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Hybrid and Multifield Inflation Evangelos I. Sfakianakis

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Hybrid and Multifield Inflation Evangelos I. Sfakianakis Hybrid and Multifield Inflation by Evangelos I. Sfakianakis Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 c Massachusetts Institute of Technology 2014. All rights reserved. Author.............................................................. Department of Physics May 22, 2014 Certified by. Alan H. Guth Victor F. Weisskopf Professor of Physics Thesis Supervisor Certified by. David I. Kaiser Germeshausen Professor of the History of Science and Senior Lecturer, Department of Physics Thesis Supervisor Accepted by . Krishna Rajagopal Associate Department Head for Education 2 Hybrid and Multifield Inflation by Evangelos I. Sfakianakis Submitted to the Department of Physics on May 22, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis I study the generation of density perturbations in two classes of infla- tionary models: hybrid inflation and multifield inflation with nonminimal coupling to gravity. • In the case of hybrid inflation, we developed a new method of treating these perturbations that does not rely on a classical trajectory for the fields. A characteristic of the spectrum is the appearance of a spike at small length scales, which could conceivably seed the formation of black holes that can evolve to become the supermassive black holes found at the centers of galaxies. Apart from numerically calculating the resulting spectrum, we derived an expansion in the number of waterfall fields, which makes the calculation easier and more intuitive. • In the case of multifield inflation, we studied models where the scalar fields are coupled non-minimally to gravity. We developed a covariant formalism and examined the prediction for non-Gaussianities in these models, arguing that they are absent except in the case of fine-tuned initial conditions. We have also applied our formalism to Higgs inflation and found that multifield effects are too small to be observable. We compared these models to the early data of the Planck satellite mission, finding excellent agreement for the spectral index and tensor to scalar ratio and promising agreement for the existence of isocurvature modes. Thesis Supervisor: Alan H. Guth Title: Victor F. Weisskopf Professor of Physics Thesis Supervisor: David I. Kaiser Title: Germeshausen Professor of the History of Science and Senior Lecturer, Department of Physics 3 4 Acknowledgments I am grateful to my supervisors, Prof. Alan Guth and Prof. David Kaiser, for their support and mentoring. I thank my thesis readers, Prof. Hong Liu and Prof. Max Tegmark for their advice. I also thank my collaborators, Prof. Ruben Rosales, Prof. Edward Farhi, Prof. Iain Stewart, Dr. Mark Hertzberg, Dr. Mustafa Amin, Ross Greenwood, Illan Halpern, Matthew Joss, Edward Mazenc and Katelin Schutz for their contribution to this thesis and beyond. I would like to thank my family and friends for their continued support and Angeliki for being there for me during the last two years. I am indebted to my fellow students, the staff of the CTP, Joyce Berggren, Scott Morley and Charles Suggs and the staff of the Physics Department for creating a vibrant environment. I thank the undergraduate students who worked with us, since they reminded me why I wanted to be a physicist, at a time when I needed it the most. I would not be here without the teachers and professors, in Greece and the US, who inspired me and I hope I have met their expectations. Last but not least I would like to thank George J. Elbaum and Mimi Jensen for their generosity that gave me the opportunity to pursue a PhD at MIT. This thesis is dedicated to the memory of my beloved grandmother. 5 6 Contents 1 Introduction 25 1.1 Inflation . 25 1.1.1 Simple Single field inflation . 27 1.1.2 Inflationary Perturbations . 28 1.2 Beyond the single field . 