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

Abstract 8

Declaration 9

Acknowledgements 11

1 Introduction 13

1.1 Frames in Cosmology: A Historical Overview ...... 13

1.2 Modern Cosmology: Frames and Fine Tuning ...... 15

1.3 Outline ...... 17

2 Standard Cosmology and the Inﬂationary Paradigm 20

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

3 Classical Frame Covariance 48

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

4 Quantum Perturbations in Field Space 70

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

5 Fine Tuning in Inﬂation 88

5.1 Initial Conditions Fine Tuning ...... 88

5.2 Parameter Fine Tuning ...... 92

5.3 Trajectory Fine Tuning ...... 94

6 Models of Inﬂation 97

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

7 Beyond the Tree Level 122

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

8 Conclusions 137

A Frame-Covariant Power Spectra 140

Bibliography 147

4 List 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 frame- covariant 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 pos-

sible future measurement of αT , in the slow-roll approximation.

The parameter βiso is constrained by assuming diﬀerent non- decaying 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

5 List of Figures

1.1 Illustration of quantisation in diﬀerent frames. It is not im- JF EF mediately obvious that Γ1−loop = Γ1−loop. Figure reproduced from [14]...... 16

2.1 Schematic representation of the metric expansion of the Uni- verse. 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) 2 2 with m /(λMP ) = 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 corre-

sponds to Q∗ = 1...... 110

6 6.4 Power spectrum normalisation for the minimal two-ﬁeld model (6.42) −12 −6 with λ = 10 and m/MP = 10 at N = 60 for diﬀerent

boundary conditions in terms of ϕ0 and the corresponding hori-

zon crossing values ϕ∗. Solid lines correspond to the theoretical predictions while the horizontal line corresponds to the observed obs power spectrum PR given in (6.15)...... 111

6.5 Predictions for the inﬂationary quantities r, nR, αR, αT , fNL

and βiso in the minimal model (6.42) for boundary condition

given by ϕ0/MP = 0.496...... 112

6.6 Evolution of ω andη ¯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 −6 −12 m = 5.6 10 MP , λ = 10 , 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 −6 −12 model (6.47) with m = 5.6 10 MP , λ = 10 , and ξ = 0.01 × for diﬀerent boundary conditions in terms of ϕ0 and the corre-

sponding 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 obs PR given in (6.15)...... 116

6.10 Predictions for the inﬂationary quantities r, nR, αR, αT , fNL,

and βiso in the non-minimal two ﬁeld model (6.47) for boundary

conditions admissible under normalisation of PR to the observed obs power spectrum PR ...... 117

6.11 Sensitivity parameter Q∗ for the non-minimal two-ﬁeld model (6.47)

to boundary conditions given by ϕ0. The dashed line corre-

sponds to Q∗ = 1...... 118

6.12 Evolution of ω andη ¯ss along the observationally viable inﬂa- tionary trajectories for the non-minimal two-ﬁeld model (6.47). . 119

7 Abstract

This thesis presents the development of a fully covariant approach to scalar- tensor 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.

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8 Declaration

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.

9 Copyright Statement

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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 include Daniele Teresi, René Angeles-Martinez,´ Graeme Nail, Alex Powling, Daniel Burns, Matthew De Angelis, Kiran Ostrolenk, Kieran Finn, Chris Shep- herd, 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 real- isation 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 well- known 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.

1.2 Modern Cosmology: Frames and Fine Tuning

In hindsight, the geocentrism-heliocentrism debate was ultimately a clash between different frames and not models. It was settled by the insight that our description of Nature should be frame-independent. However, a similar issue regarding frames in the context of modified gravity theories arose in the last century. This so-called frame problem concerns the question whether the Jor- dan frame or the Einstein frame should be regarded as more physical than the other [7]. In the former, the effective Planck mass is promoted to a variable quantity, whereas in the latter, it remains constant. There has been consider- able debate as to whether the choice of frame is physically important, and while some consensus has been achieved for the classical case [8–13], there are still outstanding issues when matter couplings enter the picture. The situation becomes even more complicated when one considers radiative corrections [14–18], – Einstein versus Jordan Frame Action in Enstein Frame: SEF[g ,ϕ]= 1M 2 R + 1(∂ ϕ)2 V (ϕ) µν x − 2 P 2 µ − 16 Action in Jordan Frame: SJF[˜g , ϕ]= R CHAPTER1f(ϕ)R + 1 1.(∂ ϕ INTRODUCTION)2 V (ϕ) µν x − 2 2 µ − JF EF Frame equivalence = S [˜gµν, ϕ]R = S [gµν,ϕ] [R. H. Dicke ’62] ⇒ e e e e e e Fieldfield reparametrization parametrization SJF e SEF

“Quantization”

JF, div EF, div Γ1 loop Fieldfield reparametrization parametrization Γ1 loop − − JF EF Γ1 loop[˜gµν, ϕ] = Γ1 loop[gµν, ϕ]: Effective action is frame dependent. Figure 1.1:− Illustration6 of− quantisation in different frames. It is not immedi- ately obvious that ΓJF = ΓEF . Figure reproduced from [14]. e 1−loop 1−loop Corfu 2017 On the Cosmological Frame Problem A. Pilaftsis since it is not guaranteed that the independence of a theory on the choice of frame will necessarily persist after quantisation, as seen in Figure 1.1.

The frame problem has attracted particular attention in the context of cosmic inflation. The theory of inflation was originally proposed as a way to resolve certain long-standing issues with standard cosmology, such as the flatness and horizon problems [19–21]. Eventually, inflation was discovered to provide a generic way to generate primordial perturbations that eventually grow into observable anisotropies in the cosmic microwave background (CMB) [22–24]. Many inflationary models are inspired by modified gravity with or without additional scalar degrees of freedom, and as such, the frame problem is also present in theories of inflation where the effective Planck mass is allowed to vary by means of a non-minimal coupling to gravity. While it is possible to express inflationary models in different frames, this does not necessarily leave them manifestly invariant, which further adds to the frame problem.

Looking back at the geocentrism/heliocentrism debate, we can identify yet another relic of pre-scientific thought that somehow still manages to persist to this day despite our attempts to circumvent it. This is the idea that the Universe is somehow finely tuned to certain desirable specifications, such as allowing for life to exist. A model is said to require fine tuning when careful selection of either its initial conditions or its parameters is needed in order for its predictions to correspond sufficiently well to observations. The Ptolemaic model is a prime example of such fine tuning, as it would require an incredibly particular kind of attractive force between heavenly bodies to reproduce the complicated motion of planets that follow epicycles and deferents. Instead, 1.3. OUTLINE 17

Newton’s theory of universal attraction is not only simpler, satisfying the principle of parsimony, but can explain the elliptical orbits of the planets without having to artiﬁcially restrict their initial conditions by a severe amount.

Unfortunately, despite our attempts, many areas in modern physics suffer from some kind of fine tuning. Examples include the hierarchy problem in particle physics [25] and the cosmological constant problem in cosmology [26]. Notably, quite a few of these fine tuning problems arise as a result of attempts to reduce the amount of fine tuning of different theories. The question to ask, then, is whether inflation is one of them. A period of inflation certainly resolves at least two fine tuning problems (the aforementioned flatness and horizon problems). However, we must ask whether the conditions for inflation to occur are themselves more or less finely-tuned than a Universe without inflation. If they are, there is no point to introducing inflation as a way to “regulate” fine tuning. It has indeed been argued that for inflation to occur, the Universe must be constrained to a rather tight window of initial conditions as well as parameters [27–33]. This is often termed the initial conditions problem, but there have also been arguments as to how it may be circumvented [34–37]. The issue is not fully resoved, and, as we shall see, precisely formulating the problem of fine tuning is a challenge in and of itself. As such, it is important to have a clear understanding of what fine tuning entails as well as have a way to quantify the degree to which different models require it. This is crucial if we wish to gain further insight into how appropriate inflation is as a solution to the fine tuning problems of standard cosmology.

1.3 Outline

The structure of this thesis is as follows: in Chapter 2, we will briefly outline the historical development of standard cosmology as originating from General Relativity before turning our attention to the hot Big Bang model. We derive the equations that govern the expansion of the Universe and present the notion of a cosmological horizon. This enables us to outline the issues with standard cosmology which inflation attempts to tackle, including the flatness problem and the horizon problem. We show how an extended period of acceleration in the early Universe can address these issues generically, and further present some of the unanswered questions within the inflationary paradigm, such as the frame problem and the initial conditions problem. 18 CHAPTER 1. INTRODUCTION

In Chapter 3, we introduce the notion of frame covariance and its relation to unit transformations. We show how conformal transformations can be viewed as unit transformations, albeit one that is spacetime-dependent. We focus our attention to scalar-tensor theories, which constitute a wide class of multifield inflationary models where one or more scalar fields are non-minimally coupled to the scalar curvature in the action. We turn our attention to the dynamics pertinent to multifield inflation and derive the equations of motion. We write them in a manifestly frame-covariant form.

In Chapter 4, we work towards a frame-covariant treatment of primordial perturbations and the observable quantities that they eventually source. We extend the theory of gauge invariant perturbations to its frame-covariant with the help of the methods described in Chapter 3. We proceed to define the field space, in which the scalar fields are treated as coordinates of a manifold where inflationary trajectories reside. This enables us to distinguish between curvature and isocurvature perturbations. By constructing the perturbed equations of motion, it is possible to derive the two-point functions of the frame-covariant perturbations which are linked to the observable power spectra expressed in a frame-covariant extension of the Hubble slow-roll formalism. We then construct the potential slow-roll hierarchy before briefly discuss the super-horizon evolution of perturbations, which are relevant in the presence of isocurvature perturbations in multifield inflation. Specialising to two-field models, we examine the effects of entropy transfer on observables by deriving approximate analytical results for the transfer functions. We conclude by studying the effects that a curved field space has on the amplification and the transfer of isocurvature modes.

In Chapter 5, we turn our attention to fine tuning in inflation. We first distinguish between the problem of initial conditions and the fine tuning of parameters. We then demonstrate that there is a trade-off between the two, depending on whether we engage in model building or are attempting to verify or falsify a given model. We use an example taken from the theory of α-attractors to illustrate that parameter fine tuning can lead to the elimina- tion of the initial conditions problem. We further investigate trajectory fine tuning, which occurs only in multifield inflationary theories where the scalar fields define a hypersurface at the end of inflation, as opposed to a single point. We conclude by proposing a quantity that may be used to encode the stability of inflationary trajectories. 1.3. OUTLINE 19

In Chapter 6, we apply our formalism to both single- and multifield inflation. The single-field models we take into consideration are: (i) induced gravity inflation, (ii) Higgs inflation, and (iii) F (R) theories, which can be viewed scalar-tensor theories via the method of Lagrange multipliers. The multifield models we consider are: (iv) a simple minimal two-field model with a light scalar field and a small quartic coupling, and (v) a non-minimal model inspired by Higgs inflation. We parametrise the boundary conditions on the end-of-inflation isochrone curve, and we use the normalisation of the observed scalar power spectrum PR to select a valid inflationary trajectory. Noting that the minimal model is not observationally viable, we modify it by including a non-minimal coupling ξ between one of the light scalar fields and the Ricci scalar R. Upon choosing a nominal value for ξ, we find that isocurvature effects are significant in obtaining predictions for inflationary observables that are compatible with cosmological observations. Finally, we outline how F (ϕ, R) theories can be studied in a frame-covariant manner when written in terms of an equivalent multifield inflation model.

We depart from the tree level in Chapter 7 by demonstrating how it is possible to take into account radiative corrections in cosmological inflation in a frame-covariant manner. We first discuss the standard effective action formalism before presenting the Vilkovisky–De Witt formalism [38–42], which was originally developed in order to solve the apparent non-uniqueness problem of the effective action under field reparametrisations. Under the assumption that gravitational corrections can be neglected, we outline the fundamentals of the Vilkovisky–De Witt formalism and how it can be applied to theories of multifield inflation. We conclude by summarising our findings and presenting possible future directions for further research in the Conclusions. Chapter 2

Standard Cosmology and the Inﬂationary Paradigm

Before we delve into the intricacies of inflation, we must first take a look at standard cosmology. Before the advent of inflation, the development of modern cosmology was closely tied to General Relativity, which at the time of writing this thesis remains the best theory that we have to explain how gravity works. General Relativity holds a distinguished position among modern physics in that it concerns itself with the nature of spacetime itself, as opposed to inter- actions taking place within spacetime. Thus, it is uniquely suited to describing the Universe itself at the largest scales. As such, the development of General Relativity was instrumental in the development of cosmology as a concrete, mathematical study of the Universe. Therefore, a review of some fundamental concepts of General Relativity is in order if we are to understand how to apply it to cosmology.

2.1 General Relativity

In modern physics, spacetime is modelled as a differentiable manifold equipped with a spacetime metric [43], making it a pseudo-Riemannian manifold (the prefix “pseudo-” denotes that the metric is not necessarily positive-definite). The spacetime metric is an example of a tensor (more precisely, a tensor field since it may be spacetime dependent itself). A (m, n) tensor is rigorously defined with respect to some vector space V as a multilinear map from the

20 2.1. GENERAL RELATIVITY 21 product of m copies of V and n copies of the dual V ∗ (which is the set of all linear maps V R) to some ﬁeld F: →

×m ∗×n T : V V F. (2.1) × → For our purposes, the vector space V will be the tangent space to the manifold at some point x , denoted by Tx , and the chosen ﬁeld will be the real ∈ M M numbers:

×m ∗×n T (x): Tx Tx R. (2.2) M × M → The argument x reminds us that T (x) is a tensor field: we have assigned a ∗ tensor to each point in the manifold. Both Tx and Tx are vector spaces: M M elements of the tangent space are simply called tangent vectors, while elements ∗ of its dual Tx (the cotangent space) are called covectors or one-forms. M The definition of a tensor as a map (2.2) is completely abstract, and therefore entirely independent of the choice of basis for the vector space. However, it is often more convenient to work with the representations of tensors in some particular basis. We must first choose a basis dxµ(x) for the tangent space { } at x and a co-basis dxν(x) for its dual (where Greek indices run from 1 { } to d, the number of dimensions of the manifold ). We may then define the M components of a tensor in that basis as follows:

µ1...µm µ1 µm Tν1...νn (x) = T dx (x), . . . , dx (x), dxν1 (x), . . . , dxνn (x) . (2.3) Tensor components are numbers: we have mapped the dm+n basis elements of ×m ∗×n Tx Tx to R using (2.2). By way of abuse of terminology (especially M × M by physicists), the components of a tensor are often called “tensors” as well.

We now can examine how tensors transform under coordinate transformations xµ x0µ0 (x), which have an associated Jacobian → 0µ0 0 ∂x J µ = . (2.4) µ ∂xµ

The basis tangent vectors transform as dxµ dx0µ0 = J µ0 dxµ (and similarly → µ for covectors). Therefore, the fact that T is a multilinear map means that the 22 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION tensor components will transform as

µ0 ...µ0 0 0 1 m µ1 µm µ1...µm ν1 νn T 0 0 = J µ ...J µ Tν ...ν J 0 ...J 0 . (2.5) ν1...νn 1 m 1 n ν1 νn

In the above and throughout the course of this thesis, we employ the so-called Einstein notation, in which repeated indices are summed over. By default, we choose a chart in which Greek indices run from 0 to 3, where x0 τ is taken ≡ to be the time coordinate and xi with 1 i 3 are the spatial coordinates. ≤ ≤ In this way, an (m, n) tensor is represented in some particular coordinate

µ µ1...µm chart x by a multidimensional array Tν1...νn . The upper indices correspond to the vector space V and are referred to as contravariant, whereas the lower indices correspond to the dual space V ∗ and are referred to as covariant.

The metric tensor is a (0, 2) tensor, whereas the inverse metric is a (2, 0) tensor. µν The former is represented by gµν, whereas the latter is written as g . The metric enables us to perform “index gymnastics”, transforming contravariant ν (upper) indices to covariant via vµ gµνv . However, its main utility is that it ≡ endows spacetime with a sense of distance that does not necessarily correspond to the standard Euclidean notion. This can be seen in the deﬁnition of the 2 line element ds , which is deﬁned with the help of the metric gµν as follows:

2 µ ν ds = gµνdx dx . (2.6)

For the rest of this thesis, we employ the “mostly minus” convention, in which the metric takes on the form of the Minkowski metric ηµν = diag(1, 1, 1, 1) − − − in the limit of ﬂat spacetime. We also work in natural units with c = ~ = 1, unless otherwise noted.

General Relativity is best summarised in John Archibald Wheeler’s saying “spacetime tells matter how to move; matter tells spacetime how to curve” [44]. A point particle mass will always follow a geodesic, which is the shortest (as deﬁned by the metric) path between two points. The particular shape of geodesics is related to the intrinsic structure of spacetime (“spacetime tells matter how to move”). In order to derive the geodesic equation, we write the length of an arbitrary path as a functional L = ds. If we parametrise the µ µ path in terms of some parameter λ as x = x (λ),R the length is

µ0 ν0 L = dλ gµνx x , (2.7) Z p 2.1. GENERAL RELATIVITY 23 where the prime denotes diﬀerentiation with respect to λ. Since a geodesic corresponds to the minimal length, we extremise this functional by setting the variation δL to zero. This leads to the well-known Euler–Lagrange equations, which in this case correspond to the geodesic equations:

ρ00 ρ µ0 ν0 x + Γµνx x = 0. (2.8)

ρ The Christoﬀel symbols Γµν are deﬁned as

ρσ ρ g Γ (gµσ,ν + gσν,µ gµν,σ) , (2.9) µν ≡ 2 − where the commas denote ordinary partial diﬀerentiation with respect to that ρ coordinate. It is worth noting that Γµν is not a tensor: it does not transform according to (2.5). However, with its help, it is possible to deﬁne the covariant derivative ρ or ;ρ: ∇

µ1...µm µ1...µm αT T ∇ ν1...νn ≡ ν1...νn,α µ1 β...µm µm µ1...β (2.10) + ΓβαT ν1...νn + ... + Γβα T ν1...νn Γβ T µ1...µm ... Γβ T µ1...µm . − αν1 β...νn − − ανn ν1...β This derivative has the very important property that

µ0 ...µ0 0 0 µ1...µm 1 m µ1 µm µ1...µm α ν1 νm T 0 T 0 = J ...J T J 0 J 0 ...J 0 . (2.11) α ν1...νn α ν ...ν0 µ1 µm α ν1...νn α ν ν ∇ → ∇ 1 n ∇ 1 n

Therefore, unlike the ordinary derivative ∂ρ, it respects the covariant properties of the tensors it acts on, resulting in a (m, n + 1) tensor.

Adopting a covariant treatment ensures, amongst other things, that the redundancy inherent in the representation of the derivatives of tensorial quantities is accounted for. There are multiple ways to represent the same abstract object (such as a tensor); this is an example of a gauge freedom. All the different gauges are linked by the transformation given in (2.5). However, the ordinary derivative taken with respect to some coordinate xµ breaks this link: the ordinary derivatives of different representations of the same object are not related in a covariant manner. In a sense, the ordinary derivative does not respect the redundancy inherent in the representation of tensors. The component representation µ of the covariant derivative is still arbitrary after choosing a basis, ∇ but its definition ensures that the object resulting from its acting on a tensor is still covariant. 24 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

α Other quantities of importance are the Riemann curvature tensor R µβν, the

Ricci tensor Rµν, and the Ricci scalar R, which are given as follows in index notation:

Rα Γα Γα + Γα Γρ Γα Γρ , µβν ≡ νµ,β − βµ,ν βρ νµ − νρ βµ α Rµν R , (2.12) ≡ µαν µν R g Rµν. ≡ The Riemann tensor is particularly interesting as it contains all the information about the curvature of the manifold. When travelling on a single geodesic, an observer does not have any information about curvature. However, two nearby geodesics can be observed to diverge or converge, and it is precisely this geodesic deviation that the Riemann tensor encodes. Equivalently, the Riemann tensor controls the tidal force that a rigid body will experience as it travels along a geodesic. The Ricci tensor and scalar in general do not contain all information about the curvature of a pseudo-Riemannian manifold, but are important geometric quantities in their own right.

We have outlined how the curvature of spacetime aﬀects the motion of matter living in it. We may now examine how the presence of matter warps spacetime (“matter tells spacetime how to curve”). In order to do so, we must derive the equations of motion for the spacetime metric gµν itself. The most straightforward way to do this is by constructing an appropriate action whose vanishing variation will return the corresponding Euler–Lagrange equations. The action is a scalar, and so we construct it from the lowest derivative scalar term in gµν, the Ricci scalar R. With the help of the invariant volume element d4x √ g −2 − (where g det g) and the (reduced) Planck mass MP (8πG) , we may ≡ ≡ write the so-called Einstein–Hilbert action as follows:

2 MP 4 SG = d x √ g R. (2.13) − 2 − Z We may include a constant term denoted by Λ, known as the cosmological constant. The inclusion of this term may seem casual here, but has a long history, as we shall see in the next section. We may further modify SG by 4 adding a matter sector SM = d x √ g M . The full action then becomes − L R 2 4 MP S = d x √ g R + M + Λ . (2.14) − − 2 L Z By carefully varying the action and setting the variation to zero, it is possible 2.2. THE HOT BIG BANG MODEL 25 to arrive at the Einstein ﬁeld equations, which read as follows

1 −2 Rµν gµνR + Λgµν = M Tµν. (2.15) − 2 P The internal structure of the matter sector is a “black box” to us: we only need it to take its variational derivative in order to deﬁne the energy-momentum tensor Tµν as

2 δS T µν M . (2.16) ≡ √ g δgµν − The Einstein ﬁeld equations are our starting point for studying the evolution of the Universe at cosmic scales depending on its matter content. This is the topic of the next section.

2.2 The Hot Big Bang Model

Before we delve into the technical details of the Big Bang model and the evolution of the early Universe, a short historical review is in order. Even before Hubble’s discovery that the Universe is expanding, it was known that the Universe cannot possibly be both infinite and eternal: the dark night sky means that the Universe has to be either finite in space or finite in time. This observation is popularly known as “Olber’s paradox”, after Heinrich Wilhelm Olbers, although it was known to Johannes Kepler and Edmund Halley a few centuries prior [45]. Lord Kelvin further argued from thermodynamics that if the Universe had existed forever, it would have already reached a state of maximum entropy, making life impossible [46]. Nonetheless, the idea that the Universe is somehow unchanging and static (if not necessarily eternal) remained a silent common assumption well into the 20th century.

The development of General Relativity was a challenge to the static infinite model of the Universe, since the Einstein field equations required the arti- ficial addition of a vacuum energy density (Λ in (2.14)) for the Universe to remain static. Einstein was content to add such a cosmological constant to his field equations as seen in (2.15) in order to keep the Universe static –eternal in time but not space. Nonetheless, Alexander Friedmann [47] and Georges Lemaˆıtre[48] independently proposed a solution to the Einstein equations that challenged the picture of an unchanging and eternal Universe. Their solution, 26 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION now known as the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric, describes an expanding Universe. The full name of the metric also honors Howard P. Robertson and Arthur G. Walker [49, 50], who worked to show the uniqueness of this solution. In the interim, Edwin Hubble’s observations of redshift confirmed that the Universe was expanding, which led Einstein to famously call the cosmological constant his “greatest blunder”. Nonetheless, Hubble’s observation that the Universe is actually expanding spelled doom for static models, paving the way for the modern understanding of the Universe.

2.2.1 The Expanding Universe

The two core principles behind the construction of the FLRW metric is that the Universe looks the same irrespective of our position within in (homogeneity) and the direction we observe towards (isotropy). Robertson and Walker showed that the metric originally proposed by Friedmann and Lemaˆıtreis the only possible metric that may be constructed on a homogeneous and isotropic space. We can outline how this is achieved by employing the ADM formalism [51] in which spacetime is foliated in time slices and the spatial part of the metric is taken to be a dynamical quantity. Each slice is spanned by the shift vector N i, µ µ while motion between slices is governed by NLn . Here, n is the unit vector µ normal to the slice (nµn = 1) and the lapse function NL eﬀectively sets the passage of time. With these deﬁnitions, we may decompose the metric in terms of the projection operator qµν, which has the following properties: it vanishes ν when acting on the normal vector (qµνn = 0) and it takes on the role of the ρν ν spatial metric (qµρq = qµ). The projection operator can then be written as

qµν gµν nµnν. (2.17) ≡ − With the help of the lapse function, the shift vector, and the projection operator, the metric takes on the following form:

2 2 i 2 i i j ds = (N NiN ) dτ 2NLNi dτdx qij dx dx , (2.18) L − − − where τ stands for the time coordinate and xi for the spatial coordinates.

We now apply the FLRW speciﬁcations to the metric given in (2.18). Spatial homogeneity requires that the functions NL,Ni, and qij are functions of τ only. Isotropy further requires that there is no preferred direction on any time 2.2. THE HOT BIG BANG MODEL 27

slice, which means Ni = 0. This reduces the number of degrees of freedom from 10 to 6. The gauge freedom discussed in the previous section corresponds to xµ x0µ(x), which further reduces the degrees of freedom by another 4, → leaving us with 2 degrees of freedom total. Therefore, the spatial metric can only take on the form qij = aδij. The resulting form of the metric is then given by

µ ν 2 2 2 gµν dx dx = dt a(t) dΣ , (2.19) − where we now deﬁne NLdτ dt and a(t) a(τ(t)). The only dynamical ≡ ≡ quantity in the FLRW metric is the scale factor a(t), whose value encodes the metric expansion of the Universe relative to some reference value which, by convention, is taken to be a(ttoday) = 1.

Note that the cosmic time t is distinct from the time coordinate τ; they are related with the help of the lapse function. It is important to note that the lapse function NL is not dynamical: rescaling it simply amounts to rescaling the cosmic time t. Keeping track of it however is important when rescaling the metric via a conformal transformation, as we shall see in later chapters.

For the rest of this chapter however, we will set NL = 1 for simplicity, letting dτ = dt.

The spatial part of the line element up to the scale factor a(t) is given by dΣ2. It is homogeneous, since all the time dependence is absorbed the scale factor. Moreover, we can see that it is isotropic if the spatial subspace is of uniform 3-curvature. In polar coordinates, the spatial part of the metric takes on the form

dr2 g dxidxj = + r2 dθ2 + sin2 θ dφ2 , (2.20) ij 1 kr2 − where k is the Gaussian curvature of the Universe today. We use a convention in which k has units of (length)−2. Depending on whether k is negative, positive, or zero, we ﬁnd ourselves in an open, closed, or ﬂat Universe respectively.

As the scale factor evolves, points that would be stationary with respect to each other in a static Universe end up receding from one another. Two objects that would be at a constant distance r0 ﬁnd that their distance increases as r(t) = a(t)r0. Thus, they recede from one another with speed proportional to 28 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

Figure 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).

their current separation

v = Hr. (2.21)

This is Hubble’s law, where the Hubble parameter is deﬁned as

1 da H . (2.22) ≡ a dt This phenomenon is commonly referred to as the metric expansion of space, but this terminology is somewhat misleading as space does not expand in the usual sense. Of course, there is nowhere for space to expand into; it might be more apt to say that space is becoming “less dense”, as each each point in the Universe recedes from each other at a steady rate, as can be seen schematically in Figure 2.1.

The only way to gain information about distant objects in the Universe is from the photons they emitted in the past and that we are just now receiv- ing. However, due to the metric expansion of space, the wavelength of light that we detect from far away sources will also be “stretched”. We must take this into account in order to understand the physical meaning of our measurements. Consider a light wave originally emitted with wavelength λe that arrives at Earth and is observed today with wavelength λ0. Since the comoving wavelength of the light wave should remain constant, we have

λ λ 0 = e . (2.23) a(t0) a(te)

We can write this equality if the wavelengths are small compared to the horizon (otherwise we would need to integrate between successive peaks). Therefore, 2.2. THE HOT BIG BANG MODEL 29 the wavelength will change as

λ + δλ a = 0 , (2.24) λ a where a0 a(t0). Deﬁning the redshift as z δλ/λ, we ﬁnd ≡ ≡ a 1 + z = 0 . (2.25) a

The physical significance of redshift is that the wavelength of light that prop- agates in the Universe increases proportionally with its expansion. This effect is possible only in General Relativity, and is one of the strongest indicators for an expanding Universe. A popular analogy likens the metric expansion of the space to the gradual stretching of an infinite rubber band. This allows us to easily illustrate the distinction between comoving and proper distances. The former does not take into account the metric expansion of space, while the latter does. The two are equal today by convention (a0 = 1), but as we move backwards or forwards in time, space contracts or expands respectively, and they are no longer equal. It must be stressed that comoving lengths cannot be measured, owing purely to the fact that we as observers live within an expanding space. All measurements of distance that an observer can carry out are instead proper. Still, we may convert between proper and comoving distances with the help of the scale factor at the time that the proper distance is measured, by writing `comoving = `proper(t)/a(t).

