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String

Modern Theory Concepts from the to Cosmic Structure

Edited by Johanna Erdmenger

WILEY-VCH Verlag GmbH & Co. KGaA

String Cosmology

Edited by Johanna Erdmenger Related Titles

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Modern Concepts from the Big Bang to Cosmic Structure

Edited by Johanna Erdmenger

WILEY-VCH Verlag GmbH & Co. KGaA The Editor All books published by Wiley-VCH are carefully produced. Nevertheless, authors, Priv.-Doz. Dr. Johanna Erdmenger editors, and publisher do not warrant the Max-Planck-Institut information contained in these books, für Physik including this book, to be free of errors. Föhringer Ring 6D Readers are advised to keep in mind that 80805 München statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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Contents

Preface XI 1 Introduction to Cosmology and String Theory 1 Johanna Erdmenger and Martin Ammon 1.1 Introduction 1 1.2 Foundations of Cosmology 1 1.2.1 Metric and Einstein Equations 2 1.2.2 Energy Content of the Universe 4 1.2.3 Development of the Universe 6 1.3 Inflation 7 1.3.1 Puzzles Within the Big Bang Model 7 1.3.2 The Concept of Inflation 7 1.4 Fluctuations 9 1.4.1 Characterization of Small Fluctuations 9 1.4.2 Power Spectrum 10 1.4.3 Fluctuations and Inflation 11 1.5 12 1.5.1 Open and Closed Strings 14 1.5.1.1 Closed Strings 15 1.5.1.2 Open Strings 15 1.5.2 Quantization 16 1.5.3 String : Interactions and Scattering Amplitudes 17 1.5.4 Bosonic String Theory in Background Fields 19 1.5.5 Chan–Paton Factors 22 1.5.6 Oriented Versus Unoriented Strings 22 1.6 23 1.6.1 The RNS Formalism of Superstring Theory 23 1.6.2 Boundary Conditions for Fermions 24 1.6.2.1 Open Strings 24 1.6.2.2 Closed String 25 1.6.3 Type IIA and Type IIB Superstring 26 1.6.4 Type I Superstring 26 VI Contents

1.6.5 Heterotic Superstring 27 1.7 String Dualities and M-Theory 28 1.7.1 Low-Energy Effective Action of Superstring Theory 28 1.7.2 T-Duality 29 1.7.2.1 T-Duality of Closed Strings 30 1.7.2.2 T-Duality of Open Strings 31 1.7.2.3 T-Duality in Superstring Theory 32 1.7.3 S-Duality 32 1.7.4 Web of Dualities and M-Theory 33 1.7.4.1 Type IIA String Theory and M-Theory 34 1.7.4.2 Heterotic E8 ~ E8 String Theory and M-Theory 35 1.8 D- 36 1.8.1 Effective Action of D-Branes 36 1.8.2 D-Branes as Charged BPS Objects 38 1.9 Compactification 38 1.9.1 String Theory on Calabi–Yau Manifolds 39 1.9.1.1 Low-Energy Effective Theory 40 1.9.2 String Theory on Orbifolds 41 1.9.3 String Moduli and Their Stabilization 42 1.10 String Thermodynamics 43 1.11 Gauge–Gravity Duality 44 1.12 Summary 45 References 45

2 String Inflation I: Inflation 47 Marco Zagermann 2.1 Introduction 47 2.1.1 Inflation in String Theory and Moduli Stabilization 49 2.1.2 Brane Inflation Models 50 2.1.3 The Rest of this Chapter 51 2.2 Moduli Stabilization in Type IIB String Theory 52 2.2.1 Type IIB Calabi–Yau Orientifolds and Their Moduli 52 2.2.2 The Tree-Level Effective Action 56 2.2.3 The Volume Modulus 57 2.2.4 de Sitter Uplifting 58 2.3 Warped D3/D3-Brane Inflation (Slow-Roll) 59 2.3.1 The Warped Throat Geometry 61 2.3.2 Towards Slow-Roll Inflation 61 2.3.3 Volume Stabilization 63 2.3.4 The Inflaton Dependence of W64 2.4 D3/D7-Brane Inflation 67 2.4.1 A Compactified Example 70 2.4.2 Moduli Stabilization 71 2.5 DBI Inflation 73 2.5.1 Generalizing the Slow-Roll Conditions 76 Contents VII

