space-times, a non-singular bounce replaces the classical singularity. The bounce can be approximated as an instantaneous transition between two classical vacuum Bianchi I solutions, with simple transition rules relating the solutions before and after the bounce. These transition rules are especially simple when expressed in terms of the Misner variables: the evolution of the mean logarithmic scale factor is reversed, while the shape parameters are unaffected. As a result, the loop quantum cosmology effe…
We present a new numerical algorithm for evolving the Mixmaster spacetimes. By using symplectic integration techniques to take advantage of the exact Taub solution for the scattering between asymptotic Kasner regimes, we evolve these spacetimes with higher accuracy using much larger time steps than previously possible. The longer Mixmaster evolution thus allowed enables detailed comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate Mixmaster dynamics. In particular, we show that errors between the BKL prediction and the measured parameters early in the simulation can be elimin…
A modified semi-classical method is used to construct both ground and excited state solutions to the canonically quantized vacuum Bianchi IX (Mixmaster) cosmological models. Employing a modified form of the semi-classical Ansatz we solve the relevant Wheeler-DeWitt equation asymptotically by integrating a set of linear transport equations along the flow of a suitably chosen solution to the corresponding Euclidean-signature Hamilton-Jacobi equation. For the Moncrief-Ryan (or `wormhole') Hamilton-Jacobi solution, we compute the ground state quantum correction term associated with operator orderi…
The maximal globally hyperbolic development of non-Taub-NUT Bianchi IX vacuum initial data and of non-NUT Bianchi VIII vacuum initial data is C2 inextendible. Furthermore, a curvature invariant is unbounded in the incomplete directions of inextendible causal geodesics.
It is shown that in transitively self-similar spatially homogeneous tilted perfect fluid models the symmetry vector is not normal to the surfaces of spatial homogeneity. A direct consequence of this result is that there are no self-similar Bianchi VIII and IX tilted perfect fluid models. Furthermore the most general Bianchi VIII and IX spacetime which admit a four dimensional group of homotheties is given.
We study different aspects of the Hořava-Lifshitz inspired Bianchi-II cosmology and its relations with the mixmaster universe model. First, we present exact solutions for a toy model, where only the cubic in spatial curvature terms are present in the action; then we briefly discuss some exotic singularities, which can appear in this toy model. We study also the toy model where only the quadratic in spatial curvature terms are present in the action. We establish relations between our results and those obtained by using the Hamiltonian formalism. Finally, we apply the results obtained by studyin…
Within the scope of anisotropic non-diagonal Bianchi type-II, VIII and IX spacetime it is shown that the off-diagonal components of the corresponding metric impose severe restrictions on the components of the energy momentum tensor in general. If the energy momentum tensor is considered to be diagonal one, the spacetime, expect a partial case of BII, becomes locally rotationally symmetric.
By applying a standard solution-generating transformation to an arbitrary vacuum Bianchi type II solution, one generates a new solution with spikes commonly observed in numerical simulations. It is conjectured that the spike solution is part of the generalized Mixmaster attractor.
In this paper we present the quantum LRS Bianchi IX models or, as they may be better known, the Taub-NUT models in addition to the quantum LRS Bianchi VIII models, both coupled to a stiff matter source. We solve their Wheeler DeWitt equations for an arbitrary Hartle-Hawking ordering parameter using separation of variables. Afterwards we construct a superposition of their respective wave functions and discuss some of the interesting qualitative properties they possess.
We consider a five dimensional vacuum cosmology with Bianchi type-IX spatial geometry and an extra non-compact coordinate. Finding a new class of solutions, we examine and rule out the possibility of deterministic chaos. We interpret this result within the context of induced matter theory.
Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and in a previous article we analyzed the asymptotic behaviour of solutions in these variables. One objective of this paper is to give an asymptotic expansion for the metric. Furthermore, we relate this expans…
Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and we analyze the asymptotic behaviour of solutions in these variables. We also try to give the analytic results a geometric interpretation by analyzing how a normalized version of the Riemannian metric on th…
The Mixmaster or Bianchi IX cosmological model has become one of the archetypal settings for studying gravitational dynamics. The past decade has seen a vigourous debate about whether or not the Mixmaster's dynamics is chaotic. In this talk we review our recent work in uncovering a chaotic invariant set of orbits in the Mixmaster phase space, and how we used this discovery to prove the dynamics is chaotic.