2.2.2 The Friedmann Equations

The evolution of the Universe is controlled by the scale factor, whose evolution is in turn determined by the matter content of the Universe. Our starting point for studying the dynamics of the Universe is the Einstein ﬁeld equations, which are given in (2.15). If we assume that the matter in the Universe can be described by a perfect ﬂuid, then the energy-momentum tensor takes on the form:

µ µ Tµν = (ρ + p)u uν p δ . (2.26) − ν

The relative 4-velocity between ﬂuid and observer is denoted by uµ. In general, we are free to select a Lorentz frame in which the observer is at rest with 30 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

respect to the ﬂuid. Hence, we choose to write uµ = (1, 0, 0, 0). In this frame, the energy-momentum tensor becomes diagonal:

ρ 0 0 0 0 p 0 0  Tµν = − . (2.27) 0 0 p 0    −  0 0 0 p  −    Substituting (2.27) into the Einstein ﬁeld equations (2.15), we arrive at the well-known Friedmann equations,

2 ρ k Λ H = 2 2 + , (2.28) 3MP − a 3 ˙ ρ + 3p H = 2 , (2.29) − 6MP which encode the relative growth rate of the Universe. The overdot denotes diﬀerentiation with respect to t. We may also derive the continuity equation, which follows directly from (2.28) and (2.29):

ρ˙ = 3H(ρ + p). (2.30) −

The evolution of the scale factor is highly dependent on the nature of the matter content of the Universe. Regardless of its complicated internal structure, its features that are relevant to the growth of the scale factor can be encoded into the equation of state that links its pressure with its density as

p = wρ. (2.31)

If the value of the parameter w is known, we are able to calculate the growth of the energy density and pressure in term of the scale factor. For a constant w, we ﬁnd

ρ a−3(1+w),

∝ 2 (2.32) a t 3(1+w) . ∝ Different values for the parameter w correspond to very different evolution profiles for the Universe. For a Universe filled with non-relativistic matter (also known as “dust”), there is no pressure and hence w = 0. For relativistic matter (collectively termed “radiation”), the pressure is given by p = ρ/3. 2.2. THE HOT BIG BANG MODEL 31

In order to examine the contribution of the diﬀerent kinds of matter to the expansion of the Universe, we can rewrite the Friedmann equations, particularly (2.28), in terms of quantities that may be observed now. To this end, we deﬁne the Hubble constant H0 as the value of the Hubble parameter H today, given by

2 ρ0 k Λ H0 = 2 2 + . (2.33) 3MP − a0 3

A ﬂat Universe with a vanishing cosmological constant will have a critical density

2 2 ρc 3M H , (2.34) ≡ P 0 which we may compare to the actual density of the Universe by deﬁning the density parameter

ρ Ω 1. (2.35) ≡ ρc −

It is important to note that although we have encoded the energy density of the Universe in ρ, each component of the Universe will necessarily have a diﬀerent density. Indeed, we may parametrise the Hubble parameter as H = H0E(z), where the Hubble function E(z) is written in terms of the redshift z. Thus, the Friedmann equation can be written as

2 2 ρ k 2 Λ E = 2 (1 + z) + 2 . (2.36) ρc − H0 3H0

All that remains is to find how the energy density scales with a for different kinds of matter. We first split up ρ into matter and radiation density respectively as ρ = ρm + ρr. Using (2.32) with the appropriate values for w, we may write

−3 a 3 ρm(z) = ρm ρcΩm(1 + z) , (2.37) a0 ≡ −4 a 4 ρr(z) = ρr ρcΩr(1 + z) , (2.38) a ≡ 0 32 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

Radiation domination a(t) t1/2 ρ a−4 w = 1/3 Matter domination a(t) ∝ t2/3 ρ ∝ a−3 w = 0 Dark energy domination a(t) ∝ eHt ρ ∝ 1 w = 1 ∝ ∝ −

Table 2.1: Evolution of energy density and scalar factor for diﬀerent eras of the Universe.

where we have defined the density ratios for different kinds of matter using (2.35). We may similarly define

k Λ Ωk = 2 2 , ΩΛ = 2 . (2.39) −a0H0 3H0

Although Ωk does not physically correspond to an energy density, it is helpful to write it in this way for consistency. Using (2.37) and (2.39) into (2.36), we ﬁnally have the Hubble function:

2 4 3 2 E(z) = Ωr(1 + z) + Ωm(1 + z) + Ωk(1 + z) + ΩΛ. (2.40)

In particular, E(0) = 1 by deﬁnition, which corresponds to today. This means that 1 = Ωm + Ωr + Ωk + ΩΛ. The matter density parameter Ωm can be written as the sum of the baryonic density and the dark matter density, but for purposes of the evolution of the Universe, all that matters is that both have the same equation of state.

Looking at the evolution equation (2.40), we can see that the chronology of the Universe can be roughly divided into three eras depending on which component of the Universe dominated during its expansion. The earlier back we go, the higher the redshift z is, and the terms in (2.40) dominate in sequential order. These eras are summarised in Table 2.1. As we go further back in time, the scale factor decreases and the temperature of the Universe increases, up 4 until the energy density becomes approximately MP , beyond which General Relativity is no longer valid. For the observable Universe, the curvature domination era is usually omitted since the Universe is nearly ﬂat observationally.

The FLRW model of cosmology (whose various components are given in Ta- ble 2.2) agrees remarkably well with observations. The cosmic microwave background was famously observed by Penzias and Wilson [56]. Likewise, measurements indicating that the expansion of the Universe is accelerating indicate a constant vacuum energy density with negative pressure, and that 2.2. THE HOT BIG BANG MODEL 33

Ωm 0.316 0.014 −4 ± Ωr 10 ≈ Ωk 0.004 0.03 ± ΩΛ 0.68 0.020 ±

Table 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.

Einstein’s “greatest blunder” may be necessary to reproduce observations. In fact, since its inception, the hot Big Bang model has been further augmented with the “great unknowns”: cold dark matter in addition to a small cosmological constant. The resulting model is often referred to as the standard model of cosmology, also known as the ΛCMB model or the concordance model.

2.2.3 Horizons and Distances in Cosmology

Information has a speed limit, and that limit is the speed of light. There are certain points in space that cannot communicate because of a combination of two factors: they are too far apart and the Universe has not existed forever. The notion of a horizon in cosmology is used in order to quantify whether two points can indeed communicate. It is imperative to be precise when speaking of “distance” on a curved manifold such as our Universe, since the notion of “coordinate distance” loses its physical meaning if space is no longer ﬂat.

We ﬁrst turn our attention to the particle horizon, which corresponds to the maximum distance that photons could have traveled since the beginning of the Universe. In order to calculate an expression for it, we must remember that multiplying the age of the Universe with the speed of light is not enough, since the Universe keeps expanding as light travels throughout it. The horizon instead is an integral, which we may write down by making use of the fact that photons travel on null geodesics with dt = a(t)dr:

dr2 0 = ds2 = dt2 a(t)2 . (2.41) − 1 kr2 − We consider only radial motion since any other photon would not reach the observer at the centre of the horizon. The comoving distance between the point of emission r = rH at t = 0 and t0 at r = 0 means that there will be 34 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION an additional negative sign, and that the comoving distance to the horizon is going to be given by

t dt0 rH dr χH = . (2.42) 0 2 ≡ 0 a(t ) 0 √1 kr Z Z −

We note that in the case of a ﬂat Universe, χH = rH . It is possible to convert from the comoving horizon to the proper horizon dH by multiplying by the scale factor

H−1 ∞ dz0 d (t) = a(t)χ (t) = 0 , (2.43) H H 1 + z E(z0) Zz where we made use of the Hubble function E(z). Thus, for any given redshift z, we may use (2.43) to find the proper particle horizon. Setting z = 0 for today, we find that the horizon corresponds to the conformal time (defined through dη = dt/a) that has passed since the Big Bang.

The particle horizon is not to be confused with the Hubble horizon, which is the point beyond which points recede faster than the speed of light. The latter is given by 1/(aH) through (2.21) (we remind that c = 1) and corresponds to points that cannot communicate now, as opposed to points that are causally disconnected (and could never have communicated in the past). Both notions of a horizon are useful in diﬀerent contexts.

2.3 Problems in Standard Cosmology

The success of the concordance model in describing our Universe does not come without a cost. Even with the addition of dark matter and energy, the standard model of cosmology requires a tremendous amount of fine tuning. The allowed window in the parameter space that leads to predictions that correspond with observations is very tight. This is reflected in the flatness and horizon problems, which were the primary motivators for Alan Guth’s proposal for a period of accelerated expansion in the early Universe [19]. There are other problems with standard cosmology, the most prominent of which is the relic density problem (also known as the monopole problem). Any grand unified theory is expected to have a massive electromagnetic monopole which would be produced through a phase transition from the GUT to the Standard Model as the Universe cools down. Even if such a transition did not occur, there 2.3. PROBLEMS IN STANDARD COSMOLOGY 35 are numerous processes that should result in a rather large density of exotic particles, and yet none are observed. This is one amongst the many issues with standard cosmology (such as questions on the nature of dark matter and dark energy). However, we are primarily interested in the flatness and horizon problems since they are prime examples of the kind of fine tuning that can be remedied by inflation.

2.3.1 The Flatness Problem

The ﬂatness problem of the standard cosmological model is a ﬁne tuning problem par excellence in the sense that it requires us to severely constrain initial conditions in order to reproduce actual observations. The curvature can be measured with the help of the ratio of the observed density to the critical 2 2 density ρcrit 3H M . Using the latter, we may rewrite (2.28) (with Λ = 0) ≡ P as

ρ k 1 = 2 2 . (2.44) ρcrit − a H

We can see from the scale factor evolution equation (2.32) and the acceleration equation (2.44) that the density parameter Ω = ρ/ρcrit 1 evolves as −

2− 4 Ω t 3(1+w) (2.45) | | ∝ As long as w 1/3,a ˙ < 0 and the Universe will not accelerate. Thus, Ω 1 ≥ − | − | will increase with time as the comoving Hubble horizon 1/(aH) expands. The crux of the flatness problem appears when we consider that current measurements of the density parameter Ω indicate that the Universe is very nearly but not quite flat, as seen in Table 2.2. According to (2.45), this must mean that the Universe must have been even flatter in the past, as long as there is no exotic matter with w < 1/3. In this case, the density parameter would have − been approximately with Ω < 10−64 at the Planck era. It is an incredible | | coincidence that the Universe should be so close to the critical density at its infancy if it did not dynamically reach that value. As such, if we are to avoid fine tuning the initial curvature of the Universe to an incredibly small number, we must find a mechanism to drive it down from a more “generic” (i.e. not tightly constrained) set of initial values. 36 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

2.3.2 The Horizon Problem

The horizon problem is another problem which may be viewed in terms of ﬁne tuning. The Universe, at its largest scales, is homogeneous. Even though it was discovered by COBE that there are temperature anisotropies of the order of 10−5 K, there is no reason why patches of the Universe that were never in causal contact should have thermalised. We can formalize this by calculating the angle that corresponds to the horizon at the time the CMB was formed (or, equivalently, at the surface of last scattering where the Universe became transparent and light could travel freely). We may thus use the angular diameter distance of the horizon at the surface of last scattering, which is deﬁned as follows:

dH θH = . (2.46) dA

This relation links the observed angle θH to the physical size of the horizon dH and the angular-diameter distance dA. We can write an expression for dA by noting that the proper distance along a circle with radius rH would be

D = rH aH θ, which means that dA = rH aH (aH is the scale factor at emission).

For a ﬂat Universe, the comoving distance rH corresponds to (2.42), with the diﬀerence being that the limits correspond to today (z = 0) and the time of last scattering (z = zH ) instead. Thus, the angular diameter distance is given by

H−1 zH dz d = 0 . (2.47) A 1 + z E(z) H Z0

As such, we may substitute (2.43) (evaluated at z = zH ) and (2.47) into (2.46) to ﬁnd [57]

∞ dz/E(z) θ = zH . (2.48) H zH dz/E(z) R0 R The surface of last scattering occurred at zH 1000. At that time, the ≈ Universe was still matter dominated, which means that we may use E(z) 3 ◦ ≈ Ωm(1 + z) . Substituting this expression into (2.48) returns θH 1.7 . How- ≈ ever, when probing the CMB, patches of similar temperature are more than a few degrees apart, and there is no way to explain how they came to have the same temperature if they were outside of causal contact without ﬁnely tuning the initial conditions to explain away possible inhomogeneities. 2.4. AN ACCELERATING UNIVERSE 37

2.4 An Accelerating Universe

The flatness and horizon problems do not necessarily spell doom for the hot Big Bang model. There is no inconsistency with requiring the initial conditions of the Universe to have unnatural values. However, this fine tuning comes at a price in the form of anthropic arguments. While it might seem attractive to postulate that initial conditions had to be finely tuned because intelligent observers (humans, hopefully) would not be around to observe the Universe otherwise, there is a host of philosophical problems with this idea. The anthropic principle has no predictive power; it is entirely a posteriori argument that is almost tautological in stating that if initial conditions were different, then the Universe would, of course, look very different today. When using anthropic considerations, the likelihood of initial conditions can be calculated depending on the state of the Universe today, but this tells us nothing about the likelihood of differing initial conditions and how they may arise from a more fundamental physical principle. Thus, different approaches were sought and cosmological inflation was eventually proposed as a dynamical solution to the flatness, horizon, and monopole problems. An epoch of sustained accelerated expansion of the Universe resolves both these problems in a natural way. This means that it is possible to generate predictions that match current observations from a much more generic set of initial conditions than the ones spelled out in the previous section. There are other approaches to resolv- ing these fine-tuning issues without resorting to anthropic arguments, such as cyclic models [58,59] and string gas cosmology [60,61], but none of them have been as popular as inflation.

The speciﬁc mechanism of inﬂation that resolves the issues with standard cosmology is a period during which the comoving Hubble horizon 1/(aH) decreases, which is equivalent to a period of accelerated expansion witha ˙ > 0. If this behaviour goes on for long enough, the comoving particle horizon (2.43) can be made arbitrarily large as we go towards the past:

a da0 χH , (2.49) ≡ a0H(a) Z0 since 1/(aH) increases as we go backwards in time. This behaviour resolves the ﬂatness problem. We can see from (2.44) that as 1/(aH) increases, any initial value for Ω will eventually be driven to zero if aendHend/(a0H0) is large | | enough. This means that generic initial conditions can give rise to a nearly 38 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

ﬂat Universe today, thus avoiding any need for ﬁne tuning.

A period of accelerated expansion also resolves the horizon problem. We remind that the particle horizon and the Hubble horizon are distinct; the former separates points that are causally separated whereas the latter separates points that cannot communicate at the present time. Therefore, as the comoving Hubble horizon becomes smaller, comoving scales (which are constant by definition) that were inside it before inflation will escape it (to eventually re-enter it in the future). Thus, if we ensure that the scales the size of the observable Universe were within the Hubble horizon before inflation began, then they could have communicated even if they appear causally disconnected today. In popular lore, it suffices for the Universe to expand by a factor of approximately e60 for both of these problems to be resolved.

The most straightforward way to achieve the desired behaviour for 1/(aH) is to postulate an exponential expansion for a, given by a = eHt for an approximately constant H, which means that the Universe is described by a quasi-de Sitter background. This is a particular solution to the Einstein field equations: the “quasi-” prefix means that H is not exactly constant. This is precisely why this acceleration cannot go on forever: inflation must end at some point, at least for our observable neighbourhood. Even then, once inflation ends, the Universe will be cold and mostly empty, and so there must be a reheating mechanism that can populate the Universe with matter and most importantly radiation, such that the radiation-dominated phase can begin. Thus, any acceptable model of inflation must at the very least feature the following: (i) a sustained period of accelerated expansion of the Universe, (ii) a way for inflation to end (“graceful exit”), and (iii) a mechanism for reheating after the Universe cools down during inflation. In the course of this thesis, we will focus our attention on the mechanism that drives the de Sitter expansion of the Universe by examining models that mimic the effects of a cosmological constant.

There are two ways in general to achieve inflation, as can be seen from Ein- −2 stein’s equations Gµν = MP Tµν. We may try to modify the gravity sector by departing from GR, or we may try to modify the matter sector by adding a particle tentatively named the inflaton which can lead to the desired accelerated expansion. While inflation driven by vectors has been considered in the literature [62], the inflaton is usually taken to be a scalar in order to satisfy the isotropy requirements (which a vector field may violate). Examples of models 2.4. AN ACCELERATING UNIVERSE 39 modifying the gravitational sector include Starobinsky inflation [63] and F (R) theories [64]. Guth’s original proposal featured a scalar field that tunneled to a false vacuum with high energy density that acts as a cosmological constant [19]. The latter model later came to be called old inflation when it was superseded by slow-roll inflation [20,65], in which the accelerated expansion of the Universe is driven by the inflaton ϕ slowly rolling down a potential V (ϕ). We can see how this leads to inflation by calculating the equation of state. To do so, we consider the energy density and pressure of a scalar field by writing the matter sector in (2.14) as

1 µ M = (∂µϕ)(∂ ϕ) V (ϕ). (2.50) L 2 − By using (2.16) and (2.27), we may derive the following expressions for the energy density ρ and pressure p:

ϕ˙ 2 ϕ˙ 2 ρ = + V, p = V. (2.51) 2 2 − Thus, if we can ensure that the kinetic energy is dominated by the potential energy, we have that ρ p, meaning that the desired equation of state will be ≈ − satisfied. As the field slowly rolls down the potential, it will eventually pick up kinetic energy, at some point no longer satisfying w < 1/3, which means that − inflation will end. Eventually, the inflaton slows down due to the expansion of the Universe (“Hubble drag”), and the inflaton will oscillate. If the inflaton is coupled to the rest of the Standard Model, it will reheat, therefore producing a host of particles that eventually fill the Universe.

Inflation was viewed as a welcome solution to the problems that plagued standard cosmology. However, interest in it peaked when it was realised that it could explain the origin of anisotropies in the CMB [22–24]. Indeed, the early Universe was expected to feature primordial quantum fluctuations that eventually seed the large-scale structure that we observe today. Thus, it was realised that inflation could be the source of these primordial perturbations, and therefore should predict a Universe with slight inhomogeneities. We can see this heuristically by splitting the scalar field into its classical (background) value and its perturbation as follows:

ϕ ϕ + δϕ. (2.52) → The perturbation δϕ has dimensions of inverse length (i.e. mass), and H−1 40 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION corresponds to the horizon (with dimensions of length). Therefore, much like a particle constrained in a box, we expect that the variance of the ﬂuctuations should go as δϕ2 H2 thanks to the uncertainty principle. By performing h i ∼ a Fourier transformation, we can decompose perturbations at diﬀerent scales k, and encode the dependence in the dimensionless power spectrum P (k):

k3P (k) δϕ2 . (2.53) ∼ h ki √ If the Universe were to remain indefinitely de Sitter with a(t) e Λ t, then ∝ H would have to be exactly constant. This means that the variance would eventually be constant, and the power spectrum of perturbations P (k) kn−1 ∝ would have no k dependence, meaning that the spectral index n would be exactly equal to unity. In that case, it would correspond to the exactly scale- invariant Harrison–Zel’dovich spectrum. However, since inflation must end at some point, the spectrum will instead be scale-dependent. This corresponds to observations: even though at the largest scales the Universe is roughly homogeneous, the CMB is slightly inhomogeneous as measured by the COBE [66]. Further measurements of the inhomogeneities by the subsequent WMAP [67] and PLANCK experiments [53] have confirmed that the generic predictions of inflation: a small ratio of the amplitudes of tensor modes (gravitational waves) to scalar modes r . 0.1 and a spectral index 0.94 . ns . 0.98, further refining and solidifying the observational status of inflation as a theory.

2.5 Inﬂation: More Questions Than Answers?

While inflation as a generic framework is attractive and solves a host of problems pertaining to standard cosmology, it still has a few issues that beg to be solved. The first and foremost is that there exists a vast array of models, all of which produce similar predictions, namely a nearly scale invariant power spectrum and a small amplitude of gravitational waves. Numerous inflationary models with various theoretical motivations from particle physics have been proposed, including power-law inflation [68, 69], natural inflation [70], axion inflation [71, 72], hybrid inflation [73], false vacuum inflation [74], kinetically driven inflation [75], brane inflation [76], hilltop inflation [77], supergravity- based inflation [78], and α-attractors [79]. A large number of models of inflation have been ruled out thanks to the PLANCK 2015 survey which favours slow-roll models with an inflationary plateau [54]. Indeed, the sheer number 2.5. INFLATION: MORE QUESTIONS THAN ANSWERS? 41 of models of inflation lends credence to a common criticism of the inflationary paradigm: namely that despite its generic predictions, it ultimately lacks robustness since it seems that it is possible to construct an ad hoc potential to match any observations. Even more alarmingly, it appears that even if we sub- scribe to a particular model of inflation, there are certain “model-independent” open problems that we must still contend with. Two of these are the frame problem and the fine tuning problem, to which we now turn our attention.

2.5.1 The Frame Problem

As mentioned in the Introduction, debates about whether frames are physical have existed since antiquity. Nowadays, we understand that no observer is privileged over another, and that there is no reason to prefer a particular frame of reference apart from convenience. We thus understand that geocentrism as laid out in Tychonic system and heliocentrism in the Keplerian system are not different models, but rather different representations of the same physical picture. We simply prefer the Keplerian system because it arises solely from Newton’s law of universal attraction without the addition of fictional forces.

However, it seems that we have not yet learned our lesson. There is a conceptually similar issue in theories of gravitation which has yet to be fully resolved. We know that observers linked by Lorentz transformations see the same physics, but the behaviour of theories under conformal transformations is not as clear. Even though we will focus on conformal transformations in the context of inflationary theories, the frame problem is not unique to inflation; in fact, it was first identified in the context of conformal transformations in modified gravity theories [7].

Conformal transformations are used in General Relativity in order to transform between spacetimes with the same causal structure (as they leave the light cones unchanged). A conformal transformation is a spacetime-dependent rescaling of the metric function:

2 gµν g˜µν = Ω(x) gµν. (2.54) → This transformation is conceptually distinct from a coordinate transformation, since the coordinates themselves do not transform; instead, what changes is 2 µ ν the length and time scale of the system. Since ds = gµνdx dx , we note 42 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION that the line element will transform as ds ds˜ = Ωds after a conformal → transformation.

Intuitively, we may understand that for constant Ω, this transformation is not physical; all length scales are equally scaled which means that no observable effects should arise as a result. The transformation given in (2.54) is reminiscent of a change of units:

`P `P /Ω. (2.55) → When we perform a measurement, we are not actually measuring the length of an object L, but rather a pure number L/`P (where the denominator is the length of the base unit of our “ruler”). If we change units, the length of the ruler itself will transform as ` Ω`P and the length of the object → as L ΩL, meaning that our measurement will remain invariant. Under → this interpretation, (2.54) can be achieved by a unit change given in (2.55); the former is indeed the exact same transformation that would occur if we were to rescale the eﬀective Planck length. Moreover, since all of our measurements are of dimensionless ratios, we expect that any observable quantity will be left invariant. The laws of Nature are not unit-dependent; we know from the Buckingham π theorem that every law in physics can be nondimensionalised [80], which means that it may be written in terms of dimensionless quantities π1, π2,... as:

f(π1, π2,...) = 0. (2.56)

The dimensionless quantities πi are independent (i.e. none of them is a product of powers of the others) and they are constructed from dimensionful quantities related by the law in question. For instance, consider the energy-momentum relation E2 = p2c2 + m2c4: the dimensionful quantities are E, p, m and c, and

E p π = , π = . (2.57) 1 mc2 2 mc

Therefore, the nondimensionalised form of the energy-momentum relation is simply π2 π2 = 1. This example illuminates the Buckingham π theorem, 1 − 2 which is merely a more rigorous expression of the idea that the laws of physics are unit-independent. All πi will not scale as they are dimensionless (much like L/`P in our example above).

Our discussion up until now assumes that the conformal factor is constant. 2.5. INFLATION: MORE QUESTIONS THAN ANSWERS? 43

We now consider the case where the conformal factor is spacetime dependent. First of all, we note that even if Ω = Ω(x) is spacetime dependent, the FLRW metric will transform to

µ ν 2 2 2 2 2 2 g˜µν dx dx = Ω(x) N dt Ω(x) a(t) dΣ . (2.58) L − This metric at ﬁrst glance appears not to be FLRW, since it is neither manifestly homogeneous nor manifestly isotropic. However, it is possible to apply a particular diﬀeomorphism such that the metric is once again explicitly FLRW:

dt˜= Ω(x) dt, (2.59) dΣ = Ω(x) dΣ.

This is no trick: the statement thate the Universe is isotropic and homogeneous presupposes a particular time slicing, and therefore, we are always free to select a particular gauge in which this is the case.

We now turn our attention to the components of the FLRW metric after a conformal transformation. The lapse function NL and the scale factor a transform as follows

NL NL = ΩNL, (2.60) → a a˜ = Ωa. (2.61) →e

Therefore, a and NL transform as if they are dimensionful, while the coordinates xµ transform as if they are dimensionless. However, in the case that Ω is spacetime-dependent, a problem does arise when considering the derivatives of dimensionful quantities. For instance, the Hubble parameter will transform as follows under a conformal transformation:

H H = H + Ω˙ /Ω. (2.62) → We see that the Hubble parameter ise now dependent on the conformal factor Ω. From the viewpoint that it this is a spacetime-dependent unit change, however, this is a violation of the Buckingham π theorem, according to which, we expect the Hubble law (2.21) to be invariant. We expect the law to transform from v = Hr to

v˜ = Hr.˜ (2.63)

e 44 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION

Therefore, at ﬁrst glance, it seems that the Buckingham π theorem fails when it comes to spacetime-dependent systems of units, since the dimensionless quantity π = v/Hr is not invariant. This is one of the simplest instances of the frame problem.

The frame problem can also be seen at the level of the action. In General Rel- ativity, the standard Einstein-Hilbert action is usually written in the Einstein frame as

2 4 MP S = d x√ g R + M , (2.64) − − 2 L Z 2 where the eﬀective Planck mass MP is a dimensionful but constant parameter. After a conformal transformation, the Ricci scalar transfoms as [7]

−2 −3 µν R R = Ω R 6Ω g µ νΩ. (2.65) → − ∇ ∇ Substituting (2.65) in (2.64),e we ﬁnd

2 MP 4 2 −1 2 −2 S = d x √ g Ω R + 6Ω Ω 2M M , − 2 − ∇ − P L Z −2 2 2 (2.66) 4 Ω MP 6MP µν −4 = d x g˜ eR + g˜ ( µΩ)( νΩ) + Ω M , − − 2 Ω2 ∇ ∇ L Z p where we integrated by partse and discarded a boundary term in the second step. We observe that there is a new effective Planck mass that is now space- −1 time dependent: MP (x) = Ω (x) MP , as well as a coupling between the new scalar degree of freedom Ω and gravity. The relation between the two masses is not dynamical,f since Ω(x) is chosen beforehand. If it is not chosen to be constant and is instead left general, the theory is said to be in the Jordan frame. Whether this coupling will lead to different observational effects is one of the many ways to express the frame problem.

The frame problem also appears in inflation, particularly when we introduce a 2 non-minimally coupled light scalar field in order to promote MP to a dynamical term f(ϕ). In this case, there is some consensus that there should be no difference between the frames at the classical level, but the question of their equivalence once radiative corrections are taken into account is still an open question [8–18], especially when further matter couplings are introduced. This occurs due to the fact that the matter picture of the Universe is ostensibly different in (2.66), as the matter sector is now coupled to a spacetime-dependent 2.5. INFLATION: MORE QUESTIONS THAN ANSWERS? 45 conformal factor. The crux of the frame problem is that it is not clear whether radiative corrections should be the same in all frames, or whether a particular frame is more “physical” than others. As we enter the era of precision cosmology, such corrections could very well be observable soon. It is therefore imperative that we understand exactly what the physical implications of a change in frame are in terms of the phenomenology of the theory of inflation.

2.5.2 Fine Tuning and Initial Conditions

A mathematical model is said to be finely tuned when its parameters have to be carefully selected in order to reproduce observations. Conversely, a model which matches predictions without such a restriction is said to be robust or natural. As noted above, the flatness problem is an example of fine tuning, since in order to reproduce the almost flat Universe we live in today, the initial density parameter of the Universe needs to be severely constrained. There is nothing inherently problematic about having a finely tuned model, and so fine tuning is in a sense more of a conceptual problem than an inconsistency. It is easy to dismiss the fact that the parameters of a fundamental model happened to fall within a very tight window as a coincidence, but there is still something unsavoury about setting the parameters of a model “by hand”. Fine tuning plagues many fundamental theories of physics, not the least of which is the Standard Model itself [25]. The fine tuning problems found within the framework of standard cosmology were discussed in Section 2.3. Inflation was introduced precisely to circumvent these issues, but it was soon found that inflation does not necessarily occur in generic way [27–31,33].

