CLASSICAL AND QUANTUM FEATURES OF THE INHOMOGENEOUS MIXMASTER MODEL

CLASSICAL AND QUANTUM FEATURES OF THE INHOMOGENEOUS MIXMASTER MODEL
UNIVERSITA’ DEGLI STUDI DI BOLOGNA
“ALMA MATER STUDIORUM”
FACOLTA’ DI SCIENZE MATEMATICHE, FISICHE E
NATURALI
TESI DI DOTTORATO IN FISICA
CLASSICAL AND QUANTUM FEATURES
OF THE INHOMOGENEOUS
MIXMASTER MODEL
Settore Scientifico-Disciplinare: FIS/02
Dottorando Riccardo Benini
matricola n.6428
Relatore interno
Chiarmo Prof. Giovanni Venturi
Relatore esterno
Dr. Giovanni Montani
Coordinatore del Dottorato
Prof. Fabio Ortolani
Anno Accademico 2005/2006
Mai tornare indietro,
neanche per prendere la rincorsa
Ernesto Che Guevara
Contents
Introduzione
V
Introduction
XI
1 The Mixmaster Model and the Generic Cosmological Solution
1.1
1
Homogeneous 4-dimensional models and the Mixmaster . . . . . . . .
2
1.1.1
The Kasner solution . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.2
The type IX model . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2
The role of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
The Generic Cosmological Solution . . . . . . . . . . . . . . . . . . .
15
1.4.1
Rotation of the Kasner axes . . . . . . . . . . . . . . . . . . .
20
1.4.2
Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Appendix 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Appendix 1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Appendix 1C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2 Covariant Characterization of the Chaos in the Generic Cosmological
Solution
29
2.1
Is the Mixmaster dynamics chaotic? . . . . . . . . . . . . . . . . . . .
30
2.2
Hamiltonian formulation for a generic cosmological model . . . . . . .
33
2.3
Solution to the super-momentum constraint . . . . . . . . . . . . . .
34
2.4
The Ricci scalar near the singularity . . . . . . . . . . . . . . . . . .
35
2.5
Dynamics of the physical degrees of freedom . . . . . . . . . . . . . .
38
2.6
The Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . . . .
39
2.7
Invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.8
Role of the potential walls . . . . . . . . . . . . . . . . . . . . . . . .
45
I
2.9
On the covariance of chaos . . . . . . . . . . . . . . . . . . . . . . . .
46
2.10 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3 Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
51
3.1
Quantum Cosmology and the Wheeler-DeWitt equation . . . . . . . .
52
3.1.1
The wave function of the Universe . . . . . . . . . . . . . . . .
54
3.2
The Canonical Quantization and the Multi-Time formalism . . . . . .
56
3.3
Classical properties of the inhomogeneous Mixmaster model . . . . .
58
3.3.1
Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . .
58
3.3.2
Hamilton-Jacobi approach . . . . . . . . . . . . . . . . . . . .
59
3.3.3
The statistical Mechanical description . . . . . . . . . . . . . .
60
3.4
Quasi-classical limit of the quantum regime . . . . . . . . . . . . . . .
62
3.5
Quantum properties of the inhomogeneous Mixmaster model . . . . .
64
3.5.1
Schroedinger quantization and the eigenfunctions of the model
64
3.5.2
Boundary conditions and the energy spectrum . . . . . . . . .
65
3.5.3
The ground state . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.5.4
Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . .
70
3.6
On the effects of the square-root on the quantum dynamics . . . . . .
73
3.7
The quantum chaos in the Mixmaster model . . . . . . . . . . . . . .
75
3.8
The Inhomogeneous picture and conclusions . . . . . . . . . . . . . .
76
Appendix 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4 Multi-dimensional Mixmaster and the role of matter fields
81
4.1
Mixmaster in higher-dimensional models . . . . . . . . . . . . . . . .
82
4.2
Matter-filled spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.3
n-dimensional homogeneous models coupled to an Abelian vector field
90
4.3.1
Homogeneous cosmological models . . . . . . . . . . . . . . .
91
4.4
Kasner-like behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.5
Billiard representation: the return map and the rotation of Kasner
vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
Appendix 4A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.6
Conclusions and Outlooks
103
Table of Figures
105
Bibliografy
107
Attachments
117
Attachment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Attachment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Attachment 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Attachment 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Attachment 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Attachment 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Attachment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
III
Introduzione
Il Modello Cosmologico Standard è sicuramente fra i risultati più importanti in fisica
moderna. Le ipotesi di omogeneità ed isotropia della struttura dell’Universo su larga
scala sono supportate sia da notevoli evidenze osservative, che da ottimi riscontri fra
predizioni teoriche e dati sperimentali. Fra le prime, la più importante è certamente
la quasi assoluta omogeneità ed isotropia della Radiazione Cosmica di Fondo; tra le
seconde, si segnala l’ottimo accordo fra le predizioni della nucleosintesi primordiale
e l’attuale abbondanza degli elementi leggeri nell’Universo. Per questi, e per altre
ragioni, il dominio di validità temporale di questo schema può essere esteso da oggi
fino a circa 10−32 secondi dal Big Bang (o, equivalentemente, ad energie dell’ordine
di 1016 GeV).
Il Modello Standard, altresì, soffre di notevoli problemi interni: il paradosso degli
orizzonti e la mancanza di un meccanismo che spieghi la formazione delle strutture a
larga scala sono due fra gli esempi più rilevanti. Se alcuni di questi problemi possono
essere risolti inserendo nella storia evolutiva dell’Universo una fase inflattiva, ci sono
forti motivazioni che spingono a cercare l’estensione della cosmologia a modelli più
generici di quelli di Friedmann-Robertson-Walker (FRW). Le dinamiche omogenee ed
isotrope dei modelli FRW sono, per esempio, instabili, durante la fase di collasso,
rispetto a perturbazioni tensoriali tipiche delle onde gravitazionali; oltretutto, si suppone che l’attuale struttura classica dell’Universo sia il risultato di una precedente
fase quantistica della gravità. Quest’ultimo fatto limita fortemente la possibilità di
simmetrie globali: infatti, per tempi dell’ordine di quello di Planck, l’Universo era
composto da più orizzonti causalmente sconnessi, e le fluttuazioni topologiche, che
caratterizzano una simile fase, suggeriscono che l’Universo classico sia iniziato in condizioni di assoluta generalità e senza alcuna simmetria.
Lo studio di una soluzione cosmologica generica (generale), quindi, è di assoluto
interesse, sia per i motivi legati ai problemi del Modello Standard, sia come problema
teorico a se stante.
V
Negli anni ’70, V.A. Belinsky, I.M. Khalatnikov ed E.M. Lifshitz dimostrarono che
è possibile costruire una soluzione cosmologica generale delle equazioni di Einstein
nell’intorno della singolarità iniziale; per soluzione generale, si intende una soluzione
che ha a disposizione tutti i gradi di libertà, e che quindi permette di assegnare
un problema di Cauchy su una generica ipersuperficie spaziale. Questo modello è
caratterizzato da una omogeneità locale e dalla presenza di una singolarità non eliminabile, e la dinamica risulta causalmente non connessa su scale che superano quella
dell’orizzonte; all’interno dei singoli orizzonti, si sviluppa un’evoluzione tipica dei
modelli omogenei dei tipi VIII e IX della classificazione di Bianchi, evoluzione che esibisce un regime di avvicinamento alla singolarità dal carattere fortemente oscillatorio
che può essere descritto in maniera appropriata solo con strumenti statistici. Questo
peculiare disaccoppiamento dei punti spaziali della varietà, associata al regime caotico,
riduce la struttura dello spazio-tempo classico a quello di una “schiuma” attraverso
un meccanismo di frammentazione.
La soluzione generica propone una visione, che per quanto sofisticata ed affascinante, resta comunque assolutamente classica. Il suo limite maggiore è legato alle
scale temporali ed energetiche alle quali l’Universo dovrebbe essere descritto da una
simile dinamica: a quei regimi, stimabili in un intervallo compreso fra i 1016 e i 1019
GeV, è attendibile che l’Universo non possa più essere descritto da una teoria classica,
ma entri in un regime in cui le correzioni quantistiche siano importanti. Il fenomeno
che mappa la fase quantistica in quella classica è l’espansione del volume dell’Universo;
poiché la comparsa di una dinamica isotropa ed omogenea è prevista alla fine della
fase oscillatoria, quando le anisotropie sono ridotte a piccole perturbazioni, ha senso
ritenere che l’inizio della fase oscillatoria si sovrapponga a quella Planckiana.
L’analisi della dinamica quantistica della soluzione generale è di assoluto interesse,
sia per quello che riguarda il Modello Standard, ma anche perché è uno scenario in
cui poter studiare la natura e le proprietà della singolarità iniziale e, di conseguenza,
i meccanismi che hanno portato alla nascita dell’Universo in cui viviamo.
Il lavoro presentato in questa dissertazione di dottorato può essere diviso in due
parti, e la prima si inserisce in questa linea di ricerca, con contributi originali inerenti alle proprietà classiche, semi-classiche e quantistiche della soluzione cosmologica
generale, nota anche come modello Mixmaster inomogeneo.
Per quanto concerne il regime classico, è stato effettuato uno studio dettagliato del
VI
regime caotico che ne caratterizza la dinamica; fra gli altri risultati, il più importante è quello che riguarda il problema di una caratterizzazione covariante del caos
nella soluzione cosmologica generale, e questo è stato risolto calcolando un indicatore,
utilizzato in teoria ergodica e noto come esponente di Lyapunov, in maniera assolutamente indipendente dal sistema di riferimento introdotto, ed, in particolare, dalla
variabile temporale.
Per quanto riguarda invece i regimi semi-classici e quantistici del Mixmaster, sono
stati ottenuti una collezione di risultati che riguardano l’approccio alla HamiltonJacobi alla dinamica, la descrizione meccanico-statistica del modello, il limite semiclassico alla WKB, ed un’analisi dettagliata del regime quantistico. Fra tutti, i risultati di maggior rilievo sono l’ordinamento operatoriale ottenuto dal confronto delle
dinamiche classica e semi-classica, e la forma esplicita dello spettro (discreto) dei livelli energetici del modello Mixmaster quantistico.
La seconda parte di questa dissertazione riguarda invece il problema del caos gravitazionale nell’intorno della singolarità come fenomeno dipendente dal numero delle
dimensioni del modello, e dalla materia di cui questo è riempito. I modelli multidimensionali, omogenei e non, esibiscono proprietà diverse: i primi possono essere
caotici solo in tre dimensioni spaziali, i secondi mantengono questa proprietà, in generale, fino a dieci dimensioni spaziali. Anche la materia influenza questo aspetto
in maniera decisiva: l’inserimento di un campo scalare inibisce il comportamento
oscillatorio in ogni numero di dimensioni. Questi meccanismi possono però essere
re-innescati se nella dinamica multi-dimensionale vengono inserite delle p-forme.
Il contributo originale qui discusso riguarda lo studio di un generico modello omogeneo accoppiato ad un campo Abeliano non massivo. L’effetto di questo campo
elettromagnetico multi-dimensionale è quello di restaurare il caos in ogni numero di
dimensioni; ancora più interessante è che la mappa associata esibisce una forte dipendenza dal numero di dimensioni, e che si riduce a quella del modello Mixmaster del
caso tridimensionale.
Questi studi sono stato l’oggetto di tre pubblicazioni su riviste internazionali, una
apparsa nel 2004 su Physical Review D, e due, nel 2005 e nel 2007 rispettivamente,
apparse su Classical and Quantum Gravity.
L’esposizione del materiale è organizzata in quattro capitoli.
VII
Nel capitolo 1, dopo aver classificato gli spazi omogenei seguendo lo schema classico
di L. Bianchi, vengono discusse in dettaglio le soluzioni dei modelli del tipo I, o di
Kasner, e quella del tipo IX, o Mixmaster. L’analisi della dinamica di quest’ultimo
viene poi riproposta nel formalismo Hamiltoniano per evidenziare l’analogia esistente
fra questi spazi ed i sistemi meccanici bidimensionali. L’ultima parte del capitolo è
dedicata alla soluzione cosmologica generale: sono discussi in dettaglio la costruzione
del regime oscillatorio a partire dalla soluzione di Kasner generalizzata, l’effetto delle
disomogeneità, il fenomeno della rotazione degli assi e quello della frammentazione.
Tre appendici di carattere tecnico chiudono la discussione.
Il capitolo 2 è centrato sul problema della covarianza del caos nella soluzione cosmologica generale. Il primo paragrafo ripercorre, attraverso la discussione dei risultati
più importanti, le fasi del dibattito ventennale attorno alla possibilità di utilizzare
l’esponente di Lyapunov come indicatore caotico in sistemi relativistici, con particolare interesse per il Mixmaster. Segue quindi il materiale originale, ed è così organizzato: l’approccio Hamiltoniano è discusso nel paragrafo 2, la riduzione della dinamica
ai gradi fisici di libertà e la struttura asintotica dello scalare di curvatura tridimensionale sono discussi nei paragrafi 3-5; l’equivalenza con il biliardo ed il calcolo esplicito
dell’esponente di Lyapunov sono proposti nel paragrafo 6, la misura invariante discussa nel 7, mentre al ruolo delle pareti e alla covarianza della descrizione proposta
sono dedicati i paragrafi 8 e 9. Nell’ultimo paragrafo sono presentate le conclusioni.
Il capitolo 3 è dedicato allo studio delle proprietà semi-classiche e quantistiche del
modello Mixmaster inomogeneo. Nei paragrafi 1 e 2 sono discussi due schemi diversi
per la quantizzazione della gravità, l’equazione di Wheeler-DeWitt e l’approccio alla
Multi-Time; nel paragrafo 3 viene risolta l’equazione di Hamilton-Jacobi e discussa la
dinamica dal punto di vista meccanico-statistico, mentre nel paragrafo 4 viene analizzato il limite semi-classico alla WKB. Il modello viene quantizzato nel paragrafo
5, ed il problema della non-località dell’operatore Hamiltoniano affrontato nel paragrafo 6. Gli ultimi due paragrafi sono dedicati al cosiddetto “quantum chaos” e ad
alcune conclusioni. Un’appendice su alcuni aspetti dell’analisi armonica sul piano di
Lobachevsky chiude il capitolo.
Il capitolo 4 è centrato sul rapporto esistente fra caos, numero di dimensioni e materia. Le generalizzazioni a spazi multi-dimensionali, omogenei e non, della dinamica del
Mixmaster nel vuoto sono rapidamente esposte nel paragrafo 1, mentre nel paragrafo
2 discussi gli effetti dei campi di materia. Il lavoro originale comincia nel paragrafo 3,
VIII
dove viene formulato, in approccio Hamiltoniano, il problema dinamico di un generico
modello omogeneo multi-dimensionale accoppiato ad un campo Abeliano. Il regime
di tipo Kasneriano generalizzato e la mappa sono derivati nel paragrafo 4 e 5 rispettivamente. Nel paragrafo 6 sono presenti delle conclusioni finali, e in appendice è
proposta la classificazione di Fee delle varietà omogenee quadri-dimensionali.
L’ultimo capitolo è dedicato alle conclusioni e a possibili sviluppi futuri.
In appendice sono inserite le copie delle pubblicazioni e degli atti dei congressi,
tutte conformi agli originali.
IX
Introduction
The Standard Cosmological Model is an outstanding result in modern physics. The
homogeneity and isotropy hypotheses on the Universe large scale structure are supported both by observational evidences and by very good agreement between theoretical predictions and experimental data. Among the first, the most important is
the almost absolute homogeneity and isotropy of the Cosmic Microwave Background
Radiation; among the latter, we remember the very good agreement between predictions of the primordial nucleosynthesis and the actual abundance of light elements
in the Universe. For these and other reasons, the temporal extension of this scheme
can be extended from today up to about 10−32 seconds far from the Big Bang (or,
equivalently, energies up to 1016 GeV).
The Standard Model, however, exhibits some important internal problems: the
horizon paradox and the lack of a formation mechanism for large scale structure are
two of the most relevant example. Even though some of these problems can be solved
by adding an inflationary stage during Universe evolution, strong reasons exist for
a generalization of cosmology to models more generic than Friedmann-RobertsonWalker (FRW) ones. FRW homogeneous and isotropic dynamics are, for example,
unstable during the collapse phase with respect to tensorial perturbations, like that
of gravitational waves; furthermore, it is supposed that the actual classical structure
of the Universe is the result of a previous quantum one. This is a severe restriction
on the possibility of global symmetries: indeed, at Planck time-scales, many causally
not correlated horizons were present in the Universe, and the topological fluctuations suggest that the classical phase of the Universe began in conditions of maximal
generality and without any symmetry.
The study of the generic (general) cosmological solution is, then, of outstanding
importance, both for the problems of the Standard Model, and as a question on its
own.
In the 70’s, V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz showed that it is
XI
possible to construct a generic cosmological solution of the Einstein equations in the
neighbourhood of the initial singularity, i.e. a solution that has all its degrees of freedom available, and thus allowing to specify a Cauchy problem on a generic space-like
hypersurface. This model is characterized by local homogeneity and by the presence
of a non eliminable singularity, and the dynamics results to be causally not connected
on scales greater than the horizon size; inside each horizon, an evolution, resembling
that of the homogenous spaces of the type VIII and IX of the Bianchi classification, takes place; such an evolution exhibits an oscillatory behaviour while reaching
the singularity, that can be well described only within a statistical framework. This
particular de-coupling of the spatial points of the manifold, together with the oscillatory behaviour, reduces space-time structure to that of a “foam” by a fragmentation
process.
The generic cosmological solution proposes a picture that, even though sophisticated and fascinating, is, however, a classical one. The most important problem is
what energy and time-scale the Universe should be described by such a framework:
at that energy, in a range of about 1016 − 1019 GeV, the Universe should be badly
described by a classical theory, and quantum corrections ought to be relevant. The
phenomenon that maps the classic into the quantum phase is the volume of the Universe; as soon as an isotropic and homogeneous dynamics is attended at the end of the
oscillatory phase, i.e., when the anisotropy are only small corrections, it is believable
that the begin of the oscillatory phase overlaps the Planckian one.
The quantum analysis of the generic cosmological solution is of absolute interest,
not only for the Standard Model, but also for the analysis of the properties of the
initial singularity, and, consequently, of the mechanisms at the ground of the birth of
our Universe.
The work presented in this PhD thesis can be divided in two parts, and the first
is inserted in this line of research, with original contributions about classical, semiclassical and quantum properties of the generic cosmological solution, also known as
inhomogeneous Mixmaster model.
In the classical framework, a complete study of the chaotic dynamics has been
performed; among the others, the most important result deals with the problem of
the covariant characterization of chaos in the generic solution, and this has been faced
by a direct, time and reference frame-independent calculation of an indicator, used
in ergodic theory and known as the Lyapunov exponent.
XII
In the semi-classical and quantum framework, a collection of results has been
obtained about the Hamilton-Jacobi formulation of the dynamics, the statisticalmechanical description of the model, WKB semi-classical limit, and the quantum
regime. Among the others, the most relevant results are the operator-ordering, deduced from a comparison of the classical and the semi-classical dynamics, and the
explicit shape of the (discrete) energy spectrum of the quantum Mixmaster model.
The second half of this dissertation is about the problem of gravitational chaos in
the neighbourhood of the singularity as a phenomenon dependent on the number of
dimensions and on matter fields. Multi-dimensional homogeneous models exhibits
different properties from the inhomogeneous ones: the first can be chaotic only in
three space-like dimensions, the latter show this property, in general, up to ten spacelike dimensions. Matter has also an important influences on this feature: a scalar
field inhibits the oscillatory behaviour in any number of dimensions. This mechanism
can be restated as soon as p-forms are inserted in the multi-dimensional dynamics.
The original contribution here discussed deals with the study of an homogeneous
multi-dimensional model coupled to a mass-less Abelian field. The main effect of this
field is to restore chaos in any number of dimensions; a much more interesting feature
is that the associated map exhibits a strong dependence on the number of dimensions,
and reduces to that of the three-dimensional Mixmaster model in vacuum.
These results were the subject of three publications on international journals, one
appeared in 2004 on Physical Review D, and two, in 2005 and 2007 respectively, on
Classical and Quantum Gravity.
The subject is organized in four chapter.
In chapter 1, after Bianchi’s classification of homogeneous spaces, the solution of
type I or Kasner model and that of type IX or Mixmaster, are discussed in detail. The
Hamiltonian formulation of the latter is also illustrated to put in evidence the existent
analogy with two-dimensional mechanical system. The last part of the chapter is
about the generic cosmological solution: we discuss the construction of the oscillatory
behaviour from the generalized Kasner solution, the effects of inhomogeneity, axis
rotation phenomenon and the fragmentation. Three technical appendices close the
chapter.
Chapter 2 is centred around the question of the covariance of chaos in the generic
cosmological solution. First chapter proposes the most important phases of a debate,
XIII
lasted for twenty years, around the use of the Lyapunov exponent in relativistic
dynamical system, with particular attention to the Mixmaster. Then the original
contribution follows: Hamiltonian formulation is in section 2, the reduction of the
dynamics to the physical degrees of freedom and the asymptotic structure of threedimensional Ricci scalar are presented in sections 3-5; the equivalence with billiard
description and the explicit calculation of the Lyapunov exponent are discussed in
section 6, the invariant measure in section 7, and sections 8 and 9 are about the
role of the potential walls and on the covariance of this scheme. Concluding remarks
follow in the last section.
In Chapter 3, a detailed investigation of the semi-classical and quantum properties of the inhomogeneous Mixmaster model is presented. Two different schemes of
quantization, the Wheleer-DeWitt equation and the Multi-Time formalism, are elucidated in sections 1 and 2; in section 3 the Hamilton-Jacobi equation is solved and
the dynamics is discussed from the statistical-mechanical point of view, while the
WKB limit is presented in section 4. The full quantization of the model is proposed
in section 5, and non-locality of the Hamiltonian operator is faced in section 6. The
last two sections are about the so-called “quantum chaos” and on some concluding
remark. An appendix, that deals with harmonic analysis on the Lobachevsky plane,
closes the chapter.
Chapter 4 deals with the existing link between chaos, number of dimensions, and
matter fields. Generalization to higher-dimensional, homogeneous and inhomogeneous, models in vacuum of the Mixmaster dynamics is addressed in section 1, while
section 2 deals with the effects of matter. The original contribution is presented starting in section 3, where the dynamical problem of an homogeneous multi-dimensional
space coupled to an Abelian vector field is formulated in Hamiltonian framework.
The generalized Kasner-like behaviour and the associated map are derived in sections
4 and 5 respectively. Concluding remarks follow in section 6, and an appendix, on
Fee’s classification of homogeneous five-dimensional space-times, closes the chapter.
Conclusion and outlooks are presented in the last chapter.
Copies of publications and of proceedings of congresses are appended at the end of
this dissertation.
XIV
1 The Mixmaster Model and the
Generic Cosmological Solution
The observational data on the Cosmic Microwave Background Radiation [dB+ 00,
S+ 03] suggest that the Universe, on large scales, is homogeneous and isotropic. These
two features completely determine the metric tensor in FRW models, and the sign of
the curvature is the only free parameter; in the meanwhile, the only homogeneity hypothesis, without further symmetries, leaves a much greater arbitrariness and allows
a richer behaviour in Einstein dynamics. These homogeneous spaces were classified
[Bia98] at the end of the XIX century, but only in the 60’s and 70’s the underlying deep cosmological meaning was clarified by the works of Belinsky, Khalatnikov
and Lifshitz (BKL) [BKL70, BKL82]: the types VIII and IX homogeneous models
of Bianchi classification, indeed, provide the prototypes for the construction of the
general (non-homogeneous) solution of the Einstein equations in the neighbourhood
of the singularity.
In this chapter, we review the main existing results on the dynamics of the type IX
model of the Bianchi classification (also known as Mixmaster [Mis69b]), and on the
behaviour of the generic cosmological solution near the Big Bang.
In particular, in Section 1, we classify the homogeneous spaces, discuss their dynamics
from the field equations point of view, and then the oscillatory approach to the singularity of the type IX model [BKL70, BKL82]. The Hamiltonian framework [Mis69a]
and the reduction of the dynamics of the Mixmaster to that of a two-dimensional
particle moving in a potential is addressed (Section 2).
The role of matter is discussed in Section 3, where it is shown that behaves as a test
fluid.
The generic cosmological solution of Einstein equations in the neighbourhood of the
singularity is then constructed and its dynamics analyzed in details in Section 4.
The chapter is closed by three appendices, where the tetradic formalism (1A), the
1
The Mixmaster Model and the Generic Cosmological Solution
property of synchronous reference frame (1B) and the Arnowitt-Deser-Misner [ADM62]
Hamiltonian formulation of General Relativity (1C) are briefly discussed.
1.1 Homogeneous 4-dimensional models and the
Mixmaster
Homogeneity means that the metric properties are the same in each point of the
three-manifold. More precisely, a generic space is homogeneous if it admits a group
of movements that maps the space onto itself; since we deal with 3-dimensional spaces,
the group has 3 independent parameters.
Let’s introduce a synchronous frame of reference (see Appendix 1B), and consider a
spatial slice at a fixed instant t; the action of such a group leaves unchanged the spatial
metric tensor hαβ , and, furthermore, must leave unchanged the three independent one(a)
forms eα dxα (here the latin index a denotes the three one-forms). In the tetradic
picture (see Appendix 1A), the homogeneity can be imposed directly on vectors e(a) ,
resulting in the following condition
(c)
(c)
∂eβ
∂eα
−
β
∂x
∂xα
!
c
eα(a) eβ(b) = Cab
.
(1.1)
c
The constants Cab
are called structure constants, and their values completely deter-
mine the structure and the properties of the space 1 ; they are anti-symmetric in the
lower indexes and obey the Jacobi identity [LL71].
The standard classification of the Bianchi spaces is based on the dual of the C’s with
two indexes defined as
c
Cab
= eabd C dc ,
(1.2)
(eabc = eabc is the unitary completely antisymmetric tensor, with e123 = 1).
The resulting classification2 is summarized in Table (1.1). The type I denotes the
euclidean space, and contains both the standard galilean space and the Kasner one;
the type V contains as particular case the 3-sphere, while the type IX contains the
1
This name is taken from the theory of Lie groups: an equivalent derivation of the homogeneous
spaces exists, and it is based on this theory, where the generators are proportional to the Killing
vectors of the 3 manifold, and the structure constants are exactly the same obtained with this
2
description.
The quantities a, n1 , n2 , n3 are related to the C ab by the relation C ab = na δab eab1 a, a ≥ 0.
2
1.1 Homogeneous 4-dimensional models and the Mixmaster
Type
a
n1
n2
n3
I
0
0
0
0
II
0
1
0
0
VII
0
1
1
0
VI
0
1
-1
0
IX
0
1
1
1
VIII
0
1
1
-1
V
1
0
0
0
IV
1
0
0
1
a
0
1
1
a
0
1
-1
VII
III (a = 1)
)
VI (a 6= 1)
Table 1.1:
Standard Bianchi classification of the homogeneous spaces with the
corresponding structure constants [Bia98].
constant negative curvature case.
1.1.1 The Kasner solution
The first anisotropic solution was found by Kasner in 1922 while studying the dynamics of type I model. He obtained a particular cosmological solution describing
a flat homogeneous space where the volumes grow linearly in time, but where the
linear distances grow differently along different directions: they grows along two, and
decrease along the last. This metric is called Kasner metric, and can be summarized
as
ds2 = dt2 − t2p1 dx2 − t2p2 dy 2 − t2p3 dz 2 ,
(1.3)
p1 + p2 + p3 = p21 + p22 + p23 = 1 .
(1.4)
From (1.3) and (1.4), the determinant of the 4-metric, −g = t2 , follows. Relations
(1.4) constrain the Kasner exponents pi ’s to be all different; only two sets of values,
(0, 0, 1) and (−1/3, 2/3, 2/3), have an index equal to another. In general the pi ’s can
3
The Mixmaster Model and the Generic Cosmological Solution
vary in the following intervals (if ordered as p1 6 p2 6 p3 )
−
1
6 p1 6 0,
3
2
0 6 p2 6 ,
3
2
6 p3 6 1 .
3
(1.5)
1
By (1.5), one and only one is negative; this determines the anisotropic evolution of
the axes.
These indexes can be expressed as functions of a parameter u:
p1 (u) =
−u
,
1 + u + u2
p2 (u) =
1+u
,
1 + u + u2
p3 (u) =
u(1 + u)
.
1 + u + u2
(1.6)
In Fig (1.1), the functions pi (u) are shown.
All the admissible values for the pi ’s are obtainable as the parameter u runs in the
1
p3
0.8
2
3
0.6
p2
0.4
0.2
0.2
-0.2
0.4
0.6
p1
0.8
1/u
1
−
1
3
Figure 1.1:
The three curves show all the admissible values for the three pi ’s. The figure
clearly shows that one and only one can be negative
region u ≥ 1; the values u < 1 are in the same interval because the inversion property
holds, i.e.,
1
p1
= p1 (u) ,
u
1
p2
= p3 (u) ,
u
1
p3
= p2 (u) .
u
(1.7)
The instant t = 0 is the only singular point; the metric has a singularity that
cannot be eliminated with any coordinates transformation, and the invariants of the
four-dimensional Ricci tensor become infinite. The only exception is the case (0, 0, 1),
that describes the standard flat space-time; the Galilean metric can be obtained as
the coordinate transformation t sinh z = ζ, t cosh z = τ is considered.
4
1.1 Homogeneous 4-dimensional models and the Mixmaster
1.1.2 The type IX model
Among all the homogeneous models, the type IX stands for its very peculiar dynamics3 .
Let’s introduce three spatial vectors l, m, n, and let’s take the matrix ηab (t) (see Appendix 1A) diagonal; the spatial metric can be recast in the form
hαβ = a2 lα lβ + b2 mα mβ + c2 nα nβ ,
(1.8)
where all the dependence on the time variable is contained only in the functions a, b, c.
Even though (1.8) is not the most general case, it already contains the most important
features we want to discuss.
For the type IX model, the structure constants with two indixes take the values
C 11 = C 22 = C 33 = 1 .
(1.9)
The Einstein equations assume the form
(ȧbc)˙
1
= 2 2 2 (b2 − c2 )2 − a4 ,
abc
2a b c
1
(aḃc)˙
= 2 2 2 (a2 − c2 )2 − b4 ,
abc
2a b c
(abċ)˙
1
= 2 2 2 (a2 − b2 )2 − c4 ,
abc
2a b c
ä b̈ c̈
+ + =0;
a b c
where the dot denotes the derivative with respect to the time variable.
(1.10)
(1.11)
Let’s introduce the logarithm of the functions a, b, c, and the logarithmic time variable
τ
a = eα ,
b = eβ ,
c = eγ ,
dt = Λ dτ = abc dτ .
Then, (1.10) and (1.11) become


2α = (b2 − c2 )2 − a4 ,


 ,τ,τ
2β,τ,τ = (a2 − c2 )2 − b4 ,



2γ = (a2 − b2 )2 − c4 ,
,τ,τ
3
Most of the considerations we will develop, apply to the type VIII also (see Section 2)
5
(1.12)
(1.13)
(1.14)
The Mixmaster Model and the Generic Cosmological Solution
1
(α + β + γ),τ,τ = α,τ β,τ + α,τ γ,τ + β,τ γ,τ ,
2
from which, the following constant of the motion can be easily obtained
α,τ β,τ + α,τ γ,τ + β,τ γ,τ =
1 4
a + b4 + c4 − 2a2 b2 − 2a2 c2 − 2b2 c2 .
4
(1.15)
(1.16)
The dynamical scheme (1.10)-(1.11) cannot be solved exactly, but a detailed study of
the dynamics near the singular point can be performed, and it will unravel the BKL
mechanism.
The left-hand sides of (1.10) (or equivalently of (1.14)) play the role of a potential
term; if we assume them to be negligible at a certain instant of time τ , then a Kasner
like dynamics takes place
a ∼ tpl ,
b ∼ tpm ,
c ∼ tpn ,
(1.17)
where pl , pm and pn satisfy the Kasner relations
pl + pm + pn = p2l + p2m + p2n = 1 .
(1.18)
The Kasner dynamics cannot last forever because the right-hand sides of (1.14) exponentially grow as the singularity is approached (t → 0).
Let’s assume that, for some interval of time, the regime is Kasner-like, and that
pl = p1 < 0. As t → 0, functions b(t) and c(t) become smaller and smaller, while a(t)
grows monotonically. The potential term in (1.14) cannot be neglected forever, and
we have to retain at least the growing terms
1
α,τ,τ = − e4α ,
2
1
β,τ,τ = γ,τ,τ = e4α .
2
(1.19)
The solution to this set of equations has to describe the evolution of a metric that
starts from a Kasner-like behaviour, and then is modified by the potential term; as
this ”initial condition” is imposed, another Kasner-like solution is obtained, where the
new set of Kasner indexes (p0l , p0m , p0n ) is linked to the previous one by the BKL map
p0l =
|p1 |
,
1 − 2|p1 |
p0m = −
2|p1 | − p2 0
p3 − 2|p1 |
, pn =
.
1 − 2|p1 |
1 − 2|p1 |
(1.20)
The perturbation, i.e., the right-hand sides of (1.14), induces a transition from a
Kasner-regime to another where the negative index passes from the l direction to the
6
1.1 Homogeneous 4-dimensional models and the Mixmaster
m one: if pl was negative, now p0m < 0. During this exchange process, the function a(t)
reaches a maximum, and b(t) a minimum; then a starts decreasing, b starts growing,
and c continues to decrease.
The result is that the initial perturbation a4 (t) becomes smaller and smaller until
it is negligible, the new Kasner-regime continues until the new perturbation b4 (t) is
relevant. Then a new transition occurs with a new set of indexes, and the phenomenon
is repeated infinitely many times up to the singularity.
A suitable way to represent the replacement rule is the following: if
pl = p1 (u) ,
pm = p2 (u) ,
pn = p3 (u) ,
then
p0l = p2 (u − 1) ,
p0m = p1 (u − 1) ,
p0n = p3 (u − 1) .
(1.21)
After each replacement, the negative index becomes positive, while the minor among
the others, negative.
It is worth noting how the effects of map (1.21) on the dynamics can be analyzed in
terms of continuos fractions in the following sense. Let’s assume that the initial value
of the parameter u is u0 = k0 + x0 , where k0 is an integer and 0 < x0 < 1. Then,
the length of the first series of oscillations is equal to k0 , and the initial value of the
successive era is u1 = 1/x0 = k1 + x1 . It’s very easy to realize that the lengths of
the following eras are given by the elements k0 , k1 , k2 , .... coming from the infinite
continuos fraction
u0 + k0 +
1
k1 +
1
k2 + k
.
(1.22)
1
3 +...
An analytical description of the properties of such a dynamics is impossible as the
number of the eras increases, and the properties can be studied only by statistical
techniques.
Let’s summarize the dynamics of the type IX model while reaching the singularity.
The continuos exchange (1.21), where the negative value passes from direction l to
direction m, goes on until u < 1. This value becomes u > 1 with the law (1.7); now
pn is the smallest positive index, and the subsequent exchanges will be among l-n
or m-n. It can be shown that, for arbitrary initial values of u, the process will last
forever.
7
The Mixmaster Model and the Generic Cosmological Solution
For the case of an exact solution, the pi ’s lose their literal meaning, and this gives
them some ”indetermination”: it makes no sense to consider particular values for the
parameter u, and only generic arbitrary values have to be studied.
All this scheme means that an infinite series of oscillations occurs as the Big Bang
is reached. During each of these oscillations, the linear distances along two axes
oscillate and monotonically decrease along the third. The volume element decreases
proportionally to t. Passing from a series to the successive one, the non-oscillating
direction changes to another axis. As time goes by, this process becomes quite random
(see (1.22)), and the length of the eras changes duration too.
The point t = 0 is the only singularity; furthermore, it is also an accumulation point
for the series of oscillations.
On the non-diagonal case
We have studied in details the dynamics of the diagonal Bianchi type IX, showing how
the BKL mechanism takes place because of the behaviour of the Ricci scalar in the
neighbourhood of the singular point. Even though it describes as-well the oscillating
regime while approaching the Big-Bang, this is not the most generic case, and new
features appear as we relax the diagonal condition. It has to be said that the offdiagonal components identically vanish in the empty case because of the 0α Einstein
equations; these new features indeed are typical only of a matter-filled space-time.
In the non diagonal case, the Kasner-like evolution takes place along three directions
that do not coincide with the frame vectors e(a) ; these directions are called Kasner
axes, and we will denote them with the familiar notation l, m, n4 . The projections
of the Kasner axes along the frame vectors gives us a set of constant parameters
La , M a , N a
l = La e(a) ,
m = Ma e(a) ,
n = Na e(a) .
(1.23)
Using all the arbitrariness that one has, we can fix the value of L, M, N as
L = (1, 0, 0) ,
M = (cot θm , 1, 0) ,
N = (cot θn / sin φn , cot φn , 1) ,
4
indeed in the diagonal case the Kasner axes can be taken as frame vectors.
8
(1.24)
1.2 The role of matter
with
pm − pl
,
C3
pn − pm
tan φn = Λ
C1
C2 sin φn − C3 cos φn
cot θn =
.
Λ(pl − pn )
tan θm = Λ
(1.25)
√
Here, the Ci ’s are three constants of the motion equal to Cc =
a
hχba Ccb
, and the latter
C’s are the structure constants. In terms of these angles here above introduced, we
can summarize the full Kasner map, comprehensive of rotation of the axes,
pl = p1 (u)
p0l = p2 (u − 1) ,
,
m0 = m − l
pn = p3 (u) ,
p0m = p1 (u − 1) ,
0
tan θm
2u − 1
=−
,
tan θm
2u + 1
l0 = l ,
pm = p2 (u) ,
φ0n = φn ,
4p1 cot θm
,
p2 + 3p1
p0n = p3 (u − 1) ,
tan θn0
u−2
=−
,
tan θn
u+2
n0 = n − l
(1.26)
4p1 cot θn
.
p3 + 3p1 sin φn
0
/θm | < 1 and |θn0 /θn | < 1. This means that, with each replacement of
We see that |θm
the epochs, the Kasner axes approach each other. The asymptotic behaviour of the
general model of type IX displays thus the same properties as the diagonal case, but
a new feature is also added, a gradual drawing together of the Kasner axes.
We will not discuss the effects of matter, as it does not play any significant role,
as it is shown in the Kasner case in the next paragraph; rotation of Kasner axes is
the only important feature that it adds to the Bianchi type IX dynamics. For further
details, see [BKL70, BLK71, BKL82] and the references therein.
1.2 The role of matter
Here we discuss the time evolution of a uniform distribution of matter in the Bianchi
type I space near the singularity; it will result that it behaves as a test fluid, and,
thus, it does not affect the properties of the solution.
Let’s take a uniform distribution of matter, and let’s assume that we can neglect the
9
The Mixmaster Model and the Generic Cosmological Solution
influence of matter on the gravitational field. The hydrodynamics equations describe
the evolution [LL71]
1 ∂ √
√
−gσui = 0 ,
i
−g ∂x
∂ui
∂p
∂p
1 l ∂gkl
k
(p + )u
= − i − ui u k k .
− u
k
i
∂x
2 ∂x
∂x
∂x
(1.27)
(1.28)
Here ui is the four-velocity, and σ is the entropy density; in the neighbourhood of the
singularity, it is necessary to use ultra-relativistic equation of state p = /3, and we
have, then, σ ∼ 3/4 .
As soon as all the quantities are functions of time, we have
d
abcu0 3/4 = 0 ,
dt
4
duα
d
+ uα = 0 .
dt
dt
(1.29)
From (1.29), we get the following two integrals of motion:
abcu0 3/4 = const,
uα 1/4 = const .
(1.30)
From (1.30), all the covariant components uα are all of the same order. Among the
controvariant ones, the greatest is u3 = u3 (as t → 0). Retaining only the dominant
contribution in the identity ui ui = 1, we obtain that u20 ≈ u23 , and, from (1.30)
∼
1
a2 b 2
uα ∼
,
√
ab ,
(1.31)
or, equivalently,
∼ t−2(p1 +p2 ) = t−2(1−p3 ) ,
uα ∼ t(1−p3 )/2 .
(1.32)
As expected, diverges as t → 0, for all the values of p3 , except p3 = 1 (this is due
to the non-physical character of the singularity in this case).
The validity of the approximation is verified from a direct evaluation of the components of the energy-momentum Tik ; their dominant terms are
T00 ∼ u20 ∼ t−(1+p3 ) ,
T22 ∼ u2 u2 ∼ t−(1+2p2 −p3 ) ,
T11 ∼ ∼ t−2(1−p3 ) ,
T33 ∼ u3 u3 ∼ t−(1+p3 ) .
(1.33)
(1.34)
As t → 0, all the components grow slower than t−2 , that is the behaviour of the
dominant terms in the Kasner analysis. Thus they can be disregarded.
10
1.3 Hamiltonian formulation
1.3 Hamiltonian formulation
A different insight on the Bianchi type IX model is that of Misner. In [Mis69a,
Mis69b], he described the complex dynamics in terms of a particle motion in a potential, adopting the ADM formalism (see Appendix 1C).
In the ADM formalism, the homogeneous dynamics are described by the following
line element
1
ds2 = N 2 dΩ2 − R2 (Ω) e2β ij σi σj .
(1.35)
4
Here the time variable Ω is related to the cosmic time t of an observer by the relation
dt = −N dΩ, thus the singular point appears in the limit Ω → ∞. βij is a diagonal
traceless matrix, so e2β = diag e2β11 , e2β22 , e2β33 and det(e2β ) = e2Trβ = 1. Thus
R3 (Ω) is proportional to the volume of the Universe at time Ω. The one-forms σi are
so defined
σ1 = sin ψdθ − cos ψ sin θdφ ,
σ2 = cos ψdθ + sin ψ sin θdφ ,
(1.36)
σ3 = − (dψ + cos θdφ) .
These one-forms reflect the invariance of the metric under the group of unit quaternions, which is the covering group of the rotation group SO(3). The coordinates
(ψ, θ, φ) are Euler angle coordinates on SO(3), taken over to covering group by letting ψ have the range 0 6 ψ < 4π, while 0 6 θ 6 π and 0 6 φ < 2π as usual. The
numeric factor 1/4 is chosen so that, when βij = 0, the space part of the metric is
just the standard metric for a three-sphere of radius R and circumference 2πR. Thus
for β = 0 this metric is the Robertson-Walker positive curvature metric.
The three-dimensional Ricci scalar can be evaluated and results to be equal to
3
R=
6
(1 − V ) ,
R2
1
V (β+ , β− ) = Tr e4β − 2e−2β + 1 =
3
√
√
2
4
1
= e4β+ cosh 4 3β− − 1 + 1 − cosh 2 3β− + e−8β+ .
3
3
3
(1.37)
(1.38)
Under these assumptions, the super-Hamiltonian constraint (1C.7) becomes
H = −pΩ + p2+ + p2− + e−4Ω (V − 1) = 0 .
11
(1.39)
The Mixmaster Model and the Generic Cosmological Solution
We take the following condition on the lapse function N and on the shift vector N α
1
N = e−3Ω
H
2
3π
1/2
,
Nα = 0 .
To avoid numerical factors in the calculation we set
1/2
2
R=
e−Ω .
3π
The metric is, therefore,
2
1 −2Ω 2β 2
ds =
H −2 e−6Ω dΩ2 −
e
e ij σi σj .
3π
6π
(1.40)
(1.41)
(1.42)
As soon as we solve the super-Hamiltonian constraint with respect to pΩ , the dynamics of the Bianchi type IX model results to be equal to that of a two-dimensional
particle moving in a time-dependent potential (see Fig(1.2)) described by the following variational principle
Z
I=
(p± dβ± − N HADM ) dΩ ,
1/2
HADM = p2+ + p2− + e−4Ω (V − 1)
.
(1.43)
(1.44)
The system described by (1.43) and (1.44) cannot be analytically integrated, but it
unravels interesting features in the neighbourhood of the singularity (Ω → +∞).
The Hamilton equations dβ± /dΩ = p± /HADM can be used to evaluate the “velocity”
|β 0 | of the point Universe, obtaining
−2
β+0 2 + β−0 2 + HADM
e−4Ω (V − 1) = 1 .
(1.45)
If we analyse the time variation of HADM
dHADM
∂HADM
=
dΩ
∂Ω
⇒
2
d ln HADM
= −4(1 − β 02 ) ,
dΩ
(1.46)
we easily recognize that, towards the singular point, in first approximation,
H ' const and β 02 ' 1. A closer inspection shows that these approximations are
valid only for finite Ω intervals, roughly while |β| < 21 Ω, interrupted by epochs where
the potential V does play a role. The anisotropy potential V (β± ) is positive definite
with V ≈ 8(β+2 + β−2 ) near β = 0. The potential walls rise steeply away from β = 0,
12
1.3 Hamiltonian formulation
Figure 1.2:
The potential in which the point Mixmaster Universe moves. The lines
represent the same equipotential at different time Ω. From this figure, it can
be easily seen how the potential term behaves as a infinite well. The three
corners represent three asymptotic directions by which the Universe could
escape, but the trajectories result to be unstable. The symmetry under a
rotation of 120◦ is manifest (from [Mis69a]).
with the equipotentials forming equilateral triangles in the (β+ , 0, β− ) plane.
One of the three equivalent sides of the triangle is described by the asymptotic form
1
V ∼ e−8β+ , β+ → ∞ ,
(1.47)
3
√
which is valid in the sector |β− | < − 3β+ . As Ω → ∞, the space curvature terms
e−4Ω (V − 1) in HADM can only play a role if V 1, so we will use the asymptotic
form (1.47). The condition that V be important is easily seen from (1.45) to be
−2
2
e−4Ω V ≈ 1 or exp[−4(Ω + 2β+ )] ≈ 3HADM
HADM
or
1
1
2
β+ ≈ βwall = − Ω − ln(3HADM
).
2
8
(1.48)
Thus βwall defines an equipotential in the β plane bounding the region in which the
potential terms are significant. When β is well inside this equipotential, one has
|β 0 | = 1 and HADM = const, and, consequently, from (1.48) |βwall0 | = 1/2. Thus the
13
The Mixmaster Model and the Generic Cosmological Solution
β point moves twice as fast as the receding potential wall, and at finite intervals as
Ω → ∞, the β(Ω) trajectory will collide with the potential wall and be deflected from
one straight line motion to another.
This is exactly what happens in the BKL description: the straight line motion is
the Kasner-like evolution, while the bounces against the potential walls induce the
BKL map.
On the dynamics of type VIII model
Up to now, we have discussed in details the very peculiar dynamics of the type IX
model; now we want to discuss the properties of another homogeneous space, those
of the type VIII. This model possesses most of the features we have outlined in the
previous sections, as it can be easily seen in the ADM framework.
This model is characterized by the following structure constants
C 11 = C 22 = −C 33 = 1 ,
(1.49)
and its dynamics can be described, like those of all the homogeneous models, as that
of a two-dimensional particle moving in a potential well. The analogues of (1.36) now
read
σ1 = − sinh ψ sinh θdφ + cosh ψdθ ,
σ2 = − cosh ψ sinh θdφ + sinh ψdθ ,
(1.50)
σ3 = cosh θdφ + dψ ,
and the potential structure is very similar to that of type IX, as it can be seen in
Fig(1.3). As soon as we analyze the asymptotic structure of the potential term,
we easily realize that the two dynamics are equivalent in the neighbourhood of the
singularity; indeed, we have (1.38) can be rewritten
V =
1
D4Q1 + D4Q2 + D4Q3 − D2Q1 +2Q2 ± D2Q2 +2Q3 ± D2Q3 +2Q1 ,
2
(1.51)
where D ≡ det e−Ω+βij = e−3Ω is the determinant of the three-metric, (+) and (−)
refers respectively to Bianchi type VIII and IX, and the anisotropy parameters Qi (i =
14
1.4 The Generic Cosmological Solution
Figure 1.3:
The potential in Misner variables in which the point Bianchi type VIII
Universe moves. The lines represents equipotential curves. Like in the type IX
model, the walls exponentially grow in the neighbourhood of the singular
point, and cut a closed domain.
1, 2, 3) denote the functions [KM97a]
Q1
Q2
Q3
√
1 β+ + 3β−
−
,
=
3
3Ω
√
1 β+ − 3β−
=
−
,
3
3Ω
1 2β+
+
.
=
3
3Ω
(1.52)
In the limit D → 0, the singularity appears, and the second three terms of the
above potential turn out to be negligible with respect to the first ones; this shows the
equivalence of the dynamics of the two models in the neighbourhood of the Big Bang.
1.4 The Generic Cosmological Solution
The homogeneous models of types VIII and IX provide the prototypes for the construction of the general (non-homogeneous) solution of the Einstein equations in the
neighbourhood of the singularity. Here, with “general solution”, we mean a solution
that possesses the right number of “physical arbitrary functions”. This number must
allow to specify arbitrary initial conditions on a generic space-like hyper-surface at
15
The Mixmaster Model and the Generic Cosmological Solution
a fixed instant t, thus a solution is general if it possesses four arbitrary functions in
vacuum, or eight if some kind of matter is present [LL71].
In [LK63], it is shown that the Kasner solution can be generalized to the inhomogeneous case and near the singularity, as

