Table Of ContentFormation and In teractions of
Topological Defects
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Series B: Physics
Formation and Interactions of
Topological Defects
Edited by
Anne-Christine Davis
University of Cambridge
Cambridge, England
and
Robert Brandenberger
Brown University
Providence, Rhode Island
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Proceedings of a NATO Advanced Study Institute on
Formation and Interactions of Topological Defects,
held August 22--september 2, 1994,
in Cambridge, England
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"Proceedings of a NATO Advanced Study Institute an Formation and
Interactions of Topological Defects, held August 22-September 2,
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Includes bibliographical references and index.
ISBN 978-1-4613-5767-4 ISBN 978-1-4615-1883-9 (eBook)
DOI 10.1007/978-1-4615-1883-9
1. Topology--Congresses. 2. Cos.ology--Mathe.atics--Congresses.
3. Phase transitions (Statistical physics)--Congresses.
4. Mathematical physics--Congresses. 1. Davis, Anne-Christine.
II. Brandenberger, Robert Hans. III. North Atlantic Treaty
Organization. Scientific Affairs Oivision. IV. NATO Advanced Study
Institute an For.ation and Interactions of Topological Defects (1994
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Preface
Topological defects have recently become of great interest in condensed matter physics,
particle physics and cosmology. They are the unavoidable remnants of many symmetry
breaking phase transitions. Topological defects can play an important role in describing
the properties of many condensed matter systems (e.g. superfluids and superconduc
tors); they can catalyze many unusual effects in particle physics models and they may
be responsible for seeding the density perturbations in the early Universe which de
velop into galaxies and the large-scale structure of the Universe. Topological defects
are also of great interest in mathematics as nontrivial solutions of nonlinear differential
equations stabilized by topological effects.
The purpose of the Advanced Study Institute "Formation and Interactions of Topo
logical Defects" was to bring together students and practitioners in condensed matter
physics, particle physics and cosmology, to give a detailed exposition of the role of topo
logical defects in these fields; to explore similarities and differences in the approaches;
and to provide a common basis for discussion and future collaborative research on
common problems.
This Advanced Study Institute was part of a six-month programme on topological
defects sponsored by the Newton Institute for Mathematical Sciences in Cambridge,
England, organized by Professors T Kibble, A Bray and R Ward. The Advanced Study
Institute was held from August 22-September 2 1994. We are grateful for the generous
NATO sponsorship of this school. We wish to thank the European Union and the
Leverhulme Trust for additional support. We acknowledge our gratitude to the Newton
Institute for hosting the ASI, and to Professor P Goddard and the entire staff of the
Newton Institute for their help in organizing the workshop and for making sure it ran
smoothly.
Special thanks go to the lecturers and participants of this ASI for their enthusiasm
and for generating a stimulating atmosphere. Finally, we wish to thank Lin Hardiman,
Adrian Martin and Mark Trodden for their administrative and editorial assistance.
Anne-Christine Davis
Robert Brandenberger
v
CONTENTS
Phase Transitions in the Early Universe and Defect Formation 1
T.W.B. Kibble
The Topological Classification of Defects ............................. 27
M. Kleman
Introduction to Growth Kinetics Problems ............................ 63
G.F. Mazenko
Dynamics of Cosmological Phase Transitions: What Can We Learn
from Condensed Matter Physics? ................................ 93
N. Goldenfeld
Topological Defects and Phase Ordering Dynamics ................... 105
A.J. Bray
The Production of Strings and Monopoles at Phase Transitions ....... 139
R.J. Rivers and T.S. Evans
Geometry of Defect Scattering ....................................... 183
N.S. Manton
Theory of Fluctuating Nonholonomic Fields and Applications:
Statistical Mechanics of Vortices and Defects and
New Physical Laws in Spaces with Curvature and Torsion ...... 201
H. Kleinert
String Network Evolution ............................................ 233
E.P.S. Shellard
Global Field Dynamics and Cosmological Structure Formation ........ 255
R. Durrer
vii
Electroweak Baryogenesis ............................................ 283
N.G. Turok
Dynamics of Cosmic Strings and other Brane Models ................. 303
B. Carter
Cosmological Experiments in Superfluids and Superconductors ....... 349
W.H. Zurek
Cosmological Experiments in Liquid 4He - Problems and Prospects ... 379
P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock
and C.D.H. Williams
Index ............................................................... 389
viii
PHASE TRANSITIONS IN THE EARLY UNIVERSE AND DEFECT
FORMATION
T.W.B. Kibble
Blackett Laboratory, Imperial College, London SW7 2BZ, UK, and
Isaac Newton Institute for Mathematical Sciences,
20 Clarkson Road, Cambridge CB3 OER, UK
Abstract. The currently accepted standard models of cosmology and par
ticle physics, taken together, imply that early in its history the Universe
underwent a series of symmetry-breaking phase transitions, at which topo
logically stable defects may have been formed. I review the types of defects
that may appear - domain walls, cosmic strings, monopoles, and combi
nations of these - the conditions under which they arise and the question
of what determines their initial number density.
1. INTRODUCTION
The possibility that some of the observed large-scale features of the Universe today
may be due to events in the first fraction of a second after the Big Bang is an exciting
one. The interface between particle physics and cosmology has become one of the most
rapidly advancing areas of physics. One popular idea is that topologically stable defects,
such as cosmic strings, may have been formed at very early phase transitions, and that
some of these may have survived long enough to influence what we see today. This is a
most intriguing notion, which will be the subject not only of my lectures but of several
others.