There are two main ways in which fine tuning can be potentially realised: the fine tuning of initial conditions [81,82] and the fine tuning of fundamental parameters [33]. The two are closely related: most ways that resolve the initial conditions problem must resort to some form of parameter fine tuning. This can be seen in most currently favoured models of inflation with a single scalar field require a very flat inflationary plateau, which requires very careful selection of the couplings to allow for such a potential. For instance, in ϕ4 inflation, the appropriate value for the quartic coupling λ is of the order of 10−12, which is very difficult to realise for most realistic particle theories. In practice, the flatness of the potential is popularly achieved through via a non-minimal coupling between the scalar fields and the Ricci curvature R. This raises further 46 CHAPTER 2. STANDARD COSMOLOGY AND INFLATION questions as to the particle physics behind such couplings, since we would hope that a proper particle theory of inflation does not feature an ad hoc potential, but rather one that corresponds to some particle in Nature (even if it is yet unknown to us). Even in Higgs inflation, where the role of the inflaton is adopted by the Higgs boson, there exists a non-minimal coupling of the order of 104 [84], which is not easy to justify with our current knowledge of particle physics.

The initial condition problem comes into two flavours. It first concerns the initial value of the inflaton, which must be such that inflation will not stop before the density parameter Ωk is sufficently driven down and the flatness problem is resolved. In popular lore, it is assumed that an inflationary plateau will lead to inflation “most of the time”, since we expect that the inflaton will start on top of the inflationary plateau [37]. While it is true that in most practical applications the fields are assumed to slowly roll down the inflationary potential, realising this condition is not necessarily trivial. Even if the slow-roll solution acts as an attractor (since the kinetic energy of the field will eventually dissipate due to the so-called Hubble drag term in the equations of motion), there is always the possibility that the attractor solution is reached late enough such that inflation does not persist long enough to solve the flatness and horizon problems [85]. This can occur if, as the Universe exits the singularity and can be described by classical physics, the inflaton happens to have little potential energy and large kinetic energy, which would indicate that it has not reached the attractor yet. Therefore, quantifying the probability that sufficient inflation will occur may be a challenge in and of itself.

Even if inflation is assumed to have started with no issues, there is another fine tuning issue that appears long after inflation has begun. We will find that in the presence of multiple light scalar fields, there is still the freedom to choose boundary conditions even if the inflationary attractor has been reached. In fact, the choice of boundary conditions is a completely classical consideration that can give rise to different trajectories in the field space, which in turn leads to drastically different observables, enabling us to differentiate between trajectories. Taking this kind of fine tuning into account is important for quantifying the viability of different multifield models, especially ones with realistic couplings where the only freedom we have is the choice of trajectories.

Inflation has many subtleties, and we have outlined two of them so far: the 2.5. INFLATION: MORE QUESTIONS THAN ANSWERS? 47 frame problem and the problem of fine-tuning and initial conditions. In the next chapter, we will focus on the fundamentals of frame covariance, paving the way for a manifestly frame-covariant theory of perturbations during inflation, which we will study in subsequent chapters. Chapter 3

Classical Frame Covariance

In this chapter, we will introduce the notion of frame covariance as an extension of the notion of frame invariance. We will distinguish between conformal and Weyl transformations, and by borrowing concepts from conformal field theory, we will illustrate how conformal transformations in cosmology effectively amount to a change of units. We will move on to demonstrate how frame covariance can be used to study non-minimal multifield scalar-tensor theories. We will demonstrate that it is possible to make seemingly disparate scalar-tensor theories manifestly equivalent through the notion of frame covariance. We conclude by studying the background dynamics of these theories in a manifestly covariant manner.

3.1 Conformal and Weyl Transformations

There is much confusion in literature on cosmology between conformal transformations and Weyl transformations and their relation to scale invariance and conformal invariance. These terms have well-deﬁned meanings and are closely related, but are sometimes used loosely in a cosmological context, and so understanding their distinction is crucial before we can develop a frame-covariant formalism.

We begin by considering a generic coordinate transformation given

xµ xµ = xµ0 (x). (3.1) 7→

48 3.1. CONFORMAL AND WEYL TRANSFORMATIONS 49

Such a transformation has an associated Jacobian given by

dx0µ J µ = . (3.2) ν dxν

Using the Jacobian, the metric transforms as follows:

0 ρ σ gµν g = gρσ J J . (3.3) 7→ µν µ ν What is usually referred to as a conformal transformation in the context of cosmology diﬀers from the more common use of the term in mathematics. Mathematicians use “conformal transformation” to refer to a coordinate transformation that leaves the metric invariant up to a scalar multiplicative factor:

0 2 gµν g = Ω gµν. (3.4) 7→ µν Thus, a conformal transformation in this context is simply a particular kind of a diﬀeomorphism. On the other hand, a Weyl transformation is the scaling of the metric tensor itself:

0 2 gµν g = Ω gµν. (3.5) 7→ µν In this context, Ω is referred to as the conformal factor. To summarise, a conformal transformation is a diﬀeomorphism whose eﬀect on the metric can be reversed by applying a Weyl transformation.

We may see the distinction between a conformal transformation and a Weyl transformation by examining the following D-dimensional scalar ﬁeld theory on a manifold equipped with metric gµν and a potential V (ϕ):

D 1 µν S = d x √ g g (∂µϕ)(∂νϕ) V (ϕ) . (3.6) − 2 − Z We ﬁrst consider a scaling transformation (or dilation), which is a particular example of a conformal transformation:

xµ Λxµ, → (3.7) ϕ Λ−∆ϕ, → 50 CHAPTER 3. CLASSICAL FRAME COVARIANCE where ∆ is the classical scaling dimension of the ﬁeld. For the transforma- µ0 µ0 tion (3.7), the Jacobian is J µ = Λδµ and so the action takes on the form

D 1 µν D−2 0 0 D −∆ S = d x √ g g Λ (∂µϕ )(∂νϕ ) Λ V (Λ ϕ) . (3.8) − 2 − Z Working in Planck units, the action should be dimensionless, which indicates that the ﬁeld should have dimension

D 2 ∆ = − , (3.9) 2 otherwise the kinetic term would not transform with the desired dimension. The action (3.8) will be scale invariant if

ΛDV (Λ−∆ϕ) = V (ϕ). (3.10)

n For a monomial potential V (ϕ) = λnϕ , the dimension of the coupling λn is going to be [λ] = D n∆. In order for (3.10) to be satisﬁed, we must ensure − that the coupling is dimensionless. This occurs when

2D n = D/∆ = . (3.11) D 2 − For D = 4, we ﬁnd out that the theory becomes scale invariant when n = 4.

We now consider the behaviour of the theory under a constant Weyl transformation instead. Under a Weyl transformation (3.5) with Ω held constant, the action will transform as

D 1 2−D µν −D S = d x √ g Ω g (∂µϕ)(∂νϕ) Ω V (ϕ) . (3.12) − 2 − Z It is possible to assign a conformal weight to ϕ for this transformation as well:

ϕ ϕ0 = Ω−dϕ. (3.13) → We also ﬁnd that

D 2 d = − . (3.14) 2

The theory will be Weyl invariant if

Ω−DV (Ω−dϕ) = V (ϕ). (3.15) 3.2. CONFORMAL TRANSFORMATIONS AND UNIT CHANGES 51

We see that if a scaling transformation and a Weyl transformation both leave a system invariant, the system has no inherent scale, which explains why even if conformal transformations and Weyl transformations are conceptually distinct, they are very closely related. Indeed, Weyl transformations (3.5) are more often called conformal transformations in cosmology. For the rest of this thesis, we will adopt this terminology unless otherwise noted.

3.2 Conformal Transformations and Unit Changes

Having drawn a distinction between conformal and Weyl transformations, we now turn our attention to their relation to changes of units. We first talk about the invariance of the laws of physics under a transformation of units. It has been noted that the notion of “physical equivalence” between conformally- related theories is ill-defined in the literature [86], and so we adopt the simple viewpoint that any two theories with the exact same dimensionless ratios are physically equivalent, following the Buckingham π theorem. Since we measure only dimensionless ratios in experiments (e.g. the length of an object divided by the length of a ruler), any transformation that leaves such ratios invariant should be undetectable. Transformations of dimensionful quantities are not meaningful on their own. For instance, evidence that the fine-structure 2 constant α = e /(4π0~c) is evolving with time [87] does not tell us anything about whether the value of ~ or c or e is evolving with time. Depending on whether we use, for instance, Planck units with ~ = c = G = 1 or Stoney units with c = e = G = 1, the fine structure constant will be given by a = e2 or a = 1/~ respectively. The variation of dimensionless quantities on the other hand is completely independent of the choice of units [88–90].

It can be argued that the number of units we need is theory dependent; after all, without special relativity we would not be justiﬁed in arbitrarily introducing the speed of light c in our measurements and freely setting it to unity, and in general the number of dimensionless parameters will depend on the content of our theory. However, there are no “fundamental” dimensionful parameters: any variation in them must be accompanied by a description of how other dimensionful parameters change as well. With this information, we can calculate how dimensionless ratios πi will change, and if they do not, we can rest assured that there will be no change in the physics. 52 CHAPTER 3. CLASSICAL FRAME COVARIANCE

In this dimensional analysis context, we may view conformal transformations as changes of units. A theory that is scale invariant looks the same at all energy scales. This is a statement about the content of the theory. However, we expect any theory to be conformally-covariant in the sense that after all dimensionful quantities are transformed, it will remain form invariant. The distinction is subtle; the behaviour of a theory at diﬀerent scales is controlled by a dimensionless parameter m/mP . On the other hand, the behaviour of a theory should not depend on the system of units. The theory should be completely invariant under a rescaling of the units such that all dimensionless parameters remain the same. The distinction between a change of scale and a change of units can be schematically shown as follows:

m Ω−1m change of scale , (3.16) MP → MP m Ω−1m change of units −1 . (3.17) MP → Ω MP

The former is physically important; it changes a dimensionless parameter of a theory and therefore has the potential to change the physics. The latter is physically unimportant; the dimensionless parameters remain the same. This distinction can be made manifest when (3.6) is properly nondimensionalised. In order to do so, we use the convention that the metric is dimensionful and the coordinates are not:

D 4 1 −2 µν −4 S = d x M √ g M g ∂µ ϕ/MP ∂ν ϕ/MP M V (ϕ) . P − 2 P − P Z (3.18)

Each term in square brackets is dimensionless, making the invariance under a change of units manifest. We note that scale invariance occurs when V (ϕ) takes on the particular form prescribed in (3.10) and (3.15), but covariance under a change of units is preserved in all cases: an object X with mass

−mX dimension mX will transform as X Ω X if the Planck mass transforms −1 → as MP Ω MP . → It is very important to note that the link between conformal transformations and unit changes is closely related to the convention of Planck units common in cosmology where ~ = c = 1 . In general, the Planck mass, length, and time 3.2. CONFORMAL AND UNIT CHANGES 53 are given by

c G G M 2 = ~ , `2 = ~ , t2 = ~ . (3.19) P 8πG P c3 P c5

Since all these quantities are dimensionful, we may set them to unity without losing any physical information about the system. To do so, we would need to keep track of three diﬀerent conformal factors:

−1 −1 −1 MP Ω MP , `P Ω `P , tP Ω tP . (3.20) → M → ` → t The conformal transformation (3.5) would scale ds2, which can also be realised by scaling the base length unit `P with Ω`. This would not necessarily correspond to scaling the base mass unit, whose scaling is controlled by ΩM . However, the very choice of using Planck units ensures that a conformal transformation is directly linked to mass transformation, since in this case, a single −1 conformal factor ΩM = Ω` is suﬃcient to describe the change of units.

As we discussed in Subsection 2.5.1, we must be particularly careful in cases where there is a spacetime-dependent variation in dimensionful parameters. If we promote M 2 to a function of spacetime M 2 f(x) but in a way that P P → all masses scale as m √f m as well, then we have simply made a unit → transformation. If an observer at a point x measures the mass of an object to be m(x), then we will have

m(x ) m(x ) 1 = 2 . (3.21) f(x1) f(x2)

Thus, when measuring the changep of thep mass with respect to some dimensionless parameter Λ, we must have

d m(x ) d m(x ) 1 = 2 . (3.22) dΛ " f(x1)# dΛ " f(x2)# p p This expression essentially tells us that it is impossible to measure a change of units with any experiment, even if those units are spacetime-dependent. Expanding this expression, we ﬁnd

1 dm d ln √f m = constant. (3.23) √f dΛ − dΛ Starting from (3.23), we may motivate the deﬁnition of a “unit-covariant” 54 CHAPTER 3. CLASSICAL FRAME COVARIANCE

derivative |Λ, which respects the transformations between diﬀerent systems of units. A quantity X that has mass dimension wX transform as follows under the change M 2 f(x): P → f wX /2 X X = X. (3.24) → M 2 P Therefore, we have motivated thee deﬁnition of the unit-covariant derivative, which is given by

wX X|Λ = X,Λ (ln f),ΛX. (3.25) − 2

We may view X|Λ as a “physical” derivative not in the sense that it is invariant to changes of units, but rather covariant. When making a unit transformation, we expect that a dimensionful quantity will be multiplied by a prefactor. The unit-covariant derivative (3.25) ensures that this will be the case even if the unit transformation is spacetime dependent. Note that we can also ensure that the unit-covariant derivative is also gauge-covariant (which only matters when X is a tensor) simply by replacing the partial derivatives in (3.25) with gauge-covariant derivatives:

wX X|Λ = X;Λ (ln f);ΛX, (3.26) − 2

µ where X;Λ = X;µx,Λ. This deﬁnition makes sense only along a trajectory parametrised as xµ = xµ(Λ), but it is worth noting that Λ here does not need to be an aﬃne parameter. We may even replace Λ by xµ, as long as we use (3.26) to ensue gauge covariance.

The unit-covariant derivative applied to a quantity gives us the change of that quantity up to the choice of units. We can see this by applying it to the eﬀective Planck mass √f. We obviously expect that a base unit should be constant up to the choice of units themselves, and indeed we ﬁnd that

f |µ = 0. (3.27) p This relation is a covariant representation of the fact that there is no experiment that can differentiate between different values of the effective Planck mass. It is reminiscent to a gauge derivative where the gauge is the choice of units [90]. Indeed, much like we can define a metric-compatible gauge derivative, we have defined in (3.25) and (3.26) a unit-compatible derivative. We 3.3. FRAMES IN MULTIFIELD SCALAR-TENSOR THEORIES 55 thus understand that choosing units that are spacetime-independent is akin to a particular choice of gauge, much like the Coulomb or Lorentz gauge. We might for instance choose to work in the “constant unit gauge” by appropri- ately selecting Ω(x) such that

∂µf = 0. (3.28)

2 In this “gauge”, f is a constant, which we may of course denote by MP , leading us to the conclusion that a theory expressed this way is simply expressed in the Einstein frame.

It might seem at ﬁrst that there is no reason to adopt such a strange convention where our system of units can vary in space and time. However, we will soon see that such a convention is rather imposed on us by the introduction of a non-minimal coupling in scalar ﬁeld theories living in a curved background. We will thus see that there is a particular orbit in the space of theories that links all theories related by a simple unit transformation. This is closely linked to quantisation: the scale and conformal invariance of a classical theory is not necessarily inherited by the corresponding quantised theory thanks to the well known conformal anomaly. However, we expect that conformal covariance should translate to the quantum case as long as all quantities are properly transformed, since it is simply a more involved unit transformation.

3.3 Frames in Multiﬁeld Scalar-Tensor Theories

We have outlined the idea of conformal covariance and how it can be viewed as an invariance of the laws of physics under transformations of units. We now wish to apply this notion to inflation, and in particular to multifield scalar- tensor theories, a class of models featuring more than one scalar field contribut- ing to the inflationary expansion of the Universe [91, 92]. While the current cosmological data are well described by single-field inflation [52, 54], models of multifield inflation are of great theoretical interest, as they provide new predictions that could be tested by future observations [93, 94]. Scalar-tensor theories constitute a fairly wide class of theories: they include standard single- field inflation, non-minimally coupled inflation, and F (ϕ, R) theories (via a Legendre transformation, as we shall see in Chapter 6). In this section, we will specify the tree-level action for scalar-tensor theories and determine their 56 CHAPTER 3. CLASSICAL FRAME COVARIANCE behaviour under frame transformations, which we will use to motivate them as examples of a differentiable manifold. By introducing explicit expressions for the covariant extensions of cosmological quantities, we will be able to recast the background equations of motion in a frame-covariant form, paving the way for dealing with the perturbations that eventually seed observables in a frame-covariant manner.

The motivation for the form of the action in scalar-tensor theories is the pro- motion of the eﬀective Planck mass MP to a dynamical function of one or more scalar ﬁelds, as shown below:

4 f(ϕ)R kAB(ϕ) µν A B S d x √ g + g (∂µϕ )(∂νϕ ) V (ϕ) + M (ϕ, ψ) , ≡ − − 2 2 − L Z (3.29) where we have left the dependence of M on the metric implicit. In the 2 L action (3.29), MP has been replaced by a field dependent coupling f(ϕ), which is the non-minimal coupling to the Ricci scalar R. The kinetic term is given by kAB (sometimes known as the inflaton wavefunction), and V (ϕ) is the scalar inflaton potential. These (functional) parameters are functions of ϕ which, without any indices, collectively stands for all the scalar fields ϕA. Uppercase indices A, B, . . . run over the different fields. Finally, we use M to denote the L matter field sector, whose internal structure is not specified for now, save for the fact that it is composed of fermionic fields ψ coupled to the scalar fields.

In the previous sections, we have described how a conformal transformation can change the scale of the system via a local rescaling of the metric, given by

2 gµν g˜µν = Ω gµν, 7→ (3.30) ϕA ϕA = Ω−1ϕA. 7→ We have incorporated the dilation ofe the ﬁelds ϕA in the conformal transformation (3.30). In order for the algebra induced by the transformations to be closed, we restrict ourselves to the ﬁeld-dependent class of conformal factors, where the spacetime dependence is implicit: Ω(x) = Ω(ϕ(x)).

There is nothing inherently “fundamental” about the fields ϕA in the way we have written the action. Indeed, ϕA is nothing more than a representation of an underlying manifold, which as we shall see in the next section is known as the field space. For this precise reason, it is possible to apply a field 3.3. FRAMES IN MULTIFIELD SCALAR-TENSOR THEORIES 57 reparametrisation reminiscent of a diffeomorphism given by

ϕA ϕAe = ϕAe(ϕ), (3.31) 7→ and rewrite the action in terms of the new ﬁelds ϕAe. This ﬁeld reparametrisation has an associated Jacobian:

dϕAe = J Ae . (3.32) dϕB B

Note that we use transformed indices with a tilde in the new basis of fields and transformed fields with a tilde for the conformal transformation (3.30). The transformation (3.31) essentially corresponds to a diffeomorphism between two sets of coordinates.

Applying both a conformal transformation (3.30) and a ﬁeld reparametrisation (3.32) is referred to as a frame transformation, which has an associated Jacobian given by:

Ae dϕ −1 Ae Ae = Ω J ϕ (ϕ)(ln Ω),B dϕB B − (3.33) h i e Ω−1KAe . ≡ B Thus, the full frame transformation is given by

2 gµν g˜µν = Ω gµν , 7→ (3.34) ϕA ϕAe = Ω−1 ϕAe(ϕ) . 7→ The original action (3.29) is saide to be in the Jordan frame, where the non- minimal coupling of the inﬂaton to the Ricci scalar is explicit. Applying (3.34) allows us to transform the action from one frame to another. In particular, it is possible to transform to the Einstein frame by choosing the conformal factor Ω such that the non-minimal coupling f(ϕ) in the action becomes a constant:

4 1 2 kAB A µ B S = d x g˜ MP R + ( µϕ )( ϕ ) V (ϕ) + M . − "−2 2 ∇ ∇ − L # Z p e e e e e e e (3.35)

We may thus say that two actions belong in the same frame class if one can be transformed into the other by virtue of a frame transformation. 58 CHAPTER 3. CLASSICAL FRAME COVARIANCE

There has been much discussion in the literature about whether the Jordan frame or the Einstein frame are physically equivalent. In particular, there have been claims of both conformal independence [8–13] and conformal dependence [15–18] in the literature. We observe, however, that the way that the conformal transformation ends up scaling only dimensionful quantities reminds us of the unit transformations considered in Section 3.2. Fundamentally, there is no system of units that is preferable over any other, much like no observer enjoys an inherent privilege. Thus, we might conclude that diﬀerent frames should be physically indistinguishable. This is exactly the case for a constant conformal factor, and with the help of frame covariance, we argue that this should extend to the case for a spacetime-dependent conformal factor as well. We will devote the rest of the section to demonstrating how this may be made manifest via the techniques of frame covariance.

Spacetime tensors are gauge-covariant quantites, characterised by their transformation properties under a diffeomorphism. Similarly, we may define frame tensors that transform in a similar way. In the rigorous definition of a tensor in (2.2), we use the tangent space of some manifold which is parametrised by the fields ϕA. We will later see that this is a Riemannian manifold known as the field space, but for now, its metric structure does not matter. What matters is that it has an associated tangent space at each field configuration. Then, if A we use a particular basis dϕ and a cobasis dϕB , it is possible to express { } { } the tensor through its components. Neglecting conformal transformations, the components transform under a reparametrisation (3.31) as

Ae1...Aep Ae1 Aep A1...Ap B1 Bq X = J A ...J A XB ...B J ...J . (3.36) Be1...Beq 1 p 1 q Be1 Beq

We have transformed only the indices to indicate that this has only been a basis transformation with associated Jacobian (3.32).

When Ω = 1, the Jacobian that corresponds to the combined reparametri- 6 sation and conformal transformation (which places a tilde on both the ﬁelds themselves and their indices as ϕA ϕAe) is given by (3.33). Therefore, we 7→ expect the quantity XA1...Ap will pick up factors of Ω under a frame trans- B1...Bq formation (3.34). We therefore identifye it as a frame tensor if it obeys the 3.3. FRAMES IN MULTIFIELD SCALAR-TENSOR THEORIES 59 following transformation properties [101]:

A ...A A ...A X 1 p = Ω−wX X 1 p , (3.37) B1...Aq B1...Bq

Ae1...Aep −(p−q) Ae1 Aep A1...Ap B1 Bq X = Ω K A ...K A XB ...B K ...K . (3.38) eBe1...Beq 1 p 1 q Be1 Beq

The first transformation property gives the usual conformal weight of the quantity, and often corresponds to its mass dimension. The second property demonstrates how the frame-covariant quantity transforms under a reparamerisation when Ω = 1. We can thus see that the effects of conformal transformations 6 and field reparametrisations are linked as a consequence of our choice to assign a weight to the fields ϕA in the conformal transformation (3.30). Indeed, if we had chosen not to assign a weight to the fields, there would be no prefactor of Ω in (3.38).

A quantity that satisﬁes (3.37) and (3.38) is more rigorously termed a tensor density, but we will slightly abuse terminology and refer to it as a tensor. In order to avoid clutter, we will suppress the arguments of ϕ in the following. We note that (3.37) is similar to (3.24), except written in terms of the conformal factor Ω. The diﬀerence is that wX now stands for the conformal weight of the quantity XA1...Ap , which does not depend on its number of indices. B1...Bq

The frame transformation (3.34) simply combines the two transformation properties (3.37) and (3.38) as follows, giving the following transformation property for frame tensors:

Ae1...Aep −dX Ae1 Aep A1...Ap B1 Bq X = Ω K A ...K A XB ...B K ...K . (3.39) Be1...Beq 1 p 1 q Be1 Beq

In this expression,e dX denotes what we term the scaling dimension of X, given by

dX = wX + p q . (3.40) − The scaling dimension is in a sense the frame weight of this quantity. Every contravariant index contributes +1 to the scaling dimension, whereas covariant indices contribute 1, in addition to the weight wX induced by the conformal − transformation. For instance, according to this convention, the spacetime metric gµν (which carries no ﬁeld indices) will have conformal weight and scaling dimension both equal to 2. We note that the ﬁeld ϕA does not transform − as a vector under the transformation (3.33). Indeed, it transforms in a highly 60 CHAPTER 3. CLASSICAL FRAME COVARIANCE non-linear way, since it is dϕA that transforms as a vector (contracted with the Jacobian). This is once again similar to spacetime: only elements of the tangent space dxµ are vectors, whereas the coordinates xµ for some chart are not.

We may now use the definitions we laid out in this section to determine whether the action S in (3.29) is frame invariant. To do so, it is instructive to study how the action behaves under the most general frame transformation. Ap- plying (3.5) to the Ricci scalar R, we find that it transforms according to (2.65). Therefore, applying a conformal transformation to (3.29) followed by a redefinition of the field, the transformed action becomes

˜ k˜ (ϕ) 4 f(ϕ)R AeBe µν Ae Be S = d x g˜ + g˜ (∂µϕ )(∂νϕ ) V (ϕ) + M , − "− 2 2 − L # Z p e e e e e (3.41) where there is also an implicit gµν dependence in M . Comparing (3.41) with L (3.29), we ﬁnd that the model functions transform as follows [100]:

f˜(ϕ) = Ω−2f(ϕ), ˜ A B k (ϕ) = kAB 6f(ln Ω),A(ln Ω),A + 3f,A(ln Ω),B + 3f,A(ln Ω),B K K , AeBe − Ae Be V (ϕ) =h Ω−4V (ϕ), i −4 M = Ω M . e L L (3.42) e Given the definition of frame-covariant quantities in (3.39), we can see from (3.42) that the effective Planck mass squared f and the scalar potential V are frame- covariant, with scaling dimensions 2 and 4 respectively. We observe that kAB is not a frame-covariant quantity; however, these transformation rules may still be used to show that the action S is form invariant. This means that models linked by a frame transformation have equal actions, despite being manifestly different:

˜ ˜ S[gµν, ϕ, f, kAB,V, M ] = S[˜gµν, ϕ, f, kAB, V, M ] . (3.43) L L This equivalence is the starting point for the frame-covariante e e formalism, since it ensures that any results derived for one frame must apply to any other frame. This agrees with our argument that a frame transformation may be viewed as a ﬁeld-dependent change of units. 3.3. FRAMES IN MULTIFIELD SCALAR-TENSOR THEORIES 61

Our aim in the next section is to write the equations of motion in a manifestly covariant manner. To this end, we proceed to deﬁne the basic frame-covariant quantities [100–102]:

kAB 3 f,Af,B V GAB + ,U . (3.44) ≡ f 2 f 2 ≡ f 2

Under (3.39), these quantities transform as

2 A B G = Ω GAB K K , U = U. (3.45) AeBe Ae Be

We thus observe wGe = 0 (as it is invariant under conformale transformations), dG = 2 (thanks to its two covariant field indices) and wU = dU = 0. − Our subsequent task is to define a derivative that respects the covariant properties of the quantities on which it acts. We have an inkling as to how to proceed about this thanks to our work in Section 3.2 when it comes to conformal transformations. We consider the derivative with respect to the field ϕC . Such a derivative must satisfy the transformation property (3.39):

Ae1...Aep −(dX −1) Ae1 Aep A1...Ap C B1 Bq C X = Ω K A ...K A C XB ...B K K ...K . (3.46) ∇ e Be1...Beq 1 p ∇ 1 q Ce Be1 Beq In thee above expression, X represents the frame-covariant derivative of X ∇Ae with respect to the conformally transformed ﬁeld in the new basis, ϕAe.

Focusing on preserving property (3.37) ﬁrst, we construct the conformallye covariant ﬁeld derivative. The form is similar to (3.25):

w f XA1A2...Ap XA1A2...Ap X ,C XA1A2...Ap . (3.47) B1B2...Bq|C ≡ B1B2...Bq,C − 2 f B1B2...Bq

It is also possible to deﬁne a conformally covariant spacetime derivative as

wX f,µ X|µ X;µ X, (3.48) ≡ − 2 f where we have suppressed field space indices. Using the conformally covariant field derivative given in (3.47), we may write down a Christoffel-like connection using GAB as a metric analogue [104],

AD A G Γ = GDB|C + GCD|B GBC|D . (3.49) BC 2 − A This construction ensures that ΓBC is conformally invariant, with wΓ = 0. 62 CHAPTER 3. CLASSICAL FRAME COVARIANCE

Therefore, the conformally-covariant derivative can be extended so as to incorporate field reparametrisations, leading to a fully frame-covariant field derivative defined as

A1...Ap A1...Ap A1 D...Ap Ap A1...D C XB ...B X + ΓCDXB ...B + + ΓCDXB ...B ∇ 1 q ≡ B1...Bq|C 1 q ··· 1 q (3.50) ΓD XA1...Ap ΓD XA1...Ap . − B1C D...Bq − · · · − BqC B1...Bq It is then straightforward to verify that the definition of the frame-covariant derivative given above satisfies the covariance condition (3.46). Using (3.50), it is thus possible to define a frame-covariant derivative λ of a frame-covariant D quantity with respect to any parameter λ (including coordinates xµ) as follows:

C A1...Ap dϕ A1...Ap λX C X . (3.51) D B1...Bq ≡ dλ ∇ B1...Bq For a quantity without any field tensorial indices, this definition simply reduces to the unit-covariant derivative (3.25). We have therefore spelled out a method by which an infinite array of conformally covariant quantities may be constructed. This may be construed as a natural extension to [103], where a method for constructing an arbitrary number of conformally invariant quantities in scalar-tensor theories is presented.

