MECHANISMS OF REHEATING AFTER INFLATION
a dissertation
submitted to the department of physis
and the ommittee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
dotor of philosophy
Gary Felder
April 2001
Copyright by Gary Felder 2001
All Rights Reserved
ii
I ertify that I have read this dissertation and that in
my opinion it is fully adequate, in sope and quality, as
a dissertation for the degree of Dotor of Philosophy.
Andrei Linde
(Prinipal Adviser)
I ertify that I have read this dissertation and that in
my opinion it is fully adequate, in sope and quality, as
a dissertation for the degree of Dotor of Philosophy.
Savas Dimopoulos
I ertify that I have read this dissertation and that in
my opinion it is fully adequate, in sope and quality, as
a dissertation for the degree of Dotor of Philosophy.
Robert Wagoner
Approved for the University Committee on Graduate
Studies:
iii
Abstrat
For the last two deades the dominant paradigm in early-universe osmology has
been the theory of ination. Aording to this theory the universe underwent a
brief period of exponential expansion during whih it beame nearly homogeneous on
sales muh larger than those we an observe today. The theory also states that the
density of matter during this period nearly vanished, and all the matter we observe
in the universe today was produed after ination in a proess known as reheating.
Although the basi mehanisms of ination are well understood and aepted, many
unresolved problems remain onerning the prodution of matter after ination and
the subsequent transition to a more traditional, hot big bang model of the evolution
of the universe. These problems are partiularly diÆult beause we do not know the
orret eetive theory of physis for the energy sales relevant to inationary theory,
and beause most models that we an onsider involve non-perturbative, nonlinear
interations that an only be fully studied using numerial tehniques.
The researh desribed here onsists of a ombination of analytial and numerial
work aimed at desribing the possible mehanisms of reheating. My ollaborators
and I have studied the eets of parametri resonane, symmetry breaking and phase
transitions, gravitational partile prodution, tahyoni (aka spinodal) instability, and
a novel mehanism of reheating that we all "instant preheating." We investigated
these eets in the ontext of single eld ination models with polynomial potentials,
single eld models with no minima, and multi-eld hybrid ination models. We found
that many inationary models ould be ruled out on the basis of osmologially unaeptable onsequenes suh as domain walls, moduli elds, or exessive isourvature
utuations. Conversely we found that some seemingly impossible models ould be
iv
resued by means of a seondary stage of ination, or by instant preheating.
Muh of this work required the use of numerial alulations, and in many ases
only full sale lattie simulations ould apture the nonlinear dynamis. Over the
past several years I have been one of the two developers of a C++ lattie program
for simulating the evolution of interating salar elds. This program, alled LATTICEEASY, has been a key element of the researh desribed in this thesis. The
program is now available on the World Wide Web at
http://physis.stanford.edu/gfelder/lattieeasy/.
v
Aknowledgements
First and foremost I would like to thank my advisor Andrei Linde, who has been a
onstant soure of support to me both on an aademi and a personal level. I was
drawn to him by his love of physis, his profound intuition, his interest in the deepest
questions in our eld, and his willingness to share that interest with me and others.
Sine I rst formed those impressions I have never been disappointed by him. I also
sinerely thank Lev Kofman, who has funtioned very muh as a seond advisor to
me through muh of my graduate areer. In addition to working with me on many
projets over the years he also graiously invited me to ome work with him for a
year in Toronto, whih was an invaluable experiene for me both sientially and
personally. I thank the Canadian Institute for Theoretial Astrophysis for their
hospitality and for providing a stimulating and pleasant working environment. I also
thank Lev and CITA for oering me a postdo position, whih I was delighted to
aept.
Thanks are also due to the many other great physiists that have helped me
throughout this proess, inluding Patrik Greene, Igor Tkahev, Juan Garia-Bellido,
and Nemanja Kaloper. These people have ollaborated with me and provided useful
sounding boards for both my ideas and for my onfusion. Thanks go to Robert
Wagoner for putting up with my inessant questions, whih he was always willing to
take time for. And for making my eduation at Stanford many times more useful and
abundantly more enjoyable than it would have been without them I would like to
thank my study partners Aaron Birh, Travis Brooks, Amanda Weinstein, and Yue
Chen. Speial thanks go to Aaron Miller for a seemingly endless supply of rides and
other favors, to Yonatan Zunger for keeping my omputer running all these years,
vi
and to Danna Rosenberg for being there for me when I most needed it. Thanks go
also to Maria Keating, who helped me out in almost every aspet of learning to deal
with the bureauray and regulations at Stanford.
To Mom and Dad, Marty and Rebea, Kenny and Elena, and all the other
members of my family I want to say that I have never in my life known a more
loving and supportive family than ours. The opportunity to live near my sister Elena
has been one of the greatest joys of being in graduate shool. And to Rosemary
MNaughton, my personal andidate for the most wonderful woman in the world,
thank you for being there for me in so many ways.
I dediate this work to the memory of Johnny Charlton: Dear friend, you will
always be missed.
vii
viii
We are a bit of stellar matter gone wrong. We are physial mahinery puppets that strut and talk and laugh and die as the hand of time pulls
the strings beneath. But there is one elementary inesapable answer. We
are that whih asks the question.
-Sir Arthur Eddington
As an adolesent I aspired to lasting fame, I raved fatual ertainty, and
I thirsted for a meaningful vision of human life - so I beame a sientist.
This is like beoming an arhbishop so you an meet girls.
- Matt Cartmill
Let's start at the very beginning. A very good plae to start.
- Julie Andrews in \The Sound of Musi"
ix
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Contents
Abstrat
iv
Aknowledgements
vi
I Introdution
1
1 Motivation
2
2 Outline of the Thesis
5
3 A Primer on Ination
3.1 The Big Bang Model . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Overview of the Model . . . . . . . . . . . . . . . . .
3.1.2 Initial Condition Problems . . . . . . . . . . . . . . .
3.1.3 Reli Problems . . . . . . . . . . . . . . . . . . . . .
3.1.4 Summary of the Problems With the Big Bang Model
3.2 Ination . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Why Ination Ours . . . . . . . . . . . . . . . . . .
3.2.2 Some Basi Consequenes of Ination . . . . . . . . .
3.3 Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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II Mehanisms of Reheating
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4 Introdution to the Researh
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5 Instant Preheating
5.1 Introdution . . . . . . . . . . . . .
5.2 Instant preheating: The basi idea .
5.3 The simplest models . . . . . . . .
5.4 Fat wimpzillas . . . . . . . . . . . .
5.5 Instant Preheating and NO Models
5.6 Conlusions . . . . . . . . . . . . .
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6 Ination and Preheating in NO Models
6.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Isourvature perturbations in NO Models . . . . . . . . . . . . . . . .
6.3 Cosmologial prodution of gravitinos and moduli elds . . . . . . . .
6.4 Saving NO models: Instant preheating . . . . . . . . . . . . . . . . .
6.4.1 Initial onditions for ination and reheating in the model with
2
interation g2 2 2 . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Instant preheating in NO models . . . . . . . . . . . . . . . .
6.5 Other versions of NO models . . . . . . . . . . . . . . . . . . . . . . .
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7 Gravitational Partile Prodution and the Moduli Problem
7.1 Chapter Abstrat . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Moduli problem . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Generation of light partiles from and after ination . . . . .
7.5 Light moduli from ination . . . . . . . . . . . . . . . . . . .
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8 Ination After Preheating
8.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Theory of the phase transition . . . . . . . . . . . . . . . . . . . . .
8.3 Simulation Results and Their Interpretation . . . . . . . . . . . . .
8.4 Nonthermal phase transitions and prodution of topologial defets
8.5 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
50
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9 The Development of Equilibrium After Preheating
9.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Calulations in Chaoti Ination . . . . . . . . . . . . . . . . . .
9.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 The Output Variables . . . . . . . . . . . . . . . . . . .
9.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Other Models of Ination and Interations . . . . . . . . . . . .
9.3.1 Variations on Chaoti Ination With a Quarti Potential
9.3.2 Chaoti Ination with a Quadrati Potential . . . . . . .
9.3.3 Hybrid Ination . . . . . . . . . . . . . . . . . . . . . . .
9.4 The Onset of Chaos, Lyapunov Exponents and Statistis . . . .
9.5 Rules of Thermalization . . . . . . . . . . . . . . . . . . . . . .
9.6 Disussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Dynamis of Symmetry Breaking and Tahyoni Preheating
10.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Tahyoni Instability and symmetry breaking . . . . . . . . . .
10.2.1 Quadrati potential . . . . . . . . . . . . . . . . . . . . .
10.2.2 Cubi potential . . . . . . . . . . . . . . . . . . . . . . .
10.3 Tahyoni preheating in hybrid ination . . . . . . . . . . . . .
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III LATTICEEASY: Lattie Simulations of Salar Field
Dynamis
138
11 Introdution to LATTICEEASY
139
11.1 Lattie Simulations and Cosmology . . . . . . . . . . . . . . . . . . . 140
11.2 My Contributions to LATTICEEASY . . . . . . . . . . . . . . . . . . 141
12 Equations
143
12.1 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.1.1 Field Equations and Coordinate Resalings . . . . . . . . . . . 143
12.1.2 Sale Fator Evolution . . . . . . . . . . . . . . . . . . . . . . 144
xiii
12.2 Initial Conditions on the Lattie . . . . . . . . . . . . . . .
12.2.1 Initial Conditions for Field Flutuations . . . . . .
12.2.2 Initial Conditions for Field Derivative Flutuations
12.2.3 Standing Waves . . . . . . . . . . . . . . . . . . . .
12.2.4 The Adiabati Approximation . . . . . . . . . . . .
12.3 Renormalization . . . . . . . . . . . . . . . . . . . . . . . .
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13 Computational Methods Used by LATTICEEASY
13.1 Time Evolution: The Staggered Leapfrog
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Corretion to the Sale Fator Evolution Equation to Aount for Staggered Leapfrog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Spatial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 The Auray of the Simulations . . . . . . . . . . . . . . . . . . . .
13.4.1 The Classial Approximation . . . . . . . . . . . . . . . . . .
13.4.2 Numerial Auray . . . . . . . . . . . . . . . . . . . . . . .
155
IV Conlusion: My Philosophial Ramblings
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Bibliography
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xiv
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List of Figures
7.1 Flutuations vs. mode frequeny at a late time. The lower plots show
runs that started lose to the end of ination, with initial onditions
fk = p12k e ikt . The starting times range from 0 = 1:5Mp (lowest
urve) up through 0 = 2Mp. The highest urve shows the Hankel
funtion solutions given by eqs. (7.20) and (7.21) As we see, alulations with fk = p12k e ikt produe the orret spetrum at large k but
underestimate the level of quantum utuations at small k. . . . . . .
69
8.1 The spatial average of the ination eld as a funtion of time. The
eld is shown in units of v , the symmetry breaking parameter. Time
is shown in Plank units. . . . . . . . . . . . . . . . . . . . . . . . . .
84
8.2 The spatial average of the ination eld as a funtion of time in the
viinity of the phase transition. The left gure shows the eld just
before the phase transition and at the moment of the transition. The
eld osillates with an amplitude approahing 10 3 v . The right gure
shows the eld after the phase transition, when it osillates near the
(time-dependent) position of the minimum of the eetive potential at
v . Time is shown in Plank units. . . . . . . . . . . . . . . . . .
86
8.3 These plots show the region of spae in whih symmetry breaking has
ourred at four suessive times. . . . . . . . . . . . . . . . . . . . .
87
p
8.4 The sale fator a as a funtion of time. In the beginning a t,
whih is a urve with negative urvature, but then at some stage it
begins to turn upwards, indiating a short stage of ination. . . . . .
xv
88
8.5 The seond derivative of the sale fator, a. A universe dominated by
ordinary matter (relativisti or nonrelativisti) will always have a < 0,
whereas in an inationary universe a > 0. We see that starting from
the moment t 2 1012 (in Plank units) the universe experienes
aelerated (inationary) expansion. . . . . . . . . . . . . . . . . . . .
8.6 The ratio of pressure to energy density p=. (Values were time averaged
over short time sales to make the plot smoother and more readable.)
9.1 Number density n for V = 41 4 + 12 g 22 2 . The plots are, from bottom
to top at the right of the gure, n , n , and ntot . The dashed horizontal
line is simply for omparison. The end of exponential growth and the
beginning of turbulene (i.e. the moment t ) ours around the time
when ntot reahes its maximum. . . . . . . . . . . . . . . . . . . . . .
9.2 Evolution of the spetrum of in the model V = 14 4 + 12 g 22 2 . Red
plots orrespond to earlier times and blue plots to later ones. For blak
and white viewing: The sparse, lower plots all show early times. In
the thik bundle of plots higher up the spetrum is rising on the right
and falling on the left as time progresses. . . . . . . . . . . . . . . . .
2
9.3 Varianes for V = 41 4 + 12 g 2 2 2 . The upper plot shows h i
and the lower plot shows h( )2 i. . . . . . . . . . . . . . . . . . .
9.4 Eetive masses for V = 41 4 + 12 g 2 2 2 as a funtion of time in units
of omoving momentum. The lower plot is m and the upper one is m .
9.5 Time evolution of the eetive masses for the model V = 14 4 +
1 2 2 2 1 2 2 2
2 g + 2 h . From bottom to top on the right hand side the
plots show m , m , and m . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Evolution of varianes of elds in the model (9.25). The two elds that
grow at late times, in order of their growth, are and Im(). . . . .
9.7 The Lyapunov exponent for the elds (lower urve) and (upper
urve). The vertial axis is t. . . . . . . . . . . . . . . . . . . . . . .
9.8 The Lyapunov exponent 0 for the elds and using the normalized
distane funtion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
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9.9 The probability distribution funtion for the eld after preheating.
Dots show a histogram of the eld and the solid urve shows a best-t
Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.10 Deviations from Gaussianity for the eld as a funtion of time. The
solid, red line shows 3hÆ2i2 =hÆ4 i and the dashed, blue line shows
3hÆ _ 2i2 =hÆ _ 4 i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.11 Deviations from Gaussianity for the eld as a funtion of time. The
solid, red line shows 3hÆ2 i2 =hÆ4 i and the dashed, blue line shows
3hÆ _ 2 i2 =hÆ _ 4 i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.12 V = 1=44 . (Note that the vertial sale is larger than for the subsequent plots.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.13 V = 1=44 + 1=2g 22 2 , g 2 = = 200. The upper urve represents n . 120
9.14 V = 1=44 + 1=2g 22 2 + 1=2h22 2 , g 2 = = 200, h2 = 100g 2. The
highest urve is n . The number density of diminishes when n grows.120
9.15 V = 1=44 + 1=2g 22 2 + 1=2h2i 2 i2 , g 2 = = 200, h21 = 200g 2; h22 =
100g 2. The pattern is similar to the three-eld ase until the growth
of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.16 V = 1=44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.17 V = 1=44 + 1=2g2 2 2 , g2 = = 200. The spetra of and are nearly
idential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.18 V = 1=44 + 1=2g2 2 2 + 1=2h2 2 2 , g2 = = 200, h2 = 100g2 The and
spetra are similar, but rises in the infrared). The spetrum of is
markedly dierent from the others. . . . . . . . . . . . . . . . . . . . . . 121
9.19 V = 1=44 + 1=2g2 2 2 + 1=2h2i 2 i2 , g2 = = 200, h21 = 200g2 ; h22 = 100g2 .
All elds other than the inaton have nearly idential spetra. . . . . . .
9.20 V = 1=2m2 2 + 1=2g 22 2 , g 2 Mp2 =m2 = 2:5 105 . The upper urve
represents n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.21 V = 1=2m2 2 + 1=2g 22 2 + 1=2h2 2 2 , g 2 Mp2 =m2 = 2:5 105 , h2 =
100g 2. The highest urve is n . The eld that grows latest is . . . .
9.22 V = 1=2m2 2 + 1=2g 22 2 , g 2 Mp2 =m2 = 2:5 105 . The upper urve
represents the spetrum of . . . . . . . . . . . . . . . . . . . . . . .
xvii
121
122
122
122
9.23 V = 1=2m2 2 + 1=2g 22 2 + 1=2h2 2 2 , g 2 Mp2 =m2 = 2:5 105 , h2 =
100g 2 The and spetra are similar, while the spetrum of rises
muh higher in the infrared. . . . . . . . . . . . . . . . . . . . . . . . 122
10.1 The proess of symmetry breaking in the model (10.1) for = 10 4 .
In the beginning the distribution is very narrow. Then it spreads out
and shows two maxima that osillate about = v with an amplitude
muh smaller than v . These maxima never ome lose to the initial
point = 0. The values of the eld are shown in units of v . . . . . . .
10.2 Growth of quantum utuations in the proess of symmetry breaking
in the quadrati model (10.1). . . . . . . . . . . . . . . . . . . . . .
10.3 The development of domain struture in the quadrati model (10.1).
10.4 The proess of symmetry breaking in the model (10.1) for a omplex
eld . The eld distribution falls down to the minimum of the eetive
potential at jj = v and experienes only small osillations with rapidly
dereasing amplitude jj v . . . . . . . . . . . . . . . . . . . . . .
10.5 The distribution of strings in the model (10.1) for a omplex eld . .
10.6 Fast growth of the peaks of the distribution of the eld in the ubi
model (10.4). It should be ompared with Fig. 10.2 for the quadrati
model (10.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Histograms desribing the proess of symmetry breaking in the model
(10.4) for = 10 2 . After a single osillation the distribution aquires
the form shown in the last frame and after that it pratially does not
osillate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 The proess of symmetry breaking in the hybrid ination model (10.6)
for g 2 . The eld distribution moves along the ellipse g 22 + 2 =
g 22 from the bifuration point = , = 0. . . . . . . . . . . . . .
xviii
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136
Part I
Introdution
1
Chapter 1
Motivation
The topi of this thesis is reheating after ination. Put in a more general way, I intend
to talk about where all the matter we see in the universe (inluding dark matter) ame
from. Cosmologists today generally believe that this matter was produed about
15 billion years ago following a period of rapid expansion known as ination. The
theory of ination, inluding this period of matter prodution known as reheating, is
a modiation of the standard big bang model of osmology developed early in this
entury by Einstein, Hubble, Friedmann, and others.
Muh of the thesis is fairly tehnial and assumes a basi knowledge of general
relativity and quantum eld theory. For the sake of ompleteness, however, I have
written a self-ontained \ination primer" outlining the basi struture of the big
bang model, the motivations for ination as a modiation of that model, and some
of the key ideas in the theory of ination and reheating.
Subsequent hapters will explain work I have done exploring dierent mehanisms
by whih reheating an our. The wide variety of models being onsidered is a
reetion of our urrent state of ignorane about the early universe. On the one hand
we have no diret observational knowledge of any proesses that ourred before the
time of nuleosynthesis, i.e. the formation of nulei of the light elements suh as
deuterium, helium, and lithium. Ination was ertainly over long before this period.
On the other hand we have very little theoretial guidane as to what model to use for
desribing physis at the high energy sales that prevailed during the time of ination.
2
3
Most theorists agree that the standard model of partile physis that desribes all of
physis at the energy sales we an observe is almost ertainly not valid at energy
sales muh higher than our urrent observations. We have many ideas about possible
elements of a theory of high energy physis, inluding supersymmetry, Grand Unied
Theories, and string theory, but no spei model has been formulated and tested for
physis at high energies.1
So the physis of the early universe has not been observed, and the theory desribing it is not known. Given suh a dearth of knowledge it might seem that researh
into ination would be hopeless. It turns out, however, that many results an be
formulated that seem to be generi aross a wide spetrum of possible models. Some
of the generi preditions of ination have already been tested and others are likely to
be tested in the next 5-10 years, mostly by experiments testing in detail the properties
of the mirowave bakground radiation.
Moreover, a lot of important work has been done and ontinues to be done exploring the parameter spae of possible inationary models. On the observational side
we an use urrently available data to onstrain the spae of inationary models, and
over the last deade this tehnique has allowed us to eliminate many andidate models. On the theoretial side we an use ideas suh as supersymmetry and supergravity
to guide our searhes for inationary models and explore some of the onsequenes
of models that inorporate those theoretial ideas. As better experimental and theoretial results beome available our modeling will be more tightly onstrained. Of
ourse the hope is to eventually have a lear theoretial knowledge of what elds and
interations exist at high energies, use our knowledge of ination to make detailed,
testable preditions within that model, and have the experimental sophistiation to
test those preditions. We are still a long way from aomplishing any of those three
goals, but progress is being made on all three.
1 By
high energies I mean anything from the sale of our urrent observations, roughly 1000 GeV,
up to the Plank sale at about 1019 GeV. Within this range it is still likely that physis is desribed
by some quantum eld theory, but we don't know the spei elds and interations involved.
At energies above the Plank sale quantum eld theory itself may break down beause quantum
utuations of spaetime beome important. Unless I speify otherwise I will always use "high
energy theory" to mean something in the range from 103 to 1019 GeV. Essentially all inationary
models involve typial energy sales within this range.
4
CHAPTER 1.
MOTIVATION
Given all of this, I view the role of inationary theorists in the following way. We
need to understand what mehanisms and preditions hold generially in ination.
We need to explore the spae of models and gure out what observational signatures,
if any, would allow us to distinguish between these models. Finally we need to develop
tehniques of analysis, both analytial and numerial, that are general enough to be
of use as the theoretial and observational foundation of our work hanges. It is in
the spirit of these goals that the work presented here was done.
Chapter 2
Outline of the Thesis
This thesis is divided into three main setions. The rst setion ontains general
introdutory material (inluding this outline), followed by a short ination primer.
The material in the primer does not assume any knowledge of advaned physis and
it should serve as a general introdution for anyone not familiar with the eld of
osmology. It desribes the standard big bang theory that has held sway for most of
this entury, why many people ame to believe this theory was in need of modiation,
and how inationary theory has been able to address some of the problems with the
standard big bang osmology. Finally, the introdution ends with a brief disussion
of reheating, the proess by whih we believe all the matter in the observable universe
was generated.
The following setion, part II, onstitutes the main bulk of the thesis. The hapters
in this setion are adapted from the papers that I have o-authored as part of my
graduate researh. In general, these hapters desribe dierent mehanisms by whih
reheating an our and the onsequenes of those dierent proesses. A more detailed
outline of the material in the hapters and my spei role in the dierent aspets of
the researh is presented at the beginning of part II.
A good deal of the researh desribed here has been done with the help of lattie
simulations. Building on a program originally written by Igor Tkahev I developed a
C++ program for performing suh simulations of elds in the early universe. Muh
of my graduate work has been devoted to developing this program. This work has
5
6
CHAPTER 2.
OUTLINE OF THE THESIS
inluded aspets of physis, mathematis, and programming. The nal setion of the
thesis is thus a desription of these lattie simulations. The website I reated for the
lattie program ontains detailed doumentation on the internal funtioning and the
use of the program itself. In this thesis I onentrate instead on the physis underlying
the simulations. I also desribe in more detail there the separate ontributions that
Dr. Tkahev and I eah made to this work.
Chapter 3
A Primer on Ination
3.1 The Big Bang Model
3.1.1 Overview of the Model
For most of this entury our view of the large sale struture and history of the
universe has been dominated by the big bang model. Aording to this model the
universe at early times was a nearly homogeneous expanding olletion of high energy
partiles in thermal equilibrium. As the universe expanded and ooled very small
inhomogeneities were then amplied by gravity and ollapsed to form the strutures
we see today suh as lusters, and galaxies. Extrapolating bakwards, on the other
hand, that homogeneous reball would have had higher temperatures and densities
at earlier times, ultimately reahing innite density at a time alled the big bang,
about 15 billion years ago.
This model is in perfet aord with the theory of general relativity, whih predits
that a universe with the initial onditions speied by the model will expand and ool
in exatly that way. Moreover there have been many observational onrmations of
the big bang model. These onrmations inlude the apparent motions of distant
objets relative to us, the eletromagneti radiation emitted in the early universe
and detetable now in the form of mirowaves, and the abundanes of light elements
predited by the theory of nuleosynthesis.
7
8
CHAPTER 3.
A PRIMER ON INFLATION
I want to fous in partiular on the latter of these. Nulear theory is well tested
and understood, and by applying this theory to a homogeneous, expanding medium at
high temperature we an predit what relative abundanes of hydrogen, deuterium,
helium, and lithium nulei should have emerged when these nulei were formed in
the early universe. These preditions aurately math the observational data. This
math is partiularly important beause it provides the earliest observational evidene
we have for the big bang model. We have no diret measurements of any proesses
that ourred before nuleosynthesis. So in that sense we are free to imagine any
deviations we wish from the big bang model before that point. However, sine it
seems unlikely that a very dierent senario would give the same preditions for these
abundanes, any suh models are onstrained to redue to the big bang senario by
the time of nuleosynthesis.
Given how suessful the big bang model has been in mathing essentially every
observation to date one might legitimately wonder why we would want to modify it at
all, rather than just aepting it straight bak to the moment of the big bang. Suh
a omplete extrapolation of the theory is not possible, however, beause of ertain
limitations of our theories of high energy physis. When we talk about extrapolating
bakwards in the big bang model we are referring to running the equations of general
relativity bakwards to earlier times and higher densities. We know, however, that
general relativity eases to be valid when we try to desribe a region of spaetime
whose density exeeds a ertain value known as the Plank density, roughly 1096 mkg3 .
If we try to onsistently apply quantum mehanis and general relativity at suh a
density we nd that quantum utuations of spaetime should be important, and we
have no theory that desribes suh a situation.
So the best we an do in using the big bang model to desribe the very early
universe is the following. At some point in the past the density of the universe was
above the Plank density. We don't know what physis governs suh a ase so we an
make no preditions based on it. Somehow this super-Plankian state (sometimes
alled spaetime foam) gave rise to at least one region of sub-Plankian density with
the right initial onditions to produe the universe we urrently see. From now on
when I refer to the \initial onditions" for our universe I will mean the state that the
3.1.
THE BIG BANG MODEL
9
observable part of the universe was in after the density rst beame sub-Plankian
and the universe (or at least this region of it) ould thus be desribed lassially.
Even in this more limited sense, however, there are ertain problems with extending the big bang model bak to the beginning of the universe. These problems an
be ategorized into \initial ondition problems" and \reli problems." I dene what
eah of these are in the following two setions.
3.1.2 Initial Condition Problems
These problems onsist of a number of seemingly ne-tuned aspets of the early
universe in the big bang model. For example, we know that the universe was almost
perfetly homogeneous at early times. Suh statements are, to one way of thinking,
not problems at all. As I disussed in the previous setion, we do not know what
physis gave rise to these initial onditions, so they are free parameters of the theory.
Put another way, we an imagine that somewhere in the ultra-high energy physis
theory that we don't yet know is an explanation for why our universe emerged from
the spaetime foam with exatly the right initial onditions to produe the kind of
universe we see. It would be more satisfying if we ould nd an explanation for these
features within known physis, but nature is not obliged to satisfy us. As it happens,
we do know of suh an explanation. Before disussing this solution, however, I want
to desribe some of these initial ondition problems.
Homogeneity
The universe we observe today is very lumpy on small sales, e.g. those of people or
galaxies, but on suÆiently large sales appears to be smooth and uniform. We know
from our observations of the mirowave bakground radiation that at muh earlier
times the universe was almost ompletely uniform even on smaller sales, with a mean
variation Æ= 10 5 . 1 If the initial onditions were random it might seem strange
that there should be suh good homogeneity, but of ourse the initial onditions were
1 The
Greek letter is often used to mean density. So the meaning of this equation is that the
energy density of any two points in the universe diered on average by less than one part in 100; 000.
10
CHAPTER 3.
A PRIMER ON INFLATION
presumably not random but set by some unknown physis. It's not hard to imagine
that some physial mehanism drove the universe to a state of high homogeneity.
However, this mehanism must have been imperfet beause the universe retained a
small amount of inhomogeneity. If it weren't for these small inhomogeneities there
would have been no seeds for galaxy formation and hene no struture formation. A
mehanism that would drive the universe to near-perfet uniformity while still leaving
these small but observable utuations seems a little stranger.
Flatness
General relativity relates the urvature of spaetime to the matter in that spaetime.
In the ase of an expanding, nearly homogeneous universe suh as ours the urvature
depends on the overall density of matter and energy. If this density is above a ritial
value then the universe will be \open," meaning parallel lines will diverge. If the density is sub-ritial the universe will be \losed," meaning parallel lines will onverge.
Finally, if the universe is exatly at ritial density the universe will be \at," i.e.
Eulidean. Typially the ratio of the density in the universe to the ritial density
moves away from one over time. A perfetly at universe will always remain so but
a urved universe will beome more urved over time, whether it's open or losed.
Currently we an measure that the universe is within a fator of two or three of
ritial density. (Based on reent measurements it's probably within about 10% of
ritial density.) For this to be true now =rit must have been 1 10 59 at the
time when the universe had Plank density. If the deviation had been a few orders
of magnitude greater than that the universe would either have reollapsed long ago,
or would have a nearly vanishing energy density by now. Was there some mehanism
that drove the universe to suh near-perfet (or possibly perfet) atness?
Horizon
This problem is related to the homogeneity problem desribed above. Aording to
the theory of relativity no ausal eet an propagate faster than the speed of light.
If we believe that the universe had a beginning a nite time ago, then there is a nite
3.1.
THE BIG BANG MODEL
11
radius aross whih ausal eets ould have propagated by now. At the time when the
mirowave bakground was emitted there should have been roughly a million ausally
disonneted regions within the region of the universe we an urrently observe. What
ould have aused suh strong homogeneity over suh a vast region before any ausal
signals ould have spread information throughout this spae?
3.1.3 Reli Problems
Another set of problems with the big bang model has to do with the prodution of
exoti partiles at high energies. By \high energies" in this ontext I mean energies
below the Plank sale but above the prevalent sale at nuleosynthesis. We don't have
a lear understanding of physis at these energies, but we do know some theoretial
ideas that are likely to be part of suh high energy theory, inluding Grand Unied
Theories (\GUT's"), supersymmetry, and supergravity. All suh theories tend to
inlude speies of partiles that an only be produed at energies far above what we
an produe today or what was present during nuleosynthesis. If we believe that the
big bang model was valid bak to the Plank era then the universe must have passed
through a stage where the energies were high enough for these partiles to have been
produed.
In many ases these partiles have long lifetimes. Some of them ould, if produed,
last until nuleosynthesis and spoil the preditions of light element abundanes. Others ould survive to the present and would be expeted to dominate the urrent energy
density. Partiles in either of these two groups are known as reli partiles beause
they persist from an earlier and higher energy epoh. For those familiar with theories
of partile physis, suh partiles inlude moduli, gravitinos, monopoles, and more.
3.1.4 Summary of the Problems With the Big Bang Model
As I said before, every testable predition of the big bang model to date has been
veried, and it is almost ertain that this model gives an aurate desription of the
universe bak at least as far as the time of nuleosynthesis. The earliest it ould
possibly be applied would be the Plank era. If this extrapolation were valid then we
12
CHAPTER 3.
A PRIMER ON INFLATION
would have to suppose that all the very ne-tuned initial onditions we observe suh
as homogeneity and atness were present from the beginning, presumably as a result
of some unknown quantum gravity eets. Even given this assumption, however, it
is unlear how the theory ould avoid the prodution of reli partiles that would
destroy the suessful desription it has made of the later universe.
It would be wonderful if a theory existed that with a minimum of assumptions
ould explain the initial onditions suh as atness and homogeneity, provide a ausal
mehanism for propagating information over all the seemingly ausally disonneted
regions in the early universe, eliminate all high energy reli partiles, and then segue
into the big bang model itself by the time of nuleosynthesis. Fortunately suh a
theory exists. It's known as ination.
3.2 Ination
The basi idea of ination is extremely simple, and has to do with the rate at whih
the universe is expanding. In the standard big bang model the universe experienes
\power-law expansion," meaning the distane between any two distant objets grows
like tp where t is the time sine the big bang and p is a number that depends on what
the universe is made of. Typially p will be either 1=2 or 2=3, or possibly something
in between the two. Aording to inationary theory, before this power law expansion
there was a brief period of exponential expansion. In other words distanes during this
time grew as eHt where t is one again time and H an be any number.2 Exponential
growth an be muh faster than power-law growth. During this time the universe
expanded by at least 60 e-folds, i.e. by a fator of at least e60 (roughly 1026 , or
nearly a billion billion billion). There are two obvious questions raised by this idea:
What mehanism would ause suh an expansion to our and what would be the
onsequenes if it did? In the next setion I will explain how ination an ome
about in the ontext of basi eld theory. I will argue that ination is a very natural
ourrene that may be expeted to take plae within a wide variety of high energy
2 The
symbol e just refers to a number, roughly 2:8, that is typially used for onveniene. You
ould also write this law as 10Ht and you would simply get a dierent value for H .
3.2.
13
INFLATION
physis models. In the following setion I will desribe some of the basi onsequenes
of ination, inluding the resolution of all the problems raised in the previous setion
on the big bang model.
3.2.1 Why Ination Ours
In this setion I am going to use a little more math and physis than in the rest of
the primer. For any non-tehnial readers who nd this setion onfusing you should
be able to skip it, take my word that ination does our naturally in the ontext of
many models of physis, and go on to the setion on the onsequenes of ination.
In general relativity the expansion of a homogeneous path of the universe is
desribed by the Friedmann equations
k
8
H2 + 2 =
a
3Mp2
a =
4
( + 3p) a:
3Mp2
(3.1)
(3.2)
The sale fator a is proportional to the distane between two objets omoving with
the expansion. The onstant k indiates the urvature of the universe; k = 1, 0,
or 1 for an open, at, or losed universe respetively. The Hubble parameter H is
dened as H aa_ and the other terms in the equations are the energy density , the
p
pressure p, and the Plank mass Mp = 1= G 1:22 1019 GeV .
Starting from the rst Friedmann equation, the seond one is equivalent to the
statement of energy onservation, dE = pdV , applied to a loal path of the universe. Taking E = V and V / a3 this energy onservation equation beomes the
equation of ontinuity
_ = 3H ( + p) :
(3.3)
For an equation of state of the form p = the solution to the equation of ontinuity
is
/ a 3(1+) :
(3.4)
For the two most ommon ases of nonrelativisti partiles (\matter") and relativisti
14
CHAPTER 3.
A PRIMER ON INFLATION
partiles (\radiation") the values of are 0 and 1=3, meaning that dereases as a 3
and a 4 respetively in these two ases. Note that in both of these ases the energy
density is dereasing faster than the urvature term in equation [3.1℄. This is why in
a matter or radiation dominated universe urvature tends to inrease.
What would happen if the universe were dominated by a form of energy with
= 1, i.e. p = ? In this ase the energy density wouldn't derease at all as
the universe expanded. The urvature would quikly beome negligible and equation
[3.1℄ would redue to
8
;
(3.5)
H2 =
3Mp2
whih for a onstant gives a onstant H . Realling that H a_ this equation implies
a
a = eHt ;
(3.6)
in other words ination.
Having said this, however, we are still faed with the question of what would give
rise to suh a strange equation of state. One possibility is a \osmologial onstant,"
i.e. an energy density assoiated with the vauum. Beause this density wouldn't
hange as the universe expanded (the vauum presumably still being the same) it
would behave as we have just desribed. (There are other, more diret ways to
show that in general relativity the only onsistent way to have a frame-independent
vauum energy would be for it to have the equation of state p = .) A osmologial
onstant term would not be a useful way to have ination, however, beause in this
ase ination would never end.
