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Radiative Corrections in Curved Spacetime and Physical Implications to the Power Spectrum and Trispectrum for Different Inflationary Models

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Radiative Corrections in Curved Spacetime and Physical Implications to the Power Spectrum and Trispectrum for Different Inflationary Models Radiative Corrections in Curved Spacetime and Physical Implications to the Power Spectrum and Trispectrum for different Inflationary Models Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades Doctor rerum naturalium der Georg-August-Universit¨atG¨ottingen Im Promotionsprogramm PROPHYS der Georg-August University School of Science (GAUSS) vorgelegt von Simone Dresti aus Locarno G¨ottingen,2018 Betreuungsausschuss: Prof. Dr. Laura Covi, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Karl-Henning Rehren, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universit¨atG¨ottingen Miglieder der Prufungskommission:¨ Referentin: Prof. Dr. Laura Covi, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Korreferentin: Prof. Dr. Dorothea Bahns, Mathematisches Institut, Universit¨atG¨ottingen Weitere Mitglieder der Prufungskommission:¨ Prof. Dr. Karl-Henning Rehren, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Stefan Kehrein, Institut f¨urTheoretische Physik, Universit¨atG¨ottingen Prof. Dr. Jens Niemeyer, Institut f¨urAstrophysik, Universit¨atG¨ottingen Prof. Dr. Ariane Frey, II. Physikalisches Institut, Universit¨atG¨ottingen Tag der mundlichen¨ Prufung:¨ Mittwoch, 23. Mai 2018 Ai miei nonni Marina, Palmira e Pierino iv ABSTRACT In a quantum field theory with a time-dependent background, as in an expanding uni- verse, the time-translational symmetry is broken. We therefore expect loop corrections to cosmological observables to be time-dependent after renormalization for interacting fields. In this thesis we compute and discuss such radiative corrections to the primordial spectrum and higher order spectra in simple inflationary models. We investigate both massless and massive virtual fields, and we disentangle the time dependence caused by the background and by the initial state that is set to the Bunch-Davies vacuum at the beginning of inflation. For the investigated models, we find that the radiative corrections to the primordial spectrum result in oscillatory features that are not present at tree-level. These features are also present in higher order spectra and depend on the initial conditions of the theory. In all the investigated cases the departure from near-scale invariance and from Gaussianity is very small and it is in full agreement with the current Planck constraints. Future cosmic microwave background measurements may improve the current limits on feature-full primordial spectra, giving the hope to observe these effects in the scenario of hybrid inflation. Keywords: Primordial Power Spectrum, Trispectrum, Inflationary Perturbations, Inflaton Field, Slow-Roll Inflation, Hybrid Inflation, Chaotic Inflation, Non-Linearity Parameter, De-Sitter Spacetime, Renormalization in Curved Spacetime, Radiative Corrections, Finite Time Contributions, CTP Formalism, WKB Propagator, Hyperge- ometric Propagator vi CONTENTS 1 Introduction1 2 Cosmic inflation3 2.1 The Friedmann Lema^ıtreRobertson Walker universe . .3 2.2 Motivations for inflation . .7 2.3 Slow-roll inflation . .8 2.4 Inflationary perturbations . 10 2.5 Selection of classes of models . 15 3 Quantum field theory in curved spacetime 19 3.1 Introduction . 20 3.2 Renormalization in Minkowski spacetime . 22 3.3 Adiabatic renormalization . 24 3.4 From distribution theory to the Epstein-Glaser renormalization . 28 4 The Schwinger and Keldysh formalism 37 4.1 Introduction . 38 4.2 Perturbation theory . 40 4.3 CTP propagators . 42 5 Renormalization in the CTP formalism 49 5.1 The equal-time two-point correlation function . 50 5.2 The equal-time four-point correlation function . 59 5.3 Massive WKB counter-term in de-Sitter spacetime . 74 5.4 Counter-terms in the adiabatic regularization scheme . 76 viii CONTENTS 6 Interaction profile and the adiabatic limit 81 6.1 Profile dependence of the two-point correlation function . 82 6.2 Proper definition of the adiabatic limit of the tadpole . 89 7 Cosmological applications of renormalization in de-Sitter spacetime 93 7.1 Classical dynamics of the inflationary models . 93 7.2 Radiative corrections to the primordial power spectrum . 101 7.3 One-loop corrections to the power spectrum with a monomial interaction103 7.4 One-loop corrections to the power spectrum in hybrid inflation . 104 7.5 The non-linearity parameter τNL ...................... 112 8 Summary and conclusions 115 Appendix A Feynman rules 119 A.1 λφ4-theory . 119 A.2 Hybrid model . 121 Appendix B Useful integrals 123 Appendix C Various results 125 Bibliography 135 Acknowledgements 147 Curriculum Vitæ 155 LIST OF FIGURES 2.1 Example of a slow-roll single-field inflation . 16 3.1 Cutoff function θn(x) for different parameters . 33 3.2 Singular structure of different propagators in Minkowski spacetime . 36 4.1 Closed-time contour C representing the time-ordering TC ........ 39 4.2 Hadamard propagators derived from the hypergeometric function for different masses . 