We close this section by underlining a very important point regarding the term “non-minimal models”. We have seen that each scalar-tensor theory corresponds to a particular differentiable manifold equipped with a metric GAB. However, we must be very careful not to confuse the representation of a theory (as given by writing down a particular form of the action) with the underlying theory itself. “Non-minimality” is not an inherent feature of the model, but rather a consequence of working in an “exotic” system of spacetime-dependent units. At least at the classical level, every model can be fully classified with the help of two parameters: the kinetic term GAB and the potential V (ϕ). Even speaking of “non-canonical” models where the kinetic term is not “flat”

(i.e. GAB is not proportional to δAB) is misleading. It is possible to work in a parametrisation of the fields where the kinetic term appears non-canonical but the underlying manifold is not curved. The physically relevant quantity that encodes whether the theory is flat or not is the (conformally covariant) field space Riemann curvature tensor, defined as

RA ΓA ΓA + ΓA ΓC ΓA ΓC . (3.52) MBN ≡ NM|B − BM|N BC NM − NC BM 3.4. DYNAMICS OF MULTIFIELD INFLATION 63

If all the components vanish, then the model is indeed ﬂat. “Non-canonical” and “non-minimal” are therefore statements about the representation of a model, not the model itself.

3.4 Dynamics of Multiﬁeld Inﬂation

The form invariance of the action given in (3.43) indicates that frame covariance must be reflected at the level of the equations of motion. In this section, we will derive the equations of motion for a general multifield scalar-tensor theory. The equations of motion are commonly written in a non-covariant form, but we will show how they can be written in a manifestly covariant form. We will focus on the inflaton sector; we will assume that the matter sector is sufficiently diluted by the scalar fields such that M vanishes. L We begin by varying the action S (3.29) with respect to the fields ϕA and setting the variation to zero. This yields the inflaton equations of motion:

µ B kAB,C kCA,B kBC,A B µ C f,A kAB µ ϕ + + ( µϕ )( ϕ ) + R + V,A = 0. ∇ ∇ 2 2 − 2 ∇ ∇ 2 (3.53)

If we vary S with respect to the metric gµν, we find rise to a modified version of the Einstein field equations:

−2 (NM) Gµν = MP Tµν , (3.54)

−2 where we remind MP (8πG) . ≡ The presence of the non-minimal coupling in the action can be viewed in two equivalent ways. We may view it as a modification of the left-hand side of the Einstein field equations (the gravity sector), or as a modification to the energy-momentum tensor. The standard energy momentum tensor (in the case 2 of f = MP ) is given by

A B kAB A ρ B Tµν = kAB( µϕ )( νϕ ) ( ρϕ )( ϕ )gµν + V gµν. (3.55) ∇ ∇ − 2 ∇ ∇ Using the expression for the standard energy-momentum tensor (2.16), we may 64 CHAPTER 3. CLASSICAL FRAME COVARIANCE

(NM) ﬁnd the expression for the non-minimal energy-momentum tensor Tµν :

−2 (NM) Tµν f,AB A ρ B f,A 2 A M T = ( ρϕ )( ϕ )gµν ( ϕ )gµν P µν f − f ∇ ∇ − f ∇

f,A A f,AB A B + ( µ νϕ ) + ( µϕ )( νϕ ) . (3.56) f ∇ ∇ f ∇ ∇

We have thus derived the equations of motion that govern the evolution of scalar ﬁelds on a general curved background.

The equations of motion are of particular cosmological interest when the scalar ﬁelds ϕ are spatially homogeneous, i.e. ϕA = ϕA(τ). We have already seen that the Friedmann and acceleration equations take on the forms (2.28) and (2.29) when the metric takes on the FLRW form, which we remind is

2 2 2 2 i j ds = NL(τ) dτ a(τ) δij dx dx . (3.57) − Once again, we diﬀerentiate between the time coordinate τ and the cosmic time t. This distinction is important for frame covariance since setting NL to unity would not allow us to keep track of the conformal transformation.

We now turn our attention to cosmology by imposing the assumptions of homogeneity of the ﬁelds and the FRW form for the metric, we may eliminate the Ricci scalar R from the scalar ﬁeld equation (3.53) by taking the trace of (3.54). This results in

3f f 0 = k + ,A ,B ϕ¨B + 3H + H ϕ˙ B AB 2f L 1 f,A B C 2 2 V + kBA,C + kAC,B kBC,A + kBC + 3f,BC ϕ˙ ϕ˙ + N f . 2 − f L f 2 ,A (3.58)

In addition to the Hubble parameter, we may also deﬁne the lapse rate HL as follow: ˙ a˙ NL H ,HL . (3.59) ≡ a ≡ NL

Here and in what follows, the overdot denotes differentiation with respect to the coordinate τ, not the cosmic time t. We make this choice because even when NL = 1, it is common to define the Hubble parameter using the time co- 6 ordinate τ, not the cosmic time t. The two are identified when NL(τ) = 1, but 3.4. DYNAMICS OF MULTIFIELD INFLATION 65

we do not make this assumption since we cannot have NL constant simultaneously in diﬀerent frames (since it picks up a conformal factor). Furthermore, the Friedmann and acceleration equations may be derived via the temporal and spatial components of (3.54):

1 k ϕ˙ Aϕ˙ B Hf˙ H2 = AB + N 2V , (3.60) 3f 2 L − f A B ˙ ¨ ˙ 1 kABϕ˙ ϕ˙ Hf f H + HLH = + . (3.61) −2f 2 2f − 2f As noted before, these equations of motion appear in the literature in various 2 forms. For minimal inflation models, f = MP , whereas for single-field inflation A the fields ϕ are replaced by ϕ. Moreover, it is common to set NL = 1 in the literature, since it makes calculations easier when working in a given frame.

However, it is crucial to work with a generic lapse function NL when working in the context of frame transformations.

Note that the cosmological equations (3.58), (3.60), and (3.61) are not manifestly frame invariant as written above. It is possible (though cumbersome) to show that the equations of motion derived from the action in the Jordan and the Einstein frame are equivalent [105]. However, using our frame-covariant techniques, we will see that it is possible to write them solely in terms of frame-covariant quantities. With the aid of (3.51), we may extend the deﬁni- tion of the usual Hubble parameter H given in (2.22) to the covariant Hubble parameter and covariant lapse parameter L by promoting the ordinary H H time derivative to a covariant derivative:

ta tNL D , L D , (3.62) H ≡ a H ≡ NL where we use dt = NLdτ and from these definitions we find = (H HL)/NL H − (remember that H in (3.59) is defined with respect to τ and has no NL factor

“hidden” in it). We note that if NL = a, then = 0. This is not a problem: H the derivative is taken with respect to cosmic time (which scales with NL), not the time coordinate (which does not scale). Simply setting NL = a without compensating by scaling a is not a conformal transformation, therefore is H under no obligation to scale covariantly in this case. If the metric is conformally

ﬂat and observers use dt = NLdτ = adτ as their clock, the Hubble rate will vanish, since they would not be able to see any expansion. Relations in the literature linking the Einstein and Hubble frame parameters [10] are predicated upon the assumption that the physical clock is always the time coordinate, and 66 CHAPTER 3. CLASSICAL FRAME COVARIANCE so they reduce to (3.62) if we do not transform t between frames.

We remind that the frame-covariant derivative acting on quantities with no ﬁeld indices takes on the form

wX f,λ λX = X,λ X D − 2 f

= X,λ + dX HLX. (3.63)

Note that in (3.63), we have eliminated the dependence on f by making use of the fact that for the FLRW metric,

˙ ˙ NL f HL = = . (3.64) NL − 2f

2 This equality is derived from the observation that NLf is a constant by con- 2 struction in the Einstein frame (where dτ = dt and f = MP ). As such, it is constant in all frames (since its scaling dimension is zero).

We may now begin to rewrite the equations of motion (3.58), (3.60), and (3.61) in a manifestly covariant way. We ﬁrst write the second covariant time derivative of the ﬁelds ϕA by writing

A −1 A A tϕ = N ϕ˙ + HLϕ . (3.65) D L We note that ϕA is not frame-covariant itself, and as such, it does not strictly A make sense to apply µ to it. However, ϕ does transform as a conformally D A covariant quantity, and whereas it would be more precise to write ϕ|t instead, A we slightly abuse notation by using t to emphasise that Dtϕ is not only D conformally covariant, but also a frame-covariant (and gauge covariant) entity A A in and of itself. Using (3.51) with X = tϕ , we may write D A −1 ˙ A A B C t tϕ = N X + Γ X X . (3.66) D D L BC Using (3.66), along with the frame-covariant quantities GAB, U, and given H in (3.44) and (3.62), it is now possible to rewrite the equations of motion in a fully frame-covariant manner. In detail, (3.58) becomes

A A AB t tϕ + 3 ( tϕ ) + fG U,B = 0 . (3.67) D D H D 3.4. DYNAMICS OF MULTIFIELD INFLATION 67

X conformal weight (wX ) scaling dimension (dX ) dxµ 0 0 dϕA 0 1 dϕA 0 1 − gµν 2 2 gµν −2− 2 NL, a 1 1 −1− 1 fH 2 2 GAB 0 2 GAB 0− 2 U 0 0 XA1...Ap d d d d + p q λ B1...Bq X δλ X δλ D A1...Ap − − − X dX dX 1 + p q B1...Bq;A − − Table 3.1: Conformal weights and scaling dimensions of various frame- covariant quantities.

Correspondingly, (3.60) and (3.61) become

A B 1 GAB( tϕ )( tϕ ) 2 = D D + fU , (3.68) H 3 2 A B GAB( tϕ )( tϕ ) t = D D . (3.69) D H − 2 We may easily verify that these equations reduce to their well-known forms for single-field inflation. We finally observe that (3.67) is a geodesic equation with A metric GAB featuring two external forces: (i) a drag force proportional to tϕ D and (ii) a conservative external force proportional to U,A. This analogy to differential geometry will be further explored in Section 4. For now, it suffices to note that (3.67)–(3.69) are fully frame-covariant; each individual term in them transforms with exactly the same weight and Jacobian. The properties of the quantities that make up the equations of motion can be seen in Table 3.1.

As discussed in the simple single-field model discussed in Section 2.4, inflation occurs when the kinetic term of the inflaton is dominated by the potential such that the effective equation of state can generate enough negative pressure to cause the Universe to expand. This motivates the definition of the invariant Hubble slow-roll parameters as follows:

t t¯H ¯H D H2 , η¯H D . (3.70) ≡ − ≡ ¯H H H

These parameters are straightforward extensions of the usual parameters H = 68 CHAPTER 3. CLASSICAL FRAME COVARIANCE

˙ 2 H/H and η = ˙H /(HH ) which are not invariant. Using (3.68) and (3.69), − it is possible to demonstrate that when ¯H 1, the potential term dominates the kinetic term and inflation can occur. We also observe that ¯H acts as a deformation parameter for de Sitter space: when it is zero, inflation will never end, since there will be no kinetic term. However, if it is small but nonvanishing, inflation will eventually reach its end.

We have further defined a parameterη ¯H , which controls how quickly ¯H varies: this further motivates the definition of slow-roll approximation, which occurs when the slow-roll parameters are small but also slowly varying. We note that while the name of this approximation stems from the single-field minimal scenario, the slow-roll parameters defined in (3.70) are only implicitly dependent on the fields. We may be tempted to write the slow-roll approximation in a covariant form as follows:

A A ,A t tϕ ( tϕ ) V . (3.71) D D H D However, this particular form depends on the particular chart we choose. There have been attempts at writing the hierarchy in a way that respects the spacetime-dependent effective Planck mass [106,107], but in the end, they all are parametrisation dependent. The manifestly covariant way to ensure that slow-roll is occuring to examine whether ¯H andη ¯H are small. Asking “which” fields act as inflatons (and which as spectators, such as in the curvaton scenario [108]) only makes sense after we have selected a particular parametrisation.

We conclude our frame-covariant treatment of the dynamics of inflation by considering when inflation ends. As discussed in Section 2.4, the Universe stops accelerating when the comoving Hubble horizon stops shrinking. In the d −1 Einstein frame, this occurs when dt (aH) > 0, which may equivalently be written as H = 1. However, the non-covariance of H means that this is not a covariant expression. Indeed, it holds only in the Einstein frame. Using the frame-invariant Hubble slow-roll parameters, we may define the invariant end-of-inflation condition as follows:

¯H (tend) = 1. (3.72)

Slow-roll inﬂation carries the additional assumption that ¯H is slowly varying, which is encoded in the assumption thatη ¯H must also be a small parameter. 3.4. DYNAMICS OF MULTIFIELD INFLATION 69

This is reﬂected in the slow-roll end-of-inﬂation condition

max(¯H , η¯H ) = 1 . (3.73) | | Calculating the time at which inflation ends leads to the definition of the number of e-folds, which is defined as dN = dt (counting counting backwards −H from the end of inflation):

t N = dt0 (t0). (3.74) − H Ztend

The frame-covariant derivative N with respect to the number of e-folds is D deﬁned with the help of (3.51). While it is possible to deﬁne the number of e-folds such that it is not frame-invariant [15], doing so requires additional book-keeping: the number of e-folds between two frames related by conformal factor Ω (as f/f˜ = Ω−2) is going to be related as

N N = ln Ω. (3.75) − Moreover, this contradicts oure intuition regarding the conformal weight of dimensionless quantities, and as such, we will adopt the deﬁnition (3.74) that ensures that the number of e-folds is the same in all frames.

In this chapter, we have introduced the concept of frame covariance beginning from the point of view that frame transformations are akin to a change of units, both of which cannot be observed by any experiment. We formalised the notion of frame covariance by virtue of specifying the transformation properties of frame-covariant quantities, which we further applied explicitly to multifield scalar-tensor theories. After deriving the classical equations of motion that govern the dynamics of inflation into frame-covariant forms as given in equations (3.67) – (3.69), we have completed our treatment of classical frame covariance by presenting the frame-covariant slow-roll approximation. In the next chapter, we will further study the concept of a field space and use the background equations. This will help us determine the evolution of the perturbations which form the seeds for the observable cosmological anisotropies in the presence of multiple scalar fields. Chapter 4

Quantum Perturbations in Field Space

The generation of observable anisotropies on the surface of last scattering is fundamentally a quantum phenomenon. The correlation functions of the primordial perturbations of the metric and the scalar fields are the seeds that eventually give rise to the profile that we may see in the cosmic microwave background. In single-field inflation, perturbations “freeze” as they cross the horizon [109,110], meaning that the perturbations that are just now re-entering the horizon are relics of inflation, leaving their imprint on the CMB.

In the scenario where multiple fields drive inflation, the evolution of the fields can be described by a trajectory in a manifold known as the field space and parametrised by the fields. This geometric picture enables us to decompose the perturbations into curvature modes that are parallel to the inflationary trajectory, and isocurvature modes that are perpendicular to it [111–113]. Isocur- vature modes do not freeze after they exit the horizon [114], and as such, we need to be particularly careful in their treatment.

In this chapter, we will employ well-established results from differential geometry in order to treat multifield inflation with the help of the field space, a concept first formally identified by [115]. We will extend the field space by ensuring that our treatment of it is conformally covariant, using the notions we developed in the previous chapter. We will finally discuss the phenomenological impact of the entropy transfer between curvature and isocurvature modes before specialising to two-field inflation.

70 4.1. GAUGE INVARIANT PERTURBATIONS 71

4.1 Gauge Invariant Perturbations

In order to calculate the correlation functions due to the metric and the field, we will first need to decompose the metric. We will be using the background field method, in which the metric and the scalar fields are perturbed around their classical values. We can parametrise the first order perturbations of the metric in the so-called scalar-vector-tensor (SVT) decomposition:

µ ν 2 2 gµν dx dx = NL (1 + 2φ) dτ i + 2 (aNL)(B|i + Bi) dτ dx

2 i j a 1 2ψ δij + E|ij + Ej|i Ei|j + hij dx dx . − − − h i (4.1) In this decomposition, the four scalar perturbations are given by φ, ψ, E and B. The four vector perturbations are contained in Ai and Bi (the con- i i straints E|i = B|i = 0 reduce the number of degrees of freedom from 6 to 4).

Finally, hij contains the two tensor perturbations (thanks to the constraints j i of symmetry hj = h|j = 0 that reduce the number of degrees of freedom from 6 to 2). We note that we have used conformally-covariant derivatives as opposed to standard partial derivatives in the decomposition. This ensures that the perturbations will be conformally covariant (and, in this particular case, invariant as well, since the conformal weight is fully absorbed in the background quantities a and NL) without aﬀecting the number of remaining degrees of freedom.

We now examine how the components of the SVT decomposition transform under a diffeomorphism. A diffeomorphism is simply a transformation of the coordinates xµ, and an infinitesimal diffeomorphism can be written as

xµ xˆµ = xµ + ξµ. (4.2) → We may split this inﬁnitesimal transform in temporal and spatial components as follows:

x0 xˆ0 = x0 + , → xi xˆi = xi + ξi. (4.3) → Under this transformation, the metric and the inﬂaton will transform. As a 72 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE result, their perturbations will also transform under

A A A ϕ + MP δϕ (x) ϕˆ + MP δϕˆ (x), → 2 2 gµν + M δgµν(x) gˆµν + M δgˆµν(x). (4.4) P → P

A It is important to point out that the perturbations δϕ and δgµν are independent of their background values. This explains the diﬀerence in their conformal weights: it is more convenient to take the perturbations to be dimensionless.

The factors of MP in the above decompositions ensue dimensional consistency according to Table 3.1.

The change in the perturbation of a quantity Q after a gauge transform is given by its Lie derivative

∆(δQ) = δQˆ δQ = Lie(Q, ξ). (4.5) − Applying the Lie derivative to the inflaton perturbation and metric perturbations, we find δϕA(x) δϕÂ(x) = δϕA(x) + ξµ ϕA (x). (4.6) → |µ

In this derivation, ordinary derivatives ,µ have been replaced with their conformally covariant counterparts |µ. Similarly, for the metric perturbation, we may write

ρ ρ ρ δgµν(x) δgˆµν(x) = δgµν(x) + ξ gµν|ρ(x) + gρν(x)ξ + gµρ(x)ξ . (4.7) → |µ |ν In both (4.6) and (4.7), the covariant derivatives are taken with respect to the background. We can express (4.7) in terms of Christoﬀel symbols via covariant derivatives:

δgµν(x) δgˆµν(x) = δgµν(x) + µξν + νξµ. (4.8) → D D

Note that in (4.8), we use the frame-covariant derivative µ to replace the D covariant derivative µ, since we now desire to have both diﬀeomorphism ∇ covariance alongside conformal covariance.

We now proceed to derive the rules that the perturbations will transform under. Substituting the SVT decomposition (4.1) into (4.7) and comparing 4.1. GAUGE INVARIANT PERTURBATIONS 73 the temporal and oﬀ-diagonal components, we ﬁnd

ˆ φ = φ + NL + |η, (4.9) H ˆ 1 1 Bi = Bi + ξi|η + |i + NL ξi, (4.10) 2 2 H where dη (NL/a)dτ and we have used µ to indicate the conformally covari- ≡ µ | T ant derivative (3.48) with respect to x . We write ξi to be the transverse part T of the spatial perturbation, given by ξi = ξ,i + ξi . Comparing the spatial part of (4.7) and (4.8), we ﬁnd

By comparing terms (4.11), we may ﬁnd how the perturbations transform. The rules for the scalar perturbations are therefore summarised as follows:

φ φ + NL + |η, → H ψ ψ NL , → − H (4.12) E E + ξ|η, → B B + ξ|η → − while the rules for vector and tensor perturbations are

T Ei Ei + ξ , → i (4.13) hij hij. → We note that the tensor perturbations are gauge invariant. We also note that only the scalar part of the diffeomorphism affects the scalar perturbations, since at first order the different perturbations decouple.

We now restrict our attention to scalar perturbations only. For convenience, we may deﬁne

∆ B E. (4.14) ≡ − This quantity transforms via (4.12) as

∆ ∆ . (4.15) → − With the help of ∆, we may construct the so-called Bardeen potentials [116], 74 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE which are deﬁned as

Φ = φ + ∆ + ∆|η, H Ψ = ψ ∆. (4.16) − H The Bardeen potentials are fully gauge invariant. However, our use of conformally covariant derivatives ensures that they are frame-invariant as well. They are often used to express the spacetime metric in the Newtonian gauge, where we set ∆ = 0:

µ ν 2 2 2 i j gµνdx dx = (1 + 2Ψ)N dt a (1 2Φ)δij + hij dx dx , (4.17) L − − We may further deﬁne the frame-covariant extension of the gauge invariant Mukhanov–Sasaki variables [117,118] as follows:

A tϕ QA δϕA + D Ψ . (4.18) ≡ H We can verify using (4.12) and (4.6) that QA is indeed gauge invariant. In the Newtonian gauge, we usually write Ψ = ψ. The diﬀerence between the standard decomposition commonly found in the literature [119, 120] and the decomposition outlined above is the use of frame-covariant derivatives. This ensures that the metric perturbations are also frame-invariant and that the Mukhanov–Sasaki variables are frame-covariant (and conformally invariant, since their weights are given by wQ = wΦ = 0 and wδϕ = 0).

4.2 The Field Space in Multiﬁeld Inﬂation

As seen in the previous chapter, any scalar-tensor theory may be viewed as a particular representation of a differentiable manifold where the fields ϕA correspond to a particular coordinate chart. This motivates the concept of the field space [115, 121], which is particularly useful in studies of multifield inflation (in single-field inflation, the manifold is one-dimensional and therefore trivial). The metric may be used to define the field space line element

2 A B dσ = GAB dϕ dϕ , (4.19)

A which also naturally leads to the definition of the field-space connection ΓBC as AB given in (3.49). We assume that the metric GAB has an inverse G , and that 4.2. THE FIELD SPACE IN MULTIFIELD INFLATION 75 both can be used to raise and lower indices of vectors and covectors living in the tangent and cotangent field spaces respectively. If GAB is positive-definite, all scalar fields in the corresponding theory will have physical (non-tachyonic) kinetic terms. Moreover, we expect that the metric can be used to derive (3.67) as a geodesic equation in the absence of external forces. The simplest form of the metric that does this is GAB, given in (3.44).

There is an element of perturbation theory unique to multifield inflation that we have not considered yet: the distinction between curvature and isocurvature modes. The Sasaki-Mukhanov variables QA describe the perturbations in a frame-invariant way, but they are expressed in a chart ϕA chosen before the trajectory is calculated. We would prefer if the perturbations are decomposed in a chart defined with respect to the trajectory itself. To do so, we distinguish between perturbations that live the manifold subspace parallel to the inflationary trajectory (curvature perturbations) and perturbations perpendicular to the curvature ones (isocurvature perturbations). In order to rewrite QA in terms of curvature and isocurvature perturbations, we use a vielbein-like for- A malism. We define a set of frame fields ea that have the following properties:

α β AB A B αβ GAB e e δαβ ,G e e δ . (4.20) ≡ A B ≡ α β Using these frame ﬁelds, we may rewrite the perturbations as follows:

α A α A α A Q = Q eA ,Q = Q eα . (4.21)

In this context, Greek indices α, β, . . . run over σ, s1, s2,... : the direction σ { } corresponds to the curvature component, and s1, s2,..., correspond to the isocurvature components.

We now examine the isocurvature submanifold perpendicular to the tangent vector. This submanifold is spanned by a projection operator, given by

AB AB A B (s1) G e e . (4.22) ≡ − σ σ

A The tangent frame ﬁelds eσ can be explicitly written as

A AB A A tϕ A (s1) U,B ω eσ = D , es1 = = . (4.23) tσ − (s )ABU U ω D 1 ,A ,B We remind that σ is defined via (4.19),p which is important in normalising the 76 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE frame fields. The physical interpretation of the above frame fields can be seen by considering the trajectory as it evolves in field space. As the value of the A A fields moves along the trajectory, eσ is parallel to it, whereas the es1 gives us its component of acceleration perpendicular to the tangent space, corresponding to the “first” isocurvature perturbation. The first isocurvature perturbation is closely related to the turn rate ω, which is the field space magnitude of the acceleration vector ωA, given by

A A tϕ ω = N D , (4.24) D tσ D where N is given in (3.74). The acceleration vector simply corresponds to the A rate of change of the unit tangent vector eσ . As such, it eﬀectively measures the rate of change between geodesics in curved space. We may express the scalar ﬁeld equations (3.67) and (3.68) in terms of σ, which enables us to write the equations of motion as

,σ t tσ + 3 ( tσ) + fU = 0 , (4.25) D D H D 2 1 ( tσ) 2 = D + fU , (4.26) H 3 2 ,σ ,A σ where U = U eA.

In order to make contact with observations, we deﬁne the comoving curvature perturbation and the comoving isocurvature perturbations (i): R S

H Qσ, (i) H Qsi . (4.27) R ≡ tσ S ≡ tσ D D These quantites are gauge- and frame-invariant, and as such, they are observationally relevant. The curvature perturbation is of particular interest to us, R since it remains constant on superhorizon scales [109,110] in single ﬁeld inﬂa- tion. Moreover, sources the (dimensionless) observable scalar spectrum PR R through its two-point function:

2 2π (3) PR δ (p + q) p q , (4.28) p3 ≡ hR |R i where the two-point correlation function is taken with respect to the Bunch– Davies vacuum, which describes the quantum state of the Universe at very early times [122]. 4.2. THE FIELD SPACE IN MULTIFIELD INFLATION 77

In order to ﬁnd an expression for PR, we must solve and canonically quantise the perturbed equations of motion [104]. These can be found by varying the action to second order and integrating by parts. Eventually, we obtain

2 A A p A A B t tQ + 3 ( tQ ) + Q + M Q D D p H D p a2 p B p 3 (4.29) 1 NLa A B C 3 t ( tϕ )( tϕ ) GBC Qp = 0 . − NLa D D D H

The frame-covariant mass matrix MAB is given by

M N MAB f( A BU) + RAMBN ( tϕ )( tϕ ) , (4.30) ≡ ∇ ∇ D D and RAMNB is given in (3.52). Once again, these motions are often written in a way that is not conformally covariant way in most of the literature, but we have ensured that they are fully frame-covariant by virtue of the use of frame-covariant derivatives. Finally, by imposing the commutator relations on the ladder operators and using the ﬂat Bunch-Davies vacuum condition, we can ﬁnd the two-point function of the (dimensionless) power spectra at horizon crossing [123,124]:

2 2 2 PR = 2 H ,PT = 2 H . (4.31) 8π f(ϕ)¯H π f(ϕ)

In (4.31), PR refers to the scalar power spectrum while PT corresponds to the tensor power spectrum. In writing the spectra, we have made use of the frame-invariant Hubble slow-roll parameters ¯H andη ¯H deﬁned in (3.70). It is important to note that the expressions for the power spectra (4.31) presuppose the slow-roll approximation outlined in (3.71). For a more detailed derivation of the frame-invariant power spectra (4.31), the reader is invited to consult the Appendix.

In the above treatment, we singled out the first isocurvature mode because it directly couples to . This is not the case for the remaining isocurvature R modes, which only couple to each other [125]. This means that the “entropy transfer” between curvature and isocurvature modes (referring to the evolution of the isocurvature modes having an observational impact on the curvature modes) can be traced back to the coupling between Qσ and Qs1 . We may intuitively understand this by visualising the trajectory as it turns in field space. The first isocurvature mode is the only isocurcature mode that can couple to the curvature mode, since it is the only component that can cause 78 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE it to turn. This is simply because, by definition, it is the only mode that parallel to the acceleration vector. It is possible to define a projection operator perpendicular to the acceleration vector and the tangent vector:

AB AB A B (s2) (s1) e e . (4.32) ≡ − s1 s1 This may be used to define the second entropic mode, which will couple to the first entropic mode as the trajectory turns in the submanifold spanned AB by (s2) . If we repeat this procedure, we will find that there is a hierarchy of modes, all of which will couple to their immediate neighbours: the curvature mode couples to the first isocurvature mode, which couples to the second isocurvature mode, and so forth.

4.3 Frame-Covariant Observable Quantities

As noted above, the multifield scenario of inflation is qualitatively different from the single-field scenario for a few reasons. First, in the multifield case, isocurvature modes evolve outside the horizon. Moreover, since the different modes are coupled, the power spectrum (4.31) evaluated at horizon exit will differ from the observable spectrum at horizon re-entry [112]. As noted in Section 4.2, we expect the perturbations to form a hierarchy in which every mode explicitly couples to its immediate neighbours. Indeed, the equations of motion for the perturbations in terms of the curvature perturbations (i) R and in the long wavelength limit, where the coupling between and t S R D R vanishes [126], we find

(1) t = A , D R HS (1) (2) t = B1 , D S HS . . (4.33) (n−2) (n−1) t = Bn−2 , D S HS (n−1) (n−1) t = Bn−1 . D S HS

In the above hierarchy, A(t) and B1(t),B2(t),...,Bn−1(t) are parameters that depend on the speciﬁcs of the model. We can see that the above system of equations is fully frame-invariant.

