Classical and Quantum Features of the Mixmaster Singularity Giovanni Montani, ICRA, International Center for Relativistic Astrophysics ENEA C.R. Frascati, Dipartimento F.P.N., Frascati, Roma, Italy Dipartimento di Fisica Universit´a“Sapienza”, P.le a. Moro 5, 00185 Roma, Italy ICRANet, International Relativistic Astrophysics Network, Pescara, Italy [email protected] Marco Valerio Battisti, ICRA, International Center for Relativistic Astrophysics Dipartimento di Fisica Universit´a“Sapienza”, P.le a. Moro 5, 00185 Roma, Italy [email protected] Riccardo Benini ICRA, International Center for Relativistic Astrophysics Dipartimento di Fisica Universit´a“Sapienza”, P.le a. Moro 5, 00185 Roma, Italy [email protected] Giovanni Imponente arXiv:0712.3008v3 [gr-qc] 12 Sep 2008 Museo Storico della Fisica e Centro Studi e Ricerche “E. Fermi”, Roma, Italy Dipartimento di Fisica Universit´a“Sapienza”, P.le a. Moro 5, 00185 Roma, Italy [email protected] Abstract This review article is devoted to analyze the main properties characterizing the cosmo- logical singularity associated to the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we re- view in details the main results got along the last forty years on the Mixmaster topic. We firstly assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mix- master behavior. Finally, we extend both the classical and quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results. Keywords: Early cosmology; primordial chaos; quantum dynamics. Montani et al. 4 Contents 1 Preface 9 2 Fundamental Tools 11 2.1 EinsteinEquations ............................... 11 2.2 MatterFields..................................... 13 2.3 TetradicFormalism ............................... 15 2.4 HamiltonianFormulationoftheDynamics . ...... 16 2.4.1 CanonicalGeneralRelativity . 16 2.4.2 Hamilton-Jacobi Equations for Gravitational Field . .......... 20 2.5 SynchronousReference. 21 2.6 SingularityTheorems. 22 2.6.1 DefinitionofaSpace-timeSingularity . 22 2.6.2 Preliminarynotions . 23 2.6.3 SingularityTheorems . 25 3 Homogeneous Universes 27 3.1 HomogeneousSpaces............................... 27 3.1.1 KillingVectorFields . 27 3.1.2 DefinitionofHomogeneity . 28 3.1.3 ApplicationtoCosmology. 29 3.2 BianchiClassificationandLineElement . ...... 32 3.3 FieldEquations .................................. 34 3.4 KasnerSolution ................................... 34 3.5 Theroleofmatter................................. 37 3.6 TheDynamicsoftheBianchiModels . 38 3.7 ApplicationtoBianchiTypesIIandVII . ...... 40 3.7.1 BianchitypeII................................ 40 3.7.2 BianchitypeVII............................... 41 4 Chaotic Dynamics of the Bianchi Types VIII and IX Models 43 4.1 Constructionofthesolution . ..... 43 4.2 TheBKLoscillatoryapproach . 44 4.3 Stochastic properties and the Gaussian distribution . ............ 46 4.4 SmallOscillations ............................... 48 4.5 HamiltonianFormulationoftheDynamics . ...... 50 4.6 TheADMReductionoftheDynamics . 51 4.7 MisnervariablesandtheMixmastermodel . 53 4.7.1 MisnerApproachtotheMixmaster . 54 4.7.2 Lagrangian Formulation in the Misner Variables . ...... 56 4.7.3 ReducedADMHamiltonian . 57 5 Contents 4.7.4 MixmasterDynamics..... ...... ..... ...... ..... 57 4.8 Misner-Chitre–likevariables . ...... 60 4.8.1 TheHamiltonEquations. 61 4.8.2 DynamicsintheReducedPhaseSpace. 62 4.8.3 Billiard InducedfromtheAsymptoticPotential . ....... 63 4.8.4 The Jacobi Metric and the Billiard Representation . ....... 64 4.9 TheInvariantLiouvilleMeasure . 66 4.10 Chaoscovariance ................................ 67 4.10.1 InvariantLyapunovExponent . 69 4.10.2 OntheOccurrenceofFractalBoundaries . ..... 71 4.11 IsotropizationMechanisms . 74 4.12 Cosmological Implementation of the Bianchi Models . ......... 76 4.12.1 ExpectedAnisotropyofCMBforBianchiI . 76 4.12.2 ExpectedAnisotropyofCMBforBianchiIX . 77 4.13 TheRoleofaScalarField. 78 4.14 MultidimensionalHomogeneousUniverses . ....... 80 4.15 TheRoleofaVectorField . 83 5 Quantum dynamics of the Mixmaster 87 5.1 TheWheeler-DeWittEquation. 87 5.2 TheProblemofTime ................................ 89 5.2.1 The Brown and Kuchar˘ mechanism ................... 89 5.2.2 EvolutionaryQuantumGravity. 90 5.2.3 TheMulti-TimeApproach. 92 5.3 TheMinisuperspaceRepresentation . ..... 93 5.4 OntheScalarFieldasaRelationalTime . ...... 93 5.5 InterpretationoftheUniverseWaveFunction . ......... 95 5.6 QuantizationintheMisnerPicture . ..... 97 5.7 ThequantumUniverseinthePoincar´ehalf-plane . ......... 98 5.8 ContinuityEquationandtheLiouvilletheorem . .........100 5.9 Schr¨odingerdynamics . 102 5.10 SemiclassicalWKBlimit . 