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. (burton@starband.net); University of Leiden, The Netherlands (burton@strw.leidenuniv.nl)

Executive Committee

J. M. E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands E. P. J. VAN DEN HEUVEL, Astronomical Institute, University of Amsterdam,

The Netherlands H. VAN DER LAAN, Astronomical Institute, University of Utrecht,

The Netherlands

MEMBERS

1. APPENZELLER, Landesstemwarte Heidelberg-Konigstuhl, Gennany J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.S.A.

F. BERTOLA, Universita di Padova, Italy J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A.

C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway

R. McCRAY, University of Colorado, JILA, Boulder, U.S.A. P. G. MURDIN, Institute of Astronomy, Cambridge, U.K.

F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India

K. SATO, School of Science, The University of Tokyo, Japan F. H. SHU, University of California, Berkeley, U.S.A.

B. V. SOMOV, Astronomical Institute, Moscow State University, Russia R. A. SUNYAEV, Space Research Institute, Moscow, Russia

Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan S. TREMAINE, CITA, Princeton University, U.S.A.

N. O. WEISS, University of Cambridge, U.K.

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 in any form or by any means, electronic, mechanical, photocopying, microfilming, recording

or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered

and executed on a computer system, for exclusive use by the purchaser of the work.

I. Introduction

A. Self-similarity

II. The Theory of Dynamical Systems

Contents

1

3

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

A. Definitions and Kinematical Quantities

B. Asymptotic States of Perfect Fluid Bianchi Models

C. More Recent Work

D. Scalar Field Models

E. Harmonic Potentials

1. Analysis

2. Chaos

IV. Scalar Field Cosmologies with Barotropic Matter

27

28

30

33

36

39

40

42

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

V.

C. Stability of the Equilibrium Points and Some Global Results

D. Discussion

Physical Applications

A. Isotropisation

1. Analysis of the Bianchi VlIh Equations

B. Matter Scaling Solutions

1. Stability of the Matter Scaling Solution

55

57

59

59

60

62

63

VI. Closed Models 67

67 A. Closed Friedmann Models

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

1. Four-Dimensional Effective Action

A. Cosmological Field Equations

B. Structure of State Space and Local Analysis

1. Qualitative Analysis of Spatially Flat Cosmologies

2. Physical Interpretation

C. Discussion

1. Effects of Spatial Curvature

2. Inhomogeneous String Cosmology

3. Future Work

Acknowledgments

References

156

157

159

161

162

163

166

166

168

171

174

175

I. INTRODUCTION

In the standard cosmological paradigm the Universe is described by an expand­ing Friedmann-Robertson-Walker (FRW) model with a hot big bang. Mathematical cosmology involves the study of the early- and late- time behaviour of more gen­eral 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 cos­mological models. Dynamical systems techniques are powerful tools in such an investigation. This is particularly true for the qualitative analysis of spatially homo­geneous cosmological models whose evolution is governed by a (finite-dimensional) autonomous system of ordinary differential equations (ODE). Inhomogeneous cos­mological models can also be studied, but since the evolution equations are au­tonomous 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 cosmo­logical models at early times using dynamical systems theory techniques, thereby obtaining important information about the early Universe. 'We shall primarily as­sume 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 in­troduction 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 cos­mic nu~leosynthesis occurs up to the development of the cosmic microwave back­ground, 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 theo­ries 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 ex­pansion 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 per­fect 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 com­plementary 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 (log­arithmic) 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 interpre­tation. 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 cor­respond to dynamically evolving self-similar cosmological models. More precisely, using the dimensionless time variable and a normalised set of variables, the gov­erning 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, self­similar 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 con­siderable 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

3

study of scalar field models, with particular emphasis on Bianchi models of class B with barotropic matter, and in the next Chapter we discuss some applications. In Chapter VI we study closed models and in Chapter VII we discuss multiple scalar field models. In Chapter VIII we discuss scalar-tensor theories and in Chapter IX we investigate magnetic field models. In Chapter X we discuss string cosmologi­cal models and in the following Chapter we extend the analysis to anisotropic and curved string models. In the final Chapter, we discuss M-theory cosmology. Dy­namical systems can also be used in the analysis of other early universe models such as, for example, brane cosmological models. This will be briefly discussed in the final Section.

The notation used in this book is given in Chapter III (in particular see Section lILA, but also see Sections LA and IV.A) and at the beginning of Chapters X and XII. A number of commonly used acronyms are displayed in the following table for convenience.

GR I General Relativity KS I Kantowski-Sacks

(E)FE I (Einstein) Field Equations BD(T) I Brans-Dicke (Theory)

(DIP)DE I (Ordinary/Partial) Differential Equation(s) I CMB I Cosmic Microwave Background

(F)RW I (Friedrnann-)Robertson-Walker NS I Neveu-Schwarz

(O)SH I (Orthogonal) Spatially Homogeneous RR I Ramond-Ramand

LRS I Locally Rotationally Symmetric WE I Wainwright and Ellis [363]

A. Self-similarity

Self-similar solutions of the EFE play an important role in describing the asymp­totic properties of more general models. Let us consider a perfect fluid spacetime in which the energy-momentum tensor is given by

Tab = (J.l + P)UaUb + pgab, (1.1)

where ua is the normalized fluid 4-velocity, J.l is the density and p is the pressure satisfying a linear barotropic equation of state of the form

p = ("( - 1)J.l, (1.2)

where 'Y is a constant. The existence of a self-similarity of the first kind can be invariantly formulated in terms of the existence of a homothetic vector [69]. For a

4

general spacetime a proper homothetic vector (HV) is a vector field ~ which satisfies

'cf.g/J.V = 2g/J.v, (1.3)

where g/J.v is the metric and ,c denotes Lie differentiation along~. An arbitrary constant on the right-hand-side of (1.3) has been rescaled to unity. If this constant is zero, i.e., 'cf.guv = 0, then ~ is a Killing vector. A homothetic motion or homothety captures the geometric notion of "invariance under scale transformations" .

From (1.3) it follows that

'cf.Ra bed = 0, (1.4)

and hence

'cf.Rab = 0, 'cf.Gab = 0. (1.5)

A vector field ~ that satisfies equation (1.4) is called a curvature collineation, one that satisfies equation (1.5a) is called a Ricci collineation, and one that satisfies equation (1.5b) is called a matter collineation.

For a perfect fluid, it follows from equation (1.3) and the EFE that the physical quantities transform according to

'cf.ua = _ua, (1.6)

and

'cf./-L = -2/-L, 'cf.P = -2p, (1.7)

where

'cf.Tab = 0. (1.8)

Perfect fluid spacetimes admitting a homothetic vector within GR have been com­prehensively studied by Eardley [135). In such spacetimes equations (1.6) and (1.7) imply that all physical quantities transform according to their respective dimensions, so "geometrical" and "physical" self-similarity coincide. The question of whether the matter field exhibits the same symmetries as the geometry within GR is called the symmetry "inheritance" problem [100). If a perfect fluid spacetime admits a non-trivial homothetic vector and there is an equation of state of the form P = p(/-L), then necessarily p//-L is constant [69); i.e., equation (1.2) results from equation (1.7). We note that homothetic initial data is preserved by the EFE [135). In the case of radiation (with P = i /-L), the existence of a homothetic vector implies the existence of a conserved current, pa = ~ (4UaUbe + ~a) (i.e., pa ;a = 0). Vacuum spacetimes admitting a homothetic vector were studied by McIntosh [273).

The geometric properties of homothetic vectors were studied by Yano [380). The totality of homothetic vectors on a spacetime form a Lie algebra Hn of dimension n which (if Hn is non-trivial) contains an (n - 1) dimensional sub algebra of Killing

5

vectors, Gn - I . Except when the spacetime is conformal to a "generalized plane­wave" spacetime, it follows that if the orbits of Hn are r-dimensional, then the orbits of Gn - I are (r - I)-dimensional [135]. If, in addition, the spacetime is not conformally flat, then it is conformally related to a spacetime for which the Lie algebra Hn is the Lie algebra of Killing vectors [127]. In the trivial case of a (locally) flat spacetime, the dimension of the homothetic algebra is eleven and that of its associated Killing subalgebra is ten. The orbits of the homothetic group and the isometry group can coincide only if they are 4-dimensional or 3-dimensional and null; the resulting spacetime is consequently either locally flat or is a special type of "generalized plane-wave" spacetime [170].

Self-similar models are often related to the asymptotic states of more general models [190]. In particular, self-similar models play an important role in the asymp­totic properties of spatially homogeneous models [138, 221], spherically symmetric models [77], G2 models [363] and silent universe models [62]. We note that the self-similar Bianchi models of relevance below are transitively self-similar (in the sense that the orbits of the H4 are the whole spacetime).

Let us briefly review some exact self-similar solutions that are of particular im­portance in cosmology:

• Minkowski space (M):

ds2 -de + dx2 + dy2 + dz2, (1.9)

~ 8 8 8 8

t 8t + x 8x + y 8y + z 8z . (1.10)

Alternative forms of Minkowski space are given in Wainwright and Ellis [363].

The (RW) line element for a spatially homogeneous and isotropic spacetime has the form

[ dr2 ] ds2 = -dt2 + R2(t) 1 _ kr2 + r2(dfj2 + sin2((})d¢2) , (1.11)

where k = +1, -1,0 determines whether the model is closed (positive-curvature), open (negative-curvature), or flat (zero-curvature). All FRW models admit a time­like HV in the special case of matter satisfying J.L + 3p = O. Otherwise, only the k = 0 models admit a HV, and this occurs for all such models in which p = (-y-l)J.L and hence the scale function has power-law dependence on time .

• k = 0, FRW (F):

ds2 -dt2 + t 4 / 3-y (dx2 + dy2 + dz2),

8 ( 2)(8 8 8) ~ = t- + 1 - - x- + y- + z- , 8t 3)' 8x 8y 8z

where

4 -2 ( _~) p = (-y -1)J.L; J.L = 3)'2t U - at .

(1.12)

(1.13)

(1.14)

6

All transitively self-similar orthogonal spatially homogeneous perfect fluid solu­tions (with ~ < 'Y ::::: 2) and spatially homogeneous vacuum solutions are summa­rized in Hsu and Wainwright [190]. In particular, the Kasner vacuum solution is self-similar .

• Kasner (vacuum) (K):

ds2 -dt2 + t 2P1 dx2 + ep2 dy2 + t2P3 dz2 ,

~ a a a a

t- + (1 - Pl)X- + (1 - P2)Y- + (1 - P3)Z-, at ax ay az

where

~Pi = ~p~ = 1.

(1.15)

(1.16)

(1.17)

II. THE THEORY OF DYNAMICAL SYSTEMS

We shall begin by presenting a brief review of the qualitative theory of ODE [169, 181, 182, 373], with particular emphasis on those aspects of the non-linear theory of importance in cosmology. Let us consider the system of ODE, or more briefly the DE, of the form

x' = f(x), (2.1)

where x' == ~~, x == (Xl,"', X n) E JRn and f : JRn -+ JRn, where we shall assume that f is (at least) CIon JRn. Since the right-hand-side of (2.1) does not depend on t explicitly, the DE is called autonomous. In general f will be a non-linear function, but if f is linear, i.e.,

f(x) = Ax, (2.2)

where A is an n x n matrix of real numbers, then the DE is said to be linear. The vector x E JRn is called the state vector of the system, and JRn is called the state space or phase space. The function f can be interpreted as a vector field on JRn, since it associates with each x E JRn an element f(x) on JRn, which can be interpreted as a vector f(x) = (h(x), ... , fn(x)) at x. A solution of the DE (2.1) is a function 'Ij; : JR -+ JRn which satisfies

'Ij;'(t) = f('Ij;(t)), (2.3)

for all t E JR in the domain of'lj; (which may be a finite open interval). The image of the solution curve 'Ij; in JRn is called an orbit of the DE. Equation (2.3) implies that the vector field f at x is tangent to the orbit through x. The evolution of the system in time is described by the motion of a point x E JRn representing the state of the physical system along an orbit of the DE in JRn.

In general we cannot hope to find exact solutions of a non-linear DE (2.1) for n ~ 2. Consequently qualitative methods, perturbative methods, or numerical methods, must be utilized to deduce the behaviour of the physical system. The aim of qualitative analysis is to understand the qualitative behaviour (such as, for exam­ple, the long-term behaviour as t -+ 00) of typical solutions of the DE. Exceptional solutions such as equilibrium solutions or periodic solutions, and their stability, are also of interest and can be important for determining the long-term behaviour of typical solutions.

The zeros of the vector field, or equilibrium points of the DE (2.1), are points a E JRn such that

f(a) = O. (2.4)

Equilibrium points are also referred to as singular points or fixed points. If f(a) = 0, then 'Ij;(t) = a, for all t E JR, and it is a solution of the DE, since

'Ij;'(t) = f('Ij;(t)) (2.5)

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

8

is satisfied trivially for all t E JR. A constant solution 'lj;(t) = a describes an equi­librium state of the physical system. In order to study the stability of equilibrium states it is necessary to study the behaviour of the orbits of the DE close to the equilibrium points, and hence we consider the linear approximation of the vector field 1 : JRn -t JRn at an equilibrium point. We shall consequently assume that the partial derivatives of 1 exist and are continuous functions on JRn (Le., that the function 1 is of class C 1 (JR n ) ) .

The derivative matrix of I: JRn -t JRn is the n x n matrix D/(x) defined by

D/(x) = (;~;), i,j = 1, ... , n, (2.6)

where the Ii are the component functions of I. The linear approximation of 1 is written in terms of the derivative matrix:

I(x) = I(a) + D/(a)(x - a) + Rl (x, a), (2.7)

where D/(a)(x - a) denotes the n x n derivative matrix evaluated at a, acting on the vector (x - a), and R 1 (x, a) is the error term such that if 1 is of class C 1 then the magnitude of the error IIR1(x,a)11 tends to zero faster than the magnitude of the displacement IIx - all (where the Euclidean norm on JRn is defined by IIxll = "';X1 2 + ... + x n 2 ). If a E JRn is an equilibrium point of the DE (2.1), using (2.7) the DE can be written in the form

x' = D I(a)(x - a) + Rl (x, a). (2.8)

Defining u = x - a, the linear DE

u' = D/(a)u, (2.9)

which is called the linearization 01 the non-linear DE at the equilibrium point a E JRn ,

can be associated with the non-linear DE. In general solutions of the linear DE approximate the solutions of the non-linear DE near x = a (although in special situations the approximation can fail).

A. Linear Autonomous Differential Equations

The matrix series, called the exponential of A, is defined by

A 12 1 3 2:001k e =I+A+-A +-A + ... = -A 2! 3! k! '

k=O

(2.10)

where A is an n X n real matrix, I is the n x n identity matrix and A 2 = A A (matrix product), etc. A matrix series is said to converge if the n 2 infinite series correspond­ing to the n 2 entries converge in R The exponential matrix, eA , converges for all n x n matrices A [181].

9

A similarity transformation (which corresponds to a change of basis), defined by

B = p- l AP, (2.11)

where P is a non-singular matrix, can be effected to simplify A by writing it in a (Jordan) canonical form. Indeed, for any 2 x 2 real matrix A, there exists a non-singular matrix P such that

J = p- l AP,

and J is one of the following matrices:

(AI 0) (A 1) (0: (3) o A2 ' 0 A ' -(3 0: .

(2.12)

(2.13)

Noting that if B = p- l AP then eB = p- l eA P, we can now calculate eA for any matrix. In particular, we now have a complete algorithm for calculating eA for any 2 x 2 real matrix A.

Fundamental Theorem for Linear Autonomous DE. Let A be an n x n real matrix. Then the initial value problem

x' = Ax, x(O) = a E JRn , (2.14)

has the unique solution

x(t) = etA a, for all t E JR. (2.15)

The unique solution of the DE (2.14) is given by (2.15) for all t. Thus, for each t E JR, the matrix etA maps

a ---t etAa (2.16)

(where a is the state at time t = 0 and etA is the state at time t). The set {etA hEJR is a I-parameter family of linear maps of JRn into JRn , and is called the linear flow of the DE, denoted by

t g = etA. (2.17)

The flow describes the evolution in time of the physical system for all possible initial states. As the physical system evolves in time, one can think of the state vector x as a moving point in state space, its motion being determined by the flow gt = etA. The linear flow gt = etA satisfies the properties gO = I and gil +t2 = gil 0 gt2 (which also hold for non-linear flows), which imply that the flow {gt} tEJR forms a group under composition of maps. The flow gt of the DE (2.14) partitions the state space JRn into subsets called orbits, defined by

"((a) = {gtalt E JR}. (2.18)

The set "((a) is called the orbit of the DE through a and is the image in JRn of the solution curve x(t) = etAa. It follows from the uniqueness of solutions that for a, bE JRn , either "((a) = "((b) or "((a) n "((b) = 0.

Orbits of a DE can be classified as follows:

lO

1. If gta = a for all t E JR, then ")'(a) = {a} and it is called a point orbit. Point orbits correspond to equilibrium points.

2. If there exists aT> 0 such that gT a = a, then ")'(a) is called a periodic orbit. Periodic orbits describe a system that evolves periodically in time.

3. If gta "I a for all t "I 0, then ")'(a) is called a non-periodic orbit.

We note that non-periodic orbits can be of great complexity for linear DE if n > 3 and for non-linear DE if n > 2. In addition, since a solution curve of a DE is a parameterized curve it contains information about the flow of time t. The orbits are paths in (or subsets of) state space and orbits which are not point orbits are directed paths with the direction defined by increasing time. Hence, the orbits do not provide detailed information about the flow of time. For an autonomous DE, the slope of the solution curves depend only on x and hence the tangent vectors to the solution curves define a vector field f(x) in x-space and infinitely many solution curves may correspond to a single orbit. A non-autonomous DE does not define a flow or a family of orbits.

Given a linear DE x' = Ax in ll~n, we can introduce new coordinates by y = Px, where P is a non-singular matrix, and a new time variable T = kt, where k is a positive constant. It follows that y' = By, where B = iP AP-l, and the linear DE x' = Ax and x' = Bx are said to be linearly equivalent if

A = kP-1 BP. (2.19)

This condition ensures that the linear map P maps each orbit of the flow etA onto an orbit of the flow etB . Two linear flows etA and etB on JRn are said to be linearly equivalent if there exists a non-singular matrix P and a positive constant k such that

P etA = ektB P, for all t E JR. (2.20)

We can now consider the three cases corresponding to the three Jordan canonical forms for any 2 x 2 real matrix A.

CASE I: If A has two independent eigendirections, then there exists a matrix P such that J = P AP-1 , where

J = (A1 0) o A2 .

It follows that the given DE is linearly equivalent to y' = Jy. The flow is

etJ = (eA1 to) o eA2t '

where the ei~envectors are e1 = (1,0)T and e2 = (0,1)T. The solutions are y(t) = etJb, b E JR , i.e., Y1 = eA1tb1 and Y2 = eA2t b2 . By eliminating t, we obtain

11

1 1

( f ) Xl = (f r2, if bl b2 :f. 0 or YI = 0 if bl = 0, Y2 = 0 if b2 = O. These

equations define the orbits of the DE y' = Jy. The various cases are illustrated in Fig. (1): la. Al = A2 < 0: Attracting Focus. lb. Al < A2 < 0: Attracting Node. Ie. Al < A2 = 0: Attracting Line. ld. Al < 0 < A2: Saddle. Ie. Al = 0 < A2: Repelling Line (time reverse of Fig. (Ie)). If. 0 < Al < A2: Repelling Node (time reverse of Fig. (lb)). 19. 0 < Al = A2: Repelling Focus (time reverse of Fig. (Ia)).

Ia Y2 Ib Y2

Y1

Ie Y2 Id Y2

~j~ Y1

~1r7 FIG. 1: The phase portrait of the 2-dimensionallinear autonomous DE in (Jordan) canon­ical form which depicts orbits close to the equilibrium point at the origin in case I of two independent eigendirections: (a) Attracting Focus, Al = A2 < 0 (the portrait in the case of a Repelling Focus with 0 < Al = A2 is the time reverse of this portrait). (b) Attracting Node, Al < A2 < 0 (the time reverse is the case of a Repelling Node with 0 < Al < A2)' (c) Attracting Line, Al < A2 = 0 (the time reverse is the case of a Repelling Line with Al = 0 < A2)' (d) Saddle, Al < 0 < A2.

Y1

Y1

12

CASE II: If A has one eigendirection, then there exists a matrix P such that J = P Ap-l, where

J=(>'l) o >. .

It follows that the given DE is linearly equivalent to y' = Jy. The flow is

etJ = eAt (1 t) o 1 .

and the single eigenvector is el = (l,O)T. We note that if >. i- 0, the orbits are

given by Yl = Y2 [~ + t log ~], if b2 i- 0, or Y2 = 0 if b2 = O. These cases are

illustrated in Fig. (2): lIa. >. < 0: Attracting Jordan Node. lIb. >. = 0: Neutral Line. lIc. >. > 0: Repelling Jordan Node (time reverse of Fig. (lIa)).

IIa Y2 vertical lIb Y2

)

Y1

FIG. 2: The phase portrait of the 2-dimensionallinear autonomous DE in (Jordan) canon­ical form in case II of one independent eigendirection: (a) Attracting Jordan Node, >. < 0 (the case of a Repelling Jordan Node with>' > 0 is the time reverse of this portrait). (b) Neutral Line, >. = o.

CASE III: If A has no eigendirections, then there exists a matrix P such that J = P AP-1 , where

J=(O: 13). -13 0:

It follows that the given DE is linearly equivalent to y' = Jy. The simplest way to find the orbits is to introduce polar coordinates (r,O): Yl = r cos 0, and Y2 =

Y1

13

rsinO. The DE becomes r' = ar and 0' = -(3. It follows that ~9 = -~r which

can be integrated to yield r = roe-~(6-6o). Without loss of generality, we can assume (3 > 0, since the DE is invariant under the changes «(3, yt) -t (-(3, -Yl). Thus limt-+oo 0 = -00 (counterclockwise rotation as t increases). These cases are illustrated in Fig. (3): IlIa. a < 0: Attracting Spiral. Illb. a = 0: Centre. I1Ic. a > 0: Repelling Spiral (time reverse of Fig. (IlIa)).

lila

( vertical isocline Y2 =- ~ Y1

~

\ \

Y2 IIIb Y2

Y1

FIG. 3: The phase portrait of the 2-dimensionallinear autonomous DE in (Jordan) canon­ical form in case III with no independent eigendirections: (a) Attracting Spiral, 0 < 0 (the case of a Repelling Spiral with 0 > 0 is the time reverse of this portrait). (b) Centre, 0=0.

1. Topological Equivalence

Under linear equivalence, two dimensional flows can be simplified in that they can be parameterized by one real-valued parameter and several discrete parameters (e.g., the number of independent eigenvectors). Linear equivalence thus acts as a filter, which retains only certain essential features of the flow, such as the behaviour of the orbits near the equilibrium point (0,0). On the other hand, if one is primarily interested in long-term behaviour, one can use a finer filter, which eliminates more features, and hence leads to a much simpler (but coarser classification). This is the notion of Topological Equivalence of linear flows.

For example, cases la, Ib, I1a, and IlIa have the common characteristic that all orbits approach the origin (an equilibrium point) as t -t 00. We would like these flows to be "equivalent" in some sense. For all of these flows, the orbits of one

Y1

14

flow can be mapped onto the orbits of the simplest flow la, using a (non-linear) map h : ]R2 ---+ ]R2: which is a homeomorphism (i.e., h is one-to-one and onto, his continuous, and h-1 is continuous) on ]R2. Two linear flows etA and etB on ]Rn are said to be topologically equivalent if there exists a homeomorphism h on ]Rn and a positive constant k such that

h(etAx) =ektBh(x), for all x E ]Rn and for all t E lR. (2.21)

In addition, a hyperbolic linear flow in ]R2 is one in which the real parts of the eigenvalues are all non-zero (Le., ~e(>'i) :I 0, i = 1,2). The following result is well-known: Any hyperbolic linear flow in ]R2 is topologically equivalent to the linear flow etA, where A is one of the following matrices:

1. A = ; standard sink. ( -1 ° ) ° -1

2. A = ( ~ ~); standard source.

(-1 0) 3. A = ° 1 ; the standard saddle.

For non-hyperbolic linear flows in ]R2, it is clear that none of the 5 canonical flows [Le., the centre, the attracting and repelling lines, the neutral line and the neutral 2-space (A = 0)] are topologically equivalent (their asymptotic behaviour as t ---+ 00 differs). Thus, two non-hyperbolic linear flows in ]R2 are topologically equivalent if and only if they are linearly equivalent. Therefore, any non-hyperbolic linear flow in]R2 is linearly (and hence topologically) equivalent to the flow etA, where A is one of the following matrices:

( 00) (0 -1) (0 1) (-10) (10) 00' 10 ' 00' ° 0' 00·

(2.22)

These five flows are topologically inequivalent.

2. Linear Stability

It is important to determine whether a physical system that is disturbed from an equilibrium state remains close to (stable) or approaches (asymptotically stable) the equilibrium state as time evolves (t ---+ 00). First, we need the following definitions

1. The equilibrium point ° of a linear DE x' = Ax in ]Rn is stable if for all neighbourhoods U of 0, there exists a neighbourhood V of ° such that gtv ~ U for all t 2: 0, where gt = etA is the flow of the DE.

15

2. The equilibrium point 0 of a linear DE x' = Ax in IRn is asymptotically stable if it is stable and if, in addition, for all x E V, limt---+oo IIgtxll = O.

If A E Mn(IR), it follows that

lim etAa = 0, t--+oo

for all a E IRn, (2.23)

if and only if ~e(A) < 0 for all eigenvalues of A. This means that if ~e(A) < 0 then all solutions x(t) of the DE x' = Ax approach the equilibrium point 0 in the long term; i.e., x(t) -+ 0 E ]R.n, as t -+ 00. Thus if A E Mn(lR) is such that ~e(A) < 0 for all eigenvalues, then we say that the equilibrium point 0 of the DE x' = Ax is a sink in ]R.n. If we replace A by -A and t by -t, we obtain the time reverse. Thus if A E Mn(lR) is such that ~e(A) > 0 for all eigenvalues, then we say that the equilibrium point 0 of the DE x' = Ax is a source in IRn. Finally, we note that if A E Mn(lR) and if there exists a constant k such that all eigenvalues of A satisfy ~e(A) < -k < 0, then there exists a positive constant M such that

IletAxl1 :::; Me-ktllxll, for all x E ]R.n, all t 2: O. (2.24)

We note that if the equilibrium point 0 E ]R.n is a sink of the DE x' = Ax, then 0 is an asymptotically stable equilibrium point.

B. Non-Linear Differential Equations

For non-linear DE, the flow usually cannot be written down explicitly. Indeed, the aim of dynamical systems is to describe the qualitative properties of a non-linear flow without knowing the flow explicitly. Let us first state the standard existence­uniqueness theorem for the initial value problem (IVP) for a DE in ]R.n.

Theorem (Existence-Uniqueness) [181] . Consider the initial value problem

x' = f(x), x(O) = a E IRn. (2.25)

If f : IRn -+ IRn is of class C 1 (IRn), then for all a E ]R.n, there exists an interval (-8,8) and a unique function 'l/Ja : (-8,8) -+ IRn such that

'l/J~(t) = f('l/Ja(t»), 'l/Ja(O) = a. (2.26)

We note that if the hypothesis that f be of class C 1 is weakened, then unique­ness may fail. The existence-uniqueness theorem is a local result - it guarantees existence of a solution in some interval (-8,8) centered at t = O. Since we are interested in the long-term behaviour of solutions, we would like the solutions to be defined for all t 2: o. We can extend the interval of definition of the solution 'l/Ja(t) by successively reapplying the theorem, and in this way obtain a maximal interval of definition of the solution 'l/Ja(t). We shall denote this maximal interval

16

by (a, fJ). We say that the solution ¢a(t) has finite escape time fJa if II¢a(t)11 -+ +00 as t -+ fJ;;.

Theorem (Maximality). Let ¢a(t) be the unique solution ofthe DE x' = f(x), where f E CI(JRn), which satisfies, ¢a(O) = a, and let (aa,fJa) denote the maximal interval on which ¢a(t) is defined. If fJa is finite, then [181]

lim II¢a(t)11 = +00. t-tf3- a

(2.27)

Therefore, for the DE (2.1), f E CI(JRn), if a solution ¢a(t) is bounded for t ~ 0, then the solution is defined for all t ~ O.

A given DE x' = f(x), x E JRn, and f E CI(JRn), can always be modified so that the orbits are unchanged, but such that all solutions are defined for all t E JR, by re-scaling the vector field f (the velocity of the state point x):

f(x) -+ >.(x)f(x), (2.28)

where >.(x) : JRn -+ JR is a CI-function (a scalar) which is positive on m.n (in order to preserve the direction of time). This rescaling does not change the direction of the vector field, hence the orbits are unchanged. However, one can choose>. so that IIVII is bounded; e.g., >.(x) = (1 + Ilf(x)II-I) [284].

Let us consider the DE x' = f (x), where f is of class C I, and the unique maximal solution which satisfies ¢a(O) = a. The flow of the DE is defined to be the one­parameter family of maps {gthEm. such that gt : m.n -+ m.n and gta = ¢a(t) for all a E m.n . The flow {gt} is defined in terms of the solution function ¢a(t) of the DE by

gta = ¢a(t). (2.29)

It is important to understand the difference between ¢a(t) and gta: For a fixed a E m.n , ¢a : JRn -+ JRn gives the state of the system ¢a(t) for all t E m., with ¢a(O) = a initially. For a fixed t E m., gt : m.n -+ m.n gives the state of the system gta at time t for all initial states a.

The solution function ¢a(t) satisfies ¢~(t) = f(¢a(t)) , ¢a(O) = a. Hence ¢~(O) = f(a). By definition of the flow, it follows that

! (gta) It=o = f(a), (2.30)

which is simply a restatement of the fact that the vector field f is tangent to the orbits of the DE. If {gt} is the flow of a DE x' = f(x), then the following two properties are satisfied:

gO = I gtl +t2 = gtl 0 gt2

(identity map) (composition)

(2.31)

17

Theorem (Smoothness of a Flow)[181]. IT f E Cl(]Rn), then the flow {gt} of the DE x' = f(x) consists of C l maps.

Therefore, the solutions of the DE depend smoothly on the initial state. The orbit through a, denoted 'Y( a), is defined to be

'Y(a) = {x E ]Rnlx = gta, for all t E ]R}. (2.32)

As in the linear case, orbits are classified as point orbits, periodic orbits, and non­periodic orbits. Sometimes it is convenient to work with the positive orbit through a denoted 'Y+ (a) and defined by

'Y+(a) = {x E ]Rnlx = gta, for all t ~ o}. (2.33)

Let us consider a physical system with initial state vector x E ]Rn, whose evolution is described by a DE x' = f(x) which determines a flow {gt}tE]R. It is of interest to determine the long-term behaviour of the system as t ---+ 00, starting at an initial state a when t = O. The simplest behaviour is (i) the system, starting at state a, approaches an equilibrium state as t ---+ 00; i.e., limt---HX) gta = p. The next simplest behaviour is (ii) the system, starting at state a, approaches periodic evolution; i.e., the orbit approaches a periodic orbit 'Y. In this situation, limt--+oo gta does not exist, since the orbit does not approach a unique point. However, for any point p E 'Y, we can choose a sequence of times {tn}, with limn--+oo tn = 00, such that limn--+oo gtn a = p. This motivates the important notion of a limit set.

Consider the DE (2.1) in ]Rn, and the associated flow {gthE]R. Given an initial point a E ]Rn, a point p E ]Rn is said to be an w-limit point of a if there exists a sequence {tn} with limn--+oo tn = 00 such that limn--+oo gtn a = p. The set of all w­points of a is called the w-limit set of a, denoted by w(a). In cases (i) and (ii) above the w-limit sets of the initial point a is the equilibrium point p and the periodic orbit 'Y, i.e., w(a) = {p} and w(a) = 'Y, respectively. Determining the subsets of ]Rn which can be w-limit sets for a flow {gt} is a difficult question, and is unsolved if n > 2. The following two results are useful for identifying w(a): (1) An w-limit set w(a) of a flow {gt} is a whole orbit of the flow, or is the union of more than one whole orbit. (2) If the positive orbit through a, 'Y+(a) = {gtalt ~ O} is bounded, then w(a) t=- 0.

1. Liapunov Theory

Given a DE x' = f(x) in ]Rn, a set S ~ ]Rn which is the union of whole orbits of the DE, is called an invariant set for the DE. For example, for a Hamiltonian DE in ]R2 the level sets H (Xl, X2) = k are invariant sets, since H is constant along any orbit. More generally, a function H : ]Rn ---+ ]R of class C 1, that is not constant on any open subset ofl[~n, is called a first integral of the DE x' = f(x) if H is constant on every orbit; i.e.,

!H(X(t)) = 'YH(x(t)) . f(x(t)) = 0, (2.34)

18

for all t (using the chain rule and the DE). It follows that H(x) is a first integral of the DE x' = I(x) if and only if \7 H(x) ·/(x) = 0, for all x E ]Rn, and H(x) is not identically constant on any open subset of ]Rn. If there is a first integral (e.g., a Hamiltonian function) then the orbits of the DE are contained in the one-parameter family of level sets H(x) = k.

It sometimes happens that there is a function F : ]Rn -+ ]R such that only a particular level set 01 F is an invariant set. Given a DE x' = I(x) in ]Rn, and a function G:]Rn -+]R of class C1, if \7G(x)· I(x) = 0 for all x such that G(x) = k, then the level set G(x) = k is an invariant set of the DE. These invariant sets play a major role in determining the portrait of the orbits.

Given a DE x' = I(x) in ]Rn, with flow gt, a subset S c ]Rn is said to be a trapping set of the DE if it satisfies: (i) S is a closed and bounded set, and (ii) a E S implies gta E S for all t ~ O. Hence if an orbit enters a trapping set S it never leaves it, and for all a E S, the w-limit set w(a) is non-empty and is contained in S. Let V: ]Rn -+ ]R be C l(]Rn). We can calculate the rate of change of V along a solution of the DE (2.1) by:

d 8V dXl 8V dXn () ( ) . -V(x(t)) = --+ ... + -- = \7V x(t) . I x(t) == Vex), & 8~& 8~&

(2.35)

where (.) is the scalar product in ]Rn. Suppose that Vex) ~ 0 for all x E ]Rn. Then for any orbit rea) in a trapping set S, Vex) will keep decreasing along rea) until the orbit approaches its w-limit set w(a). Strong restrictions on the possible w-limit sets can therefore be obtained.

Theorem (Global Liapunov Theorem). Consider the DE x' = I(x) in ]Rn, and let V : ]Rn -+ ]R be a C 1 function. If S c ]Rn is a trapping set, and Vex) ~ 0

for all xES, then for all a E S, w(a) ~ {x E SIV(x) = O} [169).

A function V : ]Rn -+ ]R which satisfies the above theorem for xES C ]Rn is called a Liapunov function on S. In applying the theorem, we note that we simply

have to find whole orbits that are contained in the set {x E SIV(x) = O} to obtain

the w-limit set w(a).

The stability of an equilibrium point can be determined by studying the lineariza­tion of the DE. The basic definitions are the same as in the linear case, with the linear flow etA being replaced by gt:

1. The equilibrium point x of a DE x' = I(x) in ]Rn is stable iffor all neighbour­hoods U of x, there exists a neighbourhood V of x such that gtv ~ U for all t ~ 0, where gt is the flow of the DE.

2. The equilibrium point x of a DE x' = I(x) in ]Rn is asymptotically stable if it is stable and if, in addition, for all x E V, limt-+oo Ilgtx - xii = 0

Now, consider a non-linear DE (2.1) in ]Rn. Let V : ]Rn -+ ]R be C 1 (]Rn). The rate of change of V along a solution ofthe DE is given by V = \7V(x(t)) . l(x(t))V(x).

19

Thus, if V(x) < 0 for all t then V(x) decreases with time along the corresponding orbit. From a geometrical point of view, the orbits cut the level sets V(x) = k in the direction away from V'V(x). Suppose that x is an equilibrium point of the DE. If V(x) = 0 and V(x) > 0 for all x E U - {x}, where U is a neighbourhood of x, then we expect the level sets of V in U to be concentric curves (n=2) or concentric spheres (n=3); consequently when V < 0 for all x E U - {x}, any orbit in U - {x} will cut the level sets of V in the inward direction, and we expect that this will continue until the orbit is forced to approach the equilibrium point x as t -t 00,

showing that the equilibrium point is asymptotically stable. If, instead, V ::; 0 for all x E U - {x}, then U may contain periodic orbits, and we only obtain the weaker conclusion that x is stable. Finally, if V> 0 for all x E U - {x}, then the orbits are forced away from x, which is thus an unstable equilibrium point.

Theorem (Liapunov Stahility Theorem). Let x be an equilibrium point of the DE x' = f(x) in jRn. Let V : jRn -t jR be a C1 function such that V(x) = 0, V(x) > 0 for all x E U - {x}, where U is a neighbourhood of x.

1. If V(x) < 0 for all x E U - {x}, then x is asymptotically stable.

2. If V::; 0 for all x E U - {x}, then x is stable.

3. If V(x) > 0 for all x E U - {x}, then x is unstable.

A function V : jRn -t jR which satisfies V(x) = 0, V(x) > 0 for all x E U - {x}, and V(x) ::; 0 (respectively < 0) for all x E U - {x}, is called a Liapunov function (respectively, a strict Liapunov function) for the equilibrium point x. Hence we obtain the following Criterion for Asymptotic Stability: Let x be an equilibrium point of the DE x' = f(x) in jRn. If all of the eigenvalues of the derivative matrix Df(x) satisfy lRe(oX) < 0, then the equilibrium point x is asymptotically stable.

2. Linearization and the Hartman-Grobman Theorem

In general the linearizations give a reliable description of the non-linear orbits near the equilibrium points.

Theorem (Hartman-Grohman). Let x be an equilibrium point of the DE x' = f(x) in jRn, where f : jRn -t jRn is of class C1 . If all of the eigenvalues of the matrix D f (x) satisfy lRe (oX) -I- 0, then there is a homeomorphism h : U -t (j of a neighbourhood U of 0 onto a neighbourhood (j of x which maps orbits of the linear flow etDf(x) onto orbits of the non-linear flow gt of the DE, preserving the parameter t [172].

Two flows gt and gt on jRn are said to be topologically equivalent if there is a homeomorphism h : jRn -t jRn which maps orbits of gt onto orbits of gt, and preserves the direction of the parameter t. An equilibrium point x of a non-linear

20

DE is said to be hyperbolic if all eigenvalues of the matrix Df(x) satisfy iRe('\) ¥- o. The Hartman-Grobman Theorem can therefore be stated more concisely: if x is a hyperbolic equilibrium point, then the flow of the DE x' = f(x) and the flow of its linearization u' = Df(x)u, are locally topologically equivalent.

An equilibrium point x of a DE (2.1) in Rn is a saddle point if the real parts of the eigenvalues of the matrix Df(x) are all non-zero, and not all of one sign (Le., a saddle point is a hyperbolic (all iRe('\) ¥- 0) equilibrium point which is neither a sink (all iRe('\) < 0) nor a source (all iRe('\) > 0)). If x is a saddle point of the DE (2.1) in R n , and U is a neighbourhood of x, then the local stable manifold of x in U is defined by

WB(X, U) = {x E U lim gtx = x, gtx E U for all t 2:: O} . t-+=

(2.36)

Theorem (Stable Manifold Theorem). Let x be a saddle point of x' = f(x) in R n , where f is of class C I , and let ES be the stable subspace of the linearization at x. Then there exists a neighbourhood U of x such that the local stable manifold W B (x, U) is a smooth (C I ) curve which is tangent to ES at x.

We can define in an analogous way the local unstable manifold of x in U, denoted W"(x, U), and similarly there is an "Unstable Manifold Theorem."

Let us give a more detailed description of the local behaviour of non-linear orbits near a non-linear sink. Suppose that x = (Xl, X2) E R2 is an asymptotically stable equilibrium point of the DE x' = f(x). In order to describe the orbits near x, we can introduce polar coordinates Xl - Xl = r cos 0, X2 - X2 = r sin O. Since x is asymptotically stable,

lim r(t) = 0, t-++=

(2.37)

if r(O) is sufficiently close to zero. We say that the equilibrium point x is a non-linear spiral if

lim O(t) = ±oo, t-++=

(2.38)

for any solution (r(t), O(t)) for which (2.37) holds. A number of results are known [90]. Consider the DE

x' = f(x), (2.39)

in R2, where f is of class C I , and consider the linearization

u' = Df(x)u, (2.40)

at the equilibrium point x. If 0 is an attracting spiral point of the linearization of the DE, then x is an attracting spiral point of the non-linear DE. Similarly, if 0 is an attracting node (or Jordan node) of the linearization of the DE then x is an attracting node (respectivley, Jordan node) of the non-linear DE [90]. An

21

asymptotically stable equilibrium point x is said to be an attracting non-linear focus if all orbits sufficiently close to x approach x in a definite direction as t -+ 00, and given any direction there exists an orbit which tends to x in this direction. Note that if 0 is a focus of the linearization of the DE, it does not necessarily follow in general that x is a non-linear focus of the non-linear DE. Finally, suppose that the vector field f is of class C 2 • If 0 is an attracting focus of the linearization of the DE then x is an attracting focus of the non-linear DE [90]. A stable equilibrium point x is said to be a non-linear centre if in some neighbourhood of x, the orbits are periodic orbits which enclose x. Recall that the Hartman-Grobman theorem does not apply if 0 is a centre of the linearization of the DE; i.e., one cannot conclude that x is a centre of the non-linear DE. However, if 0 is a centre of the linearization of the DE, then x is either a centre, an attracting spiral, or a repelling spiral of the non-linear DE [90].

c. Periodic Orbits and The Poincare-Bendixson Theorem in the Plane

A linear DE can admit a family of periodic orbits. Of greater interest is the case where a DE admits an isolated periodic orbit, i.e., the orbit has a neighbourhood U which contains no other periodic orbits, which is only possible for a non-linear DE. In this situation, the periodic orbit 'Y may attract neighbouring orbits, thereby describing a physical system which has an oscillatory steady state which is stable. The question of the existence of periodic orbits is a difficult one. However, a criterion for excluding periodic orbits for a DE in II~? was given by Dulac based on Green's theorem.

Dulac's Criterion. If D ~ ]R2 is a simply connected open set and div(Bf) = 8~1 (BII) + 8~2(Bh) > 0 « 0), for all xED where B is a C 1 function, then the DE x' = f(x) where f E 0 1 has no periodic orbit which is contained in D. The function B (Xl, X2) is called a Dulac function for the DE in the set D.

A second criterion for excluding periodic orbits, which is valid in ]Rn, n 2:: 2, follows from the observation that if a function Vex) is monotone decreasing along an orbit of a DE, then that orbit cannot be periodic.

Monotone Criterion. Let V : ]Rn -+ ]R be a 0 1 function. If Vex) = VV(x) . f(x) :::; 0 on a subset D ~ ]Rn, then any periodic orbit of the DE x' = f(x) which

lies in D, belongs to the subset {xIV(x) = o} n D.

An isolated periodic orbit 'Y of a DE x' = f(x) in II~?, is called a stable limit cycle if there exists a neighbourhood U of'Y such that w(a) = 'Y for all a E U.

A local section of the flow of a DE in ]R2 is a smooth curve segment E such that the vector field f of the DE satisfies n . f ¥ 0 on E, where n is normal to E. Note that this implies that no equilibrium points of f lie on E, and by continuity, that orbits pass through E in one direction only. If x is a regular (i.e., non-equilibrium)

22

point of the flow (i.e., f(x) i= 0) and E is a local section through x, then a flow-box

for x is a neighbourhood of x of the form N = {gtElltl < 8} for some fJ > O.

Finally we recall the following properties of w-limit sets: (i) w(a) is the union of whole orbits, (ii) w(a) is a closed set, (iii) if w(a) is bounded, then w(a) is connected (Le., is not the union of disjoint sets). Then we obtain the following Lemma (which is not valid for a flow in ]R3).

Lemma (Fundamental Lemma on w-limit sets in ]R2). Let w(a) be an w-limit set of a DE in ]R2. If y E w(a), then the orbit through y, l'(y), cuts any local section E in at most one point [169, 181].

Theorem (Poincare-Bendixson). Let w(a) be a non-empty w-limit set of the DE x' = f(x) in ]R2, where fECi. If w(a) is a bounded subset of]R2 and w(a) contains no equilibrium points, then w(a) is a periodic orbit [181].

In applications it is often convenient to use the following Corollary of the Poincare­Bendixson theorem.

Corollary. Let K be a positively invariant subset of the DE x' = f(x) in ]R2, where fECi. If K is a closed and bounded set, then K contains either a periodic orbit or an equilibrium point.

The fundamental property of an w-limit set is that it consists of one whole orbit, or that it is the union of more than one whole orbit. The simplest situations are: (i) w(a) is an equilibrium point; i.e., the system approaches an equilibrium state as t -t +00, (ii) w(a) is a periodic orbit; i.e., the system approaches an oscillatory steady state as t -t +00. Other examples of invariant sets (i.e., unions of orbits) can arise as w-limit sets in ]R2, such as, for example, cycle graphs. Let a(x) denote the a-limit set of the point x, which is the set of past limit points of x; i.e.,

a(x) = {yly = nl~~ gtn x, and tn -t -00 } . (2.41)

A cycle graph S of a DE x' = f(x) in ]R2 is a connected union of orbits such that: (i) For all XES, w(x) = {p} and a(x) = {q}, where p and q are equilibrium points in S. (ii) For all equilibrium points pES, there exists points x, yES such that w(x) = {p}, a(y) = {q}, and the number of equilibrium points in S is finite. (iii) The orientations of the orbits define a continuous closed path in S. If we consider the DE x' = f(x) in ]R2 and let a E ]R2 be an initial point such that {gtalt 2': O} lies in a closed bounded subset K c ]R2, it then follows that if K contains only a finite number of equilibrium points then one of the following holds: 1. w(a) is an equilibrium point. 2. w(a) is a periodic orbit. 3. w(a) is a cycle graph [169, 227]. Unfortunately, this theorem does not generalize to DE in ]Rn, n 2': 3, or to DE on the 2-torus. Indeed, the problem of describing all possible w-limit sets in ]Rn, n 2': 3, is presently unsolved.