dl2 = hαβ dxα dxβ ,
h = a2 l l + b2 m m + c2 n n ,
αβ
α β
α β
α β
(1.53)
where
a ∼ tpl ,
b ∼ tpm ,
c ∼ tpn ,
(1.54)
and pl, , pm , pn are functions of spatial coordinates subjected to the conditions
pl (xγ ) + pm (xγ ) + pn (xγ ) = p2l (xγ ) + p2m (xγ ) + p2n (xγ ) = 1 .
(1.55)
In distinction of the homogeneous models, the frame vectors l, m, n are now arbitrary
functions of the coordinates (more exactly, subjected only to the conditions imposed
by the 0α of the Einstein equations [LK63]).
The behaviour (1.53) cannot last up to the singularity, unless a further condition is
imposed on the vector l (i.e., the one that correspond to the negative index p1 ), i.e.,
l · curll = 0 .
(1.56)
This condition reduces the number of arbitrary functions to three, one less than the
general solution: in fact, (1.53) possesses twelve arbitrary functions of the coordinates
(nine components of the Kasner axes and three indexes pi (xγ )), and is subjected to
the two Kasner relations (1.55), the three 0α Einstein equations, the three conditions
from the invariance under three-dimensional coordinate invariance of the theory in
the synchronous frame of reference, and (1.56).
Starting from (1.55), however, it is possible to construct the general cosmological
solution, and this is achieved in the following steps:
• construction of the general solution for the individual Kasner epoch;
• a general description of the alternation of two successive epochs.
The answer to the first question is given by the generalized Kasner solution (1.53);
there is no need to impose in this solution any additional condition, since the individual Kasner epoch lasts only through a finite interval of time, and hence contains the
16
1.4 The Generic Cosmological Solution
entire set of four arbitrary functions in vacuum (eight if the space is matter-filled).
The answer to the second question is given below; we shall see that the replacements
of epochs in the general solution does indeed occur in close analogy to the replacement
in the homogeneous model5 .
We will assume that the factors that determine the order of magnitude of the
components of the spatial metric tensor (1.53) be included in the functions a, b, c.
These functions change with time according to the laws (1.54), i.e., the vectors l, m, n
also define the directions of the Kasner axes. For non homogeneous spaces there is, of
course, no reason to introduce a fixed set of frame vectors, which would be independent
of Kasner axes.
The (time) interval of applicability of solution (1.53) is determined by conditions
which follow from the Einstein equations. Near the singularity, the matter energymomentum tensor in the 00- and αβ-components of these equations may be neglected
1
1
R00 = χ̇αα + χαβ χβα = 0 ,
2
4
1 ∂ √ β β
Rα = √
hχα + Pαβ = 0 .
2 h ∂t
(1.57)
(1.58)
Solution (1.53)-(1.55) is obtained neglecting the three-dimensional Ricci tensor Pαβ
in (1.58). The condition for such neglect to be valid is easy formulated in terms
of the projections of the tensors along the directions l, m, n [LK63]. The diagonal
projections of the Ricci tensor must satisfy the conditions
Pll , Pmm , Pnn t−2 ,
Pll Pmm , Pmn .
(1.59)
The off-diagonal projections of (1.58) determine the off-diagonal projections of the
metric tensor (hlm , hln , hmn ), which should be only small corrections to the leading
terms of the metric, as given by (1.53). In the latter, the only non-vanishing projections are the diagonal (hll , hmm , hnn ), and the fact that the off-diagonal projections
are small means that
hlm 5
p
hll hmm ,
hln p
hll hnn ,
hmn p
hmm hnn .
(1.60)
We also remind that the regular evolution of the homogeneous model may be violated by the
occurrence of a series of small oscillations [BLK72]. Although the likelihood of occurrence of
such violations tends to zero as t → 0 [LLK71], a construction of the general solution must also
include this case; the answer to this part of the problem is in [BKL70]
17
The Mixmaster Model and the Generic Cosmological Solution
This leads to the following conditions on the Ricci tensor:
Plm ab/t2 ,
Pln ac/t2 ,
Pmn bc/t2 .
(1.61)
If these conditions are satisfied, the off-diagonal components of (1.58) may be completely disregarded (in the leading order). For the metric (1.53), the Ricci tensor Pαβ
is given in Appendix D of [LK63].
The diagonal projections Pll , Pmm , Pnn contain the terms
1
a lcurl(a l)
k 2 a2
k 2 a4
∼ 2 2 = 22 ,
2 a b c(l · [m × n])
bc
Λt
(1.62)
and analogous terms with a l replaced by b m and c n (1/k denotes the order of magnitude of spatial distances over which the metric changes substantially, and Λ is the
same as in (1.13)). In (1.62), all the vectorial operations are intended to be performed
as in the Euclidean case.
The requirement that these terms be small leads, according to (1.59), to the inequalities
a
p
k/Λ 1 ,
b
p
k/Λ 1 ,
c
p
k/Λ 1 .
(1.63)
It is remarkable that these inequalities are not only necessary, but also sufficient
conditions for the existence of the generalized Kasner solution. In other words, after
conditions (1.63) are satisfied, all other terms in Pll , Pmm , Pnn , as well as all terms in
Plm , Pln , Pmn , satisfy conditions (1.59) and (1.61) automatically.
An estimate of these terms leads to the conditions
k2 2 2
3
2
a
b
,
...,
a
b
,
...,
a
bc
,
..
1.
Λ2
(1.64)
Inequalities (1.64) contain on the left the products of powers of two or three of the
quantities which enter the inequalities (1.63), and therefore are a fortiori true if the
latter are satisfied.
Inequalities (1.64) represent a natural generalization of those which already appear
in the construction of the oscillatory regime in the homogeneous case: the Kasner
regime arises when the second terms in (1.10) can be neglected compared to the first
ones; this requires a2 , b2 , c2 1.
As t decreases, an instant ttr may eventually occur when one of the conditions (1.63)
may be violated (the case when two of these conditions are violated simultaneously,
and this can happen when the exponents p1 and p2 are close to zero, corresponds to
18
1.4 The Generic Cosmological Solution
the above mentioned small oscillations). Thus, if during a given Kasner epoch the
negative exponent refers to the function a(t), i.e., pl = p1 , then, at the instant ttr , we
will have
r
k
∼ 1.
(1.65)
Λ
Since during that epoch the functions b(t) and c(t) decrease with the decrease of t,
atr =
the other two inequalities of (1.63) remain valid and at t ∼ ttr we shall have
btr atr ,
ctr atr .
(1.66)
It is remarkable that at the same time all the conditions (1.64) continue to hold. This
means that all off-diagonal projections of (1.58) may be disregarded as before. In the
diagonal projections (1.62), only terms of one type become important, terms which
contain a4 /t. We draw attention to the fact that in these remaining terms we have
(a l · curl (a l)) = a (l · [∇a × l]) + a2 (l · curl l) = a2 (l · curl l) ,
(1.67)
i.e., the spatial derivatives of a drop out.
As a result the following equations are obtained for the process of replacement of
two Kasner epochs
− Rll =
(ȧbc).
a2
+ λ2 2 2 = 0 ,
abc
2b c
(aḃc).
a2
− λ2 2 2 = 0 ,
abc
2b c
.
(abċ)
a2
− λ2 2 2 = 0 ,
− Rll =
abc
2b c
m
− Rm
=
(1.68)
ä b̈ c̈
+ + =0.
a b c
These equations differ from the corresponding equations of the homogeneous model
− R00 =
(1.10) only through the fact that the quantity
λ=
l · curl l
,
l · [m × n]
(1.69)
is no longer a constant, but a function of the space coordinates. Since, however, (1.68)
is a system of ordinary differential equations with respect to time, this difference does
not affect at all the solution of the equations and the map which follows from this
solution. Thus the law of alternation of exponents derived for homogeneous models
remains valid in the general inhomogeneous case.
19
The Mixmaster Model and the Generic Cosmological Solution
1.4.1 Rotation of the Kasner axes
We pass now onto the question of the rotation of the Kasner axes with the replacement
of epochs.
If in the initial epoch the spatial metric was given by (1.53), then in the final epoch
we shall have
hαβ = a2 lα0 lβ0 + b2 m0α m0β + c2 n0α n0β ,
(1.70)
with the new functions a, b, c given by a new set of Kasner indexes, and some new
vectors l0 , m0 , n0 . If we project all tensors (including hαβ ) in both epochs onto the
same directions l, m, n, it is clear that the turning of the Kasner axes can be described
as the appearance, in the final epoch, of off-diagonal projections hlm , hln , hmn , which
behave in time like linear combinations of the functions a2 , b2 , c2 . It is possible to
show that such projections do indeed appear [BKL82], and that these projections
induces the rotation of the Kasner axes. Without going deeper in these calculations,
we go directly to the conclusions.
The main effects can be reduced to a rotation of the second Kasner axis by a large
(∼ 1) angle, and a rotation of the first axis by a small angle (∼ b2tr /a2tr 1). Taking
only the big rotations into account, we find that the new Kasner axes are related to
the old ones by relations of the form
l0 = l ,
m0 = m + σm l ,
n0 = n + σn l ,
(1.71)
where the σm , σn are of the order of the unity, and are given by the following expressions
p1 2p1
1
[l × m] · ∇ +
m · curl l
,
λ
λ
l · [m × n]
2
p1 2p1
1
σn =
[n × l] · ∇ −
n · curl l
,
p2 + 3p1
λ
λ
l · [m × n]
2
σm = −
p2 + 3p1
(1.72)
that reduces to those of the homogeneous case (1.26).
It’s quite important to underline that the rotation of the Kasner axes (that appears
only for a matter-filled homogeneous space) is inherent in the inhomogeneous solution
already in the vacuum case. The role played by the matter energy-momentum tensor
can be imitated, in the Einstein equations, by the terms due to inhomogeneity of
the spatial metric. Here again (as in the generalized Kasner solution), in the general
inhomogeneous approach to the singularity the presence of matter is manifested only
in the relations between the arbitrary spatial functions which appear in the solution.
20
1.4 The Generic Cosmological Solution
1.4.2 Fragmentation
We will now qualitatively discuss a further mechanism that takes place in the inhomogeneous Mixmaster model in the limit towards the singular point: the so called
“fragmentation” process [KK87, Kir93b, Mon95].
The extension of the BKL mechanism to the general inhomogenous case contains
an important physical restriction which consists of a “local homogeneity” hypothesis.
In fact, the general derivation of this behaviour is based on the assumption that the
spatial variation of all the spatial metric components possess the same “characteristic
length”, described by a unique parameter k, which can be regarded as an average
wavenumber. There are reliable indications that such local homogeneity ceases to be
valid as a natural consequence of the asymptotic evolution of the system towards the
singularity.
We start by noting that conditions (1.55) do not require that the functions pa (xγ )’s
have the same ordering in all the points of space. Indeed, such functions can vary
their order throughout space even an infinite number of times without violating the
conditions (1.55), in agreement with an oscillatory-like behaviour of their spatial
dependence.
Furthermore, the most important property of the BKL map is the strong dependence
on initial values, a feature which produces an exponential divergence of the trajectories
which results from its iteration.
Given a generic initial condition p0a (xγ ), continuity of the three-manifold requires that,
at two points of the space very close to each other, the Kasner index functions assume
correspondingly close values. But because of the previously mentioned property, the
trajectories emerging from these two values diverge exponentially, and, since the
functions pa (xγ ) can vary only within the interval [−1/3; 1], we conclude that spatial
dependence effectively acquires an increasingly oscillatory like behaviour.
Let’s illustrate some more details of this process by reviewing the simplest case.
Let’s assume that, at a fixed instant of time t0 , all the points of the manifold can be
described by a generalized Kasner metric, that the Kasner index functions have the
same order point by point, and that the precise values p1 (xγ ), p2 (xγ ) and p3 (xγ ) can be
described, for all the points, by a narrow interval of u-values, let’s say u ∈ [K, K + 1]
for a generic integer K. We will refer to such a situation saying that the manifold is
composed by one “island”.
21
The Mixmaster Model and the Generic Cosmological Solution
We introduce the remainder part of the function u(xγ )
X(xγ ) = u(xγ ) − [u(xγ )] ∈ [0, 1),
(1.73)
where the square-brackets indicate the integer part of the expression enclosed by them.
Then the values of the narrow interval we chose can be written u0 (xγ ) = K 0 +X 0 (xγ ).
As the evolution takes place, the BKL mechanism induces a transition from an epoch
to another; the n-th epoch is characterized by a narrow interval [K − n, K − n + 1],
until K − n = 0; then the era comes to an end, and a new era begins. What now
happens, is that the new u1 (xγ ) starts from u1 (xγ ) ∈ 1/X 0 (xγ ), i.e., takes value in the
much wider interval [0, ∞). Only points that are very close can still be in the same
“island” of u values; distant points in space will now be described by very different
integer K 1 , and will experience eras of very different duration. As the singular point
is reached, many and many eras take place, causing the formation of a greater and
greater number of smaller and smaller “islands”. This is the so called “fragmentation”
process. The reason why such a process is interesting is that the value of the parameter
k, which describes the characteristic length, increases as the islands get smaller. This
means the progressive increase of the spatial gradients, and could, at least in principle,
destroy the BKL mechanism.
It can be argued that this is not the case, and a qualitative explanation is the following:
the progressive increase of the spatial gradients produces the same qualitative effects
on all the terms present in the three-dimensional Ricci tensor, including the dominant
ones. In other words, for each single value of k, in any island, a condition of the form
inhomogeneous term
k 2 t2Ki f (t)
∼
1,
dominant term
k 2 t4p1
Ki = 1 − pi ≥ 0, f (t) = O(ln t, ln2 t), t 1 ,
(1.74)
(1.75)
is still valid, where we call inhomogeneous those terms which contain spatial gradients of the scale factors, since they are evidently absent from the dynamics of the
homogeneous cosmological models.
From this point of view, the fragmentation process does not produce any behaviour
capable of stopping the iterative scheme of the oscillatory regime.
22
Appendix 1A
Appendix 1A
Vierbein representation of Einstein equations
Let’s introduce four linearly independent vectors eµ(a) (a is the vector index) that
satisfy the only condition:
eµ(a) e(b)µ = ηab ,
(1A.1)
where ηab is a given, symmetric, constant matrix with signature (+, −, −, −) ; let
η ab be the inverse of ηab . Furthermore, let e(a)µ be the reciprocal vectors, such that
(a)
(a)
eµ eµ(b) = δba . Relations eµ eν(a) = δµν are also true.
This way, the vector index is lowered and raised by the matrix ηab , and the metric
tensor gµν assumes the form
(b)
µ
ν
gµν = ηab e(a)
µ eν dx dx .
(1A.2)
We denote with “vierbein components” of the vector Aµ the projections along the
four e(a) :
A(a) = eµ(a) Aµ ,
ab
A(a) = e(a)
µ = η A(b) .
(1A.3)
The generalization to a tensor of any number of covariant or controvariant indexes is
quite natural.
We introduce now a couple of quantities which will appear directly in the Einstein
equations, γacb , and their linear combinations, λabc ( ; denotes the covariant derivative)
γacb = e(a)i;k ei(b) ek(c) ,
(1A.4)
λabc = γabc − γacb .
(1A.5)
With some little algebra, the Riemann and the Ricci tensors can be easily calculated
with the aid of the γabc and of the Ricci’s coefficients λabc
R(a)(b)(c)(d) = γabc,d − γabd,c + γabf (γ fcd − γ fdc ) + γaf c γ fbd − γaf d γ fbc ,
R(a)(b)
(1A.6)
1 c
= − λab, c + λba,c c + λcca,b + λccb,a +
2
1 cd
d
c
d
c
cd
cd
. (1A.7)
+ λ b λcda + λ b λcda − λb λacd + λ cd λab + λ cd λba
2
23
Appendix 1B
Appendix 1B
Einstein equations in the synchronous frame
In this appendix, it is shown how to separate the time-like from the space-like components in the field equations by the introduction of a synchronous frame.
Let us define χαβ
∂hαβ
,
(1B.1)
∂t
is the 3-dimensional metric tensor such that ds2 = dt2 − hαβ dxα dxβ . The
χαβ =
where hαβ
standard lowering and raising operations for the tensor χαβ are achieved by the metric
hαβ .
Here below, we list the expressions of the Christoffel symbols, and of the components
of the Ricci tensor.
Γ000 = Γα00 = Γ00α = 0 ,
(1B.2)
1
1
Γ0αβ = χαβ ,
Γα0β = χαβ ,
Γαβγ = λαβγ ,
(1B.3)
2
2
1∂ α 1 β α
1
R00 = −
χα − χα χβ ,
R0α = (χβα;β − χββ;α ),
(1B.4)
2 ∂t
4
2
1∂
1
Rαβ = Pαβ +
χαβ + (χαβ χγγ − 2χγα χβγ ) ,
(1B.5)
2 ∂t
4
where λαβγ are the three-dimensional Christoffel symbols, and Pαβ is the three-dimensional
Ricci tensor, both constructed from the metric hαβ
(a)
P(b) =
1
{2C bd Cad + C db Cad + C bd Cda +
2η
− C dd (C ba + Ca b ) + δab (C dd )2 − 2C df Cdf .
(1B.6)
Finally, the structure for the field equations is
R00 = −
2
1
1∂ α 1 α β
χα − χβ χα = (T00 − T ),
2 ∂t
4
k
2
2
1
Rα0 = (χβα;β − χββ;α ) = Tα0 ,
2
k
√
1 ∂
2
1
Rαβ = −Pαβ − √
( hχβα ) = (Tαβ − δαβ T ) ,
k
2
2 h ∂t
where k is a constant such that k = (16πG)−1 .
24
(1B.7)
(1B.8)
(1B.9)
Appendix 1C
Appendix 1C
The Arnowitt-Deser-Misner formalism and the
Hamiltonian formulation of General Relativity
In this appendix, a short review of the ADM (Arnowitt-Deser-Misner) formalism
[ADM62] is given (this topic is discussed in much more details in standard textbooks,
like [MTW73] or [Wal84]).
In the usual Lagrangian formulation of General Relativity, the fundamental field
is the metric tensor gµν , and the formulation is manifestly space-time covariant. The
Einstein-Hilbert action is usually expressed as
Z
√
SE−H = −k d4 x −gR(g) ,
(1C.1)
where R(g) is the Ricci scalar, which is function of gµν and of its first and second
derivatives. The Einstein equations follow from requiring the action to be stationary
under variations of the metric.
In contrast, the Hamiltonian formulation is not manifestly space-time covariant: it
requires a 3+1 splitting of the metric, and the dynamical degrees of freedom are the
spatial components of the metric. The usual 3+1 splitting is accomplished through
the ADM decomposition of space-time metric gµν in terms of a lapse function N , a
shift vector Ni , and an induced spatial metric hij .
Let Σt be a family of hypersurfaces labeled by a parameter t such that the whole
space-time M is the direct product M = Σ ⊗ <. The spatial components of gµν on
Σt induce a spatial metric (i.e., a three-dimensional Riemannian metric), given by
hij = −gij + ni nj , where nµ is the vector field normal to the hypersurface.
The construction of the lapse function N and of the shift vector Ni is shown in Fig
(1.4), and, in what follows, refer to it for notation.
Pick a point P1 on Σt and construct the normal at that point. The normal intersects
Σt+dt at a point P2 . The proper distance between points P1 and P2 defines the lapse
function N : dτ = N dt. In general, the normal vector will take the spatial coordinates
xi on Σt to some other spatial coordinates on Σt+dt . Let P3 denote the point on Σt+dt
with the same spatial coordinates as the point P1 on Σt . The vector from P2 to P3
defines the shift vector N i .
This way, the space-time metric can be constructed from N , N i and hij . Take some
25
Appendix 1C
Figure 1.4:
The construction of the lapse function N and of the shift vector Ni (from
[KT90]).
other point, P4 , on Σt+dt with spatial coordinates xi + dxi . The proper length from
P1 to P4 is defined in terms of the space-time metric gµν , that in terms of these new
quantities is
ds2 = gµν dxµ dxν = (N dt)2 − hij (N i dt + dxi )(N j dt + dxj ) .
From (1C.2) it is easy to obtain the form of the metric tensor


!
1
Nj
2
ij
−
N − Ni Nj h
−Nj
2


N2
gµν =
, g µν =  NN
 .
i j
N
N
i
ij
−Ni
−hij
− 2
−h
N
N2
√
√
Here hij is the inverse of hij , and Ni = hij N j ; moreover −g = N h.
(1C.2)
(1C.3)
One way to express the Ricci scalar in term of the lapse function, of the shift vector
and of the induced metric is to use the Gauss-Codacci formula [Wal84]
R = K 2 − Kij K ij −3 R ,
(1C.4)
where 3 R is the 3-dimensional Ricci scalar and Kij is the extrinsic curvature tensor
[MTW73]
1
∂hij
Kij ≡
Ni;j + Nj;i −
.
2N
∂t
(1C.5)
The standard Legendré transformation yields the Hamiltonian principle.
The momenta conjugate to N and N i vanish, and these relations are referred as the
“primary” constraints; much more, it is clear that the lapse and the shift are not
26
Appendix 1C
dynamical variables.
The final result is (π ij being the momentum conjugate of hij )
Z
H = d3 x π ij ∂t hij − N H − Ni H i ,
(1C.6)
where the “super-Hamiltonian” H and the “super-momentum” H i read explicitly
√ 3
1
√ (hik hjl + hil hjk − hij hkl ) π ij π kl − k h R ≡
2k h
√ 3
≡ −k −1 Gijkl π ij π kl − k h R ,
H=
H i = −2kπ;jij .
(1C.7)
(1C.8)
Gijkl is also referred as the DeWitt super-metric [DeW67].
27
2 Covariant Characterization of
the Chaos in the Generic
Cosmological Solution
From the beginning of the 80’s till the end of the last century, a puzzling question
attracted the interest of many scientists: is the Mixmaster dynamics really chaotic?
In this chapter, we focus our attention on this topic generalized to a generic cosmological model after a quite wide review of the principal works on the problem of
Mixmaster chaos in Section 1. Then the original work of this PhD thesis starts in
Section 2, where the dynamics of a generic cosmological model is formulated in a
Hamiltonian framework.
In Sections 3, 4 and 5 the reduction to the four physical degrees of freedom is performed together with a detailed discussion on the asymptotic behaviour to the singular
point of the three-dimensional Ricci scalar. The characterization of the chaotic dynamics is addressed in Sections 6, 7 and 8 by using the Lyapunov exponent as chaos
indicator, by deriving the invariant measure associated to such a model and by studying the effects of the boundary on chaos.
A detailed discussion of the covariance of this scheme then follows in Section 9.
In the last section some concluding remarks are presented.
This work was published on the international journal Physical Review D in the late
2004 [BM04]; the publication is appended at end of this PhD thesis (Attachment 1).
29
Covariant Characterization of the Chaos in the Generic Cosmological Solution
2.1 Is the Mixmaster dynamics chaotic?
The challenge over the relativistic chaos is well demonstrated in the very intense
debate started at the beginning of the 80’s over the chaoticity of the Mixmaster Universe. In the first works by Barrow [Bar81, Bar82b, CB83], the chaotic properties of
a discrete approximation to the full dynamics, known as the Gauss map, are studied,
and the question of the chaos by using some special indicator as the Lyapunov exponent, the metric entropy and the topological entropy is faced. In such works, the
properties of the dynamics is reduced to those of the following map:
∂α
s(1 + u)
= sp2 (u) =
,
∂τ
1 + u + u2
su
∂β
= sp1 (u) = −
,
∂τ
(1 + u + u2 )
∂γ
su
= sp3 (u) =
,
∂τ
(1 + u + u2 )
α=0,
Σx
,
(x + 1 + u)
Σ(1 + u)
,
γ=
x+1+u
β=

(u − 1, x/(1 + x)) if ∞ > u > 1 oscillations
0
0
(u , x ) =
(1/u − 1, 1 + 1/x) if 1 > u > 0 bounces .
(2.1)
(2.2)
Map (2.1)-(2.2) is a simple generalization of the well-studied Baker transformation
(which is given as (w0 , y 0 ) = (2w − [2w], (y + [2w])/2) for w, y ∈ (0, 1) [AA68]): this
means that it has dense periodic orbits and that there are no integral of the motion;
furthermore, it is ergodic and strongly mixing. This is a strong evidence that the
exact equations of motion have a very complex dynamics.
The dispute began when numerical experiments, run in different coordinate systems,
revealed that the Lyapunov exponents vanish [FM88, Rug90, Ber91, HBWS91]. Furthermore, the Gauss map itself corresponds to a specific time slicing. The ambiguity
of time was manifest.
In [HBWS91], the vanishing behaviour of the Lyapunov exponent for the Bianchi
type IX in vacuum under certain conditions and for a finite interval of time is shown,
as well as generation of entropy. Their limited results show that the application of
numerical methods to the Einstein equations often requires a careful demonstration
of consistency with known analytic results.
In [Ber91], the dependence of the Lyapunov exponent on the choice of the time variable is demonstrated. Numerical simulations evaluated the Lyapunov exponents along
30
2.1 Is the Mixmaster dynamics chaotic?
some trajectories in the (β+ , β− ) plane for different choices of the time variable;
more precisely, three different time variables, τ (BKL), Ω (Misner) and λ , the “minisuperspace” time variable(dλ = | − p2Ω + p2+ + p2− |1/2 dτ ) were adopted to evolve the
trajectories indicated in Fig.(2.1).
It is emphasized that the same trajectory, which clearly gives zero Lyapunov exponent
for τ or Ω-time, clearly fails to give zero for λ-time (even if the time duration of the
simulation is limited in extension). This can be seen in Fig.(2.2).
More recently, further efforts [BT93, HBC94, KM97a, CL97a, CL97b, WIB98, IM01,
Figure 2.1:
The trajectory in the anisotropy plane (β+ , 0, β− ) evolving towards the
singularity. Figures (a) - (d) represent the same trajectory with the scale
increased in each case. In (a), the origin is at the center with the range on β±
equal to ±20. The trajectory begins at the maximum of expansion (very near
the origin). In (b), the origin is at the centre with the scale ±800. Each
straight line segment is an epoch with the labelled epochs indicating the sense
of the trajectory. In (c), only the upper left quadrant of the anisotropy plane is
shown. The length of each axis is 5 × 105 . In (d), the entire anisotropy plane is
again shown with the scale ±7 × 106 (from [Ber91]).
ML01, Mot03] were made in order to decide and characterize by some covariant (timeindependent) indicator if the Mixmaster model were chaotic or not.
Two convincing arguments, appeared in [CL97a, CL97b] and [IM01], support the idea
that the Mixmaster chaos remains valid in any system of coordinates.
31
Covariant Characterization of the Chaos in the Generic Cosmological Solution
Figure 2.2:
Lyapunov exponents depend on time variable. (a) The solid line indicates
mini-superspace proper time λ vs. coordinate time τ for epochs a through h of
the trajectory displayed in Fig.(2.1). The dashed line indicates the three
positive Lyapunov exponents and are shown in a log-log plot vs. the coordinate
time τ used to compute them for the same epochs. (b) The same exponents
computed with mini-superspace proper time λ are shown by the dashed line
(from [Ber91]).
In [CL97a, CL97b], Cornish and Levin used coordinate-independent fractal methods
to show that the Mixmaster Universe is indeed chaotic. By exploiting techniques
originally developed to study chaotic scattering, they gained a new perspective on
the evolution of the Mixmaster cosmology. They found a fractal structure, namely
the strange repellor (see Fig.(2.3)), that describes the chaos well. The strange repellor
is the collection of all Universes periodic in (u, v). A typical, aperiodic Universe will
experience a transient age of chaos if it brushes against the repellor. The fractal was
exposed in both the exact Einstein equations and in the discrete map that they did
use to approximate the solution. The most important feature of their work is that
their fractal approach is independent of the time coordinate used. Thus the chaos
reflected in the fractal weave of Mixmaster Universes is unambiguous.
In [IM01], the issue of the dependence of the Lyapunov exponent on the time variable is faced: addressing an Hamiltonian point of view in treating the dynamical
equations of the Bianchi type IX model, they succeeded in defining it in a covariant
way. The key point of their work relies on the choice of a generic function as time
variable, avoiding this way all the problems related to the definition of the Lyapunov
32
2.2 Hamiltonian formulation for a generic cosmological model
Figure 2.3:
The numerically generated basin boundaries in the (u, v) plane are built of
Universes which ride the repellor for many orbits before being thrown off.
Similar fractal basins can be found by viewing alternative slices through the
phase space, such as the (β, β̇) plane. The overall morphology of the basins is
altered little by demanding more strongly anisotropic outcomes. (From
[CL97b])
exponent.
Our work [BM04] is inserted in this line of research.
2.2 Hamiltonian formulation for a generic
cosmological model
A generic cosmological solution is represented by a gravitational field having available
all its degrees of freedom and, therefore, allowing to specify a generic Cauchy problem.
As we have already discussed in the previous sections, it must contain four physical
arbitrary functions of the coordinates (in the field equations formulation), that means
8 independent functions in the phase-space; the real physical degrees of freedom are
two (as the possible polarizations of a gravitational wave), that means four in the
phase space.
In the ADM formalism, the metric tensor corresponding to such a generic model takes
33
Covariant Characterization of the Chaos in the Generic Cosmological Solution
the form
dΓ2 = N 2 dt2 − hαβ (dxα + N α dt)(dxβ + N β dt) ,
(2.3)
where N and N α denote the lapse function and the shift-vector respectively, and
hαβ (α, β = 1, 2, 3) the 3-metric tensor of the spatial hyper-surfaces Σ3 for which
t = const.
The spatial metric possesses six degrees of freedom, and we choose the following
parametric representation:
hαβ = eqa δad Oba Ocd ∂α y b ∂β y c ,
a, b, c, d, α, β = 1, 2, 3 .
(2.4)
Here q a = q a (x, t) and y b = y b (x, t) are six scalar functions, and Oba = Oba (x) is a
non-dynamical SO(3) matrix.
By the metric tensor (2.4) and (1C.7)-(1C.8), the action for the gravitational field is
Z
S=
dtd3 x pa ∂t q a + Πd ∂t y d − N H − N α Hα ,
(2.5)
Σ(3) ×<
where
#
"
X
1 X
1
H=√
pb )2 − h(3) R ,
(pa )2 − (
2
h a
b
(2.6)
Hα = Πc ∂α y c + pa ∂α q a + 2pa (O−1 )ba ∂α Oba .
(2.7)
In (2.6) and (2.7), pa and Πd are the momenta conjugate to the variable q a and y b
respectively, and
(3)
R is the Ricci 3-scalar which plays the role of a potential term.
2.3 Solution to the super-momentum constraint
From (2.7), we note that the super-momentum constraint can be diagonalized and
explicitly solved by choosing the functions y a ’s as special coordinates; in fact, as soon
as we take the transformation

