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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

Abstract

The inationary paradigm makes a compelling case for providing the initial conditionsof the universe. Despite being consistent with cosmological observations, the theoreticalfoundation 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 itsUV-sensitivity can be addressed. In particular, we show how inationary model buildingworks in the context of string theory and supergravity. Furthermore, models of inationinvolving axions are treated and we comment on their validity, both from a bottom-upand top-down perspective. We show that such models are consistent from an eective eldtheory point of view, but are in tension with general properties of quantum gravity.

Acknowledgements

First of all, I like to acknowledge Jan Pieter. Jan Pieter, thank you for supervising me thepast year and introducing me to the various topics we have looked at. While I may havebeen 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, youhave learned me a lot and not once have I left your oce without feeling inspired. Yourenthusiasm is inspiring and I am looking very much forward to working with you in thecoming years.

Second of all, I would like to thank Ben for being my second reader. I have enjoyed thequestions you asked during my presentation, which allowed me to show my audience someaspects 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 ofcosmology.

Of course, I also want to thank my fellow students. Adri, thank you for being my partnerin crime as a student of Jan Pieter and the useful discussions we had. Vincent, Jonas andJorrit, thank you for being friends and fellow students the past years. Physics would notbe the same without you. Also, all other master students get a big thank you for creatingthe unique atmosphere C4.273A has.

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

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

Lars AalsmaJuly 2015

Contents

1 Introduction 1

2 Physics of ination 4

2.1 Introduction to cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . 72.3 How natural are the initial conditions of our universe? . . . . . . . . . . . . 92.4 Addressing the initial conditions of the universe with ination . . . . . . . 122.5 How to drive ination? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Quantum uctuations during ination . . . . . . . . . . . . . . . . . . . . . 162.7 The energy scale and UV-sensitivity of ination . . . . . . . . . . . . . . . 22

3 The eective eld theory of ination 25

3.1 Constructing an eective action . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Symmetries of ination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 The eective action of ination . . . . . . . . . . . . . . . . . . . . . . . . 273.4 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Eective N = 1 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Axion ination 44

5.1 Natural ination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2 Saving natural ination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.3 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 . 596.2 Supergravity formulation of a non-minimal coupling . . . . . . . . . . . . . 636.3 Cosmological attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.4 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 . . . . . . . . . 807.2 Generalization to p-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.3 Applying the Weak Gravity Conjecture to ination . . . . . . . . . . . . . 827.4 Contribution of gravitational instantons and a loophole . . . . . . . . . . . 86

iv

Contents

8 Conclusions and outlook 88

Bibliography 91

Appendix A The eective eld theory of ination 97

Appendix B Perturbative quantum gravity 101

v

Conventions

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

M−2p ≡ 8πGN = 2.4× 108 GeV.

Mostly, we will be working in 3 + 1 dimensions. If this is not the case we will denote thespacetime dimension by d. The Hubble slow-roll parameters are indicated with a tilde andare dened as

ε ≡ − H

H2and η ≡ −1

2

H

HH.

Here, the dot denotes a derivative with respect to time. The potential slow-roll parametersare dened as

ε ≡M2

p

2

(V ′(φ)

V (φ)

)and η ≡M2

p

V ′′(φ)

V (φ),

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

ε ' ε and η ' η − ε.

1 Introduction

In the 20th century, cosmological observations have led to a description of the origin andevolution of the universe, known as the hot Big Bang model. In this model, the originof the universe is described as an extremely hot and dense state that expanded in a `BigBang', 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 itsinitial conditions. When we follow the evolution of an arbitrary initial state, the resultinguniverse will not look like our own. More specically, the isotropy of the Cosmic MicrowaveBackground and the atness of the universe can not be explained. In order to obtain auniverse 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 someunderlying mechanism that creates more natural initial conditions. Alan Guth realized in1981 that a period of exponential expansion, preceding the hot Big Bang phase, will drivean arbitrary state towards homogeneity, isotropy and atness [1]. This naturally sets thestage for the evolution of the universe during the hot Big Bang phase, as we can startfrom 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 inatesduring this phase), could not be smoothly connected to the regular hot Big Bang phase ofthe universe, making this model not realistic. Nevertheless, it inspired the idea that theregular 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-exponentialexpansion could be driven by a scalar eld with a potential energy that dominates overits kinetic energy [2].

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

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

1

1 Introduction

is nowadays well-established, because all cosmological observations are consistent withit. Nevertheless, the origin of ination and the existence of a multiverse are still openquestions.

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 scalareld that drives ination (the inaton). These perturbations source small matter inho-mogeneities that are visible in the Cosmic Microwave Background as small temperatureuctuations [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 ofination in the early universe.

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

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

As we will motivate at the end of chapter 2, this UV-sensitivity can only be properly dealtwith in a UV-complete theory, i.e. a theory that complements the eective theory at highenergy. In the case of ination, this sensitivity reveals itself in terms of operators thatare suppressed by the cuto, but can become active during ination. This implies thatin order to have theoretical control over these operators, ination needs to be describedin a theory of quantum gravity. Because the best candidate for such a theory is stringtheory, this motivates us to study ination in the context of string theory. Unfortunately,string theory is rather technical and only partially understood. Therefore, simplifyingassumptions 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-energylimit, i.e. supergravity. This opens up a theoretically accessible landscape, which also isnon-trivial enough to allow for interesting dynamics. Above all, we know that supergravityis UV-completed into string theory, which makes it (in principal) possible to check if anylow-energy assumptions are valid up to high energy.

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

2

The contents of this thesis are as follows. In chapter 2-4, we give a basic introduction tocosmology, an overview of the physics of ination, its description as an eective theoryand the motivation for embedding ination into string theory. Continuing, chapters 5-7 give recent advances in particular well-motivated models, such as axion ination. Inaddition, 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 theend.

3

2 Physics of ination

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

2.1 Introduction to cosmology

The foundation of modern cosmology is the assumption that the universe is homogeneousand 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

+ r2dθ2 + r2 sin2(θ)dφ2

). (2.1)

Here, a(t) is a scale factor that measures how much the universe has expanded in a certainamount 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 FRWmetric can be rewrittenin 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 theuniverse. 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)[−dτ 2 + dx2 + dy2 + dz2

]. (2.4)

If we assume the energy content of the universe to have the stress-energy tensor of a perfectuid, which is given by

T µν = diag (ρ, P, P, P ) , (2.5)

4

2.1 Introduction to cosmology

we can derive the Friedmann equations from the Einstein equations.

H2 =ρ

3M2p

− k

a2(2.6)

a

a= − 1

6M2p

(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 theuniverse at a certain time. Furthermore, ρ is the energy density of the uids in theuniverse, P the pressure and k the same parameter as in the FRW metric. The eect ofthe value of k can be most intuitively seen by introducing the density parameter Ω, whichis dened as

Ω ≡ ρ

ρcrit= ρ

8πGN

3H2. (2.9)

The critical energy density (ρcrit) corresponds to the situation where the energy density ofthe universe is just right to sustain a at universe, while it evolves. Any small deviationof |Ω| > 1 results in a curved universe. To be more specic, Ω < 1 corresponds to negativespatial curvature and Ω > 1 to positive spatial curvature, see gure 2.1. In terms of thedensity 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, wewould also like to know how it evolves. For this, the relation between the energy densityand the scale factor is needed. By using conservation of energy, the following conservationequation 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

P = wρ. (2.13)

5


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 toan open universe. Figure from wikipedia.

The equation of state is bounded by energy conditions, which constrain w. For constantw, (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 thefollowing values1.

w =

13

(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 byradiation. By taking w = 1

3and 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 theexpansion rate of the universe.

a(t) ∝ t12 , (2.17)

1Observations tell us that dark energy has w ' −1, so values slightly larger than -1 are also allowed.

6

2.2 The Cosmic Microwave Background

This calculation can also be straightforwardly applied to other uids, which will give adierent evolution, see gure 2.2. Generally, when given an equation of state, we canderive a relation between the scale factor and the energy density. Plugging this into theFriedmann equation tells us the evolution of the universe. Of course, it is very well possiblethat multiple components contribute to the energy density of the universe, complicatingthe 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 asingle component. Secondly, if one does not want to make this assumption, the Friedmannequation 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 goodapproximation.

t

a(t)

Radiation : a(t)∼ t

Matter: a(t)∼t2

3

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 universefrom being opaque to transparent. At a redshift of z . 1100, the universe was in a hotand dense state, which consisted of photons, electrons and baryons. As photons wereconstantly Thomson scattering on electrons, their mean free path was small, causing theuniverse 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 themean free path of the photons dramatically (decoupling). As a consequence, photons werenow 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 CosmicMicrowave Background (CMB), is still measurable today as an isotropic radiation that is

7


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

visible when all foreground light is subtracted. Due to the expansion of the universe, it issignicantly redshifted with respect to its original wavelength, such that the CMB has thehighest intensity at a temperature of T0 = 2.7K. The accidental discovery of the CMB byPenzias 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 predictedthat the CMB would have the spectrum of a black body. This spectrum was rst measuredby the COBE satellite, see gure 2.4. Due to the large amount of information the CMBcarries about the early universe, the initial discovery of the CMB initiated the launch ofmore surveys, such as WMAP and Planck that measured the CMB with an even higherprecision. These surveys showed that, on top of black body spectrum, the CMB has tinytemperature 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 importantphysical relevance of these uctuations.

8

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 theoreticalprediction of a black body with a temperature of T0 = 2.7K agrees excellently with theobservations (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 dierencecorresponds 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 andspatially at on large scales, is quite curious. Starting from an arbitrary state, and letting

9


it evolve according to the Friedmann equation, would result in a dierent universe than theone we are living in. Therefore, a logical question to ask ourselves is: how natural are theinitial conditions of our universe? In this section, we will specify two problems concerningthe 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 horizondenes the maximal distance that is in causal contact with us. Any signal separated anyfurther will not be able to reach us due to the expansion of the universe. Using comovingcoordinates, this causal structure can be conveniently explored, because for ds2 = 0,dx2 = dτ 2. Thus, the maximum distance a light-like signal can have travelled between thesingularity and a particular time t is given by the conformal time elapsed.

τ =

t∫0

dt′

a(t′)=

a(t)∫a(0)

da(t′)

a(t′)2H, (2.19)

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

τ =

log(a(t))∫0

d log(t′)

a(t′)H=

log(a(t))∫0

d log(t′)RH , (2.20)

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

During radiation and matter domination (which was the dominant energy contributionduring 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 observationis problematic if we want to explain the observed isotropy of the CMB. This can be seen asfollows. 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 beforedecoupling. If this was not the case, there is no reason why dierent patches would havethe same temperature. However, the fact that the largest proper time contribution comesfrom late times, implies that there was not enough conformal time between the singularityand the surface of last scattering (the surface at redshift z = 1100 from which the CMBoriginates) for all dierent patches of the CMB to have been in causal contact with eachother, see gure 2.6.

10

2.3 How natural are the initial conditions of our universe?

Figure 2.6: Not enough conformal time has elapsed since the singularity for dierentpatches of the CMB on the surface of last scattering to have been in causal contact witheach 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 factthat 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 largedegree of ne-tuning is usually undesirable and not considered a solution. This issue isknown as the horizon problem. To resolve this problem, we would like a mechanism thatprovides 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 witheach 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 initialconditions of such a universe are. Problematically, any initial deviation from atness, willgrow under the inuence of gravity, such that an arbitrary initial state will evolve into auniverse with large curvature, very dierent from our own. This can be seen from the rstFriedmann equation (2.6).

H2 =ρ

3M2p

− k

a2(2.21)

11


During the largest part of the history of the universe, the energy density was dominatedby 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,rada4

+ρi,mata3

+ρi,curva2

, (2.22)

where the subscript i denotes an initial value. Since curvature only scales with a−2, it willdominate the energy density of the universe for large a(t), unless we ne-tune the initialcurvature of the universe to be extremely close to zero. Thus, again we have to deal witha 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 initialconditions of our universe, as we do not consider ne-tuning a solution. Of course, somepeople might take the point of view that the issues we treated are no problems at all, butsimply a reection of our ignorance about the physics that is relevant at the energy scalejust after the singularity. Indeed, to properly describe this period, a theory of quantumgravity is needed, of which we have limited knowledge. Nevertheless, we will argue thatthis is not a very reasonable point of view to take.