A simple way to mimi the eets of a osmologial onstant, however, would be
to have a potential dominated salar eld. The energy density and pressure for a
salar eld with potential V minimally oupled to gravity in a FRW universe are
1
1
= _ 2 + jrj2 + V
2
2
(3.7)
1
p = _ 2
2
(3.8)
1
jrj2
6
V:
3.2.
15
INFLATION
So if the dominant energy term is V the equation of state will be p = and the
eld will at like a osmologial onstant, maintaining a onstant energy density as
the universe expands. This behavior an be intuitively understood simply by noting
that for a given potential funtion V (), if is not hanging the energy density V
should not hange.
The term \ination" was oined by Alan Guth [1℄, who developed a model where
a salar eld was trapped in a loal minimum of its potential with V > 0. In suh a
ase the eld would remain potential dominated until it tunneled to its true minimum,
thus ending ination. Unfortunately this model suered from a number of problems.
For example, it failed to produe the right level of inhomogeneities. It was also far
from obvious what would have driven the eld to be trapped in this minimum in
the rst plae, so to some degree the model simply traded one set of ne-tunings for
another.
Sine Guth's paper ame out there have been many dierent versions of ination
devised, inluding \new ination," \haoti ination," \hybrid ination," \extended
ination," and on and on. Most plausible models, however, are based on some variant
of the following idea, rst developed by Andrei Linde in 1983 [2℄.
The evolution equation for a homogeneous salar eld in an expanding universe is
V
 + 3H _ +
= 0:
(3.9)
Note that H plays the role of a frition term in this equation, slowing the motion of
. If starts out at a value with a large potential, then by equation [3.1℄ H will be
large and will tend to roll very slowly. In this ase the potential term will dominate
over the kineti term and both V and H will vary slowly. The universe will thus
expand quasi-exponentially, i.e. ination will our. Eventually, however, will roll
to a small enough value that H will no longer overdamp the system, _ will beome
important, and ination will end.
Linde showed that for simple potentials suh as V / 2 or V / 4 suh ination
an our. If you assume that the eld had haoti initial onditions when the
universe rst began with sub-Plankian energy density then there will in general be
16
CHAPTER 3.
A PRIMER ON INFLATION
some path where the potential energy of dominates over the kineti and gradient
terms enough to ause ination. If in the entire universe there is one path, even a
Plank size one, with these onditions, then ination will drive the size of this path
to exponentially large values. Suh inationary regions will quikly ome to dominate
the volume of the universe. In fat, using the assumption of haoti initial onditions
an inationary region will typially expand by at least about 10107 times, and often
muh more.
So the requirements for having ination are fairly simple. Provided the quantum
eld theory governing sub-Plankian physis has a salar eld with a reasonable potential and somewhere in the universe a region exists where the potential energy of
that eld is large and dominates the overall energy density, ination will our in that
region. Regions where ination ours will then ll nearly all of spae. The salar
eld that drives ination in this way is often referred to as the \inaton". Note that
a \reasonable potential" inludes the polynomial potentials mentioned above as well
as others. At present we have never diretly observed a salar eld, but at least one
suh eld must exist in nature if the standard model is orret, and supersymmetry
predits the existene of many salar elds. So although we don't know the orret
fundamental physis model that would give rise to the inaton, it seems at the very
least reasonable to suppose that suh a eld might exist.
3.2.2 Some Basi Consequenes of Ination
The impat of ination on the initial ondition problems mentioned before is quite
simple. If the universe undergoes exponential expansion then any loal path of
it will ome to be very homogeneous and at, like the surfae of a balloon being
blown up very rapidly. Mathematially this an be seen from the argument above
explaining why urvature tends to grow in a matter or radiation dominated universe
but derease in an inationary universe. Given the enormous expansion that ours
during ination, the growth of the urvature that has ourred sine then should be
ompletely negligible and we would expet to see an essentially perfetly at universe
now.
3.2.
INFLATION
17
Ination an also solve the horizon problem. The statement that the observable
universe at the Plank era onsisted of many ausally disonneted regions omes from
extrapolating bak assuming matter and/or radiation domination. During ination,
however, a single, Plank size, ausally onneted region an expand to beome many
times bigger than the urrent observable universe, thus reating ausal orrelations
on sales muh bigger than we an hope to observe.
Ination also solves the problem of reli partiles beause during ination the
density of all partiles will be exponentially suppressed. When ination ends and the
universe moves towards a state of thermal equilibrium, the temperature may be low
enough to avoid reproduing these reli partiles.
All of these results of ination are theoretially attrative as explanations of the
features we observe in the universe at large sales. Probably the most important
suess of ination, however, is its explanation of the origin of inhomogeneities in the
universe. At rst this idea might seem ontraditory given that I said ination attens
the universe and redues to eetively zero the density of anything that was in it. It
turns out, however, that small inhomogeneities an be generated during ination. The
utuations that are generated in this way early in ination are typially strethed out
by the exponential expansion to sales muh greater than we an observe (although
see hapter 7 for some possible onsequenes of these large sale utuations). The
utuations generated at the last stages of ination, however, form the seeds of the
inhomogeneities that we observe in the CMB, and whih subsequently gave rise to
the strutures we observe in the universe.
The origin of these inhomogeneities is in quantum utuations that are strethed
out during ination. As the wavelength of these utuations grows larger than the
Hubble radius H 1 they ease osillating and get frozen in as lassial waves. The
utuations on the sales we an observe today were all generated during the last 60
or so e-folds of ination. See [3℄ for more details on this proess.
The overall amplitude of these utuations depends on the parameters in the inaton potential, so these parameters must be adjusted to math the observed CMB
anisotropy. The form of the utuations, however, is a robust predition of ination.
18
CHAPTER 3.
A PRIMER ON INFLATION
In almost all versions of ination these utuations have the form of Gaussian, adiabati perturbations with a nearly sale-invariant spetrum.3 So far the preditions
of ination have t the CMB data exellently, and no other theory has been able to
reprodue these preditions.
3.3 Reheating
Although ination produes a universe with many of the features we observe in our
own, suh as near-homogeneity with small utuations, the universe immediately after
ination is in one sense almost ompletely empty. While ination gets rid of all reli
partiles suh as gravitinos and monopoles, it also gets rid of all other partiles as
well. After ination nearly all the energy density in the universe is in the homogeneous
inaton eld. One ination ends this eld must somehow deay and give its energy
to other elds, eventually giving rise to the panoply of partiles we see around us
today. The proess by whih the inaton deays into other forms of energy is known
as reheating.
The issue of reheating has been explored for almost as long as inationary theory
itself (see e.g. [4℄). In the earliest papers on the subjet it was noted that after
ination ends the homogeneous inaton eld would osillate about its minimum.
Assuming the inaton were oupled to other elds it would then deay, leading to a
asade of energy density into many dierent forms. Several methods were developed
for perturbatively investigating this deay.
In 1994 it was realized that there is in general a muh more eÆient mehanism
by whih the inaton ould deay, namely parametri resonane [5℄. To see how this
works onsider the simple ase of a eld oupled to the inaton via an interation
term
1
Vinteration = g 2 2 2 :
(3.10)
2
The equation of motion for will be that of an osillator whose frequeny depends
on . As osillates this frequeny will hange, ausing utuations of to be
3 There
are many introdutions and reviews about the CMB available. See for example http://fawww.harvard.edu/ mwhite/htmlpapers.html.
3.3.
REHEATING
19
rapidly amplied in bands of resonant frequenies. (For a more omplete disussion
of parametri resonane in general see [6℄ and for a disussion of parametri resonane
during reheating see [7, 8℄.) This eet is known as preheating beause it rapidly
transfers muh of the inaton's energy to utuations of the eld before the slower,
perturbative mehanisms of reheating an our.
Sine the disovery of preheating a great deal of work has been done to understand
the onsequenes of nonperturbative eets in reheating. These eets an inlude
phase transitions, the prodution of topologial defets, the prodution and/or elimination of various kinds of reli partiles, and seondary stages of ination. Suh
nonperturbative eets often involve strong, nonlinear interations between elds far
from thermal equilibrium in an expanding universe. Thus progress in this eld has
in many ases required large-sale numerial simulations.
The work desribed in this thesis forms part of the ongoing investigation into
dierent mehanisms by whih reheating an our and the onsequenes of those
mehanisms within various models of inationary osmology.
Part II
Mehanisms of Reheating
20
Chapter 4
Introdution to the Researh
This part of the thesis is adapted from the papers I have published in the ourse of
my graduate researh. All of these papers deal with probing mehanisms of reheating
after ination. Colletively they, along with my work on lattie simulations desribed
in part III, form the essential ontent of this thesis.
Before presenting this researh it is appropriate that I should say a few words
about my ontributions to the work presented here. Of ourse most of the work done
in sienti ollaboration is the result of numerous disussions, arguments, and other
forms of give and take that make it impossible to separate out individual ontributions
in a meaningful way. Nonetheless I an note in a general sense that all of the lattie
alulations reported in these hapters were done by me. I also performed various
other numerial alulations suh as the linear alulations desribed in hapter 7.
The interpretation of these results was ollaborative. The development of the lattie
simulations, and in partiular the physis (as opposed to omputational) aspets
of them, has ontinually evolved over the last several years as a result of disussions
between me and several of my ollaborators. The program itself was originally written
by Igor Tkahev and has subsequently been rewritten and extended by me. Part III
ontains a more detailed disussion of our respetive ontributions to the simulations.
The theoretial parts of the researh, inluding the disussions and onlusions, was
worked out in so many exhanges between so many people that I don't think individual
ontributions an meaningfully be attributed.
21
22
CHAPTER 4.
INTRODUCTION TO THE RESEARCH
The rest of this introdution onsists of a brief summary of the subjet matter
of the hapters in this part. This summary is not intended as a full, pedagogial
aount of the ontent of the researh, and it will be presented in a tehnial form
most suitable for people familiar with inationary osmology. The details of the work
are ontained in the hapters themselves and this setion is merely intended as a
quik referene to those results.
Summary of Papers
Chapter 5, \Instant Preheating" explains a novel mehanism by whih the deay of
the inaton an produe partiles many orders of magnitude heavier than the inaton
partiles themselves. My ollaborators and I all this mehanism instant preheating
beause it an lead to essentially omplete deay within a single osillation. The basi
idea is that if the inaton eld is oupled to another eld , e.g. through a g 22 2
oupling, then the mass of the partiles will depend on . In partiular, as passes
through zero the partiles will be momentarily massless. The resulting nonadiabati
hange of the mass will lead to the prodution of a ertain number density of partiles. All of this has been understood for many years. What wasn't appreiated,
however, was the eet that the ontinued movement of the eld would have on the
partiles. In partiular, as moves away from zero these partiles would beome
massive. In most typial inationary models will ontinue to roll to a value of the
order of a Plank mass, meaning that if g 2 = O(1) the partiles will aquire masses
of the order of the Plank mass, irrespetive of the mass of . If then rolls bak and
settles near zero this hanging mass will be relatively unimportant. If, however, an in turn deay to another speies of massive partiles then it an eetively drain
all the energy from the eld in a single osillation while produing partiles with
near Plankian masses. If doesn't osillate but ontinues rolling (as for example in
some quintessene models) then this mehanism will be even more eÆient, an issue
that is taken up in more detail in hapter 6.
Chapter 6 on NO models desribes a lass of inationary models in whih the
inaton eld has no minimum, but rather ontinues rolling in an asymptotially at
23
potential. Suh models generated a lot of interest for a while beause of their possible
relevane to the theory of quintessene. Beause the usual mehanisms of reheating
don't work in these models it had been proposed by several authors that reheating
ould our in these ases via gravitational partile prodution. My ollaborators and
I showed that suh senarios suer from serious problems involving the prodution
of isourvature utuations and reli partiles suh as moduli and gravitinos. We
explained how these problems an be avoided, however, by introduing a oupling
between the inaton and one or more light elds and invoking the instant preheating
mehanism desribed above.
Chapter 7 on moduli elds reonsiders an old problem, namely the prodution of
large values of moduli elds during ination. Moduli, i.e. light, weakly oupled elds,
appear generially in string and supersymmetry motivated models of partile physis.
In this hapter I show that the problems assoiated with their prodution in the early
universe are more severe than had been previously realized. In partiular, even if
no lassial displaement of the moduli elds our there will be long wavelength
quantum utuations of these elds amplied during ination. The amplitude of
these utuations will depend on the duration of ination and for typial haoti
ination models these will be large enough to disrupt nuleosynthesis and in many
ases even ause a new stage of ination dominated by these light elds. Ination
driven by suh light elds would be in serious onit with CMB observations. I
disuss how these eets an depend on the details of the models, but do not present
a general solution to the problem.
Chapter 8, \Ination After Preheating," also deals with a seondary stage of
ination, but one of a very dierent sort. It was known from previous work that
if the inaton has a symmetry breaking potential and ouples to another eld then
utuations of this seond eld reated during preheating an temporarily lead to
symmetry restoration, followed by a rst order phase transition. It was speulated
that suh symmetry restoration ould lead to a very brief seondary stage of ination,
perhaps alleviating some problems of reli prodution during ination and preheating.
In this hapter I report the results of lattie simulations in whih my ollaborators
and I veried this predition. In partiular we found that after symmetry was restored
24
CHAPTER 4.
INTRODUCTION TO THE RESEARCH
in our simulations there was a brief period where the pressure was negative and the
expansion of the universe was aelerating, i.e. a period of ination.
Chapter 9 on the development of equilibrium onsiders what happens to the inaton and the produts of its deay after the explosive stage of partile prodution
assoiated with preheating. I report here the results of a series of lattie simulations
in dierent models. My ollaborator and I used these results to derive a set of empirial rules that seem to govern the proess of thermalization. Most notably, we argued
that in most models of ination there is some mehanism that leads to rapid partile
prodution, with the exitation onentrated in the infrared modes of the produed
elds. Following this initial exitation all elds that are diretly or indiretly oupled
to the exited setor also get exited exponentially rapidly. The elds then form into
groups, in whih all of the elds within a partiular group have idential spetra and
time evolution. Within eah group, the elds then thermalize by spreading their energy to the ultraviolet end of the spetrum. The omposition of these groups depends
strongly on the detailed interations between the elds.
Finally, hapter 10 on tahyoni preheating desribes a new mehanism by whih
the inaton an rapidly deay. If the inaton setor inludes a diretion in eld spae
with negative urvature then utuations an be rapidly produed via a spinodal
instability. Suh potentials our generially in models of hybrid ination. Essentially
the proess being desribed is just spontaneous symmetry breaking. The existene of
spinodal instabilities has long been known in ondensed matter physis, and of ourse
spontaneous symmetry breaking has been studied extensively in ondensed matter
and partile physis. However, so far as I know nobody had previously realized that
in the ontext of the sudden appearane of a negative urvature (as ours in hybrid
ination) this mehanism of deay is so eÆient as to lead to a omplete deay of the
homogeneous eld(s) within a single osillation. We found this result to be generi
for a wide variety of models and parameters.
Chapter 5
Instant Preheating
Note: This hapter is based on a paper by Gary Felder, Lev Kofman, and Andrei
Linde, available on the Los Alamos eprint server as hep-ph/9812289. The full itation
appears in the bibliography [9℄.
Chapter Abstrat
I desribe here a new eÆient mehanism of reheating. Immediately after rolling
down the rapidly moving inaton eld produes partiles , whih may be either
bosons or fermions. This is a nonperturbative proess that ours almost instantly;
no osillations or parametri resonane are required. The eetive masses of the
partiles may be very small at the moment when they are produed, but they
\fatten" when the eld inreases. When the partiles beome suÆiently heavy,
they rapidly deay to other, lighter partiles. This leads to an almost instantaneous
reheating aompanied by the prodution of partiles with masses whih may be as
large as 1017 1018 GeV. This mehanism works in the usual inationary models
where V () has a minimum, where it takes only a half of a single osillation of the
inaton eld , but it is espeially eÆient in models with eetive potentials slowly
dereasing at large as in the theory of quintessene.
25
26
CHAPTER 5.
INSTANT PREHEATING
5.1 Introdution
As disussed in the introdution to the thesis, the rst stages of preheating are typially governed by nonperturbative eets. In partiular, the most eÆient mehanism
of reheating known before the publiation of the researh in this hapter was based
on the theory of the nonperturbative deay of the inaton eld due to the eet of
broad parametri resonane [5, 7, 8℄. To distinguish this stage of nonperturbative
partile prodution from the stage of partile deay and thermalization whih an
be desribed using perturbation theory [10, 11, 12℄ (see also [13, 4℄), it was alled
preheating.
This proess an rapidly transfer the energy of a oherently osillating salar eld
to the energy of other elds or elementary partiles. Beause of the nonperturbative
nature of the proess, it may lead to many unusual eets, suh as nonthermal osmologial phase transitions [14, 15, 16, 17℄. Another unusual feature of preheating
disovered in [5, 7, 8℄ is the possibility of the prodution of a large amount of superheavy partiles with masses one or two orders of magnitude greater than the inaton
mass. In the simplest versions of haoti ination with the inaton mass m 1013
GeV this an lead to the opious prodution of partiles with masses up to 1014 1015
GeV [5, 7, 8, 18, 19, 20, 21, 22, 23℄. This issue is rather important sine interations
and deay of superheavy partiles may lead to baryogenesis at the GUT sale [18, 19℄.
However, GUT baryogenesis was only marginally possible in previously studied
models of preheating beause the masses of produed partiles just barely approahed
the GUT sale. Moreover, in some models the partiles reated by the resonane
strongly interat with eah other, or rapidly deay. This may take them out of the
resonane band, in whih ase parametri resonane does not last long or does not
happen at all. Also, there are some models where the eetive potential does not have
a minimum, but instead slowly dereases at large [24, 25, 26, 27, 28, 29℄. In these
models the salar eld does not osillate at all after ination, so neither parametri
resonane nor the standard perturbative mehanism of inaton deay works there.
Suh models are disussed at more length in hapter 6.
In this hapter I show how my ollaborators and I turned these potential problems
5.2.
INSTANT PREHEATING: THE BASIC IDEA
27
into an advantage. I desribe a new mehanism of preheating, whih works even
in the models where parametri resonane annot develop. The new mehanism is
also nonperturbative but very simple. It leads to an almost instantaneous reheating
aompanied by the prodution of superheavy partiles with masses whih may be as
great as 1017 1018 GeV. In some ases it may even lead to the prodution of blak
holes of a Plankian mass, whih immediately evaporate.
5.2 Instant preheating: The basi idea
To explain the main idea of the new senario I will onsider the simplest model of
2
haoti ination with the eetive potential m2 2 or 4 4 and assume that the inaton
eld interats with some other salar eld with the interation term V = 12 g 2 2 2 .
In these models ination ours at jj >
0:3Mp [3℄. Suppose for deniteness that
initially is large and negative, and ination ends at 0:3Mp . After that the
eld rolls to = 0, then it grows up to 10 1 Mp 1018 GeV, and nally rolls bak
and osillates about = 0 with a gradually dereasing amplitude. If the oupling
onstant g is large enough (g >
10 4), then, aording to [5, 7, 8℄, the prodution of
partiles ours for the rst time when the salar eld reahes the point = 0
after the end of ination. With eah subsequent osillation, partile reation ours
as rosses zero. This mehanism of partile prodution is desribed by the theory
of preheating in the broad resonane regime [5, 7, 8℄. But now I will onentrate on
the rst instant of this proess. Remarkably, in ertain ases this is all that we need
for eÆient reheating.
Usually only a small fration of the energy of the inaton eld 10 2 g 2 is transferred to the partiles at that moment (see Eq. (5.7) in the next setion). The role
of parametri resonane was to inrease this energy exponentially within several osillations of the inaton eld. But suppose that the partiles interat with fermions
with the oupling h . If this oupling is strong enough, then partiles may
deay to fermions before the osillating eld returns bak to the minimum of the
eetive potential. If this happens, parametri resonane does not our. However,
as I will show, something equally interesting may our instead of it: The energy
28
CHAPTER 5.
INSTANT PREHEATING
density of the partiles at the moment of their deay may beome muh greater
than their energy density at the moment of their reation.
Indeed, prior to their deay the number density of partiles, n , remains pratially onstant [5, 7, 8℄, whereas the eetive mass of eah partile grows as m = g
when the eld rolls up from the minimum of the eetive potential. Therefore their
total energy density grows. One may say that partiles are \fattened," being fed by
the energy of the rolling eld . The fattened partiles tend to deay to fermions
at the moment when they have the greatest mass, i.e. when reahes its maximal
value 10 1 Mp , just before it begins rolling bak to = 0.
At that moment partiles an deay to two fermions with mass up to m g 10 1 M , whih an be as large as 5 1017 GeV for g 1. This is two orders of
p
2
magnitude greater than the masses of the partiles that an be produed by the usual
mehanism based on parametri resonane [5, 7, 8℄. As a result, the total energy
density of the produed partiles also beomes two orders of magnitude greater than
their energy density at the moment of their prodution. Thus the hain reation
! ! onsiderably enhanes the eÆieny of transfer of energy of the inaton
eld to matter.
More importantly, superheavy partiles (or the produts of their deay) may
eventually dominate the total energy density of matter even if in the beginning their
energy density was relatively small. For example, the energy density of the osillating
inaton eld in the theory with the eetive potential 4 4 dereases as a 4 in an
expanding universe with a sale fator a(t). Meanwhile the energy density stored in
the nonrelativisti partiles (prior to their deay) dereases only as a 3 . Therefore
their energy density rapidly beomes dominant even if originally it was small. A
subsequent deay of suh partiles leads to a omplete reheating of the universe.
Sine the main part of the proess of preheating in this senario (prodution of and partiles) ours immediately after the end of ination, within less than one
osillation of the inaton eld, we alled it instant preheating. I should emphasize
that instant preheating is a ompletely nonperturbative eet, whih an lead to the
prodution of partiles with momenta and masses many orders of magnitude greater
than the inaton mass. This would be impossible in the ontext of the elementary
5.3.
29
THE SIMPLEST MODELS
theory of reheating developed in [10, 11, 12℄. In what follows I will give a more
detailed desription of the instant preheating senario.
5.3 The simplest models
Consider rst the simplest model of haoti ination with the eetive potential
2
V () = m2 2 , and with the interation Lagrangian 21 g 2 2 2 h . I will take m =
10 6 Mp , as required by mirowave bakground anisotropy [3℄, and in the beginning I
will assume for simpliity that partiles do not have a bare mass, i.e. m () = g jj.
Reheating in this model is eÆient only if g >
10 4 [5, 7, 8℄1, whih implies gMp >
102 m for the realisti value of the mass m 10 6 Mp . Thus, immediately after the
end of ination, when Mp =3, the eetive mass g jj of the eld is muh greater
than m. It dereases when the eld moves down, but initially this proess remains
adiabati, jm_ j m2 .
The adiabatiity ondition beomes violated and partile prodution ours when
jm_ j gj_ j beomes greater than m2 = g22. For a harmoni osillator one has
j_ 0j = m, where j_ 0j is the veloity of the eld in the minimum of the eetive
potential, and 10 1 Mp is the amplitude of the rst osillation. This implies
that the proess beomes nonadiabati for g2 <
m, i.e. for < < , where
q
mg [5, 7, 8℄. Here 10 1 Mp is the initial amplitude of the osillations of
the inaton eld. Note that under the ondition g 10 4 whih is neessary for
eÆient reheating, the interval <
< is very narrow: . As a result,
the proess of partile prodution ours nearly instantaneously, within the time
t _ j0j
(gm)
1=2 :
(5.1)
This time interval is muh smaller than the age of the universe, so all eets related
to the expansion of the universe an be negleted during the proess of partile prodution. The unertainty priniple implies in this ase that the reated partiles will
1 Note
that if one takes g > 10 3 , radiative orretions to the eetive potential may onsiderably hange its shape [3℄. However, this does not happen in supersymmetri theories where the
ontributions of fermions and bosons nearly anel eah other.
30
CHAPTER 5.
INSTANT PREHEATING
have typial momenta k (t ) 1 (gm)1=2 . The oupation number nk of partiles with momentum k is equal to zero all the time when it moves toward = 0.
When it reahes = 0 (or, more exatly, after it moves through the small region
<
< ) the oupation number suddenly (within the time t ) aquires the
value [5, 7, 8℄
!
k2
nk = exp
;
(5.2)
gm
and this value does not hange until the eld rolls to the point = 0 again.
A detailed desription of this proess inluding the derivation of Eq. (5.2) was
given in Ref. [7℄; see in partiular Eq. (55) there. This equation (5.2) an be written
in a more general form. First of all, the shape of the eetive potential does not
play any role in its derivation. The essential point of the derivation of Eq. (5.2) is
that partiles are produed in a small viinity of the point = 0, when (t) an be
represented as (t) _ 0 (t t0 ). The only thing whih one needs to know is not V (),
m or , but the veloity of the eld at the time when it passes the point = 0.
Therefore one an replae m by j_ 0 j in this equation. Also, the same equation is
valid for massive partiles as well, if one replaes k2 by k2 + m2 , where m is the
bare mass of the partiles at = 0. (A similar result is valid for fermions and for
vetor partiles.) Therefore Eq. (5.2) in a general ase (for any m and V ()) an
be written as follows:
!
(k2 + m2 )
nk = exp
:
(5.3)
g j_ 0j
This an be integrated to give the density of partiles
1
1 Z
(g _ )3=2
n = 2 dk k2 nk = 0 3 exp
2 0
8
!
m2
:
g j_ 0j
(5.4)
2
Numerial investigation of ination in the theory m2 2 with m = 10 6 Mp gives j_ 0 j =
10 7 Mp2 , whereas in the theory 4 4 with = 10 13 one has a somewhat smaller value
j_ 0j = 6 10 9Mp2 . This implies, in partiular, that if one takes g 1, then in the
2
theory m2 2 there is no exponential suppression of prodution of partiles unless
their mass is greater than m 2 1015 GeV. This agrees with a similar onlusion
5.3.
31
THE SIMPLEST MODELS
obtained in [5, 7, 8, 18, 19, 20, 21, 22, 23℄.
Let us now onentrate on the ase m2 <
gj_ 0j, when the number of produed
partiles is not exponentially suppressed. In this ase
n (g _ 0)3=2
:
8 3
(5.5)
Aording to Eq. (5.3), a typial initial energy (momentum) of eah partile at
the moment of their prodution is (g j_ jj= )1=2 , so their total energy density is
(g _ 0 )2
:
8 7=2
(5.6)
The ratio of this energy to the total energy density = _ 20 =2 of the salar eld at
this moment gives
5 10 3 g2:
(5.7)
This result is pratially model-independent, given the interation term 12 g 2 2 2 .
2
In partiular, it does not depend on the inaton mass m in the theory m2 2 . The
same result an be obtained in the theory 4 4 independently of the value of .
An interesting possibility appears if one has m2 g j_ 0 j. Then the probability of
prodution of suh partiles is not exponentially suppressed during the rst osillation, but it is exponentially suppressed during all subsequent osillations beause j_ j
dereases due to the expansion of the universe, and the ondition m2 <
gj_ j beomes
violated. In this ase new partiles are not reated. However, as we already explained, these new partiles may not even be neessary. For example, in the theory
4 the energy density of the inaton eld dereases as a 4 , whereas the energy
4
density stored in the nonrelativisti partiles (prior to their deay) dereases only
as a 3 . Therefore their energy density rapidly beomes dominant even if originally it
was small. Their subsequent deay makes the proess of reheating omplete.
But preheating in this model beomes muh more eÆient if one uses the mehanism desribed in the beginning of this hapter. Indeed, let us assume that the
partiles survive until the eld rolls up from = 0 to the point 1 from whih
32
CHAPTER 5.
INSTANT PREHEATING
2
it returns bak to = 0. In the theory m2 2 one has 1 0:07 Mp, whereas in the
theory 4 4 one has 1 0:12 Mp. I will take 1 0:07 Mp in my estimates. At that
time the mass of eah partile will be g1 10 1 gMp, they will be nonrelativisti,
2
and their total energy density (for the ase of the theory m2 2 ) will be
(g _ )3=2
= m n 10 1 gMp 0 3
8
10 14 g5=2Mp4 :
(5.8)
Therefore the ratio of the energy density of partiles to the energy density of the
inaton eld _ 20 =2 will be
10 3j_ 0j
1=2
Mp g 5=2 2 g 5=2 :
(5.9)
The last result follows from the relation j_ 0 j 10 1mMp 10 7Mp2 for m 10 6 Mp .
Under the ondition g >
10 4, whih is the standard ondition for eÆient preheating
[5, 7, 8℄, this ratio is muh greater than the one in Eq. (5.7).
If the partiles do not deay when the eld reahes 1 , then their energy will
derease again in parallel with jj, until it reahes the value given by Eq. (5.7). Thus,
preheating is most eÆient if all partiles an deay at the moment when the eld reahes its maximal value 1 . This is possible if the lifetime of the partiles reated
at the moment t0 is lose to t m 1 =4. Partiles in this model an deay to
fermions, with the deay rate [5, 7, 8℄
( !
)=
h2 m h2 g jj
=
:
8
8
(5.10)
Note that the deay rate grows with the growth of the eld jj, so partiles tend to
deay at large jj. One an easily hek that the partiles deay when the eld reahes its maximal value jj 0:07Mp if
h2 g 500m
Mp
5 10 4:
(5.11)
At the moment when jj reahes 0:07Mp , the partiles have eetive mass m =
5.3.
THE SIMPLEST MODELS
33
g jj 0:07gMp. Suh partiles an deay to two fermions if m < 0:035 gMp.
This implies that after the rst half of an osillation, the salar eld an produe
fermions with mass up to 0:035 gMp. For example, in the theory with g 10 1 ,
h 7 10 2 one an produe fermions with mass up to m 4 1016 GeV, and
in the theory with g 1, h 2 10 2 one an produe partiles with mass up to
4 1017 GeV.
As we have found, initially the ratio is suppressed by the fator 2g 5=2 , see Eq.
(5.9). But this suppression is not very strong, and if the energy density of the
partiles during some short period of the evolution of the universe dereases not as
fast as the energy density of the inaton eld and other produts of its subsequent
deay, then very soon the universe will be dominated by the produts of deay of the
partiles , and reheating will be omplete.
If h2 g 510 4 , the partiles may deay before the osillating eld reahes its
maximal value 1 10 1 Mp . This an make our mehanism somewhat less eÆient.
However, the deay annot our until m = g jj beomes greater than 2m . If,
for example, the fermions have mass 0:03 gMp, then the deay ours only when
the eld reahes its maximal value 1 even if h2 g 5 10 4. This preserves the
eÆieny of our mehanism even for very large h2 g .
On the other hand, for h2 g 5 10 4 , the partiles do not deay within a
single osillation. In this ase the parametri resonane regime beomes possible,
whih again leads to eÆient preheating aording to [5, 7, 8℄. Moreover, superheavy
fermions still will be produed in this regime, beause the osillating eld will spend
a ertain amount of time at 1 . During this time superheavy partiles will be
produed, and their number may not be strongly suppressed.
The mehanism of partile prodution desribed above an work in a broad lass of
2
theories. For example, one an onsider models with the interation g2 2 (+v )2 . Suh
interation terms appear, for example, in supersymmetri models with superpotentials
of the type W = g2( + v ) [30℄. In suh models the mass m vanishes not at 1 = 0,
but at 1 = v , where v an take any value. Correspondingly, the prodution of partiles ours not at = 0 but at = v . When the inaton eld reahes the
minimum of its eetive potential at = 0, one has m gv , whih may be very
34
CHAPTER 5.
INSTANT PREHEATING
large. If one takes v Mp , one an get m gMp , whih may be as great as 1018
GeV for g 10 1 , or even 1019 GeV for g 1. If, however, one takes v Mp , the
density of partiles produed by this mehanism will be exponentially suppressed
by the subsequent stage of ination. This possibility will be disussed in the next
setion.
Sine parametri ampliation of partile prodution is not important in the ontext of the instant preheating senario, it will work equally well if the inaton eld
ouples not to bosons but to fermions [31, 32℄. Indeed, the reation of fermions with
mass g jj also ours beause of the nonadiabatiity of the hange of their mass at
= 0. The theory of this eet at g >
10 4 is very similar to the theory of the
reation of partiles desribed above; see in this respet [32℄. The eÆieny of
preheating will be enhaned if the fermions with a growing mass g jj an deay
into other fermions and bosons, as in the senario desribed in the previous setion.
It is amazing that osillations of the eld with mass m = 1013 GeV an lead to
the opious prodution of superheavy partiles with masses 4 - 5 orders of magnitude
greater than m. The previously known mehanism of preheating was barely apable
of produing partiles of mass 1015 GeV, whih is somewhat below the GUT sale,
and even that was possible only in the strong oupling limit g = O(1). This new
mehanism allows for the prodution of partiles with mass greater than 1016 GeV
even if the oupling onstants are relatively small. This fat may play an important
role in the theory of baryogenesis in GUTs.
5.4 Fat wimpzillas
There have been a number of papers published on the possibility of the prodution of
superheavy WIMPS after ination [33, 34, 35, 36, 37, 38, 23℄. Suh partiles (whih
have been proudly alled WIMPZILLAS [39℄) ould be responsible for the dark matter
ontent of the universe, and, if they have a very large but nite deay time, they an
also be responsible for osmi rays with energies greater than the Greisen{Zatsepin{
Kuzmin limit [40, 41℄. The fous of these works in a ertain sense was opposite to that
of the theory of preheating: It was neessary to nd a mehanism for the prodution
5.4.
FAT WIMPZILLAS
35
of stable (or nearly stable) partiles whih would survive until now. For that purpose,
the mehanism of their prodution must be extremely ineÆient, sine otherwise the
present density of suh relis would be unaeptably large.
As one ould expet, it is muh easier to make the mehanism ineÆient rather
than the other way around. For example, Eq. (5.4) implies that the probability
of
m2
prodution of superheavy partiles is suppressed by a fator of exp gj_ 0 j . In
2
the theory m2 2with m =10 6 Mp we have j_ 0 j = 10 7 Mp2 , so this suppression fator
7
2
is given by exp 10gMmp2 . This implies that for g 1 the prodution of partiles
with m 1016 GeV is suppressed approximately by 10 10 , and this suppression
beomes as strong as 10 40 for m = 2 1016 GeV. This same level of suppression
an be ahieved, for example, with g = 10 2 and m = 2 1015 . Thus, by ne-tuning
of the parameters m and g one an obtain any value of the density of WIMPS at
the present stage of the evolution of the universe. This result agrees with the result
obtained in [23℄ by a dierent method.
This suppression mehanism is equally operative for the proess ! ! disussed here. If the partiles are heavy at = 0, their number will be exponentially
suppressed. When the eld grows, their masses grow as follows: m2 () = m2 + g 2 2 ,
At the moment of their deay these partiles an have mass of the order 1017 1018
GeV. The main advantage of this new mehanism is that the proess of fattening of
the partiles desribed above allows for the prodution of partiles whih an
be 102 times heavier than their ousins disussed in [33, 34, 35, 36, 37, 38, 23℄. In
the absene of established terminology, one an all suh superheavy partiles FAT
WIMPZILLAS.
Another way to produe an exponentially small number of superheavy WIMPS
is to produe them at the last stages of ination. This is possible in theories with
2
the interation term g2 2 ( + v )2 , as desribed in the previous setion. If one takes
v >
of Mp, then the partiles will be reated during ination. The number
2
partiles produed during ination in the simplest theory with V () = m2 2 does not
p
depend on v beause _ does not depend on and on v in this senario: _ = mM
2 [3℄.