η = t ,
y a = y a (t, x) ,
we get the expression
Πb = −pa
a
∂q a
−1 c ∂Oc
−
2p
(O
)
.
a
a
∂y b
∂y b
34
(2.8)
(2.9)
2.4 The Ricci scalar near the singularity
This way, the solution to this constraint is trivial. Let’s study more in detail this
transformation: starting by (2.6) and (2.7), the following transformation law holds



q a (t, x) → q a (η, y) ,





0

pa (t, x) → pa (η, y) = pa (η, y)/|J| ,
∂
∂y b ∂
∂

→
,
+

b

∂t
∂t
∂y
∂η




∂
∂y b ∂

 α →
,
∂x
∂xα ∂y b
(2.10)
where |J| denotes the Jacobian of the transformation. The first relation holds in
general for all the scalar quantities, while the second one for all the scalar densities.
From direct substitution of (2.9) in action (2.5), we get
Z
S=
Σ(3) ×<
dηd3 y pa ∂η q a + 2pa (O−1 )ca ∂η Oca − N H .
(2.11)
The change (2.8) not only allows to easily solve the super-momentum constraint, but
also “promotes” the SO(3) matrices (which can be expressed, for example, with the
Euler angles) to dynamical variables. This is not surprising at all: indeed Hα = 0
are 3 relations that can be used to eliminate three variables (Πd ) from the dynamics,
but, this way, we “kill” the three components of the shift-vector N α also. In some
sense, this phenomenon corresponds to the conservation of the number of dynamical
variables.
2.4 The Ricci scalar near the singularity
Before going on in the dynamical analysis of the Einstein equation, let’s have a look
at the behaviour of the Ricci scalar near the singular point. This term plays the role
of a potential term, and its analytical structure unravels the peculiar BKL dynamics.
From direct but quite tedious calculation, the exact form of the Ricci scalar can be
35
Covariant Characterization of the Chaos in the Generic Cosmological Solution
obtained:
−h
(3)
R=
X1
l
2
e2q
l
X
X
~ kl T~ kj )2 −
(Ojl ∇O
kj
l
m
X
eq +q [
lm,mn,nl
~ kl T~ kξ )+
(Oξl ∇O
k,ξ,σ,θ
~ m T~ kξ )(∇O
~ l T~ σθ ) − Ol Om [(∇O
~ m T~ kξ )(Ol ∇q
~ l T σθ )+
− 2Oξl Oθm (∇O
k
σ
ξ θ
k
σ
~ m T~ kξ )(∇q
~ l T~ σθ )]+
~ l T~ σθ )(Om ∇q
~ m T~ kξ )] − 1 Ol Om Om Ol (∇q
+ (∇O
ξ θ
k
σ
σ
k
2
X
X
1
∇h
l +q m
l
m
q
~ ∇q
~
~ m+
~ m − Om ∇O
~ l ) + 1 (Ol Om ∇q
+2
(∇q
−
e
)[ [(Oξl ∇O
ξ k
k
ξ
k
2 h
2
lm,mn,nl
k,ξ
X
X
X
l
m
~ l )]T~ kξ ][
~ l ∇O
~ m − ∇O
~ m ∇O
~ l+
+ Okl Oξm ∇q
Oσl Oθm T~ σθ ] + 2
eq +q [
[(∇O
ξ
k
ξ
k
θσ
lm,mn,nl
k,ξσθ
1 ~ l m
l~
m ~ m
l ~
m ~ l
~ l m
+ Oξl ∇2 Okm − Oξm ∇2 Okl ) + [(∇O
ξ Ok + Oξ ∇Ok )∇q + (∇Ok Oξ + Ok ∇Oξ )∇q +
2
X
~ m+
~ m − Om ∇O
~ l ) + 1 (Ol Om ∇q
Oξl Okm ∇2 q m + Okl Oξm ∇2 q l ]]T~ kξ Oσl Oθm T~ θσ + [ [(Oξl ∇O
ξ k
k
ξ
k
2
k,ξ
X
~ l )]T~ kξ ][ (Oθm ∇O
~ σl + Oσl ∇O
~ θm )T~ σθ ]] . (2.12)
+ Oξm Okl ∇q
σθ
Here a symbolic notation is adopted in order to reduce the extension of the analytical
expression; in (2.12), vectorial operations are to be performed like in the Euclidean
case, all the indexes run from 1 to 3, and
~ k ∧ ∇y
~ j
T~ kj = ∇y
(2.13)
~ k and ∇y
~ j.
is the Euclidean external product of the vectors ∇y
This is quite a complicated expression, but, as soon as we restrict our analysis to the
neighbourhood of the singular point, we can recognize the leading terms. In fact, we
can model (2.12) near the Big-Bang as
X
X
D (3)
2 2Qa
DQb +Qc O ∂q, (∂q)2 , y, ∂y ,
R
=
λ
D
+
U=
a
2
|J|
a
b6=c
X
D ≡ exp
qa,
(2.14)
(2.15)
a
qa
Qa ≡ P b
bq
λ2a ≡
X
~ c ∧ ∇y
~ b )2
~ ca (∇y
Oba ∇O
(2.16)
.
(2.17)
kj
Assuming the functions y a (t, x) smooth enough, (2.19), then all the gradients appearing in the potential U are regular. This assumption is justified by the following two
reasons:
36
2.4 The Ricci scalar near the singularity
1. in [Kir93b], it is shown that the spatial gradients increase logarithmically in
the proper time along the billiard geodesic, and therefore result to be of higher
order
2. as it can be seen from the variations with respect to pa , Πa in the action (2.5),
the functions y a are linked to the system of co-ordinates. These provides six
relations that can be re-arranged in order to relate the variables y d with the
lapse function and the shift vector
∂t y d = N α ∂α y d ,
√
h
N=P
a pa
(2.18)
!
N α ∂α
X
q b − ∂t
b
X
qb .
(2.19)
b
This implies, for example, that if we impose the condition Nα = 0, fixing this
way a synchronous frame of reference, than ∂t y d = 0, i.e., they are frozen variables with no dynamics: the request of the smoothness corresponds to requiring
regularity of the frame of reference.
It’s worth noting that the quantity D is proportional to the determinant of the threeP
metric, being det(hαβ ) = |J| exp a q a = |J|D, and controls its vanishing behaviour;
thus, as D → 0, the spatial curvature
(3)
R diverges and the cosmological singularity
appears. In this limit, the first terms of the potential U dominates all the remaining
ones and can be modeled by the potential wall
U=
X
a
Θ(Qa ) ,

+∞
Θ(x) =
0
if x < 0,
(2.20)
if x > 0.
This means that the Universe dynamics evolves independently in each space point
because they are no more related among them by the spatial gradients; furthermore,
this potential cuts a dynamically-closed domain in the configuration space (ΓQ , see
figure (2.4)) that corresponds to the following conditions on the so-called anisotropy
parameters Qa
Qa ≥ 0 ,
a = 1, 2, 3.
(2.21)
Since in ΓQ the potential U vanishes asymptotically, near the singularity we have
∂pa
=0;
∂η
(2.22)
thus the term 2pa (O−1 )ca ∂η Oca in (2.11) behaves as an exact time-derivative and can
be ruled out of the variational principle.
37
Covariant Characterization of the Chaos in the Generic Cosmological Solution
2.5 Dynamics of the physical degrees of freedom
We will now solve the super-Hamiltonian constrain H = 0 by introducing the so-called
Misner-Chitré like variables [Chi72, Kir93b, Mon00, IM01] as

p
√

q l = eτ [ ξ 2 − 1(cos θ + 3 sin θ) − ξ] ,



p
√
q m = eτ [ ξ 2 − 1(cos θ − 3 sin θ) − ξ] ,



q n = −eτ (ξ + 2pξ 2 − 1 cos θ) .
(2.23)
The reason way those variables are so suitable in treating the Mixmaster model relies
in the τ -independent expressions for the Qa ’s (2.16)
p
√
ξ2 − 1
1
Q1 = −
(cos θ + 3 sin θ) .
3
3ξ
p
√
ξ2 − 1
1
Q2 = −
(2.24)
(cos θ − 3 sin θ) ,
3
3ξ
p
1 2 ξ2 − 1
Q3 = +
cos θ .
3
3ξ
The quantity D, defined in (2.15), that controls the potential well behaviour of the
Ricci scalar now explicitly reads
D = exp[−3ξeτ ] .
(2.25)
We remember that the singularity appears in the limit D → 0, and by (2.25), it
happens as τ approach infinite values.
The super-Hamiltonian constraint can be solved in the domain ΓQ :
s
p2
− pτ ≡ HADM ≡ = (ξ 2 − 1)p2ξ + 2 θ ,
ξ −1
and the reduced action reads as
Z
δSΓQ = δ dηd3 y(pξ ∂η ξ + pθ ∂η θ − ∂η τ ) = 0 .
(2.26)
(2.27)
Before starting the analysis of the dynamical system (2.27), we would like to stress
the feature of : by the asymptotic limit (2.20) and the Hamilton equations associated
with (2.27), is a constant of motion, i.e.,
d
∂
=
= 0 ⇒ = E(y a ) .
dη
∂η
38
(2.28)
2.6 The Lyapunov exponent
5
4
3
2
"
1
0
1
!
-1
10
Figure 2.4:
The limited region ΓQ (ξ, θ) of the configuration space where the dynamics of
the point Universe is restricted by mean of the curvature term.
It corresponds to a infinite potential well, and it is the region where the
conditions Qa ≥ 0 are fulfilled.
2.6 The Lyapunov exponent
We will now perform the analysis of the dynamics of the generic cosmological solution,
first using the technique of the Jacobi metric [Arn78] to reduce the Hamilton equations
to a geodesic problem on a Riemannian surface, and then showing the positiveness of
the Lyapunov exponent for such a system.
The Jacobi metric
Being a constant of motion, the term ∂η τ = E(y a )∂η τ in (2.27) behaves like an
exact time derivative; hence the variational principle rewrites:
Z
δ d3 y(pξ dξ + pθ dθ) = 0 ,
(2.29)
coupled with the constraint (2.26). As a consequence of the asymptotic behaviour of
Ricci scalar, the variational principle (2.29) is the direct product of infinite independent dynamical systems, one for each point of the space; for this reason, we will be
interested in only one of them.
In order to derive the Jacobi line element, we set xa0 ≡ g ab pb , and by the Hamiltonian
39
Covariant Characterization of the Chaos in the Generic Cosmological Solution
equations
∂
∂η τ
(∂η τ HADM ) =
ξ 2 − 1 pξ ,
∂pξ
HADM
∂
∂η τ
pθ
=
(∂η τ HADM ) =
,
2
∂pθ
HADM (ξ − 1)
ξ0 =
θ0
(2.30)
we obtain the metric
∂η τ
ξ2 − 1 ,
a
E(y )
∂η τ
1
=
.
a
2
E(y ) ξ − 1
g ξξ =
g θθ
(2.31)
By these and by the fundamental constraint relation
pθ 2
ξ 2 − 1 pξ 2 + 2
= E(y a )2 ,
ξ −1
(2.32)
we get
∂η τ
gab x x =
E(y a )
a0 b 0
By the definition xa0 =
pθ 2
ξ − 1 pξ 2 + 2
ξ −1
2
= E(y a )∂η τ .
ds
dxa ds
≡ ua , (2.33) rewrites as
ds dη
dη
2
ds
a b
= E(y a )∂η τ ,
gab u u
dη
(2.33)
(2.34)
which leads to the key relation
s
dη =
gab ua ub
ds .
E(y a )∂η τ
(2.35)
Indeed, expression (2.35), together with pξ ξ 0 + pθ θ0 = E(y a )∂η τ , allows us to put the
variational principle (2.29) in geodesic form:
Z
Z q
Z p
a
b
a
δ ∂η τ E dη = δ
gab u u E(y )∂η τ ds = δ
Gab ua ub ds = 0 ,
(2.36)
where the metric Gab ≡ E(y a )∂η τ gab satisfies the normalization condition Gab ua ub = 1
and therefore
1
∂s
= E(y a )∂η τ .
∂η
1
(2.37)
We take the positive root since we require that the curvilinear coordinate s increase monotonically
with increasing value of τ , i.e., when approaching the initial cosmological singularity.
40
2.6 The Lyapunov exponent
Summarizing, in the region ΓQ , the considered dynamical problem reduces to a
geodesic flow on a two dimensional Riemannian manifold described by the line element
2
2
a
ds = (E(y )∂η τ )
2
dξ 2
2
+ ξ − 1 dθ .
ξ2 − 1
(2.38)
Now, it is easy to check that the metric above has negative curvature, since the
associated curvature scalar reads R = −2/ (E(y a )∂η τ )2 ; therefore the point-universe
moves over a negatively-curved bidimensional space on which the potential wall (2.20)
cuts the region ΓQ . Independently of the temporal gauge, we gave a satisfactory
representation of the system as isomorphic to a billiard on a Lobachevsky plane
[Arn78]. The so-obtained Jacobi metric is valid independently of the point, as the
space coordinates behave like external parameters and the evolution is spatially notcorrelated.
Fermi basis
Let us construct a Fermi basis {uµ , wµ }, (µ = 1, 2), i.e., a parallel transported frame
of reference. The geodesic vector uµ can be taken equal to
uµ =
dξ dθ
,
ds ds
p
=
ξ2
−1
E(y a )∂η τ
cos φ(s),
sin φ(s)
p
ξ2 − 1
E(y a )∂η τ
!
,
(2.39)
where s denotes the curvilinear coordinate, while φ(s) is an angular variable (0 ≤
φ < 2π), whose dynamics is obtained by requiring that the geodesic equation hold
dφ(s)
ξ
p
=−
sin φ(s) .
ds
E(y a )∂η τ ξ 2 − 1
(2.40)
The vector wµ is determined by the properties of the Fermi basis; it is orthonormal,
and it explicitly reads
p
wµ =
ξ2 − 1
cos φ
p
−
sin φ;
a
E(y )∂η τ
E(y a )∂η τ ξ 2 − 1
!
.
(2.41)
Let Z µ be the connecting vector between a generic couple of close geodesics; let us
project it over the Fermi basis defined above
Z µ = Zu (s)uµ + Zw (s)wµ ,
41
(2.42)
Covariant Characterization of the Chaos in the Generic Cosmological Solution
The dynamics of Z µ is given by the geodesic deviation equation, that results in a
couple of differential equations, one for each component of the vector
 2
d Zu


 2 =0,
ds
d2 Zw
Zw


.
 ds2 =
(E(y a )∂η τ )2
The solution to system (2.43) is given by

Zu = As + B, A, B = const ,
Z = c e E(yas)∂η τ + c e− E(yas)∂η τ ,
w
1
2
(2.43)
(2.44)
c1 , c2 = const .
The value of E given by the constraint (2.26), involved in the line element (2.38),
is determined by the initial conditions and cannot vanish. By the first of solutions
(2.44), no geodesic deviation takes place along the geodesic vector (as expected); on
the contrary, from the second solution, we get a non-zero Lyapunov exponent of the
form [Pes77]
λ(y a ) = lim sup
s→∞
1
ln(Zw2 + (dZw /ds)2 )
=
> 0.
a
2s
E(y )∂η τ
(2.45)
For the validity of this analysis, we have to verify that, in the limit to the initial
singularity, the curvilinear coordinate s approaches infinity. This can be easily seen
from (2.37),
s = E(y a )τ + f (y a ) ⇒ lim s = ∞ ,
τ →∞
(2.46)
f (y a ) being a generic function of the space coordinates. This ensures that the curvilinear coordinate s has the requested behaviour towards the singularity (we remember
that the singularity appears as and D = exp[−3ξeτ ] → 0 ⇒ τ → ∞).
2.7 Invariant measures
To better characterize the chaoticity of the obtained billiard, we show that, in each
point of the space, this system admits an invariant measure, which, in the present
variables, is uniform over the admissible phase space.
From statistical-mechanical point of view, such a system admits, point by point in
space, an “energy-like” constant of motion which corresponds to the kinetic part of
the ADM Hamiltonian = E(y a ). The point-universe randomizes within the closed
42
2.7 Invariant measures
domain ΓQ , and it is represented by a dynamics which allows for an ensemble representation; in view of the existence of the “energy-like” constant of motion, the system
evolution is appropriately described by a microcanonical ensemble. Therefore, the
stochasticity of this system is governed by the Liouville invariant measure
dρ ∝ δ (E(y a ) − ) dξdθdpξ dpθ ,
(2.47)
where δ (x) denotes the Dirac functional.
Since the particular value taken by the function ( = E(y a )) cannot influence the
stochastic property of the system and must be fixed by the initial conditions, then
we must integrate (in the functional sense) over all admissible form of . To do this,
it is convenient to introduce the natural variables (, ϕ) instead of (pξ , pθ ) by
pξ = p
pθ
cos ϕ ,
ξ2 − 1
p
= ξ 2 − 1 sin ϕ ,
(2.48)
where 0 6 ϕ < 2π. By integrating over all functional forms of , we remove the
Dirac delta functional, which leads in each point of space to the uniform normalized
invariant measure
1
.
(2.49)
8π 2
Since the space points are dynamically decoupled, the whole invariant measure of the
dµ(y a ) = dξdθdϕ
system corresponds to the infinite product of (2.49) through the space domain.
Let us now consider the following choice of coordinates on the 2-surface:
(1 + ξ)
~y = (y1 , y2 ) = p
(cos θ, sin θ) .
ξ2 − 1
(2.50)
On the basis of (2.50), the new line element and the anisotropy parameters read
ds2 =
4E 2 d~y 2
, y<1,
(1 − y 2 )2
Qa = [(~y 2 + Aa )2 + 1 − (Aa )2 ] , a = 1, 2, 3 ,
(2.51)
(2.52)
√ √
being A1a (− 3, 3, 0) and A2a (1, 1, −2). We introduce the Poincaré model of the
Lobachevsky plane in the form of the upper-half plane by using well-known Poincaré
variables
~η = 2
~y + ~b
− ~b ,
2
~
(~y + b)
43
(2.53)
Covariant Characterization of the Chaos in the Generic Cosmological Solution
where ~b denotes a point on the absolute (b2 = 1). In terms of these new variables,
the metric (2.50) takes the form
ds2 =
(d~η )2
,
(~η · ~b)2
(2.54)
and the available domain |y| ≤ 1 transforms into the half-plane (~η · ~b) ≥ 0, while the
absolute represents the line (~η · ~b) = 0. Now, if we write
2
~η = √
3
1 ~⊥
u+
b + v~b ,
2
v≥0,
(2.55)
where ~b = (0, 1), ~b⊥ = (1, 0), then it is easy to verify that, in term of these coordinates,
the anisotropy functions have the form
Q1 (u, v) = −u/d ,
Q2 (u, v) = (1 + u)/d ,
(2.56)
Q3 (u, v) = (u(u + 1) + v 2 )/d ,
d = u2 + u + 1 + v 2 .
This is a very suitable expression for the boundary; in fact geodesic on this half-plane
are semicircles having centers on the absolute (i.e. u(s) = A + R cos s, v(s) = R sin s
being (A, 0) the coordinate of the semicircle centre and R the corresponding radius)
and rays perpendicular to the absolute; the billiard is bounded by the geodesic triangle
u = 0, u = −1, and (u + 1/2)2 + v 2 = 1/4; the new domain is shown in Fig.(2.5).
The billiard has a finite measure, and its open region at infinity, together with the
two points on the absolute (0, 0) and (−1, 0), correspond to the three cuspids of the
potential in Fig.(2.4).
It is quite easy to show, that in the u, v plane, (2.49) becomes
dµ =
1 du dv dφ
π v 2 2π
(2.57)
It is worth stressing that, with the use of the invariant measure here introduced, and
the Artins theorem [Art65], the complete equivalence between the BKL piece-wise
description and the Misner-Chitré continuos one can be shown [KM97a].
The existence of the stationary probability distribution in ΓQ (2.57), outlines the
chaotic properties associated to the point-like billiard resulting from this analysis.
44
2.8 Role of the potential walls
3
2.5
v
2
1.5
1
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
u
Figure 2.5:
ΓQ (u, v) is the available portion of the configuration space in the Poincaré
upper half-plane. It is bounded by three geodesic u = 0, u = −1, and
(u + 1/2)2 + v 2 = 1/4, it has a finite measure µ = π (it is “dynamically”
bounded).
2.8 Role of the potential walls
Up to now, we have involved the role of the potential walls in order to cut a billiard
on the Lobachevsky plane. Here, we discuss a notion of Lyapunov exponents, which
include both the feature of the geodesic flow, and the structure of the bounding potential walls, and we arrive to show that even in this more complete framework this
system is a chaotic one. To this end, we will show that our system meets all the
hypothesis at the ground of the Wojtkowski Theorem (see [Woj85]):
Theorem: Let (τ, A) be a measurable cocycle with values in S F , i.e. A : X →
S F . Then the maximal Lyapunov exponent is positive almost everywhere.
Here (X, µ) is a probability space, τ : X → X a measure-preserving transformation, and A : X → S F a measurable mapping to a special set of the n × n real
matrices.
We can easily construct in the (u, v) variables the cocycle and verify that this theorem applies: we identify the transformation τ with the geodesic flow and the map
A with the matrix of the dynamics M defined starting from the tangent field to the
geodesics. The tangent field to the geodesic flow takes the explicit form:
tµ ≡ (−v, u − A) ,
45
(2.58)
Covariant Characterization of the Chaos in the Generic Cosmological Solution
and the associated matrix of the dynamics Mµν ≡ ∂µ tν is constant and orthonormal.
Both the geodesic flow and the matrix of the dynamics M satisfy the hypothesis of
the theorem.
The second condition is given by the construction of an invariant bundle of sectors
and by a “dispersing” profile of the boundary of the billiard (see [Woj85] for a detailed
discussion). This is possible because the following two properties hold:
1. because of the constant negative curvature of the surface, a one-parameter family of geodesics with negative curvature is invariant under the evolution;
2. the bounding potential walls consist of two straight lines and a semicircle of
negative curvature; the first ones do not affect the structure of the cones during
the bounces of the geodesic, while the latter one, as a dispersing profile, ensures
that, after reflection against it, the cones will evolve in themselves.
After this discussion, we can claim that the largest Lyapunov exponent has positive
sign almost everywhere.
2.9 On the covariance of chaos
Our dynamical scheme relies on the use of Misner-Chitré-like variables and therefore the covariance of the Lyapunov exponent is invariant with respect to space-time
coordinates; however it could be sensible of the choice of configurational variables,
and thus the result here obtained calls attention to be extended to any choice of the
configurational variables.
In this respect, we compare our result with the analysis presented in [Mot03], according to which, given a dynamical system of the form
dx/dt = F(x) ,
(2.59)
then the positiveness of the associated Lyapunov exponents are invariant under the
following diffeomorphism: y = φ(x, t), dτ = λ(x, t)dt, as soon as the four hypotheses
hold:
1. the system is autonomous
2. the relevant part of the phase space is bounded
46
2.9 On the covariance of chaos
3. the invariant measure is normalizable
4. the domain of the time parameter is infinite
To show that such a covariance criterion is here fulfilled, we observe that the variables
x can be identified with τ, ξ, θ, and the time variable with our curvilinear coordinate
s. On the other hand, the diffeomorphism above in its time-independent form can
match a phase-space coordinate transformation; then we underline also that:
1. in the considered asymptotic limit, our dynamical system is autonomous because
the Hamiltonian coincides with the constant of motion = E(y a ) and the
potential walls are fixed in time.
2. apart from sets of zero-measure which cannot be explored by the system [BKL70],
the phase-space ΓQ is a compact domain.
3. the system admits, in each space point, a normalized invariant measure over
the phase space.
4. the curvilinear coordinate s admits an infinite domain because τ ∈ (−∞, ∞)
(2.46).
Thus, on the base of [Mot03], we can claim that the Lyapunov exponent calculated
in (2.45) provides an appropriate chaos indicator only when the effects of the boundary are taken into account in agreement with our discussion. Furthermore, such an
indicator is covariant with respect to any configurational coordinate transformation
which preserves requirements 1-4. Strikingly, we have to stress that, if we adopt
Misner-like variables, the Lyapunov exponent (2.45) is no longer a good indicator;
in fact, the anisotropy parameters (2.16) in Misner-like variables depend not only on
β+ , β− , but also on Ω, and therefore, after the ADM reduction, on the curvilinear
coordinate s. As a consequence, the conditions 1 and 3 are no longer fullfilled for
this choice, because the potential walls move with time. However, for any generic
transformation of coordinates, which involves only ξ and θ, the chaoticity of the Mixmaster model is preserved. From the analysis developed in the previous sections,
the chaoticity of the inhomogeneous Mixmaster dynamics is ensured if ΓQ is a closed
domain2 , and follows from the independence of the Lyapunov exponent of the lapse
2
Indeed the compact phase space is constituted by ΓQ ⊗ Sφ1 , being Sφ1 the unit φ-circle
47
Covariant Characterization of the Chaos in the Generic Cosmological Solution
function and the shift vector. In fact, N and Nα are fixed (in turn) by choosing the
form of the quantities y a , and the latter can be generic functions subjected only to
the condition to be smooth enough.
The covariance of this picture is equivalent to the covariance of the inhomogeneous
Mixmaster chaos because it is well known [IM01, KM97a] that the obtained billiard
has stochastic properties (see also [Bar82a]), and this is mainly due to the negative curvature of the Lobachevsky plane that makes unstable the geodesic flow. The
potential walls, furthermore, have the role of replacing a given geodesic with a different one (whose tangent vector is related to the previous one by a reflection rule
[Bar82a]), and, as we showed, their structure influences the chaotic properties of the
system dynamics. Eq.(2.45) provides the form of the Lyapunov exponent in the whole
space domain, but we stress how its value depends on the choice of Misner-Chitré like
variables; the independence of this scheme on the shift vector is ensured by the asymptotic behaviour of the potential term, but, to get as constant of motion, allowing
the Jacobi metric representation, Misner-Chitré like variables are needed.
2.10 Concluding remarks
The main issue of the present chapter consists in the proof that the chaotic behaviour,
singled out by a generic inhomogeneous model near a singularity, has a covariant
nature. This result has been obtained by a “gauge” independent ADM reduction of
the dynamics to the physical degrees of freedom, which, for the Universe, correspond
to the anisotropy degrees, i.e., to the functions ξ and θ.
We describe the evolution as independent of the lapse function and the shift vector
form by adopting the variables y a as the new spatial coordinates. However their
degrees of freedom do not disappear from the problem because they are transferred to
the SO(3) matrices Oba which acquire a dependence on time in the new variables; such
a dependence is then eliminated from the dynamics by solving the super-momentum
√ (3)
constraint and using some implications deriving by the approximation of h R as
an infinite potential wall.
The potential behaviour (2.20) is crucial for the existence of an "energy-like" constant
of motion ≡ E(y a ), and therefore is at the basis of the chaos description.
From this point of view, the approximation is naturally induced for D → 0, due to
the potential structure; the only assumption required is that the functions y a ’s and
48
2.10 Concluding remarks
Oba (y c ) be smooth, in the sense that their presence does not affect the asymptotic
behaviour of the potential term. This restriction is natural (having a good degree of
generality) because the smoothness of the functions y a is ensured by the smoothness
of the lapse function and the shift vector (see (2.18)-(2.19)), i.e., by the choice of a
regular reference frame. The initial smoothness of the matrices Oba (when are taken
on the spatial coordinates xa ) is preserved by the coordinate transformation.
Concluding, the cosmological meaning of this concept corresponds to independent
"horizons"; neglecting in (2.26) the potential term with respect to the value of ”” is
equivalent to requiring that the scale of the inhomogeneities be super-horizon sized
(see [BKL82, Kir93b, Mon95]).
49
3 Semi-classical and Quantum
Properties of the Inhomogeneous
Mixmaster Model
In the previous chapters, we have discussed in details the classical aspects of the inhomogeneous Mixmaster model in vacuum. However, this classical description is in
conflict with the requirement of a quantum behaviour of the Universe through the
Planck era; in fact, there are reliable indications [KM97b] that the Mixmaster dynamics overlaps the quantum Universe evolution, requiring an appropriate analysis
of the transition between these two different regimes.
The dynamics of the very early Universe corresponds, indeed, to a very peculiar situation with respect to the link existing between the classical and quantum regimes,
because the expansion of the Universe is the crucial phenomenon which maps into
each other these two stages of the evolution. As shown in [KM97b], the appearance of
a classical background takes place essentially at the end of the Mixmaster phase, when
the anisotropy degrees of freedom can be treated as small perturbations. This result
indicates that the oscillatory regime takes place almost during the Planck era, and,
therefore, it is a problem of quantum dynamics. However, the end of the Mixmaster (and in principle the quantum to classical transition phase) is fixed by the initial
conditions on the system, and, in particular, it takes place when the cosmological
horizon reaches the inhomogeneity scale of the model. Therefore, the question of an
appropriate treatment for the semiclassical behaviour arises when the inhomogeneity
scale is so larger than the Planck scale that the horizon can approach it only in the
classical limit.
The chapter is organized as follows: we review the Wheeler-DeWitt equation
51
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
[DeW67] and some aspects of the quantum cosmology in Section 1. The analysis of
the Dirac scheme of quantization for gravity, and the “Multi-time” formalism [Ish92]
is presented in Section 2. The original contribution begins in Section 3, with a quite
complete analysis of the semi-classical properties of the inhomogeneous Mixmaster
model: we will illustrate the mechanical statistical description of the dynamics, and
solve the Hamilton-Jacobi functional for the model. In Section 4 the W KB limit to
the quantum dynamics is analyzed; the main point we want to stress right now is
that we are able to fix a proper ordering for the quantum operators by comparing the
dynamics in the classical regime and in the quasi-classical one.
The complete multi-time quantization is performed in Section 5. We derive the eigenfunctions and the spectrum of the Mixmaster model; boundary conditions are taken
into account in some approximations. Other results on the quantum dynamics, both
analytical and numerical, are also presented. The problem of non-locality of the
Hamilton operator is then discussed in Section 6, by analyzing the effects of the
square root on the time evolution; some comments on the “quantum chaos” are in
Section 7. A discussion on the role of the inhomogeneity can be found in Section 8,
together with some concluding remark.
In Appendix 3A, a brief mathematical review on the automorphic functions and on
the harmonic analysis on the Poincaré plane is presented.
The material presented here is published on the international journal Classical and
Quantum Gravity [BM07]. The publication is appended at the end of this PhD Thesis
(Attachment 3).
3.1 Quantum Cosmology and the Wheeler-DeWitt
equation
How the Universe did begin is one of the fundamental cosmological question, and
in order to to answer one must treat the gravitational degrees of freedom and interactions quantum mechanically. A huge literature exists on this and related topics,
and very different points of view, like that of string theory [Pol98a, Pol98b] and loop
quantum gravity [AL04]. We limit our discussion here to the quantum cosmology
point of view.
52
3.1 Quantum Cosmology and the Wheeler-DeWitt equation
Several promising and not-unrelated approaches to quantizing the gravitational degrees of freedom fall under this general category. One direction that has been pursued,
is to derive, using canonical quantum procedure, the analogy of a Schroedinger equation (more accurately, a Klein-Gordon equation) for the wave function of the Universe,
a wave function governing both the matter fields and the space-time geometry. The
equation governing the wave function of the Universe is known as the Wheeler-DeWitt
equation [DeW67].
Another aspect of quantum cosmology is that of third quantization. Third quantized operators create and destroy complete Universes with second quantized fields.
While today a single-Universe approximation seems to be quite adeguate, the quantum effects of emitting or absorbing small Universes could play an important role
in determining the fundamental parameters that we can measure in our Universe,
i.e., the value of the cosmological constant, particle masses, and coupling constants
[Col88].
We will now develop the theory of the WDW equation.
The starting point is the Hamiltonian framework for gravity; these are the fundamental results we need (from Appendix 3A):
Z
H = d3 x π ij ∂t hij − N H − Ni H i ,
√ 3
1
√ (hik hjl + hil hjk − hij hkl ) π ij π kl − k h R ≡
2k h
√ 3
≡ k −1 Gijkl π ij π kl − k h R ,
(3.1)
H=
H i = −2kπ;jij ,
(3.2)
(3.3)
where Gijkl is often referred to as the super-metric.
Identifying the Hamiltonian constraint H = 0 as the zero-energy Schroedinger equation: H[πij , hij ]Ψ[hij ] = 0, where the state vector Ψ is the wave function of the
Universe, is the first step towards quantization.
The second step is the canonical quantization procedure in which the momenta are
replaced by derivatives of their corresponding coordinates:
δ
π ij → −ik 3/2
.
δhij
(3.4)
Following the canonical procedure, we obtain the Wheeler-DeWitt equation [DeW67]
√ 3
δ δ
2
k Gijkl
+ k h R Ψ[hij ] = 0 .
(3.5)
δhij δhkl
53
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
The generalization of the Wheeler-DeWitt equation with cosmological constant Λ and
matter field (denoted generically by φ) is straightforward:
√ 3
δ δ
2
k Gijkl
+ k h( R − 2Λ) − T Ψ[hij , φ] = 0 ,
δhij δhkl
(3.6)
where T = T00 (φ, −i∂/∂φ), and Tµν is the stress-energy tensor of the matter fields.
There are many subtle questions about this technique, first of all the question and
the effect of the factor-ordering of quantum operators associated with π ij hij terms.
For many choices of the factor-ordering, the effect can be parametrized by a constant
a, and the corresponding Hamiltonian is obtained by the substitution
∂ a∂
2
−a
π → −q
q
.
∂q ∂q
(3.7)
The independence of time of the wave functions is a much more puzzling and interesting feature. In fact Ψ depends upon only the three-geometry hij and the matter
field content φ. Because of this, the interpretation of Ψ is a matter of intense debate
[Har88]. One possible interpretation is that Ψ[hij , φ] represents a measure of the
probabilistic correlation between hij and φ. In this way, one can imagine to use some
function of φ as a surrogate of the time variable (a similar point of view is that of
the relational time used in recent years in loop quantum cosmology [APS06]). The
question of time or even whether time has a role to play in quantum cosmology is an
issue far from being settled.
The only statement that most practitioners would agree with is that, when Ψ ∼
exp[iSE−H /~], classical behaviour ought to pertain.
It should also be emphasized that having an equation for the wave function of the
Universe no more resolves the issues of the quantum evolution of the Universe than the
Schroedinger equation for an electron resolves the issue of the quantum evolution of
the said electron. To be specific about the quantum evolution, one needs information
about the initial quantum state, and this is quite puzzling in quantum cosmology.
3.1.1 The wave function of the Universe
The wave function Ψ[hij , φ] is defined in an infinite-dimensional space of all possible
three-geometries and matter field configurations, known as superspace [MTW73]. The
number of degrees of freedom is infinite, and this makes the problem untreatable in
general; the only way to face the problem is to freeze out all but a small number
54
3.1 Quantum Cosmology and the Wheeler-DeWitt equation
of degrees of freedom. The resulting finite-dimensional superspace is known as minisuperspace. An example is the following: let’s take in consideration a FRW model with
line element ds2 = dt2 − R2 (t)dΩ23 , dΩ23 being the line element for the unit 3-sphere,
and R the radius of the Universe; in this model, the shift vector Ni vanishes, and
the equation follows from the canonical quantization of H = 0. Our mini-superspace
model is governed by the following Hamiltonian (here we consider also a cosmological
constant term)
G 2
Λ
3π
πR +
R(1 − R2 ) ,
(3.8)
3πR
4G
3
here πR is the momentum conjugate to R, and they are the only two degrees of freedom
H=−
for such a system. Following the canonical quantization prescription, πR → i∂/∂R,
the WDW HΨ(R) = 0 equation becomes
2
Λ 4
∂
9π 2
2
R − R
=0.
−
∂R2 4G2
3
(3.9)
The factor ordering here chosen is a = 0, and the equation resembles the onedimensional Schroedinger equation for a particle with zero total energy moving in
the potential
9π 2 R02
U (R) =
4G2
"
R
R0
2
−
R
R0
4 #
.
(3.10)
The potential U (R) admits a classicaly forbidden region 0 < R < R0 , and an allowed
one R > R0 ; R0 is a turning point. From the quantum-mechanical point of view,
there is a probability that the “particle” can tunnel through the barrier and emerge
at the classical turning point, beyond which it evolves classically.
Let’s now discuss the effects of the boundary condition on the Wheleer-DeWitt equation:
a) purely expanding solution [Vil88] i.e. iΨ−1 ∂Ψ/∂R > 0, corresponding to a purely
outgoing wave; this solution is referred as “creation of the Universe from nothing”,
where “nothing” corresponds to the initial state R = 0:
ΨV (R) = Ai[z(R)]Ai[z(R0 )] + iBi[z(R)]Bi[z(R0 )],
(3.11)
b) “no-boundary” boundary condition [HH83]:
ΨHH (R) = Ai[z(R)] .
(3.12)
The potential U (R) and these two solution are shown in Fig.(3.1).
By this simple application, the initial conditions have shown their importance in
55
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
Figure 3.1:
The potential U (R) and the two solution (3.11)-(3.12) are plotted. Solution
(3.11) is complex, the two dashed curves represents the real and the imaginary
part. Solution (3.12), instead, is a real function (from [KT90]).
the context of quantum cosmology and, in particular, in the WDW approach to the
problem.
3.2 The Canonical Quantization and the
Multi-Time formalism
We briefly discuss here another point of view on the possible way to a quantized
version of General Relativity, i.e., the multi-time formalism [Kuc72, Ish92].
The multi-time formalism is based on the idea that many gravitational degrees
of freedom appearing in the classical geometrodynamics have to be not quantized
because are not real physical ones; indeed we have to do with 10 ×∞3 variables, i.
e., the values of the functions N N i hij in each point of the hypersurface Σ3 , but it
is well-known that the gravitational field possesses only 4 × ∞3 physical degrees of
freedom (in fact the gravitational waves have, in each point of the space, only two
independent polarizations and satisfy second order equations).
The first step is therefore to extract the real canonical variables by the transformation
hij π ij
→ {ξ µ πµ } , {Hr P r } ,
56
µ = 0, 1, 2, 3 r = 1, 2
(3.13)
3.2 The Canonical Quantization and the Multi-Time formalism
Here Hr , P r are the four real degrees of freedom, while ξ µ πµ play the role of embedding
variables.
In terms of this new set of canonical variables, action (3.1) rewrites
Z
gφ
S =
πµ ∂t ξ µ + P r ∂t Hr − N H − N i Hi d3 xdt ,
(3.14)
M4
where H = H(ξ µ , πµ , Hr , P r ), and Hi = Hi (ξ µ , πµ , Hr , P r ).
Now we provide an ADM reduction of the dynamical problem by solving the Hamiltonian constraint for the momenta πµ
πµ + hµ (ξ µ , Hr , P r , φ, πφ ) = 0 .
(3.15)
Hence the above action takes the reduced form
Z
S=
{P r ∂t Hr − hµ ∂t ξ µ } d3 xdt .
(3.16)
M4
Finally, the lapse function and the shift vector are fixed by the Hamiltonian equations
lost with the ADM reduction, as soon as, the functions ∂t ξ µ are assigned. A choice
of particular relevance is to set ∂t ξ µ = δ0µ which leads to
Z
{P r ∂t Hr − h0 } d3 xdt .
S=
(3.17)
M4
The canonical quantization of the model follows by replacing all the Poisson brackets
with the corresponding commutators; if we assume that the states of the quantum
system are represented by a wave functional Ψ = Ψ(ξ µ , Hr ), then the evolution is
described by the equations
i~
δΨ
= ĥµ Ψ ,
δξ µ
(3.18)
where ĥµ are the operator version of the classical Hamiltonian densities.
We will now apply this formalism in the case of the Bianchi type IX Universe.
To set up the multi-time approach, we have to preliminarily perform an ADM reduction, as in (1.44); we find the relation
− pΩ ≡ hADM =
q
p2β+ + p2β− + V
.
Therefore action (1.43) rewrites as
Z n
o
S=
pβ+ β˙+ + pβ− β˙− − Ω̇hADM dt .
57
(3.19)
(3.20)
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
Thus we see how Ω plays the role of an embedding variable (indeed it is related to
the Universe volume), while β± are the real gravitational degrees of freedom (they
describe the Universe anisotropy).
By one of the Hamiltonian equation lost in the ADM reduction, we get
Ω̇ = −2N e3Ω pΩ = 2N e−3Ω hADM .
(3.21)
Hence, by setting Ω̇ = 1, we fix the lapse function as
e−3Ω
N=
.
2hADM
(3.22)
The quantum dynamics in the multi-time approach is summarized by the equation
q
(3.23)
i~∂Ω Ψ = −~2 (∂β2+ + ∂β2− + V )Ψ , Ψ = Ψ(Ω, β± ) .
We stress that, in this approach, the variable Ω, i.e., the volume of the Universe,
behaves as a “time”-coordinate, and therefore the quantum dynamics can not avoid
the Universe reaches the cosmological singularity (Ω → +∞).
3.3 Classical properties of the inhomogeneous
Mixmaster model
3.3.1 Hamiltonian formulation
The starting point of our analysis is the classical framework developed in the previous
chapter.
We have seen that the generic cosmological model in vacuum can be described in a
generic gauge by the following variational principle
Z
δSΓQ = δ dηd3 y(pξ ∂η ξ + pθ ∂η θ − ∂η τ ) = 0 .
(3.24)
Here ΓQ (Fig.(2.5)) is a reduced portion of the configuration space; it is a bi-dimensional
surface characterized by constant negative curvature. (3.24) implies that the generic
cosmological solution, in the neighbourhood of the singular point, is the sum of infinite
not-correlated point Universes, which move on a closed portion of the Lobachevsky
space.
We choose the gauge ∂η τ = 1, as soon as we are no more interested in a covariant
58
3.3 Classical properties of the inhomogeneous Mixmaster model
description in what follows.
Among the possible representations for ΓQ , we choose the so-called Poincaré model
in the complex upper half-plane [KM97a] that can be introduced with the following
coordinate transformation