23

D. More General Non-Linear Behaviour

The motion of an undamped symmetric 2-mass oscillator, whose orbits lie in invariant 2-tori, depends on the values of two constants WI and W2, which are the two natural frequencies of oscillation of the physical system. If mWI = nW2, where m, n are positive integers without common factors, the solutions are periodic with period T = 2wll"m - 211"n. The corresponding orbit on one of the invariant tori is

1 W2

thus periodic, and eventually closes up as it winds around the torus. If, on the other hand, ~ is irrational, then the orbits are not periodic, and hence do not close up as they wind around the invariant tori. As the orbit winds around the torus, it passes arbitrarily close to each point of the torus, and the orbit is said to be everywhere dense on the torus [11]. In this latter case, w(a) is the union of an uncountable infinity of whole orbits, including the orbit through a, and the invariant 2-tori do not attract neighbouring orbits. However, an attracting 2-torus can be created, giving rise to so-called quasiperiodic motion, in a similar fashion to that in which non-linearity can be used to create an attracting periodic orbit. This example illustrates the richness of the possible dynamical behaviour in non-linear systems and motivates the idea of attracting sets in describing long-term behaviour. Given a DE x' = f(x) in ]Rn, a closed invariant set A c ]Rn is said to be an attracting set if there exists a neighbourhood U of A such that gtU ~ U for all t ~ 0 and w(a) ~ A for all a E U, where gt is the flow of the DE and w(a) is the w-limit set of the point a. Intuitively, an attracting set is a generalization of an asymptotically stable equilibrium point or periodic orbit. The basin of attraction of an attracting set A is the subset of]Rn defined by p(A) = {x E ]Rnlw(x) ~ A}.

If a DE has an attracting set A, then for all initial states a in the basin of attrac­tion the physical system approaches some kind of "steady state". The nature of the steady state is determined by the orbits which form the attractor. In the case that the attracting set is an equilibrium point or a periodic orbit, the long term steady state behaviour is an equilibrium state or periodic motion, respectively. In the case that the attracting set is an invariant 2-torus (or k-torus in general) with dense orbits the long-term steady state behaviour is 2-quasiperiodic (k-quasiperiodic). Finally, there is the possibility of "strange" attractors and chaotic behaviour which is the subject of current research. A strange attractor is not a piecewise smooth surface, and can have a structure like that of a Cantor set. Chaotic behaviour occurs when neighbouring orbits diverge (separate) from each other at an exponen­tial rate, while remaining bounded, a phenomenon that is referred to as "sensitive dependence on initial conditions" [14, 275].

1. Higher Dimensions

There are many of new features possible in higher dimensions. Although, in general, a qualitative analysis is much more difficult there are some cases, such as conservative and gradient systems, which have special characteristics that make

24

their analysis possible (e.g., limit sets of orbits in gradient systems are necessarily part of the set of equilibria). In addition, Hamiltonian systems often occur in physical applications. Completely integrable systems can be analyzed successfully. However, an analysis of general Hamiltonian systems for n ~ 4 is currently out of reach.

Unlike a linear DE, a non-linear system allows for singular structures which are more complicated than that of equilibrium points, fixed lines or periodic orbits, particularly in higher dimensions (n > 2). These structures include, but are not limited to, such things as heteroclinic and/or homo clinic orbits and non-linear in­variant sub-manifolds [373]. Sets of non-isolated equilibrium points often occur in applications (and particularly in cosmology) and therefore their stability needs to be examined more carefully. A set of non-isolated equilibrium points is said to be normally hyperbolic if the only eigenvalues with zero real parts are those whose corresponding eigenvectors are tangent to the set. Since by definition any point on a set of non-isolated equilibrium points will have at least one eigenvalue which is zero, all points in the set are non-hyperbolic. The stability of a set which is nor­mally hyperbolic can, however, be completely classified by considering the signs of the eigenvalues in the remaining directions (Le., for a curve, in the remaining n-l directions) [13].

The local dynamics of an equilibrium point may depend on one or more arbitrary parameters. parameter When small continuous changes in a parameter results in dramatic changes in the dynamics, the equilibrium point is said to undergo a bi­furcation [181]. The values of the parameter{s) which result in a bifurcation at the equilibrium point can often be located by examining the linearized system. Equi­librium point bifurcations will only occur if one (or more) of the eigenvalues of the linearized system are a function of a parameter and the bifurcations are located at the parameter values for which the real part of an eigenvalue is zero.

There are a variety of possible future and past asymptotic states of a non-linear system. In the case of a plane system the possible asymptotic states can be given explicitly via the Poincare-Bendixson Theorem due to the limited degrees of freedom and the fact that the flows (or orbits) in any 2-dimensional phase space cannot cross. This theorem has a very important consequence in that if the existence of a closed (Le., periodic, heteroclinic or homo clinic) orbit can be ruled out it follows that all asymptotic behaviour is located at an equilibrium point. As noted earlier the existence of a closed orbit can be ruled out by many methods. When the phase space is of a higher dimension (than two) the requirement that orbits cannot cross does not result and the decisive Poincare-Bendixson theorem does not follow. The behaviour in such higher-dimensional spaces is very complicated, with the possibility of phenomena such as recurrence and strange attractors occurring [165]. For this reason the analysis of non-linear systems in spaces of three or more dimensions cannot in general progress much further than the local analysis of the equilibrium points (or non-isolated equilibrium sets). However, one tool which does allow for some progress in the analysis of higher dimensional systems is the possible existence of monotone functions.

25

Theorem (LaSalle Invariance Principle) [332]. Consider a DE x = f(x) on ]R.n. Let S be a closed, bounded and positively invariant set of the flow, and let Z be a C 1 monotonic function. Then for all Xo E S,

w(xo) C {x E SIZ = a},

where w(xo) is the w-limit set for the orbit with initial value Xo.

This principle has been generalized to the following result:

Theorem (Monotonicity Principle)[226]. Let (Pt be a flow on ]R.n with S an invariant set. Let Z : S -t lR be a C 1 function whose range is the interval (a, b), where a E lR U {-oo}, b E ]R. U {oo} and a < b. If Z is decreasing on orbits in S, then for all xES,

w(x) ~ {s E S \ Silim Z(y) -I b}, y-ts

a(x) ~ {s E S \ Silim Z(y) -I a}, y-ts

where w(x) and a(x) are the w- and a-limit sets of x, respectively.

As noted earlier, in most cases the eigenvalues of the linearized DE will have eigenvalues with both positive, negative and/or zero real parts. In these cases it is important to identify which orbits are attracted to the equilibrium point, and which are repelled away, as the independent variable tends to infinity. For a linear DE (2.2), the phase space Rn is spanned by the eigenvectors of A. These eigenvectors divide the phase space into three distinct subspacesj namely, the stable subspace ES = span(sl,s2, ... Sns), the unstable subspace EU = span(ul,u2, ... Unu), and the centre subspace E e = span(cl, C2, ... cne ), where Si are the eigenvectors who's asso­ciated eigenvalues have negative real part, Ui those who's eigenvalues have positive real part, and Ci those who's eigenvalues have zero eigenvalues. Flows (or orbits) in the stable subspace asymptote in the future to the equilibrium point, and those in the unstable subspace asymptote in the past to the equilibrium point.

In the non-linear case, the topological equivalence of flows allows for a similar classification of the equilibrium points. The equivalence only applies in directions where the eigenvalue has non-zero real parts. In these directions, since the flows are topologically equivalent, there is a flow tangent to the eigenvectors. The phase space is again divided into stable and unstable subspaces (as well as centre subspaces). The stable manifold W S of an equilibrium point is a differential manifold which is tangent to the stable subspace of the linearized system (E S ). Similarly, the unstable manifold is a differential manifold which is tangent to the unstable subspace (EU) at the equilibrium point. The centre manifold, we, is a differential manifold which is tangent to the centre subspace EC. It is important to note, however, that unlike the case of a linear system the centre manifold we will contain all the dynamics not classified by linearization (i.e., in the non-hyperbolic directions). In particu­lar, this manifold may contain regions which are stable, unstable or neutral. The classification of the dynamics in this manifold can only be determined by utilizing

26

more sophisticated methods, such as the Centre Manifold Theorem or the theory of normal forms [373).

III. SPATIALLY HOMOGENEOUS MODELS

Many people have studied self-similar spatially homogeneous models, both as ex­act solutions and in the context of qualitative analyses (see WE [363] and Coley [91] and references therein). Exact spatially homogeneous solutions were first displayed in early papers [135]; however, it was not until after 1985 that many of them were recognized by Wainwright [358] and Rosquist and Jantzen [202, 309] as being self­similar. The complete set of self-similar orthogonal spatially homogeneous perfect fluid and vacuum solutions were given by Hsu and Wainwright [190] and they have also been reviewed in WE. Kantowski-Sachs models were studied by Collins [111].

Spatially homogeneous models have attracted considerable attention since the governing equations reduce to a relatively simple finite-dimensional dynamical sys­tem, thereby enabling the models to be studied by standard qualitative techniques. Planar systems were initially analyzed by Collins [108, 110] and a comprehensive study of general Bianchi models was made by Bogoyavlenski and Novikov [57] and Bogoyavlenski [58] and more recently (using automorphism variables and Hamilto­nian techniques) by Jantzen and Rosquist [202, 309-311]. Perhaps the most illumi­nating approach has been that of Wainwright and collaborators [178, 190, 365], in which the more physically or geometrically natural expansion-normalized (dimen­sionless) configuration variables are used. In this case, the physically admissible states typically lie within a bounded region, the dynamical system remains analytic both in the physical region and its boundaries, and the asymptotic states typically lie on the boundary represented by exact physical solutions rather than having singular behaviour. We note that the physically admissible states do not lie in a bounded region for Bianchi models of types VIIo, VIII and IX (see WE for details).

Wainwright utilizes the orthonormal frame method [138] and introduces expansion-normalized (commutation function) variables and a new "dimension­less" time variable to study spatially homogeneous perfect fluid models satisfying p = b - 1)/L. The equations governing the models form an N-dimensional system of coupled autonomous ODE. When the ODE are written in expansion-normalized variables, they admit a symmetry which allows the equation for the time evolution of the expansion or the Hubble parameter H (the Raychaudhuri equation) to de­couple. The reduced N -I-dimensional dynamical system is then studied. At all of the equilibrium points of the reduced system, if is proportional to H2 and hence all such points correspond to transitively self-similar cosmological models [190]. This is why the self-similar models play an important role in describing the asymptotic dynamics of the Bianchi models. For orthogonal Bianchi models of class A, the resulting reduced state space is five-dimensional [365]. Orthogonal Bianchi cos­mologies of class B were studied by Hewitt and Wainwright [178] and are governed by a five-dimensional system of analytic ODE with constraints.

Two perfect-fluid models were studied by Coley and Wainwright [107]. In further work, imperfect fluid Bianchi models were studied under the assumption that all physical quantities satisfy "dimensionless equations of state" , thereby ensuring that the equilibrium points of the resulting reduced dynamical system are represented

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

28

by exact self-similar solutions. Models satisfying the linear Eckart theory of irre­versible thermodynamics were studied by Burd and Coley [67] and Coley and van den Hoogen [102], those satisfying the truncated causal theory of Israel-Stewart by Coley and van den Hoogen [103], and those satisfying the full (Le., non-truncated) relativistic Israel-Stewart theory by Coley et al. [106]. Self-similar solutions also play an important role in describing the dynamical behaviour of cosmological models close to the Planck time in GR with scalar fields [45, 116], in scalar-tensor theories of gravity [92], and particularly in the low-energy limit in supergravity theories from string theory and other higher-dimensional gravity theories.

A. Definitions and Kinematical Quantities

Let ua be the 4-velocity of a normalized timelike congruence (uaua = -1), rep­resenting the average motion of matter at each point. The projection tensor is defined as hab = gab + UaUb, which determines the orthogonal properties of the instantaneous rest-spaces of particles moving with 4-velocity ua . Two derivatives are defined relative to ua ; the covariant time derivative along ua (denoted by an overdot) and a fully orthogonally projected covariant derivative (spatial derivative, denoted by V). The first covariant derivative of U a can be split into its irreducible parts, defined by their symmetry properties [137]:

- 1 VaUb = - Ua Ub + VaUb = - Ua Ub + 3 (j hab + Uab + Wab , (3.1)

where the trace (j = Vaua is the (volume) rate of expansion of the fluid (and H = (j/3 is the Hubble parameter); Uab = V(aUb) is the trace-free symmetric (de­noted by angled brackets) rate of shear tensor (Uab = U(ab) , Uabub = 0, uaa = 0), describing the rate of distortion of the matter flow; and Wab = V[aUb] is the skew­symmetric vorticity tensor (Wab = W[ab] , Wab ub = 0), describing the rotation of the matter relative to a non-rotating (Fermi-propagated) frame. We can define a rela­tive position vector by 1]1 = hab1]b, where 1]a is a deviation vector for the family of fundamental worldlines, Le. ubV b1]a = 1]bV bUa. Writing 1]1 = 8£ ea, eaea = 1, we then find the relative distance 8£ obeys the propagation equation

(8£)" = ~ (j + (Uabeaeb) 8£ 3

(3.2)

(the generalised Hubble law), and the relative direction vector ea obeys the propa­gation equation

e(a) = (Uab - (ucdeced) hab - Wab) eb , (3.3)

from which the interpretation of the kinematical quantities can be deduced. Finally, ua = UbVbUa is the relativistic acceleration vector, representing the degree to which the matter moves under forces other than gravity plus inertia. The acceleration vanishes for matter in free fall.

29

We can also define the the vorticity vector

1 wa = "2 ryabc Wbc * Wa ua = 0 , (3.4)

where ryabc denotes the volume element for the rest-spaces, the magnitudes

1 1 w2 = - (w wab ) > 0 0"2 = - (0" O"ab) > 0 2 ab -, 2 ab -, (3.5)

and the average length scale R determined by

!l=~() R 3 '

(3.6)

so the volume of a fluid element varies as R3.

In analogy to the Maxwell Field strength, the Weyl conformal curvature tensor Cabcd can be split relative to ua into "electric" and "magnetic" Weyl curvature parts according to

Eab = CacM U C U d

H - 1 Cde c ab - "2 ryade bc U

* E a a = 0 , Eab = E(ab) , Eab ub = 0 ,

* H a a = 0 , Hab = H(ab) , Hab ub = 0 .

(3.7)

(3.8)

These represent the "free gravitational field", enabling gravitational action at a distance, and influence the motion of matter and radiation through the geodesic deviation equation for timelike and null vectors, respectively. Together with the Ricci tensor Rab (determined locally at each point by the matter tensor through the EFE), these quantities completely represent the space-time Riemann curvature tensor Rabcd.

There are evolution and constraint equations resulting from the EFE and its associated integrability conditions. The Ricci identities for the vector field ua give rise to propagation equations for () or H (the Raychaudhuri equation) [137), for w (the vorticity propagation equation) and for 0", and a number of constraint equations. The remaining Bianchi identities give rise to propagation equations for E and H (and two further constraint equations). In particular, the energy conservation law gives rise to an evolution equation for JL.

We can define "expansion-normalized" variables, by dividing quantities by an appropriate power of H (for H > 0) in order for the new variables to be dimension­less. For example, the density parameter is denoted by n = ~, and a normalized rate of shear can be defined by (3 = 2 viH. In addition, we define a dimension-

less time by ~~ = H, so that T -t - 00 as t -t 0+. The evolution of the new "expansion-normalized" variables can be then be derived.

The equation of state is given by equation (1.2), where 1 ::; 'Y ::; 2 for normal matter and models with 0 ::; 'Y < ~ are of interest in connection with inflationary

30

models of the universe (see, for example, Wald [367]). The weak energy condition implies that

The constraint,

/-L ~ o.

1 9H2 = - - 3R + 30-2 + 3/-L,

3

called the Hamiltonian constraint and referred to here as the generalized Friedmann equation, plays a key role in determining the physical phase space. Indeed, when the Ricci 3-curvature 3R is not positive, this constraint often leads to a compact phase space.

B. Asymptotic States of Perfect Fluid Bianchi Models

We now discuss the asymptotic states of Bianchi models, again assuming the linear equation of state (1.2). We will summarize the work of Wainwright and Hsu [365] and Hewitt and Wainwright [178], who studied the asymptotic states of orthogonal spatially homogeneous models in terms of attractors of the associated dynamical system for class A and class B models, respectively. The specialization diagram for Bianchi models is presented in Fig. (4) [264]. Due to the existence of monotone functions, it is known that there are no periodic or recurrent orbits in class A models. Although "typical" results can be proved in a number of Bianchi type B cases, these are not "generic" due to the lack of knowledge of appropriate monotone functions. In particular, there are no sources or sinks in the Bianchi invariant sets B~ (VIII) or B± (IX).

The key results are as follows:

• A large class of orthogonal spatially homogeneous models (including all class B models) are asymptotically self-similar at the initial singularity and are approximated by exact perfect fluid or vacuum self-similar power-law models. Examples include self-similar Kasner vacuum models or self-similar locally rotationally symmetric (class III) Bianchi type II perfect fluid models [108, 113,129].

However, this behaviour is not generic; general orthogonal models of Bianchi types IX and VIII have an oscillatory behaviour with chaotic-like characteristics, with the

A

j\r)/~' VI' VII'

\/~~ //NJ 11--

j ,.,V

-"" I"

-"";

,,-"" ;;-""

31

B

FIG. 4: Specialization diagram for Bianchi group types. A broken arrow indicates a change of group class (from class B to class A), while an unbroken arrow does not.

matter density becoming dynamically negligible as one follows the evolution into the past towards the initial singularity. Ma and Wainwright [260] show that the orbits of the associated cosmological dynamical system are negatively asymptotic to a lower 2-dimensional attractor. This is the union of three ellipsoids in ~5 consisting of the Kasner ring joined by Taub separatrices; the orbits spend most of the time near the Kasner vacuum equilibrium points. Clearly the self-similar Kasner models play a primary role in the asymptotic behaviour of these models .

• Exact self-similar power-law models can also approximate general Bianchi models at intermediate stages of their evolution (e.g., radiation Bianchi VUh models [129]). Of special interest are those models which can be approximated by an isotropic solution at an intermediate stage oftheir evolution (e.g., those models whose orbits spend a period of time near to a flat Friedmann equilib­rium point).

This last point is of particular importance in relating Bianchi models to the real Universe, and is discussed further in general terms in WE and specifically in a com­prehensive study of Bianchi VUh models in Wainwright et al. [362]. In particular, a quasi-isotropic epoch is universal in that the flat Friedmann equilibrium point is contained in the state space of each Bianchi type. Isotropic intermediate behaviour has also been found in tilted Bianchi V models [177], and it appears that many tilted models have isotropic intermediate behaviour (indeed, it appears that the presence of tilt increases the likelihood of intermediate isotropization [179]).

32

• Self-similar solutions can describe the behaviour of Bianchi models at late times (i.e., as t --t 00). Examples include self-similar flat space and self-similar homogeneous vacuum plane waves [108, 358).

All models expand indefinitely except for the Bianchi type IX models. Bianchi type IX models obey the "closed universe recollapse" conjecture [235). All orbits in the Bianchi invariant sets B±(IX) are positively departing; in order to analyse the fu­ture asymptotic states of such models it is necessary to compactify phase-space. The description of these models in terms of conventional expansion-normalized variables is only valid up to the point of maximum expansion (where H = 0). Alternative normalized variables were suggested in WE, and an analysis of the LRS Bianchi IX models was presented. Recently closed spatially homogeneous models with a scalar field and a barotropic perfect fluid have been investigated [96).

The question of which Bianchi models can isotropize was addressed in the famous paper by Collins and Hawking [112), in which it was shown that, for physically reasonable matter, the set of homogeneous initial data that give rise to models that isotropize asymptotically to the future is of zero measure in the space of all homogeneous initial data. All vacuum models of Bianchi (B) types IV, V, Vlh and VIIh are asymptotic to plane wave states to the future. Type V models tend to the Milne form of flat spacetime [178). Typically, and perhaps generically [178), non­vacuum models are asymptotic in the future to either plane-wave vacuum solutions [129) or non-vacuum Collins type Vlh solutions [108).

All orbits in the invariant sets B';(VIIo) (0 > 0) and B';(VIII) and B±(IX) are also positively departing. Although Bianchi (A) models of types VIIo (non­vacuum) and VIII expand indefinitely, they are found to have oscillatory (though non-chaotic) behaviour in the Weyl curvature. The late time behaviour of Bianchi type VIIo models with a non-tilted perfect fluid source has been studied recently [364). Most significantly, in contrast to models of Bianchi type VIIh and due to the fact that the Bianchi VlIo state space is unbounded, the Bianchi type VIIo models were shown not to be asymptotically self-similar at late times in general. This breaking of asymptotic self-similarity is characterized by oscillations in the dimensionless shear scalar (the shear parameter) that become increasingly rapid at late times, and consequently leads to Weyl curvature dominance, in which a certain expansion-normalized scalar formed from the Weyl tensor (the Weyl parameter) becomes unbounded. This contrasts with the early time behaviour in Mixmaster (Bianchi types VIII and IX) models, which also are not asymptotically self-similar since they oscillate indefinitely as the initial singularity is approached into the past (although, as discussed above, exact self-similar Kasner models are important in describing the evolution of these models). A comprehensive and rigorous treatment of the late time behaviour of radiation models has recently been presented [285), completing the earlier work. The late time isotropization in these models is effected in a significant way in that the models isotropize as regards the shear but not as regards the Weyl curvature. The Bianchi VIII models, which exhibit similar behaviour, are currently under investigation [180).

In summary, due to the non-existence of periodic, recurrent and homo clinic orbits

33

in the Bianchi state space (deduced from the existence of monotone functions), the dynamical behaviour of Bianchi models is dominated by equilibrium points and heteroclinic sequences (or heteroclinic cycles contained in the Mixmaster attractor for class A models). This is why self-similar models, which correspond to equilibrium points, playa dominant role in the dynamics of Bianchi cosmological models. These issues are further discussed in WE, wherein a variety of heteroclinic sequences in Bianchi models are depicted. The Bianchi VUh state space and the corresponding heteroclinic sequences are illustrated in Figures (5) and (6), respectively (taken from [362]). However, as noted above, not all Bianchi models are asymptotically self­similar. In addition, one can generalize the above analysis to two-fluid models [107] and also to inflationary models with 0 ::; 'Y < 2/3 (see later). The case 'Y = 2, which is a bifurcation value, has also been discussed. The case of stiff C'Y = 2) perfect fluid models, which are formally equivalent to spatially homogeneous massless scalar field models, will be discussed in detail later. The exceptional class of Bianchi VIL1/ 9

models is currently being studied, addressing one of the remaining open problems in the study of non-tilting perfect fluid cosmological models [180].

c. More Recent Work

Recent work [285, 362, 364] demonstrates the fact that numerical studies comple­ment analytical work and is crucial for a complete understanding of the dynamics of the models. In addition, dimensionless variables, which are often bounded, lead to evolution equations that are very well suited for numerical studies of Bianchi mod­els. For example, in order to understand the observational properties of cosmologi­cal models, and particularly the temperature of the cosmic microwave background (CMB) radiation, it is necessary to study the behaviour of their null geodesics. In [286] dynamical systems techniques were utilized to augment the EFE with the geodesic equations, all written in appropriately normalized dimensionless form, ob­taining an extended system of first-order ODE that simultaneously describes the evolution of the gravitational field and the behaviour of the associated null and timelike geodesics. The extended system is a powerful tool for investigating the ef­fect of spacetime anisotropies on the temperature of the CMB radiation in Bianchi models, generalizing and improving upon previous analyses. Essentially, by inte­grating the null geodesic equations the temperature patterns on the celestial sphere can be determined in different models, and the observed bounds on the multiple moments can be used to constrain the parameters (e.g., for the shear and the Weyl curvature) in the models.

Numerical simulations have been carried out to determine the present day CMB temperature pattern in the class of spatially homogeneous non-tilting Bianchi type

34

plane wave equilibrium

points

A+

shear2

I spatial curvature

Milne

~ open FL orbit (~ =0)

typical orbit

-- shear1

Bianchi I orbit

Kasner equilibrium points

FIG. 5: The Bianchi VIh state space, with one dimension suppressed, showing the past attractor A -, which is a subset of the arc of Kasner equilibrium points, and the future attractor A +, which is a subset of the arc of plane-wave equilibrium points. F is the flat FRW equilibrium point, M is the Milne equilibrium point and the orbit F -+ M represents the open FRW models. All equilibrium points correspond to exact self-similar models.

VIlo dust models [287J, whose late time dynamical behaviour was described above [364J. It was shown that the observational bounds on the quadrupole and oc­tupole moments do not imply that the Weyl parameter is necessarily small (i.e., in the class of spatially homogeneous models considered there are models for which the anisotropy of the CMB temperature is within the current observational limits but whose Weyl curvature is not negligible, and hence these models are not close to isotropy.) These models therefore illustrate the fact that, contrary to what is widely believed, an almost isotropic CMB temperature does not imply that the universe is "close to a Friedmann-Lemaitre universe" [287J. In further work [234J the existence of a class of (locally rotationally symmetric) non-tilted dust models of Bianchi type VIII, whose CMB temperature is exactly isotropic at one instant of time (as measured by all fundamental observers) but whose rate of expansion is highly anisotropic, was explicitly demonstrated. The existence of these spatially homogeneous but anisotropic cosmological models again emphasizes that the obser­vation of a highly isotropic CMB temperature cannot alone be used to infer that

35

VII hvac (3)

I VII, '~3)PW.t A-------~- F

Vll h(2)

1 Vll h(2)

A+

n vac (2) D (1)

Kinf I(2)

CS Vll h(1)

n (3)

FIG. 6: Heteroclinic sequences in Bianchi VIIh models; this figure shows the skeleton of orbits determined by the stable and unstable manifolds of the saddle points. A-and A + denote the past and future attractors, respectively, F denotes the flat FRW equilibrium point, OS denotes the Collins-Stewart equilibrium point, Kint denotes the arc of Kasner saddle points and PWint denotes the arc of plane-wave saddle points.

the universe is close to a Friedmann-Lemaitre model.

There continues to be much work done on the qualitative study of cosmologi­cal models with sources more general than a non-tilting perfect fluid with a linear equation of state. Tilting perfect fluid models of Bianchi type II have been compre­hensively studied [179] and recently the problem of late-time isotropization in irro­tational Bianchi type V cosmological models, when the source of the gravitational field consists of two non-interacting perfect fluids - one tilted and one non-tilted, has been considered [156]. It is interesting, as noted in WE, that tilting Bianchi models will generically exhibit an initial oscillating regime, with both "standard mixmaster" bounces and, in addition, so-called "centrifugal bounces". More work also continues to be done on the qualitative study of cosmological models with a non-linear equation of state or an imperfect fluid, or multiple fluids or multiple sources. In particular, oscillatory behaviour also occurs in cosmological models with a magnetic field (and in Einstein-Yang-Mills theory in general). Further work on spatially homogeneous models with a magnetic field and a non-tilted perfect fluid has been carried out recently [39, 224-226]. In particular, Weaver [370] has gener­alized (to the non-polarized solutions) the work of Leblanc, Kerr and Wainwright [226], and rigorously shown that the evolution toward the singularity is oscillatory in Bianchi VIo vacuum models. Spatially homogeneous models with a magnetic field and a scalar field have also been studied [88], and this will be discussed further

36

in Chapter IX.

In the study of inhomogeneous models, the EFE (regarded as evolutionary equa­tions) are PDE, and the resulting state space is thus an infinite dimensional function space. The special class of diagonal O2 cosmologies, in which governing equations reduce to a finite-dimensional invariant subset, were described in WE. Other simple classes of inhomogeneous models have been studied, particularly spherically sym­metric models [352] (and the self-similar subcase, in which the governing equations again reduce to ODE [58, 78, 79, 157, 158]). The study of infinite-dimensional dy­namical systems, and particularly the role of inertial manifolds in describing their asymptotic behaviour, is of current interest [333]. Clearly the study of inhomoge­neous cosmological models is very difficult, but it is also of fundamental importance. In the future it will be of interest to study the asymptotic properties of inhomoge­neous cosmological models and, in particular, address the question of whether past and future asymptotic states are self-similar and investigate the local stability of certain important cosmological solutions, such as the Kasner models and isotropic and homogeneous models, in an infinite-dimensional state space.

D. Scalar Field Models

A variety of theories of fundamental physics predict the existence of scalar fields [44, 164, 291]' motivating the study of the dynamical properties of scalar fields in cosmology. Indeed, scalar field cosmological models are of great importance in the study of the early Universe, particularly in the investigation of inflation (during which the universe undergoes a period of accelerated expansion) [167, 237, 291]. Recently there has also been great interest in the late-time evolution of scalar field models. "Quintessential" scalar field models (or slowly decaying cosmological con­stant models) [15, 70] give rise to a residual scalar field which contributes to the present energy-density of the universe that may alleviate the dark matter problem and can predict an effective cosmological constant which is consistent with obser­vations of the present accelerated cosmic expansion [296, 305]. In addition, a scalar field has been proposed as a candidate for gravitational lensing [270, 355] as well as for dark matter at cosmological scales [16, 83, 271].

The energy-momentum tensor describing a minimally coupled scalar field is given by

Tab = ¢;a¢;b - gab (~¢;c¢;c + V(¢)) . (3.9)

Models with a self-interaction potential V with an exponential dependence on the scalar field, ¢, of the form

V = Aek </> , (3.10)

where A and k are positive constants, have been the subject of much interest and arise naturally from theories of gravity such as scalar-tensor theories or string the-

37

ories [44J. Recently, it has been argued that a scalar field with an exponential potential is a strong candidate for dark matter in spiral galaxies [168J and is con­sistent with observations of current accelerated expansion of the universe [194J.

A number of authors have studied scalar field cosmological models with an expo­nential potential within GR. Homogeneous and isotropic FRW models were studied by Halliwell [l71J using phase-plane methods. Homogeneous but anisotropic mod­els of Bianchi types I and III (and Kantowski-Sachs models) were studied by Burd and Barrow [66J, Bianchi type I models were studied by Lidsey [231J and Aguirre­gabiria et al. [3J, and Bianchi models of types III and VI were studied by Feinstein and Ibanez [140J. A qualitative analysis of Bianchi models with k 2 < 2 (including standard matter satisfying standard energy conditions) was completed by Kitada and Maeda [217J. The governing DE in spatially homogeneous Bianchi cosmologies containing a scalar field with an exponential potential reduce to a dynamical system when appropriate expansion- normalized variables are defined [99J.

One particular solution that is of great interest is the flat, isotropic power-law inflationary solution which occurs for k 2 < 2. This power-law inflationary solution is known to be an attractor for all initially expanding Bianchi models (except a subclass of the Bianchi type IX models which will recollapse) [99, 217J. Therefore, all of these models inflate forever; there is no exit from inflation and no conditions for conventional reheating.

The governing DE in spatially homogeneous Bianchi cosmologies containing a scalar field with an exponential potential exhibit a symmetry [56, 101]' and when appropriate expansion-normalized variables are defined, the governing equations reduce to a dynamical system, which was studied qualitatively in detail in [99J. In particular, the question of whether the spatially homogeneous models inflate and/or isotropize, thereby determining the applicability of the so-called cosmic no-hair conjecture in homogeneous scalar field cosmologies with an exponential potential, was addressed. The relevance of the exact solutions (of Bianchi types III and VI) found by Feinstein and Ibanez [140], which neither inflate nor isotropize, was also considered. In follow up work [349J the isotropization of the Bianchi VIIh cosmological models possessing a scalar field with an exponential potential was further investigated; in the case k2 > 2, it was shown that there is an open set of initial conditions in the set of anisotropic Bianchi VIlh initial data such that the corresponding cosmological models isotropize asymptotically. Hence, scalar field spatially homogeneous cosmological models having an exponential potential with k 2 > 2 can isotropize to the future. However, in the case of the Bianchi type IX models having an exponential potential with k 2 > 2 the result is different in that there do not exist any expanding Bianchi type IX models that isotropize to the future; the analysis of [351 J indicates that if k 2 > 2, then the model recollapses.

Recently cosmological models which contain both perfect fluid matter and a scalar field with an exponential potential have come under heavy analysis [144, 145, 368, 371, 372J. One of the exact solutions found for these models has the property that the energy density due to the scalar field is proportional to the energy density of the perfect fluid, hence these models have been labelled scaling

38

cosmologies [116, 371]. With the discovery of these scaling solutions, it has become imperative to study spatially homogeneous Bianchi cosmologies containing a scalar field with an exponential potential and an additional matter field consisting of a barotropic perfect fluid. The scaling solutions studied in [116, 371], which are spa­tially flat isotropic models in which the scalar field energy density tracks that of the perfect fluid, are of particular physical interest. For example, in these models a significant fraction of the current energy density of the Universe may be contained in the scalar field whose dynamical effects mimic cold dark matter.

In [45] the stability of these cosmological scaling solutions within the class of spa­tially homogeneous cosmological models with a perfect fluid subject to the equation of state p = b - l)p (with 0 < 'Y < 2) and a scalar field with an exponential po­tential was studied. It is known that the scaling solutions are late-time attractors (i.e., stable) in the subclass of flat isotropic models [116, 371]. In [45] it was found that that the scaling solutions are stable (to shear and curvature perturbations) in generic anisotropic Bianchi models when 'Y < 2/3. However, when 'Y > 2/3, and particularly for realistic matter with 'Y ~ 1, the scaling solutions are unstable; es­sentially they are unstable to curvature perturbations, although they are stable to shear perturbations. Although these solutions are unstable, since they correspond to equilibrium points of the governing dynamical system the universe model can spend an arbitrarily long time near these scaling solutions, and hence they may still be of physical importance.

In addition to the scaling solutions described above, curvature scaling solu­tions and anisotropic scaling solutions are also possible. In [350] homogeneous and isotropic spacetimes with non-zero spatial curvature were studied in detail and three possible asymptotic future attractors in an ever-expanding universe were found. In addition to the zero-curvature power-law inflationary solution and the zero-curvature scaling solution alluded to above, there is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature, which also acts as a possible future asymptotic attractor. In [98] spatially homogeneous models with a perfect fluid and a scalar field with an exponential po­tential were also studied and the existence of anisotropic scaling solutions was also discovered; the stability of these anisotropic scaling solutions within a particular class of Bianchi type models was discussed.

Although the exponential potential models are interesting models for a variety of reasons, they have some shortcomings as inflationary models [237, 291]. While Bianchi models generically asymptote towards the power-law inflationary model in which the matter terms are driven to zero for k 2 < 2, there is no graceful exit from this inflationary phase. Furthermore, the scalar field cannot oscillate and so reheating cannot occur by the conventional scenario. In recent work [53] interaction terms were included, through which the energy of the scalar field is transferred to the matter fields. These terms were found to affect the qualitative behaviour of these models and, in particular, lead to interesting inflationary behaviour.

Scalar-tensor theories of gravitation, in which gravity is mediated by a long-range scalar field in addition to the usual tensor fields present in Einstein's theory, are

39

perhaps the most natural alternatives to GR. In the simplest Brans-Dicke theory of gravity (BDT; [60]), a scalar field, ¢, with a constant coupling parameter, wo, acts as the source for the gravitational coupling. More general scalar-tensor theories have a non-constant parameter w (¢), and a non-zero self-interaction scalar potential, V (¢). BDT (and other theories of gravity, such as, for example, more general scalar­tensor theories and quadratic Lagrangian theories and also theories undergoing dimensional reduction to an effective 4-dimensional theory [171]), are known to be conformally equivalent to GR plus a scalar field having exponential-like potentials [42,43,171].

Scalar-tensor gravity theory is currently of particular interest since such theories occur as the low-energy limit in supergravity theories from string theory and other higher-dimensional gravity theories [162]. Lacking a full non-perturbative formula­tion which allows a description of the early Universe close to the Planck time, it is necessary to study classical cosmology prior to the GUT epoch by utilizing the low-energy effective action induced by string theory. To lowest order in the inverse string tension the tree-level effective action in four dimensions for the massless fields includes the non-minimally coupled graviton, the scalar dilaton and an antisymmet­ric rank-two tensor, hence generalizing GR (which is presumably a valid description at late, post-GUT, epochs) by including other massless fields; hence the massless bosonic sector of (heterotic) string theory reduces generically to a 4-dimensional scalar-tensor theory of gravity. As a result, BDT includes the dilaton-graviton sec­tor of the string effective action as a special case (w = -1) [164]. String cosmology has recently been investigated by various authors [136, 212, 213] and, in particular, Billyard et al. [47] presented a qualitative analysis for spatially fiat, isotropic and homogeneous cosmologies derived from the string effective action when a cosmo­logical constant term is included. A discussion of how exponential potentials arise in effective 4-dimensional theories (in the sO-called conformal Einstein frame) after dimensional reduction from higher-dimensional theories such as string theory and M-theory is given in [166].

E. Harmonic Potentials

Let us consider a cosmological model with the action

s = J d4xH {R + ~g/LVa/L¢aV¢ - V(¢)} , (3.11)

where V(¢) is the (arbitrary) self-potential and matter can also be included. Many potentials have been studied in early universe cosmology [291]' and especially con­vex potentials such as the harmonic potential !m2¢2 and the quartic potential i},4¢4. Exponential potentials, which are extensively studied in this book, are also of physical importance and are of particular interest mathematically due to the existence of a symmetry in the evolution equations which leads to the decoupling of the Raychaudhuri equation. As noted above, this makes a qualitative analysis

40

of such models particularly illuminating. Indeed, we focus on exponential poten­tials because such potentials occur in early universe string/ M-theory cosmological models (in the Einstein frame), the subject of Chapters X, XI and XII. However, a qualitative analysis of scalar field cosmological models with non-exponential poten­tials is still fruitful. In an analysis of inflation in scalar field models it is usually the dynamics at intermediate times that are of importance; for example, for massive scalar field models a slow roll regime is entered and inflation results and contin­ues until oscillations in the scalar field develop, after which the scalar particles decay reheating the plasma. Nevertheless, a qualitative analysis can still be useful; in particular the question of the degree of generality of solutions that possess an inflationary stage in such models was addressed in [33-35].

1. Analysis

The spatially homogeneous field equations for the harmonic or quadratic potential V = !m2¢2 become

iI

¢ fJ,

2 2 2 1. 2 1 2 2 1 --(J" - 3H - -¢ + -m ¢ - -(3, - 2)

3 3 6 6 -3H¢ - m 2¢

-3H,Ji-

and an evolution equation for the shear scalar (J". Interaction terms can also be included [53]. The generalized Friedmann equation is

3H2 = (J"2 + !j,2 + !m2-+.2 + /I. _ ! 3R 2'1' 2 'I' fA' 2 ' (3.12)

where 3 R is the three-Ricci scalar which we assume is non-positive. We then define

3H D VI +9H2

E2 3(J"2 3Ji-

1 +9H2 n = 1 + 9H2

U2 3¢2 w-/[~

2(1 + 9H2) - 2 v'I+9H2

so that equation (3.12) becomes

D2 _ E2 - U2 - W2 - n = 3R

6(1 + 9H2) ~ 0,

and since by definition -1 < D < 1,

1 ~ D2 ~ E2 + U2 + W2 + n ~ 0,

41

and all of the variables E, U, W, 0 are bounded.

dT / Defining a new time variable, T, by dt = V 1 + 9H2, the evolution equations

become

D' = (1- D2)3 (3.13)

0' = O{ -,D - 2D3} (3.14)

W' = mVl - D2U - DW3 (3.15)

U' = -DU-mVI-D2W-UD3 (3.16)

and an equation for E', where differentiation is denoted by a prime, and

3 == {_~E2 _ ~D2 _ ~U2 + ~W2 _ (3,- 2)0} 3 3 3 3 6 .

Note that for 3 R ::; 0, 3 < 0, and so D is a monotonically decreasing function. This simplifies the dynamics condiserably. Also, for D > 0 (expanding models with H > 0) D ~ 1 at early times and D ~ 0 at late times. Note that 3 = 0 only when E = U = 0 = 0, D2 = W 2, and 3 R = o.

These equations are relatively simple. However, since the equation for D' does not decouple, the system is of a higher dimension than the corresponding exponential potential case. Also, because of the square root term (VI - D2) the system is not analytic. In addition, the important equilibrium point at the origin is highly degenerate.

Let us summarize the equilibrium points in the isotropic case with E = 0 ({ i=- 2):

(i) D = U = W = 0 = 0 [3 = 0, 3 R = OJ (3.17)

(iia) D2=I,W=0;U=0,0=1[3=~,3R=0] (3.18)

(iib) D2 =I,W=0;0=0,U2 =1[3=-1,3R=OJ (3.19)

(iic) D2 = 1, W = 0;0 = O,U = 0 [3 = -~, 3R i=- 0] (3.20)

(iii) D2 =1,0=0,U=0 W 2 =1[3=0,3R=OJ. (3.21)

In the absence of matter (0 = 0), in the zero-curvature case D2 = U2 + W 2 and we obtain the plane system

W' = mvi - U2 - W2 U + VU2 + W2 WU2

U' = _VU2+W2 U[I-U2J-mVI-U2-W2 W

or

D' -(1 - D2)U2 < 0

U' -DU(I- U2) + mVl- D2VD2 - U2.

42

In the latter case the equilibrium points are {D2, U2} = {O, O}, {I, O}, {I, I} and D is a decreasing function. However, even in this case the analysis is not straightforward.

An alternative approach is to compactify using a Poincare transformation; isotropic and spatially homogeneous massive scalar field models (but with no mat­ter; i.e., /-L = 0) were studied in [34, 35], and this was generalized to Bianchi I models in [33]. In particular, the highly degenerate equilibrium point (i) at the origin was studied and it was proven that it is a late-time attractor (a focus) in D, U, W space (corresponding to coherent oscillations of the scalar field). Hence to study the equilibrium points in higher-dimensional systems (e.g., with shear) we only need to determine the stability in the "extra-dimensions".

Since for 3 R ::; 0, H ::; 0, so that H > 0 and the models expand forever, we can also obtain bounded variables by normalizing with H. That is, defining

0"2 -2 ~ = 3H2

·2 -2 ¢ U = 6H2

o=L 3H2

W=m¢ 6H

and a new time coordinate, T, by : = 3H, we obtain

-2 -2 -2 -l-~ -U -W -n~o

(for 3 R ::; 0), and hence

Defining

-2 -2 -2 -o ::; {~ ,U ,W ,n} ::; 1.

;::; I -2 -2 -2 I -~=3{-2~ -1-2U +W -2(31'-2)n,

the evolution equations become

0' W'

U'

-0(1' - 23)

mU-WS - mW --

-U- 3H -us. Unfortunately, H (and I / H) are not bounded using this normalization. However, we note that in some physical applications compactification may not be necessary; a local analysis of the dynamics in a neighbourhood of an equilibrium point repre­senting an important physical solution (such as the origin) may be sufficient.

2. Chaos

The case 3 R > 0 is much more complicated. The chaotic dynamics of closed isotropic (FRW) cosmological models with a minimally coupled massive scalar field

43

has been extensively studied [29, 31, 36]. It was found that this model allows the existence of a discrete set of periodic trajectories [173] and aperiodic infinitely bouncing trajectories (without singularities) having a fractal nature [294]. In [119] the set of periodic trajectories was studied from the viewpoint of dynamical chaos theory, and it was shown that the dynamics of a closed universe with a massive scalar field is chaotic and an important invariant of the chaos, the topological en­tropy, was calculated. The mathematics of the chaotic behaviour of closed FRW models with a minimally coupled massive scalar field (with or without matter) has also been discussed by other authors [72, 292] (see also [182]). The chaotic behaviour is only possible in positive-curvature models. Indeed, the dynamics of the positive-curvature models is much richer than the simpler zero- and negative­curvature models. As well as chaotic behaviour and the possibility that models can recollapse [97], the space of initial conditions that lead to inflation in positive­curvature scalar field models is broader [335].

The results on chaos in FRW cosmology with a massive scalar field have been extended to other scalar field potentials, and it was shown that for sufficiently steep potentials, such as exponential-like potentials that can arise in modern scenarios based on ideas of the string theory and compactification, the chaos disappears [214, 335]. The steepness of the potential apparently changes the possibilities of escaping the singularities and alters the structure of infinitely bouncing trajectories. Under some conditions the chaotic behaviour can completely disappear, and in other cases potentials less steep than quadratic can give rise to a chaotic dynamics which differs qualitatively from that described in [119].

IV. SCALAR FIELD COSMOLOGIES WITH BAROTROPIC MATTER

In this Chapter we shall present a comprehensive study of the qualitative proper­ties of spatially homogeneous models with a barotropic fluid and a non-interacting scalar field with an exponential potential in the class of Bianchi type B models (except for the exceptional case Bianchi VL1/ 9 ), using the Hewitt and Wainwright formalism [178, 363]. In particular, we shall study the generality of the scaling solu­tions. We define the governing equations, which are modified from those developed in [363], and discuss the invariant sets and the existence of monotonic functions. We then classify and list all of the equilibrium points, and their local stability is discussed. We present a detailed analysis and conclude with a discussion.

A. Models of Bianchi Class B: The Equations

We shall assume that the matter content is composed of two non-interacting components where the total energy-momentum tensor is the sum of the energy­momentum tensors (1.1) and (3.9). The first component is a separately conserved barotropic fluid with a gamma-law equation of state, i.e., p = b - 1)J.t, where I is a constant with 0 ~ I ~ 2, while the second is a noninteracting scalar field ¢ with an exponential potential (3.10), V(¢) = Aek ¢, where A and k are positive constants (recall that we use units in which 87rG = c = 1). By non-interacting we mean that the energy-momentum of the two matter components will be separately conserved.

