Table Of ContentPreface
A seminal paper, dated 1981, marked the birth of what was to become the
mostsuccessfulparadigminmoderncosmologyfollowingthatofthebigbang
itself: inflation. Its 25th birthday offered a welcome opportunity to celebrate
the phenomenologicalsuccessofinflationandto gather the worldleading sci-
entists engaged in forefront research in this field. Such was the objective of
the XXII IAP colloquium, which took place at “Institut d’Astrophysique de
Paris” (IAP) in June 2006. During this meeting and the immediately follow-
ing two-week workshop, scientists from the world over and from both obser-
vational and theoretical communities gathered to discuss the present status,
the achievements and the shortcomings as well as the future of the theory
of inflation. The numerous discussions that took place offered solid ground
for the publication of regular proceedings. However, inflationary cosmology
encompasses different disciplines of physics, from high energy physics to ob-
servational astrophysics, and it has also become a field of researchin its own
right. Therefore it was felt that a more pedagogical text, containing exhaus-
tive discussions of the ins and outs of inflation, would be more useful. This is
precisely what this present volume of the Lecture Notes in Physics series is
aiming at.
As is by now well known, cosmic inflation corresponds to an episode of
accelerated expansion in the very early Universe which solves the handful of
puzzles that plague the standard hot big bang cosmology, namely the flat-
ness, horizon, monopole excess problems, and, in some models, the problem
oftheprimordialsingularity.Theseachievementsevencomewithabonus:the
production of density perturbations to the level needed to explain the origin
of large scale structure of the Universe. The first chapter of this volume, by
A.Linde,introducesthisframework,offersahistoricaloverviewofthissubject
and develops the present status of the theory. This is followed L. Kofmann’s
discussion on preheating which describes how matter and radiation can have
been producedduring this period which smoothly connects inflation with the
standard big bang phase.
VI Preface
Any cosmological model needs to be implemented in a particle physics
context. The contribution of D. Lyth shows how this can be done in the
most reasonable extensions of the standard particle physics model, namely
those based on supersymmetry. This chapter is followed by the discussion of
R. Kallosh on the embedding of inflation in string theories.
As of today, there are various ways of implementing inflation. One such
framework is “eternal inflation”, in which different parts of the Universe un-
dergo an episode of inflation at different times, the Universe being eternally
inflating and self-reproducing. This particular scenario is discussed in length
by S. Winitzki
As shownby J.Martinin asubsequentchapter,the productionof density
perturbations during inflation is akin to the production of charged particles
out of the vacuum in a strong electric field. This analogy is developed in full
detailinordertoexplainthe inflationaryoriginofprimordialdensity fluctua-
tions.Thenumericalimplementationofthecalculationoftheseperturbations,
whichisrequiredinordertocomparetheseresultstothehighaccuracydataof
cosmic microwavebackground fluctuations, is then discussed by C. Ringeval.
Thenextchapter,byD.Wands,discussesthemodelscontainingmorethan
onescalarfield,inparticulartheirdynamicsandtheobservationalpredictions;
the curvatonmodelis herereviewedasanalternativeto the pure inflationary
productionof perturbations. Then, A. Riotto shows that the measurementof
non-Gaussianities in the spectrum of inflationary perturbations could offer a
way of discrimating the different models.
Finally the possibility of finding alternative scenariosto inflation is a ma-
jor but unanswered issue. The old contender, in which topological defects
seedthe primordialdensity fluctuations has been shownto disagreewith cos-
mic microwave background data. However, as M. Sakellariadou argues, such
topological defects might still be present in our Universe as they should be
producedinconvincingmodelsofinflation.Theircontributiontotheobserved
fluctuations might open a window on physics of an otherwise inaccessible en-
ergy scale. R. Brandenberger concludes this volume by presenting a radically
different perspective in which string gas cosmology plays the main role and
by pointing out some shortcomings of inflation which may argue for the need
of a broader conceptual framework.
Paris, Martin Lemoine, J´eroˆme Martin & Patrick Peter.
