Can Inflation Be Made Safe?
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MASTER DISSERTATION IMPERIAL COLLEGE LONDON DEPARTMENT OF PHYSICS THEORETICAL PHYSICS GROUP Can Inflation Be Made Safe? Author: Supervisor: Evita Gamber Prof. Kellogg Stelle Submitted in partial fulfillment of the requirements for the degree of MSc Quantum Fields and Fundamental Forces October 2020 Abstract It is claimed that an accelerated expansion era drove the universe to near flatness, ho- mogeneity and isotropy on large scales and stretched quantum fluctuations to today’s large scale structure. Many models have been developed in the past decades and the predictions made by inflation have been validated by cosmological measurements. In this thesis we will begin with its motivation and development. We will see that the theory (class) of inflation suffers from major problems. In the second part we dis- cuss the perturbative non-renormalisability of gravity, the asymptotic safety scenario and its renormalisation group treatment. Finally, we will explore whether and how inflation can be made safe. Acknowledgements Hereby, I would like to thank the Imperial Theory Group for the deep and broad knowl- edge I gained in the Qfff course and also thanks to my fellow peers for exciting dis- cussions and support. Flattening the curve wasn’t only important in inflation, we will all remember this year and probably proudly look back. It increased my curiosity in theoretical physics even more. I want to thank Prof Stelle and Prof Litim for their ad- vice in the past months. I am looking forward to proceeding this fascinating research during my PhD. Herzlichen Dank für die Unterstützung der Studienstiftung des deutschen Volkes. I would also like to thank my mum and all others that have supported me in the past years to fulfill my dream despite the difficulties outside the academic life and supported my curiosity and meticulosity. Lastly, I want to thank the science community from yesterday, today and tomorrow. Maths and physics have always been the best medicine . Contents 1 Introduction 10 1.1 Background and Aim of This Thesis................... 10 1.2 History of Inflation (and Asymptotic Safety)............... 12 1.3 Conventions................................ 13 2 Motivation 14 2.1 Cosmological Principle.......................... 14 2.2 Horizon Problem............................. 15 2.3 Flatness Problem............................. 16 2.4 Monopole Problem............................ 17 2.5 Large Scale Structure........................... 18 2.6 The Need for Quantum Gravity (?).................... 18 2.6.1 Asymptotic Safety......................... 18 3 Physics of Inflation 20 3.1 Theory................................... 20 3.2 How to Inflate - Constraints on Inflation................. 20 3.2.1 Horizon and Flatness Problem as First Constraints....... 21 3.2.2 Slow-Roll Condition........................ 22 3.2.3 LSS................................. 23 3.2.4 Monopole Problem........................ 24 3.3 Models of Inflation............................ 24 3.3.1 De Sitter model.......................... 25 3.3.2 Old Inflation............................ 25 3.3.3 New Inflation........................... 27 3.3.4 Chaotic Inflation.......................... 27 3.3.5 Eternal Inflation.......................... 28 3.3.6 Starobinsky Inflation....................... 30 3.4 How to End Inflation........................... 34 3.4.1 Reheating............................. 34 3.5 Observations................................ 38 3.5.1 Cosmological Perturbations.................... 38 3.6 Inflation’s Extensions and Possible Alternatives............. 51 3.6.1 Variable Speed of Light...................... 51 3.6.2 Global Structure.......................... 52 3.6.3 Inflation in Loop Quantum Gravity................ 56 3.7 Dark Energy and Inflation......................... 57 2 4 Problems of Inflation 61 4.1 Fine Tuning Again............................. 61 4.1.1 Penrose’s Alternative....................... 65 4.2 ’Multimess’................................. 68 4.3 Physics/Mechanism not Clearly Understood............... 70 4.3.1 ISL Criticism (Again)....................... 72 4.4 Probability Problem............................ 73 4.4.1 Measure Problem......................... 74 4.4.2 Fokker-Planck treatment..................... 78 4.5 Observations................................ 79 4.6 Falsifiable?................................. 82 4.7 Need for Quantum Gravity........................ 84 5 Asymptotic Safety and Inflation 88 5.1 Asymptotic Safety and Renormalisation Group............. 88 5.2 Renormalisation.............................. 93 5.3 Coupling constants............................ 95 5.4 The Advantage of Asymptotic Safety................... 96 5.4.1 Effective Field Theory....................... 