We can now solve the system of equations given in (4.33) in the scenario of 4.3. FRAME-COVARIANT OBSERVABLE QUANTITIES 79

two-ﬁeld inﬂation. For times t > t∗, where t∗ is the time of horizon exit for a pivot scale of cosmological interest (usually taken to have a wavenumber −1 k∗ 0.002 Mpc ), the perturbations evolve according to [112,127,128] ≈

(t) 1 TRS ∗ R = R , (4.34) (t)! 0 TSS ! ∗ ! S S where ∗ = (t∗) and ∗ = (t∗). The two functions TRS and TRS are referred R R S S to as the transfer functions, whose explicit form depends on the particulars of the model in question. To ﬁrst order in perturbation theory, the corrections to the power spectrum that are of the order of the slow-roll parameters can be neglected [129]. We may hence write

2 −2 PR(t) = 1 + TRS (t∗, t) PR(t∗) = PR(t) cos Θ , (4.35) h i where we have encoded the eﬀect of the coupling between curvature and isocurvature modes into the transfer angle Θ. Moreover, we may deﬁne the isocurvature fraction βiso [93,94]:

2 PS TSS βiso = 2 2 , (4.36) ≡ PR + PS 1 + TRS + TSS where PS is defined through a relation similar to (4.28). The isocurvature fraction encodes the strength of the effects that are “genuinely multifield”; if it is much smaller than unity, then we may study the theory at hand as an effectively single-field model, with the equations of motion given by (4.25) and (4.26).

Up until now, we have focused on the generic features of the entropy transfer between curvature and isocurvature modes. We will apply our analysis to a generic two-ﬁeld scenario in the next section and provide numerical results for concrete models in Chapter 6, but now, we will write down concise and fully frame-invariant expressions for a number of relevant inﬂationary observables.

The power spectra PR and PT are directly linked to standard inﬂationary observables, known as the scalar spectral index nR, the tensor spectral index nT , and tensor-to-scalar ratio r:

d ln PR d ln PT PT nR 1 , nT , r . (4.37) − ≡ d ln k ≡ d ln k ≡ P k=aH k=aH R

These parameters must be evaluated at the time that the perturbations cross 80 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE the horizon since, as we noted, curvature perturbations freeze after horizon exit, which is given by k = a . We may also deﬁne the runnings of the H spectral indices:

dnR dnT αR , αT . (4.38) ≡ d ln k ≡ d ln k k=aH k=aH

We may also write down the non-linearity parameter fNL, which encodes the three point correlation function for [130–132]. Its form is given by [121,133]: R ,A ,B 5 N N ( A BN) fNL = ∇,A∇2 . (4.39) 6 (N,AN )

The expression in (4.39) is given with respect to the number of e-folds N, which is conformally invariant thanks to the definition given in(3.74). Note that in the minimal case, the field space is not curved and the covariant field derivatives in (4.39) simply reduce to ordinary field derivatives [134].

Applying the deﬁnitions for the observables (4.37) and (4.38) to the power spectra (4.31) leads to expressions for the observables to ﬁrst slow-roll order:

2 nR = 1 2¯H η¯H N ln 1 + T , nT = 2¯H , − − − D RS − 2 r = 16¯H cos Θ , ¯ 2 αR = 2¯H η¯H η¯H ξH + N N ln 1 + T , αT = 2¯H η¯H . (4.40) − − D D RS − ¯ We have defined ξH as part of a hierarchy of Hubble parameters which extends the definitions given in (3.70) and whose terms generally appear in calculations of higher-order runnings of the inflationary observables. In detail, the hierarchy is given is

¯H,1 ¯H ,..., ¯H,n N ln H,n−1 , (4.41) ≡ ≡ −D ¯ with ¯H,2 =η ¯H and ¯H,3 = ξH . This hierarchy differs from the formal Hamilton– Jacobi hierarchy introduced by Liddle [135, 136], which uses the Hamilton flow equations to derive the evolution of the Hubble parameter as a function of the fields ϕA. The expressions given in (4.40) are exactly similar to the ones reported in the literature; the only difference is the use of manifestly frame-invariant slow-roll parameters, which ensures that the expressions for the observables reduce to the standard parameters in the Einstein frame.

We now employ the hierarchy given in (4.41) for the inflationary parameters in order to find a general relation for all multifield scalar-tensor models of 4.3. FRAME-COVARIANT OBSERVABLE QUANTITIES 81 inflation:

2 2 1 + nT nR N N ln 1 + T r = 8αT cos Θ . (4.42) − − D D RS − h i This is a consistency relation which can be a significant observational test for both single and multifield inflation in the case that a large fraction r of tensor perturbations is detected. Similar expressions have been reported in [129] and [137].

4.3.1 The Potential Slow-Roll Hierarchy

We now proceed to examine the potential slow-roll approximation, in which we may transform all time-dependent quantities to quantities dependent on fields. In order to do so, we first write the multifield inflationary equations of motion (3.67) and (3.68) in terms of the slow-roll parameters as follows:

A A ,A N N ϕ N ϕ = (ln U) + D D , D 3(1 ¯H /3) − 1 A 1 ¯H η¯H ¯H = (ln U),A(DN ϕ ) , (4.43) 2 − 6 (1 ¯H /3) − DN ¯U η¯H = . − ¯U

At zeroth order, the ﬁeld is completely stationary and the spectra are completely Zel’dovich invariant, we have:

A (0) [ N ϕ ] = 0 , D (0) ¯U = 0 , (4.44) (0) η¯U = 0 .

We may therefore deﬁne the potential slow-roll hierarchy in a recursive way:

A (i−1) B (i−1) ( N ϕ ) ( N ϕ ) A (i) ,A ;B [ N ϕ ] = (ln U) + D D , D (i−1) 3(1 ¯U /3) − (i−1) (i−1) (i) 1 U,A A (i) 1 ¯U η¯U ¯U = [(DN ϕ )] , (4.45) 2 U − 6 1 ¯(i−1)/3 − U (i) A (i) [ N ϕ ] η¯(i) = U,A D . U − (i) ¯U 82 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE

We can see that at ﬁrst order, this returns:

A (1) ,A [ N ϕ ] = (ln U) , (4.46) D 1 U U ,A ¯(1) = ,A , (4.47) U 2 U 2 (1) (ln U),A η¯(1) = U;A . (4.48) U − (1) ¯U

At the i-order approximation, we wish to expand the right sides of (4.45) so that it has no more than (i) terms. Thus, using O 1 ¯ ¯2 = 1 + H + H + ..., (4.49) 1 ¯H /3 3 9 − we may write the approximation

i−1 (i−1) k A (i) ,A 1 A (i−1) B (i−1) ¯U [ N ϕ ] = (ln U) + ( N ϕ ) ( N ϕ ) , D 3 D ;B D 3 k=0 " # X i−1 (i−1) k (i) 1 U,A A (i) 1 (i−1) (i−1) ¯U ¯ = (DN ϕ ) ¯ η¯ , (4.50) U 2 U − 6 U U 3 k=0 " # X (i) A (i) [ N ϕ ] η¯(i) = U;A D . U − (i) ¯U

Therefore, as long as the slow-roll approximation (3.71) holds, we may approximate the Hubble slow-roll parameters to any desired order. The potential (i) (i) slow-roll parameters ¯U andη ¯U have as their defining feature that they reduce to the Hubble slow-roll parameters in the slow-roll approximation. The first order inflaton equation is commonly known as the inflationary attractor; it essentially defines a class of trajectories that the scalar fields will approach towards regardless of initial conditions. Our recursive expression (4.45) may be used to approximate the inflationary attractor to any desired order.

We now focus on the first order of the slow-roll approximation. To this order, the deviation from the geodesic in field space is small, which is equivalent to A setting t tϕ = 0 in (3.67). The first order inflationary attractor therefore D D becomes

A ,A 3 (Dtϕ ) + fU = 0 . (4.51) H 4.3. FRAME-COVARIANT OBSERVABLE QUANTITIES 83

Indeed, the first equation in (4.50) is the more general version of the inflationary attractor to arbitrary order. The inflationary attractor equation defines a class of trajectories which will be approached regardless of initial conditions. Using (4.51), the first order potential slow-roll parameters are given by

AB 1 G U,AU,B (¯U ),A AB U,B ¯ (¯ηU ),A AB U,B ¯U 2 , η¯U G , ξU G . (4.52) ≡ 2 U ≡ − ¯U U ≡ − η¯U U

Given the deﬁnitions of the potential slow-roll parameters in (4.52), we may easily write down concise expressions for the cosmological observables in terms ¯ of ¯U ,η ¯U , and ξU :

2 nR = 1 2¯U +η ¯U N ln 1 + T , nT = 2¯U , − − D RS − 2 r = 16¯U (cos Θ) , (4.53) ¯ 2 αR = 2¯U η¯U η¯U ξU + N N ln 1 + T , αT = 2¯U η¯U . − − D D RS − Likewise, the scalar and tensor power spectra read as follows:

1 U −2 2 PR = 2 cos Θ ,PT = 2 U. (4.54) 24π ¯U 3π

In the same vein, a simple expression for fNL may be obtained by substituting the following expression for N,A in (4.39):

(ln U),A N,A = ,B . (4.55) (ln U),B (ln U)

We have thus deﬁned two hierarchies of manifestly frame-invariant slow-roll parameters which we may use to write down expressions for the cosmological observables.

4.3.2 Isocurvature Eﬀects in Two-Field Models

We will now analyze the eﬀects of entropy transfer between the curvature and isocurvature modes in two-ﬁeld models. In this case, the isocurvature modes are fully encoded in , and the superhorizon equations of motion (4.33) can S be written as N = A(N) , D R − S (4.56) N = B(N) , D S − S 84 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE

where B1(t) = B(t). The solution to (4.56) is given by means of the transfer functions, which for two-ﬁeld inﬂation take on the form

N 0 0 0 TRS (N∗,N) = dN A(N ) TSS(N∗,N ) , − N∗ Z N (4.57) 0 0 TSS (N∗,N) = exp dN B(N ) . − ZN∗ In the two-ﬁeld scenario, the transfer parameters A and B are [126]

A(ϕ) = 2ω , (4.58) 4 2 B(ϕ) = 2¯H η¯ss +η ¯σσ ω . − − − 3 In (4.58), we have used the frame-invariant parameters. We may ﬁnd the magnitude ω of the acceleration vector ωA through (4.24). The directional slow-roll parameters η¯AB are deﬁned with the help of the mass matrix given in (4.30),

MAB A BU 2¯U η¯AB ∇ ∇ + RAσBσ . (4.59) ≡ fU ≈ U 3

A The parameter ηAB is given in the “global” (orthogonal) basis of ﬁelds ϕ .

In order to switch to the local basis ηαβ, we must use the frame ﬁelds given in (4.23). This returns

A B ,A ,B A BU η¯σσ = MABe e (ln U) (ln U) ∇ ∇ , (4.60) σ σ ≈ U A B A B ω ω A BU ¯U η¯ss = MABe e ∇ ∇ + S, (4.61) s s ≈ ω2 U 3 where S = 2Rsσsσ is the Ricci scalar of the two-dimensional ﬁeld space.

A In arriving at the last equality in (4.60), we have used the fact that eσ given in (4.23) becomes eA (ln U),A in the slow-roll approximation. Similarly, we σ ≈ may obtain the slow-roll approximation on the right hand side of (4.61) by A A A noticing that es = ω /ω. In the slow-roll regime, the acceleration vector ω can be approximated to second order in a recursive way similar to that we used to derive (4.50). In this way, rewriting the RHS of (4.24) by substituting A the fist order expression for DN ϕ and converting the time derivatives to field 4.3. FRAME-COVARIANT OBSERVABLE QUANTITIES 85 derivatives, we find

,A A ,B (ln U) ω = (ln U) B . (4.62) ∇ √2¯ U In the same slow-roll regime, we can also compute the single and double covariant derivatives of the transfer function TRS with respect to N by simply diﬀerentiating their integral forms (4.57):

N TRS = A∗ + B∗TRS , D (4.63) 2 N N TRS = A∗B∗ + B TRS . D D ∗ In this way, we can express all inﬂationary observables entirely in terms of the model functions of the theory.

We may analytically perform the integrals in (4.57) by assuming that the slow-roll parameters vary slowly after horizon exit in the so-called constant slow-roll approximation, such as in [140]. This is a crude approximation to the full solution, as the slow-roll parameters do evolve beyond the horizon. For most realistic inﬂation models, the turn rate ω peaks at 20 to 30 e-folds, which is well beyond the horizon exit at N = 60. Therefore, this assumption will generally underestimate the amount of entropy transfer. Hence, assuming constant slow-roll, we may evaluate the parameters A and B at horizon crossing using (4.57):

−B∗(N−N∗) TSS (N∗,N) = e . (4.64)

Substituting the last expression for TSS in (4.57), we ﬁnd

A∗ −B∗(N−N∗) TRS (N∗,N) = e 1 . (4.65) B∗ − h i We may further improve the constant slow-roll approximation by modelling the evolution of A∗ as a linearly increasing function from zero to its maximum value Amax in the interval: N [0,Nmax], and a linearly decreasing function ∈ from Amax to A∗ in the interval N [Nmax,N∗], where A(Nmax) = Amax. ∈ Substituting this approximate form for A in (4.57) along with (4.64), we ﬁnd 86 CHAPTER 4. QUANTUM PERTURBATIONS IN FIELD SPACE that the transfer function at the end of inﬂation N = 0 becomes

1 B∗N∗ 2 TRS (N∗, 0) = 2 e 1 B∗ N∗ − × B∗N∗ B∗N∗ 2Amax e 2 1 + A∗ B∗N∗ + e 2 (B∗N∗ 2) + 2 , (4.66) − − n h io where Nmax N∗/2 is a good approximation for most models of interest. ≈ In addition, we observe that in the limit Amax A∗, the transfer function → in (4.66) reduces to one of (4.65). We note that for scales whose horizon crossing N = N∗ coincides with the end of inﬂation N = 0, the transfer angle Θ vanishes, as there is no time to generate enough entropy transfer.

We conclude this subsection and chapter by commenting on the amplification and transfer of isocurvature perturbations in models with a curved field space. Looking at (4.57) and (4.58), we observe that entropic transfer will occur only if ω = 0. The turn rate ω is related to the external forces, as we can see by 6 rewriting ωA in (4.24) using the inflaton equations of motion (3.67) and the single-field effective equation of motion (4.25):

A A eA U ,si A 1 ( t tϕ ) tσ tϕ ( t tσ) 1 si ω = D D D − D2 D D = . (4.67) − ( tσ) tσ H D H D We can thus see that a non-zero turn rate occurs if and only if the force generated by the frame-invariant potential U has a non-vanishing component perpendicular to the inﬂationary trajectory. Such a force can only exist due to A a potential, since the drag force 3 ( tϕ ) is always in the direction of the − H D tangent vector. As such, entropic transfer will not be aﬀected by a non-zero

field-space Riemann curvature RAσBσ. This is because neither the conservative force fU,A nor the field-space geodesics are directly influenced by RAσBσ. − Unlike the entropic transfer, however, the amplification of isocurvature modes does depend on the field-space curvature RAσBσ. For two-field models, we observe that even if the acceleration vector ωA vanishes, the field-space Ricci scalar S = 2Rsσsσ will necessarily contribute to the generation of isocurvature perturbations throughη ¯ss. For a small turn rate ω, the generation of isocurvature modes will generically be amplified when the inflationary trajectory goes over a hilltop in the potential, for whichη ¯ss < 0. This corresponds to a tachyonic instability. The isocurvature modes will instead be suppressed when the trajectory passes through a valley withη ¯ss > 1. In the case that 0 < η¯ss < 1, 4.3. FRAME-COVARIANT OBSERVABLE QUANTITIES 87 the generation of isocurvature modes is highly dependent on the initial conditions [93,94]. In general, we observe in (4.61) that in the deep slow-roll regime where ¯U , η¯U 1, the generation of entropic modes will be almost independent of S. As we approach the end of inflation, ¯U approaches unity and the effect of a negative field-space curvature S can no longer be neglected, which may give rise to an enhanced entropy production forη ¯ss < 0. Finally, we re- mark that both the generation of entropy and its transfer are controlled by the frame-covariant parameters GAB and U; they are not the result of conformal transformations to any particular frame. Chapter 5

Fine Tuning in Inﬂation

In an ironic twist of fate, inflation, a paradigm devised explicitly in order to deal with the fine tuning problems of standard cosmology, is very finely tuned itself. Most predictions derived from inflation assume slow-roll inflation, which requires a “perfect storm” of coincidences in order to be realised. This issue has have been extensively discussed in the literature [28–32], and many ways have been proposed to deal with them, both generically and for specific models. In this chapter, we will distinguish fine tuning in inflation in two broad categories: (i) initial condition fine tuning and (ii) parameter fine tuning. By taking the α-attractor class of models as an example, we will demonstrate how there is a careful balance between these flavours of fine tuning. Moreover, after considering multifield models of inflation, we will demonstrate that even if the usual fine tuning problems are resolved, we are still faced with the ambiguity of selecting a particular inflationary trajectory in field space. Such a choice is distinct from the initial conditions that allow for inflation to occur, and we refer to it as trajectory fine tuning. As we shall see, the presence of multiple scalar degrees of freedom can provide an avenue from parameter fine tuning to trajectory fine tuning, which may be more appealing for realistic models of inflation.

5.1 Initial Conditions Fine Tuning

We are still a long way from a complete theory of quantum gravity. As such, we must contend with our ignorance about the state of the Universe before

88 5.1. INITIAL CONDITIONS FINE TUNING 89

4 the time at which the energy density was close to MP (around tP after the singularity at t = 0). Therefore, any discussion of initial conditions must necessarily regard the time when the Universe has departed from GUT scales (which is often referred to as “exiting the singularity”). Inflation promises that the Universe will eventually evolve to resemble what we observe today without constraining these initial conditions within a narrow window. We do not need to assume that the Universe looked incredibly flat at tP , nor do we need to assume that it was homogeneous; for most initial conditions, inflation ensures that the Universe will end up looking as it does today. For this to happen, however, the Universe must have been homogeneous to some extent in the past as well, otherwise the energy density would have been dominated by inhomogeneities over the Hubble horizon [33] and negative pressure would not be realized at a large enough scale for inflation to occur. This is the first example of the fine tuning that inflation itself requires.

The evolution of the Universe can be described as a trajectory through some infinite dimensional phase space (thanks to the fact that quantum gravity is a field theory with infinite degrees of freedom). In effect, we are comparing current observations to what a randomly chosen history of the Universe should look like: if most trajectories of the Universe through phase space allow for inflation, then inflation can truly be said to be generic. The question that we must then ask is whether a “randomly” chosen initial seed for the inflaton in phase space is likely to be consistent with this assumption. This is an ill-posed question unless we assign a particular measure to the phase space [99]. It is in fact possible to construct a “canonical” measure on the various trajectories in phase space using the theory of Hamilton flows. This is usually done within the minisuperspace approximation, in which the phase space of the Universe is reduced to the scale factor a and its conjugate momentum (related to the Hubble parameter H). With this measure, the notion of randomly selecting a trajectory becomes formal, and the probability of inflation can be determined [95,97]. The mini-superspace approximation does impose the restriction that the Universe is homogeneous, which at first sight appears to be circular reasoning: if the initial conditions are homogeneous, then inflation will necessarily occur. However, the slow-roll trajectory has been numerically shown to act as a local attractor [27] even in the presence of large inhomogeneities, which partly justifies this approximation.

It is not necessary to delve into the intricacies of the canonical approach to note that even if we define a measure, there still is an ambiguity in quantifying 90 CHAPTER 5. FINE TUNING IN INFLATION the degree to which inflation is finely tuned. It has been argued that inflation is both very likely [96] and very unlikely [98]: this stems from the application of the canonical measure to either the phase space today or the phase space right after the singularity. The latter approach is essentially frequentist: if the creation of the Universe could be repeated, the answer that it provides is the probability that inflation will occur. On the other hand, the former approach has a Bayesian flavour: it is essentially a calculation of the conditional probability that the Universe evolved in an inflationary manner given its current state today.

We may illustrate how parameter fine tuning and initial condition fine tuning are related even if the slow-roll attractor solution has been reached and the mini-superspace approach is valid. We know that currently-favoured slow-roll models of inflation feature an inflationary plateau. In particular, in order to achieve slow-roll inflation, we expect the inflaton to emerge from the singularity with a sufficiently high enough value such that it reaches the inflationary attractor sufficiently early. We will focus on single-field α-attractors [79], which have an action specified by

2 4 MP R 1 1 µ S = d x √ g + (∂µϕ)(∂ ϕ) V (ϕ) . (5.1) − − 2 2 ϕ2 − 1 6αM 2 Z − P

We use natural units with MP = 1 for the rest of the section for notational simplicity. Thanks to the poles present in the kinetic term, the ﬁeld ϕ is assumed to be constrained within the interval ( √6α, √6α). However, the − canonically normalised ﬁeld ϕ can take on any real value:

ϕ ϕ b= √6α tanh . (5.2) √6α b Therefore, no matter what the form of the potential, when expressed in the canonical ﬁeld ϕ, it will be “stretched” as shown below:

ϕ b U(ϕ) = V √6α tanh (5.3) √6α b b This potential plateaus at a value given by Umax = V (√6α), assuming that it is symmetric. Choosing V = m2ϕ2/2 for concreteness, we ﬁnd

ϕ U(ϕ) =b 3αm2 tanh2 . (5.4) √6α b b 5.1. INITIAL CONDITIONS FINE TUNING 91

In order to quantify whether inﬂation will occur at all, we must calculate the slow-roll parameter

4 2 ¯ = csch2 ϕ . (5.5) U 3α 3α r ! b We observe that the inﬂation will not occur when ¯U > 1, which corresponds to

3α 2 ϕ . ϕcrit(α) = asinh . (5.6) | | 2 √3α At first glance, it may seemb thatb this condition is generic: after all, the canonically normalised field can take any real value and only a finite range of values does not lead to inflation. “Common wisdom” dictates that if the scalar field is “dropped” on this plateau has no choice but to eventually inflate [37], and this is true if its value is large enough. However, it is fallacious to argue that it is more likely to be dropped at a larger value. This is because it is impossible to impose a uniform measure on an infinite interval. Moreover, we remind that the parametrisation ϕ of the inflaton field is no more “physical” than ϕ or any other parametrisation. Therefore, any argument about the naturalness of initial conditions cannotb be formulated in terms of the representation of the field (or even in terms of the conformal frame: see [141] for an example where the required initial field value, as well as the kinetic and potential energy den- sities, are dependent on whether the model is studied in the Einstein or Jordan frame).

A uniform distribution is desirable insofar as we have no information about the value of the field as it exits the singularity. For the case of a bounded potential, we have an obvious candidate to impose a uniform distribution: the potential energy U(ϕ) itself. Thus, if U is weighted uniformly, the probability of finding ourselves in the range (5.6) is b U(ϕcrit) 2 1 2 P ( ϕ ϕcrit) = = tanh asinh . (5.7) | | ≤ Umax 2 √3α e b We display P ( ϕ ϕcrit) = 1 P ( ϕ ϕcrit) = P (U 1) in Figure 5.1. | | ≥ − | | ≤ ≤ This is the probability that the slow-roll condition is satisfied for a (uniformly) randomly selectedb value for the inflationaryb potential (by virtue of said value being above ϕcrit). 92 CHAPTER 5. FINE TUNING IN INFLATION

0.8

0.6 1) ≤ U (

P 0.4

0.2

0 0 1 2 3 4 5 α Figure 5.1: Probability that a randomly selected value for the α-attractor potential corresponds to an initially inﬂationary trajectory in the slow-roll approximation.

It can be argued that the above derivation is weakened by the fact that we had to assume the slow-roll approximation. Indeed, if the field does exit the singularity with a bounded potential and dropped on the inflationary plateau at ϕi for which V (ϕi) 1, we expect its kinetic energy to dominate for a while before it settles into the inflationary attractor thanks to the Hubble drag. In- stead, (5.7) should be interpreted as the probabiltiy that the slow-roll approximation is self-consistent. Still, even if not fully rigorous, the above argument shows that the severity of the initial conditions problem can be dependent on the particular values of the parameters of the model under consideration.

5.2 Parameter Fine Tuning

It can be argued that the generic nature of inflation as a framework does more harm than good to its predictive power. While the general framework inflation could be falsified by observations of large-scale vorticity in the CMB or evidence of non-trivial topology [142], it is possible to construct a nearly unlimited number of ad-hoc potential that can fits observations. We can, for instance, examine the phenomenology of a simple single-field model whose potential is given by

λϕn V (ϕ) = . (5.8) (1 + ξϕn/2)2 5.2. PARAMETER FINE TUNING 93

2 This potential features an inflationary platau at V = λn/ξn. Computing the three main inflationary observables with (4.53), we find:

8n2 r = , (5.9) ϕ2 (1 + ξϕn/2)2 2 n 2 ϕ (1 + ξϕ 2 ) 2n n nR = − − , (5.10) ϕ2 (1 + ξϕn/2) 2λ P = ϕn+2. (5.11) R n2

The end-of-inflation condition along with the desirable number of e-folds corresponding to the pivot scale will set the value of ϕ = ϕ∗, and we are left with a system of three equations and three unknowns. If we were to numerically solve it, we would have at our hands yet another inflationary model that is entirely compatible with predictions and with no insight as to what exactly caused inflation.

The fact that the above model is unrealistic is the point in and of itself: it is possible to estimate the values of any inflationary model, no matter how bizarre or exotic simply by matching its predictions to observations. From an instrumentalist point of view, this is not troubling: there is no “fine tuning” taking place, only an empirical estimation of parameters. However, when it comes to realistic model building, one must serve two masters that are opposed to each other: particle physics and observational cosmology. It is telling that minimal Higgs inflation produces “bad” cosmology but “good” particle physics, while the opposite happens when the potential is changed to incorporate the well non-minimal coupling [84], as we shall see in the next chapter.

Our examination of initial conditions in the α-attractor model demonstrates that, to a certain extent, the initial conditions problem is model-dependent. As we can see in (5.1), if α is chosen to be sufficiently large, the slow-roll solutions are self-consistent. Finely tuning the initial conditions is therefore not necessary if the parameters of a model are chosen carefully to match observations. Therefore, when not working with a class of models as opposed to a model with fixed parameters, parameter fine tuning comes into play: if slow-roll inflation is to be achieved, we have to carefully select the parameters of the model in question (such as α) in order for inflation to be achieved generically. There is often a trade-off between the two: the question is which one we are more comfortable with. 94 CHAPTER 5. FINE TUNING IN INFLATION

5.3 Trajectory Fine Tuning

We have seen that fine-tuning can be thought of as an inherent quality of a theory; we can attempt to “hide” it in the initial conditions or the parameters, but it will still exist. Still, having places to hide our fine tuning is not necessarily problematic, as this may correspond to further degrees of freedom that we have yet to consider in a theory . For instance, take single field inflation. In single-field inflation, the end-of-inflation condition imposes a particular field value ϕ = ϕ0 at N = 0. Using this value, it is possible to determine N(ϕ) through (3.74), which we may invert and substitute in the first order expressions for the cosmological observables, leading to expressions entirely in terms of the number of e-folds N.

However, in the presence of n evolving scalar fields, we may identify another kind of fine tuning, one that may persist even if we have carefully selected the parameters of our model and specified initial conditions such that the fields have time to settle into the inflationary attractor. The end of inflation in multifield inflation specifies not a particular field value ϕ0, but rather an end-of-inflation hypersurface, which we may parametrise by a set of n 1 pa- − rameters λI = (λ1, . . . , λn−1). There is a one-to-one correspondence between the boundary values of the fields on this locus and the possible slow-roll trajectories. In fact, there is a one-to-one correspondence between each trajectory and any hypersurface in field space. Thus, instead of switching between initial conditions after the fields exit the singularity, we now switch between boundary conditions on some hypersurface or, equivalently, between trajectories.

We expect observable quantities to change as we select end-of-inflation field values on the N = 0 isochrone. Thus, there are many possible inflationary trajectories and we require some criterion in order to select the observationally viable ones. In this respect, an important constraint is the normalisation of the theoretical prediction for PR to the observed scalar power spectrum, which may be used to select trajectories that only fall within its experimental bounds. Another question that naturally arises is how sensitive a given inflationary model is to the boundary conditions. Slightly changing the boundary conditions on the end-of-inflation isochrone might lead to a large change on the horizon crossing isochrone N = N∗, which might greatly affecting the values of the observable quantities. Therefore, it is useful to introduce a parameter that quantifies the stability of inflationary trajectories between any two 5.3. TRAJECTORY FINE TUNING 95 hypersurfaces.

In order to define such a quantity, we first consider the space of possible trajectories. We can parametrise different trajectories on any hypersurface; hence, we first consider an arbitrary isochrone at a given number of e-folds I A A N, parametrised by λ and described by ϕ = ϕN (λ). The extrinsic geometry of this hypersurface embedded in the field space is encoded in the induced metric [ΓIJ ]N , given by

A B [ΓIJ ]N = GAB ϕN;I ϕN;J , (5.12) where I,J are indices which correspond to the vectors living on the tangent space of this particular (n 1)-dimensional surface. All frame-covariant quan- − tities on this surface are generated using the induced metric [ΓIJ ]N , which inherits the conformal weight and scaling dimension of GAB.