Firstly, hoping that some unknown property of quantum gravity would take care of theinitial conditions of the universe seems rather naive. Because we do not have a rm graspon quantum gravity, it seems not very convincing to try to use it to address the raisedissues. For this reason, any explicit mechanism that addresses the initial conditions ispreferable. Secondly, the conventional mechanism used to resolve the horizon and atnessproblem (ination) not only addresses the initial conditions, it also makes predictions. Byquantizing the theory of ination, a spectrum of perturbations from elds present in theearly universe is obtained (see section 2.6). These perturbations were also present at thetime that the photons in the early universe did not yet decouple. In particular densityperturbations (originating from the inaton) will lead to small temperature anisotropiesin the CMB. On top of that, the density perturbations also plant the seeds for the largescale structure. These predictions have been conrmed, which are very dicult to explainwithout ination. For these two reasons, we will take ination to be the best solution toaddress 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 achievedby introducing a period before the hot Big Bang phase of the universe where RH wasdecreasing, such that (2.20) receives a dominant contribution from early times. In thisway, enough conformal time will have elapsed for the light cones of dierent patches ofthe CMB to have overlapped in the past, see gure 2.7.

12

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 phaseof 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 ε.

ε ≡ − H

H2< 1 (2.24)

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

H +H2 = − 1

6M2p

(ρ+ 3P ), (2.25)

allows us to combine this expression with the rst Friedmann equation (2.6) in at space.

H = −3H2

2

(1 +

P

ρ

). (2.26)

13


Thus, a decreasing comoving Hubble sphere implies

ε = − H

H2=

3

2

(1 +

P

ρ

)< 1. (2.27)

This leads to the following equation of state.

P < −ρ3⇐⇒ w < −1

3(2.28)

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

requiring w < −13also solves the atness problem. That is, if a uid with an equation

of state w < −13dominates 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 statewith 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=

√ρ

3M2p

, (2.29)

which has the solution

a(t) ∼ e

√ρ

3M2pt

= 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 ofefolds N , which is dened as the number of Hubble times ination lasts2.

N =

tend∫tbegin

dt′ H(t′) (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, weobserved that if a uid with an equation of state of w < −1

3dominates the energy density,

RH will decrease, leading to a period of ination. An example of such a uid is a scalareld with a potential energy that dominates over its kinetic energy. The action of a scalareld, minimally coupled to gravity, is given by

S =

∫d4x√−g[M2

p

2R +

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 valuethat is accessible to the CMB.

14

2.5 How to drive ination?

The equation of motion of this scalar eld in a FRW spacetime is given by

φ+ 3Hφ+ V ′(φ) = 0. (2.33)

Remember that we should satisfy

ε = − H

H2< 1. (2.34)

Combining (2.33) with the rst Friedmann equation (2.6) results in

3M2pH

2 =1

2φ2 + V (φ). (2.35)

Using this expression, we can rewrite (2.34) as

ε = − H

H2=

12φ2

M2pH

2< 1, (2.36)

which is satised when

V (φ) 1

2φ2, (2.37)

such that

3M2pH

2 ' V (φ). (2.38)

Thus, a scalar eld with a potential energy that dominates over its kinetic energy can leadto a period of ination.

Furthermore, to allow for a prolonged period of ination, the kinetic energy has to remainsmall during some time, which requires the acceleration φ to be small with respect to theHubble friction 3Hφ. By neglecting the Hubble friction in the equation of motion, ε canbe written as

ε =M2

p

2

(V ′

V

)2

1. (2.39)

We refer to ε as the potential slow-roll parameter (note that ε = ε). Neglecting theacceleration in the equation of motion, requiring a prolonged period of ination can becaptured 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 − ε).

|η| ≡M2p

|V ′′|V 1 (2.41)

15


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

Finally, at the end of ination, a smooth transition to the regular hot Big Bang phaseof the universe must be made. At this phase, known as reheating, the energy densityof the inaton needs to be transferred to standard model particles that are heated to asucient temperature to start the hot Big Bang phase of the universe. Reheating is arather model-dependent process and beyond the scope of this thesis. Therefore, will nottreat 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 howevernecessary to take quantum eects into account, as ination couples the smallest to thelargest scales. Furthermore, because we observe deviations from homogeneity and isotropyin the universe, we also want ination to provide the perturbations that are responsible forgenerating these deviations. Quantizing ination leads to spectra of perturbations. Theaverage 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 bothof these. The metric has both scalar and tensor perturbations, the inaton only has scalarperturbations. In this section, we will show how these spectra can be obtained and whattheir observational consequences are. We will only describe the most important aspectsand 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 beperformed 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 tensorperturbations and obey the following equation of motion.

h+ 2Hh+ k2h = 0 (2.42)

When quantizing h, we obtain the following expansion in terms of creation and annihilationoperators [10].

h(~k, τ) =

√2

aMp

[v(k, τ)a~k + v(k, τ)a†~k

](2.43)

16


Here, the hat indicates that we are dealing with operators and the bar indicates complexconjugation. When we use the fact that H evolves slowly, the v's satisfy the followingdierential equation.

v +

(k2 − 2

τ 2

)v = 0 (2.44)

Now, the variance is given by

〈h†(~k, τ)h(~k′, τ)〉 =2

a2M2p

|v(~k, τ)|2(2π)3δ(3)(~k − ~k′)

≡ (2π)3Ph(k)δ(3)(~k − ~k′).(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 willsolve this equation for two cases. First, we solve it for the case when the wavelength of theperturbation is inside the horizon. After that, we consider the case when the wavelengthof 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 tookthis limit in (2.44), we would have obtained the equation of motion of a simple harmonicoscillator. 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 anyeect 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

−ikτ. (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 xesthe tensor power spectrum.

Ph(k) =4H2

?

M2pk

3(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 ≡

k3

2π2Ph(k) =

2H2?

π2M2p

. (2.50)

17


Deriving the power spectrum for scalar perturbations is a bit more involved, as we needto 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 scalarpower spectrum in terms of gauge-invariant quantities. Often, the comoving curvatureperturbation 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 thissubtlety, the same logic as for the tensor power spectrum applies. Again, it is observedthat the scalar perturbations vanish outside of the horizon. The variance of the scalarperturbations is given by

〈R(~k, τ)R(~k′, τ)〉 = (2π)3PR(k)δ(~k − ~k′), (2.52)

and the scalar power spectrum is given by

PR(k) =H2?

4ε?M2p

1

k3. (2.53)

We can also dene a dimensionless power spectrum.

∆2R(k) ≡ k3

2π2PR(k) =

H2?

8π2ε?M2p

(2.54)

2.6.2 Observational signatures of cosmological perturbations

Importantly, we saw that the expressions for the dimensionless power spectrum did notdepend explicitly on the wavelength of a perturbation, but only weakly via H and ε on ascale (both parameters should vary slowly). Therefore, a robust prediction of ination isthe fact that it generates a nearly scale-invariant spectrum of perturbations. Deviationsfrom scale invariance can be captured by the spectral indices, which are dened as

d log(∆2R)

d log(k)≡ ns − 1

d log(∆2h)

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

ns − 1 = 2η − 6ε (2.56)

nt = −2ε (2.57)

r = 16ε, (2.58)

18


where r is the tensor-to-scalar ratio that is dened as

r =∆2R(k)

∆2h(k)

. (2.59)

While it was nice that we saw that perturbations freeze outside of the horizon, this doesnot yet have observational consequences. Only when the comoving Hubble sphere growsduring the regular hot Big Bang phase, perturbations can re-enter the horizon again, seegure 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 havea profound eect. The small matter inhomogeneities have grown under the inuence ofgravity after the photons decoupled, which has led to the large scale structure we observetoday.

Additionally, photons in the early universe could also feel the presence of these matterinhomogeneities. This resulted in temperature anisotropies in the CMB that, on largescales, can be directly attributed to primordial perturbations. Complementary, the po-larization of the CMB also gives primordial information. Local quadrupole anisotropiespresent 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 perturbationscan result in a divergence-free polarization signal (B-modes), see gure 2.9.

19


Figure 2.9: Schematic illustration of possible polarization signals of the CMB. E-modescorrespond 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 revealthat the temperature anisotropies of the CMB are nearly scale-invariant. Furthermore,polarization measurements of the CMB should determine the size of the primordial tensorperturbations.

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 usedat which all observable quantities are evaluated. In eld space, this scale φ? correspondsto the point after which ination lasted 50-60 of efolds given by

N? =

φ?∫φend

dφ

Mp

1√2ε. (2.60)

While the scalar power spectrum has been measured via the temperature uctuations of theCMB, there has not yet been a measurement of the tensor power spectrum. Consistent withination, the spectrum of scalar perturbations is indeed nearly scale-invariant. Indeed, themost 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.

3The pivot scale at which Planck measured these quantities is k? = 0.05Mpc−1

20


2.6.3 A simple example: V (φ) = 12m

2φ2

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

V (φ) =1

2m2φ2 (2.63)

The slow-roll parameters for (2.63) are given by

ε = η = 2M2

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) = 12m2φ2 while driving

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

A reasonable estimate of the number of efolds required for ination to address the initialconditions of the universe is N? = 60 [14]. From (2.60), we then nd that φ? = 16Mp.As the end of ination (which corresponds to ε = 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 modelsand have some interesting properties. Most importantly, they are intimately related tofundamental physics, a point which we will come back to in section 2.7. The spectralindex 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 indisagreement with this particular model, but it shows how computing observables froman inationary model can be performed. An overview of some popular ination modelscompared with Planck 2015 data is given in gure 2.11.

21


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 theoreticalpredictions of inationary models to observations. However, this leaves a large parameterspace with models consistent with observations. In particular, the energy scale of ination,which is given by

Einflation = V 1/4 = (3H2M2p )1/4, (2.66)

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

Einflation = (3H2M2p )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 onlypin down the scale of ination to be high (Einflation ' Emax), it also implies that inationwas superplanckian. This can be seen from the bound(

∆φ

Mp

)2

&r

0.01, (2.69)

22

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 thatpredict an observable gravitational wave signal for two reasons. Firstly, (2.67) impliesthat such a model is tied to interesting fundamental physics, as ination occurred ata scale where new physics is expected to be relevant. Secondly, upcoming polarizationmeasurements of the CMB will constrain r further or measure its value, allowing suchmodels 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 buildingrather 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 eldination models suer in a more dramatic way4. This sensitivity to high-energy physicsis known as the UV-sensitivity of ination, which makes constructing large-eld inationmodels a challenge.

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

δm2 ∼ Λ2. (2.70)

Because Λ > H, the slow-roll parameter η will receive a large correction.

δη = M2p

δV ′′

V

=δm2

3H2

∼ Λ2

H2> 1.

(2.71)

This is known as the η-problem and needs to be addressed in every model of slow-rollination. One way of preventing large corrections to the inaton mass, is by appealing toa 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 weaklybreak this shift-symmetry, such that the inaton obtains a small mass. Still, one shouldbe 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 globalsymmetries (see for example [17]), while the low energy eective theory may exhibit thissymmetry. Therefore, high-energy physics might still induce corrections to the inatonmass that will lead to the η-problem. Determining whether such corrections will be largeis a subtle question that can only be properly answered by knowing the theory whichcompletes 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 moreoptimistic theorist may view this as an opportunity to use ination to probe a regime of unknownphysics.

23


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

are now no longer suppressed and contribute to the potential. An innite amount ofoperators should be ne-tuned to keep the atness of the potential, which is clearly notnatural. From an eective eld theory perspective the atness of the potential is notsustainable. Therefore, to properly construct a large-eld model, it is necessary to knowif an eective theory admits a UV-completion in which the size of the corrections can bechecked.

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

24

3 The eective eld theory of ination

The eective eld theory of ination is a powerful principle, that allows for the systematicstudy of ination in an eective and model independent way. This means that, in acertain energy regime, only the relevant degrees of freedom are studied, while degrees offreedom that are important at a dierent scale will decouple. An EFT description hasmany benets. For example, an eective eld theory only incorporates degrees of freedomup 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 possibletheories 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 physicsat a certain energy regime. When the full theory (SUV ), 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 onlyvalid up to the cuto and consists of the low-energy action (SIR) and higher dimensionalcorrections. The eective action in d spacetime dimensions can then be schematicallywritten as

Seff = SIR +∑i

∫ddx ci

OniΛni−d

. (3.1)

Here, Oni denotes an operator of dimension [mass]ni . This gives an ordering principle ofthe dierent operators in the eective action.

• Operators with ni > d are called irrelevant; they are important in the UV (highenergy) and can be neglected in the IR (low energy).

• Operators with ni < d are called relevant; they are unimportant in the UV andbecome important in the IR.

• Operators with ni = d are called marginal.