The state of any Bianchi type B model with the above matter content can be described by the evolution of the variables

(H,(1+,if,8,ii,n+,~,¢) E ]R8, (4.1)

where the evolution of the state variables are given by equations (5.8) and (7.8) in Wainwright and Ellis [363] with the addition of the Klein-Gordon equation for the scalar field,

¢+3H~+kV(¢)=O. (4.2)

By introducing dimensionless variables, the evolution equation for H decouples and the resulting reduced system has one less dimension. Defining [99, 363]

E+ (1+ - if 8 - ii H' E= H2' ~= H2' A= H2'

N+ n+ w=~ <P = JV(¢) n=~ H' V6H' v'3H' 3H2'

(4.3)

the DE for the quantities

X = (E+, i;,~,.4, N+, W, <p) E ]R7 (4.4)

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

45

then reduces to:

~~ = (q - 2) ~+ - 2N, (4.5)

t' = 2 (q - 2) t - 4 ~ N+ - 4 ~+ A, (4.6)

~' = 2 (q + ~+ - 1) ~ + 2 (t - N) N+, (4.7)

A' = 2 (q + 2 ~+) A, (4.8)

N~ = (q + 2~+)N+ + 6~, (4.9)

'11' = (q - 2) '11 -! v'6k<l>2, 2

(4.10)

1 <1>' = (q + 1 + 2 v'6kw) <1>, (4.11)

where a prime denotes differentiation with respect to the time T, where dtldT = H. The deceleration parameter q is defined by q == -(1 + H' I H) and given by

2 - 1 q = 2 ~+ + 2 ~ + 2 (3'Y - 2) 0 + 2 '112 - <1>2, (4.12)

and both N (a curvature term) and 0 (the matter term) are obtained from first integrals:

N = !N2 - !lA 3 + 3 '

2 2 2 - - -o 1 - '11 - <I> - ~+ - ~ - N - A.

The evolution of 0 is given by the auxiliary equation

0' = 0 (2 q - 3'Y + 2).

(4.13)

(4.14)

(4.15)

The parameter l = I/h, where h is the group parameter, is equivalent to h in [363]. If l < 0 and A > 0, then the model is of Bianchi type VIh . If l > 0 and A > 0 and N+ i- 0, then the model is of Bianchi type VIIh . If l = 0, then the model is either Bianchi type IV or type V. If A = 0 then the model is either a Bianchi type I or a Bianchi type II model. line)

There is one constraint equation that must also be satisfied:

G(X) = tN - ~2 - A~! = o. (4.16)

Therefore the state space is six-dimensional; the seven evolution equations (4.5)­(4.11) are subject to the constraint equation (4.16). We shall refer to the seven­dimensional state space (4.4) as the extended state space.

By definition A is non-negative, which implies from equations (4.13) and (4.16) that t and N are also non-negative. Thus we have

A20, t 2 0, N20. (4.17)

46

In addition, from the physical constraint n 2: 0 together with equation (4.14), we find that the state space is compact. Indeed, we have that

{ 2- 2-- 2} 0::; ~+,~, A ,A, N, ill ,<I> ::; 1. (4.18)

Since both A and N are bounded, we have from equation (4.13) that N+ is bounded. In principle, there exists negative and positive values for <1>, but from the definition (4.3) a negative <I> implies a negative H and hence H < 0 for all time; i.e., the models are contracting. Since the system is invariant under <I> -t -<1>, without loss of generality we shall only consider <I> 2: o.

1. Invariant Sets, Monotone Functions and The Constraint Surface

There are a number of important invariant sets [50], which are presented in Ta­bles I and II. Recall that the state space is constrained by equation (4.16) to be a six-dimensional surface in the seven-dimensional extended space. Taking the con­straint equation (4.16) into account we calculate the dimension of each invariant set. These invariant sets can be classified into various classes according to Bianchi type and/or according to their matter content. Some invariant sets (notably the Bianchi invariant sets) have lower-dimensional invariant subsets. Equilibrium points and orbits occurring in each Bianchi invariant set correspond to cosmological models of that Bianchi type. Various lower-dimensional invariant sets can be constructed by taking the intersection of any Bianchi invariant set with the various Matter invari­ant sets. For example, B(J) n M is a 3-dimensional invariant set describing Bianchi type I models with a massless scalar field.

An analysis of the dynamics in the invariant sets V and :F has been presented by Wainwright and Hewitt [178]. Equilibrium points and orbits in the invariant set M correspond to models with a massless scalar field; i.e., scalar field models with vanishing potential. These models are equivalent to models with a stiff perfect fluid (i.e., "I = 2) equation of state [178]. Equilibrium points and orbits in the invariant set :F M can be interpreted as representing a two-perfect-fluid model with "12 = 2 [107]. A partial analysis of the isotropic equilibrium points in the invariant set S was completed by van den Hoogen et al. [349]. We note that the so-called scaling solutions [116, 368, 371] are in the invariant set :FS.

The isotropic and spatially homogeneous models are found in the invariant sets S±(V Jh) uS(I) if l -=I- 0, and S(V) US(I) if l = O. In particular, the zero curvature isotropic models are found in the two dimensional set S(J), while the negative curvature models are found in the 3-dimensional sets S±(VJh) or S(V) depending upon the value of l.

We note that in the invariant set B(J) there exists the invariant set 1:+ ~~ + i11 2 < 1, A = A = N+ = <P = 0, which may be directly integrated to yield

1: + ~~ + i11 2 = [1 + (e3(2-')')T] -1, (= constant, (4.19)

47

Bianchi Type Notation Dimension Restrictions

Bianchi I B(I)

S(1) Bianchi II B± (I I)

S±(II)

Bianchi IV B±(IV)

Bianchi V B(V)

S(V)

Bianchi Vh B(Vh)

S(Vh) S±(II1)

Bianchi VIIh B±(VIIh) S±(VIh)

4

2

5 4

6

4

3

6

4

5

6

3

A=~=N+ =0

A = E+ = i; = ~ = N+ = 0

A=O, N+ > 0 or N+ < 0 A=O, - 2

E = 3E+, ~=E+N+

l = 0, A> 0, N+ > 0 or N+ < 0

l = 0, A>O, E+ =~=N+ =0 l = 0, A> 0, E+ = i; = ~ = N+ = 0

l < 0, A>O

l < 0, A>O, 2 -3E+ +lE = 0, N+ =~ =0

l = -1, A> 0, 2 -3E+ - E = 0, ~=E+N+

l > 0, A> 0, N+ > 0 or N+ < 0 l > 0, A> 0, E+ = i; = ~ = 0, N~ = lA > 0

TABLE I: Bianchi Invariant Sets. We note that B(I) and B±(II) are class A Bianchi invariant sets which occur in the closure of the appropriate higher-dimensional Bianchi type B invariant set (see Fig. (4)). In addition, if l is non-negative, N+ > 0 and N+ < 0 define disjoint invariant sets (indicated by a superscript ± in the table). Due to the discrete symmetry ~ --t -~, N+ --t -N+, these pairs of invariant sets are equivalent.

Matter Content Notation Dimension Restrictions

Scalar Field S 5 n=Oj wlO, <I> 10 Massless Scalar Field M 4 n=Oj wlO, <I> =0 Vacuum V 3 n=Oj w=o, <1>=0

Perfect Fluid + Scalar Field FS 6 nlOj wlO, <I> 10 Perfect Fluid + Massless Scalar Field F M 5 nlOj wlO, <I> = 0 Perfect Fluid F 4 nlOj w=O, <I> =0

TABLE II: Matter Invariant Sets.

where T is the time parameter. This solution asymptotes into the past towards the paraboloid K (Section IV B.l), and asymptotes to the future towards the point "~+ = t = 6. = A = N+ = "iji = <) = 0" P(I). This solution belongs to the matter invariant set :F M, asymptoting into the past towards the set M.

The existence of strictly monotone functions, W(X) : ]R7 -+ lR, on any invariant set, S, proves the non-existence of periodic or recurrent orbits in S and can be used to provide information about the global behaviour of the dynamical system in S. Some monotone functions are presented in Table III. Hewitt and Wainwright found a number of monotone functions in the invariant sets of dimension less than four in the perfect fluid case (i.e., in lower-dimensional subsets of the perfect fluid invariant set) and these are summarized in an Appendix in Hewitt and Wainwright [178, 363]. However, they were not able to find a monotonic function in the full perfect fluid invariant set for 2/3 < 'Y < 2.

48

Function: Wi(X) Derivative: WJ(X) Region of Monotonicity

WI =(1 + ~+)2 -..4: W{ = -2(2 - q)WI Monoton. approaches zero

+3 (1 + ~+) (2<I>2 + (2 -,) n) in the invar. set M U V.

Monoton. decreasing to zero 1 - n - <I>2 - 1}i2 ,

W2= ~ W2 =-W2(2-3,) in (F8 U F M U F)\S(I)

when 0 ::::: , ::::: 2/3

W3 = f;

..4:2 W4= N+

1 (2 -) - n ~+ +~

W~ = -2(2 - q)W3

-4(~N+ + ~+..4:)

Monoton. decreasing to zero

in the invar. sets B(I)\S(I) and B(V)\S(V).

Monoton. approaches zero in

W~ = 3W4 (q + 2~+~+ - ~) the set S±(IIl)\(8 U F8),

when, > 2/3.

TABLE III: Functions, their derivatives and the sets in which they are monotonic.

The constraint equation G(X) = 0 and the Implicit Function Theorem can gener­ally be used to eliminate one of the variables at any point in the extended state-space provided the constraint equation is not singular there, i.e., grad(G(X)) f=- o. The constraint surface is singular for all points in the invariant sets 8(1), B(V) and 8(V1h). Therefore, we cannot determine the local stability of equilibrium points in the sets 8(1), B(V) or 8(V1h) within the six-dimensional state-space, and hence we are required to determine the local stability of these equilibrium points in the extended space, due to the singular nature of the constraint surface. This leads to further complications. If these equilibrium points are stable in the extended state space, then they are stable in the six-dimensional constrained surface. However, if these equilibrium points are saddles in the extended state-space, then one must use the Stable Manifold Theorem to determine the dimension of the stable manifold within the constraint surface.

B. Classification of the Equilibrium Points

From equation (4.15) we find that at the equilibrium points either

0=0, (4.20)

49

or

3 q = "21' - 1. (4.21)

In the scalar field case (4.20) there is no perfect fluid present. This is the scalar field invariant set S. The equilibrium points and their stability will be studied in Subsection IV B 1. These models include the massless scalar field case in which <p = 0 (V = 0), but not the vacuum case <p = W = 0 which will be dealt with as a subcase of the perfect fluid case (see below). These equilibrium points also include the isotropic Bianchi VIh models studied in [349].

If, on the other hand, equation (4.21) is satisfied, assuming that l' < 2 so that q -=I- 2, from equations (4.9) and (4.10) we have that either

or

W = O,<P = 0

W = -J31' v'2k '

<p2 = 31'(2 - 1') 2k2

(4.22)

(4.23)

In the former case, in which both equations (4.21) and (4.22) are valid, there is no scalar field present. The perfect fluid subcase, which was studied by Hewitt and Wainwright [178], will be dealt with in Subsection IVB 2. Note that from equation (4.10) <p = 0 is an invariant set, denoted M. The latter case, in which equation (4.23) is valid and neither the scalar field nor the perfect fluid is absent, corresponds to the scaling solutions when l' > O. If we define

1 . p¢ == "2¢2 + V(¢),

1 '2 PcP =="2¢ - V(¢), (4.24)

then from equation (4.23) we find that

_ p¢ + P¢ 2W2 1'¢= P¢ =W2 +<p2 =1', (4.25)

so that the scalar field "inherits" the equation of state of the fluid. It can be shown that there are exactly three equilibrium points corresponding to scaling solutions; the flat isotropic scaling solution described in [116], and whose stability was dis­cussed within Bianchi type VIIh models in [45], and two anisotropic scaling solutions [98]. This will be further discussed in Subsection IV B 3.

Hereafter, we shall assume that 0 < l' < 2. The value l' = 0 corresponds to a cosmological constant and the model can be analyzed as a scalar field model with the potential V = Va + Aek ¢ [46]. The value l' = 2, corresponding to the stiff fluid case, is a bifurcation value and will not be considered here.

50

1. Scalar Field Models

There are seven equilibrium points and one equilibrium set in the scalar field invariant set S in which n = O. results of" to The first three equilibrium points represent isotropic models (I:+ = E = if = ~ = 0);

1) Ps(I): I:+ = E = ~ =.A = N+ = 0, 'II = -k/v6,cp = y'1- k2/6

This equilibrium point, for which q = -1 + k 2/2 and which exists only for k2 ::; 6, is in the Bianchi I invariant set B(I). This point represents a flat FRW power-law inflationary model [99, 171]. The corresponding eigenvalues in the extended state space are (we shall not explicitly display the corresponding eigenvectors):

1 --(6-k2)

2 ' 1

-2(6-k2 ), -(6-k2), -(4-k2), -(2-k2), 1

--(2-k2) 2 '

± . _ - _ _ - _ (k2 _2) _ Jl(kL 2) _ A 2)Ps (Vlh). I:+-I:-~-O,A-~,N+-± k ,W--,;3k'

if,. _ 2 'J!' - ,;3k

k 2 -3,,/.

(4.26)

These two equilibrium points (the indices "±" correspond to the ± values for N+), which occur in the Bianchi VIIh invariant set S(Vlh) (since .A ~ 0, then k2 ~ 2 and therefore l > 0), have q = O. These equilibrium points represent an open FRW model [349]. The corresponding eigenvalues in the extended state space are:

V3i 2 - 3,,/, -1 ± Ty'k 2 - 8/3,

-2 ± V; J k 2 - 4(k2 - 2)l ± V[k2 - 4(P - 2)l]2 + 16l (P - 2)2 + k4. (4.27)

- - ~ A 2a)Ps (V): I:+=I:=~=O,A= k;- ,N+=o,w=-d,cp= ~ v3k v3k

This case corresponds to points 2) for l = 0 and belongs to the set S(V). The corresponding eigenvalues in the extended state space are:

V3i Ik2 / 2 - 3,,/, -1 ± TV. - 8 3, -2, -2, 0, -4. (4.28)

± k 2 -2 - 2 -3) Ps(II): I:+ = -k2+l6,I: = 3I:+,~ = I:+N+,A = 0, _ .J-(k2-2)(k2-8) __ 3v'6k _ y8-k2

N+ - ±3 k2+16 ' 'II - k2+16'cp - 6 k2+16

51

These two equilibrium points, for which q = 8(k2 - 2)/(k2 + 16) > 0, exist only for 2 ::::: k 2 ::::: 8. These two points represent Bianchi type II models analogous to those found in [178). The corresponding eigenvalues are:

k 2 - 2 k 2 - 8 12 6--

k2 -8 6-­

k 2 + 16' (k2 - 8) ± V(13k2 - 32)(k2 - 8) 3 _ _ ,

k 2 + 16' k 2 + 16 ' k2

-37+ 18 k2 + 16' (4.29)

4) P (VI ). ~ - -1(k2 -2) i' - -3E2 /l A - ° A - 9(k2 -21)(k2 -2) N - ° S h· L,,+ - n ' L" - +, u - , - n2 , + - , \If = v'6k(l-l) cf> = 2v'3v'(k2 -21)(1-1)

n' n' where n == k2(l- 3) +4l. Since E > 0, we have that l < ° and hence this equilibrium point occurs in the Bianchi VIh invariant set. The deceleration parameter is given by q = 2l(k2 - 2)/[k2(l - 3) + 4l) ~ 0, where k2 ~ 2, and this point corresponds to a Collins Bianchi type VIh solution [108) . The corresponding eigenvalues are:

k2 - 2,l k2(1 -l) 6 [k2 (l _ 3) + 4l) , -37 - 6 [k2 (l - 3) + 4l)'

(k2 - 2l) ± V(k2 - 2l)2 + 8l(1 -l)(k2 - 2) 3 [k2(l _ 3) + 4l) ,

(k2 - 2l) ± V(k2 - 2l)[(k2 - 2l) - 4(1-l)(k2 - 2)) (430) 3 [k2(l _ 3) + 4l) "

Let us next consider the Massless Scalar Field Invariant Set M: there is one equilibrium set which generalizes the work in [178) to include scalar fields:

5) /CM: E + E~ + \lf2 = 1, ~ = A = N+ = cf> = 0, \If 1= ° This paraboloid, for which q = 2, generalizes the parabola /C in [178) defined by

E + E~ = 1 to include a massless scalar field, and represents Jacobs' Bianchi type I non-vacuum solutions [108) . However, the eigenvalues are considerably different from those found in [178), and so we list them all here (the variables which define the subspaces in which the corresponding eigendirections reside are included below in curly braces):

~ v'6 2[(1 + E+) ± V 3E], 0, _ 0: 3(2 -:: 7), 4(1 ~ E_+), 3 + _2k\lf. (4.31)

{~,N+} {E+,E}{E+,E,\If}{E+,E,\If}{E+,E,A,\If} {E,cf>}

52

2. Perfect Fluid Models

As mentioned earlier, the perfect fluid invariant set F in which W = <I> = 0 was studied by Hewitt and Wainwright [178]; hence this Subsection generalizes their results which are summarized in Table IV by including a scalar field with an expo­nential potential. We shall use their notation to label the equilibrium points/sets. There are five such invariant points/sets. In all of these cases the extra two eigen­values associated with wand <I> are (respectively)

3 3 -2(2 - 'Y) < 0, 2'Y > O. (4.32)

1) P(I): ~+ = f; = ~ = A = N+ = w = <I> = 0

This equilibrium point, for which n = 1, is a saddle for 2/3 < 'Y < 2 in F [178] (and is a sink for 0 ::; 'Y < 2/3), which corresponds to a flat FRW model.

2) p±(II): ~+ = -116 (3'Y - 2), f; = 3~~, ~ = ~+N+, A = 0, N+ = ±h/(3'Y - 2)(2 - 'Y), w = <I> = 0

This equilibrium point, for which n = 136 (6 - 'Y), is a saddle in the perfect fluid invariant set [178].

3) P(Vh): ~+ = -~(3'Y - 2), f; = -3~~/l, ~ = 0, A = -1~1 (3'Y - 2)(2 - 'Y), N+ = w = <I> = 0

Since f; > 0 and A > 0, this equilibrium point occurs in the Bianchi VIh invariant set and corresponds to the Collins solution [108] , where n = £(2 - 'Y) + -ft(3'Y - 2) (and therefore 2/3::; 'Y::; 2(-l-1)/(3 -l) and so l::; -1). In [178] this was a sink in F, but is a saddle in the extended state space due to the fact that the two new eigenvalues have values of different sign.

There are also two equilibrium sets, which generalize the work in [178] to include scalar fields:

4) £t: f; = -~+(1 + ~+),~ = o,A = (1 + ~+)2, N+ = ±J(l + ~+)[l(l + ~+) - 3~+], w = <I> =0

For this set n = O. The local sinks in this set occur when [178]

(a) l < 0 (Bianchi type VIh) for -~(3'Y - 2) < ~+ < l/(3 - l) and l > -(3'Y-2)/(2 - 'Y) < 0,

(b) l = 0 (Bianchi type IV) for -i(3,}, - 2) < :E+ < 0,

(c) l > 0 (Bianchi type VIIh) for -i(3,}, - 2) <:E+ < O.

The additional two eigenvalues for the full system are:

1 - 2:E+, -2(1 + :E+).

Finally, let us consider the Massless Scalar Field Invariant Set :F M:

5) K: E + :E~ = 1, ~ = .A = N+ = <I> = w = 0

53

(4.33)

This parabola, for which q = 2, is the special case of KM in which W = 0 and corresponds to the parabola Kin [178]. However, the eigenvalues are considerably different from those found in [178] and so we list them all here (the variables define the subs paces in which the corresponding eigendirections reside are included below in curly braces):

0, 0, 2[(1 + :E+) ± V3E], {~,N+} {:E+, E} {w}

Eigenvalues

P(l) -~(2 - ")')

p±(II) ~(3")'-2) -~(2-")')

P(Vh) -~(2-")')(1 ± v'l-r2 )

3(2 - '}'),

{:E+, E}

K o 2(1 + I:+) 2(2-")') 2[1 + I:+ ±

4(1 + :E+),

{:E+,E,.A}

3.

{<I>}

Comments

'E eliminated

'E eliminated

1-D invar. set

(4.34)

v o 0 2[1+I:+ ± v3'El 2(1+I:+) 2-D invar. set, ")'=2

£1 o - 4I:+ - (3")' - 2) - 2 [(1 + I:+) ± 2iN+l 'E eliminated

Fl -2 4 - 2 0 0 120, ")' = 2/3

TABLE IV: Equilibrium sets found by Hewitt and Wainwright [26], and the corresponding eigenvalues in the extended space. In the table r2 == 2(3")' - 2)(1 -Ie/I), q2 == 2r2/(2 - ")') and Ie == -(3")' - 2)/(2 - ")').

Defining

3. Scaling Solutions

Ws == -JI~, <I>~ == 3'}'(2 - '}') 2k2

(4.35)

and recalling that 0 < '}' < 2, there are three equilibrium points corresponding to scaling solutions. Because the scalar field mimics the perfect fluid with the exact

54

same equation of state (1'4> = 1') at these equilibrium points, one may combine these two "fluids", via Ptot = P4> + p, fltot = fl4> + fl, Ptot = (1' - l)fltot; therefore, all of these equilibrium points will correspond to exact perfect fluid models analogous to the equilibrium points found in [178].

The flat isotropic FRW scaling solution [368, 371]:

1) ;:s(1): E+ = E = ~ = A = N+ = 0, W = ws,<I> = <I>s

The eigenvalues for these points in the extended space, for which n = 1 - 31'/k2 (and therefore k 2 ?: 31'), are:

3 -2(2-1')' -3(2-1'), 31'-4, 31'-2,

133 -2(31' - 2), - 4(2 - 1') ± 4 \1'(2 - 1')(2 - 91' + 241' /k2) (4.36)

There are also two anisotropic scaling solutions:

2) As(II): E+ = -116 (31' - 2), E = 3E~, ~ = E+N+, A = 0,

N+ = ±i \1'(31' - 2)(2 - 1'), W = W s, <I> = <I>s

The eigenvalues for these points, for which n = 136 (6 - 1') - 31'/ k2 (and therefore k 2 ?: 161'/[6 - 1']), are:

333 4 (31' - 2), -2(2 - 1'), -4 [(2 - 1') ±

'2 - 0' - ~ (2 0) { 2 (30 - 2) + 0(6k~ 0) (k'- 6' : 00) ± y'E,}] ,(4.37)

where E1 == [2 (31' - 2) - '"Y(~;'"Y) (k2 - ~ ) ] 2 + ~ (31' - 2) '"Y(~;'"Y) (k2 - ~ ) .

3)As(Vh): E+ = -~ (31' - 2), E = -3E~/l, ~ = 0, - 9 A = -161 (2 - 1') (31' - 2), N+ = 0, W = W s, <I> = <I>s

These points occur in the Bianchi V1h invariant set (l < ° since E > 0) for which n = !(2 - 1') + tz(31' - 2) - 31'/k2 (and therefore _l-1 ::; (2 - 1')/(31' - 2) and k2 ?: 41'/[(2 - 1') + (31' - 2)/l] ) and correspond to the Collins Bianchi V1h perfect fluid solutions [108] . The eigenvalues for these equilibrium points are:

-~ [(2-1')± (2 - 1')2 - 4(31' - 2)2 -- + - ,-- (2 - 1')± ( 2-1' 1)] 3[ 31'-2 l 4

55

(2 - ,)2 - (2 -,) [4, (1- !~) + (3, - 2) (32,-_'2 + ~) ± JE;"J] ,(4.38)

where E2 == [4, (1- ~) - (3, - 2) (32'"(-_'"(2 + t) r -128~. C. Stability of the EquilibriUIn Points and Some Global Results

The stability of the equilibrium points listed in the previous Section can be easily determined from the eigenvalues displayed. Often the stability can be determined by the eigenvalues in the extended state space, otherwise the constraint must be utilized to determine the stability in the six-dimensional state space (Le., within the constraint surface). In the cases in which this is not possible, we must analyse the eigenvalues in the extended seven-dimensional state-space. Employing local stability results and utilizing the monotone functions found in Table III, we are able to prove some global results. In the absence of monotone functions, and in the same spirit as Refs. [363] and [178], we conjecture plausible results which are consistent with the local results and the dynamical behaviour on the boundaries and which are substantiated by numerical experiments.

1. The Case n = 0: If n = 0 and <I> = 0, then the function W 1 in Table III monotonically approaches zero. The existence of the monotone function W 1 implies that the global behaviour of models in the set M U V can be determined by the local behaviour of the equilibrium points in M U V. Consequently, a portion of the equilibrium sets K and KM (corresponding to local sources) represent the past asymptotic states while the future asymptotic state is represented by L/, or in the case of Bianchi types I and II, by a point on K. Therefore, all vacuum models and all massless scalar field models are asymptotic to the past to a Kasner state and are asymptotic to the future either to a plane wave solution (Bianchi types IV, VIh and Vllh), or to a Kasner state (Bianchi types I and II), or to a Milne state (Bianchi type V).

If n = 0 and <I> -I- 0, then the models only contain a scalar field. It was proven in [216, 217] that all Bianchi models evolve to a power-law inflationary state (repre­sented by Ps(I)) when k2 < 2. If k2 > 2, then it was shown in [349] that a subset of Bianchi models of types V and Vllh evolve towards negatively curved isotropic mod­els represented by points Ps(V) and P~(Vlh). In [349] it was shown that when k 2 > 2 the future state of a subset of Bianchi type VIh solutions is represented by the point P s(V h). It can be seen here that the future state of a subset of Bianchi type II models is represented by the point P~(II). Therefore, all scalar field models with n = 0 evolve to a power-law inflationary state if k2 < 2. If k2 > 2, then the future asymptotic state for all Bianchi type IV, V and VIh models is conjectured to be a negatively-curved, isotropic model and the future asymptotic state for all Bianchi type VIh models is conjectured to be the Feinstein-Ibanez anisotropic scalar field model [140]. If 2 < k 2 < 8, then the future asymptotic state for all Bianchi

56

type II models is the anisotropic Bianchi type II scalar field model, and if k 2 2: 8 then the future asymptotic state is that of a Kasner model. If 2 < k2 < 6, then the Bianchi type I models approach a non-inflationary, isotropic (Le., the point Ps(I)); if k2 2: 6, then they evolve to a Kasner state in the future.

2. The Case n =I- 0, 0 $ 'Y $ 2/3: If n =I- 0 and 0 $ 'Y $ 2/3, then the function W2 in Table III is monotonically decreasing to zero. Therefore, we conclude that the w-limit set of all non-exceptional orbits (Le., those orbits excluding equilibrium points, heteroclinic orbits, etc.) of the dynamical system (4.5)-(4.11) is a subset of S(I). This implies that all non-exceptional models with n =I- 0 evolve towards the zero-curvature spatially-homogeneous and isotropic models in S(I) and hence isotropize to the future. In [350], it was shown that the zero-curvature spatially­homogeneous and isotropic models evolve towards the power-law inflationary model, represented by the point Ps(I) when k2 < 3'Y or towards the isotropic scaling solution, represented by the point Fs (I) , when k2 > 3'Y. Using WI, we also conclude that the past asymptotic state(s) of all non-exceptional models (including models in S(I)) is characterized by n = o. In other words, matter is dynamically unimportant as these models evolve to the past. It was shown in [350] that all models evolve in the past to some portion of K or KM (the Kasner models) which are local sources.

3. The Case n =I- 0, ~ < 'Y < 2: Table V lists the local sinks for ~ < 'Y < 2. The function W3 is monotonically decreasing to zero in B(I)\S(I) and B(V)\S(V).

Sink Bianchi type

Ps(I) I Ps(Vlh)t I Ps(VL!) III

.ct(VL!) III

Ps(V) V Ps(V1h) VIh .c~(Vh) VIh As(Vh) VIh P~(Vlh) VIIh

k

k 2 S 2 k 2 = 2

k2 ? 2

all

k 2 > 2

Other constraints

"I > k2 /(k2 +1), l=-l "1>1, ~+=-1/4

k 2 ? 2 "I> 2k2(1-l)/[k2(l- 3) + 4l]

all "I > 4/3, ~+ < -1/2 k 2 > 4")' l < -(3J-2)

- [(2-,,()+(3,,(-2)//[ - 2-"( 2 k7 2

k ? 2 l > 4(kL2)~4-k2} for 2 < k S 4

l < .".? :,,,OJ " for k 2 > 4

TABLE V: All of the sinks in the various Bianchi invariant sets for 2/3 < "I < 2 are presented. A subset of ICM acts as a source for all Bianchi class B models. t Note: in this case N+ = 0 (i.e., Ps=I1) and so corresponds to a Bianchi I model.

This implies that there do not exist any periodic or recurrent orbits in these sets and, furthermore, the global behaviour of the Bianchi type I and V models can be determined from the local behaviour of the equilibrium points in these sets. We conjecture that there do not exist any periodic or recurrent orbits in the entire phase space for 'Y > 2/3, whence it follows that all global behaviour can be determined from Table V. We note that a subset of KM with (1 + E+)2 > 3~, 111 > -.j6/k acts as a source for all Bianchi class B models. For k2 < 2, P s (I) is the global

57

attractor (sink). From Table V we see that there are unique global attractors (both past and future) in all invariant sets and hence the asymptotic properties are simple to determine. The sinks and sources for a particular Bianchi invariant set, which may appear in that invariant set or on the boundary corresponding to a (lower­dimensional) specialization of that Bianchi type, can be easily determined from Table V and Fig. (4) which lists the specializations of the Bianchi class B models [264].

The most general models are those of Bianchi types VIh and VIh. The Bianchi type VIh models are of particular physical interest since they contain open FRW models as special cases. From Table V and Fig.(4) we argue that generically these models (with a scalar field) isotropize to the future, a result which is of great significance. The Bianchi type VIh models are also of interest since they contain a class of anisotropic scaling solutions that act as attractors for an open set of Bianchi type B models. We note that generically Bianchi type VIh models do not isotropize for k2 ~ 2.

D. Discussion

We have discussed the qualitative properties of Bianchi type B cosmological mod­els containing a barotropic fluid and a scalar field with an exponential potential. The most general models are those of type VIh, which include the anisotropic scal­ing solutions, and those of type VIIh, which include the open FRW models. In cases in which we have been able to find monotone functions we have been able to prove global results. Otherwise, based on the local analysis of the stability of equilibrium points and the dynamics on the boundaries of the appropriate state space, we have presented plausible global results (this is similar to the analysis of perfect fluid models in [363] and [178] in which no monotone functions were found in the Bianchi type VI and VII invariant sets). In all cases, however, our results are further justified by numerical experimentation.

Let us summarize the main results [50]:

• All models with k2 < 2 asymptote toward the flat FRW power-law inflationary model [99, 171], corresponding to the global attractor Ps(I), at late times; i.e., all such models isotropize and inflate to the future.

• Fs(J) is a saddle and hence the flat FRW scaling solutions [368, 371] do not act as late-time attractors in general [45].

• A subset of KM acts as a source for all Bianchi type B models; hence all models are asymptotic in the past to a massless scalar field analogue of the Jacobs anisotropic Bianchi I solutions.

• For k 2 ~ 2, Bianchi type VIlh models generically asymptote towards an open FRW scalar field model, represented by one of the local sinks Ps (V) or P}(VJh), and hence isotropize to the future.

58

• For k 2 ~ 2, Bianchi type VIh models generically asymptote towards either an anisotropic scalar field analogue of the Collins solution [108J , an anisotropic vacuum solution (with no scalar field) or an anisotropic scaling solution [98], corresponding to the local sinks P§'(V1h), Lk(Vh) or As(Vh), respectively (see Table V). These models do not generally isotropize .

• In particular, the equilibrium point As(V h) is a local attractor in the Bianchi Vh invariant set and hence there is an open set of Bianchi type B models containing a perfect fluid and a scalar field with exponential potential which asymptote toward a corresponding anisotropic scaling solution at late times.

It is also of interest to determine the intermediate behaviour of the models. In order to do this, we need to investigate the saddles, determine the dimension of their stable submanifolds, and construct possible heteroclinic sequences. This could then be used, in conjunction with numerical work, to establish the physical properties of the models. For example, we could investigate whether intermediate isotropization can occur in Bianchi type VIh models [362J. There are many different cases to consider depending upon the various bifurcation values and the particular Bianchi invariant set under investigation. As an example, the heteroclinic sequences in the 4-dimensional invariant set S (V h) was studied in [50J.

We should stress that our analysis and results are applicable to a variety of other cosmological models in, for example, scalar-tensor theories of gravity (which are formally equivalent to GR containing a scalar field with an exponential potential) [25, 46, 230, 277J, theories with multiple scalar fields with exponential potentials [229J and string theory [47, 210], some of which will be discussed later. We shall also discuss spatially homogeneous models with positive spatial curvature in Chapter VI [97J.

v. PHYSICAL APPLICATIONS

Let us discuss some of the more physically important results of the last Chapter in more detail.

A. Isotropisation

In the famous paper by Collins and Hawking [112] it was proven that within the set of spatially homogeneous cosmological models (which satisfy reasonable energy conditions) those which approach isotropy at infinite times is of measure zero; that is, in general anisotropic models do not isotropize as they evolve to the future. Since we presently observe the Universe to be highly isotropic, we therefore need an explanation of why our Universe has evolved the way it has. This problem, known as the isotropy problem, can be solved with an idea popularized by Guth [167]. If the early Universe experiences a period of inflation, then all anisotropies are essentially pushed out of our present observable light-cone and are therefore not presently observed. The Cosmic No-Hair Conjecture asserts that under appropriate conditions, any universe model will undergo a period of inflation and will consequently isotropize.

A significant amount of work on the Cosmic No-Hair Conjecture has already been done for spatially homogeneous (Bianchi) cosmologies [175, 198, 204, 216, 367]. For instance, Wald [367] has proven a version of the Cosmic No-Hair Conjecture for spatially homogeneous spacetimes with a positive cosmological constant; namely, he has shown that all initially expanding Bianchi models asymptotically approach a spatially homogeneous and isotropic model except for the subclass of Bianchi type IX models which recollapse.

Anisotropic models with scalar fields and with particular forms for the scalar field potential have also been investigated. Heusler [175] has analyzed the case in which the potential function passes through the origin and is concave up and, like Collins and Hawking [112], has found that the only models that can possibly isotropize to the future are those of Bianchi types I, V, VII and IX. As noted in the previous Chapter, all initially expanding scalar field Bianchi models with an exponential potential except possibly those of type IX must isotropize when k < .;2. Let us consider what happens in the case k > .;2. In [198] it was proven, using results from [175], that the only models that can possibly isotropize when k > .;2 are again those of Bianchi types I, V, VII, or IX. Since the Bianchi I, V and VIIo models are restricted classes of models, the only general spatially homogeneous models that can possibly isotropize are consequently of types VIIh or IX. Here we shall discuss the possible isotropization of the Bianchi type VIIh models when k > .;2 in more detail.

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

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1. Analysis of the Bianchi Vlh Equations

The Bianchi type VIIh models belong to Bianchi class B [138]. The equations describing the evolution of these models with a minimally coupled scalar field ¢ with an exponential potential (3.8) were discussed in the previous Chapter. The energy-momentum tensor was given by (3.9) where, for a homogeneous scalar field, ¢ = ¢(t). In this case we can formally treat the energy-momentum tensor as a perfect fluid with velocity vector ua = ¢;a / J -¢;b¢;b, where the energy density, J.liP, and the pressure, PiP' are given by equations (4.24). The variables were defined earlier; in particular, \]! == ¢ / ..j6H, CI> == JV /..j3H describe the scalar field. We note that niP == J.liP/3H2 = \]!2 + Cl>2.

The dimensionless evolution equations are therefore given by (4.11) (with n = 0), where the evolution equation for \]! is given by equation (4.10), and q is now given by

q = 2~! + 2f; + 2\]!2 - Cl>2, (5.1)

and iT is given by equation (4.13). There also exists the constraint (4.16), and the variables are subject to the conditions A :::: 0, f; :::: 0, iT :::: 0. The generalized Friedmann equation, written in dimensionless variables, now becomes

2 2 - - - 2 CI> = 1 - ~+ - ~ - A - N - \]! , (5.2)

which serves to define CI>, where the evolution of CI> is governed by equation (4.11). Equations (4.10) and (4.11) are equivalent to the Klein-Gordon equation (4.2) writ­ten in dimensionless variables. The parameter e = * defines the group parameter h in the Bianchi VIh models. The variables ~+ and f; describe the shear anisotropy. The variables A, N+ and iT describe the spatial curvature of the models. The variable Ll describes the relative orientation of the shear and spatial curvature eigenframes.

We are not interested here in the complete qualitative behaviour of the cosmo­logical models [99] but simply whether the Bianchi VIIh models isotropize to the future when k > v'2. This question can be easily answered by examining the sta­bility of the isotropic equilibrium points of the six-dimensional dynamical system (4.5-4.10). All of the isotropic equilibrium points lie in the invariant set FRW de­fined by {~+ = 0, f; = 0, iT = 0, Ll = O}. Therefore, we need to examine all of the isotropic equilibrium points from Section V.B (with n = 0) and determine whether any are stable attractors or sinks [349].

The equilibrium points ct(~+ = 0; n = 0) have ~+ = 0, f; = 0, Ll = 0, A = 1, N+ = Vi, \]! = 0, which implies CI> = 0, and the eigenvalues are given by (0,2, -2, -4, -2 + 4R, -2 + 4R), and therefore the equilibrium points which represent negatively curved Milne vacuum models are saddles. The equi­librium points KM (~+ = f; = 0; n = 0) have ~+ = 0, f; = 0, Ll = 0, A = 0, N+ = 0, \]! = 1 or - 1, which imply that CI> = 0, and represent flat non­inflationary FRW models. The eigenvalues in both cases are (0,0,2,2,4,6 + ..j6k),

61

and these equilibrium points are saddles. The equilibrium point P s(I)(n = 0) has ~+ = o,t = O,~ = O,A = O,N+ = 0, \}! = -k/.;6, which implies ~ = J1- k2 /6. The eigenvalues are given by (4.26) (excluding the final eigenvalue (k2 - 3'Y) dis­played). For k < -../2, this equilibrium point represents the usual power-law in­flationary attractor. If -../2 < k < 2, then the equilibrium point has an unstable manifold of dimension 4. If 2 < k < .;6, then the equilibrium point has an unstable manifold of dimension 3. This equilibrium point does not exist if k > .;6.

The equilibrium point P~(Vlh) with n = 0 (denoted F here)

F: ~+ = o,t = O,~ = O,A = 1- ~,N+ = Ie (1- 2) \}! = _ .;6 k2 k2 ' 3k'

(5.3)

has ~ = 2;;; and represents a non-inflationary negatively curved FRW model. The six eigenvalues are given by (4.27) (excluding the first eigenvalue (2-3'Y) displayed). It can be shown that if k > -../2 (note that e> 0 in the Bianchi VIIh models) then all of the eigenvalues have negative real parts. Therefore, if k > -../2, then this equilibrium point is a stable attractor. In other words, there exists an open set of initial conditions in the set of anisotropic Bianchi VIIh (with a scalar field and exponential potential) initial data for which the corresponding cosmological models asymptotically approach an isotropic and negatively curved FRW model.

Consequently it follows that within the set of all spatially homogeneous initial data, there exists an open set of initial data describing the Bianchi type VIh mod­els (having a scalar field with an exponential potential and k > -../2) such that the models approach isotropy at infinite times. This compliments the results of Kitada and Maeda [216, 217], who first showed that all ever-expanding spatially homoge­neous models (including the Bianchi VIIh models) with k < -../2 approach isotropy to the future. In other words, there exists an open set of spatially homogeneous initial data for which models will isotropize to the future for all positive values of k. Of course, there also exists an open set of spatially homogeneous initial data for which models will not isotropize to the future when k > -../2 (e.g., the Bianchi VIII models).

If k < -../2, then all models will inflate as they approach the power-law inflationary attractor represented by Ps(I). For k > -../2, the stable equilibrium point F given by equation (5.3), which does not exist for k < -../2, is isotropic and resides on the surface q = o. This means that the corresponding exact solution is marginally non-inflationary. However, this does not mean that the corresponding cosmological models are not inflating as they asymptotically approach this singular state. As orbits approach F they may have q < 0 or q > 0 (or even q = 0) and consequently the models mayor may not be inflating. If they are inflating, then the rate of inflation is decreasing as F is approached (i.e., q -+ 0). When -../2 < k < V873, we find that F is node-like, hence there is an open set of models that inflate as they approach F and an open set which do not. When k > V873, F is found to be spiral-like, and so it is expected that orbits experience regions of both q < 0 and q > 0 as they wind their way towards F. As in Kitada and Maeda [216, 217],

62

the inclusion of matter in the form of a perfect fluid is not expected to change the results of the analysis provided the matter satisfies appropriate energy conditions.

B. Matter Scaling Solutions

Spatially homogeneous scalar field cosmological models with an exponential po­tential and with barotropic matter may also be important even if the exponential potential is too steep to drive inflation. For example, there exist 'scaling solutions' in which the scalar field energy density tracks that of the perfect fluid (so that at late times neither field is negligible) [371]. In particular, in [368] a phase-plane analysis of the spatially flat FRW models showed that these scaling solutions are the unique late-time attractors whenever they exist. The cosmological consequences of these scaling models have been further studied in [145]. For example, in such models a significant fraction of the current energy density of the Universe may be contained in the homogeneous scalar field whose dynamical effects mimic cold dark matter; the tightest constraint on these cosmological models comes from primor­dial nucleosynthesis bounds on any such relic density [145, 368, 371]. Clearly these matter scaling models are of potential cosmological significance. It is consequently of prime importance to determine the genericity of such models by studying their stability in the context of more general spatially homogeneous models.

Let us first discuss the matter scaling solution. The governing equations for a scalar field with an exponential potential (3.10) evolving in a flat FRW model con­taining a separately conserved perfect fluid which satisfies the barotropic equation of state p = Ci- l)Jl (where 0 < "( < 2 here) are given explicitly by

H

it ¢

subject to the Friedmann constraint

1 ·2 -2CiJl +¢ ),

-3"(HJl,

-3H¢-kV,

2 1 1 ·2 H = 3(Jl+ 2¢ + V),

(5.4)

(5.5)

(5.6)

(5.7)

where again an overdot denotes ordinary differentiation with respect to time t

Defining \11 and <1> by equations (4.3) and again using the logarithmic time variable, T, equations (5.4) - (5.6) can be written explicitly as the plane-autonomous system [368]:

\11' (3 2 3 [ 2 2 2 -3\11- V 2k<1> + 2\11 2\11 + "((1- \11 - <1> )], (5.8)

<1>' ~<1> [Ii k\l1 + 2\112 + "((1 - \112 - <1>2) 1 ' (5.9)

63

where for reference we note that

0= J.t - 3H2'

o = J.t", = W2 + c)2. '" - 3H2 '

0+0", = 1, (5.10)

which implies that 0 ~ W2 + c)2 ~ 1 for 0 ~ 0, so that the phase-space is bounded.

A qualitative analysis of this plane-autonomous system was given in [368]. The well-known power-law inflationary solution for k 2 < 2 [217, 291] corresponds to the equilibrium point W = -k/J6, c) = (1 - k2/6)1/2 (0", = 1, 0 = 0) of the system (5.8)/(5.9), which is known to be stable (Le., attracting) for k 2 < 3, in the presence of a barotropic fluid. As noted in the previous Chapter, when k 2 < 2 this power-law inflationary solution is a global attractor in spatially homogeneous models in the absence of a perfect fluid (except for a subclass of Bianchi type IX models which recollapse) .

For, > 0, there exists a scaling solution corresponding to the equilibrium point

w = Wo = - J[ ~, c) = c)o = 3(2 - ,h 2k2

(5.11)

whenever k2 > 3" corresponding to Fs(J) given by (4.36). The linearization of system (5.8)/(5.9) about the equilibrium point (5.11) yields the two eigenvalues with negative real parts

3 3 -4 (2 -,) ± 4k /(2 - ,)[24,2 - k2 (9f - 2)], (5.12)

when, < 2. The equilibrium point is consequently stable (a spiral for k 2 > 24,2/(9, - 2), else a node) so that the corresponding cosmological solution is a late-time attractor in the class of flat FRW models in which neither the scalar field nor the perfect fluid dominates the evolution. The effective equation of state for the scalar field is given by

_ (J.t", + p",) 2W~ ,'" = II = w2 + c)2 ="

,.,.'" 0 0

which is the same as the equation of state parameter for the perfect fluid. The solution is referred to as a matter scaling solution since the energy density of the scalar field remains proportional to that of the barotropic perfect fluid according to 0/0", = k2 /3, - 1 [371]. Since the scaling solution corresponds to an equilibrium point of the system (5.8)/(5.9) we note that it is a self-similar cosmological model [77].

1. Stability of the Matter Scaling Solution

To further study the significance of the scaling solution it is important to deter­mine its stability within a general class of spatially homogeneous models such as

64

the (general) class of Bianchi type Vlh models (which are perhaps the most physi­cally relevant models since they can be regarded as generalizations of the negative­curvature FRW models). The Bianchi Vlh models are sufficiently complicated that a simple coordinate approach is not desirable. In Chapter IV the Bianchi VIIh spa­tially homogeneous models with a minimally coupled scalar field with an exponential potential were studied by employing a group-invariant orthonormal frame approach with expansion-normalized state variables governed by a set of dimensionless evo­lution equations (constituting a "reduced" dynamical system) with respect to a dimensionless time subject to a non-linear constraint [178]. The reduced dynamical system is seven-dimensional (subject to a constraint). The scaling solution, which only exists for k2 > 3'1', is an equilibrium point of this seven-dimensional system and has two eigenvalues given by (5.12) which have negative real parts for 'I' < 2, two eigenvalues (corresponding to the shear modes) proportional to C'Y - 2) which are also negative for 'I' < 2, and two eigenvalues (essentially corresponding to cur­vature modes) proportional to (3'1' - 2) which are negative for 'I' < ~ and positive for 'I' > ~ [45]. The remaining eigenvalue (which also corresponds to a curvature mode) is equal to 3'1' - 4. Hence for 'I' < ~ (k2 > 3'1') the scaling solution is again stable. However, for realistic matter C'Y 2: 1) the corresponding equilibrium point is a saddle with a four- or five-dimensional stable manifold (depending upon whether 'I' > 4/3 or 'I' < 4/3, respectively). It is illustrative to study this in some special cases in a more transparent way.