We consider the most general form of Bianchi types VIII and IX space-times for studying proper affine symmetry by using holonomy and decomposability, the rank of the 6X6 Riemann matrix and direct integration techniques. Studying proper affine symmetry in the above space-times it is shown that there exists only one possibilty when the above space-times admit proper affine vector field.
Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of $ξ^a(x^b)$ is different from zero and (ii) two components of $ξ^a(x^b)$ are different from zero, for $a,b=1,2,3,4$, are presented. In subcase (i.b), which is $ξ^a= (0,ξ^2(x^a),0,0)$, some known solutions are found, and in subcase (i.d), which is $ξ^a =(0,0,0,ξ^4(x^a))$, choosing $S(t)=const.\times R(t)$, the Bianchi type II, VIII, and IX space-times is reduced to the Robertson-Walker metric.
In this paper we study the exact solutions for a viscous fluid distribution in Bianchi II, VIII, and IX models. The metric is simplified by assuming a relationship between the coefficients and the metric tensor. Solutions are obtained in two special cases: in one, an additional assumption is made where the matter density and the expansion scalar have a definite relation and in the other a barotropic equation of state between the matter density and the thermodynamic pressure is assumed. While the Bianchi II solutions are already found in the literature the other two classes of solutions are app…
This paper has excessive overlap with the following papers also written by the authors or their collaborators: gr-qc/0607110, gr-qc/0607115, gr-qc/0607119, gr-qc/0607109, gr-qc/0607103, gr-qc/0607089, gr-qc/0602012, gr-qc/0608024, and others.
This paper is a study of the effects of anisotropic matter sources on the qualitative evolution of spatially homogenous cosmologies of Bianchi type VIII. The analysis is based on a dynamical system approach and makes use of an anisotropic matter family developed by Calogero and Heinzle which generalises perfect fluids and provides a measure of deviation from isotropy. Thereby the role of perfect fluid solutions is put into a broader context. The results of this paper concern the past and future asymptotic dynamics of locally rotationally symmetric solutions of type VIII with anisotropic matter…
The asymptotic behaviour of vacuum Bianchi models of class A near the initial singularity is studied, in an effort to confirm the standard picture arising from heuristic and numerical approaches by mathematical proofs. It is shown that for solutions of types other than VIII and IX the singularity is velocity dominated and that the Kretschmann scalar is unbounded there, except in the explicitly known cases where the spacetime can be smoothly extended through a Cauchy horizon. For types VIII and IX it is shown that there are at most two possibilities for the evolution. When the first possibility…
We study the Hamiltonian dynamics of the dust-Bianchi IX universe in dust time gauge. This model has three physical metric degrees of freedom, with evolution determined by a time-independent physical Hamiltonian. This approach gives a new physical picture where dust-Bianchi IX dynamics is described by oscillations between dust-Kasner solutions, rather than between vacuum-Kasner solutions. We derive a generalized transition law between these phases, which has a matter component. Sufficiently close to a singularity, we show that this law reduces to the vacuum Belinski-Khalatnikov-Lifshitz map. W…
This paper has been removed by arXiv administrators because it plagiarizes gr-qc/0303009, hep-th/0405047, gr-qc/0412120, gr-qc/0602107, hep-th/0311050, and gr-qc/0508005. This paper has excessive overlap with the following papers also written by the authors or their collaborators: gr-qc/0607119, gr-qc/0607103, gr-qc/0607126, gr-qc/0606080, gr-qc/0607102, gr-qc/0510123, gr-qc/0607110, and others.
Previous studies \cite{berger98a} have provided strong support for a local, oscillatory approach to the singularity in U(1) symmetric, spatially inhomogeneous vacuum cosmologies on $T^3 \times R$. The description of a vacuum Bianchi type IX, spatially homogeneous Mixmaster cosmology (on $S^3 \times R$) in terms of the variables used to describe the U(1) symmetric cosmologies indicates that the oscillations in the latter are in fact those of local Mixmaster dynamics. One of the variables of the U(1) symmetric models increases only at the end of a Mixmaster era. Such an increase therefore yields…
Description of the magnetic Bianchi VI$_0$ cosmologies of LeBlanc, Kerr, and Wainwright in the formalisms both of Belinskii, Khalatnikov, and Lifshitz, and of Misner allows qualitative understanding of the Mixmaster-like singularity in those models.
