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University of Amsterdam

MSc Physics

Track: GRAPPA

Master Thesis

Going beyond the eective theory of ination

by

Lars Aalsma

10001465

July 2015

48EC

September 2014 - July 2015

Supervisor:

Dr. Jan Pieter van der Schaar

Second reader:

Dr. Ben Freivogel

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Abstract

The inationary paradigm makes a compelling case for providing the initial conditions of the universe. Despite being consistent with cosmological observations, the theoretical foundation of ination is not yet well-understood. Specically, ination is UV-sensitive, which means that one cannot ignore the eect of UV-physics on the eective description. In this thesis, we motivate that ination needs to be embedded in string theory, in which its UV-sensitivity can be addressed. In particular, we show how inationary model building works in the context of string theory and supergravity. Furthermore, models of ination involving axions are treated and we comment on their validity, both from a bottom-up and top-down perspective. We show that such models are consistent from an eective eld theory point of view, but are in tension with general properties of quantum gravity.

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Acknowledgements

First of all, I like to acknowledge Jan Pieter. Jan Pieter, thank you for supervising me the past year and introducing me to the various topics we have looked at. While I may have been a bit worried halfway through my project about the direction it would be heading, you ensured me that everything would fall into place, which it did. Furthermore, you have learned me a lot and not once have I left your oce without feeling inspired. Your enthusiasm is inspiring and I am looking very much forward to working with you in the coming years.

Second of all, I would like to thank Ben for being my second reader. I have enjoyed the questions you asked during my presentation, which allowed me to show my audience some aspects of my thesis in a bit more detail.

On top of that, I also thank Volkert van der Willigen. I am very grateful for your nan-cial support and I hope this thesis might add some knowledge to your understanding of cosmology.

Of course, I also want to thank my fellow students. Adri, thank you for being my partner in crime as a student of Jan Pieter and the useful discussions we had. Vincent, Jonas and Jorrit, thank you for being friends and fellow students the past years. Physics would not be the same without you. Also, all other master students get a big thank you for creating the unique atmosphere C4.273A has.

Besides closing the chapter of my life as a student, the past year has oered me much more. Thank you, my three sisters, for giving me the pleasure to see a new generation grow up. Even though you may not always hear much from me, I think of you. On the same note, I would like to thank my parents for supporting me in their own ways and being there for me when I need them.

Finally, thank you Jolijn for making my life more interesting, exciting and enjoyable. I hope that the past year has set the stage for everything that will follow.

Lars Aalsma July 2015

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1 Introduction 1

2 Physics of ination 4

2.1 Introduction to cosmology . . . 4

2.2 The Cosmic Microwave Background . . . 7

2.3 How natural are the initial conditions of our universe? . . . 9

2.4 Addressing the initial conditions of the universe with ination . . . 12

2.5 How to drive ination? . . . 14

2.6 Quantum uctuations during ination . . . 16

2.7 The energy scale and UV-sensitivity of ination . . . 22

3 The eective eld theory of ination 25 3.1 Constructing an eective action . . . 25

3.2 Symmetries of ination . . . 26

3.3 The eective action of ination . . . 27

3.4 Why should we go beyond the eective eld theory of ination? . . . 35

4 From string theory to cosmology 36 4.1 Aspects of string theory . . . 36

4.2 Eective N = 1 supergravity . . . 40

5 Axion ination 44 5.1 Natural ination . . . 44

5.2 Saving natural ination . . . 45

5.3 Statistical generality of axion ination . . . 49

6 Non-minimal coupling 59 6.1 Description of a non-minimal coupling in the Jordan and Einstein frame . 59 6.2 Supergravity formulation of a non-minimal coupling . . . 63

6.3 Cosmological attractors . . . 65

6.4 Radiative generation of a non-minimal coupling . . . 73

7 The Weak Gravity Conjecture and axion ination 79 7.1 The Weak Gravity Conjecture for particles and gauge elds . . . 80

7.2 Generalization to p-forms . . . 81

7.3 Applying the Weak Gravity Conjecture to ination . . . 82

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Contents

8 Conclusions and outlook 88

Bibliography 91

Appendix A The eective eld theory of ination 97

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Throughout this thesis, we will work in natural units, such that c = ~ = 1. The metric convention we use is mostly plus, i.e. (-,+,+,+), unless stated otherwise. Furthermore we will use the reduced Planck mass which is dened as

Mp−2 ≡ 8πGN = 2.4 × 108 GeV.

Mostly, we will be working in 3 + 1 dimensions. If this is not the case we will denote the spacetime dimension by d. The Hubble slow-roll parameters are indicated with a tilde and are dened as ˜  ≡ − H˙ H2 and η ≡ −˜ 1 2 ¨ H H ˙H.

Here, the dot denotes a derivative with respect to time. The potential slow-roll parameters are dened as  ≡ M 2 p 2  V0(φ) V (φ)  and η ≡ Mp2V 00(φ) V (φ) ,

where the prime denotes a derivate with respect to φ. These slow-roll parameters are related to each other as

˜

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1 Introduction

In the 20th century, cosmological observations have led to a description of the origin and

evolution of the universe, known as the hot Big Bang model. In this model, the origin of the universe is described as an extremely hot and dense state that expanded in a `Big Bang', after which it evolved under the inuence of gravity to today's observed universe. Whereas this view gave new insights into the history of the universe, it did not address its initial conditions. When we follow the evolution of an arbitrary initial state, the resulting universe will not look like our own. More specically, the isotropy of the Cosmic Microwave Background and the atness of the universe can not be explained. In order to obtain a universe with the observed properties, the initial state has to be extremely ne-tuned. This situation, in which a large amount of ne-tuning is needed, hints towards some underlying mechanism that creates more natural initial conditions. Alan Guth realized in 1981 that a period of exponential expansion, preceding the hot Big Bang phase, will drive an arbitrary state towards homogeneity, isotropy and atness [1]. This naturally sets the stage for the evolution of the universe during the hot Big Bang phase, as we can start from an arbitrary state and still end up with a universe with the correct properties. Unfortunately, this period of exponential expansion, known as ination (as space inates during this phase), could not be smoothly connected to the regular hot Big Bang phase of the universe, making this model not realistic. Nevertheless, it inspired the idea that the regular hot Big Bang phase of the universe was preceded by a period of exponential expan-sion. A step forward came when Andrei Linde suggested that a period of quasi-exponential expansion could be driven by a scalar eld with a potential energy that dominates over its kinetic energy [2].

This mechanism, known as slow-roll ination, does allow for a smooth transition to the regular hot Big Bang phase. However, it was shown that in this new inationary scenario there still would be a problem with its initial conditions, i.e. the universe was not likely to live long enough for ination to start [3]. At the time, it seemed that one problem with initial conditions was simply swapped for a dierent one.

The resolution to this problem came with the introduction of chaotic ination [4], which in its essence states that there may exist dierent initial conditions in dierent parts of the universe or in dierent universes. This implies that there will always be some patch in which ination can occur, from which our universe originates. Of course, this notion was (and still is) quite controversial. Steinhardt and Vilenkin observed that this implies that our universe may be part of a multiverse where ination perhaps always takes place in some part (eternal ination) [5]. Despite its controversy, the inationary paradigm

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is nowadays well-established, because all cosmological observations are consistent with it. Nevertheless, the origin of ination and the existence of a multiverse are still open questions.

One of the predictions of ination that is conrmed by measurements is a nearly scale-invariant spectrum of perturbations, originating from quantum uctuation in the scalar eld that drives ination (the inaton). These perturbations source small matter inho-mogeneities that are visible in the Cosmic Microwave Background as small temperature uctuations [6]. These temperature uctuations have been measured with very high ac-curacy by satellites such as Planck (see gure 2.5) and are in agreement with a period of ination in the early universe.