1 + u + u2 + v 2


√
,
ξ =
3v
√

3(1 + 2u)

θ = − arctan
.
−1 + 2u + 2u2 + 2v 2
The line element for this 2-dimensional surface reads
du2 + dv 2
.
v2
In the (u, v) scheme, the ADM Hamiltonian assumes the expression
p
HADM ≡ = v p2u + p2v .
ds2 =
(3.25)
(3.26)
(3.27)
3.3.2 Hamilton-Jacobi approach
We shall now derive the Hamilton-Jacobi (HJ) [Arn78] equations for the system (3.27)
and we will solve them. The HJ prescribes to change the momenta into derivatives
of a functional S with respect to the configurational variable: implementing the HJ
technique, the Hamiltonian relation (3.27) leads to the functional differential equation
in ΓQ
δS
−
=v
δτ
s
δS
δu
2
+
δS
δv
2
.
(3.28)
Because of the infinite-walls structure of the potential, the Hamiltonian (3.27) is
“time-independent”; thus, we look for a solution of the type
Z
3
a
d y(y ) τ ,
S = S0 (u, v) −
(3.29)
where S0 satisfies the equation
2 = v 2
δS0
δu
2
+
δS0
δv
2 !
.
(3.30)
Equation (3.30) can be solved by separation of constant, i.e. by splitting it in a couple
of equations

δS0


=k,
δu
r
2
δS

 0 = − k2 .
δu
v2
59
(3.31)
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
This is the Hamilton-Jacobi function for the point Universe
p
+
S0 (u, v) = k(y a )u + 2 − k 2 (y a )v 2 − ln 2
!
p
2 − k 2 (y a )v 2
+ c(y a ) . (3.32)
2 v
In (3.32), k(y a ) is the separation constant, and c(y a ) is the integration constant.
The expression (3.29), together with (3.32) and the features of the potential wall
(2.56), summarizes the classical dynamics of a generic inhomogeneous Universe.
3.3.3 The statistical Mechanical description
We pointed out that, for the Mixmaster inhomogeneous dynamics, the spatial points
decouple while approaching the singularity, and that an energy-like constant of motion
appears. As the point Universe randomizes within a closed domain at fixed energy, we
can treat it from the statistical mechanical point of view, characterizing its dynamics
as a microcanonical ensemble.
The physical properties of a stationary ensemble are described by a distribution
function ρ = ρ(u, v, pu , pv ) [HR]; this function represents the probability of finding
the system in a precise infinitesimal interval of the phase-space (u, v, pu , pv ), and it
obeys the continuity equation
∂(u̇ρ) ∂(v̇ρ) ∂(p˙u ρ) ∂(p˙v ρ)
+
+
+
=0.
∂u
∂v
∂pu
∂pv
(3.33)
Here the dot denotes the time derivative and these are given by the Hamilton equations associated to (3.27), i.e.,
∂u
∂HADM
v2
=
= pu ,
∂τ
∂pu
∂v
∂HADM
v2
=
= pv ,
v̇ ≡
∂τ
∂pv
∂pu
∂HADM
=−
=0,
∂τ
∂u
∂pv
∂HADM
ṗv ≡
=−
=− .
∂τ
∂v
v
u̇ ≡
ṗu ≡
(3.34)
From (3.33) and (3.34), we obtain
v 2 pu ∂ρ v 2 pv ∂ρ ∂ρ
+
−
=0.
∂u
∂v
v ∂pv
(3.35)
We stress how the continuity equation provides an appropriate representation for the
ensemble associated to the Mixmaster only when we are sufficiently close to the initial
singularity, and, therefore, the infinite-potential-wall approximation works. Such a
60
3.3 Classical properties of the inhomogeneous Mixmaster model
model for the potential term, indeed, corresponds to deal with the energy-like constant of the motion, and fixes the microcanonical nature of the ensemble. From a
dynamical point of view, this picture arises naturally because the Universe volume
element vanishes monotonically (for non stationary correction to this scheme in the
Misner-Chitré-like variables, see [Mon01]).
We are interested in studying the distribution function in the (u, v) space, and, thus,
we will reduce the dependence on the momenta by integrating ρ(u, v, pu , pv ) in the
momentum space. By using the HJ solution (3.32), and assuming ρ to be a regular,
vanishing at infinity in the phase-space, limited function of its arguments, we can
integrate over (3.35) in the momentum space, and work out the following equation
R
for w̃ = w̃(u, v; k) = ρ(u, v, pu , pv )dpu dpv
s 2
∂ w̃
∂ w̃ E 2 − 2k 2 v 2
w̃
E
p
+
−1
+
=0.
(3.36)
2
∂u
kv
∂v
kv
E 2 − (kv)2
In this new equation, a constant k appears; this is due to the analytic expression
of the HJ solution, and expresses in some way the role of the initial conditions in
that solution. However, we are dealing with a distribution function that can not take
in considerations these initial conditions, and we have to eliminate it from the final
result.
Eq. (3.36) is a linear partial differential equation of the first order, and the solution
can be obtained with the aid of the integrals of motion, solving an “associated” dynamical problem (see, for example [BRS93] for further details on solving such kind of
equations). We obtain the following solution in term of a generic function g
q
E2
g u + v k2 v 2 − 1
√
.
w̃(u, v; k) =
v E 2 − k2v2
(3.37)
As we have said, the distribution function cannot contain the constant k, and the
final result is obtained after the integration over the constant k. Therefore we define
the reduced distribution w(u, v) as
Z
w(u, v) ≡
w̃(u, v; k)dk ,
(3.38)
A
where the integration is taken over the classical available domain for pu ≡ k
E E
A≡ − ,
.
v v
61
(3.39)
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
To obtain more precise information, we have to make some considerations on the
expected distribution of the model, and compare them with our result. We demonstrated in the second chapter, (2.57), that the measure associated to such a domain
is the Liouville measure ; it is easy to verify that this measure wmc (after integration
over the admissible values of φ) corresponds to the case g = const
Z
E
v
1
wmc (u, v) =
−E
v
kv 2
q
E2
k2 v 2
dk =
−1
π
.
v2
(3.40)
Summarizing, we have derived the generic expression of the distribution function
for this model, fixing its form for the microcanonical ensemble which, in view of
the energy-like constant of the motion , is the most appropriate to describe the
Mixmaster system when the picture is restricted to the configuration space.
3.4 Quasi-classical limit of the quantum regime
In this section, we will fix the proper operator-ordering [Kuc81] by comparing the
classical and the semi-classical dynamics; this will allow us to face the problem of the
full quantization of the Mixmaster model in the next section.
If we consider the WKB limit for ~ → 0, the coincidence between the classical
distribution function wmc (u, v) and the quasi-classical probability function r(u, v)
takes place for a precise choice of the operator-ordering only [IM06] (for a connected
topic, see [KM97b], and, for a general discussion, see [Gut90]).
We implement the canonical variables into operators
v̂ → v , û → u ,
∂
∂
∂
,
p̂u → −i~
, p̂τ → −i~
.
∂v
∂u
∂τ
The quantum dynamics for the state function Φ = Φ(u, v, τ ) obeys, in each point of
p̂v → −i~
the space, the Schrödinger equation
∂Φ
i~
= ĤADM Φ = ~
∂τ
s
−v 2
∂2
∂
− v 2−a
2
∂u
∂v
∂
v
Φ,
∂v
a
(3.41)
where we have adopted a generic operator-ordering for the position and momentum
operators parametrized by the constant a [KT90].
62
3.4 Quasi-classical limit of the quantum regime
In the above equation, the Hamiltonian operator has a non-local character as a consequence of the square-root function; in principle, this constitutes a subtle question
about the quantum implementation though. As we will demonstrate later, the presence of the square root does not produce non-local effects (see [Puz94]); thus, we will
2
assume that the operators ĤADM and ĤADM
have the same set of eigenfunctions with
eigenvalues E and E 2 , respectively1
If we take the following integral representation for the wave function Φ
Z ∞
Φ(u, v, τ ) =
Ψ(u, v, E)e−iEτ /~ dE ,
(3.42)
−∞
the eigenvalues problem reduces to
2
2 ∂
a ∂
2
2
2−a ∂
v
Ψ = E 2Ψ ,
Ĥ Ψ = ~ −v
−v
∂u2
∂v
∂v
(3.43)
where Ψ = Ψ(u, v, E) In order to study the WKB limit of equation (3.43), we separate
the wave function into its phase and modulus
p
Ψ(u, v, E) = r(u, v, E)eiσ(u,v,E)/~ .
(3.44)
In this scheme, the function r(u, v) represents the probability density, and the quasiclassical regime appears as we take the limit for ~ → 0; substituting (3.44) in (3.43),
and retaining only the lowest order in ~, we obtain the system
 " 2 #
2

∂σ
∂σ


+
= E2 ,
v 2
∂u
∂v
2
2

∂r
a
∂
σ
∂
σ
∂r
∂σ
∂σ
∂σ


+
+r
+ 2 + 2 =0.

∂u ∂u ∂v ∂v
v ∂v
∂v
∂u
(3.45)
In view of the HJ equation, and of Hamiltonian (3.27), we can identify the phase σ
to the functional S0 of the HJ approach.
Now we turn our attention to the equation for r = r(u, v). Taking (3.32) into account,
second of (3.45) reduces to
s 2
∂r
E
∂r a(E 2 − k 2 v 2 ) − E 2
√
r=0.
k
+
− k2
+
∂u
v
∂v
v2 E 2 − k2v2
(3.46)
Comparing (3.46) with (3.36), we easily see that they coincide (as expected) for a = 2
only.
1
The problems discussed in this respect by [KMV06] do not arise here because in the domain ΓQ
our ADM Hamiltonian has a positive sign (the potential vanishes asymptotically)
63
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
It is worth noting that this correspondence is expectable once a suitable choice for the
configurational variables is taken; however, here it is remarkable that it arises only
if the above operator-ordering is addressed. It is just in this result the importance
of this correspondence, whose request fixes a particular quantum dynamics for the
system.
Summarizing, we have demonstrated from our study that it is possible to get a WKB
correspondence between the quasi-classical regime and the ensemble dynamics in the
configuration space, and we provided the operator-ordering when quantizing the inhomogeneous Mixmaster model
v̂ 2 p̂2v
∂
→ −~
∂v
2
∂
v
∂v
2
.
(3.47)
3.5 Quantum properties of the inhomogeneous
Mixmaster model
3.5.1 Schroedinger quantization and the eigenfunctions of the
model
Our starting point is the point-like eigenvalue equation (3.43), which, together with
the boundary conditions, completely describes the quantum features of the model,
i.e.,
"
∂2
∂2
∂
v 2 2 + v 2 2 + 2v
+
∂u
∂v
∂v
2 #
E
Ψ(u, v, E) = 0 ,
~
Ψ(∂ΓQ ) = 0 .
(3.48)
In equation (3.48) we can recognize a well known operator: by redefining
Ψ(u, v, E) = ψ(u, v, E)/v, we can reduce (3.48) to the eigenvalue problem for the
Laplace-Beltrami operator in the Poincaré plane [Ter85]
2
∂2
∂
2
∇LB ψ(u, v, E) ≡ v
+
ψ(u, v, E) = Es ψ(u, v, E) ,
∂u2 ∂v 2
(3.49)
which is central in the harmonic analysis on symmetric spaces and has been widely
investigated in terms of its invariance under SL(2, C) (see Appendix 3A).
Its eigenstates and eigenvalues are described as
ψs (u, v) = av s + bv 1−s +
√ P
v n6=0 an Ks−1/2 (2π|n|v)e2πinu , a, b, an ∈ C ,
64
3.5 Quantum properties of the inhomogeneous Mixmaster model
∇LB ψs (u, v) = s(s − 1)ψs (u, v) ,
(3.50)
where Ks−1/2 (2πnv) are the modified Bessel functions of the third kind [AS65], and
s denotes the index of the eigenfunction. This is a continuous spectrum, and the
summation runs over every real value of n.
The eigenfunctions for our model, then, read
Ψ(u, v, E) = av s−1 + bv −s +
X
n6=0
an
Ks−1/2 (2π|n|v) 2πinu
√
e
,
v
(3.51)
with eigenvalues
E 2 = s(1 − s)~2 .
(3.52)
3.5.2 Boundary conditions and the energy spectrum
The spectrum and the explicit eigenfunctions are obtained by imposing the boundary
conditions (3.48); in particular we require Dirichlet boundary conditions, i.e., we
require that equation (3.51) vanish on the edges of the geodesic triangle of Fig.(2.5).
Since, from our analysis, no way arose to impose exact boundary conditions, we
approximate the domain with the simpler one in Fig.(3.2); the value for the horizontal
line y = 1/π provides the same measure for the exact and the approximate domain:
Z
Z
dudv
dudv
=
=π.
(3.53)
2
v2
ΓQ v
Approximate domain
The reason why we did not succeeded in imposing the exact boundary conditions
relies in the very sophisticated number theory which is linked to these functions;
furthermore, the circle, that bounds from below the domain, mixes solutions with
different indexes s.
We note that the Laplace-Belatrami operator and the exact boundary conditions are
invariant under the parity transformation u → −u − 1; however, it is clear that the
full symmetry group is C3v . C3v has two one-dimensional irreducible representations
and one two-dimensional representation. Those eigenstates transforming according
to one of the two dimensional representations are twofold degenerate, while the other
are non-degenerate. These non-degenerate eigenstates can be divided in two classes.
Either they satisfy Neumann boundary conditions, or Dirichlet ones instead. We will
be interested only in the latter case. The choice of the line v = 1/π approximates
65
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
Figure 3.2:
The approximate domain where we impose the boundary conditions.
The choice v = 1/π for the straight line preserves the measure µ = π.
symmetry lines of the original billiard and corresponds to one-dimensional irreducible
representations.
The conditions on the vertical lines u = 0, u = −1 require to disregard the first two
terms in (3.51) (a = b = 0); furthermore, we get the condition on the last term
X
e2πinu →
∞
X
sin(πnu) ,
n=1
n6=0
n being an integer. As soon as we restrict to only one of the two one-dimensional
representations, we get
X
n6=0
2πinu
e
→
∞
X
sin(2πnu) ,
n=1
while the condition on the horizontal line implies
X
an Ks−1/2 (2n) sin(2nπu) = 0 ,
∀u ∈ [−1, 0] ,
(3.54)
n>0
which is satisfied by requiring Ks−1/2 (2n) = 0 only. This last condition, and the form
of the spectrum (3.52), ensure the discreteness of the energy levels, because of the
following result: the zeros of the Bessel functions are a discrete set.
The functions Kν (x) are real and positive for real argument and real index, therefore
1
the index must be imaginary, i.e., s = + it. In this case, these functions have (only)
2
real zeros, and the corresponding eigenvalues turn out to be real and positive.
66
3.5 Quantum properties of the inhomogeneous Mixmaster model
n
8
6
4
2
2
4
6
8
10
12
t
Figure 3.3:
The intersections between the straight lines and the curves represents the roots
of the equation Kit (n) = 0, where K is the modified Bessel function.
n=2
n=4
n=6
n=8
4.425
7.016
9.434
11.768
6.333
9.353
12.076
14.655
7.947
11.313
14.283
17.059
9.410
16.264
16.263
19.212
10.774
17.742
18.096
21.203
Table 3.1:
The first roots for the first value of n are indicated.
(E/~)2 = t2 + 1/4 .
(3.55)
We remark that eigenfunctions (3.51) exponentially vanish as infinite values of v are
approached.
The conditions (3.54) cannot be solved analytically for all the values of n and t, and
the roots must be worked out numerically for each n. There are several results on
their distribution that allow us to find at least the first levels: a theorem on the zeros
of these functions states that Kiν (νx) = 0 ⇔ 0 < x < 1 (for a proof see [Pal57]);
furthermore, the energy levels (3.55) depends monotonically on the zeros. These two
observations allow us to search the lowest levels by solving equation (3.54) for the
first n. In Fig.(3.3) and Tab. (3.1), we plot the first roots, and, in Fig.(3.4) and in
67
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
2
E
1
= t2 +
~
4
Ε2 " t2 # 1 ! 4
140
19.831
40.357
120
49.474
100
63.405
80
87.729
89.250
60
116.329
40
128.234
20
138.739
0
146.080
Table 3.2:
Figure 3.4:
The first ten energy eigenvalues,
The lines represent in a typical spectral
ordered from the lowest one.
manner the first levels.
Tab.(3.2), we list the first ten "energy" levels2 .
Completeness of the spectrum.
The problem of completeness can be faced by studying first the sinus functions, and
then the Bessel ones.
sin(2πnu) is not a complete basis on the interval [−1, 0]; as soon as we request the wave
function to satisfy the symmetry of the problem, we obtain that this is a complete
basis.
Pick a value for n > 0; functions (3.51) have the form Φ(u, v) = sin(2πnu)g(v).
Substituting into the eigenfunction equation,v 2 (∂v2 + (2πn)2 )) g(v) = s(1 − s)g(v).
The solutions to this equation are exactly the Bessel functions .
This property, plus the condition on the line v = 1/π, form a Sturm-Liouville problem;
thus this set is complete.
2
for a detailed numerical investigation of the energy spectrum of the standard Laplace-Beltrami
operator, especially with respect to the high-energy levels, see[CGS91, GHSV91], where it is also
numerically analyzed the effects on the level spacing of deforming the circular boundary condition
towards the straight line
68
3.5 Quantum properties of the inhomogeneous Mixmaster model
Scalar product.
The found eigenfunctions define a function space where we can introduce a scalar
product. This is naturally induced by the metric of the Poincaré plane
Z
dxdy
(ψ, φ) = ψ(x, y)φ∗ (x, y) 2 ,
y
where
∗
(3.56)
denotes complex conjugation.
These last observations complete the discussion on the general properties of the
discrete spectrum of the inhomogeneous Mixmaster model.
3.5.3 The ground state
Let us describe the properties of the ground state level with a major accuracy, starting
from the result of its existence with a non-zero energy E0 , i.e., E02 = 19.831~2 .
In Fig.(3.5) we plot the wave function Ψgs in the (u, 0, v) plane, and in Fig.(3.6) the
corresponding probability distribution |Ψgs |2 , (the normalization constant is equal to
N = 739.466). The knowledge of the ground-state eigenfunction allows us to estimate
!gs (u,v)
|!gs|2 (u,v)
1.5
1
8
0.5
6
0
4
-0.5
2
-1
-1.5
0
1.25
1.25
1
-1
0.75
-0.75
-0.5
u
1
v
-1
0.5
-0.25
0.75
-0.75
-0.5
0
u
Figure 3.5:
-0.25
v
0.5
0
Figure 3.6:
The ground state wave function of the
The probability distribution in
Mixmaster model is sketched.
correspondence to the ground state of
the theory, according to the boundary
conditions.
the average values of the anisotropy variables u, v and the corresponding root mean
square, i.e.,
hui = −1/2 ,
hvi = 0.497 ,
69
(3.57)
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
∆u = 0.266 ,
∆v = 0.077 .
(3.58)
This result tells us that in the ground state the Universe is not exactly an isotropic
one, and it fluctuates around the line of the Misner plane β− = 0. However, we
observe that it remains localized in the center of the Misner space far from the corner
at v → ∞ (the other two equivalent corners were cut out from our domain by the
approximation we considered on the boundary conditions, but, because of the potential symmetry, they have to be unaccessible too). Thus, we can conclude that the
Universe, approaching the minimal energy configuration, conserves a certain degree of
anisotropy and lives in the region where the latter can be treated as a small correction
to the full isotropy. Such a behaviour is a consequence of the zero-point energy associated to the ground state which prevents the absence of oscillation modes. This feature
has been numerically derived, but it can be inferred on the basis of general considerations about the Hamiltonian structure. The Hamiltonian, indeed, contains a term
v 2ˆp2 which has positive definite spectrum and cannot admit vanishing eigenvalue.
v
3.5.4 Asymptotic expansions
To study the distribution of the highest energy levels, let us take into account the
asymptotic behavior of the zeros for the modified Bessel functions of the third kind.
We will discuss asymptotic regions of the plane (t, 0, n) in both the cases i) t n
and ii) t ' n 1.
i) For t n, the Bessel functions admit the following representation:
√
∞
X
(−1)k
1
2πe−tπ/2
p
Kit (n) = 2
[
sin
a
u
(
)+
2k
2k
2
(t − n2 )1/4
t
1
−
p
k=0
+ cos a
where a = π/4 −
√
∞
X
(−1)k
k=0
t2k+1
1
u2k+1 ( p
1 − p2
)] .
(3.59)
t2 − n2 + t arccosh(t/n), p ≡ n/t and uk are the following polyno-
mials


 u0 (t) = 1 ,
Z
(3.60)
1 2
1 1
2 0

(1 − 5t2 )uk (t)dt .
 uk+1 (t) = t (1 − t )uk (t) +
2
8 0
Retaining in the expression above only those terms of o( nt ), the zeros are fixed by the
following relation
π
1
π
sin( − t + t(log(2) − log(p))) −
cos( − t + t(log(2) − log(p))) = 0 .
4
12t
4
70
(3.61)
3.5 Quantum properties of the inhomogeneous Mixmaster model
In the limit n/t 1, (3.61) can be recast as follows
t log(t/n) = lπ ⇒ t =
lπ
,
productlog( lπ
)
n
(3.62)
where productlog(z) is a generalized function that gives the solution of the equation
z = wew , and, for real and positive domain, is a monotonic function of its argument.
In (3.62), l is an integer number that must be taken much greater than 1 in order to
verify the initial assumptions n/t 1.
ii) In case the difference between 2n and t is o(n1/3 ) (t, n 1), we can evaluate
the first zeros ks,ν by this relations [Bal67]
ks,ν ∼ ν +
∞
X
(−1)r sr (as )
r=0
ν −(2r−1)/3
2
,
(3.63)
where as is the s-th zero of Ai (2/z)1/3 , and si are some polynomials. From this
expansion it results that, at the lowest order,
t = 2n + 0.030n1/3 .
(3.64)
(3.64) provides the lowest zero (and therefore the energy) for a fixed value of n and
then the relation for the eigenvalues for high occupation numbers
2
E
∼ 4n2 + 0.12n4/3 .
~
(3.65)
In this region of the spectrum, the condition t ∼ 2n, reduces the 2 quantum numbers
to a single one, and the whole wave function is then assigned by n.
The analysis above shows that, as an effect of Dirichlet boundary condition and in
the limit of high occupation numbers, analytical expressions for the discrete structure
of the spectrum can be obtained. In fact, for large values of t, it was possible to
give analytical representations for the position of the zeros, but we emphasize that
the request of dealing with these approximations causes the loss of a large number
of levels, and prevents a complete discussion of the quantum chaos associated to the
model. However, the expressions above are of interest because allow us to compare
these results with the corresponding spectrum provided by Misner in his original
work [Mis69a]. Of course, such a comparison of the two results can take place only
on a qualitative level; in fact, between Misner (Ω, β+ , β− ) and Misner-Chitré-like
(τ, u, v) variables, a crucial difference exists, and it has to be recognized into the
71
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
correspondingly different behaviour of the potential walls. In the Misner scheme, the
domain available to the point Universe increases as Ω → ∞, and, therefore, we deal
with a non stationary infinite well; the Misner-Chitré-like variables allow to fix the
infinite potential walls into a time-independent configuration in the (u, 0, v) plane.
However, we can at least compare the behaviour of the energy eigenvalues with respect
to the occupation number n.
In his original treatment, Misner replaces, for fixed Ω values (indeed he makes use
of an adiabatic approximation, see [IM06]), the triangular well by a square of equal
measure and determines the energy spectrum in the form
EnM
π
=√
~
S
q
n2+ + n2− ≡
A
|n| ,
Ω
(3.66)
where S denotes the triangular well measure S = Ω2 /A2 , A being a numerical factor.
Above, n± denote the occupation numbers relative to β± respectively.
In our approach, for sufficiently large n, we get the dominant behaviors

lπ
i)
MC
,
En
productlog( lπ
)
n
∼
ii)2n + O(n2/3 ) .
~
(3.67)
Thus we see that, in case ii), apart from a numerical factor (which is different because
of the different approximation made on the real domains), the only significant difference relies on the term Ω−1 ; in the case i) the situation is a bit different because 2
quantum numbers explicitly remain, and we get a linear relation as far as we require
l ∝ n by a factor much greater than the unity (indeed the function productlog(lπ/n)
provides a smooth contributions in the considered region l n). The difference of
the factor Ω−1 with respect to Misner case can be easily accounted as soon as we
observe that the following relations hold:
β± = Ωb± (u, v) ,
(3.68)
where the functions b± can be calculated from the coordinates transformations

