String theory in the early universe Rhiannon Gwyn Physics Department McGill University Montreal, Quebec June 2009 arXiv:0911.2782v1 [hep-th] 14 Nov 2009 A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Abstract String theory is a rich and elegant framework which many believe furnishes a UV- complete unified theory of the fundamental interactions, including gravity. However, if true, it holds at energy scales out of the reach of any terrestrial particle accelerator. While we cannot observe the string regime directly, we live in a universe which has been evolving from the string scale since shortly after the Big Bang. It is possible that string theory underlies cosmological processes like inflation, and that cosmology could confirm or constrain stringy physics in the early universe. This makes the intersection of string theory with the early universe a potential window into otherwise inaccessible physics. The results of three papers at this intersection are presented in this thesis. First, we address a longstanding problem: the apparent incompatibility of the experimentally constrained axion decay constant with most string theoretic realisations of the axion. Using warped compactifications in heterotic string theory, we show that the axion decay constant can be lowered to acceptable values by the warp factor. Next, we move to the subject of cosmic strings: linelike topological defects formed during phase transitions in the early universe. It was realised recently that cosmic superstrings are produced in many models of brane inflation, and that cosmic superstrings are stable and can have tensions within the observational bounds. Although they are now known not to be the primary generators of primordial density perturbations leading to structure formation, the evolution of cosmic string networks could have important consequences for astrophysics and cosmology. In particular, there are quantitative differences between cosmic superstring networks and GUT cosmic string networks. We investigate the properties of cosmic superstring networks in warped backgrounds, where they are expected to be produced at the end of brane inflation. We give the tension and properties of three-string junctions of different kinds in these networks. Finally, we examine the possibility that cosmic strings in heterotic string theory could be responsible for generating the galactic magnetic fields that seeded those observed today. We were able to construct suitable strings from wrapped M5-branes. Demanding that they support charged zero modes forces us into a more general heterotic M–theory picture, in which the moduli of a large moduli space of M-theory compactifications are time dependent and evolve i cosmologically. Thus a string theory solution of this problem both implies constraints on the string theory construction and has cosmological implications which might be testable with future observations. The breadth of topics covered in this thesis is a reflection of the importance of the stringy regime in the early universe, the effects of which may be felt in many different contexts today. The intersection of string theory with cosmology is thus a complex and exciting field in the study of fundamental particle physics. ii Acknowledgements I would like to thank my supervisor, Keshav Dasgupta, for his seemingly boundless time and help. I am indebted to him for patiently teaching me string theory and guiding my work in all the projects undertaken during my Ph.D. and for his support and encouragement throughout. I would also like to thank the other faculty members in the high energy theory group at McGill, from whom I have learnt a great deal. I am indebted to Jim Cline, Alex Maloney and Guy Moore, and especially Robert Brandenberger. My Master’s supervisor Robert de Mello Koch’s support and encouragement have been indispensable. During my Ph.D. I received financial support from the Physics department at McGill University, my supervisor Keshav Dasgupta, a McGill Major’s Chalk-Rowles fellowship and a Schulich fellowship. Anke Knauf has been a source of support, an inspiration and a font of wisdom. Thanks also to Hassan Firouzjahi, Andrew Frey, Omid Saremi and Bret Underwood. I would also like to thank my collaborators Stephon Alexander, Josh Guffin and Sheldon Katz; my officemates and peers Neil Barnaby, Aaron Berndsen, Simon Carot-Huot, Racha Cheaib, Lynda Cockins, Rebecca Danos, Paul Franche, Johanna Karouby, Nima Lashkari, Dana Lindemann, Subodh Patil, Natalia Shuhmaher, James Sully, Aaron Vincent, Alisha Wissanji, Hiroki Yamashita and especially Ra’ad Mia; the lecturers and organisers of the Jerusalem winter school, PiTP, Les Houches and TASI; and my fellow students there Michael Abbott, Murad Alim, Ines Aniceto, Chris Beem, Adam