In addition, quantum uctuations of spacetime itself will lead to a nearly scale-invariant spectrum of tensor perturbations, also known as primordial gravitational waves. These gravitational waves should be visible in the polarization of the Cosmic Microwave Back-ground [6]. A measurement of this polarization signal would establish the energy scale of ination. Currently, observations have only been able to put an upper bound on the amplitude of the gravitational waves, but upcoming measurements will push this bound further down or measure the amplitude of the gravitational waves.

Despite the fact that ination is a compelling mechanism, it is theoretically not yet fully under control. It is well-known that when ination is treated in an eective way, which oers simplicity, the eect of new physics above the cuto of the theory can have a large eect on the eective theory, rendering ination dicult. This problem is known as the UV-sensitivity of ination.

As we will motivate at the end of chapter 2, this UV-sensitivity can only be properly dealt with in a UV-complete theory, i.e. a theory that complements the eective theory at high energy. In the case of ination, this sensitivity reveals itself in terms of operators that are suppressed by the cuto, but can become active during ination. This implies that in order to have theoretical control over these operators, ination needs to be described in a theory of quantum gravity. Because the best candidate for such a theory is string theory, this motivates us to study ination in the context of string theory. Unfortunately, string theory is rather technical and only partially understood. Therefore, simplifying assumptions have to be made in order to have control over the theory.

When considering string theory, such a simplication occurs when looking at its low-energy limit, i.e. supergravity. This opens up a theoretically accessible landscape, which also is non-trivial enough to allow for interesting dynamics. Above all, we know that supergravity is UV-completed into string theory, which makes it (in principal) possible to check if any low-energy assumptions are valid up to high energy.

However, the situation is a bit more subtle. String theory is not yet fully understood, such that not all assumptions (e.g. the existence of de Sitter vacua) can be straightforwardly checked. In summary, a better understanding of string theory is needed, before we can truly embed ination in it. For now, the best we can do is pursue an understanding of ination in the parts of string theory that are well under control or make assumptions in a theory of supergravity.

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The contents of this thesis are as follows. In chapter 2-4, we give a basic introduction to cosmology, an overview of the physics of ination, its description as an eective theory and the motivation for embedding ination into string theory. Continuing, chapters 5-7 give recent advances in particular well-motivated models, such as axion ination. In addition, chapter 6 also contains some original work. Finally, we conclude in chapter 8. Complementary, some appendices and a summary for non-physicists can be found at the end.

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The goal of this chapter is to give a self-contained introduction to the basic concepts and formalism relevant for this thesis. Tools and techniques necessary to describe the evolution of the universe are presented and we will motivate how the initial conditions of the universe can be properly addressed. Continuing, we will show how theory and observations can be connected, paving the way for subsequent chapters.

2.1 Introduction to cosmology

The foundation of modern cosmology is the assumption that the universe is homogeneous and isotropic on large scales. From this assumption, the Einstein equations can be solved, which results in the Friedmann-Robertson-Walker (FRW) metric. In spherical coordinates, this metric is given by

ds2 = −dt2+ a(t)2  dr2 √ 1 − kr2 + r 2 dθ2+ r2sin2(θ)dφ2  . (2.1)

Here, a(t) is a scale factor that measures how much the universe has expanded in a certain amount of time and k is a parameter that determines the spatial curvature of the universe. Mostly, we will work in a at spacetime (k = 0) for which the FRW metric can be rewritten in Cartesian coordinates as

ds2 = −dt2+ a2(t)dx2+ dy2+ dz2 . (2.2)

Furthermore, we will also use comoving coordinates (τ, ~x), that x the expansion of the universe. Here, τ is conformal time, dened as

dt

dτ = a(t). (2.3)

In terms of conformal time, (2.2) can be written as

ds2 = a2(τ )−dτ2+ dx2 + dy2+ dz2 . (2.4) If we assume the energy content of the universe to have the stress-energy tensor of a perfect uid, which is given by

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2.1 Introduction to cosmology we can derive the Friedmann equations from the Einstein equations.

H2 = ρ 3M2 p − k a2 (2.6) ¨ a a = − 1 6M2 p (ρ + 3P ) (2.7)

Here, H is the Hubble parameter which is dened as H ≡ ˙a

a. (2.8)

As can be seen from (2.8), the Hubble parameter describes the expansion rate of the universe at a certain time. Furthermore, ρ is the energy density of the uids in the universe, P the pressure and k the same parameter as in the FRW metric. The eect of the value of k can be most intuitively seen by introducing the density parameter Ω, which is dened as

Ω ≡ ρ ρcrit

= ρ8πGN

3H2 . (2.9)

The critical energy density (ρcrit) corresponds to the situation where the energy density of

the universe is just right to sustain a at universe, while it evolves. Any small deviation of |Ω| > 1 results in a curved universe. To be more specic, Ω < 1 corresponds to negative spatial curvature and Ω > 1 to positive spatial curvature, see gure 2.1. In terms of the density parameter, (2.6) can be written as

Ω − 1 = k

H2a2, (2.10)

which allows us to make the following identication.

Ω =      > 1 ⇐⇒ k > 1 ⇐⇒Closed universe 1 ⇐⇒ k = 0 ⇐⇒Flat universe < 1 ⇐⇒ k < 1 ⇐⇒Open universe (2.11)

Now that we know how the geometry of the universe depends on its energy content, we would also like to know how it evolves. For this, the relation between the energy density and the scale factor is needed. By using conservation of energy, the following conservation equation can be derived [7].

˙ ρ

ρ = −3(1 + w) ˙a

a (2.12)

Here, w determines the equation of state of a particular uid, which is given by

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Figure 2.1: The value of the density parameter Ω determines the geometry of the universe. Positive spatial curvature corresponds to a closed universe, negative spatial curvature to an open universe. Figure from wikipedia.

The equation of state is bounded by energy conditions, which constrain w. For constant w, (2.12) can be integrated to obtain

ρ ∝ a−3(1+w). (2.14)

In the standard model of the universe (ΛCDM), which is well-established by observa-tions, it is dominated at dierent epochs in time by three dierent uids; radiation, (non-relativistic) matter and a component known as dark energy. For these uids, w takes the following values1. w =      1 3 (radiation) 0 (matter) −1 (dark energy) (2.15) Given a particular uid, (2.14) can be used to relate the energy density to the scale factor. For example, consider a universe that is spatially at (Ω = 1, k = 0) and dominated by radiation. By taking w = 1

3 and plugging it into (2.14), we obtain

ρ ∝ a−4. (2.16)

Using this relation in the rst Friedmann equation (2.6) and integrating it results in the expansion rate of the universe.

a(t) ∝ t12, (2.17)

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2.2 The Cosmic Microwave Background This calculation can also be straightforwardly applied to other uids, which will give a dierent evolution, see gure 2.2. Generally, when given an equation of state, we can derive a relation between the scale factor and the energy density. Plugging this into the Friedmann equation tells us the evolution of the universe. Of course, it is very well possible that multiple components contribute to the energy density of the universe, complicating the process of solving the Friedmann equation. There are two possible ways to proceed. Firstly, it is often reasonable to approximate a certain epoch as being dominated by only a single component. Secondly, if one does not want to make this assumption, the Friedmann equation can be solved numerically.

Using these techniques, the ΛCDM model tells us that the universe is dominated in dif-ferent periods of time by respectively radiation, matter and dark energy. On top of that, measurements have shown that the universe is very at, such that Ω = 1, k = 0 is a good approximation.

t a(t)

Radiation : a(t)∼ t Matter: a(t)∼t23

Dark Energy: a(t)∼et

Figure 2.2: The evolution of the scale factor for dierent uids.