However, their density will subsequently be exponentially suppressed by ination.
This is exatly what we need if the partiles or the produts of their deay are
36
CHAPTER 5.
INSTANT PREHEATING
2
WIMPS. For example, in the theory with V () = m2 2 the universe inates by
a fator of exp(2v 2 =Mp2 ) after the reation of partiles [3℄, so their density at
the end of ination beomes smaller by a fator of exp(6v 2 =Mp2 ). This leads to a
desirable suppression for v 2Mp . (The exat number depends on the subsequent
thermal history of the universe.) Meanwhile the masses of WIMPS produed by this
mehanism an be extremely large, of order gMp. If the partiles are stable, they
themselves may serve as superheavy WIMPS with nearly Plankian mass. If they
deay to fermions, then the fermions may play a similar role.
5.5 Instant Preheating and NO Models
The mehanism of instant preheating works most eÆiently in models where the
inaton potential beomes at at large values of . Suh non-osillatory (NO) models
have been the subjet of a great deal of interest reently beause of their potential
appliation to the theory of quintessene, i.e. the possibility that a rolling salar
eld with a nearly at potential ould mimi the role of a osmologial onstant in
the urrent universe. The appliation of instant preheating in these models will be
disussed in detail in hapter 6. Here I will simply note that if the mass of the
partiles grows with and has a at potential then the mass of an beome
essentially arbitrarily large.
5.6 Conlusions
The theory of reheating after ination is already rather old. For many years it was
thought that the lassial osillating inaton eld ould be represented as a olletion
of salar partiles of mass m <
1013 GeV, that eah partile deayed to partiles of
smaller mass, and that the nal goal was to alulate the reheating temperature Tr .
During the last deade we have learned that this simple piture in ertain ases an
be very useful, but typially one must use the nonperturbative theory of reheating for
the desription of the rst stages of reheating. The main ingredient of this theory was
the theory of broad parametri resonane. Partile prodution in this senario ould
5.6.
CONCLUSIONS
37
be represented as a series of suessive ats of reation, during whih the number of
produed partiles inreased exponentially. It seems now that this was only a rst
step towards a omplete understanding of nonperturbative mehanisms of reheating
after ination. It may be suÆient to onsider a single at of reation, espeially if
one takes into aount the relative inrease of energy of produed partiles during the
subsequent evolution of the lassial inaton eld, and the possibility of the hain
reation ! ! . This new mehanism is apable of produing partiles of
nearly Plankian energy, whih was impossible in the previous versions of the theory
of reheating.
One of the key ingredients of the nonperturbative mehanism of preheating desribed above is a nonadiabati hange of the mass m () near the point where it
vanishes (or at least strongly dereases). Suh situations our very naturally in supersymmetri theories of elementary partiles if one identies and with moduli
elds that orrespond to at diretions of the eetive potential. Indeed, in supersymmetri theories the eetive potential often has several at diretions, whih may
interset. When one of the moduli elds (the inaton) moves along a at diretion
and reahes the intersetion, the mass of another eld vanishes. A simple example of
this situation was desribed in setion 5.3. The hange of the number of massless degrees of freedom is a generi phenomenon whih is under intense investigation in the
ontext of supersymmetri gauge theories, supergravity and string theory, where it is
assoiated with the points of enhaned gauge symmetry, see e.g. [42, 43, 44, 45, 46℄.
Masses of elementary partiles may also hange nonadiabatially during osmologial phase transitions. At the moment of a phase transition masses of some partiles
vanish and may even temporarily beome tahyoni. In this ase partile prodution
may beome even more intense. See hapter 10.
The main onlusion of this work is that with an aount taken of the new possibilities disussed above the senario of preheating beomes more robust. In the
ases where parametri resonane may our, it provides a very eÆient mehanism
of preheating. Now we have found that eÆient preheating is possible even in models
where parametri resonane does not happen beause of the rapid deay of produed
38
CHAPTER 5.
INSTANT PREHEATING
partiles. Instant preheating ours in the usual inationary models where the inaton eld osillates near the minimum of its eetive potential. But this mehanism
works espeially well in models with eetive potentials whih slowly derease at large
, as in the theory of quintessene. The onversion of the energy of the inaton eld
to the energy of elementary partiles in these models ours very rapidly, and it is
always 100% eÆient. The next hapter disusses suh senarios.
Chapter 6
Ination and Preheating in NO
Models
Note: This hapter is based on a paper by Gary Felder, Lev Kofman, and Andrei
Linde, available on the Los Alamos eprint server as hep-ph/9903350. The full itation
appears in the bibliography [47℄.
Chapter Abstrat
In this hapter I disuss inationary models in whih the eetive potential of the
inaton eld does not have a minimum, but rather gradually dereases at large . In
suh models the inaton eld does not osillate after ination, and its eetive mass
beomes vanishingly small, so the standard theory of reheating based on the deay of
the osillating inaton eld does not apply. For a long time the only known mehanism
of reheating in suh non-osillatory (NO) models was based on gravitational partile
prodution in an expanding universe. This mehanism is very ineÆient. I will
show that it may lead to osmologial problems assoiated with large isourvature
utuations and overprodution of dangerous relis suh as gravitinos and moduli
elds. These problems an be resolved in the ontext of the senario of instant
preheating desribed in hapter 5 if there exists an interation g 22 2 of the inaton
eld with another salar eld . I show that the mehanism of instant preheating in
39
40
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
NO models is muh more eÆient than the usual mehanism of gravitational partile
prodution even if the oupling onstant g 2 is extremely small, 10 14 g 2 1.
6.1 Introdution
Usually it is assumed that the inaton eld after ination rolls down to the minimum of its eetive potential V (), osillates, and eventually deays. The stage
of osillations of the inaton eld is a neessary part of the standard mehanism of
reheating of the universe [10, 11, 12, 5, 7, 8℄.
However, there exist some models where the inaton potential V () gradually
dereases at large and does not have a minimum. In suh theories the inaton eld
does not osillate after ination, so the standard mehanism of reheating does not
work there.
Investigation of inationary models of this type has been rather sporadi [24,
25, 26, 27, 28℄, and eah new author has given them a new name, suh as deation
[25℄, kination [26, 27℄, and quintessential ination [28℄. However, the universe does
not deate in these models, and in general they are not related to the theory of
quintessene. The main distinguishing feature of inationary models of this type is the
non-osillatory behavior of the inaton eld, whih makes the standard mehanism
of reheating inoperative. Therefore we deided to all suh models \non-osillatory
models," or simply \NO models." In addition to desribing the most essential feature
of this lass of theories whih makes reheating problemati, this name reets the
rather negligent attitude towards these models whih existed until reently.
One of the reasons why NO models have not attrated muh attention was the
absene of an eÆient mehanism of reheating. For a long time it was believed that
the only mehanism of reheating possible in NO models was gravitational partile
prodution [24, 25, 26, 27, 28℄, whih ours beause of the hanging metri in the
early universe [48, 49, 50, 51, 52, 53℄. This mehanism is very ineÆient, whih may
lead to ertain osmologial problems.
However, reently the situation hanged. The mehanism of instant preheating
desribed in the previous hapter is very eÆient, and it works in NO models even
6.2.
ISOCURVATURE PERTURBATIONS IN NO MODELS
41
better than in the models where V () has a minimum.
In this hapter I will desribe various features of NO models. First of all, I will
disuss the problem of initial onditions in these models, whih had not been properly
addressed before. The standard assumption made in [24, 25, 26, 27, 28℄ is that at
the end of ination in NO models one has a large and heavy inaton eld that
rapidly hanges and reates light partiles minimally oupled to gravity from a
state where the lassial value of the eld vanishes. We showed that this setting of
the problem needed to be reonsidered. If the elds and do not interat (whih
was the standard assumption of Refs. [24, 25, 26, 27, 28℄), then at the end of ination
the eld typially does not vanish. Usually the last stages of ination are driven by
the light eld rather than by the heavy eld . But in this ase reheating ours
due to osillations of the eld , as in the usual models of ination.
In addition to reexamining the problem of initial onditions, we will point out potential diÆulties assoiated with isourvature perturbations and gravitational prodution of gravitinos and moduli elds in NO models.
In order to provide a onsistent setting for the NO models one needs to introdue
interation between the elds and . This resolves the problem of initial onditions
in these models and makes it possible to have a non-osillatory behavior of the inaton
eld after ination. We show that all of these problems an be resolved in the ontext
of the reently proposed senario of instant preheating [9℄ if there is an interation
g2 2 2 of the inaton eld with another salar eld , with g 2 10 14 . In this ase
2
the mehanism of instant preheating in NO models is muh more eÆient than the
usual mehanism of gravitational partile prodution studied in [24, 25, 26, 27, 28℄.
6.2 Isourvature perturbations in NO Models
In hapter 7 I show that if ination begins with a large value of the inaton and
a vanishing value of a light eld then typially large utuations of the eld will be generated by the end of ination. These utuations are largest at long
wavelengths, well outside the horizon, and thus appear eetively homogeneous on
subhorizon sales. For a suÆiently long period of ination these utuations will
42
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
grow large enough to initiate a new inationary stage driven by the light eld . Thus
the standard assumption that reheating begins with a light eld having vanishing
amplitude is generially inorret.
Here I will onsider a more general question: If the two elds and do not
interat, then why should we assume that one of them should vanish at the beginning
of ination? And if it does not vanish, then how does it hange the whole piture?
Suppose for example that the eld is a Higgs eld [28℄ with a relatively small
mass and with a large oupling onstant . The total eetive potential in
this theory (for < 0) is given by
V (; ) =
4 2
+ (
4
4
v 2 )2 :
(6.1)
Here v is the amplitude of spontaneous symmetry breaking, v Mp . During ination
and at the rst stages of reheating this term an be negleted, so we will study the
simplied model
V (; ) = 4 + 4 :
(6.2)
4
4
This model was rst analyzed in [54℄. It is diretly related to the Peebles-Vilenkin
model [28℄ if the eld is the Higgs boson eld with a small mass m.
In general, at the beginning of ination one has both 6= 0 and 6= 0. Thus we
will not assume that = 0, and instead of studying quantum utuations of this eld
whih an make it large, I will assume that it ould be large from the very beginning.
Even though the elds and do not interat with eah other diretly, they move
towards the state = 0 and = 0 in a oherent way. The reason is that the motion
of these elds is determined by the same value of the Hubble onstant H .
The equations of motion for both elds during ination look as follows:
3H _ = 3 :
(6.3)
3H _ = 3 :
(6.4)
6.2.
ISOCURVATURE PERTURBATIONS IN NO MODELS
These equations imply that
d
d
=
;
3
3
43
(6.5)
whih yields the general solution
1
1
1
=
+
2
2
20
1
;
20
(6.6)
Sine the initial values of these elds are muh greater than the nal values, at the
last stages of ination one has [54℄
v
u
u
= t :
(6.7)
Suppose . In this ase the \heavy" eld rapidly rolls down, and then
from the last equation it follows, rather paradoxially, that the Hubble onstant at
the end of ination is dominated by the \light" eld . Thus we an onsistently
onsider the reation of utuations of the eld ( partiles) at the end of and after
the last inationary stage driven by the eld. But now these utuations our on
top of a nonvanishing lassial eld .
To study the behavior of the lassial elds and and their utuations analytially,
remember that during the inationary stage driven by one has
q one should
2
2
H = 3 Mp . In this ase, as before, the solution for the equation of motion of the
eld is given by [2℄
0 s
1
= 0 exp M tA :
(6.8)
6 p
Meanwhile, aording to Eq. (6.7),
v
u
u
= 0 t 0 s
1
A
exp Mp t ;
6
(6.9)
44
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
whereas for perturbations of the eld one has:
0
Æ = Æ0 exp s
1
3 Mp tA :
6
(6.10)
Let us onsider, for example, the behavior of the elds and their utuations at
the end of ination, starting from the moment = i . One may take, for example,
i 4Mp , whih orresponds to a point approximately 60 e-folds before the end of
ination. The utuations Æi H (i )=2 derease aording to (6.10), and at the
end of ination one gets
0
H (i)
exp Æ
=
2i
s
2
q
1
A p i
Mp t =
6
2e
:
6Mp 2i
(6.11)
Here e 0:3Mp orresponds to the end of ination.1 After that moment the
elds and begin osillating, and the ratio of Æ to the amplitude of osillations
of the eld remains approximately onstant. This gives the following estimate for
the amplitude of isourvature perturbations in this model:
ÆV ()
V ()
q
Æ
2
4 = 2 10 :
(6.12)
Initially perturbations of V () give a negligibly small ontribution to perturbations of the metri beause V () V (); that is why they are alled isourvature
perturbations. However, the main idea of preheating in NO models is that eventually
elds or the produts of their deay will give the dominant ontribution to the
energy-momentum tensor beause the energy density of the eld rapidly vanishes
due to the expansion of the universe ( a 6 ). However, beause of the inhomogeneity of the distribution of the eld (whih will be imprinted in the density
distribution of the produts of its deay on sales greater than H 1), the period of the
dominane of matter over the salar eld will happen at dierent times in dierent
2
are grateful to Peebles and Vilenkin for pointing out that the fator e2 should be present
i
in this equation.
1 We
6.2.
ISOCURVATURE PERTURBATIONS IN NO MODELS
45
parts of the universe. In other words, the epoh when the universe begins expanding
p
as a t or a t2=3 instead of a t1=3 will begin at dierent moments t (at different total densities) in dierent parts of the universe. Starting from this time the
isourvature utuations (6.12) will produe metri perturbations, and, as a result,
perturbations of CMB radiation.
Note that if the equation of state of the eld or of the produts of its deay
oinided with the equation of state of the salar eld after ination, utuations
of the eld would not indue any anisotropy of CMB radiation. For example, these
utuations would be harmless if the eld deayed into ultrarelativisti partiles
with the equation of state p = =3 and if the equation of state of the eld at that
time were also given by p = =3. However, in our ase the eld has equation of
state p = , whih is quite dierent from the equation of state of the eld or of its
deay produts.
Isourvature utuations lead to approximately 6 times greater large sale anisotropy
of the osmi mirowave radiation as ompared with adiabati perturbations. To avoid
osmologial problems, one would need to have ÆVV (()) <
5 10 6. If is the Higgs
eld with >
10 7, then the perturbations disussed above will be unaeptably
large. This may be a rather serious problem. Indeed, one may expet to have many
salar elds in realisti theories of elementary partiles. To avoid large isourvature
utuations eah of these elds must be extremely weakly oupled, with <
10 7,
The general onlusion is that the theory of reheating in NO models, as well as
their onsequenes for the reation of the large-sale struture of the universe, may
be quite dierent from what was antiipated in the rst papers on this subjet. In the
simplest versions of suh models ination typially does not end in the state = 0,
and large isourvature utuations are produed.
46
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
6.3 Cosmologial prodution of gravitinos and moduli elds
If the inaton eld is sterile, not interating with any other elds, the elementary
partiles onstituting the universe should be produed gravitationally due to the
variation of the sale fator a(t) with time. This was one of the basi assumptions
of all papers on NO models [24, 25, 26, 27, 28℄. Not all speies an be produed this
way, but only those whih are not onformally invariant. Indeed, the metri of the
Friedmann universe is onformally at. If one onsiders, for example, massless salar
partiles with an additional term 121 2 R in the Lagrangian (onformal oupling),
one an make onformal transformations of simultaneously with transformations
of the metri and nd that the theory of partiles in the Friedmann universe is
equivalent to their theory in at spae. That is why suh partiles would not be
reated in an expanding universe.
Sine onformal oupling is a rather speial requirement, one expets a number
of dierent speies to be produed. An apparent advantage of gravitational partile
prodution is its universality [35, 36, 37℄. There is a kind of \demoray" rule for all
partiles non-onformally oupled to gravity: the density of suh partiles produed
at the end of ination is X X H 4 , where X 10 2 is a numerial fator spei
for dierent speies and H is the Hubble parameter at the end of ination.
Unfortunately, demoray does not always work; there may be too many dangerous relis produed by this universal mehanism. One of the potential problems is
related to the overprodution of gravitons mentioned in [28℄. In order to solve it one
needs to have models with a very large number of types of light partiles. This is
diÆult but not impossible [28℄. However, even more diÆult problems will arise if
NO models are implemented in supersymmetri theories of elementary partiles.
For example, in supersymmetri theories one may enounter many at diretions
of the eetive potential assoiated with moduli elds. These elds usually are very
stable. Moduli partiles deay very late, so in order to avoid osmologial problems
the energy density of the moduli elds must be many orders of magnitude smaller
than the energy density of other partiles [55, 56, 57, 58, 59, 60℄.
6.3.
47
COSMOLOGICAL PRODUCTION OF GRAVITINOS AND MODULI FIELDS
Moduli elds typially are not onformally invariant. There are several dierent
eets that add up to give them masses CH during expansion of the universe, with
C = O(1) (in general, C is not a onstant) [55, 56, 57, 58, 59℄. This is very similar to
what happens if, for example, one adds a term 2 R2 to the lagrangian of a salar
eld. Indeed, during ination R = 12H 2, so this term leads to the appearane of a
ontribution to the mass of the salar eld m2 = 12H 2. Conformal oupling would
orrespond to m2 = 2H 2.
Aording to [24℄, the energy density of salar partiles produed gravitationally
at the end of ination is given by 10 2 H 4 (1 6 )2 . Thus, unless the onstant C
is ne-tuned to mimi onformal oupling, we expet that in addition to the energy
of lassial osillating moduli elds, at the end of ination one has gravitational
prodution of moduli partiles with energy density 10 2 H 4 , just as for all other
onformally noninvariant partiles.
In usual inationary models one also enounters the moduli problem if the energy
of lassial osillating moduli elds is too large [55, 56, 57, 58, 59℄. Here we are disussing an independent problem whih appears even if there are no lassial osillating
moduli. Indeed, in NO models all partiles reated by gravitational eets at the end
of ination will have similar energy density 10 2 H 4 . But if the energy density
of moduli elds is not extremely strongly suppressed as ompared with the energy
density of other partiles, then suh models will be ruled out [55, 56, 57, 58, 59, 60℄.
See hapter 7 for further disussion of the moduli problem in ination.
A similar problem appears if one onsiders the possibility of gravitational (nonthermal) prodution of gravitinos. Usually it is assumed that gravitinos have mass
m3=2 = 102 103 GeV, whih is muh smaller than the typial value of the Hubble
onstant at the end of ination. Therefore naively one ould expet that gravitinos,
just like massless fermions of spin 1=2, are (almost exatly) onformally invariant and
should not be produed due to expansion of the Friedmann universe.
However, in the framework of supergravity, the bakground metri is generated
by inaton eld(s) j with an eetive potential onstruted from the superpotential
W (j ). The gravitino mass in the early universe aquires a ontribution proportional
to W (j ). Depending on the model, the gravitino mass soon after the end of ination
48
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
may be of the same order as H or somewhat smaller, but typially it is muh greater
than its present value m3=2 .
A general investigation of the behavior of gravitinos in the Friedmann universe
shows that the gravitino eld in a self-onsistent Friedmann bakground supported
by salar elds is not onformally invariant [61℄. For example, the eetive potential
p
4 an be obtained from the superpotential 3 in the global supersymmetry limit.
p
This leads to a gravitino mass 3 =Mp2 . At the end of ination Mp , and
p
therefore the gravitino mass is omparable to the Hubble onstant H 2 =Mp .
This implies strong breaking of onformal invariane.
The theory of gravitational prodution of gravitinos is strongly model-dependent,
and in some models it might be possible to ahieve a ertain suppression of their
prodution as ompared to the prodution of other partiles. The problem is that, just
like in the situation with the moduli elds, this suppression must be extraordinarily
strong. Indeed, to avoid osmologial problems one should suppress the number of
gravitinos as ompared to the number of other partiles by a fator of about 10 15
[62, 63, 64, 65, 66, 67℄. For a more detailed disussion of the osmologial prodution
of gravitinos see [61℄.
The gravitino/moduli problem and the problem of isourvature perturbations are
interrelated in a rather nontrivial way. Indeed, the gravitino and moduli problems are
espeially severe if the density of gravitinos and/or moduli partiles produed during
reheating is of the same order of magnitude as the energy density of salar elds .
In my previous alulations I assumed, following [24, 28℄, that the energy density of
the elds after ination is O(10 2H 4 ). But this statement is not always orret.
It was derived in [24℄ under an assumption that partile prodution ours during
a short time interval when the equation of state hanges. Meanwhile in inationary
osmology the long-wavelength utuations of the eld minimally oupled to gravity
are produed during ination all the time when the Hubble onstant H (t) is smaller
than the mass of the partiles m . The energy density of partiles produed
2
during ination will ontain a ontribution 0 = m2 h2 i, whih may be many orders
of magnitude greater than 10 2 H 4.
For the sake of argument, one may onsider ination in the theory 4 4 and take
6.4.
SAVING NO MODELS: INSTANT PREHEATING
49
p
m equal to the value of H at the end of ination, m Mp . Then, aording to
2 6
6
Eq. (7.29), after ination one has 0 36mMp40 10 2 H 4 M0p 10 2 H 4 , beause
0 Mp .
This is the same eet that I disuss in hapter 7: If 0 is large enough, we
may even have a seond stage of ination driven by the large energy density of the
utuations of the eld . But even if 0 is not large enough to initiate a seond
stage of ination, it still must be muh greater than Mp to drive the rst stage of
ination, whih makes the standard estimate 10 2 H 4 inorret [68℄.
There is one more eet that should be onsidered, in addition to gravitational
p
partile prodution. The eetive mass of the partiles at < 0 is given by 3.
At the end of ination, at Mp , this mass is of the same order as the Hubble
p
onstant 2 =Mp. Then, within the Hubble time H 1 the eld rolls to the
valley at > 0 and its mass vanishes. This is a non-adiabati proess; the mass of
the salar eld hanges by O(H ) during the time O(H 1). As a result, in addition to
gravitational partile prodution there is an equally strong prodution of partiles due to the nonadiabati hange of their mass [68, 69℄.
This may imply that the fration of energy in gravitinos will be muh smaller than
previously expeted, simply beause the fration of energy in the utuations of the
eld will be muh larger. But there is no free lunh. For example, the prodution of
large number of nearly massless partiles may lead to problems with nuleosynthesis.
Large inationary utuations of the eld an reate large isourvature utuations.
One an avoid this problem if one assumes, for example, that the elds aquire
eetive mass O(H ) in an expanding universe. Then their utuations will not be
produed during ination. But in suh a ase their density after ination will be
given by 10 2 H 4 , and therefore we do not have any relaxation of the gravitino and
the moduli problems.
6.4 Saving NO models: Instant preheating
As we will see, the problems disussed above will not appear in theories of a more
general lass, where the elds and an interat with eah other. I will onsider a
50
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
2
model with the interation g2 2 2 . First I will show that in this ase it really makes
sense to study preheating assuming that = 0. Then I will disuss how the instant
preheating mehanism desribed in hapter 5 allows eÆient energy transfer from the
eld to partiles of the eld .
6.4.1 Initial onditions for ination and reheating in the model
2
with interation g2 22
Consider a theory with an eetive potential dominated by the term V (; ) =
g2 2 2 . This means that the onstant g is large enough for me to temporarily neglet
2
the terms 4 4 + 4 4 in the disussion of initial onditions.
In this ase the Plank boundary is given by the ondition
g2 2 2
Mp4 ;
2
(6.13)
whih denes a set of four hyperbolas
g jjjj Mp2 :
(6.14)
At larger values of and the density is greater than the Plank density, so the
standard lassial desription of spae-time is impossible there. On the other hand,
the eetive masses of the elds should be smaller than Mp , and onsequently the
urvature of the eetive potential annot be greater than Mp2 . This leads to two
additional onditions:
jj < g 1Mp; jj < g 1Mp:
(6.15)
I assume that g 1. Suppose for deniteness that initially the elds and belong
to the Plank boundary (6.14) and that jj is half-way towards its upper bound (6.15):
jj g 1Mp=2. The hoie of the oeÆient 1=2 here is not essential; we only want
to make sure that the eld initially is of order Mp , but it an be slightly greater
than Mp , This allows for an extremely short stage of ination when the eld rolls
down towards = 0.
6.4.
SAVING NO MODELS: INSTANT PREHEATING
51
The equations for the two elds are
and
 + 3H _ = g 2 2 :
(6.16)
 + 3H _ = g 2 2 :
(6.17)
The urvature of the eetive potential in the diretion initially is g 2 2 g 2 Mp2 ,
whih is very small ompared to the initial value of H 2 Mp2 . Thus the eld will
move very slowly, so one an neglet the term  in Eq. (6.16).
3H _ = g 2 2 :
(6.18)
2
If the eld hanges slowly, then the eld behaves as in the theory m2 2 with
m g jj being slightly smaller than Mp and with the initial value of being slightly
greater than Mp . This leads to a very short stage of ination that ends within a few
Plank times. After this short stage the eld rapidly osillates. During this stage
the energy density of the osillating eld drops down as a 3 , the universe expands as
a t2=3 , and H = 32t . Thus the square of the amplitude of the osillations of the eld
dereases as follows: 2 20 a 3 t 2 . This leads to the following equation for
the eld :
_
g2
:
(6.19)
t
g2
The solution of this equation is = 0 tt0
with t0 Mp 1 , and 0 Mp =g .
(The ondition t0 Mp 1 follows from the fat that the initial value of H = 32t is not
muh below Mp .) This gives
Mp
(Mp t)
g
g2
:
(6.20)
The inaton eld beomes equal to Mp after the exponentially large time
1
t Mp 1
g
!g
2
:
(6.21)
52
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
During this time the energy of osillations of the eld beomes exponentially small,
and the small term 4 4 that I negleted until now beomes the leading term driving
the salar eld . At this stage we will have the usual haoti ination senario with
jj > Mp and with the elds evolving along the diretion = 0.
2
Thus in the presene of the interation term g2 2 2 one an indeed onsider ination and reheating with = 0. As we have seen, this possibility was rather problemati in the models where and interated only gravitationally.
The eetive mass of thepeld during ination is g jj, whih is muh greater
2
2
2
than the Hubble onstant Mp for g M p2 . In realisti versions of this model one
has 10 13 [3℄, and g 2 . Therefore long-wavelength utuations of the eld are not produed during the last stages of ination, when Mp .
A similar onlusion is valid if at the last stages of ination the eetive potential
2
of the eld is quadrati, V () = m2 2 . In this ase H m
Mp , and inationary
utuations of the eld are not produed for g Mmp . In realisti versions of this
model one has m 10 6 Mp [3℄, and utuations Æ are not produed if g 2 10 12 .
This means that the problem of isourvature utuations does not appear.
6.4.2 Instant preheating in NO models
To explain the main idea of the instant preheating senario in NO models, I will
2
onsider for simpliity a model where V () = m2 2 for < 0, and V () vanishes for
> 0. I will disuss a more general situation later. I will assume that the eetive
2
potential ontains the interation term g2 2 2 , and that partiles have the usual
Yukawa interation h with fermions . For simpliity, we will assume here that
partiles do not have any bare mass, so that their eetive mass is equal to g jj.
In hapter 5 I showed that for this ase partiles are produed at the moment
_ 3=2
when = 0 with number density (g80)3 , whih then dereases as a 3 (t) to give
n =
(g _ 0 )3=2
:
8 3 a3 (t)
(6.22)
(See equation (5.5).) Here we take a0 = 1 at the moment of partile prodution.
6.4.
53
SAVING NO MODELS: INSTANT PREHEATING
Partile prodution ours only in a small viinity of = 0. Then the eld ontinues rolling along the at diretion of the eetive potential with > 0, and
the mass of eah partile grows as g. Therefore the energy density of produed
partiles is
(g _ )3=2 g(t)
= 0 3
:
(6.23)
8
a3 (t)
The energy density of the eld drops down muh faster, as a 6 (t). The reason
is that if one neglets bakreation of produed partiles, the energy density of the
eld at this stage is entirely onentrated in its kineti energy density 12 _ 2 , whih
orresponds to the equation of state p = . I will study this issue now in a more
detailed way.
The equation of motion for the inaton eld after partile prodution looks as
follows:
 + 3H _ = g 2 h2 i
(6.24)
I will assume for simpliity that the eld does not have bare mass, i.e. m = g. As
soon as the eld beomes greater than (and this happens pratially instantly,
when partile prodution ends), the partiles beome nonrelativisti. In this ase
h2i an be easily related to n :
1 Z
2
h i 22
2
_ 3=2
n
(g 0 )
pkn2k +k dk
:
2
2
g 8 3 ga3(t)
g
(6.25)
Therefore the equation for the eld reads
(g _ )3=2
 + 3H _ = gn = g 3 0 3 :
8 a (t)
(6.26)
To analyze the solutions of this equation, I will rst neglet bakreation. In this ase
one has a t1=3 , H = 31t , and
=
where t0 = 3H1 0 = 2pM3p_ 0
p35m .
Mp
p
2 3
log
t
;
t0
(6.27)
54
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
One an easily hek that this regime remains intat and bakreation is unim3
portant for t < t1 p85 _ , until the eld grows up to
g 0
1 5Mp
p log 1g :
4 3
(6.28)
This equation is valid for g 1. For example, for g = 10 3 one has 1 3Mp. For
g = 10 1 one has 1 Mp . Note that the terms in the left hand side of the Eq. (6.26)
derease as t 2 when the time grows, whereas the bakreation term goes as t 1 . As
soon as the bakreation beomes important, i.e. as soon as the eld reahes 1 ,
it turns bak, and returns to = 0. When it reahes = 0, the eetive potential
beomes large, so the eld annot beome negative, and it bounes towards large
again.
Now let us take into aount interation of the eld with fermions. This interation leads to deay of the partiles with the deay rate [5, 7, 8℄
( !
)=
h2 m h2 g jj
=
:
8
8
(6.29)
Note that the deay rate grows with the growth of the eld jj, so partiles tend to
deay at large . In our ase the eld spends most of the time prior to t1 at Mp
(if it does not deay earlier, see below). The deay rate at that time is
( !
)
h2 gMp
:
8
(6.30)
3
If 1 < t1 g58=2 _ 0 , then partiles will deay to fermions at t < t1 and the fore
driving the eld bak to = 0 will disappear before the eld turns bak. In this
ase the eld will ontinue to grow, and its energy density will ontinue dereasing
anomalously fast, as a 6 . This happens if
h2 gMp g 5=2 _ 10=2
>
:
8 8 3
(6.31)
p
6
Taking into aount that in our ase _ 0 mM
10 and m 10 Mp , one nds that this
6.4.
SAVING NO MODELS: INSTANT PREHEATING
55
ondition is satised if h >
5 10 3g3=4. This is a very mild ondition. For example,
it is satised for h > 5 10 3 if g = 1, and for h > 5 10 7 if g = 10 4 .
This senario is always 100% eÆient. The initial fration of energy transferred
to matter at the moment of partile prodution is not very large, about 10 2 g 2 of
the energy of the inaton eld [9℄. However, beause of the subsequent growth of
this energy due to the growth of the eld , and beause of the rapid derease of
kineti energy of the inaton eld, the energy density of the partiles and of the
produts of their deay soon beomes dominant. This should be ontrasted with the
usual situation in the theories where V () has a minimum. As was emphasized in
[5, 7, 8℄, eÆient preheating is possible only in a sublass of suh models. In many
models where V () has a minimum the deay of the inaton eld is inomplete, and it
aumulates an unaeptably large energy density ompared with the energy density
of the thermalized omponent of matter. The possibility of having very eÆient
reheating in NO models may have signiant onsequenes for inationary model
building.
It is instrutive to ompare the density of partiles produed by this mehanism
to the density of partiles reated during gravitational partile prodution, whih is
given by 10 2 H 4 Mp4 , where is the energy density of the eld at
the end of ination. In the model 4 4 one has 10 16 Mp4 , and, onsequently,
Mp4 10 16 . Meanwhile, as we just mentioned, at the rst moment after
partile prodution in our senario the energy density of produed partiles is of
the order of 10 2 g 2 [9℄, and then it grows together with the eld beause of
the growth of the mass g of eah partile. Thus, for g 2 >
10 14 the number
of partiles produed during instant preheating is muh greater than the number of
partiles produed by gravitational eets. Therefore one may argue that reheating of
the universe in NO models should be desribed using the instant preheating senario.
Typially it is muh more eÆient than gravitational partile prodution. This means,
in partiular, that prodution of normal partiles will be muh more eÆient than
the prodution of gravitinos and moduli.
In order to avoid the gravitino problem altogether one may onsider versions of
NO models where the partiles produed during preheating remain nonrelativisti
56
CHAPTER 6.
INFLATION AND PREHEATING IN NO MODELS
for a while. Then the energy density of gravitinos during this epoh dereases muh
faster than the energy density of usual partiles. New gravitinos will not be produed
if the resulting temperature of reheating is suÆiently small.
6.5 Other versions of NO models
In the previous setion we onsidered the simplest model where V () = 0 for > 0.
However, in general V () may beome at not at = 0, but only asymptotially, at
>
Mp. Suh theories have beome rather popular now in relation to the theory of
quintessene; for a partial list of referenes see e.g. [70, 71, 72, 73, 74, 75, 76, 29, 77,
78, 79℄. In suh a ase the bakreation of reated partiles may never turn the salar
eld bak to = 0. Therefore the deay of the partiles may our very late, and
one an have very eÆient preheating for any values of the oupling onstants g and
h.
On the other hand, if the partiles are stable, and if the eld ontinues rolling
for a very long time, one may enounter a rather unusual regime. If the partile
masses g jj at some moment approah Mp , the partiles may onvert to blak holes
and immediately evaporate.
Indeed, in onventional quantum eld theory, an elementary partile of mass M
has a Compton wavelength M 1 smaller than its Shwarzshild radius 2M=Mp2 if
M>
Mp. Therefore one may expet that as soon as m = gjj beomes greater than
Mp , eah partile beomes a Plank-size blak hole, whih immediately evaporates
and reheats the universe. If this regime is possible, it should be avoided. Indeed,
blak holes of Plank mass may produe similar amounts of all kinds of partiles,
inluding gravitinos. Therefore if reheating ours beause of blak hole evaporation,
then we will return to the gravitino problem again.
Thus, the best possibility is to onsider those versions of the instant preheating
senario that do not lead to the reation of stable partiles of Plankian mass. It may
seem paradoxial that one needs to be areful about this onstraint. Several years ago
it would have seemed impossible to produe partiles of mass greater than 51012 GeV
during the deay of an inaton eld of mass m 1013 GeV. Here I have desribed
6.5.
OTHER VERSIONS OF NO MODELS
57
a nonperturbative mehanism of preheating that may produe partiles 5 orders of
magnitude heavier than m . It is interesting that the mehanism of instant preheating
works espeially well in the ontext of NO models where all other mehanisms are
rather ineÆient.
Chapter 7
Gravitational Partile Prodution
and the Moduli Problem
Note: This hapter is based on a paper by Gary Felder, Lev Kofman, and Andrei
Linde, available on the Los Alamos eprint server as hep-ph/9909508. The full itation
appears in the bibliography [68℄.