We may now deﬁne the density of trajectories n(N) for a neighbourhood (λI , λI + I dλ ) (which has a corresponding area element dSN on the isochrone) as follows:

n−1 1 d SN n(N) n−1 . (5.13) ≡ SN d λ

Note that this deﬁnition of the density of trajectories ensures that the surface n−1 integral of n(N) over any isochrone N is 1. The area element d SN that corresponds to the neighbourhood (λI , λI + dλI ) under consideration is given by

n−1 n−1 d SN = d λ det[ΓIJ ]N . (5.14) | | p Therefore, the density of trajectories takes on the form:

det[ΓIJ ]N n(N) = | | . (5.15) n−1 d λ det[ΓIJ ]N p | | The above expression makes senseR onlyp for ﬁnite hypersurfaces. In the case of inﬁnite hypersurfaces, we must assign a non-uniform statistical weight to them, although we will focus only on the former case.

We call a trajectory stable at N = N2 with respect to its boundary conditions at N = N1 if the density of trajectories increases as we move from N1 to N2. Similarly, an unstable trajectory is one along which the density of trajectories 96 CHAPTER 5. FINE TUNING IN INFLATION Trajectories in ﬁeld space

dλ1 dλ2 N(ϕ) = N1 dS1

dS2

N(ϕ) = N2

Different trajectories and isochrones in n-dimensional field space. Figure 5.2: Trajectory flow between two isochrone surfaces N(ϕ) = N1 and N(ϕ) = N2 in field space. Sotirios Karamitsos Collabor8.2 Initial Conditions and Fine-Tuning in Multifield Inflation decreases. Schematically, this is shown in Figure 5.2. Thus, we may define the sensitivity parameter Q(N1,N2) for a trajectory at the N = N2 isochrone to the boundary conditions at the N = N1 isochrone as follows:

n(N2) Q(N1,N2) , (5.16) ≡ n(N1) where n(N1) and n(N2) must be evaluated along the trajectory of interest. For our purposes, we are interested in the sensitivity of the field values at horizon crossing N2 = N∗ to the boundary conditions at N1 = 0, since it is the values of the fields at horizon exit that impact the inflationary observables. To this end, we define Q∗ Q(N∗, 0), observing that Q∗ < 1 corresponds to stable ≡ trajectories with respect to boundary conditions, whereas Q∗ > 1 corresponds to unstable trajectories.

We note that boundary condition fine tuning represents yet another place to “hide” the fine-tuning of our theory. It may be more appealing conceptually. Suppose that we believe that the Standard Model Higgs boson cannot have the form of the potential required to generate cosmological observables. Normally, we would dismiss it as ruled out at this point. However, the addition of additional non-spectator scalars adds degrees of freedom that we already know have the possibility to impact the observables of inflation. Therefore, the model is “rescued”: instead of finely tuning couplings (which for an established particle theory are difficult to justify), we can determine which trajectory was followed in field space during inflation. In the next Chapter, we will demonstrate how we can do this in specific two-field models of inflation. Chapter 6

Models of Inﬂation

In the previous chapters, we have studied the phenomenology of inflation in general terms. For this chapter, we will apply our findings a few typical scalar- tensor models of inflation, both single-field and multifield. Single-field models that we will examine include induced gravity inflation, Higgs inflation, and Starobinsky-like F (R) models of inflation, the last of which which can be shown to be equivalent to scalar-tensor theories via a Legendre transform. Multifield models that we will examine will include a minimal model that can be made phenomenologically viable by virtue of the addition of a minimally coupled scalar field. We will further consider F (ϕ, R) models of inflation, which can similarly be written as a generic multifield scalar-tensor theory. We will assume that the slow-roll approximation describes the inflationary dynamics sufficiently well, such that we can use the results presented in the previous chapter to derive analytical expressions for all relevant cosmological observables.

6.1 Single-Field Models

Single-field models are considerably simpler to deal with than models with multiple scalar degrees of freedom. There is only one possible trajectory induced by the inflationary attractor, which means that there is no need to specify the (classical) initial conditions. The prototypical single-field model was outlined in the Introduction and is described by an action where a scalar field is minimally coupled to the Ricci scalar.

97 98 CHAPTER 6. MODELS OF INFLATION

6.1.1 Induced Gravity Inﬂation

The framework of induced gravity postulates that the value of the eﬀective

Planck mass MP is induced by the non-zero vacuum expectation value of a scalar field due to spontaneous symmetry breaking. In the Jordan frame, induced gravity inflation is described by a non-minimal coupling f(ϕ) = ξϕ2 to the Ricci scalar R, a canonical kinetic term and a Mexican hat potential V (ϕ) = λ(ϕ2 M 2 /ξ2)2 [143,144]. − P Using the forms model functions f(ϕ), k(ϕ) and V (ϕ), we may easily evaluate ¯ the slow-roll parameters ¯U ,η ¯U and ξU defined in (4.52). Then, with the aid of these parameters, we can analytically calculate all relevant inflationary parameters in the slow-roll approximation, as functions of the inflaton field ϕ. In detail, we find

128M 4 ξ r = P , (6.1) 2 2 2 (1 + 6ξ)(MP ξϕ ) 4 − 2 2 2 4 MP (1 10ξ) 2MP ξ(1 + 14ξ)ϕ + ξ (1 + 6ξ)ϕ nR = − − , (6.2) 2 2 2 (1 + 6ξ)(MP ξϕ ) 4 3 2 2 2 − 128MP ξ ϕ (3MP + ξϕ ) αR = , (6.3) 2 2 2 4 − (1 + 6ξ) (MP ξϕ ) 6 3 −2 256MP ξ ϕ αT = , (6.4) 2 2 2 4 −(1 + 6ξ) (MP ξϕ ) 2 2 2 −2 2 10MP ξ (MP + ξ ϕ ) fNL = , (6.5) 3 (M 2 ξ2ϕ2)2 P − with nT = r/8, since this is a single-ﬁeld model. The power spectrum is − given through (4.54):

2 2 4 1 λ (1 + 6ξ)(MP ξϕ ) PR = 2 4 5 4− . (6.6) 24π 4MP ξ ϕ

In the same slow-roll approximation, the number of e-folds N is found to be

2 2 2 (1 + 6ξ) 2M ln √ξϕ/MP + M ξϕ N = P P − . (6.7) 8ξM 2 − P

Here we used the fact that induced gravity inﬂation ends exactly at ϕ = ϕend =

MP /√ξ.

In order to express the observables in terms of N, we must invert the relation 6.1. SINGLE-FIELD MODELS 99

N = N(ϕ) to ϕ = ϕ(N), something which is challenging for most models where the equation is transcendental such as (6.7). Therefore, we will instead expand N(ϕ), truncate the series to lowest order, and then invert this trun- cated relation. This will help us make contact with already established results in the literature, while simultaneously allowing for more accurate predictions to be extracted simply by including more terms in the series expansion.

Induced gravity inflation can be studied in two separate domains: (i) the scenario of small-field inflation, in which the inflaton starts at small field values ξϕ2 M 2 , and (ii) the scenario of standard chaotic large-field inflation, in P which ξϕ2 M 2 . Depending on which scenario we study, we will expand the P expression for N(ϕ) about ϕ = 0 for small-field inflation, and about ϕ = for ∞ large-field inflation. It is worth noting that “large” and “small” field inflation is a frame-invariant concept, but the criterion that determines in which domain we are does depend on which frame we are in. The definition above holds for the Jordan frame; in the Einstein frame, we would need to transform the

“critical value” ϕcrit = ϕend = MP /√ξ that separates the small-ﬁeld and large-

ﬁeld domains to ϕcrit by using ϕ(ϕ). While this is not possible in closed form, it does underline the notion that the particular parametrisation we use does not change the featurese of the model.e

For small-field (SF) inflation, a good approximation is that the scales exited the horizon long before the end of inflation, and that the cosmological observables listed in (6.1)–(6.4) and the number of e-folds N in (6.7) are expanded about ϕ = 0. Thus, for SF values of ϕ, the number of e-folds N becomes

1 + 6ξ ξϕ2 N = ln + 1 . (6.8) − 8ξ M 2 P From this last expression, we see that a large number of e-folds N corresponds to small values of ϕ. Inverting the relation N = N(ϕ) given in (6.8), i.e. as 100 CHAPTER 6. MODELS OF INFLATION

ϕ = ϕ(N), enables us to write the cosmological observables in terms of N:

128ξe2βξN+2 rSF = 2 , (1 + 6ξ)(eβξN+1 1) − 1 + 6ξ 2(1 + 14ξ)eβξN+1 + (1 10ξ)e2βξN+2 nR,SF = − −2 , (1 + 6ξ)(eβξN+1 1) − 128ξ2e2βξN+2 3eβξN+1 + 1 αR,SF = 4 , (6.9) − (1 + 6ξ)2 (eβξN+1 1) − 256ξ2e3βξN+3 αT,SF = 4 , −(1 + 6ξ)2 (eβξN+1 1) − 10ξ2 ξeβξN−1 + 1 fNL,SF = 2 , 3 (1 ξeβξN−1) − with βξ = 8ξ/(1 + 6ξ). In particular, for a large number N of e-folds, we ﬁnd

128ξ 16ξ rSF , nR,SF 1 . (6.10) ' 1 + 6ξ ' − 1 + 6ξ

We thus observe that the tensor-to-scalar ratio r and the scalar spectral index nR are not sensitive to N in the gravity induced scenario of SF inﬂation.

In the large-field (LF) scenario, we follow a similar procedure, and expand the field ϕ about infinity. In this LF limit, the number of e-folds simplifies to

(1 + 6ξ)ϕ2 N = 2 . (6.11) 8MP

Substituting this expression in (6.1)–(6.4), the inﬂationary observables in the same LF limit become

128 ξ (1 + 6ξ) r = , LF [(8N 6)ξ 1]2 − − 4 (16N 2 56N 15) ξ2 4(4N + 1)ξ + 1 nR,LF = − − − , [(8N 6)ξ 1]2 − − 1024Nξ3[2(4N + 9)ξ + 3] αR,LF = , (6.12) − [(6 8N)ξ + 1]4 − 2048Nξ3(1 + 6ξ) αT,LF = , − [(6 8N)ξ + 1]4 − 10ξ2(1 + 6ξ) (6ξ + 8ξ2N + 1) fNL,LF = 3 (6ξ 8ξ2N + 1)2 − Upon expanding for a large number N of e-folds, the above expressions simplify 6.1. SINGLE-FIELD MODELS 101 to:

3 1 1 1 rLF = + + , 4 8ξ N 2 O N 3 2 1 nR,LF = 1 + , − N O N 2 2 1 (6.13) αR,LF = + , −N 2 O N 3 1 1 1 αT,LF = 3 + + . − 2ξ N 3 O N 4 5(6ξ + 1) 1 fNL,LF = + 12N O N 2

Finally, we evaluate the phenomenologically admissible value of λ by the normalisation of the power spectrum given in (6.6). In terms of N:

1 λ [1 + (6 8N)ξ]4 PR = − . (6.14) 24π2 512N 2ξ5(1 + 6ξ)

The power spectrum is normalised by [54]

P obs = (6.41 0.18) 10−9, (6.15) R ± × sometimes given as

(0.0267)4 P obs . (6.16) R ≈ 8π2 Finally, it is possible to estimate the value of the quartic coupling λ in terms of the non-minimal coupling ξ as follows:

ξ(1 + 6ξ) λ (0.0267)4 , (6.17) ≈ × 2N 2 where N 60 is the scale at which the largest cosmological scales have ≈ presently re-entered the horizon. To leading order in 1/N, our predictions for the cosmological observables reproduce the results known from the literature for both the scenarios of SF and LF induced gravity inﬂation [145] . 102 CHAPTER 6. MODELS OF INFLATION

6.1.2 Higgs Inﬂation

Higgs inflation [83, 84] is based on the suggestion that the Standard Model Higgs boson could act as the inflaton, since it is the only known physical scalar field since its observation at the CERN Large Hadron Collider (LHC).

Higgs inflation in the Jordan frame features a non-minimal coupling function 2 2 f(ϕ) = MP +ξϕ , a canonical kinetic term for the inflaton and the SM quartic potential V (ϕ) = λ(ϕ2 v2)2, where v is the VEV of the Higgs boson. We may − use the slow-roll expressions in a way analogous to induced gravity inflation to analytically compute the cosmological observables:

4 128MP r = 2 2 2 , ϕ [MP + ξ(1 + 6ξ)ϕ ] 1 2 2 6 2 2 4 nR = ξ (1 + 6ξ) ϕ 2M ξ 48ξ + 2ξ 1 ϕ 2 2 2 2 P ϕ [MP + ξ(1 + 6ξ)ϕ ] − − h + M 4 1 40ξ 192ξ2 ϕ2 24M 6 , P − − − P 4 6 4 2 2 2 2 4 3 2 6 64MP [3MP + 9MP ξ(1 + 6ξ)ϕ + 8MP ξ (1 + 6ξ) ϕ + 2iξ (1 + 6ξ) ϕ ] αR = 4 2 2 4 2 2 − ϕ [MP + ξ(1 + 6ξ)ϕ ] / (MP + ξϕ ) 6 2 2 2 2 128MP (MP + ξϕ )[MP + 2ξ(1 + 6ξ)ϕ ] αT = , 4 2 2 3 − ϕ [MP + ξ(1 + 6ξ)ϕ ] 2 2 2 2 2 10MP (MP + ξϕ )(MP + 2ξ(6ξ + 1)ϕ ) fNL = , 2 2 2 2 3ϕ [(MP + ξ(1 + 6ξ)ϕ ] (6.18) with nT = r/8. If we assume that the field value ϕ at horizon exit is much − larger than that at the end of inflation, i.e. ϕ ϕend, the number N of e-folds becomes (1 + 6ξ)ϕ2 6 M 2 N = + ln P . (6.19) 8M 2 8 M 2 + ξϕ2 P P Further assuming ξϕ2 M 2 throughout the inflationary phase and invert- P ing (6.19), we obtain the following result to first order:

8M 2 N 1/2 ϕ = P . (6.20) 1 + 6ξ 6.1. SINGLE-FIELD MODELS 103

Substituting this last expression in the observables (6.18) returns

16(1 + 6ξ) r = 8ξN 2 + N 2 1 1 = 12 + + , ξ N 2 O N 3 64ξ2N 3 + (1 40ξ 192ξ2) N 16ξ(8ξ 1)N 2 3(1 + 6ξ) nR = − − − − − N(1 + 8ξN)2 2 1 = 1 + , − N O N 2 1 3 2 2 4 αR = 2048ξ N 4N + 15N + 9 ξ + 12(8N + 3)ξ −N 2(1 + 8ξN)4 + 32N 160N 2 + 300N + 81 + 4 272N 2 + 252N + 27 ξ2 + 3 2 1 = + , −N 2 O N 3 2(1 + 6ξ) (32N(4N + 3)ξ2 + 6(4N + 1)ξ + 1) αT = − N 2(8ξN + 1)3 1 1 1 = 3 + + , − 2ξ N 3 O N 4 5(1 + 6ξ) (16ξN + 1) [8ξN/(1 + 6ξ) + 1] fNL = 12N (8ξN + 1)2 5(12ξ 1) 5 1 = − + + . 96ξB2 6B O N 3 (6.21)

In the above, we also quote approximate results for large values of e-folds N, assuming that the non-minimal parameter ξ is not too small (ξ & 1).

We may now estimate the value for ξ using the normalisation of the dimensionless power spectrum PR. The power spectrum in terms of the inﬂaton ﬁeld ϕ is given by 1 λϕ6 [M 2 + ξ(1 + 6ξ)ϕ2] PR = . (6.22) 24π2 4 2 2 2 4MP (MP + ξϕ ) This expression may be translated in terms of N as follows:

1 128λN 3(8ξN + 1) P = . (6.23) R 24π2 (1 + 6ξ)[(8N + 6)ξ + 1]2

Setting λ = 0.129 as the value for the quartic coupling (corresponding to a SM Higgs-boson mass of 125 GeV) and N = 60 as the nominal number of e-folds for the horizon exit, we may match PR with the normalisation (6.15) to deduce 104 CHAPTER 6. MODELS OF INFLATION the known result [84]:

N √λ ξ = 17, 000 . (6.24) √3 (0.0267)2 ≈

6.1.3 F (R) Models

An interesting class of possible inflationary models occurs when the Planck mass is promoted to a function not of some scalar field, but rather a function of the curvature. In this scenario, the Universe self-accelerates without the direct presence of a scalar field [146]. These models are known as F (R) theories, and are described by the following action:

F (R) S = d4x √ g . (6.25) − − 2 Z These theories may be recast in a form equivalent to the scalar-tensor theories by introducing an auxiliary ﬁeld Φ:

4 1 S = d x√ g F (Φ) + F (Φ),Φ(R Φ) . (6.26) − − 2 − Z h i It is not diﬃcult to check that the equation of motion for Φ, δS/δΦ = 0, implies Φ = R, provided F (Φ),ΦΦ does not vanish in the domain of interest. Consequently, the action in (6.26) is equivalent to the original action of F (R) theories given in (6.25).

We may now introduce another ﬁeld f, such that

f = F (Φ),Φ , (6.27) which will play the role of the eﬀective Planck mass squared. In order to see this, we write (6.26) as

fR S = d4x√ g + V (f) , (6.28) − − 2 Z where V (f) is given by

1 1 V (f) = f Φ(ϕ) F Φ(f) . (6.29) 2 − 2 Here, the expression for Φ = Φ(f) comes from inverting the functional relation 6.1. SINGLE-FIELD MODELS 105 in (6.27). This action is similar to that of Brans-Dicke models [147], which are a subclass of scalar-tensor theories with a vanishing kinetic term.

We will now present some typical results that can be obtained for a simple class of F (R) theories. We consider a modiﬁed version of the Starobinsky model [146], in which the function F (R) takes on the form

n F (R) = αR + βnR , (6.30)

where α and βn are arbitrary parameters and n 2. ≥ Given the form of F (R) in (6.30), (6.27) yields

n−1 f(Φ) = α + βnnΦ , (6.31) which is easily inverted to

f α 1/(n−1) Φ(f) = − . (6.32) β n n The potential thus becomes

n 1 f α n/(n−1) V (f) = − βn − . (6.33) 2 β n n

Employing the potential slow-roll approximation, we may obtain the following analytic expressions for the cosmological observables:

16 [(n 2)f 2α(n 1)]2 r = − − − , 3 (n 1)2(f α)2 − − (n2 + 2n 5) f 2 2α (n2 + 4n 5) f 5α2(n 1)2 nR = − − − − − , 3(n 1)2(f α)2 − − 8αnf(3α(n 1) + f)[(n 2)f 2α(n 1)] αR = − − − − , (6.34) 9(n 1)3(f α)4 − − 8αnf[(n 2)f 2α(n 1)]2 αT = − − − , − 9(n 1)3(f α)4 − − 5αfn f = . NL 9(n 1)(f α)2 − − with nT = r/8. As before, the inﬂaton ﬁeld value f must be evaluated at the − point of horizon crossing. We note that generically, the value of f is small and becomes even smaller as we go further back in time and N increases. Hence, 106 CHAPTER 6. MODELS OF INFLATION

we calculate the number of e-folds N by expanding f about fend to lowest order: 3 (n 1)(f fend)(fend α) N = − − − . (6.35) − 2 fend[(n 2)fend 2α(n 1)] − − − At the end of inﬂation, we expect that the theory returns to standard General 2 Relativity. This imposes the constraint that F (R) = MP R, i.e.

n 2 F (Rend) = αRend + βnRend = MP Rend . (6.36)

Since Φ = R, we ﬁnd

2 1/(n−1) MP α Φend = − . (6.37) β n From (6.31), it is then possible to calculate the ﬁnal value of f:

2 fend = nM (n 1)α . (6.38) P − − The number of e-folds N in (6.35) becomes

3(n 1) (α M 2 )[α(n 1) nM 2 + f] N = − − P − − P . (6.39) 2 [M 2 (n 2) α(n 1)] [M 2 n α(n 1)] P − − − P − − Solving (6.39) for f, substituting its expression into (6.34), and expanding the latter for large N to order 1/N, we derive the following approximate analytic expressions for the cosmological observables:

16(n 2)2 16¯α(n 2)n (¯α 1) 1 r − − − , ≈ 3(n 1)2 − (n 1) [(n 2) α¯(n 1)] [n α¯(n 1)] N − − − − − − − n2 + 2n 5 2¯αn (¯α 1) 1 nR − − , ≈ 3(n 1)2 − (n 1) [¯α2(n 1)2 2¯α(n 1)2 + (n 2)] N − − − − − − 4¯α(n 2)n (¯α 1) 1 αR − − , ≈ 3(n 1)2 [(n 2) α¯(n 1)] [n α¯(n 1)] N − − − − − − 4¯α(n 2)2n (¯α 1) 1 αT − − , ≈ −3(n 1)2 [(n 2) α¯(n 1)] [n α¯(n 1)] N − − − − − − 5¯αn (¯α 1) 1 fNL − , ≈ 6 [(n 2) α¯(n 1)] (n α¯(n 1)) N − − − − − (6.40) whereα ¯ α/MP . We observe that βn does not enter the expressions for the ≡ 6.2. MULTIFIELD MODELS 107 observables, save for the strength of the spectrum which is given by

n 3 2 n−1 βn(n 1) (f α) f α PR = 2 2 − − 2 − . (6.41) 16π f [f(n 2) 2α(n 1)] βnn − − −

In fact, all other inﬂationary observables are independent of βn to all orders in 1/N. Instead, there is strong dependence on the power n in (6.30), and forα ¯ = 1, we see that the expressions listed in (6.40) are independent of the number of e-folds N through order 1/N. Finally, we note that, for α = M 2 , 6 P the runnings of the spectral indices αR and αT start at order 1/N, and so they turn out to be at least one order of magnitude larger than those found in the models of induced gravity and Higgs inﬂation.

6.2 Multiﬁeld Models

As we have seen, multifield models have features that are qualitatively different from single-field inflation, the most prominent of which include the presence of multiple admissible classical trajectories as well as entropic transfer. The latter is often taken to be negligible, but for certain values of the parameters of a model may severely affect the observables. In order to illustrate, we examine two multifield models in this section: (i) a two-field minimal model where the entropy transfer is negligible and (ii) a two-field non-minimal model inspired by Higgs inflation where entropic effects are significant. We will study the different inflationary trajectories admissible in each theory by normalising to obs the observable power spectrum PR , which will allow us to present numerical predictions for the inflationary observables.

6.2.1 Minimal Two-Field Inﬂation

We ﬁrst examine a simple minimal two-ﬁeld model described by the Lagrangian

M 2 R 1 1 λϕ4 m2χ2 = P + ( ϕ)2 + ( χ)2 , (6.42) L − 2 2 ∇ 2 ∇ − 4 − 2 where m is a mass parameter and λ is the quartic coupling [113]. We note that this model is distinct from hybrid inflation [73], where one field acts like a “waterfall” field. Instead, both fields slowly roll down the inflationary potential and they can both act as inflaton fields. Using the expressions for the 108 CHAPTER 6. MODELS OF INFLATION observables in terms of the potential slow-roll parameters, we may calculate the general form of the observables in terms of ϕ and χ as

128M 2 (λ2ϕ6 + m4χ2) r = P , (λϕ4 + 2m2χ2)2

4 2 2 −2 2 6 4 2 −1 4 2 2 2 2 6 4 2 nR = λϕ + 2m χ λ ϕ + m χ λϕ + 2m χ λ ϕ + m χ

8M 2 3λ4ϕ12 + 4m8χ4 λm6χ2ϕh4 + 12λ2m4χ2ϕ6 6λ3m2χ2ϕ8 , − P − − i 4 4 2 2 −4 2 6 4 2 −2 8 24 16 8 αR = 64M λϕ + 2m χ λ ϕ + m χ 3λ ϕ + 8m χ − P 8λm14χ6ϕ4 + 68λ2m12χ6ϕ6 12λ3m10χ4ϕh8 6χ2 + ϕ2 − − 4 8 2 10 4 4 2 2 5 6 2 14 2 2 + λ m χ ϕ 48χ + ϕ + 80χ ϕ 2λ m χ ϕ 24χ + 7ϕ − + 12λ6m4χ2ϕ16 χ2 + 5ϕ2 36λ7m 2χ2ϕ20 , − i 4 4 12 8 4 6 2 4 2 4 2 6 3 2 2 8 128MP (λ ϕ + 2m χ λm χ ϕ + 8λ m χ ϕ 6λ m χ ϕ ) αR = − − , − (λϕ4 + 2m2χ2)4

2 2 2 4 14 8 6 6 4 4 3 2 2 10 10λm MP ϕ (λ ϕ + 6m χ 5λm χ ϕ 6λ m χ ϕ ) fNL = − − . − 3 (λϕ4 + 2m2χ2)2 (λ2ϕ6 + m4χ2)2 (6.43)

If the values of the ﬁelds ϕ∗ and χ∗ are known at the time of horizon crossing for the scale of interest, the cosmological observables can be all evaluated through (6.43).

In order to find the values of the fields at the end-of-inflation, we require ¯U = 1 in the slow-roll approximation. This gives

8M 2 (λ2ϕ6 + m4χ2) P 0 0 = 1 . (6.44) 4 2 2 2 (λϕ0 + 2m χ0)

2 2 Solving (6.44) for m /(λMP ) = 1 yields

1/2 4 6 1 4 ϕ0 ϕ0 χ0(ϕ0)/MP = 1 (ϕ0/MP ) + 1 + 2 . (6.45)  − 2 − M M  s P P   This equation defines the end-of-inflation contour ¯U = 1 in field space, which can be seen in be seen in Figure 6.1. We note that it corresponds to N = 0, and ϕ0 varies from 0 to √8MP . As we choose different boundary conditions on this isochrone, the field trajectories change as shown in Figure 6.2. We further observe that there exists some critical region for ϕ0 in which the trajectories 6.2. MULTIFIELD MODELS 109

4 U = 1 max( , η ) = 1 U | U | 3 P 2 χ/M

0 0 1 2 3 4

ϕ/MP

Figure 6.1: End-of-inﬂation curve for the minimal two-ﬁeld model (6.42) with 2 2 m /(λMP ) = 1.

25 14 ϕ0 = 0.25 ϕ0 = 0.458 12 ϕ0 = 0.461 20 ϕ0 = 1 10 N = 60 15 P 8 P N = 40 χ/M χ/M 6 10 N = 20

4 5 2 N = 0

0 0 0 2 4 6 8 10 12 14 0 5 10 15 20 25

ϕ/MP ϕ/MP

Figure 6.2: Field space trajectories and isochrone curves for the minimal two- ﬁeld model (6.42). 110 CHAPTER 6. MODELS OF INFLATION

2 ∗

Q 0 ln

2 − 4 − 0 0.5 1 1.5 2 2.5

ϕ0/MP

Figure 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.

are unstable. It is useful to parametrise the horizon crossing curve at N = N∗ in terms of ϕ0:

χ∗ = χ∗(ϕ0), ϕ∗ = ϕ∗(ϕ0). (6.46)

Then, we may use (5.12) in order to compute the values of the induced met- rics [ΓIJ ]N on the two isochrones N = 0 and N = N∗ and substitute them in (5.16). In this case, the sensitivity parameter Q∗ expressed in terms of ϕ0 is shown in Figure 6.3. We observe that trajectories with ϕ0/MP [0.391, 0.775] ∈ are unstable at horizon crossing with respect to the end-of-inflation isochrone. This indicates that most trajectories at N = 60 (which is a finite isochrone) end inflation in that interval. The sensitivity parameter Q∗ is maximised around the trajectory specified by ϕ0 = ϕcrit = 0.458 MP .

obs The power spectrum normalisation PR = PR may in principle be used to re- late ϕ0 to the parameters of the theory and further restrict the parameter space if we have a particular inflationary trajectory in mind. However, following the discussion of the previous chapter, we have moved from parameter fine tuning to trajectory fine tuning. This allows us to single out a particular trajectory that is observationally allowed by matching the value of PR to the scalar power spectrum at the 68% confidence level given in (6.15). We first numerically cal- −12 culate PR in terms of ϕ0 for the minimal model for parameter values λ = 10 −6 and m = 10 MP . We display the predicted power spectra PR∗ at horizon crossing and PR at the end of inflation (taking into account entropy transfer) as functions of ϕ0 in Figure 6.4. We find that the entropy transfer has a negligible effect on the boundary condition ϕ0/MP = 0.495 0.001 compatible obs ± with PR . After numerically solving for ϕ(N) and χ(N) with ϕ(0) = ϕ0 and 6.2. MULTIFIELD MODELS 111

8 5 10− · 8 4 10− · 3 10 8 P P − R ≈ R∗ · P obs 8 R 2 10− · 8 1 10− ·

0 0.5 1 1.5 2 2.5

ϕ0/MP 8 5 10− · P P R R∗ 8 obs≈ 4 10− P · R 8 3 10− · 8 2 10− · 8 1 10− ·

2 4 6 8 10 12 14 16 18

ϕ /MP ∗ Figure 6.4: Power spectrum normalisation for the minimal two-ﬁeld −12 −6 model (6.42) with λ = 10 and m/MP = 10 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 hori- obs zontal line corresponds to the observed power spectrum PR given in (6.15). 112 CHAPTER 6. MODELS OF INFLATION

1 0.96

0.8 0.94

0.6 0.92 R r n 0.4 0.9

0.2 0.88

0 0.86 40 50 60 70 80 90 40 50 60 70 80 90 N N ∗ ∗ 0

2 6 10− 3 · 1 10− − · 2 4 10− 3 2 10− NL · − · f

2 3 2 10− 3 10− · − · α R αT 3 4 10− 0 − · 40 50 60 70 80 90 40 50 60 70 80 90 N N ∗ ∗ 3 4 10− ·

3 3 10− ·

iso 3 2 10− β ·

3 1 10− ·

0 40 50 60 70 80 90 N ∗

Figure 6.5: Predictions for the inﬂationary quantities r, nR, αR, αT , fNL and βiso in the minimal model (6.42) for boundary condition given by ϕ0/MP = 0.496.