2.2 The Cosmic Microwave Background

One of the greatest sources of cosmological information is the transition of the universe from being opaque to transparent. At a redshift of z . 1100, the universe was in a hot and dense state, which consisted of photons, electrons and baryons. As photons were constantly Thomson scattering on electrons, their mean free path was small, causing the universe to be opaque. Only when the universe cooled suciently enough at z ∼ 1100, neutral atoms were able to form (an event known as recombination), which increased the mean free path of the photons dramatically (decoupling). As a consequence, photons were now able to travel freely, which made the universe transparent, see gure 2.3.

After decoupling, the radiation emitted by the opaque universe could travel freely, car-rying information about the universe at z ∼ 1100. This radiation, known as the Cosmic Microwave Background (CMB), is still measurable today as an isotropic radiation that is

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Figure 2.3: When the universe cools down, neutral atoms are able to form, increasing the mean free path of the photons. Eventually, the photons are able to travel freely, making the universe transparent.

visible when all foreground light is subtracted. Due to the expansion of the universe, it is signicantly redshifted with respect to its original wavelength, such that the CMB has the highest intensity at a temperature of T0 = 2.7K. The accidental discovery of the CMB by

Penzias and Wilson [8] conrmed this idea, which earned them the Nobel prize of 1978.

Furthermore, because the early universe was in thermal equilibrium, it was also predicted that the CMB would have the spectrum of a black body. This spectrum was rst measured by the COBE satellite, see gure 2.4. Due to the large amount of information the CMB carries about the early universe, the initial discovery of the CMB initiated the launch of more surveys, such as WMAP and Planck that measured the CMB with an even higher precision. These surveys showed that, on top of black body spectrum, the CMB has tiny temperature uctuations of the order

δT T ∼ 10

−4

, (2.18)

which are nearly scale-invariant, see gure 2.5. Later, we will comment on the important physical relevance of these uctuations.

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2.3 How natural are the initial conditions of our universe?

Figure 2.4: Spectrum of the CMB as measured by the COBE satellite. The theoretical prediction of a black body with a temperature of T0 = 2.7K agrees excellently with the

observations (the true error bars are even smaller). Figure from Wikipedia.

Figure 2.5: All-sky map of the CMB taken by the Planck satellite. The colour dierence corresponds to small temperature uctuations. Figure from NASA.

2.3 How natural are the initial conditions of our

universe?

As we mentioned in the introduction, a universe that is very homogeneous, isotropic and spatially at on large scales, is quite curious. Starting from an arbitrary state, and letting

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it evolve according to the Friedmann equation, would result in a dierent universe than the one we are living in. Therefore, a logical question to ask ourselves is: how natural are the initial conditions of our universe? In this section, we will specify two problems concerning the initial conditions of the universe, known as the horizon and atness problem.

2.3.1 The horizon problem

As in any expanding geometry, our universe has a horizon. This cosmological horizon denes the maximal distance that is in causal contact with us. Any signal separated any further will not be able to reach us due to the expansion of the universe. Using comoving coordinates, this causal structure can be conveniently explored, because for ds2 = 0,

dx2 = dτ2. Thus, the maximum distance a light-like signal can have travelled between the

singularity and a particular time t is given by the conformal time elapsed.

τ = t Z 0 dt0 a(t0) = a(t) Z a(0) da(t0) a(t0)2H, (2.19)

which can be rewritten in terms of the comoving Hubble radius RH ≡ (aH)−1.

τ = log(a(t)) Z 0 d log(t0) a(t0)H = log(a(t)) Z 0 d log(t0)RH, (2.20)

The comoving Hubble radius is the radius of the Hubble sphere, the sphere that contains the observable universe.

During radiation and matter domination (which was the dominant energy contribution during the largest part of the history of the universe) RH increases. Hence, we can see from

(2.20) that the biggest proper time contribution comes from late times. This observation is problematic if we want to explain the observed isotropy of the CMB. This can be seen as follows. The fact that we observe that every patch of the CMB has the same temperature (up to tiny uctuations) implies that they were in causal contact with each other before decoupling. If this was not the case, there is no reason why dierent patches would have the same temperature. However, the fact that the largest proper time contribution comes from late times, implies that there was not enough conformal time between the singularity and the surface of last scattering (the surface at redshift z = 1100 from which the CMB originates) for all dierent patches of the CMB to have been in causal contact with each other, see gure 2.6.

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2.3 How natural are the initial conditions of our universe?

Figure 2.6: Not enough conformal time has elapsed since the singularity for dierent patches of the CMB on the surface of last scattering to have been in causal contact with each other. This can be seen from the fact that their light cones do not overlap.

Because dierent patches of the CMB were space-like separated before decoupling, the fact that they all have the same temperature seems like a remarkable coincidence. Of course, we could ne-tune every patch of the CMB to have the same temperature, but such a large degree of ne-tuning is usually undesirable and not considered a solution. This issue is known as the horizon problem. To resolve this problem, we would like a mechanism that provides more conformal time between the singularity and the surface of last scattering, such that dierent patches of the CMB have had the time to be in causal contact with each other.

2.3.2 The atness problem

In addition to the homogeneity and isotropy of the universe, we also observe it to be excep-tionally at. Following the previous discussion, we again examine how natural the initial conditions of such a universe are. Problematically, any initial deviation from atness, will grow under the inuence of gravity, such that an arbitrary initial state will evolve into a universe with large curvature, very dierent from our own. This can be seen from the rst Friedmann equation (2.6). H2 = ρ 3M2 p − k a2 (2.21)

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During the largest part of the history of the universe, the energy density was dominated by radiation and matter and the density respectively scaled as ρrad ∼ a−4 and ρmat∼ a−3.

Furthermore, if we interpret curvature as a uid, it scales as ρcurv ∼ a−2. Schematically,

we can therefore write

H2 = ρi,rad a4 + ρi,mat a3 + ρi,curv a2 , (2.22)

where the subscript i denotes an initial value. Since curvature only scales with a−2, it will

dominate the energy density of the universe for large a(t), unless we ne-tune the initial curvature of the universe to be extremely close to zero. Thus, again we have to deal with a ne-tuning problem, which is known as the atness problem.

2.4 Addressing the initial conditions of the universe with

ination

The issues raised in the previous section require a mechanism that explains the initial conditions of our universe, as we do not consider ne-tuning a solution. Of course, some people might take the point of view that the issues we treated are no problems at all, but simply a reection of our ignorance about the physics that is relevant at the energy scale just after the singularity. Indeed, to properly describe this period, a theory of quantum gravity is needed, of which we have limited knowledge. Nevertheless, we will argue that this is not a very reasonable point of view to take.

Firstly, hoping that some unknown property of quantum gravity would take care of the initial conditions of the universe seems rather naive. Because we do not have a rm grasp on quantum gravity, it seems not very convincing to try to use it to address the raised issues. For this reason, any explicit mechanism that addresses the initial conditions is preferable. Secondly, the conventional mechanism used to resolve the horizon and atness problem (ination) not only addresses the initial conditions, it also makes predictions. By quantizing the theory of ination, a spectrum of perturbations from elds present in the early universe is obtained (see section 2.6). These perturbations were also present at the time that the photons in the early universe did not yet decouple. In particular density perturbations (originating from the inaton) will lead to small temperature anisotropies in the CMB. On top of that, the density perturbations also plant the seeds for the large scale structure. These predictions have been conrmed, which are very dicult to explain without ination. For these two reasons, we will take ination to be the best solution to address the initial conditions of the universe.

As we mentioned, in order to solve the horizon problem, additional conformal time be-tween the singularity and the surface of last scattering is needed. This can be achieved by introducing a period before the hot Big Bang phase of the universe where RH was

decreasing, such that (2.20) receives a dominant contribution from early times. In this way, enough conformal time will have elapsed for the light cones of dierent patches of the CMB to have overlapped in the past, see gure 2.7.

