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Quantum Fluctuations of Vector Fields and the Primordial Curvature Perturbation in the Universe

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Quantum Fluctuations of Vector Fields and the Primordial Curvature Perturbation in the Universe Quantum Fluctuations of Vector Fields and the Primordial Curvature Perturbation in the Universe Mindaugas Karčiauskas MSc, BSc December 2009 This thesis is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy. No part of this thesis has been previously submitted for the award of a higher degree. arXiv:1009.1779v1 [astro-ph.CO] 9 Sep 2010 “What he [a scientist] is really seeking is to learn something new that has a cer- tain fundamental kind of significance: a hitherto unknown lawfulness in the order of nature, which exhibits unity in a broad range of phenomena. Thus, he wishes to find in the reality in which he lives a cer- tain oneness and totality, or wholeness, constituting a kind of harmony that is felt to be beautiful. In this respect, the scientist is perhaps not basically differ- ent from the artist, the architect, the music composer, etc., who all want to create this sort of thing in their work.” David Bohm Acknowledgments This thesis is a culmination of more than three years spend in England as a PhD student, which signify not only my scientific achievements but also people I met and moments experienced together. Although written by me, these pages include contributions from all these people without whom this would be just a collection of blank paper. Some of contributions are too subtle to put into words, but each of them is essential and invaluable. It is impossible to name everyone, so I sincerely apologize those whose name is not mentioned here. First and foremost I want to say thank you to Milda Beišyt˙e. Using letters and words it is impossible to express my infinite gratitude to you. You filled my every moment with life. Without you everything would be so much different. I will always be indebted to my family: my father Algirdas Karčiauskas, my mother Aldona Karčiauskien˙e,my brother Algirdas and his family as well as my grandmothers. Although they were far away, I always felt their unconditional support very closely. Ači¯u jums labai. I feel extremely lucky to have Konstantinos Dimopoulos as my supervisor. First of all, I am very grateful for a PhD project he assigned to me. It allowed to experience a real thrill, despair and excitement of research and discovery. I want to thank him for endless discussions about cosmology, science and life in general. They enriched my knowledge enormously and made me rethink many things which I took for granted. His guidance through the maze of all aspects of becoming a doctor was vital in many cases. Following the example of his previous student, I want to say as well ευχαριστ! πoλυ, K!στα! My PhD years, and especially the last one, would have been much poorer without David Lyth. I learned so many things from him and not only about cosmology. I am very grateful for his patience with my intelligent and not at all intelligent questions and for his encouragements. I am very much obliged to each member of the Cosmology and Astroparticle Physics Group for a particularly friendly and supportive atmosphere. Thank you John, Anupam, Kaz, Narendra, Juan, Chia-Min, Rose, Jacques, Francesca and Philip. A very special thank you I would like to say to Rasa Beišyt˙e. I cannot imagine how I v would have gone through my thesis writing if not you. I want to thank two dear friends I made in Lancaster. Thank you Ugn˙eGrigait˙efor many unforgettable moments we shared together exploring the English life and beauties of the Lake District. And thank you Art¯urasJasiukevičius, you were a close and much needed companion in experiencing cultural shocks during my first year in England. It is difficult to express my gratefulness to Paul Taylor and Rev. Wilfrid Powell for sharing their path. It was an island when drowning seemed the only option and it is a constant source of hope. In my last year I met a person with endless optimism and enthusiasm - Annmarie Ryan. Thank you for your drumming classes and thanks for all the group with whom we have been creating and sharing moments of beauty with African rhythms. It would be unfair if I didn’t mention my friends and comrades back in Lithuania. Knowing and feeling their presence was a great support while in England and it was always a great fun to visit them. Unfortunately, thickness limitations for this thesis does not allow to name each of you separately. But I am sure you know who you are. Thank you very much! The final touches and improvements to this thesis were made due to my examiners Dr. John McDonald and Prof. Anne-Christine Davis. Thank you for this and for approving me to become a Doctor of Philosophy. And finally I want to thank Physics Department of Lancaster University. Without their financial support this thesis and these acknowledgments would not even exist. I also want to thank the Faculty of Science and Technology and the William Ritchie travel fund for supporting my travels to conferences. 