In this paper we give, for the first time, a complete description of the late-time evolution of non-tilted spatially homogeneous cosmologies of Bianchi type VIII. The source is assumed to be a perfect fluid with equation of state $p = (γ- 1)μ$, where $γ$ is a constant which satisfies $1 \leq γ\leq 2$. Using the orthonormal frame formalism and Hubble-normalized variables, we rigorously establish the limiting behaviour of the models at late times, and give asymptotic expansions for the key physical variables. The main result is that asymptotic self-similarity breaking occurs, and is accompanied …
In this work we explore the quantum Bianchi type IX-model, its semi-classical features, and its relevance in early cosmology to tentatively explain inflation and production of primordial structures. We specially focus on the analytical and numerical exploration of the dynamical system derived from the phase-space portraits. Afterwards we investigate the reliability of our results with regard to inflation and post inflation scenarii commonly accepted nowadays.
We analyze the quantum dynamics of the Bianchi Type IX model, as described in the so-called polymer representation of quantum mechanics, to characterize the modifications that a discrete na- ture in the anisotropy variables of the Universe induces on the morphology of the cosmological sin- gularity. We first perform a semiclassical analysis, to be regarded as the zeroth-order approximation of a WKB (Wentzel-Kramers-Brillouin) approximation of the quantum dynamics, and demonstrate how the features of polymer quantum mechanics are able to remove the chaotic properties of the Bianchi IX dynamics.…
The Bianchi IX cosmological model is analyzed in a generalized uncertainty principle framework. The Arnowitt-Deser-Misner reduction of the dynamics is performed and a time-coordinate, namely the volume of the Universe, naturally arises. Such a variable is treated in the ordinary way while the anisotropies (the physical degrees of freedom) are described by a deformed Heisenberg algebra. The analysis of the model (passing through Bianchi I and II) is performed at classical level by studying the modifications induced on the symplectic geometry by the deformed algebra. We show that, the Universe c…
We present a quantum version of the vacuum Bianchi IX model by implementing affine coherent state quantization combined with a Born-Oppenheimer-like adiabatic approximation. The analytical treatment is carried out on both quantum and semiclassical levels. The resolution of the classical singularity occurs by means of a repulsive potential generated by our quantization procedure. The quantization of the oscillatory degrees of freedom produces a radiation energy density term in the semiclassical constraint equation. The Friedmann-like lowest energy eigenstates of the system are found to be dynam…
We analyze the semiclassical and quantum behavior of the Bianchi IX Universe in the Polymer Quantum Mechanics framework, applied to the isotropic Misner variable, linked to the space volume of the model. The study is performed both in the Hamiltonian and field equations approaches, leading to the surprising result of a still singular and chaotic cosmology, whose Poincaré return map asymptotically overlaps the standard Belinskii-Khalatnikov-Lifshitz one. In the quantum sector, we reproduce the original analysis due to Misner, within the revised Polymer approach and we arrive to demonstrate that…
This paper begins with a short presentation of the Bianchi IX or ``Mixmaster'' cosmological model, and some ways of writing the Einstein equations for it. There is then an interlude describing how I came to a study of this model, and then a report of some mostly unpublished work from a Ph.\ D. thesis of D. M. (Prakash) Chitre relating approximate solutions to geodesic flows on finite volume negative curvature Riemannian manifolds, for which he could quote results on ergodicity. A final section restates studies of a zero measure set of solutions which in first approximation appear to have only …
Using a qualitative analysis based on the Hamiltonian formalism and the orthonormal frame representation we investigate whether the chaotic behaviour which occurs close to the initial singularity is still present in the far future of Bianchi VIII models. We describe some features of the vacuum Bianchi VIII models at late times which might be relevant for studying the nature of the future asymptote of the general vacuum inhomogeneous solution to the Einstein field equations.
Global existence to the coupled Einstein-Maxwell-Massive Scalar Field system which rules the dynamics of a kind of charged pure matter in the presence of a massive scalar field is proved, in Bianchi I-VIII spacetimes; asymptotic behaviour, geodesic completeness, energy conditions are investigated in the case of a cosmological constant bounded from below by a strictly negative constant depending only on the massive scalar field.