1Heavy degrees of freedom are degrees of freedom that are only probed at a high energy scale

25


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

Often, SUV is not known and the procedure of integrating out heavy modes cannot beexecuted. 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 makingassumptions about the symmetries of the UV-theory and allowing all operators that areconsistent with these symmetries, the same eective action can be obtained. This allowsus to construct an eective action, even though we do not have information about theUV-theory. An eective theory is said to be UV-completed when there exists a SUV thatcomplements SIR above Λ. An example that shows that these dierent ways of obtainingthe 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 symmetriesof ination. In chapter 2, we saw that ination corresponds to a period that satises

ε = − H

H2< 1. (3.2)

In the limit that H → 0, the inationary background can be described by a de Sitterspace; a maximally symmetric vacuum solution of the Einstein equations with a positivecosmological constant. However, ination cannot be described by a pure de Sitter space, asination does not end in a de Sitter space. Said dierently, the isometries of de Sitter spacecontain time-translations, which do not allow ination to end because this would introducean explicit time dependence. This implies that we can describe ination as a quasi-de Sitterphase where time dieomorphisms are spontaneously broken by a time-dependent scalarφ(t) (the inaton). This scalar acts as a clock that measures the amount of inationelapsed. Perturbations of the inaton around the inating background transform thefollowing 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 containan associated Goldstone boson [18] that we will denote by π, which parametrizes δφ. Wecan use the gauge freedom of choosing our coordinates in a way we prefer to let x0 coincidewith the slicing of spacetime introduced by φ(t). This is known as unitary gauge and hasthe 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 describedin terms of an additional degree of freedom of the graviton. Hence, the choice of unitarygauge is convenient, because the theory of cosmological perturbations can be described

26

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.


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

S =

∫d4x√−g[

1

2M2

pR− c(t)g00 − Λ(t) +1

2M2(t)4(δg00)2

+1

6M3(t)4(δg00)3 − M1(t)3

2(δg00)δKµ

µ

− M2(t)2

2δ(Kµ

µ)2 − M3(t)2

2δKµ

ν δKνµ + . . .

].

(3.6)

Here,Kµν is the extrinsic curvature: the curvature of constant time slices. Furthermore, weexpanded the eective action in perturbations of the metric and derivatives. All quantitiesare written in terms of perturbations, while the unperturbed quantities are absorbed in

√−g[−c(t)g00 − Λ(t)

]. (3.7)

For example

g00 = g00FRW + δg00 = −1 + δg00, (3.8)

where the factor −1 can be absorbed in Λ(t). In Appendix A, we show that allowing alloperators 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 timedependence. Since ination only breaks time dieomorphism invariance weakly (the slow-roll parameters that break the time dieomorphism invariance have to be small), we expectthat it is also natural for all other time-dependent parameters to not vary signicantly perHubble time.

3.3.1 Restoring full dieomorphism invariance

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

27


restore the full dieomorphism invariance. This reintroduction can be realized by usingthe 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, whichsimplies 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)) =∂x0(x)

∂xµ∂x0(x)

∂xνgµν(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∫d4x√−g[A(t) +B(t)g00(x)

], (3.13)

transforms as∫d4x√−g(x(x))

∣∣∣∣∂x∂x∣∣∣∣ [A(t) +B(t)

∂x0

∂xµ(x)

∂x0

∂xν(x)gµν(x(x))

], (3.14)

which can be simplied by changing the integration variable to x, such that the Jacobianexactly cancels the |∂x/∂x| term.∫

d4x√−g(x)

[A(t− ξ0(x(x))) +B(t− ξ0(x(x)))

∂(t− ξ0(x(x)))

∂xµ(x)

∂(t− ξ0(x(x)))

∂xν(x)gµν(x)

](3.15)

Next, introduce π, by using (3.10) and drop all tildes to obtain∫d4x√−g(x)

[A(t+ π(x)) +B(t+ π(x))

∂(t+ π(x))

∂xµ(x)

∂(t+ π(x))

∂xν(x)gµν(x)

]. (3.16)

By evaluating the second term

∂(t+ π(x))

∂xµ(x)

∂(t+ π(x))

∂xν(x)gµν(x) =(1 + π(x))2g00 + 2(1 + π(x))(∂iπ(x))g0i

+ (∂iπ(x))(∂jπ(x))gij,

(3.17)

28


we can rewrite (3.6) in terms of the Goldstone boson as

S =

∫d4x√−g[

1

2M2

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 invarianceis 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 =

∫d4x√−g[

1

2M2

pR− Λ(t)− c(t)g00

], (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 andc by 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 cancellation [18]. We can nd the equationsof motion of (3.20) by deriving the stress-energy tensor and plugging it into the Einsteinequations. The stress energy tensor is given by2

Tµν ≡ −2√−g

δS

δgµν(3.21)

The variation of (3.20) is given by

δS =

∫d4x

[−δ√−gΛ− δ

√−g cg00 −

√−g c δ0

µδ0νδg

µν], (3.22)

which can be rewritten using the identity

δ√−g =

1

2

√−ggµνδgµν . (3.23)

2By denition, only the matter terms contribute to Tµν , so the Ricci scalar gives no contribution.

29


We then obtain

δS =

∫d4x√−g[

1

2gµνΛ +

1

2gµνc g

00 − c δ0µδ

0ν

]δgµν , (3.24)

so the stress-energy tensor is given by

Tµν = −gµν(Λ + c g00

)+ 2c δ0

µδ0ν . (3.25)

The Einstein equations can be written as

Gµν =1

M2p

Tµν . (3.26)

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

Gµν =

G00 = 3H2

Gµµ = −12H2 − 6H.

(3.27)

Comparing this with the stress-energy tensor results in

3H2(t) =1

M2p

[Λ + c] (3.28)

−12H2 − H =1

3M2p

[Λ− 2c] , (3.29)

where we used (3.28) to derive (3.29). Solving these equations for Λ and c results in

c = −M2p H and Λ = M2

p

[3H2 + H

](3.30)

3.3.3 Decoupling limit

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

In our case, the gauge boson is analogous to the graviton. Similar to the gauge theoryexample, we can neglect metric uctuations at suciently high energy. Essentially, themetric 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,

30


we can estimate the energy at which this simplication occurs by canonically normalizingπ. From (3.18) we see that the kinetic term of π is given by

M2p H∂µπ∂νπg

µν . (3.31)

Thus, the canonically normalized Goldstone boson is dened as

πc ≡MpH1/2π. (3.32)

The kinetic term for δg00 is

M2p (∂µδg

00)(∂νδg00)gµν , (3.33)

such that

δg00c ≡Mpg

00. (3.34)

If, in analogy with the gauge theory, we then identify

MpH1/2 =

m

g, (3.35)

where m = H1/2 is the `mass' of the Goldstone boson and g = 1/Mp the coupling togravity. By this identication, the Goldstone self-interaction gets strongly coupled at ascale

Λ2 ∼MpH1/2. (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 thetheory is given by

H1/2 E M1/2p H1/4. (3.38)

As a consistency check, we can look at the leading order mixing term between δg00 and πfor simple slow-roll ination (M2 = M3 = 0).

M2p Hπδg

00 = H1/2πcδg00c , (3.39)

from which we see that metric uctuations indeed decouple at an energy E H1/2.

31


3.3.4 The simplest slow-roll eective action

When taking the decoupling limit, the eective action simplies greatly. Take for examplethe eective action describing the simplest3 scenario of slow-roll ination (M2 = M3 = 0).By expanding (3.18) and taking the decoupling limit, we obtain

S =

∫d4x√−g[

1

2M2

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 =

∫d4x√−g[

1

2M2

p − Λ− Λπ + c

(π2 − ∂iπ∂

iπ

a2

)+ 2cπ + cπ

]. (3.41)

Now, we apply integration by parts to the the term containing π (only with respect totime).

2

∫d4x√−g cπ = −2

∫d4x√−g [3Hcπ + cπ] (3.42)

Plugging this into (3.41) gives

S =

∫d4x√−g[

1

2M2

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, aswe promised, and yields the following eective action for the Goldstone boson.

Sπ =

∫d4x√−g[M2

p H

(−π2 +

∂iπ∂iπ

a2

)](3.44)

This simple action describes all possible simple slow-roll models of ination to leadingorder.

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 simplestslow-roll scenario by turning on other operators (settingM2 andM3 non-zero). This allowsus 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

M42∂µπ∂νπg

µν . (3.45)

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

32


Thus, the canonically normalized Goldstone boson is dened as

πc ≡M22π, (3.46)

from which we can identify m = M22/Mp. The energy scale at which the theory becomes

strongly coupled becomes

Λ2 ∼M22 . (3.47)

Thus, in the same manner as for the simple slow-roll scenario, we see that the energy scaleat 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

M42 πδg

00 =M2

2

Mp

πcδg00c , (3.49)

this is indeed consistent. This results (up to cubic order in π and derivatives) in

Sπ =

∫d4x√−g[M2

p H

(π2 − ∂iπ∂

iπ

a2

)+2M4

2

(π2 + π3 − π ∂iπ∂

iπ

a2

)− 4

3M4

3 π3

].

(3.50)

The fact that M2 is turned on has some interesting consequences. For example, it resultsin a dierent coecient for the time and spatial kinetic term of π

(2M42 −M2

PlH)π2 and M2PlH

∂iπ∂iπ

a2, (3.51)

which leads to a non-trivial speed of sound (cs 6= 1) of π.

c−2s ≡ 1− 2M4

2

M2PlH

. (3.52)

Rewriting (3.50) in terms of cs yields

Sπ =

∫d4x

[M2

PlH

c2s

(π2 + c2s

∂iπ∂iπ

a2) +M2

p H(1− c−2s )(π3 − π ∂iπ∂

iπ

a2)

−4

3M4

3 π3

].

(3.53)

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

s in front of theterms cubic in π (which contribute to the bispectrum).

33


3.3.6 Observables in the power spectrum

A careful reader might be worried that the energy scale at which the simplication ofthe eective action occurred (the decoupling limit) is not the energy scale relevant forus, because the observations we do today are at much lower energies than the scale atwhich decoupling takes place. Luckily, we saw in chapter 2 that perturbations that exitthe comoving Hubble sphere are conserved outside of the horizon and we therefore haveto 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 V

1/4? , 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 thecomoving curvature perturbation. For the simple slow-roll scenario with the Goldstoneaction

Sπ =

∫d4x√−g[M2

p H

(−π2 +

∂iπ∂iπ

a2

)], (3.55)

this of course results in the same power spectrum we saw earlier, as it should.

∆2R(k) ≡ k3

2π2PR(k) =

H2

8π2εM2p

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

Sπ =

∫d4x

[M2

PlH

c2s

(π2 + c2

s

∂iπ∂iπ

a2

)](3.57)

Calculating the power spectrum of R from this action shows that it depends on the speedof sound [20].

∆2R =

H2

8π2csεM2p

. (3.58)

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

ns − 1 = −6ε+ 2η − s, (3.59)

34

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 ≡ csHcs

(3.60)

The Goldstone actions (3.55) and (3.57) do not generate a non-trivial speed of sound forthe tensor perturbations, thus the tensor power spectrum is the same as our original result(2.50).

∆2h ≡

k3

2π2Ph(k) =

2H2

π2M2p

(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 ination in a simple andsystematic way and turning on dierent operators controls deviations from the simplestslow-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 cosmologicalperturbations can be straightforwardly taken into account, as the symmetry structure ofthe action only allows for a nite number of terms. This implies that renormalizationwill not generate new operators. Conversely, the normal action of φ contains an innitenumber of operators. Taking loop corrections into account is therefore possible.

Despite the benets of the EFT of ination, it also has shortcomings. As we discussedin 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, becausepreviously 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 gravitythat 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 beenprogress over the years. In the next chapter, we will treat aspects of string theory relevantfor ination and show how realistic models of ination might come about.

35

4 From string theory to cosmology

At the end of the previous chapter we argued that, to construct a proper model of ination,it is necessary to go beyond the eective theory of ination. In particular, it would beinteresting to construct a model of ination in string theory, for a number of reasons. Firstof all, we saw that in order to check the assumptions about the symmetries of ination,it is necessary to have knowledge about the UV-completion. Only then, it can be seenif low-energy symmetries are completed in the UV. More specically, quantum gravitydoes not necessarily respect the symmetries of the eective action. Furthermore, in large-eld models, ination took place at a relatively high energy scale at which new physicsis expected to be relevant. String theory can oer a description of this regime. Finally,string theory oers an inspiration for cosmologists to construct well-motivated eectiveactions. For example, models of axion ination originate from string theory, but alsoadmit a phenomenological low-energy description.

In this chapter, we will treat some aspects of string theory and supergravity, withoutmaking any pretences of being thorough or complete, as this is beyond the scope of thisthesis. Instead, we skip technical details and focus on conceptual aspects relevant forination. For a more in-depth introduction to string theory in the context of ination, werefer the reader to chapter 3 of the excellent book [14].

4.1 Aspects of string theory

String theory starts from the idea that at a fundamental level, elementary particles canbe described as vibrating strings. This string can be dened by a two dimensional world-sheet action which leads to a target space that describes the spacetime. In order forthis description to be consistent, the target space has to have more dimensions than ourordinary four-dimensional spacetime. Specically, superstring theory (string theory withsupersymmetry on the target space) gives rise to a ten-dimensional spacetime [21].