+ u2 + v 2

τ 1 + u√

Ω = −e
,



3v

+ 2u2 + 2v 2
τ −1 + 2u √
(3.69)
β
=
e
,
+


2
3v



 β− = −eτ 1 + 2u ,
2v
72
3.6 On the effects of the square-root on the quantum dynamics
but their form is not relevant in what follows. On the base of (3.69), the measure of
a domain D in the β± plane reads
Z
Z
2
dβ+ dβ− = Ω
|J(u, v)|dudv ,
(3.70)
D0
D
J being the Jacobian of the transformation associated to b± , while D0 is the image of
D onto the (u, 0, v) plane. As a consequence, we see that between a measure s in
the β± plane and a similar one (even not exactly its image), there is a difference for
a factor Ω2 which immediately provides an explanation for the difference in the two
energy spectra.
3.6 On the effects of the square-root on the
quantum dynamics
Now we will discuss if the presence of a non-local function, like the square-root of
a differential operator, can give rise to non-local phenomena. In particular, we will
study the case of a wavepacket which is non zero in a finite region of the domain and
far from the infinite, and we will let the true operator act on such a function. We
will show that the wave packet fails to run out to infinity in a finite time, i.e., the
probability exponentially goes to zero. We will develop the following analysis for the
approximate domain. Since the difference between the model and the exact domain
lies in the region around v = 1/π, it should have little or no effect on the behaviour
at infinity.
To that end, let Ψ(u, v) be a function which vanishes when v > M for some constant
M > 1. Then we can expand Ψ in a Fourier series over the u,
Ψ(u, v) =
∞
X
ψk (v) sin(2πku) .
(3.71)
k=0
Let ψ̂k (q) denote the Fourier transform of ψk (v) with respect to v; then we have that
the whole wave-function can be expressed as
∞ Z +∞
X
Ψ(u, v) =
sin(2πku)eiqv ψ̂k (q)dq .
k=0
(3.72)
−∞
If all the functions satisfy some regularity conditions, then we can invert the expansion
Z 0 Z +∞
ψ̂k (q) =
Ψ(u, v) sin(2πku)e−iqv dvdu .
(3.73)
−1
1/π
73
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
Let’s find an upper bound to the modulus of ψ̂k (q); as we choose the wave-function
Ψ to be different from zero only in a region v < M , then we have
Z 0Z ∞
|ψ̂k (q)| 6
|Ψ(u, v)||e−iqv || sin(2πku)|dvdu 6
−1
1/π
Z
0
Z
M
dvdu =
6sup Ψ
−1 1/π
1
= M−
sup Ψ 6 M sup Ψ .
π
(3.74)
Now let’s study the action of the square-root:
Z +∞ p
∞
X
p
2
2
v ∂u + ∂v Ψ(u, v) =
−q 2 − k 2 ψ̂k (q)eiqv dq .
sin(2πku)
(3.75)
−∞
k=0
To evaluate this integral, we pass from the real q to the complex q 0 , and we close the
integration path from above (for the convergence of the exponential). We define a
branch of the square root by a cut from ik to infinity and a cut from −ik to infinity.
The contour of integration can be deformed to wrap about the cut from ik to infinity,
thus giving
Z
∞
X
p
2
2
v ∂u + ∂v Ψ(u, v) =2
sin(2πku)
k=0
∞
X
=2i
i∞
0
eiq v v
p
−k 2 − q 2 ψ̂k (q 0 )dq 0 =
ik
∞
Z
e−ξv v
sin(2πku)
p
ξ 2 − k 2 ψ̂k (iξ)dξ .
(3.76)
k
k=0
Now we can estimate from above the value that the wave-function assumes under the
action of the square-root:
|v
p
∂u2 + ∂v2 Ψ(u, v)| 62 M sup Ψ
∞ Z
X
k=0
=2 M sup Ψ
∞
X
∞
e−ξv v
p
ξ 2 − k 2 dξ =
k
kK1 (k v) .
(3.77)
k=0
The Bessel functions of the third kind are positive monotonic decreasing functions
and go to zero faster than an exponential, thus the series absolutely converges to a
finite value for every v > 1/π.
We will calculate an upper bound on the probability of finding this new function
in the region v > M . For great values of v, we have
r
r
r
π
π
π −kv
3
1
−kv
K1 (k v) ∼ e
+
+O
<
e
.
3/2
2
2k v
2 8(vk)
v
2
74
(3.78)
3.7 The quantum chaos in the Mixmaster model
Now we can sum the series
∞ r
X
π
k=0
2
−kv
ke
r
=
π
ev
∼
2 (−1 + ev )2
r
π −v
e .
2
(3.79)
Finally, we can calculate the probability of finding the packet far away, P (v > M ):
Z 0Z ∞ p
dv du
|v ∂u2 + ∂v2 Ψ(u, v)|2 2 <
P (v > M ) =
v
−1 M
r
Z 0Z ∞
dv du
π
e−2v 2 =
(sup Ψ)2
< 4 M2
2
v
−1 M
r
−2M
π
e
2
2
= 4M
(sup Ψ)
+ Ei(−2M ) <
2
M
r
π
(sup Ψ)2 M e−2M
(3.80)
<4
2
R∞
(Ei(z) = − −z e−t /tdt being the exponential integral function). It can be seen that
the functions that are localized in a finite region v < M decay at infinity exponentially
fast when hit with the square-root operator. Thus, these functions fall off more rapidly
than any exponential.
We have succeeded in showing that the probability of finding a wavepacket which
starts out localized will decay at infinity as v → ∞. This allows us to conclude
that nevertheless the square root is a non-local function, non-local phenomena don’t
appear (like a wavepacket that starting from a localized zone fall out to infinity).
3.7 The quantum chaos in the Mixmaster model
In [Gra94], the structure of the high energy levels is also connected to the so-called
quantum chaos of the Mixmaster model, which was studied from the wave functional
point of view in [Fur86] and from the the path integral one in [Ber89] using Misner
variables. Our analysis implicitly contains the information about the quantum chaos
of the considered dynamics; in fact we can take a generic state of the system ξ(τ, u, v)
in the form
Z
ξ(τ, u, v) =
dt
X
√1 2
ct,n Ψn (τ, u, v)δ(Kit (2n))e−i 4 +t τ ,
(3.81)
n
and its evolution from a generic initial condition ξ0 (u, v) ≡ ξ(τ0 , u, v) at an initial
instant τ0 provides all the quantum properties of the system. The quantum chaos is
75
Semi-classical and Quantum Properties of the Inhomogeneous Mixmaster Model
recognized in [Fur86] by numerically integrating the Wheeler-DeWitt equation from a
gaussian like initial packet and outlining the appearance of a fractal structure in the
profile of the resulting wave function; in this approach (as the infinite potential walls
approximation works), the dynamics is provided by (3.81) and after the evolution of
the dynamics from an initial localized wave packet, the quantum chaos has to arise.
About the information on the quantum chaos emerging from the high-energy spectrum, we emphasize the following two important points: i) the this-stationary corrections due to the real potential term are expected to be simply small perturbation
to our result rapidly decaying as the singularity is approached and ii) the analytic
expressions we provided for large values of t cannot be used to fix the existence of the
quantum chaos because they explore limited regions of the plain (t, 0, n) only, but
they are very useful to clarify the morphology of the spectrum and its dependence on
two different quantum numbers.
3.8 The Inhomogeneous picture and conclusions
At the end of our analysis, we wish to bring the reader’s attention to some physical
aspects of the inhomogeneous Mixmaster.
First of all, the obtained dynamics regime is indeed a generic one. In a synchronous
reference (for which ∂t y a = 0), indeed, the inhomogeneous Mixmaster contains four
independent (physically) arbitrary functions of the spatial coordinates (y a (xi ) and
E(xi )), available for the Cauchy data on a non-singular hyper-surface.
Let us now come back to the full inhomogeneous problem, in order to understand
the structure of the quantum space-time near the cosmological singularity. Since the
spatial gradients of the configurational variables play no relevant dynamical role in the
asymptotic limit τ → ∞ (indeed the spatial curvature simply behaves as a potential
well), then the quantum evolution takes place independently in each space point, and
the total wave function of the Universe can be represented as follows
Ξ(τ, u, v) = Πxi ξxi (τ, u, v) ,
(3.82)
where the product is (heuristically) taken over all the points of the spatial hypersurface.
However, it is worth recalling that, in the spirit of the "long-wavelength approximation" adopted here, the physical meaning of a space point must be recovered on the
76
3.8 The Inhomogeneous picture and conclusions
notion of a cosmological horizon; in fact, we are dealing with regions over which the
inhomogeneity effects are negligible, and this statement corresponds to super-horizon
sized spatial gradients. On a classical point of view, this request is at the ground of
the BKL approximation and it is well confirmed on the statistical level (see [Kir93b]);
however, on a quantum level, it can acquire a precise meaning if we refer the dynamics
to a kind of lattice space-time in which the spatial gradients of the configurational
variables become real potential terms. In this respect, it is important to observe that
the geometry of the space-time is expected to acquire a discrete structure on the
Planck scale, and we believe that a regularization of this approach could arrive from
a "loop quantum gravity" treatment [Boj03].
Despite this local homogeneous framework of investigation, the appearance near the
singularity of high spatial gradients and of a space-time foam (like outlined in the
classical dynamics, see subsection 1.3.2 on the fragmentation process) can be easily
recognized in the above quantum picture too. In fact, the probability that in n different space points (horizons) the variables u and v take values within the same narrow
interval, decrease with n as pn , p being the probability in a single point; in fact,
these probabilities are all identical to each other and no interference phenomenon
takes place. From a physical point of view, this simple consideration indicates that a
smooth picture of the large scale Universe is forbidden on a probabilistic level, and
different causal regions are expected to be completely disconnected from each other
during their quantum evolution. Therefore, if we start with a strongly correlated
initial wave function Ξ0 (u, v) ≡ Ξ(τ0 , u, v), its evolution towards the singularity induces increasingly irregular distributions, approaching (3.82) in the asymptotic limit
τ → ∞.
The main result of this presentation can be recognized in the clear correspondence
established between the classical space-time foam and the quantum one. We have outlined how this link takes place naturally for a precise choice of the operator-ordering
only and how the "energy spectrum" is a discrete one, due to the billiard structure of
the point-like Hamiltonian. Finally, we fixed as a new feature the zero-point "energy"
for the ground state associated to the anisotropy degrees.
77
Appendix 3A
Appendix 3A
The Poincaré plane and the automorphic functions
In this section, the basic notions and the most important results on a particular
mathematics topic, the automorphic functions (for a wider and more than exhaustive
introduction, see [Ter85]), will be reviewed.
Automorphic functions are the eigenfunctions of the Laplace operator on the Poincaré
plane, also known as Laplace-Beltrami operator
2
∂
∂2
2
∇LB ψ(x, y) ≡ v
ψ(x, y) = λs ψ(x, y) .
+
∂x2 ∂y 2
(3A.1)
A quite important feature of these functions is that they are invariant under Moebius transformations. Let’s introduce the complex variable z = x + iy; than it is
easy to prove that, if ψ(z) is an automorphic function, than ψ(z 0 ) is an automorphic
function too, where z 0 = γz = (az + b)/(cz + d), a, b, c, d being integer numbers, and
γ is an element of the Moebius group. This allows one to study their properties only
in what is called a fundamental domain of the symmetry group, i.e., a portion of the
Poincaré upper half-plane that can be mapped via the element of the groups onto the
whole plane. In the Fig (3.7) it is indicated the fundamental domain of SL2 (Z). For
this reason they are also called “modular invariant” functions.
For these function can be defined an inner product, that naturally induced by the
Poincaré metric; that is
Z
dxdy
.
(3A.2)
y2
Let’s now discuss the eigenvalues of the Laplace-Beltrami operator. First, the
(ψ, φ) =
ψ(x, y)φ∗ (x, y)
spectrum has both a discrete and a continuous part. If the eigenvalue equation is
1
written in the form ∇LB ψ(x, y) = s(1 − s)ψ(x, y) with s = + it, t real, then, for
2
every value of t, there exists an eigenfunction Gs (x, y) called Eisenstein series, while
for certain discrete values of s there can also be a second eigenfunction, called a cusp
form, which we shall denote by vs (x, y).
The Eisenstein series can be defined by the series
X
Gs (z) =
(=(γz))s ,
(3A.3)
γ∈SL2 (Z)/Γ
where Γ stands for the translation subgroup of SL2 (Z) (i.e., the group of linear
fractional maps of the form z → z± n). Although the expression above converges
78
Appendix 3A
absolutely and uniformly in s, and makes it clear that Gs (z) is indeed a modular
invariant eigenfunction with eigenvalue s(1 − s), a different expression for Gs (z) turns
out to be more useful practically. It is
Λ(1 − s) 1−s
y +
Λ(s)
2 X s−1/2
√
+
|n|
σ1−2s (n) yKs−1/2 (2π|n|y)e2πinx ,
Λ(s) n6=0
X
σs (n) =
ds ,
Gs (x, y) =y s +
(3A.4)
(3A.5)
divisorsn
Λ(s) =π −s Γ(s)ζ(2s) ,
(3A.6)
where K is the Bessel function, and ζ is the Riemann zeta function. It should be
mentioned that Λ(s)Gs (z) = Λ(1 − s)G1−s (z), so there is indeed only one linearly
independent Eisenstein series for each eigenvalue. Also, the above expression shows
that, for large y, these functions grow as a power of y. Because of this growth, these
functions are not square integrable.
As mentioned earlier, for certain discrete values of s, there may appear other eigenfunctions, known as cusp forms. Unlike the Eisenstein series, for which one has the
explicit formulas given above, there is no explicit formula for the cusp forms. However,
if a cusp form is in the form of a series
vs (z) =
X
an Ks−1/2 (2π|n|y)e2πinx ,
(3A.7)
n6=0
√
then the coefficients an will satisfy the inequality an = O( n). The expansion above
also reveals that, as y → ∞, vs ∼ ∞. Although there is no exact formula for the
location of the eigenvalues s(1 − s), it is known that, for large X, the number of
linearly independent cusp forms with eigenvalues s(1 − s) < X goes as X/12 (this
last result can be seen via the Selberg trace formula [Gut90]). Finally, it is possible
to numerically compute the eigenvalues of the cusp forms.
There is an important last result we want to bring to attention: there exists an
expansion of any modular invariant function which is square integrable on the fundamental domain of SL2 (Z), known as the Roelcke-Selberg formula:
Z
X
1
Ψ(x, y) =
(f, vn ) +
(f, Gs ) Gs (x, y)ds ,
4πi
<(s)=1/2
n≥0
79
(3A.8)
Appendix 3A
3
2.5
2
1.5
1
0.5
-1
-0.5
0.5
1
Figure 3.7:
The dashed region (open above) represents the fundamental domain of the
modular group SL2 (Z) in the Poincaré upper half-plane. The boundaries are
three geodesics that are mapped one onto each other by a Möbius
transformation. By these symmetries, all the Poincaré plane can be covered
starting from this domain.
Z
ψ(x, y)φ∗ (x, y)
(ψ, φ) :=
fundamental domain
where (, ) is just the Hermitian L2 inner product.
80
dxdy
,
y2
(3A.9)
4 Multi-dimensional Mixmaster
and the role of matter fields
We turn our attention now to the chaotic properties that the generalizations of Mixmaster show in the higher-dimensional cases and in presence of matter. We will briefly
describe how chaos is a dimension-number sensitive phenomenon, and how even the
presence of spatial inhomogeneity or matter fields can change this picture.
A short review of homogeneous and inhomogeneous higher dimensional models in
vacuum is presented in Section 1. We will see that chaos disappears in the homogeneous case as the number of spatial dimensions is greater than three, and how it
characterizes the dynamics of inhomogeneous models till nine spatial dimensions.
Multi-dimensional matter-filled space-times are then discussed in Section 2, with particular attention to the dumping effects of a scalar field; then very little is said about
the more general case of the p-forms.
The original contribution constitutes the remaining of the chapter.
We consider a generic n-dimensional homogeneous model coupled to an Abelian vector field: through an ADM analysis (Section 3), a generalized Kasner behaviour is
worked out as first approximation to the full dynamics (Section 4).
The standard BKL analysis can be reproduced, and, as soon as the growing terms in
the potential are taken in consideration, the billiard representation is obtained (Section 5), together with both the map for the generalized Kasner exponents, both the
rotation law of the Kasner axes.
Concluding remarks and one appendix follow.
The results here described were published on the international Journal Classical
and Quantum Gravity during 2005 [BKM05]; the publication is appended at end of
this PhD thesis (Attachment 2).
81
Multi-dimensional Mixmaster and the role of matter fields
4.1 Mixmaster in higher-dimensional models
Homogeneous models
In a number of spatial-dimensions greater than three, homogeneous models lose their
chaotic dynamics, as can be seen in the four dimensional case already (see Appendix
4A for notations and more).
The question of chaos in higher dimensional cosmologies has been widely investigated, and many authors [BSS85, Fur86, Hal03, Hal02] showed that none of higherdimensional extensions of the Bianchi IX model possesses chaotic features. The main
difference relies in the finite number of oscillations characterising the dynamics near
the singularity.
Among the five-dimensional homogeneous space-times, G13 is the analogous of the
Bianchi type IX: the structure constants are the same for both the models.
Einstein equations can be written down, and they read
2
(b − c2 )2 − a4 d2 ,
= (a2 − c2 )2 − b4 d2 ,
= (b2 − a2 )2 − c4 d2 ,
2α,τ,τ =
2β,τ,τ
2γ,τ,τ
(4.1)
δ,τ,τ = 0 ,
α,τ,τ + β,τ,τ + γ,τ,τ + δ,τ,τ = 2α,τ β,τ + 2α,τ γ,τ +
= 2α,τ δ,τ + 2β,τ γ,τ + 2β,τ δ,τ + 2γ,τ δ,τ .
(4.2)
If we assume that the BKL approximation is valid, i.e., that the right-hand sides of
eqns (4.1) are negligible, then the asymptotic solution for τ → ∞ is five-dimensional
Kasner-like
ds2 = dt2 −
4
X
t2pi (dxi )2 ,
(4.3)
i=1
with the Kasner exponents pi that satisfy the generalized Kasner relations
4
X
i=1
pi =
4
X
p2i = 1 .
(4.4)
i=1
This regime can hold only until the BKL approximation works; however, as τ approaches the singularity, one or more of the terms may increase. Let’s assume p1 to
82
4.1 Mixmaster in higher-dimensional models
be the smallest index; then a = exp(α) is the greatest contribution and we can neglect
all the other terms, obtaining
1
α,τ,τ = − exp(4α + 2δ) ,
2
1
β,τ,τ = γ,τ,τ = exp(4α + 2δ) ,
2
δ,τ,τ = 0 .
(4.5)
As soon as the asymptotic limits for τ → ±∞ are considered, the following map
follows from solution to (4.5)
p1 + p4
,
1 + 2p1 + p4
p02 =
p2 + 2p1 + p4
,
1 + 2p1 + p4
p3 + 2p1 + p4
,
1 + 2p1 + p4
p04 =
p4
;
1 + 2p1 + p4
p01 = −
p03 =
abcd = Λ0 t ,
Λ0 = (1 + 2p1 + p4 ) .Λ
(4.6)
(4.7)
What makes this dynamical scheme different from the four dimensional case are the
conditions needed to undergo a transition: a quite simple analysis of the behaviour
of the potential terms in (4.1) shows that two of the four parameters must satisfy
1 − 3p21 − 3p22 − 2p1 p2 + 2p1 + 2p2 ≥ 0 ,
(4.8)
and one of the following
3p21 + p22 + p1 − p2 − p1 p2 < 0 ,
3p22 + p21 + p2 − p1 − p1 p2 < 0 ,
(4.9)
3p21 + p22 − 5p1 − 5p2 + 5p1 p2 + 2 < 0 ,
for a transition to occur. In Fig. (4.1), the existence of a region where (4.8) is satisfied
but none of (4.9) is shown. This way, the Universe undergoes a certain number of
transitions and Kasner epochs and eras; as soon as the Kasner indexes p1 , p2 take
values in the shaded region, then no more transitions take place, and the evolution
remains Kasner like until the singular point is reached.
Type G14 is quite similar to G13: for this model, the structure constants are the
same as Bianchi type VIII, and, under the same hypothesis, only a finite sequence of
epochs occurs.
83
1
Multi-dimensional Mixmaster and the role of matter fields
1
p2
0.8
0.6
0.4
0.2
p1
-0.4
-0.2
0.2
0.4
0.6
0.8
1
-0.2
-0.4
Figure 4.1:
The shaded region corresponds to all the couples (p1 , p2 ) that do not satisfy
(4.9): as soon as a couple (p1 , p2 ) takes value in that portion, then the BKL
mechanism breaks down, and the Universe experiences a last Kasner epoch till
the singular point
Inhomogeneous models.
Let us consider a (d + 1)-dimensional space-time (d ≥ 3), whose associated metric
tensor obeys to a dynamics described by a generalized Einstein-Hilbert action
(d+1)
Rik = 0 ,
(i, k = 0, 1, ..., d) ,
(4.10)
where the (d + 1)-dimensional Ricci tensor takes its natural form in terms of the
metric components gik (xl ).
In [DHH+ 86], it is shown that, within the framework of the Einstein theory, the
inhomogeneous Mixmaster behaviour finds a direct generalization in correspondence
to any value of d. It is also shown that, in correspondence to d > 9, the generalized
Kasner solution acquires a generality character (in the sense of the correct number of
arbitrary functions)
In a synchronous reference (described by usual coordinates (t, xγ )), the time-evolution
of the d-dimensional spatial metric tensor hαβ (t, xγ ) singles out, near the cosmological
singularity (t = 0), an iterative structure. Each single stage consists of intervals of
time (Kasner epochs) during which tensor hαβ takes the generalized Kasner form
γ
hαβ (t, x ) =
d
X
i=1
84
t2pi lαi lβi ,
(4.11)
4.1 Mixmaster in higher-dimensional models
where the Kasner index functions pi (xγ )’s have to satisfy the conditions
d
X
γ
pi (x ) =
i=1
d
X
p2i (xγ ) = 1 ,
(4.12)
i=1
and l1 (xγ ), ..., ld (xγ ) denote d linear independent forms li = mij dxj , whose components
are arbitrary functions of the spatial coordinates.
In each point of the space, the conditions (4.12) define a set of ordered indexes {pi }
(p1 ≤ p2 ≤ ... ≤ pd ) which, from a geometrical point of view, fixes one point in
Rd , lying on a connected portion of a (d − 2)-dimensional sphere. We note that the
validity of the conditions (4.12) requires p1 ≤ 0, pd−1 ≥ 0, where the equality takes
place only for the values p1 = ... = pd−1 = 0, pd = 1.
As shown in [DHS85, HGJSS87] (see also [Kir93a, KM95]), each single step of this
iterative solution results to be stable, in a given point of the space, if
lim t2 d R = 0 .
(4.13)
t→0
Eq. (4.13) is a sufficient condition that allows us to disregard the spatial gradients in
the Einstein equations.
An elementary computation indicates that the only “dangerous terms” in t2 d R contain
the powers t2αijk , where the exponents αijk are simply related to the Kasner exponents
through
αijk = 2pi +
X
pl ,
(i 6= j, i 6= k, j 6= k),
(i, j, k = 1, ..., d) .
(4.14)
l6=i,j,k
For generic functions l, all possible powers t2αijk appear in t2 d R.
This leaves two possibilities for t2 Rba to vanish as t → 0. Either the Kasner exponents
can be chosen in an open region of the Kasner sphere defined above (4.12), so as to
make αijk positive for all triples i, j, k. Or the conditions
∀(x1 , ..., xd ) : αijk (xγ ) > 0 (i 6= j, i 6= k, j 6= k) (i, j, k = 1, ..., d) ,
(4.15)
are in contradiction with (4.12), and one must impose “extra” conditions on the functions l and their derivatives. As we discussed in details in previous chapters, the
second possibility occurs when d = 4, since αijk is then given by 2pi , and one Kasner
exponent is always negative. This implies that (4.12) is a solution of the vacuum
85
Multi-dimensional Mixmaster and the role of matter fields
Einstein equations to leading order if and only if the vector l1 , associated with the
negative Kasner exponent p1 , obeys the “extra” condition
l1 · curl l1 = 0 ,
(4.16)
and this kills one arbitrary function.
It can be shown [DHS85, HGJSS87] that, for 3 ≤ d ≤ 9, at least the smallest of the
quantities (4.14), i.e. α1,d−1,d results to be always negative (excluding isolated points
{pi } in which it vanishes); for d ≥ 10, on the contrary, there exists an open region of
the (d − 2)-dimensional Kasner sphere where this same quantity takes positive values,
the so-called Kasner Stability Region (KSR).
As a consequence, for 3 ≤ d ≤ 9, the evolution of the system to the singularity consists
of an infinite number of Kasner epochs; instead for d ≥ 10, the existence of the KSR,
implies a profound modification in the asymptotic dynamics. In fact, the indications,
presented in [DHH+ 86, KM95] in favor of the “attractivity” of the KSR, imply that
in each space point (excluding sets of zero measure) a final stable Kasner-like regime
appears.
Finally we stress that, in correspondence to any value of d, the considered iterative
scheme contains the right number of (d + 1)(d − 2) physically arbitrary functions
of the spatial coordinates, required to specify generic initial conditions (on a nonsingular spacelike hypersurface). This calculation is very simple: we have d2 functions
from the d vectors l and d − 2 Kasner exponents; from these functions we have to
eliminate d, because of invariance under spatial reparametrizations, and d because of
the 0a Einstein equations. Therefore, this piecewise solution describes the asymptotic
evolution of a generic inhomogeneous multidimensional cosmological model.
4.2 Matter-filled spaces
The case of the scalar field
Let ds2 be the line element of the diagonal Bianchi type IX cosmology, that in Misner
variables (Ω, β+ , β− ) reads
ds2 = −dt2 + e2Ω (e2β )ij σ i σ j .
86
(4.17)
4.2 Matter-filled spaces
Furthermore, let φ(t) be a scalar field minimally coupled to gravity, and V(φ) a generic
potential. All the dynamics is contained in super-Hamiltonian H, that explicitly reads
2H = −p2Ω + p2+ + p2− + p2φ + V (Ω, β± ) + e6Ω V(φ) .
Here V (Ω, β± ) is proportional to the curvature: V (Ω, β± ) =
3
(4.18)
h 3 R, and pi are the
momenta conjugate to the generalized positions i, i = β± , Ω, φ.
The singularity occurs as Ω → −∞; this implies that the potential V(φ) is not
important during the asymptotic phase, unless it depends exponentially on Ω. We
will assume that it can be ignored, and we will retain only the curvature term.
As soon as we make the BKL assumptions on the curvature term, the solution to this
model can be easily found, and the metric tensor so obtained is Kasner-like, even if
the indexes pi ’s satisfy different properties
2
2
ds = dt −
3
X
t2pi (dxi )2 ,
(4.19)
i=1
p1 + p2 + p3 = 1 ,
p21 + p22 + p23 = 1 − q 2 ,
(4.20)
where q is related to the evolution of the scalar field: φ(t) = q ln(t) + φ0 .
Even if the difference is very small, the effect is really interesting: the insertion of
q in the second Kasner relation completely destroys the Mixmaster dynamics. This
can be easily realized as soon as the possible values for the indexes pi ’s are studied.
In order to perform this analysis, we introduce a parametric representation [BK73]
−u
,
1+u
+ u2
p
1+u
u−1
2
p2 =
u−
1− 1−β
,
1 + u + u2
2
p
1+u
u−1
2
p3 =
1−
1− 1−β
,
1 + u + u2
2
p1 =
β≡
√ (1 + u + u2 ) q
2
.
(u2 − 1)
(4.21)
(4.22)
The usual indexes (1.4) are obtained as q is taken equal to zero. We observe also that
the inversion property is conserved
1
1
p1
= p1 (u) , p2
= p3 (u) ,
u
u
87
1
p3
= p2 (u) ;
u
(4.23)
Multi-dimensional Mixmaster and the role of matter fields
thus the analysis can be restricted to region −1 6 u 6 1.
From (4.20) we obtain that the admissible values for u, q are to be taken in the region
defined by
β2 = 2
(1 + u + u2 )2 q 2
61.
(u2 − 1)2
(4.24)
As it was discussed in the first chapter, the BKL mechanism is based on the structure
1
of the curvature term: it grows like ∼ t2pi as t → 0, and because one of the three
Kasner exponents is negative, there is always a growing term. As a scalar field is
inserted in the dynamics, this last result does not hold anymore.
The region defined by the relations p1 ≥ 0, p2 ≥ 0, p3 ≥ 0 is shown in Fig (4.2): as
1
soon as the map takes value in the dashed region, then the Kasner era lasts until the
singularity is reached . The existence of this region inhibits the BKL dynamics from
q
p3
1
0.75
0.8
0.5
0.6
0.25
p2
0.4
-1
-0.8
-0.6
-0.4
-0.2
u
0.2
-0.25
-1
-0.5
-0.5
0.5
-0.2
-0.75
p1
1
u
Figure 4.2:
In the graphic on the left it is shown the existence of a region where all the pi ’s
are greater than zero. A Kasner era, characterized by a couple (u, v) in that
region, lasts until the singularity is reached.
On the right instead the different curves of the Kasner indexes are plotted for
several values of the parameter q.
lasting forever. The Universe undergoes changes until the Kasner indexes take values
out of this domain; as the pi ’s are mapped in the dashed region, the potential stops
to grow and the following Kasner era results to be stable.
A different point of view on the question of chaos is that of “consistent potential
(MCP)”, originally developed in [GcvacM93] and used in [BGS97, Ber99] to explain
how a minimally coupled classical scalar field can suppress Mixmaster oscillations in
the approaching to the singularity of generic cosmological spacetimes. It is possible
88
4.2 Matter-filled spaces
to retain the Mixmaster behaviour, according to the MCP, if V(φ) is such that
e6Ω V(φ) = a2 eαφ + b2 e−ζφ ,
(4.25)
where α, ζ > 0. Coupling between scalar field and gravitational degrees of freedom
in e6Ω V(φ) could lead to very complicated behaviour (potentials like (4.25) can arise
in string theory [LP96]).
The case of the p-forms
We complete this review on the effects of matter in higher-dimensional models with
a brief discussion about the insertion of the p-forms and dilatons in the gravitational
dynamics. This topic is quite wide, and a good review is, for example, [DHN03].
The inclusion of massless p-forms, in a generic multi-dimensional model [DH00a,
DH00b], can restore chaos when it is otherwise suppressed. In particular, even though
pure gravity is non-chaotic in 11 spacetime dimensions, the three-form of D = 11
supergravity renders the system chaotic (those p-forms are part of the low-energy
bosonic sector of superstring/M-theory models).
We will not enter in any detail, but we want to stress that the billiard description
we gave in the four dimensional case is quite general, and can be extended to higher
spacetime dimensions, with p-forms and dilatons [IMK94, DH01]. If d ≡ D − 1 is the
number of spatial dimensions, and if there are n dilatons, the billiard is a region of
the hyperbolic space Hd+n−1 , each dilaton being equivalent, in the Hamiltonian, to
the logarithm of a new scale factor.
The other ingredients, that enter billiard definition, are the walls that bound it, that
in these case can be of different types: symmetry walls related to the off-diagonal
components of the spatial metric, gravitational walls related to the spatial curvature,
and p-form walls (electric and magnetic); all these walls are hyperplanar, and the
billiard is a convex polyhedron with finitely many vertices, some of which are at
infinity.
89
Multi-dimensional Mixmaster and the role of matter fields
4.3 n-dimensional homogeneous models coupled to
an Abelian vector field
We will discuss this model in an ADM framework; let’s set down the basic equations.
Let us consider a vector field Aµ = (ϕ, Aα ) , (α = 1, 2, . . . , n), and let’s adopt for the
metric the standard ADM representation generalized to n-dimensional manifold
ds2 = N 2 dt2 − hαβ (dxα + N α dt) dxβ + N β dt .
(4.26)
Lagrangian of this vector-tensorial system is the sum of the “electromagnetic” term
plus the gravitational part
L = LG + LEM ,
Z
√
LG = − dn xdt −g
1
LEM = − F µν Fµν ,
4
(4.27)
n+1
R,
(4.28)
µ, ν = 0, .., n ,
(4.29)
F µν = ∂µ Aν − ∂ν Aµ being the electromagnetic n-dimensional tensor.
A standard Legendre transformation yields the Hamiltonian of the model
Z
n
αβ ∂
α ∂
α
α
I = d xdt π
hαβ + π
Aα + ϕDα π − N H − N Hα .
∂t
∂t
(4.30)
Here,
1
H=√
h
πβα παβ −
1
1
(παα )2 + hαβ π α π β + h
n−1
2
Hα = −∇β παβ + π β Fαβ ,
1
Fαβ F αβ − R
4
,
(4.31)
(4.32)
denote, respectively, the super-Hamiltonian and super-momentum; Fαβ is the spatial
electromagnetic tensor, h ≡ det(hαβ ) is the determinant of the n-metric, R is the
n-scalar of curvature constructed by the metric hαβ , and Dα ≡ ∂α + Aα . π α and
π αβ are the momenta conjugate to the electromagnetic field and to the n-metric,
respectively; they result to be a vector and a tensorial density of weight 1/2, because
their expressions contain the square root of the determinant of the spatial metric
√
h αβ
αβ
(K − hαβ Tr(K)) ,
(4.33)
π =
N
√ h ∂Aβ αβ
α
α
β
π =
h − N Fβ
,
(4.34)
N
∂t
90
4.3 n-dimensional homogeneous models coupled to an Abelian vector field
(here K αβ denotes the n-dimensional extrinsic curvature tensor in the synchronous
frame). Variation with respect to the lapse function N yields the Super-Hamiltonian
constraint
H = 0,
(4.35)
while its variation with respect to ϕ provides the constraint ∂α π α = 0.
We will deal with a source-less Abelian vector field; it is then enough to consider only
the transverse (or Lorentz) components for Aα and π α . Therefore, we take the gauge
conditions ϕ = 0 and Dα π α = 0, and this will be enough to exclude the longitudinal
parts of the vector field from the action.
It is worth noting how, in the general case, i.e., either in presence of the sources, or
in the case of non-abelian vector fields, this simplification can no longer take place
in such explicit form, and the terms ϕ(∂α + Aα )π α must be considered in the action
principle.
4.3.1 Homogeneous cosmological models
As homogeneity is considered, the whole spatial dependence of the model can be integrated out from the action, and the dynamical variables become only time dependent.
In general, neglecting in the total Hamiltonian the spatial derivatives contained in
Fαβ and R, is equivalent to the generalized Kasner approximation.
When going over the homogeneous case, we choose the gauge N α = 0 and, within
the Kasner approximation, we get equations of motion for the vector field having the
form
∂
N
Aα = √ hαβ π α ,
∂t
h
∂ α
π = 0.
∂t
The field equations which describe the n-metric dynamics read as follows
∂
2N
1
γ
hαβ = √
παβ −
hαβ πγ ,
∂t
n−1
h
Eα =
(4.36)
(4.37)
(4.38)
∂ α
N
πβ = − √ π α πβ .
(4.39)
∂t
2 h
This dynamical scheme is completed by adding to the above Hamiltonian equations
the super-Hamiltonian constraint (4.35).
91
Multi-dimensional Mixmaster and the role of matter fields
4.4 Kasner-like behaviour
We propose a generalized Kasner-like treatment of the model by introducing a spatial
vielbein lα , and projecting all vectors and tensorial quantities along them
hαβ = δab lαa lβb ,
παβ = pab lαa lβb .
(4.40)
The vielbein can be chosen in such a way that the matrix pab is diagonal:
pab = diag (p1 , . . . , pn ).
A dual basis Lα can be defined by requiring
Lαa = hαβ lβa ,
Lαa lαb = δab ,
(4.41)
Lαa lβa = δβα .
Let’s project (4.38) along the Kasner vectors defined in (4.40) and (4.41); in this way
we get the following dynamical system:
!
∂
1 X
N
La la = √
pa −
pb ,
∂t
n−1 b
h
∂
∂
La lb + Lb la = 0,
a 6= b.
∂t
∂t
(4.42)
(4.43)
Here (La lb ) = Lαa lbα denotes the ordinary vector product, treating the vector components as Euclidean ones.
In close analogy with above, from (4.39), we find the additional equations
N
∂
pa = − √ λ2a ,
λa = (π α lαa ) ,
∂t
2 h
N λa λ b
∂
,
a 6= b.
Lb la = − √
∂t
2 h pa − pb
(4.44)
(4.45)
In particular, we see that (4.45) is anti-simmetric under permutation of the indexes;
thus it already contains (4.43).
By combining together both systems, we obtain (4.44) as the first independent equation; the equation for the Kasner vectors takes place in the form (for the sake of
simplicity, we neglect the vector index α)
!
(
)
∂
N
1 X
1 X λ a λb
la = √
pa −
pb la −
lb .
∂t
n−1 b
2 b6=a. pa − pb
h
92
(4.46)
4.4 Kasner-like behaviour
Kasner dynamics is characterized by an anisotropic evolution of the linear distances
along different directions; in order to unravel that behaviour, let us distinguish scale
functions q a in the vielbein
la = exp (q a /2) `a ,
(4.47)
La = exp (−q a /2) La .
(4.48)
The vectors `a will be the Kasner axes.
Using this new set of dynamical variables, (4.44) and (4.46) become
∂
N e2
a
pa = − √ λ
a exp (q ) ,
∂t
2 h
∂ a 2N
q =√
∂t
h
ea = (π α `a ) ,
λ
α
(4.49)
!
1 X
pa −
pb ,
n−1 b
(4.50)
ea λ
eb
N X λ
∂
`a = − √
exp q b `b =
∂t
2 h b6=a. pa − pb
e
X λa
∂
pb
∂t
e
b6=a. λb (pa − pb )
`b .
(4.51)
The last two equations are obtained by substituting variables (4.47) and then projecting (4.46) on the vectors Lβc ; we get (4.50) as soon as the index c is taken equal to
a, otherwise we get (4.51). We have to stress that the second equation is an approximated one: we have in fact ignored an higher-order term with respect to the Kasner
approximation.
The dynamical scheme is complete as the Hamiltonian constraint is considered
N X 2
1 X 2 1 X qa e 2
H=0= √
pa −
pa +
e λa .
(4.52)
n−1
2
h
ea be constants implies that vectors `a do not depend
Requiring the quantities λ
on time in turn; therefore no Kasner vectors rotation can take place. Such a situation corresponds exactly to the n-dimensional Kasner behavior [DHHS85, DHH+ 86].
Since the rotation of Kasner vectors is induced by an higher order term, for h → 0,
ea are near to be
with respect to the pure Kasner dynamics, we can assume that λ
constant, and, therefore, near the singularity, (4.49), (4.50) give the billiards on
(n − 1) −dimensional Lobachevsky space, exactly like in the 3-dimensional Mixmaster case.
By other words, in this scheme, the evolution is not simply Kasner like, but we get a
dynamical picture in which the point Universe moves according to a piecewise Kasner
93
Multi-dimensional Mixmaster and the role of matter fields
solution, as we will see later.
In order to analyze the time-evolution of Kasner vectors, let’s project them on the
time independent quantities π α , and split them in two parts
~`a = ~`ak + ~`a⊥ ,
e
~`ak = λa ~π ,
~
~π `a⊥ = 0 .
(4.53)
π2
Here, for the sake of simplicity, we use vector notation, which is useful when treating
the components of these vectors as Euclidean ones.
Hence, we can split (4.51) into the two independent components
∂
e
X ∂t pb λa
∂e
e ,
λ
λa =
eb (pa − pb ) b
∂t
λ
b6=a.
∂
e
X ∂t pb λa
∂~
~`b⊥ = Aa (t) ~`b⊥ .
`a⊥ =
b
e
∂t
b6=a. λb (pa − pb )
Matrix Aab does not depend on ~`a⊥ ; then the following formal solution holds
 t

Z

~`a⊥ (t) = T exp
Aab (t0 ) dt0 ~`b⊥ (t0 ) .


(4.54)
(4.55)
(4.56)
t0
The remaining equations (4.54), together with (4.49) and (4.50), provide a selfconsistent dynamical system.
We assume that is possible to neglect the contributions of the n-dimensional Ricci
scalar to the dynamics. This means that the limit in which all the terms exp (q a )
become of higher order has to be taken. The dynamics is then described by the
following simplified system
pa = const,
ea = const,
λ
~`a⊥ = const,
!
∂
2N
1 X
qa = √
pa −
pb ,
∂t
n−1 b
h
P 2
1 X 2 1 X qa e 2
pa −
pa +
e λa = 0 .
n−1
2
94
(4.57)
4.5 Billiard representation: the return map and the rotation of Kasner vectors
We choose the gauge N = 1. In the neighbourhood of the singular point, the solution
of (4.57) is Kasner-like

P

hαβ = a t2sa `aα `aβ ,
pa

,
sa = 1 − (n − 1) P
b pb
(4.58)
where the Kasner indexes sa satisfy the generalized Kasner-like identities
n
X
a=1
sa =
n
X
s2a = 1.
(4.59)
a=1
4.5 Billiard representation: the return map and the
rotation of Kasner vectors
We will now unravel the BKL mechanism in this n-dimensional homogeneous model.
Let’s take the Kasner indexes in the increasing order
s1 6 s2 6 . . . 6 sn .
(4.60)
This implies that s1 < 0 and sn ≥ sn−1 ≥ 0. The singular point appears as soon as
the limit for t → 0 is taken. Like in the standard four-dimensional Mixmaster model,
there exists an instant ttr from which we cannot neglect anymore all exp(qa ), but we
have to retain in (4.49)-(4.51) at least the greatest term, i.e., t2s1
The field equations (4.49) and (4.50) rewrite

∂e


λ1 = 0 ,


∂t



∂

ea

p1 λ


∂

∂t

ea =

λ
, a 6= 1 ,


(pa − p1 )
 ∂t
∂
pa = 0 , a 6= 1,


∂t


∂
N e2

1

√
λ
exp
q
,
p
=
−

1
1


∂t
2
h
!



X

∂
2N
1


pb .

 ∂t qa = √h pa − n − 1
b
(4.61)
e1 = const, while the second admits the solution
The first of equations (4.61) gives λ
ea (pa − p1 ) = const.
λ
95
(4.62)
Multi-dimensional Mixmaster and the role of matter fields
The remaining part of the dynamical system allows us to determine the return map
governing the replacements of Kasner epochs. This is obtained in the standard way,
i.e., imposing a Kasner-like evolution on the solution of system (4.61) and matching
the asymptotic behaviour. This procedure yields the following map

−s1


s01 =
,

2

1 + n−2
s1





s + 2 s

s0a = a n−2 1 ,
2
1 + n−2
s1


λ
e0 = λ
e1 ,

1




(n
−
1)
s
1

0
ea 1 − 2
e =λ

.
λ
a
(n − 2) sa + ns1
We definine the quantities ka = pa −
(4.63)
1 X
pb , and find for them the following
n−1
iteration law
k10 = −k1 ,
2
k1 ,
n−2
n
n
X
X
2
0
ki =
ki +
k1 .
n−2
i=1
i=1
ka0 = ka +
(4.64)
(4.65)
(4.66)
Our analysis is completed by investigating the rotation of Kasner vectors ~`a through
the epochs replacements; by (4.55) we get the following system