2.6 Gravitational Waves and Inflaton Field Range 77 2.6.1 D3-Branes on a Symmetric Torus 79 2.6.2 D3-Branes in a Warped Throat 79 2.6.3 DBI Inflation 80 2.6.4 Wrapped Branes 81 2.6.5 Other Approaches and Related Work 82 2.7 Conclusions 83 References 84 3 String Inflation II: Inflation from Moduli 89 C.P. Burgess 3.1 Introduction 89 3.1.1 Closed-String Moduli as Inflatons 90 3.1.2 A Brief Roadmap 91 3.1.3 Justifying the Approximations 92 3.2 Accelerated Expansion in 92 3.2.1 Accelerated Expansion in Higher Dimensions 92 3.2.2 Acceleration in 4D Supergravity 93 3.3 Type IIB Moduli and Their Stabilization 94 3.3.1 Leading-Order Expressions 94 3.3.2 Corrections to the Leading Approximation 95 3.3.2.1 Leading α Corrections 96 3.3.2.2 String-Loop Corrections 96 3.3.2.3 Superpotential Corrections 97 3.3.3 Breaking Potentials 98 3.4 Inflation from Kähler Moduli 99 3.4.1 Racetrack Inflation 100 3.4.2 Blow-Up Mode Inflation 103 3.4.3 Fiber Inflation 107 3.4.4 K3-Fibration Calabi–Yaus 107 3.4.5 The Scalar Potential 108 3.4.5.1 Kinetic Terms 111 3.4.5.2 Inflationary Slow Roll 111 3.5 What We’ve Learned so far 116 References 118 4 Cosmic Superstrings 121 Robert C. Myers and Mark Wyman 4.1 Introduction 121 4.1.1 Symmetry Breaking and Topological Defects 122 4.1.2 A Brief Review of Cosmic-String Networks 124 4.1.2.1 Small-Scale Structure 126 4.2 Superstring Theory on Cosmological Scales 127 4.2.1 Low String Tensions? 129 4.2.2 Strings After Inflation? 131 4.2.3 Stability of Cosmic Superstrings? 134 VIII Contents

4.2.3.1 Breakage on Space-Filling Branes 134 4.2.3.2 Confinement by Axion Domain Walls 136 4.2.3.3 “Baryon Decay” 136 4.2.3.4 Tachyon Condensation 137 4.2.3.5 An Example: The KKLMMT Model 137 4.2.4 Novel Physics from Cosmic Superstrings 140 4.2.4.1 Potential Problems for Superstring Networks 142 4.3 Observing Cosmic Superstrings 143 4.3.1 Experimental Limits and Observational Tests 144 4.3.1.1 Current Limits 144 4.3.1.2 Signatures Testable by Near-Term Observations 146 4.3.2 Novel Physics from Cosmic Superstrings: Observational Aspects 149 4.3.2.1 Reduced Intercommutation Rates 149 4.3.2.2 Cosmic (p, q)-Strings 150 4.3.3 Monopoles and Beads 150 4.3.4 Semilocal Strings 151 4.3.5 Miscellaneous Observations 151 4.4 Conclusion 152 References 152 5 The CMB as a Possible Probe of String Theory 157 Gary Shiu 5.1 Introduction 157 5.2 String Theory and Inflation 158 5.3 Example 1: Initial State of Inflation 160 5.3.1 Initial State Effects in the CMB and Their Relation to New Physics 163 5.3.2 Corrections to the Primordial Spectrum from Scale-Invariant Initial Conditions 165 5.3.3 Corrections to the Primordial Spectrum from Boundary EFT 165 5.3.4 Observable Parameters and Physical Quantities 167 5.4 Example 2: Non-Gaussianities in the CMB 169 5.4.1 The Shape of Non-Gaussianities and Experimental Constraints 173 5.5 Example 3: Probing the Shape of Extra Dimensions 177 5.5.1 The Warped Deformed Conifold and Other Warped Throats 178 5.5.2 Observing Warped Geometries via the CMB 182 5.6 Summary and Future Directions 186 References 188 6 String Gas Cosmology 193 Robert H. Brandenberger 6.1 Introduction 193 6.1.1 Motivation 193 6.1.2 The Current Paradigm of Early Universe Cosmology 193 Contents IX

