PARTICLE PHYSICS AND INFLATIONARY COSMOLOGY1 Andrei Linde Department of Physics, Stanford University, Stanford CA 94305-4060, USA 1 This is the LaTeX version of my book “Particle Physics and Inflationary Cosmology”(Harwood, Chur, Switzerland, 1990). Contents Preface to the Series ix Introduction x CHAPTER 1 Overview of Unified Theories of Elementary Particles and the Infla- tionary Universe Scenario 1 1.1 The scalar field and spontaneous symmetry breaking 1 1.2 Phase transitions in gauge theories 6 1.3 Hot universe theory 9 1.4 Some properties of the Friedmann models 13 1.5 Problems of the standard scenario 16 1.6 A sketch of the development of the inflationary universe sce- nario 25 1.7 The chaotic inflation scenario 29 1.8 The self-regenerating universe 42 1.9 Summary 49 CHAPTER 2 Scalar Field, Effective Potential, and Spontaneous Symmetry Break- ing 50 2.1 Classical and quantum scalar fields 50 2.2 Quantum corrections to the effective potential V(ϕ) 53 2.3 The 1/N expansion and the effective potential in the λϕ4/N theory 59 2.4 The effective potential and quantum gravitational effects 64 CHAPTER 3 Restoration of Symmetry at High Temperature 67 3.1 Phase transitions in the simplest models with spontaneous symmetry breaking 67 3.2 Phasetransitionsinrealistictheoriesoftheweak, strong, and electromagnetic interactions 72 3.3 Higher-order perturbation theory and the infrared problem in the thermodynamics of gauge fields 74 CHAPTER 4 Phase Transitions in Cold Superdense Matter 78 4.1 Restoration of symmetry in theories with no neutral currents 78 CONTENTS vii 4.2 Enhancement of symmetry breaking and the condensation of vector mesons in theories with neutral currents 79 CHAPTER 5 Tunneling Theory and the Decay of a Metastable Phase in a First- Order Phase Transition 82 5.1 General theory of the formation of bubbles of a new phase 82 5.2 The thin-wall approximation 86 5.3 Beyond the thin-wall approximation 90 CHAPTER 6 Phase Transitions in a Hot Universe 94 6.1 Phasetransitionswithsymmetry breakingbetween theweak, strong, and electromagnetic interactions 94 6.2 Domain walls, strings, and monopoles 99 CHAPTER 7 General Principles of Inflationary Cosmology 108 7.1 Introduction 108 7.2 The inflationary universe and de Sitter space 109 7.3 Quantum fluctuations in the inflationary universe 113 7.4 Tunneling in the inflationary universe 120 7.5 Quantumfluctuationsandthegenerationofadiabaticdensity perturbations 126 7.6 Are scale-free adiabatic perturbations sufficient to produce the observed large scale structure of the universe? 136 7.7 Isothermal perturbations and adiabatic perturbations with a nonflat spectrum 139 7.8 Nonperturbative effects: strings, hedgehogs, walls, bubbles, ... 145 7.9 Reheating of the universe after inflation 150 7.10 The origin of the baryon asymmetry of the universe 154 CHAPTER 8 The New Inflationary Universe Scenario 160 8.1 Introduction. The old inflationary universe scenario 160 8.2 The Coleman–Weinberg SU(5) theory and the new inflationary universe scenario (initial simplified version) 162 8.3 Refinement of the new inflationary universe scenario 165 8.4 Primordial inflation in N = 1 supergravity 170 8.5 The Shafi–Vilenkin model 171 8.6 Thenewinflationaryuniversescenario: problemsandprospects176 CHAPTER 9 The Chaotic Inflation Scenario 179 9.1 Introduction. Basic features of the scenario. The question of initial conditions 179 CONTENTS viii 9.2 The simplest model based on the SU(5) theory 182 9.3 Chaotic inflation in supergravity 184 9.4 The modified Starobinsky model and the combined scenario 186 9.5 Inflation in Kaluza–Klein and superstring theories 189 CHAPTER 10 Inflation and Quantum Cosmology 195 10.1 The wave function of the universe 195 10.2 Quantum cosmology and the global structure of the inflationary universe 207 10.3 The self-regenerating inflationary universe and quantum cos- mology 213 10.4 The global structure of the inflationary universe and the problem of the general cosmological singularity 221 10.5 Inflation and the Anthropic Principle 223 10.6 Quantum cosmology and the signature of space-time 232 10.7 Thecosmologicalconstant,theAnthropicPrinciple, andredu- plication of the universe and life after inflation 234 CONCLUSION 243 REFERENCES 245 Preface to the Series The series of volumes, Contemporary Concepts in Physics, is addressed to the professional physicist and to the serious graduate student of physics. The subjects to be covered will includethoseattheforefrontofcurrent research. Itisanticipatedthatthevariousvolumes in the series will be rigorous and complete in their treatment, supplying the intellectual tools necessary for the appreciation of the present status of the areas under consideration and providing the framework upon which future developments may be based. Introduction With the invention and development of unified gauge theories of weak and electromag- netic interactions, a genuine revolution has taken place in elementary particle physics in the last 15 years. One of the basic underlying ideas of these theories is that of sponta- neous symmetry breaking between different types of interactions due to the appearance of constant classical scalar fields ϕ over all space (the so-called Higgs fields). Prior to the appearance of these fields, there is no fundamental difference between strong, weak, and electromagnetic