7.1 Chapter Abstrat
A theory of gravitational prodution of light salar partiles during and after ination
is investigated. In the most interesting ases where long-wavelength utuations of
light salar elds an be generated during ination, these utuations rather than
quantum utuations produed after ination give the dominant ontribution to partile prodution. In suh ases a simple analytial theory of partile prodution an
be developed. Appliation of these results to the theory of quantum reation of moduli elds demonstrates that if the moduli mass is smaller than the Hubble onstant
then these elds are opiously produed during ination. This gives rise to the osmologial moduli problem even if there is no homogeneous omponent of the lassial
moduli eld in the universe. To avoid this version of the moduli problem it is neessary for the Hubble onstant H during the last stages of ination and/or the reheating
temperature TR after ination to be extremely small.
58
7.2.
INTRODUCTION
59
7.2 Introdution
Reently there has been a renewal of interest in gravitational prodution of partiles in
an expanding universe. This was a subjet of intensive study many years ago, see e.g.
[48, 49, 50, 51, 52, 53℄. However, with the invention of inationary theory the issue
of the prodution of partiles due to gravitational eets beame less urgent. Indeed,
gravitational eets are espeially important near the osmologial singularity, at
the Plank time. But the density of the partiles produed at that epoh beomes
exponentially small due to ination. New partiles are produed only after the end of
ination when the energy density is muh smaller than the Plank density. Prodution
of partiles due to gravitational eets at that stage is typially very ineÆient.
There are a few exeptions to this rule that have motivated the reent interest in
gravitational partile prodution. First of all, there are some models where the main
mehanism of reheating during ination is due to gravitational prodution. Even
though this mehanism is very ineÆient, in the absene of other mehanisms of reheating it may do the job. For example, in the NO models disussed in the previous
hapter the inaton eld does not osillate after ination, so the standard mehanism of reheating does not work [24, 25, 26, 27, 28, 47℄. Usually gravitational partile
prodution in suh models lead to dangerous osmologial onsequenes, suh as large
isourvature utuations and overprodution of gravitinos. In order to overome these
problems, it was neessary to modify the NO models and to use the non-gravitational
mehanism of instant preheating for the desription of partile prodution. See hapters 5 and 6 for more details.
There are some other ases where even very small but unavoidable gravitational
partile prodution may lead either to useful or to atastrophi onsequenes [35,
37, 57, 61, 80℄. For example, it has reently been found that the prodution of
gravitinos by the osillating inaton eld is not suppressed by the small gravitational
oupling. As a result, gravitinos an be opiously produed in the early universe
even if the reheating temperature always remains smaller than 108 GeV [61, 80℄.
Another important example is related to moduli prodution. 15 years ago Coughlan
et al realized that string theory and supergravity give rise to a osmologial moduli
60
CHAPTER 7. THE MODULI PROBLEM
problem assoiated with the existene of a large homogeneous lassial moduli eld in
the early universe [81℄. Soon afterwards Gonharov, Linde and Vysotsky showed that
quantum utuations of moduli elds produed at the last stages of ination lead to
the moduli problem even if initially there were no lassial moduli elds [57℄. Thus
the osmologial moduli problem may appear either beause of the existene of a large
long-living homogeneous lassial moduli eld or beause of quantum prodution of
exitations (partiles) of the moduli elds. In [47℄ it was pointed out that the problem
of moduli prodution is espeially diÆult in the ontext of NO models, where moduli
are produed as abundantly as usual partiles.
Reently the problem of moduli prodution in the early universe was studied by
numerial methods in [80℄, with onlusions similar to those of Ref. [57℄. As I am
going to demonstrate, the main soure of gravitational prodution of light moduli
in inationary osmology is very simple, and one an study the theory of moduli
prodution not only numerially but also analytially by the methods developed in
[57, 47℄. This fat allowed us to generalize and onsiderably strengthen the results of
Refs. [57, 80℄.
In partiular, we will see that in the leading approximation the problem of overprodution of light moduli partiles is equivalent to the problem of large homogeneous
lassial moduli elds [57℄. I will show that the ratio of the number density of light
moduli produed during ination to the entropy of the universe after reheating satises the inequality
n TR H02
>
:
(7.1)
s 3mMp2
Here m is the moduli mass, Mp 2:4 1018 GeV is the redued Plank mass, and
H0 is the Hubble onstant at the moment orresponding to the beginning of the last
60 e-foldings before the end of ination.
In the simplest versions of inationary theory with potentials M 2 2 =2 or 4 =4
one has H0 1014 GeV. In suh models our result implies that in order to satisfy the
osmologial onstraint n < 10 12 one needs to have an abnormally small reheating
s
temperature TR <
1 GeV. Alternatively one may onsider inationary models where
the Hubble onstant at the end of ination is very small. But I will argue that even
7.3.
61
MODULI PROBLEM
this may not help, so one may need either to invoke thermal ination or to use some
other mehanisms that an make the moduli problem less severe, see e.g. [82, 83, 84℄.
In the next setion I outline the lassial and quantum versions of the moduli
problem and explain how eah of them an arise in inationary theory. In setion
7.4 I desribe the results of our numerial simulations of gravitational prodution
of light salar elds during and after preheating. In partiular these results verify
our predition that the dominant ontribution to partile prodution omes from
long-wavelength modes that are indistinguishable from homogeneous lassial moduli
elds. Finally in setion 7.5 I analytially ompute the prodution of these long wavelength modes and derive Eq.(7.1). This setion also ontains a onluding disussion.
7.3 Moduli problem
String moduli ouple to standard model elds only through Plank sale suppressed
interations. Their eetive potential is exatly at in perturbation theory in the
supersymmetri limit, but it may beome urved due to nonperturbative eets or
beause of supersymmetry breaking. If these elds originally are far from the minimum of their eetive potential, the energy of their osillations will derease in an
expanding universe in the same way as the energy density of nonrelativisti matter,
m a 3 (t). Meanwhile the energy density of relativisti plasma dereases as a 4 .
Therefore the relative ontribution of moduli to the energy density of the universe
may quikly beome very signiant. They are expeted to deay after the stage of
nuleosynthesis, violating the standard nuleosynthesis preditions unless the initial
amplitude of the moduli osillations 0 is suÆiently small. The onstraints on the
energy density of the moduli eld and the number of moduli partiles n depend
on details of the theory. The most stringent onstraint appears beause of the photodissoiation and photoprodution of light elements by the deay produts of the
moduli elds. For m 102 103 GeV one has
n
< 10 12
s 10 15 :
(7.2)
62
CHAPTER 7. THE MODULI PROBLEM
see [60, 85, 86℄ and referenes therein. In this hapter I use a onservative version of
this onstraint, ns <
10 12.
If the eld is a lassial homogeneous osillating salar eld, then this onstraint
applies to it as well if one denes the orresponding number density of nonrelativisti
partiles by the following obvious relation:
n =
m2
=
:
m
2
(7.3)
Let us rst onsider moduli with a onstant mass m 102 103 GeV and
assume that reheating of the universe ours after the beginning of osillations of the
moduli. This is indeed the ase if one takes into aount that in order to avoid the
gravitino problem one should have TR < 108 GeV. I will also assume for deniteness
that the minimum of the eetive potential for the eld is at = 0; one an always
ahieve this by an obvious redenition of the eld .
Independent of the hoie of inationary theory, at the end of ination the main
fration of the energy density of the universe is onentrated in the energy of an
osillating salar eld . Typially this is the same eld that drives ination, but in
some models suh as hybrid ination this may not be the ase. I will onsider here
the simplest (and most general) model where the eetive potential of the eld after
ination is quadrati,
M2 2
:
(7.4)
V () =
2
After ination the eld osillates. If one keeps the notation for the amplitude of
osillations of this eld, then one an say that the energy density of this eld is given
2
by () = M2 2 .
To simplify my notation, I take the sale fator at the end of ination to be a0 = 1.
The amplitude of the osillating eld in the theory with the potential (7.4) hanges
as
(t) = 0 a 3=2 (t):
(7.5)
The eld does not osillate and almost does not hange its magnitude until the
7.3.
63
MODULI PROBLEM
moment t1 when H 2 (t) = 3Mp2 beomes smaller than m2 =3. At that time one has
2 2
2
0 0
6Hm2(t)M
2 2M 2
p
p
(7.6)
This ratio, whih an also be obtained by a numerial investigation of osillations of
the moduli elds, does not hange until the time tR when reheating ours beause
and derease in the same way: they are proportional to a 3 .
At the moment of reheating one has (tR ) = 2 N (T )TR4 =30, and the entropy of
produed partiles s = 2 2 N (T )TR3 =45, where N (T ) is the number of light degrees of
freedom. This yields
20 TR
n (7.7)
s ms 3mMp2
Usually one expets TR m 102 GeV. Then in order to have ns < 10 12
one would need 0 10 6 Mp . However, it is hard to imagine why the value of the
moduli eld at the end of ination should be so small. If one takes 0 Mp , whih
looks natural, then one violates the bound ns < 10 12 by more than 12 orders of
magnitude. This is the essene of the osmologial moduli problem [81℄.
In general, the situation is more omplex. During the expansion of the universe
the eetive potential of the moduli aquires some orretions. In partiular, quite
often the eetive mass of the moduli (the seond derivative of the eetive potential)
an be represented as
m2 = m2 + 2 H 2 ;
(7.8)
where is some onstant and H is the Hubble parameter [55, 56℄. Higher derivatives
of the eetive potential may aquire orretions as well. This leads to a dierent
version of the moduli problem disussed in [57℄, see also [58, 59℄. The position of the
minimum of the eetive potential of the moduli eld in the early universe may our
at a large distane from the position of the minimum at present. This may x the
initial position of the eld and lead to its subsequent osillations.
64
CHAPTER 7. THE MODULI PROBLEM
A simple toy model illustrating this possibility was given in [82, 83℄:
1
2
V = m2 2 + H 2 ( 0 )2 :
2
2
(7.9)
At large H the minimum appears at = 0 ; at small H the minimum is at = 0.
Thus one would expet that initially the eld should stay at 0 , and later, when H
dereases, it should osillate about = 0 with an initial amplitude approximately
equal to 0 . The only natural value for 0 in supergravity is 0 Mp . This may lead
to a strong violation of the bound (7.2).
A more detailed investigation of this situation has shown [84℄ that one should
distinguish between three dierent possibilities: 1, 1 and 1.
If > O(10), the eld is trapped in the (moving) minimum of the eetive
potential, its osillations have very small amplitudes, and the moduli problem does
not appear at all [84℄. This is the simplest resolution of the problem, but it is not
simple to nd realisti models where the ondition > O(10) is satised.
The most natural ase is 1. It requires a omplete study of the behavior of
the eetive potential in an expanding universe. There may exist some ases where
the minimum of the eetive potential does not move in this regime, but in general
the eets of quantum reation of moduli in this senario [57, 80℄ are subdominant
with respet to the lassial moduli problem disussed above [57, 58, 59℄, so I will not
disuss this regime.
Here I study the ase 1. In this ase the eetive mass of the moduli at
H m is always muh smaller than H , so the eld does not move towards its
minimum, regardless of its position. Thus if there is any lassial eld 0 it simply
stays at its initial position until H beomes smaller than m, just as in the ase
onsidered above, and the resulting ratio ns is given by Eq. (7.7).
The moduli problem in this senario has two aspets. First of all, in order to avoid
q
the lassial moduli problem one needs to explain why 0 10 6 TmR Mp , whih is
neessary (but not suÆient) to have n =s < 10 12 . Then one should study quantum
reation of moduli in an expanding universe and hek whether their ontribution to
n violates the bound n =s < 10 12 . This last aspet of the moduli problem was
7.3.
MODULI PROBLEM
65
studied in [57, 80℄.
In inationary osmology these two ontributions (the ontributions to n from
the lassial eld and from its quantum utuations) are almost indistinguishable.
Indeed, the dominant ontribution to the number of moduli produed in an expanding
universe is given by the utuations of the moduli eld produed during ination.
These utuations have exponentially large wavelengths and for all pratial purposes
they
have the same onsequenes as a homogeneous lassial eld of amplitude 0 =
q
2
h i.
To be more aurate, these utuations behave in the same way as the homogeneous lassial eld only if their wavelength is greater than H 1 . During ination
this ondition is satised for all inationary utuations, but after ination the size of
the horizon grows and eventually beomes larger than the wavelength of some of the
modes. Then these modes begin to osillate and their amplitude begins to derease
even if at that stage m < H . To take this eet into aount one may simply exlude
from onsideration those modes whose wavelengths beome smaller than H 1 prior
to the moment t m 1 when H drops down to m. It an be shown that in the
ontext of our problem
this is a relatively minor orretion, so one an use the simple
q
estimate 0 = h2 i.
In order to evaluate this quantity I will assume that 1 and m H during
and after ination. This redues the problem to the investigation of the prodution of
massless (or nearly massless) partiles during and after ination. In the next setion
I study this issue and show that in the most interesting ases where inationary
long-wavelength utuations of a salar eld an be generated during ination, they
give the dominant ontribution to partile prodution. This allowed us to redue a
ompliated problem of gravitational partile prodution to a simple problem that
ould be easily solved analytially.
66
CHAPTER 7. THE MODULI PROBLEM
7.4 Generation of light partiles from and after
ination
In this setion I will present the results of a numerial study of the gravitational
reation of light salar partiles in the ontext of ination. Consider a salar eld with the potential
1
V () = m2 R 2
(7.10)
2
where R is the Rii salar. In a Friedmann universe R = a62 (aa + a_ 2 ). The salar
eld operator an be represented in the form
1 Z 3 h
d k a^k k (t)eikx + a^yk k (t)e
(x; t) =
3
=
2
(2 )
ikx
i
(7.11)
where the eigenmode funtions k satisfy
2 !2
a_
k
k + 3 _k + 4
+ m2
a
a
3
R5 k
= 0;
(7.12)
By introduing onformal time and eld variables dened as (7.12) an be simplied to
fk00 + !k2 fk = 0
R dta ; fk ak eq.
(7.13)
where primes denote dierentiation with respet to and
1
2
2
2
2
!k = k + a m +
6
a
:
a
(7.14)
The growth of the sale fator is determined by the evolution of the inaton eld with potential V (). In onformal time
a =
a_ 2
a
8a 02
3
a2 V ()
a_
V ()
00 + 2 0 + a2
= 0:
a
(7.15)
(7.16)
7.4.
GENERATION OF LIGHT PARTICLES FROM AND AFTER INFLATION
67
For initial onditions for the modes fk , in the rst approximation one an use
positive frequeny vauum utuations fk = p12k e ikt , see e.g. [37℄. However, when
desribing utuations produed at the last stages of a long inationary period, one
should begin with utuations that have been generated during the previous stage of
ination. For example, for massless salar elds minimally oupled to gravity instead
of fk = p12k e ikt one should use Hankel funtions [3℄:
!
!
ia(t) H
k
i k Ht
fk (t) = p
1 + e H t exp
e
;
iH
H
k 2k
(7.17)
where H is the Hubble onstant at the beginning of alulations. To make the alulations even more aurate, one should take into aount that long-wavelength
perturbations were produed at early stages of ination when H was greater than
at the beginning of the alulations. If the stage of ination is very long, then the
nal results do not hange muh if instead of the Hankel funtions (7.17) one uses
fk = p12k e ikt . However, if the inationary stage is short, then using the funtions
fk = p12k e ikt onsiderably underestimates the resulting value of h2 i.
At late times the solutions to Eq. (7.13) an be represented in terms of WKB
solutions as
( ) R ( ) R fk ( ) = pk e i !k d + pk e+i !k d ;
(7.18)
2!k
2!k
where k ( ) and k ( ) play the role of oeÆients of a Bogolyubov transformation.
This form is often used to disuss partile prodution beause the number density of
partiles in a given mode is given by nk = jk ( )j2 and their energy density is !k nk .
As we will see, though, the main ontribution to the number density of partiles at
late times omes from long-wavelength modes that are far outside the horizon during
reheating. As long as they remain outside the horizon these modes do not manifest
partile-like behaviour, i.e. the mode funtions do not osillate. In this situation the
oeÆients and have no lear physial meaning. I therefore present our results in
terms of the mode amplitudes jfk ( )j2 , whih as I will show ontain all the information
relevant to number density and energy density at late times.
68
CHAPTER 7. THE MODULI PROBLEM
00
At late times when aa H < m the long wavelength modes of will be nonrelativisti and their number density will simply be given by Eq. (7.3). Moreover the
very long wavelength modes whih are still outside the horizon at late times (e.g. at
nuleosynthesis) will at like a lassial homogeneous eld whose amplitude is given
by
Z
h2i = 212a2 dkk2jfk j2:
(7.19)
It is these very long wavelength modes whih will dominate and therefore the quantity
of interest for us is the amplitude of these utuations.
In our alulations we assumed that m2 = 2 H 2 with 1; the results shown
are for = 0 but we also did the alulations with = :01 and found that the results
were independent of in this range.
Figure 7.1 shows the results of solving Eq. (7.13) for a model with the inaton potential V () = 14 4 . These data were taken after ten osillations of the inaton eld.
The vertial axis shows k2 jfk j2 as a funtion of the momentum k. The momentum is
shown in units of the Hubble onstant at the end of ination.
The dierent plots represent runs with dierent starts of ination, i.e. with dierent initial values of . They all oinide in the ultraviolet part of the spetrum, but
the runs that started towards the end of ination show a signiant suppression in
the infrared. This shows that utuations produed during ination are the primary
soure of the infrared modes, whih in turn dominate the number density.
The urve on top shows the Hankel funtion solutions (7.17), whih give
2 2
jfk j2 = a2kH3
(7.20)
for de Sitter spae, i.e. for a onstant H . In the gure, we have orreted this
expression by using for eah mode the value of the Hubble onstant at the moment
when that mode rossed the horizon. For the 4 model shown here the appropriate
Hubble parameter for eah mode an be approximated as
s
2 2
Hk =
e
3
1
k
ln
He
!!
(7.21)
7.4.
GENERATION OF LIGHT PARTICLES FROM AND AFTER INFLATION
69
k2 fk 2
1012
106
1
10-4
10-2
1
k
Figure 7.1: Flutuations vs. mode frequeny at a late time. The lower plots show
runs that started lose to the end of ination, with initial onditions fk = p12k e ikt .
The starting times range from 0 = 1:5Mp (lowest urve) up through 0 = 2Mp. The
highest urve shows the Hankel funtion solutions given by eqs. (7.20) and (7.21) As
we see, alulations with fk = p12k e ikt produe the orret spetrum at large k but
underestimate the level of quantum utuations at small k.
where e and He are the values of the inaton and the Hubble parameter respetively
at the end of ination.
Note that if the Hankel funtion solutions (7.17) are used as initial onditions
for a numerial run then they do not hange as the modes ross the horizon, so
the upper urve of the plot an also be obtained from suh a run. Relative to this
upper urve it's easy to see how the numerial runs show suppression in the infrared
due to starting ination at late times and hoosing the initial onditions in the form
fk = p12k e ikt , and suppression in the ultraviolet due to the end of ination. The
latter suppression is physially realisti. The infrared suppression should our at a
wavelength orresponding to the Hubble radius at the beginning of ination.
The dierent regions of the graph illustrate eets that ourred at dierent times.
During ination long wavelength modes rossed the horizon at early times. Thus the
far left portion of the plot shows the modes that rossed earliest. They have the
highest amplitudes both beause they were frozen in earliest and beause the Hubble
onstant was higher at earlier times when they were produed. The lower plots don't
70
CHAPTER 7. THE MODULI PROBLEM
show these high amplitudes beause ination began too late for these modes to ross
outside the horizon and be amplied. Farther to the right the urve shows modes
that were only slightly if at all amplied during ination. The far right modes were
produed during the fast rolling and osillatory stages. These modes are not frozen
and an be desribed meaningfully in terms of and oeÆients. The regularized
expression
R
h
i
jfk j2 = !1 jk j2 + Re k ke 2i !kd
(7.22)
k
shows why the amplitudes of these modes osillate as a funtion of k.
In short inationary utuations are primarily responsible for produing infrared
modes and post-inationary eets aount for ultraviolet modes, but it is the infrared
modes that were outside the horizon at the end of ination that dominate the number
density at late times. The earlier ination began the farther this distribution will
extend into the infrared, and the long wavelength end of this spetrum will always
give the greatest ontribution to the number density of partiles.
Our numerial alulations were similar to those of Kuzmin and Tkahev [37℄.
However, they took a rather small initial value of the lassial salar eld , whih
resulted in less than 60 e-folds of ination. As initial onditions for the utuations
they used fk = p12k e ikt . They pointed out that the results of suh alulations
an give only a lower bound on the number of partiles produed during ination.
Consequently, similar alulations performed in [80℄ ould give only a lower bound on
the number of moduli elds produed in the early universe.
Our goal is to nd not a lower bound but the omplete number of partiles produed at the last stages of ination in realisti inationary models, where the total
duration of ination typially is muh greater than 60 e-foldings. One result revealed
by our alulations is that the eets of an arbitrarily long stage of ination an be
mimiked by the orret hoie of \initial" onditions hosen for the modes k after
ination. Instead of using the Minkowski spae utuations fk = p1k e ikt used in [37℄
as well as in our numerial alulations, one should use the de Sitter spae solutions
(7.17), with H orreted to the value it had for eah mode at horizon rossing. Using
these modes at the end of ination is equivalent to running a simulation with a long
7.5.
LIGHT MODULI FROM INFLATION
71
stage of ination.
Our numerial alulations onrmed the result that I am going to derive analytially in the next setion: The number of partiles (n mh2 i) produed during
the stage of ination beginning at = 0 in the simplest model M 2 2 =2 is proportional to 40 , whereas in the model 4 =4 it is proportional to 60 . Thus the total
number of partiles produed during ination is extremely sensitive to the hoie of
initial onditions. If one onsiders 0 orresponding to the beginning of the last 60
e-folds of ination, the total number of partiles produed at that stage appears to be
muh greater than the lower bound obtained in [37℄. As we will see, this allowed us to
put a muh stronger onstraint on the moduli theories than the onstraint obtained
in [80℄.
7.5 Light moduli from ination
The numerial results obtained in the previous setion onrmed our expetation that
in the most interesting ases where long-wavelength inationary utuations of light
salar elds an be generated during ination, they give the dominant ontribution
to partile prodution. In partiular, in the ase of 1, m H most moduli eld
utuations are generated during ination rather than during the post-inationary
stage. These utuations grow at the stage of ination in the same way as if the
moduli eld were massless [3℄:
dh2 i H 3
= 2:
dt
4
(7.23)
If the Hubble onstant does not hange during ination, one obtains the well-known
relation
3
h2i = H42t :
(7.24)
However, in realisti inationary models the Hubble onstant hanges in time, and
utuations of the light elds with m H behave in a more ompliated way.
As an example, let us onsider the ase studied in the last setion. Here ination
is driven by a eld with an eetive potential V () = 4 4 at > 0. This potential
72
CHAPTER 7. THE MODULI PROBLEM
ould be osillatory or at for < 0. We onsider a light salar eld that is not
oupled to the inaton eld , and that is minimally oupled to gravity.
The eld during ination obeys the following equation:
3H _ = 3 :
(7.25)
Here
s
1 2
H=
:
2 3 Mp
These two equations yield the solution [3℄
0
= 0 exp (7.26)
1
s
2 Mp tA ;
3
(7.27)
where 0 is the initial value of the inaton eld . In this ase Eq. (7.23) reads:
s
p
dh2 i
60
= p 2 3 exp 12 Mp tA
dt
3
96 3 Mp
0
1
(7.28)
The result of integration at large t onverges to
!
30 2
2
h i = 2 24M 2 :
p
(7.29)
This result agrees with the results of our numerial investigation desribed in the
previous setion.
From the point of view of the moduli problem, these utuations lead to the same
onsequenes as a lassial salar eld that is homogeneous on the sale H 1 and
that has a typial amplitude
0 =
q
s
30
h2i = 2 24M
2:
p
(7.30)
2
A similar result an be obtained in the model V () = M2 2 . In this ase one has
7.5.
73
LIGHT MODULI FROM INFLATION
[3℄
s
2
M Mt:
3 p
The time-dependent Hubble parameter is given by
(t) = 0
H=
whih yields
0 =
q
pM (t);
6M
(7.31)
(7.32)
p
2
p 0 2:
h2 i = 8M
3M
p
(7.33)
As we see, the value of 0 depends on the initial value of the eld . This result
has the following interpretation. One may onsider an inationary domain of initial
size H 1 (0 ). This domain after ination beomes exponentially large. For example,
2
its size in the model with V () = M2 2 beomes [3℄
l H 1 (0 ) exp
!
20
:
4Mp2
(7.34)
In order to ahieve 60 e-folds of ination in this model one needs to take 0 15Mp .
This implies that a typial value of the (nearly) homogeneous salar eld in a
universe that experiened 60 e-folds of ination in this model is given by
0 =
q
h2 i 5M:
In realisti versions of this model one has M
tution of this result into Eq. (7.7) gives
n
s
2 10
5 10 6Mp 1013 GeV [3℄.
10 TR :
m
(7.35)
Substi(7.36)
This implies that the ondition n =s <
10 12 requires that the reheating temperature
in this model should be at least two orders of magnitude smaller than m. For example,
for m 102 GeV one should have TR <
1 GeV, whih looks rather problemati.
74
CHAPTER 7. THE MODULI PROBLEM
This result onrms the basi onlusion of Ref. [57℄ that the usual models of ination do not solve the moduli problem. Our result is similar to the result obtained in
[80℄ by numerial methods, but it is approximately two orders of magnitude stronger.
The reason for this dierene is that the authors of Ref. [80℄ used a muh smaller
value of 0 in their numerial alulations. Consequently, they took into aount only
the partiles produed at the very end of ination, whereas the leading eet ours
at earlier stages of ination, i.e. at larger .
q
In general one an get a simple estimate of 0 = h2 i by assuming that the
universe expanded with a onstant Hubble parameter H0 during the last 60 e-folds
of ination. To make this estimate more aurate one should take the value of the
Hubble onstant not at the end of ination but approximately 60 e-foldings before it,
at the time when the utuations on the sale of the present horizon were produed.
The reason is that the largest ontribution to the utuations is given by the time
when the Hubble onstant took its greatest values. Also, at that stage the rate of
hange of H was relatively slow, so the approximation H = H0 = onst is reasonable.
Thus one an write
0 =
This gives
q
q
h2 i > H20 H0t H20
n TR H02
>
:
s 3m Mp2
p
60 H0 :
(7.37)
(7.38)
In the simplest versions of haoti ination with potentials M 2 2 =2 or 4 =4 one has
H0 1014 GeV, whih leads to the requirement TR <
1 GeV. But this equation shows
that there is another way to relax the problem of the gravitational moduli prodution:
one may onsider models of ination with a very small value of H0 [57℄. For example,
one may have n =s 10 12 for TR H0 107 GeV.
However, this ondition is not suÆient to resolve the moduli problem; the situation is more ompliated. First of all, it is very diÆult to nd inationary models
where ination ours only during 60 e-foldings. Typially it lasts muh longer, and
the utuations of the light moduli elds will be muh greater. This is espeially
7.5.
LIGHT MODULI FROM INFLATION
75
obvious in the theory of eternal ination where the amplitude of utuations of the
light moduli elds an beome indenitely large [3℄. In partiular, if the ondition
m2 m2 + 2 H 2 with 1 remains valid for >
Mp, then one may expet the
generation of moduli elds > Mp . This should initiate ination driven by the light
moduli [47℄. Then the situation would beome even more problemati: we would
need to nd out how one ould produe the baryon asymmetry of the universe after
the light moduli deay and how one ould obtain density perturbations Æ= 10 4
in suh a low-sale inationary model.
One may expet that the region of values of where its eetive potential has
small urvature m2 H 2 may be limited, and may even depend on H . Then the
existene of a long stage of ination would push the utuations of the eld up
to the region where its eetive potential beomes urved, and instead of our results
for 0 one should substitute the largest value of for whih m2 < H 2 . In suh a
situation one would have a mixture of problems that our at 1 and at 1.
Finally, I should emphasize that all our worries about quantum reation of moduli
appear only after one makes the assumption that for whatever reason the initial value
of the lassial eld in the universe vanishes, i.e. that the lassial version of the
moduli problem has been resolved. We do not see any justiation for this assumption
in theories where the mass of the moduli eld in the early universe is muh smaller
than H . Indeed, in suh theories the lassial eld initially an take any value,
and this value is not going to hange until the moment t H 1 m 1 . The main
purpose of the researh being reported on here was to demonstrate that even if one
nds a solution to the light moduli problem at the lassial level, the same problem
will appear again beause of quantum eets.
This does not mean that the moduli problem is unsolvable. One of the most
interesting solutions is provided by thermal ination [82, 83℄. The Hubble onstant
during ination in this senario is very small, and the eets of moduli prodution
there are rather insigniant. Another possibility is that moduli are very heavy in the
early universe, m2 = m2 + 2 H 2, with > O(10), in whih ase the moduli problem
does not appear [84℄. The main question is whether we really need to make the theory
so ompliated in order to avoid the osmologial problems assoiated with moduli.
76
CHAPTER 7. THE MODULI PROBLEM
Is it possible to nd a simpler solution? One of the main results of our investigation
is to onrm the onlusion of Ref. [57℄ that the simplest versions of inationary
theory do not help us to solve the moduli problem but rather aggravate it.
In onlusion I would like to note that the methods disussed here apply not only
to the theory of moduli prodution but to other problems as well. For example,
one may study the theory of gravitational prodution of superheavy salar partiles
after ination [35, 37℄. If these partiles are minimally oupled to gravity and have
mass m H during ination, then one an use Eqs. (7.30), (7.33) to alulate
the number of produed partiles. These equations imply that the nal result will
strongly depend on 0 , i.e. on the duration of ination. If ination ours for more
than 60 Hubble times, the prodution of partiles with m H is muh more eÆient
than was previously antiipated. As I just mentioned, if 0 is large enough then the
prodution of utuations of the eld may even lead to a new stage of ination
driven by the eld [47℄. On the other hand, if m is greater than the value of the
Hubble onstant at the very end of ination, then quantum utuations are produed
only at the early stages of ination (when H > m). These utuations osillate
and derease exponentially during the last stages of ination. In suh ases the nal
number of produed partiles will not depend on the duration of ination and an be
unambiguously alulated.
Chapter 8
Ination After Preheating
Note: This hapter is based on a paper by Gary Felder, Lev Kofman, Andrei Linde,
and Igor Tkahev, available on the Los Alamos eprint server as hep-ph/0004024. The
full itation appears in the bibliography [87℄.
Chapter Abstrat
Preheating after ination may lead to nonthermal phase transitions with symmetry
restoration. These phase transitions may our even if the total energy density of utuations produed during reheating is relatively small as ompared with the vauum
energy in the state with restored symmetry. As a result, in some inationary models
one enounters a seondary, nonthermal stage of ination due to symmetry restoration after preheating. Suh seondary ination ould also provide a partial solution
to the moduli problem. We review the theory of nonthermal phase transitions and
make a predition about the expansion fator during the seondary inationary stage.
We then present the results of lattie simulations that verify these preditions, and
disuss possible impliations of our results for the theory of formation of topologial
defets during nonthermal phase transitions.
77
78
CHAPTER 8.
INFLATION AFTER PREHEATING
8.1 Introdution
The theory of osmologial phase transitions is usually assoiated with symmetry
restoration due to high temperature eets and the subsequent symmetry breaking
that ours as the temperature dereases in an expanding universe [88, 89, 90, 91,
92, 93, 3℄. A partiularly important version of this theory is the theory of rst order
osmologial phase transitions developed in [93℄. It served as a basis for the rst
versions of inationary osmology [1, 13, 94℄, as well as for the theory of eletroweak
baryogenesis [95, 96℄.
Reently it was pointed out that preheating after ination [5, 7℄ may rapidly
produe a large number of partiles that for a long time remain in a state out of
thermal equilibrium. These partiles may lead to spei nonthermal osmologial
phase transitions [14, 15℄. In some ases these phase transitions are rst order [16,
97℄; they our by the formation of bubbles of the phase with spontaneously broken
symmetry inside the metastable symmetri phase. If the lifetime of the metastable
state is large enough for the energy density of utuations to be diluted, one may
enounter a short seondary stage of ination after preheating [14℄. Suh a seondary
ination stage, if it ours late enough, ould be important in solving the moduli
and gravitino problems. In this respet seondary \nonthermal" ination due to
preheating may be an alternative to the \thermal ination" [82℄, suggested for solving
these problems.
In this hapter I will briey present the theory of suh phase transitions and then
give the results of numerial lattie simulations that diretly demonstrated the possibility of suh brief ination. I will also disuss possible impliations of these results
for the theory of formation of topologial defets during nonthermal phase transitions.
The lattie alulations reported here were done using the program LATTICEEASY,
desribed in part III of this thesis.
8.2.
79
THEORY OF THE PHASE TRANSITION
8.2 Theory of the phase transition
Consider a set of salar elds with the potential
V (; ) = (2
4
v 2 )2 +
g2 2 2
:
2
(8.1)
The inaton eld has a double-well potential and interats with an N -omponent
P
salar eld ; 2 Ni=1 2i . For simpliity, the eld is taken to be massless and
without self-interation. The elds ouple minimally to gravity in a FRW universe
with a sale fator a(t).
The potential V (; ) has minima at = v , = 0 and a loal maximum in
the diretion at = = 0 with urvature V; = v 2 . The eetive potential
aquires orretions due to quantum and/or thermal utuations of the salar elds
[88, 89, 93, 3℄,
3
g2
g2
V = h2 i2 + h2 i2 + h2 i2 + :::;
(8.2)
2
2
2
where I have written only the leading terms depending on and . The eetive
mass squared of the eld is given by
m2 = m2 + 32 + 3h2 i + g 2 h2 i;
(8.3)
where m2 = v 2 . Symmetry is restored, i.e. = 0 beomes a stable equilibrium
point, when the utuations h2i; h2 i beome suÆiently large to make the eetive
mass squared positive at = 0.
For example, one may onsider matter in thermal equilibrium. Then, in the large
2
temperature limit, one has h2 i = h2i i = T12 : The eetive mass squared of the eld
m2;eff = m2 + 32 + 3hÆ2i + g 2 h2 i
(8.4)
is positive and symmetry is restored (i.e. = 0 is the stable equilibrium point) for
2
T > T , where T2 = 312+mNg2 m2 . At this temperature the energy density of the gas
80
CHAPTER 8.
INFLATION AFTER PREHEATING
of ultrarelativisti partiles is given by
24 m4 N (T) 2
2
= N (T ) T4 =
:
30
5 (3 + Ng 2 )2
(8.5)
Here N (T ) is the eetive number of degrees of freedom at large temperature, whih
in realisti situations may vary from 102 to 103 . We will assume that Ng 2 , see
2
below. For g 4 < 96N5(NT2 ) the thermal energy at the moment of the phase transition
4
is greater than the vauum energy density V (0) = m4 , whih means that the phase
transition does not involve a stage of ination.
In fat, the phase transition with symmetry breaking ours not at T > T , but
somewhat earlier [93℄. To understand this eet let us ompare the temperature
p
T m=( Ng ) and the mass m = g of the partiles in the minimum of the zerop
temperature eetive potential at = v = m= . One an easily see that m T
for Ng 4 . This means that for Ng 4 the temperature T is insuÆient to exite
perturbations of the elds i at = v . As a result, these perturbations do not hange
the shape of the eetive potential = v . Thus the potential at T slightly above T
has its old zero-temperature minimum at = v , as well as the temperature-indued
minimum at = 0. Symmetry breaking ours as a rst-order phase transition due to
formation of bubbles of the phase with v at some temperature above T when the
minimum at = v beomes deeper than the minimum at = 0, and the probability
of bubble formation beomes suÆiently large. A more detailed investigation in the
ase N = 1 shows that the phase transition is rst order under a weaker ondition
g 3 [93℄.