6.1.3 Challenges for String Cosmology 195 6.1.4 Preview 197 6.2 Basics of String Gas Cosmology 197 6.2.1 Principles of String Gas Cosmology 197 6.2.2 Dynamics of String Gas Cosmology 199 6.3 Moduli Stabilization in String Gas Cosmology 203 6.3.1 Principles 203 6.3.2 Stabilization of Geometrical Moduli 204 6.3.3 Stabilization 207 6.4 String Gas Cosmology and Structure Formation 209 6.4.1 Overview 209 6.4.2 String Thermodynamics 212 6.4.3 Spectrum of Cosmological Fluctuations 217 6.4.4 Spectrum of Gravitational Waves 219 6.4.5 Discussion 221 6.5 Conclusions 222 References 225

7 Gauge–Gravity Duality and String Cosmology 231 Sumit R. Das 7.1 Introduction 231 7.2 Null Singularities and 232 7.2.1 and Matrix Membrane Theory 232 7.2.1.1 Matrix String Theory 232 7.2.1.2 Matrix Membrane Theory 235 7.2.2 Matrix Big Bangs 236 7.2.2.1 IIB Big Bangs 237 7.2.2.2 pp-Wave Big Bangs 239 7.2.2.3 Issues 241 7.3 Cosmological Singularities and the AdS/CFT Correspondence 242 7.3.1 Time-Dependent Sources in Gauge Theory and Their Dual 243 7.3.1.1 Solutions with Null Singularities 243 7.3.1.2 Solutions with Space-Like Singularities 245 7.3.1.3 Energy–Momentum Tensors 247 7.3.1.4 General Properties of the Dual Gauge Theory 248 7.3.1.5 The Wavefunctional 250 7.3.1.6 Energy Production 253 7.3.1.7 Particle Production 254 7.3.1.8 The Fate of the System 255 7.3.1.9 Summary 256 7.3.2 AdS Cosmologies with Unstable Potentials and Their Duals 256 7.3.2.1 The Bulk Cosmology 257 7.3.2.2 The Dual Gauge Theory 258 7.3.2.3 Self Adjoint Extensions 259 X Contents

7.3.2.4 Evolution of the Homogeneous Mode 260 7.3.2.5 The Inhomogeneous Mode and Particle Production 262 7.4 Conclusions 263 References 264 8 Heterotic M-Theory and Cosmology 267 Axel Krause 8.1 Introduction 267 8.2 Heterotic M-Theory Flux Compactifications 269 8.2.1 Geometry 269 8.2.2 G-Flux 271 8.2.3 SU(3) Structure 272 8.3 Heterotic Cosmic Strings 272 8.3.1 Wrapped M2 and M5-Branes 273 8.3.2 Tensions 275 8.3.3 Stability 278 8.3.4 Production 281 8.3.5 Relation to Other Types of Cosmic Strings 283 8.4 Towards Dark Energy from M-Theory 284 8.4.1 The Dark Energy Enigma 285 8.4.2 Fine-Tuning Problem and Two-Step Strategy 287 8.4.3 Dark Energy and Supergravity 288 8.4.4 Heterotic versus Type IIB String Theory 289 8.4.5 Vacuum Energy in 291 8.4.6 Vacuum Energy in Heterotic M-Theory 293 8.5 Multibrane Inflation and Gravitational Waves 296 8.5.1 Multi M5-Brane Inflation 296 8.5.2 Detectable Gravitational Waves 299 References 300 Index 305 XI

Preface

Cosmology has made huge steps forward over the last twenty-five years, both through new observations as well as through phenomenological models. Impor- tant cosmological parameters have been measured with unprecedented accuracy. For instance, measurements of the cosmic microwave background have severely constrained the possible models describing the early universe. At the same time, string theory has progressed as a most promising candidate for a quantum theory of gravity. Simultaneously, it provides a unified framework for all four fundamental interactions, including the standard model of elementary particles in addition to gravity. Nevertheless, important questions remain. On one hand, theorists aim for a mi- croscopical understanding of the effective theories describing the early universe. Also, the physics close to the initial singularity of the universe remains to be un- derstood. This requires a full quantum theory of gravity. On the other hand, new and forthcoming precision measurements, such as of the fluctuations in the cos- mic microwave background, will provide possibilities for further detailed tests of theories describing the early universe. Recently, string theory has taken up the challenge of deriving experimentally or observationally testable predictions. This applies in particular to cosmology, as the examples in this book show. This is in particular due to the fact that cosmology allows one to access very high-energy scales in the early universe. An important activity in recent years has been to obtain inflation, that is a period of accelerated exponential expansion in the early universe, within string-theoretical models. The forthcoming experiments may potentially discriminate between different classes of these models. For instance, some of these models predict new structures in the power spectrum of cosmic microwave background fluctuations in a very natural way. Moreover, recent new developments in string theory have shed new light on the possibility that microscopic superstrings created in the early universe could have been magnified to macroscopic size during the cosmic expansion, possibly leading to astronomically observable consequences. Also, string theory may be use- ful in gaining new understanding of the origin of the cosmological constant and of the nature of dark energy. It may also provide mechanisms by which primordial gravitational waves can be generated. XII Preface

