Frame Covariance and Fine Tuning in Inflationary Cosmology

FRAME COVARIANCE AND FINE TUNING IN INFLATIONARY COSMOLOGY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 By Sotirios Karamitsos School of Physics and Astronomy Contents <ul style="display: flex;"><li style="flex:1">Abstract </li><li style="flex:1">8</li></ul><ul style="display: flex;"><li style="flex:1">9</li><li style="flex:1">Declaration </li></ul>Copyright Statement Acknowledgements 1 Introduction 10 11 13 1.1 Frames in Cosmology: A Historical Overview . . . . . . . . . . . 13 1.2 Modern Cosmology: Frames and Fine Tuning . . . . . . . . . . 15 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 <ul style="display: flex;"><li style="flex:1">2 Standard Cosmology and the Inflationary Paradigm </li><li style="flex:1">20 </li></ul>2.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The Hot Big Bang Model . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 The Expanding Universe . . . . . . . . . . . . . . . . . . 26 2.2.2 The Friedmann Equations . . . . . . . . . . . . . . . . . 29 2.2.3 Horizons and Distances in Cosmology . . . . . . . . . . . 33 2.3 Problems in Standard Cosmology . . . . . . . . . . . . . . . . . 34 2.3.1 The Flatness Problem . . . . . . . . . . . . . . . . . . . 35 2.3.2 The Horizon Problem . . . . . . . . . . . . . . . . . . . . 36 2 2.4 An Accelerating Universe . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Inflation: More Questions Than Answers? . . . . . . . . . . . . 40 2.5.1 The Frame Problem . . . . . . . . . . . . . . . . . . . . 41 2.5.2 Fine Tuning and Initial Conditions . . . . . . . . . . . . 45 <ul style="display: flex;"><li style="flex:1">3 Classical Frame Covariance </li><li style="flex:1">48 </li></ul>3.1 Conformal and Weyl Transformations . . . . . . . . . . . . . . . 48 3.2 Conformal Transformations and Unit Changes . . . . . . . . . . 51 3.3 Frames in Multifield Scalar-Tensor Theories . . . . . . . . . . . 55 3.4 Dynamics of Multifield Inflation . . . . . . . . . . . . . . . . . . 63 <ul style="display: flex;"><li style="flex:1">4 Quantum Perturbations in Field Space </li><li style="flex:1">70 </li></ul>4.1 Gauge Invariant Perturbations . . . . . . . . . . . . . . . . . . . 71 4.2 The Field Space in Multifield Inflation . . . . . . . . . . . . . . 74 4.3 Frame-Covariant Observable Quantities . . . . . . . . . . . . . . 78 4.3.1 The Potential Slow-Roll Hierarchy . . . . . . . . . . . . . 81 4.3.2 Isocurvature Effects in Two-Field Models . . . . . . . . . 83 <ul style="display: flex;"><li style="flex:1">5 Fine Tuning in Inflation </li><li style="flex:1">88 </li></ul>5.1 Initial Conditions Fine Tuning . . . . . . . . . . . . . . . . . . . 88 5.2 Parameter Fine Tuning . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Trajectory Fine Tuning . . . . . . . . . . . . . . . . . . . . . . . 94 <ul style="display: flex;"><li style="flex:1">6 Models of Inflation </li><li style="flex:1">97 </li></ul>6.1 Single-Field Models . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.1 Induced Gravity Inflation . . . . . . . . . . . . . . . . . 98 6.1.2 Higgs Inflation . . . . . . . . . . . . . . . . . . . . . . . 102 6.1.3 F(R) Models . . . . . . . . . . . . . . . . . . . . . . . . 104 3 6.2 Multifield Models . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2.1 Minimal Two-Field Inflation . . . . . . . . . . . . . . . . 107 6.2.2 Non-minimal Two-Field Inflation . . . . . . . . . . . . . 114 6.2.3 F(ϕ, R) Models . . . . . . . . . . . . . . . . . . . . . . . 120 <ul style="display: flex;"><li style="flex:1">7 Beyond the Tree Level </li><li style="flex:1">122 </li></ul>7.1 The Conventional Effective Action . . . . . . . . . . . . . . . . 123 7.2 The Vilkovisky Effective Action . . . . . . . . . . . . . . . . . . 127 7.3 The De Witt Effective Ection . . . . . . . . . . . . . . . . . . . 130 7.4 The Conformally Covariant Vilkovisky–De Witt Formalism . . 132 <ul style="display: flex;"><li style="flex:1">8 Conclusions </li><li style="flex:1">137 </li></ul>140 147 A Frame-Covariant Power Spectra Bibliography 4List of Tables 2.1 Evolution of energy density and scalar factor for different eras of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 Density parameters for different components of the Universe [53, 55]. The total radiation density is the sum of the photon density and the neutrino density. . . . . . . . . . . . . . . . . . 33 3.1 Conformal weights and scaling dimensions of various framecovariant quantities. . . . . . . . . . . . . . . . . . . . . . . . . 67 6.1 Observable inflationary quantities for the minimal two-field model at N = 60. Note that the running of the tensor spectral index αT is not quoted in [54], as no tensor modes were measured by PLANCK. It is derived from the consistency relation (4.42) with transfer angle Θ = 0, and serves as a constraint on a possible future measurement of αT , in the slow-roll approximation. The parameter βiso is constrained by assuming different nondecaying isocurvature modes: (i) the cold dark matter density isocurvature mode (CDI), (ii) the neutrino density mode (NDI), and (iii) the neutrino velocity mode (NVI). . . . . . . . . . . . 