∂


 ~`1⊥ = 0 ,
∂t
ea
λ
 ∂ ~` = ∂ p
~` ,

1
 ∂t a⊥
e1 (pa − p1 ) 1⊥
∂t
λ
(4.67)
admitting the integral
e1 ~`a⊥ − λ
ea ~`1⊥ = const.
λ
(4.68)
Putting together (4.53), (4.63) and (4.67), we arrive to the final iteration law
~`0 = ~`a + σa ~`1 ,
a
σa =
ea
e0 − λ
ea
λ
(n − 1) s1
λ
a
= −2
.
e1
e1
(n − 2) sa + ns1 λ
λ
which completes our dynamical scheme.
96
(4.69)
4.6 Concluding remarks
Thus the homogeneous Universe here discussed approaches the initial singularity
being described by a metric tensor with oscillating scale factors and rotating Kasner
vectors. Passing from one Kasner epoch to another one, the negative Kasner index
s1 is exchanged between different directions (for istance ~`1 and ~`2 ) and, at the same
time, these directions rotate in the space according to the law (4.69). The presence
of a vector field is crucial because, independently of the considered model, it induces
a (dynamically [BM04]) closed domain on the configuration space.
In correspondence to these oscillations of the scale factors, the Kasner vectors ~`a rotate, and, at the lowest order in q a , the quantities σa remain constant along a Kasner
epoch; in this sense, the vanishing behaviour of the determinant of the spatial metric
tensor h approaching the singularity, does not affect significantly the rotation law
(4.69).
4.6 Concluding remarks
The analysis here presented shows how chaos is restored in multi-dimensional homogeneous spaces as soon as an Abelian vector field is coupled to gravity. Furthermore,
the presence of chaos is not a dimensional phenomenon, in this case.
This result is a consequence of the capability that a vector field has to generate a
billiard configuration in the asymptotic evolution; such a billiard-ball representation
of the Universe dynamics coincides with the BKL piecewise approach only in the
4-dimensional space-time, while, in higher dimensions, new features appear. In particular, the obtained oscillatory regime characterizes all the homogeneous models,
disregarding their potential term; furthermore, the map (4.63), acquires a direct dependence on the number of dimensions.
It has to be remarked once more that the validity of the Kasner approximation is
based on the possibility to neglect the n-dimensional Ricci scalar with respect to the
terms containing time derivatives. In the Kasner solution this picture holds along the
whole system evolution to the initial singularity. In the piecewise Kasner solution, on
the contrary, the n-dimensional Ricci tensor is negligible only for finite time intervals,
ending with a bounce against the potential terms arising from the spatial curvature.
The scheme here proposed relies on the choice of a synchronous (or Gaussian) reference frame, which corresponds to have N = 1 and N α = 0; however, the results
97
Multi-dimensional Mixmaster and the role of matter fields
obtained have to remain valid for other gauge choices in view of the general covariance.
A valuable issue of this Hamiltonian dynamics relies on fixing the rule of the Kasner
vectors rotation. (4.69) is relevant to connect the Cauchy problem with later stages
of the system dynamics. More precisely, rotation of the Kasner vectors is sensitive
to the boundary conditions on the “matter” fields, and has to be taken into account
when using the studied homogeneous models within a cosmological “picture”.
We infer that the results here presented can be extended to a generic inhomogeneous
cosmological model, in the same spirit as the Bianchi type VIII and IX model oscillatory regime is upgraded in four dimensions, as soon as the spatial gradients are taken
into account. Such an extension will reliably show how, in the presence of a vector
field, the generic cosmological solution is described by an oscillatory approach to the
Big-Bang in any number of dimensions.
98
Appendix 4A
Appendix 4A
Homogeneous 5-dimensional models
The Fee’s classification [Fee79] divides the five-dimensional homogeneous spaces in 15
types, called G0 − G14; it is based on the analysis of the four dimensional Lie groups
and it is reported below in Tab. (4.6) (the right- and left-invariant vector fields and
forms can be found in Fee’s master thesis):
Table 4.1: Four-dimensional Lie algebras
G0
All zero
G1
[x, z] = w
G2
[x, z] = w
[y, z] = w
G3
[x, z] = w
[y, z] = y
G4
[w, y] = w
[x, z] = x
G5
[w, y] = −x
[w, z] = w
[x, y] = w
G6
[w, z] = w
[x, z] = w + x
[y, z] = x + y
G7
[w, z] = w
[x, z] = w + x
[y, z] = py
G8
[w, z] = w
[x, z] = px
[y, z] = qy
G9
[w, z] = pw + x
[x, z] = −w + px
[y, z] = qy
G10 [w, z] = 2w
[x, y] = w
[x, z] = x
[y, z] = x + y
G11 [w, z] = (1 + p)w
[x, y] = w
[x, z] = x
[y, z] = py
G12 [w, z] = 2pw
[x, y] = w
[x, z] = px + y
[y, z] = −x + py
G13 [w, x] = y
[w, y] = −x
[x, y] = w
G14 [w, x] = y
[w, y] = −x
[x, y] = −w
[x, z] = w
The line element can be written using the Cartan basis of left-invariant forms, and
reads explicitly
ds2 = dt2 − gij ω i ω j ,
gij = gij (t) .
(4A.1)
1 i s
C st ω ∧ ω t , where the C ist are the
2
structure constants of the considered group. We will limit our discussion, without
The one-forms ω i obey the relation dω i =
loss of generality, to the diagonal metric case, i.e.,
gij = diag(a2 , b2 , c2 , d2 ) .
99
(4A.2)
Appendix 4A
The Einstein equations are easily obtained:
ä b̈ c̈ d¨
+ + + =0,
a b c d
(ȧbcd)˙
R11 =
+ S 11 = 0 ,
abcd
(aḃcd)˙
+ S 22 = 0 ,
R22 =
abcd
(abċd)˙
R33 =
+ S 33 = 0 ,
abcd
˙
(abcd)˙
R44 =
+ S 44 = 0 ,
abcd
ẋn ẋm
0
Rn =
−
C mmn = 0 ;
xn xm
R00 =
(4A.3)
(4A.4)
(4A.5)
(4A.6)
(4A.7)
(4A.8)
here xn (n = 1, 2, 3, 4) denotes the scale factors a, b, c, d. The S nn are functions of the
scale functions and of the structure constants.
The analysis developed for the Bianchi type IX model can be straightforwardly generalised to the 5 dimensional case, obtaining the following set of equations
α,τ,τ = −Λ2 S 11 ,
β,τ,τ = −Λ2 S 22 ,
γ,τ,τ = −Λ2 S 33 ,
(4A.9)
δ,τ,τ = −Λ2 S 44 ,
α,τ,τ + β,τ,τ + γ,τ,τ + δ,τ,τ =2α,τ β,τ + 2α,τ γ,τ +
+2α,τ δ,τ + 2β,τ γ,τ + 2β,τ δ,τ + 2γ,τ δ,τ ,
(4A.10)
and
Rn0 = 0 .
(4A.11)
These dynamical scheme (4A.9)-(4A.11) is valid for any of the 15 models as soon as
the corresponding S nn are taken into account.
Exact homogeneous 5- dimensional solutions
The simpler group to consider is the G0; for this model the functions S nn are all
equal to zero and the solution simply generalises to 4 spatial dimensions the Kasner
dynamics.
100
Appendix 4A
Equations (4A.9) for the G1 model become
− α,τ,τ = β,τ,τ = γ,τ,τ =
1
exp 4α + 2δ ,
2
δ,τ,τ = 0 .
(4A.12)
The solution is easily obtained, and reads
a2 = Aωsech(ωτ ) exp(−k1 τ ) ,
b2 = Bcosh(ωτ ) exp(k2 τ ) ,
c2 = Ccosh(ωτ ) exp(k3 τ ) ,
d2 = 1/A2 exp(2k1 τ ) ,
(4A.13)
where ω 2 = −2k12 + k1 k2 + k1 k3 + k2 k3 and A, B, C are constants; we observe that the
asymptotic behaviour of (4A.13) also is a generalised Kasner like.
Quite interesting is the G3 dynamics in that it does not admit four nonzero scale
b2
factors; the functions S nn are S 11 = − 4 4 = −S 44 ; S 22 = −(a2 b2 c4 d2 )−1 ; S 33 =
2a c
S 11 + S 22 ; its solution is the four dimensional Kasner behaviour:
a = Atp ,
with p = −
b = c = Btq ,
(4A.14)
d = 0,
(4A.15)
1
2
and q = .
3
3
G12 is the last model of the Fee’s classification we will discuss. It has a different
structure and no four-dimensional analogue, but it undergoes the same “fate” of an
asymptotic Kasner evolution after a first phase of BKL transitions.
101
Conclusions and Outlooks
The central object of my studies has been the inhomogeneous Mixmaster model. I’ve
been interested in both the classical and the quantum properties that this dynamical system exhibits in the neighbourhood of the initial singularity. To conclude this
PhD thesis, I want to underline once more the main results discussed, and illustrate
possible developments.
On the classical level, the most important achieved result is the proof that the
chaotic behaviour of the dynamics near the Big Bang can be characterized in a covariant way. The proof is based on the reduction of the dynamics to the physical
degrees of freedom, that was obtained solving the super-Hamiltonian and the supermomentum constraints without imposing any gauge condition on the lapse function
or on the shift vector. Then the dynamics of the generic cosmological solution was
reduced to the sum of infinite not-correlated identical dynamical system, one for each
point of the space. Finally, chaos was quantified by direct calculation of the Lyapunov
exponent, and the chaotic properties granted by the shape of the potential walls and
by the finite measure of the domain. I want to underline once more that the mathematical concept of “point Universe” corresponds to the much more physical concept
of cosmological horizon, and this is due to the hypothesis of local homogeneity at the
ground of the construction of the generic cosmological solution.
On the quantum level, instead, two are the results that I’d like to remember here.
First of all, the correct operator-ordering. Indeed, the choice of an ordering when
facing the quantization of a dynamical system is, most of the times, a very puzzling
question; in the scheme I discussed, the correct choice was not imposed ab initio, but
obtained requiring that the classical phase overlap the semi-classical one.
Then, the wide analysis of the quantum dynamics. I derived the eigenfunctions of the
model, and, even if in some approximations, I imposed Dirichlet boundary condition,
obtaining this way the energy spectrum for the quantized Mixmaster model; this
103
Conclusions and Outlooks
spectrum exhibits two very interesting properties, i.e., it is a discrete spectrum, and
the ground-state energy is greater than zero.
The last result I will discuss here concerns with the multi-dimensional generalization of the Mixmaster model. I was able to show that, in a generic n-dimensional
homogeneous model, chaos is restored as soon as an Abelian vector field is coupled to
the gravitational dynamics; the key result absolutely is the BKL-like map that characterizes the dynamics, and that links the Kasner exponents and the Kasner vectors
of two successive epochs. This map is valid for any model, and exhibits a very peculiar dependence on the number of dimensions; furthermore, it reduces to the standard
Mixmaster map in vacuum and in three spatial dimensions.
Starting from these results, there are many things that can still be done.
First of all, a complete characterization of the quantum chaos of the Mixmaster model:
starting from the eigenfunctions, it is possible to perform numerical simulations of the
evolution of a generic gaussian wave-packet, in order to verify if some fractal structure
arises in the shape of the wave, or not.
An interesting possibility is that of quantization a la Wigner: this is a quantization
scheme in the whole phase-space, and yields, if one succeeds, a “quasi distribution
function”. This would be very interesting in order to improve the analogy between
classical and semi-classical dynamics.
Another possible idea is that of the use Ashtekar connection variables in the billiard
framework. In Ashtekar description, the super-Hamiltonian constrain is a polynomial
one without any potential term, and this, in principle, means a billiard without any
wall. Always in this scheme, a complete discussion of the role of inhomogeneity would
be really interesting.
In the multi-dimensional case, there are at least two way to improve and generalize
my results. One is the inclusion of inhomogeneity, and the consequent discussion on
the generic cosmological solution coupled to an electromagnetic field. The other is
the generalization to a non-Abelian vector field, and this would be really interesting
because fundamental interactions are described by this kind of fields. Further developments are surely related to String theory and super-gravity, or to some Kaluza-Klein
cosmological model.
104
List of Figures
1.1
pi (u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Bianchi type IX potential term . . . . . . . . . . . . . . . . . . . . .
13
1.3
Bianchi type VIII potential term . . . . . . . . . . . . . . . . . . . .
15
1.4
The construction of the lapse function and of the shift vector . . . . .
26
2.1
Numerical simulations of a trajectory in the anisotropy plane . . . . .
31
2.2
Numerical evaluations of Lyapunov exponents for different choice of
the time variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.3
Fractal structure in Mixmaster dynamics . . . . . . . . . . . . . . . .
33
2.4
The reduced configurational space ΓQ (ξ, θ) . . . . . . . . . . . . . . .
39
2.5
The triangular domain ΓQ (u, v) in the Poincaré plane . . . . . . . . .
45
3.1
The Hartle-Hawking and the Vilkenin wave functions. . . . . . . . . .
56
3.2
The approximate domain for the quantum dynamics . . . . . . . . . .
66
3.3
Roots of the eq. Kit (n) = 0 . . . . . . . . . . . . . . . . . . . . . . .
67
3.4
The first energy levels of the Mixmaster model . . . . . . . . . . . . .
68
3.5
The wave function of the ground-state . . . . . . . . . . . . . . . . .
69
3.6
The probability distribution associated to the ground state . . . . . .
69
3.7
The fundamental domain of SL2 (Z) . . . . . . . . . . . . . . . . . . .
80
4.1
The couple (p1 , p2 ) that causes the end of the BKL mechanism. . . .
84
4.2
The region where all the pi ’s are greater than zero . . . . . . . . . . .
88
105
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Attachments
117
Attachment 1
119
Attachment 1
PHYSICAL REVIEW D, VOLUME 70, 103527
Frame independence of the inhomogeneous mixmaster chaos via Misner-Chitré-like variables
Riccardo Benini1,* and Giovanni Montani2,3,†
1
Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
2
Dipartimento di Fisica, Università di Roma ‘‘La Sapienza,’’ Piazza Aldo Moro, 5 00185 Rome, Italy
3
ICRA–International Center for Relativistic Astrophysics, c/o Dipartimento di Fisica (G9),
Università di Roma ‘‘La Sapienza,’’ Piazza Aldo Moro, 5 00185 Rome, Italy
(Received 2 March 2004; revised manuscript received 26 October 2004; published 30 November 2004)
We outline the covariant nature, with respect to the choice of a reference frame, of the chaos
characterizing the generic cosmological solution near the initial singularity, i.e., the so-called
inhomogeneous mixmaster model. Our analysis is based on a gauge independent Arnowitt-DeserMisner reduction of the dynamics to the physical degrees of freedom. The resulting picture shows how
the inhomogeneous mixmaster model is isomorphic point by point in space to a billiard on a
Lobachevsky plane. Indeed, the existence of an asymptotic (energylike) constant of the motion allows
one to construct the Jacobi metric associated with the geodesic flow and to calculate a nonzero
Lyapunov exponent in each space point. The chaos covariance emerges from the independence of our
scheme with respect to the form of the lapse function and the shift vector; the origin of this result relies
on the dynamical decoupling of the space points which takes place near the singularity, due to the
asymptotic approach of the potential term to infinite walls. At the ground of the obtained dynamical
scheme is the choice of Misner-Chitré-like variables which allows one to fix the billiard potential walls.
DOI: 10.1103/PhysRevD.70.103527
PACS numbers: 98.80.Jk, 04.20.Jb, 05.45.–a, 98.80.Bp
I. BASIC STATEMENTS
The homogeneous and isotropic Friedmann-LemaitreRobertson-Walker (FLRW) metric provides a valuable
framework to describe the history of the Universe up to
its very early stages of evolution. Indeed the very good
agreement between the light element abundances predicted for the primordial nucleosynthesis and the observed one allows one to extrapolate backward in time
for the FLRW dynamics up to 102 –103 s [1]; furthermore, recent observations of the cosmic microwave background radiation [2,3] suggest that an inflationary
scenario took place and that the Universe was homogeneous and isotropic on the horizon scale up to O1032 s.
In spite of such an experimental evidence in favor of
the FLRW model, there are mainly two well-grounded
reasons to believe that, when the Universe temperature
was above the grand unification scale [O1015 GeV]
and below the Planck mass [O1019 GeV], the Universe
was appropriately described by a generic inhomogeneous
cosmological model [4–6]; indeed we have to stress
that
(1) as shown in [7] the Universe in an expanding
picture is stable with respect to tensorial perturbations;
in fact the amplitude of a gravitational wave decays like
the inverse of the scale factor [8,9]. Therefore reversing
the expanding behavior into a collapsing one, the homogeneity and the isotropy of the Universe become unstable
with respect to wavelike perturbations. In particular in
[9] is shown how a Bianchi type IX model far from the
*Electronic address: [email protected]
†
Electronic address: [email protected]
1550-7998= 2004=70(10)=103527(7)$22.50
singularity can be represented (in terms of exact solution)
by a closed FLRW model plus small gravitational ripples;
instead, close to the singularity this model has a fully
developed anisotropy which results into a chaotic behavior. This issue is valuable because the Bianchi type IX
cosmology presents features extensible (point by point in
space) to the generic cosmological solution.
(2) It is commonly believed [1,10] that the actual
Universe came from a quantum regime which was fully
developed during the Planckian era and approached a
classical limit (see [11]) only in a later stage of evolution.
Since the early Universe contained more than a single
horizon (see also [12]), then during the Planckian era the
metric and topology quantum fluctuations had to take
place independently over two causally disconnected regions, and hence the survival of any global symmetry is
prevented. In view of this, we infer that the quantum
behavior of the early Universe has to be properly analyzed
only in terms of a generic inhomogeneous model. With
respect to these two points, it is worth noting that the
bridge between the generic evolution and the FLRW
dynamics is provided on the horizon scale just by an
inflationary scenario, as outlined in [13].
As is well known, the generic cosmological solution [4]
is characterized near the big bang by a chaotic evolution
which reduces the space-time to the structure of a foam
[5,6]. When approaching the cosmological singularity, the
space points dynamically decouple and a time evolution
resembling the so-called mixmaster dynamics of the
Bianchi VIII and IX models takes place independently
in each of them [14,15] (for further discussions on inhomogeneous cosmological models see [16,17]); from a
physical point of view, a space neighborhood is here
70 103527-1
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RICCARDO BENINI AND GIOVANNI MONTANI
PHYSICAL REVIEW D 70 103527
considered at a horizon size (for recent discussions on
mixmaster covariance see [18–20]).
In recent years, the chaoticity of the homogeneous
mixmaster model has been widely studied in the literature (see [18–23]) in view of understanding the features
of its covariant nature.
Two convincing arguments, appearing in [20,22], support the idea that the mixmaster chaos (described by the
invariant measure introduced in [23,24]) remains valid in
any system of coordinates.
The main issue of the present work is to show that the
property of space-time covariance can be extended to the
inhomogeneous mixmaster model. In Sec. II we provide a
gauge independent analysis (i.e., independent of the
choice of the lapse function as well as of the shift vector)
for the dynamics of the gravitational degrees of freedom.
In Sec. III we discuss the asymptotic behavior of the
potential term associated with the Ricci 3-scalar, showing
how it can be modeled in terms of a potential wall
surrounding a (dynamically) closed domain. In Sec. IV
in view of the existence of an energylike constant of
motion, we construct the Jacobi metric associated with
the resulting billiard on a two-dimensional Lobachevsky
plane point by point in space, and finally we calculate the
covariant nonzero Lyapunov exponent. The concluding
remarks presented in Sec. V are devoted to outline how
the approximation used for the potential term is completely self-consistent being dynamically induced. It is
worth noting how we show that for any choice of lapse
function and the shift vector, taking Misner-Chitré-like
variables, the system results to be chaotic; however, in
this sense, our analysis states the independence of the
chaos on the choice of the space-time coordinates, but the
questions of covariance remains open when using different configuration coordinates because the Lyapunov exponent is not a good indicator for all of them.
Oab
and
a SO3 matrix. By the metric tensor (2),
the action for the gravitational field is
Z
dtd3 xpa @t qa d @t yd NH N H ; (3)
S
3 <
where
2
d N dt dx N dtdx N dt;
(1)
where N and N denote (respectively) the lapse function
and the shift vector, (; 1; 2; 3) the 3-metric
tensor of the spatial hypersurfaces 3 for which t const, being
eqa ad Oab Odc @ yb @ yc ;
a; b; c; d; ; 1; 2; 3;
(4)
H c @ yc pa @ qa 2pa O1 ba @ Oab :
(5)
@t yd N @ yd ;
(6)
p X
X NP
N @ qb @t qb
pa
b
b
(7)
a
take place.
III. ADM REDUCTION OF THE DYNAMICS
We use the Hamiltonian constraints H H 0 for
the reduction of the dynamics to the physical degrees of
freedom; from (4) and (5), we note that the supermomentum constraints can be diagonalized and explicitly solved
by choosing the function ya as special coordinates, i.e.,
taking the transformation t, and ya ya t; x. In fact
starting by (4) and (5) we get the expression
b pa
A generic cosmological solution is represented by a
gravitational field having available all its degrees of freedom and, therefore, allowing one to specify a generic
Cauchy problem. In the Arnowitt-Deser-Misner (ADM)
formalism, the metric tensor corresponding to such a
generic model takes the form
2
2
1 X
1 X
H p
pa 2 pb 3 R ;
a
2 b
In (4) and (5) pa and d are the conjugate momenta of the
variables qa and yb respectively, and the 3R is the Ricci
3-scalar which plays the role of a potential term.
The ten independent components of a generic metric
tensor are represented by the three scale factors qa , the
3 degrees of freedom ya , the lapse function N, and the
three components of the shift vector N a ; it is worth
noting that, by the variation of the variables pa , a in
the action (3), the relations:
II. HAMILTONIAN FORMULATION
2
Oab x
(2)
@qa
@Oa
2pa O1 ca bc :
b
@y
@y
Furthermore, in the new coordinates we have
8 a
q t; x ! qa ; y;
>
>
>
>
>
0
>
< pa t; x ! pa ; y pa ; y=jJj;
b
@y @
@
@
>
> @t ! @t @yb @
>
>
>
>
: @ ! @yb @b ;
@x
@x @y
(8)
(9)
where jJj denotes the Jacobian of the transformation. The
first relation holds in general for all the scalar quantities,
while the second one for all the scalar densities; hence the
action (3) rewrites as
Z
S
dd3 ypa @ qa 2pa O1 ca @ Oac NH:
3 <
while qa qa x; t and yb yb x; t six scalar functions,
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(10)
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FRAME INDEPENDENCE OF THE INHOMOGENEOUS . . .
PHYSICAL REVIEW D 70 103527
IV. THE POTENTIAL WALL AND THE REDUCED
VARIATIONAL PRINCIPLE
5
4
The potential term appearing in the super-Hamiltonian
reads, in obvious notation, as
3
X
X
D
U 2 R 2a D2Qa DQb Qc [email protected]; @q2 ; y; ;
jJj
a
bc
2
(11)
where
D exp
X
qa ;
1
0
(12)
a
qa
Qa P b ;
q
1
X
~ c ^ ry
~ b 2 :
~ ac ry
Oab rO
FIG. 1.
(14)
kj
Assuming the functions ya t; x smooth enough (which
implies by (6) and (7) that the coordinates system is
smooth ‘‘itself ’’), then all the gradients appearing in the
potential U are regular. Indeed this notion of regularity is
not to be intended in absolute sense. In fact what really
matters here is not that the gradient increases but simply
that their behavior is not so strongly divergent to destroy
the billiard representation (see next paragraph). In [5] it
was shown that the spatial gradients increase logarithmically in the proper time along the billiard’s geodesic and
therefore result to be of higher order. Thus, as D ! 0 [25]
the spatial curvature 3R diverges and the cosmological
singularity appears; in this limit, the first term of the
potential U dominates all the remaining ones and can be
modeled by the potential wall
X
U Qa ;
(15)
a
being
x 1 if x > 0;
0
if x < 0:
-1
10
ξ
(13)
b
2a ϑ
(16)
Q &; '.
The way in which the anisotropy parameters Qa (13)
are rewritten is an important feature of these variables
since we easily get the %-independent expressions:
p
p
&2 1
1
Q1 cos' 3 sin';
3
3&
p
p
&2 1
1
(18)
Q2 cos' 3 sin';
3
3&
p
1 2 &2 1
Q3 cos':
3
3&
When expressed in terms of such variables the superHamiltonian constraint can be solved in the domain Q :
s
p2
(19)
p% * &2 1p2& 2 '
& 1
and the reduced action reads as
Z
SQ dd3 yp& @ & p' @ ' *@ % 0:
(20)
By (15) the Universe dynamics evolves independently in
each space point; the point Universe can move only
within a dynamically closed domain Q (see Fig. 1) (indeed the three corners at infinity are open but they correspond to a set of zero measure in the space of initial
conditions). Since in Q the potential U asymptotically
vanishes, near the singularity we have @pa [email protected] 0. Then
the term 2pa O1 ca @ Oac in (10) behaves as an exact time
derivative and can be ruled out of the variational principle. The ADM reduction is completed by introducing the
so-called Misner-Chitré-like variables [5,22,26,27] as
8
p
> ql e% &2 1cos' p3 sin' &;
>
>
<
p
p
(17)
qm e% &2 1cos' 3 sin' &;
>
>
>
: qn e% & 2p
2
& 1 cos':
By the asymptotic limit (15) and the Hamilton equations
associated with (20) it follows that * is a constant of
motion, i.e., d*=d @*[email protected] 0 ) * Eya .
V. THE JACOBI METRIC AND THE LYAPUNOV
EXPONENT
Being * a constant of motion, the term *@ % Eya @ % in (20) behaves as an exact time derivative;
hence the variational principle rewrites
Z
d3 yp& d& p' d' 0;
(21)
coupled with the constraint (19). This dynamical scheme
allows one to construct the Hamilton-Jacobi metric [28]
corresponding to the dynamical flow. Indeed for each
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RICCARDO BENINI AND GIOVANNI MONTANI
PHYSICAL REVIEW D 70 103527
point of the space it can be reproduced the same as the
analysis developed in [22] for the homogeneous mixmaster model; in particular, all the spatial gradients are
dumped and the space points dynamically decouple in
the asymptotic limit to the singularity. In fact, in each
space point,
R the system dynamics is replaced by a geodesic flow ds 0, with
d&2
ds2 E2 ya 2
(22)
&2 1d'2 ;
& 1
corresponding to the Jacobi line element. The Jacobi
metric is valid independently in each of such point (here
the space coordinates behave like external parameter)
since the evolution is spatially uncorrelated. The Ricci
scalar takes the value R 2=E2 ; hence such a metric
describes a two-dimensional Lobachevsky space. The role
of the potential wall (15) consists of cutting a closed
domain Q on such a negative curved surface. Thus,
summarizing, the system obtained is isomorphic to a
billiard on a Lobachevsky plane.
A precise information about the dynamical stability of
the geodesic flow associated with the line element (22)
arises by the calculation of the Lyapunov exponent. Let us
project the connecting vector Z- between two close geodesics on a Fermi basis fu- ; w- g, - 1; 2; the geodesic
vector u- is taken in the form
p
&2 1
d& d'
sin0s
;
u- cos0s; p ; (23)
ds ds
E
E &2 1
where s denotes the curvilinear coordinate, while 0s is
an angular variable (0 0 < 21) whose dynamics is
obtained by requiring that the geodesic equation is verified, i.e.,
&
d0s
p sin0s:
ds
E &2 1
(24)
The vector w- is determined by the property of the Fermi
basis to be orthonormal, and it reads explicitly
p
&2 1
cos0
w- sin0; p :
(25)
E
E &2 1
Projecting the connecting vector Z- over the Fermi basis
defined above
Z- Zu su- Zw sw- ;
(26)
from the geodesic deviation equation we get that the
dynamics is described by the following equations:
8 2
>
< d Z2u 0;
ds
(27)
2
>
: d Zw Zw :
ds2
E2
The solution for the system (27) is given by
A; B const;
(28)
The value of E given by the constraint (19) and involved
in the line element (22) is determined by the initial
conditions and cannot vanish. By the first of solutions
(28) no geodesic deviation takes place along the geodesic
vector (as expected); instead from the second solution we
get a nonzero Lyapunov exponent of the form
ya limsup
s!1
lnZ2w dZw =ds2 1
> 0: (29)
2s
Eya For the validity of this analysis we have to verify that in
the limit to the initial singularity the curvilinear coordinate s approaches infinity. Repeating the same procedure of [22], formulas (31) appearing there, in our
inhomogeneous case is replaced by
@s
Eya @ %;
@
(30)
i.e., [being fya a generic function of the space coordinates]
s Eya % fya ) lim s 1:
%!1
(31)
This ensures that the curvilinear coordinate s behaves in
the appropriate manner in the limit toward the singularity
(% ! 1).
Hence the chaoticity of the inhomogeneous mixmaster
dynamics is ensured by Q to be a closed domain [29],
and the covariance of such a description follows from the
independence of the Lyapunov exponent with respect to
the lapse function and the shift vector. In fact, in (6) and
(7) N and N are fixed (in turn) by choosing the form of
the quantities ya and the latter can be generic functions
subjected only to the condition to be smooth enough.
Equation (29) provides the form of the Lyapunov exponent in the whole space domain but we stress how its value
depends on the choice of Misner-Chitré-like variables.
The independence of our scheme on the shift vector is
ensured by the asymptotic behavior of the potential term,
but to get * as a constant of motion, allowing the Jacobi
metric representation, we need a Misner-Chitré-like
variable.
The covariance of our picture is equivalent to the
covariance of the inhomogeneous mixmaster chaos because it is well known [22,23] that the obtained billiard
has stochastic properties (see also [30]). In fact the negative curvature of the Lobachevsky plane makes the geodesic flow unstable; the potential walls have the role of
replacing a given geodesic with a different one (whose
tangent vector is related to the previous one by a reflection
rule [30]), and as we will show in Sec. VI their structure
will influence the chaotic properties of the system
dynamics.
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124
Zu As B;
Zw c1 es=E c2 es=E ; c1 ; c2 const:
Attachment 1
FRAME INDEPENDENCE OF THE INHOMOGENEOUS . . .
To better characterize the chaoticity of the obtained
billiard, we show that in each point of the space our
system admits an invariant measure which in the present
variable is uniform over the admissible phase space.
From the point of view of statistical mechanics, such a
system admits, point by point in space, an ‘‘energylike’’
constant of motion which corresponds to the kinetic part
of the ADM Hamiltonian * Eya . The point Universe,
randomizing within the closed domain Q , is represented
by a dynamics which allows for an ensemble representation. In view of the existence of the energylike constant of
motion, the system evolution is appropriately described
by a microcanonical ensemble. Therefore the stochasticity of this system is governed by the Liouville invariant
measure
d5 / Eya *d&d'dp& dp' ;
(32)
where x denotes the Dirac functional.
Since the particular value taken by the function * [* Eya ] cannot influence the stochastic property of the
system and must be fixed by the initial conditions, then
we must integrate (in functional sense) over all admissible forms of *. To do this it is convenient to introduce
the natural variables *; ’ in place of p& ; p' by
q
*
p& p cos’;
p' * &2 1 sin’; (33)
&2 1
where 0 ’ < 21. By integrating over all functional
forms of *, we remove the Dirac delta functional, which
leads in each point of space to the uniform normalized
invariant measure [31]
d-ya d&d'd’
1
:
812
(34)
The existence of the above stationary probability distribution in Q outlines the chaotic properties associated
with the pointlike billiard resulting from our analysis.
As we stressed, our dynamical scheme relies on the use
of Misner-Chitré-like variables and therefore the covariance of the Lyapunov exponent is invariant with respect
to space-time coordinates, but it could be sensitive to the
choice of configurational variables. Thus the result here
obtained calls attention to be extended to any choice of
the configurational variables. In this respect we compare
our result with the analysis presented in [19] according to
which, given a dynamical system of the form,
dx=dt Fx:
(35)
Then the positiveness of the associated Lyapunov exponents is invariant under the following diffeomorphism:
y 0x; t; d% x; tdt, as soon as the four requirements hold:
(1) the system is autonomous,
(2) the relevant part of the phase space is bounded,
(3) the invariant measure is normalizable,
PHYSICAL REVIEW D 70 103527
(4) the domain of the time parameter is infinite.
To show that such a covariance criterion is here fulfilled, we observe that the variables x can be identified
with %, &, ', and ’, and the time variable with our
curvilinear coordinate s. On the other hand the above
diffeomorphism relation in its time independent form
can match a phase-space coordinates transformation;
then we underline also that
(1) in the considered asymptotic limit our dynamical
system is autonomous because its Hamiltonian coincides with the constant of motion * Eya and
the potential walls are fixed in time.
(2) Apart from sets of zero measure which cannot be
explored by the system [14], the phase space Q is a
compact domain.
(3) As shown by the above analysis which leads to (34)
the system admits, in each space point, a normalized invariant measure over the phase space.
(4) We showed by (31) that the curvilinear coordinate
s admits an infinite domain because the variables
% 2 1; 1.
Thus, on the base of [19], we can claim that the
Lyapunov exponent calculated in (29) provides an appropriate chaos indicator only when the effects of the boundary are taken into account in agreement with our
discussion of the next section. Furthermore such an indicator is covariant with respect to any configurational
coordinate transformation which preserves the requirements 1– 4. Strikingly, we have to stress that if we adopt
Misner-like variables [15] the Lyapunov exponent (29) is
no longer a good indicator; in fact the anisotropy parameters (13) in Misner-like variables depend not only on
; , but also on , and therefore after the ADM
reduction on the curvilinear coordinate s. As a consequence, the conditions 1 and 3 are no longer fulfilled for
this choice because the potential walls move with time.
However for any generic transformation of coordinates
which involves only & and ', the chaoticity of the mixmaster model is preserved. We conclude this section by
observing that if in different configurational coordinates
the inhomogeneous mixmaster would not appear as chaotic, then the stochasticity has to be transferred to the
coordinate transformation which links them to the
Misner-Chitré-like variable.
VI. THE ROLE OF THE POTENTIAL WALLS
Now we involve the role of the potential walls in order
to cut a billiard on the Lobachevsky plane. Here we
discuss a notion of Lyapunov exponents which include
either the feature of the geodesic flow or the structure of
the bounding potential walls, and we arrive to show that
even in this more complete framework our system is a
chaotic one. To this end we are showing below that our
system meets all the hypothesis at the ground of the
Wojtkowski Theorem; see [32]. Let us consider the fol-
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RICCARDO BENINI AND GIOVANNI MONTANI
PHYSICAL REVIEW D 70 103527
3
lowing new choice of coordinates on the 2-surface:
1 &
y~ y1 ; y2 p cos'; sin':
&2 1
2.5
(36)
4E2 dy~ 2
;
ds2 1 y2 2
y < 1;
(37)
Qa y~ 2 Aa 2 1 Aa 2 ;
a 1; 2; 3 (38)
p
p
being A1a 3; 3; 0 and A2a 1; 1; 2
Since here we introduce the Poincaré model of the
Lobachevsky plane in the form of the upper-half plane,
this is reached by using the well-known Poincaré variables
y~ b~
~
~ 2
b;
(39)
~2
y~ b
where b~ denotes a point on the absolute (b2 1). In terms
of these new variables the metric (36) takes the form
d
~ 2
ds2 (40)
~2
~ b
and the available domain jyj 1 transforms into the half~ 0 while the absolute represents the line
plane ~ b
~ 0. Now if we write
~ b
2
1
~ p u b~? vb~ v 0;
(41)
2
3
where b~ 0; 1; b~? 1; 0, then it is easy to verify that
in terms of these coordinates the anisotropy functions
have the form
8
u
>
> Q1 u; v u2 u1v2 ;
<
1u
Q2 u; v u2 u1v2 ;
(42)
>
>
2
:
Q3 u; v uuu1v
:
2 u1v2
This is a very suitable expression for the boundary; in
fact geodesic on this half-plane are semicircles having
centers on the absolute [i.e., us A R coss, vs R sins being A; 0 the coordinate of the semicircle center
and R the corresponding radius] and rays being perpendicular to the absolute. The billiard is bounded by the
geodesic triangle u 0, u 1, and u 1=22 v2 1=4. The new domain is shown in Fig. 2.
The billiard has a finite measure, and its open region at
infinity together with the two points on the absolute 0; 0
and 1; 0 correspond to the three cuspids of the potential in Fig. 1. The tangent field to the geodesic flow takes
the explicit form:
t- v; u A
v
2
On the basis of (36) the new line element and the
anisotropy parameters read as
(43)
1
0.5
0
-2
-1.5
-1
-0.5
u
0
0.5
1
FIG. 2. Q u; v.
9
and the associated matrix of the dynamics M @- t9 is
constant and orthonormal.
Referring to the notation of [32] we can easily construct in this variables the cocycle of the dynamics and
verify that theorem 2.2 applies. Since the manifold is
connected, has finite volume, and the matrix of the cocycle is constant, we have only to find an invariant bundle
of sectors. This is possible because the following two
properties hold:
(1) because of the constant negative curvature of the
surface, a one-parameter family of geodesics with
negative curvature is invariant under the evolution,
(2) the bounding potential walls are constituted by
two straight lines and a semicircle of negative
curvature; the former ones do not affect the structure of the cones during the bounces of the geodesic, while the latter one, being a dispersing profile,
ensures that after reflection against it, the cones
will evolve in themselves.
After this discussion we can claim that the largest
Lyapunov exponent has a positive sign almost everywhere. The covariance of this result is ensured in view
of the discussion developed in the previous section based
on [19]. The analysis of this section completes our analysis about the covariant nature of the chaos associated to
the billiard representation of the mixmaster model.
VII. CONCLUDING REMARKS
The main issue of the present work consists of the proof
that the chaotic behavior singled out by a generic inhomogeneous model near a singularity has a covariant
nature. This result has been obtained by a ‘‘gauge’’ independent ADM reduction of the dynamics to the physical
degrees of freedom, which for the Universe corresponds
to the anisotropy degrees, i.e., to the functions & and '.
We describe the evolution as independent of the lapse
function and the shift-vector form by adopting the variables ya as the new spatial coordinates. However their
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1.5
Attachment 1
FRAME INDEPENDENCE OF THE INHOMOGENEOUS . . .
PHYSICAL REVIEW D 70 103527
a
degrees of freedom do not disappear from the problem
because they are transferred to the SO(3) matrices Oab
which acquire a dependence on time in the new variables.
Such a dependence is then eliminated from the dynamics
by solving the supermomentum constraint and using
some implications deriving by the approximation of
p
gR3 as an infinite potential wall. The potential behavior (15) is crucial for the existence of an energylike
constant of motion * Eya , and therefore is at the basis
of the chaos description. In this view, this approximation
is naturally induced for D ! 0, due to the potential
structure; the only assumption required is for the functions ya and Oab yc to be smooth ones, in the sense that
their presence does not affect the asymptotic behavior of
the potential term. This restriction is a natural one having
a good degree of generality, because the smoothness of
the functions y is ensured by the smoothness of the lapse
function and the shift vector [see (6) and (7)], i.e., by the
choice of a regular reference frame. The initial smoothness of the matrices Oab (when taken on the spatial coordinates xa ) is preserved by the coordinates
transformation. Concluding, the cosmological meaning
of this concept corresponds to deal with independent
‘‘horizons’’; the approximation neglecting in (19) the
potential term with respect to the value of ‘‘*’’ is equivalent to require the scale of the inhomogeneities to be
superhorizon sized (see [4 –6]).
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Hobill, A. Burd, and A. Coley (World Scientific,
Singapore, 1994).
G. P. Imponente and G. Montani, Phys. Rev. D 63, 103501
(2001).
A. A. Kirillov and G. Montani, Phys. Rev. D 56, 6225
(1997).
D. F. Chernoff and J. D. Barrow, Phys. Rev. Lett. 50, 134
(1983).
It is worth noting that the quantity D is proportional to
the determinant
of the 3-metric, being det P
jJj exp a qa jJjD, and controls its vanishing behavior
G. Montani, Classical Quantum Gravity 17, 2205 (2000).
D. M. Chitré, Ph.D. thesis, University of Maryland, 1972.
V. I. Arnold, Mathematical Methods of Classical
Mechanics (Springer-Verlag, Berlin, 1989).
Indeed the compact phase space is constituted by Q S10 , being S10 the unit 0 circle.
J. D. Barrow, Phys. Rep. 85, 1 (1982).
Since the space points are dynamically decoupled the
whole invariant measure of the system would correspond
to the infinite product of (34) through the space domain.
M. Wojtkowski, Ergod. Theory Dynam. Sys. 5, 145
(1985).
ACKNOWLEDGMENTS
We thank Carlangelo Liverani for his very valuable
comments on this subject.
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
103527-7
127
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INSTITUTE OF PHYSICS PUBLISHING
CLASSICAL AND QUANTUM GRAVITY
doi:10.1088/0264-9381/22/7/018
Class. Quantum Grav. 22 (2005) 1483–1491
Oscillatory regime in the multidimensional
homogeneous cosmological models induced by
a vector field
R Benini1, A A Kirillov2 and Giovanni Montani3
1 Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, via Irnerio 46,
40126 Bologna, Italy
2 Institute for Applied Mathematics and Cybernetics, 10 Ulyanova str, Nizhny Novgorod,
603005, Russia
3 Dipartimento di Fisica (G9) and International Center for Relativistic Astrophysics,
Università di Roma, I-00185 Roma, Italy
Received 23 November 2004
Published 17 March 2005
Online at stacks.iop.org/CQG/22/1483
Abstract
We show that in multidimensional gravity, vector fields completely determine
the structure and properties of singularity. It turns out that in the presence of
a vector field the oscillatory regime exists in all spatial dimensions and for
all homogeneous models. By analysing the Hamiltonian equations we derive
the Poincaré return map associated with the Kasner indexes and fix the rules
according to which the Kasner vectors rotate. In correspondence to a fourdimensional spacetime, the oscillatory regime here constructed overlaps the
usual Belinski–Khalatnikov–Liftshitz one.
PACS numbers: 04.20.Jb, 98.80.−k
1. Introduction
The wide interest attracted by the homogeneous cosmological models of the Bianchi
classification [1] relies over all in allowing for their anisotropic dynamics. Among them,
types VIII and IX stand because of their chaotic evolution towards the initial singularity [2].
The stochastic properties outlined by such models were widely studied in order to describe
their detailed features [3–6] and in recent years a new line of research has been developed
to establish if the chaos is covariant and survives in each system of spacetime coordinates
[7–11] (see also references therein). For approaches via Ashtekar variables which, providing
a non-negative Hamiltonian function, could have interesting applications in the analysis of
chaotic anisotropic cosmologies, see the formulation presented in [12].
The cosmological interest in the Bianchi type VIII and IX universe dynamics comes out
because they correspond to the maximum degree of generality allowed by the homogeneity
constraint; as a consequence it was shown [13–17] that the generic cosmological solution can
0264-9381/05/071483+09$30.00 © 2005 IOP Publishing Ltd Printed in the UK
131
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be described properly, near the big bang, in terms of the homogeneous chaotic dynamics as
referred to each cosmological horizon (recent interest in a special homogeneous model such
as the Taub one arose because, in [18], it was shown that it admits an accelerating stage in the
dynamics as the actual universe seems to do).
It is just the dynamical decoupling of the space points which takes place asymptotically
that allows such an inhomogeneous extension; indeed the increase in the spatial gradients
is controlled by the oscillatory behaviour of their time dependence, so that their effects
are negligible at horizon size [14, 15]. However, the correspondence existing between the
homogeneous dynamics and the generic inhomogeneous one holds only in four spacetime
dimensions. In fact, a generic cosmological inhomogeneous model remains characterized by
chaos near the big bang up to a ten-dimensional spacetime [19–23] while the homogeneous
models show a regular (chaos free) dynamics beyond four dimensions [24, 25].
Here we address a Hamiltonian point of view and show how the homogeneous models
(of each type) single out, near the singularity, an oscillatory regime in correspondence to
any number of dimensions, as soon as an electromagnetic field is included in the dynamics
(for a study of the dynamics induced by an electromagnetic field in the framework of the
quasi-isotropic solution, see [26]). In fact the presence of an electromagnetic field restores a
closed potential domain for the point-universe evolution; thus, even if the considered model
does not possess a sufficient numbers of potential walls to determine the oscillatory regime, its
asymptotic evolution is dominated by the matter field and randomizes within the corresponding
billiard (for a connected analysis which studies the disappearance of chaos when a scalar field
is included in the dynamics, though in the presence of a p-form, see [27]).
We stress how the electromagnetic billiard and its associated Poincaré return map coincide
with usual ones in four dimensions. However, for higher dimensional spaces they differ from
the corresponding vacuum dynamics; in particular, the map we derive outlines an interesting
dependence on the number of dimensions. A relevant outcome of our analysis consists in
fixing the law for the rotation of the n-independent oscillating directions.
The procedure followed here calls for attention to be extended to more general contexts
such as the studies on the appearance of chaos in superstring dynamics [28, 29]. More
precisely, in section 2 we review the Hamiltonian formalism for general relativity; in section 3
we derive the Kasner solution for homogeneous models. In section 4 we introduce the Kasner
parametrization in our dynamical scheme. Finally, in section 5, we derive the solution in the
Kasner approximation, and in section 6 we show how this dynamical system can be described
in terms of the return map for the Kasner indexes and the law for Kasner vector rotation. In
section 7 brief concluding remarks follow.
2. Hamiltonian formulation
We start by reviewing the dynamical framework of our analysis.
Let us consider a vector field Aµ = (ϕ, Aα )(α = 1, 2, . . . , n) and adopt for the metric
the standard ADM representation [30, 31]
ds 2 = N 2 dt 2 − gαβ (dx α + N α dt)(dx β + N β dt).
Then the action which describes the dynamics of the model takes the form
n
αβ ∂
α ∂
α
α
I = d x dt gαβ + π
Aα + ϕDα π − NH0 − N Hα ,
∂t
∂t
where
1
1 α 2 1
1
α + gαβ π α π β + g
Fαβ F αβ − R ,
H0 = √ αβ βα −
g
n−1
2
4
132
(1)
(2)
(3)
Attachment 2
Oscillatory regime in the multidimensional homogeneous cosmological models
1485
Hα = −∇β βα + π β Fαβ ,
(4)
denote respectively the super-Hamiltonian and super-momentum; here Fαβ ≡ ∂β Aα − ∂α Aβ
is the electromagnetic tensor, g ≡ det(gαβ ) is the determinant of the n-metric, R is the n-scalar
of curvature constructed by the metric gαβ and Dα ≡ ∂α + Aα .
π α and αβ are the conjugate momenta of the electromagnetic field and the n-metric
respectively; they result in a vector and a tensorial density respectively of weight 1/2 because
√
their expressions contain g, and are defined via the relations
√
−g αβ
αβ =
(5)
(K − g αβ Tr(K))
N
√
−g ∂Aβ αβ
α
,
(6)
πα =
g − N β Fβ
N
∂t
where K αβ denotes the extrinsic curvature in the synchronous frame.
When varying this action with respect to the lapse function N we obtain the superHamiltonian constraint
H0 = 0,
(7)
while its variation with respect to ϕ provides the constraint ∂α π = 0. Since we are dealing
with an Abelian vector field (i.e. corresponding to an Abelian group of symmetry as an
electromagnetic field does) whose sources (charged particles) are absent, it is enough to
consider only the transverse (or Lorentz) components for Aα and π α . Therefore, we take the
gauge conditions ϕ = 0 and Dα π α = 0 and this will be enough to exclude the longitudinal
parts of the vector field from the action.
It is worth noting how, in the general case, i.e. in the presence of the sources, or in the
case of non-Abelian vector fields, this simplification can no longer take place in such explicit
form and, therefore, we have to retain the term ϕ(∂α + Aα )π α in the action principle.
In what follows we consider the behaviour of homogeneous cosmological models in the
asymptotic limit towards the initial singularity.
α
3. Homogeneous cosmological models: basic equations
When considering the homogeneity constraint, the whole spatial dependence of the models
can be integrated out from the action and the dynamical variables become only time dependent.
In general neglecting the total Hamiltonian the spatial derivatives contained in Fαβ and R is
equivalent to the so-called (generalized) Kasner approximation.
When going over the homogeneous case, we choose the gauge N α = 0 and, within the
Kasner approximation, we get equations of motion for the vector field having the form
∂
N
(8)
Eα = Aα = √ gαβ π α ,
g
∂t
∂ α
π = 0.
∂t
The field equations which describe the n-metric dynamics read
2N
1
∂
gαβ = √ αβ −
gαβ γγ ,
∂t
g
n−1
N
∂ α
= − √ π α πβ .
∂t β
2 g
(9)
(10)
(11)
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R Benini et al
This dynamical scheme is completed by adding to the above Hamiltonian equations the superHamiltonian constraint (7).
4. Kasner parametrization
To develop the analysis below, it turns out very convenient to adopt the so-called Kasner
parametrization which is based on the metric and conjugate momentum decomposition along
the spatial n-bein:
gαβ = δab lαa lβb ,
αβ = pab lαa lβb ,
(12)
where the n-bein is chosen in such a way that the matrix pab = diag (p1 , . . . , pn )
(for a discussion of this Hamiltonian structure in correspondence to an inhomogeneous
multidimensional model see [32]); this diagonal form of the conjugate momenta is a
consequence of requiring the canonical nature of the adopted transformation. We also define
a dual basis Lαa = g αβ lβa , such that Lαa lαb = δab and Lαa lβa = δβα .
To face our goal we have to project (10) along the Kasner vectors defined in (12); in this
way we get the following dynamical system:
N
∂
1 La l a = √
pb ,
(13)
pa −
∂t
g
n−1 b
∂
∂
La lb + Lb la = 0,
∂t
∂t
a = b.
(14)
Here (La lb ) = Lαa lbα denotes the ordinary vector product, treating the vector components as
Euclidean ones.
In close analogy with the above, from equation (11), we find the additional equations
∂
N
pa = − √ λ2a ,
(15)
λa = π α lαa
∂t
2 g
N
λa λb
∂
Lb l a = − √
,
a = b.
(16)
∂t
2 g p a − pb
In particular, we see that (16) already contains (14).
By combining together both such systems, we obtain (15) as the first independent equation;
the equation for the Kasner vectors takes place in the form (for the sake of simplicity we neglect
the vector index α)