In the ase Ng 4 > 102 the phase transition ours after a seondary stage of
ination. In this regime radiative orretions are important. They lead to the reation
of a loal minimum of V (; ) at = 0 even at zero temperature, and the phase
transition ours from a strongly superooled state [93℄. That is why the rst models
of ination required superooling at the moment of the phase transition [1, 13, 94℄.
In supersymmetri theories one may have Ng 4 102 and still have a potential
that is at near the origin due to anellation of quantum orretions of bosons and
fermions [82℄. In suh ases the thermal energy beomes smaller than the vauum
8.2.
THEORY OF THE PHASE TRANSITION
81
energy at T < T0 , where T04 = 2N152 m2 v 2 . Then one may have a short stage of
ination that begins at T T0 and ends at T = T . During this time the universe
may inate by the fator
g 4 1=4
a T0
1
= 10
:
(8.6)
a0 T
Similar phase transitions may our muh more eÆiently prior to thermalization,
due to the anomalously large utuations h2 i and h2 i produed during preheating
[14, 15℄. These utuations an hange the shape of the eetive potential and lead
to symmetry restoration. Afterwards, the universe expands, the values of h2 i and
h2i drop down, and the phase transition with symmetry breaking ours.
An interesting feature of nonthermal phase transitions is that they may our
even in theories where the usual thermal phase transitions do not happen. The main
reason an be understood as follows. Suppose reheating ours due to the deay of
a salar eld with energy density . If this energy is instantly thermalized, then one
obtains relativisti partiles with energy density O(T 4) that in the rst approximation
an be represented as E 2 (h2 i + h2 i). Here E T 1=4 is a typial energy
of a partile in thermal equilibrium. After preheating, however, one has partiles and with muh smaller energy but large oupation numbers. As a result, the same
energy release may reate muh greater values of h2 i and h2 i than in the ase of
instant thermalization. This may lead to symmetry restoration after preheating even
if the symmetry breaking ours on the GUT sale, v 1016 GeV [14, 15℄.
The main onlusions of [14, 15℄ have been onrmed by detailed investigation
using lattie simulations in [16, 17, 98, 97℄. One of the main results obtained in [16℄
was that for suÆiently large g 2 nonthermal phase transitions are rst order. They
our from a metastable vauum at = 0 due to the reation of bubbles with 6= 0.
This result is very similar to the analogous result in the theory of thermal phase
transitions [93℄. Aording to [16℄, the neessary onditions for this transition to
our and to be of the rst order an be formulated as follows:
(i) At the time of the phase transition, the point = 0 should be a loal minimum
of the eetive potential. From (8.3), we see that this means that Ng 2 h2i i > v 2 .
(ii) At the same time, the typial momentum p of i partiles should be smaller
82
CHAPTER 8.
INFLATION AFTER PREHEATING
than gv . This is the ondition of the existene of a potential barrier. Partiles with
momenta p < gv annot penetrate the state with jj v , so they annot hange the
shape of the eetive potential at jj v . Therefore, if both onditions (i) and (ii)
are satised, the eetive potential has a loal minimum at = 0 and two degenerate
minima at v .
(iii) Before the minima at v beome deeper than the minimum at = 0, the
inaton's zero mode should deay signiantly, so that it performs small osillations
near = 0. Then, after the minimum at jj v beomes deeper than the minimum
at = 0, utuations of drive the system over the potential barrier, reating an
expanding bubble.
The investigation performed in [16℄ onrmed that for suÆiently large g 2 and
N these onditions are indeed satised and the phase transition is rst order. One
may wonder whether for g 2 one may have a stage of ination in the metastable
vauum = 0.
Analytial estimates of Ref. [14℄ suggested that this is indeed the ase, and the
degree of this ination for N = 1 is expeted to be
a g 2 1=4
;
a0
(8.7)
4 1=4
whih is muh greater than the number 10 1 g
in the thermal ination senario.
One ould also expet that the duration of ination, just like the strength of the
phase transition, inreases if one onsiders N elds i with N 1.
However, the theory of preheating is extremely ompliated, and there are some
fators that ould not be adequately taken into aount in the simple estimates of
[14℄. The most important fator is the eet of resattering of partiles produed
during preheating [99, 100℄. This eet tends to shut down the resonant prodution
of partiles and thus shorten or prevent entirely the ourrene of a seondary stage of
ination. Thus the estimates above reet the maximum degree of ination possible
for a given set of parameter values, but in pratie the expansion fator will be
somewhat smaller than these preditions. The only way to fully aount for all the
eets of bakreation and expansion is through numerial lattie simulations. In the
8.3.
SIMULATION RESULTS AND THEIR INTERPRETATION
83
next setion I will desribe the basi features of our method and desribe our main
results.
8.3 Simulation Results and Their Interpretation
I take 10 13 , whih gives the proper magnitude of inationary perturbations of
density [3, 101℄. I assume that g 2 , and onsider v 1016 GeV, whih orresponds
to the GUT sale. A numerial investigation of preheating in the model (8.1) was
rst performed in [16℄. The authors found a strongly rst order phase transition.
The strength of the phase transition inreased with an inrease of g 2 = and of the
number N of the elds i . However, for the parameters of the model studied in
[16℄ (g 2 = 200) there was no ination during symmetry restoration. This is not
unexpeted beause the estimates disussed above indiated that the expansion of
the universe during the short stage of nonthermal ination annot be greater than
2
( g )1=4 .
Keeping in mind that in this model is extremely small, one would expet that
in realisti versions of this model one may have g 2= as large as 1010 , whih ould
lead to a relatively long stage of ination. However, for very large g 2 = our analytial
estimates are unreliable, and lattie simulations beome extremely diÆult: One
needs to have enormously large latties to keep both infrared and ultraviolet eets
under ontrol.
To mimi the eets of large g 2, we onsidered a large number of the elds i .
We performed simulations for g 2 = = 800 and N = 19. With these parameters the
strength of the phase transition beame muh greater, and there was a short stage of
ination prior to the phase transition.
The details of our lattie alulations an be found in part III. Here I present
only our main results for this model. These results were obtained using a grid of 1283
1 where 0 = :342Mp is the value of the
points and a grid spaing of dx = :133 p
0
inaton eld at the end of ination, and hene at the beginning of our simulation.
These parameters orrespond to a total box size roughly equal to 10 Hubble radii at
the end of ination.
84
CHAPTER 8.
INFLATION AFTER PREHEATING
3
2
1
0
-1
-2
0
12
1´10
12
12
2´10
3´10
12
4´10
12
5´10
t
Figure 8.1: The spatial average of the ination eld as a funtion of time. The eld
is shown in units of v , the symmetry breaking parameter. Time is shown in Plank
units.
The simulation showed that the osillations of the inaton eld dereased until the
eld was trapped near zero. It remained there until the moment of the phase transition
when it rapidly jumped to its symmetry breaking value, as shown in Fig. 8.1.
The trapping of the eld ourred beause of the orretions to the eetive potential indued by the partiles and produed during preheating, just like in the
theory of high-temperature phase transitions. In this ase, however, this eet has
some unusual features.
To rst order in g 2 , the leading ontribution to the equation of motion  = V 0
is given by g 2 h2 i, where
1
N Z nk k2 dk
2
h i 22 ! () :
k
0
q
(8.8)
Here !k = k2 + g 2(2 + h2 i) is the energy of i partiles with momentum k and
nk is their oupation number; is the homogeneous omponent of the eld. For
8.3.
SIMULATION RESULTS AND THEIR INTERPRETATION
q
85
h2i, one has
1
N Z q nk k2 dk
:
(8.9)
=0 2 2 0 k2 + g 2 h2i
This quantity does not depend on ; it an be evaluated using our lattie simulations
when the eld osillates near = 0. It leads to the usual quadrati orretion to the
eetive potential, see Eq. (8.2). This orretion adequately desribes the hange of
the shape of the eetive potential for smaller than the amplitude of the osillations
of this eld, beause most of the time
prior to the moment of the phase transition
q
this amplitude is muh smaller than h2 i.
However, if we want to evaluate the eetive potential at all values of jj from 0 to
v , rather than for similar to the amplitude of the osillations, then one shouldqtake
into aount that for suÆiently large jj the term g jj beomes greater than g h2 i
and than the typial momentum k of partiles i . In this ase the main ontribution
to h2 i is given by nonrelativisti partiles with !k +g jj, and one has
h2 i
1
N Z nk k2 dk n
2
h i 22
=
;
g jj
0 g jj
(8.10)
where n is the total density of all types of i partiles. This implies that at large jj
the eetive potential aquires a orretion
ÆV
gjjn:
(8.11)
Thus, instead of being quadrati or ubi in jj, as one ould expet from the analogy
with the high-temperature theory [93, 97℄, the orretions to the eetive potential
at large jj are proportional to jj [5, 7℄.
The ombination of these two types of orretions to the eetive potential (quadrati
at small jj and linear at large jj) leads to the symmetry restoration that we have
found in our lattie simulations.
It is instrutive to look in a more detailed way at the small region near the time
of the phase transition. The rst of the graphs in Fig. 8.2 shows the osillations
of the eld soon before the phase transition, whereas the seond one shows these
86
CHAPTER 8.
0.01
INFLATION AFTER PREHEATING
1
0.0075
0.95
0.005
0.0025
0.9
0
0.85
-0.0025
0.8
-0.005
0.75
-0.0075
12
3.´10
12
3.5´10
t
12
4.´10
0.7
12
3.8´10
4.3´10
t
12
4.8´10
12
Figure 8.2: The spatial average of the ination eld as a funtion of time in the
viinity of the phase transition. The left gure shows the eld just before the phase
transition and at the moment of the transition. The eld osillates with an amplitude
approahing 10 3 v . The right gure shows the eld after the phase transition,
when it osillates near the (time-dependent) position of the minimum of the eetive
potential at v . Time is shown in Plank units.
osillations soon afterwards.
First of all, one an see that just before the phase transition the eld osillates
with an amplitude three orders of magnitude smaller than v , whih is a lear sign of
symmetry restoration. Another interesting feature is that the frequeny of osillations
does not vanish as we approah the phase transition, but remains nearly onstant.
Moreover, this frequeny is only about two times smaller than the frequeny of osilp
lations after the phase transition, whih is equal to 2m. Note that the frequeny
of the osillations is determined by the eetive mass of the salar eld, whih is
given by the urvature of the eetive potential: m2 = V 00 . This means that at the
moment of the phase transition the eetive potential has a deep minimum at = 0
with urvature V 00 +m2 , i.e. the phase transition is strongly rst order. Suh phase
transitions should our due to the formation of bubbles ontaining nonvanishing eld
.
Indeed, we found that this transition ourred in a nearly spherial region of the
lattie that quikly grew to enompass the entire spae. The growth of this region
of the new phase is shown in Fig. 8.3. The nearly perfet spheriity of this region is
an additional indiation that the transition was strongly rst-order. In omparison,
8.3.
87
SIMULATION RESULTS AND THEIR INTERPRETATION
1282.18
100
50
0
1282.68
100
100
50
0
50
1283.67
100
100
50
0
0
50
0
50
1286.67
100
100
100
0
0
50
50
0
50
100
100
0
0
50
100
50
0
0
50
100
Figure 8.3: These plots show the region of spae in whih symmetry breaking has
ourred at four suessive times.
the bubble observed in the lattie simulations of [16℄ for g 2 = 200 was not exatly
spherially symmetri.
The rst order phase transition and bubble formation seen in our simulations an
be understood as a result of gradual aumulation of lassial utuations Æ(t; ~x).
These utuations stohastially limb up from = 0 towards the loal maximum
of the eetive potential. Consider the regions in whih Æ(t; ~x) > . If the
probability of formation of suh regions is small beause they orrespond to high
peaks of the random eld , then these regions will have a nearly spherial shape
and an be represented by spherial surfaes of radius R (bubbles). If the radius R
is small, gradient terms will prevent the eld inside the region from rolling down
towards the global minimum at = v (subritial bubble). If R is large enough,
88
CHAPTER 8.
INFLATION AFTER PREHEATING
800
600
400
200
0
0
12
1´10
12
2´10
12
3´10
12
4´10
12
5´10
p
Figure 8.4: The sale fator a as a funtion of time. In the beginning a t, whih
is a urve with negative urvature, but then at some stage it begins to turn upwards,
indiating a short stage of ination.
the gradient terms annot push the eld bak to the metastable state = 0 and
the eld inside the bubble rolls towards the global minimum, forming a bubble of
ever inreasing radius. This proess an be desribed within the stohasti approah
to tunneling proposed in [102℄. Typially, the gradient terms annot win over the
potential energy terms if R > O(jm 1j), where m orresponds to the eetive mass
of the salar eld in the interior of the bubble. This provides an estimate for the
initial size of the bubble R O(jm 1 j).
The phase transition ours from a state with energy density dominated by the
vauum energy density V (0). Figure 8.4 shows the sale fator a as a funtion of time.
The urvature beomes slightly positive at the time before the phase transition, whih
indiates a short stage of exponential growth of the universe. Beause the urvature
is hard to see in gure 8.4 I have also plotted the seond derivative a in gure 8.5.
While the inaton is trapped in the false vauum state, a beomes positive, indiating
a brief stage of ination.
Another signature of ination is an equation of state with negative pressure. Figure 8.6 shows the parameter = p=, whih beomes negative during the metastable
phase. At the moment of the phase transition the universe beomes matter dominated
8.4.
NONTHERMAL PHASE TRANSITIONS AND PRODUCTION OF TOPOLOGICAL DEF
-23
3´10
-23
2´10
-23
1´10
0
-23
-1´10
-23
-2´10
0
12
1´10
12
2´10
12
3´10
12
4´10
12
5´10
Figure 8.5: The seond derivative of the sale fator, a. A universe dominated by
ordinary matter (relativisti or nonrelativisti) will always have a < 0, whereas in an
inationary universe a > 0. We see that starting from the moment t 2 1012 (in
Plank units) the universe experienes aelerated (inationary) expansion.
and the pressure jumps to nearly 0.
From the beginning of this inationary stage (roughly when the pressure beomes
negative) to the moment of the phase transition the total expansion fator is 2.1. As
expeted
this is of the same order but somewhat lower than the predited maximum,
2 1=4
g
5:3. We an thus onlude that it is possible to ahieve ination for
parameters for whih this would not have been possible in thermal equilibrium (g 4 ). In our simulation we showed the ourrene of a very brief stage of ination.
This stage may be muh longer for larger (realisti) values of g 2 =. However, to
hek whether this is indeed the ase one would need to perform a more detailed
investigation on a lattie of a muh greater size.
8.4 Nonthermal phase transitions and prodution
of topologial defets
In this setion I would like to disuss possible impliations of our investigation for the
theory of prodution of topologial defets after preheating [16, 17, 98℄.
90
CHAPTER 8.
INFLATION AFTER PREHEATING
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
12
1´10
12
2´10
12
3´10
12
4´10
12
5´10
Figure 8.6: The ratio of pressure to energy density p=. (Values were time averaged
over short time sales to make the plot smoother and more readable.)
The bubbles that appear after the phase transition an ontain either positive
or negative eld, = v . If bubbles of either type are formed with omparable
probability, then after the phase transition the universe beomes divided into nearly
equal numbers of domains with = v , separated by domain walls. Suh domain
walls would lead to disastrous osmologial onsequenes, whih would rule out the
models where this may happen [14, 15℄.
In general, the number of bubbles with = +v may be muh greater (or muh
smaller) than the number of bubbles with = v . Then the domain wall problem
does not appear beause the bubbles with = +v would rapidly eat all their ompetitors with = v (or vie versa). This may happen, for example, if the moment
of the bubble prodution is determined by the oherently osillating salar eld .
In suh a ase, after osillating a bit near the top of the eetive potential, the eld
may wind up in the same minimum of the eetive potential everywhere in the
universe.
To investigate the domain wall problem in our model one would need to repeat the
alulation many times with slightly dierent initial onditions or to make them in a
box of a muh greater size that would allow one to see many bubbles simultaneously.
Fortunately, the results obtained in our study may be suÆient to give an answer to
8.4. TOPOLOGICAL DEFECTS
91
this question without extremely large simulations.
First of all, aording to the stohasti approah [102℄ to the theory of tunneling
with bubble formation [103, 104℄, the bubbles of the eld are reated as a result of
the aumulation of long-wavelength utuations of the salar eld with momenta k
smaller than the typial mass sale m assoiated with this eld, see Setion 8.3. In
our ase this mass sale is related to the frequeny of osillations of the salar eld
at the moment of the phase transition. At that moment the leading ontribution to
the utuations h2 i is given by utuations with momenta muh
smaller than m .
q
2
We alulated the value of the long-wavelength omponent of h i and found that
it is (approximately) of the same order as the amplitude of osillations of the eld at the moment of the phase transition. The existene of a rst-order phase transition suggests that the probability of bubble formation must have been exponentially
suppressed during the metastable stage. Suh suppression would only our only if
the amplitude
of utuations required to form a bubble of the new phase was muh
q
larger than h2 i [102℄, whih would in turn mean the required amplitude was muh
greater than the amplitude of osillations of . This suggests that the probability of
the bubble formation is almost entirely determined by the inoherent utuations of
the eld rather than by the small oherent osillations of this eld. Consequently,
the probability of formation of bubbles ontaining = +v in the rst approximation
must be equal to the probability of formation of bubbles ontaining = v .
To make this statement more reliable one would need to estimate the amplitude
of the long-wavelength utuations of the eld in a more preise way, whih would
involve using latties of a greater size. However, there is additional evidene suggesting that the number of bubbles with positive and negative must be approximately
equal to eah other.
Indeed, as we have seen, the urvature of the eetive potential remained approximately onstant during dozens of osillations of the eld prior to the moment of
the phase transition. This suggests that the shape of the eetive potential and,
onsequently, the probability of the tunneling, did not hange muh during a single
osillation. Therefore one may expet that, within a single osillation, the probability
of a bubble forming when the osillating eld was negative was approximately the
92
CHAPTER 8.
INFLATION AFTER PREHEATING
same as the probability of the bubble forming when it was positive.
If the number of bubbles with positive and negative are approximately equal to
eah other, the phase transition leads to the formation of dangerous domain walls,
whih rules out our model [105℄. If orret, this is a rather important onlusion
that shows that the investigation of nonthermal phase transition may rule out ertain
lasses of inationary models that otherwise would seem quite legitimate [14, 15℄.
But this onlusion does not imply that all theories where the nonthermal phase
transition is strongly rst order are ruled out. For example, one may onsider a model
(8.1) with being not a real but a omplex eld, = p12 (1 + i2 ), jj2 = 12 (21 + 22 ).
Sine the main ontribution to the eetive potential of the eld in the theory (8.1)
is given not by the eld(s) but by the elds i , we expet that this generalization
will not lead to a qualitative modiation of our results. In partiular, we expet that
for suÆiently large N and g 2 = the phase transition will be strongly rst order and
there will be a short stage of ination after preheating. However, in the new model
we will have strings instead of domain walls.
A similar model in the absene of interation of the elds with the elds was studied in [17, 98℄. It was argued that even in this ase innite strings may
be formed. The theory of galaxy formation due to osmi strings is urrently out of
favor, but it is ertainly true that osmi strings produed after ination may add new
interesting features to the standard theory of formation of the large-sale struture
of the universe [106, 107, 108, 109℄
The possibility of strongly rst order phase transitions indued by preheating in
models with g 2 adds new evidene that innite strings an be produed after
nonthermal phase transitions. Indeed, innite strings may not be produed if the
diretion in whih the eld falls from the point = 0 at the moment of the phase
transition is determined by the osillations of the eld . If, just as in qthe ase
disussed above, the amplitude of these osillations are muh smaller than h2i at
the moment of the phase transition, then innite strings are indeed formed.
8.5.
CONCLUSIONS
93
8.5 Conlusions
The results of our lattie simulation onrmed our expetations that preheating may
lead to nonthermal phase transitions even in those theories where spontaneous symmetry breaking ours at the GUT sale, v 1016 GeV. Some time ago this question
was intensely debated in the literature. Some authors laimed that nonthermal phase
transitions indued by preheating are impossible, and the notion of the eetive potential after preheating is useless. I believe that Figs. 1, 2 and 3 give a lear answer
to this question. In partiular, Fig. 1 shows that 90% of the time from the end of
ination to the moment of symmetry breaking the eld osillates about = 0 with
an amplitude muh smaller than v . This ould happen only beause the orretions
to the eetive potential indued by partiles and hange the shape of V () near
= 0, turning its maximum into a deep loal minimum.
In some theories, this eet may lead to prodution of superheavy strings, whih
may have important osmologial impliations for the theory of formation of the large
sale struture of the universe. In some other theories, these phase transitions may
lead to exessive prodution of monopoles and domain walls. This may rule out a
broad lass of otherwise aeptable inationary models.
In this paper we have shown that under ertain onditions a nonthermal phase
transition may lead to a short seondary stage of ination. It would be interesting
to study the possibility that a seondary stage of ination indued by preheating
ould help solve the moduli and gravitino problems. The answer to this question will
be strongly model-dependent beause gravitinos an be produed by the osillating
salar eld even after the seondary ination [61, 80℄. Independently of all pratial
impliations, the possibility of a seondary stage of ination indued by preheating
seems very interesting beause it learly demonstrates the potential importane of
nonperturbative eets in post-inationary osmology.
Chapter 9
The Development of Equilibrium
After Preheating
Note: This hapter is based on a paper by Gary Felder and Lev Kofman, available
on the Los Alamos eprint server as hep-ph/0011160. The full itation appears in the
bibliography [110℄.
Chapter Abstrat
In this hapter I present the results of a fully nonlinear study of the development of
equilibrium after preheating. The rapid transfer of energy from the inaton to other
elds during preheating leaves these elds in a highly nonthermal state with energy
onentrated in infrared modes. We performed lattie simulations of the evolution
of interating salar elds during and after preheating for a variety of inationary
models. We formulated a set of generi rules that govern the thermalization proess
in all of these models. Notably, we found that one one of the elds is amplied
through parametri resonane or other mehanisms it rapidly exites other oupled
elds to exponentially large oupation numbers. These elds quikly aquire nearly
thermal spetra in the infrared, whih gradually propagate into higher momenta.
Prior to the formation of total equilibrium, the exited elds group into subsets with
almost idential harateristis (e.g. group eetive temperature). The way elds
94
9.1.
INTRODUCTION
95
form into these groups and the properties of the groups depend on the ouplings
between them. We also studied the onset of haos after preheating by alulating the
Lyapunov exponent of the salar elds.
9.1 Introdution
Reheating typially involves some form of rapid, non-perturbative preheating phase.
The harater of preheating may vary from model to model, e.g. parametri exitation
in haoti ination [7℄ or tahyoni preheating in hybrid ination (see hapter 10),
but its distint feature remains the same: rapid ampliation of one or more bosoni
elds to exponentially large oupation numbers. This ampliation is eventually shut
down by bakreation of the produed utuations. The end result of the proess is
a turbulent medium of oupled, inhomogeneous, lassial waves far from equilibrium
[99℄.
Despite the development of our understanding of preheating after ination, the
transition from this stage to a hot Friedmann universe in thermal equilibrium has
remained relatively poorly understood. A theory of the thermalization of the elds
generated from preheating is neessary to bridge the gap between ination and the
Hot Big Bang. The details of this thermalization stage depend on the onstituents
of the fundamental Lagrangian L(i ; i ; i ; A ; h ; :::) and their ouplings, so at rst
glane it would seem that a desription of this proess would have to be strongly
model-dependent. We found, however, that many features of this stage seem to hold
generially aross a wide spetrum of models. This fat is understandable beause
the onditions at the end of preheating are generally not qualitatively sensitive to the
details of ination. Indeed, at the end of preheating and beginning of the turbulent
stage (denoted by t ), the elds are out of equilibrium. We examined many models
and found that at t there is not muh trae of the linear stage of preheating and
onditions at t are not qualitatively sensitive to the details of ination. We therefore
found that this seond, highly nonlinear, turbulent stage of preheating seemed to
exhibit some universal, model-independent features.
Although a realisti model would inlude one or more Higgs-Yang-Mills setors,
96
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
we treated the simpler ase of interating salars. Within this ontext, however, we
onsidered a number of dierent models inluding several haoti and hybrid ination
senarios with a variety of ouplings between the inaton and other matter elds.
There are many questions about the thermalization proess that we set out to
answer in our work. Could the turbulent waves that arise after preheating be desribed
by the theory of (transient) Kolmogorov turbulene or would they diretly approah
thermal equilibrium? Could the relaxation time towards equilibrium be desribed by
the naive estimate (nint ) 1 , where n is a density of salar partiles and int
is a ross-setion of their interation? If the inaton were deaying into a eld
, what eet would the presene of a deay hannel for the eld have on the
thermalization proess? For that matter, would the presene of signiantly alter
the preheating of itself, or even destroy it as suggested in [111℄? How strongly
model-dependent is the proess of thermalization; are there any universal features
aross dierent models? Finally there's the question of haos. It is known that
Higgs-Yang-Mills systems display haoti dynamis during thermalization [112℄. The
possibility of haos in the ase of a single, self-interating inaton was mentioned in
passing in [99℄, but when we began our work it was unlear at what stage of preheating
haos might appear, and in what way.
Beause the systems we studied involve strong, nonlinear interations far from
thermal equilibrium it is not possible to solve the equations of motion using linear
analysis in Fourier spae. Instead we solved the salar eld equations of motion
diretly in position spae using lattie simulations. These simulations automatially
take into aount all nonlinear eets of sattering and bakreation. Using these
numerial results we were able to formulate a set of empirial rules that seem to govern
thermalization after ination. These rules qualitatively desribe thermalization in a
wide variety of models. The features of this proess are in some ases very dierent
from our initial expetations.
Setions 9.2 and 9.3 desribe the results of our numerial alulations. Setion
9.2 desribes one simple haoti ination model that we hose to fous on as a lear
illustration of our results, while setion 9.3 disusses how the thermalization proess
ours in a variety of other models. Setion 9.4 desribes the onset of haos during
9.2.
CALCULATIONS IN CHAOTIC INFLATION
97
preheating and inludes a disussion of the measurement and interpretation of the
Lyapunov exponent in this ontext. Setion 9.5 ontains a list of empirial rules that
we formulated to desribe thermalization after preheating. Setion 9.6 disusses these
results and other aspets of non-equilibrium salar eld dynamis.
9.2 Calulations in Chaoti Ination
In this setion I present the results of our numerial lattie simulations of the dynamis of interating salars after ination. I disuss in detail one simple model that
illustrates the general properties of thermalization after preheating. The next setion
will disuss thermalization in the ontext of other models.
9.2.1 Model
The example I have hosen to fous on is haoti ination with a quarti inaton
potential. The inaton has a four-legs oupling to another salar eld , whih in
turn an ouple to one or more other salars i . The potential for this model is
1
1
1
V = 4 + g 22 2 + h2i 2 i2 :
4
2
2
(9.1)
The equations of motion for the model (9.1) are given by
a_
 + 3 _
a
1 2 2 2 2
r + + g = 0
a2
(9.2)
a_
1 2
22 + h2 2 = 0
 + 3 _
r
+
g
(9.3)
i i
a
a2
a_
1
2 i + h2 2 i = 0:
i + 3 _i
=
r
(9.4)
i
a
a2
We also inluded self-onsistently the evolution of the sale fator a(t). The model
desribed by these equations is a onformal theory, meaning that the expansion of the
universe an be (almost) eliminated from the equations of motion by an appropriate
hoie of variables [8℄.
98
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
A detailed explanation of our lattie simulations is in part III of this thesis. For
the haoti ination potential desribed in this setion we resaled the variables from
Plank units in the following ways
fpr =
p
p dt
a
f ; ~xpr = 0~x; dtpr = 0 ;
0
a
(9.5)
where the subsript pr (for \program") indiates the variables used in our alulations.
The variable 0 = :342Mp is the initial value of in the simulations. All plots in this
hapter are shown in these program variables.
Preheating in this theory in the absene of the i elds was desribed in [8℄. For
2
g >
the eld will experiene parametri ampliation, rapidly rising to exponentially large oupation numbers. In the absene of the eld (or for suÆiently
small g ) will be resonantly amplied through its own self-interation, but this selfampliation is muh less eÆient than the two-eld interation. The results shown
here are for = 9 10 14 (for COBE normalization) and g 2 = 200. When I add a
third eld I take h21 = 100g 2 and when I add a fourth eld I take h22 = 200g 2.
9.2.2 The Output Variables
There are a number of ways to illustrate the behavior of salar elds, and dierent
ones are useful for exploring dierent phenomena. The raw data is the value of the
eld f (t; ~x), or equivalently its Fourier transform fk (t). One of the simplest quantities
one an extrat from these values is the variane
Z
h f (t) f(t) 2 i = (21 )3 d3kjfk (t)j2 ;
(9.6)
where the integral does not inlude the ontribution of a possible delta funtion at
~k = 0, representing the mean value f.
One of the most interesting variables to alulate is the (omoving) number density
of partiles of the f -eld
1 Z 3
nf (t) d knk (t) ;
(9.7)
(2 )3
9.2.
CALCULATIONS IN CHAOTIC INFLATION
99
where nk is the (omoving) oupation number of partiles
nk (t) 1 _ 2 !k 2
jf j + 2 jfk j
2!k k
q
!k k2 + m2eff
2V
m2eff 2 :
f
For the model (9.1) this eetive mass is given by
8
>
3h2i + g 2 h2 i
>
>
<
m2eff = > g 2h2 i + h2i hi2 i
>
>
: h2 h2 i
(9.8)
(9.9)
(9.10)
(9.11)
i
for , , and i respetively. For the lassial waves of f that we are dealing with, nk
orresponds to an adiabati invariant of the waves. Formula (9:8) an be interpreted
as a partile oupation number in the limit of large amplitude of the f -eld. As
we will see below this oupation number spetrum ontains important information
about thermalization. Notie that the eetive mass of the partiles depends on the
varianes of the elds and may be signiant and time-dependent. The momenta of
the partiles do not neessarily always exeed their masses, meaning the interating
salar waves are not neessarily always in the kineti regime. In partiular this means
R
that in general we annot alulate the energies of the elds simply as d3 k !k nk
beause interation terms between elds an be signiant.
From here on I will use n without a subsript to denote the total number density
for a eld, and will use the subsript only to speify a partiular eld, e.g. n . I use
ntot to mean the sum of the total number density for all elds ombined. Oupation
number will always be written nk and it should be lear from ontext whih eld is
being referred to.
In pratie it is not very important whether you onsider the spetrum fk and
the variane of f or the spetrum nk and the number density. Both sets of quantities
qualitatively show the same behavior in the systems we onsidered. The variane and
100
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
number density grow exponentially during preheating and evolve muh more slowly
during the subsequent stage of turbulene. Most of our results are shown in terms of
number density nf and oupation number nk beause these quantities have obvious
physial interpretations, at least in ertain limiting ases. I shall oasionally show
plots of variane for omparison purposes.
In what follows I disuss the evolution of n(t) and nk (t). The evolution of the
total number density ntot is an indiation of the physial proesses taking plae. In
the weak interation limit the sattering of lassial waves via the interation term
1 g 2 2 2 an be treated using a perturbation expansion with respet to g 2 . The
2
leading four-legs diagrams for this interation orresponds to a two-partile ollision
( ! ), whih onserves ntot . The regime where suh interations dominate orresponds to \weak turbulene" in the terminology of the theory of wave turbulene
[113℄. If we see ntot onserved it will be an indiation that these two-partile ollisions onstitute the dominant interation. Conversely, violation of ntot (t) = onst
will indiate the presene of strong turbulene, i.e. the importane of many-partile
ollisions. Suh higher order interations may be signiant despite the smallness of
the oupling parameter g 2 (and others) beause of the large oupation numbers nk .
Later, when these oupation numbers are redued by resattering, the two-partile
ollision should beome dominant and ntot should be onserved.
For a bosoni eld in thermal equilibrium with a temperature T and a hemial
potential the spetrum of oupation numbers is given by
nk =
1
e
!k T
1
:
(9.12)
(I use units in whih h = 1.) Preheating generates large oupation numbers for
whih equation (9.12) redues to its lassial limit
nk !k
T
;
(9.13)
whih in turn redues to nk / 1=k for k m; and nk onst: for k m; . We
ompared the spetrum nk to this form to judge how the elds were thermalizing. Here
9.2.
CALCULATIONS IN CHAOTIC INFLATION
101
I onsider the hemial potential of an interating salar elds as a free parameter.
Unless otherwise indiated all of our results are shown in omoving oordinates
that, in the absene of interations, would remain onstant as the universe expanded.
Note also that for most of the disussion I onsider eld spetra only as a funtion of
j~kj, dened by averaging over spherial shells in k spae. For a Gaussian eld these
spetra ontain all the information about the eld, and even for a non-Gaussian eld
most useful information is in these averages. This issue is disussed in more detail in
setion 9.4.
9.2.3 Results
The key results for this model are shown in Figures 9.12-9.19, whih show the evolution of n(t) with time for eah eld and the spetrum nk for eah eld at a time long
after the end of preheating. These results are shown for runs with one eld ( only),
two elds ( and ), and three and four elds (one and two i elds respetively). I
will begin by disussing some general features ommon to all of these runs, and then
omment on the runs individually.
All of the plots of n(t) show an exponential inrease during preheating, followed
by a gradual derease that asymptotially slows down. See for example Figure 9.1.
This exponential inrease is a onsequene of explosive partile prodution due to
parametri resonane. This regime is fairly well understood [8℄. After preheating the
elds enter a turbulent regime, during whih n(t) dereases. This initial, fast derease
an be interpreted as a onsequene of the many-partile interations disussed above;
as nk shifts from low to high momenta the overall number dereases. Realistially,
however, the onset of weak turbulene should be aompanied by the development of
a ompensating ow towards infrared modes, whih we were unable to see beause
of our nite box size. Thus the ontinued, slow derease in n(t) well into the weak
turbulent regime is presumably a onsequene of the lak of very long wavelength
modes in our lattie simulations.
To see why this shift is ourring look at the spetra nk (Figures 9.16-9.19, see
also [111, 114℄). Even long after preheating the infrared portions of some of these
102
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
n
14
6´10
14
5´10
14
4´10
14
3´10
14
2´10
14
1´10
500
1000
1500
2000
t
Figure 9.1: Number density n for V = 14 4 + 12 g 2 2 2 . The plots are, from bottom
to top at the right of the gure, n , n , and ntot . The dashed horizontal line is simply
for omparison. The end of exponential growth and the beginning of turbulene (i.e.
the moment t ) ours around the time when ntot reahes its maximum.
spetra are tilted more sharply than would be expeted for a thermal distribution
(9.13). Even more importantly, many of them show a uto at some momentum k,
above whih the oupation number falls o exponentially. Both of these features,
the infrared tilt and the ultraviolet uto, indiate an exess of oupation number
at low k relative to a thermal distribution. This exess ours beause parametri
resonane is typially most eÆient at exiting low momentum modes, and beomes
ompletely ineÆient above a ertain uto k . A lear piture of how the ow to
higher momenta redues these features an be seen in Figure 9.2, whih shows the
evolution of the spetrum nk for in the two eld model.