113 6.2 Observable inflationary quantities for the non-minimal model at N = 60. The limits on these quantities from 2015 PLANCK data [54] are the same as in Table 6.1. . . . . . . . . . . . . . . 118 5List of Figures 1.1 Illustration of quantisation in different frames. It is not im- JF 1−loop EF 1−loop <ul style="display: flex;"><li style="flex:1">mediately obvious that Γ </li><li style="flex:1">= Γ </li><li style="flex:1">. Figure reproduced </li></ul>from [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1 Schematic representation of the metric expansion of the Universe. As time passes, the “density” uniformly decreases such that points recede from each other at a speed proportional to their distance (Hubble’s law). . . . . . . . . . . . . . . . . . . . 28 5.1 Probability that a randomly selected value for the α-attractor potential corresponds to an initially inflationary trajectory in the slow-roll approximation. . . . . . . . . . . . . . . . . . . . . 92 5.2 Trajectory flow between two isochrone surfaces N(ϕ) = N1 and N(ϕ) = N2 in field space. . . . . . . . . . . . . . . . . . . . . . . 96 6.1 End-of-inflation curve for the minimal two-field model (6.42) with m2/(λMP2 ) = 1. . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Field space trajectories and isochrone curves for the minimal two-field model (6.42). . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 Sensitivity parameter Q for the minimal model (6.42) at N = ∗60 to boundary conditions given by ϕ0. The dashed line corresponds to Q = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 110 ∗6 6.4 Power spectrum normalisation for the minimal two-field model (6.42) with λ = 10−12 and m/MP = 10−6 at N = 60 for different boundary conditions in terms of ϕ0 and the corresponding horizon crossing values ϕ . Solid lines correspond to the theoretical ∗predictions while the horizontal line corresponds to the observed power spectrum PRobs given in (6.15). . . . . . . . . . . . . . . . 111 6.5 Predictions for the inflationary quantities r, n , α , αT , fNL <ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li></ul>and βiso in the minimal model (6.42) for boundary condition given by ϕ0/MP = 0.496. . . . . . . . . . . . . . . . . . . . . . . 112 6.6 Evolution of ω and η¯ss along the inflationary trajectory with ϕ0 = 0.495 for the minimal two-field model (6.42). . . . . . . . 113 6.7 End-of-inflation curve for the non-minimal model (6.47)with m = 5.6 × 10−6 MP , λ = 10−12, and ξ = 0.01. . . . . . . . . . . . 115 6.8 Field space trajectories and isochrone curves for the non-minimal two-field model (6.47). . . . . . . . . . . . . . . . . . . . . . . . 115 6.9 Power spectrum normalisation for the non-minimal two-field model (6.47) with m = 5.6 × 10−6MP , λ = 10−12, and ξ = 0.01 for different boundary conditions in terms of ϕ0 and the corresponding horizon crossing values ϕ . Solid lines correspond to ∗the theoretical predictions while the horizontal dashed lines correspond to the allowed band for the observed power spectrum PRobs given in (6.15). . . . . . . . . . . . . . . . . . . . . . . . . 116 <ul style="display: flex;"><li style="flex:1">6.10 Predictions for the inflationary quantities r, n , α , αT , fNL </li><li style="flex:1">,</li></ul><ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li></ul>and βiso in the non-minimal two field model (6.47) for boundary conditions admissible under normalisation of P to the observed Rpower spectrum PRobs. . . . . . . . . . . . . . . . . . . . . . . . . 117 6.11 Sensitivity parameter Q for the non-minimal two-field model (6.47) ∗to boundary conditions given by ϕ0. The dashed line corresponds to Q = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 118 ∗6.12 Evolution of ω and η¯ss along the observationally viable inflationary trajectories for the non-minimal two-field model (6.47). . 119 7Abstract This thesis presents the development of a fully covariant approach to scalartensor theories of gravity in the context of inflation, as well as a covariant treatment of trajectory fine tuning in multifield models. Our main result is the introduction of frame covariance as a way to confront the frame problem in inflation. We treat the choice of a gravitational frame in which a theory is presented as a particular instance of gauge fixing. We take frame covariance beyond the tree level by virtue of the Vilkovisky–De Witt formalism, which was originally developed with the aim of removing the gauge and reparametrisation ambiguities from the path integral. Adopting an analogous approach, we incorporate conformal covariance to