Topological defects appear in a wide variety of physical systems. This Advanced
Study Insitute brings together field theorists, condensed-matter physicists, particle
physicists and cosmologists. Professor KIernan and others will be describing the for
mation of defects in condensed matter systems. My job is to set the scene so far as
cosmology is concerned. In these lectures I shall try to explain why we think there was
Formation and Interactions oj Topological Dejects. Edited by
A.-C. Davis and R. Brandenberger, Plenum Press, New York, 1995
a sequence of phase transitions in the early universe, and to answer questions such as:
What sorts of defects may appear in these transitions? How are they formed? And in
what numbers? Their subsequent evolution will be covered by later lecturers.
The first lecture will be largely introductory. I shall begin by summarizing very
briefly the presently accepted standard models of cosmology and particle physics. Then
I will show how putting them together leads inevitably to the conclusion that very
shortly after the Big Bang the Universe underwent a sequence of high-temperature
phase transitions, in which an original symmetry is broken in successive stages. To lead
into the formation of defects, I will discuss first the simplest possible case, involving
a single real scalar field, with a symmetry-breaking transition at which domain walls
form. I will explain how the way in which the defects form is governed by the nature
of the phase transition, in particular whether it is first- or second-order.
In my second lecture, I shall consider the problem of defect formation in the
more general context of a non-Abelian gauge theory spontaneously broken by a scalar
field in some given representation, showing how the topology of the vacuum manifold
controls the possible types of defects - domain walls, cosmic strings or monopoles.
This discussion parallels much of what Professor Kleman had to say in the condensed
matter context. 1 I shall also discuss how one can estimate the likely density of defects
immediately after the transition, using the particular example of cosmic strings.
My third lecture will be concerned with somewhat more exotic types of defects, in
particular composite defects that may be formed when there are two or more successive
transitions, such as strings that form the boundaries of domain walls or monopoles that
become joined by strings. Similar objects are known to appear in condensed-matter
systems. I shall also discuss some very interesting recent work on 'semi-local' strings,
stable defects that may form despite the fact that the usual topological condition for
stability is not met. Similar objects may playa role in the electro-weak theory, especially
in connection with the problem of baryogenesis - understanding the origin of the
observed matter-antimatter asymmetry. Whether there is any analogue in condensed
matter I do not know.
2. BIG BANG COSMOLOGY
Modern cosmology is founded on two observational pillars, the cosmic red-shift and the
microwave background.2
In 1926 Edwin Hubble observed that there is an approximately linear relation
between the red-shift of light from distant galaxies and their estimated distance. If this
red-shift is interpreted in the natural way as a Doppler shift, this means that galaxies
are receding from us with a speed proportional to distance:
v=Hr. (1)
The Hubble parameter H (often called the Hubble constant, but that is misleading since
it changes with time) is usually expressed in terms of a dimensionless number h as
H = 100h km S-1 Mpc-l. (2)
Observationally, h lies between 0.5 and 1. The inverse of H defines a time scale for
expansion:
(3)
2
Since the galaxies are receding from us (and equally of course from each other), it follows
that some ten billion years ago the Universe was much denser and therefore hotter than
it now is. Following its evolution back in time, one would see the galaxies coming
together and merging into hot gas. The early Universe contained mainly hydrogen gas,
expanding and cooling more or less adiabatically.
This brings us to the second observational pillar. Hot gas radiates, and the radia
tion emitted when it was dense and hot should still be present today. Since it has been
subjected to adiabatic expansion, it should still have a blackbody spectrum, red-shifted
to the microwave region of the spectrum. This cosmic microwave background radiation
(CMBR) had been predicted, but was found accidentally in 1965 by Arno Penzias and
Robert Wilson, who were trying to discover the source of the ubiquitous interference
on a new microwave antenna. The spectrum of the CMBR has recently been measured
with unparalleled accuracy by the COBE satellite. They found a beautiful blackbody
spectrum with temperature3
T = 2.726 ± 0.010 K. (4)
This observation provides very convincing confirmation of the idea that the early
Universe was hot. The near-isotropy of the CMBR temperature (to within a few parts
in 105 once the relative motion of the Earth has beeen allowed for) also confirms that on
a large scale the Universe was nearly homogeneous and isotropic. On galactic scales of
course matter is very inhomogeneous, clumped in stars, galaxies and galactic clusters.
But the density contrast grows steadily less as one goes to larger scales, so it seems a
good approximation on the largest scales to assume homogeneity and isotropy.
If we do make that approximation, the space-time can be described by the
Robertson-Walker metric,
(5)
Here K is a constant determining the curvature of the three-metric; the expression
within large parentheses is the metric of a three-space of uniform curvature. The func
tion a depends on time via Einstein's equations, here called the Friedmann equation,
H2 = a2 = 87rG P _ K (6)
a2 3 a2'
where p is the mass or energy density of the Universe. (I use units in which c = Ii =
kB = 1.)
This equation must be supplemented by an equation giving the rate of change of
p:
p = -3~(p + p), (7)
a
where p is the pressure. This is really the first law of thermodynamics - it may also
be written as
d (3) d 3
dt pa = -p dt (a ). (8)
We also need an equation of state relating p to p.
The nature of the solutions is largely determined by the sign of K (see Fig. 1). If
K:::; 0, a will never vanish, so the Universe will continue to expand for ever. For K < 0,
a
reaches a limiting value, ../-K, while if K = 0, the Universe continues to expand but
3
