This page intentionally left blank ELEMENTS OF STRING COSMOLOGY The standard cosmological picture of our Universe emerging from a “big bang” leaves open many fundamental questions: Is the big bang a true physical singularity? What happens to the Universe at ultra-high energy densities when even gravity should be quantized? Has our cosmological history a finite or infinite past extension? Do we live in more than four space-time dimensions? String theory, a unified theory of all forces of nature, should be able to answer these questions This book contains a pedagogical introduction to the basic notions of string theory and cosmology It describes the new possible scenarios suggested by string theory for the primordial evolution of our Universe It discusses the main phenomenological consequences of these scenarios, stresses their differences from each other, and compares them with the more conventional models of inflation The first book dedicated to string cosmology, it summarizes over 15 years of research in this field and introduces current advances The book is self-contained so it can be read by astrophysicists with no knowledge of string theory, and highenergy physicists with little understanding of cosmology Detailed and explicit derivations of all the results presented provide a deeper appreciation of the subject MAURIZIO GASPERINI is Professor of Theoretical Physics in the Physics Department of the University of Bari, Italy He has authored several publications on gravitational theory, high-energy physics and cosmology, and has twice received one of the Awards for Essays on Gravitation from the Gravity Research Foundation (1996 and 1998) ELEMENTS OF STRING COSMOLOGY MAURIZIO GASPERINI University of Bari, Italy CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521868754 © M Gasperini 2007 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2007 eBook (NetLibrary) ISBN-13 978-0-511-33545-7 ISBN-10 0-511-33545-8 eBook (NetLibrary) hardback ISBN-13 978-0-521-86875-4 hardback ISBN-10 0-521-86875-0 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To Patty with gratitude Contents Preface Acknowledgements Notation, units and conventions page xi xiii xiv A short review of standard and inflationary cosmology 1.1 The standard cosmological model 1.2 The inflationary cosmological model References 1 20 35 The basic string cosmology equations 2.1 Tree-level equations 2.2 The Einstein-frame representation corrections 2.3 First-order Appendix 2A Differential forms in a Riemannian manifold References 37 38 45 52 64 70 Conformal invariance and string effective actions 3.1 Strings in curved backgrounds and conformal anomalies 3.2 Higher-curvature and higher-genus expansion Appendix 3A The massless multiplet of the bosonic string in Minkowski space Appendix 3B Superstring models and effective actions References 71 72 84 91 104 130 Duality symmetries and cosmological solutions 4.1 Scale-factor duality and the pre-big bang scenario 4.2 Duality with matter sources 4.3 Global O(d, d) symmetry Appendix 4A A non-local, general-covariant dilaton potential 132 134 142 157 174 vii viii Contents Appendix 4B References Examples of regular and self-dual solutions 181 192 Inflationary kinematics 5.1 Four different types of inflation 5.2 Dynamical equivalence of super-inflation and accelerated contraction 5.3 Horizons and kinematics References 194 195 The string phase 6.1 High-curvature fixed points of the string cosmology equations 6.2 Strong coupling corrections and the curvature “bounce” 6.3 String gas cosmology Appendix 6A Birth of the Universe in quantum string cosmology References 210 213 223 231 239 250 The cosmic background of relic gravitational waves 7.1 Propagation of tensor perturbations 7.2 Parametric amplification and spectral distribution 7.3 Expected relic gravitons from inflation 7.4 Sensitivities and cross-correlation of gravitational detectors Appendix 7A Higher-derivative corrections to the tensor perturbation equations References 253 254 266 287 310 Scalar perturbations and the anisotropy of the CMB radiation 8.1 Scalar perturbations in a cosmological background 8.2 The anisotropy spectrum of the CMB radiation 8.3 Adiabatic metric perturbations from the string theory axion Appendix 8A Photon–dilaton interactions and cosmic magnetic fields Appendix 8B Seeds for the CMB anisotropy References 334 335 362 389 408 416 424 Dilaton phenomenology 9.1 Spectral intensity of a massive dilaton background 9.2 Interaction with gravitational antennas 9.3 Dilaton dark energy and late-time acceleration Appendix 9A The common mode of interferometric detectors References 428 429 445 463 478 482 10 Elements of brane cosmology 10.1 Effective gravity on the brane 10.2 Warped geometry and localization of gravity 484 487 496 198 202 209 323 331 ... one of the Awards for Essays on Gravitation from the Gravity Research Foundation (1996 and 1998) ELEMENTS OF STRING COSMOLOGY MAURIZIO GASPERINI University of Bari, Italy CAMBRIDGE UNIVERSITY PRESS. .. Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge. .. provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2007