Brown, Alejandra Castro, Paul Cook, Sophia Domokos, Lisa Dyson, Damien George, Manuela Kuraxizi, Louis Leblond, Nelia Mann, Arvind Murugan, Jonathan Pritchard, Rakib Rahman, Sarah Shandera, Alex Sellerholm, Jihye Seo, Julian Sonner, David Starr, Linda Uruchurtu, Amanda Weltman Murugan, Ketan Vyas, and Navin Sivanandam in particular. Thanks are also due to my friends outside of string theory - listed elsewhere - and to my family. I’d like to thank Rhys and Lludd, Fiona and Marianne Ackerman, Cathleen Mawdsley-Inggs, and especially my parents Gwyn Campbell and Judith Inggs, to whom I owe everything. This thesis is dedicated to Nannie and Grandad, Mamgu, and Iago and Iestyn Grwndi. iii Table of Contents Abstract ........................................... i Acknowledgements .................................... iii List of Figures ....................................... vii List of Tables ........................................viii 1 Introduction: String theory and cosmology .................. 1 1.1 Overview .................................... 1 1.2 Stringtheory .................................. 2 1.2.1 Motivation................................ 2 1.2.2 Stringtheorybasics........................... 9 1.3 Intersection of string theory with cosmology . ........ 10 1.3.1 Choosing a string theory vacuum: cosmological inputs to string theory................................. 11 1.3.2 Inflation ................................. 14 2 The Throat: Warped Compactifications .................... 15 2.1 Calabi-Yau compactifications . 15 2.2 Warpedcompactifications . 16 2.2.1 Phenomenology of warped compactifications . 16 2.2.2 TheKlebanov-Strasslerthroat . 17 2.2.3 Flux compactifications . 20 3 Axions in string theory .............................. 23 3.1 ThePeccei-QuinnAxion ............................ 24 3.1.1 ThestrongCPproblem . 24 3.1.2 ThePeccei-Quinnmechanism . 25 3.1.3 Constraintsontheaxion . 27 3.2 WarpedHeteroticAxions ........................... 28 3.2.1 AxionsinStringTheory . 28 3.2.2 The effect of warping on fa ...................... 29 3.3 Warped Heterotic Construction . 32 iv 3.3.1 Heterotic Compactification on a non-K¨ahler Manifold ........ 32 3.3.2 An AdS-type Background in Heterotic Theory . 32 3.3.3 The Axion Decay Constant . 38 3.4 ConstantCouplingBackground . 40 3.5 “Model-dependent”axions. 41 3.6 Discussionandconclusions. 42 4 Cosmic Strings .................................... 45 4.1 Introduction................................... 45 4.2 Topologicaldefects .............................. 46 4.2.1 Topological classification . 46 4.2.2 Domainwalls .............................. 48 4.2.3 Vortices-Cosmicstrings . 50 4.2.4 Monopoles................................ 51 4.3 Cosmicstrings ................................. 52 4.3.1 Phase transitions in the early universe . 52 4.3.2 Network evolution . 52 4.4 Cosmological implications of cosmic strings . ........ 53 4.4.1 Structureformation. 53 4.4.2 Gravitational waves . 57 4.4.3 Gravitationallensing . 57 4.4.4 Observational constraints . 58 4.5 Cosmicstringsfromstringtheory . 58 5 Cosmic strings in warped geometries ...................... 60 5.1 Introduction................................... 60 5.2 Stringsproducedinbraneinflation . 61 5.2.1 F-andD-strings ............................ 61 5.2.2 (p,q)-strings............................... 61 5.2.3 Semilocalstrings ............................ 64 5.3 (p,q)-stringsinthethroat . .. .. .. .. .. .. .. .. 64 5.3.1 D3-brane construction of (p,q)-strings in the throat . 65 5.4 Three-stringjunctionsinthethroat. ...... 68 5.4.1 Three-stringjunctions . 68 5.4.2 D3-brane construction of a three-string junction in the throat . 69 5.5 Beadsinthethroat............................... 74 5.5.1 Beads .................................. 74 5.5.2 D3-brane construction of beads in the throat . ..... 75 5.6 Cosmological implications . 76 5.6.1 Evolution of (p,q)-stringnetworks. 76 5.6.2 Evolution of cosmic necklaces . 78 5.6.3 Conclusions ............................... 80 v 6 Primordial magnetic fields from superconducting heterotic cosmic strings ........................................ 81 6.1 Introduction................................... 81 6.2 Galacticmagneticfields . .. .. .. .. .. .. .. 82 6.2.1 Primordial Magnetic Fields . 82 6.2.2 Thedynamomechanism . 83 6.2.3 Seed fields and the coherence length . 87 6.3 Magnetic fields from cosmic strings . 88 6.3.1 Generalarguments . .. .. .. .. .. .. .. .. 88 6.3.2 Pionstrings ............................... 90 6.3.3 Superconductionmechanisms . 93 6.4 Heteroticcosmicstrings . 95 6.4.1 Theaxionicinstability . 95 6.4.2 Loopholes via M theory and the BBK construction . 96 6.5 Superconductivity ............................... 102 6.5.1 Fermioniczeromodes. .102 6.5.2 Coupling to Electromagnetism