Table Of Content

Preface 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 scientists 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 observational 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 observational 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 fluctuations.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 major but unanswered issue. The old contender, in which topological defects seedthe primordialdensity fluctuations has been shownto disagreewith cosmic 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 energy 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, Jéroˆme 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 Jéroˆme 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 immediately 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 perturbations 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 expansion 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 inflationary 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 equilibrium 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 exponentially [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