45 4.3 Comparison of the massive Hadamard propagator reconstructed from the hypergeometric function and the WKB propagator . 47 5.1 Tree-level equal-time two-point correlation function . 50 5.2 Tadpole diagram and its counter-term for a quartic self-interacting scalar field theory in Minkowski spacetime . 51 5.3 Tadpole diagram and its counter-term for a quartic self-interacting scalar field theory in de-Sitter spacetime . 55 5.4 Cosine integral and sine integral . 58 5.5 Time dependence of the renormalized two-point correlation function in de-Sitter spacetime . 59 5.6 Tree-level contributions T1, T2 and T1 +T2 from the massive propagators in Minkowski spacetime . 61 5.7 Tree-level contributions T1, T2 and T1 +T2 from the massless propagators in de-Sitter spacetime . 63 5.8 Fish diagrams A1 and A2 .......................... 64 5.9 Tree-level and one-loop contributions to the four-point function . 73 ix x List of Figures 5.10 Mass dependence of the amputated tadpole diagram after renormaliza- tion with the WKB couter-term . 75 5.11 Relative difference between the Hypergeometric and the WKB propaga- tor for a scalar field of mass 60H ..................... 76 6.1 Interaction profiles with different extensions during the transition . 82 6.2 Renormalized two-point function using different continuous interaction profiles λ(t)................................. 84 6.3 Switching-on profile constructed from the arc-tangent function with different parameters . 84 6.4 Renormalized two-point function using the arc-tangent interaction profile 85 6.5 Interaction profile dependence of the renormalized tadpole diagram in de-Sitter spacetime . 88 6.6 One-parameter family of test functions fs(t) for the adiabatic limit . 90 6.7 Time-evolution of the tadpole amplitude for different test functions fs(t) 90 6.8 Tadpole amplitude as a function of the continuous parameter s in fs(t) for a fixed time . 91 7.1 Relative difference and power spectrum for a scalar field theory with monomial interaction . 104 7.2 Oscillatory contribution arising from the external propagators . 106 7.3 Relative difference and renormalized power spectrum for hybrid inflation108 7.4 Relative difference and renormalized power spectrum for the hybrid supersymmetric model . 109 7.5 Relative difference and renormalized power spectrum for the spectator field model . 111 7.6 Non-linearity parameter jτNLj calculated from the contribution T1 + T2 for a λφ4-theory . 113 LIST OF TABLES 2.1 Solutions of the Einstein equation for a spatially flat universe dominated by a de-Sitter era or by a non-relativistic matter or radiation epoch . .6 5.1 Inequivalent Feynman diagrams for the tree-level contributions to the four-point function from a massive scalar field in Minkowski spacetime 60 5.2 Inequivalent Feynman diagrams for the tree-level contributions to the four-point function from a (nearly) massless scalar field in de-Sitter spacetime . 62 5.3 Inequivalent Feynman diagrams for the one-loop correction to the four- point function . 66 A.1 Feynman rules for the λφ4 theory . 120 A.2 Feynman rules for a two scalar field theory . 122 xi xii List of Tables CHAPTER 1 INTRODUCTION Cosmic inflation [1{3] is a theory that describes an accelerating period of expansion of the primordial universe. It is celebrated as one of the most successful paradigms in cosmology. The success has come through relating the primordial fluctuations [4, 5] to observational data as the temperature anisotropies of the cosmic microwave background or the observed large scale structure of the universe. Moreover, inflation provides a solution to many issues that existed before the 1980s, e.g. to understand the homogeneity and the flatness of our universe. The spectrum of the primordial perturbations of the curvature tensor is predicted to be nearly scale invariant in the slow-roll scenario and the current observations [6] are consistent with the predictions of a simple single-field inflationary model [7,8]. The classical dynamics of different inflationary scenarios has been studied in the last 40 years. Many effects were already addressed in the past for different models [9{16], in- cluding non-gaussianities [17, 18] or oscillatory features in the primordial spectrum [19]. Less attention has been devoted to the understanding of the radiative corrections to the correlation functions of the inflaton field. Since Lorentz invariance is broken because of the expansion of the universe, one expects an intrinsic time dependence in the quantum corrections arsing from the evolving background. The main goal of this thesis is to investigate the time dependence of the one-loop radiative corrections to the two- and four-point correlation functions that arises both because of the background evolution and the initial conditions. The natural theoretical framework is the Schwinger and Keldysh formalism, where the time-evolution of expectation values is easily calculated, including quantum effects. In this scenario the system is supposed to be in the Bunch-Davies [20] vacuum at the beginning of inflation, giving periodic contributions to the physical observables. The initial time dependence 2 Introduction is investigated by testing the correlation functions with different interaction profiles and by addressing the question of the adiabatic limit.
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