It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical trajectories in the backward movement of time. This behaviour (Mixmaster Model of the Universe) can be encoded by the shift of two-sided continued fractions. Exactly the same shift encodes the sequences of intersections of hyperbolic geodesics with purely imaginary axis in the upper complex half-plane, that is geodesic flow on an appropriate modular surface. A physical interpretation of this coincidence was suggested in arXiv:1402.2158: namely,…
We rigorously verify that in the spatially homogeneous spacetimes (Bianchi VIII and IX), the presence of matter does not affect the oscillatory behavior of the solutions to the Einstein field equations as first conjectured by Belinski, Khalatnikov, and Lifshitz (BKL) in 1970. This paper is an extension of the paper by Reiterer and Trubowitz (arXiv 1005.4908) and uses the same formalism. We use the ultrarelativistic equation of state.
Usual approaches to quantisation of a 3-Toda lead to numerical calculations requiring many steps that can be time consuming to insure their reliability. In order to reduce as much as possible the numerical part of the EKB quantisation procedure, and then to ease numerical calculations, we propose a reformulation of the mathematical framework with more adapted variables. The resulting equations and procedure might be easily implemented in a short Mathematica code. This more explicit framework will be useful for studying quantum Toda-Bianchi IX models in quantum cosmology where the true Bianchi …
We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. The `Bianchi type IX attractor theorem' states that the past asymptotic behavior of generic type IX solutions is governed by Bianchi type I and II vacuum states (Mixmaster attractor). We give a comparatively short and self-contained new proof of this theorem. The proof we give is interesting in itself, but more importantly it illustrates and emphasizes that type IX is special, and to some extent misleading when one considers the broader context…
Following our previous papers concerning the quantization of the vacuum Bianchi-IX model within or beyond the Born-Oppenheimer and adiabatic approximation, we develop a more elaborate analysis of the dynamical properties of the model based the vibronic approach utilized in molecular physics. As in the previous papers, we restrict our approach to the harmonic approximation of the anisotropy potential in order to obtain resoluble analytical expressions.
In the present work we revisit the axisymmetric Bianchi VIII and IX models. At the classical level we reproduce the known analytic solution, in a novel way making use of two quadratic integrals of motion, the constraint equation, as well as a linear non-local integral of motion. These quantities correspond to two second rank Killing tensors and a homothetic vector field existing on the relevant configuration space. On the corresponding phase space the two quadratic charges commute with the Hamiltonian constraint but not among themselves. Thus, after turning these charges into operators we obta…
New non trivial solutions to the Lorentzian-signature symmetry reduced Bianchi VIII Wheeler DeWitt equation for Hartle Hawking ordering parameters $ \pm 2\sqrt{33} $ are obtained using an Euclidean-signature semi classical method. Using the aforementioned method we define 'excited' states for the Bianchi VIII wave function and construct leading order 'excited' states which appear to be a hybrid of scattering and bound states. Also six new solutions for the $Λ\ne 0$ Bianchi VIII Euclidean-signature Hamilton Jacobi equation are presented which can be used to construct semi-classical states corre…
We use the dynamical systems approach to investigate the Bianchi type VIII models with a tilted $γ$-law perfect fluid. We introduce expansion-normalised variables and investigate the late-time asymptotic behaviour of the models and determine the late-time asymptotic states. For the Bianchi type VIII models the state space is unbounded and consequently, for all non-inflationary perfect fluids, one of the curvature variables grows without bound. Moreover, we show that for fluids stiffer than dust ($1<γ<2$), the fluid will in general tend towards a state of extreme tilt. For dust ($γ=1$), or for …
Global existence to the coupled Einstein-Maxwell system which rules the dynamics of a kind of charged matter with a pseudo-tensor of pressure is proved, in Bianchi I-VIII spacetimes. We study the geodesics completeness, the asymptotic behavior, the positivity conditions, and we prove that the problem is well-posed in the sense of Hadamard.