interactions. Their spontaneous appearance over all space essentially signifies a restructuring of the vacuum, with certain vector (gauge) fields acquiring high mass as a result. The interactions mediated by these vector fields then become short- range, and this leads to symmetry breaking between the various interactions described by the unified theories. The first consistent description of strong and weak interactions was obtained within the scope of gauge theories with spontaneous symmetry breaking. For the first time, it became possible to investigate strong and weak interaction processes using high-order perturbation theory. A remarkable property of these theories — asymptotic freedom — also made it possible in principle to describe interactions of elementary particles up to center-of-mass energies E M 1019 GeV, that is, up to the Planck energy, where P ∼ ∼ quantum gravity effects become important. Here we will recount only the main stages in the development of gauge theories, rather than discussing their properties in detail. In the 1960s, Glashow, Weinberg, and Salam proposed a unified theory of the weak and electromagnetic interactions [1], and real progress was made in this area in 1971–1973after the theories were shown to be renormal- izable [2]. It was proved in 1973 that many such theories, with quantum chromodynamics inparticularserving asadescription ofstrong interactions, possess thepropertyofasymp- totic freedom (a decrease in the coupling constant with increasing energy [3]). The first unified gauge theories of strong, weak, and electromagnetic interactions with a simple symmetry group, the so-called grand unified theories [4], were proposed in 1974. The first theories to unify all of the fundamental interactions, including gravitation, were proposed in 1976 within the context of supergravity theory. This was followed by the development of Kaluza–Klein theories, which maintain that our four-dimensional space-time results from the spontaneous compactification of a higher-dimensional space [6]. Finally, our most recent hopes for a unified theory of all interactions have been invested in super- string theory [7]. Modern theories of elementary particles are covered in a number of xi excellent reviews and monographs (see [8–17], for example). The rapid development of elementary particle theory has not only led to great ad- vances in our understanding of particle interactions at superhigh energies, but also (as a consequence) to significant progress in the theory of superdense matter. Only fifteen years ago, in fact, the term superdense matter meant matter with a density somewhat higher than nuclear values, ρ 1014–1015 g cm 3 and it was virtually impossible to − ∼ · conceive of how one might describe matter with ρ 1015 g cm 3. The main problems − ≫ · involved strong-interaction theory, whose typical coupling constants at ρ > 1015 g cm 3 − · were large, making standard perturbation-theory predictions of the pro∼perties of such matter unreliable. Because of asymptotic freedom in quantum chromodynamics, how- ever, the corresponding coupling constants decrease with increasing temperature (and density). This enables one to describe the behavior of matter at temperatures approach- ing T M 1019 GeV, which corresponds to a density ρ M4 1094 g cm 3. ∼ P ∼ P ∼ P ∼ · − Present-day elementary particle theories thus make it possible, in principle, to describe the properties of matter more than 80 orders of magnitude denser than nuclear matter! The study of the properties of superdense matter described by unified gauge theories began in 1972 with the work of Kirzhnits [18], who showed that the classical scalar field ϕ responsibleforsymmetry breaking shoulddisappear atahighenoughtemperatureT.This means that a phase transition (or a series of phase transitions) occurs at a sufficiently high temperature T > T , after which symmetry is restored between various types of c interactions. When this happens, elementary particle properties and the laws governing their interaction change significantly. This conclusion was confirmed in many subsequent publications [19–24]. It was found that similar phase transitions could also occur when the density of cold matter was raised [25–29], and in the presence of external fields and currents [22, 23, 30, 33]. For brevity, andtoconformwith current terminology, we willhereafter refer tosuch processes asphase transitions in gauge theories. Such phase transitions typically take place at exceedingly high temperatures and densities. The critical temperature for a phase transition in the Glashow–Weinberg– Salam theory of weak and electromagnetic interactions [1], for example, is of the order of 102 GeV 1015 K. The temperature at which symmetry is restored between the strong ∼ and electroweak interactions in grand unified theories is even higher, T 1015 GeV c ∼ ∼ 1028 K. For comparison, the highest temperature attained in a supernova explosion is about 1011 K. It is therefore impossible to study such phase transitions in a laboratory. However, the appropriate extreme conditions could exist at the earliest stages of the evolution of the universe. According to the standard version of the hot universe theory, the universe could have expanded from a state in which its temperature was at least T 1019 GeV [34, 35], ∼ cooling all the while. This means