χ(0) = χ0(ϕ0), we may substitute the resulting solutions in (6.43) in order to plot the observables in terms of N for diﬀerent values of ϕ0 in Figure 6.5.

We summarise the predictions of this model in Table 6.1, where we also compare them to currently observed values. We further see in Figure 6.6 that the turn rate ω achieves its maximum at around 20 e-folds, but the generation of isocurvature modes is suppressed sinceη ¯ss does not achieve a negative value throughout the inﬂationary trajectory. Therefore, the entropy generation in this model is negligible, something which is reﬂected in the smallness of βiso as can be seen in the last panel in Figure 6.5). We further note that the values of r and nR lie far outside the PLANCK bounds, which is why we extend this model by adding a non-minimal parameter to the Lagrangian in the next subsection. 6.2. MULTIFIELD MODELS 113

2 8 10− · 0.8

2 6 10− · 0.6

2 ss ω 0.4 ¯

4 10− η ·

2 0.2 2 10− · 0 0 10 20 30 40 50 60 10 20 30 40 50 60 N N

Figure 6.6: Evolution of ω andη ¯ss along the inﬂationary trajectory with ϕ0 = 0.495 for the minimal two-ﬁeld model (6.42).

ϕ0/MP = 0.495 PLANCK 2015 r 0.501 0.12 (95% CL) ≤ nR 0.906 0.968 0.006 (68% CL) ± αR 0.00288 0.003 0.008 (68% CL) − − ± αT 0.0019 0.000167 0.000167 (68% CL) − − ± fNL 0.0129 0.8 5.0 (68% CL) ± βiso 0.000717 0.08 (CDI), 0.27 (NDI), 0.18 (NVI) (95% CL) ≤ Table 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 non-decaying isocurvature modes: (i) the cold dark matter density isocurvature mode (CDI), (ii) the neutrino density mode (NDI), and (iii) the neutrino velocity mode (NVI). 114 CHAPTER 6. MODELS OF INFLATION

6.2.2 Non-minimal Two-Field Inﬂation

The most straightforward extension to the model described in the previous subsection is to introduce a coupling between one of the ﬁelds and the scalar curvature R. The Lagrangian for such a model model is given by

(M 2 + ξϕ2)R 1 1 λ(ϕ2 v2)2 m2χ2 = P + ( ϕ)2 + ( χ)2 − , (6.47) L − 2 2 ∇ 2 ∇ − 4 − 2 where once again we assume that the VEV v is negligible in the inflationary regime ϕ MP . Unlike in the minimal scenario discussed in Subsection 6.2.1, ∼ 2 2 −6 −12 we deviate from m /(λM ) = 1 by choosing m = 5.6 10 MP and λ = 10 , P × as well as ξ = 0.01 for the non-minimal parameter. This model is drastically different from Higgs inflation, where ξ is of the order 104, but still features a curved field space, as we can see by calculating the field space Ricci scalar:

2ξ [M 4 ξ2(1 + 6ξ)ϕ4] S = P − . (6.48) 2 2 2 [MP + ξ(1 + 6ξ)ϕ ]

Assuming ξ > 0, we observe that the curvature is positive, S 2ξ in the ≈ regime where ϕ is small, given by ϕ/MP 1/√ξ. Conversely, in the regime where ϕ is large, the curvature becomes negative, S 2ξ/(1 + 6ξ). For ξ of ≈ − order 103, such as in [94], the trajectory is in the region of the field space with negative curvature for most of inflation, which contributes greatly towards the generation of isocurvature perturbations. In our case with ξ = 0.01, the curvature is positive but small, which means that isocurvature perturbations will not necessarily be generated towards the end of inflation.

We proceed much like the minimal model in the previous section. We have illustrated the end-of-inflation contour in Figure 6.7. In Figure 6.8, we display the field space trajectories and in Figure 6.11 we display the sensitivity parameter Q∗ for all possible trajectories on the boundary conditions. We observe that Q∗ is small for most values of the boundary conditions, shooting up as we approach the purely χ2 trajectory. This agrees well with the left panel of Figure 6.8, where trajectories with different boundary conditions converge as we go back in time. The critical value is given by ϕcrit/MP = 2.694, close to the end of the N = 0 isochrone.

In Figure 6.9, we give predictions for the values of the power spectrum PR. From the left panel in Figure 6.8, we observe that an increasing boundary 6.2. MULTIFIELD MODELS 115

6 U = 1 max(U , ηU ) = 1 5 | |

4 P 3 χ/M

0 0 1 2 3 4 5 6

ϕ/MP

Figure 6.7: End-of-inﬂation curve for the non-minimal model (6.47)with m = −6 −12 5.6 10 MP , λ = 10 , and ξ = 0.01. ×

25 ϕ0/MP = 1 20 ϕ0/MP = 1.5 ϕ0/MP = 2.5 20 ϕ0/MP = 2.72 15 ϕ0/MP = 2.74 N = 60 15

P P N = 40

χ/M 10 χ/M 10 N = 20

5 5 N = 0 0 0 0 2 4 6 8 10 12 14 0 5 10 15 20 25

ϕ/MP ϕ/MP

Figure 6.8: Field space trajectories and isochrone curves for the non-minimal two-ﬁeld model (6.47). 116 CHAPTER 6. MODELS OF INFLATION

9 8 10− · P 9 R 7.5 10− P · R∗ obs 9 P 7 10− · R 9 6.5 10− · 9 6 10− · 9 5.5 10− · 9 5 10− · 0 0.5 1 1.5 2 2.5 ϕ0/MP 9 8 10− · P 9 R 7.5 10− P · R∗ obs 9 P 7 10− · R 9 6.5 10− · 9 6 10− · 9 5.5 10− · 9 5 10− · 0 0.1 0.2 0.3 0.4 ϕ /MP ∗ Figure 6.9: Power spectrum normalisation for the non-minimal two-ﬁeld −6 −12 model (6.47) with m = 5.6 10 MP , λ = 10 , 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 obs spectrum PR given in (6.15). 6.2. MULTIFIELD MODELS 117

0.97 0.22 ϕ0/MP = 1.048 0.2 ϕ /M = 1.634 0 P 0.96 0.18

0.16 R r 0.95 n 0.14 0.12 0.94 ϕ0/MP = 1.048 0.1 ϕ0/MP = 1.634 0.08 0.93 40 50 60 70 80 90 40 50 60 70 80 90 N N ∗ ∗ 2 0 1.4 10− · ϕ0/MP = 1.048 2 4 1.2 10− ϕ0/MP = 1.634 5 10− · − · 2 1 10− 3 · 1 10− NL − · f 3 8 10− · 3 α (ϕ0/MP = 1.048) 1.5 10− R 3 − · α (ϕ0/MP = 1.634) 6 10− R · αT 3 3 2 10− 4 10− − · 40 50 60 70 80 90 · 40 50 60 70 80 90 N N ∗ ∗ 1

0.9

iso 0.8 β

0.7 ϕ0 = 1.048 ϕ0 = 1.634 0.6 40 50 60 70 80 90 N ∗

Figure 6.10: Predictions for the inﬂationary quantities r, nR, αR, αT , fNL, and βiso in the non-minimal two ﬁeld model (6.47) for boundary conditions obs admissible under normalisation of PR to the observed power spectrum PR .

value ϕ0 up to ϕ0/MP 2.73 corresponds to a progressively sharper turn in ≈ field space, leading to an increased entropy transfer. After this value, entropy transfer effects become small again. These effects have been studied in more detail in Figure 6.9. Specifically, solid lines correspond to the full computation

of PR, where the eﬀect of entropy transfer is included. Instead, the dotted

lines give the predictions for PR∗, in which entropy transfer eﬀects have been ignored. Observationally viable values for the boundary condition belong to

the interval ϕ0/MP (1.048, 1.634), as well as to ϕ0/MP = 2.72. For the ∈ interval (1.048, 1.634), the ﬁeld value ϕ0/MP = 1.392 corresponds to the mean obs −9 value P = 6.41 10 . We observe that the predicted value for PR is more R × sensitive to the value of ϕ0 than in the minimal model.

In Figure 6.10, we show the dependence of the observables r, nR, αR, αT , fNL, 118 CHAPTER 6. MODELS OF INFLATION

2 ∗

Q 0 ln 2 − 4 − 0 0.5 1 1.5 2 2.5

ϕ0/MP

Figure 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.

+0.243 ϕ0/MP = 1.391−0.343 PLANCK 2015 +0.0053 r 0.1204−0.0049 0.12 (95% CL) +0.005 ≤ nR 0.955−0.002 0.968 0.006 (68% CL) +0.00005 ± αR 0.0004−0.00006 0.008 0.008 (68% CL) − +0.000003 − ± αT 0.000276−0.000003 0.000155 0.00016 (68% CL) − +0.00003 − ± fNL 0.0693−0.00002 0.8 5.0 (68% CL) +0.003 ± βiso 0.939 0.08 (CDI), 0.27 (NDI), 0.18 (NVI) (95% CL) −0.003 ≤ Table 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.

and βiso on the number of e-folds N. Taking N = 60 to be the point of horizon exit for the largest cosmological scales, we summarise our results in Table 6.2 for the non-minimal two-ﬁeld model. We focus on the interval

ϕ0/MP (1.048, 1.634), as the trajectory for ϕ0/MP = 2.72 generates pre- ∈ dictions similar to λϕ4 inflation, which is not observationally viable. We note that this model agrees well with current observations within their uncertainties at the 2σ level displayed in Table (6.2), except for βiso, which could be close to unity (see the last panel in Figure 6.10). However, the observables for our model were calculated at the end of inflation, not at the present epoch. As such, we have not taken into account reheating effects, which may cause the isocurvature perturbations to decay [148]. This could cause the empirically- determined value of βiso to be much higher at the end of inflation, thus bring- ing our predictions within observational bounds. This behaviour, of course, is highly model-dependent, depending on how the isocurvature perturbations evolve between the end of inflation and today in our chosen reheating model. Nonetheless, reheating could potentially provide a mechanism by which the 6.2. MULTIFIELD MODELS 119

2 3 10− 0.2 · ϕ0/MP = 1.048 ϕ0/MP = 1.048 ϕ0/MP = 1.634 ϕ0/MP = 1.634 2 2 10− 0.1 · ss ω ¯ η 2 1 10− 0 ·

0 0.1 10 20 30 40 50 60 − 10 20 30 40 50 60 N N

Figure 6.12: Evolution of ω andη ¯ss along the observationally viable inﬂationary trajectories for the non-minimal two-ﬁeld model (6.47).

generation and transfer of isocurvature perturbations is brought down to observationally acceptable levels.

Notice that even the addition of a small non-minimal parameter ξ modifies the cosmological observables by a significant amount and amplifies the effects

of the entropy transfer on PR. The turn rate ω in this model does not vary considerably from the minimal case, as calculated from (4.62). To lowest order 2 2 in ξϕ /MP , the turn rate ω is given by

2 2 2 ϕ (MP ξ 4ξ χ ) ω = −2 . (6.49) MP χ

We see from Figure 6.12 that ω is of order 10−3 throughout inﬂation, i.e. one order of magnitude smaller than the one found in the minimal model. However, the ampliﬁcation of isocurvature modes in this model is much larger than in the minimal model. According to our discussion in Subsection 4.3.2, this is because

the parameterη ¯ss is negative for most of the two inflationary trajectories, as seen in Figure 6.12. The effect of the positive field-space Ricci scalar S on this model is negligible, since S 2ξ = 0.02 is relatively small and also ≈ multiplied by ¯H 1 in (4.61), giving rise to a suppressed contribution toη ¯ss. Towards the end of inflation only, we have ¯H 1 and a positive S could ∼ suppress entropy production. But, the final stage of the inflationary era is too brief to affect the growth of isocurvature perturbations. This is in contrast to other models with non-minimal couplings in which entropy perturbations are amplified towards the end of inflation [93]. Thus, in our non-minimal model, the amplification of entropy is mainly driven by a concave potential U in the isocurvature direction, whereas the turn rate ω, albeit small, is sufficiently large to cause sizeable entropy transfer. 120 CHAPTER 6. MODELS OF INFLATION

6.2.3 F (ϕ, R) Models

We conclude the chapter with an overview of F (ϕ, R) models. These models are an extension of the F (R) models considered in the previous section, and are described by the action

4 F (ϕ, R) kAB µν A B S = d x √ g + g ( µϕ )( νϕ ) V (ϕ) . (6.50) − − 2 2 ∇ ∇ − Z Our goal is to write the action (6.50) in terms of a multiﬁeld theory such that our formalism may be applied. The standard way to achieve this is to introduce a non-dynamical auxiliary degree of freedom Φ as a Lagrange multiplier and rewrite the action in (6.50) as

4 1 S = d x √ g F (ϕ, Φ) + F (ϕ, Φ),Φ R Φ − − 2 − Z (6.51) kAB µν A B + g ( µϕ )( νϕ ) V (ϕ) . 2 ∇ ∇ − Varying S in (6.51) with respect to the auxiliary field Φ, we find Φ = R, implying that this action is equivalent to (6.50). Treating Φ as an independent scalar field, we may express the action as

4 1 kAB µν A B S = d x √ g F (ϕ, Φ),ΦR + g ( µϕ )( νϕ ) − − 2 2 ∇ ∇ Z F (ϕ, Φ) 1 + F (ϕ, Φ),ΦΦ V (ϕ) , − 2 2 − (6.52) from which we may read oﬀ the new model functions in terms of ϕM = (ϕA, Φ):

f(ϕ, Φ) = F (ϕ, Φ),Φ,

kMN (ϕ, Φ) = diag(kAB, 0), (6.53) F (ϕ, Φ) F (ϕ, Φ),ΦΦ V (ϕ, Φ) = V (ϕ) + − . 2 where the indices M and N run over 1 to n as well as 0, the latter of which corresponds to the scalar curvature Φ = R. These new model functions have the same transformation properties as the ones in (3.42) and may be used to calculate the potential slow-roll parameters for the theory, leading to analytic predictions in terms of Φ and ϕ. It is also possible to write the ﬁeld space 6.2. MULTIFIELD MODELS 121

metric GAB and the invariant potential U:

kAB 3 3 + (ln F,Φ),A(ln F,Φ),B (ln F,Φ),Φ(ln F,Φ),B G = FΦ 2 2 (6.54) MN 3 3 2 2 (ln F,Φ),A(ln F,Φ),Φ 2 (ln F,Φ),Φ !

V (ϕ) F (ϕ, Φ) F (ϕ, Φ),ΦΦ U(ϕ, Φ) = 2 + − 2 (6.55) F (ϕ, Φ),Φ 2F (ϕ, Φ),Φ

2 The function F (ϕ, R) must reduce to MP R at the vacuum expectation value

(ϕ, Φ)VEV induced by U(ϕ, Φ) such that the Einstein gravity limit is reached. If there is no such value, we expect the theory to be unstable. Note that even though kAB is singular, the theory has no redundant degrees of freedom: the

ﬁeld space metric GMN is non-singular as expected.

We note that even in the presence of a single scalar degree of freedom ϕA = ϕ, isocurvature modes may still be generated as the perturbations δΦ and δϕ are independent. For F (ϕ, R) models with only one scalar degree of freedom, it is possible to write down the only independent component of the Riemann curvature tensor. For instance, for large values of ϕ, we assume that F (ϕ, R) may be expanded as

a2(R) 4 F (ϕ, R) = a0(R) + + (1/ϕ ) . (6.56) ϕ2 O

For this generic class of F (ϕ, R) theories, the ﬁeld-space Ricci scalar S is found to be

S = 2a2(R) . (6.57) −

If the coefficient a2(R) in the expansion (6.56) of F (ϕ, R) is positive, then the field-space Ricci scalar S is negative, giving rise to a negative contribution toη ¯ss [cf. (4.61)]. As a result, if the frame-invariant potential U happens to be concave, a positive a2(R) will lead to further amplification of entropy perturbations, which in turn can affect the adiabatic perturbations and so produce relevant effects on the observable CMB power spectrum. Chapter 7

Beyond the Tree Level

In the previous chapters, we developed a covariant formalism applicable to multifield theories of inflation in the Born approximation. However, the question of whether frame covariance, which we have shown leads to frame-invariant classical predictions, may be extended beyond the tree level remains open. In most models of monomial inflation, the radiative corrections generated from the quantum fluctuations of the scalar fields to the matter sector are argued to be suppressed. Still, their phenomenological impact may have significant ramifications for cosmology, and so in this respect, it is natural to extend the concept of frame covariance to incorporate radiative corrections.

The one-particle irreducible (1PI) effective action formalism in quantum field theory is a valuable tool in the study of radiative corrections as it allows us to calculate the effective equations for the mean field [149]. Moreover, by defining an action that inherently incorporates all quantum effects beyond the tree level, the symmetries and covariance (or lack thereof) of the system under study are made manifest in the effective Lagrangian. However, the standard definition of the effective action does not behave as a scalar under field reparametrisations. In order to rectify this issue, Vilkovisky defined a effective action motivated by covariance at the level of the path integral formalism [38]. This “unique” action (in the sense that it is does not depend on the parametrisation of the fields) was further improved by De Witt, who noted that the modification proposed by Vilkovisky is inconsistent in the case of theories admitting a curved configuration space, resulting in what is now known as the Vilkovisky– De Witt effective action. In this chapter, we will review the standard effective action formalism before discussing the modifications made by Vilkovisky and

122 7.1. THE CONVENTIONAL EFFECTIVE ACTION 123

De Witt [39–42] and discussing how frame covariance may be applied to it.

7.1 The Conventional Eﬀective Action

We will first outline the derivation of the effective action in order to highlight the issues with its non-covariance. We begin by considering a physical system described by a classical action S[ϕ], where ϕ without indices collectively stands for an array of classical fields ϕA, where A runs over (1, . . . , n). The generating functional Z[J] in the presence of an external source Ja JA(xA) is defined ≡ as an integral over all possible configurations of the quantum fields φA (up to a normalisation factor that drops out of observable quantities and may be absorbed into the measure):

i a (S[φ] + J φa) Z[J] = [ φ] [φ] e ~ . (7.1) D M Z 1 n The functional integral element is [ φ] φ (x1) ... φ (xn) and the mea- D ≡ D D sure [φ] of the configuration space for the quantum fields φa depends on the M form of the classical action. We suppress indices of Ja when it appears as an argument and we use the generalised Einstein summation convention, where repeated configuration space indices that carry both field space and spacetime information are summed and integrated over:

a 4 A Jaφ d xA JA(xA) φ (xA). (7.2) ≡ Z

The classical action S[ϕ] has an associated generating functional W [J] for all 1PI diagrams in the presence of external sources. The generating functional is given as an integral in the configuration space by the following functional integral over the configuration space of the quantum fields φa:

i i a exp W [J] = [ φ] [φ] exp S[φ] + Jaφ , (7.3) h D M ~ Z The eﬀective action Γ[ϕ] is then deﬁned by a Legendre transform:

a Γ[ϕ] = W [J] Jaϕ , (7.4) − 124 CHAPTER 7. BEYOND THE TREE LEVEL

a where the ϕ now take on the role of the mean ﬁelds and Ja = Ja[ϕ] is considered to be a functional of ϕ. In the presence of the source terms Ja, the mean ﬁelds and the sources are related as follows:

a δW [J] δΓ[ϕ] ϕ = ,Ja = a . (7.5) δJa − δϕ

Note that the average of the quantum ﬁeld can also be written as

a a i a i a a ϕ = φ exp Γ[ϕ] [ φ] [φ] φ exp S[ϕ] (ϕ φ )Ja , h i ≡ − D M − − ~ Z ~ h (7.6)i simply by applying the ﬁrst equation of (7.5) to (7.3) and using the Legendre transform (7.4). In general, the expected value of any function F (φ) of the quantum ﬁelds can be written as

a i i a a F exp Γ[ϕ] [ φ] [φ] F (φ) exp S[ϕ] (ϕ φ )Ja . h i ≡ − D M − − ~ Z ~ h i(7.7)

The effective action Γ[ϕ] is the quantum-corrected version of the classical action in the sense that it generates the quantum-corrected equations of motion under the condition that the effective action is stationary at the mean field ϕ:

Γ,a[ϕ] = 0. (7.8)

We may ﬁnd the inverse propagator ∆ab = W ,ac by functionally diﬀerentiat- − ing the Legendre transform (7.4) and (7.5) and solving the resulting functional equations:

a bc δc = Γ,ab ∆ , (7.9) mn,l Γ,abc = Γ,amΓ,bnΓ,cl ∆ , (7.10)

a ,a where Γ,a δΓ/δϕ and W δW/δJa. The one-particle irreducible rela- ≡ ≡ tions can be found by evaluating both sides at the background ϕ and on-shell

(Γ,a[ϕ] = 0). The effective action satisfies the following functional integro- differential equation, which we may derive by substituting (7.4) and (7.5) 7.1. THE CONVENTIONAL EFFECTIVE ACTION 125 in (7.1):

i i a a exp Γ[ϕ] = [ φ] [φ] exp S[φ] + Γ,a[ϕ](ϕ φ ) , (7.11) h D M ~ − Z h i a where Γ,a denotes the functional derivative of the action with respect to ϕ .

It is possible to solve (7.11) perturbatively with the help of the background field method, where we split the quantum field φa in its background component, as specified by the mean field ϕa (which coincides with the classical field in the mean-field approximation), and a quantum component λψa, where λ ~1/2: ∼ φa ϕa + λψa. (7.12) ≡ We may then expand the effective action as follows:

(1) 2 (2) Γ[ϕ] = S[ϕ] + ~Γ [ϕ] + ~ Γ [ϕ] . (7.13) ···

To order (λ), we may write Γ,a S,a through (7.5) and [ϕ + λψ] [ϕ]. O ≈ M ≈ M Thus, equations (7.12) and (7.13) used in conjunction with (7.11) return:

i i a a exp S[ϕ] + ~Γ1[ϕ] = [ φ] [ϕ] exp S[ϕ + λψ ] λS,a[ϕ]ψ . D M − ~ Z ~ (7.14)

We now perform a functional Taylor expanding for integrand on the right-hand side of (7.14) and shifting the integration variable to ﬁnd

(1) i a b exp iΓ [ϕ] = [ ψ] [ϕ] exp S,ab ψ ψ . (7.15) D M 2 Z Using the functional analogue of Gaussian integration where the normalisation constant has been absorbed by the functional measure leads us to the following expression for Γ(1):

(1) i Γ [ϕ] = i ln [ϕ] ln detS,ab[ϕ] . (7.16) M − 2 We have thus arrived at an expression for the eﬀective action at one-loop level.

Finally, we note that for a conﬁguration space equipped with a metric Gab, the natural measure that leaves the volume element [dϕ] [ϕ] invariant is M 126 CHAPTER 7. BEYOND THE TREE LEVEL

[ϕ] = √det Gab, which leads to the following expression: M

(1) i Γ [ϕ] = i ln det Gab ln detS,ab − 2 i p ,a = ln det S , (7.17) −2 ,b where S and its functional derivatives are evaluated at the mean ﬁeld ϕ. We may thus employ the above expression for Γ(1) in order to extract quantum corrections to the parameters of the theory in the standard way. Computing expressions at higher orders is possible simply by diagramatically evaluating 1PI graphs and using the inverse propagator, which is given by writing (7.9) at the one-loop level:

ab a ∆ S,bc = δc . (7.18)

The two-loop correction is given by

(2) 1 am bn cl 1 ab cd Γ [ϕ] = ∆ ∆ ∆ S,(abc)S,(mnl) ∆ ∆ S,abcd, (7.19) 12 − 8 where again all expressions are evaluated at the background and brackets denote symmetrisation with a factor of 1/N!.

We would expect that the features of a theory would not be affected by a reparametrisation of the mean fields. As such, any reparametrisation ϕA 7→ ϕA(ϕ) should leave all observable quantities invariant. While the effective action is not an observable quantity, its second derivatives enter the S-matrix observables.e Furthermore, given the effective equations of motion Γ,a[ϕ] = 0 and the definition of the mean fiend ϕa, it is desirable to have an expression for Γ[ϕ] that transforms as a scalar. However, the effective action given in (7.11) as well as the perturbative expressions (7.17) and (7.19) suffer from a manifest reparametrisation dependence. This occurs because the difference ϕa φa in (7.11) does not transform as a vector in configuration space, spoil- − ing the covariance of the current term. Similarly, the presence of standard functional derivatives in (7.17) induces extra terms in the expression for Γ(1), which means that the expression for the effective action does not transform as a scalar. We may also observe this in the Legendre transform (7.4), which fails to be covariant due to the presence of ϕa. These issues together indicate that the conventional approach to the effective action must be re-evaluated. 7.2. THE VILKOVISKY EFFECTIVE ACTION 127

7.2 The Vilkovisky Eﬀective Action

The Vilkovisky–De Witt effective action was originally developed by Vilko- visky and subsequently modified by De Witt as a response to the apparent parametrisation dependence of the standard effective action outlined in the previous section. After realising that this problem stems from unduly privileg- ing a particular parametrisation of the fields, Vilkovisky proposed a redefinition of the effective action which has no explicit dependence on the coordinates. The way he achieved this was by redefining the mean field such that it lives in the tangent space, leading to a manifestly covariant formulation of the effective action. De Witt’s contribution was the extension of the formalism for theories with a curved configuration space. We will first examine Vilkovisky’s original construction before looking at the modification made by De Witt.

Vilkovisky’s proposal was to to treat the configuration space as a manifold a equipped with an appropriate affine connection Γbc. The connection may then be used to define covariant derivatives in the standard way:

c S;ab = S,ab Γ S,c. (7.20) − ab Thus, replacing the standard derivatives with their covariant counterparts in (7.17) and (7.19), we observe that Γ(1) and Γ(2) (and subsequently higher orders) transform as required: the contravariant and covariant indices each contribute a multiplicative factor of det K and (det K)−1 respectively, which cancel out for fully contracted expressions. As noted by Vilkovisky [38], the a connection Γbc must be determined by the classical action and the connection should be “ultralocal” (proportional to undifferentiated delta functions and fields). Furthermore, if the fields are free, the effective action must coincide with the classical action in a parametrisation in which the action is kinetically quadratic:

1 A µ B = GAB(∂µϕ )(∂ ϕ ) V (ϕ). (7.21) L 2 − This means that the connection should vanish for such a parametrisation in a kinetically quadratic theory. In order for a theory to be free, the Riemann tensor Rambn associated with the metric Gab must vanish, which implies that 128 CHAPTER 7. BEYOND THE TREE LEVEL the connection must be given by the standard Christoﬀel symbols:

a 1 ad Γ = G Gdc,b + Gbd,c Gbc,d , (7.22) bc 2 − where

Gab = GAB δ(xA xB). (7.23) −

Since Gab is the only ultralocal tensor deﬁned through the Lagrangian, (7.22) is the only appropriate connection for the theory. With this form for the connection, the eﬀective action can be written simply by replacing standard functional derivatives with covariant derivatives:

i Γ(1)[ϕ] = ln det S;a . (7.24) − 2 ;b

At the level of the functional integral, Vilkovisky’s proposal was to replace the difference φa ϕa with a two-point quantity σa[φ, ϕ] which transforms as a − vector with respect to the mean field ϕ and a scalar with respect to the quantum field φ. An example of such an appropriate quantity is the tangent vector to the geodesic connecting ϕa and φa evaluated at ϕa. Thus, we observe that fields ϕa are no longer given a special status: they are simply a particular set of coordinates parametrising the configuration space, thus explaining the apparent loss of covariance in the conventional approach. The (affinely normalised) tangent vector σa[ϕ, φ] can be found by solving the following equation:

a b a σ [ϕ, φ];b σ [ϕ, φ] = σ [ϕ, φ], (7.25) where all covariant derivatives are with respect to the ﬁrst argument ϕ, along with the boundary conditions

a a a σ [ϕ, φ] ϕ=φ = 0, σ [ϕ, φ];b ϕ=φ = δb . (7.26)

a a It is possible to expand σ [ϕ, φ] in terms of the connection Γbc as

1 σa[ϕ, φ] = (ϕa φa) + Γa (ϕb φb)(ϕc φc) + . (7.27) − − − 2 bc − − ··· It is further possible to express scalars in the conﬁguration tangent space 7.2. THE VILKOVISKY EFFECTIVE ACTION 129 through a covariant functional Taylor expansion:

∞ ( 1)n a1 an S[φ] = − S[ϕ];a ...a σ (ϕ, φ) . . . σ (ϕ, φ), (7.28) n! 1 n n=0 X This expansion is possible due to the properties given in (7.25) and (7.26).