April 2007
Contents
1 Inflationary Cosmology
Andrei Linde .................................................... 1
1.1 Brief History of Inflation ..................................... 1
1.2 Chaotic Inflation............................................ 3
1.3 Hybrid Inflation............................................. 9
1.4 Quantum Fluctuations and Density Perturbations .............. 10
1.5 Creation of Matter After Inflation: Reheating and Preheating..... 13
1.6 Eternal Inflation ............................................ 15
1.7 Inflation and Observations ................................... 18
1.8 Alternatives to Inflation?..................................... 21
1.9 Naturalness of Chaotic Inflation............................... 26
1.10 Chaotic Inflation in Supergravity.............................. 28
1.11 Towards Inflation in String Theory ............................ 30
1.12 Scale of Inflation, the Gravitino Mass, and the Amplitude
of the GravitationalWaves ................................... 35
1.13 Initial Conditions for the Low-Scale Inflation and Topology
of the Universe ............................................. 38
1.14 Inflationary Multiverse, String Theory Landscape
and the Anthropic Principle .................................. 40
1.15 Conclusions ................................................ 46
References ...................................................... 47
2 Preheating After Inflation
Lev Kofman..................................................... 55
2.1 Generalities: Reheating the Universe........................... 55
2.2 Pair Creation by an Electric Field............................. 58
2.3 Linear Resonant Preheating .................................. 59
2.4 Non-linear Dynamics of Resonant Preheating ................... 61
2.5 Inflaton Fragmentation ...................................... 64
2.6 Equation of State During Preheating .......................... 68
2.7 Effects of Trilinear Interactions ............................... 69
VIII Contents
2.8 Modulated Fluctuations from Preheating....................... 71
2.9 Reheating After String Theory Inflation........................ 73
2.10 Gravitational Waves from Preheating .......................... 73
2.11 Looking Toward the Future................................... 74
References ...................................................... 78
3 Particle Physics Models of Inflation
David H. Lyth ................................................... 81
3.1 Introduction................................................ 81
3.2 Beyond the Standard Model .................................. 82
3.3 The Initial Condition for Observable Inflation................... 83
3.4 Slow-roll inflation ........................................... 85
3.5 Modular Inflation ........................................... 93
3.6 Small-Field Models.......................................... 95
3.7 Supersymmetry: General Features ............................. 97
3.8 Supersymmetry: Form of the Potential......................... 99
3.9 One-Loop Correction ........................................100
3.10 Small-Field Models: Moving Away from the Origin ..............102
3.11 Moving Toward the Origin; Power-LawPotential................103
3.12 F and D Term Inflation......................................105
3.13 Tree-Level Hybrid Inflation...................................107
3.14 Running Mass Models .......................................109
3.15 Large-Field Models..........................................111
3.16 Warm Inflation .............................................112
3.17 Present Status and Outlook ..................................113
References ......................................................115
4 Inflation in String Theory
Renata Kallosh ..................................................119
4.1 Introduction................................................119
4.2 Cosmology and Particle Physics Phenomenology ................122
4.3 String Theory Inspired Supergravity Models and Cosmology......124
4.4 Brane Inflation in String Theory .............................131
4.5 Modular Inflation in String Theory ............................136
4.6 N-flation/Assisted Inflation ..................................147
4.7 Discussion..................................................150
References ......................................................153
5 Predictions in Eternal Inflation
Sergei Winitzki ..................................................157
5.1 Eternal Inflation ............................................157
5.2 Stochastic Approach to Inflation ..............................166
5.3 Predictions and Measure Issues ...............................177
References ......................................................187
Contents IX
6 Inflationary Perturbations: The Cosmological
Schwinger Effect
J´eroˆme Martin ..................................................193
6.1 Introduction................................................193
6.2 The Schwinger Effect ........................................195
6.3 Quantization of a Free Scalar Field in Curved Space–Time .......204
6.4 Inflationary Cosmological Perturbations of Quantum-Mechanical
Origin .....................................................219