In the 90's, it was discovered that the ve existing superstring theories1 were are allrelated to each other by dualities and the limit of a new eleven-dimensional theory; M-theory. In order to obtain a four-dimensional theory which can describe our universe, thesesuperstring theories need to be compactied. This process reduces the ten dimensions ofthe target space to four, by making six dimensions compact. Only at very high energy,the compact dimensions are probed [14].

1Type I, Type IIA, Type IIB, heterotic E8 × E8 and heterotic SO(32) string theory.

36


4.1.1 Obtaining a four-dimensional eective action from string

theory

In order to connect string theory to our four-dimensional world by compactifying thetheory, it is convenient to rst obtain the ten-dimensional low-energy limit of string theory;supergravity. Integrating out the heavy degrees of freedom (massive excitations of thesuperstring) results in a ten-dimensional supergravity theory.

The next step is the compactication of the supergravity theory. While the ten-dimensionaltheory enjoys supersymmetry, the four-dimensional eective theory that one obtains aftercompactifying does not necessarily. However, it was shown that compactifying on Calabi-Yau manifolds can preserve supersymmetry, such that we obtain a four-dimensional ef-fective supergravity action with supersymmetry preserved. Calabi-Yau manifolds have avery rich geometry and topology, with some interesting phenomenological consequences.On the Calabi-Yau, one can perform deformations which leave it topologically invariant.Such deformations are controlled by parameters that are known as moduli. In the four-dimensional theory, moduli appear as massless scalar elds. An example of a modulus isthe volume of the compact space. The moduli need to be stabilized (given a mass), suchthat they do not destabilize the four-dimensional theory. For example, if the moduluscorresponding to the volume of the compact space would not be stabilized, the compactspace could decompactify. Typically, Calabi-Yau compactications contain many moduliwhich have to be stabilized in order to obtain a stable four-dimensional theory [14].

If we can generate a potential for these moduli, such that they obtain a large mass, theydecouple from the low-energy theory. The process of generating a potential for the moduliis known as moduli stabilization. Summarizing, the process of obtaining a four-dimensionaleective supergravity theory from a string theory is illustrated in gure 4.1. In this gure,the rst arrow depicts integrating out the massive string excitations and the second arrowillustrates compactication and moduli stabilization. All compactication data (such asmoduli etc.) can be stored by two functions, which specify the supergravity theory. Theseare the real Kähler potential K and the holomorphic superpotential W , which we willcome back to in section 4.2.

4.1.2 De Sitter vacua from string theory

Readers with little prior knowledge of string theory, may get the impression that obtaininga stable four-dimensional vacuum from string theory is rather straightforward. This is farfrom the truth. The fact that we have swept all technical details under the rug obscuresthe diculty of these constructions. Obtaining stable vacua from string theory, that canbe used for cosmology (de Sitter vacua), is an extremely complicated task that is onlyexecuted by the bravest of physicists. As a matter of fact, it is currently not even clearif string theorists have really succeeded in this task. Because we think that ination anddark energy domination can be described by an approximate de Sitter vacuum, this isa crucial question to answer. In addition, we would like the minimum of the potential

37


Figure 4.1: All ten-dimensional superstring theories are related to each other by dualitiesand to eleven-dimensional M-theory. The low energy limit of a superstring theory is aten-dimensional supergravity theory. By performing a compactication (on a Calabi-Yaumanifold), a four-dimensional eective theory of supergravity is obtained.

of the inaton to be only slightly positive, to explain the smallness of the cosmologicalconstant (as Vmin(φ) = Λ), see gure 4.2. As we mentioned before, in order to obtain astable vacuum we have to employ some moduli stabilization scheme. The moduli appearas at directions in the potential of the supergravity theory and can be stabilized in typeIIB string theory by perturbative and non-perturbative corrections to the Kähler andsuperpotential2 [14].

To make matters even more dicult, when it is possible to stabilize the moduli using(non)perturbative corrections, stable vacua appear only at very non-typical places3. As isdescribed in section 3.3.3 of [14], competition between at least two corrections is needed,

2The Kähler potential receives perturbative corrections while the rst correction to the superpotentialis non-perturbative, due to a famous non-renormalizable theorem [22].

3This is known as the Dine-Seiberg problem. For a more precise formulation of this statement, we referthe reader to [23].

38


ΛdS

ΛAdS

ϕ

V(ϕ)

Figure 4.2: Two vacua from a compactication of string theory. The minimum of thepotential determines the size of the cosmological constant.

to prevent an instability from destroying the stability of the vacuum. Moreover, evenwhen a stable vacuum is obtained this way, it will always be an anti-de Sitter (AdS) space(negative cosmological constant) or Minkowski space (vanishing cosmological constant)instead of a de Sitter (dS) space. This result is the Maldacena-Nunez no-go theorem[24].

A possible evasion of this no-go theorem was published in 2003, when Kachru, Kallosh,Linde and Trivedi (KKLT) proposed a mechanism to take a stable AdS vacuum and `uplift'it to a meta-stable dS vacuum by adding a new source to the background; anti-branes [25].It was claimed that in this way, a landscape of stable dS vacua could be constructed. Incontrast, this mechanism has received a large amount of critique as it was observed thatthe anti-branes used for the uplifting procedure create a singularity [26, 27, 28]. Dierentattempts have been made to resolve this singularity [29, 30], but to no avail. Concluding,it is currently unclear if it is possible to resolve the singularity in order to obtain a meta-stable dS vacuum from string theory, despite the fact that a lot of hard work has beendone to proof or disproof their existence4.

More philosophically, one might wonder why it seems so dicult to obtain a stable dSvacuum from string theory. After all, we think that string theory should describe natureand that nature can be approximately described by a dS vacuum. How can we resolvethis tension? On the one hand, if we believe that it should be possible to obtain dSvacua from string theory, the current absence of convincing constructions might simplybe due to our limited creativity and calculational power. On the other hand, perhaps theframework of string theory cannot really be applied to cosmology in the way that we have

4Recently, dierent approaches of obtaining dS vacua have been presented, such as perturbing super-symmetric Minkowski vacua [31, 32].

39


tried. Currently, we are only guided by our theoretical intuition and have no observationalsignatures of string theory. Nevertheless, here we take the optimistic point of view thatthese issues will be settled in the future and embedding ination in string theory is aninteresting direction to explore.

4.2 Eective N = 1 supergravity

While the eective four-dimensional supergravity theory does not have the complete infor-mation of string theory, it is a whole lot simpler. Its simplicity makes constructing modelsof ination more easily realizable than in string theory. Furthermore, we are equipped withthe knowledge that supergravity admits a UV-completion, in contrast with an ordinaryeective theory. In this section, we give a brief introduction to supersymmetry, followingthe primer by Martin [33], which turns into a theory of supergravity when it is gauged[34]. Here, we will only discuss some aspects relevant for ination and refer the interestedreader to Martins solid introduction into the subject.

The foundation of supersymmetry lies in the idea that bosons and fermions should berelated by a symmetry (supersymmetry) to each other. An operator Q that realizes sucha transformation should be a spinor that transform bosons into fermions and vice versa.

Q |boson〉 = |fermion〉 and Q |fermion〉 = |boson〉 (4.1)

The generators Q and Q† should satisfy the following (anti)commutation relations.Q,Q†

= P µ (4.2)

Q,Q =Q†, Q†

(4.3)

[P µ, Q] = [P µ, Q†] = 0 (4.4)

Here, P µ is the generator of spacetime translations, which implies that supersymmetry is aspacetime symmetry. A supersymmetric theory can have more supersymmetry generatorsthan just the set

Q,Q†

we considered here. In this thesis, we will only consider one set

of supersymmetry generators, which is denoted by N = 1.

Due to the relation between bosons and fermions by supersymmetry, it is convenient towork with supermultiplets; states that contain an equal number of bosonic and fermionicdegrees of freedom. The components of the supermultiplet are related to each other bythe supersymmetry generators. Satisfying the demand that a supermultiplet contains anequal amount of bosonic and fermionic degrees of freedom allows for dierent occupationsof the supermultiplet. In the simplest case, a supermultiplet contains one Weyl fermionwith two spin helicity states and two real scalars that can be combined in a single complexeld. Such a supermultiplet is known as a chiral supermultiplet. Moving on, we can alsotake a massless spin 1 vector boson (with two helicity states) accompanied by a spin 1/2Weyl fermion. This combination is known as a vector multiplet. Finally, we can alsoincorporate the graviton in a supermultiplet together with a spin 3/2 Weyl fermion.

40


A supersymmetric theory can be conveniently reformulated in so-called superspace insteadof ordinary four-dimensional spacetime. Superspace has four (bosonic) coordinates xµ

(the usual spacetime coordinates) and four (fermionic) coordinates θα, θ†α with dimension[mass]1/2, where θ is a two-component spinor and α, α spinor indices. In this language,the components of the supermultiplets can be written as superelds, which depend on thecoordinates of the full superspace.

Any supereld S(x, θ, θ†) can then be expanded in a nite power series of θ, θ† with coef-cients that depend on xµ, such that it is invariant under supersymmetry. We can nowmake a distinction between two dierent types of elds. If a supereld depends on allfermionic superspace coordinates, the term is referred to as a D-term. Contrary, a super-eld can also only depend on half of the fermionic superspace coordinates. Such a termis known as a F-term. Typically, chiral superelds only depend on half of the fermioniccoordinates and vector superelds depend on all fermionic coordinates. To obtain an ac-tion in four dimensions, the dierent contributions to the action are integrated over thefermionic coordinates. As the inaton is a scalar eld, for a description of ination it issucient to only consider F-term contributions. Some relevant contributions are:

• The superpotential W (Φ); an arbitrary holomorphic function of the chiral super-elds. It has dimension [mass]3.

• The real Kähler potential K(Φ, Φ), which depends on both the chiral and anti-chiral(complex conjugate of the chiral supereld) superelds. It has dimension [mass]2.

As we mentioned before, all data of a string compactication can be captured by thesetwo functions. Therefore, one can explicitly construct a model of ination that is inspiredby string theory. For N = 1 supergravity, this results in the following general action fora set of chiral superelds [14].

S =

∫d4x√−g[KAB∂µΦA∂µΦB − VF

](4.5)

Here, KAB is the Kähler metric and VF is the F-term potential. The F-term potential isgiven by [14]

VF = eK/M2p

[KABDAWDBW − 3

|W |2

M2p

], (4.6)

where DA denotes the Kähler covariant derivative

DAW = ∂AW +1

M2p

(∂AK)W (4.7)

and a subscript indicates a derivative with respect to the superelds.

∂AW =∂W

∂ΦA(4.8)

41


4.2.1 An example of ination in supergravity

We will now construct a simple inationary model in supergravity to illustrate some issuesone encounters in trying to do so. The most straightforward choice of Kähler potentialone can consider is

K = ΦΦ. (4.9)

This makes the Kähler metric rather simple

KAB = δAB, (4.10)

such that the kinetic term in (4.5) is canonical when the superelds are normalized. Itwas however quickly realized that this is a particular unfortunate choice, as this inducesan exponential steepening of the potential by

eK/M2p = eΦΦ/M2

p , (4.11)

in the potential (4.6). A resolution to this problem was proposed in [35]. The authorsargued that instead of using (4.10), another type of Kähler potential can be used that alsoleads to a canonical kinetic term. When one considers

K =1

2

(Φ + Φ

)2, (4.12)

the Kähler potential has a shift symmetry

Φ→ Φ + ic, (4.13)

where c is a constant. This creates a at direction in the potential for the combination Φ−Φthat can be used for ination. This circumvents the problem of exponential steepening,as the role of the inaton will be played by the combination of Φ and Φ that does notappear in the Kähler potential.

This observation allows for a construction of a model of quadratic ination in supergravity,a model that we already encountered in chapter 2. Consider a Kähler potential andsuperpotential of the form

K =1

2

(Φ + Φ

)2+ SS (4.14)

W = mSg(Φ). (4.15)

Here, we added an extra supereld S, m is a real number and g(Φ) is an arbitrary holo-morphic function. If we assume that we can stabilize the elds S and Re(Φ) at zero [36],the potential becomes

V = eK/M2p

[KABDAWDBW − 3

|W |2

M2p

]∣∣∣∣Re(Φ)=S=0

= ∂SW∂SW

= m2 |g(Φ)|2

(4.16)

42


This gives us a large freedom of obtaining almost any potential we like, because g(Φ) isan arbitrary holomorphic function. For example, if we take the simple choice

g(Φ) = Φ, (4.17)

we obtain

V = m2 Im(Φ)2 (4.18)

which, as promised, is the familiar quadratic ination potential when we parametrizeIm(Φ) = φ/

√2.

Although we saw in chapter 2 that this model is in disagreement with observations, itillustrates how a model of ination in supergravity might come about. In chapter 5 and6, we will see other well-motivated models of ination in supergravity. Because we knowthat supergravity is UV-completed by string theory, such models are a step in the rightdirection to embed ination in string theory.