We now revisit the Legendre transform which is used to deﬁne Γ[ϕ]. General- ising (7.4) and (7.5), we write [42]

a Γ[ϕ] = W [J] Jav [ϕ], (7.29) −

a δW [J] v [ϕ] = , Γ;a = Ja. (7.30) δJa −

We have thus replaced the mean field ϕa with va[ϕ], which now transforms as a vector under redefinitions of the mean field ϕa. We may now replace the problematic term in the definition of the generating functional (7.3) as follows:

a a a Jaφ Ja v [ϕ] σ [ϕ, φ] , (7.31) → − leading to the following expression for the new generating functional W [J]:

i i a a exp W [J] = [ φ] [φ] exp S[φ] + Ja v [ϕ] σ [ϕ, φ] . h D M − Z ~ h i(7.32)

Finally, we use (7.31) in order to derive the modiﬁed version of (7.11) expression for the Vilkovisky eﬀective action Γ:

i i a exp Γ[ϕ] = [ φ] [φ] exp S[φ] + Γ;a[ϕ] σ [ϕ, φ] . (7.33) h D M ~ Z This expression ensures that Γ[ϕ] is going to be a scalar, since the contraction of the last two terms (both of which now transform as tensors in conﬁgura- tion space) is a scalar, thus avoiding the reparametrisation dependence found in (7.11). 130 CHAPTER 7. BEYOND THE TREE LEVEL

7.3 The De Witt Eﬀective Ection

It is worth noting that Vilkovisky’s formulation of the effective action does not have to be postulated a priori in case of a flat field space [41, 42]. If the a Riemann curvature tensor R mbn vanishes, the Vilkovisky action can be derived by transforming from the conventional effective action written in terms of the initial parametrisation of the fields (for which the Christoffel symbols vanish) to a reparametrisation for which Γa = 0. It is then possible to recover the form bc 6 of the effective action given in (7.33). In the presence of curvature, however, the generalisation is not as straightforward. In replacing the non-covariant term a a a Γ,a[ϕ](ϕ φ ) term with Γ;a[ϕ]σ [ϕ, φ], we have not taken into account the − curvature of the configuration space, since the tangent vector σa only encodes information about geodesics.

De Witt noted that there is no reason why we could not write the expression for the generating functional and the eﬀective action more generally as

i i a a exp W [J] = [ φ] [φ] exp S[φ] + Ja v [ϕ] Σ [φ] , h D M − Z ~ h i(7.34)

i i a exp Γ[ϕ] = [ φ] [φ] exp S[φ] + Γ;a[ϕ]Σ [ϕ, φ] . (7.35) h D M ~ Z In the above expressions, Σ[ϕ, φ] is a functional that may be Taylor expanded in terms of σa[ϕ, φ]. We may extend the functional average ... deﬁned in (7.6) h i to ... to h iΣ i f a[ϕ, φ] exp Γ[ϕ] [ φ] [φ] f a[ϕ, φ] h iΣ ≡ − D M ~ Z (7.36) i a exp S[ϕ] + Σ [ϕ, φ]Γ;a[ϕ] . × ~ h i Covariantly diﬀerentiating (7.36), we arrive at the following relation:

b b a Σ [φ];a δ Γ;b[ϕ] + Σ [φ] Γ;ab[ϕ] = 0. (7.37) h iΣ − a h iΣ h i Restricting ourselves to the on-shell eﬀective action, (7.37) gives us

a Σ [φ] Σ = 0. (7.38) h i Γ;b=0

7.3. THE DE WITT EFFECTIVE ECTION 131

For Vilkovisky’s original choice Σa = σa we ﬁnd

σa [ϕ, φ] = δa. (7.39) h ;b iσ b

We require that Σ[ϕ, ϕ] = 0 in order for the eﬀective action Γ[ϕ] to reduce to the classical action S[ϕ] as ~ 0, as can be seen by expanding (7.35) → using (7.13). As such, the simplest non-trivial extension of σa to Σa is

Σa[ϕ, φ] F a σb[ϕ, φ]. (7.40) ≡ b Similarly, the generalised Legendre transform is

a Γ[ϕ] = W [J] Jav [ϕ], (7.41) −

a where the mean ﬁelds ϕ in the presence of the source terms Ja are given by

a δWΣ[J] b δΓΣ[ϕ] v [ϕ] = , ( au ) Jb = a . (7.42) δJa ∇ − δϕ where va[ϕ] takes on the role of the mean field. We may simplify the sec- a a ond equality in (7.42) by assuming that v ,b = δb , which holds only in flat configuration space, may be covariantly extended to

a a bv = δ . (7.43) ∇ b

This assumption allows us to write Ja aΓΣ is crucial in ensuring that − ≡ ∇ the action still acts as a generator of connected graphs through the generalisation of (7.9), after covariantly diﬀerentiating the covariant Legendre transform (7.41) and (7.42):

a ;ab δ = (Γ )(W;bc), (7.44) c − ;mnl Γ;abc = Γ;amΓ;bnΓ;cl (W ), (7.45) − where all derivatives are evaluated at the background.

If we assume that this expression holds even when bΓ = 0 (as required, for ∇ 6 instance, to show that a gauge-fixing condition is independent even off-shell), we may substitute (7.38) into (7.37) to find:

Σa [φ] = δa. (7.46) h ;b iΣ b 132 CHAPTER 7. BEYOND THE TREE LEVEL

Taking the average of (7.40) and setting it equal to (7.46), it follows that

a −1 a F b[ϕ] = (C ) b[ϕ], (7.47) where

a a C [ϕ] = bσ [ϕ, φ] , (7.48) b h∇ iΣ with Σ given by (7.40). This expression for Σa leads to the following implicit expression for the Vilkovisky–De Witt eﬀective action Γ:

i i −1 a b exp Γ[ϕ] = [ φ] [φ] exp S[φ] + Γ[ϕ];a (C [ϕ]) σ [ϕ, φ] . h D M b Z ~ h (7.49)i

a We may recover the Vilkovisky eﬀective action by expanding C b as follows:

1 σa [ϕ, φ] = δa Ra [ϕ] σm[ϕ, φ] σn[ϕ, φ] + ..., (7.50) ;b b − 3 mbn

a where the Riemann curvature tensor R mbn is defined as usual through the metric Gab and vanishes when the configuration space is flat. This indicates a the need for the use of C b[ϕ] when the configuration is not flat; Vilkovisky’s original prescription Σa = σa is inconsistent with (7.47) in the presence of curvature, in which case the correct functional integro-differential equation that returns the effective action is (7.49).

7.4 The Conformally Covariant Vilkovisky–De Witt For- malism

So far, we have stayed silent on the topic of conformal transformations. The Vilkovisky–De Witt formalism as originally conceived did not take them into account; although it later found use in quantum gravity [150,151], the original inception was related to ﬁeld theory. In order to extend the Vilkovisky–De Witt formalism to include conformal transformations in the presence of gravitons, we must necessarily extend it to include gauge transformations. This was one of the original motivators for its development, but we will not delve deeper into this topic here. The issue of whether radiative corrections beyond the tree-level are frame invariant does not require us to work with perturbations 7.4. COVARIANT VILKOVISKY–DE WITT FORMALISM 133 of the metric: we can address the issue even while assuming a classical curved background.

Our treatment here almost mirrors that in Section 3.3. After all, both the field space and the configuration space are manifolds: they have different dimen- sionality, but the parallels between the two are rather straightforward. Indeed, we expect the configuration space to inherit all frame-covariant properties of the field space of the classical theory. This is because a vector dϕa = dϕA(x) in configuration space is a scalar with respect to its argument, but it also carries a field space index. Therefore, it is straightforward to define frame-covariant quantities in configuration space in a way similar to (3.39):

a ...a a ...a b e1 ep −dX ea1 eap 1 p b1 q X = Ω K a ...K a Xb ...b K ...K . (7.51) eb1...ebq 1 p 1 q eb1 ebq

Here, the scalinge dimension dX and the conformal weight wX are related through the expression dX = wX +p q, which remains unchanged from (3.40). −a˜ The conﬁguration space Jacobian K b is given by

a˜ −1 Ae K Ω K δ(x xB) . (7.52) b ≡ B Ae −

Deﬁning a frame-covariant functional derivative is similarly easy: the conformally- covariant functional derivative takes on the form

w f Xa1...ap Xa1...ap X ,c Xa1...ap . (7.53) b1...bq|c ≡ b1...bq,c − 2 f b1...bq

This derivative is now covariant with respect to field-dependent unit transformations as encoded in f = f(ϕa). This further enables us to write the conformally covariant configuration space connection, which replaces the definition given in (7.22)

¯a 1 ad Γ = G Gdc|b + Gbd|c Gbc|d . (7.54) bc 2 − The two expressions (7.22) and (7.54) coincide in the Einstein frame where units are constant. We have therefore eliminated all dependencies on the system of units, and can write the fully frame-covariant functional derivative as:

¯c ∆a∆b = S|ab Γ S,c. (7.55) − ab 134 CHAPTER 7. BEYOND THE TREE LEVEL

The one-loop eﬀective action can therefore be written in a manifestly frame- invariant form by replacing the ﬁeld-covariant functional derivatives a with ∇ their frame-covariant counterparts:

i Γ1[ϕ] = i ln [ϕ] + ln det ∆a∆b S[ϕ] . (7.56) − M 2 We may verify that Γ1[ϕ] is invariant under a frame transformation due to the property (7.53) of the frame-covariant derivative and the property for [ϕ]: M [ϕ] = Ωn det K [ϕ] . (7.57) M | | M where n is the number off scalare ﬁelds. The property (7.57) is a consequence of the fact that the path element measure [ φ] [φ] remains invariant under a D M frame transformation. The explicit form of the measure [ϕ] is given by the M determinant of the metric of the conﬁguration space

[ϕ] det ab , (7.58) M ≡ G p which satisﬁes (7.57). Substituting (7.58) in (7.56), we arrive at the frame- invariant one-loop eﬀective action

i i i a Γ1[ϕ] = tr ln ab + tr ln ∆a∆b S[ϕ] = tr ln ∆ ∆b S[ϕ] . − 2 G 2 2 (7.59)

This concludes our frame-covariant treatment of radiative corrections for the case of scalars [38, 41, 42]. While it might seem that not much changed by the introduction of conformal covariance, we can see how it aﬀects matter couplings by very brieﬂy looking at fermions minimally to gravitation under conformal transformations. The form of the Dirac action is:

S = dDx√ g iψ¯ ∂/ m ψ , (7.60) − − Z h i ¯ † µ where ψ ψ γ0 and ∂/ γ ∂µ. The conformal transformation in this case ≡ ≡ takes on the form

2 gµν Ω gµν → (7.61) ψ Ω−∆ψ ψ, → where we have assigned some weight to the fermions to be determined later.

The anticommutation relationship for the matrices is γµ, γν = 2gµνI, which { } 7.4. COVARIANT VILKOVISKY–DE WITT FORMALISM 135 indicates that the gamma matrices will transform as

γµ Ω γµ, → (7.62) γµ Ω−1γµ. → Hence, the fully transformed action will be

D ¯ D−2∆ψ −1 µ S = d x√ g iψΩ Ω γ ∂µ m ψ . (7.63) − − Z h i If we wish for the action to be invariant, we must have e

m = Ω−1m, (7.64) ∆ψ = (D 1)/2, e − which returns ∆ψ = 3/2 for 4-dimensional spacetime as usual. The requirement that the mass is scales with this weight agrees with the nondimensional- isation of the action that was given in

¯ D D ψ µ ψ S = d x(√ gMP ) i ∆ (γ /MP )∂µ (m/MP ) ∆ . (7.65) − ( M ψ − M ψ ) Z P h i P In the example of matter coupled coupled to nonminimally coupled gravitation, the master action can be written as follows:

4 f(ϕ)R kAB A µ B ¯ S = d x√ g + (∂µϕ )(∂ ϕ ) V (ϕ) + iψ / m(ϕ) ψ , − − 2 2 − D − Z (7.66) where we again abuse notation to write µψ ψ|µ, since the only rescaling D ≡ of the fermions can come from the ﬁeld-dependent unit transformation of the eﬀective Planck mass. We denote

µ 3 / ψ γ µψ = ∂/ (∂/ ln f)ψ. (7.67) D ≡ D µ − 4 µ

The gamma matrix ensures that / picks up a conformal factor. The coupling D between the fermion and the scalar will transform as

m(ϕ) = Ω−1(ϕ)m(ϕ). (7.68)

In order to transform to thee Einstein frame, Ω = MP /√f, and the action 136 CHAPTER 7. BEYOND THE TREE LEVEL becomes:

2 4 MP R GAB A µ B S = d x√ g + (∂µϕ )(∂ ϕ ) U(ϕ) − − 2 2 − Z (7.69) m(ϕ) + iψ¯ ∂/ ψ , − √f where the frame-covariant derivative / has been reduced to the standard par- D tial operator ∂/. Thus, we reaffirm the idea of frame covariance: introducing both a varying fermionic coupling and a varying effective Planck mass is redundant, exactly like scalar-tensor theories are fully specified by the metric GAB and invariant potential U. The only relevant physical value is the dimensionless coupling m(ϕ)/√f, which fully classifies the model and contains the true physics of the theory, as a varying dimensionless parameter is certainly observable. In some sense, we may identify the Einstein frame as more “intuitive”: we are not accustomed to working with spacetime dependent units, and it is more convenient to work in a system where the unit of mass is homogeneous. It could be desirable for some reason to work in a system where the fermion coupling is constant while the effective Planck mass varies; that is possible as well by transforming to an appropriate frame. What is important and must be stressed is that these different pictures are just that; pictures of the same underlying theory. Both frames are perfectly acceptable, and while some may be more convenient, they are in no sense more “physical” than the others. Chapter 8

Conclusions

We are entering a new era of precision cosmology, and so developing new theoretical methods for computing observable inflationary quantities with in- creasingly higher accuracy is of utmost importance. In particular, radiative corrections which might be subleading in several models of inflation may become observationally significant not too far in the future. Hence, any formalism that aspires to remain relevant should describe quantum loop effects in a frame-covariant manner. This is a particularly pressing problem, as frame covariance of field theories beyond the tree level is still an open question. With this motivation, we have developed a fully frame-covariant formalism of inflation for multifield scalar-tensor theories at the classical level, which may be extended to include radiative corrections.

We began by overviewing the idea of frames and how their physical meaning has been a contentious point since antiquity. The heliocentrism/geocentrism debate and the subsequent gradual shift from a global to a local formulation of physics were instrumental in the understanding that the underlying fundamental laws of physics should be independent of the way we parametrise them. In a sense, the Copernicean principle was the seed of thought that led to the requirements of homogeneity and isotropy that were codiﬁed in the FLRW metric. With the development of general relativity, it became possible to qualitatively study the evolution of the Universe at the largest scales, leading to the development of concordance cosmology. We discussed its general features as well as problems, which we saw could be resolved with a period of accelerated expansion, known as inﬂation. Still, nearly 40 years after it was originally

137 138 CHAPTER 8. CONCLUSIONS proposed, inﬂation has a few conceptual problems, including the frame problem and the question of its initial conditions. It has been argued that inﬂation poses more problems than it solves, but it is observationally favoured enough that much research has been devoted to rescuing it.

We examined conformal transformations and their relation to transformations of units, which we expect to not change the laws of physics. From the Buck- ingham π theorem, we argued that any law in physics should be written in terms of dimensionless ratios, and argued that a physical derivative of a dimensionful quantity should be no different. From this argument, we derived the form that a physical derivative has to follow; it must be conformally covariant when acting on quantities that transform with a non-vanishing weight. Using this conformally covariant approach in conjunction with a field-covariant formalism, we examined the dynamics of a fairly general class of theories (non- minimally coupled multifield inflation) in a fully frame-covariant manner.

With an eye towards studying the evolution of perturbations, we employed well-known ideas from differential geometry, adopting an approach in which the scalar fields take on the role of generalised coordinates of a manifold. The equations of motion thus can be seen as describing a trajectory within the field space. By perturbatively expanding and quantising the resulting frame- covariant equations of motion, we showed how manifestly frame-invariant expressions for inflationary observables may be obtained. These include the tensor-to-scalar ratio r, the spectral indices nR and nT , their runnings αR and αT , the non-Gaussianity parameter fNL, and the isocurvature faction βiso. Examining the inflationary attractor, we further defined approximate potential slow-roll parameters that can be used to approximate the Hubble slow-roll parameters to arbitrary order. We further examined the effects of the field space curvature on the generation and transfer of isocurvature modes in a frame-covariant way.

We distinguished the different kinds of fine tuning one may encounter when working in inflation, including initial condition fine tuning, which may put the self-consistency of the slow-roll inflationary paradigm in trouble, and parameter fine tuning, which, while acceptable to some degree to the dedicated model builder, might irk the particle physicist. We demonstrated that there is often a trade-off between the two, using α-attractors as an explicit example. We further identified another kind of fine tuning which occurs when multiple fields can be used to “hide” the fine tuning of a theory. We introduced a criterion to 139 discriminate between stable and unstable trajectories, and noted that the more degrees of freedom we have in selecting an inflationary trajectory, the better a rigid model may be rescued without the need to fine tune its fundamental parameters.

In order to illustrate our approach, we considered a few example models. We examined induced gravity inflation and Higgs inflation, as well as a specific example of f(R) inflation (Starobinsky inflation), analytically presenting their predictions. We further examined multifield models, such as simple minimal two-field model, as well as a more realistic non-minimal extension inspired by Higgs inflation. We observed that after selecting nominal values for the model parameters, the normalisation of the power spectrum of scalar perturbations can be used in order to select an appropriate boundary condition for the fields at the end of inflation. We also found that, for certain values of the mass and coupling parameters, the calculation of the entropy transfer becomes crucial in selecting the appropriate trajectory. Furthermore, we briefly discussed how F (ϕ, R) theories may be incorporated into our formalism by re- casting them in terms of a multifield theory through the method of Lagrange multipliers.

Finally, in the last chapter we outlined how the Vilkovisky–De Witt formalism may be applied to multifield inflation. We extended the notion of classical frame covariance to the configuration field space in the path integral formulation under the assumption that quantum gravity effects can be ignored. By replacing the functional derivatives with their covariant counterparts, the so-defined effective action is unique and becomes invariant under frame transformations, which includes both conformal transformations and field reparametrisations. This ensures that scalar-tensor theories related by frame transformations are phenomenologically equivalent. This can be extended to include matter corrections in a frame-covariant manner by similarly understanding that a conformal transformation can be viewed as a change of units in this context. This independence of the way we choose to parametrise our units and its relation to frame covariance thus becomes a celebration of the idea that the laws of Nature, intractable though they may seem at times, are indeed independent of how Man chooses to express them. Appendix A

Frame-Covariant Power Spectra

In this appendix, we derive the frame-covariant two-point function of the comoving curvature perturbation , which can be used to write down the power R spectrum of scalar perturbations in the CMB. We do so by writing the second- order action in terms of the comoving curvature perturbation, and quantising the resulting equations of motion, which have the form of a simple harmonic oscillator with time-varying mass. As a result, the vacuum solution varies with time as well, and we must impose a particular choice for the vacuum. Using the Bunch–Davies condition which corresponds to a Minkowski comoving observer in the far past, we arrive at a simple expression for the scalar power spectrum. We also outline how a similar result for the tensor power spectrum may be derived.

We begin from the general scalar-tensor action (without matter couplings) in the Jordan frame, given by (3.29):

4 f(ϕ)R kAB(ϕ) µν A B S d x √ g + g (∂µϕ )(∂νϕ ) V (ϕ) . (A.1) ≡ − − 2 2 − Z Finding the second-order action δ(2)S is equivalent to varying the equations of motion. This is normally a rather cumbersome procedure, but thankfully, the notions of frame covariance can help. The usual procedure is to vary along δϕa and then substitute the equations of motions in order to arrive at δ(2)S. However, it is possible to drastically cut down on the required number of steps by varying along ∆ϕa instead. This “covariant variation” is deﬁned

140 141 such that

a a b ∆X = ∆bX ∆ϕ , (A.2)

where ∆b is the frame-covariant functional derivative used in (7.55). Explicitly, the frame-covariant derivative for a quadratic action is

∂ ∂ ∆aS = µ L a La , (A.3) D ∂( µϕ ) − ∂ϕ D where we remind that the frame-covariant partial derivative µ is given by (3.51) a D and that µϕ is deﬁned similarly to (3.65). D We now turn our attention to the perturbed metric. We choose to work in the Newtonian gauge (4.17) in which

µ ν 2 2 2 i j gµνdx dx = (1 + 2Φ)N dt a (1 2Ψ)δij + hij dx dx . (A.4) L − − We can show that Ψ = Φ by considering the oﬀ-diagonal spatial part of the perturbed Einstein equation (3.54), where we have used the equations of motion to eliminate the background:

(NM) ∆Gµν = ∆Tµν . (A.5)

The non-minimal energy-momentum tensor is given in (3.56) and we have covariantly perturbed both sides, as per our prescription above. As a consequence, the usual derivative terms of f that we might expect when vary- (NM) ing Tµν in the standard way do not appear. For i = j, we ﬁnd: 6

i j(Ψ Φ) = 0. (A.6) D D − Since this occurs for all possible combinations of i, j, this expression indicates that Ψ Φ can only be a pure function of f(ϕ). However, due to its dimension- − lessness, Ψ Φ can only be constant if this is the case. Since it must vanish at − the boundary, we conclude it must be zero. Therefore, Φ = Ψ, and as a result, there is only one gauge-invariant degree of freedom in the metric.

With the form of the metric given in (A.4), we are able to covariantly vary the 142 APPENDIX A. FRAME-COVARIANT POWER SPECTRA action twice, leading to the second-order action as follows:

3 (2) 4 NLa A B −2 A i B ∆ S = d x GAB( tQ )( tQ ) a GAB(∂iQ )(∂ Q ) 2 D D − Z 3 (A.7) A B 1 NLa A B C MABQ Q + 3 t ( tϕ )( tϕ ) GBC Q , − NLa D D D H where the mass matrix is given in (4.30). Equivalently, we may arrive at the second-order action by varying the equations of motion given in (3.67), and performing a covariant integration by parts. This is possible in general for any operator that behaves as a derivation, i.e. (AB) = A B + B A, as long D D D D as (AB) vanishes at the boundary. D R Each term in (A.7) is frame-covariant, something which is ensured by the use of covariant derivatives. We could have instead varied the action non- covariantly, and the resulting action would have been equal to (A.7). Indeed, ∆(2)S = δ(2)S, since the action has no frame weight. However, in this case, non-covariant terms would appear in the second-order action. We now turn our attention to the comoving curvature perturbation, given in (4.27) as

σ A σ H Q = H Q eA. (A.8) R ≡ tσ tσ D D A A σ B Further using e = tϕ / tσ and e = GAB tϕ / tσ, we have σ D D A D D

A B = H 2 Q GABDtϕ . (A.9) R ( tσ) D 2 A B Using (Dtσ) = GAB( tϕ )( tϕ ), we may write the second-order action as D D 3 2 (2) 4 NLa ( tσ) 2 −2 i ∆ S = d x D ( t ) a (∂i )(∂ ) , (A.10) 2 2 D R − R R Z H where we have also used the (derivatives of) the equations of motion in order to eliminate the terms. We now deﬁne the Mukhanov–Sasaki variable

v z , (A.11) ≡ R

2 2 2 2 2 where z a ( tσ) / = 2a ¯H . With this deﬁnition, the second-order ≡ D H action (A.10) can be written as

(2) 3 2 i η ηz 2 ∆ S = dηd x ( ηv) (∂iv)(∂ v) + D D v , (A.12) D − z Z 143

where the conformal time η is given as adη = dt = NLdτ. Fourier transforming the ﬁeld v, we can write the equation of motion for its Fourier components as

3 ik·x vk(η) = d xe v(η, x). (A.13) Z Substituting this into the equations of motion for (A.12) yields

2 η ηz η ηvk + k D D vk = 0, (A.14) D D − z where vk is isotropic since the time-dependent mass only depends on the magnitude of k. This is the well-known Mukhanov–Sasaki equation. Once again, note the presence of conformal derivatives: even if η is scaled under a conformal transformation, the conformal derivatives will ensure that any additional terms do not spoil the form of the equation. Of course, in the Einstein frame, we may replace η with ordinary derivatives as usual, and indeed this is more D convenient as we shall see below. Note, however, that frame covariance is in no way spoiled by our choice to work in a particular frame. In fact, it is the opposite: after we demonstrate frame covariance, we are guaranteed that results derived in one frame are applicable to all frames (even if parametrised with non-invariant quantities).

In order to calculate the two-point function of = zv, our next step is to R quantise the Mukhanov–Sasaki equation (A.14). To do so, we simply promote the variable v to an operator by promoting its the Fourier components to operators:

d3k vˆ = v aˆ (η)eik·x + v∗(η)â† e−ik·x . (A.15) (2π)3/2 k k k k Z h i † The operators ak and ak correspond to the creation and annihilation operators for the field σ. In order to recover the canonical commutation relation † [âk, aˆ ] = δ(k q), we must impose q −

∗ ∗ ( ηvk)v vk( ηv ) = i. (A.16) D k − D k This is one of the two required boundary conditions for the mode functions. The second one comes from our choice of vacuum viaa ˆ 0 0. Our choice is | i ≡ the Bunch–Davies vacuum [152], characterised by the requirement that the vacuum is ﬂat (Minkowski) in the far past, which occurs when η ηz/z in (A.14) D D 144 APPENDIX A. FRAME-COVARIANT POWER SPECTRA vanishes. Solving the Mukhanov–Sasaki equation in this (free) regime returns

−ikη uk(η = ) e . (A.17) −∞ ∝ Further imposing the normalisation condition (A.16) gives us

e−ikη vk(η = ) = , (A.18) −∞ √2k where η now is deﬁned in the Einstein frame, such that the covariant derivatives reduce to ordinary derivatives. As alluded to above, even if the equations are frame-covariant, it may be more expedient to work in a particular frame. In this case, departing from the Einstein frame would result in additional (non- dynamical) factors that would make solving (A.14) prohibitively cumbersome even in the very early time limit.

Returning to the general Mukhanov–Sasaki equation (A.14), we can see that ¯H acts as a deformation parameter relating de Sitter space (in which the scale factor is a = 1/( η) and is exactly constant) to quasi-de Sitter space, in − H H which we have that

1 a = . (A.19) − η(1 ¯H ) H − 2 2 In order to solve (A.14), we ﬁrst use z = 2a ¯H , which to ﬁrst slow-roll order is (in the de Sitter limit)

2 η ηz ν 1/4 D D = − z η2 (A.20) 1 3 = 2 ¯H + η¯H . η2 − 2 Therefore, the Mukhanov–Sasaki equation becomes

2 2 η ηvk + k vk = 0. (A.21) D D − η2 The unique solution to this equation that satisﬁes the Bunch–Davies condition (A.18) is

e−ikη i vk(η) = 1 . (A.22) √2k − kη 145

We can ﬁnally compute the two-point function of by using R

−2 k q = z vk vq . (A.23) hR |R i h | i Substituting the Fourier expansion given in (A.15) into (A.23) and using the canonical commutation relationships along with (A.19), we ﬁnd

2 δ(k q) k q = H −3 . (A.24) hR |R i ¯H 2k

Observable cosmological quantities measured in the CMB are linked to the primordial perturbations through transfer functions which induce a multiplicative multipole contribution to the power spectrum of scalar perturbations [153]:

k3 PR δ(k q) k q . (A.25) − ≡ 4π2f(ϕ) hR |R i

The above relation is usually written in the Einstein frame, but we have pro- 2 moted MP to the effective Planck mass squared f(ϕ) in order to keep PR frame-invariant (this multiplicative factor will not affect the spectral index). Finally, comparing (A.25) with (A.24), we find

PR = 2 H . (A.26) 8π f(ϕ)¯H

This expression may then be evaluated at any given time in order to ﬁnd the amplitude of scalar perturbations.

A similar relation can be found for tensor perturbations hij, which can be split into two modes, h+ and h×. Computing the second-order action for the two tensor modes returns an equation very similar to (A.14), except that z = a. Since the form of the Mukhanov–Sasaki equation does not change, the mode functions (A.15) remain the same. The only diﬀerence is that the two point function is given by

2 + + −2 h h = a vk vq = H , (A.27) h k | q i h | i k3 and similarly for h×. In this case, the relation between the two-point function and the spectrum is

3 2k + + PT δ(k q) h h . (A.28) − ≡ f(ϕ) h k | q i 146 APPENDIX A. FRAME-COVARIANT POWER SPECTRA

The factor of two comes from the fact that there are two tensor modes which equally contribute to the spectrum. Once again, we compare (A.27) and (A.28) to ﬁnd the following expression for the tensor power spectrum:

2 2 PT = H . (A.29) π2 f(ϕ)

Notably, ¯H does not appear in this expression. Therefore, tensor perturbations are not ampliﬁed by the Universe being close to a de Sitter space, unlike scalar perturbations. Bibliography

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