Anisotropic Bianchi I models are the simplest spatially homogeneous generaliza­tions of the flat FRW models and have non-zero shear but zero three-curvature. The governing equations in the Bianchi I models are equations (5.5) and (5.6), and equation (5.7) becomes

H2 = ~ (Il + ~¢2 + V) + ~ ~2 , (5.13)

where ~2 == 3~~R-6 is the contribution due to the shear, where ~o is a constant and R is the scale factor. Equation (5.4) is replaced by the time derivative of equation (5.13).

Using normalized variables (4.3) we can deduce the governing ODE. Due to the ~2 term in (5.13) we can no longer use this equation to substitute for Il in the remaining equations, and we consequently obtain the 3-dimensional autonomous system:

\I! ,

q,'

0'

f3 2 3 2 -3\I! - V 2kq, + 2\I![2 + ('I' - 2)0 - 2q, ],

~q, { jik\I! + 2 + C'Y - 2)0 - 2q,2 } ,

30{C'Y - 2)(0 - 1) - 2q,2},

where equation (5.13) yields

1 - 0 - \I!2 - q,2 = 9~2 H-2 2: 0,

(5.14)

(5.15)

(5.16)

(5.17)

65

so that we again have a bounded phase-space.

The matter scaling solution, corresponding to the flat FRW solution, is now represented by the equilibrium point

37 q, = q,o, ~ = ~o, 0 = 1 - k2 ' (5.18)

The linearization of system (5.14) - (5.16) about the equilibrium point (5.18) yields three eigenvalues, two of which are given by (5.12) and the third has the value -3(2 - 7), all with negative real parts when 7 < 2. Consequently the scaling solution is stable to Bianchi type I shear perturbations.

Let us next consider the class of curved FRW models, which have curvature but no shear. Again equations (5.5) and (5.6) are valid, but in this case equation (5.7) becomes

2 1 1 '2 H = 3(IL+ 2¢ + V) - kR-2 , (5.19)

where k is a constant that can be scaled to 0, ±l. Equation (5.4) is again replaced by the time derivative of equation (5.19).

As in the previous case we cannot use equation (5.19) to replace IL, and using the normalized variables (4.3) we obtain the 3-dimensional autonomous system:

, ~ 2 3 [( 2) 2 ( 2 2 ] q, = -3q, - 2k~ + 2q, 7 - 3 0 + 3 1 + 2q, - ~) , (5.20)

~' = ~~ { v/ikq, + (7 - ~) 0 + ~(1 + 2q,2 _ ~2)}, (5.21)

0' = 30{(7-~)(0-1)+~(2q,2-~2)}, (5.22)

where

1- 0 - q,2 _ ~2 = -9kR-2H-2 • (5.23)

The phase-space is bounded for k = 0 or k = -1, but not for k = +l.

The matter scaling solution again corresponds to the equilibrium point (5.18). The linearization of system (5.20) - (5.22) about this equilibrium point yields the two eigenvalues with negative real parts given by (5.12) and the eigenvalue (37-2). Hence the scaling solution is only stable for 7 < ~. For 7 > ~ the equilibrium point (5.18) is a saddle with a 2-dimensional stable manifold and a one-dimensional unstable manifold.

Consequently the scaling solution is unstable to curvature perturbations in the case of realistic matter h ~ 1); i.e., the scaling solution is no longer a late-time attractor in this case. However, the scaling solution does correspond to an equi­librium point of the governing autonomous system of ODE and hence there are

66

cosmological models that can spend an arbitrarily long time "close" to this solu­tion. Moreover, since the curvature of the Universe is presently constrained to be small by cosmological observations, it is possible that the scaling solution could be important in the description of our actual Universe. That is, not enough time has yet elapsed for the curvature instability to have effected an appreciable deviation from the flat FRW model (as in the case of the standard perfect fluid FRW model). Hence the scaling solution may still be of physical interest.

VI. CLOSED MODELS

Cosmological models with positive spatial curvature are also of interest [96]. These models have attracted less attention since they are more complicated mathemati­cally. Positive-curvature FRW models [171, 350, 361, 363], Kantowski-Sachs models [71] and Bianchi type IX models [216, 345, 351, 363] have been studied using quali­tative methods, although more rigorous analyses using bounded variables have not been carried out until recently [96, 155]. The Bianchi type IX models are known to have very complicated dynamics, exhibiting the characteristics of chaos [182, 363], and are hence beyond the scope of the present analysis. In this Chapter we shall discuss the qualitative properties of the positive-curvature FRW models and the Kantowski-Sachs models with a barotropic fluid and a non-interacting scalar field with an exponential potential [96].

The matter content of the models is taken to be a perfect fluid and a scalar field with exponential potential (3.10). The corresponding energy-momentum tensor is consequently Tab = (T:: + T~O, where T:: is given by (1.1) and T~t is given by (3.9), and since the matter components are assumed to be non-coupled, they are separately conserved: i.e., V' aT;! = 0 = V' aTsrb. The pressure is given by p = b - 1)/L, with the equation-of-state parameter in the range 1 ::::: 'Y ::::: 2, and k is a non-zero constant. The fluid energy density /L and the scalar field ¢ are functions of a timelike coordinate t, and a dot denotes differential with respect to t. For convenience, we define

1 . X = yf2¢. (6.1)

A. Closed Friedmann Models

Let us begin by studying the closed FRW cosmological models. The line element for these models is given by (1.11) (with k=l). The expansion of the fluid congruence is given by 3H = 3RJ R, and the evolution equation for the curvature K == 1/(H2 R2) is

:. 2K K = -If(H +H2).

The conservation equations yield

From the EFE we obtain

jJ,

X

-3'YH/L, k

HX -V. -3 - yf2

/L = 3(1+K)H2-X2-V,

(6.2)

(6.3)

(6.4)

(6.5)

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

68

if = -!{3H2+!D2+~[3X2-V+('Y-l)Jt]}. 362

(6.6)

Assuming I' ~ 0, the Friedmann equation (6.5) shows that D = 3V(1 + K)H2 is a dominant quantity. Thus, bounded variables can be defined according to

3H Qo=V' U = V3X

D'

Note also that the curvature is given by

The Friedmann equation becomes

K = l-Q~ Q~ .

W = v'3V D·

_ 31' 2 2 !lD = D2 = 1 - U - W .

(6.7)

(6.8)

(6.9)

Defining a new independent variable, I = d/dr = ! d/dt, the evolution equation for D

D' -3Qo (U2 + ~!lD) D

decouples. Thus, a reduced set of evolution equations is obtained:

Q~ = (1 - Q~) [1 - 3 (U2 + ~!lD )] ,

U' = 3QoU [-1 + (U2 + ~!lD)] - y'IkW2 ,

W' = 3Qo (U2 + ~!lD) W + y'IkUW.

There is also an auxiliary evolution equation

!l~ = -3Qo [(1 - !lDh - 2U2] !lD,

(6.10)

(6.11)

(6.12)

(6.13)

(6.14)

and it is straight-forward to consider the set of variables (Qo, U,!lD), rather than (Qo,U, W) [343]. Note that by setting k = O,U = 0, and identifying !lA = W 2 ,

the evolution equations corresponding to closed FRW models with a cosmological constant are obtained [155].

It is also useful to consider the deceleration parameter, given by

_ (U~fVaH) 1 [ (2 'Y )] qpf = - 1 + H2 = - Q~ 1 - 3 U + 2!lD , (6.15)

for Qo f O. From this expression, we can see that there is an inflationary region (qpf < 0) in the state space whenever !lD < 3~(1 - 3U2 ). However, as will be seen

69

below, it is only when k2 < 2 that there exist attractors that are inflationary. We also note that for Qo :j:. 0,

Q~ = -(1 - Q~)Q~qpf' (6.16)

so that Q~ < 0 whenever qpf > 0 in which case Qo is itself monotonic. When qpf < 0, that is in the inflationary region, Qo need not be monotonic (for example, see the orbits close to +q; in Figs. (7) and (8)).

The dynamical system (6.11)-(6.13) is symmetric under the transformation

(7, Qo, u, W) --t (-7, -Qo, -U, W). (6.17)

Thus, it is sufficient to discuss the behaviour in one part of the state space, the dynamics in the other part being obtained via (6.17). Furthermore, we note that

M:= 1 ~~~; M' = -(31' - 2)QoM, (6.18)

is a monotonic function in the regions Qo < 0 and Qo > 0 for OD :j:. O. As there are no equilibrium points with Qo = 0 when l' > 2/3, M acts as a monotonic function in the interior of the state space. Consequently there can be no periodic or recurrent orbits in the interior state space and global results can be deduced. In addition, from the expression for the monotonic function M we can see immediately that either Q3 --t 1 or OD --t 0 asymptotically.

1. Qualitative Analysis

In Table VI, the equilibrium points of the dynamical system (6.11)-(6.13) are summarized. The subscripts on the labels have the following significance: The left subscript gives the sign of Qo (denoted by f = ± 1) and indicates whether the corresponding model is expanding (+) or contracting ( - ). The right subscript gives the sign of U; i.e., the sign of (p. 0", = U2 + W 2 is a density parameter associated with the scalar field.

The equilibrium points labeled ±F correspond to the flat Friedmann solution. For these points, the scalar field vanishes (U = 0 = W). There is an orbit from +F to _F and this orbit represents the closed FRW solution with no scalar field, starting from a big bang at +F and recollapsing to a "big crunch" at _F. The k points represent exact solutions with a massless scalar field (W = 0). As the fluid is negligible (OD = 0), these solutions are dominated by the kinetic term U. They correspond to Jacobs analogues of Kasner solutions in which U takes on the role of a shearing mode [363].

There are equilibrium points with non-vanishing potential, where the scalar field dominates (OD = 0). These points ±q; are only physical when k 2 < 6. For k 2 = 6, +q; coincides with +K+, and _q; with _K_. For k 2 > 6, they are outside the

70

Interpretation

±F Flat Friedmann

±K± Kinetic dom.

±<I> Scalar-field dom.

±X Curvature scaling

±FS Flat matter scaling

S± Static

Qo

f

E

E

-k y'2E

f

U

o ±1

w o o

Note

-k E Jl- k2 k2 <6 vI6 6

--L fi k 2 < 2 v'3 V3

-jhf -jiiv'~[~(2---[~) k2 > 3[

o ±J 3~;-!7) 0 [ < 2/3

TABLE VI: Equilibrium points of the closed FRW models.

physical part of the state space. The equilibrium point with f = +1 (i.e., + ~) is a sink, and for k 2 < 2 it corresponds to the power-law inflationary attractor solution (cf. the expression for qpf in Table VII).

There are also points ±X for which the matter is unimportant, but the curvature is non-vanishing (Q~ 1= 1) and tracks the scalar field. The corresponding solutions are called curvature scaling solutions [350]. These solutions only exist when k2 < 2. For k2 = 2, +X coincides with +~, and _x with _~. Above this value, these equilibrium points are outside the physical part of the state space. When k 2 > 31', there is a flat matter scaling solution, for which both the fluid and the scalar field are dynamically important. The corresponding equilibrium points are denoted ±FS, and for k2 = 31' they coincide with ± ~.

Finally, there are equilibrium points S± corresponding to static solutions, anal­ogous to the Einstein static universe. These are only physical when l' < 2/3. For ,= 2/3, a set of equilibrium points appears along the line U = 0 = W, signaling a change of stability when the points S± leave the physical state space. In what follows, we will only consider equations of state for which 1 ~ l' ~ 2, hence we will not consider these points further.

OD Oq, qp£ Eigenvalues

±F 1 0 ~(3[- 2)2 E -~(2 -[)E ¥E ±K± 0 1 2 4f 3(2 -[)f 3f + jiksgn(U)

±<I> 0 1 Hk2 - 2) -(3[- k2)E -~(6-k2)E -(2 - k2)E

±X 0 1 0 ~(3[- 2)E ~(kE ± VB - 3k2)

±FS -b(3[ ~ k2) ~ H3[- 2) (3[- 2)f -~ [(2 ~ [) T E] E S± 4 ~ 3~i=~) Not defined ..1£.. J2-3J sgn(U) ±v'2v2 - 3[ 3(2-,) y'2 2-,

TABLE VII: The physical quantities OD, Oq" and qpf for the different equilibrium points for the closed FRW models are given and the corresponding eigenvalues are displayed. In the table E == iv'(2 -[)[24[2 - 9[- 2)k2].

Past attractors

Expanding from a singularity (Qo > 0) +iL Always - 2

+K+ k < 6 Contracting from a dispersed state (Qo < 0) _<1> k 2 < 2 (and k 2 < 3,)

F1lture attractors

Contracting to a singularity (Qo < 0) _ K+ Always - 2 _K_ k < 6

Expanding to a dispersed state (Qo > 0) +<1> k 2 < 2 (and k 2 < 3,)

TABLE VIII: Summary of sources and sinks for the closed FRW models.

71

Table VII presents some physical quantities for the various equilibrium points, and their eigenvalues are listed. The sources and sinks of the dynamical system when'Y > 2/3 are listed in Table VIII, and the global behaviour for different values of k can be summarized as follows. The state space when 0 < k2 < 2/3 is depicted in Fig. (7), where the features in the rear part of the state space have been suppressed. Dashed and full curves represent orbits in the boundary submanifolds, while dotted curves represent orbits in the interior. Both +K_ and +K+ are past attractors, while _K+ and +<1> act as future attractors. There are also orbits from +K_, whose future attractor is _K_ at the rear ofthe figure. Note that orbits future asymptotic to + <1> correspond to solutions that exhibit power-law inflation (-1 < qpf < 0, see Table VII). Observe that the outgoing eigenvector directions from the saddle point +F span a separatrix surface in the interior of the state space. Similarly, +X is a saddle for which the ingoing eigenvector directions span another separatrix surface. These separatices confine orbits in the interior state space to specific regions. For example, there is one region where all orbits are past asymptotic to +K+ and future asymptotic to _K+.

When k2 > 2/3, the separatrix surface associated with +X changes structure (see Fig. (8) and Fig. 3 in [96]); there is one separatrix orbit from _<1> to +X, and one from _X to +<1>. For k 2 = 2/3, these two orbits coalesce into a single orbit from _X to +X. This orbit corresponds to a special "Bouncing Universe" solution, existing only for this particular value of k. When k2 > 2/3, the separatrix orbit to +X starts at +K_, and the orbit from _X goes to _K+. Thus, there is a bifurcation for k2 = 2/3; however, we note that there is no stability change of equilibrium points involved. A similar behaviour of separatrix surfaces has been found for Bianchi type IX models [345].

When k increases, the equilibrium point +X approaches +<1>. For k 2 = 2 these two points coincide, and the state space for k 2 > 2 is depicted in Fig. (9). Both +K_ and +K+ still act as past attractors, while _K+ is a future attractor. However,

72

'It!

w

Ou

_K+

FIG. 7: The state space for closed FRW models with a scalar field when k 2 < 2/3. Dashed curves and white arrows and circles are screened. Dotted orbits are in the interior of the state space.

'" ... ~./ I ..

w +K_

Ou u

_K+

FIG. 8: The state space for closed FRW models with a scalar field when 2/3 < e < 2. See caption to Fig. (7).

the stability of + <1> has changed; it has become a saddle. There is still a separatrix surface associated with +F. Thus, orbits having +K+ as their past attractor all end at _K+.

For k2 > 31', the equilibrium point +FS, corresponding to the matter-scaling solution, appears from +<1>, see Fig. (10). The point +FS is a spiral sink with an out-going eigenvector direction entering the interior state space. Note that this equilibrium point thus is stable in the flat (Qo = 1) submanifold, but unstable to curvature perturbations (Le., perturbations in the Qo direction). The scalar-field dominated point + <1> still is a saddle, and now there is a separatrix surface spanned by the out-going eigenvector directions there. Thus, there are two separatrix sur­faces, both of which are spiraling around the out-going eigenvector direction of +FS.

73

---- --- - - " -- -t.._/ I --

_K+

FIG_ 9: The state space for closed FRW models with a scalar field when 2 < k 2 < 3"(- See caption to Fig_ (7)_

When k increases, the +cI> equilibrium point comes closer and closer to +K+, and for k 2 = 6 they coincide_ The state space when k 2 > 6 is given in Fig_ (11). There is only one past and one future attractor, namely +K_ and _K+, respectively.

In summary, when k 2 > 2 all solutions start from and recollape to a singularity (K ----t K). Thus, in this case solutions can neither expand forever nor inflate. When k 2 < 2, there are also ever-expanding (K ----t cI» (and ever-collapsing cI> ----t K) solutions in addition to the recollapsing solutions. Inflation occurs when R > 0, i.e., H + H2 > 0, which leads to the condition

-(3"( - 2) - 3(2 - "()U2 + 3,,(W2 > O. (6.19)

This corresponds to a parabolic region along the ridge of the state spaces depicted in Figs. (7) - (11). The only equilibrium points within this region are ±cI> for k2 < 2, corresponding to power-law inflation. Consequently, in the case k 2 < 2 there is a subclass of solutions that inflate.

B. Kantowski-Sachs Models

Let us now discuss the Kantowski-Sachs (KS) models. The line element can be written

ds2 = -dt2 + Dl(t)2dx2 + D2(t)2dQ2, (6.20)

where Dl = exp [,8°(t) - 2,8+ (t)] , D2 = exp [,8°(t) + ,8+ (t)] , and the kinematic quantities of the fluid congruence are related to the Misner variables (,80, ,8+) by

74

/

'. £::,.. / . I. I .

:·~·~·~'~··~··~·~_~~I~~i;jJ'i~t", . _K+

FIG. 10: The state space for closed FRW models with a scalar field when 3, < k 2 < 6. See caption to Fig. (7).

•.• f::,... / I.

I I

I .

/

- - - - - [>....::.:.~:.;..~.- .......... ·":"·:"·~·-l.n:;t ", - - - --"- '. '. - _r'I ,. -v-- ~. '-...A... / - - - ". ". .....4 ~ - -':":~:':':.: . ...::~ ...... '"

-"~ _K+

FIG. 11: The state space for closed FRW models with a scalar field when k2 > 6. See caption to Fig. (7).

H = /Jo, a+ = 3/J+. The evolution equations for the metric functions Bl == Dl1 and B2 == D:;l then become

. 1 Bl = -"3(3H + a+)B1 ,

. 1 B2 = --(3H + a+)B2.

3

The conservation equations give

and the EFE yield

It

if

j1

X

-3"(HIt,

k V -3HX- V2 '

3H2 - !a2 + B22 - X2 - V 3 + '

1 ( 2 2 2 2 1( ) ) -"3 3H +"3a + + 2X - V + 2" 3"( - 2 It ,

(6.21)

(6.22)

(6.23)

(6.24)

(6.25)

75

0-+ 2 1 2 2 3H - -(J" - H(J"+ - X - V - II. 3 + "". (6.26)

The Friedmann equation (6.24), together with the assumption I-l ~ 0, shows that D = J9H2 + 3B~ is a dominant quantity. Consequently, compact variables are introduced according to

3H QO=15'

(J"+

Q+=J5' u = v'3x D'

W = v'3V D· (6.27)

The curvature variable k = tB~ H-2 = (1 - Q5)/Q5 shows that the flat solutions correspond to Q5 = 1. The Friedmann equation becomes

nD = ~ = 1- Q! - U2 - W 2 • (6.28)

By introducing a new independent variable, T, where I djdT -! djdt, the evolution equation for D,

D' = - [Q+(1- Q~) + 3Qo (Q! + U2 + ~nD)] D, (6.29)

decouples, and a reduced set of evolution equations is obtained:

Q~ = (1 - Q5) [1 + QoQ+ - 3 (Q! + U 2 + ~nD)] , (6.30)

Q~ = -(1- Q5)(1 - Q!) + 3QoQ+ [-1 + (Q! + U 2 + ~nD)] , (6.31)

U' = U {(I - Q5)Q+ + 3Qo [-1 + (Q! + U2 + ~nD)]} - JIkW2 ,(6.32)

W' = W {(I - Q5)Q+ + 3Qo ( Q! + U2 + ~nD ) } + JIkUW (6.33)

There is also an auxiliary evolution equation:

n~ -nD {3,Qo - 2 [Q+(l- Q5) + 3Qo (Q! + U2 + ~nD)]}. (6.34)

Note that by setting k = 0, U = 0, and identifying nA = W 2 , the evolution equa­tions corresponding to Kantowski-Sachs models with a cosmological constant are obtained [155]. The deceleration parameter is given by

qpf - ~5 [1 - 3 (Q! + U2 + ~nD)] . (6.35)

We note that the dynamical system (6.30)-(6.33) is symmetric under the trans­formation

(T, Qo, Q+, U, W) ---> (-T, -Qo, -Q+, -U, W). (6.36)

76

Thus, it is sufficient to discuss the dynamical behaviour in a restricted part of the state space. The function

M

M'

Q:;:2(3')'-2)(1_ Q5)-3(2-')')0:b;

2Q:;:1 [(3, - 2)(1- Q5) + 3(2 - ,)Q!J M, (6.37)

is monotonic in the regions Q+ > 0 and Q+ < 0, since 2/3 <, < 2. Noting that

Q~IQ+=o = -(1 - Q5) < 0, (6.38)

we conclude that the submanifold Q + = 0 is not invariant, but acts as a membrane. Thus, the existence of M rules out any periodic or recurrent orbits in the interior of the state space and again global results are possible. From the expression for the monotonic function M we can immediately see that asymptotically Q+ ----+ 0, Q~ ----+ 1 or OD ----+ o.

1. Qualitative Analysis

The equilibrium points of the dynamical system (6.30)-(6.33) are displayed in Table IX. As before, f = ±1 denotes the sign of Qo, O</> == U 2 + W 2 , and the left subscript gives the sign of Qo and indicates whether the corresponding model is expanding or contracting. The values of OD, O</>, and qpf for each of the equilibrium points and the corresponding eigenvalues are displayed in Table X. Note that all of the equilibrium points correspond to exact self-similar cosmological models [50, 363].

Interpretation Qo Q+ U W I Note

±F Flat Friedmann E 0 0 0

±K Kinetic dom. E ±V1 - U5 Uo 0 Ring

±<P Scalar-field dom. E 0 -k -L.J6 - k 2 k 2 < 6 ,J6E ,J6

±B Curvature scaling 1+k2 2_k2 -,J6k v'6 .J2 + k2 k2 < 2 24 + k2 E - 4+k 2 E 4+k 2 E 1(2

±FS Flat matter scaling 10 0 -/f7i E - HV,(2-,) k2 > 3, ±SSKS Self-similar KS _2_10 3J-2 0 0 , < 2/3 3')'-4 3')'-4 E

TABLE IX: Equilibrium points of the KS models.

As in the case of the closed FRW models, ±F denotes the flat Friedmann solu­tion. Note that the closed FRW solution without a scalar field does not appear as a submanifold of the KS models without a scalar field. Consequently, there is no orbit connecting +F with _F. There are two sets ±K of vacuum (OD = 0) equilibrium points, parameterized by the constant Uo, corresponding to kinetic dominated so­lutions. These sets are analogues of the "Kasner rings" that are present for various Bianchi models.

77

The flat scalar-field dominated points ±<1>, already encountered for the closed FRW models, appear in the KS case as well. As in the case of the closed FRW models, they are physical when k 2 < 6 and inflationary when k 2 < 2. There are also equilibrium points ±3, corresponding to curvature scaling solutions (Le. they have OD = 0, Q5 < 1) which are physical when k2 < 2. They are also inflationary, but in other respects they resemble the points ±X of the closed FRW models. As in the case of the closed FRW models, the equilibrium points ±FS, corresponding to the flat matter-scaling solution, enter the physical part of the state space when k 2 > 31'. Finally, there are also equilibrium points corresponding to the self-similar KS solution. This solution is only physical when l' < 2/3, and so we will not consider them further.

Ov 04> qpf Eigenvalues

±F 1 0 ~(3, - 2) (3, - 2)£ -~(2-,)c: -~(2-,)c: ~c: ±K 0 UJ 2 4c: - 2Q+ 3(2 - ,)c: 0 3£ - j"ikUo

±<P 0 1 ~(k2_2) -~(6 - k2)c: -~(6 - k2)c: -(2 - k2)c: -(3, - k2)c:

±2 0 12(1+ki 1 k2 _2 -3 2+k2 c: -6 '"(+(-y_l)k2 c: -!! 2+k2 ±E, c: (4+k2 ) "2 1+k2 4+k2 4+k2 2 4+k 2

±FS k21:.}'1 !!:r. ~(3, - 2) -~(2-,)c: (3, - 2)c: :it [-(2 - ,)k ± E2J c: k2

SSKS 12(1-'"(~ 0 H3, - 2) -3~£ ....kLc: 3 2-),±E3 ± (3'"(-4) 3'"(-4 3'"(-4 -"2 3'"(-4 £

TABLE X: The physical quantities Ov, 04>, and qpf for the different equilibrium points for the KS models are given and the corresponding eigenvalues are displayed. In the table El == ";(2 + k2)(18 - 7k2), E2 == ";(2 - ,)(24,2 - (9, - 2)k2 ), E3 == ";(2 - ,)(24,2 - 41, + 18).

Past attractors

Expanding from a singularity (Qo > 0) +K kUo < v'6 Contracting from a dispersed state (Qo < 0) _ <P k 2 < 2

Future attractors

Contracting to a singularity (Qo < 0) _K kUo < v'6 Expanding to a dispersed state (Qo > 0) +<P k 2 < 2

TABLE XI: Summary of sources and sinks for the KS models.

The sources and sinks of the dynamical system when l' > 2/3 are listed in Table XI (all ofthe other equilibrium points are saddles). Thus, there is always two segments of the equilibrium set +K that act as sources for orbits. Similarly there are two segments on _K that are sinks. When k2 > 2, these are the only attractors, and all solutions start from and recollapse to a singularity (K -+ K). When k 2 < 2, which then implies that k 2 < 31' for l' > 2/3, the equilibrium points ±<1> are attractors. Thus, for k 2 < 2, there are also ever-expanding (+K -+ +<1» and ever-collapsing (_<1> -+ _K) solutions.

78

From the expression for the monotonic function M we deduce that all orbits asymptotically have Q+ -+ 0, Q6 -+ 1 or nD -+ O. Indeed, the existence of the

monotonic function ensures that there are no periodic orbits and that generically orbits asymptote towards the local attractors (sinks and sources). Therefore, we can determine the global dynamics of the models.

In summary, when k2 > 2 all solutions start from and recollape to a singularity (K -+ K). Thus, in this case solutions can neither isotropize nor inflate. When k 2 < 2, there are also ever-expanding (K -+ <I» (and ever-collapsing <I> -+ K) solutions in addition to the recollapsing solutions. Again, the <I> points correspond to power-law inflation when k2 < 2. Consequently, in this case there is a subclass of solutions that isotropize and inflate.

The global asymptotic dynamics is similar to that in the case of the positive­curvature FRW models. However, due to the presence of shear, the intermediate or transient dynamics can be quite different. In the KS case the state space is 4-dimensional and so we cannot display the phase portraits graphically (as in the FRW case). However, as an illustration we can present the phase portraits in the 3-dimensional massless scalar field invariant set, in part to study the intermediate behaviour of the models.

The massless case corresponds to the invariant submanifold W = 0, which leads to a 3-dimensional system in (Qo, Q+, U):

Q~ 2 [3, -2 3 2 2 ] -(1- Qo) -2- - QoQ+ + 2(2 - ,)(Q+ + U) , (6.39)

Q~ 2 2 3 ( ) -(1 - Qo)(1- Q+) - 2 2 -, QoQ+nD, (6.40)

U' U [(1- Q~)Q+ - ~(2 - ,)QonD] . (6.41)

From Table IX, it is immediately seen that the equilibrium points that are contained in this submanifold are ±K and ±F. The state space is depicted in Fig. (12).

J" Qo

FIG. 12: The state space for KS models with fluid and massless scalar field, W = O. See caption to Fig. (7). The horizontal plane is the U = 0 invariant submanifold (compare with [132]).

79

The fluid vacuum (OD = 0) is an invariant submanifold, as seen from equation (6.34). Using the Friedmann equation to eliminate W, we obtain a 3-dimensional dynamical system in (Qo, Q+, U):

Q~

Q~

U'

(1 - Q~) [1 + QoQ+ - 3(Q~ + U2 )] ,

-(1 - Q~)(1 - Q~) - 3QoQ+(1- Q~ - U2 ),

(1 - Q~)Q+U - ( /fk + 3QoU) (1 - Q~ - U2 ).

(6.42)

(6.43)

(6.44)

From Table X, it is immediately seen that the equilibrium points that are contained in this submanifold are ±K, ±cJ>, and ±2. Note that Z = 0, where Z is defined by

Z == Q+ - ~Qo _ J6 2 2k U, (6.45)

is an invariant submanifold, and that both ±cJ> and ±2 are contained in this sub­manifold. It is also possible to display the phase portraits in this case (see Figs. 7-9 in [96)).

80

c. Discussion

We have studied closed cosmological models with a perfect fluid satisfying a linear equation of state with 2/3 < 'Y < 2 and a scalar field with an exponential potential. We have utilized a new set of normalised variables which lead to the compactification of state space, enabling us to apply the theory of dynamical systems to determine the qualitative properties of the models. In all cases we have been able to find monotonic functions which, together with a local analysis of the equilibrium points, enables us to determine the global properties of the models.

We first studied the closed FRW cosmological models. We found that when k 2 > 2, all solutions start from and recollape to a singularity (K -t K). In this case solutions generically do not inflate. When k 2 < 2, solutions can either recollapse (K -t K) or expand forever (K -t q,) towards power-law inflationary solutions (or collapse forever q, -t K); consequently, in this case there is a subclass of solutions that inflate. These results generalise previous qualitative work on positive-curvature FRW models with a scalar field (only) [171] and with a scalar field plus a barotropic perfect fluid [350] in which compactified variables were not utilized, and rigorous analyses of perfect fluid (only) models using compactified variables [361, 363], and completes and generalises more recent work using different compactified variables [343]. We also note that positive-curvature FRW models with a perfect fluid and a positive cosmological constant have been investigated recently using qualitative methods and utilizing compactified variables [155].

In the case of the Kantowski-Sachs (KS) models we again found that when k2 > 2 all solutions start from and recollape to a singularity (K -t K) and can consequently neither isotropize nor inflate. When k 2 < 2, there are also ever-expanding (K -t q,) (and ever-collapsing q, -t K) solutions in addition to the recollapsing solutions, where again the q, points correspond to the flat FRW power-law inflationary solu­tion. Consequently, in this case there is a subclass of solutions that isotropize and inflate. The investigation of KS models complements the study of Bianchi mod­els [50] and completes the analysis of spatially homogeneous models. Collins [111] studied perfect fluid KS models qualitatively using expansion-normalised variables (for which the state space was non-compact) and showed that all models start at a big bang and recollapse to a final "big crunch" singularity. This work was gener­alised recently by Goliath and Ellis [155] in which KS models with a perfect fluid and a cosmological constant were investigated using qualitative methods and utiliz­ing the compactified variables of Uggla and Zur-Muhlen [345]; particular attention was focussed upon whether the models isotropize. In addition, KS models with a scalar field and an exponential potential, but without barotropic matter, have been studied qualitatively [66], although compactified variables were not utilized.

To conclude an analysis of positive-curvature spatially homogeneous cosmological models with a perfect fluid and a scalar field with an exponential potential, Bianchi type IX models would need to be studied. Unfortunately, such a study is beyond the scope of the current work. For example, Bianchi type IX models are known to have very complicated dynamics, exhibiting the characteristics of chaos [182, 363].

81

However, partial results are known. Bianchi type IX models with a scalar field (only) have been studied qualitatively, with an emphasis on whether these models can isotropize [351]. Scalar-field models with matter have also been studied [216]. For example, it has been shown that the power-law inflationary solution is an attractor for all initially expanding Bianchi type IX models except for a subclass of the models which recollapse [99, 216]. However, compact variables have not been utilized and the analyses were not rigorous.

A more rigorous treatment of the class of Bianchi type IX models with a non-tilted perfect fluid (only) using compactified variables is possible [363]. Although an ap­propriately defined normalised Hubble variable is found to be monotonic, enabling some results to be obtained, several problems remain open. More rigorous global results are possible. For example, Bianchi type IX models with matter have been shown to obey the "closed universe recollapse" conjecture [235], whereby initially expanding models enter a contracting phase and recollapse to a future "big crunch" . In addition, Ringstrom has proven that a curvature invariant is unbounded in the incomplete directions of inextendible null geodesics for generic vacuum Bianchi mod­els [306], and rigorously shown that the Mixmaster attractor is the past attractor of Bianchi type IX models with an orthogonal perfect fluid [307]. A complete quali­tative analysis of the special class of locally rotationally symmetric Bianchi type IX perfect fluid models, which do not exhibit oscillatory or chaotic behaviour near to the initial or final singularities, has been given in [345], based upon an appropriately defined set of bounded variables.

The KS models exhibit similar global properties to the positive-curvature FRW models; in particular, for k2 > 2 all initially expanding models reach a maximum expansion and thereafter recollapse, whereas for k 2 < 2 models generically recol­lapse or expand forever towards a flat isotropic power-law inflationary solution. The Bianchi type IX models share these qualitative properties. However, the interme­diate behaviour of these models can be quite different.

VII. MULTIPLE SCALAR FIELDS

Inflation is generally considered to be a reasonable solution to many of the funda­mental problems within the standard cosmological model [167, 236, 327]. There are a variety of inflationary models which include scalar fields [291], and in scalar field models with an exponential potential [372] the universe inflates at a power-law rate, R( t) <X tP , where p > 1 [249]. As we have seen, all ever-expanding scalar field models with an exponential potential experience power-law inflation when the parameter k 2 < 2; i.e., when the potential is sufficiently flat. Exponential potentials arise in many theories of the fundamental interactions including superstring and higher­dimensional theories [164, 291]. Typically, "realistic" supergravity theories predict steep exponential potentials [164] (Le., k 2 > 2), effectively eliminating the possi­bility of power-law inflation. However, dimensionally reduced higher-dimensional theories also predict numerous scalar fields, and so it is of interest to study models with multiple scalar fields.

The effect of n scalar fields with independent exponential potentials in a spatially flat FRW universe was considered in [229]. It was found that an arbitrary number of scalar fields with exponential potentials evolve towards a novel inflationary scal­ing solution, which was denoted assisted inflation, in which all of the scalar fields scale with one another (and are hence all non-negligible asymptotically) with the result that inflation occurs even if each of the individual potentials is too steep to support inflation on its own. The existence of multiple uncoupled scalar fields could therefore, through a combined (or assisted) effort, be a source for power-law inflation, and hence might lead to compatibility with supergravity theory. In [269] it was shown that this assisted inflationary solution is a late-time attractor in the class of zero-curvature FRW models. This was done by choosing a redefinition of the fields (a rotation in field space) which allows the effective potential for field variations orthogonal to this solution to be written down; in analogy with models of hybrid inflation [238] it was then shown that this potential has a global mini­mum along the attractor solution. Also, analytic solutions describing homogeneous and inhomogeneous perturbations about the attractor solution without resorting to slow-roll approximations were presented in [269], and curvature and isocurvature perturbation spectra produced from vacuum fluctuations during assisted inflation were discussed.

In this Chapter we shall present a qualitative analysis of models with the action

8= J d4XA[R-~t(V'¢D-Ateki¢il +8m , (7.1)

where 8m is the matter contribution. We extend previous analyses to include both non-zero curvature and matter [104]. First, we shall present the governing equations for n scalar fields with exponential potentials and matter. We shall then study the two-scalar field model with no matter and, in particular, discuss the stability of the two-field assisted inflationary model. In the subsequent Section we shall study the

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

83

two-scalar field model with barotropic matter. Finally, we shall discuss three and multi-scalar field models and present some conclusions.

1. The Model

We shall assume that the spacetime is spatially homogeneous and isotropic. The line element for such a spacetime is given by (1.11). We shall consider n scalar fields ¢i, where 1 ~ i ~ n, in which the effective potential has the form

n

Veff == L Aeki <l>i , i=l

where the k i are real non-zero positive constants. We also assume that there ex­ists a non-interacting perfect fluid with density J.L and pressure which satisfies an equation of state (1.2). The Einstein field equations, the conservation equations, together with the Klein-Gordon equations for the scalar fields, yield the following autonomous system of ODE:

2 1 (~ .2) 1 1 3R H -6 ~¢i -"3Veff-"3J.L=-6'

.=1

. 2 1 (~.2) 1 1 H = -H -"3 {;;:¢i + "3Veff - 6(31' - 2)J.L,

jJ, = -31'HJ.L,

¢i + 3H¢i + kiAeki<l>; = 0,

(7.2)

(7.3)

(7.4)

(7.5)

where 3R = kl R2 is the curvature of the spacelike hypersurfaces, H = RI R is the Hubble expansion and again an overdot represents differentiation with respect to coordinate time t.

To analyze the dynamical system given by equations (7.2)-(7.5) we choose expan­sion normalized variables of the form

0= J.L 3H2'

<P- _ VAeki <l>;/2 .---V3H ;Pi

'I1i = V6H' dt 1 dT = H· (7.6)

Denoting differentiation with respect to T by a prime, the resulting dynamical sys­tem describing these perfect fluid multiple scalar field models becomes [104]

0' = O(2q - 31' + 2), (7.7)

, ( ) V6 k 2 '11- = '11- q - 2 - - -<P-., 2 • p (7.8)

, ( V6) <Pi = <Pi q + 1 + Tk i 'l1 i , (7.9)

84

for (1 :S i :S n), where the deceleration parameter has the following form

n n

q = (3, - 2) 0 + 2 L q,~ - L cpr, 2 i=1 i=1

and

3R n n

6H2 = -1 + 0 + L q,~ + L cpr i=1 i=1

Assuming a non-negative energy density (i.e., 0 2: 0) and if 3R :S 0 (i.e., in the negative and zero-curvature cases) the phase space for the dynamical system in the expansion normalized variables (0, CPi, q, i) is compact. If 3 R > 0 (i.e., in the positive curvature case) then the transformation given by equation (7.6) becomes singular when H = O. Here we shall only make some partial comments with regards to the asymptotic behaviour of the positive curvature models. All of the equilibrium points correspond to self-similar cosmological models and hence to power-law solutions [50].

A. Qualitative Analysis of the Two-Scalar field Model

We shall first discuss the dynamics of the model with only two minimally coupled scalar fields and with no matter. We obtain this model by setting n = 2 and 0 = 0 in (7.7)-(7.9). In this case we obtain the 4-dimensional dynamical system given by:

I () v'6 2 q,l = q,l q - 2 - TkI CPI (7.10)

I () v'6 2 q,2 = q,2 q - 2 - Tk2CP2 (7.11)

cpl v'6 (7.12) 1 CPI(q + 1 + TkIq,d

cp' v'6 (7.13) 2 CP2(q + 1 + Tk2q,2)

where

q = 2q,i + 2q,~ - cpi - CP~

and

3 R 1 q,2 q,2 cp2 cp2 6H2 = - + 1 + 2 + 1 + 2'

It is possible to choose simplified variables as in [242] via a rotation in field space; although this would simplify the analysis of the assisted inflationary solution, it would perhaps be more difficult to describe all of the qualitative properties of the models and relate this analysis to previous work.

85

1. Assisted Inflation

The flat Assisted Inflation model [229) corresponds to the equilibrium point A of the system (7.10)-(7.13) given by

A _ K2 K2 ..jK2(6---:::-K2) ..jK2(6 - K2) {'lit, 'Ii 2 , 4h, ~2} = {- . Iii ,- Iii' Vii ' Vii }, (7.14)

V 6k1 v 6k2 6k1 6k2

where

-2 _ 1 1 K =-2+-2.

kl k2

The deceleration parameter for this solution is given by

K2_2 qA =

2

and hence this solution has

R(t) ex tP , k1¢>1 = k2<P2'

and is inflationary (qA < 0) if

211 p= 2L-2 = 2K-2 = -1-- > 1;

i=l k i + qA

(7.15)

2~K2. (7.16)

Since a single scalar field can only give rise to an inflationary power-law solution if ~ > ! for i = 1 or 2 [50, 372), this means that the two-scalar field model can be inflationary even when the each of the individual potentials is too steep for the corresponding single scalar field model to inflate (and hence the terminology assisted inflation). The eigenvalues corresponding to the equilibrium point A are given by

K2_2, K2-6

2 ~ ((K2 - 6) ± ..j(K2 - 6)2 + 8K2(K2 - 6)) (7.17)

Hence this equilibrium point is stable when (7.16) is satisfied, and so the corre­sponding assisted inflationary solution is a late-time attractor [269).

2. Stability of Equilibria and Discussion

We note that several of the equilibrium points occur in the 3-dimensional invariant set corresponding to the zero-curvature models defined by

1 = 'Ii~ + 'Ii~ + ~~ + ~~.

86

When matter is included, there exists a monotonic function so that in the full dynamical phase space there can be no periodic or recurrent orbits and the global dynamics can be determined. This implies that the qualitative features described in this Section can be more rigorously proven. All of the equilibrium points and their corresponding eigenvalues are listed in Table XII. Using this Table let us discuss the local stability of these equilibrium points.

As noted above the equilibrium point A, given by (7.14), corresponds to the assisted inflationary solution. It exists for all parameter values satisfying

1 1 2 1 2

6 < k1 + k2 ' (7.18)

and is a sink (late-time attractor) for all parameter values satisfying (7.16) (else it is a saddle).

There are two equilibrium points, denoted by P1 and P2 , whose coordinate val­ues and associated eigenvalues are given in Table XII, which correspond to zero­curvature power-law solutions in which one scalar field (either (P1 or ¢2, respectively) is negligible; these solutions exist if ~ < -b and are inflationary if, in addition,

% < k12 (for each i = 1,2, respectively) and correspond to the well-known single

scalar 'field power-law solutions [249, 371]. From Table XII we see that each Pi has two negative eigenvalues and one positive eigenvalue for all relevant parame­ter values and an additional eigenvalue which is negative if k; < 2 (and positive for 2 < k; < 6); hence these points are saddles and have a one- or 2-dimensional unstable manifold depending upon whether k; < 2 or k; > 2, respectively.

There also exist equilibrium points, denoted by CS1 , CS2 and CS, whose coor­dinate values and the associated eigenvalues are given in Table XII. The solutions correspond to power-law solutions in which the curvature scales with the first scalar field, the second scalar field or both, respectively. The single-field curvature scaling equilibrium points C Sl and C S2 are both saddles. The two-field curvature scaling equilibrium point C S is a sink whenever % > 12 + -b (otherwise a saddle). When-

1 2

ever the two-field curvature scaling solution is stable, it necessarily has negative curvature.

There is an equilibrium point, denoted by M, corresponding to the Milne form of flat spacetime, which is always a saddle. Finally, there is a one-dimensional set of equilibrium points parametrized by wo, denoted by MSF, corresponding to zero­curvature massless scalar field models (in which both potentials are zero). There is one zero eigenvalue corresponding to the fact that there is a one-dimensional set of equilibrium points. There are values for Wo for which the remaining three eigenvalues are positive and hence a subset of MSF are sources (the remainder are saddles). These correspond to the well-known early-time attracting massless scalar field models [50].

Let us now discuss the dynamics. From the analysis above we conclude that the

87

TABLE XII: Equilibrium points in the two-scalar field model with no matter. In the Table, A and CS correspond to the two-field assisted inflationary solution and the two-field curvature scaling solution, respectively.

Solution/Label Coordinates Decel. Curvature Eigenvalues

{\jIl, \jI2,<P1,<P2} Para., q 3R/H2

Assist. Inflation, A (see (7.14» K2_2

0 (see (7.17» -2-

kl H k l2 - 2 k 2 k l2 k l2 - 6 ( ) Power-Law, Pl {-- 0 1-~ O} 0 1 -2'2'--2- x2 .,/6' , 6 ' 2

k2 H k22 - 2 k 2 kl k22 - 6 ( ) Power-Law, P2 {O,- .,/6,0, I--t} 0 2 -2'2'--2- x2 2

Curv. Scaling, CS .,/6 .,/6 2 2

-1 ± v'3i,-l± {-3kl'-3k2'v'3kl 'v'3k2}

0 -1+ 2(k 2 + k 2) .

~ 2k 22 VI + 4[2(k12 + k,2) - 1]) 1 2

. .,/6 2 0

2 - k l2 -2,1, -1± Curv. Scalmg, CSl {--,O, v'3 ,O}

~ 3kl 3kl

Vl+4k12 (2-k, 2 )

. .,/6 2 0

2 - k22 -2,1, -1± Curv. Scaling, CS2 {O,--,O, v'3 }

~ 3k2 3k2

VI + 4k,2 (2 - k22)

Milne, M {O,O,O,O} 0 -1 -2, -2, 1, 1

Massless SF, MSF {\jIo, Vl- \jI02,0,0} 2 0 .,/6

0,4,3+ 2 k l \jlo,

where 0 ~ \jIo2 ~ 1 .,/6 ~ 3 + 2k2 1- \jIo .

two-field assisted inflationary solution A is the global attractor when 2::7=1 k:;2 > ~ and the two-field curvature scaling solution CS is the global attractor when

2::7=1 k:; 2 < ~. The massless scalar field solutions M SF are always the early-time attractors.

In all cases both scalar fields are non-negligible in generic late-time behaviour. This is contrary to the commonly held belief that in multi-field models with expo­nential potentials the scalar field with the shallowest potential (i.e., smallest value of k) would dominate at late times. Indeed, we have shown that the single field power­law inflationary models always correspond to saddles, so that we have the rather surprising result that generically a single scalar field never dominates at late-times.

We note that both the assisted inflationary solution and the massless scalar field early-time attractors correspond to zero-curvature models. However, the curvature is not always dynamically negligible asymptotically because the two-field curvature

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scaling solution has non-zero curvature.