43

5 Axion ination

In chapter 2, we saw that large-eld ination is extremely UV-sensitive. This requiresthat we impose some symmetry to suppress dangerous corrections. In chapter 4, wemotivated why we need to account for this symmetry in string theory. In large-eldmodels, only a shift symmetry is able to suppress all dangerous corrections. Thus, wewould like to construct a model of ination which has this symmetry built-in naturally.The most popular ination model that has these features is axion ination. Axions areshift-symmetric particles that arise naturally in string theory [37]. The shift symmetryof the axions is weakly broken to a discrete symmetry by non-perturbative corrections(instantons) [38]. This makes axions radiatively stable, such that they can accommodatelarge-eld ination.

5.1 Natural ination

Phenomenologically, we can describe ination as driven by a pseudo nambu-goldstoneboson from the breaking of a shift symmetry and this observation has led to a model ofination known as natural ination [39]. We can think of the pseudo-nambu goldstoneboson as an axion with the following potential.

V (φ) = Λ4

[1− cos

(φ

f

)](5.1)

Here, f is the axion decay constant, which determines the periodicity of the eld (andtherefore the maximal eld range available for ination) and Λ is some generated energyscale.

Of course, we also want to know if this model can be embedded in string theory. Therefore,we mention how a similar potential as (5.1) can be obtained in four-dimensional eectiveN = 1 supergravity. We consider two complex elds; the chiral supereld Φ (which canarise as a Kähler modulus) and a supereld S (for example a goldstino, which is associatedwith supersymmetry breaking). If we then take the following Kähler and superpotential

K =Φ + Φ

2+ SS − g(SS)2 W =

Λ2

√2S(1− e−aT ), (5.2)

44

5.2 Saving natural ination

the role of the inaton is played by the linear combination of Φ and Φ that does not appearin the Kähler potential. It will be useful to parametrize Φ and S in terms of real elds.

Φ =1√2

(β + iφ) S =1√2

(α + is) (5.3)

By imposing certain constraints on the parameter g, φ can play the role of the inaton,while the other elds are heavy and decouple [40]. This leads via the standard potential(4.6) to

V = Λ4

[1− cos

(aφ√

2

)]. (5.4)

Modifying the Kähler and superpotential can lead to similar natural ination-like poten-tials.

Whereas natural ination is a well-motivated model from string theory, it unfortunatelyhas a fatal aw. In order to be consistent with observations, the axion decay constant fhas to be larger than the Planck mass. This is problematic, because string theory doesnot allow for such large axion decay constants.

To be more precise, whenever one nds a superplanckian axion decay constant in stringtheory, there are also unsuppressed corrections that destroy the superplanckian eld dis-placement, by generating additional maxima along the axion potential [41]. For a moreelaborate discussion of this point, see section 7.2. We conclude that natural ination,in its original form, is not compatible with observations, as it is not possible to obtainsuperplanckian decay constants in string theory.


Even though natural ination is not a good model for obtaining large-eld ination instring theory, there have been attempts at modifying natural ination, such that it can berealized in string theory and also is compatible with observations. The leading proposalsfor modifying natural ination such that it keeps its features (well-motivated from stringtheory, radiatively stable and having a large eld range) and also is consistent with obser-vations, can be divided into three dierent categories: N-ation, alignment mechanismsand axion monodromy ination. We will only focus on the rst two proposals and referthe interested reader to [38] for a review of axion monodromy ination (and other modelsof axion ination).

The idea of N-ation is based on the following observation: while individual axions musthave a subplanckian axion decay constant, it might be possible to use many subplanckianaxions and travel along a diagonal in eld space, where a superplanckian direction arises.Alignment mechanisms are inspired by the fact that generally the kinetic term of theaxions is not canonical. In order to make the kinetic term canonical, a non-trivial rotationof the axions is required, which brings the axions into a new basis. In this new basis,

45

5 Axion ination

a large direction in eld space may be present, while it was not manifest in the originalbasis. We will see some examples of both of these approaches and comment on the issuesand features these models have.

5.2.1 N-ation

Dimopoulos, Kachru, McGreevy and Wacker were the rst ones to present the idea ofN-ation; using multiple axions to obtain a superplanckian eld range [42]. If we considerN copies of the potential (5.1), each of which breaks a dierent shift symmetry, we obtainthe following action.

S =

∫d4x√−g

N∑i=1

[−1

2∂µφ

i∂µφi − Λ4i

(1− cos

(φi

fi

))](5.5)

Of course, it is necessary to make sure that no other corrections will change the form of(5.5), a point which is (to some extent) addressed in the original paper. In order to see ifits possible to obtain a large at direction in eld space, we introduce the notion of thefundamental domain, which is a N -dimensional space constrained by the periodicity ofthe axions.

|θi| ≤ π ∀i (5.6)

In the case of (5.5), the fundamental domain is a hypercube with N sides of 2π. For large-eld ination to occur, we have to nd a large direction in eld space, while satisfyingthe constraints (5.6). In the language of the fundamental domain, this means nding asuperplanckian distance in the fundamental domain. If we now dene the inaton Φ tobe the linear combination of the individual axions that is parallel to a diagonal of thefundamental domain, we can obtain an enhancement of the eld displacement ∆Φ. Ifthe dierent axion decay constants are comparable, the eld displacement is enhanced bya factor of

√N , with respect to the single-eld case. The diameter of the fundamental

domain is given by1

D = 2π√f 2

1 + . . .+ f 2N ≈ 2πf

√N. (5.7)

This situation is illustrated for two axions in gure 5.1. Summarizing, even when we startwith all fi < Mp, a superplanckian eld displacement can arise in a particular direction ineld space. Nevertheless, it can be questioned how general this is, as it requires pickingout a special direction in the fundamental domain along which ination occurs.

1Note that only part of the total diameter of the fundamental domain can be used for ination.

46


θ1 = 2π

θ2

=2π~φ1 = f1πφ1

~ φ2

=f 2πφ

2

Φ=π√ f

21+f22

Figure 5.1: Picking out a direction in eld space parallel to the diagonal of the fundamentaldomain to be used for ination, leads to a

√N enhancement of the eld range, when all

axion decay constants are of the same order. Here, this situation is illustrated for twoaxions, for which the fundamental domain is a square.

5.2.2 Kinetic alignment

A priori, there is no reason to expect the kinetic term of the axions to be canonical.Therefore, the more general version of the action of the axions is given by

S =

∫d4x√−g

[−1

2Kij∂µθ

i∂µθj −N∑i=1

Λ4i

(1− cos

(θi))]

, (5.8)

where Kij is the metric on eld space. In the case that Kij is diagonal, it is given by

K = diag(f 2i ), (5.9)

and we can rescale

~φ = diag(fi)~θ, (5.10)

to obtain the same action as in (5.5). In the case that Kij is not diagonal, we have toperform a basis transformation that diagonalizes Kij and then rescale

~φ = diag(fi)RT~θ. (5.11)

Here, R is a rotation matrix that diagonalises K.

RTKR = diag(f 2i ) (5.12)

Performing the rotation and rescaling, we obtain

S =

∫d4x√−g

N∑i=1

[−1

2∂µφ

i∂µφi − Λ4i

(1− cos

((R−1~φ)i

fi

))]. (5.13)

47

5 Axion ination

Due to the rotation R, necessary to bring the kinetic term into a canonical form, therenow is a non-trivial relation between the axions that dene the fundamental domain andthe canonical axions. If one of the canonical axions φi is parallel to a diagonal of thefundamental domain, we can again obtain an enhancement of the eld displacement by afactor

√N , but now only one axion decay constant needs to be relatively large, see gure

5.2. The maximal diameter is now obtained when the largest axion decay constant fmaxcorresponds to an axion that is parallel to a diagonal in the fundamental domain. This isknown as kinetic alignment [43]. The diameter is now given by

D = 2πfmax√N. (5.14)

θ1 = 2π

θ2

=2π~φ 1

=f 1πφ

1~φ2 =

f2 πφ

2

Φ=πf

1

√ 2

Figure 5.2: The canonical axions ~φ are related by a non-trivial rotation to the axions ~θthat dene the fundamental domain. If a canonical axion with a relatively large axiondecay constant is parallel to a diagonal in the fundamental domain, an enhancement ofthe eld displacement by a factor

√N is obtained.

5.2.3 Lattice alignment

Another type of alignment that leads to a large axion decay constant is known as lat-tice alignment. This mechanism was introduced by Kim, Nilles and Peloso [44]. Theyconsidered two axions θ1 and θ2 appearing in the action as follows.

S =

∫d4x√−g

[2∑i=1

−1

2∂µθi∂

µθi − Λ41

(1− cos

(θ1

f1

+θ2

g1

))+ Λ4

2

(1− cos

(θ1

f2

+θ2

g2

))](5.15)

For simplicity, take f = f1 = f2. If in addition we take g1 ' g2, the same linear combina-tion of θ1 and θ2 appears in the cosines. Thus, a at direction appears for the orthogonal

48

5.3 Statistical generality of axion ination

combination

θ1

g1

− θ2

f. (5.16)

When changing to the normalized elds

Ψ =fg1√f 2 + g2

1

(θ1

f+θ2

g1

)Φ =

fg1√f 2 + g2

1

(θ2

f− θ1

g1

),

(5.17)

it can be seen that the eld Φ has an axion decay constant of

fΦ =g2

g1 − g2

√f 2 + g2

1. (5.18)

If we dene

g1 − g2 = ε, (5.19)

a parametrically large axion decay constant can be obtained in the limit ε → 0, whereasthe original decay constants f, g are all subplanckian. Again, the question arises howgeneral the situation is where alignment occurs 2.


In the previous section, we found that obtaining a superplanckian direction in eld spaceis in principle no problem, when multiple axions are used. Nevertheless this does not proofthat such a situation is very generic in string theory. It can be the case that the situationswe considered above are extremely rare and, in realistic setups, not realizable. Due to thetechnical nature of explicit constructions in string theory, checking this point is not an easytask. Therefore, a more accessible approach of nding the occurrence of superplanckiandirections is to make use of statistical arguments in the large N limit (when many axionsare present). Using statistical arguments in the string landscape is not new [45] and hasalso been applied to axion ination by McAllister et al., which led to some interestingobservations [43, 46, 47].

5.3.1 Emergent properties of N-ation

The statistical properties of models of N-ation with a large number of axions were rstderived by Easther and McAllister [46]. They showed that in the large N limit, the

2See chapter 7 for another issue with lattice alignment.

49

5 Axion ination

spectrum of masses of the axions is an emergent property, which does not depend on theexact details of the stringy origin of the axions.

Taylor expanding the axion potential obtained from supergravity (see [46] for the details)around its minimum, we obtain

V = (2π)2Mijθiθj +O(θ3), (5.20)

where Mij is the mass matrix. Now, the action is given by

S =

∫d4x√−g[−1

2Kij∂µθ

i∂µθj − (2π)2Mijθiθj]

(5.21)

As before, we can perform a change of basis to render the kinetic term canonical and absorbthe eigenvalues of Kij (the axion decay constants squared) with a eld redenition.

S =

∫d4x√−g[−1

2∂µφ

i∂µφi − (2π)2Mij

fifjφiφj

](5.22)

To characterize the behaviour of the axions, information about the mass matrix and inparticular its eigenvalues (the squared masses of the axions) is needed. However, a fullmicroscopic consideration is rather complicated. Luckily, when Mij is large enough, it isreasonable to approximate it by a random matrix. In particular, Easther and McAllisterhave shown that is is reasonable to approximate Mij by a Wishart matrix: a matrix ofthe form

M = ATA, (5.23)

where A is a (N + P ) × N random matrix with entries that can be (but are not limitedto be) Gaussian random variables. It is well-known that the spectrum of eigenvalues of alarge Wishart matrix is described by the Marcenko-Pastur probability distribution [48].

P (m2) =

√(b−m2)(m2 − a)

2πm2βσ2, (5.24)

where

a = σ2(

1− β12

)2

(5.25)

b = σ2(

1 + β12

)2

(5.26)

β =N

N + P. (5.27)

Since the variance σ2 has to be tuned such that the average mass scale matches observa-tions, β is the only free parameter. Even this parameter is not entirely free, because Ncan be interpreted as the number of axions and N + P as the total number of moduli.Surprisingly, the dimension of the moduli space determines the dynamics of the theory.

50


In gure 5.3, 5.4 and 5.5, we show the eigenvalue distribution of a set of mass matricesM = ATA with dierent values for β, where the entries of A were randomly picked from anormal distribution. It can be seen that this spectrum is well-tted by a Marcenko-Pasturdistribution.

500 1000 1500 2000 2500m 2

Entries

Figure 5.3: Spectrum of eigenvalues of 100 300 × 300 matrices with β = 13, tted with a

Marcenko-Pastur distribution.