the Vilkovisky–De Witt formalism, demonstrating that the choice of a conformal frame is not physically important. This makes it possible to define a unique action even in the presence of matter couplings. We therefore show that even if the matter picture of the Universe may appear different in conformally related models, the underlying theory is independent of its frame representation. We further examine the relation between parameter fine tuning and initial condition fine tuning. Even though conceptually distinct, they both adversely affect the robustness of using established particle physics models to drive inflation. As a way to remedy this, we note that the presence of additional scalar degrees of freedom can “rescue” particular models that are ruled out observationally by shifting the burden of fine tuning from the parameters to the choice of slow-roll trajectory. We refer to this uniquely multifield phenomenon as “trajectory fine tuning”, and we propose a method to quantify the sensitivity of multifield models to it. We illustrate by presenting examples of both single-field and multifield models of inflation, as well as F(R) and F(ϕ, R) theories. Wordcount: 28617 8Declaration I declare that no portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 9Copyright Statement The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. The ownership of certain Copyright, patents, designs, trademarks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see <a href="/goto?url=http://documents.manchester.ac.uk/ DocuInfo.aspx?DocID=24420" target="_blank">http://documents.manchester.ac.uk/DocuInfo. </a><a href="/goto?url=http://documents.manchester.ac.uk/ DocuInfo.aspx?DocID=24420" target="_blank">aspx?DocID=24420</a>), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see <a href="/goto?url=http://www.library.manchester.ac.uk/about/regulations/" target="_blank">http: </a><a href="/goto?url=http://www.library.manchester.ac.uk/about/regulations/" target="_blank">//www.library.manchester.ac.uk/about/regulations/</a>) and in The Uni- versity’s policy on Presentation of Theses. 10 Acknowledgements I would first like to thank my supervisor Apostolos Pilaftsis for his guidance and support throughout my doctorate. His experience, insight, and attention to detail have been invaluable during my stay in Manchester and my first steps in research. I wholeheartedly appreciate both his patience during the early stages of my PhD and the many skills he has since imparted to me. I would also like to thank Fedor Bezrukov, whose work on Higgs inflation was the catalyst for my first venture in research and with whom I have had plenty of stimulating discussions. I would like to thank everybody in the Theory Office for both the physics discussions as well as always keeping the office lively and interesting. These ínclude Daniele Teresi, René Angeles-Martinez, Graeme Nail, Alex Powling, Daniel Burns, Matthew De Angelis, Kiran Ostrolenk, Kieran Finn, Chris Shepherd, and Jack Holguin. I would especially like to thank Daniel Burns for the countless questions he so patiently answered during my first couple of years, Kieran Finn for the incisive and thought-provoking questions he asked me in turn, and the both of them for the amazing collaboration opportunities they provided me with. Finally, I would like to thank the entire Manchester Particle Physics Group for making the department an especially friendly and welcoming environment to work in. 11 12 I dedicate this thesis to my parents. If they hadn’t gifted me a book titled Physics for Children when I was eight years old, I may have never set forth to write it. Chapter 1 Introduction Humans have always been fascinated with the Great Questions, and there is arguably no question greater than that of our place in the Universe. In their attempts to formulate an answer as to why and how we came to be here, ancient cultures devised a wide range of imaginative narratives known as cosmogonic myths. Examples include the creation of the world by an all-powerful being, its birth by a pair of “world parents” or a cosmic “egg”, or even its creation from the primordial chaos itself. These myths permeated almost every facet of cultural life at the time [1], which made them uniquely anthropocentric: they begin and end with humanity and its place in the Universe in mind. As a result, many early pictures of cosmology placed Earth in a prominent position at the very center of the cosmos. This stands in sharp contrast to the modern view that there is nothing special about our position in the Universe, which came to be after several paradigm shifts spanning millennia of history. 