1 September 2010 vi Abstract The successes and fine-tuning problems of the Hot Big Bang theory of the Universe are briefly reviewed. Cosmological inflation alleviates those problems substantially and give rise to the primordial curvature perturbation with the properties observed in the Cosmic Microwave Background. It is shown how application of the quantum field theory in the exponentially expanding Universe leads to the conversion of quantum fluctuations into the classical field perturbation. The δN formalism is reviewed and applied to calculate the primordial curvature perturbation ζ for three examples: single field inflation, the end-of-inflation and the curvaton scenarios. The δN formalism is extended to include the perturbation of the vector field. The latter is quantized in de Sitter space-time and it is found that in general the particle production process of the vector field is anisotropic. This anisotropy is parametrized by introducing two parameters p and q, which are determined by the conformal invariance breaking mechanism. If any of them are non-zero, generated ζ is statistically anisotropic. Then the power spectrum of ζ and the non-linearity parameter fNL have an angular modulation. This formalism is applied for two vector curvaton models and the end-of-inflation sce- nario. It is found that for p 6= 0, the magnitude of fNL and the direction of its angular modulation is correlated with the anisotropy in the spectrum. If jpj & 1, the anisotropic part of fNL is dominant over the isotropic one. These are distinct observational signa- tures; their detection would be a smoking gun for a vector field contribution to ζ. In the first curvaton model the vector field is non-minimally coupled to gravity and in the second one it has a time varying kinetic function and mass. In the former, only statistically anisotropic ζ can be generated, while in the latter, isotropic ζ may be realized too. Parameter spaces for these vector curvaton scenarios are large enough for them to be realized in the particle physics models. In the end-of-inflation scenario fNL have similar properties to the vector curvaton scenario with additional anisotropic term. vii Contents 1. The Hot Big Bang and Inflationary Cosmology1 1.1. Kinematics of HBB . .1 1.2. Dynamics of the HBB . .3 1.3. Big Bang Nucleosynthesis . .6 1.4. The Problem of Initial Conditions of the Hot Big Bang . 11 1.4.1. The Flatness Problem . 11 1.4.2. The Horizon Problem . 14 1.4.3. The Origin of Primordial Perturbations . 15 1.5. Inflation . 17 1.5.1. The Accelerated Expansion . 17 1.5.2. The Scalar Field Driven Inflation . 19 1.5.3. The End of Inflation and Reheating . 22 2. The Origin of the Primordial Curvature Perturbation 25 2.1. Statistical Properties of the Curvature Perturbation . 25 2.1.1. Random Fields . 25 2.1.2. The Curvature Perturbation and Observational Constraints . 29 2.1.2.1. The Power Spectrum . 30 2.1.2.2. The Bispectrum . 31 2.2. Scalar Field Quantization . 33 2.2.1. Quantization in Flat Space-Time . 33 y 2.2.1.1. Interpretation of a^m and a^m ................ 36 2.2.2. Quantization in Curved Space-Time . 40 2.2.2.1. From FST to CST . 40 2.2.2.2. Bogolubov Transformations . 42 2.2.2.3. Quantization in Spatially Homogeneous and Isotropic Back- grounds . 43 2.2.2.4. The Vacuum State in FRW Background . 47 2.2.2.5. The Field Perturbation in the Inflationary Universe . 48 ix Contents 2.2.2.6. Quantum to Classical Transition . 51 2.3. The Primordial Curvature Perturbation . 53 2.3.1. Gauge Freedom in General Relativity . 54 2.3.2. Smoothing and The Separate Universe Assumption . 55 2.3.3. Conservation of the Curvature Perturbation . 57 2.3.4. The δN Formalism . 60 2.3.5. The Power Spectrum and Non-Gaussianity of ζ ........... 62 2.3.6. Density Perturbations . 63 2.4. Mechanisms for the Generation of the Curvature Perturbation . 65 2.4.1. Single Field Inflation . 65 2.4.2. At the End of Inflation . 69 2.4.3. The Curvaton Mechanism . 72 3. The Primordial Curvature Perturbation from Vector Fields 79 3.1. Vector Fields in Cosmology . 79 3.1.1. Conformal Invariance . 80 3.1.2. Large Scale Anisotropy . 81 3.1.3. The Physical Vector Field . 82 3.2. Vector Field Quantization and the Curvature Perturbation . 83 3.2.1. δN Formula with the Vector Field . 83 3.2.2. The Vector Field Quantization . 84 3.2.3.
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