λa λb
N
1
1
∂
la = √
pb l a −
lb .
(17)
pa −
∂t
g
n−1
2
p a − pb 
b=a.
b
We want to put in evidence the oscillatory regime that the bein vectors possess; to better
analyse this dynamics let us distinguish scale functions in the following way:
la = exp(q a /2)
a .
(18)
La = exp(−q /2)La ,
(19)
a
a being the so-called Kasner vectors.
Thus instead of (15) and (17) we rewrite
N 2
∂
pa = − √ λ exp(q a ),
∂t
2 g a
λa = π α aα ,
134
(20)
Attachment 2
Oscillatory regime in the multidimensional homogeneous cosmological models
∂ a
2N
q =√
∂t
g
1 pb ,
pa −
n−1 b
1487
λa ∂t∂ pb
λa
λb
∂
N b
a = − √
b .
exp(q )
b =
∂t
2 g b=a pa − pb
λ (pa − pb )
b=a b
(21)
(22)
The last two equations are obtained by substituting variables (18) and then projecting (17)
β
on the vectors Lc ; when the index c is equal to a, we get (21), otherwise we get (22) (in which
we neglected a term vanishing in the exact Kasner-like solution, and therefore, in our analysis,
of higher order).
To this system we should also add the Hamiltonian constraint
N 2
1 2 1 qa 2
(23)
λa .
H0 = 0 = √
pa −
pa +
e g
n−1
2
To require that the quantities λa are constants implies that vectors a do not depend on
time in turn and therefore no Kasner vector rotation takes place (see section 6); such a situation
corresponds exactly to the n-dimensional Kasner behaviour [20, 21, 33]. Since the rotation of
Kasner vectors is induced by a higher order term for g → 0 with respect to the pure Kasner
dynamics, we can assume that λa are near to constant and therefore near the singularity (20),
(21) give, in the asymptotic limit to the cosmological singularity, the billiards [10, 16, 3]
on (n − 1)-dimensional Lobachevsky space, exactly as in the three-dimensional mixmaster
case [34]. In other words, in our scheme the evolution is not simply Kasner-like, but we
get a dynamical picture in which the point universe moves according to a piecewise Kasner
solution. The features predicted by equations (20) and (21) will be discussed in sections 5
and 6.
We remark that the validity of the Kasner approximation is based on the possibility of
neglecting the n-dimensional Ricci scalar with respect to the term containing time derivatives.
In the Kasner solution this picture holds along the whole system evolution to the initial
singularity [33]; instead, in the piecewise Kasner solution, the n-dimensional Ricci tensor
is negligible only for finite time intervals ending with a bounce against the potential terms
arising from the spatial curvature [2]. However, in both these cases the validity of the Kasner
approximation is confirmed by a large number of theoretical and numerical investigations,
concerning homogeneous and inhomogeneous cosmological models near the singularity [2–6,
10, 13, 14, 16, 17, 34]
Our scheme relies on the choice of a synchronous (or Gaussian) reference frame which
corresponds to having N = 1 and N α = 0. This system has many interesting properties,
among which outstands its geodesic nature [33]; in fact the line x α = constant results in a
geodesic of the manifold and, since the normal to spatial hypersurfaces reads nµ = (1, 0),
free particles are co-moving to this system. From a cosmological point of view the Gaussian
frame is relevant because the galaxies are (almost) free particles in the actual universe and
therefore our phenomenology relies on a synchronous gauge. However, the results obtained
in this work have to remain valid for other gauge choices in view of the general covariance.
5. Kasner solution
To analyse the time evolution of the Kasner vectors and to obtain their rotation law, it is
convenient to project them on the time-independent quantities π α and to decompose them into
135
Attachments
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R Benini et al
the two parts:
λa
a = 2 π,
π
a = a + a⊥ ;
(π a⊥ ) = 0.
(24)
Here, for the sake of simplicity, we use the vector notation which is useful when treating the
components of these vectors as Euclidean ones.
Hence we can split (22) into the two independent components
∂ λa
p ∂
∂t b
λa =
λb ,
(25)
∂t
λ
−
pb )
(p
a
b=a. b
∂ λa
p ∂ ∂t b
b⊥ = Aab (t) b⊥ .
a⊥ =
∂t
λ
−
p
(p
)
b
a
b
b=a.
Since the above matrix Aab does not depend on a⊥ , then we get the formal solution
t
a Ab t dt b⊥ (t0 ).
a⊥ (t) = T exp
(26)
(27)
t0
The remaining equations (25) together with (20), (21) and (23) provide a self-consistent
dynamical system.
Since the Kasner solution corresponds to neglecting the contributions of the n-dimensional
Ricci scalar to the dynamics, then to get such behaviour we have to take the limit in which
all the terms exp (q a ) become of higher order; under this assumption we get the following
simplified dynamical system:
λa = const,
a⊥ = const,
∂
2N
1 pb ,
qa = √
pa −
∂t
g
n−1 b
pa = const,
pa2 −
(28)
1 2 1 qa 2
λa = 0,
pa +
e n−1
2
whose solution, in the gauge N = 1 and towards the cosmological singularity (g → 0), takes
the form
pa
gαβ =
t 2sa aα aβ ,
sa = 1 − (n − 1) ,
(29)
b pb
a
where the Kasner indexes sa satisfy the identities
sa =
sa2 = 1.
(30)
Let us take the Kasner indexes in the increasing order
s1 s2 · · · sn .
(31)
In this way we always have s1 < 0 and sn sn−1 0. Therefore the largest increasing term
(as t → 0 t s1 → ∞) among the neglected ones comes from s1 and it is to be taken into account
to construct the oscillatory regime towards the cosmological singularity.
136
Attachment 2
Oscillatory regime in the multidimensional homogeneous cosmological models
1489
6. Billiard representation: the return map and the rotation of Kasner vectors
To construct the oscillatory regime we retain just the leading term corresponding to exp(q 1 ),
so the field equations (20) and (21) can be rewritten as
∂
λ1 = 0,
∂t
∂ λa
p1 ∂
λa = ∂t
,
a = 1,
∂t
(pa − p1 )
∂
pa = 0,
a = 1,
∂t
∂
N 2
λ exp(q 1 ),
p1 = − √ ∂t
2 g 1
2N
1 ∂
qa = √
pb .
pa −
∂t
g
n−1 b
(32)
The first of equations (32) gives λ1 = const, while the second admits the solution
λa (pa − p1 ) = const.
(33)
The remaining part of the dynamical system allows us to determine the return map governing
the replacements of Kasner epochs (i.e. intervals of time during which the evolution is Kasnerlike)
s1 =
−s1
,
2
1 + n−2
s1
λ1 = λ1 ,
sa =
1+
λa = λa 1 − 2
1
Defining the quantities ka = pa − n−1
k1
sa +
2
s
n−2 1
,
2
s
n−2 1
(n − 1) s1
(n − 2) sa + ns1
(34)
,
(35)
pb we find for them the following iteration law:
= −k1
(36)
2
k1
n−2
2
k1 .
k =
k+
n−2
ka = ka +
(37)
(38)
Our analysis is completed by investigating the rotation of Kasner vectors a through the
epoch replacements; by (26) we get the system
∂ λa
p ∂ ∂ ∂t 1
(39)
1⊥ ,
a⊥ =
1⊥ = 0,
∂t
∂t
λ1 (pa − p1 )
admitting the integral
λ1 a⊥ − λa 1⊥ = const.
(40)
Putting together (24), (35) and (39) we arrive at the final iteration law
a = a + σa 1 ,
σa =
λa − λa
λa
(n − 1) s1 = −2
.
(n − 2) sa + ns1 λ1
λ1
which completes our dynamical scheme.
137
(41)
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R Benini et al
Thus the homogeneous universes discussed here approach the initial singularity being
described by a metric tensor with oscillating scale factors and rotating Kasner vectors. Passing
from one Kasner epoch to another, the negative Kasner index s1 is exchanged between different
directions (for instance 1 and 2 ) and at the same time these directions rotate in the space
according to the law (41). The presence of a vector field is crucial because, independently
of the considered model, it induces a (dynamically [16]) closed domain on the configuration
space. The amplitudes of q a -oscillations increase approaching the initial singularity and their
minimum value approaches −∞.
In correspondence with this oscillation of the scale factor the Kasner vectors a rotate
and, at the lowest order in q a , the quantities σa remain constant along a Kasner epoch; in this
sense the vanishing behaviour of the determinant g approaching the singularity does not affect
significantly the rotation law (41). We conclude by stressing that the duration of a Kasner
epoch decrease as the singularity is approached in this scheme.
7. Concluding remarks
By the study developed above, we have shown how any multidimensional homogeneous
cosmological model acquires, near the singularity, an oscillatory regime when an
electromagnetic field is included in the dynamics. This result is a consequence of the capability
of a vector field to generate a billiard configuration in the asymptotic evolution; such a
‘billiard-ball’ representation of the universe dynamics coincides with the BKL (Belinski–
Khalatnikov–Liftshitz) piecewise approach only in the four-dimensional spacetime while
in higher dimensions new features appear. In particular, the oscillatory regime obtained
characterizes all the homogeneous models, disregarding their potential term; furthermore,
the map by which the Kasner indexes evolve acquires a direct dependence on the dimension
number.
A valuable issue of our Hamiltonian dynamics relies on fixing the rule of the Kasner
vector rotation. Such a law of rotation is relevant to connect the Cauchy problem with later
stages of the system dynamics. More precisely, the rotation of the Kasner vectors is sensitive
to the boundary conditions on the ‘matter’ fields and has to be taken into account when using
the studied homogeneous models within a cosmological ‘picture’.
Once the spatial gradients are taken into account, our result can be extended to a generic
inhomogeneous cosmological model, in the same spirit as the Bianchi type VIII and IX model
oscillatory regime is upgraded in four dimensions.
Such an extension will reliably show how, in the presence of a vector field, the generic
cosmological solution is described, in correspondence with any dimension number, by an
oscillatory approach to the big bang.
Investigations in this direction, as well as those extended to the more general frameworks
of superstrings [28, 29] and brane dynamics [35], will be a subject for further development.
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Attachment 3
INSTITUTE OF PHYSICS PUBLISHING
CLASSICAL AND QUANTUM GRAVITY
doi:10.1088/0264-9381/24/2/007
Class. Quantum Grav. 24 (2007) 387–404
Inhomogeneous quantum Mixmaster: from classical
towards quantum mechanics
Riccardo Benini1,2 and Giovanni Montani2,3
1 Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, via Irnerio 46,
40126 Bologna, Italy
2 ICRA—International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9),
Università di Roma ‘La Sapienza’, Piazza A.Moro 5 00185 Roma, Italy
3 ENEA C.R. Frascati (Dipartimento FTP), via Enrico Fermi 45, 00044 Frascati, Roma, Italy
E-mail: [email protected] and [email protected]
Received 8 June 2006, in final form 23 October 2006
Published 8 December 2006
Online at stacks.iop.org/CQG/24/387
Abstract
Starting from the Hamiltonian formulation for the inhomogeneous Mixmaster
dynamics, we approach its quantum features through the link of the quasiclassical limit. We fix the proper operator-ordering which ensures that the
WKB continuity equation overlaps the Liouville theorem as restricted to the
configuration space. We describe the full quantum dynamics of the model
in some detail, providing a characterization of the (discrete) spectrum with
analytic expressions for the limit of high occupation number. One of the
main achievements of our analysis relies on the description of the ground state
morphology, showing how it is characterized by a non-vanishing zero-point
energy associated with the universe anisotropy degrees of freedom.
PACS numbers: 04.20.Jb, 98.80.Dr, 83C
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Einstein’s equations, despite their nonlinearity, admit a generic solution asymptotically to the
cosmological singularity having a piece-wise analytic expression. This dynamical regime
was derived by Belinski, Khalatnikov and Lifshitz (BKL) [1, 2] and it has an oscillatory-like
behaviour which extends point by point in space the homogeneous Mixmaster dynamics of
the Bianchi type IX model [3]. When a generic initial condition is evolved towards the initial
singularity, a spacetime foam [4, 5] appears: this is a direct consequence of the dynamical
decoupling which characterizes space points close enough to the singularity and which can be
appropriately described by the Mixmaster point like measure [4, 6] (a wide literature exists
0264-9381/07/020387+18$30.00 © 2007 IOP Publishing Ltd Printed in the UK
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about the chaotic properties of the homogeneous Mixmaster model; see, for instance, [1, 7–10]
and the references therein).
However, this classical description is in conflict with the requirement of a quantum
behaviour of the universe through the Planck era; in fact there are reliable indications [8] that
the Mixmaster dynamics overlap the quantum universe evolution, requiring an appropriate
analysis of the transition between these two different regimes.
Indeed the dynamics of the very early universe correspond to a very peculiar situation with
respect to the link existing between the classical and quantum regimes, because the expansion
of the universe is the crucial phenomenon which maps these two stages of the evolution into
each other. As shown in [8] the appearance of a classical background takes place essentially
at the end of the Mixmaster phase when the anisotropy degrees of freedom can be treated as
small perturbations; this result indicates that the oscillatory regime takes place almost during
the Planck era and therefore it is a problem of quantum dynamics. However the end of the
Mixmaster (and in principle the quantum to classical transition phase) is fixed by the initial
conditions on the system and in particular it takes place when the cosmological horizon reaches
the inhomogeneity scale of the model; therefore the question of an appropriate treatment for
the semiclassical behaviour arises when the inhomogeneity scale is so larger than the Planck
scale that the horizon can approach it only in the classical limit. Some interesting features of
the quantum Mixmaster have been developed by [11–14] and [15, 16] for the homogeneous
and the inhomogeneous cases, respectively.
Here we refine this analysis both connecting some properties of the quantum behaviour
to the ensemble representation of the model and describing the precise effect of the boundary
conditions on the structure of the quantum states.
We start by an Arnowitt–Deser–Misner (ADM) [17] representation of the system dynamics
which allows us to disregard the contributions of the spatial gradients relative to the
configurational variables, thus reducing the dynamics to a number of ∞3 independent pointlike Mixmaster models. Such representation can take place independently of the gauge choice
[18], nevertheless here we require one of the Misner–Chitré-like variables to play the role of
time for the system. In this way, we can describe the whole evolution in terms of a triangular
billiard on the Poincaré upper half-plane where the point universe randomizes.
Then we apply the Liouville theorem restricted to the configuration space using the
Hamilton–Jacobi solution of the model, and we require the continuity equation to match the
WKB limit. Such a requirement leads us to determine the proper operator-ordering associated
with the ADM operator in a unique way. We emphasize that the existence of an energy-like
constant of motion not only provides a microcanonical measure for the statistical dynamics but
also induces a quantum dynamics completely described by a time-independent Schrödinger
equation on the Poincaré plane. However, the effective Hamiltonian associated with such an
eigenvalue problem is non-local, characterized by the presence of a square root function; in
agreement with the analysis developed by [19], we make the well-grounded hypothesis that
the eigenfunctions’ form is independent of the presence of the square root, since its removal
implies the square of the eigenvalues only.
A crucial step relies on recognizing how our squared equation can be restated as the
Laplace–Beltrami eigenvalue problem. Hence a characterization of the ‘energy’ spectrum
comes out from imposing the triangular boundary conditions (the semicircle is replaced by a
straight line preserving the measure of the domain).
We outline that the spectrum is discrete and that admits a point-zero energy which implies
an intrinsic anisotropy of the quantum Mixmaster universe. We numerically investigate the
first ‘energy’ levels and finally provide an analytic expressions for high occupation numbers.
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The structure of the presentation is the following: section 2 is devoted to fixing the
classical Hamiltonian formulation of the system dynamics, based on the use of Misner–
Chitré-like variables which allow the ADM reduction of the degrees of freedom. In
section 3 we fix the solution of the Hamilton–Jacobi equation within the domain allowed
by the vanishing of the potential. Such a solution is at the ground level of the analysis
presented in section 4; in fact here we restrict the Liouville theorem to the configuration space,
eliminating the momentum dependence by virtue of an integration along the Hamilton–Jacobi
dynamics. In section 5 we describe the Schrödinger equation for the Mixmaster model and
construct the WKB limit; by the comparison of this semiclassical behaviour with the equation
induced by the Liouville theorem, we get a unique choice for the operator-ordering of the
super-Hamiltonian kinetic term.
Finally, in section 6 we provide a detailed description of the Mixmaster spectrum and
eigenfunctions; particular attention is devoted to the ground state in view of the idea that it is
the expected universe configuration during the Planck era. Section 7 is devoted to concluding
remarks on the physical issues of our discussion, and particular attention is devoted to fix the
full inhomogeneous quantum picture.
Over the whole paper we adopt units such that c = 16π G = 1.
2. Hamiltonian formulation
In this section we introduce the Hamiltonian formalism for a generic inhomogeneous model
in vacuum in terms of the Misner–Chitré-like variables [3, 20].
In the ADM approach, the line element associated with such a model can be written in
the form
ds 2 = N 2 dη2 − γαβ (dx α + N α dη)(dx β + N β dη),
(1)
α
where N and N denote the lapse function and the shift vector, respectively, while γαβ
(α, β = 1, 2, 3) is the 3-metric tensor of the spatial hyper-surfaces 3 η = const, and we
take [4]
a
γαβ = eq δad Oba Ocd
∂y b ∂y c
,
∂x α ∂x β
a, b, c, d, α, β = 1, 2, 3,
(2)
where q a = q a (x, η) and y b = y b (x, η) are six scalar functions, and Oba = Oba (x) an SO(3)
matrix (repeated indexes are summed).
Using the metric tensor (2), the action for the gravitational field reads
S=
dη d3 x(pa ∂η q a + d ∂η y d − NH − N α Hα ),
(3)
(3) ×
with


2
1 
1
H =√
(pa )2 −
pb − γ (3) R 
γ
2
a
b
Hα = c ∂α y c + pa ∂α q a + 2pa (O −1 )ba ∂α Oba ;
(4)
(5)
H and Hα being the super-Hamiltonian and the super-momentum, respectively.
In (4) and (5) pa and c play the role of conjugate momenta to the variables q a and y b ,
respectively, and (3) R is the Ricci 3-scalar which behaves like a potential term.
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No sooner should we adopt y a like new spatial variables, than the super-momentum
constraint Hα = 0 (provided by the variation of (3) with respect to N α ) can be solved, and the
action rewrites as [18]
dη d3 y pa ∂η q a + 2pa (O −1 )ca ∂η Oca − N H .
(6)
S=
(3) ×
Furthermore the potential term appearing in the super-Hamiltonian (4) can be approximated
towards the singularity as
(Qa ),
(7)
U=
a
where
(x) =
+∞
0
if x > 0,
if x < 0,
(8)
qa
Qa = a .
aq
(9)
This picture arises from the vanishing of the metric tensor determinant close to the singularity,
and the quantities Qa , known in the literature as anisotropy parameters, cut a closed domain
Q to which the dynamics is restricted.
By virtue of the potential structure (7), the universe dynamics evolves independently in
each space point inducing a corresponding factorization for the phase space of the model.
Since in Q the potential U asymptotically vanishes, close to the singularity the relation
∂pa /∂η = 0 holds and then the term 2pa (O −1 )ca ∂η Oca in (6) behaves like an exact timederivative and can be ruled out of the variational principle.
The ADM reduction is completed by introducing the so-called Misner–Chitré-like
variables [4, 10, 20, 21], whose relevance consists in making the anisotropy parameters
independent of τ , which will behave as the time variable:

√

q 1 = eτ ξ 2 − 1(cos θ + 3 sin θ ) − ξ


√
(10)
q 2 = eτ ξ 2 − 1(cos θ − 3 sin θ ) − ξ


 3
q = −eτ ξ + 2 ξ 2 − 1 cos θ .
In terms of these new variables the super-Hamiltonian rewrites
H = −pτ2 + pξ2 (ξ 2 − 1) +
and the Qa become
pθ2
−1
ξ2
(11)
√
ξ2 − 1
(cos θ + 3 sin θ )
3ξ
√
ξ2 − 1
1
(cos θ − 3 sin θ )
Q2 = −
(12)
3
3ξ
1 2 ξ2 − 1
cos θ.
Q3 = +
3
3ξ
Let us solve the constraint H = 0 (obtained variating N in (3)) with respect to pτ in the domain
Q in order to perform the ADM reduction, thus obtaining
p2
(13)
−pτ ≡ = (ξ 2 − 1)pξ2 + 2 θ .
ξ −1
1
Q1 = −
3
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Then, taking the time gauge ∂η τ = 1, the reduced action explicitly reads as
SQ = dτ d3 y pξ ∂τ ξ + pθ ∂τ θ − .
391
(14)
As we approach the singularity, behaves like a constant of motion, i.e. d/dτ = ∂/∂τ =
0 ⇒ = E(y a ).
The dynamics of such a model is equivalent to (that of) a billiard ball on a Lobatchevsky
plane [7, 14, 18, 22]; this can be shown by the use of the Jacobi metric, i.e. a geometric
approach that reduces the dynamical equations of a generic system to a geodesic problem on
a manifold. Such a technique applied to (14) produces a geodesic equation corresponding to
the line element
dξ 2
2
2 a
2
2
(15)
+ (ξ − 1) dθ .
dl = E (y )
(ξ 2 − 1)
The manifold described by (15) turns out to have a constant negative curvature, where the
Ricci scalar is given by R = −2/E 2 : the complex dynamics of the generic inhomogeneous
model results in a collection of decoupled dynamical systems, one for each point of the space,
and all of them equivalent to a billiard problem on a Lobatchevsky plane.
Among the possible representations for it, we choose the so-called Poincaré model
in the complex upper half-plane [6] that can be introduced with the following coordinate
transformation:

1 + u + u2 + v 2



√
ξ =
3v
(16)
√

3(1 + 2u)


θ = −arctan
.
−1 + 2u + 2u2 + 2v 2
The line element for this two-dimensional surface now reads
du2 + dv 2
.
(17)
v2
Figure 1 shows how, in terms of these coordinates, the anisotropy functions cut a stripe in
the plane (u, 0, v) characterized by a finite measure µ = π and three unstable directions to
asymptotically escape.
The boundary is given by the set of points where one of the anisotropy parameters is equal
to zero; it is a geodesic triangle, where the edges are given by the vanishing of the numerators
of Qa , now reading

Q1 (u, v) = −u/δ
Q2 (u, v) = (1 + u)/δ
(18)

Q3 (u, v) = (u2 + u + v 2 )/δ
ds 2 =
δ = u2 + u + 1 + v 2 .
In the (u, v) scheme, the ADM Hamiltonian (13) assumes the expression
HADM = = v pu2 + pv2 ,
(19)
which will be the starting point of our analysis.
We conclude by observing how [1, 2] the present representation of the Lobatchevsky plane
is suitable to link this Hamiltonian formulation to the piece-wise representation provided by
Belinski, Khalatnikov and Lifshitz map, as it is easy to recognize that for v = 0, the functions
Qa reduce to the familiar Kasner indexes.
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3
2.5
v
2
1.5
1
0.5
0
-2
-1.5
-1
-0.5
u
0
0.5
1
Figure 1. The domain Q (u, v) in the Poincaré upper half-plane where the dynamics of the
space-point universe is asymptotically restricted by the potential. The measure is finite and equal
to π .
3. Hamilton–Jacobi approach
We shall now derive the Hamilton–Jacobi (HJ) [23] equations for the system (19) in view of
the following developments.
The HJ prescribes to change the momentum variables into the derivatives of a functional S
with respect to the configurational variable: implementing the HJ technique, the Hamiltonian
relation (19) leads to the functional differential equation in Q :
2 2
δS
δS
δS
=v
−
+
.
(20)
δτ
δu
δv
To obtain the solution of this dynamical equation, we take S in the form
S = S0 (u, v) −
d3 y (y a ) τ,
where S0 satisfies the equation
δ S0 2
δ S0 2
2
2
+
=v
δu
δv
(21)
(22)
and it is therefore provided by
S0 (u, v) = k(y )u +
a
2
−
k 2 (y a )v 2
− ln 2
+
2 − k 2 (y a )v 2
2v
+ c(y a ),
(23)
where k(y a ), c(y a ) are integration constants.
The expression (21), together with (23) and the features of the potential wall (18),
summarizes the classical dynamics of a generic inhomogeneous universe.
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4. The statistical mechanic description
In section 2 we pointed out that, for the Mixmaster inhomogeneous dynamics, the spatial
points decouple approaching the singularity and an energy-like constant of motion appears;
let us discuss the problem from a statistical mechanics point of view by treating the system as
a microcanonical ensemble.
The physical properties of a stationary ensemble are described by a distribution function
ρ = ρ(u, v, pu , pv ), obeying the continuity equation defined in the phase space (u, v, pu , pv )
as
∂(u̇ρ) ∂(v̇ρ) ∂(p˙u ρ) ∂(p˙v ρ)
+
+
+
= 0,
∂u
∂v
∂pu
∂pv
(24)
where the dot denotes the time derivative
∂u
∂HADM
v2
=
= pu
∂τ
∂pu
∂v
∂HADM
v2
v̇ ≡
=
= pv
∂τ
∂pv
u̇ ≡
∂HADM
∂pu
=−
=0
∂τ
∂u
∂HADM
∂pv
=−
=− .
ṗv ≡
∂τ
∂v
v
ṗu ≡
(25)
We stress how the above continuity equation provides an appropriate representation for the
ensemble associated with the Mixmaster only when we are sufficiently close to the initial
singularity and therefore the infinite-potential wall approximation works; in fact, such a model
for the potential term corresponds to deal with the energy-like constant of the motion and fixes
the microcanonical nature of the ensemble. From a dynamical point of view this picture arises
naturally because the universe volume element vanishes monotonically (for non-stationary
correction to this scheme in the Misner–Chitré-like variables see [24]).
We are interested in studying the distribution function in the (u, v) space, and thus we
will reduce the dependence on the momentum variables by integrating ρ(u, v, pu , pv ) in the
momenta space and using the informations contained in the HJ approach. If we assume ρ to be
a regular, vanishing at the infinity of the phase space, limited function
of its arguments, and if
we use (23), we work out the following equation for w̃(u, v; k) = ρ(u, v, pu , pv ) dpu dpv :
2
E
∂ w̃(u, v; k) E 2 − 2k 2 v 2 w̃(u, v; k)
∂ w̃(u, v; k)
+
+
−1
=0
(26)
∂u
kv
∂v
kv 2
E 2 − (kv)2
admitting a solution in terms of a generic function g of the form
2
g u + v kE2 v2 − 1
.
w̃(u, v) =
√
v E2 − k2v2
(27)
The distribution function in (u, v) is obtained after the integration over the constant k. Indeed,
this constant expresses the freedom of choosing the initial conditions (25), which cannot affect
the chaotic properties of the model. Therefore we define the reduced distribution w(u, v) as
w̃(u, v; k) dk,
(28)
w(u, v) ≡
A
where the integration is over the classical available domain for pu ≡ k:
E E
.
A≡ − ,
v v
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It is easy to verify that the microcanonical Liouville measure [1, 6, 7, 10, 18, 24, 25] wmc
(after integration over the admissible values of ) corresponds to the case g = const, i.e. we
get the normalized distribution
E
v
1
π
wmc (u, v) =
(30)
dk = 2 .
2
v
E
− Ev kv 2
−
1
2
2
k v
Summarizing, we have derived the generic expression of the distribution function for our
model, fixing its form for the microcanonical ensemble which, in view of the energy-like
constant of the motion , is the most appropriate to describe the Mixmaster system when the
picture is restricted to the configuration space.
5. Quasi-classical limit of the quantum regime
Let us underline some common features between the classical and the semi-classical dynamics
with the aim of fixing the proper operator-ordering [26] in treating the quantum approach.
In fact, considering the WKB limit for h̄ → 0, the coincidence between the classical
distribution function wmc (u, v) and the quasi-classical probability function r(u, v) takes place
for a precise choice of the operator-ordering [16] only (for a connected topic see [8] and for a
general discussion see [27]).
As soon as we implement the canonical variables into operators, as in the canonical
quantization, i.e.
v̂ → v,
û → u
∂
∂
∂
p̂v → −ih̄ , p̂u → −ih̄ , p̂τ → −ih̄ ,
∂v
∂u
∂τ
the quantum dynamics for the state function (u, v, τ ) obeys, in each point of the space, the
Scrhödinger equation
∂(u, v, τ )
∂2
∂
∂
ih̄
= Ĥ ADM (u, v, τ ) = h̄ −v 2 2 − v 2−a
va
(u, v, τ ),
(31)
∂τ
∂u
∂v
∂v
where we have adopted a generic operator-ordering for the position and momentum operators
parametrized by the constant a [28].
In the above equation the Hamiltonian operator has a non-local character as a consequence
of the square root function; in principle this constitutes a subtle question about the quantum
implementation though. We will make the well-grounded ansatz [19] that the operators Ĥ ADM
and Ĥ 2ADM have the same set of eigenfunctions with eigenvalues E and E 2 , respectively4 .
If we take the following integral representation for the wavefunction ,
∞
(u, v, τ ) =
(u, v, E) e−iEτ/h̄ dE,
(32)
−∞
the eigenvalues problem reduces to
∂2
∂
∂
Ĥ 2 (u, v, E) = h̄2 −v 2 2 − v 2−a
va
(u, v, E) = E 2 (u, v, E).
(33)
∂u
∂v
∂v
In order to study the WKB limit of equation (33), we separate the wavefunction into its phase
and modulus
(u, v, E) = r(u, v, E) eiσ (u,v,E)/h̄ .
(34)
4
The problems discussed in this respect by [29] do not arise here because in the domain Q our ADM Hamiltonian
has a positive sign (the potential vanishes asymptotically).
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In this scheme the function r(u, v) represents the probability density, and the quasi-classical
regime appears as we take the limit for h̄ → 0; substituting (34) in (33) and retaining only the
lower order in h̄, we obtain the system
 2 2 ∂σ
∂σ


v 2
+
= E2

∂u
∂v
(35)


a ∂σ ∂ 2 σ ∂ 2 σ
∂r ∂σ ∂r ∂σ


= 0.
+
+r
+
+
∂u ∂u ∂v ∂v
v ∂v ∂v 2 ∂u2
In view of the HJ equation and of the Hamiltonian (19), we can identify the phase σ to the
functional S0 of the HJ approach.
Now we turn our attention to the equation for r(u, v); taking (23) into account it reduces
to
2
E
∂r(u, v, E) a(E 2 − k 2 v 2 ) − E 2
∂r(u, v, E)
+
+
− k2
r(u, v, E) = 0.
(36)
k
√
∂u
v
∂v
v2 E2 − k2v2
Comparing (36) with (26), we easily see that they coincide (as expected) for a = 2 only.
It is worth noting that this correspondence is expectable once a suitable choice for the
configurational variables is taken; however here it is remarkable that it arises only if the above
operator-ordering is addressed. It is just in this result the importance of this correspondence
whose request fixes a particular quantum dynamics for the system.
Summarizing, we have demonstrated from our study that it is possible to get a
WKB correspondence between the quasi-classical regime and the ensemble dynamics
in the configuration space, and we provided the operator-ordering when quantizing the
inhomogeneous Mixmaster model to be [28]
2 2
2 ∂
2 ∂
v
.
(37)
v̂ p̂v → −h̄
∂v
∂v
By means of these results, we now face the full quantum dynamical approach.
6. Canonical quantization and the energy spectrum
Our starting point is the point-like eigenvalue equation (33) (for a discussion of this same
eigenvalue problem in the Misner variables see [3]) which, together with the boundary
conditions, completely describes the quantum features of the model, i.e.
2 2
2
E
∂
2 ∂
2 ∂
v
+
+v
+ 2v
(u, v, E) = 0
∂u2
∂v 2
∂v
h̄
(38)
(∂Q ) = 0.
In equation (38) we can recognize a well-known operator: by redefining (u, v, E) =
ψ(u, v, E)/v, we can reduce (38) to the eigenvalue problem for the Laplace–Beltrami operator
in the Poincaré plane
2
∂
∂2
ψ(u, v, E) = Es ψ(u, v, E),
+
(39)
∇LB ψ(u, v, E) ≡ v 2
∂u2 ∂v 2
which is central in the harmonic analysis on symmetric spaces and has been widely investigated
in terms of its invariance under SL(2, C)(for a detailed discussion, see [30] and the
bibliography therein).
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Figure 2. The approximate domain where we impose the boundary conditions. The choice
v = 1/π for the straight line preserves the measure µ = π .
Its eigenstates and eigenvalues are described as
√ an Ks−1/2 (2π |n|v) e2πinu ,
ψs (u, v) = av s + bv 1−s + v
a, b, an ∈ C
n=0
(40)
∇LB ψs (u, v) = s(s − 1)ψs (u, v),
where Ks−1/2 (2π nv) are the modified Bessel functions of the third kind, and s denotes the
index of the eigenfunction. This is a continuous spectrum, and the summation runs over every
real value of n.
The eigenfunctions for our model read as
Ks−1/2 (2π |n|v)
(u, v, E) = av s−1 + bv −s +
an
(41)
e2πinu
√
v
n=0
with eigenvalue
E 2 = s(1 − s)h̄2 .
(42)
The spectrum and the explicit eigenfunctions are obtained by imposing the boundary conditions
(38), i.e. requiring that equation (41) vanishes on the edges of the geodesic triangle of
figure 1.
Since from our analysis no way arose to impose exact boundary conditions we approximate
the domain with the simpler one in figure 2; the value for the horizontal line y = 1/π provides
the same measure for the exact and the approximate domain.
The conditions on the vertical lines u = 0, u = −1 require to disregard the first two terms
in (41) (a = b = 0); furthermore we get the condition on the last term
n=0
e2πinu →
∞
sin(2π nu),
n=1
n being an integer, while the condition on the horizontal line implies
an Ks−1/2 (2n) sin(2nπ u) = 0, ∀u ∈ [−1, 0],
n>0
which is satisfied by requiring Ks−1/2 (2n) = 0 only.
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397
n
8
6
4
2
2
4
6
8
10
12
t
Figure 3. The intersections between the straight lines and the curves represent the roots of the
equation Kit (n) = 0, where K is the modified Bessel function.
Table 1. We numerically investigated the roots of Kit (n) = 0 with respect to t for different values
of n. The values in the columns are the first t’s in the above figure.
n=2
n=4
n=6
n=8
4.425
6.333
7.947
9.410
10.774
7.016
9.353
11.313
16.264
17.742
9.434
12.076
14.283
16.263
18.096
11.768
14.655
17.059
19.212
21.203
The functions Kν (x) are real and positive for real argument and real index, therefore the
index must be imaginary, i.e. s = 12 + it. In this case these functions have (only) real zeros,
and the corresponding eigenvalues turn out to be real and positive
(E/h̄)2 = t 2 + 1/4.
(44)
We remark that our eigenfunctions naturally vanish as infinite values of v are approached.
The conditions (43) cannot be solved analytically for all the values of n and t, and the
roots must be worked out numerically for each n; there are several results on their distribution
that allow us to find at least the first levels. A theorem on the zeros of these functions states
that Kiν (νx) = 0 ⇔ 0 < x < 1 (for a proof see [33]); by this theorem and the monotonic
dependence of the energy (44) on the zeros, we can easily search the lowest levels by solving
equation (43) for the first n. In figure 3 and table 1 we plot the first roots, and in figure 4 and
in table 2 we list the first ten ‘energy’ levels5 .
Below we will provide an analytical treatment of the high-energy levels associated with
our operator in correspondence to certain region of the plane (t, 0, n). In [14] the structure
of the high-energy levels is also connected to the so-called quantum chaos of the Mixmaster
model, which was studied from the wavefunctional point of view in [12] and from the path
integral one in [13] using Misner variables. Our analysis implicitly contains the information
about the quantum chaos of the considered dynamics; in fact we can take a generic state of
5 For a detailed numerical investigation of the energy spectrum of the standard Laplace–Beltrami operator, especially
with respect to the high-energy levels, see [31, 32], where the effects on the level spacing of deforming the circular
boundary condition towards the straight line are also numerically analysed.
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2
t2 1 4
140
120
100
80
60
40
20
0
Figure 4. The lines represent in a typical spectral manner the first levels.
Table 2. The first ten energy eigenvalues, ordered from the lowest one.
E 2
h̄
= t2 +
1
4
19.831
40.357
49.474
63.405
87.729
89.250
116.329
128.234
138.739
146.080
the system ξ(τ, u, v) in the form
√1 2
ct,n n (τ, u, v)δ(Kit (2n)) e−i 4 +t τ
ξ(τ, u, v) = dt
(45)
n
and its evolution from a generic initial condition ξ0 (u, v) ≡ ξ(τ0 , u, v) at an initial instant τ0
provides all the quantum properties of the system. The quantum chaos is recognized in [12] by
numerically integrating the Wheeler-De Witt equation from a gaussian like initial packet and
outlining the appearance of a fractal structure in the profile of the resulting wavefunction;
in our approach (as the infinite potential walls approximation works) the dynamics is
provided by (45) and evolving it from an initial localized wave packet the quantum chaos has
to arise.
About the information on the quantum chaos emerging from the high-energy spectrum
we emphasize the following two important points: (i) the non-stationary corrections due to the
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399
Ψgs (u,v)
1.5
1
0.5
0
-0.5
-1
-1.5
1.25
1
-1
0.75
-0.75
-0.5
u
v
0.5
-0.25
0
Figure 5. The ground state wavefunction is sketched.
|Ψgs|2 (u,v)
8
6
4
2
0
1.25
1
-1
0.75
-0.75
u
-0.5
v
0.5
-0.25
0
Figure 6. We show the probability distribution in correspondence to the ground state of the theory,
according to the boundary conditions.
real potential term are expected to be simply small perturbation to our result rapidly decaying
as the singularity is approached; (ii) the analytic expressions we are going to provide for large
values of t cannot be used to fix the existence of the quantum chaos because they explore
limited regions of the plane (t, 0, n) only, but they are very useful to clarify the morphology
of the spectrum and its dependence on two different quantum numbers.
6.1. The ground state
Let us describe the properties of the ground state level with a major accuracy, starting from
the result of its existence with a non-zero energy E0 , i.e. E02 = 19.831h̄2 .
In figure 5 we plot the wavefunction gs in the (u, 0, v) plane and in figure 6 the
corresponding probability distribution |gs |2 , whose normalization constant is equal to
N = 739.466. The knowledge of the ground state eigenfunction allows us to estimate
the average values of the anisotropy variables u, v and the corresponding root mean square,
i.e.
u = −1/2,
v = 0.497
(46)
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u = 0.266,
v = 0.077.
(47)
This result tells us that in the ground state the universe is not exactly an isotropic one, and it
fluctuates around the line of the Misner plane β− = 0. However we observe that it remains
localized in the centre of the Misner space far from the corner at v → ∞ (the other two
equivalent corners were cut out from our domain by the approximation we considered on the
boundary conditions, but, because of the potential symmetry, they have to be unaccessible
too). Thus we can conclude that the universe, approaching the minimal energy configuration,
conserves a certain degree of anisotropy but lives in the region where the latter can be treated
as a small correction to the full isotropy. Such a behaviour is a consequence of the zeropoint energy associated with the ground state which prevents the absence of oscillation modes
concerning the anisotropy degrees of freedom; we numerically derived this feature but it can
be inferred on the basis of general considerations about the Hamiltonian structure. In fact
the Hamiltonian contains a term v 2ˆpv2 which has positive definite spectrum and cannot admit
vanishing eigenvalue.
6.2. Asymptotic expansions
In order to study the distribution of the highest energy levels, let us take into account the
asymptotic behaviour of the zeros for the modified Bessel functions of the third kind.
We will discuss asymptotic regions of the plane (t, 0, n) in both the cases (i) t n and
(ii) t n 1.
(i) For t n the Bessel functions admit the following representation:
√
∞
1
2π e−tπ/2
(−1)k
u2k sin a
Kit (n) = 2
(t − n2 )1/4
t 2k
1 − p2
k=0
∞
(−1)k
1
+ cos a
u2k+1 (48)
t 2k+1
1 − p2
k=0
√
where a = π/4 − t 2 − n2 + t arccosh(t/n), p ≡ n/t and uk are the following polynomials:

u0 (t) = 1
(49)
1
1 1
uk+1 (t) = t 2 (1 − t 2 )uk (t) +
(1 − 5t 2 )uk (t) dt.
2
8 0
Retaining in the above expression only those terms of o nt , the zeros are fixed by the following
relation:
π
π
1
− t + t (log(2) − log(p)) −
cos
− t + t (log(2) − log(p)) = 0,
(50)
sin
4
12t
4
that in the limit for n/t 1, can be recast as follows:
t log(t/n) = lπ ⇒ t =
lπ
,
productlog lπn
(51)
where productlog(z) is a generalized function that gives the solution of the equation z = w ew
and for real and positive domain, it is a monotonic function of its argument. In (51) l is an
integer number that must be taken much greater than 1 in order to verify the initial assumptions
n/t 1.
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(ii) In case the difference between 2n and t is o(n1/3 )(t, n 1), we can evaluate the first
zeros by the expansion (worked out from formula (9) of [34])
t = 2n + 0.030n1/3 ,
(52)
providing the lowest zero (and therefore the energy) for a fixed value of n and then the relation
for the eigenvalues for high occupation numbers as
2
E
∼ 4n2 + 0.12n4/3 .
(53)
h̄
In this region of the spectrum the condition t ∼ 2n, reduces the two quantum numbers
characterizing the system to a single one, the whole wavefunction becoming assigned
by n.
The above analysis shows that, as effect of Dirichlet boundary condition and in the limit
of high occupation numbers, we get an analytical expressions for the discrete structure of the
spectrum. In fact for large values of t it was possible to give analytical representations for
the position of the zeros, but we emphasize that the request to deal with these approximations
causes the loss of a large number of levels and prevents a complete discussion of the quantum
chaos associated with the model. However the above expressions are of interest because they
allow us to compare these results with the corresponding spectrum provided by Misner in
his original work [11]. Of course, such a comparison of the two results can take place only
on a qualitative level; in fact, between Misner (α, β+ , β− ) and Misner–Chitré like (τ, u, v)
variables a crucial difference exists and it has to be recognized into the correspondingly
different behaviour of the potential walls. In the Misner scheme the domain available to the
point universe increases as α → ∞ (α being the isotropic Misner variable) and, therefore, we
deal with a non-stationary infinite well; the Misner–Chitré like variables allow us to fix the
infinite potential walls into a time-independent configuration in the (u, 0, v) plane. However
we can at least compare the behaviour of the energy eigenvalues with respect to the occupation
number n.
In his original treatment, Misner replaces, for fixed α values (indeed he makes use of
an adiabatic approximation, see [16]), the triangular well by a square of equal measure and
determines the energy spectrum in the form
π
A
EnM
=√
(54)
n2+ + n2− ≡ |n|,
h̄
α
S
where S denotes the triangular well measure S = α 2 /A2 , A being a numerical factor; n±
denote the occupation numbers relative to β± respectively.
In our approach, for sufficiently large n, we get the dominant behaviours

lπ (i)
MC
En
productlog lπn
(55)
∼
(ii) 2n + O(n2/3 ).
h̄
Thus we see that, in case (ii), a part from a numerical factor (which is different because of
the different approximation made on the real domains), the only significant difference relies
on the term α −1 ; in the case (i) instead the situation is a bit different because two quantum
numbers explicitly remain and we get a linear relation as far as we require l ∝ n by a
factor much greater than the unity (indeed the function productlog(lπ/n) provides a smooth
contributions in the considered region l n). The difference of the factor α −1 with respect
to the Misner case can be easily accounted as soon as we observe that the following relations
hold:
β± = αb± (u, v),
(56)
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where the functions b± can be calculated from the coordinates transformations