98 5.5 Einstein-Hilbert Truncation........................ 101 5.5.1 RG Flow.............................. 102 5.5.2 Quantum Einstein Gravity..................... 105 5.5.3 FRG and Perturbation Theory................... 107 5.6 Inflation in ASG.............................. 111 5.6.1 RG Improvement......................... 111 5.6.2 Asymptotically Safe Starobinsky Inflation............ 117 5.6.3 Consistency Conditions...................... 119 5.6.4 Further Higher Derivative AS Inflation.............. 123 5.6.5 Ghosts Once More......................... 125 5.6.6 Finite Action............................ 126 5.6.7 Scale Invariance.......................... 128 5.6.8 Entropy Production........................ 129 5.6.9 Mathematical Tools........................ 129 5.7 Some Remarks............................... 129 6 Conclusion 133 6.1 Outlook.................................. 133 6.2 A Summary of Important Questions................... 135 7 Appendix 137 7.1 A Brief History of the Universe...................... 137 7.2 Scale Overview.............................. 137 7.3 Cosmological Standard Model...................... 139 7.3.1 SR................................. 142 7.3.2 Energy Conditions......................... 144 7.4 Scalar Fields in Cosmology........................ 145 7.5 Dicke’s Fine-Tuning............................ 146 3 7.6 Cosmological No-Hair Conjecture..................... 147 7.7 Further Calculations............................ 147 7.7.1 How Many Causally Unconnected Regions Were There?.... 147 7.7.2 Energy Scale at Inflation..................... 148 7.7.3 Magnetic Monopoles in Cosmology................ 148 7.7.4 Slow-Roll Parameters....................... 150 7.7.5 Size of the Universe........................ 152 7.7.6 CMB Temperatures........................ 152 7.7.7 Weyl Curvature.......................... 153 7.7.8 Starobinsky Powerspectrum.................... 155 7.8 VSL Constraints.............................. 157 7.9 The Wavefunction of the Universe.................... 158 7.10 The Top-Down Approach......................... 160 7.11 Dark Energy - Attractor in Quintessence................. 161 7.12 Quantum Gravity in Brief......................... 161 7.12.1 Linearised Gravity......................... 163 7.12.2 Quantum Corrected Newton Potential.............. 167 7.13 ASG.................................... 168 7.13.1 Example.............................. 168 7.13.2 ’Derivation’ ERGE......................... 171 7.13.3 Example β-functions....................... 172 7.13.4 Effective potential......................... 174 7.13.5 Ghost mass............................. 175 4 List of Figures 2.1 CMB 2018 Planck, apod.nasa.gov/apod/ap180722.html ....... 14 2.2 Horizon problem [12] past light cone from today and initial light cone from the Big Bang don’t meet at recombination. .................. 15 2.3 Flatness problem, Ω = 1 is very unstable. In order to have spatial flatness today, Ω had to be very close to 1 from the beginning onwards (astro.umd.edu 2015 class23) ................................ 16 3.1 Horizon problem, is solved. Upper: temporal evolution of the physical length (blue) vs Hubble radius (red). Lower: temporal evolution of the same, but comoving scales. During Inflation the Hubble radius is behind the growing a, structures initially inside it grow beyond that. Lidde’s lectures(ned.ipac. caltech.edu/level5/Liddle) ...................... 21 3.2 Slow-roll potentials that are possible. For SR single field inflation a flat long plateau, linear potential Vi = V0 ´ cφi (a), a more generic potential with different regions where on the saddles and maxima SR is possible (b), plateau with potential moving uphill at φ ă 0 (c) and hilltop with potential moving 1 2 2 down for φ ă 0 Vi = V0 ´ 2 m φi (d). ’Multi-field inflation on the landscape’, 2009. .................................... 24 3.3 Piston explanation, PS6 MIT ’The Early Universe’, Guth, 2011. ....... 25 3.4 Comparison between the old and new inflationary model. Former has a local minimum as false vacuum, latter has a local maximum for which both quantum and classical treatment is metastable. ............... 26 3.5 Eternal universe, ’Inflationary Models and Connections to Particle Physics’, Guth, 2000. ................................. 28 3.6 Starobinsky potential. The potential has a minimum at 0(V 2(0) ¡ 0), sat- 3m2 urates as φ Ñ at V0 = 32πG and diverges for φ Ñ ´ .The RHS is dom- inated by the R2 behaviour, the LHS the usual Einstein-Hilbert term. ’Semi- classical analysis1 of the tensor power spectrum in the Starobinsky1 inflationary model’, 2020. ................................ 33 3.7 Eternal chaotic universe: quantum fluctuations are superimposed on the clas- 1 2 2 sical motion, here for V = 2 m φ . For values greater than the Planck density quantum fluctuations dominate that we cannot describe without