There is a range of parameter values for which the assisted inflationary solution is the global late-time attractor (when the solution is non-inflationary it corresponds to a saddle). For all of these parameter values the single field power-law solutions PI and P2 are saddles. However, there are allowable parameter values for which either PI and P2 are both inflationary, or one is inflationary while the other is not, or both are non-inflationary. This might give rise to some new interesting physical scenarios. For example, a model could asymptote towards an inflationary single field solution Pi, stay close to Pi for an arbitrarily long period (since Pi is an equilibrium point) inflating all the time, and then eventually leave Pi and evolve towards the stable attracting inflationary solution A. (Note from (7.16) that if either of PI or P2 are inflationary, then A is necessarily inflationary). This is akin to a double­inflationary model [325] in which the density fluctuations on large and small scales decouple (Le., the scale invariance of the spectrum is broken) thereby allowing the possibility of more power on large scales which is perhaps in better accord with observations.

B. Qualitative Analysis of the Two-Scalar Field Model with Matter

To investigate the dynamics of the model with matter (i.e., with 0 -I- 0) we shall study the model with two minimally coupled scalar fields together with matter hav­ing energy density J.L with the barotropic equation of state given by (1.2). This model is obtained by setting n = 2 in (7.7)-(7.9), whence we obtain the five-dimensional {O, \[I, \[12, <lh, cI>2} dynamical system given by (7.7) and (7.10)-(7.13), where now

63;2 = -1 + 0 + \[I~ + \[I§ + cI>~ + cI>§,

and

3')' - 2 2 2 2 2 q = -2-0 + 2\[11 + 2\[12 - cI>I - cI>2·

1. Invariant Sets, Monotonic Functions and Stability of Equilibria

The zero-curvature models constitute a 4-dimensional invariant set. The models with no matter also constitute a 4-dimensional invariant set. The function

has derivative

0 2 W = --:---~,-------;~-,,---=--:-::-

- (0 + \[Ii + \[I~ + cI>i + cI>~ - 1)2'

dW = 2(2 - 3')')W. dT

(7.19)

(7.20)

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TABLE XIII: Equilibrium points with n = 0 in the two-scalar field model with matter. Each equilibrium point has n = 0 and the coordinates given in Table XII. The additional fifth eigenvalue is displayed.

Label A P, P 2 CS CS, CS2 M MSF

Eigenvalue k 2k 2 2 2 ~k -3"( k, -3"( k2 -3"( 2-3"( 2-3"( 2-3"( 2-3"( 3(2-"() k, + 2

We observe that this function is monotonic when 0 -# 0 (Le., non-zero matter) and (0 + -w~ + -w~ + <P~ + <P~ - 1) -# 0 (Le., non-zero curvature). We also observe that the sign of 31' - 2 significantly changes the dynamics of these models. For example, in the case of interest here 31' - 2 > 0, whence W is a decreasing function of time 7. This immediately implies that

• There exist no periodic or recurrent orbits in the full five-dimensional phase space (this does not preclude the existence of closed orbits in the invariant sets 0 = 0 and 3 R = 0; however, we shall be primarily concerned with the dynamics of the models in the complete phase-space with matter and non-zero curvature).

• The future asymptotic state lies within the invariant set 0 = o. Matter becomes dynamically unimportant to the future.

• The past asymptotic state lies within the set of zero-curvature models.

The equilibrium points can be classified into two sets; those with 0 = 0 and those with 0 -# o. All equilibrium points listed in Table XII exist in the case with o = 0, and Table XIII lists the equilibrium points with 0 = 0 together with the additional eigenvalue due to the addition of matter. Using the function W above, we can further conclude that those equilibrium points in the set 0 -# 0 necessarily must have zero-curvature. In Table XIV the equilibrium points together with their eigenvalues in the invariant set 0 -# 0 are listed.

Let us focus on the attractors in the full physical phase space. All late-time attractors (sinks) occur in the invariant set 0 = o. In the previous Section we found that A and C S are the only sinks in the invariant set 0 = 0 (clearly, all of the saddles remain saddles in the full five-dimensional phase space). The additional eigenvalue for the equilibrium point A in the full physical phase space is given in Table XIII and is negative if 2:7=1 k;:2 > 3\. But this is always satisfied when

2:7=1 k;:2 > ~ and l' > ~, and hence A is a sink and assisted inflation is a global attractor. Similarly, from Table XIII the equilibrium point CS is always a sink for 2:7=1 k;:2 < ~ and hence the two-field curvature scaling solution remains the global attractor in this case.

The early-time attractors lie in the zero-curvature invariant set and consist of massless scalar field models. From Table XIII we see that the massless scalar field models corresponding to the repelling equilibrium points MSF are always sources (for l' < 2).

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TABLE XIV: Equilibrium points with n f= 0 in the two-scalar field model with matter. Note that in each case 3 R = 0 and q = (3')' - 2)/2.

Solution/Label Coordinates

{fl, WI, W2, <1>1, <l>2}

FRW,F {l,O,O,O,O}

Matter Scaling, MS1 {I _ 31' _ V61' ° ",,61'(2 - 1') kl2 ' 2kl" 2kl ' O}

Matter Scaling, MS2 {I _ 31' ° _ V61' ° y'&y(2 =-:yJ k22" 2k2" 2k2 }

Matter Scaling, M S {I - 31' (k -2 + k -2) _ V61' I 2 , 2kl '

_ V61' ""61'(2 - 1') \1"61'(2 - 1') } 2k2' 2kl ' 2kl

Eigenvalues

31' - 2, i(1' - 2), iC1' - 2), h, h

3 3 3 f. 2 C1' - 2), 21',31' - 2, 4: \C1' - 2)±

V C1' - 2)2 + 81'(1' - 2)[1 - 31'kl- 2 ])

3 3 3 f. 2C1' - 2), 21',31' - 2, 4: \C1' - 2)±

V C1' - 2)2 + 81'(1' - 2)[1 - 31'k2 -2])

31' - 2, ~ ((1' - 2)±

VC1'-2)2 +81'C1'- 2)[1-31'(kl- 2 +k2 -2)]) ~ ((1' - 2) ± VC1' - 2)2 + 81'C1' - 2»)

2. Matter Scaling Solutions

In the case of a single scalar field there exist zero-curvature FRW "matter scaling" solutions when the exponential potential is too steep to drive inflation, in which the scalar field energy density tracks that of the perfect fluid so that at late times neither field is negligible [372]. It was shown earlier that whenever these matter scaling solutions exist they are the unique late-time attractors within the class of flat FRW models [116]. The cosmological consequences of these scaling models have been discussed in [144, 145, 372]. For example, in these models the scalar field energy density tracks that of the perfect fluid and a significant fraction of the current energy density of the Universe may be contained in the homogeneous scalar field whose dynamical effects mimic cold dark matter; the tightest constraint on these cosmological models comes from primordial nucleosynthesis bounds on any such relic density [117, 144, 145, 372]. The stability of these fiat, isotropic matter scaling solutions within the class of spatially homogeneous cosmological models with a barotropic perfect fluid and a scalar field with an exponential potential was discussed earlier in Section V.B [45]. It was shown that while the matter scaling solutions are stable to shear perturbations, for realistic matter with 'Y ~ 1 they are unstable to curvature perturbations.

Returning to the models under investigation here, none of the equilibrium points with n -I- 0 can be late-time attractors for 'Y > ~. Indeed, from Table XIV all such equilibrium points are seen to be saddles. In particular, the two-field matter scaling solution corresponding to the equilibrium point MS, which exists for L~=l k;2 < 3~' is a saddle. From Table XIV we see that the first eigenvalue associated with M S

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is positive, while the real parts of the remaining four eigenvalues are all negative. This is consistent with the stability analysis of matter scaling solutions in models with a single scalar field which found that the models were unstable to curvature perturbations when I > ~ [45]. However, these two-field matter scaling solutions may still be of physical import. We note that when the curvature is zero, the two-field matter scaling solution is an attractor (all four eigenvalues of MS in the 4-dimensional zero-curvature invariant set have negative real parts - so that M S is a sink in this invariant set), as in the case for the matter scaling solution in a single field model. Note also from Table XIV that both of the single-field matter scaling solutions, corresponding to the equilibrium points MS1 and MS2 , have two positive eigenvalues, so that again the solution with multiple scalar fields is the "stronger" attractor.

c. Qualitative Analysis of the Three-Scalar Field Model

Let us now consider models with more than two scalar fields. For simplicity, we shall exclude a matter term here. However, from the previous Section we can easily determine the essential properties resulting from the inclusion of a matter field. In particular, in this case a monotonic function exists and this enables us to prove the qualitative results outlined below. We begin with the three-scalar-field model, obtained by setting n = 3 and n = 0 in (7.7)-(7.9). In this case the resulting six-dimensional {WI, '112, '113, <PI, <P2, <P3} dynamical system is given by (7.7)-(7.9) (i = 1,2,3) where

q = 2W~ + 2W~ + 2W~ - <P~ - <I>~ - <I>~,

3R 6H2 = -1 + W~ + W~ + W~ + <I>~ + <I>~ + <I>~.

We shall not present a complete qualitative analysis similar to that of Section VII A since the essential features are the same. Rather, we shall describe the main effects of including a third scalar field on the inflationary solutions. There exists a zero-curvature assisted inflationary solution which now corresponds to the equilib­rium point given by

{ W. = _ K2 m.. _ VK2(6 - K2) t r,:;' '*'t - } v6ki r;:;,

where

K-2 == k12 + k;;2 + k3 2.

In this solution all of the three scalar fields scale together at late times. The corresponding eigenvalues are

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K2 _ 2, K2~- 6, ~ ((K2 _ 6) ± J(K2 - 6)2 + 8K2(K2 - 6)) ,

~ ((K2 - 6) ± J(K2 - 6)2 + 8K2(K2 - 6)).

It is known [229J that this solution is a late-time attractor for all parameter values for which it is inflationary (Le., K2 < 2; recall the solution does not exist if K2 > 6).

There are three solutions in which two scalar fields scale together asymptotically and the third is negligible. Assuming that the third scalar field is zero (W3 = <1>3 = 0), the coordinates of the corresponding equilibrium point, denoted by P120, are given by

klk22 {- J6(kI2 + k22) ,

k12k2 0 k J6(k12 + k22) - k12k22 J6(kI2 + k22) , ,2 J6(kI2 + k22) ,

k J6(k12 + k22) - k12k22 } 1 J6(kI2 + k22) ,0 .

(7.21 )

Four of the eigenvalues are given by (7.17), which all have negative real parts. There are three solutions in which one scalar field scale dominates asymptotically and the remaining two are negligible. Assuming that the first scalar field is non-zero (lit 1 =f 0 =f <1>1), the coordinates of the corresponding equilibrium point, denoted by P100, are given by

{ kl g2 - ,r,,' 0, 0, 1 - _1 0 O}

v6 6 " .

Two of the eigenvalues are negative, one is positive, and there is an additional eigenvalue which is negative if kr < 2 and positive if 2 < kr < 6. In both of these cases the additional (remaining) two eigenvalues can be calculated and are given by

{q - 2 < 0, q + 1 > O},

where q is the deceleration parameter evaluated at the equilibrium point. Hence, the equilibrium point P120 is a saddle with one eigenvalue with positive real part. The equilibrium points denoted by P103 and P 023 are also saddles with one eigenvalue with positive real part. In additional, the point P 100 is a saddle with two eigenvalues with positive real parts (if k; < 2, and three eigenvalues with positive real parts if k; > 2). The same is true for the equilibrium points denoted by P 020 and P 003 .

Consequently there is a nested set of equilibrium points. At the top level is the stable three-scalar field assisted inflationary solution. In the next level there are three two-scalar field models which are saddles with one eigenvalue with positive real part. In the final level there there are three one-scalar field models which are saddles

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with two eigenvalues with positive real parts (or three eigenvalues with positive real parts). Associated with this dynamical nesting are cosmological models with very interesting physical properties. This will also be true in the case of n scalar fields. There will be a unique stable n-scalar field assisted inflationary solution. There will then be n of the (n - 1)-scalar field models which are saddles with one eigenvalue with positive real part. There will be ~n(n - 1) of the (n - 2)-scalar field models which are saddles with two eigenvalues with positive real parts. And so on. Finally, there will be n of the (1 )-scalar field models which are saddles with n - 1 (or n - 2) eigenvalues with positive real parts. As one "goes up" the nested structure the equilibrium points respectively become "stronger attractors" (Le., the stable manifold of the equilibrium points increases in dimension).

There is also a three-field curvature scaling solution corresponding to the equi­librium point given by

2 2 { 'T' cJ>. - -}

'l'i = - V6ki ' • - V3ki

whose associated eigenvalues are given by

-1 ± V1 + 4K-2(2 - K2), -1 ± V3i, -1 ± V3i.

This equilibrium point is a sink whenever K2 > 2, in which case it represents an FRW model with negative curvature (2K- 2 - 1) (else it is a saddle and represents a positive curvature model). Finally, there are saddle equilibrium points correspond­ing to the Milne model and the one- and two-field curvature scaling solutions, and a set of equilibrium points with n=~=l Wi2 = 1, cJ>i = O} corresponding to massless scalar field models, a subset of which are sources.

A complete qualitative analysis can be done for n-scalar field models. All results can be proven by induction (see, for example, [269]). The n-scalar field assisted inflationary solution is given by [229]

and

R(t) <X tP; n 1

P =2'"'->1 - ~ 2 ' i=l k i

ki<Pi = kj<pj; V1 ::; i i= j ::; n.

We note that in the two-scalar field model, although inflation can occur for poten­tials that are steeper than in the single-field case, it cannot occur for arbitrarily steep potentials. For example, if kl = k2 == k, then inflation occurs if k2 < 4. However, for n-fields, if k i = k for all i, then inflation occurs if k2 < 2n (e.g., k2 < 8 for four scalar field models).

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D. Discussion

We have studied multi-scalar field FRW cosmological models with exponential potentials and barotropic matter. We have used dynamical systems techniques, and by establishing a monotonic function in the complete dynamical phase space (which includes both matter and curvature), we have been able to deduce global results.

In Section VII A a comprehensive qualitative analysis was presented in the case of two scalar fields with no matter. We concluded that the two-field assisted in­flationary solution A is the global attractor when 2:;=1 k;2 > ! and the two-field curvature scaling solution as is the global attractor when 2:;=1 k;2 < !. A subset of the massless scalar field solutions M SF are always the early-time attractors. Con­sequently, we found that in all cases both scalar fields are non-negligible in generic late-time behaviour, contrary to popular belief. We note that both the assisted inflationary solution and the massless scalar field early-time attractors correspond to zero-curvature models. However, the curvature is not dynamically negligible asymptotically in the two-field curvature scaling solution. The zero-curvature as­sisted inflationary FRW scaling solutions [229] are of particular physical importance since, through the combined effect of multiple uncoupled scalar fields, each having an exponential potential, power-law inflation is possible even when each individual scalar field need not be a source for inflation. For the parameter values in which the assisted inflationary solution is the global late-time attractor the two single field power-law solutions were shown to be saddles, and we showed that there are allow­able parameter values for which either or both are inflationary, perhaps leading to new interesting physical scenarios.

In Section VII B we studied the two-scalar field model with barotropic matter. A monotonic function was established in the resulting phase space, thereby proving that the matter must be negligible at late times. We found that A and as are the only global sinks and that consequently assisted inflation and the two-field curvature scaling solution are the global late-time attractors in their appropriate respective parameter ranges. This confirmed the result that both scalar fields must be dynam­ically non-negligible in generic late-time behaviour, and establishes the stability of the two-field assisted inflationary model when matter is included. The monotonic function also implies that the early-time attractors lie in the zero-curvature invari­ant set, and we showed that they consist of a subset of the massless scalar field models. For 'Y > ~, all of the equilibrium points with n -:F 0 were shown to be sad­dles (see Table XIV). The two-field matter scaling solution MS was shown to be a "stronger" attractor than the single-field matter scaling solutions MS1 and MS2 .

We note that when the curvature is zero, the two-field matter scaling solution is the late-time attractor, consistent with the stability analysis in [45]. These one- and two-field matter scaling solutions give rise to new transient dynamical behaviour and may be of physical import. For example, there are solutions which spend a pe­riod of time with the scalar field mimicking the barotropic fluid in which there is a non-negligible scalar field energy density (corresponding to a matter scaling saddle

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equilibrium point) and subsequently evolve towards a scalar-field dominated power­law inflationary epoch (corresponding to a single-field saddle equilibrium point or a two-field assisted inflationary attractor) with an accelerated expansion, perhaps explaining current high redshift data.

In Section VII C we discussed three- and multi-scalar field models (where, for simplicity, a matter term was excluded). In the three-scalar field model we again established that the assisted inflationary solution and the three-field curvature scal­ing solutions are the stable late-time attractors. We then considered n-scalar field models, and established a nested structure for the m-field scaling (assisted infla­tionary) solutions. The n-scalar field assisted inflationary solution is again the late-time attractor. All of the m-field (with m < n) scaling solutions are saddles and in general the equilibrium points corresponding to the m-field scaling solutions will have n - m eigenvalues with positive real parts so that the equilibrium points corresponding to the greater number of non-negligible scalar fields are, respectively, the "stronger attractors" .

Finally, from previous investigations [50] of spatially homogeneous scalar field cosmological models with an exponential potential and barotropic matter and from the above analysis, we can conclude that the assisted inflationary solution is a global attractor for all ever-expanding spatially homogeneous multi-field cosmolog­ical models with exponential potentials provided I:~=1 k;2 > 1/2. We can also conclude that the multi-field curvature scaling solution is a global attractor for models of Bianchi types V and Vlh provided I:~=1 k;2 < 1/2 and a multi-field generalization of the Feinstein-Ibanez anisotropic single-field solution is the global attractor for models of Bianchi types I II and V h if I:~=1 k;2 < 1/2 [50]. Indeed, there will be n-field generalizations corresponding to all equilibrium points of the single-field Bianchi Type B models.

VIII. SCALAR TENSOR THEORIES OF GRAVITY

Scalar-tensor theories of gravity are currently of great interest, partially due to the fact that such theories occur as the low-energy limit in superstring theory (see [188] and references therein). The first scalar-tensor theories to appear (with w = wo) were due to Jordan [205, 206], Fierz [146] and Brans and Dicke [60] and the most general scalar-tensor theories were formulated by Bergmann [40], Nordtvedt [289] and Wagoner [356]. The observational limits on scalar-tensor theories include solar system tests [2, 59, 64, 189] and cosmological tests such as Big Bang nucleosynthesis constraints [17, 322].

The action for the general class of scalar-tensor theories (in the so-called Jordan frame) is given by [40, 356],

s= J Fu[4>R-W~)gab4>,a4>'b-2V(4))+2.cm]d4x. (8.1)

However, under the conformal transformation and field redefinition [25, 230, 277]

* gab dcp

d4>

4>gab ± ";'-w-:-c( 4>-:-) -+ -3 /-;-2

4>

the action becomes (in the so-called Einstein frame)

(8.2)

(8.3)

s* = J ../-g* [R* - gMbcp,acp,b - 2 V~t) + 2~~] d4x, (8.4)

which is the action for GR containing a scalar field cp with the potential

V( 4>( cp)) V*(cp) = 4>2(cp) . (8.5)

The simplest scalar-tensor theory of gravity is the Brans-Dicke theory (BDT) [60], in which the scalar field, 4>, acts as the source for the gravitational coupling with a varying Newtonian gravitational constant G ,...., 4>-1, and was essentially motivated by apparent discrepancies between observations and the weak-field predictions of GR [19]. More general scalar-tensor theories with a non-constant BD parameter, w(4)), and a non-zero self-interaction scalar potential, V(4)), have been formulated, and the solar system and astrophysical constraints on these theories, and partic­ularly on BDT, have been widely studied [26, 374]. Observational limits on the present value of Wo need not constrain the value of w at early times in more general scalar-tensor theories (than BDT). Hence, more recently there has been greater focus on the early Universe predictions of scalar-tensor theories of gravity, with particular emphasis on whether cosmological models exist in which the scalar field acts as a source for inflation [223] and whether models have an initial singularity.

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

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There are many exact cosmological solutions known in BDT, including the flat isotropic and homogeneous exact vacuum solutions of O'Hanlon and Tupper [290] and the special class of power-law perfect fluid solutions of Nariai [283] (more general solutions are reviewed in [183]). A phase-space analysis of the class of FRW models was performed by Kolitch and Eardley [219] and was improved upon by Holden and Wands [183] who presented all FRW models in a single phase plane (including those at "infinite" values via compactification). It was found that typically at early times (t -t 0) the BDT solutions are approximated by vacuum solutions (i.e., the O'Hanlon-Tupper FRW vacuum solutions) and at late times (t -t 00) by matter­dominated solutions, in which the matter is dominated by the BD scalar field (e.g., the power-law Nariai solutions). A variety of exact spatially homogeneous but anisotropic BDT solutions have also been found [92]. Exact perfect fluid solutions in scalar-tensor theories of gravity with a non-constant BD parameter w(¢) have been obtained by various authors (see [92, 276] and references cited within). However, it has been argued that scalar-tensor theories with a "free" scalar field are perhaps not well motivated since, often, quantum corrections produce interactions resulting in a non-trivial potential V(¢).

1. Stiff Perfect Fluids in General Relativity

The qualitative properties of orthogonal spatially homogeneous (OSH) stiff per­fect fluid models with an equation of state p = (r -1)1L with the value [ = 2 within GR is of interest here. Since this is a bifurcation value for [, models with [ = 2 may have different qualitative properties to models with [ < 2. The finite equilib­rium points (and their stability) have been investigated, and all non-tilting spatially homogeneous solutions of the EFE with a perfect fluid with [ = 2 (and IL > 0) as source which admit a 4-dimensional similarity group acting simply transitively on spacetime are listed in Table 9.2 in WE; the flat isotropic [ = 2 solution (FL), the Jacobs stiff perfect fluid solutions .J and the one-parameter family of Bianchi type VIIh plane wave solutions are given explicitly therein.

The known asymptotic properties of stiff perfect fluid models (r = 2) in GR can be summarized as follows [92]. For all models (Bianchi models of classes A and B), a subset of the Jacobs Disc, which consists of exact self-similar Jacobs stiff perfect fluid solutions (corresponding to equilibrium points of the governing system of autonomous ODE), is the past attractor.

As regards future evolution, all stiff models behave like vacuum models with the following exceptions: (i) Bianchi I models, all of which are exact Jacobs solutions. (ii) Bianchi II models, which are future asymptotic to another subset of the Jacobs Disc. For Bianchi models of types VIo and VIIo the future asymptote is a flat Kasner model, as in the case of vacuum models. The Bianchi VIII models do not have a self-similar future asymptote; these ever-expanding stiff models are the only models for which this is the case.

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A. Scalar-Tensor Theories of Gravity with No Potential

Since the qualitative properties of spatially homogeneous stiff perfect fluid and minimally coupled massless scalar field models within GR are known, the formal equivalence under conformal transformations and field redefinitions of certain classes of theories of gravity can be exploited and the asymptotic properties of spatially homogeneous models in a class of scalar-tensor theories of gravity that includes the BDT can be determined.

In the absence of a potential (V = 0) the action (8.1) is equivalent to the action for GR minimally coupled to a massless scalar field 'P and matter (Lm). In the spatially homogeneous case, ¢ = ¢(t), and hence under the conformal transformation the Bianchi type of the underlying model is invariant. Also, in all of the applications presented here this transformation is non-singular and so the asymptotic behaviour of the scalar-tensor theories can be determined directly from the corresponding behaviour of the GR models (cf. [46])

In the scalar-tensor theory (8.1), the energy-momentum of the matter fields is separately conserved. In the Einstein frame this is no longer the case (although the overall energy-momentum of the combined scalar field and matter field is, of course, conserved), and [276]

,rT. = -! ¢,b T ab 2 ¢ , (8.6)

where T == T;:. When T -# 0, these equations indicate energy-transfer between the matter and scalar field. T = 0 when 3p - p, = 0; this can occur either for vacuum (p, = p = 0; i.e., no matter present) or in the case of radiation (-y = 4/3). We shall assume that the matter satisfies a linear equation of state (1.2). Finally, defining

so that (8.6) yields

1 .2 p,<p = PIp = 2"'P ,

{l<p + 3(p,<p + p<p)H = Qt.j;,

(8.7)

(8.8)

where Q == [2(3 + 2w)]-!T, we see that the massless scalar field is equivalent to a stiff perfect fluid (-y<p = 2). Hence the model is equivalent to an interacting two-fluid model, one fluid of which is stiff [276].

1. Application: Brons-Dicke Theory

To study the qualitative properties of OSH perfect fluid models with a linear equation of state within scalar-tensor theories (with no potential), and particularly within BDT, expansional-normalized variables can be introduced and the resulting

99

system of ODE can be investigated [46). In BDT it can be shown that again one differential equation decouples and the "reduced" finite-dimensional system of ODE can be analysed; the equilibrium points of the reduced dynamical system again correspond to exact self-similar solutions [101). However, here we shall present some of the more important qualitative properties by utilizing the earlier results and applying conformal transformations to known exact solutions in GR, and noting that solutions corresponding to equilibrium points of the governing dynamical system can act as future and past attractors.

• For the flat isotropic stiff FRW metric, the BD metric becomes (after a constant rescaling of the spatial coordinates)

2(1-3w) dS~L = -dT2 + T 3(1-w) (dX2 + dy2 + dZ2). (8.9)

We deduce that this exact, flat (non-inflationary) isotropic BD solution (where the scalar field can be derived from equations (8.2) and (8.3) above) is an attractor in the class of isotropic models in BDT (see [183)). This vacuum BD solution was first obtained by O'Hanlon and Tupper [290). We note that for large values of wo,

2(1-3w)/3(I-w) ~ ~ (1 ± 3~J ~~; indeed, aswo -t 00, W -t 0 and we formally

recover the GR solution in this limit.

• For the Jacobs stiff perfect fluid solutions, the associated BD metric becomes (after a constant rescaling of each spacelike coordinate)

ds~ = -dT2 + T 2ql dX2 + T 2q2 dy2 + T 2q3 dZ2 , (8.10)

where the qa«a = 1,2,3) are constants given in [32, 60). These Bianchi type I BD solutions therefore act as attractors for a variety of OSH Bianchi models. In particular, all non-exceptional, initially expanding Bianchi type B BD models are asymptotic in the past to this BD solution. This metric reduces to the isotropic metric when all of the qa< are equal.

• All Bianchi models of type B in BDT are asymptotic to the future to a vacuum plane wave state. For example, we can obtain the following BD Bianchi type VUh plane wave solution (after a constant rescaling of the 'y' and 'z' coordinates)

ds~w = -dT2 + D2(T2dx2 + T2(~:::~) e2rx {e.B[cosv dY - sinv dZ)2 + e-.B[cosv dZ - sin v dy)2}), (8.11)

where r,(J,D,v are constants (defined in [92)). This metric can be simplified by a redefinition of the 'x' coordinate. The BD scalar field can again be derived immediately from the conformal transformation rules.

All of the asymptotic results in GR reviewed earlier have BDT analogues, and we can now deduce the asymptotic properties of spatially homogeneous cosmological models in BDT. For example, all orthogonal Bianchi type B BDT models, except for a set of measure zero, are asymptotic to the future to a vacuum plane-wave state. One immediate consequence of this result, since Bianchi models of type B constitute

100

a set of positive measure in the set of spatially homogeneous initial data, is that a recent conjecture that the initial state of the pre-big-bang scenario within string theory generically corresponds to the Milne (flat spacetime) universe [353] is un­likely to be valid (recall that past asymptotic behaviour in pre-big-bang cosmology corresponds to future asymptotic states in classical cosmological solutions).

However, some care is needed in interpreting these results and they must be ap­plied in concert with special exact solutions and the analysis of specific but tractable classes of models to build up a complete cosmological picture. First, the GR anal­ysis (in the Einstein frame) is incomplete in that the phase-space in some Bianchi classes is not compact and the Hubble parameter can become zero (and hence the expansion-normalized variables become ill-defined). Second, in scalar-tensor theo­ries of gravity (in the Jordan frame) it is known that there exist solutions which do not have an initial singularity but have a "bounce" (at which H = 0). Since there is always an initial singularity in the Einstein frame, such an "avoidance of a singularity" is due to the properties of the conformal transformations; for example, Mimosa and Wands [276] describe a set of models that reach an anisotropic singu­larity in a finite time in the Einstein frame which correspond to non-singular and shear-free evolution in infinite proper time in the Jordon frame. Consequently, the asymptotic results, including their global features and their physical interpretation, must be determined from the properties of the transformations and how solutions are matched together.

B. Scalar Tensor Theories with a Non-Zero Potential

The asymptotic properties of more general scalar-tensor theory models can be studied in a similar way (cf. [46]). In particular, the asymptotic behaviour of spa­tially homogeneous cosmological models in a class of scalar-tensor theories which are conformally equivalent to general relativistic Bianchi cosmologies with a scalar field and an exponential potential (whose qualitative features were discussed earlier) can also be investigated. In these scalar-tensor theory cosmological models, self-similar solutions again play an important role in describing the asymptotic behaviour of more general models (e.g., these cosmological models can act as early-time and late-time attractors). Scalar-tensor theories of gravity with action (8.1) transform to GR with a scalar field with the exponential potential (3.10), V* = Aekcp , un­der the transformations (8.2, 8.3). Since we know the asymptotic behaviour of spatially homogeneous scalar field Bianchi models with an exponential potential, we can again deduce the asymptotic properties, including their possible isotropiza­tion and inflation, of the corresponding scalar-tensor theories under the conformal transformations (8.2, 8.3) (so long as the transformations are not singular).

101

1. Application

In particular, let us discuss the explicit example of BDT with a power-law po­tential, viz.,

w(¢)

V Wo {3¢Ot

(where (3 and a are positive constants), then (8.3) integrates to yield

¢=¢o exp(tp-;tpo),

where

w == ±vwo + 3/2,

(8.12)

(8.13)

(8.14)

(8.15)

and hence from the action (8.1) and equation (8.5) we obtain the exponential po­tential (3.10), where the parameter k is given by

a-2 k ---- . w (8.16)

We can now determine the asymptotic behaviour of the models in the Einstein frame, for a given model with specific values for a and w (and hence a particular value for k) (99). The possible isotropization and inflationary behaviour of the given scalar-tensor theory can now be obtained directly. Let us further discuss the asymptotic behaviour of the corresponding scalar-tensor theories (in the Jordan frame). From equations (8.2), (8.3) and (8.14) we have that asymptotically

¢ = ¢o [±(t - to) (1 + k~) ] -2/(1+kW) , (8.17)

where the ± sign is determined from the DO-component of (8.2) (Le., dt* = ±J(fx1t). Both this sign and the signs of wand 1 + kw are crucial in determining the relation­ship between t* and tj Le., as t* -+ 00 either t -+ ±oo or t -+ to and hence either ¢ -+ 0 or ¢ -+ 00, respectively, as tp -+ -00.

Let us exploit the formal equivalence of the class of scalar-tensor theories with w(¢) and V(¢) given above with that of GR containing a scalar field and an ex­ponential potential. Since the conformal transformation (8.2) is well-defined in all cases of interest, the Bianchi type is invariant under the transformation and we can deduce the asymptotic properties of the scalar-tensor theories from the correspond­ing behaviour in the Einstein frame. We recall that at the finite equilibrium points in the Einstein frame we have that

(J*

tp(t*)

(Jot;!, 2

tpo - k In(t*),

(8.18)

(8.19)

102

where

k2 ()* = 1 + _ekr.po o 2 . (8.20)

Integrating equation (8.3) we obtain

<p(t*) = dexp (w-1rp(t*)) = <POt";2/kw, (8.21)

where the constant <Po == dexp(rpo/w) and we recall that t and t* are related by equation (8.2), which can be written as gab = <p-lg~b'

We can summarize the main result. All initially expanding scalar field Bianchi models with an exponential potential with 0 < k2 < 2 within G R (except for a subclass of models of type IX) isotropize to the future towards the power-law inflationary flat FRW model [216], whose metric is given by

ds2 = -dt; + t!/k2 (dx2 + dy2 + dz2) . (8.22)

In the scalar-tensor theory (in the Jordan frame), <p is given by equation (8.21) and from (8.2) we have that

dS~T = <Po1t;/kw {ds2} .

Defining a new time coordinate by

l+kw T = ct*kw j

kw _1.

C == 1 + kw <Po 2

(8.23)

(8.24)

(where kw + 1 -I- OJ Le., a -I- 1), we obtain after a constant rescaling of the spatial coordinates

where

dS~T = -dT2 + T2k (dX2 + dy2 + dZ2) ,

k2 + 2kw Ii == k2(1 + kw)'

Finally, the scalar field is given by 2 -2 - 2

<p = <PoC1+kwTl+kw = <PoT1 -".

(8.25)

(8.26)

(8.27)

Therefore, all initially-expanding spatially homogeneous models in this class of scalar-tensor theories with 0 < (a - 2)2 < 2wo + 3 (except for a subclass of Bianchi IX models which recollapse) will asymptote towards the exact power-law flat FRW model given by equations (8.25) and (8.27), which will always be inflationary since K = ,1+?,,~2w'l., > 1. Note that this solution is self-similar [46].

Finally, we note that when k2 > 2, the models in the Einstein frame cannot inflate and mayor may not isotropize (see [46] for details). This work can be generalized in a number of ways. In particular, more general scalar-tensor theories can be considered and more general (than spatially homogeneous) geometries can be studied.

103

c. Inhomogeneous Models

Inhomogeneous G2 cosmologies have been studied since there is some evidence that the class of self-similar G2 models plays an important role in describing the asymptotic behaviour of more generic general relativistic scalar field models with an exponential potential. It is possible to use the techniques above to find special scalar-tensor G2 cosmological models that describe the asymptotic properties of more general scalar-tensor cosmologies. G2 scalar-field cosmological models with an exponential potential have been studied by Ibanez and Olasagasti [195, 196].

Let us study the scalar field models in the inhomogeneous G2 geometry with two commuting spacelike Killing vectors (tx and ty ) and in which the metric is given by

ds2 = eF (-dt2 + dz2 ) + G (ePdx2 + e-Pdy2) , (8.28)

where all metric functions depend upon t and z [221]. To preserve the G2 geometry the scalar field is assumed to be of the form,

¢ = ¢(t, z), (8.29)

so that providing the transformation (8.2) is non-singular the corresponding GR metric g* is also a G2 metric.

In previous work on G2 perfect fluid models [176], in which the EFE in expansion­normalized variables take on the form of a quasi-linear hyperbolic system of au­tonomous PDE, it was shown that the equilibrium points of the corresponding infinite-dimensional dynamical system are represented by exact self-similar G2 cos­mological models. Thus, it is reasonable to assume that there will be a general class of scalar field general relativistic G2 cosmological models with an exponential potential that will be asymptotic in the past or to the future to an exact self­similar G2 cosmology, since the corresponding EFE again have the structure of an infinite-dimensional dynamical system in which the equilibrium points correspond to self-similar models.

In [196] scalar field G2 cosmologies with an exponential potential were studied. In this work the metric components were assumed to be separable in the variables t and z and of the form

G(t, z) eF(t,z)

eP(t,z)

T(t)Z(z), ef(t)eft(z) , Q(t)Z(z)n,

(8.30)

(8.31 )

(8.32)

where n is a constant. This form of the metric has been used by several authors [313J in different contexts and it allows the z-dependence of the field equations to be completely determined, leaving a set of ODE for the unknown functions of t, which can be analyzed using dynamical systems techniques. In the case studied in [195], where the function G was homogeneous, Le., Z(z) = const., it was found

104

that with a linear inhomogeneity isotropization depends solely on the parameter k (as in the case of the homogeneous Bianchi model subclass); when k2 < 2 all solutions isotropize and homogenize, but for k 2 > 2 only a subclass of solutions of measure zero isotropize, although all models homogenize. For more general solutions the analysis performed in [196] showed that most models asymptote towards an inhomogeneous class of solutions, except for a subclass of massless scalar field (Yo =

0) models of measure zero for which the late-time attractor is a homogeneous Bianchi type I model with a scalar field. The early-time attractors are inhomogeneous models which are Kasner-like in their temporal dependence.

1. Inhomogeneous Models in Scalar-Tensor Theories of Gravity

Exact self-similar spatially inhomogeneous G2 cosmological models can be found in a class of scalar-tensor theories of gravity by exploiting the formal equivalence of this class of theories (under a conformal transformation and field redefinition) to GR minimally coupled to a scalar field with an exponential potential. It is plausible that these exact self-similar solutions play an important role in describing the asymptotic behaviour (both at early and at late times) of more general scalar-tensor G2 models. In recent years there has been an interest in studying inhomogeneous cosmological solutions of low energy string theory [23]. Since the low-energy action for bosonic string theory is identical to the BDT action with w = -1 it is of interest to extend the analysis done in [46] to inhomogeneous scalar-tensor models.

We again consider the subclass of scalar-tensor theories of gravity in which the arbitrary functions are given by w(¢) = Wo and V = f3¢Ci., where Wo, Q and f3 are constants (i.e., the potential V is of power-law form), so that equation (8.3) inte­grates to yield ¢ = ¢o exp( <e-;;<eO) (all constants are defined above). We can obtain the exact G2 solutions in scalar-tensor theories by the conformal transformations in a straightforward way [49].

Let us display the scalar-tensor counterparts of the exact GR solutions. In what follows

A2 - fa2 A2+;w2 A2 nw2

ds2 = eC3t E C±(-dt2+dz2)+e tE1+ndx2+e~tEl-ndy2 GR 2Vo Z Z Z'

(8.33) where Ez and all of the constants are defined in [49] (Ez = E(z) is given in terms of hyper-trigometric, linear or trigometric functions depending on k2 <, =, > 2). The scalar field is given by

1 ¢ = -k[C3t + C±ln(Ez )]. (8.34)

Through the conformal transformations, we find that the early-time attracting scalar-tensor models are described by the metric

t(k/2-m±C2)/w zC±/kw dS~T = ~ (dS~R)' (8.35)

105

and by the scalar field

¢ = ¢ot(m±C2 -k/2)/w Z-C±/kw. (8.36)

We note here that the corresponding transformations are singular at z = 0 (t = 0 corresponds to a physical singularity).

Similarly, the late-time GR attractors are transformed to the scalar-tensor solu­tions described by:

(i) the metric

and the scalar field

(ii) and the metric

and the scalar field

r 2k/[w(2-k2 )] zC±/kw dS~T = "- (dsZm ) ,

¢ = ¢o t 2k/[w(2-k 2 )]

Qa ds2 ekwt EC±/kw

ST = z ¢o

(dS~R) ,

¢= ¢o Qat e- kw E-C±/kw z .

(8.37)

(8.38)

(8.39)

(8.40)

It is plausible that these exact solutions in scalar-tensor theories are attractors (past and future) for a more general class of scalar-tensor models. Indeed, we conjecture that these self-similar G2 scalar-tensor models play an important role in describing the asymptotic behaviour of more general inhomogeneous scalar-tensor models. This conjecture may be proven by setting up the governing equations in the scalar-tensor theory as a dynamical system and determining the stability of the equilibrium points corresponding to these solutions, or from a straightforward perturbation analysis of these solutions.

In addition, there is a massless scalar field "saddle point" scalar-tensor model given in [49). This special V = 0 solution corresponds to a Brans-Dicke theory solution [92). In this case the theory is formally equivalent to GR plus a minimally coupled massless scalar field, and we can deduce the possible asymptotic behaviour of the Brans-Dicke theory cosmological models not from the general relativistic scalar field models with an exponential potential but from the general relativistic G2 stiff perfect fluid models.

Since the conformal transformations depend in general on both z and t, these transformations will typically be singular for a particular value of z. However, the transformation is well defined for z > 0 (for example) and scalar-tensor G2 solutions can be obtained formally by analytic continuation. The possible isotropization and the homogenization of these of these models is discussed in [50). In addition, we note that this work can be generalized to scalar-tensor theories with non-constant w [46)

IX. MAGNETIC FIELD COSMOLOGY

There are many observations that imply the existence of magnetic fields in the Universe. Recent observations place an upper bound on the strength of a magnetic field in interstellar and intergalactic space [222, 378], but are not conclusive as to whether such a cosmic field exists [346J. Although cosmologists have investigated the possible existence of a homogeneous intergalactic magnetic field of primordial origin both from a theoretical and an observational point of view for many years, research on cosmological magnetic fields has been rather marginal despite their potential importance. The reasons could be the perceived weakness of the field effects or the lack, as yet, of a consistent theory explaining the origin of cosmic magnetism. However, this situation has changed considerably recently (cf. [161]).

Primordial magnetic fields would affect the temperature distribution of the mi­crowave background radiation, primeval nucleosynthesis and galaxy formation. But its most direct observational effect is the Faraday rotation it would cause in linearly polarized radiation from observed extragalatic radio sources. Fundamental proper­ties of magnetic fields include their vectorial nature, which inevitably couples the field to the spacetime geometry [338J, leading to magneto-curvature effects which tend to accelerate positively curved perturbed regions, while it decelerates regions with negative local curvature, with the result that it can reverse the pure magnetic effect on density perturbations [339, 340J.

In a recent analysis [272J it was shown how a cosmological magnetic field can modify the expansion rate of an almost-FRW universe. The effects of the interac­tion between magnetism and geometry in cosmology are subtle, and it was argued that even weak magnetic fields can lead to appreciable effects, provided that there is a strong curvature contribution. It was found that in spatially open cosmological models containing "matter" with negative pressure, the phase of accelerated expan­sion, which otherwise would have been inevitable, may not even happen. This leads to the question of the efficiency of inflationary models in the presence of primordial magnetism.

We shall study open spatially homogeneous cosmological models containing both a uniform magnetic field and a scalar field using dynamical systems methods. A qualitative analysis of simple anisotropic cosmological models with matter and a uni­form magnetic field has been presented previously (cf. [109,201]). In LeBlanc et al. [226J the Einstein-Maxwell field equations for orthogonal Bianchi VIo cosmologies with a 'Y-law perfect fluid and a pure, homogeneous source-free magnetic field were written as an autonomous differential equation in terms of expansion-normalized variables. A complete analysis of the stability properties of the equilibrium points of the differential equation and a description of the bifurcations that occur as the equation of state parameter 'Y varies was given. In particular, oscillatory behaviour occurs in cosmological models with a magnetic field (and in Einstein-Yang-Mills theory in general). Further work on spatially homogeneous models with a magnetic field and a non-tilted perfect fluid has been carried out recently [225, 370J.

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

107

In this Chapter we shall investigate the class of Bianchi VIo models with barotropic matter, a scalar field with an exponential potential and a uniform mag­netic field. The reasons for concentrating on the Bianchi VIo models are ones primarily of mathematical simplicity. It is well known [191] that a pure magnetic field is only possible in Bianchi cosmologies of types I, II, VIo, VIIo (in class A) and type III (in class B). For types VIo and VIIo, the algebraic constraints that arise from the EFE imply that the shear eigenframe is Fermi-propagated, which in turn implies that the remaining field equations reduce to an autonomous differen­tial equation with a polynomial vector field. Finally, for type VIo, but not for type VIIo, the physical region of state space is compact. We note that since a magnetic field is not compatible with Bianchi types VIII and IX (for example), the class of Bianchi VIo magnetic cosmologies is of the same generality as the Bianchi VIII/IX models (without a magnetic field) [360]. We shall discuss some general properties of the complete class of models, and then investigate the special case of the Bianchi I system in detail. We note that this is not the general class of Bianchi I cosmological models; we are only considering those Bianchi I models that occur as subcases of the Bianchi VIo models with one magnetic degree of freedom. The Bianchi I system in the absence of a scalar field was first qualitatively analysed by Collins [109]. A qualitative analysis of the Bianchi I models in the absence of a magnetic field was given earlier; recall that the well-known power-law inflationary solution was shown to be a stable attractor for an appropriate parameter range in the presence of a barotropic fluid in all Bianchi class B models.

A. Bianchi VIo Models

We shall follow the approach of LeBlanc et al. [226] who qualitatively studied the evolution of orthogonal Bianchi cosmologies of type VIo with a perfect fluid and a magnetic field as source. An invariant orthonormal frame of vector fields on the spacetime is introduced in which one vector is aligned along the fluid flow vector so that the remaining three spatial vectors (triad) span the tangent space orthogonal to the fluid flow at each point of the group orbits. The commutation functions of this frame are then taken as the basic variables. The Einstein-Maxwell field equations for a pure magnetic field are then derived in the orthonormal frame for­malism, giving rise to a set of evolution equations for the shear variables ((Y a.{3), the curvature variables (na.{3), the energy density (conservation equation), the magnetic field (Maxwell equation) and a first integral (the Friedmann constraint), where the remaining freedom in the choice of spatial tetrad was used to simultaneously diago­nalize (Ya.{3 and na.{3. Expansion-normalized variables are then introduced, leading to a reduced set of evolution equations (the dynamical system) for the dimensionless shear variables E±, the spatial curvature variables N a., the dimensionless magnetic field variables and the density parameter (defined as usual). Defining a dimension­less time variable, the reduced evolution equations in the special case of a Bianchi VIo were then established (in this case there is a single dimensionless magnetic field variable, M2). We again note that all of the equilibrium points of the DE

108

correspond to self-similar exact solutions of the EFE.

Let us now discuss the governing equations of the models presently under investi­gation. We assume a scalar field with an exponential potential (equation (3.10)) and a separately conserved perfect which satisfies the barotropic equation of state (1.2) (where the constant "( satisfies 0 < "( < 2). The energy conservation and Klein­Gordon equations are given by equations (5.5) and (5.6), respectively. We also assume a uniform magnetic field (in the 'x' direction) which satisfies the Maxwell equations. The form of this magnetic field, which we shall denote here by M2, is given in Collins [109] or LeBlanc et al. [226], but its precise form is unimpor­tant since we use the generalized Friedmann constraint to eliminate M2 from the evolution equations.

