UNIVERSITY of CALIFORNIA Santa Barbara Analytic Approaches to the Study of Small Scale Structure on Cosmic String Networks A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics by Jorge V. Rocha arXiv:0812.4020v1 [gr-qc] 20 Dec 2008 Committee in charge: Professor Joseph Polchinski, Chair Professor Tommaso Treu Professor Donald Marolf September 2008 The dissertation of Jorge V. Rocha is approved: Professor Tommaso Treu Professor Donald Marolf Professor Joseph Polchinski, Chair August 2008 Analytic Approaches to the Study of Small Scale Structure on Cosmic String Networks Copyright 2008 by Jorge V. Rocha iii To my parents, Carlos and Isabel iv Acknowledgements In writing these lines I think of the people without whom I would not be writing these lines. To the person who taught me the inner workings of science, Joe Polchinski, I am deeply grateful. Thank you, Joe, for sharing your knowledge and insight with me, for showing me the way whenever I got off track and for your powerful educated guesses. Over these years I have also had the opportunity of collaborating with Florian Dubath. For this and for his enthusiasm I am thankful. I was fortunate enough to have Don Marolf and Tommaso Treu as committee members. I learned much about physics from Don, and always with a good feeling about it. I am grateful for your constructive criticism as well. From Tommaso I appreciate him always keeping an open door for me and also showing so much interest. I thank both of you for all that. Having spent these years in UCSB I have benefited from the excellence of the local Physics Department, either through classes, gravity lunches or simple conversations. I wish to thank Professors Ski Antonucci, David Berenstein, Matt Fisher, Steve Giddings, Sergei Gukov, Jim Hartle, Gary Horowitz, Peng Oh, Mark Srednicki, Bob Sugar and Tony Zee. Now going back further in time, I am grateful to Professors Jo˜ao Pimentel Nunes and Jos´eMour˜ao, who advised me during the last years of undergraduate studies in Lisbon. Thank you for introducing me to the wonders of string theory and for all your support that led me here. I have also benefited from the presence of many former and fellow students of the high energy theory and gravity groups at UCSB, as well as postdocs. I would like to thank those who taught me one way or another, either through journal club talks and discussions, interactions in classes (as classmates or TAs), or simply by sharing the office with me. These people include Aaron Amsel, Curtis Asplund, Dan Balick, Charlie Beil, Johannes Br¨odel, Geoffrey Comp`ere, Keith Copsey, Richard Eager, Henriette Elvang, Ted Erler, Scott Fraser, Mike Gary, Misato Hayashida, Sungwoo Hong, Matt Lippert, Anshuman Maharana, Nelia Mann, Ian Morrison, Jacopo Orgera, Jo˜ao Penedones, Matt Pillsbury, Sam Pinansky, Matt Roberts, Sam Vazquez, Amitabh Virmani and Sho Yaida. My friends in Santa Barbara also helped making these years more enjoyable. Among the ones I haven’t already mentioned are Biliana, Carlos, Cec´ılia, Coral, Daniel, Eduardo, German, Gokhan, James, Jo˜ao, Jonathan, Larisa, Lu, Melissa, Pedro, Ricardo, Rishi, Sabine, Scott, Sofia and Susana. Furthermore, I would like to thank our neighbors at Family Student Housing who contributed to make it such a nice place to live at, and my soccer buddies for times well spent on the v field. On the financial side, I acknowledge support from Funda¸c˜ao para a Ciˆencia e a Tecnologia through grant SFRH/BD/12241/2003, which allowed me to focus more on research, rather than worrying about more mundane issues. I saved for last the people I love and care for most deeply. I would not have reached this point in my life if it had not been for the ones who brought me up and educated me. Thank you Mother, thank you Father, for your constant support and encouragement. I am also very happy (and proud, I admit) to say thank you to my daughter Sara, who keeps on showing me how joyful life can be, specially when she displays the most beautiful smile I have ever seen. Finally, to you, Susana, I am more grateful than I can express. Thanks for sharing my life over all these years, and for sharing your life with me. Thank you for putting up with me through all the bad moments (helping me keep my mental sanity!) and for inflating the good ones. Thank you for your contagious happiness. And thank you for giving me Sara. vi Curriculum Vitæ Jorge V. Rocha Education 2008 PhD in Physics, University of California, Santa Barbara 2002 Licenciado in Technological Physics Engineering, Instituto Superior T´ecnico, Lisbon, Portugal Publications “Analytic study of small scale structure on cosmic strings”, J. Polchinski and J. V. Rocha, published in Phys.Rev.D74:083504,2006, e-print: hep-ph/0606205. “Cosmic string structure at the gravitational radiation scale”, J. Polchinski and J. V. Rocha, published in