500 1000 1500m 2

Entries

Figure 5.4: Spectrum of eigenvalues of 100 300 × 300 matrices with β = 12, tted with a

Marcenko-Pastur distribution.

51

5 Axion ination

0 200 400 600 800 1000 1200m 2

Entries

Figure 5.5: Spectrum of eigenvalues of 100 300 × 300 matrices with β = 1, tted with aMarcenko-Pastur distribution.

From these gures, we can make some interesting observations. The Marcenko-Pasturdistribution provides an excellent t to the eigenvalue spectrum of the mass matrix. Sincethe shape of the distribution is only determined by β, it is (in good approximation)possible to characterize the dynamics of N-ation by the dimension of the moduli space.Nevertheless, determining if it is possible for a sucient amount of ination to occurwith N-ation is still a rather dicult question to answer. As the spectrum of masses istypically broad3, the axions will not all evolve in the same manner. Therefore, to obtainthe full dynamics, we have to follow the evolution of each of the axions, which is a verylaborious task with a large number of axions. Thus, from this analysis, it is not conclusiveif it is possible to construct a model of N-ation in which a superplanckian eld rangeoccurs naturally.

5.3.2 Eigenvector delocalization in models with kinetic mixing

At rst sight, models with kinetic alignment seem less ne-tuned than N-ation, due tothe fact that the large direction in the fundamental domain is generated by only a singleaxion (with a relatively large axion decay constant). To nd a more thorough answer howgeneral kinetic alignment is, we will determine how likely it is for a canonical axion to beparallel to a diagonal in the fundamental domain. This can be found out by looking atthe eigenvectors of the kinetic matrix.

As was remarked by Bachlechner, Dias, Frazer and McAllister in [43], kinetic alignment isa rather generic occurrence. This can be intuitively seen from the fact that at large N , thefundamental domain has only N sides, but 2N diagonals. It is therefore statistically morelikely for an eigenvector to be parallel to a diagonal than not, if we expect the eigenvectorsto be randomly distributed. This can be complemented by mathematical statements aboutrandom matrix theory. If the kinetic matrix K can be approximated by a Wishart matrix,

3Easther and McAllister argued that, by considering the renormalization of the Planck mass in thepresence of a large number of species, β = 1

2 is favoured.

52


there exists a proof that the entries of the eigenvectors of K are normally distributed [49].This implies that at large N the eigenvectors are all parallel to a diagonal with very highprobability. This is known as eigenvector delocalization.

Furthermore, eigenvector delocalization is not very sensitive to the type of distributionthat is chosen. Therefore, we are not only limited to the case when K is a Wishartmatrix. This was also cross-checked with explicit string constructions by Bachlechner,Long and McAllister in [47], who showed that in Calabi-Yau compactications of stringtheory the kinetic matrix is well-approximated by a random matrix from an ensemblethat exhibits eigenvector delocalization. In particular, they compared their results to anexplicit compactication [50] and found agreement with their statistical results.

5.3.3 Diameter of the fundamental domain of the most general

axion action

In order to nd more realistic examples of kinetic alignment, we consider a generalizationof (5.8),which is inspired by explicit string constructions [47]. This generalization takesinto account the fact that there can be more instantons (P ) than axions (N) in the action.This leads to a superposition of axions in the cosine terms. The action is now given by

S =

∫d4x√−g

[−1

2Kij∂µθ

i∂µθj −P∑i=1

Λ4i

(1− cos(Qijθj)

)]. (5.28)

Here, Q is a matrix with integer entries. It will proof useful to introduce the decomposi-tion

Q =

(Q

QR

), (5.29)

where Q is a square N × N matrix and QR a (P − N) × N matrix. This allows us torewrite (5.28) as

S =

∫d4x√−g

[−1

2Kij∂µθ

i∂µθj −N∑i=1

Λ4i

(1− cos(Qi

jθj))−

P−N∑i=1

Λ4i

(1− cos((QR)ijθ

j))].

(5.30)

In order to dene the fundamental domain, we perform a basis transformation that rendersthe fundamental domain hypercubic, as we did before.

Q~θ = ~φ with (Q−1)TKQ−1 ≡ Ξ (5.31)

Now, (5.30) becomes

S =

∫d4x√−g

[−1

2Ξij∂φ

i∂φj −N∑i=1

Λ4i

(1− cos(φi)

)−

P−N∑i=1

Λ4i

(1− cos((QRQ

−1~φ)i))]

.

(5.32)

53

5 Axion ination

In this basis, it becomes manifest that the fundamental domain is a hypercube with sides2π, cut by (P − N) additional constraints. Summarizing, the fundamental domain isdened by

|φi| ≤ π ∀i and |(QRQ−1~φ)i| ≤ π ∀i. (5.33)

Lastly, in order to determine the eld range of the canonical eld Φ in this fundamentaldomain, a last basis transformation that diagonalizes Ξ is needed. The canonical eldsare then dened as

~Φ = diag(ξi)ST ~φ with STΞS = diag(ξ2

i ). (5.34)

We are now ready to estimate the typical eld range for a number of dierent situations.

Case #1: Equal number of instantons as axions

Firstly, we look at the case when there are an equal number of instantons as axions(P = N), such that Q is square. We then obtain

S =

∫d4x√−g

[−1

2Ξij∂φ

i∂φj −N∑i=1

Λ4i

(1− cos(φi)

)], (5.35)

such that the fundamental domain is a hypercube with sides 2π. This is similar to thesituation we encountered for the action (5.8), with the important dierence that K and Ξare non-trivially related. The diameter is now given by

D2 = ~ΦT · ~Φ= ~φTSdiag(ξ2

i )ST ~φ

= ~φTΞφ

(5.36)

If we denote the maximal possible diameter squared by r2max, we see from (5.36) that the

diameter is dened by an ellipsoid in the fundamental domain, where the principal axes aregiven by the normalized eigenvectors of Ξ and with a length determined by 1/ξi. Findingthe maximal diameter then amounts to nding the largest r2 for which the entire ellipsoidintersects the fundamental domain. The maximal possible displacement is obtained whenthe eigenvector ~Ψmax of Ξ with the largest eigenvalue ξ2

max (corresponding to the shortestaxis of the ellipse) is parallel to a diagonal in the fundamental domain, see gure 5.6.

54


φ1 = 2π

φ2

=2π~Ψm

ax

~Ψmin

Figure 5.6: Finding the maximal eld displacement amounts to nding the maximal r2

for which the ellipse intersects the fundamental domain. The maximal diameter is thengiven by the shortest axis.

Summarizing, to obtain the largest possible diameter, we have to rotate the ellipse in sucha way that the eigenvector ~Ψmax corresponding to the largest eigenvalue ξ2

max is parallelto a diagonal and then nd the largest possible r2, such that the ellipsoid intersects thefundamental domain. The maximal possible diameter is then given by

D = 2πξmax√N, (5.37)

which is similar to (5.14) with the dierence that there is a non-trivial relation betweenthe eigenvalues of Ξ and K.

Case #2: More instantons than axions

Secondly, there can also be more instantons than axions (P > N), such that the funda-mental domain is a hypercube, cut by P −N additional constraints, see (5.33). It is nowno longer possible to obtain a closed expression for the maximal eld displacement as wewere able to in the previous case. Nevertheless, Bachlechner, Long and McAllister showedin [47] that it is still possible to compute the eld displacement in an arbitrary directionwhen the matrices Q and K are known. Furthermore, even when these matrices are notknown we can use the fact that, when they are large, they reach a universal limit in whichtheir properties are statistically determined. This makes it still possible to obtain boundson the maximal eld range.

55

5 Axion ination

The strongest constraint on the fundamental domain is given by the P − N constraintsin (5.32), that cut the hypercube. Therefore, we can obtain the eld displacement in aparticular direction v by introducing an operator that rescales the ellipse on which v ends,such that it intersects the fundamental domain.

W (v) =2π

Maxi((QRQ−1φ)i)v (5.38)

It will be convenient to parametrize Ψ as a linear combination of some of the eigenvectorsmultiplied by the square root of its corresponding eigenvalue.

~v =∑i

ξi~Ψi (5.39)

The maximal diameter along the direction v is then given by (using (5.34) and (5.38))

D = ||diag(ξi)STW (v)||

= ||W (v)|| × ||diag(ξi)ST∑j

ξj~Ψj||

= ||W (v)|| × ||diag(ξi)ST ~ξ|| ( ~Ψj's are orthonormal)

= ||W (v)|| ×√∑

i

ξ4i ,

(5.40)

where ~ξ is a vector that contains the ξi's that correspond to the eigenvectors we usedto construct ~v, the remaining entries are zero. This gives us an analytical expression tocompute the diameter in an arbitrary direction, if Q and the eigenvalues of Ξ are known.

5.3.4 Statistical bound on the eld range

In the case that Q and K are not known, but can be approximated by a random matrixfrom some ensemble, it is possible to obtain a lower bound on the eld displacement. Tocalculate this, an estimate of ||W (v)|| is needed.

In the case that Q is square (P = N), the operator W (v) simply evaluates to

W (v) =2π

Maxi(vi)× v, (5.41)

as the fundamental domain is now a hypercube with sides 2π. Now, if it is reasonable toapproximate Ξ by a random matrix that is rotationally invariant, its eigenvectors will haveentries that are normally distributed, such that they will exhibit eigenvector delocalization.The median size of the largest entry can then be approximated by [47]

Maxi(vi) =

√2 erf−1(2−1/N)√

N, (5.42)

56


such that

||W (v)|| = 2π√N√

2 erf−1(2−1/N)(P = N). (5.43)

Furthermore, in the case that Q is not square, it was shown (numerically) by Bachlechner,Long and McAllister that

Maxi((QRQ−1φ)i) ' 2 erf−1(2−

1P−N ) ' 2

√log(P −N), (5.44)

such that

||W (v)|| = π√log(P −N)

(P > N). (5.45)

In the case that P = N , the enhancement of ||W (v)|| grows with N , see gure 5.7, butwhen P > N a large enhancement only occurs when P is slightly bigger than N anddecreases at large N , see gure 5.8. It is now possible to obtain a bound on the maximal

100 200 300 400 500N

15

20

25

30

35

40

45

||W(v)||

Figure 5.7: ||W (v)|| for the case P = N . The enhancement grows with N .

diameter when we consider dierent ensembles of random matrices. For example, if weconsider the kinetic matrix to be diagonal (K = diag(f 2

i )), Ξ becomes

Ξ = f 2(QQT )−1. (5.46)

By diagonalizing Ξ with S, we obtain the relation

diag(ξ2i ) = f 2diag(

1

Q2i

), (5.47)

from which it follows that the largest eigenvalue of Ξ is given by

ξmax =f

Min(Qi). (5.48)

57

5 Axion ination

0 100 200 300 400 500N

2

4

6

8

10||W(v)||

PN=1.1

PN=1.4

PN=1.7

PN=2

Figure 5.8: ||W (v)|| for dierent values of P/N . A large enhancement is possible for asmall P/N , which decreases with N .

By using some properties of random matrix theory, (5.43) and (5.45), it was shown thatthe maximal diameter is bounded by [47]

D .

fN3/2 (P = N)

fN (P > N).(5.49)

Interestingly, this exceeds the naive enhancement of√N we derived earlier. This shows

that for generic congurations, obtaining superplanckian diameters is no problem. Incontrast, its realization in string theory is something that is currently under discussion,see chapter 7.

58

6 Non-minimal coupling

Up till now, we have only considered scalar elds that were minimally coupled to gravity.This means that the Einstein-Hilbert term in the action had the canonical form of

S =

∫d4x√−g

M2p

2R. (6.1)

In contrast, one might add some function f(φ) multiplying the Ricci scalar, such that theaction deviates from the canonical Einstein-Hilbert form.

S =

∫d4x√−gf(φ)R. (6.2)

Adding a non-minimal coupling can be motivated in dierent ways. For example, it iswell-known that there exists interesting models of ination that require a non-minimalcoupling, such as Higgs ination [51] and cosmological attractor models [52]. Whereasthe non-minimal coupling is sometimes considered a pathology in Higgs ination, it is afeature in the attractor models. Furthermore, even when a non-minimal coupling is absentin the classical action, it will be generated by quantum corrections [53].

Despite being an asset in some inationary models, a non-minimal coupling can also spoilthe atness of the potential [54]. We therefore have to make sure that the non-minimalcoupling is theoretically under control. In this chapter, we will treat some importantaspects that one encounters when including a non-minimal coupling and show how a non-minimal coupling can be introduced in supergravity. Continuing, we treat some interestingmodels and show how a non-minimal coupling is radiatively generated in the presence ofN scalars.