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2.4 Addressing the initial conditions of the universe with ination

Figure 2.7: By introducing a period between the singularity and the hot big bang phase of the universe where RH was decreasing, additional conformal time is generated. Now,

dierent patches in the CMB have been in causal contact with each other. A decreasing RH implies the following condition.

˙ RH = d dt  1 aH  = −1 a " 1 + ˙ H H2 # < 0 (2.23)

This can be captured by the Hubble slow-roll parameter ˜. ˜

 ≡ −H˙

H2 < 1 (2.24)

Rewriting the second Friedmann equation (2.7) using the denition of ˙H as ˙

H + H2 = − 1 6M2

p

(ρ + 3P ), (2.25)

allows us to combine this expression with the rst Friedmann equation (2.6) in at space. ˙ H = −3H 2 2  1 + P ρ  . (2.26)

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Thus, a decreasing comoving Hubble sphere implies ˜  = −H˙ H2 = 3 2  1 + P ρ  < 1. (2.27)

This leads to the following equation of state. P < −ρ

3 ⇐⇒ w < − 1

3 (2.28)

If we now remember that curvature scales as ρcurv ∼ a−2, which corresponds to w = −13,

requiring w < −1

3 also solves the atness problem. That is, if a uid with an equation

of state w < −1

3 dominates the energy density, curvature eects are subleading and the

universe is driven towards atness. In the limit that ˙H → 0, we obtain an equation of state with w = −1, which leads to a constant energy density. The rst Friedmann equation (2.6) for a spatially at universe (k = 0) is then given by

˙a a = r ρ 3M2 p , (2.29)

which has the solution

a(t) ∼ e

r

ρ 3M 2pt

= eHt. (2.30)

Thus, we see that ination corresponds to exponential expansion in the limit that ˙H → 0. It is necessary that H evolves slowly for ination to end, which corresponds to quasi-exponential expansion. The amount of ination elapsed is measured by the number of efolds N, which is dened as the number of Hubble times ination lasts2.

N =

tend

Z

tbegin

dt0 H(t0) (2.31)

2.5 How to drive ination?

Now that it is clear how ination can properly address the initial conditions of the universe, we need to know how we can drive a period of ination. In the previous section, we observed that if a uid with an equation of state of w < −1

3 dominates the energy density,

RH will decrease, leading to a period of ination. An example of such a uid is a scalar

eld with a potential energy that dominates over its kinetic energy. The action of a scalar eld, minimally coupled to gravity, is given by

S = Z d4x√−g M2 p 2 R + 1 2∂µφ∂νφg µν − V (φ)  . (2.32)

2Measurements of the CMB only give access to the last 50-60 efolds of ination, so tbegin has a value that is accessible to the CMB.

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2.5 How to drive ination? The equation of motion of this scalar eld in a FRW spacetime is given by

¨

φ + 3H ˙φ + V0(φ) = 0. (2.33) Remember that we should satisfy

˜

 = −H˙

H2 < 1. (2.34)

Combining (2.33) with the rst Friedmann equation (2.6) results in 3Mp2H2 = 1

2 ˙

φ2+ V (φ). (2.35)

Using this expression, we can rewrite (2.34) as ˜  = −H˙ H2 = 1 2φ˙ 2 M2 pH2 < 1, (2.36)

which is satised when

V (φ)  1 2 ˙ φ2, (2.37) such that 3Mp2H2 ' V (φ). (2.38) Thus, a scalar eld with a potential energy that dominates over its kinetic energy can lead to a period of ination.

Furthermore, to allow for a prolonged period of ination, the kinetic energy has to remain small during some time, which requires the acceleration ¨φ to be small with respect to the Hubble friction 3H ˙φ. By neglecting the Hubble friction in the equation of motion, ˜ can be written as  = M 2 p 2  V0 V 2  1. (2.39)

We refer to  as the potential slow-roll parameter (note that ˜ = ). Neglecting the acceleration in the equation of motion, requiring a prolonged period of ination can be captured by a second Hubble slow-roll parameter

˜ η ≡ −1 2 ¨ H ˙ HH  1, (2.40)

which can also be equivalently captured by a potential slow-roll parameter (˜η ' η − ). |η| ≡ M2

p

|V00|

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Thus, when some potential V (φ) is specied, we can check if this potential can sustain ination by calculating the slow-roll parameters  and η. Ination occurs as long as these parameters are small.

Finally, at the end of ination, a smooth transition to the regular hot Big Bang phase of the universe must be made. At this phase, known as reheating, the energy density of the inaton needs to be transferred to standard model particles that are heated to a sucient temperature to start the hot Big Bang phase of the universe. Reheating is a rather model-dependent process and beyond the scope of this thesis. Therefore, will not treat reheating, but refer the interested reader to [9].

2.6 Quantum uctuations during ination

The previous description of ination was executed completely classically. It is however necessary to take quantum eects into account, as ination couples the smallest to the largest scales. Furthermore, because we observe deviations from homogeneity and isotropy in the universe, we also want ination to provide the perturbations that are responsible for generating these deviations. Quantizing ination leads to spectra of perturbations. The average of the perturbations is zero, but the variance (the average of the squared ampli-tude) is non-zero. This leads to observational signatures of the primordial perturbations, generated by ination.

Ination involves a metric and a scalar eld, so we need to consider perturbations to both of these. The metric has both scalar and tensor perturbations, the inaton only has scalar perturbations. In this section, we will show how these spectra can be obtained and what their observational consequences are. We will only describe the most important aspects and refer the reader to [6, 10] for a more in depth-discussion.

2.6.1 Power spectrum of perturbations

Here, we will consider tensor perturbations to the metric, as computing the scalar pertur-bations is a bit more involved. After that, we comment on how the computation can be performed for the scalar perturbations, and show its result.

When expanding the metric in small uctuations two functions (h+ and h×) that

cor-respond to a dierent polarization are obtained. These functions describe the tensor perturbations and obey the following equation of motion.

¨

h + 2H ˙h + k2h = 0 (2.42) When quantizing h, we obtain the following expansion in terms of creation and annihilation operators [10]. ˆ h(~k, τ ) = √ 2 aMp h v(k, τ )ˆa~k+ ¯v(k, τ )ˆa † ~k i (2.43)

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2.6 Quantum uctuations during ination Here, the hat indicates that we are dealing with operators and the bar indicates complex conjugation. When we use the fact that H evolves slowly, the v's satisfy the following dierential equation. ¨ v +  k2− 2 τ2  v = 0 (2.44)

Now, the variance is given by

hˆh†(~k, τ )ˆh(~k0, τ )i = 2 a2M2 p |v(~k, τ )|2(2π)3δ(3)(~k − ~k0 ) ≡ (2π)3Ph(k)δ(3)(~k − ~k0). (2.45) Here, Ph(k) is referred to as the power spectrum of tensor perturbations. To determine

the power spectrum, we need to know |v(~k, τ)|2, for which we need to solve (2.44). We will

solve this equation for two cases. First, we solve it for the case when the wavelength of the perturbation is inside the horizon. After that, we consider the case when the wavelength of the perturbation is longer than the horizon. The general solution of (2.44) is

v = e −ikτ √ 2k  1 − i kτ  . (2.46)

Inside the horizon, k|τ|  1, such that we can neglect the i/kτ term. If we directly took this limit in (2.44), we would have obtained the equation of motion of a simple harmonic oscillator. The solution becomes

v = e

−ikτ

2k. (2.47)

Comparing this with (2.45), we see that the amplitude of the power spectrum decreases. Therefore, if this would remain the case during ination, we would not be able to see any eect of the perturbations.

However, because the comoving Hubble sphere shrinks, something remarkable occurs. When the perturbation exits the horizon, k|τ| < 1, such that the solution becomes

v = e −ikτ √ 2k −i kτ. (2.48)

Comparing this with (2.45) and using τ ' −1/(aH), we see that outside of the horizon, the power spectrum becomes constant! Taking both polarizations into account, this xes the tensor power spectrum.