Figure 9.2 illustrates the initial exitation of modes in partiular resonane bands,
followed by a rapid smoothing out of the spetrum. The ultraviolet uto is initially
at the momentum k where parametri resonane shuts down, but over time the
uto moves to higher k as more modes are brought into the quasi-equilibrium of the
infrared part of the spetrum. Meanwhile the infrared setion is gradually attening
as it approahes a true thermal distribution. During preheating the exitation of the
infrared modes drives this slope to large, negative values. From then on it gradually
9.2.
103
CALCULATIONS IN CHAOTIC INFLATION
nk
11
1. ´ 10
1. ´ 108
100000.
100
2
5
10
20
50
k
Figure 9.2: Evolution of the spetrum of in the model V = 14 4 + 12 g 2 2 2 . Red
plots orrespond to earlier times and blue plots to later ones. For blak and white
viewing: The sparse, lower plots all show early times. In the thik bundle of plots
higher up the spetrum is rising on the right and falling on the left as time progresses.
approahes thermal equilibrium (i.e. a slope of 1 to 0 depending on the hemial
potential and the mass). The relaxation time for the equilibrium is signiantly
shorter than that given by formula 1=nint . This estimate is valid for dilute gases
of partiles, but in our ase the large oupation numbers amplify the sattering
amplitudes [7℄.
2
Figure 9.3 shows the evolution of the varianes h f f i for the two eld model.
As indiated above it shows all the same qualitative features as the evolution of n for
that model.
I an now go on to point out some dierenes between the models, i.e. between
runs with dierent numbers of elds. The one eld model (pure 4 ) shows the basi
features disussed above, but the tilt in the spetrum is still very large at the end
of the simulation and n is dereasing very slowly ompared to the spetral tilt and
hange in n we see in the two eld ase. This dierene ours beause the interations
between and greatly speed up the thermalization of both elds. In the one eld
ase an only thermalize via its relatively weak self-interation.
The spetra in the two eld run also show a novel feature, namely that the spetra
104
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
0.01
0.0001
1. ´ 10-6
1. ´ 10-8
1. ´ 10-10
500
1000
1500
2000
t
2
Figure 9.3: Varianes for V = 41 4 + 12 g 2 2 2 . The upper plot shows h i and
the lower plot shows h( )2 i.
for and are essentially idential, whih means among other things
n n :
(9.14)
This mathing of the two spetra ours shortly after preheating and from then on the
two elds evolve identially (exept for the remaining homogeneous omponent of ).
A posteriori this result an be understood as follows. Looking at the potential 4 +
g 2 2 2 , the seond term dominates beause of the hierarhy of oupling strengths
g 2 = 200. So the potential V g 2 2 2 is symmetri with respet to the two elds,
and therefore they at as a single eetive eld.
Figures 9.14 and 9.18 show the eets of adding an additional deay hannel for
. The interation of and does not aet the preheating of , but does drag
exponentially quikly into an exited state. The eld is exponentially amplied
not by parametri resonane, but by its stimulated interations with the amplied eld. Unlike ampliation by preheating, this diret deay nearly onserves partile
number, with the result that n dereases as grows, and the spetra of and are
9.2.
CALCULATIONS IN CHAOTIC INFLATION
105
no longer idential. Instead and develop nearly idential spetra,
n n < n ;
(9.15)
and they both thermalize (together) muh more rapidly than did in the absene
of . There is a looser relationship n n + n , whose auray depends on the
ouplings. The inaton, meanwhile, thermalizes muh more slowly; note the low k
of the uto in the spetrum in Figure 9.18. By ontrast, there is no visible uto
in the spetra of and and the tilt is relatively mild. The most striking property
of this hain of interation is the grouping of elds; and behave identially to
eah other and dierently from . This again an be understood by the hierarhy of
oupling onstants, h2 = 100g 2 = 20; 000. The term h2 2 2 is dominant and puts and on an equal footing.
Varying the oupling h did not hange the overall behavior of the system, but it
hanged the time at whih grew. In the limiting ase h g , grew with during
preheating and remained indistinguishable from it right from the start. (We found
this, for example, for h2 = 10; 000g 2.)
When we added a seond eld we found that the eld most strongly oupled
to would grow very rapidly and the more weakly oupled one would then grow
relatively slowly. Note for example that n2 in Figure 9.19 grows more slowly than
n in Figure 9.18 despite the fat that they have the same oupling to . In the four
eld ase n is redued when the more strongly oupled eld grows and this slows
the growth of the more weakly oupled one. Nonetheless, the addition of another eld one again sped up the thermalization of and the elds. The three elds ,
1 , and 2 one again have idential spetra
n n1 n2 < n ;
(9.16)
but in the four eld ase by the end of the run they look indistinguishable from
thermal spetra. If there is an ultraviolet uto for these spetra it is at momenta
higher than an be seen on the lattie we were using. Again, we notied a loose
relationship n n + n1 + n2 . in this ase.
106
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
Field masses
14
12
10
8
6
4
2
500
1000
1500
2000
Figure 9.4: Eetive masses for V = 14 4 + 12 g 2 2 2 as a funtion of time in units of
omoving momentum. The lower plot is m and the upper one is m .
I'll lose this setion with a few words about the eetive masses of the elds,
equation (9.10). All the masses are saled in the omoving frame, i.e. I onsider
a2 m2eff . Figure 9.4 shows the evolution of the eetive masses in the two eld model.
Note that the vertial axis of these plots is in the same omoving units as the horizontal (k) axes of the spetra plots. Sine the momentum uto was of order k 5 10
(see Figure 9.2) the mass of was onsistently smaller than the typial momenta
of the eld. By ontrast m started out muh larger and only gradually dereased.
The utuations of remained massive through preheating (although with a physial mass 1=a) and for quite a while afterwards the typial momentum of these
utuations was k m.
Figure 9.5 shows the evolution of the eetive masses for the three eld model.
One again m remains small. Although m grows large briey it quikly subsides.
However, m , with ontributions from and , remains relatively large. Note, however, that the spetrum of has no lear uto after has grown, so it is diÆult to
say whether this mass exeeds a \typial" momentum sale or not.
9.3.
107
OTHER MODELS OF INFLATION AND INTERACTIONS
Field masses
14
12
10
8
6
4
2
500
1000
1500
2000
Figure 9.5: Time evolution of the eetive masses for the model V = 14 4 + 21 g 2 2 2 +
1 h2 2 2 . From bottom to top on the right hand side the plots show m , m , and m .
2
9.3 Other Models of Ination and Interations
The model (9.1) was hosen to illustrate our basi results beause 4 ination and
preheating is relatively simple and well studied. Our main interest, however, was in
universal features of thermalization. In this setion I therefore more briey disuss our
results for a variety of other models. First I ontinue with 4 ination by disussing
variants on the interation potential desribed above. Next I disuss thermalization
in m2 2 models of haoti ination. Finally I disuss hybrid ination.
9.3.1 Variations on Chaoti Ination With a Quarti Potential
We looked at several simple variants of the potential (9.1). We onsidered a model
with a further deay hannel for so that the total potential was
1
1
1
1
V = 4 + g 22 2 + h21 2 2 + h22 2 2 :
4
2
2
2
(9.17)
Setting h1 = h2 we found that for this four eld model the evolution of the eld
utuations, spetra, and number density were qualitatively similar to those in the
108
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
four eld model (9.1). We found that at late times
n n n < n :
(9.18)
The elds , , and formed a group with nearly idential spetra and evolution
and rapid thermalization, while remained distint and thermalized more slowly.
Compare these results to the four eld model results in Figures 9.15 and 9.19.
We also onsidered parallel deay hannels for 1
1
1
1
V = 4 + g12 2 2 + g22 2 2 + h2 2 2 :
4
2
2
2
Setting g1 = g2 and h2 = 100g12 we found that at late times
n n > n n :
(9.19)
(9.20)
In other words the four elds formed into two groups of two, with eah group having
a harateristi number density evolution.
Finally we looked at adding a self-interation term for 1
1
1
V = 4 + g 2 2 2 + 4
4
2
4
(9.21)
with = g 2 and found the results were essentially unhanged from those of the two
eld runs with no 4 term. The self-oupling aused the spetra of and to
deviate slightly from eah other, but their overall evolution proeeded very similarly
to the ase with no self-interation term.
9.3.2 Chaoti Ination with a Quadrati Potential
We also onsidered haoti ination models with an m2 2 inaton potential. Figures 9.20-9.23 show results for the model
1
1
1
V = m2 2 + g 2 2 2 + h2 2 2 ;
2
2
2
(9.22)
9.3.
OTHER MODELS OF INFLATION AND INTERACTIONS
109
with m = 10 6Mp 1:22 1013 GeV (for COBE), g 2 = 2:5 105 m2 =Mp2 , and h2 =
100g 2. The resalings used for this model were
fpr =
a3=2
f ; ~xpr = m~x; dtpr = mdt;
0
(9.23)
and one again all plots here are shown in these resaled variables. The initial value
0 was set to :193Mp.
We onsidered separately the ase of two elds and and three elds , , and .
This model exhibits parametri resonane similar to the resonane in quarti ination
[7℄, whih results in the rapid growth of n seen in these gures. The spetra produed
in this way are one again tilted towards the infrared. In the two eld ase, and
do not have idential spetra as they did for quarti ination. This is beause the
oupling term 1=2g 22 2 redshifts more rapidly than the mass term 1=2m2 2 , so the
latter remains dominant in the potential, whih is therefore not symmetri between and . In the three eld ase we again see similar spetra for and , although they
are not as indistinguishable as they were in 4 theory. The basi features of rapid
growth of n, high oupation of infrared modes, and then a ux of number density
towards ultraviolet modes and a slow derease in ntot are all present as they were for
4 theory. The shape of the spetrum does not appear thermal, but it is unlear
if this spetrum is ompatible with Kolmogorov turbulene.
9.3.3 Hybrid Ination
Hybrid ination models involve multiple salar elds. The simplest potential for
two-eld hybrid ination is
V (; ) = ( 2
4
g2
v 2 )2 + 2 2 :
2
(9.24)
Ination in this model ours while thep homogeneous eld slow rolls from large towards the bifuration point at = g v (due to the slight lift of the potential in
diretion). One (t) rosses the bifuration point, the urvature of the eld,
m2 2 V= 2 , beomes negative. This negative urvature results in exponential
110
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
growth of utuations. Ination then ends abruptly in a \waterfall" manner.
In hapter 10 I disuss preheating in hybrid ination, whih typially ours via a
mehanism alled \tahyoni preheating." Although this mehanism is very dierent
in some ways from parametri resonane the result is quite similar; utuations of
the elds are rapidly driven to a state with exponentially large oupation numbers
up to some uto in momentum. I won't disuss tahyoni preheating in any detail
here. Rather my onern here is to note how thermalization ours after preheating.
One reason to be interested in hybrid ination is that it an be easily implemented
in supersymmetri theories. In partiular, for illustration I will use supersymmetri
F-term ination as an example of a hybrid model. The potential is
j
4 v 2 j2 + 4jj2 jj2 + j j2 + h2 2 jj2 ;
(9.25)
4
where = 2:5 10 5 and h2 = 2. Here , and are the omplex salar elds
of the inaton setor and is an additional matter eld. Ination ours along the
diretion for hi v , when = = 0. When the magnitude of the slow-rolling
eld reahes the value hj ji = v2 spontaneous symmetry breaking ours and the
elds beome exited. It an be shown that at the end of ination and the start
of symmetry breaking the ompliated potential (9.25) an be eetively redued to
the simple two eld potential (9.24) (where and are ombinations of and , and g 2 = 21 ) plus the oupling term with h2 2 jj2 .
V=
Figure 9.6 show the evolution of the six degrees of freedom of the inaton setor
as well as the eld . We see that all of the inaton elds exept Im() are exited
very quikly. Later the elds and Im() are dragged into exited states as well.
This dragging orresponds to preheating in the non-inaton setor. The elds and
Im() are exited by their stimulated interations with the rest of the elds. The
result of this ampliation is a turbulent state that evolves towards equilibrium very
similarly to the haoti models. Although the details of ination and preheating are
very dierent in hybrid and haoti models, we found that one a matter eld has
been amplied, the thermalization proess proeeds in the same way.
9.4.
THE ONSET OF CHAOS, LYAPUNOV EXPONENTS AND STATISTICS
111
Field Variances: F-Term model
0.1
0.01
0.001
0.0001
0.00001
2000
4000
6000
8000
t
10000
Figure 9.6: Evolution of varianes of elds in the model (9.25). The two elds that
grow at late times, in order of their growth, are and Im().
9.4 The Onset of Chaos, Lyapunov Exponents and
Statistis
Interating waves of salar elds onstitute a dynamial system, meaning there is no
dissipation and the system an be desribed by a Hamiltonian. Dynamial haos is
one of the features of wave turbulene. In this setion I address the question if, how
and when the onset of haos takes plae after preheating.
The salar eld utuations produed during preheating are generated in squeezed
states [115, 7℄ that are haraterized by orrelations of phases between modes ~k and
~k . Beause of their large amplitudes we an onsider these utuations to be standing lassial waves with denite phases. During the linear stage of preheating, before
interations between modes beomes signiant, the evolution of these waves may
or may not show haoti sensitivity to initial onditions. Indeed, for wide ranges of
oupling parameters parametri resonane has stohasti features [7, 8℄, and prior to
this work the issue of the numerial stability of parametri resonane had not been
investigated. When interation (resattering) between waves beomes important, the
112
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
waves beome deoherent. At this stage the waves have well dened oupation numbers but not well dened phases, and the random phase approximation an be used to
desribe the system. This transition signals the onset of turbulene, following whih
the system will gradually evolve towards thermal equilibrium.
To investigate the onset of haos in this system we followed the time evolution of
two initially nearby points in the phase spae, see e.g. [112℄. Consider two ongurations of a salar eld f and f 0 that are idential exept for a small dierene of
the elds at a set of points xA . I use f (t; ~xA ); f_(t; ~xA ) to indiate the unperturbed
eld amplitude and eld veloity at the point ~xA and f 0 (t; ~xA ); f_0 (t; ~xA ) to indiate
slightly perturbed values at this point. In other words, the eld ongurations with
f (t; ~xA ); f_(t; ~xA ) and f 0 (t; ~xA ); f_0 (t; ~xA ) are initially lose points in the eld phase
spae. We independently evolved these two systems (phase spae points) and observe how the perturbed eld values diverge from the unperturbed ones. Chaos an
be dened as the tendeny of suh nearby ongurations in phase spae to diverge
exponentially over time. This divergene is parametrized by the Lyapunov exponent
for the system, dened as
1 D(t)
log
(9.26)
t
D0
where D is a distane between two ongurations and D0 is the initial distane at
time 0. Here we dened the distane D(t) simply as
D(t)2 X
A
(jfA0
fA j)2 + (jf_A0
f_A j)2 ;
(9.27)
where fA f (t; ~xA ) and the summation is taken over all the points where the ongurations initially diered.
For illustration I present the alulations for the model V = 14 4 + 12 g 2 2 2 . We
did two lattie simulations of this model with initial onditions that were idential
exept that in one of them we multiplied the amplitude of by 1 + 10 6 at 8 evenly
spaed points on the lattie. Figure 9.7 shows the Lyapunov exponent for both elds
and . Note that the vertial axis is t rather than just . During the turbulent
stage the parameter D(t) is artiially saturated to a onstant beause of the limited
phase spae volume of the system. Fortunately, the most interesting moment around
9.4.
THE ONSET OF CHAOS, LYAPUNOV EXPONENTS AND STATISTICS
113
Λ t
25
20
15
10
5
100
200
300
400
500
t
Figure 9.7: The Lyapunov exponent for the elds (lower urve) and (upper
urve). The vertial axis is t.
t , where the haoti motion begins, is overed by this simple approah. Certainly,
the eld dynamis ontinue to be haoti in subsequent stages of the turbulene, and
one an use more sophistiated methods to alulate the Lyapunov exponent during
these stages [116, 112℄. However, this issue is less relevant for our study.
Both elds show roughly the same rate of growth of , but grows muh earlier
than and therefore reahes a higher level. The reason for this is simple. The
amplitude of is initially very small and grows exponentially, so even in the absene
of haos we would expet that during preheating the
dierene 0 (t; ~xA ) (t; ~xA )
R
must grow exponentially, proportionally to e dt(t) itself. So this exponential
growth is not a true indiator of haos.
To get around this problem and dene the onset of haos in the ontext of preheating more meaningfully we introdued a normalized distane funtion
(t) X
A
!
!2
fA0 fA 2
f_A0 f_A
+ _0 _
fA0 + fA
fA + fA
(9.28)
114
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
Λ t
20
15
10
5
100
200
300
400
500
t
Figure 9.8: The Lyapunov exponent 0 for the elds and using the normalized
distane funtion .
that is well regularized even while the eld is being amplied exponentially. Fig(t) for . In this ase we see the
ure 9.8 shows the Lyapunov exponent 0 1t log (
t0 )
onset of haos only at the end of preheating. The plot for the eld is nearly idential. The Lyapunov exponents for the elds were 0 0 0:2 (in the units of time
adopted in the simulation). This orresponds to a very fast onset of haos.
Thus we see that haoti turbulene starts abruptly at the end of preheating.
Initially wave turbulene is strong and resattering does not onserve the total number
of partiles ntot . The fastest variation in ntot ours at the same time as the onset of
haos, t 100 200. We onjeture that the entropy of the system of interating
waves is generated around the moment t . As the partile oupation number drops,
the turbulene will beome weak and ntot will be onserved. Figure 9.1 learly shows
this evolution of the total number of partiles ntot in the model.
We also onsidered the statistial properties of the interating lassial waves in
the problem. The initial onditions of our lattie simulations orrespond to random
gaussian noise. In thermal equilibrium, the eld veloity f_ has gaussian statistis,
while the eld f itself departs from that unless it has high oupation numbers.
9.4.
THE ONSET OF CHAOS, LYAPUNOV EXPONENTS AND STATISTICS
115
0.014
0.012
0.01
0.008
0.006
0.004
0.002
50
100
150
200
250
Figure 9.9: The probability distribution funtion for the eld after preheating. Dots
show a histogram of the eld and the solid urve shows a best-t Gaussian.
Figure 9.9 shows the probability distribution of the eld during the weak turbulene
stage after preheating, and indeed the distribution is nearly exatly gaussian. Thus, at
this stage we an treat the superposition of lassial salar waves with large oupation
numbers and random phases as random gaussian elds.
During preheating, however, this gaussian distribution is altered. A simple measure of the gaussianity of a eld omes from examining its moments. For a gaussian
eld there is a xed relationship between the two lowest nonvanishing moments,
namely
3hÆ2i2 = hÆ4 i ;
(9.29)
where Æ hi and angle brakets denote ensemble averages or, equivalently,
large spatial averages. We measured the ratio of the left and right hand sides of this
equation for and and their time derivatives using spatial averages over the lattie.
The results are shown in Figures 9.10 and 9.11. As expeted, the elds are initially
gaussian, deviate from it during preheating, and rapidly return to it afterwards. The
plots for the moments of the eld veloities are similar, although the eld veloities
remain loser to gaussianity.
116
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
Gaussianity Plot for Φ
2
1.75
1.5
1.25
1
0.75
0.5
0.25
500
1000
1500
2000
Figure 9.10: Deviations from Gaussianity for the eld as a funtion of time. The
solid, red line shows 3hÆ2i2 =hÆ4i and the dashed, blue line shows 3hÆ _ 2i2 =hÆ _ 4i.
Gaussianity Plot for Χ
2
1.75
1.5
1.25
1
0.75
0.5
0.25
500
1000
1500
2000
Figure 9.11: Deviations from Gaussianity for the eld as a funtion of time. The
solid, red line shows 3hÆ2 i2 =hÆ4 i and the dashed, blue line shows 3hÆ _ 2 i2 =hÆ _ 4 i.
9.5.
RULES OF THERMALIZATION
117
It is quite important to notie that gaussianity is broken around the end of preheating and the beginning of the strong turbulene. In partiular, it makes invalid
the use of the Hartree approximation beyond this point.
9.5
Rules of Thermalization
This researh was primarily empirial. We numerially investigated the proesses of
preheating and thermalization in a variety of models and determined a set of rules
that seem to hold generially. These rules an be formulated as follows:
1. In many, if not all viable models of ination there exists a mehanism for exponentially amplifying utuations of at least one eld . These mehanisms tend to
exite long-wavelength exitations, giving rise to a highly infrared spetrum.
The mehanism of parametri resonane in single-eld models of ination has been
studied for a number of years. Contrary to the laims of some authors, this eet is
quite robust. Adding additional elds (e.g. our elds) or self-ouplings (e.g. 4 )
has little or no eet on the resonant period. Moreover, in many hybrid models a
similar eet ours due to other instabilities. The qualitative features of the elds
arising from these proesses seem to be largely independent of the details of ination
or the mehanisms used to produe the elds.
2. Exiting one eld is suÆient to rapidly drag all other light elds with whih interats into a similarly exited state.
We saw this eet when multiple elds were oupled diretly to and when hains
of elds were oupled indiretly to . All it takes is one eld being exited to rapidly
amplify an entire setor of interating elds. These seond generation amplied elds
will inherit the basi features of the eld, i.e. they will have spetra with more
energy in the infrared than would be expeted for a thermal distribution.
3. The exited elds will be grouped into subsets with idential harateristis (spetra,
oupation numbers, eetive temperatures) depending on the oupling strengths.
We saw this eet in a variety of models. For example in the models (9.1) and
(9.17) the and elds formed suh a group. In general, elds that are interating
118
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
in a group suh as this will thermalize muh more quikly than other elds, presumably beause they have more potential to interat and satter partiles into high
momentum states.
4. One the elds are amplied, they will approah thermal equilibrium by sattering
energy into higher momentum modes.
This proess of thermalization involves a slow redistribution of the partile oupation number as low momentum partiles are sattered and ombined into higher
momentum modes. The result of this sattering is to derease the tilt of the infrared
portion of the spetrum and inrease the ultraviolet uto of the spetrum. Within
eah eld group the evolution proeeds identially for all elds, but dierent groups
an thermalize at very dierent rates.
9.6 Disussion
We investigated the dynamis of interating salar elds during post-inationary preheating and the development of equilibrium immediately after preheating. We used
three dimensional lattie simulations to solve the non-linear equations of motion of
the lassial elds.
There are a number of problems both from the point of view of realisti models
of early universe preheating and from the point of view of non-equilibrium quantum
eld theory that I have not yet addressed. In this setion I shall disuss some of them.
Although we onsidered a series of models of ination and interations, we mostly
restrited ourselves to four-legs interations. (The sole exeption was the hybrid
ination model, whih develops a three-legs interation after symmetry breaking.)
This meant we still had a residual homogeneous or inhomogeneous inaton eld.
In realisti models of ination and preheating we expet the omplete deay of the
inaton eld. (There are radial suggestions to use the residuals of the inaton
osillations as dark matter or quintessene, but these require a great deal of ne
tuning.) The problem of residual inaton osillations an be easily ured by three-legs
interations. In the salar setor three-legs interations of the type g 2 v2 may result
9.6.
DISCUSSION
119
in stronger preheating. Yukawa ouplings h will lead to parametri exitations
of fermions [32, 117℄.
There are subtle theoretial issues related to the development of preise thermal
equilibrium in quantum and lassial eld theory due to the large number of degrees
of freedom, see e.g. [118℄. In our simulations we see the attening of the partile
spetra nk and we desribe this as an approah to thermal equilibrium, but in light
of these subtleties we should larify that we mean approximate thermal equilibrium.
Often lassial salar elds in the kineti regime display transient Kolmogorov
turbulene, with a asade towards both infrared and ultraviolet modes [119, 113℄. In
our systems it appears that the ux towards ultraviolet modes is ourring in suh a
way as to bring the elds loser to thermal equilibrium (9.13). Indeed, the slope of the
spetra nk at the end of our simulations is lose to 1. However, given the size of the
box in these simulations we were not able to say muh about the phase spae ux in
the diretion of infrared modes. This question ould be addressed, for example, with
the omplementary method of hains of interating osillators, see [119℄. This is an
interesting problem beause an out-of-equilibrium bose-system of interating salars
with a onserved number of partiles an, in priniple, develop a bose-ondensate.
It would be interesting to see how the formation of this ondensate would or would
not take plae in the ontext of preheating in an expanding universe. One highly
speulative possibility is that a osmologial bose ondensate ould play the role of a
late-time osmologial onstant.
The highlights of our study for early universe phenomenology are the following.
The mehanism of preheating after ination is rather robust and works for many
dierent systems of interating salars. There is a stage of turbulent lassial waves
where the initial onditions for preheating are erased. Initially, before all the elds
have settled into equilibrium with a uniform temperature, the reheating temperature
may be dierent in dierent subgroups of elds. The nature of these groupings is
determined by the oupling strengths.
120
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
Number Density vs. Time
n
n
Number Density
18
Number Density
1. ´ 10
1. ´ 1014
16
1. ´ 10
1. ´ 1012
1. ´ 1014
1. ´ 1012
1. ´ 1010
1. ´ 1010
1. ´ 108
1. ´ 108
1. ´ 10
6
0
500
1000
1500
2000
t
1. ´ 106
0
500
1000
1500
2000
t
Figure 9.12: V = 1=44. (Note that the Figure 9.14: V = 1=44 + 1=2g 22 2 +
vertial sale is larger than for the subse- 1=2h22 2 , g 2 = = 200, h2 = 100g 2. The
quent plots.)
highest urve is n . The number density
of diminishes when n grows.
n
Number Density
1. ´ 1014
n
1. ´ 1012
1. ´ 1013
1. ´ 1010
1. ´ 10
8
1. ´ 10
6
Number Density
1. ´ 1011
1. ´ 109
0
500
1000
1500
2000
t
1. ´ 107
100000.
0
500
1000
1500
2000
t
Figure 9.13: V = 1=44 + 1=2g 22 2 ,
g 2 = = 200. The upper urve represents Figure 9.15: V = 1=44 + 1=2g 22 2 +
1=2h2i 2 i2 , g 2 = = 200, h21 = 200g 2; h22 =
n .
100g 2. The pattern is similar to the threeeld ase until the growth of 2 .
9.6.
121
DISCUSSION
Oupation Number vs. Momentum
Occupation Number: t=2400.
Occupation Number: t=2400.
1. ´ 1013
1. ´ 109
1. ´ 1012
1. ´ 107
1. ´ 1011
100000.
1. ´ 1010
0.2
0.5
1
2
k
5
2
Figure 9.16: V = 1=44 .
5
10
20
50
k
Figure 9.18: V = 1=44 + 1=2g2 2 2 +
1=2h2 2 2 , g2 = = 200, h2 = 100g2 The and spetra are similar, but rises in the
infrared). The spetrum of is markedly
dierent from the others.
Occupation Number: t=2400.
1. ´ 1010
1. ´ 108
1. ´ 106
10000
Occupation Number: t=2400.
1. ´ 1011
100
2
5
10
20
50
k
1. ´ 1010
1. ´ 109
Figure 9.17: V = 1=44 + 1=2g2 2 2 ,
g2 = = 200. The spetra of and are
nearly idential.
1. ´ 108
1. ´ 107
1. ´ 106
2
5
10
20
k
Figure 9.19: V = 1=44 + 1=2g2 2 2 +
1=2h2i 2 i2 , g2 = = 200, h21 = 200g2 ; h22 =
100g2 . All elds other than the inaton have
nearly idential spetra.
122
CHAPTER 9. THE DEVELOPMENT OF EQUILIBRIUM
Number Density vs. Time
n
Number Density vs. Momentum
Number Spectra: t=499.996643
Number Density
1. ´ 1013
5. ´ 109
1. ´ 1011
1. ´ 109
5. ´ 108
1. ´ 109
1. ´ 108
1. ´ 107
100000.
5. ´ 107
0
100
200
300
400
500
t
1. ´ 107
2
5
10
20
Figure 9.20: V = 1=2m2 2 + 1=2g 22 2 , Figure 9.22: V = 1=2m2 2 + 1=2g 22 2 ,
g 2 Mp2 =m2 = 2:5 105 . The upper urve g 2M 2 =m2 = 2:5 105 . The upper urve
p
represents n .
represents the spetrum of .
n
Number Density
Number Spectra: t=499.996643
1. ´ 1013
2. ´ 108
1.5 ´ 108
1. ´ 1011
1. ´ 108
1. ´ 109
7. ´ 107
5. ´ 107
1. ´ 107
3. ´ 107
100000.
2. ´ 107
1.5 ´ 107
0
100
200
300
400
500
t
1. ´ 107
2
5
10
20
Figure 9.21: V = 1=2m2 2 + 1=2g 22 2 +
9.23: V = 1=2m2 2 + 1=2g 22 2 +
1=2h2 2 2 , g 2 Mp2 =m2 = 2:5 105 , h2 = Figure
2 2 2 2 2 2
5 2
100g 2. The highest urve is n . The eld 1=2h2 , g Mp =m = 2:5 10 , h =
100g The and spetra are similar,
that grows latest is .
while the spetrum of rises muh higher
in the infrared.
Chapter 10
Dynamis of Symmetry Breaking
and Tahyoni Preheating
Note: This hapter is based on a paper by Gary Felder, Juan Garia-Bellido, Patrik
Greene, Lev Kofman, Andrei Linde, and Igor Tkahev, available on the Los Alamos
eprint server as hep-ph/0012142. The full itation appears in the bibliography [120℄.
Chapter Abstrat
We reonsidered the old problem of the dynamis of spontaneous symmetry breaking
using 3d lattie simulations, and developed a theory of tahyoni preheating, whih
ours due to the spinodal instability of the salar eld. Tahyoni preheating is
so eÆient that symmetry breaking typially ompletes within a single osillation
of the eld distribution as it rolls towards the minimum of its eetive potential.
As an appliation of this theory we onsidered preheating in the hybrid ination
senario, inluding SUSY-motivated F-term and D-term inationary models. We
showed that preheating in hybrid ination is typially tahyoni and the stage of
osillations of a homogeneous omponent of the salar elds driving ination ends
after a single osillation. Our results may also be relevant for the theory of the
formation of disoriented hiral ondensates in heavy ion ollisions.
123
124
CHAPTER 10. TACHYONIC PREHEATING
10.1 Introdution
Spontaneous symmetry breaking is a basi feature of all realisti theories of elementary
partiles. In the simplest models, this instability appears beause of the presene
of tahyoni mass terms suh as m2 2 =2 in the eetive potential. As a result,
long wavelength quantum utuations k of the eld with momenta k < m grow
p
exponentially, k exp(t m2 k2 ), whih leads to spontaneous symmetry breaking.
This proess may our gradually, as in the theory of seond order phase transitions, when the parameter m2 slowly hanges from positive to negative and the degree
of symmetry breaking gradually inreases in time [88, 89, 92, 90, 91℄. Sometimes the
symmetry breaking ours disontinuously, due to a rst order phase transition [93℄.
But there is also another possibility, whih I disuss in this hapter: The tahyoni
mass term may appear suddenly, on a time sale that is muh shorter than the time
required for symmetry breaking to our. This may happen, for example, when the
hot plasma reated by heavy ion ollisions in the `little Big Bang' suddenly ools down
[121, 122, 123, 124, 125℄. A more important appliation from the point of view of
osmology is the proess of preheating in the hybrid ination senario [126, 127, 128℄,
where ination ends in a `waterfall' regime triggered by tahyoni instability.
The proess of symmetry breaking has been studied before by advaned methods of
perturbation theory, see e.g. [129, 130℄ and referenes therein. However, spontaneous
symmetry breaking is a strongly nonlinear and nonperturbative eet. It usually leads
to the prodution of partiles with large oupation numbers inversely proportional to
the oupling onstants. As a result the perturbative desription, inluding the Hartree
and 1=N approximations, has limited appliability. It does not properly desribe
resattering of reated partiles and other important features suh as prodution
of topologial defets. For these reasons we addressed the problem using lattie
simulations. In addition to aurately apturing the full dynamis of resattering,
these simulations allowed us to have a lear visual piture of all the proesses involved.
At several points in this hapter I will make referene to omputer generated movies
available on the World Wide Web that illustrate dierent aspets of spontaneous
symmetry breaking.
10.2.
TACHYONIC INSTABILITY AND SYMMETRY BREAKING
125
I will show here that tahyoni preheating an be extremely eÆient. In many
models it leads to the transfer of the initial potential energy density V (0) into the
energy of salar partiles within a single osillation. Contrary to some expetations,
the rst stage of preheating in hybrid ination is typially tahyoni, whih means
that the stage of osillations of a homogeneous omponent of the salar elds driving
ination either does not exist at all or ends after a single osillation.
10.2 Tahyoni Instability and symmetry breaking
Symmetry breaking ours due to tahyoni instability and may be aompanied by
the formation of topologial defets. Here I will onsider two toy models that are
prototypes for many interesting appliations, inluding symmetry breaking in hybrid
ination.
10.2.1 Quadrati potential
The simplest model of spontaneous symmetry breaking is based on the theory with
the eetive potential
V () = (2
4
v 2 )2 m4
4
m2 2 4
+ ;
2
4
(10.1)
where 1. V () has a minimum at = v and a maximum at = 0 with
urvature V 00 = m2 .
The development of tahyoni instability in this model depends on the initial onditions. I will assume that initially the symmetry is ompletely restored so that the
eld does not have any homogeneous omponent, i.e. hi = 0. But then hi remains
zero at all later stages, and for the investigation of spontaneous symmetry breaking
one needs to nd the spatial distribution of the eld (x; t). To avoid this ompliation, many authors assume that there is a small but nite initial homogeneous
bakground eld (t), and even smaller quantum utuations Æ(x; t) that grow on
top of it. This approximation may provide some interesting information, but quite
126
CHAPTER 10. TACHYONIC PREHEATING
often it is inadequate. In partiular, it does not desribe the reation of topologial defets, whih, as we will see, is not a small nonperturbative orretion but an
important part of the problem.
For deniteness, I assume that in the symmetri phase = 0 there are usual
quantum utuations of the massless eld with the mode funtions p12k eikt+i~k~x and
then at t = 0 we `turn on' the term m2 2 =2 orresponding to the negative mass
squared m2 . The modes with k = j~kj < m grow exponentially so the dispersion of
these utuations an be estimated as
m
Z
p
hÆ2i = 41 2 dk k e2t m2
0
k2
:
(10.2)
To get a qualitative understanding of the proess of spontaneous symmetry breaking, instead of many growing waves with momenta k < m in (10.2) let us onsider
rst a single sinusoidal wave Æ = (t) os kx with k m and with initial amplitude
2m in one-dimensionalpspae. The amplitude of this wave grows exponentially until
its beomes O(v ) m= . This leads to the division of the universe into domains
of size O(m 1 ) in whih the eld hanges from O(v ) to O( v ). The gradient energy density of domain walls separating areas with positive and negative will be
k2 Æ2 = O(m4=). This energy is of the same order as the total initial potential
energy of the eld V (0) = m4 =4. This is one of the reasons why any approximation
based on perturbation theory and ignoring topologial defet prodution annot give
a orret desription of the proess of spontaneous symmetry breaking.
