Syracuse University SURFACE Dissertations - ALL SURFACE June 2018 Cosmological Perturbations in the Early Universe Gizem Sengor Syracuse University Follow this and additional works at: https://surface.syr.edu/etd Part of the Physical Sciences and Mathematics Commons Recommended Citation Sengor, Gizem, "Cosmological Perturbations in the Early Universe" (2018). Dissertations - ALL. 902. https://surface.syr.edu/etd/902 This Dissertation is brought to you for free and open access by the SURFACE at SURFACE. It has been accepted for inclusion in Dissertations - ALL by an authorized administrator of SURFACE. For more information, please contact [email protected]. Abstract One essence in capturing the history of primordial fluctuations that arise during inflation and eventually lead to formation of large scale structures in the universe, relies on quantizing general relativity coupled to a scalar field. This is a system that respects diffeomorphism invariance, the free- dom of choosing the coordinate system to work with without changing the physics. Hence the way to handle its quantization requires an understand- ing of how to quantize diffeomorphisms. Deciding on suitable coordinate choices and making sure that this gauge fixing, which is unavoidable in any calculation, does not effect the end result, is tricky. In this thesis we make full use of the effects of diffeomorphism invariance of the theory on the primordial fluctuations to pursue two different approaches that focus on treating perturbations after gauge fixing. On one hand we work towards developing our understanding of how to handle quantization in terms of Dirac quantization and Becchi, Rouet, Stora, Tyutin (BRST) quantiza- tion. On another, we focus on how to generalize the allowed interactions and understand the scales they bring in the era of preheating that fol- lows inflation, with effective field theory (EFT) methods on cosmological backgrounds. Cosmological Perturbations in the Early Universe by Gizem Şengör B.Sc. (Physics) Boğaziçi University, Istanbul, 2011 M.Sc. (Physics) Boğaziçi University, Istanbul, 2013 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Syracuse University June 2018 Copyright 2018 Gizem Şengör All rights reserved To Meral Çakır Menna’ya sevgilerle Acknowledgments I am grateful to Scott Watson for being an encouraging and understanding advisor. As much as he has shown me good techniques to work with, he has also left a door open for all else that is out there and he has always been easy to reach when needed. I acknowledge many conversations both in physics and on a good range of other topics. I especially thank him for allowing me to find my own way and being supportive for me to work with other professors. It has been a great pleasure to learn from and work with Cristian Armendariz- Picon. I thank him for a lot of skills and knowledge he has imparted. He has been a second advisor to me and I sincerely thank him for taking me on, and working with me through a lot of things which I believe have given me solid grounds to stand on and his approach has been inspirational. I also thank Professors Kameshwar Wali and Carl Rosenzweig for supporting me at all times. I have had the chance to listen to very nice stories on the world of Physics from them and I thank Carl Rosenzweig for his enlightening advices. It was a great opportunity to join the University of Amsterdam as a visiting student for a semester. I thank Jan Pieter van der Schaar for being my advisor there and making this visit possible. It was a great joy to work with him. I have had many great discussions with him as well as Dionysios Anninos and other post docs and students there. Patty and Yudy have always been welcoming and throughly helpful with any of the administrative issues and I sincerely appreciated running into them at any point in the Physics building. I owe special thanks to Boqian, Leyla and Jerome for being there since the first day I walked onto campus. It would have been hard to survive initially without the logistical support of my neighbors Duygu and Ergin, since then I continue to spend many joyful and memorable times with them. I thank Ogan for the crucial logistics he shared with me even before my arrival and it has been an v interesting experience to share an advisor with him. I appreciate a lot of technical things I learned from Jayanth and thank him for always finding the time to share them when needed. I have also taken a lot of traveling advice from him, which were very useful. I thank Swetha for joining me on many a journey both within physics and across new countries. I cherish a lot of enjoyable end of the week coffee time and dinners with Lorena and conversations with Suraj. I thank Brandon for the journal club discussions