Ph(k) = 4H2 ? M2 pk3 (2.49) Here, the ? indicates that a quantity should be evaluated at horizon crossing. It is conve-nient to also introduce a dimensionless power spectrum, dened as

∆2h ≡ k 3 2π2Ph(k) = 2H2 ? π2M2 p . (2.50)

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Deriving the power spectrum for scalar perturbations is a bit more involved, as we need to account for scalar perturbations of the metric, as well as the inaton. As it turns out, it is useful to work in dierent gauges to compute perturbations and dene the scalar power spectrum in terms of gauge-invariant quantities. Often, the comoving curvature perturbation R is used to parametrize the scalar perturbations. In the comoving gauge, this quantity appears in the spatial part of the metric as follows.

gij = a2e2Rδij (2.51)

Hence, we see that R generates small spatial curvature perturbations. Apart from this subtlety, the same logic as for the tensor power spectrum applies. Again, it is observed that the scalar perturbations vanish outside of the horizon. The variance of the scalar perturbations is given by

hR(~k, τ )R(~k0, τ )i = (2π)3PR(k)δ(~k − ~k0), (2.52)

and the scalar power spectrum is given by PR(k) = H2 ? 4?Mp2 1 k3. (2.53)

We can also dene a dimensionless power spectrum. ∆2R(k) ≡ k3 2π2PR(k) = H2 ? 8π2 ?Mp2 (2.54)

2.6.2 Observational signatures of cosmological perturbations

Importantly, we saw that the expressions for the dimensionless power spectrum did not depend explicitly on the wavelength of a perturbation, but only weakly via H and  on a scale (both parameters should vary slowly). Therefore, a robust prediction of ination is the fact that it generates a nearly scale-invariant spectrum of perturbations. Deviations from scale invariance can be captured by the spectral indices, which are dened as

d log(∆2 R) d log(k) ≡ ns− 1 d log(∆2 h) d log(k) ≡ nt, (2.55) where ns and nt are the spectral indices of respectively the scalar and tensor power

spec-tra. The limit ns → 1 and nt → 0 corresponds to scale invariance. The power spectra

are related to the theory by the slow-roll parameters. By using the expression of the dimensionless power spectra we obtain

ns− 1 = 2η − 6 (2.56)

nt= −2 (2.57)

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2.6 Quantum uctuations during ination where r is the tensor-to-scalar ratio that is dened as

r = ∆ 2 R(k) ∆2 h(k) . (2.59)

While it was nice that we saw that perturbations freeze outside of the horizon, this does not yet have observational consequences. Only when the comoving Hubble sphere grows during the regular hot Big Bang phase, perturbations can re-enter the horizon again, see gure 2.8.

Figure 2.8: Perturbations exit RH during ination after which they freeze until they

re-enter at a later time. Figure from [11].

In particular, it is expected that the perturbations that re-entered in the early universe, have left an imprint. Because the scalar perturbations generated small curvature pertur-bations, these resulted in small matter inhomogeneities in the early universe, which have a profound eect. The small matter inhomogeneities have grown under the inuence of gravity after the photons decoupled, which has led to the large scale structure we observe today.

Additionally, photons in the early universe could also feel the presence of these matter inhomogeneities. This resulted in temperature anisotropies in the CMB that, on large scales, can be directly attributed to primordial perturbations. Complementary, the po-larization of the CMB also gives primordial information. Local quadrupole anisotropies present before the CMB decoupled, can result in a polarization of the CMB. Scalar pertur-bations result in a curl-free polarization signal (E-modes), while the tensor perturpertur-bations can result in a divergence-free polarization signal (B-modes), see gure 2.9.

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Figure 2.9: Schematic illustration of possible polarization signals of the CMB. E-modes correspond to scalar perturbations and B-modes to tensor perturbations. Figure from [12].

Therefore, in order to put ination to the test, measurements of the CMB should reveal that the temperature anisotropies of the CMB are nearly scale-invariant. Furthermore, polarization measurements of the CMB should determine the size of the primordial tensor perturbations.

By measurements of the CMB, we only have access to the last 50-60 efolds of ination. Typically, a reference scale k?(called the pivot scale) which is accessible to the CMB is used

at which all observable quantities are evaluated. In eld space, this scale φ? corresponds

to the point after which ination lasted 50-60 of efolds given by

N? = φ? Z φend dφ Mp 1 √ 2. (2.60)

While the scalar power spectrum has been measured via the temperature uctuations of the CMB, there has not yet been a measurement of the tensor power spectrum. Consistent with ination, the spectrum of scalar perturbations is indeed nearly scale-invariant. Indeed, the most recent measurements [13] have measured ns and put a bound on r 3.

ns = 0.968 ± 0.006 (2.61)

r < 0.11, (2.62)

in agreement with ination. Now, in order to compare theoretical predictions with obser-vations, we have to calculate ns and r for particular potentials.

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2.6 Quantum uctuations during ination

2.6.3 A simple example: V (φ) =

1 2

m

2

φ

2

We will illustrate how inationary predictions are made by considering the following simple potential.

V (φ) = 1 2m

2φ2 (2.63)

The slow-roll parameters for (2.63) are given by  = η = 2M

2 p

φ2 , (2.64)

and ination is possible as long as φ >√2Mp, see gure 2.10.

ϵ =1

ϕ V(ϕ)

Figure 2.10: A scalar eld rolls slowly down its potential V (φ) = 1 2m

2φ2 while driving

ination as long as , η < 1. The shaded area corresponds to the region where ination is possible.

A reasonable estimate of the number of efolds required for ination to address the initial conditions of the universe is N? = 60 [14]. From (2.60), we then nd that φ? = 16Mp.

As the end of ination (which corresponds to  = 1) occurs at φend =

2Mp, we observe

an interesting property of this model; the eld displacement ∆φ during ination is super-planckian. Models which exhibit this property are known as large-eld ination models and have some interesting properties. Most importantly, they are intimately related to fundamental physics, a point which we will come back to in section 2.7. The spectral index and the tensor-to-scalar ratio of this model are given by (2.56) and (2.58).

ns= 0.97

r = 0.13, (2.65)

where we used φ? = 16Mp. Unfortunately the bound on r by the Planck 2015 data is in

disagreement with this particular model, but it shows how computing observables from an inationary model can be performed. An overview of some popular ination models compared with Planck 2015 data is given in gure 2.11.

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Figure 2.11: Planck 2015 constraints in the ns-r plane on various inationary models.

Figure from [15].

2.7 The energy scale and UV-sensitivity of ination

Using the tools developed in the previous sections, it is now possible to connect theoretical predictions of inationary models to observations. However, this leaves a large parameter space with models consistent with observations. In particular, the energy scale of ination, which is given by

Einf lation = V1/4= (3H2Mp2)1/4, (2.66)

is not yet well-constrained by measurements, because in the scalar power spectrum the value of H during ination is obscured by , see (2.54). In contrast, a measurement of the tensor power spectrum does establish the scale of ination, as the tensor power spectrum directly depends on H, see (2.50). In terms of r, the energy scale is given by [14]

Einf lation = (3H2Mp2)

1/4= 8 × 10−3 r

0.1 1/4

Mp. (2.67)

The upper bound on r (2.62) therefore gives an upper bound on the scale of ination Emax = 10−2Mp. (2.68)

As was noted by Lyth [16], a detection of primordial gravitational waves would not only pin down the scale of ination to be high (Einf lation' Emax), it also implies that ination

was superplanckian. This can be seen from the bound  ∆φ

Mp

2 & r

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2.7 The energy scale and UV-sensitivity of ination which is known as the Lyth bound. This motivates us to consider models of ination that predict an observable gravitational wave signal for two reasons. Firstly, (2.67) implies that such a model is tied to interesting fundamental physics, as ination occurred at a scale where new physics is expected to be relevant. Secondly, upcoming polarization measurements of the CMB will constrain r further or measure its value, allowing such models to be falsiable, which promotes a healthy way of doing physics. Problematically, (2.69) implies that such a model should be superplanckian, which makes model building rather dicult, as we will see.