6.5 The Classical Limit of Quantum Perturbations..................229
6.6 Conclusions ................................................237
References ......................................................239
7 The Numerical Treatment of Inflationary Models
Christophe Ringeval ..............................................243
7.1 Motivations ................................................243
7.2 Multifield Inflation ..........................................247
7.3 Numerical Method ..........................................253
7.4 Application to CMB Data Analysis............................262
7.5 Conclusion .................................................269
References ......................................................271
8 Multiple Field Inflation
David Wands....................................................275
8.1 Introduction................................................275
8.2 Homogeneous Scalar Field Dynamics ..........................276
8.3 Primordial Perturbations from Inflation........................280
8.4 Perturbations from Two-Field Inflation ........................287
8.5 Non-Gaussianity ............................................293
8.6 Curvaton Scenario...........................................296
8.7 Conclusions ................................................301
References ......................................................302
9 The Quest for Non-gaussianity
Antonio Riotto ..................................................305
9.1 Introduction:WhyIsitsoInterestingtoMeasureNon-gaussianity
in CosmologicalPerturbations? ...............................305
9.2 What Is Non-gaussianity? ....................................309
9.3 Linear Perturbations on Large Scales ..........................316
9.4 Non-linear Perturbation on Large Scales: Generation of NG ......318
9.5 What Do We Learn from What We Have Done so Far? ..........323
9.6 Observational Constraints on NG in the CMB ..................324
9.7 Towards the Second-Order Transfer Function ...................355
References ......................................................358
X Contents
10 Production of Topological Defects at the End of Inflation
Mairi Sakellariadou ..............................................359
10.1 Introduction................................................359
10.2 CosmologicalInflation .......................................360
10.3 Topological Defects in GUTs .................................367
10.4 BraneworldCosmology ......................................374
10.5 Observational Consequences ..................................377
10.6 Conclusions ................................................389
References ......................................................390
11 ConceptualProblemsofInflationary CosmologyandaNew
Approach to Cosmological Structure Formation
Robert H. Brandenberger..........................................393
11.1 Introduction................................................393
11.2 Problems of Scalar Field-Driven Inflation.......................394
11.3 String Gas Cosmology .......................................400
11.4 String Gas Cosmology and Structure Formation.................408
11.5 Discussion..................................................418
References ......................................................419
Index..........................................................425
1
Inflationary Cosmology
Andrei Linde
Department of Physics, Stanford University,Stanford, CA 94305, USA
[email protected]
Abstract. This chapter presents a general review of the history of inflationary
cosmology and of its present status.1
1.1 Brief History of Inflation
Since the inflationary theory is more than 25 years old, perhaps it is not
inappropriate to start this chapter with a brief history of its development,
and some personal recollections.
Severalingredientsofinflationarycosmologywerediscoveredinthebegin-
ning ofthe 1970s.The firstrealizationwasthatthe energydensityofa scalar
field plays the role of the vacuum energy/cosmological constant [1], which
was changing during the cosmological phase transitions [2]. In certain cases
these changes occur discontinuously,due to first-orderphase transitions from
a supercooled vacuum state (false vacuum) [3].
In 1978, we with Gennady Chibisov tried to use these facts to construct
a cosmological model involving exponential expansion of the universe in the
supercooled vacuum as a source of the entropy of the universe, but we im-
mediately realized that the universe becomes very inhomogeneous after the
bubble wall collisions. I mentioned our work in my review article [4], but did
not pursue this idea any further.
The first semi-realistic model of inflationary type was proposed by Alexei
Starobinsky in 1979–1980 [5]. It was based on the investigation of a confor-
mal anomaly in quantum gravity. His model was rather complicated, and its
goalwassomewhatdifferentfromthe goalsofinflationarycosmology.Instead
of attempting to solve the homogeneity and isotropy problems, Starobinsky
considered the model of the universe which was homogeneous and isotropic
from the very beginning, and emphasized that his scenario was “the extreme
opposite of Misner’s initial ‘chaos’.”
1 Based on a talk given at the 22nd IAP Colloquium, “Inflation+25”, Paris, June
2006.
