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Fluctuation and Dissipation Dynamics in the Early Universe Jo¨elMabillard I V N E R U S E I T H Y T O H F G E R D I N B U Doctor of Philosophy The University of Edinburgh May 2019 2 Abstract The aim of this thesis is to study the effects of fluctuation and dissipation dynamics in Early Universe cosmology. The formal description of the Early Universe relies on cosmological fields that, in general, are not completely isolated from and, therefore, interact with their environment. These interactions might lead to two non-negligible effects. Fluctuations, acting as stochastic forces, tend to perturb the motion of the field. In addition, some fraction of the energy is transferred from the field to other degrees of freedom, corresponding to dissipation. We are interested in three situations where fluctuation and dissipation dynamics plays a significant role for cosmology. After a brief review of the prevailing model of cosmology, the Standard Big Bang Model, we study the formation of embedded defects. This particular realization of topological defects is not stable by construction. While considering one of the simplest examples, the pion string, we show that the interactions with a thermal and dense medium might, in some circumstances, provide a stabilization mechanism. We then turn our interest to the warm realization of inflation. Ideas borrowed from the renormalization group are applied to warm inflation in order to define universality classes among the different models of inflation. Beyond the identification of universality, this approach is well-suited for an analytical treatment of warm inflation and helps in the characterization of the possible smooth transition to the radiation-dominated regime. Finally, we extend the Kramers problem to quantum field theory. In the presence of fluctuation and dissipation dynamics, there is a non-vanishing probability for a field initially located at a minimum of its potential to escape from the well. We define and derive the escape rate for a scalar field, due to thermal fluctuations, and discuss the applications for cosmology. i ii Lay Summary Fluctuation and dissipation dynamics is a common phenomenon in physics. One of the simplest illustrations appears in Brownian motion, which describes, for example, the random motion of a particle suspended in a liquid or a gas. The interactions between the particle and its surrounding, the fluid, generate two competing effects. Fluctuations, acting as random forces, perturb the motion of the particle. In addition, a damping leads to the dissipation of a fraction of the particle energy. Such phenomena are ubiquitous in physics. Problems described by Brownian motion, or related fluctuation and dissipation dynamics, appear in subject areas ranging from condensed matter to astrophysics. In the present work, we are interested in the consequences of fluctuation and dissipation dynamics in Early Universe cosmology. The formal description of the Early Universe is based on cosmological fields. In general, a field is not entirely isolated from and, therefore, interacts with its environment, which can lead to fluctuation and dissipation dynamics. Despite its prevalence in the subject of cosmology, the influence of this dynamics has not been extensively studied so far. We are particularly interested in three situations where fluctuation and dissipation dynamics plays a significant role : phase transition and formation of topological defects, warm inflation, and in the escape problem. iii iv Declaration I declare that this thesis was composed by myself, that the work contained herein is my own except where explicitly stated otherwise in the text, and that this work has not been submitted for any other degree or professional qualification except as specified. Parts of this work have been published in [1, 2]. (Jo¨elMabillard, May 2019 ) v vi Acknowledgements First of all, I would like to express my deep gratitude to my supervisor, Prof. Arjun Berera. The uninterrupted care, guidance and encouragements given throughout my Phd have been extremely helpful, precious and appreciated. I also thank Prof. Rudnei Ramos for his constant help and for making the visit to the UERJ possible. I am specially grateful to Dr. Mauro Pieroni, for being such a great collaborator and friend. I wish to thank also the people I had the chance to work with, in particular, Angelo, Bruno and Francesco. To the members of the PPT at the University of Edinburgh I would like to say thanks, I have much appreciated being a member of the group. To James, Julia and Saad, who have been great officemates. To Alex, Andreas, Andries, Jos´e and Vincenzo, for the interesting discussions, especially at the coffee breaks. I am specially grateful to Christian and Gianluca, who made the time spent in Edinburgh the most enjoyable. I would like to thank the other people I have met in Edinburgh and that I forget to cite here. Je souhaite remercier ma famille et en particulier mes parents, Joseph et Marie- Th´er`ese,qui ont toujours ´et´ed'un soutien inconditionnel, notamment pendant mes ´etudes. Ce sont eux qui m'ont transmis cette curiosit´epour le monde qui nous entoure, et qui, de fil en aiguille, m'a conduit vers la physique. Je remercie aussi Lo¨ıc,M´elanie,Jicky et Claude pour leur soutient constant. Je remercie tout particuli`erement Aurelia, qui a toujours ´et´epr´esente pour moi. Ses encouragements, son soutien et son aide, en particulier dans les moments plus difficiles, ont ´et´eextr^emement pr´ecieux.J'ai beaucoup de chance de t'avoir dans ma vie. Finally, I have a special thought for Prof. Pierre Bin´etruy, who unfortunately left us too soon. Pierre was not only one of the greatest physicists I have ever met, he was also a very kind person. Pierre was my Master thesis supervisor, and I am specially thankful to him for encouraging me to pursue my studies with a Phd. vii viii Contents Abstract i Lay Summary iii Declaration v Acknowledgements vii Contents ix List of Figures xiii List of Tables xv 1 Introduction 1 2 The Standard Big Bang Model 9 2.1 FLRW Cosmologies............................................................. 10 2.1.1 The Cosmological Principle and the FLRW Metric............ 10 2.1.2 Friedmann Equations.................................................. 12 2.1.3 Energy Contributions and Associated Solutions................ 14 2.2 Thermal History of the Hot Big Bang Model............................. 16 2.3 Observational Cosmology and the ΛCDM Model........................ 19 2.3.1 Observing the Universe............................................... 19 ix 2.3.2 ΛCDM Model........................................................... 22 3 Stabilization of Embedded Topological Defects 25 3.1 Introduction...................................................................... 25 3.2 Review of Topological Defects in Cosmology.............................. 27 3.2.1 Spontaneous Symmetry Breaking and Phase Transitions..... 28 3.2.2 Kibble Mechanism..................................................... 29 3.2.3 Existence and Classification of Topological Defects............ 30 3.2.4 Consequences for Cosmology........................................ 32 3.2.5 Embedded Defects..................................................... 34 3.3 The Pion String in the Linear Sigma Model.............................. 35 3.3.1 LSMq at Zero Temperature.......................................... 35 3.3.2 Chiral Phase Transition at Finite Temperature and Chem- ical Potential............................................................ 38 3.3.3 Pion String Solution and its Stability in Vacuum.............. 44 3.4 Stabilization of the Pion String in a Thermal and Dense Medium... 48 3.4.1 Pion String Solution at Finite-Temperature..................... 48 3.4.2 Stability of the Pion String.......................................... 50 3.4.3 Stability in the Physical Case h = 0............................... 54 6 3.5 Discussion and Conclusions................................................... 57 4 Universality in Warm Inflation 61 4.1 Introduction...................................................................... 61 4.2 Review of Cosmic Inflation.................................................... 64 4.2.1 Shortcomings of the SBBM.......................................... 65 4.2.2 Realization of Inflation................................................ 69 x 4.2.3 Structure Formation from Inflation................................ 70 4.2.4 Warm Realization of Inflation....................................... 73 4.3 Universality in Warm Inflation............................................... 77 4.3.1 Identifying Universality with the β-Function Formalism...... 78 4.3.2 β-Function Formalism for Warm Inflation......................