Nevertheless, quantum corrections are not exclusively an issue for large-eld ination. All models of slow-roll ination are sensitive to high-energy physics, but large-eld eld ination models suer in a more dramatic way4. This sensitivity to high-energy physics

is known as the UV-sensitivity of ination, which makes constructing large-eld ination models a challenge.

For example, consider quantum corrections to an eective (only valid up to a cuto scale Λ) theory of ination. In the absence of any symmetry, the mass of the inaton will receive a correction of the order [14]

δm2 ∼ Λ2. (2.70)

Because Λ > H, the slow-roll parameter η will receive a large correction. δη = Mp2δV 00 V = δm 2 3H2 ∼ Λ 2 H2 > 1. (2.71)

This is known as the η-problem and needs to be addressed in every model of slow-roll ination. One way of preventing large corrections to the inaton mass, is by appealing to a symmetry. Specically, imposing a shift symmetry

φ → φ + constant (2.72)

forbids any term other than the kinetic term in the action. The potential may only weakly break this shift-symmetry, such that the inaton obtains a small mass. Still, one should be careful as physics above the cuto scale may not respect this symmetry. In particular, it is well known that any consistent theory of quantum gravity should not contain global symmetries (see for example [17]), while the low energy eective theory may exhibit this symmetry. Therefore, high-energy physics might still induce corrections to the inaton mass that will lead to the η-problem. Determining whether such corrections will be large is a subtle question that can only be properly answered by knowing the theory which completes the eective theory above the cuto scale (the UV-completion).

4From the perspective of a model builder, this pessimistic view is understandable. In contrast, a more optimistic theorist may view this as an opportunity to use ination to probe a regime of unknown physics.

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The situation is even worse in large-eld ination. When the eld displacement during ination (∆φ) is superplanckian, all operators that were previously suppressed by Mp

are now no longer suppressed and contribute to the potential. An innite amount of operators should be ne-tuned to keep the atness of the potential, which is clearly not natural. From an eective eld theory perspective the atness of the potential is not sustainable. Therefore, to properly construct a large-eld model, it is necessary to know if an eective theory admits a UV-completion in which the size of the corrections can be checked.

These issues have inspired theorists to work in high-energy theories such as string theory and theories that are known to admit a UV-completion (such as supergravity) to construct models of ination. Summarizing, the UV-sensitivity of ination is both a blessing and a curse. While it makes explicit constructions dicult, it also brings about the hope that one day ination can be used to probe physics at an energy scale where we expect new physics to be relevant.

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3 The eective eld theory of ination

The eective eld theory of ination is a powerful principle, that allows for the systematic study of ination in an eective and model independent way. This means that, in a certain energy regime, only the relevant degrees of freedom are studied, while degrees of freedom that are important at a dierent scale will decouple. An EFT description has many benets. For example, an eective eld theory only incorporates degrees of freedom up to a cuto scale, which oers a simplication with respect to the full theory. Moreover, an EFT description that assumes a particular set of symmetries encompasses all possible theories with the same symmetry structure.

3.1 Constructing an eective action

There are two ways of obtaining an eective action of a theory that describes the physics at a certain energy regime. When the full theory (SU V), which is valid up to high energy,

is known, heavy degrees of freedom1 above the cuto scale Λ can be integrated out.

Integrating out the heavy degrees of freedom results in an eective action that is only valid up to the cuto and consists of the low-energy action (SIR) and higher dimensional

corrections. The eective action in d spacetime dimensions can then be schematically written as Sef f = SIR+ X i Z ddx ci Oni Λni−d. (3.1)

Here, Oni denotes an operator of dimension [mass]

ni. This gives an ordering principle of

the dierent operators in the eective action.

• Operators with ni > d are called irrelevant; they are important in the UV (high

energy) and can be neglected in the IR (low energy).

• Operators with ni < d are called relevant; they are unimportant in the UV and

become important in the IR.

• Operators with ni = dare called marginal.

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By ordering the eective action according to this criteria, the dominant contributions can be identied.

Often, SU V is not known and the procedure of integrating out heavy modes cannot be

executed. This brings us to the second way of obtaining the eective action. As we saw, ignorance about UV-physics will be parametrized by irrelevant operators. By making assumptions about the symmetries of the UV-theory and allowing all operators that are consistent with these symmetries, the same eective action can be obtained. This allows us to construct an eective action, even though we do not have information about the UV-theory. An eective theory is said to be UV-completed when there exists a SU V that

complements SIR above Λ. An example that shows that these dierent ways of obtaining

the eective action lead to the same result is given in chapter 2 of [14].

3.2 Symmetries of ination

In order to nd the eective action that describes ination, we have to nd the symmetries of ination. In chapter 2, we saw that ination corresponds to a period that satises

˜  = −

˙ H

H2 < 1. (3.2)

In the limit that ˙H → 0, the inationary background can be described by a de Sitter space; a maximally symmetric vacuum solution of the Einstein equations with a positive cosmological constant. However, ination cannot be described by a pure de Sitter space, as ination does not end in a de Sitter space. Said dierently, the isometries of de Sitter space contain time-translations, which do not allow ination to end because this would introduce an explicit time dependence. This implies that we can describe ination as a quasi-de Sitter phase where time dieomorphisms are spontaneously broken by a time-dependent scalar φ(t) (the inaton). This scalar acts as a clock that measures the amount of ination elapsed. Perturbations of the inaton around the inating background transform the following way under time-dieomorphisms (given by x0 → x0+ ξ0(x0, ~x)).

δφ(x0, ~x) → δφ(x0, ~x) + ˙φ(x0)ξ0(x0, ~x) (3.3) Because the inaton spontaneously breaks time dieomorphisms, the theory will contain an associated Goldstone boson [18] that we will denote by π, which parametrizes δφ. We can use the gauge freedom of choosing our coordinates in a way we prefer to let x0 coincide

with the slicing of spacetime introduced by φ(t). This is known as unitary gauge and has the feature that the scalar perturbations (and thus the Goldstone boson) vanish.

δφ = π = 0 (3.4)

Of course, the Goldstone boson degree of freedom does not disappear, but can be described in terms of an additional degree of freedom of the graviton. Hence, the choice of unitary gauge is convenient, because the theory of cosmological perturbations can be described

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3.3 The eective action of ination only in terms perturbations of the metric. Furthermore, this constrains the eective action, because only operators that are invariant under spatial dieomorphisms.

xi → xi+ ξi(t, x) (3.5)

are allowed.

3.3 The eective action of ination

Working in unitary gauge, we now want to write down the most general action consistent with spatial dieomorphism invariance. This was rst done by Cheung et al. in [19]. The eective action is given by

S = Z d4x√−g 1 2M 2 pR − c(t)g 00− Λ(t) +1 2M2(t) 4(δg00)2 + 1 6M3(t) 4(δg00)3 M¯1(t)3 2 (δg 00)δKµ µ − M¯2(t) 2 2 δ(K µ µ) 2M¯3(t)2 2 δK µ νδK ν µ+ . . .  . (3.6)

Here, Kµν is the extrinsic curvature: the curvature of constant time slices. Furthermore, we

expanded the eective action in perturbations of the metric and derivatives. All quantities are written in terms of perturbations, while the unperturbed quantities are absorbed in

−g−c(t)g00− Λ(t) . (3.7)

For example

g00 = g00F RW + δg00= −1 + δg00, (3.8) where the factor −1 can be absorbed in Λ(t). In Appendix A, we show that allowing all operators consistent with spatial dieomorphism invariance will lead to (3.6).