Phys.Rev.D75:123503,2007, e-print: gr-qc/0702055. “Periodic gravitational waves from small cosmic string loops”, F. Dubath and J. V. Rocha, published in Phys.Rev.D76:024001,2007, e-print: gr-qc/0703109. “Scaling solution for small cosmic string loops”, J. V. Rocha, published in Phys. Rev.Lett.100:071601,2008, e-print: arXiv:0709.3284 [gr-qc]. “Cosmic String Loops, Large and Small”, F. Dubath, J. Polchinski and J. V. Rocha, published in Phys.Rev.D77:123528,2008, e-print: arXiv:0711.0994 [astro-ph]. “Evaporation of large black holes in AdS: coupling to the evaporon”, J. V. Rocha, e-print: arXiv:0804.0055 [hep-th]. Honors and Awards 2003 Broida fellowship award 2003 Doctoral fellowship from Funda¸c˜ao para a Ciˆencia e a Tec- nologia (Portugal) vii Abstract Analytic Approaches to the Study of Small Scale Structure on Cosmic String Networks by Jorge V. Rocha We present an analytic model specifically designed to address the long standing issue of small scale structure on cosmic string networks. The model is derived from the microscopic string equations, together with a few motivated assumptions. The resulting form of the correlation between two points on a string is exploited to study smoothing by gravitational radiation, loop formation and lensing by cosmic strings. In addition, the properties of the small loop population and the possibility of detecting gravitational waves generated by their lowest harmonics are investigated. Whenever possible, we compare the predictions of the model to the most recent numerical simulations of cosmic string networks. viii Contents Contents ix List of Figures xi List of Tables xiii 1 Introduction 1 1.1 Cosmicstrings:whatarethey? . 1 1.2 Cosmic strings in our universe? . 3 1.3 Cosmicsuperstrings.......................... 5 1.4 Networksofcosmicstrings ...................... 6 1.4.1 The scaling regime . 8 1.4.2 Smallscalestructure ..................... 9 1.4.3 Analyticapproaches ..................... 10 1.5 Outline................................. 11 2 Model for small scale structure 13 2.1 Preliminaries: Basic ingredients . 14 2.2 Assumptionsofthemodel . .. .. 19 2.3 Two-pointfunctionsatshortdistance . 20 2.3.1 Small fluctuation approximation . 24 2.4 Discussion............................... 25 3 Gravitational radiation from cosmic strings 30 3.1 Radiation from long strings and back-reaction . 31 3.2 Radiationandback-reactionimproved . 35 3.3 The long-string two-point function revisited . 38 3.4 Smallscalestructureandcusps . 41 ix 4 Loop formation 43 4.1 General results: loops, cusps and kinks . 43 4.2 Therateofloopproduction . .. .. 45 4.2.1 Preliminary considerations . 45 4.2.2 Calculation of the rate . 47 4.2.3 The normalization . 51 4.3 Discussion............................... 53 4.3.1 Comparison with simulations . 54 4.3.2 Effect of loop production on the two-point function . 56 4.3.3 Fragmentation......................... 57 4.3.4 Largeloops .......................... 58 5 Properties of the small loop population 60 5.1 Phenomenologyofsmallloops . 60 5.1.1 Gravitational radiation from small loops and their lifetimes 60 5.1.2 Loop velocities and cusps . 62 5.1.3 Amodelloop ......................... 63 5.2 Scalingoftheloopnumberdensity . 66 5.2.1 Preliminaries . 66 5.2.2 Evolution neglecting gravitational radiation . 67 5.2.3 Evolution including gravitational radiation . 68 6 Some implications for observation 71 6.1 Lensing ................................ 71 6.1.1 Distortion of images . 72 6.1.2 Alignment of lenses and non-Gaussianity . 75 6.2 Periodic gravitational waves from small cosmic string loops . 77 6.2.1 EmissionofGW ....................... 78 6.2.2 Visibility of the small loops . 81 6.2.3 Loop number density and expected number of detections . 85 6.2.4 Summaryanddiscussion . 88 7 Final Remarks 90 A Gaussian averages 92 B The matter-radiation transition 96 Bibliography 98 x List of Figures 1.1 The usual quartic “Mexican hat” potential for the complex Higgs field φ(x) is shown on the left. A vortex string configuration maps spacetime points belonging to the core of the string (on the right side of the figure) to the local maximum of the potential V (φ) and large circles surrounding the string to the ring of degenerate minima of V ................................... 2 2.1 The top panel shows an intercommutation event in which two seg- ments of string exchange partners. The bottom panel displays a single cosmic string self-intersecting and chopping off a loop. In such processes pairs of kinks are formed on the intervening seg- ments and travel in opposite directions along the string. 18 2.2 Comparison of the model (dashed line) with the data provided by C. Martins and E. P. S. Shellard (solid red line) in the radiation- dominated era, for which the correlation length is ξ 0.30t. ... 27 2.3 Comparison of the model (dashed line) with the data≃ provided by C.