We define

0.==~ 3H2'

¢ W== V6H'

v cP == 3H2' (9.1)

two normalized (dimensionless) shear variables E+ and E_ and the new logarithmic time variable r by dr/dt == H. The evolution equations for the quantities

x = (E+,E_, W,cp,0.,N+,N_) E lle, (9.2)

are then as follows:

E~ (q - 2)E+ + 2(1 - Nl- w2 - cP - E! - E~ - 0.) - 2N~, (9.3)

E~ (q - 2)E_ - 2N+N_, (9.4)

W' V6k (q - 2)W - -2-CP, (9.5)

cP' = (2q + 2 + v6kW)CP, (9.6) 0.' (2q - 3"( + 2)0., (9.7)

N' + (q + 2E+)N+ + 6E_N_, (9.8)

N'- (q + 2E+)N_ + 2E_N+, (9.9)

where a prime denotes differentiation with respect to the time r and the deceleration parameter q is given by

1 q = 1 - N~ + E! + E~ + "2(3,,( - 4)0. + w2 - 2CP. (9.10)

The physical state space is restricted to 0. 2: 0 and Ni ~ 3N~ (where we include equality to include models of Bianchi types I and II [226]). Since N_ = 0 implies N'-- = 0, the subsets N _ > 0 and N _ < 0 are invariant. Due to a discrete symmetry in the evolution equations we can, without loss of generality, restrict attention to the subset N _ ~ O.

The magnetic field, defined through the first integral, is given by

~ M2 = 1 - N 2 - w2 - cP - E2 - E2 - 0. 2 - + - , (9.11)

109

and since M2 ~ 0, all of the variables are bounded:

° $ {E~, E~, '112, CP, n, N;/3, -N_} $ 1. (9.12)

Without loss of generality, we can restrict attention to the subset M ~ 0. There is also an auxiliary equation for the magnetic field:

M' = (q - 1 - 2E+ )M. (9.13)

1. Discussion of Qualitative Properties

A number of invariant sets of the physical state space can be identified. In addition, there are a number of monotonic functions that exist in various invariant sets (particularly on the boundary; cf. p.521 [226)). However, we shall not present them here. Indeed, there are many (over fifty) equilibrium points, the vast majority of which are saddles, and hence in this Section we shall simply comment on some of the more physically interesting local properties of the models. In the next Section we shall investigate the Bianchi I models more rigorously, and a monotonic function in this invariant set will be given explicitly, in order to discuss some aspects of the intermediate or transient behaviour of the models .

• The equilibrium point P: (E+, E_, '11, cp,n,N+, N_)=(O,O, ~, 1- k: ,0,0, 0), M = ° corresponds to the power-law inflationary zero-curvature (flat) FRW model. The eigenvalues ofP are {k2-4, k2-3" !(k2-2)(x2), !(k2-6)(x3)}. P exists for k2 $ 6 and is a sink for k2 < 4 and k2 < 3" and is inflationary (q < 0) for k 2 < 2. In fact it is the global attractor for this range of parameter values; this result generalizes previous work by including a cosmic magnetic field (cf. [50)) .

• We are primarily interested in whether there are any sinks (or sources) with M I=- ° in the Bianchi YIo state space. We know from [226] that there are no attractors in the absence of a scalar field, so hence we must have M =I- 0, '112 I=- ° (and cP I=- 0, otherwise M = 0). From equation (9.7) we see that n approaches zero at late times, so we also take n = ° (and q < 2). From equations (9.5), (9.6) and (9.13) we have that (for M I=- 0)

'11 = -4 v'6k (1 + E+),

4 cP = 3k2 (1 + E+)(1 - 2E+); q = 1 +2E+. (9.14)

In particular, let us consider the inflationary case in which q < 0; i.e., -1 < E+ < -!. If N_,N+ (E_) are not zero, equations (9.8) and (9.9) yield N+ = ±v'3N-, E_ = =f2~(1 + 4E+), whence equation (9.4) then yields

N'!.. = l2(1 - 2E+)(1 + 4E+), which is negative for the range of E+ under consideration! (There do exist equilibrium points with N _ =I- 0, N + I=- ° and E_ I=- 0, but these all have q ~ ° (i.e., are non-inflationary) and are saddles.)

110

Hence we consider N_ = N+ = E_ = 0. Equations (9.3) and (9.10) then yield (since E+ i= !), E+ = -1 + 3k2 /(8 + k 2 ). This equilibrium point, denoted V in Subsection IX B, is physical and in the state space when 4 < k2 < 8 and is always a saddle (Ni is increasing as orbits evolve away from V to the future). However, since q < ° only for k2 < ~ (E+ < -!), this point is never inflationary.

We conclude that inflation is not fundamentally affected by the presence of pri­mordial magnetism in the models under study here. This latter result is not neces­sarily inconsistent with the conclusions of Matravers and Tsagas [272], since only a uniform magnetic field was considered here. Indeed, it is the non-uniform magnetic field gradients that give rise to the magneto-curvature effects which modify the cosmological expansion rate of an almost-FRW universe that can have undesirable implications for inflationary models (and may even prevent inflation from taking place in the presence of primordial magnetism) .

• There is an equilibrium set K: (E+, E_, W, 0, 0, 0, 0) where 1-E~ - E:' - w2 = ° (and M = 0, q = 2). (These points reduce to the equilibrium set KM in the absence of a magnetic field consisting of Jacobs type I solu­tions with no matter and a massless scalar field [50]). The eigenvalues are {O, 0, 2(1 - 2E+), 3(2 - 1'),2(1 + E+ ± .J3E_), 6(1 + :% n. There are two zero eigenvalues (corresponding to the fact that K is a 2-parameter family of equi­librium points). An analysis shows that there is always a subset of K that acts as sources (these are, in fact, global sources); all other points are saddles.

This contrasts with the situation in the absence of a scalar field in which it was shown that there are no equilibrium points that are sources and it was argued that generically an orbit with n > ° will pass through a transient stage and then ap­proach the Kasner circle into the past (consisting of Kasner, Bianchi type I vacuum models) [226]. Since these equilibrium points are saddles, the orbit subsequently leaves along a uniquely determined heteroclinic orbit and then approaches the Kas­ner circle again. This process then continues indefinitely, and the orbit follows an infinite heteroclinic sequence of Rosen orbits and Taub orbits joining Kasner equilib­rium points undergoing Mixmaster-like chaotic oscillations (the past attractor here consists of the union of the Kasner circle and a family of Rosen magneto-vacuum type I orbits and Taub vacuum type II orbits - see also [370]) .

• In the absence of a scalar field [226] it was shown that for n > 0, 1 < l' < 2 the global sink is the equilibrium point PM (VIo) , so that at late times all such magnetic Bianchi VIo cosmologies are approximated by a self-similar magneto-vacuum Bianchi VIo model [133]. This equilibrium point has an analogue here with W = <I> = 0, but is now a saddle. Indeed, it can be easily shown that none of the equilibrium points with vanishing scalar field can be sinks. In particular, the sink in the case l' = 1 in [226] (subset of £M(VIo)) becomes a saddle.

111

• All equilibrium points corresponding to models with a non-zero matter fluid, including those corresponding to the flat FRW perfect fluid (Einstein-de Sit­ter) solution and the flat FRW scaling solution, are saddles.

We have discussed some of the general properties of the class of open Bianchi VIo universe models with barotropic matter, a scalar field with an exponential potential and a uniform magnetic field by utilyzing dynamical systems techniques. The equilibrium point P (with M = 0), corresponding to the power-law inflationary flat FRW model, is a global sink for k2 < 2. Hence all models are future aymptotic to this inflationary attractor for these parameter values. There is an equilibrium set K with 1 - E~ - E=- - \Il2 = 0 (and M = 0, q = 2), a subset of which are global sources. Hence all models are past asymptotic to massless scalar field models with no matter and no magnetic field. A (partial) analysis of the saddles was undertaken in order to determine some of the transient features of the models. In particular, we found that there are no equilibrium points with M -# 0 in the Bianchi VIo state space which are inflationary, and hence concluded that inflation is not fundementally affected by the presence of a uniform primordial magnetic field in these models.

We have also made some passing comments concerning the intermediate be­haviour of the models, but we have not discussed this in detail. The transient behaviour is important for assessing the physical significance of the models. For example, LeBlanc et al. [226] also considered whether any of the Bianchi VIo mag­netic cosmologies are compatible with observations of a highly isotropic universe. For a subset of the Bianchi VIo cosmologies, the orbits will initially be attracted to the equilibrium point :F corresponding to the flat FLRW model, but will even­tually be repelled by it. For such a model there will be an epoch during which the model is close to isotropy and hence will be compatible with observations. This is the phenomenon of intermediate isotropization. It will not occur for a randomly selected magnetic Bianchi VIo model, but there will be a finite probability that an arbitrarily selected model will undergo intermediate isotropization. In the next Subsection we will discuss the intermediate behaviour of the (subset) of Bianchi I models in more detail.

B. Bianchi I Models

The evolution equations for the Bianchi I models are obtained by restricting the above equations to the zero-curvature case (Le., N± = 0), whence we obtain

X = (E+, E_, \II, q>, 0) E ]R5, (9.15)

and the evolution equations are as follows:

E~ E~

\II'

(q - 2)E+ + 2(1 - \Il2 - q> - E! - E=- - 0), (q - 2)E_,

y'6k (q - 2)\II - -2-<I>,

(9.16)

(9.17)

(9.18)

112

<1'>' = (2q + 2 + V6k"lll)<1'>, 0' = (2q - 37 + 2)0,

(9.19)

(9.20)

where a prime denotes differentiation with respect to the time 7" and the deceleration parameter q is now given by

1 q = 1 + E~ + E:' + 2(37 - 4)0 +"1112 - 2<1'>.

The magnetic field satisfies

~ M2 = 1 - "1112 - <1'> - E2 - E2 - 0 2 + - ,

(9.21)

(9.22)

and since M2 ~ 0, all of the variables are bounded: 0 ~ {E~, E:', "1112, <1'>, O} ~ 1.

We have followed the dynamical systems theory approach of [226] to analyse the evolution of orthogonal Bianchi cosmologies of type VIo with a uniform magnetic field. This approach uses an invariant orthonormal frame in which the commutation functions are the basic variables. Expansion-normalized variables are then intro­duced leading to a reduced set of evolution equations for the dimensionless shear variables, the spatial curvature variables, the dimensionless magnetic field variables and the density parameter. However, a coordinate approach is also possible. For ex­ample, the metric for the class of anisotropic Bianchi I models which have non-zero shear but zero three-curvature is given by

ds2 = -dt2 + X 2dx2 + 'y2dy2 + Z 2dz2,

where X, Y, Z are functions of t only. The expansion rate is given by

X Y Z o =- 3H = X + Y + Z'

(9.23)

(9.24)

and when Y = Z (the LRS subcase) there is only one independent rate of shear, which is given by

a=~(~-;). (9.25)

Normalized (dimensionless) variables and a logarithmic time variable 7" can be de­fined, which then lead to the evolution equations (9.16)-{9.21) in coordinate form, analogous to the system studied by Collins [109].

1. Qualitative Analysis

The existence of monotonic functions rule out periodic and recurrent orbits in the phase space and enables global results to be obtained. When 0 =1= 0, we can define Z =- OE=2, whence from the evolution equations above we find that

Z' = 3(2 - 7)Z, (9.26)

113

so that Z is a monotonically increasing function for "( < 2 [226]. From the mono­tonicity principle we can deduce that n --> 0 at early times and ~~ --> 0 at late times. When n = 0, we note that

q - 2 = 3 (M2 +~! + ~~ + '112 - 1) :S 0, (9.27)

and so from equation (9.17) the variable ~_ is itself a monotonic function.

There are a number of invariant sets. M = 0 is an invariant set corresponding to the absence of a magnetic field. The dynamics in this invariant set was studied in [50]. '11 = !I> = 0 is an invariant set corresponding to the absence of a scalar field. The dynamics in this invariant set was studied in [109]. n = 0 is an invariant set corresponding to the absence of barotropic matter. This invariant set is impor­tant for studying the the early time asymptotic behaviour. Finally, ~_ = 0 is an invariant set which is important in the study of late time asymptotic behaviour.

Let us next present all of the equilibrium points and their associated eigenvalues. We shall assume here that 1 :S "( < 2. The case of the bifurcation value "( = 2 will be discussed in [301]. All equilibrium points correspond to self-similar cosmological models.

Zero magnetic field: Equilibrium Points {~+, ~_, w,!I>, n}.

• P - Power-law inflation (flat FRW):

{ 0,0, - kv6, 1 - k 26, o}, M = 0, q = 12k2 - 1.

Eigenvalues· [k2 - 4 k 2 - 3"( !k2 - 3 !k2 - 3 !k2 - 3] • , , 2 ' 2 ' 2 Exists provided k 2 :S 6. Sink if k 2 < 4 and k 2 < 3"(; saddle otherwise. Inflationary if k 2 < 2 (note that this is the only equilibrium point that can be inflationary) .

• Fs - Matter-scaling (flat FRW [116]):

{ v'6 "( 3"( k2 - 3"( } 0,0'-2k' 2k2 (2-"(), k 2 '

3 M = 0, q = "2"( - 1 > O.

Eigenvalues: [~("( - 2), ~("( - 2),3"( - 4,

~ ("( - 2) ± 4t J(2 - "()(24"(2 + k2 [2 - 9"(])]

Exists provided k2 > 3"(. Sink if "( < ~; saddle otherwise.

• F - Einstein-de-Sitter (flat FRW):

3 {0,0,0,0,1}, M=O, q="2,,(-1>0.

Eigenvalues: [~("( - 2) , ~ ("( - 2) , ~ ("( - 2) , 3,,(, 3"( - 4] Saddle.

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• K - Kasner vacuum/massless scalar field:

{~~ +~:. + W2 = 1, <I> = n = O}, M = 0, q = 2.

Eigenvalues: [0,0,3(2 - ,),),2 - 4~+, 6 + V6kWJ

This is a 2-parameter family of equilibrium points. Sources when ~+ < ~; saddles otherwise.

Non-zero magnetic field: These all correspond to anisotropic models (~~ +~:.. i= 0).

• J - Matter and no scalar field (Jacobs magnetic field model [201]):

{I 3 ( I)} 2 1. 3 4(3,),-4),0,0,0'2 1- 4 ')' ,M =8(2-')')(3')'-4), q= 2')'-1 >0.

Eigenvalues: [~(')' - 2), ~ (')' - 2) ,3,)" ~ {(')' - 2) ± J(2 - ')') (3')'2 - 17')' + 18) }]

Exists provided')' > 1 (same as F for,), = 1)' Saddle.

• S - Matter and scalar field:

{ I V6')' 3')' 3 (2 2 ) } 4(3,),-4),0,- 2:k' 2k2 (2-,),), 8k2 4k -k ')'-8')' ,

2 1 3 M = - (2 - ')')(3')' - 4), q = -')' - 1 > 0.

8 2

Eigenvalues: [~(')' - 2), ~ {2 (')' - 2) ± kJ2 (2 - ')')(a ± b)}] where a 2412 + 312 k 2 - 26k21 + 20k2, and b J(18k2,), - 4k2 + a - 32k')') (18k2')' - 4k2 + a + 32k')').

Exists provided')' > 1 (same as Fs for,), = 1) and ')' ~ k~~8' Sink.

• V - Scalar field and no matter:

{ 2(k2 - 4) ° 2V6k 12(8 - k2) o}

k2 + 8 ' ,- k 2 + 8' (P + 8)2' ,

M2 _ 2(k2 - 4)(8 - k 2) _ 5k2 - 8 - (k 2 + 8) , q - k2 + 8 > 0.

E· I' [3(k2-8) 3(k2-8) 3(4k2-k21'-81') 3 {k2-8 V 17k2-72}] Igenva ues. k2+8' k2+8' k2+8 ':2 k2+8 ± k2-8 .

Exists provided 4 ~ k 2 ~ 8. Sink when,), > k1~8; saddle otherwise.

We shall now discuss the qualitative properties of the models. In Table XV we summarize the equilibrium points and their stability for different values of the parameters (,)" k2 ). The equilibrium point P represents the power-law FRW solution

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(a) 1::0; "I ::0; ~

I k 2 1 3"1 4 6 8

P SINK I SADDLE I DOES NOT EXIST

Fs DOES NOT EXIST 1 SINK

F SADDLE

V DOES NOT EXIST I SADDLE I DOES NOT EXIST

(b) ~ < "I < 2

I k 2 1 4 3"1 6 8"1/(4 - "I) 8 -_ .. _------ - -------

P SINK I SADDLE I DOES NOT EXIST

Fs DOES NOT EXIST I SADDLE

F SADDLE

:r SADDLE

S DOES NOT EXIST I SINK

V DOES NOT EXIST I SINK I SADDLE I DOES NOT EXIST

TABLE XV: Summary of all of the equilibrium points (as defined in the text) and their nature for different {T, k2 }-parameter values in the Bianchi I models with (a) 1 :::; , :::; ~ and (b) , > ~. Note that for ~ <, < 2, we have that 4 < 3, < 8,(4 _,)-1 < 8, and in (b) we have identified 8,(4 _,)-1 > 6 (Le., , > ¥) to be specific. For all values of" a subset of K: are sources (the remainder are saddles).

with no matter and no magnetic field, and is inflationary for k 2 < 2. It is a global attractor for k2 < 3"( and k 2 < 4 in the Bianchi I invariant set. Note that P is a sink in the full class of Bianchi VIo models, unlike all other sinks which are only attractors in the Bianchi I invariant set. Also note that the monotonic function Z in the Bianchi I invariant set indicates that n ~ 0 at early times (at the sources) and E_ ~ 0 at late times (at the sinks), consistent with the results in Table XV. The equilibrium points S and V, in which all of the matter fields are non vanishing and in which the matter field (only) vanishes, respectively, correspond to new exact self­similar magnetic field cosmological models. The new solutions were given explicitly in [88].

From the the Table we can deduce both the asymptotic and the intermediate behaviour of the Bianchi I models. Heteroclinic sequences (and hence the transient behaviour of the Bianchi I models) can easily be deduced from the Table. As a particular example, for "( = 5/3 and 5 < k2 < 40/7, a subset of K constitute sources (the remainder are saddles), V is a sink, J is a saddle (that lies in the 'II = 0, <I> = 0 (M = 0) submanifold) and F, P and Fs are saddles (that lie in the isotropic E+ = 0, E_ = 0 (M = 0) submanifold - in this submanifold K is a source, F and P are saddles, and Fs is a sink). Note that as "( increases, the magnetic field becomes increasingly important at late times, consistent with the observation of Wainwright [360] (in the absence of a scalar field).

X. STRING COSMOLOGY

In conventional cosmology in which the universe is expanding, as we extrapolate back in time towards an initial big bang singularity ever higher energies are reached until the classical theory breaks down and a more fundamental theory is necessary. Superstring theory represents the most promising candidate for a unified theory of the fundamental interactions, including gravity [164, 298]. There are known to be five consistent anomaly-free, supersymmetric perturbative string theories in ten dimensions [164], known as the type I, type IIA, type lIB, SO(32) heterotic and Es x Es heterotic theories. There is now evidence that these theories are related by a set of dualities and may in fact represent different manifestations of a more fundamental quantum theory, often termed M-theory [376]. Supersymmetry implies that the quantization of the string is only consistent if spacetime is ten-dimensional. On the other hand, M -theory, defined originally in terms of the strongly coupled limit of the type IIA superstring, is an eleven-dimensional theory. It is widely believed that eleven-dimensional supergravity represents the low-energy limit of M-theory [337, 376].

The cosmological implications of string theory are currently receiving consider­able attention, inspired in part by the recent advances that have been made towards a non-perturbative formulation of the theory. In the low-energy limit, to lowest­order in both the string coupling and the inverse string tension all massive modes in the superstring spectrum decouple and only the massless sectors remain, which are determined by the corresponding supergravity actions. A definitive prediction of string theory is the existence of a scalar field, known as the dilaton, which couples directly to matter in the Neveu-Schwarz/Neveu-Schwarz (NS-NS) sector. There are two further massless excitations that are common to all five perturbative string the­ories, namely the metric tensor field (the graviton) and a rank two anti-symmetric tensor field. The cosmological consequences of the dilaton field, which can be treated as a massless particle in perturbative string theory and determines the value of the string coupling parameter in the weak coupling regime, are very important. We note that the dilaton-graviton sector may be interpreted as a Brans-Dicke the­ory [60], where the coupling between the dilaton and graviton is specified by the Brans-Dicke parameter w = -1. However, such a theory of gravity is not consistent with experiments, so that the dilaton is one of the massless fields in the low-energy effective action which must evolve and attain an acceptable value at present.

The interactions between these fields lead to significant deviations from the stan­dard, hot big bang model based on conventional Einstein gravity [353] and a study of the cosmological consequences of superstring theory is therefore important. Indeed, early-universe cosmology provides one of the few environments where the predictions of the theory can be quantitatively investigated. One of the strongest constraints that a unified theory of the fundamental interactions must satisfy is that it leads to realistic cosmological models. It is also important to study the low-curvature regime and hence test the validity of the low-energy limit. In particular, we can in­vestigate whether cosmological models isotropize and/or homogenize to the future, and qualitative techniques are ideally suited for such a study, especially when the

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

117

low-energy effective action of the theory essentially reduces to GR plus massless scalar fields. In addition, we can investigate the issues of initial conditions and the singularity problem, particularly within the pre-big bang scenario [151, 353]. It is also of interest to study, in light of recent cosmological high redshift observations [239, 296, 305] which indicate that a vacuum energy density may be dominating the large-scale dynamics of the universe at the present epoch, whether the scalar field (dilaton) can give rise to "quintessence" in a string theory context [150]. Fi­nally, if sufficient cosmological inflation occurs, in which the universe undergoes an epoch of accelerated expansion in the past (see, for example, the recent reviews [228, 232, 259]), many of the problems of the hot big bang model can in principle be resolved and a causal mechanism for generating a primordial spectrum of density inhomogeneities is provided. Therefore, one of the most crucial questions that must be addressed in string cosmology is whether the theory admits realistic inflation­ary solutions. Indeed, it is known that supergravity corrections make it difficult to obtain sufficiently flat potentials to drive conventional slow-roll inflation [259].

There exist five supersymmetric string theories that each have a consistent weak coupling expansion in perturbative theory (introductory reviews to perturbative string theories and a full bibliography can be found in [164, 233, 298]). The (ten­dimensional, D = 10) type I theory and the two heterotic theories have N = 1 supersymmetry. The type II theories have N = 2 supersymmetry. The effective bosonic action of the type IIA and lIB superstrings is N = 2, D = 10 (non-chiral and chiral, respectively) supergravity. The Ramond-Ramond (RR) sector contains antisymmetric form potentials, which do not couple directly to the dilaton. The NS-NS sector for both type II theories has the same form. Heterotic and type II superstrings are oriented, unbreakable and closed, and consequently open strings are only possible in the type I theory. There is no freedom for introducing a super Yang-Mills gauge group in the type II theories and the only gauge group that can be consistently introduced in the type I theory is SO(32). The heterotic theories can admit both SO(32) and Es x Es (only). The heterotic and type I theories have the same particle content. Their effective actions differ, however, because all bosonic degrees of freedom couple directly to the dilaton field in the heterotic theory, whereas the 2-form potential is a RR degree offreedom in the type I theory. Eleven-dimensional (D = 11) N = 1, supergravity is closely related to the ten­dimensional theories discussed above [122-124]. Supersymmetry leads us to an upper limit D ~ 11 on the dimensionality of a Lorentzian spacetime [282]. The unique theory in eleven dimensions is N = 1 supergravity, in which the graviton and 3-form potential constitute the entire bosonic sector and the only parameter in the theory is the eleven-dimensional Newton constant which is related to the Planck length [81]. A Chern-Simons term may also arise as a direct consequence of the supersymmetry [124], although such a term can be neglected for the cosmological backgrounds we consider here.

In order to discuss conventional cosmological models in these theories, we must first consider their compactification down to four dimensions. In Kaluza-Klein di­mensional reduction, the universe is a product space M = .:J x /C, where .:J is the 4-dimensional space-time and the internal space is denoted by /C. If /C is a d-dimensional torus, T d , topologically it is the Cartesian product S1 x S1 X .•. X S1

118

and it is metrically flat. When the higher-dimensional metric is compactified on T d ,

the higher-dimensional graviton produces a graviton and d(d + 1)/2 spin-O fields (and d vector fields). In addition, an NS-NS p-form compactified on Td produces a total of d!/[P!(d-p)!] scalar moduli fields from its internal components. The toroidal compactification of the NS-NS sector of the string effective action which contains the dilaton field, the graviton and a 2-form potential and is common to both the type II and heterotic theories, was discussed in [233]. Finally, in four dimensions there exists a duality between the NS-NS 3-form field strength and a one-form which may be interpreted as the gradient of a pseudo-scalar axion field. Applying this duality to the toroidally compactified NS-NS action the field equations may then be derived from the dual effective action in terms of the pseudo-scalar ax­ion field, a "shifted" dilaton field (which parametrizes the effective D-dimensional string coupling) and ,p moduli fields arising from the internal degrees of freedom (which act collectively as a set of massless scalar fields). For 4-dimensional spa­tially homogeneous cosmologies, the dynamics of the external spacetime can often be determined in terms of a single modulus field, (3, which is the radius or "breath­ing mode" representing the volume of the internal dimensions (formally obtained from the compactification of the higher-dimensional effective NS-NS action onto an isotropic d-torus, where the components of the 2-form potential on the internal space are assumed to be trivial). Non-trivial RR fields can also have significant effects on the dynamics of the universe, and the truncated versions of type IIA and type IIB effective actions in four dimensions has also been derived. The effective actions for all superstring theories (and the resulting FE) are given explicitly in [233] (where the original references are cited).

In the rest of this book we shall adopt the following conventional notation. The spacetime metric has signature ( -, +, ... , +) and variables in eleven dimensions are represented with a circumflex accent. Upper case, Latin indices with circumflex accents take values in the range A = (0,1, ... ,10), and those without a circumflex accent vary from A = (0,1, ... ,9), lower case Greek indices span J.t = (0,1,2,3) and lower case Latin indices represent spatial dimensions only. A totally antisymmetric p-form is defined by Ap = (I/p!)AA1 ... ApdxA1 /\ ... /\ dxAp and the corresponding field strength is given by Fp+l = dAp = [I/(P + I)!]FA1 ... Ap+1 dXA1 /\ ... /\ dxAp+1. The coordinate of the eleventh dimension is denoted by Y. The eleven-dimensional Planck mass is the only dimensional parameter in this theory [81], and units are chosen such that 1671"0 = 1.

1. Low-Energy Effective Action

In a string theory context the macroscopic and large scale gravitational inter­actions are described by a low-energy effective action which necessarily contains a scalar field, the dilaton, which controls the strength of the gravitational cou­pling and, in superstring models of unification, the strength of other interactions. The evolution of the very early universe below the string scale is determined by ten-dimensional supergravity theories [71, 148, 164, 241]. All theories of this type

119

contain a scalar dilaton, a graviton and an antisymmetric 2-form potential in the NS-NS bosonic sector [233]. If we consider a Kaluza-Klein compactification from ten dimensions onto an isotropic six-torus of radius ef3, the low-energy limit of string theory can be described by an effective field theory whose action is given by

s = J d4xFge-q, [R + (V{fI)2 - 6 (V {3)2 - 112 H",v>..H"'V>"] , (10.1)

where the moduli fields arising from the compactification of the form-fields on the internal dimensions and the graviphotons originating from the compactification of the metric have been neglected [76, 266, 375]. In the action (10.1), R is the Ricci curvature of the spacetime with metric g",v and 9 == detg",v, the dilaton field, (fI,

parametrizes the string coupling, g; == e<I>, and H",v>.. == B[",Bv>..) is the field strength of the 2-form potential B ",v' The volume of the internal dimensions is parametrized by the modulus field, {3.

In four dimensions, the 3-form field strength is dual to a one-form:

H"'v>" == e<I> €",v>"K.V K.a, (10.2)

where €",v>"K. is the covariantly constant 4-form. In this dual formulation, the field equations (FE) can be derived from the action

s= J d4xFge-<I> [R+(V{fI)2_6(V{3)2_~e2<I>(Va)2], (10.3)

where a is interpreted as a pseudo-scalar "axion" field. It can be shown that the action (10.3) is invariant under a global SL(2, R) transformation on the dilaton and axion fields [320, 323]. The general FRW cosmologies derived from equation (10.3) have been found by employing this symmetry [114]. However, the symmetry is broken when a cosmological constant is present [215] and the general solution is not known in this case. We shall determine the general structure of the solutions for string cosmologies that contain a cosmological constant in the effective action.

A cosmological constant may arise in a number of different contexts. The NS-NS sector of massless bosonic excitations is common to both the type II and heterotic superstring theories. The low-energy limit of the massive type IIA supergravity theory in ten dimensions [308] has been the subject of renewed interest recently [41, 162, 192, 274, 298, 300). In this theory, the NS-NS 2-form potential becomes massive. In the absence of such a field, the action is given in the string frame by [41]

f 10 ~ { <I> [ )2] 1 AB 1 ABCD 1 2} S= d Xv -glO e- 10 RIO + (V{fI1O -4FABF - 48FABCDF -2m, (lOA)

where {fI1O represents the ten-dimensional dilaton field and m2 is the cosmological constant. The antisymmetric field strengths FAB and FABCD for the one-form

120

and 3-form potentials represent RR degrees of freedom because they do not couple directly to the dilaton field [164]. The massless theory is recovered when m = O. Maharana and Singh [267] have employed Kaluza~Klein techniques [266, 318] to compactify the ten~dimensional theory (lOA) onto a six~dimensional torus. We consider a truncation of the dimensionally reduced 4-dimensional action, where we neglect the moduli fields arising from the compactification of the form fields and the gauge fields originating from the higher~dimensional metric. The effective 4-dimensional action is then given by

s = I crxyCg {e~<p [R + (\7q»2 - ~ (\7,8)2] - :8eV3,B FILVA/<FILVA/< - ~m2eV3,B }, (10.5)

where the 4-dimensional dilaton, «1>, is defined in terms of the ten~dimensional dilaton by «I> == «1>10 - ..;3,8 and ,8 is a rescaled modulus field.

The Bianchi identity for the 3-form potential is trivially satisfied in four dimen­sions and its field equation is solved by the ansatz FILVA/< = Qe~6"1 EILVA/<, where Q is an arbitrary constant. Applying this duality, together with the conformal transformation

- -2 glLV = n glLv , 02 == e~<P (10.6)

implies that the effective 4-dimensional action may be expressed in the Einstein frame as

s = I d4xR [R - ~ (V «I> r -~ (V,8 r -~Q2e2<P~V3,B - ~m2e2<P+V3,B] .

(10.7) Without loss of generality, linear translations on the dilaton and modulus fields can be performed such that the parameters Q and m become effectively equal. Thus, the last two terms in the action (10.7) may be written in the form, - AMe2<P cosh( ..;3,8), where AM is a positive-definite constant. It follows that the 4-form and ten~ dimensional mass parameter together provide an effective potential for the modulus field with a global minimum located at ,8 = O. Consequently, the internal space can become stabilized for this specific compactification and a consistent truncation of the effective action is therefore given by specifying ,8 = 0 in the above action. Applying the inverse of the conformal transformation (10.6) then implies that the effective string frame action is given by equation (10.5) when the axion field is trivial. Thus, a truncated form of equation (10.5) is relevant to the type IIA theory.

Therefore, we consider the effective action

s = I d4xyCg {e~<p [R + (\7«1»2 - 6 (\7,8)2 - ~e2<P (\70')2 - 2A] - AM}'

(10.8) The constant, A, is determined by the central charge deficit of the string theory and may be viewed as a cosmological constant in the gravitational sector of the theory. In principle, it may take arbitrary values if the string is coupled to an appropriate conformal field theory. Such a term may also have an origin in terms

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of the reduction of higher degree form-fields [210]. The constant, AM, represents a phenomenological cosmological constant in the matter sector. Since it does not couple directly to the dilaton field, it may be viewed in a stringy context as a RR degree offreedom (a O-form) [251] and it may be interpreted as the potential energy of a scalar field that is held in a false vacuum state.

We shall include the combined effects of the axion, modulus and dilaton fields, thereby extending previous qualitative analyses where one or more of these terms was neglected [136, 159, 212, 213]. We shall rewrite the FE in terms of a set of bounded variables and perform a stability analysis. In FRW string models the spatial gradient of the pseudo-scalar axion field is zero due to spatial isotropy. In spatially homogeneous and anisotropic cosmologies, namely Bianchi and Kantowski­Sachs models, this is no longer necessarily true, although the components of the 3-form field strength must depend only on time in order for the energy-momentum tensor to be spatially homogeneous. In the simple spatially flat Bianchi type I model, in which the non-trivial components of the 2-form potential and the space­time metric are independent of the spatial coordinates, an effective one-dimensional action can be derived by integrating the action over the spatial variables.

A. Cosmological Field Equations

When AM = 0, the spatially flat FRW cosmological FE derived from action (10.8) are given by

2ii - 2o.cj; - &2 e2'P+60! = 0 (10.9)

2cp - cj;2 - 30.2 - 6/32 + ~&2e2'P+60! + 2A = 0 2

(10.10)

~ - /3cj; = 0 (10.11)

a+&(cj;+6o.) = 0, (10.12)

where r.p == q, - 30: defines the "shifted" dilaton field, R(t) == eO! is the scale factor of the universe and a dot denotes differentiation with respect to cosmic time, t. The generalized Friedmann constraint equation is

30.2 - cj;2 + 6/32 + ~&2e2'P+60! + 2A = o. 2

1. Exact Solutions

(10.13)

A number of physically important exact solutions to equations (10.9)-(10.13) are known when one or more of the degrees of freedom are trivial; these solutions lie in the invariant sets of the full phase space. The "dilaton-vacuum" solutions, where

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only the dilaton field is dynamically important, are given by [151]

R = R*jtj1/P±, e4> = e4>'jtjP±-l, (10.14)

where p± == ±v'3 and {R*, cl>*} are arbitrary constants. There is a curvature singularity in these solutions at t = O. In the pre-big bang inflationary scenario, the pre-big bang phase corresponds to the range t < 0 and the post-big bang phase to the solution for t > O.

The "dilaton-moduli-vacuum" solutions have II = A = 0, and are given by

R = R* jtj±h., e4> = e4>· jtj±3h. -1 ,ef3 = ef3. jtj±y'(1-3h;)/6 , (10.15)

where {h*,,8*} are constants. The general solution with A = 0 is the "dilaton­moduli-axion" solution [114]

R ~ R.I:.I'i' [1:.1' + I:. n 'i',

e4> = e;. [I~ Ir + /~ /-r] , _ ± -4>. [jT/T*j-r -jT/T*jr]

(Y - (Y* e j / j / ' T T* -r + jT T* jr

ef3 = ef3./ ~ / q , (10.16)

where (Y* is an arbitrary constant and T is conformal time; this solution asymptot­ically approaches a dilaton-moduli-vacuum solution (10.15) in the limits of high and low spacetime curvature. The axion field, which is only dynamically important for a finite time period, induces a smooth transition between these two power-law solutions.

The solutions where only the axion field is trivial and A > 0 are special cases of the "rolling radii" solutions [278]

R ef3

e-4>

R* jtanh(At/2)jm ,

ef3• jtanh(At/2)jn ,

e-4>'jcosh(At/2)j2k-6n jsinh(At/2)j21+6n ,

where A == ...j2A and the real numbers {k, I, m, n} satisfy the constraints

3m2 + 6n2 = 1, 3m + 6n = k - I, k + 1= 1.

(10.17)

(10.18)

The corresponding solutions for A < 0 are related by a redefinition. Finally, there exists the "linear dilaton-vacuum" solution where A > 0 [5, 6, 281]. This solution is static and the dilaton evolves linearly with time:

R=O, /3 = 0, cl> = ±V2At. (10.19)

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2. Pre-Big Bang Cosmology

Veneziano and Gasperini were the first to develop a comprehensive cosmological scenario based on the underlying string symmetries [151, 353). In this scenario, which is called pre-big bang cosmology since in this approach the very early universe before the big bang can be discussed, there are inflationary solutions that are driven by the kinetic energy associated with the massless fields rather than any interaction potential. There are many massless fields present in the pre-big bang scenario, each producing their own spectrum of perturbations, but it has been shown that the models may be consistent with the current constraints derived from observations of large-scale structure and cosmic microwave background anisotropies. However, there are a number of new problems that also appear in the pre-big bang scenario [233).

When the cosmological constant is zero, the governing FE derived from the string effective action are symmetric under time reversal and invariant under a discrete Z2 transformation (when the axion field is trivial), leading toa scale factor dual­ity [353). Applying the duality transformation simultaneously with time reversal implies that the Hubble expansion parameter H remains invariant while its first derivative changes sign, with the result that a decelerating, post-big bang solution is therefore mapped onto a pre-big bang phase of inflationary expansion, which is super-inflationary since the Hubble radius H- 1 decreases with increasing time (within the pre-big bang context, there is accelerated expansion in the weak cou­pling regime in the (-)-branch (Le., taking h* < 0 in (10.15)) of the dilaton-moduli­vacuum solutions). The solutions in each branch have semi-infinite proper lifetimes. The time derivative of the shifted dilaton is always positive on the (+ )-branch and always negative on the (-)-branch [63). On either of the branches the solutions (a one-parameter family of dilaton-moduli-vacuum cosmologies) become singular as the conformally invariant time parameter approaches zero. However, it is not clear how the transition between the pre- and post-big bang phases might proceed due to the fact that the two branches are separated by a curvature singularity. This is the graceful exit problem of the pre-big bang scenario [63).

There are also a number of important (and currently unresolved) issues concerning fine-tuning of initial conditions in the pre-big bang scenario [84, 211, 233, 344). There must be enough inflation in a homogeneous patch in order for the horizon and flatness problems to be solved, and hence the dilaton driven inflation must survive for a sufficiently long period of time. Assuming for simplicity that the moduli fields are trivial, the fundamental postulate of the scenario is that the initial data for inflation lies well within the perturbative regime of string theory, where the curvature and coupling are very small [151). Inflation then proceeds for sufficiently homogeneous initial conditions [65, 354), where time derivatives are dominant with respect to spatial gradients, and the universe evolves into a high curvature and strongly-coupled regime. Thus, the initial pre-big bang universe should correspond to a cold, empty and flat vacuum state and would have been very large relative to the quantum scale, and hence should have been adequately described by classical solutions to the string effective action. A critical look at the initial conditions in

124

the pre-big bang universe has been made and it has been argued that the Milne universe is an unlikely past attractor for the pre-big bang scenario [85] and that plane wave backgrounds represent a more generic initial state [84].

Therefore, in the pre-big bang cosmological solutions the universe starts very close to the cold, empty and flat perturbative vacuum. This is in contrast to the initial state in the standard cosmology in which the early universe was in a hot, dense, and high curvature state (which is believed to provide a very good description of the recent evolution of the universe). Thus, it is necessary to understand how the the initial conditions associated with the pre-big bang could evolve naturally into those of the standard scenario at some later time, smoothing out the big bang singularity. The issue of the graceful transition from the dilaton-driven kinetic inflation phase to the subsequent standard radiation dominated evolution has been addressed by a number of authors (see references in [233] and [152]).

Of course, the ultimate test of a cosmological scenario is whether it is consistent with observations. While the solutions for the homogeneous dilaton, axion and scale factor may lead to interesting behaviour in the early universe, the success of the standard big bang model suggests that the evolution should closely approach the conventional general relativistic evolution at least by the time of nucleosynthesis. Any trace of the earlier evolution will be in the primordial spectrum of inhomo­geneities present on large-scales that we observe today. Large-scale structure can be generated in a pre-big bang epoch. Hence it is important to determine whether the pre-big bang scenario is consistent with the current constraints derived from ob­servations of large-scale structure and cosmic microwave background anisotropies. The production of scalar and tensor metric perturbations in the pre-big bang sce­nario has been studied by various authors (see, for example, references in [233] and [152]) using the Mukhanov formalism to describe inhomogeneous vacuum fluctua­tions about a homogeneous background [279, 280].

B. Qualitative Analysis of the NS-NS Sector

For an arbitrary central charge deficit, the FE (10.9)-(10.13) may be written as an autonomous system of ODE:

iI ¢ N P

3H2

'Ij} + H'lj; - 3H2 - N - 2A

3H2 +N

2N'lj;

-6Hp 1

'lj;2 + N + 2 p + 2A = 0,

where we have defined the new variables

N == 6~2, p == &2 e2<p+6a 'lj; == <p,

(10.20)

(10.21)

(10.22)

(10.23)

(10.24)

(10.25)

125

and H = 0: is the Hubble parameter. The variable p may be interpreted as the effective energy density of the pseudo-scalar axion field [212]. It follows from equa­tion (10.21) that 'IjJ is a monotonically increasing function of time and this implies that the equilibrium points of the system of ODE must be located either at zero or infinite values of 'IjJ. In addition, due to the existence of a monotone function, it follows that there are no periodic or recurrent orbits in the corresponding phase space [169, 363]. The sets A = 0 and p = 0 are invariant sets. In particular, the exact solution for A = 0 divides the phase space and orbits can not cross from positive to negative A.

We must consider the cases where A < 0 and A > 0 separately. In the case where the central charge deficit is negative, A < 0, it proves convenient to employ the generalized Friedmann constraint equation (10.24) to eliminate the modulus field. We may compactify the phase space by normalizing with J 'ljJ2 - 2A and we define a new time variable by

d 1 d dT = J'ljJ2 - 2Adt'

(10.26)

The governing equations reduce to a 3-dimensional system of autonomous ODE. The equilibrium points all lie on one of the two lines of non-isolated equilibrium points (or one-dimensional equilibrium sets) L±. On the line L + the equilibrium points are either saddles or local sinks, and on the line L - the equilibrium points are local sources or saddles. The dynamics is very simple due to the existence of (two) monotonically increasing functions. A full analysis is given in [47]. Henceforward, let us consider the case A > O.

1. Models with Positive Central Charge Deficit

In the case where the central charge deficit is positive, A > 0, we choose the normalization

( 1 ) 1/2

10 == 3H2 + "2/ + N + 2A (10.27)

The generalized Friedmann constraint equation (10.24) now takes the simple form

'ljJ2 = 1 102

(10.28)

and may be employed to eliminate 'IjJ. Since by definition 10 2 0, specifying one of the roots 1/J/€ = ±1 corresponds to choosing the sign of 'IjJ. However, it follows from the definition in equation (10.25) that changing the sign of'IjJ is related to a time reversal of the dynamics. In what follows, we shall consider the case 1/J/€ = +1; the case 1/J / 10 = -1 is qualitatively similar.

126

Introducing the new normalized variables

x=: V3H E '

and a new dynamical variable

O=:~ 2E2 '

dId dT = V3E dt'

N z =: E2 (10.29)

(10.30)

equations (10.20)-(10.23) are transformed to the 3-dimensional autonomous system [47]:

dx - x dT = 0 + V3 [1 - x2 - z] (10.31)

dO -[ 1 ] dT = -20 x+ V3(Z+X2) (10.32)

dz 2 dT = V3 z (1 - x2 - z) . (10.33)

The phase space variables are bounded, 0 ::; {x2 , 0, z} ::; 1, and satisfy x2 + 0 + z ::; 1. The sets 0 = 0 and z = 0 are invariant sets corresponding to p = 0 and /J = 0, respectively. In addition, x2 + 0 + z = 1 is an invariant set corresponding to A = O. We note that the right-hand side of equation (10.33) is positive-definite and this simplifies the dynamics considerably.

The equilibrium points of the system (10.31)-(10.33) consist of the isolated equi­librium point

C: x = 0 = z = 0, (10.34)

and the line of non-isolated equilibrium points

L + : 0 = 0, z = 1 - x2 , (10.35)

where x is arbitrary. The eigenvalues associated with C are Al = 1/V3, A2 = 2/V3 and A3 = O. Since C is an isolated equilibrium point, it is therefore non-hyperbolic. However, a simple analysis shows that it is a global source. The eigenvalues associ­ated with L+ are:

AI=-2(X+ ~), 2 A2 = - V3 (10.36)

and the third eigenvalue is zero since L + is a one-dimensional set. Therefore, on L+ the equilibrium points are saddles for x E [-1, -1/V3) and local sinks for x E (-1/V3, 1]. The phase portrait is given in Fig. (13).

It is also instructive to consider the dynamics on the boundary corresponding to z = 0, since the case N = 0 is of physical interest in its own right as a 4-dimensional

x

z

i L+ ~ ~ Yi> o o

.-- 0 -=---=-

c

• •

----

127

---. Q

FIG. 13: The phase portrait of the system (10.31)-(10.33) corresponding to the ten­dimensional NS-NS model with positive central charge deficit (A > 0). The root 'Ij;/E = +1 of equation (10.28) has been chosen. Equilibrium points are denoted by dots and the labels in all figures correspond to those equilibrium points (and hence the exact solutions they represent) discussed in the text. We adopt the following convention: large black dots represent sources (Le., repellers), large grey-filled dots represent sinks (i.e., attractors), and small black dots represent saddles. Arrows on the trajectories have been suppressed since the direction of increasing time is clear using this notation. Grey lines represent typical trajectories found within the 2-dimensional invariant sets, dashed black lines are those trajectories along the intersection of the invariant sets, and solid black lines are typical trajectories within the full 3-dimensional phase space. Note that L± denote lines of non-isolated equilibrium points.

model. In this case the ODE reduce to the 2-dimensional system:

dx - x dT = 0 + y'3 (1 - x2) (10.37)

dO -[ 1] dT = -2xO 1 + y'3x . (10.38)

128

L+ (-)

g

i

c L + --1x (+)

FIG. 14: Phase portrait of the system (10.37)-(10.38) corresponding to the 4-dimensional NS-NS model with positive central charge deficit (A> 0). The TOot 'lj;/E = +1 of equation (10.28) has been chosen. See caption to Fig. (13).

The equilibrium points and their corresponding eigenvalues are:

x= n =0; 1

(10.39) C: Al = - A2 = 0 y'3'

Lt_): x = -l,n = 0; Al = - ~'A2 = 2 (1- ~) (10.40)

L+ . (+) . x = 1,n = 0; Al = - ~'A2 = -2 (1 + ~). (10.41)

Point C is a non-hyperbolic equilibrium point; however, by changing to polar coor­dinates we find that C is a repeller with an invariant ray 3H = tan- 1( -y'3). The saddle L t-) and the attractor L t+) lie on the line L +. The phase portrait is given in Fig. (14).