The coecients c(t), Λ(t), M2(t), M3(t), ¯M1(t), ¯M2(t) and ¯M3(t) all have a generic time

dependence. Since ination only breaks time dieomorphism invariance weakly (the slow-roll parameters that break the time dieomorphism invariance have to be small), we expect that it is also natural for all other time-dependent parameters to not vary signicantly per Hubble time.

3.3.1 Restoring full dieomorphism invariance

Going to unitary gauge allowed us to write down the most general action compatible with spatial dieomorphism invariance in an expansion in perturbations and derivatives. It is however convenient to explicitly reintroduce the Goldstone boson, which will non-linearly

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restore the full dieomorphism invariance. This reintroduction can be realized by using the so-called Stückelberg trick, see Appendix A. The advantage of this method is that, at suciently high energy, the Goldstone boson decouples from the metric perturbations. This allows us to describe the EFT of ination only in terms of the Goldstone boson, which simplies the action. This statement is formalized in Goldstone's equivalence theorem [18].

The Goldstone boson can be made explicit again by performing a broken time dieomor-phism.

t → ˜t = t + ξ0(x) (3.9) We then substitute

ξ0(x(˜x)) → −˜π(˜x). (3.10) Under the time dieomorphism, the 00 component of the metric transforms as

g00(x) → ˜g00(˜x(x)) = ∂ ˜x 0(x) ∂xµ ∂ ˜x0(x) ∂xν g µν(x) (3.11)

and the the determinant of the metric transforms as g → ˜g(˜x(x)) ∂ ˜x(x)

∂x 2

. (3.12)

Using these transformation rules one can see that an action of the form Z d4x√−gA(t) + B(t)g00(x) , (3.13) transforms as Z d4xp−˜g(˜x(x)) ∂x ∂ ˜x  A(t) + B(t) ∂x 0 ∂ ˜xµ(x) ∂x0 ∂ ˜xν(x)g˜ µν (˜x(x))  , (3.14) which can be simplied by changing the integration variable to ˜x, such that the Jacobian exactly cancels the |∂x/∂˜x| term.

Z d4xp−˜g(˜x)  A(˜t − ξ0(x(˜x))) + B(˜t − ξ0(x(˜x)))∂(˜t − ξ 0(x(˜x))) ∂ ˜xµ(x) ∂(˜t − ξ0(x(˜x))) ∂ ˜xν(x) ˜g µνx)  (3.15) Next, introduce ˜π, by using (3.10) and drop all tildes to obtain

Z d4xp−g(x)  A(t + π(x)) + B(t + π(x))∂(t + π(x)) ∂xµ(x) ∂(t + π(x)) ∂xν(x) g µν(x)  . (3.16) By evaluating the second term

∂(t + π(x)) ∂xµ(x) ∂(t + π(x)) ∂xν(x) g µν(x) =(1 + ˙π(x))2g00+ 2(1 + ˙π(x))(∂ iπ(x))g0i + (∂iπ(x))(∂jπ(x))gij, (3.17)

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3.3 The eective action of ination we can rewrite (3.6) in terms of the Goldstone boson as

S = Z d4x√−g 1 2M 2 pR − Λ(t + π) − c(t + π)(1 + ˙π)2g00+ 2(1 + ˙π)(∂ iπ)g0i+ (∂iπ)(∂jπ)gij  + 1 2!M2(t + π) 4(1 + ˙π)2g00+ 2(1 + ˙π)(∂ iπ)g0i+ (∂iπ)(∂jπ)gij + 1 2 +1 3!M3(t + π) 4(1 + ˙π)2g00+ 2(1 + ˙π)(∂ iπ)g0i+ (∂iπ)(∂jπ)gij + 1 3 + ....  , (3.18) where the x dependence of π was omitted. Summarizing, full dieomorphism invariance is now restored, because π transforms as

π(x) → π(x) − ξ0(x) (3.19) under t → t + ξ0(x). Before we show how the action simplies at suciently high energy,

we rst x the coecients c(t) and Λ(t) by using the equations of motion.

3.3.2 Equations of motion

The leading terms in (3.6) are given by S = Z d4x√−g 1 2M 2 pR − Λ(t) − c(t)g 00  , (3.20)

which determine the unperturbed background evolution. Because we know that the un-perturbed evolution is described by a FRW metric, we are able to x the coecients Λ and cby comparing the equations of motion of (3.20) to the Friedmann equation. Additionally, this cancels any tadpoles (terms linear in π that result in a shift of the vacuum expecta-tion value), a procedure known as tadpole cancellaexpecta-tion [18]. We can nd the equaexpecta-tions of motion of (3.20) by deriving the stress-energy tensor and plugging it into the Einstein equations. The stress energy tensor is given by2

Tµν ≡ − 2 √ −g δS δgµν (3.21)

The variation of (3.20) is given by δS = Z d4x−δ√−g Λ − δ√−g cg00−g c δ0 µδ 0 νδg µν , (3.22)

which can be rewritten using the identity δ√−g = 1

2 √

−ggµνδgµν. (3.23)

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We then obtain δS = Z d4x√−g 1 2gµνΛ + 1 2gµνc g 00− c δ0 µδ0ν  δgµν, (3.24) so the stress-energy tensor is given by

Tµν = −gµν Λ + c g00 + 2c δµ0δ 0

ν. (3.25)

The Einstein equations can be written as Gµν =

1 M2

p

Tµν. (3.26)

We can now determine Λ and c by calculating the components of the Einstein tensor of a at FRW metric and plugging it into the left hand side of the Einstein equations and the derived stress-energy tensor in the right hand side. The relevant components of the Einstein tensor are

Gµν = ( G00 = 3H2 Gµ µ = −12H2− 6 ˙H. (3.27) Comparing this with the stress-energy tensor results in

3H2(t) = 1 M2 p [Λ + c] (3.28) −12H2− ˙H = 1 3M2 p [Λ − 2c] , (3.29)

where we used (3.28) to derive (3.29). Solving these equations for Λ and c results in c = −Mp2H˙ and Λ = Mp2h3H2 + ˙Hi (3.30)

3.3.3 Decoupling limit

Now that we have xed the background evolution of the eective action, we continue to see how it simplies in a certain energy regime. In a general gauge theory with a spontaneously broken symmetry, Goldstones equivalence theorem tells us that, at high energy, the interaction of a longitudinally polarized gauge boson can be described in terms of the interaction of the Goldstone boson, see Appendix A.

In our case, the gauge boson is analogous to the graviton. Similar to the gauge theory example, we can neglect metric uctuations at suciently high energy. Essentially, the metric uctuations decouple from the theory, which allows us to describe the entire dynam-ics of the theory in terms of the Goldstone boson. Following the gauge theory example,

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3.3 The eective action of ination we can estimate the energy at which this simplication occurs by canonically normalizing π. From (3.18) we see that the kinetic term of π is given by

Mp2H∂˙ µπ∂νπgµν. (3.31)

Thus, the canonically normalized Goldstone boson is dened as

πc≡ MpH˙1/2π. (3.32)

The kinetic term for δg00 is

Mp2(∂µδg00)(∂νδg00)gµν, (3.33)

such that

δgc00≡ Mpg00. (3.34)

If, in analogy with the gauge theory, we then identify MpH˙1/2 =

m

g , (3.35)

where m = ˙H1/2 is the `mass' of the Goldstone boson and g = 1/M

p the coupling to

gravity. By this identication, the Goldstone self-interaction gets strongly coupled at a scale

Λ2 ∼ MpH˙1/2. (3.36)

Thus, the energy scale at which metric uctuations decouple is given by

E  m = ˙H1/2. (3.37)

Hence, the energy scale at which metric uctuations decouple and we can still trust the theory is given by

˙

H1/2  E  M1/2 p H˙

1/4. (3.38)

As a consistency check, we can look at the leading order mixing term between δg00 and π

for simple slow-roll ination (M2 = M3 = 0).