102 5.11 TheSpectrumoftheMixmaster . 103 5.11.1 Eigenfunctionsandthevacuumstate . 103 5.11.2 DistributionoftheEnergyLevels. 106 5.12 BasicElementsofLoopQuantumGravity . 109 5.13 IsotropicLoopQuantumCosmology . 113 5.14 MixmasterUniverseinLQC . 117 5.15 OntheGUPandtheMinisuperspaceDynamics . 120 5.16 QuantumChaos................................... 123 6 Inhomogeneous Mixmaster Model 125 6.1 Formulationofthegenericcosmologicalproblem . .........125 6.1.1 TheGeneralizedKasnerSolution. 125 6.1.2 InhomogeneousBKLindices . .126 6.1.3 RotationoftheKasneraxes . 128 6.2 Thefragmentationprocess. 129 6 Contents 6.3 Hamiltonianformulationanddryturbulence . ........130 6.3.1 Solutionofthesuper-momentumconstraint . 131 6.3.2 TheRicciscalar ...............................131 6.4 TheIwasawadecomposition. 132 6.5 InhomogeneousquantumMixmaster . 134 6.6 Multidimensionaloscillatoryregime . .......135 7 Conclusions 139 Bibliography 143 7 Contents 8 1 Preface The formulation by Belinski, Khalatnikov and Lifshitz (BKL) at the end of the Sixties about the general character of the cosmological singularity represents one of the most important contributions of the Landau school to the development of Theoretical Cosmol- ogy, after the work by Lifshitz on the isotropic Universe stability. The relevance of the BKL work relies on two main points: on one hand, this study provided a piecewise an- alytical solution of the Einstein equations, improving and implementing the topological issues of the Hawking-Penrose theorems; on the other hand, the dynamical features of the corresponding generic cosmological solution exhibit a chaotic profile. Indeed the idea that the behavior of the actual Universe is some how related to an inho- mogeneous chaotic cosmology appears very far from what physical insight could suggest. In fact, the Standard Cosmological Model is based on the highly symmetric Friedman- Robertson-Walker (FRW) model, and observations agree with these assumptions, from the nucleosynthesis of light elements up to the age when structures formation takes place in the non-linear regime. Furthermore a reliable (but to some extent, model-dependent) correspondence between theory and data exists even for an inflationary scenario and its implications. Several ques- tions about the interpretation of surveys from which inferring the present status of the matter distribution and dynamics of the Universe remain open. However, in the light of an inflationary scenario, a puzzling plan appears unless the observation of early cosmo- logical gravitational waves become viable, since such dynamics cancels out much of the pre-existent information thus realizing local homogeneity and isotropy. The validity of the BKL regime must be settled down just in this pre-inflationary Universe evolution, although unaccessible to present observations. These considerations partly determined the persistent idea that this field of investigation could have a prevalent aca- demical nature, according to the spirit of the original derivation of the BKL oscillatory regime. In this paper, we review in details and under an updated point of view the main fea- tures of the Mixmaster model. Our aim is to provide a precise picture of this research line and how it developed over the last four decades in the framework of General Relativity and canonical quantization of the gravitational field. We start with a pedagogical formulation of the fundamental tools required to under- stand the treated topics and then we pursue a consistent description which, passing through the classical achievements, touches the review of very recent understandings in this field. The main aim of this work is to attract interest, especially from researchers working on fundamental physics, to the fascinating features of the classical and quantum inhomogeneous Mixmaster model. Our goal would be to convince the reader that this dynamical regime is physically well motivated and is not only mathematical cosmology, effectively concerning the very early stages of the Universe evolution, which matches the Planckian era with the inflationary behavior. The main reason underlying the relevance of the Mixmaster dynamics can be identified 9 1 Preface as follows: the pre-inflationary evolution fixes the initial conditions for all the subsequent dynamical stages as they come out from the quantum era of the Universe, and hence the oscillatory regime concerns the mechanism of transition to a classical cosmology provid- ing information on the origin of the actual notion of space-time. Moreover, we emphasize three points strongly supporting that the very early