that in its earliest stages, the symmetry between the strong, weak, and electromagnetic interactions should have been intact. In cooling, the universe would have gone through a number of phase transitions, breaking the symmetry between the different interactions [18–24]. This result comprised the first evidence for the importance of unified theories of ele- xii mentary particles and the theory of superdense matter for the development of the theory of the evolution of the universe. Cosmologists became particularly interested in recent theories of elementary particles after it was found that grand unified theories provide a natural framework within which the observed baryon asymmetry of the universe (that is, the lack of antimatter in the observable part of the universe) might arise [36–38]. Cos- mology has likewise turned out to be an important source of information for elementary particle theory. The recent rapid development of the latter has resulted in a somewhat unusual situation in that branch of theoretical physics. The reason is that typical el- ementary particle energies required for a direct test of grand unified theories are of the order of1015 GeV,anddirect tests ofsupergravity, Kaluza–Kleintheories, andsuperstring theory require energies of the order of 1019 GeV. On the other hand, currently planned accelerators will only produce particle beams with energies of about 104 GeV. Experts estimate that the largest accelerator that could be built on earth (which has a radius of about 6000 km) would enable us to study particle interactions at energies of the order of 107 GeV, which is typically the highest (center-of-mass) energy encountered in cosmic ray experiments. Yet this is twelve orders of magnitude lower than the Planck energy E M 1019 GeV. P P ∼ ∼ The difficulties involved in studying interactions at superhigh energies can be high- lighted by noting that 1015 GeV is the kinetic energy of a small car, and 1019 GeV is the kinetic energy of a medium-sized airplane. Estimates indicate that accelerating par- ticles to energies of the order of 1015 GeV using present-day technology would require an accelerator approximately one light-year long. It would be wrong to think, though, that the elementary particle theories currently being developed are totally without experimental foundation — witness the experiments on a huge scale which are under way to detect the decay of the proton, as predicted by grand unified theories. It is also possible that accelerators will enable us to detect some of the lighter particles (with mass m 102–103 GeV) predicted by certain versions of ∼ supergravity and superstring theories. Obtaining information solely in this way, however, would be similar to trying to discover a unified theory of weak and electromagnetic inter- actions using only radio telescopes, detecting radio waves with an energy E no greater γ E E than 10 5 eV (note that P W, where E 102 GeV is the characteristic energy in − W E ∼ E ∼ W γ the unified theory of weak and electromagnetic interactions). The only laboratory in which particles with energies of 1015–1019 GeV could ever exist and interact with one another is our own universe in the earliest stages of its evolution. At the beginning of the 1970s, Zeldovich wrote that the universe is the poor man’s accelerator: experiments don’t need to be funded, and all we have to do is collect the experimental data and interpret them properly [39]. More recently, it has become quite clear that the universe is the only accelerator that could ever produce particles at energies high enough to test unified theories of all fundamental interactions directly, and in that sense it is not just the poor man’s accelerator but the richest man’s as well. These days, most new elementary particle theories must first take a “cosmological validity” test — and only a very few pass. xiii Itmight seem atfirst glancethatitwouldbedifficult togleananyreasonably definitive or reliable information from an experiment performed more than ten billion years ago, but recent studies indicate just the opposite. It has been found, for instance, that phase transitions, which should occur in a hot universe in accordance with the grand unified theories, should produceanabundance ofmagnetic monopoles, thedensity of which ought to exceed the observed density of matter at the present time, ρ 10 29 g cm 3, by − − ∼ · approximately fifteen orders of magnitude [40]. At first, it seemed that uncertainties inherent in both the hot universe theory and the grand unified theories, being very large, would provide an easy way out of the primordial monopole problem. But many attempts to resolve this problem within the context of the standard hot universe theory have not led to final success. A similar situation has arisen in dealing with theories involving spontaneous breaking of a discrete symmetry (spontaneous CP-invariance breaking, for example). In such models, phase transitions ought to give rise to supermassive domain walls, whose existence would sharply conflict with the astrophysical data [41–43]. Going to more complicated theories such as N = 1 supergravity has engendered new problems rather thanresolving theoldones. Thus ithas turnedout inmost theories based onN = 1 supergravity that the decay ofgravitinos (spin = 3/2superpartners of the graviton) which existed