We present a quantum model of the vacuum Bianchi-IX dynamics. It is based on four main elements. First, we use a compound quantization procedure: an affine coherent state quantization for isotropic variables and a Weyl quantization for anisotropic ones. Second, inspired by standard approaches in molecular physics, we make an adiabatic approximation (Born-Oppenheimer-like approximation). Third, we expand the anisotropy potential about its minimum in order to deal with its harmonic approximation. Fourth, we develop an analytical treatment on the semiclassical level. The resolution of the classic…
In this paper we give, for the first time, a complete description of the dynamics of tilted spatially homogeneous cosmologies of Bianchi type II. The source is assumed to be a perfect fluid with equation of state $p = (γ-1) μ$, where $γ$ is a constant. We show that unless the perfect fluid is stiff, the tilt destabilizes the Kasner solutions, leading to a Mixmaster-like initial singularity, with the tilt being dynamically significant. At late times the tilt becomes dynamically negligible unless the equation of state parameter satisfies $γ> {10/7}$. We also find that the tilt does not destabili…
We apply the dynamical systems approach to ever-expanding Bianchi type VIII cosmologies filled with a tilted $γ$-fluid undergoing velocity diffusion on a scalar field. We determine the future attractors and investigate the late-time behaviour of the models. We find that at late times the normalized energy density $Ω$ tends to zero, while the scalar potential $Φ$ approaches 1 and dominates the evolution. Moreover, we demonstrate that in presence of diffusion fluids with $γ<3/2$, which includes physically important cases of dust $(γ=1)$ and radiation $(γ=4/3)$, are asymptotically non-tilted; the…
We apply the scale factor duality transformations introduced in the context of the effective string theory to the anisotropic Bianchi-type models. We find dual models for all the Bianchi-types [except for types $VIII$ and $IX$] and construct for each of them its explicit form starting from the exact original solution of the field equations. It is emphasized that the dual Bianchi class $B$ models require the loss of the initial homogeneity symmetry of the dilatonic scalar field.
We analyze the Bianchi IX dynamics (Mixmaster) in view of its stochastic properties; in the present paper we address either the original approach due to Belinski, Khalatnikov and Lifshitz (BKL) as well as a Hamiltonian one relying on the Arnowitt--Deser--Misner (ADM) reduction. We compare these two frameworks and show how the BKL map is related to the geodesic flow associated with the ADM dynamics. In particular, the link existing between the \textit{anisotropy parameters} and the \textit{Kasner indices} is outlined.
In this paper we present an analysis to determine the existence of singularities in spatially homogeneous anisotropic universes filled with nonlinear electromagnetic radiation. These spaces are conformal to Bianchi spaces admitting a three parameter group of motions G$_3$. For these models we study geodesic completeness. It is shown that with nonlinear electromagnetic field some of the Bianchi spaces are geodesically complete, like G$_3$IX and G$_3$VIII; however, completeness depends on the curvature of the space. When certain topology is assumed, Bianchi G$_3$IX presents geodesics that are im…
We rigorously construct and control a generic class of spatially homogeneous (Bianchi VIII and Bianchi IX) vacuum spacetimes that exhibit the oscillatory BKL phenomenology. We investigate the causal structure of these spacetimes and show that there is a "particle horizon".
We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge …
Recent work on $N=2$ supersymmetric Bianchi type IX cosmologies coupled to a scalar field is extended to a general treatment of homogeneous quantum cosmologies with explicitely solvable momentum constraints, i.e. Bianchi types I, II, VII, VIII besides the Bianchi type IX, and special cases, namely the Friedmann universes, the Kantowski-Sachs space, and Taub-NUT space. Besides the earlier explicit solution of the Wheeler DeWitt equation for Bianchi type IX, describing a virtual wormhole fluctuation, an additional explicit solution is given and identified with the `no-boundary state'.
We investigate the quantum cosmologies of the Bianchi IX and VIII models when a cosmological constant, aligned electromagnetic field and free scalar field are present. The conserved quantity $p_ϕ$ associated with our free scalar field results in $ϕ$ classically being a quantity which monotonically increases with respect to time, thus allowing it to fulfill the role of an 'emergent' internal clock for our constrained quantum systems. We embark on this investigation to better understand how matter sources can affect general anisotropic quantum cosmologies. To aid us we use the Euclidean-signatur…
Bianchi type IX, 'Mixmaster' universes are investigated in low-energy-effective-action string cosmology. We show that, unlike in general relativity, there is no chaos in these string cosmologies for the case of the tree-level action. The characteristic Mixmaster evolution through a series of Kasner epochs is studied in detail. In the Einstein frame an infinite sequence of chaotic oscillations of the scale factors on approach to the initial singularity is impossible, as it was in general relativistic Mixmaster universes in the presence of a massless scalar field. A finite sequence of oscillatio…
In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI$_{-1/9}$. This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous $G_{2}$ cosmologies. Using the orthonormal frame formalism and Hubble-normalized variables, …
The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following: (i) There exis…