Mp2H ˙πδg˙ 00 = ˙H1/2˙πcδgc00, (3.39)

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3.3.4 The simplest slow-roll eective action

When taking the decoupling limit, the eective action simplies greatly. Take for example the eective action describing the simplest3 scenario of slow-roll ination (M

2 = M3 = 0).

By expanding (3.18) and taking the decoupling limit, we obtain S = Z d4x√−g 1 2M 2 pR − Λ − ˙Λπ −(c + ˙cπ)  −(1 + ˙π)2+∂iπ∂iπ a2  . (3.40)

By expanding (3.40) up to two π's or derivatives we get S = Z d4x√−g 1 2M 2 p − Λ − ˙Λπ + c  ˙π2− ∂iπ∂ iπ a2  + 2c ˙π + ˙cπ  . (3.41) Now, we apply integration by parts to the the term containing ˙π (only with respect to time). 2 Z d4x√−g c ˙π = −2 Z d4x√−g [3Hcπ + ˙cπ] (3.42) Plugging this into (3.41) gives

S = Z d4x√−g 1 2M 2 p − Λ − ˙Λπ + c  ˙π2 −∂iπ∂ iπ a2  − ˙cπ − 6Hcπ  . (3.43) Inserting the derived expressions for Λ and c into (3.43) cancels the terms linear in π, as we promised, and yields the following eective action for the Goldstone boson.

Sπ = Z d4x√−g  Mp2H˙  − ˙π2+∂iπ∂iπ a2  (3.44) This simple action describes all possible simple slow-roll models of ination to leading order.

3.3.5 Moving beyond the simple slow-roll scenario

The true power of the EFT of ination becomes visible when we go beyond the simplest slow-roll scenario by turning on other operators (setting M2 and M3 non-zero). This allows

us to describe deviations from the simple slow-roll scenario. For example, if M2 is large,

the leading order kinetic term of π is given by

M24∂µπ∂νπgµν. (3.45)

3With simple we mean a minimal coupling to gravity, canonical kinetic term and a potential that satises the slow-roll conditions.

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3.3 The eective action of ination Thus, the canonically normalized Goldstone boson is dened as

πc≡ M22π, (3.46)

from which we can identify m = M2

2/Mp. The energy scale at which the theory becomes

strongly coupled becomes

Λ2 ∼ M2

2. (3.47)

Thus, in the same manner as for the simple slow-roll scenario, we see that the energy scale at which metric uctuations decouple while the theory is still valid is

M22 Mp

 E  M2. (3.48)

Because the leading order mixing term is given by M24˙πδg00 = M

2 2

Mp

˙πcδgc00, (3.49)

this is indeed consistent. This results (up to cubic order in π and derivatives) in Sπ = Z d4x√−g  Mp2H˙  ˙π2− ∂iπ∂ iπ a2  +2M24  ˙π2+ ˙π3− ˙π∂iπ∂ iπ a2  − 4 3M 4 3˙π 3  . (3.50)

The fact that M2 is turned on has some interesting consequences. For example, it results

in a dierent coecient for the time and spatial kinetic term of π (2M24− M2 PlH) ˙π˙ 2 and M2 PlH˙ ∂iπ∂iπ a2 , (3.51)

which leads to a non-trivial speed of sound (cs6= 1) of π.

c−2s ≡ 1 − 2M 4 2 M2 PlH˙ . (3.52)

Rewriting (3.50) in terms of cs yields

Sπ = Z d4x " MPl2H˙ c2 s ( ˙π2+ c2s∂iπ∂ iπ a2 ) + M 2 pH(1 − c˙ −2 s )( ˙π3− ˙π ∂iπ∂iπ a2 ) −4 3M 4 3˙π 3  . (3.53)

Interestingly, a small speed of sound can enhance non-gaussianties (three point correlation functions, known in cosmology as the bispectrum) due to the factor c−2

s in front of the

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3.3.6 Observables in the power spectrum

A careful reader might be worried that the energy scale at which the simplication of the eective action occurred (the decoupling limit) is not the energy scale relevant for us, because the observations we do today are at much lower energies than the scale at which decoupling takes place. Luckily, we saw in chapter 2 that perturbations that exit the comoving Hubble sphere are conserved outside of the horizon and we therefore have to evaluate all inationary observables at the scale at which horizon crossing took place, which we denoted with a star (?). Therefore, the relevant scale for ination is V1/4

? , which

is safely above the decoupling scale, see (2.66).

Because the Goldstone boson is directly related to the comoving curvature pertubation R (see chapter 2) via the relation [20]

R = −Hπ, (3.54)

we can relate correlation functions of the Goldstone boson to correlation functions of the comoving curvature perturbation. For the simple slow-roll scenario with the Goldstone action Sπ = Z d4x√−g  Mp2H˙  − ˙π2+∂iπ∂ iπ a2  , (3.55)

this of course results in the same power spectrum we saw earlier, as it should. ∆2R(k) ≡ k 3 2π2PR(k) = H2 8π2M2 p . (3.56)

Moving away from simple slow-roll ination by turning on the parameter M2 in (3.18),

results in a non-trivial speed of sound (cs 6= 1) of π. In addition to the non-zero

three-point correlation functions that are generated when M2 6= 0, the power spectrum of R

also undergoes a change. By only looking at the quadratic terms in π (which are the ones needed to calculate the power spectrum), we obtain from (3.50) the following action

Sπ = Z d4x " M2 PlH˙ c2 s  ˙π2+ c2s∂iπ∂ iπ a2 # (3.57) Calculating the power spectrum of R from this action shows that it depends on the speed of sound [20]. ∆2R= H 2 8π2c sMp2 . (3.58)

Using the denitions of the spectral tilt (2.55), one can see that this parameter also undergoes a change with respect to (2.56).

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3.4 Why should we go beyond the eective eld theory of ination? where we dened a new parameter s which measures the change of the speed of sound.

s ≡ c˙s Hcs

(3.60) The Goldstone actions (3.55) and (3.57) do not generate a non-trivial speed of sound for the tensor perturbations, thus the tensor power spectrum is the same as our original result (2.50). ∆2h ≡ k 3 2π2Ph(k) = 2H2 π2M2 p (3.61) Hence, the expression for the tensor spectral index (2.57) is unchanged.

3.4 Why should we go beyond the eective eld theory

of ination?

The EFT of ination has many advantages as compared to the usual formulation of ina-tion in terms of φ. It can incorporate all models of single-eld inaina-tion in a simple and systematic way and turning on dierent operators controls deviations from the simplest slow-roll scenario. Moreover, the size of these operators can be constrained by mea-surements. On top of that, one-loop corrections to correlation functions of cosmological perturbations can be straightforwardly taken into account, as the symmetry structure of the action only allows for a nite number of terms. This implies that renormalization will not generate new operators. Conversely, the normal action of φ contains an innite number of operators. Taking loop corrections into account is therefore possible.

Despite the benets of the EFT of ination, it also has shortcomings. As we discussed in section 2.7, ination and in particular large-eld ination is sensitive to UV-physics. specically, quantum gravity need not to respect the symmetries of the eective theory. This would brutally violate the simplicity of the Goldstone action that we derived, because previously forbidden corrections now have to be taken into account.

This urges us to go beyond the eective theory and take a more direct approach in consid-ering quantum gravity eects. Because the best candidate for a theory of quantum gravity that we know of is string theory, it is of a particular interest to construct inationary mod-els in string theory. In practice, this is technically very dicult, although there has been progress over the years. In the next chapter, we will treat aspects of string theory relevant for ination and show how realistic models of ination might come about.

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