6.1 Description of a non-minimal coupling in the Jordan

and Einstein frame

In the action, a non-minimal coupling admits a dual description. If we describe thenon-minimal coupling in the Jordan frame, it appears as a function that multiplies theEinstein-Hilbert term in the action. Alternatively, we can perform a conformal transfor-mation on the metric that renders the Einstein-Hilbert term canonical, but changes theaction for the scalar eld. This description is known as the Einstein frame. These two dif-ferent descriptions are physically equivalent, since they are simply related by a conformaltransformation. Nevertheless, the non-trivial relation between the two frames can hide

59


interesting physics in one frame, while it is manifest in the other frame and vice versa.We will show this below.

Consider a scalar eld non-minimally coupled to gravity and with a canonical kinetic termin d spacetime dimensions.

SJordan =

∫ddx√−g[f(φ)R− 1

2gµν∇µφ∇νφ− V (φ)

](6.3)

Here, Md is the d-dimensional reduced Planck mass. We can perform a conformal trans-formation on the metric gµν .

gµν = Ω2gµν (6.4)

The Ricci scalar R constructed from gµν is then given by [55]

R =1

Ω2

[R− 2(d− 1)

ΩΩ− (d− 1)(d− 4)

Ω2

], (6.5)

with

Ω = gµν∇µ∇νΩ =1√−g

∂µ(√−ggµν∂νΩ

). (6.6)

Furthermore, the determinant of the metrics are related as√−g = Ωd

√−g. (6.7)

Using these identities, the gravitational part of (6.3) can be written as∫ddx√−gf(φ)R =

∫ddx

√−g

Ωdf(φ)

[Ω2R +

2(d− 1)

ΩΩ

+(d− 1)(d− 4)

Ω2gµν∇µΩ∇νΩ

].

(6.8)

From the rst term in (6.8), it is clear that in order to render the gravitational part of theaction canonical, we have to make the identication

f(φ)Ω2−dR =Md−2

d

2R. (6.9)

Now, the second term in (6.8) can be integrated by parts, which results in1∫ddx√−g2(d− 1)

Ωd+1f(φ)Ω = −

∫ddx√−gMd−2

d

(d− 1)(d− 3)

Ω2gµν∇µΩ∇νΩ. (6.10)

1Note that there is a dierence between ∇ and ∇. However, we are only acting on scalars, such that∇Ω = ∇Ω.

60

6.1 Description of a non-minimal coupling in the Jordan and Einstein frame

The last term in (6.8) can be rewritten as.∫ddx√−gM

d−2d

2

(d− 1)(d− 4)

Ω2gµν∇µΩ∇νΩ (6.11)

Adding these contributions, the gravitational part of (6.3) can be written in quantitiesconstructed from gµν as

S =

∫ddx√−gM

d−2d

2

[R− (d− 1)(d− 2)


]. (6.12)

The scalar part of (6.3) can be written as follows.∫ddx√−g[−1

2Ω2−d∇µφ∇νφ−

V (φ)

Ωd

](6.13)

Finally, we can write the complete action in terms of quantities constructed from gµν as

S =

∫ddx√−g[Md−2

d

2R− Md−2

d

2

(d− 1)(d− 2)


−1

2Ω2−dgµν∇µφ∇νφ− V (φ)

] (6.14)

where we dened V (φ) ≡ V (φ)/Ωd. This action can be conveniently rewritten in terms off(φ) by using (6.9). The Einstein frame action is then given by

SEinstein =

∫ddx√−g[Md−2

d

2R− 1

2K(φ)gµν∇µφ∇νφ− V (φ)

]. (6.15)

We dened the kinetic function K(φ) as

K(φ) ≡Md−4d

(d− 1

d− 2

)f 2φ

f(φ)2+Md−2

d

2f(φ), (6.16)

where

fφ = ∇φf. (6.17)

In four dimensions, (6.15) can be written as

SEinstein =

∫d4x√−g[M2

p

2R− 1

2K(φ)gµν∂µφ∂νφ− V (φ)

]. (6.18)

We conclude that a scalar non-minimally coupled to gravity with a canonical kinetic termin the Jordan frame is equivalent to a minimally coupled scalar with a non-canonical kineticterm in the Einstein frame. We can straightforwardly generalize the above derivation tothe case of multiple scalars. If we take the Jordan frame action for multiple scalars

SJordan =

∫ddx√−g[f(φA)R− 1

2δAB g

µν∇µφA∇νφ

B − V (φA)

], (6.19)

61


the corresponding Einstein frame action in four dimensions is

SEinstein =

∫d4x√−g[M2

p

2R− 1

2KAB(φA)gµν∂µφ

A∂νφB − V (φA)

], (6.20)

with

KAB(φ) = M2p

[3

2

fAfBf 2

+δAB2f

](6.21)

and

fA = ∇Af. (6.22)

6.1.1 Canonical normalization of the kinetic term

We just saw that we could describe a non-minimal coupling in the Einstein frame by acanonical Einstein-Hilbert term, at the expense of having a non-canonical kinetic term forthe scalar eld(s). However, in the case of a single scalar eld, we can always performa eld redenition that renders the kinetic term canonical. In other words, we want todene a eld φ such that

−1

2gµν∂µφ∂νφ = −1

2K(φ)gµν∂µφ∂νφ. (6.23)

Dividing by ∂µφ∂µφ gives (

dφ

dφ

)2

= K(φ). (6.24)

Thus, the canonically normalized eld can be obtained by integrating

dφ = ±[

3

2

f 2φ

f 2+

1

2Mpf

]1/2

dφ. (6.25)

We conclude that (6.18) can be simplied to

SEinstein =

∫d4x√−g[M2

p

2R− 1

2gµν∂µφ∂νφ− V (φ)

]. (6.26)

One should be careful in thinking of this action as being equivalent to the simple singlescalar eld action (2.32) rst used to describe ination. Due to the conformal transfor-mation and eld redenition to bring it into this form, (6.26) can show a very dierentbehaviour than when we start with the simple action (2.32).

In the multield case, the situation is a bit more subtle. Because the kinetic term KAB canbe interpreted as a metric on eld space, the statement that there exists a set of canonical

62

6.2 Supergravity formulation of a non-minimal coupling

elds φA is equivalent to the statement that there exists a conformal transformationthat renders

KAB → KAB = δAB. (6.27)

Thus, in order to obtain a canonical kinetic matrix, KAB must be conformally at [55]. Inorder to be conformally at, all components of the Riemann tensor constructed from themetric must vanish [56].

Rabcd = 0 (6.28)

The number of independent components of the Riemann tensor is given by [56]

1

12N2(N2 − 1), (6.29)

where N is the number of elds. We see that it becomes increasingly more dicult forKAB to be conformally at with an increasing number of elds2.

We can deal with this diculty in the following way. On a curved manifold, it is alwayspossible to dene local inertial coordinates, for which the metric is at, but only in somenite region. Therefore, this method can only be applied when ination does not takethe elds to far from their original value. When this condition is satised, new slow-rollparameters can be dened and we can continue in the same manner as usual, see AppendixA of [57].

6.2 Supergravity formulation of a non-minimal coupling

A non-minimal coupling can also be formulated in supergravity. Some years ago, a newformulation of supergravity was discovered, which makes it possible to write the N = 1supergravity action in the Jordan frame [58, 59]. This inspired new models of ination insupergravity [60]. To show the relation between the Einstein and Jordan frame, we startfrom the Einstein frame supergravity action (in the absence of gauge interactions) with aset of superelds

zA, zA

.

SEinstein =

∫d4x√−g[

1

2M2

p R−1

2KAB g

µν∂µzA∂ν z

B − V (z)

](6.30)

Here, the Einstein frame potential is given by the standard supergravity F-term potential,see (4.6). In supergravity, the kinetic matrix KAB is the Kähler metric. If we then denethe conformal transformation between gµν and gµν as

gµν = Ωgµν , (6.31)

2For N = 2, N = 3, it is sucient that the components of respectively the Ricci scalar and Ricci tensorvanish.

63


we can relate quantities constructed from both metrics in the following way.√−g = Ω2

√−g (6.32)

R =1

Ω

[R− log Ω +

3

2

gµν

Ω2∂µΩ∂νΩ

](6.33)

Using these identities, the action can be written in the Jordan frame.

SJordan =

∫d4x√−g[M2

p

2ΩR + 3M2

pΩABgµν∂µz

A∂ν zB − V (z)

], (6.34)

where V (z) = V (z)/Ω2 and the Kähler potential can be related to Ω as follows.

K = −3 log Ω (6.35)

We can now choose Ω such that the kinetic term in the Jordan frame is canonical [61].

Ωpol = 1− 1

3

(δAB

zAzB

M2p

+ J(z) + J(z)

), (6.36)

where J(z) is an arbitrary holomorphic function. Alternatively, the authors of [61] alsohave considered an exponential Ω, given by

Ωexp = exp

[−1

3

(δAB

zAzB

M2p

+ J(z) + J(z)

)]. (6.37)

This makes it clear how a non-minimal coupling can be introduced in supergravity.

6.2.1 Examples of non-minimal supergravity models

The models considered in [61] have two superelds (zA = (S,Φ)) and the function J(z)was specied as

J(z) = −3χ

4

Φ2

M2p

, (6.38)

where χ is some real number. The strength of the non-minimal coupling can be measuredby ξ, which is dened as

ξ = −1

6+χ

4. (6.39)

Using this parameter and (6.38) we can rewrite (6.36) and (6.37) as

Ωpol = 1− 1

3M2p

[SS − 3ξ

2(Φ + Φ)2 − 1

2(3ξ + 1)(Φ− Φ)2

](6.40)

Ωexp = exp

[−1

3M2p

(SS − 3ξ

2(Φ + Φ)2 − 1

2(3ξ + 1)(Φ− Φ)2

)]. (6.41)

64

6.3 Cosmological attractors

If we assume that it is possible to stabilize S = Im(Φ) = 0, we can describe ination byonly a single eld Re(Φ) = φ/

√2.

Ωpol|S=Im(Φ)=0 = 1 + ξφ2

M2p

(6.42)

Ωexp|S=Im(Φ)=0 = eξ φ

2

M2p (6.43)

Thus, the action in the Jordan frame is given by

SpolJordan =

∫d4x√−g[

1

2(M2

p + ξφ2)R− 1


](6.44)

SexpJordan =

∫d4x√−g[M2

p

2eξ φ

2

M2pR− 1


]. (6.45)

We see that ξ indeed determines the strength of the non-minimal coupling of the inatonto gravity. Specically for SpolJordan, χ = 0 corresponds to a canonical Kähler potentialwhich leads to a value of ξ = −1

6. When ξ takes this value, the inaton is conformally

coupled, which means that it does not get renormalized by quantum corrections [62].

Now that we know the form of the action of these models in the Jordan frame, we canconstruct models of ination with a non-minimal coupling in supergravity. Rememberfrom section 4.2.1 that a shift symmetry in the Kähler potential leads to a at direction,suitable for ination. As we can see from (6.40) and (6.41), the non-minimal coupling ξbreaks this symmetry. When choosing the superpotential as

W = Sg(Φ), (6.46)

this leads to the Einstein frame potential V (Φ) = eK/M2p |g(Φ)|2. Thus for (6.40) and (6.41)

we get

Vpol|S=Im(Φ)=0 =

(1 + ξ

φ2

M2p

)−3 ∣∣∣g(φ/√

2)∣∣∣2 (6.47)

Vexp|S=Im(Φ)=0 = e−3ξ φ

2

M2p

∣∣∣g(φ/√

2)∣∣∣2 , (6.48)

which for an appropriate choice of parameters, can sustain ination.


The most interesting behaviour of a non-minimal coupling becomes explicit in the contextof cosmological attractors. These attractors were discovered when it was observed that

65


dierent types of models had the same cosmological predictions to leading order [63]. Theattractor models known as ξ-attractors have predictions that converge to [64, 65]

ns = 1− 2

N

r =12

N2,

(6.49)

for large values of ξ. Here, N is the number of e-folds. Similarly, there exist a class ofα-attractors which has the predictions [66]

ns = 1− 2

N

r =12α

N2,

(6.50)

for small values of α. These models encompass already existing models of ination (suchas Starobinsky [67] and Higgs [68, 51] ination).

Furthermore, it was shown that the ξ and α-attractors overlap for specic values of the rel-evant parameters [69] (see also gure 6.1) and that the α-attractors can also be embeddedin supergravity [66, 70].

Figure 6.1: Classication of the dierent attractor models. Figure from [69].

In this section, we will present the dierent attractors models and show how these can beembedded in supergravity.

6.3.1 ξ-attractors

The ξ-attractors have the following action in the Jordan frame.

SJordan =

∫d4x√−g[M2

p

2Ω(φ)R− 1

2KJ(φ)gµν∂µφ∂νφ− V (φ)

](6.51)

66


By taking KJ = 1 and using the conformal transformation (6.31) this action can bebrought into the Einstein frame.