and proofreading sections of the thesis, and to Cem for also lending a second eye on some parts. I dearly thank Cem and Francesco for the hikes across the US. It has been very memorable to perform, dance and play African drums with Biboti Ouikahilo and Wacheva Cultural Arts and to continue on ballet with Karen Mentor and Ruth Arena at Ballet and Dance Center. I have also been inspired a lot from my professors in Bogazici University. I would especially like to thank İbrahim Semiz from whom I first learned about gravitation, Yemliha Bilal for grouping a bunch of us together and making this first introduction possible, to my Master’s advisor Metin Arık for all he has shown me about research both within cosmology and beyond, and Teoman Turgut, Tonguç Rador, Muhittin Mungan, and Ali Kaya for many technical skills. I acknowledge the presence of three good friends Cem, Emre, Mert from my first week at a university to moving to Syracuse and staying in touch with the world beyond Syracuse. I thank all my family, especially my mom and sister for making the separation both in distance and in time manageable, and my dad for drawing my attention to details in a work. I appreciate all that I have learned from Yıldız Alpar, Aylin Kalem, Oya Barbara Karanis about learning, understanding and presenting what lies underneath any concept. Last, but for sure not least, I thank Cynde Sadler who has opened up a lot of doors by teaching me English. I thank for the welcoming hospitality of the Onondaga Nation on whose lands this thesis has been written. This thesis is dedicated to Meral Çakır who has played an important role for me to develop much of my creativity. vi List of Publications This dissertation includes the following three peer-reviewed and published articles: 1. “Nonthermal WIMPs and primordial black holes”, Julian Georg, Gizem Şengör, Scott Watson, Phys.Rev. D93 (2016) no.12, 123523, arXiv:1603.00023 [hep-ph] 2. “BRST Quantization of Cosmological Perturbations”, Cristian Armendariz-Picon, Gizem Şengör, JCAP 1611 (2016) no.11, 016, arXiv:1606.03823 [hep-th] 3. “Toward an Effective Field Theory Approach to Reheating”, Ogan Özsoy, John T. Giblin, Eva Nesbit, Gizem Şengör, Scott Watson, Phys.Rev. D96 (2017) no.12, 123524, arXiv:1701.01455 [hep-th] vii Contents 1 Review of Cosmological Perturbation Theory1 1.1 Introduction: what we know so far and what we would like to learn next about the universe.........................1 1.2 Diffeomorphisms and Gauge Transformations..............7 1.3 Cosmological Perturbations....................... 12 1.3.1 ADM Formalism and Constraints................ 18 1.4 Time Evolution and Quantization of Perturbations.......... 27 1.5 Gauge Invariant Variables and Quantization.............. 30 1.5.1 The Mukhanov Variable v .................... 32 1.6 Power spectra and the Long Wavelength Limit............. 39 1.6.1 The Long Wavelength Limit................... 42 1.7 An Application: Primordial Black Hole Constraints and the Spectral Index.................................... 47 1.7.1 A Short Review of Non-thermal Histories............ 48 1.7.2 Jean’s Scale and Horizon Size.................. 51 1.7.3 Mass Fraction of Primordial Black Holes............ 54 1.7.4 Density Perturbations...................... 55 1.7.5 Why the pressure effects are negligible............. 57 1.7.6 Mass Fraction in the Pressureless Case for a Radiation Domi- nated Universe.......................... 59 viii 1.7.7 The Matter Dominated Era................... 61 1.7.8 Pressure considerations...................... 70 1.8 Spontaneously broken symmetries.................... 71 1.8.1 For cosmological backgrounds:.................. 75 2 BRST Quantization of Cosmological Perturbations 77 2.1 Introduction................................ 77 2.2 Action................................... 81 2.2.1 Perturbations........................... 83 2.3 BRST Symmetry............................. 89 2.3.1 Ghosts and Antighosts...................... 89 2.3.2 BRST Transformations...................... 91 2.3.3 Time Evolution.......................... 93 2.3.4 BRST symmetry......................... 94 2.3.5 Gauge Fixing........................... 95 2.4 Quantization............................... 98 2.4.1 Expectation Values........................ 99 2.5 Free Theory................................ 103 2.5.1 Einstein-Hilbert Action...................... 103 2.5.2 Constraints............................ 105 2.5.3 Classical BRST Symmetry.................... 106 2.5.4 Quantization: Vectors...................... 113 2.5.5 Quantization: Scalars....................... 115 2.6 Interacting Theory............................ 124 2.6.1 Dirac Quantization........................ 125 2.6.2 Derivative Gauges......................... 126 2.7 Summary and