A.Linde:Inflationary Cosmology,Lect.NotesPhys.738,1–54(2008)
DOI10.1007/978-3-540-74353-8 1 (cid:2)c Springer-VerlagBerlinHeidelberg2008
2 A. Linde
Ontheotherhand,theStarobinskymodeldidnotsufferfromthegraceful
exit problem, and it was the first model to predict gravitational waves with
a flat spectrum [5]. The first mechanism of production of adiabatic per-
turbations of the metric with a flat spectrum, which are responsible for
galaxy production, and which were found by the observations of the CMB
anisotropy,wasproposedbyMukhanovandChibisov[6]inthecontextofthis
model.
A much simpler inflationary model with a very clear physical motivation
was proposed by Alan Guth in 1981 [7]. His model, which is now called “old
inflation,” was based on the theory of supercooling during the cosmological
phase transitions [3]. Even though this scenario did not work, it played a
profoundrolein the developmentofinflationarycosmologysince it contained
a very clear explanation of how inflation may solve the major cosmological
problems.
According to this scenario, inflation is described by the exponential ex-
pansion of the universe in a supercooled false vacuum state. False vacuum is
ametastablestatewithoutanyfieldsorparticlesbutwithalargeenergyden-
sity. Imagine a universe filled with such “heavy nothing.” When the universe
expands, empty space remains empty, so its energy density does not change.
The universe with a constant energy density expands exponentially, thus we
have inflation in the false vacuum. This expansion makes the universe very
bigandveryflat.Thenthefalsevacuumdecays,thebubblesofthenewphase
collide, and our universe becomes hot.
Unfortunately,thissimpleandintuitivepictureofinflationinthefalsevac-
uum state is somewhatmisleading.If the probabilityof the bubble formation
is large,bubbles ofthe new phase are formed near eachother, inflation is too
shortto solveany problems, andthe bubble wallcollisions make the universe
extremelyinhomogeneous.Iftheyareformedfarawayfromeachother,which
is the case if the probability of their formation is small and inflation is long,
each of these bubbles represents a separate open universe with a vanishingly
smallΩ.Bothoptionsareunacceptable,whichhasleadtotheconclusionthat
this scenario does not work and cannot be improved (graceful exit problem)
[7, 8, 9].
The solution was found in 1981–1982with the invention of the new infla-
tionary theory [10], see also [11]. In this theory, inflation may begin either in
the false vacuum, or in an unstable state at the top of the effective potential.
Then the inflaton field φ slowly rolls down to the minimum of its effective
potential. The motion of the field away from the false vacuum is of crucial
importance:density perturbations produced during the slow-rollinflationare
inversely proportional to φ˙ [6, 12, 13]. Thus the key difference between the
new inflationary scenario and the old one is that the useful part of inflation
inthe new scenario,whichis responsiblefor the homogeneityofouruniverse,
does not occur in the false vacuum state, where φ˙ =0.
Soon after the invention of the new inflationary scenario it became so
popular that even now most of the textbooks on astrophysics incorrectly
1 Inflationary Cosmology 3
describe inflation as an exponential expansion in a supercooled false vacuum
state during the cosmological phase transitions in grand unified theories.
Unfortunately,thisscenariowasplaguedbyitsownproblems.Itworksonlyif
the effective potential of the field φ has a very flat plateau near φ=0, which
is somewhat artificial. In most versions of this scenario the inflaton field has
an extremely small coupling constant, so it could not be in thermal equilib-
rium with other matter fields. The theory of cosmological phase transitions,
whichwasthebasisforoldandnewinflation,didnotworkinsuchasituation.
Moreover, thermal equilibrium requires many particles interacting with each
other. This means that new inflation could explain why our universe was so
large only if it was very large and contained many particles from the very
beginning [14].
