Dynamical Systems and Cosmology

203 Pages·2003·9.32 MB·English

by  A. A. Coley (auth.)

DYNAMICAL SYSTEMS AND COSMOLOGY
ASTROPHYSICS AND
SPACE SCIENCE LIBRARY
VOLUME 291
EDITORIAL BOARD
Chainnan
W.B. BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A.
(
DYNAMICAL SYSTEMS AND
COSMOLOGY
by
A.A. COLEY
Dalhousie University, Halifax, Canada
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6329-8 ISBN 978-94-017-0327-7 (eBook)
DOI 10.1007/978-94-017-0327-7
Printed on acid-free paper
All Rights Reserved
© Springer Science+Business Media Dordrecht 2003
Originally published by Kluwer Academic Publishers in 2003.
Softcover reprint of the hardcover I st edition 2003
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Contents
I. Introduction 1
A. Self-similarity 3
II. The Theory of Dynamical Systems 7
A. Linear Autonomous Differential Equations 8
1. Topological Equivalence 13
2. Linear Stability 14
B. Non-Linear Differential Equations 15
1. Liapunov Theory 17
2. Linearization and the Hartman-Grobman Theorem 19
C. Periodic Orbits and The Poincare-Bendixson Theorem in the Plane 21
D. More General Non-Linear Behaviour 23
1. Higher Dimensions 23
III. Spatially Homogeneous Models 27
A. Definitions and Kinematical Quantities 28
B. Asymptotic States of Perfect Fluid Bianchi Models 30
C. More Recent Work 33
D. Scalar Field Models 36
E. Harmonic Potentials 39
1. Analysis 40
2. Chaos 42
IV. Scalar Field Cosmologies with Barotropic Matter 44
A. Models of Bianchi Class B: The Equations 44
1. Invariant Sets, Monotone Functions and The Constraint Surface 46
B. Classification of the Equilibrium Points 48
1. Scalar Field Models 50
2. Perfect Fluid Models 52
3. Scaling Solutions 53
VI
C. Stability of the Equilibrium Points and Some Global Results 55
D. Discussion 57
V. Physical Applications 59
A. Isotropisation 59
1. Analysis of the Bianchi VlIh Equations 60
B. Matter Scaling Solutions 62
1. Stability of the Matter Scaling Solution 63
VI. Closed Models 67
A. Closed Friedmann Models 67
1. Qualitative Analysis 69
B. Kantowski-Sachs Models 73
1. Quatative Analysis 76
C. Discussion 80
VII. Multiple Scalar Fields 82
1. The Model 83
A. Qualitative Analysis of the Two-Scalar Field Model 84
1. Assisted Inflation 85
2. Stability of Equilibria and Discussion 85
B. Qualitative Analysis of the Two-Scalar Field Model with Matter 88
1. Invariant Sets, Monotonic Functions and Stability of Equilibria 88
2. Matter Scaling Solutions 90
C. Qualitative Analysis of the Three-Scalar Field Model 91
D. Discussion 94
VIII. Scalar Tensor Theories of Gravity 96
1. Stiff Perfect Fluids in General Relativity 97
A. Scalar-Tensor Theories of Gravity with No Potential 98
1. Application: Brans-Dicke Theory 98
B. Scalar Tensor Theories with a Non-Zero Potential 100
VB
l. Application 101
C. Inhomogeneous Models 103
l. Inhomogeneous 104
IX. Magnetic Field Cosmology 106
A. Bianchi VIo Models 107
l. Discussion of Qualitative Properties 109
B. Bianchi I Models III
l. Qualitative Analysis ll2
X. String Cosmology ll6
l. Low-Energy Effective Action ll8
A. Cosmological Field Equations 121
l. Exact Solutions 121
2. Pre-Big Bang Cosmology 123
B. Qualitative Analysis of the NS-NS Sector 124
l. Models with Positive Central Charge Deficit 125
C. Qualitative Analysis of the Matter Sector 129
l. Positive Cosmological Constant 130
2. Discussion 132
D. Cyclical Behaviour in Early Universe Cosmologies 134
l. Non-Zero Central Charge Deficit 135
2. Discussion 138
XI. Anisotropic and Curved String Cosmologies 140
A. Non-Zero Central Charge Deficit 141
l. The Case A > 0, K > ° 142
2. The Case A> 0, K < ° 146
3. The Case I\. < 0, k > 0 148
4. The Case A < 0, k < 0 150
B. Discussion 152
viii
XII. M -Theory 156
1. Four-Dimensional Effective Action 157
A. Cosmological Field Equations 159
B. Structure of State Space and Local Analysis 161
1. Qualitative Analysis of Spatially Flat Cosmologies 162
2. Physical Interpretation 163
C. Discussion 166
1. Effects of Spatial Curvature 166
2. Inhomogeneous String Cosmology 168
3. Future Work 171
Acknowledgments 174
References 175
I. INTRODUCTION
In the standard cosmological paradigm the Universe is described by an expand-
ing Friedmann-Robertson-Walker (FRW) model with a hot big bang. Mathematical
cosmology involves the study of the early- and late- time behaviour of more gen-
eral classes of cosmological models as well as the detailed investigation of special
exact cosmological solutions with symmetries such as the spatially homogeneous
and isotropic models with a Robertson-Walker (RW) geometry and self-similar cos-
mological models. Dynamical systems techniques are powerful tools in such an
investigation. This is particularly true for the qualitative analysis of spatially homo-
geneous cosmological models whose evolution is governed by a (finite-dimensional)
autonomous system of ordinary differential equations (ODE). Inhomogeneous cos-
mological models can also be studied, but since the evolution equations are au-
tonomous partial differential equations (PDE) in this case the resulting state space
is infinite-dimensional (i.e., a function space), and the analysis is considerably more
complicated.
The primary aim of this book is to study the qualitative properties of cosmo-
logical models at early times using dynamical systems theory techniques, thereby
obtaining important information about the early Universe. 'We shall primarily as-
sume that the evolution of the universe is governed by Einstein's theory of General