30 1.2.1 Hybrid Inflation . 31 1.2.2 Non-minimal coupling . 31 1.3 Organization of the thesis . 33 2 Density Perturbations in Hybrid Inflation Using a Free Field Theory Time-Delay Approach 37 2.1 Introduction . 38 2.2 Model . 41 2.2.1 Field set-up . 41 2.2.2 Fast Transition . 44 2.2.3 Mode expansion . 45 2.2.4 Solution of the mode function . 46 2.3 Supernatural Inflation models . 52 2.3.1 End of Inflation . 55 2.4 Perturbation theory basics . 56 2.4.1 Time delay formalism . 56 2.4.2 Randall-Soljacic-Guth approximation . 57 2.5 Calculation of the Time Delay Power Spectrum . 59 7 2.6 Numerical Results and Discussion . 64 2.6.1 Model-Independent Parameter Sweep . 66 2.6.2 Supernatural Inflation . 71 2.7 Conclusions . 77 2.8 Acknowledgements . 77 2.9 Appendix . 78 2.9.1 Zero mode at early times . 78 2.9.2 Initial Conditions . 80 3 Hybrid Inflation with N Waterfall Fields: Density Perturbations and Constraints 87 3.1 Introduction . 88 3.2 N Field Model . 90 3.2.1 Approximations . 92 3.2.2 Mode Functions . 94 3.3 The Time-Delay . 95 3.4 Correlation Functions . 98 3.4.1 Two-Point Function . 99 3.4.2 Three-Point Function . 102 3.4.3 Momentum Space . 105 3.5 Constraints on Hybrid Models . 107 3.5.1 Topological Defects . 107 3.5.2 Inflationary Perturbations . 108 3.5.3 Implications for Scale of Spike . 109 3.5.4 Eternal Inflation . 111 3.6 Discussion and Conclusions . 112 3.7 Acknowledgements . 113 3.8 Appendix . 114 3.8.1 Series Expansion to Higher Orders . 114 3.8.2 Two-Point Function for Even Number of Fields . 115 8 3.8.3 Alternative Derivation of Time-Delay Spectra . 116 4 Primordial Bispectrum from Multifield Inflation with Nonminimal Couplings 123 4.1 Introduction . 123 4.2 Evolution in the Einstein Frame . 126 4.3 Covariant Formalism . 137 4.4 Adiabatic and Entropy Perturbations . 144 4.5 Primordial Bispectrum . 154 4.6 Conclusions . 162 4.7 Acknowledgements . 164 4.8 Appendix . 164 4.8.1 Field-Space Metric and Related Quantities . 164 5 Multifield Dynamics of Higgs Inflation 175 5.1 Introduction . 175 5.2 Multifield Dynamics . 177 5.3 Application to Higgs Inflation . 182 5.4 Turn Rate . 187 5.5 Implications for the Primordial Spectrum . 197 5.6 Conclusions . 201 5.7 Acknowledgements . 202 5.8 Appendix . 203 5.8.1 Angular Evolution of the Field . 203 6 Multifield Inflation after Planck: The Case for Nonminimal Couplings 211 6.1 Introduction . 211 6.2 Multifield Dynamics . 213 6.3 The Single-Field Attractor . 217 6.4 Observables and the Attractor Solution . 219 9 6.5 Isocurvature Perturbations . 221 6.6 Conclusions . 221 6.7 Acknowledgements . 222 7 Multifield Inflation after Planck: Isocurvature Modes from Non- minimal Couplings 229 7.1 Introduction . 229 7.2 Model . 232 7.2.1 Einstein-Frame Potential . 233 7.2.2 Coupling Constants . 236 7.2.3 Dynamics and Transfer Functions . 238 7.3 Trajectories of Interest . 243 7.3.1 Geometry of the Potential . 243 7.3.2 Linearized Dynamics . 247 7.4 Results . 251 7.4.1 Local curvature of the potential . 252 7.4.2 Global structure of the potential . 254 7.4.3 Initial Conditions . 256 7.4.4 CMB observables . 257 7.5 Conclusions . 263 7.6 Acknowledgments . 265 7.7 Appendix . 265 7.7.1 Approximated Dynamical Quantities . 265 7.7.2 Covariant formalism and potential topography . 267 10 List of Figures 1-1 A pictorial form of the hybrid potential. 32 2-1 Mode functions for different comoving wavenumbers as a function of 1 time in efolds. The model parameters are µ = 10 and µφ = 10. We can see the modes following our analytic approximation for the growth rate. Our analysis gives Ndev(1=256) ≈ −7:9 and Ntr(256) ≈ 4:76, which are very close to the values that can be read off the graph. 51 2-2 Mode functions for different comoving wavenumbers as a function of 1 time in efolds. The model parameters are µ = 2 and µφ = 1. The horizontal line corresponds to the asymptotic value of the growth fac- tor λ. We can see how.
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