Old and new inflation represented a substantial but incomplete modifica-
tion of the big bang theory. It was still assumed that the universe was in a
state of thermal equilibrium from the very beginning, that it was relatively
homogeneousandlargeenoughtosurviveuntilthebeginningofinflation,and
that the stage of inflation was just an intermediate stage of the evolution of
the universe. In the beginning of the 1980s these assumptions seemed most
naturalandpracticallyunavoidable.Onthebasisofallavailableobservations
(CMB,abundanceoflightelements)everybodybelievedthattheuniversewas
createdinahotbigbang.Thatiswhyitwassodifficulttoovercomeacertain
psychologicalbarrierand abandonall of these assumptions. This wasdone in
1983 with the invention of the chaotic inflation scenario [15]. This scenario
resolved all problems of old and new inflation. According to this scenario,
inflationmaybeginevenif therewas nothermalequilibriumin the earlyuni-
verse, and it may occur even in the theories with simplest potentials such as
V(φ) ∼ φ2. But it is not limited to the theories with polynomial potentials:
chaotic inflation occurs in any theory where the potential has a sufficiently
flat region, which allows the existence of the slow-roll regime [15].
1.2 Chaotic Inflation
1.2.1 Basic Model
Consider the simplest model of a scalar field φ with a mass m and with the
potential energy density V(φ) = m2φ2. Since this function has a minimum
2
at φ = 0, one may expect that the scalar field φ should oscillate near this
minimum. This is indeed the case if the universe does not expand, in which
casetheequationofmotionforthescalarfieldcoincideswiththeequationfor
the harmonic oscillator, φ¨=−m2φ.
However, because of the expansion of the universe with Hubble constant
H =a˙/a,anadditionalterm3Hφ˙ appearsintheharmonicoscillatorequation:
φ¨+3Hφ˙ =−m2φ. (1.1)
4 A. Linde
The term 3Hφ˙ can be interpreted as a friction term. The Einstein equation
for a homogeneous universe containing a scalar field φ looks as follows:
k 1
H2+ = (φ˙2+m2φ2). (1.2)
a2 6
Here k =−1,0,1for anopen, flat or closeduniverse respectively.We workin
units M−2 =8πG=1.
pl
If the scalar field φ initially was large, the Hubble parameter H was large
too,accordingtothesecondequation.Thismeansthatthefrictionterm3Hφ˙
wasverylarge,andthereforethescalarfieldwasmovingveryslowly,asaball
in a viscous liquid. Therefore at this stage the energy density of the scalar
field, unlike the density of ordinary matter, remained almost constant, and
the expansion of the universe continued at a much greater speed than in the
old cosmological theory. Due to the rapid growth of the scale of the universe
and the slow motion of the field φ, soon after the beginning of this regime
one has φ¨(cid:3)3Hφ˙, H2 (cid:4) k , φ˙2 (cid:3)m2φ2, so the system of equations can be
a2
simplified: (cid:2)
a˙ mφ 2
H = = √ , φ˙ =−m . (1.3)
a 6 3
The first equation shows that if the field φ changes slowly, the size of the
universe in this regime grows approximately as eHt, where H = m√φ. This
6
is the stage of inflation, which ends when the field φ becomes much smaller
than M = 1. The solution to these equations shows that after a long stage
Pl
of inflation the universe initially filled with the field φ (cid:4) 1 grows exponen-
tially [14],
a=a eφ2/4 . (1.4)
0
Thus, inflation does not require an initial state of thermal equilibrium,
supercooling and tunneling from the false vacuum. It appears in the theories
that can be as simple as a theory of a harmonic oscillator [15]. Only when it
was realized, it became clear that inflation is not just a trick necessary to fix
problems of the old big bang theory, but a generic cosmologicalregime.
1.2.2 Initial Conditions
But what is about the initial conditions requiredfor chaotic inflation? Let us
consider first a closed universe of initial size l ∼ 1 (in Planck units), which
emerges from the space–time foam, or from singularity, or from “nothing” in
astatewiththe Planckdensityρ∼1.Onlystartingfromthismoment,i.e.at
ρ(cid:2)1,canwedescribethisdomainasaclassicaluniverse.Thus,atthisinitial
moment the sum of the kinetic energy density, gradient energy density, and
the potential energy density is of the order unity: 1φ˙2+ 1(∂ φ)2+V(φ)∼1
2 2 i
(Fig. 1.1).
We wish to emphasize, that there are no a priori constraints on the
initial value of the scalar field in this domain, except for the constraint