Thus a substantial part of the false vauum energy V (0) is transferred to the
gradient energy of the eld when it rolls down to the minimum of V (). Beause
the initial state ontains many quantum utuations with dierent phases growing at
a dierent rate, the resulting eld distribution is very ompliated, so it annot give
all of its gradient energy bak and return to its initial state = 0. This is one of
the reasons why spontaneous symmetry breaking and the main stage of preheating
in this model may our within a single osillation of the eld .
Meanwhile if one were to make the usual assumption that initially there exists
a small homogeneous bakground eld v with an amplitude greater than the
10.2.
TACHYONIC INSTABILITY AND SYMMETRY BREAKING
127
p
amplitude of the growing quantum utuations Æ, so that m=2 < m= ,
one would nd out that when falls to the minimum of the eetive potential the
gradient energy of the utuations remains relatively small. One would thus ome
to the standard onlusion that the eld should experiene many utuations before
it relaxes near the minimum of V (). To avoid this error, we performed a omplete
study of the growth of all tahyoni modes and their subsequent interation without
making this simplifying assumption about the existene of the homogeneous eld .
Consider the tahyoni growth of all utuations with k < q
m, i.e. those that
ontribute to hÆ2 i in Eq. (10.2). This growth ontinues until hÆ2 i reahes the
p
value v=2, sine at v= 3 the urvature of the eetive potential vanishes and
instead of tahyoni growth one has the usual osillations of all the modes. This
2
happens within the time t 21m ln . The exponential growth of utuations up to
that moment an be interpreted as the growth of the oupation number of partiles
with k m. These oupation numbers at the time t grow up to
!
2
2
nk exp(2mt ) exp ln
=
1:
(10.3)
One an easily verify that t depends only logarithmially on the hoie of the initial
distribution of quantum utuations. For small the utuations with k m have
very large oupation numbers, and therefore they an be interpreted as lassial
waves of the eld .
The dominant ontribution to hÆ2i in Eq. (10.2) at the moment t is given
p
by the modes with wavelength l 2k 1 2m 1 ln1=2 (C 2 =) > m 1 , where
C = O(1). As a result, at the moment
when the utuations of the eld reah the
q
minimum of the eetive potential, h2i v , the eld distribution looks rather homogeneous on a sale l <
l. On average, one still has hi = 0. This implies that the
universe beomes divided into domains with two dierent types of spontaneous sym2
metry breaking, v . The typial size of eah domain is l =2 p2 m 1 ln1=2 C ,
whih diers only logarithmially from our previous estimate m 1 . At later stages the
domains grow in size and perolate (eat eah other up), and spontaneous symmetry
breaking beomes established on a marosopi sale.
128
CHAPTER 10. TACHYONIC PREHEATING
Of ourse, these are just simple estimates that should be followed by a detailed
quantitative investigation. When the eld rolls down to the minimum of its eetive
potential, its utuations satter o eah other as lassial waves. It is diÆult to
study this proess analytially, but fortunately one an do it using lattie simulations.
We performed our simulations on latties with either 1283 or 2563 gridpoints. The
details of these lattie simulations are given in part III.
Figure 10.1 illustrates the dynamis of symmetry breaking in the model (10.1).
It shows the probability distribution P (; t), whih is the fration of the volume
ontaining the eld at a time t if at t = 0 one begins with the probability distribution
onentrated near = 0, with the quantum mehanial dispersion (10.2).
As we see from this gure, after the rst osillation the probability distribution
P (; t) beomes narrowly onentrated near the two minima of the eetive potential
orresponding to = v . In this sense one an say that symmetry breaking ompletes
within one osillation. These results hold for a wide range of values of the oupling
onstant ; this gure is shown for a run with = 10 4 . Note that only when
the distribution stabilizes and the domains beome large an one use the standard
language of perturbation theory desribing salar partiles as exitations on a (loally)
homogeneous bakground. That is why the use of the nonperturbative approah based
on lattie simulations was so important for our investigation.
The growth of utuations in this model is shown in Fig. 10.2. It shows how
utuations grow in a two-dimensional slie of 3D spae. Maxima orrespond to
domains with > 0, minima orrespond to domains with < 0.
The dynamis of spontaneous symmetry breaking in this model is even better
illustrated by the omputer generated movie that an be found at
http://physis.stanford.edu/gfelder/hybrid/1.gif. It onsists of an animated sequene
of images similar to the one shown in Fig. 10.2. These images show the whole proess
of spontaneous symmetry breaking from the growth of small gaussian utuations of
the eld to the reation of domains with = v .
Figure 10.1 shows a lopsided distribution where one domain is notieably larger
than the other. This asymmetry is the result of small sampling due to the nite box
size. To illustrate this proess on a larger sale we also did a 2D simulation in a 10242
10.2.
TACHYONIC INSTABILITY AND SYMMETRY BREAKING
t=0
-2 -1
0
t=8
1
2
t=10
-2 -1
0
0
-2 -1
0
1
2
1
2
1
2
t=13
1
2
t=14
-2 -1
129
-2 -1
0
t=19
1
2
-2 -1
0
Figure 10.1: The proess of symmetry breaking in the model (10.1) for = 10 4 . In
the beginning the distribution is very narrow. Then it spreads out and shows two
maxima that osillate about = v with an amplitude muh smaller than v . These
maxima never ome lose to the initial point = 0. The values of the eld are shown
in units of v .
box. Figure 10.3 shows the development of domains for this run. This gure shows a
run with = 10 2 .
Similar results are valid for the theory of a multi-omponent salar eld i with the
potential (10.1). For example, the behavior of the probability distribution P (1; 2 ; t)
p
in the theory of a omplex salar eld = (1 +i2 )= 2 is shown in Fig. 10.4. For this
gure I used = 10 4 . As we see, after a single osillation this probability distribution
has stabilized at jj v . A omputer generated movie illustrating this proess an
be found at http://physis.stanford.edu/gfelder/hybrid/2.gif. Symmetry breaking of
a omplex eld gives rise to strings. Figure 10.5 shows the distribution of strings
in a 3D lattie after the eld has fallen down to the minimum. (For omputational
purposes a string was dened as the olletion of gridpoints for whih jj2 < :02v .)
130
CHAPTER 10. TACHYONIC PREHEATING
Figure 10.2: Growth of quantum utuations in the proess of symmetry breaking in
the quadrati model (10.1).
10.2.2 Cubi potential
Another important example of tahyoni preheating is provided by the theory
V=
3 4 4
v + + v :
3
4
12
(10.4)
This potential is a prototype of the potential that appears in desriptions of symmetry
breaking in F-term hybrid ination [131, 132℄.
The rst question to address onerns the initial amplitude of the tahyoni modes
in this model. This is nontrivial beause m2 () = 2v + 32 vanishes at =
0. However, eq. (10.2) implies that salar eld utuations with momentum k
have initial amplitude
hÆ2i 8k22 . They enter a self-sustained tahyoni regime if
q
v
k2 < jm2e j = 2v hÆ2i vk
2 , i.e. if k < 2 . The average initial amplitude of the
growing tahyoni utuations with momenta smaller than v
2 is
Ærms v
:
4 2
(10.5)
These utuations grow until the amplitude of Æ beomes omparable to 2v=3, and
10.2.
TACHYONIC INSTABILITY AND SYMMETRY BREAKING
131
Figure 10.3: The development of domain struture in the quadrati model (10.1).
the eetive tahyoni mass vanishes. At
that moment the eld an be represented
q
as a olletion of waves with dispersion hÆ2 i v , orresponding to oherent states
2 2
of salar partiles with oupation numbers nk 4 1:
Beause of the nonlinear dependene of the tahyoni mass on , a detailed desription of this proess is more involved than in the theory (10.1). Indeed, even
though the typial amplitude of the growing utuations is given by (10.5), the speed
of the growth of the utuations inreases onsiderably if the initial amplitude is
somewhat bigger than (10.5). Thus even though the utuations with amplitude a
few times greater than (10.5) are exponentially suppressed, they grow faster and may
therefore have greater impat on the proess than the utuations with amplitude
(10.5). Low probability utuations with Æ Ærms orrespond to peaks of the
initial Gaussian distribution of the utuations of the eld . Suh peaks tend to
be spherially symmetri [133℄. As a result, the whole proess looks not like a uniform growth of all modes, but more like bubble prodution (even though there are
no instantons in this model). The results of our lattie simulations for this model
are shown in Fig. 10.6. These results are for = 10 2. The bubbles (high peaks
of the eld distribution) grow, hange shape, and interat with eah other, rapidly
dissipating the vauum energy V (0). A omputer generated movie illustrating this
132
CHAPTER 10. TACHYONIC PREHEATING
Figure 10.4: The proess of symmetry breaking in the model (10.1) for a omplex
eld . The eld distribution falls down to the minimum of the eetive potential
at jj = v and experienes only small osillations with rapidly dereasing amplitude
jj v.
proess an be found at http://physis.stanford.edu/gfelder/hybrid/3.gif.
Fig. 10.7 shows the probability distribution P (; t) in the model (10.4). As we
see, in this model the eld also relaxes near the minimum of the eetive potential
after a single osillation.
One should note that numerial investigation of this model involved spei ompliations due to the neessity of performing renormalization. Lattie simulations
involve the study of modes with large momenta that are limited by the inverse lattie
spaing. These modes give an additional ontribution to the eetive parameters of
the model. In the simple model (10.1) these orretions were relatively small, but
in the ubi model they indue an additional linear term vh2i. This term should
be subtrated by the proper renormalization proedure, whih brings the eetive
potential bak to its form (10.4). For more details on this proedure see setion 12.3.
For ompleteness I should mention that in the theory with the quarti potential
V = V (0) 14 4 the deay of the symmetri phase ours via tunneling and the
formation of bubbles, even though there is no barrier between = 0 and 6= 0
10.3.
TACHYONIC PREHEATING IN HYBRID INFLATION
133
11.040157
100
50
0
100
50
0
0
50
100
Figure 10.5: The distribution of strings in the model (10.1) for a omplex eld .
[104, 3℄. In this ase quantum tunneling an be heuristially interpreted as a building
up of stohasti utuations Æ [102℄. In this respet the harater of tahyoni
instability for the ubi potential is intermediate between the quadrati and quarti
potentials.
10.3 Tahyoni preheating in hybrid ination
The results obtained in the previous setion have important impliations for the
theory of reheating in the hybrid ination senario. The basi form of the eetive
potential in this senario is [126, 127℄
V (; ) = ( 2
4
v 2 )2 +
g2 2 2 1 2 2
+ m :
2
2
(10.6)
p
The point where = = M=g and = 0 is a bifuration point. Here M v .
The global minimum is loated at = 0 and j j = v . However, for > the squares
of the eetive masses of both elds m2 = g 22 v 2 + 3 2 and m2 = m2 + g 2 2
are positive and the potential has a valley at = 0. Ination in this model ours
134
CHAPTER 10. TACHYONIC PREHEATING
Figure 10.6: Fast growth of the peaks of the distribution of the eld in the ubi
model (10.4). It should be ompared with Fig. 10.2 for the quadrati model (10.1).
while the eld rolls slowly in this valley towards the bifuration point. When reahes , ination ends and the elds rapidly roll towards the global minimum at
= 0, j j = v . If is a real one-omponent salar, this may lead to the formation
of domain walls. To avoid this problem, we assume that is a omplex eld. In this
ase symmetry breaking after ination produes osmi strings instead of domain
walls [126, 127℄.
In realisti versions of this model the mass m is extremely small, as is the initial
veloity _ . The elds fall down along a ertain trajetory (t); (t) in suh a way
that initially this trajetory is absolutely at, then it rapidly falls down, and then
it beomes at again near the minimum of V (; ). This implies that the urvature
of the eetive potential along this urve is initially negative. Therefore the elds
should experiene tahyoni instability along the way.
The deay of the homogeneous inaton eld and preheating in hybrid ination
were onsidered in two papers: for the simplest non-supersymmetri senario with a
variety of parameters [128℄ and for a SUSY F-term model [132℄. Both papers were
foused on the possibility of parametri resonane. However, in [128℄ it was also
pointed out that for g 2 the eld falls down only when the eld reahes some
10.3.
TACHYONIC PREHEATING IN HYBRID INFLATION
t=0
0
t=40
1
0
t=44
0
1
t=48
1
0
t=50
0
135
1
t=70
1
0
1
Figure 10.7: Histograms desribing the proess of symmetry breaking in the model
(10.4) for = 10 2 . After a single osillation the distribution aquires the form shown
in the last frame and after that it pratially does not osillate.
point . As a result, the motion of the eld ours just like the motion of
the eld in the theory (10.1). In this ase one has a tahyoni instability and the
elds relax near the minimum of V (; ) within a single osillation [128℄. For all other
relations between g 2 and the elds follow more ompliated trajetories. One might
expet that the elds would in general experiene many osillations, whih might or
might not lead to parametri resonane [128, 132℄.
We performed an investigation of preheating in hybrid ination in the model (10.6)
with two salar elds (one real and one omplex) and in SUSY-motivated F-term and
D-term ination models with three omplex elds. We used methods similar to those
disussed in the previous setion, inluding 3D lattie simulations. We found that
eÆient tahyoni preheating is a generi feature of the hybrid ination senario,
whih means that the stage of osillations of the quasi-homogeneous omponents
of the salar elds driving ination is typially terminated by the bakreation of
utuations.
Fig. 10.8 shows the proess of spontaneous symmetry breaking in the theory (10.6)
136
CHAPTER 10. TACHYONIC PREHEATING
for g 2 = 10 4 , = 10 2 , M = 1015 GeV. The probability distribution osillates along
the ellipse g 2 2 + 2 = g 2 2 . As before, it relaxes near the minimum of the eetive
potential within a single osillation. A omputer generated movie illustrating this
proess an be found at http://physis.stanford.edu/gfelder/hybrid/4.gif.
Figure 10.8: The proess of symmetry breaking in the hybrid ination model (10.6)
for g 2 . The eld distribution moves along the ellipse g 2 2 + 2 = g 2 2 from the
bifuration point = , = 0.
The theory of preheating in D-term ination for various relations between g 2 and
[134, 135℄ is very similar to the theory disussed above. Meanwhile, in the ase
g 2 = 2 the eetive potential (10.6) has the same features as the eetive potential
of SUSY-inspired F-term ination [131℄. In this senario the elds and fall down
along a simple linear trajetory [132℄, so that instead of following eah of these elds
one may onsider a linear ombination of them and nd the eetive potential in this
diretion. This eetive potential has exatly the same shape as our ubi potential
(10.4). Thus all of the results that we obtained for tahyoni preheating in the theory
(10.4) should be valid for F-term ination as well, with minor modiations due to
the presene of additional degrees of freedom that an be exited during preheating.
Indeed, we were able to onrm these onlusions by lattie simulations of the F-term
and D-term models.
10.3.
TACHYONIC PREHEATING IN HYBRID INFLATION
137
Thus we see that tahyoni preheating is a typial feature of hybrid ination. The
prodution of bosons in this regime is nonperturbative, very fast, and eÆient, but it
is usually not related to parametri resonane. Instead it is related to the prodution
and sattering of lassial waves of the salar elds. Of ourse, one should keep
in mind that there may exist some partiular versions of hybrid ination in whih
tahyoni preheating is ineÆient, e.g. beause of fast motion of the eld near the
bifuration point.
The tahyoni nature of preheating in hybrid ination implies that instead of the
prodution of gravitinos by a oherently osillating eld [61, 136, 80, 137℄, in hybrid
ination one should study gravitino prodution due to the sattering of lassial
waves of the salar elds produed by tahyoni preheating. Our results may also be
important for the theory of the generation of the baryon asymmetry of the universe
at the eletroweak sale [114℄.
From a more general point of view, however, the most important appliation of
our results is to the general theory of spontaneous symmetry breaking. This theory
onstitutes the basis of all models of weak, strong and eletromagneti interations.
By applying the methods of lattie simulations we developed for the study of preheating we have been able to atually see the proess of spontaneous symmetry breaking
and to reveal some of its rather unexpeted features.
Part III
LATTICEEASY: Lattie
Simulations of Salar Field
Dynamis
138
Chapter 11
Introdution to LATTICEEASY
Muh of the work reported in this thesis has involved the use of numerial alulations, and in partiular of lassial lattie simulations of eld dynamis. These
simulations were done using a program developed by Igor Tkahev and me, whih we
have made available on the Web under the name LATTICEEASY. A great deal of
my ontribution to this researh has involved working out both the physis and the
programming required for these simulations. This part of the thesis desribes these
simulations, with an emphasis on the physis behind the ode.
This hapter ontains a motivational introdution explaining the need for lattie
simulations and the role they play in osmologial researh suh as this, followed by
a setion desribing my ontribution to the development of the simulations. The following hapter desribes the equations solved by the program, inluding the evolution
equations for the elds and for the expansion of the universe, as well as the initial
onditions. The initial onditions are determined by quantum utuations present
at the end of ination. The hapter on equations ends with a brief disussion of
renormalization of eld theories in lattie alulations. The nal hapter deals with
the omputational methods employed by LATTICEEASY. These inlude the methods used for time evolution (staggered leapfrog) and spatial derivatives (seond order
nite dierening). This hapter also disusses issues related to the stability and
auray of the solutions obtained by LATTICEEASY.
139
140
CHAPTER 11.
INTRODUCTION TO LATTICEEASY
11.1 Lattie Simulations and Cosmology
Studying the early universe requires desribing the evolution of interating elds in
a dense, high-energy environment. The study of reheating after ination and the
subsequent thermalization of the elds produed in this proess typially involves
non-perturbative interations of elds with exponentially large oupations numbers
in states far from thermal equilibrium. Various approximation methods have been
applied to these alulations, inluding linearized analysis and the Hartree approximation. These methods fail, however, as soon as the eld utuations beome large
enough that they an no longer be onsidered small perturbations. In suh a situation linear analysis no longer makes sense and the Hartree approximation neglets
important resattering terms. What we have learned in the last several years is that
in many models of ination preheating an amplify utuations to these large sales
within a few osillations of the inaton eld. Moreover, suh large ampliation appears to be a generi feature, arising in virtually all known inationary models due
to either parametri resonane, tahyoni instability, or both.
The only way to fully treat the nonlinear dynamis of these systems is through
lattie simulations. These simulations diretly solve the lassial equations of motion for the elds. Although this approah involves the approximation of negleting
quantum eets, these eets are exponentially small one preheating begins. So in
any inationary model in whih preheating an our lattie simulations provide the
most aurate means of studying post-inationary dynamis.
Over the past several years Igor Tkahev and I have developed a program for
performing suh lattie simulations. The program, whih we all LATTICEEASY, is
freely available on the World Wide Web at
http://physis.stanford.edu/gfelder/lattieeasy. This website ontains detailed doumentation on how to use the program for dierent inationary models. In this thesis
I will fous instead on disussing the physis behind the simulations.
11.2.
MY CONTRIBUTIONS TO LATTICEEASY
141
11.2 My Contributions to LATTICEEASY
Beause LATTICEEASY was reated jointly by Igor Tkahev and me it is appropriate
that for this thesis I should delineate what aspets of it were my ontribution.
The original version of the lattie program was written by Dr. Tkahev. The
idea of solving the lassial equations of motion for salar elds in the early universe
on a lattie, using quantum mehanial dispersions as initial onditions, and using a
staggered leapfrog algorithm were all in plae before I began working on the projet.
Moreover the justiation of using these lassial equations was worked out in detail
by Dr. Tkahev and others.
Nonetheless, all of the equations derived in the lattie simulation part of this
thesis were derived by me. Dr. Tkahev had orretly adjusted the initial modes to
ompensate for the box size and lattie spaing as desribed here (setion 12.2.1),
but so far as I have been able to nd he never wrote down his reasons for using
the equations he did. He orretly negleted the initial time dependene of !k in
setting the mode derivatives, but never explained why; see setion 12.2.4. Likewise
the ombination of the Einstein equations used by the program was used in Dr.
Tkahev's original version, but the derivation presented here is due to me. In the
ourse of deriving these equations I found a number of minor errors in the program.
For example neither Dr. Tkahev, nor to the best of my knowledge anyone else, had
ever notied that the Gaussian distribution of the omplex eld modes required a
Rayleigh distribution for their amplitudes. I don't intend my mentioning the presene
of this and other errors to be a ritiism of Dr. Tkahev. The lattie program is a very
large and ompliated piee of ode and it is not unusual for small bugs to our in
suh ases. I mention these orretions simply by way of noting that in ollaborating
with Dr. Tkahev I had to rederive from srath all of the equations used by the
program.
Some of the physis desribed here was added after Dr. Tkahev wrote his original
version. In partiular, the orretion of the initial modes to preserve isotropy (setion
12.2.3) and the introdution of renormalization (setion 12.3) were both implemented
by me with the help of disussion and advie from Drs. Linde and Kofman. Also the
142
CHAPTER 11.
INTRODUCTION TO LATTICEEASY
denitions of oupation number and energy spetra as adiabati invariants of the
elds were mine.
Computationally and organizationally the basi onept of the program as desribed above was Dr. Tkahev's, and everything sine then has been mine. He
originally provided me with a program in Fortran and I rewrote it in C++. Later,
when I had derived all of the equations and understood the workings of the program
better I dropped the original version entirely and rewrote the entire thing with a
dierent struture. It was after this rewrite that Dr. Tkahev and I jointly released
the program under the name LATTICEEASY. It was our intent to make it available
as a tool for the osmology ommunity, although to the best of my knowledge it has
not been used by anyone other than us.
It was in the ourse of rewriting the program that I devised the orretions to
the sale fator evolution equations (setion 13.2). More importantly, I developed a
method for systematially resaling the elds and spaetime oordinates to allow the
alulations to be feasible for a broad variety of models. This subjet is disussed
briey in this thesis but forms a large part of the doumentation for LATTICEEASY.
In summary, I would say that the basi onepts behind these lattie simulations
preeded my involvement in the projet and were largely due to Dr. Tkahev, whereas
the systemati derivations of the equations and methods used was primarily my work.
Chapter 12
Equations
12.1 Evolution Equations
12.1.1 Field Equations and Coordinate Resalings
The equation of motion for a salar eld in an expanding universe is (see e.g. [3℄)
a_
f + 3 f_
a
1 2
V
r
f+
= 0:
2
a
f
(12.1)
In priniple this equation, ombined with an appropriate evolution equation for the
sale fator a, ould simply be solved diretly in this form. In pratie, however, there
are several ways in whih the proess of solving suh equations an be made easier
by resaling the variables. In partiular, by resaling the eld and time variables by
appropriate powers of the sale fator, the rst derivative term an be eliminated from
the equation of motion. Other useful riteria for hoosing resaling parameters inlude
keeping the dominant time sale of the problem xed as the universe expands and
setting the natural time and distane sales to be of order unity. The LATTICEEASY
doumentation explains these resalings in detail. Beause they are model-spei and
largely used for omputational onveniene, I will not disuss them further here. All
equations for the rest of the thesis will be given in physial variables and it should be
understood that in the atual program they are solved in a resaled form.
143
144
CHAPTER 12.
EQUATIONS
12.1.2 Sale Fator Evolution
The equation for the sale fator a is derived from the Friedmann equations for a at
universe
4a
a =
( + 3p)
(12.2)
3
2
8
a_
= :
(12.3)
a
3
For a set of salar elds fi in an FRW universe
=T +G+V; p=T
1
G V
3
(12.4)
where T , G, and V are kineti (time derivative), gradient, and potential energy
respetively, given by
1
1
(12.5)
T = f_i2 ; G = 2 jrfi j2 :
2
2a
Equations (12.2) and (12.3) and the eld evolution equations form an overdetermined
system. In priniple either sale fator equation ould be used but in pratie it is
easiest to ombine them so as to eliminate the time derivative term T beause in the
staggered leapfrog algorithm f and f_ are known at dierent times. Eliminating T we
get
3 a_ 2
T=
G V
(12.6)
8 a
4a
a_ 2
2
a_ 2 8 1
2
2
a =
(4T 2V ) = 2 + 8a G + V = 2 +
jrf j + a V :
3
a
3
a
a 3 i
(12.7)
This equation still poses problems for the leapfrog algorithm beause of the presene
of a_ in the equation. Setion 13.2 disusses this problem and how to solve it.
The program an be set to not use expansion, in whih ase only the eld evolution
equations are solved with a = 1 and H = 0. Finally, there is also an option in the
program to impose a xed power-law expansion. This option is disussed in the
doumentation but beause it wasn't used for any of the researh disussed in this
thesis I won't say anything more about it here.
12.2.
INITIAL CONDITIONS ON THE LATTICE
145
12.2 Initial Conditions on the Lattie
There are three kinds of initial onditions that need to be set for the lattie alulations. The rst onsists of the homogeneous values of the elds and derivatives.
The seond onsists of the utuations of the elds and eld derivatives. Finally, in
an expanding universe the derivative of the sale fator, or equivalently the Hubble
onstant, needs to be set.
The homogeneous values an simply be set to whatever is appropriate for a given
problem. For example in studies of preheating after haoti ination the inaton
zeromode is typially set to a value orresponding to the end of ination (e.g. Mp =3
for 4 ination).
There is a small problem involved in setting the initial value of the Hubble onstant. As we will see below this value is needed in order to determine the initial eld
utuations. Stritly speaking, however, these eld utuations would be needed
to determine the total energy used to alulate the Hubble onstant. In pratie,
though, we set the initial value of the Hubble onstant based on the homogeneous
eld values, negleting the initial utuations. In fat this method is more aurate
than if the full eld distribution were used beause the utuations represent vauum
utuations whose energy should properly be eliminated by renormalization anyway.
The inhomogeneous modes are determined by quantum utuations of the elds.
In setion 12.2.1 we derive equations for the eld utuations and in setion 12.2.2
we derive similar expressions for the utuations of the derivatives. These equations
have to modied, however, to preserve isotropy on the lattie. This modiation is
disussed in setion 12.2.3 Finally setion 12.2.4 gives a justiation for an approximation used in setting the utuations of the derivatives.
12.2.1 Initial Conditions for Field Flutuations
Although the eld equations are solved in onguration spae with eah lattie point
representing a position in spae, the initial onditions are set in momentum spae and
then Fourier transformed to give the initial values of the elds and their derivatives
146
CHAPTER 12.
EQUATIONS
at eah grid point. The Fourier transform Fk is dened by
1 Z 3
f (x) =
d kFk eikx
3
=
2
(2 )
(12.8)
It is assumed that no signiant partile prodution has ourred before the beginning of the program, so quantum vauum utuations are used for setting the
initial values of the modes. The probability distribution for the ground state of a real
salar eld in a FRW universe is given by [115, 99℄
P (Fk ) / exp( 2a2 !k jFk j2 )
(12.9)
where
!k2 = k2 + a2 m2 ;
(12.10)
2V
(12.11)
m2 = 2 :
f
Although f is a real eld the Fourier transform is of ourse omplex, so this probability distribution is over the omplex plane. The phase of Fk is uniformly randomly
distributed and the magnitude is distributed aording to the Rayleigh distribution
P (jFk j) / jFk jexp( 2a2 !k jFk j2 ):
(12.12)
Note that this distribution gives the mean-squared value
< jFk j2 >=
1
:
2a2 !k
(12.13)
There are two adjustments to equation [12.12℄ that must be made in order to
normalize these modes on a nite, disrete lattie. First this denition has to be
adjusted to aount for the nite size of the box. This is neessary in order to keep
the eld values in position spae independent of the box size. To see this onsider
the spatial average < f 2 >.
f2 =
1 Z Z 3 3 0
0
d kd k Fk Fk0 ei(k+k )x
3
(2 )
(12.14)
12.2.
INITIAL CONDITIONS ON THE LATTICE
147
1 Z 3
1 Z Z Z 3 3 3 0
0 )x
i
(
k
+
k
d xd kd k Fk Fk0 e
= 3 d kjFk j2
(12.15)
L3 (2 )3
L
where L3 is the volume of the region of integration. So in order to keep < f 2 >
onstant as L is hanged the modes Fk must sale as L3=2 .
< f 2 >=
Aounting for the disretization of the lattie is even easier. From the denition
of a disrete Fourier transform (denoted here as fk ) in three dimensions
fk 1
F:
dx3 k
(12.16)
Note that values suh as < f 2 > will be aeted by hanges in the lattie spaing, but
this is reasonable sine this spaing determines the ultraviolet uto of the theory.
Without suh a uto < f 2 > would be divergent. See setion 12.3 for further
disussion of this eet.
Putting these eets together gives us the following expression for the rms magnitudes, whih we denote by Wk .
q
Wk < jfk j2 > =
s
L3
2a2 !k dx6
(12.17)
At a point (i1 ; i2 ; i3 ) on the Fourier transformed lattie the value of k is given by
2 q 2 2 2
jkj = L i1 + i2 + i3:
(12.18)
Finally it remains to implement the Rayleigh distribution
P (jfk j) / jfk jexp( jfk j2 =Wk2 )
(12.19)
Normalizing this distribution gives
P (jfk j) =
2
2
2
j
fk je jfk j =Wk :
2
Wk
(12.20)
To generate this distribution from a uniform deviate (i.e. a random number generated
with uniform probability between 0 and 1) rst integrate it and then take the inverse
148
(see [138℄), whih gives
CHAPTER 12.
jfk j =
q
Wk2 ln(X )
EQUATIONS
(12.21)
where X is a uniform deviate.
There are two more points to note in setting the initial onditions for the utuations. Before saying what they are I want to say that if anyone has atually read
this far in my thesis please let me know. The rst person to tell me they've seen this
sentene will get eleven dollars; I'll explain why I hose eleven at the time. The rst
is simply that the sale fator is set to 1 at the beginning of the alulations and may
thus be dropped from the equations. The seond is that the phases of all modes are
random and unorrelated, so they are eah set randomly. The expression for the eld
modes is thus
q
fk = ei Wk2 ln(X )
(12.22)
where
Wk =
3=2
p2L! dx3
k
(12.23)
and is set randomly between 0 and 2 . The frequeny !k for a given point (i1 ; i2 ; i3 )
on the momentum spae lattie is given by
2 d2 V
2
2
i21 + i22 + i23 + 2 :
!k =
L
df
(12.24)
There is one more eet that needs to be aounted for in setting the initial
utuations. This eet is disussed in setion 12.2.3 below, in whih the equations
for these utuations are written in their nal form.
12.2.2 Initial Conditions for Field Derivative Flutuations
To alulate the eld derivatives it is neessary to know the time dependene of the
vauum utuations being onsidered. The full time dependene omes from several
soures. First, there is an osillatory term ei!k t . I'll disuss in setion 12.2.3 below
the use of the plus or minus sign in this term. Next, all the modes have an extra fator
of 1=a relative to their Minkowski spae values. This an be seen for example from
12.2.
INITIAL CONDITIONS ON THE LATTICE
149
equation (12.13). This extra fator ensures that the physially meaningful quantity
< f (x)2 > depends on the physial rather than the omoving momenta of the modes
in the box. Finally, there is an additional time dependene that arises from the fat
that !k is time dependent, both beause of the sale fator multiplying the mass term
and beause of possible time dependene of the eetive mass itself.
Here I will alulate the eld derivatives taking all of these eets into aount
exept the time dependene of !k . Setion 12.2.4 explains the justiation for this
approximation.
Using this approximation the full time dependene of the mode fk is given by
fk / a 1 ei!k t :
Thus
f_k =
i!k aa_ fk = (i!k H ) fk :
(12.25)
(12.26)
12.2.3 Standing Waves
Equation (12.25) tells us the frequeny of osillation of the mode fk , but the question still remains whether we should use the plus or minus sign in the exponential.
The answer is that we must use both. This fat arises from a simple property of
Fourier transforms, namely that the Fourier transform of a real eld f must obey the
symmetry
f k = fk :
(12.27)
(It doesn't matter if you are onsidering a omplex eld sine you must still then set
initial onditions for its real and imaginary parts, and their Fourier transforms will
be onstrained to obey this same symmetry relation.) We an ignore the expansion
of the universe for a moment and imagine that for some mode fk we have hosen to
use the plus sign in the exponential, i.e.
f_k = i!k fk :
(12.28)
150
CHAPTER 12.
EQUATIONS
However, sine both f and f_ are real elds it must be true that
f_ k = f_k = i!k fk = i!k f k :
(12.29)
In other words hoosing the plus sign for a given momentum k neessarily means using
the minus sign for the momentum k. Reall that a mode fk translates into a funtion
f (x) with spatial dependene e ikx . So if you use the plus sign in the exponential
for some positive k and the minus sign for k you have eetively initialized the two
osillatory modes
f (x) = e i(kx !k t) + ei(kx !k t) :
(12.30)
In other words you have reated a right moving wave. Likewise hoosing the minus
sign in the exponential for a positive value of k orresponds to setting up a left
moving wave. Of ourse there is no physially preferred diretion on the lattie, so
in reality your initial onditions should ontain equal omponents of right and left
moving utuations.
In pratie the signs you use for the exponential time dependene of dierent
modes has a negligible eet on the evolution one preheating begins. Even if every
mode is initialized to be left-moving, the total momentum this imparts to the eld is
unnotieable by the late stages of the evolution in every problem we have onsidered.
Nonetheless it is presumably desirable to enfore Lorentz invariane, at least in an
averaged sense. You ould do this by randomly initializing eah mode with either a
plus or a minus sign. Instead, I hose to set up both left and right moving waves with
equal amplitude at eah value of k. In other words
1
fk = p (fk;1 + fk;2 )
2
(12.31)
1
f_k = p i!k (fk;1 fk;2 ) Hfk :
(12.32)
2
where fk;1 and fk;2 are two modes with separate random phases but equal amplitudes
determined by equation [12.21℄. This means that in the free eld limit the initial
utuations orrespond to standing waves.
12.2.
151
INITIAL CONDITIONS ON THE LATTICE
By now it may have struk you that I seem to be determining these initial onditions based on issues of onveniene, symmetry, and so on. What about whatever is
the physially orret form for vauum utuations, as given by their quantum mehanial probability distributions? Shouldn't those distributions provide an answer
to all of these questions as to the orret form of the equations? The answer is no.
Although equation (12.20) gives the orret quantum distribution for the mode amplitudes, it is not orret to use this distribution and then use equation (12.26) to set
the values of the eld derivatives. The problem is that quantum mehanially fk and
f_k are nonommuting operators and an not be simultaneously set. Although this
unertainty presents a problem in priniple it is unimportant in pratie. One parametri resonane begins the oupation numbers of the modes fk beome large and
their quantum unertainty beomes irrelevant. Moreover the rapid growth that ours
during this resonane eetively destroys all information about the initial values of
the modes so that the nal simulation results are insensitive to the details of how the
initial onditions are set. In our experiene runs that use the probability distribution
of equation (12.20) give essentially the same results as ones that use the exat value
of equation (12.23) for eah mode, or virtually any other initial distribution for that
matter.
12.2.4 The Adiabati Approximation
We noted in setion 12.2.2 that the time dependene of the modes omes from their
expliit time dependene fk / ei!k t , from the fator of 1=a in the initial amplitude
of the modes, and from the time dependene of !k itself. The full time dependene
of the modes is given by
1
fk / p ei!k t :
(12.33)
a !k
Thus the derivative is given by
f_k =
i!k i!_ k t
1 !_ k
2 !k
a_ f = !k
a k
"
i
1 !_ k
2 !k2
#
a_
f
!k k
(12.34)
152
CHAPTER 12.