1.1 Frames in Cosmology: A Historical Overview There are many ways to motivate the idea that the Earth does not enjoy a privileged place in the Universe. Ultimately, it may be argued that it stems from the idea that the laws of Nature should be identical under different observers. Indeed, if Nature does not unduly privilege any particular observer over any other, it follows that there should be no preferred frame of reference. Thus, it makes no sense to ascribe any special importance to the position of the Earth, much less put it in the very center of the Universe. As such, we 13 14 CHAPTER 1. INTRODUCTION understand today that physical laws must be frame independent. The journey from a frame-dependent to a frame-independent formulation of laws of Nature arguably began with the geocentrism-heliocentrism debate in ancient Greece, which is one of the earliest examples of a debate on whether a particular frame of reference is inherently better suited to describing the world. The very realisation that both geocentrism and heliocentrism are simply different frames of reference was one of the key steps in understanding the notion that there can be different but physically equivalent ways to describe the Universe. The geocentric model was first proposed by Plato in the Timaeus, and subsequently refined by Aristotle [2]. It described a system in which the Earth is stationary at the centre of the Universe. In the Aristotelean model, earth (the heaviest element) is surrounded by layers of water, air, and fire, while the other planets (made of aether) are confined in celestial spheres concentric with the Earth. This model was later further refined by Ptolemaeus in the Almagest, which superseded previous work on astronomy [3]. In an attempt to explain retrograde motion, Ptolemaeus suggested that the trajectories of celestial bodies are confined to epicycles, whose centers further orbited the Earth in concentric spheres known as deferents. However, in this model, the centers of the epicycles moved at a constant speed, contrary to observations. As such, Ptolemy went on to suggest that planets moved in a circle around the Earth, but their motion was uniform around a point named the equant. This model was quite unwieldy, but it was consistent with Aristotelean philosophy and its predictions sufficiently matched observations. Thus, the Ptolemaic model became the gold standard for cosmology throughout the Middle Ages and most of the Renaissance. Heliocentric models had been proposed by the Pythagoreans as early as the 4th century BC and by Aristarchus in the 3rd century BC [4], but they did not gain widespread acceptance until the so-called Copernican Revolution. Nicolaus Copernicus explained the apparent retrograde motion of the planets by making the radical suggestion that they orbit the Sun along with the Earth. This proposal was met with resistance by the Church, and so the Ptolemaic system remained as the dominant cosmological model. Tycho Brahe, motivated in part by scripture, attempted to reconcile the Copernican and the Ptolemaic models by suggesting that the Earth is orbited by Mercury, Venus, and the Sun, which is in turn orbited by the rest of the planets [5]. This so-called Tychonic model was not popular with astronomers until Galileo’s observation of the phases of Venus confirmed that it does orbit the Sun. The Catholic 1.2. MODERN COSMOLOGY: FRAMES AND FINE TUNING 15 Church had been reluctantly tolerant of heliocentric ideas up until that point, as long as the implication was that they were only mathematical tools to help study orbital mechanics. However, they would not abide the suggestion the Earth is not actually at the centre of the Universe, leading to Galileo’s wellknown trial and his recanting of heliocentrism. It would not be until Newton’s law of universal attraction that geocentric models were finally eclipsed and the heliocentric system saw widespread acceptance. The long history of the debate between the geocentric and heliocentric systems was characterised by the gradual acceptance of the idea that the Earth does not occupy a special place in the Universe and no observer is more privileged than any other. This concept is known as the Copernicean principle, and can be gleaned by comparing the Ptolemaic, Copernicean and Tychonic systems: their descriptions of the motion of celestial bodies are identical. They are entirely equivalent in the sense that it is impossible to distinguish between them via observations [6]. Thus, in a certain sense, these systems are not physically distinct models but rather distinct frames of the same underlying model. Certain frames might be more physically intuitive than others: the non-inertial Earth frame of reference in the Ptolemaic system requires the addition of fictitious forces in order to constrain the movement of the planets on the epicycles, for instance. However, there is no inherent reason to prefer one frame over the other apart from convenience and ease of calculation.

Frame Covariance and Fine Tuning in Inflationary Cosmology

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