1 + u + u2 + v 2


α = −eτ

√


3v



−1
+
2u
+ 2u2 + 2v 2
(57)
β+ = eτ
√


2
3v





β− = −eτ 1 + 2u
2v
but their form is not relevant in what follows. On the basis of (56), the measure of a domain
D in the β± plane reads
dβ+ dβ− = α 2
|J (u, v)| du dv,
(58)
D
D
J being the Jacobian of the transformation associated with b± , while D is the image of D onto
the (u, 0, v) plane. As a consequence we see that between a measure s in the β± plane and a
similar one (even not exactly its image), there is a difference for a factor α 2 which immediately
provides an explanation for the difference in the two energy spectra.
7. The inhomogeneous picture and conclusions
At the end of our analysis, we wish to bring the reader’s attention to some physical aspects of
the inhomogeneous Mixmaster.
First of all, the obtained dynamics regime is indeed a generic one; in fact, in a synchronous
reference (for which ∂t y a = 0), the inhomogeneous Mixmaster contains four independent
(physically) arbitrary functions of the spatial coordinates (y a (x i ) and E(x i )), available for the
Cauchy data on a non-singular hyper-surface.
Let us now come back to the full inhomogeneous problem, in order to understand the
structure of the quantum spacetime near the cosmological singularity. Since the spatial
gradients of the configurational variables play no relevant dynamical role in the asymptotic
limit τ → ∞ (indeed the spatial curvature simply behaves as a potential well) then the
quantum evolution above outlined takes place independently in each space point and the total
wavefunction of the universe can be represented as follows:
(τ, u, v) = xi ξxi (τ, u, v)
(59)
where the product is (heuristically) taken over all the points of the spatial hypersurface.
However, it is worth to recall that, in the spirit of the ‘long-wavelength approximation’
adopted here, the physical meaning of a space point must be recovered on the notion of a
cosmological horizon; in fact we are dealing with regions over which the inhomogeneity
effects are negligible and this statement corresponds to super-horizon sized spatial gradients.
On a classical point of view, this request is at the ground of the BKL approximation and it is
well confirmed on the statistical level (see [4]); however on a quantum level it can acquire a
precise meaning if we refer the dynamics to a kind of lattice spacetime in which the spatial
gradients of the configurational variables become real potential terms. In this respect it is
important to observe that the geometry of the spacetime is expected to acquire a discrete
structure on the Planck scale and we believe that a regularization of our approach could arrive
from a ‘loop quantum gravity’ treatment [35].
Despite this local homogeneous framework of investigation, the appearance near the
singularity of high spatial gradients and of a spacetime foam (like outlined in the classical
dynamics see [4, 5]) can be easily recognized in the above quantum picture too. In fact the
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403
probability that in n different space points (horizons) the variables u and v take values within
the same narrow interval, decrease with n as pn , p being the probability in a single point; in
fact, these probabilities are all identical to each other and no interference phenomenon takes
place. From a physical point of view, this simple consideration indicates that a smooth picture
of the large scale universe is forbidden on a probabilistic level and different causal regions are
expected completely disconnected from each other during their quantum evolution. Therefore,
if we start with a strongly correlated initial wavefunction 0 (u, v) ≡ (τ0 , u, v), its evolution
towards the singularity induces increasingly irregular distributions, approaching (59) in the
asymptotic limit τ → ∞.
The main result of our presentation can be recognized in the clear correspondence
established between the classical spacetime foam and the quantum one. We have outlined how
this link takes place naturally for a precise choice of the operator-ordering only and how the
‘energy spectrum’ is a discrete one, due to the billiard structure of the point-like Hamiltonian.
Finally, we fixed as a new feature the zero-point ‘energy’ for the ground state associated with
the anisotropy degrees.
Acknowledgments
We would like to thank Orchidea Maria Lecian and Gian Paolo Imponente for revising the
manuscript.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
Belinski V A, Khalatnikov I M and Lifshitz E M 1970 Adv. Phys. 19 525
Belinski V A, Khalatnikov I M and Lifshitz E M 1982 Adv. Phys. 31 639
Misner C W 1969 Phys. Rev. Lett. 22 1071
Kirillov A A 1993 Zh. Eksp. Theor. Fiz. 103 721–9
Montani G 1995 Class. Quantum Grav. 12 2505
Kirillov A A and Montani G 1997 Phys. Rev. D 56 6225
Chernoff D F and Barrow J D 1983 Phys. Rev. Lett. 50 134
Kirillov A A and Montani G 1997 JETP Lett. 66 475
Cornish N J and Levin J J 1997 Phys. Rev. Lett. 78 998
Cornish N J and Levin J J 1997 Phys. Rev. D 55 7489
Imponente G P and Montani G 2001 Phys. Rev. D 63 103501
Misner C W 1969 Phys. Rev. 186 1319
Furusawa T 1986 Progr. Theor. Phys. 75 59
Berger B K 1989 Phys. Rev. D 39 2426
Graham R 1995 Chaos So1itons Fractals 5 1103
Kirillov A A 1992 JEPT Lett. 55 561
Imponente G P and Montani G 2006 Proc. ‘Albert Einstein’s Century’ Conf. (New York: AIP) (Preprint gr-qc/
0607009)
Arnowitt R, Deser S and Misner C W 1962 Gravitation: an Introduction to Current Research ed I Witten (New
York: Wiley)
Benini R and Montani G 2004 Phys. Rev. D 70 103527
Puzio R 1994 Class. Quantum Grav. 11 609–20
Chitré D M 1972 PhD Thesis University of Maryland
Montani G 2000 The Chaotic Universe ed V G Gurzadyan and R Ruffini (Singapore: World Scientific) p173
Barrow J D 1982 Phys. Rep. 85 1–49
Arnold V I 1989 Mathematical Methods of Classical Mechanics (Berlin: Springer)
Montani G 2001 Nuovo Cimento 116 B 1375
Imponente G and Montani G 2003 Int. J. Mod. Phys. D 12 977 (Preprint gr-qc/0404106)
Kuchar K 1981 Quantum Gravity II, a Second Oxford Symposium ed C J Isham et al (Oxford: Clarendon)
Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer)
Kolb E W and Turner M S 1990 The Early Universe (Reading, MA: Adison-Wesley)
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[30]
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[32]
[33]
[34]
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R Benini and G Montani
Kheyfets A, Miller W and Vaulin R 2006 Class. Quantum Grav. 23 4333–51
Terras A 1985 Harmonic Analysis on Symmetric Spaces and Applications I (New York: Springer)
Csordás A, Graham R and Szépfalusy P 1991 Phys. Rev. A 44 1491
Graham R, Hübner R, Szépfalusy P and Vattay G 1991 Phys. Rev. A 44 7002
Palm E 1957 Q. J. Mech. Appl. Math. 10 500
Balogh C B 1967 SIAM J. Appl. Math. 15 1315
Bojowald M 2003 Class. Quantum Grav. 20 2595
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Covariant Description of the Inhomogeneous Mixmaster Chaos
R. Benini12† and G. Montani23‡
1 Dipartimento
di Fisica - Università di Bologna and INFN
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
2 ICRA—International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9)
Università di Roma “La Sapienza”, Piazza A.Moro 5 00185 Roma, Italy
3 ENEA C.R. Frascati (U.T.S. Fusione),
via Enrico Fermi 45, 00044 Frascati, Rome, Italy
† [email protected][email protected]
We outline the covariant nature of the chaos characterizing the generic cosmological
solution near the initial singularity. Our analysis is based on a ”gauge” independent
ADM-reduction of the dynamics to the physical degrees of freedom, and shows that the
dynamics is isomorphic point by point in space to a billiard on a Lobachevsky plane. The
Jacobi metric associated to the geodesic flow is constructed and a non-zero Lyapunov
exponent is explicitly calculated. The chaos covariance emerges from the independence
of the form of the lapse function and the shift vector.
In recent years, the chaoticity of the homogeneous Mixmaster model has been
widely studied in the literature (see1–4 ) in view of understanding the features of its
covariant nature. Two convincing arguments, appeared in,1,4 support the idea that
the Mixmaster chaos (described by the invariant measure introduced in3,5 ) remains
valid in any system of coordinates.
The main issue of the present work is to show that the property of space-time covariance can be extended to the inhomogeneous Mixmaster model.
A generic cosmological solution is represented by a gravitational field having available all its degrees of freedom and, therefore, allowing to specify a generic Cauchy
problem. In the Arnowitt-Deser-Misner (ADM) formalism, the metric tensor corresponding to such a generic model takes the form
dΓ2 = N 2 dt2 − γαβ (dxα + N α dt)(dxβ + N β dt)
(1)
where N and N α denote (respectively) the lapse function and the shift-vector, γαβ
the 3-metric tensor of the spatial hyper-surfaces Σ3 for which t = const, being6
γαβ = eqa δad Oba Ocd ∂α y b ∂β y c
a
a
b
a, b, c, d, α, β = 1, 2, 3;
b
Oba
(2)
Oba (x)
q = q (x, t) and y = y (x, t) are six scalar functions, and
=
a SO(3)
matrix. By the metric tensor (2), the action for the gravitational field is
Z
(3)
dtd3 x pa ∂t q a + Πd ∂t y d − N H − N α Hα ,
S=
Σ(3) ×ℜ
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1 X 2
1 X
pb ) − γ (3) R]
H = √ [ (pa )2 − (
γ a
2
(4)
b
Hα = Πc ∂α y c + pa ∂α q a + 2pa (O−1 )ba ∂α Oba ;
(5)
in (4) and (5) pa and Πd are the conjugate momenta to the variables q a and y b respectively, and the (3) R is the Ricci 3-scalar which plays the role of a potential term.
We use the Hamiltonian constraints H = Hα = 0 to reduce the dynamics to the
physical degrees of freedom; the super-momentum constraints can be diagonalized
and explicitly solved by choosing the function y a as special coordinates:
Πb = −pa
S=
Z
Σ(3) ×ℜ
a
∂q a
−1 c ∂Oc
−
2p
(O
)
.
a
a
∂y b
∂y b
(6)
dηd3 y pa ∂η q a + 2pa (O−1 )ca ∂η Oca − N H .
(7)
The potential term appearing in the super-Hamiltonian behaves as an infinite potential wall as the determinant of the 3-metric goes to zero, and we can model it as
follows
(
X
+∞ if x < 0,
√ (3)
Θ(Qa )
Θ(x) =
U = γ R=
(8)
0
if x > 0.
a
where the Qa ’s are called anisotropy parameters. By (8) the Universe dynamics
evolves independently in each space point and the point-Universe can move within
a dynamically-closed domain ΓQ only6 (see figure (1)). Since in ΓQ the potential U
asymptotically vanishes, near the singularity we have ∂pa /∂η = 0; then the term
2pa (O−1 )ca ∂η Oca in (7) behaves as an exact time-derivative that can be ruled out of
the variational principle.
The ADM reduction is completed by introducing the so-called Misner-Chitré like
(τ, ξ, θ) variables,1 in terms of which the anisotropy parameters Qa become τ independent: When expressed in terms of such variables the super-Hamiltonian con-
5
1
Q1 = −
3
p
√
ξ2 − 1
(cos θ + 3 sin θ)
3ξ
4
3
2
p
√
1
ξ2 − 1
Q2 = −
(cos θ − 3 sin θ) (9)
3
3ξ
1
0
ξ
1
p
1 2 ξ2 − 1
cos θ
Q3 = +
3
3ξ
Fig. 1.
164
ϑ
ΓQ (ξ, θ)
-1
10
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straint can be solved in the domain ΓQ according to the ADM prescription:
s
p2
−pτ ≡ ǫ = (ξ 2 − 1)p2ξ + 2 θ
ξ −1
δSΓQ = δ
Z
dηd3 y(pξ ∂η ξ + pθ ∂η θ − ǫ∂η τ ) = 0 .
(10)
(11)
By virtue of the asymptotic limit (8) and the Hamilton equations associated with
(11) it follows that ǫ is a constant of motion, i.e. dǫ/dη = ∂ǫ/∂η = 0 ⇒ ǫ = E(y a );
furthermore ǫ∂η τ behaves as an exact time derivative.
This dynamical scheme allows us to construct the Jacobi metric corresponding to
the dynamical flow, and the line element reads
dξ 2
2
2
.
(12)
+
(ξ
−
1)dθ
ds2 = E 2 (y a )
(ξ 2 − 1)
Here the space coordinates behaves like external parameters since the evolution is
spatially uncorrelated. The Ricci scalar takes the value R = −2/E 2 , so that (12)
describes a two-dimensional Lobachevsky space; the role of the potential wall (8)
consists of cutting a closed domain ΓQ on such a negative curved surface.
In this scheme the Lyapunov exponent associated to the geodesic deviation along
the direction orthonormal to the geodesic 4-velocity can be evaluated, and it results
to be equal to
λ(y a ) =
1
> 0.
ǫ(y a )
(13)
Hence the chaotic behavior of the inhomogeneous Mixmaster model can be described
in a generic gauge (i.e. without assigning the form of the lapse function and of the
shift vector) as soon as Misner-Chitré like variables are adopted. The value (13) is
a positive definite function of the space point, making this calculation extendible
point by point to the whole space.
References
1.
2.
3.
4.
5.
6.
G. Imponente and G. Montani, Phys. Rev. D 63 (2001) 103501
G. Montani, Class. Quant. Grav. 18 (2001) 193
A. A. Kirillov and G. Montani, Phys. Rev. D 56 (1997) 6225.
N. J. Cornish and J. J. Levin, Phys. Rev. Lett. 78, 998 (1997)
D.F. Chernoff and J.D. Barrow, Phys. Rev. Lett, 50, 134,(1983).
R. Benini, G. Montani and R. Benini, Phys. Rev. D 70 (2004) 103527
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1
VECTOR FIELD INDUCED CHAOS IN MULTI-DIMENSIONAL
HOMOGENEOUS COSMOLOGIES
R. Benini12† , A. A. Kirillov3‡ , G. Montani24⋄
1 Dipartimento
di Fisica - Università di Bologna and INFN
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
2 ICRA—International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9)
Università di Roma “La Sapienza”, Piazza A.Moro 5 00185 Roma, Italy
3 Institute for Applied Mathematics and Cybernetics
10 Ulyanova str., Nizhny Novgorod, 603005, Russia
4 ENEA C.R. Frascati (U.T.S. Fusione), Via Enrico Fermi 45, 00044 Frascati, Roma, Italy
† [email protected][email protected][email protected]
We show that in multidimensional gravity vector fields completely determine the structure and properties of singularity. It turns out that in the presence of a vector field
the oscillatory regime exists for any number of spatial dimensions and for all homogeneous models. We derive the Poincaré return map associated to the Kasner indexes
and fix the rules according to which the Kasner vectors rotate. In correspondence to
a 4-dimensional space time, the oscillatory regime here constructed overlap the usual
Belinski-Khalatnikov-Liftshitz one.
1. Introduction
The wide interest attracted by the homogeneous cosmological models of the Bianchi
classification relies over all in the allowance for their anisotropic dynamics; among
them the types VIII and IX stand because of their chaotic evolution toward the
initial singularity1 that correspond to the maximum degree of generality allowed
by the homogeneity constraint; as a consequence it was shown2–4 that the generic
cosmological solution can be described properly, near the Big-Bang, in terms of the
homogeneous chaotic dynamics as referred to each cosmological horizon. However
the correspondence existing between the homogeneous dynamics and the generic
inhomogeneous one holds only in four space-time dimensions. In fact a generic
cosmological inhomogeneous model remains characterized by chaos near the BigBang up to a ten dimensional space-time5–7 while the homogeneous models show a
regular (chaos free) dynamics beyond four dimensions.8,9
Here we address an Hamiltonian point of view showing how the homogeneous models
(of each type) perform, near the singularity, an oscillatory regime in correspondence
to any number of dimensions, as soon as an electromagnetic field is included in the
dynamics.
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2. The Standard Kasner Dynamics
Let us consider the standard n + 1-dimensional vector-tensor theory in the ADM
representation:
Z
∂
∂
(1)
I = dn xdt Παβ gαβ + π α Aα + ϕDα π α − N H0 − N α Hα ,
∂t
∂t
1
H0 = √
g
β
Πα
β Πα −
1
1
2
(Πα ) + gαβ π α π β + g
n−1 α
2
1
Fαβ F αβ − R
4
Hα = −∇β Πβα + π β Fαβ ,
,
(2)
(3)
Here H0 and Hα denote respectively the super-Hamiltonian and super-momentum,
Fαβ ≡ ∂β Aα − ∂α Aβ is the electromagnetic tensor, g ≡ det(gαβ ) is the determinant
of the n-metric, R is the n-scalar of curvature and Dα ≡ ∂α + Aα .
Since the sources are absent, it is enough to consider only the transverse components
for Aα and π α ; therefore, we take the gauge conditions ϕ = 0 and Dα π α = 0. When
going over the homogeneous case, we choose the gauge N = 1 and N α = 0.
Let’s adopt the Kasner parameterization,that is based on the metric and conjugate
momentum decomposition along spatial n-bein:
gαβ = δab lαa lβb ,
Παβ = pab lαa lβb ,
(4)
αβ a
b
b
α a
α
We also define a dual basis Lα
lβ , such that Lα
a = g
a lα = δa and La lβ = δβ .
We want to put in evidence the oscillatory regime that the bein vectors possess and
so we distinguish scale functions and the parallel from the transverse component
ea = (π α ℓa ))
(λ
α
la = exp (q a /2) ℓa ,
La = exp (−q a /2) La .
(5)
~π~ℓa⊥ = 0.
(6)
e
~ℓak = λa ~π ,
π2
~ℓa = ~
ℓak + ~ℓa⊥ ;
The standard Kasner solution is obtained as soon as the limit in which all the terms
exp (q a ) become of higher order is taken
pa = const,
gαβ =
~
e
λa = const,
ℓa⊥
P ∂
1
2N
√
p
−
p
q
=
a
b b ,
∂t a
g
n−1
P
P 2
2
1 P qa e2
1
e λa =
pa − n−1 ( pa ) + 2
X
t2sa ℓaα ℓaβ ,
a
The Kasner indexes sa satisfy the identities
170
= const,
(7)
0,
pa
sa = 1 − (n − 1) P
,
b pb
P
sa =
P
s2a = 1.
(8)
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3. Billiard representation: the return map and the rotation of
Kasner vectors
If we order the sa ’s, the largest increasing term (as t → 0 ts1 → ∞) among the
neglected ones comes from s1 and it is to be taken into account to construct the
oscillatory regime toward the cosmological singularity.
∂
ea
p1 λ
∂e
∂e
λ1 = 0 ,
λa = ∂t
∂t
∂t
(pa − p1 )
∂
N e2
∂
p1 = − √ λ1 exp q 1 ,
pa = 0,
(9)
∂t
2 g
∂t
!
∂
1 X
2N
pa −
qa = √
pb .
∂t
g
n−1
b
e1 = const, while the second admits the solution
The first of equations (9) gives λ
ea (pa − p1 ) = const.
λ
(10)
The remaining part of the dynamical system allows us to determine the return map
governing the replacements of Kasner epochs and the rotation of Kasner vectors ~ℓa
through these epochs
s′1 =
e′ = λ
e1 ,
λ
1
~
ℓ1 ,
ℓ′a = ~ℓa + σa ~
−s1
,
2
s1
1 + n−2
s′a =
′
e
e
λa = λa 1 − 2
σa =
sa +
1+
2
n−2 s1
,
2
n−2 s1
(n − 1) s1
(n − 2) sa + ns1
(11)
,
ea
e′ − λ
ea
(n − 1) s1
λ
λ
a
= −2
.
e1
e1
(n − 2) sa + ns1 λ
λ
(12)
(13)
Thus the homogeneous Universes here discussed approaches the initial singularity being described by a metric tensor with oscillating scale factors and rotating
Kasner vectors. The presence of a vector field is crucial because, independently on
the considered model, it induces a closed domain on the configuration space.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
V.A. Belinski, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys., 19, 525, (1970).
V A Belinskii, I M Khalatnikov and E M Lifshitz, Adv. Phys. 31 (1982) 639.
A.A. Kirillov, Zh. Eksp. Teor. Fiz. 103, 721 (1993). [Sov. Phys. JETP 76, 355 (1993)].
G. Montani, Class. Quantum Grav. 12, 2505 (1995).
J. Demaret, M. Henneaux and P. Spindel, Phys. Lett., 164B, 27, (1985).
J. Demaret et al., BPhys. Lett., 175B, 129 (1986).
Y.Elskens, M.Henneaux, Nucl. Phys.290B(1987) 111
P.Halpern, Phys. Rev.D66 (2002) 027503
P.Halpern, Gen. Rel. Grav. 35 (2003) 251-261
171
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173
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1
Classical and Quantum Aspects of the Inhomogeneous Mixmaster
Chaoticity
R. Benini1 2† and G. Montani2 3‡
1 Dipartimento
di Fisica - Università di Bologna and INFN
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
2 ICRA—International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9)
Università di Roma “La Sapienza”, Piazza A.Moro 5 00185 Roma, Italy
3 ENEA C.R. Frascati (U.T.S. Fusione), Via Enrico Fermi 45, 00044 Frascati, Roma, Italy
† [email protected][email protected]
We refine Misner’s analysis of the classical and quantum Mixmaster in the fully inhomogeneous picture; we both connect the quantum behavior to the ensemble representation,
both describe the precise effect of the boundary conditions on the structure of the quantum states.
Near the cosmological singularity, the dynamics of a generic inhomogeneous cosmological model is reduced by an ADM procedure to the evolution of the physical
degrees of freedom, i.e. the anisotropies of the Universe. In fact asymptotically to
the Big-Bang the space points dynamically decouple1 because the spatial gradients
of the dynamical variables become of higher order2 and we can model the inhomogeneous Mixmaster via the reduced action:
I=
Z
d3 xdτ (pu ∂τ u + pv ∂τ v − ǫ) ,
ǫ=v
p
p2u + p2v ,
(1)
The dynamics of such a model is equivalent to (the one of) a billiard-ball on a
Lobachevsky plane; this can be shown by the use of the Jacobi metric.1 The manifold described turns out to have a constant negative curvature, where the Ricci
scalar is given by R = −2/E 2 : the complex dynamics of the generic inhomogeneous
model results in a collection of decoupled dynamical systems, one for each point of
the space, and all of them equivalent to a billiard problem on a Lobachevsky plane.
We want to investigate the relation existing between the classical and the semiclassical dynamics, and from this analysis we will derive the correct operator ordering
to be used when quantizing the system.
Let’s write down the Hamilton-Jacobi equation for the system
2
ǫ =v
2
δS0
δu
2
+
175
δS0
δv
2 !
(2)
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This can be explicitly solved by separation of constants
!
p
p
ǫ + ǫ2 − k 2 (y a )v 2
a
2
2
a
2
+ c(y a ) . (3)
S0 (u, v) = k(y )u + ǫ − k (y )v − ǫ ln 2
ǫ2 v
The semiclassical analysis can be developed furthermore, and the stationary continuity equation for the distribution function can be worked in order to obtain
informations about the statistical properties of the model; as soon as we restrict
the dynamics to the configuration space, we get the following equation for the distribution function w̃
s 2
∂ w̃(u, v; k)
∂ w̃(u, v; k) E 2 − 2k 2 v 2 w̃(u, v; k)
E
p
=0
(4)
+
+
−1
∂u
kv
∂v
kv 2
E 2 − (kv)2
This can be solved, and the exact distribution function can be obtained as soon as
we eliminate by integration the constant k
q
2
Z Ev g u + v kE2 v2 − 1
√
dk
(5)
w̃(u, v) =
v E 2 − k2 v2
−E
v
It is worth nothing how in the case g = const, the microcanonical Liouville measure
wmc (u, v) = vπ2 is recovered.
We expect that the distribution function w̃(u, v) is re-obtained as soon as the quantum dynamics is investigated to the first order in ~. This can be easily done as a
WKB approximation to the quantum dynamics is constructed; as soon as we retain
only the lower order in ~, we obtain that: i) the phase S(u, v) coincides with the
Hamilton-Jacobi function, and ii) the probability density function r(u, v) obeys the
following equation:
s
2
∂r(u, v) a(E 2 − k 2 v 2 ) − E 2
E
∂r(u, v)
√
r(u, v) = 0 .
(6)
− k2
+
+
k
∂u
v
∂v
v2 E 2 − k2 v2
that coincides with (4) for a particular choice for the operator-ordering only, i.e.
∂
∂
v̂ 2 p̂2v → −~2
v2
.
(7)
∂v
∂v
With this result, the problem of the full quantization of the system can be taken
in consideration. The main problem is the presence of the root square function
in the definition of the Hamiltonian (1), but well grounded motivations exist3 to
assume that the real Hamiltonian and the squared one have same eigenfunctions
and squared eigenvalues. This way the solution of the eigenvalue equation can be
obtained
X
Ψ(u, v) =
an Ks−1/2 (2nπv) sin(2nπu)
(8)
n>0
The spectrum is obtained as Dirichelet boundary conditions on the domain ΓQ (see
Fig.1) are taken into account. The condition on the vertical lines can be imposed
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3
exactly, but the one on the semicircle cannot be solved exactly; so we approximate
it as in Fig.2 with a straight line v = 1/π (it preserves the domain area µ = π).
All these imply that s = 1/2 + t, with t ∈ ℜ, and that the spectrum assumes the
3
2.5
v
2
1.5
1
0.5
0
-2
-1.5
-1
Fig. 1.
-0.5
u
0
0.5
1
The real domain ΓQ
Fig. 2.
The approximate domain
following form
1
Et2 = ( + t2 )~2
(9)
4
The values of the real parameter t have to be numerically evaluated by solving the
equation Kit (2n) = 0 for every natural n. This condition implies a discrete but
quite complicated shape for the spectrum.
Asymptotic expansions for high occupation numbers can be derived for different
regions of the parameters (t, n)4
References
1.
2.
3.
4.
R.
A.
R.
R.
Benini, G. Montani, Phys. Rev. D 70, 103527 (2004)
A. Kirillov, Zh. Eksp. Theor. Fiz. 103, 721-729 (1993)
Puzio, Class. Quantum Grav. 11, 609-620 (1994)
Benini, G. Montani, Class. Quantum Grav. in pubblication
177
Attachment 7
179
Attachment 7
IL NUOVO CIMENTO
Vol. ?, N. ?
?
Multi-Time approach to the Generic Quantum Cosmology
R. Benini(1 )(2 )(∗ ) and G. Montani(2 )(3 )(∗∗ )
(1 ) Dipartimento di Fisica - Università di Bologna and INFN
Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
(2 ) ICRA-International Center for Relativistic Astrophysics c/o Dipartimento di Fisica (G9) Università
di Roma “La Sapienza”, Piazza A.Moro 5 00185 Roma, Italy
(3 ) ENEA C.R. Frascati (Dipartimento F.P.N.),
via Enrico Fermi 45, 00044 Frascati, Rome, Italy
Summary. — Starting from the Hamiltonian formulation for the inhomogeneous Mixmaster
dynamics, we approach its quantum features through the link of the quasi-classical limit. We
fix the proper operator-ordering which ensures that the WKB continuity equation overlaps
the Liouville theorem as restricted to the configuration space. We describe the full quantum
dynamics of the model within the multi-time scheme, providing a characterization of the
(discrete) spectrum.
98.80.Cq 98.80.Qc
1. – Classical Dynamics
Hamiltonian Formulation.
Near the cosmological singularity, the dynamics of a generic inhomogeneous cosmological model is
reduced by an Arnowitt-Deser-Misner (ADM) procedure to the evolution of the physical degrees
of freedom, i.e. the anisotropies of the Universe. In fact asymptotically to the Big-Bang the
space points dynamically decouple [1] because the spatial gradients of the dynamical variables
become of higher order [2] and we can model the inhomogeneous Mixmaster[3, 4] via the reduced
action:
Z
p
(1)
I=
d3 xdτ (pu ∂τ u + pv ∂τ v − ) ,
= v p2u + p2v ,
ΓQ
where ΓQ is only a limited portion (see Fig.1) of the configuration-space because the asymptotic
behaviour of the curvature term is like the one of a infinite potential walls. The dynamics of such
a model is equivalent to (the one of) a billiard-ball on a Lobachevsky plane; this can be shown
(∗ ) [email protected]
(∗∗ ) [email protected]
1
c Società Italiana di Fisica
181
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2
R. BENINI AND G. MONTANI
3
2.5
v
2
1.5
1
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
u
Fig. 1.: The domain ΓQ (u, v) in the Poincaré upper half-plane where the dynamics of the spacepoint Universe is asymptotically restricted by the potential.
The measure is finite and equal to π.
by the use of the Jacobi metric [1]. The manifold described turns out to have a constant negative
curvature, where the Ricci scalar is given by R = −2/E 2 , E being a fixed value of : the subtle
dynamics of the generic inhomogeneous model results in a collection of decoupled dynamical
systems, one for each point of the space, and all of them equivalent to a billiard problem on a
Lobachevsky plane.
Hamilton-Jacobi approach.
The Hamilton-Jacobi (HJ) function in a fixed space point can be analytically obtained in this
scheme; let’s write down and solve the equation, i.e.
2
(2)
(3)
=v
a
S0 (u, v) = k(y )u +
p
2
−
2
∂S0
∂u
k 2 (y a )v 2
2
+
∂S0
∂v
2 !
+
!
p
2 − k 2 (y a )v 2
+ c(y a ) .
2 v
− ln 2
This result will be relevant to analize the semi-classical analysis below presented.
The statistical mechanics point of view .
Let us discuss the problem from a statistical mechanics point of view by treating the system as
a microcanonical ensemble. The physical properties of a stationary ensemble are described by
a distribution function ρ = ρ(u, v, pu , pv ), obeying the continuity equation defined in the phase
space (u, v, pu , pv ).
We stress how this equation provides an appropriate representation for the ensemble associated
to the Mixmaster only when we are sufficiently close to the initial singularity and therefore
the infinite-potential wall approximation works; in fact, such a model for the potential term
corresponds to deal with the energy-like constant of the motion and fixes the microcanonical
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Attachment 7
3
MULTI-TIME APPROACH TO THE GENERIC QUANTUM COSMOLOGY
nature of the ensemble. From a dynamical point of view this picture arises naturally because
the Universe volume element vanishes monotonically .
We are interested in studying the distribution function in the (u, v) space, and thus we will
reduce the dependence on the momentum variables by integrating ρ(u, v, pu , pv ) on the momenta
space. If we assume ρ to be a regular,
limited function and, using (3), then we work out the
R
following equation for w̃(u, v; k) = ρ(u, v, pu , pv )dpu dpv
(4)
∂ w̃(u, v; k)
+
∂u
s
E
kv
2
−1
∂ w̃(u, v; k) E 2 − 2k 2 v 2 w̃(u, v; k)
p
=0
+
∂v
kv 2
E 2 − (kv)2
admitting a solution in terms of a generic function g of the form
(5)
w̃(u, v) =
q
E2
k2 v 2
−1
g u+v
√
v E 2 − k2 v2
The distribution function in (u, v) is obtained after the integration over the constant k. Indeed,
this constant expresses the freedom of choosing the initial conditions, which cannot affect the
chaotic properties of the model. Therefore we define the reduced distribution w(u, v) as
E E
A≡ − ,
.
v v
Z
(6)
w(u, v) ≡
w̃(u, v; k)dk ,
A
It is easy to verify that the microcanonical Liouville measure wmc corresponds to the case g =
const, i.e. we get the normalized distribution
Z
(7)
E
v
1
wmc (u, v) =
−E
v
kv 2
q
E2
k2 v 2
dk =
−1
π
.
v2
2. – Semi-Classical Analysis
Let us underline some common features between the classical and the semi-classical dynamics
with the aim of fixing the proper operator-ordering in treating the quantum approach.
In fact, considering the WKB limit for ~ → 0, the coincidence between the classical distribution
function wmc (u, v) and the quasi-classical probability function r(u, v) takes place for a precise
choice of the operator-ordering only.
As soon as we implement the canonical variables into operators, the quantum dynamics for the
state function Φ(u, v, τ ) obeys, in each point of the space, the Scrhödinger equation
(8)
∂Φ(u, v, τ )
i~
= ĤADM Φ(u, v, τ ) = ~
∂τ
s
−v 2
∂2
∂
− v 2−a
∂u2
∂v
va
∂
Φ(u, v, τ ) ,
∂v
where we have adopted a generic operator-ordering for the position and momentum operators
parametrized by the constant a.
In the above equation the Hamiltonian operator has a non-local character as a consequence of
the square root function; we will make the well-grounded ansatz[5] that the operators ĤADM
2
and ĤADM
have the same set of eigenfunctions with eigenvalues E and E 2 , respectively.
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4
R. BENINI AND G. MONTANI
If we take a generalised Fourier transform Φ(u, v, τ ) for the wave function Ψ(u, v, E) the eigenvalues problem reduces to
(9)
∂2
∂
∂
Ĥ 2 Ψ(u, v, E) = ~2 −v 2 2 − v 2−a
va
Ψ(u, v, E) = E 2 Ψ(u, v, E) .
∂u
∂v
∂v
In order to study the WKB
p limit of equation (9), we separate the wave function into its phase and
modulus Ψ(u, v, E) = r(u, v, E)eiσ(u,v,E)/~ (the function r(u, v) representing the probability
density). The quasi-classical regime then appears as we take the limit for ~ → 0; substituting in
(9) and retaining only the lower order in ~, we obtain the system
(10)
 " 2 #
2

∂σ
∂σ

2

+
= E2
v
∂u
∂v

∂r ∂σ
a ∂σ ∂ 2 σ ∂ 2 σ
 ∂r ∂σ

= 0.
+
+r
+ 2 +

∂u ∂u ∂v ∂v
v ∂v
∂v
∂u2
Thus we can identify the phase σ with the function S0 of the HJ approach.
Now we turn our attention to the equation for r(u, v); taking (3) into account it reduces to
(11)
∂r(u, v, E)
k
+
∂u
s
E
v
2
− k2
∂r(u, v, E) a(E 2 − k 2 v 2 ) − E 2
√
+
r(u, v, E) = 0 .
∂v
v2 E 2 − k2 v2
Comparing (11) with (4), we easily see that they coincide for a = 2 only.
v̂ 2 p̂2v
(12)
∂
→ −~
∂v
2
∂
v
∂v
2
.
This unique choice of the operator-ordering allows us to face the canonical quantization of the
model which matches, in the semi-classical limit, the statistical behaviour of the Mixmaster.
3. – Canonical Quantization
The Laplace-Beltrami operator and the eigenfunctions.
Our starting point is the point-like eigenvalue equation (9) with a = 2 which, together with the
boundary conditions, completely describes the quantum features of the model.
By redefining Ψ(u, v, E) = ψ(u, v, E)/v, we can reduce (9) to the eigenvalue problem for the
Laplace-Beltrami operator in the Poincaré plane
(13)
∇LB ψ(u, v, E) ≡ v
2
∂2
∂2
+ 2
2
∂u
∂v
ψ(u, v, E) = Es ψ(u, v, E) ,
(Kν (x) are Bessel function of the third order). From the general theory of this operator, we get
the eigenfunctions and eigenvalues for our model in the form
(14)
Ψ(u, v, E) = a v s−1 + bv −s +
X
n6=0
(15)
E
2
2
= s(1 − s)~ .
184
an
Ks−1/2 (2π|n|v) 2πinu
√
e
,
v
Attachment 7
5
MULTI-TIME APPROACH TO THE GENERIC QUANTUM COSMOLOGY
The boundary conditions and the energy spectrum.
The spectrum and the explicit eigenfunctions are obtained by imposing the boundary conditions,
i.e. requiring that (14) vanishes on the boundaries of the domain ΓQ in Fig.1.
Since from our analysis no analytical way arose to impose exact boundary conditions we approximate the semicircle with the horizontal line y = 1/π.
The conditions on the vertical lines u = 0, u = −1 require to disregard the first two terms in (14)
(a = b = 0); furthermore we get the
P condition on the
P∞last term (here we disregard the odd part
because of symmetry reasons [7]) n6=0 e2πinu → n=1 sin(2πnu), n being an integer. Instead
the condition on the horizontal line y = 1/π implies
X
(16)
an Ks−1/2 (2n) sin(2nπu) = 0, ∀u ∈ [−1, 0] ,
n>0
which is satisfied by requiring Ks−1/2 (2n) = 0 only. The functions Kν (x) are real and positive
for real argument and real index, therefore we take s = 21 + it. In this case these functions have
(only) real zeros, and the corresponding eigenvalues turn out to be real and positive
(E/~)2 = t2 + 1/4 .
(17)
From (17) the energy of the ground state comes out to be always greater than zero. We remark
that our eigenfunctions naturally vanish as infinite values of v are approached.
The conditions (16) cannot be solved analytically for generic the values of n and t, and the roots
must be worked out numerically for each n. We conclude listing the first ten eigenvalues and
plotting the wave function of the ground- state.
E
~
2
= t2 +
1
4
19.831
40.357
49.474
63.405
87.729
89.250
116.329
128.234
138.739
146.080
!gs (u,v)
1.5
1
0.5
0
-0.5
-1
-1.5
1.25
1
-1
0.75
-0.75
-0.5
u
-0.25
0.5
0
Fig. 2.: The ground state wave function
REFERENCES
[1] R. Benini, G. Montani, Phys. Rev. D 70, 103527 (2004)
[2] A. A. Kirillov, Zh. Eksp. Theor. Fiz. 103, 721-729 (1993)
[3] V. A. Belinski, I. M. Khalatnikov, E. M. Lifshitz , Adv. Phys. 19, 525 (1970)
V. A. Belinski, I. M. Khalatnikov, E. M. Lifshitz , Adv. Phys. 31, 639 (1982)
[4] C. W. Misner, Phys. Rev. Lett. 22, 1071 (1969)
[5] R. Puzio, Class. Quantum Grav. 11, 609-620 (1994)
[6] R. Benini, G. Montani, Class. Quantum Grav. 24, 387-404 (2007).
[7] Graham, R., available on gr-qc/9403030
185
v
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