EQUATIONS
where the last step uses the fat that initially t = 0 and a = 1. Negleting the time
dependene of !k as we did earlier amounts to making the approximation
!_ k !k2 ;
(12.35)
whih is preisely the ondition that !k is hanging adiabatially. If this ondition is
not satised in the late stages of ination then gravitational partile prodution will
our and it will no longer make sense to take the vauum utuations of equation
(12.9) as initial onditions.
There's another way to view this ondition. Gravitational partile prodution will
our unless !k > H . Sine this ondition is automatially satised for k > H onsider
the opposite ase k H , for whih !k am. Then negleting the time dependene
of m, !_ k = a_ m = Hm when a = 1, so the ondition !_ k !k2 is equivalent to the
ondition m H . In fat !_ k !k2 is the stronger (and more aurate) ondition
beause it also speies that m shouldn't be hanging rapidly, whih would lead to
partile prodution irrespetive of the value of H . However, all partile masses should
vary slowly during ination beause they should only depend on onstants and on
the value of the inaton, whih must be hanging slowly.
In the ase of a eld with m < H during ination the approximation that the
eld ends ination in its ground state is no longer valid. In the limit m H
the utuations of the eld produed during ination an be aurately desribed
by Hankel funtions [3℄. However in this ase the elds will be opiously produed
during ination, leading to severe osmologial problems. (See hapter 7). For this
reason we do not implement these Hankel funtion solutions in the lattie program.
In order to avoid the moduli problem assoiated with light elds it's best to assume
that some mehanism must have given all salar elds large masses during ination,
in whih ase equation (12.9) is an aurate expression for the modes at the end of
ination.
12.3.
153
RENORMALIZATION
12.3 Renormalization
As I've disussed, the justiation for doing a lassial alulation for quantum elds
is that one the eld utuations are amplied suÆiently quantum eets are negligible. There are some ases, however, when these quantum eets may be important,
and in suh ases they may be (partially) aounted for through a simple form of
renormalization.
I won't review the entire theory of renormalization in quantum eld theory here.
It is disussed in many standard texts, e.g. [139℄. I will simply note that the basi idea
is as follows. There is some quantity (e.g. a mass or a oupling onstant) whose bare
value formally reeives innitely large quantum mehanial orretions. Although
the dierene between the bare and measured values of this quantity may be innite,
the dierenes between any two measurable quantities (e.g. the eetive oupling
at two dierent energy sales) should remain nite and alulable. In pratie these
alulations an be performed by adding to the Lagrangian terms that anel these
innite orretions. These anellations an be tuned so as to x the value of one
measurement, e.g. by mathing the measured oupling at some partiular energy.
Consider how this applies to the lattie alulations disussed here. Initially the
eld utuations are only those representing quantum vauum states. These utuations aet ouplings, masses, and the total energy of the system in a way that is
dependent on the lattie spaing. For example, onsider the theory
V = 4
(12.36)
and rewrite the eld as the sum of a homogeneous omponent and utuations
Æ. The eetive potential felt by the homogeneous eld will reeive a orretion
from the utuations equal to
ÆV
23 < Æ2 > 2 :
(12.37)
(The 3=2 arises from ombinatoris.) I'm assuming all odd powers of Æ vanish
on average. This orretion represents an unphysial eet in the sense that its
154
CHAPTER 12.
EQUATIONS
strength depends on the ultraviolet uto imposed by the lattie. In the limit of zero
lattie spaing where arbitrarily large momenta would be inluded on the lattie this
orretion would beome innite. This eet ould be eliminated, however, by adding
a renormalization term
Vrenormalization =
3
< Æ2 > 2
2
(12.38)
or equivalently by adding the term
3 < Æ2 > (12.39)
to the equation of motion for . Note that < Æ2 > in this ase refers to the initial
value that arises from quantum utuations, not to a dynami quantity that hanges
as the eld evolves. Suh hanges represent physial eets and should not be eliminated. In eet this orretion would x the eetive mass for (i.e. 2 V= 2 ) to
be at at = 0 when the eld is in the vauum state.
The above example illustrates how a simple form of renormalization an be implemented on the lattie. This proedure ould in priniple be used to renormalize any
mass, oupling onstant, or energy term in the theory. Ordinarily these orretions
are not important beause the quantum eets are quikly swamped as the utuations beome amplied. In some ases, however, quantum eets an hange the
initial behavior of the system in important ways. For example in the theory
1
V = 4
4
1 3 1 4
v + v
3
12
(12.40)
the ubi term ontributes a linear term to Veff () that auses the eld to initially
start rolling away from zero. In this ase we found it useful to eliminate this term by
adding
Vrenormalization = v < Æ2 >
(12.41)
to the equation of motion for . See hapter 10.
Chapter 13
Computational Methods Used by
LATTICEEASY
13.1 Time Evolution: The Staggered Leapfrog
Method
LATTICEEASY uses a method alled \staggered leapfrog" for solving dierential
equations. In order to solve a seond order (in time) equation you need to store
the value of the variable and its rst time derivative at eah step, and use these to
alulate the value of the seond time derivative. The idea of staggered leapfrog is
to store the variables (i.e. the eld values) and their derivatives at dierent times.
Speially, if the program is using a time step dt and the eld values are known at a
time t then the derivatives will initially be known at a time t dt=2. Using the eld
values the program an then alulate the seond derivative f at time t and use this
to advane f_ to t + dt=2. This value of f_ an in turn be used to advane f to t + dt,
thus restarting the proess. Shematially, this looks like
f (t) = f (t dt) + dtf_(t dt=2)
f_(t + dt=2) = f_(t dt=2) + dtf[f (t)℄
f (t + dt) = f (t) + dtf_(t + dt=2)
155
(13.1)
156
CHAPTER 13.
COMPUTATIONAL METHODS USED BY LATTICEEASY
Beause eah step advanes f or f_ in terms of its derivative at a time in the middle
of the step this method has higher order auray and greater stability than a simple
Euler method. However, the method relies on being able to alulate f in terms of
f at the time t, so both auray and stability are generally lost if f depends on the
rst derivative f_. This is the reason we hoose our resalings so as to eliminate rst
derivative terms in the equations of motion. Note that the evolution equation for the
sale fator does have a rst derivative in it. Setion 13.2 desribes how this problem
is solved in the program.
To set the initial onditions for the staggered leapfrog alulations the eld values
and derivatives must be desynhronized. The initial onditions are set at t = 0 and
then the elds are advaned by an Euler step of size dt=2 to begin the leapfrog.
Thereafter all alulations are done in full, staggered steps.
13.2 Corretion to the Sale Fator Evolution Equation to Aount for Staggered Leapfrog
Using a staggered leapfrog algorithm means that in solving for a(t) the value of a_ is
known at t d=2 where d is the time step. See setion 13.1 for more details. The
solution to this problem is to use the two equations
1
a_ + a_ + da; a_ (_a+ + a_ )
2
(13.2)
where a_ + and a_ refer to the values of a_ at t + d=2 and t d=2 respetively and all
other variables are evaluated at time t.1 Take the evolution equation to be
a_ 2
a = C1 + C2 :
a
1 Thanks
to Julian Borrill for suggesting this solution.
(13.3)
13.3.
157
SPATIAL DERIVATIVES
Plugging this form into equation (13.2) and eliminating a_ gives
a_ + a_ + d
a_ +
=
C1
dC1 2
2
(_a+ + a_ ) + C2 =
a_
4a
4a +
0
s
dC1
a_ a_
2a +
d2 C12 2 dC1
2a dC1
a_ + 1 a_ +
a_ + 1
dC1 2a
4a2
a
s
2a
2a
d2 C1 C2
2dC1
a_
a_ +
:
1+
dC1 dC1
a
a
dC1 2
a_ + dC2 + a_
4a
(13.4)
1
d2 C12 2 d2 C1 C2 dC1 A
a_ +
+
a_
4a2
a
a
(13.5)
To determine whether to use the plus or minus sign in equation (13.5) onsider the
limit as d ! 0. In this limit
s
2a
2dC1
2a
1+
a_
dC1 dC1
a
a_ + a_
a_
2a 2a
+ 2_a :
dC1
dC1
(13.6)
This suggests that the plus sign must be used in order to redue to the limit a_ + a_ .
Hene
0
1
s
2dC1
2a d2 C1 C2 A
a_ + a_
1+
a_ +
:
(13.7)
1
dC1
a
a
In the program it's useful to alulate a, whih is roughly (_a+
a 2
0
2a 1
dC1
14
2_a
d
s
a_ )=d, so
13
2dC1a_
d2 C1 C2 A5
1+
+
:
a
a
(13.8)
Thus equation (12.7) beomes
a 8
1<
2_a
d:
2
a4
1
d
9
3=
4da_
16
1
1+
+ d2 2 a 2 jrfi j2 + a2 V 5; :
a
A
3
s
(13.9)
13.3 Spatial Derivatives
There are two ontexts in whih the lattie program needs to alulate spatial derivatives. The rst is in the evolution equations, whih inlude a Laplaian term r2 f .
The seond is in alulating the eld gradients jrf j2 , both for total energy and for
158
CHAPTER 13.
COMPUTATIONAL METHODS USED BY LATTICEEASY
the sale fator evolution.
The Laplaian is alulated using a simple nearest-neighbor sheme
r2f (x; y; z) 2f 2f 2f
+
+
x2 y 2 z 2
0
1 X
f
dx2 neighbors
1
6f (x; y; z )A
(13.10)
where dx is the lattie spaing and the sum over neighbors is taken over the six lattie
points adjaent to (x; y; z ). We tried on several oasions using a higher order sheme
involving twie removed neighbors, but never found any notieable dierene in the
results.
Gradients ould be alulated using a similar formula, but it turns out there is a
more aurate way that is no more omputationally expensive.2 This simpliation is
made possible by the fat that the gradients are only needed for sums over the entire
lattie. Sine the lattie is periodi we an use integration by parts with no surfae
term, i.e.
X
X
jrf j2 =
f r2 f:
(13.11)
lattie
lattie
We an thus use the Laplaian formula above for the gradients as well. Swithing from
a diret alulation of the gradients to this indiret method improved our alulation
of the total energy so muh that the amount of energy nononservation in our runs
dereased by more than an order of magnitude!
13.4 The Auray of the Simulations
13.4.1 The Classial Approximation
As mentioned before, frequently used approximation shemes suh as linear analysis
or the Hartree approximation are inadequate for preheating problems, whih typially involve signiant resattering. Lattie simulations automatially aount for
all resattering eets, but one may legitimately ask whether these simulations involve
other, potentially dangerous approximations.
2 Thanks
to Ue-Li Pen for this suggestion.
13.4.
THE ACCURACY OF THE SIMULATIONS
159
Of ourse any numerial alulations suer from a ertain amount of inauray.
In addition to simple roundo errors lattie simulations also ontain errors due to
their inherent ultraviolet (grid spaing) and infrared (box size) utos. Setion 13.4.2
disusses how suh errors are monitored and ontrolled in our simulations.
On a more fundamental level, however, simulations suh as ours employ the lassial approximation. In other words we simply solve the lassial equations of motion
for the elds in our system, negleting all quantum mehanial eets. This approximation may seem unjustiable, partiularly given that the initial utuations
that seed reheating in inationary osmology are quantum mehanial in origin. In
preheating, however, these utuations are rapidly amplied to exponentially large
amplitudes, whih permits us to neglet quantum eets.
More preisely, the lassial limit of a eld theory ours when the oupation
number beomes muh larger than one. For a salar eld the oupation number
of a partiular mode k is given by
1
1 _ 2
2
nk = !k jk j + jk j ;
2
!k
where
p
!k k2 + m2 :
(13.12)
(13.13)
In the vauum the eld has expetation values
D
jk j2
E
D
E
j_ 2k j
and thus
=
1
2!k
(13.14)
=
!k
2
(13.15)
hnk i = 12 :
(13.16)
In all of our simulations preheating rapidly drives the elds to a state where nk 1.
In fat it turns out that not all modes on the lattie are amplied in this way, but
the ones that aren't make an exponentially small ontribution to the eld dynamis.
More speially, preheating amplies infrared modes, so for any partiular model
160
CHAPTER 13.
COMPUTATIONAL METHODS USED BY LATTICEEASY
there will be a ertain uto in momentum spae above whih the modes aren't
amplied. To ensure that all relevant physis is inluded in the simulation the lattie
spaing should be small enough to inlude all modes below this uto. Then the
small wavelength modes that remain unamplied make no appreiable dierene.
This assertion an (and should) always be heked for a given model by verifying
that the results are insensitive to the exat lattie spaing used.
Finally, one might wonder if the initial stages of the alulation are a soure for
onern sine in these stages all of the utuations have small oupation numbers.
While this onern is valid in priniple, it turns out to be irrelevant in pratie.
The nal outome of preheating in a partiular model is determined by feedbak
mehanisms that ome into play when the utuations are large and resattering
important. In all of the models we have examined to date the nal results proved to be
extremely insensitive to the details of the initial onditions, whih leads us to believe
that any potential inauraies in our treatment of the rst stages of preheating should
have little or no eet on our onlusions. Nonetheless suh onerns are ultimately
valid ones and must be reheked for eah model that one wishes to examine using
the lassial approximation.
13.4.2 Numerial Auray
There are numerous ways to monitor the numerial auray of a alulation. These
inlude straightforward, \brute-fore" methods of trial and error. For example one
should always vary quantities suh as the time step and random number seeds (used
in the initial onditions, see setion 12.2.1) to make sure that the nal results are
insensitive to hanges in these quantities.
Another useful trik is to monitor one or more onserved quantities. In the ase
of eld theory in Minkowski spae it is a straightforward task to ompute the total
energy on the lattie. All of our Minkowski spae simulations onserve total energy
to within a few perent, and virtually all of them do so to within at most a few tenths
of a perent.
13.4.
THE ACCURACY OF THE SIMULATIONS
161
Minkowski spae is only an approximation, however, useful in ases where expansion an be negleted. In an expanding universe the situation is a little more
ompliated beause energy density is not onserved, but rather dereases in a way
determined by the expansion rate and the equation of state of the elds. This redshifting of energy is desribed by the ontinuity equation
a_
_ + 3 ( + 3p) = 0:
a
(13.17)
Equation (13.17) an be derived from the Friedmann equations, so if these are being
solved exatly then the ontinuity equation will be obeyed as well. So a simple way
to hek energy onservation in an expanding universe is to use one of the Friedmann
equations, equations (12.2) and (12.3) to evolve the sale fator and then hek if the
other is being satised as well. In pratie the program uses a ombination of these
two equations so either one an be used as a hek of energy onservation. We use
equation (12.3), i.e.
2
a_
8
= :
(13.18)
a
3
So when expansion is being inluded in a simulation the program periodially alulates
8 2
H=
:
(13.19)
3
This quantity should remain lose to 1 throughout the run. We have generally found
that the deviation of this ratio from one is omparable to the lak of energy onservation for runs done with the same models without expansion. We have also heked
that using the seond Friedmann equation, equation (12.2), gives essentially the same
results as this method.
Finally, even assuming all the numeris are done aurately, there is an inherent
inauray in any lattie alulation related to the ultraviolet and infrared utos
imposed by the lattie. A ubi lattie in position spae will have a disrete Fourier
transform orresponding to a ubi lattie in momentum spae. The largest momentum modes will have a wavelength of the order of the lattie spaing, and the smallest
162
CHAPTER 13.
COMPUTATIONAL METHODS USED BY LATTICEEASY
momentum modes will have a wavelength of the order of the total box size.3 This
means that if there is signiant physis ourring at wavelengths outside this range
the lattie will simply not see it, no matter how small a time step you use.
There are a number of ways to minimize the eet of this problem. The rst
is to know the physis of the system you are onsidering. For all the problems we
have solved on the lattie the relevant wavelengths ould be approximately alulated analytially, and the lattie ould be used to verify these alulations. Note
that suh onsiderations may limit the range of parameter spae available for lattie
alulations. If a problem has two important wavelengths that dier by ten orders
of magnitude then a lattie simulation suh as ours an not possibly hope to inlude
them both. As long as all the most important wavelengths our within a ouple of
orders of magnitude of eah other, though, they an typially be treated aurately.
Seondly, as with variables suh as the time step, there's no substitute for a little
trial and error. As muh as possible we would vary the size of our box and lattie
spaing to make sure that we were in a realm where our results were insensitive to
suh hanges. This proedure doesn't guard against relevant eets ourring far
outside the range of the simulations, but it at least ensures that in the general area
of momentum spae being onsidered the lattie isn't imposing any limitations.
Finally, it's important to monitor the spetrum of the elds in order to see whih
modes are playing the most signiant roles. In the ase of preheating the spetrum
typially looks relatively at at long wavelengths and then has an exponential uto
at some point. Thus as long as we kept the lattie spaing small enough to inlude
modes up to the uto in momentum spae we felt ondent we weren't missing
important ultraviolet physis. 4
3 Stritly
speaking this is not true beause the lowest momentum mode is k = 0, orresponding
to a perfetly homogeneous eld. The lowest nonzero momentum, however, will orrespond to a
wavelength omparable to the box size.
4 Note that in the limit of zero lattie spaing these ultraviolet modes would make an innitely
large ontribution to the eld dynamis. For more disussion of this issue see setion 12.3 on
renormalization.
Part IV
Conlusion: My Philosophial
Ramblings
163
164
Working in osmology has one great advantage over work in many other areas of
physis. In all areas of physis the mathematial and other tools used an be ompliated and often provide a barrier between those working in the eld and others who
might be interested in the results of that work. It is diÆult for most working physiists to explain to someone outside the eld what they do, or sometimes even why the
questions they are working on are interesting and worth pursuing. In many ases this
diÆulty arises not beause the problems are unimportant but simply beause there
is a great deal of tehnial bakground required to understand them. In osmology
it is still true that the tools we use are often highly tehnial and mathematially
omplex. However the questions we are trying to answer are in many ases the same
questions that people have always wondered about.
Has the universe been here forever or did it have a beginning? Is it innite or
nite? Where did all the dierent things in it ome from? Will the universe be here
forever or someday be destroyed? Is our position in the universe unusual or typial?
All of these and many other questions like them seem so basi to human nature that
in general it requires no explanation for people to understand why we in the eld
onsider these problems worth pursuing.
I am not laiming that osmology has answered these questions, or even that it
ever will. I do believe, however, that whether or not we ever ahieve a omplete
understanding of these issues we an make progress on rening our understanding of
them. What will be the fate of the universe? Our urrent theories and observations
seem to indiate that it will expand forever. Where did the all the matter in the
universe ome from? If inationary theory is orret it ame from the deay of the
energy of a single eld. Where did that eld originally ome from? That is one of the
many questions we don't have an answer to. And if we ever have a omplete physial
theory of the universe from beginning to end we will still be left with the question of
why that theory should be true, and not some other. I don't believe siene an in
priniple address that question.
On the whole, however, I believe that siene is apable of making progress on
answering many of the basi questions that humans have wondered about throughout
the ages. I think that the last entury has seen tremendous progress in that diretion,
165
and the entury to ome will see even more than that. Ultimately, it is these questions
that have motivated me in all the work presented here.
\I want to know God's thoughts. The rest are details." - Albert Einstein
166
Bibliography
[1℄ A. H. Guth, Phys. Rev. D23, 347 (1981).
[2℄ A. D. Linde, Phys. Lett. B129, 177 (1983).
[3℄ A. Linde, Partile Physis and Inationary Cosmology (Harwood, Chur,
Switzerland, 1990).
[4℄ A. Albreht, P. J. Steinhardt, M. S. Turner, and F. Wilzek, Phys. Rev. Lett.
48, 1437 (1982).
[5℄ L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994),
hep-th/9405187.
[6℄ L. Landau and E. Lifshitz, Mehanis (Pergamon Press, Oxford, 1960).
[7℄ L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. D56, 3258 (1997),
hep-ph/9704452.
[8℄ P. B. Greene, L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. D56,
6175 (1997), hep-ph/9705347.
[9℄ G. Felder, L. Kofman, and A. Linde, Phys. Rev. D59, 123523 (1999), hepph/9812289.
[10℄ A. D. Dolgov and A. D. Linde, Phys. Lett. B116, 329 (1982).
[11℄ L. F. Abbott, E. Farhi, and M. B. Wise, Phys. Lett. B117, 29 (1982).
167
168
BIBLIOGRAPHY
[12℄ A. A. Starobinsky, (1984), In *Mosow 1981, Proeedings, Quantum Gravity*,
103-128.
[13℄ A. D. Linde, Phys. Lett. B108, 389 (1982).
[14℄ L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 76, 1011 (1996),
hep-th/9510119.
[15℄ I. I. Tkahev, Phys. Lett. B376, 35 (1996), hep-th/9510146.
[16℄ S. Khlebnikov, L. Kofman, A. Linde, and I. Tkahev, Phys. Rev. Lett. 81, 2012
(1998), hep-ph/9804425.
[17℄ I. Tkahev, S. Khlebnikov, L. Kofman, and A. Linde, Phys. Lett. B440, 262
(1998), hep-ph/9805209.
[18℄ E. W. Kolb, A. Linde, and A. Riotto, Phys. Rev. Lett. 77, 4290 (1996), hepph/9606260.
[19℄ E. W. Kolb, A. Riotto, and I. I. Tkahev, Phys. Lett. B423, 348 (1998),
hep-ph/9801306.
[20℄ S. Y. Khlebnikov and I. I. Tkahev, Phys. Lett. B390, 80 (1997), hepph/9608458.
[21℄ B. R. Greene, T. Prokope, and T. G. Roos, Phys. Rev. D56, 6484 (1997),
hep-ph/9705357.
[22℄ I. Zlatev, G. Huey, and P. J. Steinhardt, Phys. Rev. D57, 2152 (1998), astroph/9709006.
[23℄ D. J. H. Chung, (1998), hep-ph/9809489.
[24℄ L. H. Ford, Phys. Rev. D35, 2955 (1987).
[25℄ B. Spokoiny, Phys. Lett. B315, 40 (1993), gr-q/9306008.
[26℄ M. Joye, Phys. Rev. D55, 1875 (1997), hep-ph/9606223.
BIBLIOGRAPHY
169
[27℄ M. Joye and T. Prokope, Phys. Rev. D57, 6022 (1998), hep-ph/9709320.
[28℄ P. J. E. Peebles and A. Vilenkin, Phys. Rev. D59, 063505 (1999), astroph/9810509.
[29℄ R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998),
astro-ph/9708069.
[30℄ A. Berera and T. W. Kephart, Phys. Lett. B456, 135 (1999), hep-ph/9811295.
[31℄ J. Baake, K. Heitmann, and C. Patzold, Phys. Rev. D58, 125013 (1998),
hep-ph/9806205.
[32℄ P. B. Greene and L. Kofman, Phys. Lett. B448, 6 (1999), hep-ph/9807339.
[33℄ V. Berezinsky, M. Kahelriess, and A. Vilenkin, Phys. Rev. Lett. 79, 4302
(1997), astro-ph/9708217.
[34℄ V. A. Kuzmin and V. A. Rubakov, Phys. Atom. Nul. 61, 1028 (1998), astroph/9709187.
[35℄ D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D59, 023501 (1999),
hep-ph/9802238.
[36℄ V. Kuzmin and I. Tkahev, JETP Lett. 68, 271 (1998), hep-ph/9802304.
[37℄ V. Kuzmin and I. Tkahev, Phys. Rev. D59, 123006 (1999), hep-ph/9809547.
[38℄ D. J. H. Chung, E. W. Kolb, and A. Riotto, Phys. Rev. D60, 063504 (1999),
hep-ph/9809453.
[39℄ E. W. Kolb, D. J. H. Chung, and A. Riotto, (1998), hep-ph/9810361.
[40℄ K. Greisen, Phys. Rev. Lett. 16, 748 (1966).
[41℄ G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 (1966).
[42℄ C. M. Hull and P. K. Townsend, Nul. Phys. B451, 525 (1995), hep-th/9505073.
170
BIBLIOGRAPHY
[43℄ E. Witten, Nul. Phys. B443, 85 (1995), hep-th/9503124.
[44℄ K. Intriligator and N. Seiberg, Nul. Phys. Pro. Suppl. 45BC, 1 (1996), hepth/9509066.
[45℄ M. Shifman, Prog. Part. Nul. Phys. 39, 1 (1997), hep-th/9704114.
[46℄ J. Polhinski, String Theory (Cambridge University Press, Cambridge, 1998).
[47℄ G. Felder, L. Kofman, and A. Linde, Phys. Rev. D60, 103505 (1999), hepph/9903350.
[48℄ Y. B. Zeldovih and A. A. Starobinsky, Sov. Phys. JETP 34, 1159 (1972).
[49℄ L. P. Grishhuk, Sov. Phys. JETP 40, 409 (1975).
[50℄ L. P. Grishhuk, Nuovo Cim. Lett. 12, 60 (1975).
[51℄ S. G. Mamaev, V. M. Mostepanenko, and A. A. Starobinsky, Zh. Eksp. Teor.
Fiz. 70, 1577 (1976).
[52℄ A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Gen. Rel. Grav. 7, 535
(1976).
[53℄ A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Vauum Quantum Eets
in Strong Fields (Energoatonizdat, Mosow, 1988).
[54℄ L. A. Kofman and A. D. Linde, Nul. Phys. B282, 555 (1987).
[55℄ M. Dine, W. Fishler, and D. Nemeshansky, Phys. Lett. B136, 169 (1984).
[56℄ G. D. Coughlan, R. Holman, P. Ramond, and G. G. Ross, Phys. Lett. B140,
44 (1984).
[57℄ A. S. Gonharov, A. D. Linde, and M. I. Vysotsky, Phys. Lett. B147, 279
(1984).
[58℄ G. Dvali, (1995), hep-ph/9503259.
BIBLIOGRAPHY
171
[59℄ M. Dine, L. Randall, and S. Thomas, Phys. Rev. Lett. 75, 398 (1995), hepph/9503303.
[60℄ J. Ellis, G. B. Gelmini, J. L. Lopez, D. V. Nanopoulos, and S. Sarkar, Nul.
Phys. B373, 399 (1992).
[61℄ R. Kallosh, L. Kofman, A. Linde, and A. V. Proeyen, Phys. Rev. D61, 103503
(2000), hep-th/9907124.
[62℄ J. Ellis, A. D. Linde, and D. V. Nanopoulos, Phys. Lett. B118, 59 (1982).
[63℄ L. M. Krauss, Nul. Phys. B227, 556 (1983).
[64℄ D. V. Nanopoulos, K. A. Olive, and M. Sredniki, Phys. Lett. B127, 30 (1983).
[65℄ M. Y. Khlopov and A. D. Linde, Phys. Lett. B138, 265 (1984).
[66℄ J. Ellis, J. E. Kim, and D. V. Nanopoulos, Phys. Lett. B145, 181 (1984).
[67℄ M. Kawasaki and T. Moroi, Prog. Theor. Phys. 93, 879 (1995), hep-ph/9403364.
[68℄ G. Felder, L. Kofman, and A. Linde, JHEP 02, 027 (2000), hep-ph/9909508.
[69℄ C. Pathinayake and L. H. Ford, Phys. Rev. D35, 3709 (1987).
[70℄ M. Bronstein, Zh. Eksp. Teor. Fiz. 3, 73 (1933).
[71℄ M. Ozer and M. O. Taha, Nul. Phys. B287, 776 (1987).
[72℄ K. Freese, F. C. Adams, J. A. Frieman, and E. Mottola, Nul. Phys. B287,
797 (1987).
[73℄ C. Wetterih, Nul. Phys. B302, 668 (1988).
[74℄ B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988).
[75℄ J. A. Frieman, C. T. Hill, A. Stebbins, and I. Waga, Phys. Rev. Lett. 75, 2077
(1995), astro-ph/9505060.
172
[76℄ P. G. Ferreira and M. Joye,
ph/9707286.
BIBLIOGRAPHY
Phys. Rev. Lett. 79, 4740 (1997), astro-
[77℄ E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D57, 4686 (1998),
gr-q/9711068.
[78℄ D. Huterer and M. S. Turner, (1998), astro-ph/9808133.
[79℄ L. min Wang, R. R. Caldwell, J. P. Ostriker, and P. J. Steinhardt, Astrophys.
J. 530, 17 (2000), astro-ph/9901388.
[80℄ G. F. Giudie, I. Tkahev, and A. Riotto,
ph/9907510.
JHEP 08, 009 (1999), hep-
[81℄ G. D. Coughlan, W. Fishler, E. W. Kolb, S. Raby, and G. G. Ross, Phys. Lett.
B131, 59 (1983).
[82℄ D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. 75, 201 (1995), hep-ph/9502417.
[83℄ D. H. Lyth and E. D. Stewart, Phys. Rev. D53, 1784 (1996), hep-ph/9510204.
[84℄ A. Linde, Phys. Rev. D53, 4129 (1996), hep-th/9601083.
[85℄ T. Asaka and M. Kawasaki, Phys. Rev. D60, 123509 (1999), hep-ph/9905467.
[86℄ L. Randall and S. Thomas, Nul. Phys. B449, 229 (1995), hep-ph/9407248.
[87℄ G. Felder, L. Kofman, A. Linde, and I. Tkahev, JHEP 08, 010 (2000), hepph/0004024.
[88℄ D. A. Kirzhnits, JETP Lett. 15, 529 (1972).
[89℄ D. A. Kirzhnits and A. D. Linde, Phys. Lett. B42, 471 (1972).
[90℄ S. Weinberg, Phys. Rev. D9, 3357 (1974).
[91℄ L. Dolan and R. Jakiw, Phys. Rev. D9, 3320 (1974).
[92℄ D. A. Kirzhnits and A. D. Linde, Sov. Phys. JETP. 40, 628 (1975).
BIBLIOGRAPHY
173
[93℄ D. A. Kirzhnits and A. D. Linde, Ann. Phys. 101, 195 (1976).
[94℄ A. Albreht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).
[95℄ V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, Phys. Lett. B155, 36
(1985).
[96℄ M. E. Shaposhnikov, JETP Lett. 44, 465 (1986).
[97℄ A. Rajantie and E. J. Copeland, Phys. Rev. Lett. 85, 916 (2000), hepph/0003025.
[98℄ S. Kasuya and M. Kawasaki, Phys. Rev. D61, 083510 (2000), hep-ph/9903324.
[99℄ S. Y. Khlebnikov and I. I. Tkahev, Phys. Rev. Lett. 77, 219 (1996), hepph/9603378.
[100℄ S. Y. Khlebnikov and I. I. Tkahev, Phys. Rev. Lett. 79, 1607 (1997), hepph/9610477.
[101℄ E. Kolb and M. Turner, The Early Universe (Addison-Wesley, Redwood City,
CA, 1990).
[102℄ A. Linde, Nul. Phys. B372, 421 (1992), hep-th/9110037.
[103℄ S. Coleman, Phys. Rev. D15, 2929 (1977).
[104℄ A. D. Linde, Nul. Phys. B216, 421 (1983).
[105℄ Y. B. Zeldovih, I. Y. Kobsarev, and L. B. Okun, Phys. Lett. B50, 340 (1974).
[106℄ A. Linde and A. Riotto, Phys. Rev. D56, 1841 (1997), hep-ph/9703209.
[107℄ C. Contaldi, M. Hindmarsh, and J. Magueijo, Phys. Rev. Lett. 82, 2034 (1999),
astro-ph/9809053.
[108℄ R. A. Battye and J. Weller, Phys. Rev. D61, 043501 (2000), astro-ph/9810203.
[109℄ R. A. Battye, J. Magueijo, and J. Weller, (1999), astro-ph/9906093.
174
BIBLIOGRAPHY
[110℄ G. Felder and L. Kofman, (2000), hep-ph/0011160.
[111℄ T. Prokope and T. G. Roos, Phys. Rev. D55, 3768 (1997), hep-ph/9610400.
[112℄ T. Biro, S. Matinyan, and B. Muller, Chaos and Gauge Field Theory (World
Sienti, Singapore, 1994).
[113℄ V. Zakharov, V. L'vov, and G. Falkovih, Kolmogorov Spetra of Turbulene
(Springer-Verlag, 1992).
[114℄ J. Garia-Bellido, D. Y. Grigoriev, A. Kusenko, and M. Shaposhnikov, Phys.
Rev. D60, 123504 (1999), hep-ph/9902449.
[115℄ D. Polarski and A. A. Starobinsky, Class. Quant. Grav. 13, 377 (1996), grq/9504030.
[116℄ U. Heinz, C. R. Hu, S. Leupold, S. G. Matinian, and B. Muller, Phys. Rev.
D55, 2464 (1997), hep-th/9608181.
[117℄ P. B. Greene and L. Kofman, Phys. Rev. D62, 123516 (2000), hep-ph/0003018.
[118℄ G. Aarts, G. F. Bonini, and C. Wetterih, Nul. Phys. B587, 403 (2000),
hep-ph/0003262.
[119℄ D. V. Semikoz and I. I. Tkahev, Phys. Rev. D55, 489 (1997), hep-ph/9507306.
[120℄ G. Felder et al., (2000), hep-ph/0012142.
[121℄ A. A. Anselm and M. G. Ryskin, Phys. Lett. B266, 482 (1991).
[122℄ J. D. Bjorken, K. L. Kowalski, and C. C. Taylor, Presented at 7th Les Renontres de Physique de la Vallee d'Aoste: Results and Perspetives in Partile
Physis, La Thuile, Italy, 7-13 Mar 1993.
[123℄ K. Rajagopal and F. Wilzek, Nul. Phys. B399, 395 (1993), hep-ph/9210253.
[124℄ K. Rajagopal and F. Wilzek, Nul. Phys. B404, 577 (1993), hep-ph/9303281.
175
BIBLIOGRAPHY
[125℄ D. Boyanovsky, H. J. de Vega, and R. Holman, Phys. Rev. D51, 734 (1995),
hep-ph/9401308.
[126℄ A. Linde, Phys. Lett. B259, 38 (1991).
[127℄ A. Linde, Phys. Rev. D49, 748 (1994), astro-ph/9307002.
[128℄ J. Garia-Bellido and A. Linde, Phys. Rev. D57, 6075 (1998), hep-ph/9711360.
[129℄ D. Boyanovsky, M. D'Attanasio, H. J. de Vega, R. Holman, and D. S. Lee,
Phys. Rev. D52, 6805 (1995), hep-ph/9507414.
[130℄ D. Boyanovsky, H. J. de Vega, and R. Holman, (1999), hep-ph/9903534.
[131℄ E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart, and D. Wands, Phys.
Rev. D49, 6410 (1994), astro-ph/9401011.
[132℄ M. Bastero-Gil, S. F. King, and J. Sanderson, Phys. Rev. D60, 103517 (1999),
hep-ph/9904315.
[133℄ J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Astrophys. J. 304, 15
(1986).
[134℄ P. Binetruy and G. Dvali, Phys. Lett. B388, 241 (1996), hep-ph/9606342.
[135℄ E. Halyo, Phys. Lett. B387, 43 (1996), hep-ph/9606423.
[136℄ R. Kallosh, L. Kofman, A. Linde, and A. V. Proeyen, Class. Quant. Grav. 17,
4269 (2000), hep-th/0006179.
[137℄ G. F. Giudie, A. Riotto, and I. Tkahev,
ph/9911302.
JHEP 11, 036 (1999), hep-
[138℄ W. Press et al., Numerial Reipes in C: The Art of Sienti Computing,
Seond Edition (Cambridge University Press, Melbourne, Australia, 1992).
[139℄ M. E. Peskin and D. V. Shroeder, An Introdution to Quantum Field Theory
(Addison Wesley, Reading, Massahusetts, 1995).