in the early stages of the universe leads to results differing from the observational data by about ten orders of magnitude [44, 45]. These theories also predict the existence of so-called scalar Polonyi fields [15, 46]. The energy density that would have been accumulated in these fields by now differs from the cosmological data by fifteen orders of magnitude [47, 48]. A number of axion theories [49] share this difficulty, particularly in the simplest models based on superstring theory [50]. Most Kaluza–Klein theories based on supergravity in an 11-dimensional space lead to vacuum energies of order M4 − P ∼ 1094 g cm 3 [16], which differs from the cosmological data by approximately 125 orders − − · of magnitude... This list could be continued, but as it stands it suffices to illustrate why elementary particle theorists now find cosmology so interesting and important. An even more gen- eral reason is that no real unification of all interactions including gravitation is possible without an analysis of the most important manifestation of that unification, namely the existence of the universe itself. This is illustrated especially clearly by Kaluza–Klein and superstring theories, where one must simultaneously investigate the properties of the space-time formed by compactification of “extra” dimensions, and the phenomenology of the elementary particles. It hasnotyet beenpossibletoovercome someoftheproblems listedabove. This places important constraints on elementary particle theories currently under development. It is all the more surprising, then, that many of these problems, together with a number of others that predate the hot universe theory, have been resolved in the context of one fairly simple scenario for the development of the universe — the so-called inflationary universe scenario [51–57]. According to this scenario, the universe, at some very early stage of its evolution, was in an unstable vacuum-like state and expanded exponentially (the stage of inflation). The vacuum-like state then decayed, the universe heated up, and its subsequent evolution can be described by the usual hot universe theory. xiv Since its conception, the inflationary universe scenario has progressed from something akin to science fiction to a well-established theory of the evo- lution of the universe accepted by most cosmologists. Of course this doesn’t mean that we have now finally achieved total enlightenment as to the physical processes operative in the early universe. The incompleteness of the current picture is reflected by the very word scenario, which is not normally found in the working vocabulary of a theoretical physicist. In its present form, this scenario only vaguely resembles the simple models from which it sprang. Many details of the inflationary universe scenario are changing, tracking rapidly changing (as noted above) elementary particle theories. Nevertheless, the basic aspects of this scenario are now well-developed, and it should be possible to provide a preliminary account of its progress. Most of the present book is given over to discussion of inflationary cosmology. This is preceded by an outline of the general theory of spontaneous symmetry breaking and a discussion of phase transitions in superdense matter, as described by present-day theories of elementary particles. The choice of material has been dictated by both the author’s interests and his desire to make the contents useful both to quantum field theorists and astrophysicists. We have therefore tried to concentrate on those problems that yield an understanding of thebasic aspects ofthe theory, referringthereader tothe originalpapers for further details. In order to make this book as widely accessible as possible, the main exposition has been preceded by a long introductory chapter, written at a relatively elementary level. Ourhopeisthatbyusing thischapter asaguidetothebook, andthebookitselfasaguide to the original literature, the reader will gradually be able to attain a fairly complete and accurate understanding of the present status of this branch of science. In this regard, he mightalsobeassistedbyanacquaintancewiththebooksCosmologyof the Early Universe, by A. D. Dolgov, Ya. B. Zeldovich, and M. V. Sazhin; How the Universe Exploded, by I. D. Novikov; A Brief History of Time: From the Big Bang to Black Holes, by S. W. Hawking; and An Introduction to Cosmology and Particle Physics, by R. Dominguez- Tenreiro and M. Quiros. A good collection of early papers on inflationary cosmology and galaxy formation can also be found in the book Inflationary Cosmology, edited by L. Abbott and S.-Y. Pi. We apologize in advance to those authors whose work in the field of inflationary cosmology we have not been able to treat adequately. Much of the material in this book is based on the ideas and work of S. Coleman, J. Ellis, A. Guth, S. W. Hawking, D. A. Kirzhnits, L. A. Kofman, M. A. Markov, V. F. Mukhanov, D. Nanopoulos, I. D. Novikov, I. L. Rozental’, A. D. Sakharov, A. A. Starobinsky, P. Steinhardt, M. Turner, and many other scientists whose contribution to modern cosmology could not possibly be fully reflected in a single monograph, no matter how detailed. I would like to dedicate this book to the memory of Yakov Borisovich Zeldovich, who should by rights be considered the founder of the Soviet school of cosmology.