It should also be mentioned here that new experimental predictions from string theory are also emerging in relation to elementary particle physics. For example, cross-sections for reaction processes have been calculated under the assumption of a low string scale of about 1 TeV. These can be tested at the Large Hadron Collider (LHC) at the CERN laboratory in Geneva, whose experiments will begin to produce data soon. Also, the searches for supersymmetry may have consequences both for string theory and for cosmology. Furthermore, the correspondence between super- gravity in Anti-de Sitter spaces and conformal field theories (AdS/CFT correspon- dence) and its generalizations have provided new relations between string theo- ry and strongly coupled gauge theories, in particular gauge theories relevant for heavy-ion physics and for the quark-gluon plasma, which is expected to play a role in structure formation in the early universe. In the near future, the interrelations of string theory with both cosmology and elementary particle physics will be put to the test. In view of this background of both new theoretical ideas and new observations, this book aims at providing a snapshot of current ideas and approaches in string cosmology. The emphasis is put both on presenting theoretical ideas as well as on deriving testable predictions from them. At the same time, the book follows a pedagogical aspect of providing an introduction to present-day research topics for graduate students and scientists of neighboring fields. The book begins with an introduction to both cosmology and string theory and provides a summary and glossary for the remaining chapters. Subsequently, distin- guished string cosmologists present their areas of research. In Chapters 2 and 3, Marco Zagermann and Cliff P. Burgess introduce the important concept of string inflation, emphasizing open string and closed string aspects, respectively. In Chap- ter 4, Robert C. Myers and Mark Wyman discuss large-scale cosmic superstrings. In Chapter 5, Gary Shiu presents the non-Gaussianities in the power spectrum of cosmic microwave background fluctuations which arise in certain string infla- tion models. In Chapter 6, Robert Brandenberger introduces string gas cosmolo- gy which addresses the question of describing the earliest moments of cosmolo- gy, before the standard effective field theory approaches become valid. String gas cosmology also provides an ansatz alternative to inflation. In Chapter 7, Sumit R. Das introduces new approaches to describing spacetime singularities, which are based on gauge–gravity dualities and on matrix models. In Chapter 8, Ax- el Krause presents the cosmological implications of heterotic M-theory, in par- ticular in view of the dark energy problem and of the generation of gravitational waves. Though the areas presented are diverse, the book aims at emphasizing the cross- relations between the individual topics, and there are numerous cross-references between the different chapters. As an example consider the power spectrum of fluctuations in the cosmic microwave background: This is defined and introduced in Chapter 1. In Chapters 2 and 3, the form this spectrum assumes in string mod- els of inflation is discussed. In Chapter 5, possible non-Gaussian contributions to this spectrum arising in string inflation models are presented. In Chapter 6, this Preface XIII spectrum is discussed within string gas cosmology, where it assumes a form which differs from the inflationary models in some respects. It remains to be said that although the book provides an overview over major topics in present-day string cosmology, there are further important concepts in cosmology which, due to the diversity and wealth of the research area, it is not possible to cover here. Nevertheless two of them should be named in this preface: The first is the landscape approach to string vacua. The second is the ekpyrotic sce- nario of colliding branes, which might provide an alternative to inflation. However, after studying the present book the reader should be equipped with the necessary background information for quickly becoming acquainted with those subjects, too. Finally, I would like to thank all authors of the individual chapters for their contri- bution, Martin Ammon for his help in compiling the manuscript, René Meyer for proof-reading, Felix Rust for help with figures, and Anja Tschörtner at Wiley-VCH for her professional handling of the publishing process.

München, September 2008 Johanna Erdmenger