Let us briefly discuss the dynamics. In the case of a positive A, the isolated

129

equilibrium point C corresponds to the "linear dilaton-vacuum" solution (10.19) [212, 281]; see Figs. (13) and (14) When the modulus is frozen, all trajectories evolve away from C towards the point A and approach a superinfiationary dilaton-vacuum solution defined for t < O. Some of the orbits evolving away from C represent contracting cosmologies and the effect of the axion is to reverse the collapse in all of these cases. For the rolling modulus solutions (10.17), the orbits tend to dilaton-moduli-vacuum solutions (10.15) as they approach the attractors (the sinks on L+). The critical value x2 = 1/3 corresponds to the case where &2 = /J2, representing the isotropic, ten-dimensional cosmology (& = (3) and its dual solution (& = -(3). In the latter solution, the ten-dimensional dilaton field, <f? == <I> + 6{3, is constant. The other boundary of L+ is the point A representing the case where the kinetic energy of the modulus field vanishes. We note that the qualitative behaviour of models with 1/J < 0 is similar. The dynamics in the case A < 0 are summarized in [47]. The main feature is that the axion can be dynamically important and lead to a bouncing cosmology, in constrast to the case when no axion field is present in which there is no bounce [212, 278].

C. Qualitative Analysis of the Matter Sector

When a cosmological constant is introduced into the matter sector of equation (10.8) and the central charge deficit vanishes (Le., A = 0), the FE are given by

a &cj; + cj;2 _ 3&2 _ 6/32 _ ~ AMe'P+3Q 2

rp 3&2 + 6/32 + ~ AMe'P+3Q 2

a- -(cj; + 6&)&

jj /3cj;

and the generalized Friedmann constraint equation takes the form

3&2 _ cj;2 + 6/32 + ~&2e2'P+6Q + AMe'P+3Q - 0 2 - .

(10.42)

(10.43)

(10.44)

(10.45)

(10.46)

Equations (10.42)-(10.46) may be simplified by introducing the new time coordinate

d_ == e-('P+3Q)/2!i (10.47) dt dt

and employing the generalized Friedmann constraint equation (10.46) to eliminate the axion field. The remaining FE are then given by

a"

'P"

{3"

9 1 3 'P,2 - _a,2 + -a' 'P' - 6{3,2 - - AM

2 2 2

,2 '2 1 '2 3 " 1A 3a + 6{3 - -'P - -a 'P + - M 222

1 "2 {3' ('P' - 3a') ,

(10.48)

(10.49)

(10.50)

130

where in this Section a prime denotes differentiation with respect to i.

1. Positive Cosmological Constant

When AM > 0, we can rewrite equations (10.48)-(10.50) by defining

H::=c/, 'ljJ::= !p', N::= (3'. (10.51 )

Equation (10.46) then implies that

'ljJ2 ~ 3H2 + 6N2 + AM ~ 0, (10.52)

and consequently we may normalize using 'ljJ (compare with equations (10.27) and (10.28)). We therefore define

x::= V3H 'ljJ

6N2 z::= -

'ljJ2 AM

V ::= ---;jl2 dId

dT = "¢iE'

and assume that 'ljJ > 0 ('IjJ < 0 is again related to time-reversal).

The resulting 3-dimensional autonomous system is:

dx dT dz dT dv dT

(x + V3)[1 - x2 - Z - v] + ~v[x - V3] 2

2z { [1 - x2 - Z - v] + ~v }

2v {[I - x2 - Z - v]- ~(1 - v - V3X)}.

(10.53)

(10.54)

(10.55)

(10.56)

It follows from the definitions (10.53) that the phase space is bounded with 0 ::::: {x2,z,v} ::::: 1, subject to the constraint 1 - x2 - Z - v ~ o. The invariant set 1 - x 2 - Z - v = 0 corresponds to a zero axion field. The dynamics of the system (10.54)-(10.56) is determined primarily by the dynamics in the invariant sets z = 0 and v = o. These correspond to a zero modulus field and a zero AM, respectively. The dynamics is also determined by the fact that the right-hand side of equation (10.55) is positive-definite so that z is a monotonically increasing function. This guarantees that there are no closed or recurrent orbits in the 3-dimensional phase space.

In the invariant set z = 0, where the modulus field is trivial, the system (10.54)­(10.56) reduces to the following plane system:

dx dT dv dT

1 (x + V3)[1 - x 2 - v] + "2v[x - V3]

2v {[I - x 2 - v]- ~(1 - v - V3X)}.

(10.57)

(10.58)

131

The equilibrium points and their associated eigenvalues are given by

L+ . (-) . X = -l,v = 0; Al = 2(V3 - 1), A2 = -(1 + V3) (10.59)

L+ . (+) . x=l,v=O; Al = -2(V3 + 1), A2 = (V3 -1) (10.60)

1 i F: x = -3~'v = ~*; Al,2 ="3 ± gv'231. (10.61)

The points Lt_) and Lt+) are saddles and F is a repelling focus. The phase portrait

is given in Fig. (15). In the invariant set 1 - x 2 - V = 0, corresponding to the case of a zero axion field, equations (10.57) and (10.58) reduce to the single ODE

dx 1 dT = 2 (1 - x2 ) (x - V3) , (10.62)

which can be integrated to yield an exact solution (in terms of T time).

In the full system (10.54)-(10.56) with a non-trivial modulus field, there exists the isolated equilibrium point F: x = -1/(3V3),z = O,v = 16/27 with the associated eigenvalues Al,2 and A3 = 4/3. This implies that F is a global source. All of the remaining equilibrium points belong to the one-dimensional set

L+ : v = 0, Z = (1 - x 2 ) (10.63)

(x arbitrary), and the associated eigenvalues are given by

Al = -2V3 (x+ ~), A2 = V3 (x - ~) , (A3 = 0). (10.64)

L + lies in the invariant set v = 0 on the boundary z = 1 - x 2 • Points on L + with x E (-1/ V3, 1/ V3) are local sinks, while the remaining points are saddles in the full 3-dimensional phase space. (In the invariant set v = 0, equilibrium points with x E [-1, -1/V3) are repelling and those with x E (-1/V3, 1] are attracting). The phase portrait is given in Fig. (16).

We note that there exists an exact solution of equations (10.54)-(10.56) with non-trivial modulus field with

1 x = - 3V3'

z = _ 13 (v _ 16) 8 27'

whence

dv 9 ( 16) dT =:tv v - 27 ' (10.65)

which can be integrated in terms of T time explicitly [48].

132

v

I

+ LC-)

-----1 x +

LC+)

FIG. 15: Phase portrait of the 4-dimensional system (10.57)-(10.58), corresponding to the isotropic FRW RR model with AM > o. See caption to Fig. (13). Note that in this phase space orbits are future asymptotic to a heteroclinic cycle.

Finally, we could consider the case AM < o. The generalized Friedmann constraint equation (10.68) implies that

'l/J2 - AM ~ 3h2 ~ 0, (10.66)

and we could therefore normalize using y''l/J2 - AM. The resulting dynamical system can be analysed in a manner similar to that above (see [48]).

2. Discussion

The mathematical structure of the dynamics of the system (10.57)-(10.58) is much richer when there is a positive (RR) matter charge. The orbits are future asymptotic to a heteroclinic cycle, consisting of the two saddle equilibrium points Lt_) and Lt+) and the single (boundary) orbits in the invariant sets v = 0 and

133

v

i

I

x"- --. z

FIG. 16: Phase portrait of the system (10.54)-(10.56) corresponding to the ten-dimensional RR model with AM > o. See caption to Fig. (13). Note that W denotes a line of non­isolated equilibrium points. The dashed line represents the exact x = constant solution (10.65).

1- x 2 - V = 0 joining Lt_) and Lt+) (see Fig. (15)). The former set corresponds to the zero AM solution and the latter to the solution with constant axion field. In a given "cycle" an orbit spends a long time close to L t-) and then moves quickly to

L t+) shadowing the orbit in the invariant set v = o. It is then again quasi-stationary

and remains close to the equilibrium point Lt+) before quickly moving back to Lt_) shadowing the orbit in the invariant set 1 - x 2 - V = o. We stress that the motion is not periodic, and on each successive cycle a given orbit spends more and more time in the neighbourhood of the equilibrium points Lt_) and Lt+)" The exact solution corresponding to the equilibrium point F is power-law and is defined over the range -00 < t < 0, and represents a cosmology that collapses monotonically to zero volume at t = 0 in which the curvature and coupling are both singular. The cyclical nature of the orbits was interpreted physically in [48]. In effect, the cosmological constant resists expansion and causes the universe to recollapse (and asymptotically approach the saddle point Lt_)), whereby the collapse causes the

134

axion field to become relevant once more and a bounce ensues. The process is then repeated with the universe undergoing a series of bounces. The orbits move progressively closer towards the two saddles, Lt~), and spend increasingly more time near to these points. This behaviour may have implications in the pre-big bang inflationary scenario [48, 151J.

When a modulus field is included (z =I=- 0), F still represents the only source in the system. The orbits follow cyclical trajectories in the neighbourhood of the invariant set z = 0 and they spiral outwards monotonically around the orbit which corresponds to the exact solution given by equation (10.65), with dz/dT > 0 from equation (10.55). After a finite (but arbitrarily large) number of cycles the kinetic energy of the modulus field becomes increasingly more important until a critical point is reached where it dominates the axion and cosmological constant. The orbits then asymptote toward the dilaton-moduli-vacuum solutions corresponding to sources on the line L+ (see Fig. (16)).

A finite number of modulus fields and shear modes can be introduced into the model by defining the variable N 2 in equations (10.51) via N 2 = L~=l Nl (their inclusion could be important in the discussion of chaotic behaviour - see below). Orbits with non-trivial modulus field or shear term "shadow" the orbits in the invariant set z = 0 at early times and undertake cycles between the saddles on the equilibrium set L+ close to Lt_) and Lt+). These saddles may be interpreted as Kasner-like solutions [363J; the orbits thus experience a finite number of cycles in which the solutions interpolate between different Kasner-like states [80J.

Finally, we observe that all of the exact solutions corresponding to the equilibrium points of the governing autonomous systems of ODE are self-similar since in each case the scale factor is a power-law function of cosmic time. Therefore, exact self­similar solutions play an important role in determining the asymptotic behaviour in string cosmologies. However, we note that not all of the string solutions are asymptotically self-similar due to the existence of the heteroclinic cycle in the RR sector with trivial modulus field.

D. Cyclical Behaviour in Early Universe CosInologies

As noted above, the dynamics of the cosmologies with a non-trivial cosmological constant, AM (that does not couple directly to the dilaton field) exhibit some very interesting mathematical properties. Let us investigate this further. In particular, we shall consider the qualitative dynamics of a class of spatially flat, scalar-tensor cosmological models derived from the action (10.1) with a constant modulus field (again Bianchi type I cosmologies may also be considered by the introduction of two massless scalar fields into the reduced action (10.67) which parametrize the shear of the models; similar degrees of freedom also arise when considering the toroidal compactification of higher-dimensional theories [48]).

We assume that the metric corresponds to the spatially flat, FRW universe where

135

the action (10.8), after integrating over the spatial variables, yields the reduced action

s = J dt e3a {e-<I> [6a<i> - 6a2 - <i>2 + ~e2<I>0-2 - 2A] - AM}' (10.67)

We again assume that AM > 0, and employ the generalized Friedmann constraint equation,

3a2 - <jJ2 + 2A + ~0-2e2'P+6a + AMe'P+3a = 0, (10.68)

to eliminate the axion field from the system. The case A = 0 was considered in the previous Section. The new compact variables and time coordinate (given by (10.53)) leads to the plane system (10.57) - (10.58). The equilibrium points of this system are L t-), L t+) and F. The first two are saddles and F is a repelling focus and the functional form of these solutions was presented and discussed in [48]. The phase portrait is given in Fig. (15). As noted earlier, the orbits are future asymptotic to a heteroclinic cycle.

1. Non-Zero Central Charye Deficit

We now consider the case A -I=- O. We again employ equation (10.68) to eliminate the 0-2 term from the FE, and make the following definitions:

x== V3a e '

-2A v==~,

v == AMe'P+3a

e2

<jJ y == f.'

d d dt == edT'

(10.69)

where we define e = <jJ2 - 2A and we assume that e > 0 (the case e < 0 is related to a time-reversal of the system and the qualitative behaviour is similar), and we also assume here that A < 0 (AM> 0). The generalized Friedmann constraint equation (10.68) now yields 0 ~ x 2 + V ~ 1, so that all variables are bounded: o ::::: {x2, v, v, y2} ::::: 1. From the definition of e, v is determined from y2 + V = 1. The resulting 3-dimensional system then becomes:

dx V3 (1 - x2 - ~v ) + xy (1 - x 2 - ~v) , (10.70)

dT

dy (1 - y2) (X2 + ~v) > 0, (10.71)

dT

:; = v [y (1 - 2x2 - v) + V3x] . (10.72)

The invariant sets x 2 + v = 1, v = 0, y2 = 1 define the boundary of the phase space and it is important to note that the variable y is monotonically increasing.

136

v

f

.... .... .... 1... ....

'. /:.

L(+) ->-. :.-

x ~

.\

-·U -),...

I ' : - /-_ L+ H

/ /

......... L+

(+)

.... .... 1... .... .... ~ y

FIG. 17: Phase diagram of the system (10.70)-(10.72) for A < 0 and AM > 0, where cj; > 0 is assumed. See caption to Fig. (13).

This ensures that there are no closed or recurrent orbits in the phase space. The equilibrium points of the system are all saddles: F± (x, y, V = ~1/V27, ±1, 16/27), Lt±) (±1, 1,0) and L(±) (±1, -1,0). The points Lt±) represent power- law cos­mologies with r.j; > 0 where only the dilaton field is non- trivial; i.e., the axion field and cosmological constant terms are dynamically negligible and these solutions are the "dilaton-vacuum" solutions (10.14). The points L(±) are the corresponding solutions with r.j; < O. The phase portrait is given in Fig. (17) .

In this case there are no sinks and no sources in the full 3-dimensional phase space. Since the variable y (and hence r.j;) is monotonically increasing, solutions generically asymptote in both the past and future towards the invariant sets y = ±l. These both include a heteroclinic cycle and this implies that generically the solutions ex­hibit similar asymptotic behaviour at both early and late times (to that discussed above). The orbits interpolate between the dilaton-vacuum solutions corresponding to the saddle points L(±) in the past and the dilaton- vacuum solutions correspond-

ing to the saddle points Lt±) in the future. The effect of the cosmological constant,

137

AM, on the dynamics is significant at both early and late times. The points F± correspond to the equilibrium point F, F+ in Figs. (15), (17), but since they are saddles in the 3-dimensional phase space they do not play a primary role in the asymptotic behaviour.

Finally, we make some brief remarks on the case A > O. We can define ~ == cp and consider the subset cp ~ O. Introducing normalized variables as before yields a 3-dimensional, compact system of autonomous ODE. A full dynamical analysis ofthis system was presented in [44]. The main feature is that there is a non-hyperbolic equilibrium point, C+, which can be shown to be a (global) source (since v is a monotonically decreasing function). This point represents a static universe, where the dilaton field is evolving linearly with time and the axion field and AM are dynamically negligible. There are also two saddle points, Lt±) , which represent dilaton-vacuum solutions; these are analogues of the saddles that appear above. There is also a saddle F+, and we stress that there are no sinks in the phase space. Therefore, trajectories generically asymptote into the past towards C+, and then spiral away towards the heteroclinic cycle in the invariant set iJ = 0 containing the saddle points Lt-l and Lt+)· The phase space is depicted in Fig. (18).

v

t

"

\

/

X - • < L+'

(+)

v

FIG. 18: Phase diagram of the system with A > 0 and AM > 0, where cp > 0 is assumed. See caption to Fig. (13).

138

2. Discussion

The most important mathematical feature of the models is the cyclical behaviour that arises due to the existence of a heteroclinic cycle. This is of great physi­cal significance, and it is perhaps indicative of possible chaotic behaviour. The solutions interpolate between different "Kasner-like" dilaton-vacuum, power-law models, undertaking cycles between the saddles in the 3-dimensional phase space. This is somewhat reminiscent of the Mixmaster behaviour that occurs in the general Bianchi type VIII and IX cosmologies [182, 363J and more specialized Bianchi type cosmological models containing a magnetic field [226J. In general, the axion field is dynamically important and this is an indicator that chaos may be intrinsic in string cosmology.

There are a number of questions that are important in early universe cosmology in general and in string cosmology in particular. The questions of whether cosmological models can isotropize and/or inflate (and if they can inflate whether there is a graceful exit from inflation) are of great importance [128J. A chaotic regime due to dissipative effects or chaotic mixing [240J could possibly be an alternative to inflation as a cause of homogenization and isotropization. This might alleviate the problems of initial conditions in inflation. This last point has been addressed in [120J, where it was suggested that there would be sufficient time for a compact, negatively-curved universe to homogenize since chaotic mixing smooths out primordial fluctuations in a pre-inflationary period. On the other hand, a chaotic cosmological regime might work in tandem with an inflationary mechanism [379J to produce new interesting physical phenomena.

The question of chaos in anisotropic Bianchi type IX string cosmologies has also been considered [21 J. It was shown that since the axion and dilaton fields behave collectively as a stiff perfect fluid, the system oscillates only a finite number of times. Consequently, there is no Mixmaster chaos in these models. This is to be expected since it is known that the admission of stiff fluid matter causes chaos to cease [32J. In contrast to the anisotropic Bianchi type IX cosmologies, however, the models described above contain cosmological terms (e.g., an effective dilaton potential). Thus, this chaos has a different origin to the chaotic behaviour that arises in general relativistic models. On the other hand, there may be some connection with models that contain Yang-Mills fields. It is known that chaotic oscillations occur for such fields [24J and, moreover, it was shown in [32J that the oscillations that are suppressed by a single massless scalar field can be restored by coupling an electromagnetic field to a Brans-Dicke type field [80J. This model is related to a scalar field model with an exponential potential [45, 99J and, consequently, is also related to string theory cosmological models [242J.

There are a number of outstanding issues that need to be addressed regarding the possible existence of chaos in string cosmology. First, the chaotic behaviour depends crucially on the dimensionality of spacetime and on the product mani­fold structure of the extra dimensions [21J. In particular, superstring theories are formulated in D = 10 spacetime dimensions, while M -theory, with its low-energy

139

supergravity limit, is an eleven-dimensional theory [376]. Second, the low energy effective action is only valid in the perturbative regime of weak coupling and small curvature. In general, it may be necessary to study chaos within the context of a full non-perturbative formulation of the theory, but at present such a formulation is unknown. Nevertheless, if chaotic behaviour occurs at the level of the effective ac­tion, it is to be expected that similar behaviour should arise in the non-perturbative regime. Finally, there is the question of what will happen if inhomogeneities are introduced. Again, such effects will be most unlikely to lead to any suppression of chaotic behaviour and will perhaps make chaos even more predominant [21].

XI. ANISOTROPIC AND CURVED STRING COSMOLOGIES

Let us extend the work of the last Chapter to cosmologies containing spatial curva­ture terms [51]. This will enable us, via a qualitative analysis, to determine (among other things) whether models evolve from a low-curvature regime and thereby test the self-consistency of the low-energy effective string cosmology paradigm. In par­ticular, we consider the class of "isotropic curvature" models [265, 314], which are spatially homogeneous but contain non-trivial curvature and anisotropy and are characterized by the fact that the 3-dimensional Ricci curvature tensor is isotropic; i.e., (3) Rij is proportional to kJij on the spatial hypersurfaces and these surfaces therefore have constant curvature k [265].

The 4-dimensional line element of these cosmologies is given in [264, 314] in terms of one-forms {w1 , W 2 ,W3 } and the variable, a, which parametrizes the effective spatial volume of the universe and the shear parameter, (3, which determines the level of anisotropy. The class of isotropic curvature universes contains the Bianchi type I (k = 0) and V (k < 0) models and a special case of the Bianchi type IX (k> 0) models [314], and can be regarded as the simplest anisotropic generalizations of the flat (k = 0), open (k < 0) and closed (k > 0) FRW universes, respectively. The isotropic models are recovered when /3 = 0

We recall the effective action (10.8),

S = ! rrxFg {e-<I> [R + (V<1»2 - 6(V(3)2 - ~e2<1> (Va)2 - 2A] - AM}' (11.1)

where it is understood that either A or AM should be set to zero. We assume that all massless degrees of freedom are constant on the surfaces of homogeneity, t = constant. The FE derived from action (11.1) for the isotropic curvature metrics are then given by

.. .. 1 K- 1 A ,,",+30< 0 a - acp - 2"P + + 2" Me" =,

2 '2 1 -2tj; - <i; - 3a2 - 6(3 + 2" P - 3K + 2A = 0,

jj - /3<i; = 0,

K+2aK=0,

p+6ap = 0,

together with the generalized Friedmann constraint equation

3a2 - <i;2 + 6/32 + ! P - 3K + 2A + AMe'P+3 o< = 0, 2

(11.2)

(11.3)

(11.4)

(11.5)

(11.6)

(11. 7)

where, as before, cp == <1> - 3a defines the "shifted" dilaton field [68, 326, 353], P == &2e2'P+6o< (equation (10.25» may be interpreted as the effective energy density of the pseudo--scalar axion field [213], and

K == 2kexp( -2a) (11.8)

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

141

represents the spatial curvature term, and a dot denotes differentiation with respect to cosmic time, t.

The moduli fields that may also arise in the string effective action from the compactification of higher dimensions have not been included and it is assumed, in particular, that the internal dimensions are fixed. However, we note that in the NS-NS model (AM = 0), we can again easily include the dynamical effects of these extra dimensions in the case of a toroidal compactification by reinterpreting the shear parameter, (3, in the FE (11.2)-(11.7) [47, 266] (the moduli fields, (3m, that parametrize the radii of the circles, have the same functional form in the FE as the shear term and may therefore be combined with the shear by replacing (32 with (32 + L: (3~).

The variables K and p in equations (11.5) and (11.6) may be combined to define the new variable

3=p-K p+k·

Equations (11.5) and (11.6) then give rise to the evolution equation

,!.. 2· (1 ~2) =,==-0; -c .

(11.9)

(11.10)

Hence, all of the equilibrium points occur either for a = 0 or for 3 2 = 1, and this implies that either p = 0 (3 = -1) or K = 0 (3 = +1) asymptotically if a =I O.

In general, equations (11.2)-(11.7) define a 4-dimensional dynamical system (the constraint equation (11.7) may be employed to globally reduce the system by one dimension). In the cases in which all of the equilibrium points lie on 3 2 = 1, the asymptotic properties of the string cosmologies can be determined from the dynamics in the 3-dimensional sets p = 0 or K = O. The latter 3-dimensional dynamical system was studied in the last Chapter.

Therefore, we explicitly examine the 3-dimensional invariant set p = 0 in what follows. The only case in which there exist equilibrium points with a = 0 but 3 2 =11 occurs in the NS-NS case (AM = 0) in which k > 0 and A > O. We shall examine the full 4-dimensional system in this case, although the 3-dimensional subset p = 0 still plays a principal role in the asymptotic analysis. All solutions represented by the equilibrium points that arise in this work are presented in [51]. We shall discuss the case A =I 0, AM = 0 (the case AM =I 0, A = 0 was discussed in [51]).

A. Non-Zero Central Charge Deficit

Let us study the NS-NS string effective action (11.1) with AM = 0 for an arbitrary central charge deficit. Through equation (11.7) we eliminate the variable p from

142

the FE, and we make the following definitions:

X:::: V3a ~ , y=CP - e' Z = 6/]2

-~' U:::: ±3K

~2 '

V:::: ±2A ~2 '

d d dt :::: ~dT'

(11.11) The ± signs in the definitions for U and V are to ensure that U > 0 and V > 0 when necessary. With these definitions, all variables are bounded such that 0 :::; {X2, y2, Z, U, V} :::; 1 and equation (11.7) now reads

~pC2 = y2 ± U =f V - X2 - Z ~ O.

The variable ~ is defined in each of the following four cases by:

.A>O

(1) K > 0: e :::: 3K + cp2,

(2) K < 0: e :::: cp2,

.A<O

(3) K > 0: e :::: 3K + cp2 - 2A,

(4) K < 0: e :::: cp2 - 2A.

(11.12)

For example, consider A > 0 with K > OJ for this case y2 + U = 1 and equation (11.12) reads

1 -2 2 2P~ = 1 - V - X - Z ~ O. (11.13)

Hence, we use {X, V, Z} as the phase space variables (see case (1) below for details). We now set up the 4-dimensional dynamical system in each of these cases in turn and introduce subscripts i = (1,2,3,4) to the variables {X, y, Z, U, V} to distinguish each case. As noted above, all equilibrium points discussed below which have a "I o must have 3 2 = 1, and hence nearly all of the orbits asymptote towards the equilibrium points in one of the invariant sets p = 0 or K = O. The qualitative behaviour of the 4-dimensional phase space in each case is then examined.

1. The Case A > 0, k > 0

We define e = cp2 + 3K and utilize the positive signs for UI and VI as defined by equation (11.11). From the generalized Friedmann equation we have that

o :::; xf + ZI + VI :::; 1, y I2 + UI = 1, (11.14)

143

and therefore we may eliminate Ul (which is proportional to K), and consider the 4-dimensional system of ODE for ° ~ {Xl, Y12, Zl, VI} ~ 1:

d~l ~ (1- Xl) (2 + Y?) - V3 (Zl + VI) + XlYl (1- xl- Zl)~11.15)

dYl ( 2) ( 2 1 ) dT 1- Yl Xl + Zl + V3XlYi , (11.16)

~i 2Zl [Yl (1 - xl- Zl) + ~Xl (1 - yn] , (11.17)

~i = -2Vl [Yl (xl + Zl) - ~Xl (1 - yn] . (11.18)

The invariant sets Y[ = 1, Xl + Zl + VI = 1, Zl = ° and VI = ° define the bound­ary of the phase space. The equilibrium sets and their corresponding eigenvalues (denoted by A) are

L±: Yl = ±1, ZI = 1 - xl, VI = 0;

(Al,A2,A3,A4) = (=j= ~ [V3±Xl] ,=j=2,0,-2V3 [Xl ± ~])'11.19) 1 2) L l : Xl = 0, Zl = 0, VI = 3(2 + Yl ;

(A±,A2,A3) = (~ [Yl ± ~JI9Y12 -16] ,2Yl ,0). (11.20)

The zero eigenvalues arise because these are all lines of equilibrium points. Here, the global sources are the lines Ll (for Yi > ° or <p > 0) and L- (for Xl < I/V3). The global sinks are the lines Ll (for Yl < 0) and L+ (for Xl > -I/V3).

This case is different from the other three cases to be considered in this Chapter, since it is the only one with the line of equilibrium points, L 1 , inside the phase space. This line acts as both sink and source, and corresponds to an exact static solution which generalizes the static "linear dilaton-vacuum" solution (10.19) [281]. This solution was examined in [136] for <p > ° and was shown by a perturbation analysis to be a late-time attractor. The lines L± correspond to the spatially fiat, "dilaton-vacuum" solutions (10.14). The corresponding stable solutions are in the range h* E (-1/3, 1/V3] for L+ and h* E (-1/V3, 1/3] for L-, respectively.

Let us discuss the qualitative behaviour of these models. The invariant set K = ° for A> ° was studied in the last Chapter. It was found that Zl is a monotonically increasing function (as it is in the full 4-dimensional set). The early time behaviour of most trajectories is to asymptote towards the linear dilaton-vacuum solution, represented by the point C, where all degrees of freedom except <p are dynamically static. To the future, these solutions asymptote towards the line L + for X I > -1 V3. We note that the point C is the Yi = 1 endpoint of the line L 1 • Fig. (13) depicts this phase space.

We next consider the invariant set p = ° for A> 0, K > 0. In the p = ° case, the system reduces to the three dimensions of {Xl, YI , Zd (VI = 1 - Xl - ZI). The

144

equilibrium points are the lines L± with eigenvalues AI, A2 and A3 (from (11.19)), and the two endpoints of Ll with eigenvalues A± and ..\3 (from (11.20)). These endpoints are specified by the condition y 12 = 1 and we denote them by Li±) , where the "±" in the superscript reflects the sign of Yl . For this invariant set the entire line L + acts as a global sink and the entire line L - acts as a global source. Furthermore, we note that for Lt, A_ = 0, and so these two points are non-hyperbolic. However, the eigenvectors associated with these zero eigenvalues are both [-2/\1'3,1,0] and are located in the (Xl, Yi) plane. Hence, if we choose Zl = 0 and rotate the (Xl, Yt) axes such that

x == (Yl T 1) _ \1'3 T Xl , 2

fj == (Yi T 1) + \l'3Xl ,

we see that in the vicinity of the equilibrium point, the trajectories along x for fj = 0 are also along these eigenvectors. Hence, for fj = 0 and small x, it follows that

dx x dT ~ T"7'

Consequently, for Yl = +1, the trajectory along x asymptotes towards the equi­librium point, whereas the trajectory along x for Yi = -1 evolves away from the equilibrium point. This implies that the points Lt are saddle points. The phase space is depicted in Fig. (19). The quantity Xt/fft is monotonically decreasing. Such a monotonic function excludes the possibility of periodic or recurrent orbits in this 3-dimensional space. Therefore, solutions generically asymptote into the past towards L - and into the future towards L +. In this 3-dimensional set, spatial curvature is dynamically important only at intermediate times.

We can now discuss the the qualitative dynamics in the full 4-dimensional phase space. The past attractors are the line L- for Xl < 1/\1'3 and the line Ll for Yl > O. The future attracting sets are the line L+ for Xl > -1/\1'3 and the line Ll for Yl < O. We note that Yl / ~ is monotonically increasing and this implies that there are no periodic or recurrent orbits in the full 4-dimensional phase space. Therefore, solutions are generically asymptotic in the past to either the line L - for h* < 1/3 or to the line Ll for n > O. Similarly, solutions are generically asymptotic in the future to either L+ for h* > -1/3 or Ll for n < O. Since Yi / ~ is monotonically increasing, Yl -+ + 1 or VI -+ 0 asymptotically to the future. These limits represent global sinks on the line L +. Conversely, Y l -+ -1 or Vi -+ 0 asymptotically to the past, corresponding to the global sources on the line L-. There are also equilibrium points for finite Yl == Y* inside the phase space on the line L l ; the points Y* < 0 are global sinks and the points Y* > 0 are global sources.

We note that the reflections Xl -+ -Xl and Y l -+ -Yl are equivalent to a time reversal of the dynamics. Therefore, there are orbits starting on the line L - (for

~

• • • •

L­... .:. • •

~

1 L+

00::) o 0 0--"""""'" 0

o· ~ 0

o o

o .. _._~ 0

r ·'- .... -L(.:) -: ) ~-::..-=~ ~ 1 0 . .... ": .. :- ::::0 - _~-- .:-._ - .... _ -- r-

~ - - - 0- - ~ L(+) ---' 1

145

Y1

FIG. 19: Phase diagram of the system (11.15)-(11.18) in the NS-NS (A > 0) sector with p = 0 and k > O. See caption to Fig. (13). The labels L + and L - refer to lines of equilibrium points, and the labels Li+) and Li-) represent the equilibrium points which are the endpoints of the line Ll (for which Xl = +1 and Xl = -1, respectively). In this phase space, cp > 0 is assumed.

Xl < 1//3) and ending on the line Ll (for Y* < 0). Similarly, there are orbits which begin on the line Ll (for Y* > 0) and end on the line L+ (for Xl > -1//3) . Due to the existence of the monotonic function and the continuity of orbits in the 4-dimensional phase space, solutions cannot start and finish on L l . This is best illustrated in the invariant set Zl = O. In addition, orbits may start on the line L­(for Xl < 1//3) and end on the line L+ (for Xl > -1//3). Investigation of the invariant set Zl = 0 also indicates which sources and sinks are connected; not all orbits from L - can evolve towards L +.

Although the lines L± lie in both of the invariant sets p = 0 and k = 0, the line Ll does not. On this line, Xl = 0, and the solutions are therefore static (0: = /3 = 0). Equation (11.10) then implies that the axion field and spatial curvature can both be dynamically significant at early and late times for the appropriate orbits.

146

2. The Case A > 0, k < 0

In this case, we choose the negative sign for U2 in equation (11.11), the positive sign for V2, and the definition e = <p2. The generalized Friedmann constraint (11.7) now implies that

° :-:; xi + Z2 + U2 + V2 :-:; 1. (11.21)

For this system, yl = 1, and so the 4-dimensional system for the variables ° :-:; {X~!, Z2, U2, V2} :-:; 1 becomes:

dX2 V3 (1 - xi - Z2 - V2 - ~U2) + X 2 (1 - xi - Z2) , (11.22)

dT dZ2

2Z2 (1 - xi - Z2) > 0, (11.23) -dT dU2

-2U2 ( xi + Z2 + ~X2)' (11.24) -dT dV2

-2V2 (Xi + Z2) < 0. (11.25) -dT

The invariant sets xi+z2+u2+V2 = 1, Z2 = 0, V2 = 0, U2 = ° define the boundary of the phase space. The equilibrium sets and their corresponding eigenvalues are

s+ : 1 2 X 2 = - y'3' Z2 = 0, U2 = 3' V2 = OJ

( 2 2 4 4) (A1, A2, A3, A4) = -3' 3' 3' 3 ' (11.26)

C: X 2 = 0, Z2 = 0, U2 = 0, V2 = Ij (A1,A2,'\3,'\4) = (1,0,2,0), (11.27)

L +: Z2 = 1 - xi, U2 = 0, V2 = OJ

(A1, A2, '\3, '\4) = ( - ~ [X2 + V3] ,-2,0, -2V3 [X2 + ~] }.l1.28)

The point C represents the static "linear dilaton-vacuum" solution (10.19). The saddle S+ represents the "-" branch of the Milne solution, where only the curvature term and the scale factor are dynamic.

In the invariant set p = ° (for A > 0, k < 0), the system reduces to the three dimensions of {X2' Z2, U2} (V2 = 1 - xi - Z2 - U2). The equilibrium points are the same as above with eigenvalues '\1, A2 and '\3. We note that for this invariant set the entire line L + acts as a global sink, and C acts as a source. Although one of the eigenvalues for the point C is zero, it is shown below that this point is a source in the full 4-dimensional phase space. The variable Z2 is monotonically increasing, and as such we see that the shear term is negligible at early times, but becomes dynamically significant at late times. There are no periodic or recurrent orbits in

~--I

I I I

0-

I I

I

I

I I

U2

i /~

/ \ / \

147

\ \ ~ s+

\ '(

-. ~ L+

FIG. 20: Phase diagram of the system (11.22)-(11.25) in the NS-NS (A > 0) sector with p = 0 and k < o. See captions to Figs. (13) and (19).

this 3-dimensional phase space due to the existence of this monotonic function. In general, solutions asymptote into the past towards the static solution (point C) and asymptote into the future towards L+. Fig. (20) depicts this 3-dimensional phase space.

The qualitative behaviour for the invariant set K = 0 was discussed earlier. In the full 4-dimensional set, the point C is non-hyperbolic because of the two zero eigenvalues. However, it can be shown that this point is a source in the full 4-dimensional set by the following argument. We first note that the variable Z2 is monotonically increasing and hence orbits asymptote into the past towards the invariant set Z2 = O. Similarly, V2 is a monotonically decreasing function, and so orbits asymptote into the past towards large V2 (Le., V2 = 1). This implies that they asymptote towards the point C. Since Z2 = 0 asymptotically, let us consider

148

the invariant set Z2 = 0 where equations (11.22)-(11.25) become:

dX2

dT

dU2

dT

Y3 (l-X~ - V2 - ~ti2) +X2 (l-X~),

-2U2 (x~ + ~X2)' 2 dV2 -2V2X 2·

dT

(11.29)

(11.30)

(11.31)

It is clear from equation (11.31) that V2 increases monotonically into the past. Now, this 3-dimensional phase space is bounded by the surface xi + U2 + V2 = 1, the "apex" of which lies at V2 = 1 (and X 2 = U2 = 0). Therefore, all orbits in or on this phase space boundary lie below V2 = 1, and therefore asymptote into the past towards V2 = 1. To further illustrate that this point is indeed a source, it is helpful to consider the invariant set Z2 = U2 = 0, xi + V2 = 1. In the neighbourhood of C, equation (11.29) becomes dX2/dT = X 2(1-Xi), indicating that orbits are repelled from X 2 = O. Hence, the point C is the past attractor in the full 4-dimensional set.

The future attractor for this set is the line L+ (for X2 > -1/\13). Both C and L + lie in both of the invariant sets p = 0 and K = 0, which is consistent with the analysis of equation (11.10). We conclude, therefore, that the spatial curvature terms and the axion field are dynamically important only at intermediate times, and are negligible at early and late times. The dynamical effect of the shear becomes increasingly important because the variable Z2 increases monotonically. On the other hand, the variable V2 decreases monotonically and the dynamical effect of the central charge deficit, A, becomes increasingly negligible. In addition, the existence of monotone functions in the 4-dimensional phase space prohibits closed orbits and serves as proof of the evolution that is described above.

3. The Case A < 0, f< > 0

We choose the positive sign for U3 and the negative sign for V3 in equation (11.11) to ensure that these variables are positive definite. We also define e == rp2 + 3K - 2A for this case. The generalized Friedmann constraint equation can then be rewritten as

0:::; xj + Z3:::; 1, 132 + U3 + V3 = 1. (11.32)

We again eliminate U3 (which is proportional to K), and consider the 4-dimensional system of ODE for 0 :::; {xj, Yi, Z3, V3} :::; 1:

d~3=(1_ xj - Z3)( Y3 + X 3Y3) - ~ (1- Yl- V3)(1- xj - Z3)(11.33)

dY3 1 ( 2 ) ( 2) (2 ) dT = \l3X3Y3 1 - Y3 - V3 + 1 - 13 X3 + Z3 , (11.34)

~; = 2Z3 [~X3 (1 - Yl- 113) + Y3 (1 - xi - Z3) ] ,

~~ = 2113 [~X3 (1 - Yl- V3) - Y3 (Xi + Z3)] .

149

(11.35)

(11.36)

The invariant sets y32 + V3 = 1, Xl + Z3 = 1, Z3 = ° define the boundary of the phase space. The equilibrium sets and their corresponding eigenvalues are

L± :Y3 = ±1, Z3 = 1 - xi, 113 = 0;

(Al,A2,A3,A4) = (=f ~ [v'3±X3] ,=f2,0,=f2v'3 [~ ±X3]) (11.37)

where again the zero eigenvalues arise because these are lines of equilibrium points. Here, the global sink is the line L+ for X3 > -1/v'3 (saddle otherwise), and the global source is the line L- for X3 < 1/v'3 (saddle otherwise). Stable solutions on the line L + correspond to the range h* E (-1/3, 1/ v'3J. The stable solutions on L -arise when h* E (-I/v'3,1/3J.

The invariant set k = ° for A < ° was studied in the last Chapter. Let us briefly review the dynamics. The variables X3 and Y3 are monotonically increasing functions (corresponding to a and cp, respectively). The former implies that these trajectories represent cosmologies that are initially contracting and then reexpand. The third variable used earlier was essentially 1-xl- Z3, which is proportional to p and is only dynamically significant at intermediate times. It asymptotes to zero into the past and future, indicating that the axion field is negligible at early and late times. All the equilibrium points in this invariant set are represented by the dilaton-vacuum solutions (the lines L±). In general, orbits asymptote into the past towards the line L- (for X3 < 1/V3), and asymptote to the future towards the line L+ (for X3 > -1/V3). Fig. (21) depicts this 3-dimensional phase space.

Since a =I- ° at the equilibrium points, we next examine the invariant set p = ° (for A < 0, k > 0), where the system reduces to the three dimensions {X3, Y3, V3} (Z3 = 1 - Xl). The equilibrium points are the same as above with eigenvalues Al, A2, A3' We note that the entire line L+ acts as a global sink and that the entire line L - acts as a global source in this invariant set. The function Y3 / JV3 is monotonically increasing, eliminating the possibility of periodic orbits. Fig. (22) depicts this 3-dimensional phase space.

The qualitative dynamics in the full 4-dimensional phase space is consequently as follows. The global repellers and attractors are the lines L - and L +, respectively. Orbits generically asymptote into the past towards the line L- (for X3 < 1/V3),

150

V3 , .-)-..

/

/ I

/ / \ / I \ I

I ( I I I I \

I I \

I I

/ ~I \ . - ~ -.- - .' • 00 ' o '

X3'---- 0 0 ("). L+

.-)-.. , \

\ \

\ \ \ ,

\ , \ , ,

L - , \ .... ~ ...

~ -:: • Y3

FIG. 21: Phase portrait of the system (11.33)-(11.36) in the NS-NS (A < 0) sector for i< = 0 and p =f. o. See captions to Figs. (13) and (19).

and into the future towards L+ (for X3 > -1/v'3). In the former case the stable solutions are given by the "+" branch of equation (10.14) for h* < 1/3. The stable sinks are given by the "-" branch with h* > -1/3. Again, we see that the curvature term, axion field and the central charge deficit are dynamically important only at intermediate times. The existence ofthe monotonically increasing function Y3/.JV3 excludes the possibility of periodic orbits and serves to verify the above description of the evolution of the solutions in the 4-dimensional set.

4. The Case A < 0, i< < 0

For this case, the appropriate definition for the variable e is e = 02 - 2A and we choose the negative signs for both U4 and V4 in equation (11.11). The generalized

I I

i/I \

V3

!

I

-~ / "

I \

\

,

I J. L- L ' , I . •••••• --'--.~.. \ -\>----

- - - - >-- _.. , ..... \ ~~ ----000

151

\ \ \

\ , \ \ \ \

Y3

FIG. 22: Phase diagram of the system (11.33)-(11.36) in the NS-NS (A < 0) sector for p = 0 and k > O. See captions to Figs. (13) and (19).

Friedmann constraint equation is written as

os xl + Z4 + U4 S 1, Yi + V4 = 1. (11.38)

Treating V4 as the extraneous variable results in the 4-dimensional system for the variables 0 S {Xl, Yl, Z4, U4} S 1:

dX4 (1 - xl- Z4) (yf3 + X 4Y4) - ~U4' (11.39) -

dT dY4

(1 - Yi) (Xl + Z4) > 0, (11.40) dT

dZ4 (2) dT = 2Z4Y4 1 - X 4 - Z4 , (11.41)

dU4 [ (2 ) 1 ] dT = -2U4 Y4 X 4 +Z4 + yf3X4 . (11.42)

The invariant sets xl+Z4+U4 = 1, Z4 = 0, 1142 = 1, U4 = 0 define the boundary

152

of the phase space. The equilibrium sets and their corresponding eigenvalues are

± 1 2 S . X 4 = T_ Y,4 = ±1 Z4 = 0 U4 = -' . -, v'3' , , 3'

( 2 2 4 4) (AI, A2, A3, A4) = ±3"' 1=3"' ±3"' ±3" ' (11.43)

L ± . Y,4 - ±1 Z4 - 1 - X2 U4 - o· . - , - 4, -,

(AI, A2, A3, A4) = ( 1= ~ [V3 ± X4] ,1=2,0, 1=2V3 [ ~ ± X4]) (11.44)

The global source for this system is the line L - (for X4 < 1/v'3, h* E (-1/v'3, 1/3]) and the global sink is the line L+ (for X 4 > -1/v'3, h* E (-1/3, 1/v'3]). The saddle points, S±, represent Milne models, where S+ corresponds to the "-" solution and S- to the "+" solution.

In the invariant set p = 0 (for A < 0, k < 0), the system reduces to the three dimensions of {X4, Y4, Z4} (U4 = 1 - xl - Z4). The equilibrium points are the same as above with eigenvalues AI, A2, A3' The entire lines L+ and L - now act as a global sink and source, respectively. Recurrent orbits are forbidden by the existence of the monotonically increasing variable Y4 . Hence, solutions generically asymptote into the past (future) towards the "+" ("-") dilaton-vacuum solutions (10.14) and the curvature term and central charge deficit are dynamically significant only at intermediate times. Fig. (23) depicts this 3-dimensional phase space.

The dynamical behaviour in the invariant set k = 0 is identical to that described earlier (see Fig. (21)). The qualitative dynamics in the full 4-dimensional phase space is therefore as follows. Since Y4 is monotonically increasing, the orbits asymp­tote into the past towards Y4 = -1 and into the future towards Y4 = + 1. As in the previous cases, the existence of such a monotonic function excludes the possibility of periodic orbits in the 4-dimensional phase space. Generically orbits asymptote into the past towards the line L- (for X 4 < 1/v'3), and into the future towards the line L+ (for X4 > -1/v'3). The range of values for h* corresponding to stable solutions is given by h* > -1/3 for those on L + and h* < 1/3 for those on L -. The variable Y4 increases monotonically along orbits and this implies that rp is dynamically sig­nificant at early and late times. Conversely, we see that the curvature term, central charge deficit and axion field are dynamically important only at intermediate times and are negligible at early and late times.

B. Discussion

We have presented a complete qualitative analysis for the isotropic curvature string cosmologies derived from the effective action (11.1) when AM = 0, which

• . ~ e i

Z4

t ): . ...... . L+

c .0 • ' ~' ..- / 'v l2.1-_ -.:;-- '1 @ • " :..: ~ ".... .- -' '~~ ,~.~ ~ - !--:~. - ... ~

• v ..- ~- , 0 _"'! ,-"",,. ,,,, -0 - - < ......... ,'~ -'" s+

- _) ....:: :...~ -..;, " : . ...,.,.., . J" _"J'" ~~~ .~.i'., ----.. X4~

- ~-'~0' ---

153

Y4

FIG. 23: Phase portrait of the system (11.39)-(11.42) in the NS-NS (A < 0) sector with p = 0 and k < o. See captions to Figs. (13) and (19).

represents the action for the NS-NS fields that arise in both the type II and heterotic string theories when an arbitrary central charge deficit is present. In general a cosmological constant AM can be identified with terms that arise in the RR sector of the massive type IIA supergravity theory [51]. The subset /3 = 0 corresponds to the class of spatially isotropic FRW universes with arbitrary spatial curvature. In the positively curved case, we have extended the work of Easter et al. [136] who performed a perturbation analysis of the static, closed FRW model to show that it is a late-time attractor. More generally, the models we have considered represent Bianchi type I, V and IX universes.