Relativity (GR). However, to study the early stages of the Universe new physics,
such as string theory, is necessary close the Planck time scale. Although a complete
fundamental theory is not presently known the phenomenological consequences can
be understood by studying an effective low-energy theory, which leads to the in-
troduction of additional fields (e.g., scalar fields), or by the study of alternative
theories (to GR) such as scalar-tensor theories. Indeed, it is through observational
constraints from early Universe cosmology that these new theories can be tested.
Dynamical systems are also useful for studying the intermediate behaviour of
models in the inflationary epoch, in the pre-galactic radiation era during which cos-
mic nu~leosynthesis occurs up to the development of the cosmic microwave back-
ground, and in the galactic epoch up to the present, and is especially well-suited for
investigating the late-time behaviour of cosmological models (in some applications
inflation can be investigated as a late-time phenomena).
Scalar fields are believed to be abundant and pervasive in all fundamental theo-
ries of physics applicable in the early Universe, and we shall include scalar fields as
matter fields in all of the models we study. We shall be interested in the dynamics
in the high curvature regime at the Planck time close to a (big bang) singularity
and in the inflationary epoch during which the universe undergoes accelerated ex-
pansion and, in particular, we shall study the possible isotropization and inflation
of the models under investigation. At later times, and particularly after the epoch
of galaxy formation and later, the cosmological matter can be idealized as a per-
fect fluid. Perfect fluid cosmological models were studied in detail in the book by
Wainwright and Ellis (WE) [363], and the present treatise can be regarded as com-
plementary to this earlier work. We should note, however, that there is currently
A. A. Coley, Dynamical Systems and Cosmology
© Springer Science+Business Media Dordrecht 2003
2
some motivation from quintessence and the possible existence of dark matter to
study models with a non-trivial scalar field at late times.
The governing equations of the most commonly studied cosmological models are
a system of ODE. Since our main goal is to give a qualitative description of these
models, a dynamical systems approach is undertaken. Usually, a dimensionless (log-
arithmic) time variable, T, is introduced so that the models are valid for all times
(Le., T assumes all real values). A normalised set of variables are then chosen for
a number of reasons. First, this normally leads to a bounded dynamical system.
Second, these variables are well-behaved and often have a direct physical interpre-
tation. Third, due to a symmetry in the equations, one of the equations decouple
(in GR the expansion is used to normalize the variables in ever expanding models
whence the Raychaudhuri equation decouples) and the resulting simplified reduced
system is then studied. The equilibrium points of the reduced system then cor-
respond to dynamically evolving self-similar cosmological models. More precisely,
using the dimensionless time variable and a normalised set of variables, the gov-
erning ODE define a flow and the evolution of the cosmological models can then
be analysed by studying the orbits of this flow in the physical state space, which
is a subset of Euclidean space. When the state space is compact, each orbit will
have a non-empty a-limit set and w-limit set, and hence there will be a both a past
attractor and a future attractor in the state space.
There are two important reasons for studying exact self-similar solutions of the
Einstein field equations (EFE). First, the assumption of self-similarity reduces the
mathematical complexity of the governing differential equations (DE), often leading
to the reduction of PDE to ODE in problems of physical interest. This makes such
solutions easier to study mathematically. Indeed, self-similarity in the broadest
(Lie) sense refers to an invariance which allows such a reduction. Second, self-
similar solutions play an important role in describing the asymptotic properties of
more general models. For example, the expansion ofthe Universe from the big bang
and the collapse of a star to a singularity might both tend to self-similarity in some
circumstances. In the cosmological context, the suggestion that fluctuations might
naturally evolve from complex initial conditions via the EFE to self-similar form
has been termed the "similarity hypothesis" , although this certainly does not apply
in all circumstances [77]. In the next Subsection we discuss the important notion
of self-similarity.
A review of the theory of dynamical systems is given in Chapter II. The existence
of monotone functions in various invariant sets is very important and leads to con-
siderable simplification of the dynamics: there can be no periodic orbits, recurrent
orbits or homo clinic orbits (see definitions below) in the corresponding state space.
Consequently the dynamics is dominated by equilibrium points and heteroclinic
cycles, and hence self-similar models, which often correspond to equilibrium points
in the system, playa dominant role in the dynamical behaviour. In addition, the
monotonicity can be used to determine the past and future attractors.
The outline of the rest of the book is as follows. In Chapter III we discuss and
review spatially homogeneous perfect fluid models. In Chapter IV we begin the