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Topics in Inflation and Eternal Inflation A TOPICS IN INFLATION AND ETERNAL INFLATION A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Mahdiyar Noorbala August 2011 © 2011 by Mahdiyar Noorbala Tafti. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/bd819xz1063 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Andrei Linde, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Stephen Shenker I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leonard Susskind Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii To my parents and Leila iv Preface The theory of inflation was invented in early 1980's to solve a number of puzzles in cosmology, most notably the horizon, flatness, homogeneity and monopoles problems. The idea was an exponential phase of expansion in the early universe that could stretch out any primordial inhomogeneities, dilute the monopoles that we don't see today and make a very flat universe. In addition, different corners of the universe that are not in causal contact today, were within a causal horizon in that stage and hence the observed long-range correlations do not violate relativistic causality. Inflation has ever since become the standard paradigm in modern cosmology for solving these problems and for many other robust predictions that are confirmed by observation. There are many inflationary models that share the same robust features but differ in other details that in principle can be used to rule out some of them. In fact there is ongoing research to pinpoint a particular inflationary model arising from fundamental theories, study its detailed predictions and compare them with observations. The first chapter of this thesis is along this line of investigations. The model we study features nonminimal coupling between gravity and the scalar field (inflaton) and is embedded within the framework of supergravity. The model is characterized by a potential which itself depends on some parameters, so in fact we have a family of models and correspondingly we have a family of predictions. The key observable is the tensor-to-scalar ratio of density perturbations. We find that with a suitable choice of the parameters of the model we can reproduce a wide range of possible outcomes in future observations by, for example, Planck satellite. Soon after inflation was proposed, it was realized that in almost all models inflation is eternal. This means that there are always parts of the universe, outside our horizon, v that undergo inflation. So while inflation eventually ends at any single point there are always many other points where it's not yet terminated. When a neighborhood of points exit the inflationary phase a universe like ours emerges whose future inhabitants will never be able to communicate with the other points outside their own universe. This picture is known as the multiverse and the individual universes like ours are called pocket, bubble or baby universes. Within decades the idea that string theory has a landscape of solutions offered a suitable arena for eternal inflation and multiverse. It is to be emphasized that regardless of details of individual models, the basic behavior of eternal inflation is a direct result of some widely accepted principles of physics, namely, general relativity and semi-classical approximation to quantum mechanics of a field in a fairly generic potential. This inevitable eternal inflation has both blessings and curses. On the one hand it extends the scope of questions that we may hope to answer: If we live in only one pocket universe out of many in the multiverse, then some of the fundamental constants of our universe can be merely environmental variables that take the values they take only due to anthropic reasons. These are usually constants that are otherwise very hard to predict or explain (e.g., the cosmological constant which is extremely fine- tuned). The rest of the constants cannot be constrained anthropically but statistical statements can be made about them. Therefore, a particular theory can be falsified at some confidence level by comparing its statistical prediction with the observed values of the fundamental constants. On the other hand, there is an ambiguity in calculating the statistical distribution associated with the fundamental constants and other physical observables. This is known as \the measure problem" and is the subject of the last two chapters of this thesis. The ambiguity in defining probabilities in eternal inflation is due to the infiniteness of events occurring in spacetime. In a stochastic approach1 one can study random realizations of spacetime geometry that are weighted by some probability distribution. Almost all of these realizations have an infinite 4-volume and any given event of any kind occurs in them infinitely many times. It is the comparison of these infinities that poses the ambiguity problem. There are several competing prescriptions, called 1That is, a semi-classical approach that does not take into account quantum interferences. vi measures, that yield different results. It is hard to justify any of these measures based on any known first principle; hence a prevalent approach has been to discriminate between them based on their predictions. This is possible because some of these measures have predictions that are already inconsistent, by a large confidence level, with known experimental results (and in fact, sometime with known facts of life). In the second chapter we compare the predictions of various measures as we tran- sition from the regime of eternal to non-eternal inflation. We find that except for \the stationary measure" the other three measures we study undergo discontinuities in their predictions. Finally in the third chapter we study the issue of \Boltzmann brains" for \the scale factor cutoff measure." We find that the success of this measure highly depends on unknown details of the landscape of string theory, although with our current knowledge the situation is marginally acceptable. vii Acknowledgements I would like to thank my adviser Andrei Linde. He introduced me to research in this field and it has been a pleasure to learn from the wonderful physical intuition of a pioneer. Andrei has also been a great support during the past couple of years. I have benefited from many others at Stanford Institute for Theoretical Physics (SITP), most notably Lenny Susskind from whom I've learned a lot. I'm also grateful to Savas Dimopoulos, Shamit Kachru, Renata Kallosh, Steve Shenker, Eva Silverstein and Jay Wacker for what they taught me in classes and conversations. I'd like to thank my non-Stanford collaborators Andrea De Simone and two prominent experts in the field, Alan Guth and Alex Vilenkin. I have enjoyed the company of my nice friends and office mates: Xi Dong, Daniel Harlow and Dusan Simic. We had wonderful discussions about physics as well as many other topics. I'm also grateful to postdocs at SITP, Alex Westphal, Mike Salem and particularly Vitaly Vanchurin. I spent so many hours in Vitaly's office talking about various issues of eternal inflation. I should especially thank my two Iranian friends at SITP, Masoud Soroush and Siavosh Rezvan Behbahani who created a warm Persian environment. Masoud was a great support in my first months at Stanford and in fact in all subsequent years. I also feel so happy to be acquainted with the greater Iranian community at Stanford who were like my family abroad. Thank you all guys! Finally thanks to The Mellam Family Foundation for a year of scholarship support. *** Above all I am indebted to my parents for pretty much everything I have. I viii appreciate that they always valued education, encouraged me and inspired me with their love and affection. I am deeply grateful for all that they have provided for me in every stage of my life. I'd also like to thank my sisters, Maryam and Fateme, who were a great source of advice, kindness and friendship. At last but not least, I'm grateful to my beloved wife Leila without whose endless support, love and sacrifice I couldn't complete this work. I have been blessed by the joy of her company in every moment of life. I love you all! I cannot finish without mentioning our lovely baby daughter Zaynab who has brought delight to our life since the moment we are gifted with her. Thanks God! ix Contents Preface v Acknowledgements viii 1 Observational Consequences of Chaotic Inflation with Nonminimal Coupling to Gravity 1 1.1 Introduction . 1 1.2 Canonical Field in Einstein Frame . 2 1.3 Supergravity perspective . 5 1.4 Basic models and their extensions . 8 1.5 Quadratic Potential . 10 1.6 Quartic Potential . 12 1.7 Conclusions . 20 2 Measure Problem for Eternal and Non-Eternal Inflation 22 2.1 Introduction . 22 2.2 Eternal and Non-eternal Inflation: A Toy Model .
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