In each case we have established the existence of monotonic functions which preclude the existence of recurrent or periodic orbits. Consequently, the early- and late-time behaviour of these models can be determined by analysing the nature of the equilibrium points/lines of the system. In all cases, the spatially flat dilaton­vacuum solutions L± act as either early- or late-time attractors and, in many cases, act as both. Because these solutions lie in both the p = 0 and k = 0 invariant sets and do not contain a central charge deficit (or a non-zero AM) contribution, we may conclude that the shear and dilaton fields are dynamically dominant asymptotically. Furthermore, with the exception of the A> 0, k > 0 case, all early-time and late-

154

time attracting sets lie in either the p = 0 invariant set or the k = 0 invariant set, and a majority of these sets lie in both. Therefore, we conclude that the curvature terms and the axion field are dynamically significant only at intermediate times and are asymptotically negligible at early and late times. The exception to this general behaviour is the A > 0, k > 0 case where the generalized linear dilaton-vacuum solution, in which neither p = 0 nor k = 0, acts both as a repeller (for cp < 0) and as an attractor (for cp > 0). In these solutions, the variables p and k are proportional to the central charge deficit, A, and since asymptotically a = 0 (and /3 = 0) these models are static.

When A < 0, the central charge deficit is dynamically significant only at inter­mediate times. In fact, A is asymptotically negligible at early and late times and the only repelling and attracting sets in this case are the dilaton-vacuum solutions. When A > 0, the central charge deficit can be dynamically significant at both early and late times, and the corresponding solution is the generalized linear dilaton­vacuum solution (10.19), represented by the line L 1• When k > 0, these solutions can be repelling (cp < 0) or attracting (cp > 0). When k < 0, the endpoint of this line, C, is a repeller.

Let us now summarize what happens for AM =I=- O. When AM > 0, the cosmo­logical constant may play a significant role in the early and late time dynamics [44, 51J. For instance, although in the 4-dimensional sets there are no repelling or attracting equilibrium points in which AM is dynamically significant, the orbits which are attracted to the k = 0 invariant set end in a heteroclinic sequence which interpolates between two dilaton-vacuum solutions (see Chapter IX, Fig. (16) and [48]). During this interpolation, the orbits repeatedly spend time in a region of phase space in which AM is dynamically significant (the region v> 0 in Fig. (16)), although most time is spent near the dilaton-vacuum saddle points where AM is dynamically negligible. We note that this behaviour is only valid for solutions in which the curvature is dynamically negligible asymptotically. When AM < 0, the cosmological constant can be dynamically significant at both early and late times, since solutions typically asymptote to a solution in which the shear and axion fields are static [44, 51]. For the repelling and attracting sets in the AM < 0 cases, the shear term is only dynamically significant when the cosmological constant is not, and vice versa.

The parameter f3 measures the degree of anisotropy in the models. If we define isotropization by the condition /3 -+ 0 [112, 363], then we note that in general /3 =I=- 0 at the equilibrium points on the lines L±. Therefore, solutions asymptot­ing towards the sinks on these lines do not isotropize to the future. At all other equilibrium points, however, /3 = 0 and the corresponding string cosmologies there­fore "isotropize". This is an important result and a similar situation occurs in the time-reverse models. On the other hand, the question of isotropization of string cosmologies in a more general context remains an open question. Note that the shear in the models that we have discussed is essentially of Bianchi type 1. In GR with a perfect fluid, it is known that Bianchi type I models isotropize whereas in general spatially homogeneous models do not isotropize [112, 363].

155

For the equilibrium points corresponding to the sinks on the line L +, /32 <X r.p2 (p = k = 0) and so the energy densities of the modulus and dilaton fields are pro­portional. Hence, the corresponding dilaton-vacuum solutions (10.14) are "matter scaling" string cosmology solutions which act as local attractors, similar to the matter scaling solutions in general relativistic scalar field cosmologies [50]. Finally, for every equilibrium point the scale factor of the corresponding exact solution is a power-law function of cosmic time, and therefore again all of the corresponding exact solutions are self-similar [262]. Although we have not explicitly considered models in which r.p < 0, they are related to a time reversal of the models discussed here and many of their qualitative properties are the same.

In conclusion, we have discussed the qualitative properties of isotropic curvature string models. Typically, the curvature term is dynamically significant only at in­termediate times and is asymptotically negligible. There are only two exceptions to this. The first corresponds to the case {A > 0, k > O}, where the generalized linear dilaton-vacuum attractors and repellers have non-negligible curvature. The second case is the {AM > 0, k < O} model in which the repeller corresponds to a curvature-driven solution [51].

XII. M-THEORY

There are five anomaly-free, perturbative superstring theories [164]. It is now widely believed that these theories represent special points in the moduli space of a more fundamental, non-perturbative theory known as M-theory [376], and so no particular string theory is necessarily more fundamental since they are all related by duality symmetries (for a review see, e.g., Ref. [321]). Moreover, another point of this moduli space corresponds to eleven-dimensional supergravity. This represents the low-energy limit of M-theory [337, 376].

The original formulation of M-theory was given in terms of the strong coupling limit of the type IIA superstring. In this limit, an extra compact dimension becomes apparent, with a radius proportional to g;/3, where gs is the string coupling [376]. The compactification of M-theory on a circle, 8 1 , then leads to the type IIA su­perstring. In this framework, the dilaton field of the ten-dimensional string theory is interpreted as a modulus field parametrizing the radius of the eleventh dimen­sion. This change of viewpoint re-establishes the importance of eleven-dimensional supergravity in cosmology and has interesting consequences for the dynamics of the very early universe. An investigation into the different cosmological models that can arise in M-theory is therefore important and a number of solutions to the effective action have recently been found [37, 38, 61, 132, 143, 174, 210, 243-245, 250, 251, 253-255, 304].

The bosonic sector of eleven-dimensional supergravity consists of a graviton and an antisymmetric, 3-form potential [122-124]. The purpose of the present Chapter is to employ the theory of dynamical systems to determine the qualitative behaviour of a wide class of 4-dimensional cosmologies derived from this supergravity theory. We compactify the theory to four dimensions under the assumption that the geometry of the universe is given by the product M4 x y6 X 8 1, where M4 is the 4-dimensional spacetime, y6 represents a six-dimensional, Ricci-flat internal space and 8 1 is a circle corresponding to the eleventh dimension. We assume that the only non-trivial components of the field strength of the 3-form potential are those on the M4 x 8 1

subspace.

We shall present an effective, 4-dimensional action by employing the duality re­lationship in four dimensions between a p-form and a (4 - p )-form. The FE for the class of spatially isotropic and homogeneous FRW universes are then derived and expressed as a bounded autonomous system of ODE [52]. A complete qualita­tive analysis of the flat cosmological models is presented together with a discussion and interpretation of the results. The robustness of the models is addressed and, in particular, curvature effects are considered. Inhomogeneous models are briefly discussed. We conclude with a discussion of future work.

A. A. Coley, Dynamical Systems and Cosmology© Springer Science+Business Media Dordrecht 2003

157

1. Four-Dimensional Effective Action

The bosonic sector of the effective supergravity action for the low-energy limit of M-theory is given in component form by

A /11 ,fI;I[A 1A AAAAA 8 = d xv 191 R - _FA A A A FA1A2 A3 A4 48 A1A2A3 A4

1 €A1A2A313d~2133134C1(\C3C4 A A A 1 -124 Ji9I AA1A2A3F131132133134FC1C2C3C4 ,(12.1)

where R is the Ricci curvature scalar of the eleven-dimensional manifold with metric 9J.Lv, 9 == det9J.Lv and FA1A2A3 A4 == 48[A1 AA2A3A4J is the 4-form field strength of

the antisymmetric 3-form potential AAubA3. The topological Chern-Simons term arises as a necessary consequence of supersymmetry [124].

In deriving a 4-dimensional effective action from (12.1), we first consider the Kaluza-Klein compactification on a circle, 8 1. This results in the effective action for the massless type IIA superstring [73, 376]. The 3-form potential AA13C reduces to a 3-form potential AABC and a 2-form potential, BAB == AABY. If we ignore the one-form potential that arises from the dimensional reduction of the metric, the ten-dimensional action is given by [73, 153, 193]

8 / dlOx.JjgJ [e-'P lD ( Rs + (\i'q)1O)2 - 112HABcHABC) - :8FABCDFABCD

1 EA1A2B1B2B3B4C1C2C3C4 1 - 384 Ji9I BA1A2FB1B2B3B4Fc1C2C3C4 , (12.2)

where HABc == 38[ABBcJ and FABCD = 48[AABCDJ are the field strengths of the potentials BBc and ABcD, respectively, the ten-dimensional dilaton field, q)1O, is related to the radius ofthe eleventh dimension, e'Y, by, = !q)1O and we have affected

a conformal transformation to the string frame defined by g~1 = fi2 gAB , 0.2 == e'Y. The first bracketed term in (12.2) contains the massless excitations arising in the NS-NS sector of the type IIA superstring and the remaining terms constitute the RR sector of this theory [164].

We now consider the compactification of (12.2) to four dimensions. The simplest compactification that can be considered is on an isotropic six-torus, where the only dynamical degree of freedom is the modulus field parametrizing the volume of the internal space. We therefore assume that the string-frame metric has the form

ds 2 = g(s)dxJ.LdxV + e2{38 ·dxidxj S J.LV tJ' (12.3)

where 8ij (i, j = 1, ... ,6) is the six-dimensional Kronecker delta and (3 represents the modulus field. Moreover, we compactify the form-fields in (12.2) by assuming that the only non-trivial components that remain after the compactification are

158

those associated with the external spacetime M 4 . In particular, this implies that the Chern-Simons term is unimportant since it is proportional to F /\ F. The effective 4-dimensional action is then given in the string frame by

s = J d4x~ [e-4> (R + (V'cp)2 - 6 (V',8)2 - 112H/ow).H/LV).)

- .!..-e6f3 F. F/LV)."'] 48 /LV).'" ,

where cp == <P1O - 6,8 is the 4-dimensional dilaton field.

The FE and Bianchi identities for the form fields are

and

V' /L (e-4> H/Lv).) = 0

o[/LHv).",] = 0

V' /L (e6f3 F/Lv).",) = 0

o[/LFv).",p] = 0,

(12.4)

(12.5)

(12.6)

(12.7)

(12.8)

respectively. In four dimensions, a p-form is dual to a (4 - p)-form and equations (12.5) and (12.7) are solved by the ansiitze [149, 323]

H/LV). == e4> €/LV).'" V' ",a F/Lv).", = Qe-6f3€/Lv).""

(12.9)

(12.10)

where €/Lv).", is the covariantly constant 4-form, a is a scalar variable and Q is an arbitrary constant. Although equations (12.9) and (12.10) solve the FE (12.5) and (12.7), the Bianchi identities (12.6) and (12.8) must also be satisfied. Equation (12.10) is trivially satisfied since we are working in four dimensions, and substituting equation (12.9) into equation (12.6) implies that

V' /L (e4>V'/L a ) = o. (12.11)

Equation (12.11) may be interpreted as the FE for the pseudo-scalar axion field, a [323]. Moreover, substituting equations (12.9) and (12.10) into the remaining FE for the graviton, dilaton and modulus fields implies that they may be derived from a dual effective action

s = J d4x~ [e-4> ( R + (V'<p)2 - 6 (V',8)2 - ~e24> (V'a)2) - ~Q2e-6f3] .

(12.12) We note that the inclusion of excitations of the RR form potential on the external spacetime has introduced an effective exponential interaction potential for the 4-dimensional dilaton and modulus fields.

159

A. Cosmological Field Equations

Let us consider the 4-dimensional FRW metric on M 4 , where ea represents the scale factor and the curvature is positive (k = + 1), negative (k = -1) or zero (k = 0). The effective action (12.12), after integrating over the spatial variables and normalizing the comoving volume to unity, then reduces to

8 = J dt [e-CP (30:2 - V;2 + 6ke-2a + 6~2) + ~ecp+6a&2 - ~Q2e-6,l3+3a] ,

(12.13) where a dot denotes differentiation with respect to t and, as usual, cp == <I> - 3a defines the "shifted" dilaton field [341]. The corresponding FE are

a 1 1 O:v; + 'i P - 2ke-2a - 4Q2e- 6,l3+CP+3a , (12.14)

rjJ ~ (30:2 + V;2 + 6ke-2a + 6~2 _ ~p) (12.15)

P -60:p, (12.16)

jJ = ~v; + ~Q2e-6,l3+CP+3a , (12.17)

0 30:2 _ V;2 + 6~2 + ~P - 6ke-2a + ~Q2e-6,l3+CP+3a , (12.18)

where p e2cp+6a&2 parametrizes the kinetic energy of the pseudo-scalar axion field.

Kaloper, Kogan and Olive have considered the equivalent compactification of the M-theory effective action (12.1) directly in terms of eleven-dimensional variables in the case of the 4-dimensional spatially flat FRW spacetime [210]. In this case, the only non-trivial components of the 4-form field strength on M4 x 8 1 are essentially equivalent to the RR 4-form field strength in equation (12.12) and the NS-NS 3-form field strength. The scale factors in the string and M-theory interpretations are related by conformal transformations which provide the recipe that allows the type I1A string cosmologies to be reinterpreted in terms of eleven-dimensional M­theory models. It can be verified by direct comparison that for k = 0, equations (12.14)-(12.18) are formally equivalent to the FE derived in [210]. The advantage of employing the string-frame variables in this work is that the first derivative of the shifted dilaton field (cp) is a dominant variable and this greatly simplifies the analysis of the global dynamics.

We next define a new time variable, ry:

dry = e(-6,l3+cp+3a)/2 dt - . (12.19)

The system of equations (12.14)-(12.18) then becomes [52]:

a" = ~a' cp' + 3a' f3' - ~ (a,)2 + (cp,)2 - 6 (f3,)2 + 4ke-(5a+cp-6,l3) - ~Q2, (12.20)

160

cp" = 3 (0:')2 + 6 ((3,)2 + ~Q2 - ~ (cp,)2 + 3(3'cp' - ~a'cp" (12.21)

(3" = ~ (3' cp' + 3 ((3,)2 - ~a' (3' + ~Q2, (12.22)

~pe-(CP+3a-6!3) = (cp,)2 _ 3 (a,)2 _ 6 ((3,)2 + 6ke-(5a+cp-6!3) _ ~Q2, (12.23)

where a prime denotes differentiation with respect to T/ and the generalized Fried­mann constraint (12.18) has been employed to eliminate the axion field, p.

Since Q2 is semi-positive definite, it follows from equation (12.23) that cp' is a dominant variable in the spatially flat and negatively curved models (k ::; 0). In addition, it follows from equations (12.21) and (12.23) that cp' is a monotone function, and hence cp' is either monotonically increasing or decreasing throughout the evolution of the models. (The variable cp' plays an analogous role to that of the expansion parameter in the spatially homogeneous perfect fluid models of GR [52]). We therefore introduce the new dimensionless time variable, T, according to

dT dT/ = cp' , (12.24)

where we assume here that cp' > o. We also define the following set of dimensionless variables:

V3a' y6(3' w= Q2 6ke-(5a+cp-6!3)

(cp')2

_ e-(cp+3a-6!3) 0= p X==--, Z==--,

cp' cp' - 2(cp')2 , u == 2 (cp,)2 (12.25)

This leads to a decoupling of the equation for cp', which can be written as

dcp' (1 1 V3 2 2 y6) , dT = -2"+2"w-TX + X +z +TZ cp. (12.26)

The remaining equations can then be written in the following dimensionless form:

dx (1 - X2 - Z2 - w) (x + v3) + ~w ( x - v3) - ~ u , (12.27)

dT dz

(1- x2 - z2 - w) Z + ~w (Z + y6) , (12.28) dT dw

[w - 1 - y6z + v3x + 2 (1 - X2 - Z2 - w) ] w , (12.29) dT du -~u [2v3X + 3(w + 2X2 + 2z2)] . (12.30) dT

The variable n is given by - 2 2 o = 1 - x - Z - W - u, (12.31 )

and satisfies the auxiliary equation

~~ = - [2Z2 + w + 2x (x + v3) ] n . (12.32)

161

B. Structure of State Space and Local Analysis

In this Section we present all of the equilibrium points for the system (12.27)­(12.30). We are primarily interested in the spatially flat models. However, we also consider the stability of these models to perturbations in the spatial curvature. The local stability analysis we perform is valid for both positive and negative spatial curvature, although we explicitly consider the k ::; 0 models since in this case the condition p ~ 0 implies that all of the dimensionless variables are bounded. The physical state space is defined by

0< {x2,z2,w,U} < 1,

and a global analysis can therefore be undertaken.

(12.33)

We include the boundary of the state space in our analysis because the dynamics in the invariant boundary submanifolds is useful in determining the global properties of the orbits in the physical phase space. The boundary of the state space consists of the following invariant submanifolds of the system: (i) n = 0 (models where the axion field is trivial), (ii) u = 0 (the spatially flat models), and (iii) w = 0 (models corresponding to the case where the 4-form field strength, FJ1-VAI<' is dynamically unimportant). The system of equations also admits an invariant submanifold, K, that is not part of the boundary of state space:

K : x + V2z + V3 = 0 , u = 0 . (12.34)

The equilibrium points are:

Equilibrium set: The Line L + 2 2 -X +z =1, w=O, u=O 0=0.

A1=0, A2=-2(V3x+1) , A3=-1+V3x-V6z, A4=-2(~X+1),

where the Ai denote the eigenvalues. The zero eigenvalue corresponds to the fact that this is an equilibrium set consisting of a circle of unit radius in the (x, z) plane. We do not display the eigenvectors, but we note that the eigendirection associated with A3 points in a direction outside the submanifold w = 0 while the eigenvector of A4 extends into the u direction. The stability of these equilibria is discussed below.

Equilibrium point: The source R

5v'3 v'6 28 252-x = -1:9 ' z = -19 ' w = 361 ' u = 361 ; 0 = O.

A - 20 A _ 14 A _ 7 ± iv1I9 1 - 19' 2 - 19' 3,4-

162

Equilibrium point: The saddle S

1 x = -- z- w 0 2-;7)' - = u=_·n_o v3 '3 ' H - .

2 4 2 Al = A2 = 3 ' A3 = 3 ' A4 = -3·

The saddle point S corresponds to the Milne form of flat space. This may be mapped onto the future light cone of the origin of Minkowski spacetime and in this sense may be interpreted as the string perturbative vacuum represented in terms of non-standard coordinates.

1. Qualitative Analysis of Spatially Flat Cosmologies

Let us consider the global dynamics of the spatially flat cosmologies (k = 0, u = 0). For these models, the state space is 3-dimensional and the orbits can therefore be represented pictorially. The only equilibrium points in this case lie on the line L +. From the eigenvalues given above, it can be seen that this line is a sink for x> -1/v'3 and v'2z > x - 1/v'3. The lines x = -1/v'3 and v'2z = x - 1/v'3 intersect on L + at the point P : (x, z) = (-1/ v'3, - J273), at which all three eigenvalues are zero. Hence, P is a non-hyperbolic equilibrium point. All other points on L + are saddles.

It can be shown that the point P is a source in the 3-dimensional phase space. It follows from equations (12.27) and (12.28) that

d~ (x + hz + v'3) = (x + hz + v'3) (1 -x2 - z2 - ~w ) , (12.35)

for u = o. This implies that x + v'2z + v'3 is a monotonically increasing function in the physical phase space. The term (1 - x2 - z2 - ~w) is positive-definite in the interior region and can only be zero on the boundary, where x2 + Z2 = 1 and w = o. The term x + v'2z + v'3 is positive-definite in the physical state space and can only vanish in the extended phase space at the point P. Indeed, the line x + v'2z = -v'3 is tangent to the unit circle x2 + z2 = 1 and w = 0 and actually touches it at the point P. We may conclude, therefore, that the non-hyperbolic equilibrium point P is a source for the 3-dimensional system. This can be verified by an analysis of the equilibrium point P using spherical polar coordinates and by numerical calculations.

The dynamics on the boundary of the state space is also important when inter­preting the behaviour of the orbits. The boundary consists of the two invariant submanifolds n = 0 and w = o. The n = 0 (trivial axion field) submanifold can be solved analytically in terms of the variables of the state space and the solution is

given by

z = -V6 + (zo + V6)(x - v'3) Xo - v'3

163

(12.36)

where (xo, zo) represents the initial point of the orbit. Thus, orbits follow straight line paths in the (x, z) plane. Moreover, since by definition x < v'3, this variable is a monotonically decreasing function on this submanifold and z is a monotonically increasing function. The line L + is a source for x > -1/ v'3 and v'2z < x-I J3 and a sink otherwise.

The boundary w = 0 describes models where the 4-form field strength is dynam­ically negligible. This submanifold can also be solved exactly and the orbits follow the straight line paths:

z = zo(x + J3) Xo + J3 '

(12.37)

where (xo, zo) again represents the initial point of the orbit. In this case, the function x is monotonically increasing. The line L + is a source for x < -1/ J3 and a sink otherwise.

The time-reversed dynamics of the cp' > 0 models we have considered thus far is equivalent to the dynamics in the case where cp' < O. This follows by redefining the time variable, T:

dT d." =_cp', (12.38)

so that ." and T are both increasing or both decreasing together. If we define the other state variables as in (12.25), the variables x and z for cp' < 0 are now the reflections of the variables x and z for cp' > 0; i.e., x -+ -x and z -+ -z. With the new time variable (12.38), the evolution equations (12.27)-(12.30) will have an "overall" change in sign; i.e., dx / dT -+ -dx / dT, etc. Thus, the equilibrium points are identical in both cases, but the eigenvalues have opposite signs. Consequently, the dynamics of the cp' < 0 models is the time reversal of the cp' > 0 models, where contracting models for cp' > 0 are expanding models for cp' < 0, and vice versa.

2. Physical Interpretation

The phase space for the spatially flat models is depicted in Figs. (24) and (25), which correspond to the invariant submanifolds w = 0 and n = 0, respectively, and Fig. (26), which represents typical orbits in the full 3-dimensional phase space. The equilibrium set L + represents solutions in which the form-fields are trivial and only the dilaton and moduli fields are dynamically important known as "dilaton­moduli-vacuum" solutions (10.15). We note that the solutions represented by the line L+ correspond to t < 0 and, in the time-reverse case (cp < 0), the solutions correspond to t > o.

164

y

i

--7 x

FIG. 24: Phase portrait of the invariant submanifold w = 0, corresponding to the case where the matter 4-form field strength is trivial and the NS- NS 3-form field strength is dynamically important. The line L + represents a line of equilibrium points.

y

i

x

FIG. 25: Phase portrait of the invariant submanifold n = 0, corresponding to the case where the NS-NS 3-form field strength is trivial and the matter 4-form field strength is dynamically important.

/

x -

/ /

z

t " 'J

" .~." '.,

y

FIG. 26: Phase diagram of the spatially flat cosmologies when both the NS-NS and matter form fields are non-trivial. The trajectories in Figs. (24) and (25) are depicted in grey.

165

Let us first discuss the dynamics in the invariant submanifold w = 0, where the NS~NS axion field is non~trivial and the RR 4-form field strength vanishes. The trajectories in Fig. (24) represent "dilaton~moduli~axion" solutions (10.16). The trajectory along z = 0 corresponds to the solution where the internal dimensions are static, in which the universe is initially contracting (x < 0) but ultimately experiences a "bounce" , induced by the 2-form potential, into an expansionary phase (x > 0). It follows from equation (12.9) that the field strength of this antisymmetric tensor field is directly proportional to the volume form of the three-space, so that the axion field may be interpreted in terms of a "membrane" which gives rise to a bouncing cosmology [52]. Many solutions exhibit such a bounce, but others collapse to zero volume. In the invariant submanifold n = 0, the NS~NS 2-form potential is trivial and the RR 3-form potential is dynamical. The cosmological constant term Q2 in the effective action (12.12) may be interpreted as a O~form field strength and plays a rOle analogous to that of a "domain wall" which resists the expansion of the universe [121, 126, 246, 247] (in contrast to the "membrane" associated with the axion field). Thus, most of the solutions that are initially expanding ultimately recollapse, as shown in Fig. (25). There are also solutions where the internal space is initially evolving sufficiently rapidly that the modulus field dominates the form field and the expansion can proceed indefinitely. Solutions that are initially collapsing do not undergo a bounce. In both of these invariant submanifolds the point P corresponds to an endpoint on the line of sources. In Fig. (24), the reflection of this point in the line z = 0 represents the opposite end of the line of sources and corresponds to a dual solution, where the radius of the internal space is inverted.

We now consider the dynamics in the full 3-dimensional phase space, where both the NS~NS 2-form potential and RR 3-form potential are non-trivial. Although both fields are asymptotically negligible, since the RR field causes the universe to collapse while the NS~NS field has the opposite effect, the two fields compete against one another and their interplay has important dynamically consequences (see Fig. (26)).

The point P is the only source in the system when both form fields are present. Furthermore, it follows from the definitions (12.25) that this equilibrium point rep­resents the collapsing, isotropic, ten~dimensional cosmology, in which a = j3 and the 4-dimensional dilaton field is trivial. As the collapse proceeds, a typical orbit moves upwards in a cyclical fashion until a critical point is reached where one of the form fields is able to dominate the dynamics. The orbit then shadows the corre­sponding trajectory in the invariant submanifold w = 0 or n = O. In Fig. (26), the axion field dominates and causes the universe to bounce. By this time, however, the kinetic energy of the modulus field has become significant and the solution ul­timately asymptotes to a dilaton~moduli~vacuum solution on L +. All sinks in the phase space correspond to solutions where the internal dimensions are expanding. There is a particular point where the spatial dimensions spanning the 4-dimensional spacetime become static in the late-time limit. In general, however, solutions either collapse to zero volume in a finite time or superinflate towards a curvature singular­ity (and correspond to pre-big bang cosmologies since the co moving Hubble radius decreases [151, 353]).

166

It is of interest to reinterpret the equilibrium points of the phase space in terms of eleven~dimensional solutions. Since the eleven~dimensional 3-form potential is trivial on L +, these points represent power-law "Kasner" solutions to eleven~ dimensional, vacuum Einstein gravity. Thus, the line L + is analogous to the Kasner ring that arises in the vacuum Bianchi I models of 4-dimensional GR [363]. The source P corresponds to a special "Kasner" solution which represents the Taub form of Minkowski spacetime [363]. The relevance of this solution to the problems associated with the pre-big bang curvature singularity have recently been discussed [143, 210]. The endpoints ofthe line of sinks on L + correspond to special "Kasner" solutions in which a subset of the scale factors are static. Finally, the time-reversed dynamics of the above class of models is deduced by interchanging the sources and sinks and reinterpreting expanding solutions in terms of contracting ones, and vice­versa. Thus, the late-time attractor for the time-reversed system is the expanding, isotropic, ten~dimensional cosmology located at point P.

c. Discussion

We have presented a complete dynamical analysis of spatially flat, 4-dimensional cosmological models derived from the M-theory and type IIA string effective actions. We have shown that the models generically spiral away from a source P, undergoing bounces due to the interplay between the NS-NS 2-form potential and the RR 3-form potential, and eventually evolve towards dilaton-moduli-vacuum solutions with trivial form fields (corresponding to the sinks on L+). We note the important dynamical result that 'P' is monotonic.

Thus, the form fields that arise as massless excitations in the type IIA superstring spectrum, or equivalently from the 3-form potential of eleven-dimensional super­gravity, may have important consequences in determining initial and final conditions in string and M~theory cosmologies even though they are dynamically negligible in the early~ and late-time limits. In particular, the point P is the only source in the system. The corresponding solution can be interpreted as the isotropic, ten­dimensional solution in the string context or the Taub form of flat space in eleven dimensions.

1. Effects of Spatial Curvature

Although the compactness of the phase space depends on the fact that k :::; 0, one can assume arbitrary signs for k in order to determine the local stability of the equilibrium points in the 3-dimensional set u = 0 with respect to curvature perturbations. The eigenvalue associated with u for the equilibrium points L + is always negative. This means that the sinks on L+ (i.e., points on L+ for x> -1/V3 and v'2z > x-I / V3) remain sinks in the 4-dimensional phase-space. In addition,

167

this implies that the point P is now only a saddle; that is, P is unstable to the introduction of both positive and negative spatial curvature.

Since a portion of L + acts as sinks in the 4-dimensional phase space, there exists the global result that the corresponding dilaton-moduli-vacuum solutions (10.15) (for 3a > -cj; and 6/3 > 3a - cj;) will be attracting solutions for the spatially curved models. We may deduce further global results by restricting our attention to the negatively curved models (k < 0), in which case the 4-dimensional phase space is compact. As discussed above, the point P is a only a saddle point in this extended phase space. Moreover, it follows from the earlier analysis that the only past attracting equilibrium point is the point R. (There is an additional saddle S which will affect the possible intermediate dynamics). This source corresponds to a negatively curved model with a trivial axion field; indeed, it is a power-law, self-similar collapsing solution with non-negligible modulus and RR form-field.

We have been unable to find a monotone function on the extended 4-dimensional phase space, but it is plausible that all negatively-curved models evolve from the solution corresponding to the global source R towards the dilaton-moduli-vacuum solutions (on the attracting portion of L +). Clearly the curvature is dynamically important at early times. In the time-:reverse case, the solutions asymptote from the non-inflationary dilaton-moduli-vacuum solutions in the past and evolve to the future towards a curvature dominated model; it is plausible that they evolve towards a model which is the time-:reversal of the one represented by R. Consequently, curvature can also be dynamically important at late times.

Therefore, when the effects of spatial curvature are included we obtain the local result that the point P becomes a saddle. On the other hand, the dilaton-moduli­vacuum solutions with trivial form fields are generic attracting solutions. In the analysis of the negatively-curved models the early time attractor (the source R) has non-zero curvature, implying that spatial curvature is dynamically important at early times in these models.

The robustness of the models was further studied in [52]. The effects of generalized couplings were investigated by considering a generalization of the effective action (12.12) given by

S= J d4xF9{e-4> [R+('V~)2-6('VjJ)2-~e24>('Voi-2A] _~Q2eCi3}, (12.39)

where A represents a cosmological constant term and c is an arbitrary constant with -6 ::; c ::; 6. The former term may arise through non-perturbative corrections to the string effective action. The motivation for considering an arbitrary coupling of the modulus field to the 4-form field strength is that the generality of the dynamics discussed in the previous Section (in which c = -6) can be investigated. Equation (12.39) reduces to the action studied in [47] when c = o.

A qualitative analysis [52] indicates that the conclusions obtained for the spa­tially curved models are robust when additional physical fields (e.g., a A term) are included. In addition, the value c = -6 was found to be a bifurcation value in the

168

analysis of general models with arbitrary coupling, c. In this context, therefore, the M-theory cosmological models studied exhibit rather special dynamics.

2. Inhomogeneous String Cosmology

M-theory encompasses and unifies string theory. To lowest order (in the inverse string tension), the tree-level effective action for massless fields contains a dilaton, a form field (which in 4-dimensions is dynamically dual to pseudoscalar axion field) and a stringy cosmological constant. Even in this approximation the one-loop string equations of motion for inhomogeneous backgrounds are very difficult to solve, and it is a useful first step to consider models in which the homogeneity is broken only in one spatial direction. Metrics that admit two commuting (orthogonally-transitive) space-like Killing vectors are referred to as G 2 spacetimes.

String models admitting a 2-dimensional abelian group of isometries have a num­ber of important physical applications. The spatially homogeneous Bianchi types I-VIIh and locally rotationally symmetric (LRS) types VIII and IX admit a G2

group of isometries [334], and so the G2 cosmologies can be considered as inhomo­geneous generalizations of these Bianchi models. Non-linear inhomogeneities in the dilaton and axion fields can be investigated and, in principle, this allows density perturbations in string-inspired inflationary models such as the pre-big bang sce­nario to be studied [268, 354]. Given the potential relevance of this scenario it is important to study its generality with respect to inhomogeneities as well as with re­spect to anisotropies. The general effects of small inhomogeneities and anisotropies have been studied by Veneziano [354].

In GR the generic singularity is neither spatially homogeneous nor isotropic. Hence it is of interest to study more general models. In particular, it has been conjectured that G2 metrics represent a first approximation to the general solution of Einstein gravity in the vicinity of a curvature singularity [29-31, 36]. The high curvature regime is precisely the regime where stringy deviations from GR are expected to be significant. These models may therefore provide insight into the generic behaviour of cosmologies at very early times.

A number of exact inhomogeneous and anisotropic G2 string cosmologies have been found. Barrow and Kunze studied an inhomogeneous generalization of the Bianchi type I string cosmology [23] and Feinstein, Lazkoz and Vazquez-Mozo de­rived a closed, inhomogeneous model by applying duality transformations on the LRS Bianchi type IX cosmology [142]. Clancy et al. have found inhomogeneous generalizations of the Bianchi type VIh universe and have studied their asymp­totic behaviour [86]. Further methods for obtaining new inhomogeneous G2 string cosmologies containing a non-triviaI2-form potential by utilizing S- and T-duality symmetries and generating techniques (such as that of Wainwright, Ince and Marsh­man [366]) were discussed in Lidsey et al. [233].

In this Subsection we shall consider some separable cosmological models, whose

169

governing equations reduce to ODE, that can be studied by qualitative methods. We consider the general string action in the form [44]

S = J d4xFY {e-<I> [R + ('\7<I»2 - 6('\7,8)2 - ~e2<I>('\7a)2 - 2A] - ~Q2e-6,6 - AM}'

(12.40) The Euler-Lagrange equations then lead to the FE

1 2<I> 1 [2 2 GOl,6 - '\7 Ol '\7 {j<I> + 6'\7 Ol,8'\7 {j,8 +"2e '\7 Ola'\7,6a - "2 gOl{j ('\7 <I> ) + 6('\7,8)

11] +"2e2<I>('\7a)2 + 2A + "2Q2e<I>-6{j + AMe<I> - 20<I> , (12.41)

O<I> = ~('\7<I»2 + 3('\7,8)2 - ~e2<I>('\7a)2 + A - ~R, (12.42)

0,8 = '\7 J1.<I>'\7J1.,8 - ~Q2e<I>-6,6, (12.43)

Oa - '\7 J1.<I>'\7J1. a. (12.44)

Let us examine the FE (12.41)-(12.44) within the context of G2 cosmological models described by the line element (8.28),

ds2 = eF (-dt2 +dz2) +G(ePdx2 +e-Pdy2) , (12.45)

where the metric functions {F, G, p} and the string functions {<I>,,8, a} are all func­tions of t and z only. The local behaviour of these models is determined by the gradient BJ1. == 8J1.G, and cosmological solutions arise if BJ1. is globally time-like. The Ricci scalar is given by

R = ~e-F (82G _ 82G) 2 (82F _ (PF) _ ~ [(8G)2 _ (8G)2] G 8t2 8z2 + 8t2 8z2 G2 8t 8z

-3 [ ( :) 2 _ (:~) 2]. (12.46)

Defining the modified dilaton field,

1 'P == <I> - -F -lnG,

2 (12.4 7)

the resulting FE are given explicitly in [44, 105]. We note that when Q = 0, the modulus field ,8 and the anisotropic variable p can be combined into a single variable (as usual).

Assuming separability of the metric functions of the form

2F(t, z) InG(t,z)

p(t,z)

F(t) + f(z), G(t) + g(z), Q(t) + q(z),

170

and appropriate separability conditions on the matter fields 4>(t, z), ;3(t, z), l7(t, z), the Ricci Scalar is then given by

R = ~e-f[4G + 4F + 302 + ci - (49" + 41" + 39'2 + q,2)]e-F,

where an overdot and a prime denote ordinary differentiation with respect to t and z, respectively. If

49" + 4f" + 39'2 + q,2 = 3a2,

where a is a constant, which constrains the spatial dependence of the metric, the Ricci Scalar is given by

R = ~e-f[4G + 4F + 302 + Q2 - 3a2]e-F,

which leads to no spatial dependence in the action (after integration). After ap­plying any further separability conditions (on the matter fields), the resulting FE will be a system of ODE. Note that the effect of the spatial dependence is to add a further contribution to the cosmological constant A.

As a specific example, let us assume separability with a linear dependence in z of the form

1 2F(t, z) == F(t) + "2cz, lnG(t, z) == G(t), p(t, z) == q(t) + az

(note the change of notation in the last term), and

4>(t, z) == 4>(t) + mz, ;3(t, z) == ;3(t) + nz, l7(t, z) == l7(t) + lz,

where a, c, l, m, n are constants, and therefore

'P(t,z) = 4>(t) - F(t) - G(t) + (m - ~c) z == 'P(t) + ( m - ~c) z.

In order for the resulting FE and constraints to be independent of z, a number of further (separability) conditions must be satisfied. It can be shown [44, 105] that m = 0 and either c = 0 or c = 6n. For c = 0 we must then have that either n = 0 or Q = o. For c = 6n we must have that A = AM = o.

The governing system of ODE in the c = 6n, A = AM = 0 (m = 0) case is:

1 ( .)2 1. 2 1 2 ·2 <P = - 0 + F + -G + -q + 3;3

2 4 4

_~e2'P+2F+2G (0-2 _l2) _ ~ (a2 + 12n2) , (12.48)

jj (0 + P) /3 + ~Q2e['P+3F+G-6,61, (12.49)

17 - (0 + P + 20) 0-, (12.50)

171

ij (~+P)q, (12.51 )

p ~ (~+ p)2 _ ~G2 _ ~q2 _ 3/P + ~e2<P+2F+2G (0-2 -l) 2 4 4 4

_~Q2e[<P+3F+G-6,6] + ~(a2 + 12n2), (12.52)

G (~+ P) G + (0-2 _l2) e2<p+2F+2G

- ~Q2e[<P+3F+G-6,8J, (12.53) 2

with the constraint ·2 2 (. ) 1. 2 1 2 (. )2 o F - ~ + 6n F + ~ + 2G + 2 (q + a) + 6 {3 + n

+~ (0- + l)2 e2<p+2F+2G + ~Q2e[<P+3F+G-6,6] (12 54) 2 2 . .

These models have been studied qualitatively [105].

3. Future Work

This work can be generalized in a number of ways. We considered a specific compactification from eleven to four dimensions, where the topology of the inter­nal dimensions was assumed to be a product space consisting of a circle and an isotropic six-torus. We emphasize, however, that the analysis also applies to com­pactifications on a Calabi-Yau three-fold since the gauge fields arising from the higher-dimensional metric have been ignored [210]. The qualitative analysis may be readily extended to compactifications on a general, rectilinear torus 8 1 x ... X 8 1.

After suitable redefinitions of the additional moduli fields that subsequently arise, the dimensionally reduced action can be expressed precisely in the form of (12.12) with the inclusion of a set of massless scalar fields in the NS-NS sector. In particular, the compactification on T4 X T2 X 8 1, where Tn represents the isotropic n-torus, is relevant to compactifications involving the 4-dimensional space K3 [131, 248]. This space has played an important role in establishing various string dualities [321]. It is the simplest 4-dimensional, Ricci-flat manifold after the torus [12] and may be approximated by the orbifold K3 ~ T 4 /Z2 [293].

Moreover, the effects of spatial anisotropy in the 4-dimensional spacetime can also be considered by introducing two, uncoupled moduli fields into the NS-NS sector of the reduced action (12.13). In this context, such fields parametrize the shear in the cosmologies. When these fields are non-trivial, the models represent the class of isotropic curvature cosmologies and correspond to Bianchi type I, V and IX universes [47, 265]. It would be interesting to consider these generalizations further.

It is of great interest to further study the cosmological implications of M-theory, whose moduli space contains all five, anomoly free ten-dimensional superstrings and

172

eleven-dimensional supergravity. Of special importance is the study of inflation in string/ M-theory cosmology; a first step in this investigation has been presented here, and inflation has also been studied in the pre-big bang scenario [151, 233, 353J. In particular, it would be interesting to study inhomogeneous cosmological models, and perhaps the next step is to extend the analysis of the previous Subsection and comprehensively study the qualitative properties of G2 cosmological models. Some special G2 string theory models have been presented in [23, 86, 87, 142J (also see [233J for a brief review).

The precise region of the M-theory moduli space that describes our present-day Universe is uncertain, but from a particle physicists' point of view the most favoured location is that of Es x Es heterotic string theory. Hofava and Witten have shown that the strongly-coupled limit of this theory is M -theory on an eleven-dimensional orbifold R 10 x 8 1/Z2 [186, 187J. The weakly-coupled heterotic string theory is then recovered in the limit where the radius of 8 1 tends to zero. The orbifold 8 1/Z2

may be viewed as a segment of the real line that is bounded by two fixed points on the circle. The effect of the Z2 transformation is to reverse the orientation of the circle, y --t -y, where y is the coordinate of the eleventh dimension, and to change the sign of the 3-form potential. This latter change of sign is necessary for the eleven-dimensional supergravity theory to remain invariant. The two sets of Es gauge supermultiples are located on each of the ten-dimensional orbifold fixed planes. Despite its potential importance, relatively little work has been done thus far in deriving and analysing cosmological solutions in Hofava-Witten theory. An early study was made by Benakli, who found a class of cosmological solutions from certain p-brane configurations [37, 38J. A review of the detailed solutions can be found in [254J.

Witten subsequently considered a further compactification to four dimensions on a deformed Calabi-Yau manifold and showed that the resulting theory has N = 1 supersymmetry [377J. A comparison of the gravitational and GUT couplings then implies that the orbifold must be larger than the radius of the Calabi-Yau space; indeed, the relative sizes may differ by more than an order of magnitude. This immediately implies that the early universe may have undergone a phase where it was five-dimensional.

Motivated by these considerations, Lukas et al. [257, 258J derived an effective five­dimensional theory by a direct compactification of the Hofava-Witten theory on a Calabi-Yau space. A necessary condition that must be satisfied for a consistent compactification of Hofava-Witten theory to five dimensions is that a non-zero mode of the 4-form field strength on the Calabi-Yau three-fold has to be included. For this reason, such a compactification differs from that of the standard Kaluza­Klein reduction of eleven-dimensional supergravity. A consistent truncation of the effective, five-dimensional theory is given in [304J. Due to a non-zero mode of the 4-form field strength on the Calabi-Yau space, a self-interaction potential for the scalar field results that does not have a global minimum, which implies that flat space is not a solution to this theory. However, there does exist a static solution to the FE that may be interpreted as a pair of parallel three-branes [257, 258J. As well as three-brane solutions, solitonic p-branes are also known to exist [130J.

]73

In the Hofava-Witten theory the ten-dimensional gauge degrees of freedom live on the "branes" on the boundaries of spacetime, motivating the study of brane world models in which our universe appears as some kind of domain wall or brane embedded in a higher dimensional space-time [186]. This confinement of the gauge fields to a lower-dimensional submanifold, in contrast to the gravitational field that can propagate into the bulk, might help to explain the hierarchy between the elec­troweak and the Planck scale in four dimensions [7]. With the aim of solving the hierarchy problem, Randall and Sundrum [302, 303] put forward a proposal, inspired by the AdS/eFT correspondence, where our 4-dimensional universe is embedded in a non-factorizable way into five-dimensional anti-de Sitter (AdS) space-time. In this paradigm gravity is "trapped" on the 4-dimensional brane due to the geometry of the bulk space-time.

These recent developments have motivated the idea that gravity may be a truly higher-dimensional theory, becoming effectively 4-dimensional at lower energies. There is currently great interest in higher-dimensional gravity theories inspired by string theory in which the matter fields are confined to a 3 + I-dimensional "brane-world" embedded in higher dimensions, while the gravitational field can also propagate in the extra dimensions (i.e., in the "bulk") [186, 297, 312]. In this paradigm it is not necessary for the extra dimensions to be small or even compact, a departure from the standard Kaluza-Klein scenario. It is of considerable interest to study the classical dynamical effects in these cosmological models. A unique feature of brane cosmology is that p2 dominates at early times, which will lead to completely different behaviour to that in GR. The cosmological implications of brane world models, and especially Friedmann brane models, have been extensively investigated (for example, see [8, 10, 54, 55, 89, 115, 125, 134, 147, 199, 208, 218, 255, 256, 263, 328, 330]). In particular, there has been much emphasis on whether a period of inflation is possible in brane cosmology and whether a homogeneous and isotropic universe can emerge from generic initial conditions. It is anticipated that dynamical systems methods will again be useful in such an investigation. Brane cosmology may also provide new approaches that might solve the cosmological constant problem [329] (also see [9, 207]).

An elegant geometrical formulation and generalization of the class of Randall­Sundrum-type brane-world models has been given [317, 324]. The dynamical equa­tions on the brane differ from the GR equations in that there are local (quadratic) matter fields corrections which are significant at very high energies (and hence in the early universe), and there are nonlocal effects from the free gravitational field in the bulk, transmitted via the projection of the five-dimensional bulk Weyl tensor, that contribute further corrections to the Einstein equations on the brane [261]. Recently, Randall-Sundrum-type brane-world cosmological models with a non­vanishing bulk Weyl tensor have been analysed using dynamical systems methods [22,74,75,94,95,160,316,331,348].

174

Acknowledgments

This book is primarily based on work done in collaboration with Andrew Billyard, Chris Clarkson, Martin Goliath, Robert van den Hoogen, Jesus Ibanez, Jim Lid­sey, VIf Nilsson, Itsaso Olasagasti, Stephen Quinlan and John Wainwright, which was supported, in part, by the Natural Sciences and Engineering Research Council of Canada. I would particularly like to thank Andrew Billyard, Chris Clarkson, Clyde Clements, Maria Fe Elder and Robert van den Hoogen for help in preparing this manuscript. I would also like to thank the Institute of Physics for their kind permission to reproduce Figs. (4), (5) and (6).

175

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