SEinstein =

∫d4x√−g[M2

p

2R− 1

2K(φ)gµν∂µφ∂νφ− V (φ)

], (6.52)

with

K =3

2

(Ωφ

Ω(φ)

)2

+1

Ω(φ). (6.53)

If we now make the choice Ω = 1+ξg(φ), this model exhibits attractor behaviour for largeξ. We can see this as follows. For large ξ, the rst term in K dominates, such that (6.25)becomes

dφ = ±√

3

2

Ωφ

Ω(φ)dφ. (6.54)

Integrating this expression yields

φ = ±√

3

2log (1 + ξg(φ)) . (6.55)

One has to be careful in specifying the correct sign, as we want φ to be dierentiable atφ = 0 (at which φ vanishes when we take g(φ = 0) = 0). If g(φ) is odd, φ is well-denedfor any value of φ. In contrast, when g(φ) is even, we should pick the opposite solutionsfor φ < 0 and φ > 0 [71].

Without loss of generality, we can take the Jordan frame potential to be

V (φ) =λ

ξ2(Ω(φ)− 1)2. (6.56)

If g(φ) is odd, we can pick the positive solution of (6.55), such that the Einstein framepotential becomes

Vodd(φ) =λ

ξ2

[1− e−

√23φ]2

. (6.57)

This potential is plotted in gure 6.2. We see that, independent of the choice of g(φ), weobtain the potential (6.57), which corresponds to Starobinsky ination with the predictionsgiven by (6.49). Conversely, when g(φ) is even, the Einstein frame potential becomes

Veven(φ) =λ

ξ2

[1− e−

√23

ˆ|φ|]2

, (6.58)

which is the potential corresponding to Higgs ination. Because these two potentials dieronly for negative φ, but have the same plateau that can be used for ination, they havethe same predictions. In the previous section, we saw how this model with Ω(φ) = 1 + ξφ2

could be embedded in supergravity.

67


2 4 6 8 10ϕ

0.5

1.0

1.5

Vodd(ϕ)

Figure 6.2: Potential corresponding to (6.57). The model corresponding to this potentialis known as Starobinsky ination.

Alternatively, we can also look at models known as induced ination, with Ω(φ) = ξg(φ).For large ξ, we obtain the following canonical eld.

φ = ±√

3

2log(ξg(φ)) (6.59)

Again, the choice of sign is subtle. In this case, φ = 0 corresponds to innite |φ|. It wasargued in [71] that it is sucient to pick the positive solution with 0 < φ <∞, because itmaps to all possible values of φ. By taking the same Jordan frame potential, this resultsin the same potential as for Starobinsky ination.

Furthermore, it was also shown in [71, 72] that there exists another attractor at weakcoupling that leads to the same predictions as for quadratic ination (V (φ) ∼ m2φ2). Atweak coupling, the canonical eld is dened by the expression

dφ =1

Ω(φ)dφ. (6.60)

Because there does not exist a simple expression of the canonical eld, independent of theform of g(φ), it requires a bit more work to check that at weak coupling, there indeedexists an attractor. By expanding Ω(φ), it can be seen that the predictions of models witha weak coupling are equivalent to those of simple quadratic ination [72]. An example ofa supergravity embedding of induced ination was given in the previous section.

68


6.3.2 α-attractors

In addition to the discussed ξ-attractors, there also exist models which exhibit the samedouble attractor behaviour (and for some cases are equivalent to the ξ-attractors [69]),known as α-attractors. This attractor model can be conveniently written in the form of aT-model, which has the action [73]

ST =

∫d4x√−g[

1

2R− 1

2

gµν∂µφ∂νφ

(1− φ2/(6α))2− 1

2m2φ2

]. (6.61)

When we integrate

dφ = ± 1

1− φ2/(6α)dφ (6.62)

we obtain the canonical eld

φ = ±√

6α arctanh

(φ√6α

). (6.63)

Thus, the potential in terms of the canonical eld is given by

VT (φ) = 3αm2 tanh2

(φ√6α

). (6.64)

Interestingly, for large α this model returns to quadratic ination with its correspondingpredictions.

limα→∞

VT (φ) =1

2m2φ2 (6.65)

In contrast, for α N (where N is the number of efolds) the potential is more similar tothe Higgs and Starobinsky type potentials with the predictions (6.50), see gure 6.3.

-15 -10 -5 5 10 15ϕ

0.2

0.4

0.6

0.8

1.0

VT(ϕ)

α=1

α=2

α=3

Figure 6.3: Potential corresponding to (6.64) plotted for dierent values of α. The modelcorresponding to this potential is known as a T-model.

69


Similarly, one can also specify a somewhat dierent potential with the same kinetic term.

S =

∫d4x√−g

[1

2R− 1

2

gµν∂µφ∂νφ

(1− φ2/(6α))2− 1

2m2 φ2

(1 + φ√6α

)2

](6.66)

Because the kinetic term is the same as in the T-models, the canonical eld is again givenby (6.63). This results after rewriting the tanh's a bit in the folowing potential.

VE(φ) =3α

2

[1− e−

√32αφ]2

(6.67)

These models are known as E-models [74]. Again, in the limit of large α this modelconverges to quadratic ination. Conversely, for small α we get Starobinsky-like potentials(for α = 1 this is equivalent to Starobinsky ination), see gure 6.4.

5 10 15ϕ

0.2

0.4

0.6

0.8

1.0

VE(ϕ)

α=2

α=3

α=4

Figure 6.4: Potential corresponding to (6.67) plotted for dierent values of α. The modelcorresponding to this potential is known as an E-model.

6.3.3 Supergravity embedding of α-attractors

In [75], it was suggested how the above T-model can be embedded in supergravity. First,one takes a Kähler and superpotential of the form

K = −3 log

[1− ZZ +

α− 1

2

(Z − Z)2

1− ZZ− SS

3

](6.68)

W =√

3αmSZ(1− Z2

). (6.69)

We then need to stabilize the eld S = 0 (as we saw before) or, alternatively, take it tobe a nilpotent eld (S2 = 0). Additionally, we stabilize Im(Z) = 0 and Re(Z) = z playsthe role of the inaton. Thus, for the Kähler metric we obtain

Kzz = KZZ |S=Im(Z)=0 =3α

(z2 − 1)2(6.70)

70


The canonical eld is dened by

dφ = ±

√3α

(z2 − 1)2dz, (6.71)

from which we obtain

φ = ∓√

3α arctanh(z). (6.72)

The potential is given by the standard supergravity F-term potential

VT (z) = eK/M2p

[KABDAWDBW − 3

|W |2

M2p

]∣∣∣∣S=Im(Z)=0

=1

(1− z2)3

(Kzz∂SW∂SW

)= 3αm2z2.

(6.73)

Thus, in terms of the canonical eld we obtain

VT (φ) = 3αm2 tanh

(φ√3α

), (6.74)

which is equivalent to (6.64) up to a factor of√

2 in the tanh, which can easily be in-troduced in the Kähler potential or by a eld redenition, if we are inclined to do so.Furthermore, the α-attractor models can be extended to not only describe ination, butalso supersymmetry breaking and dark energy [76].

The attractor models of ination have many features. They oer a somewhat model-independent approach to ination, while being well-motivated from the perspective ofsupergravity. Furthermore, by modifying α, one has a large theoretical control over theinationary predictions, such that these models t comfortably inside the ns − r contour,independent of future measurements. The T-models have the same ns for any α but onlydier in r, see gure 6.3.3. In contrast, the E-models show a somewhat dierent trajectoryfor α, see gure 6.6.

71


Figure 6.5: Predictions of the T-model in the ns-r plane. Figure from [75]

Figure 6.6: Predictions of the E-model in the ns-r plane. Figure from [75]

Besides their capability of describing a large region of parameter space, it would be inter-esting to see where this behaviour originates from. In particular, we would like to know ifthe α-attractors are also realizable in string theory, which is not completely clear. Whereaswe will not answer the last question, there has been research towards the interpretation

72

6.4 Radiative generation of a non-minimal coupling

of the parameter α. From the expression for the canonical eld of the T-model

φ =√

6α tanh

(φ√6α

), (6.75)

we can see |φ| <√

6α. Thus, |φ| =√

6α describes the boundary of the moduli spaceon which the Kähler metric is dened. Additionally, to get the correct shape of thepotential, α needs to be positive. Furthermore, the curvature of this moduli space isinversely proportional to α, i.e. the Ricci scalar constructed from the Kähler metric isR = −2/(3α) [77].

We can therefore interpret large α, which converges to quadratic ination, as small curva-ture of the moduli space, while small α, which converges to the predictions of Starobinskyination, corresponds to large negative curvature.


Complementary to a formulation of the non-minimal coupling in supergravity, we canalso consider how a non-minimal coupling is generated in the eective theory by quantumcorrections. As we mentioned, in models that do not explicitly consider such an operator, anon-minimal coupling can be dangerous and spoil ination. In contrast, if a weakly brokenshift-symmetry is present, the non-minimal coupling should be suppressed. Therefore, inthe single-eld case, there is no need to worry too much. However, in the case of multipleelds (as in axion ination), there could arise a competition between the suppression anda N -dependence (where N is the number of elds).

In this section, we will show how a non-minimal coupling is radiatively generated andsuppressed by the number of elds due to the renormalization of the Planck mass in thepresence of N elds.

6.4.1 Perturbative quantum gravity with N scalars

Consider a multiplet of scalars φa (a = 1, · · · , N) minimally coupled to gravity3.

S =

∫d4x√−g[R

κ2+

1

2∂µφ

a∂νφbδabg

µν − V (φa)

](6.76)

Here, κ2 = 32πGN = 2/M2p . For simplicity, we will expand the metric around a Minkowski

background. Goldberg showed that it is convenient to introduce the tensor density [78]

gµν = gµν√−g, (6.77)

3Note that from this section on up to the end of chapter 6, we will use the mostly minus metric convention,i.e. (+,-,-,-).

73


and expand it

gµν = ηµν + κhµν

gµν = ηµν − κhµν +O(κ2),(6.78)

instead of expanding the metric directly. The gravitational part of the action can then berewritten in a convenient form.

SEH =

∫d4x

2κ2

[gρσgλµgκν −

1

d− 2gρσgµκgλν − 2δσκδ

ρλgµν

]gµκ,ρ g

λν,σ (6.79)

Furthermore, the expansion of√−g up to O(κ2) is given by

√−g = 1 +

κ

d− 2hµµ −

κ2

2(d− 2)hµνh

νµ +

κ2

2(d− 2)2(hµµ)2 +O(κ3), (6.80)

see Appendix B for its derivation. Expanding (6.79) up to O(κ) yields the followingexpressions

Sh2

EH =

∫d4x

2

[∂σhλν∂

σhλν − 1

d− 2∂σh∂

σh− 2∂λhλν∂σh

σν

](6.81)

Sh3

EH =

∫d4x

2κ[hσρ∂ρhλν∂σh

λν − 2hλν∂ρhµν∂

ρhλν

− 1

d− 2

(hρσ∂ρh

µµ∂σh

νν − 2hµκ∂ρh

µκ∂ρhνν)

+ 2hµν∂λhµσ∂σh

λν],

(6.82)

which respectively describes the graviton propagator and the the three-graviton vertex.Additionally, at higher order a four-graviton vertex is generated, but it will be irrelevantfor our purposes. Up to O(κ2), the scalar part of the action is

Sκ0

scalar =

∫d4x

2

(∂µφ

a∂µφa −m2a

2φaφa

)(6.83)

Sκ1

scalar =

∫d4x

2κ

(hµν∂µφ

a∂νφa −1

d− 2hµµm2a

2φaφa

)(6.84)

Sκ2

scalar =

∫d4x

4(d− 2)κ2m2

aφaφa

(−hµνhνµ +

1

d− 2(hµµ)2

). (6.85)

Here, we also expanded the potential

V (φa) =1

2m2aφ

aφa +O(φ3a). (6.86)

Finally, we impose the harmonic coordinate condition (also known as the de Dondergauge)

(√−ggµν),ν = 0, (6.87)

which simplies the Feynman rules. The Feynman rules (in d spacetime dimensions) aregiven in gure 6.7. The function fµνρσλδ(p1, p2, p3) is rather lengthy and can be found inAppendix B. This completes the set-up of our perturbative expansion and we are nowready to calculate quantum corrections.

74


µν ρσ = Dµνρσ(p) = i2

ηµρηνσ+ηµσηνρ−ηµνηρσp2−iε

a b = Dab(p) = i δabp2−m2

a

µν

a

b

= V abµν (p1, p2) = iκδab

2

(p1(µp2ν) − 2

d−2m2aηµν

)

µν

ρσ

a

b

= V abµνρσ = iκ

2m2aδab

2(d−2)

(ηµρηνσ + ηµσηνρ − 2

d−2ηµνηρσ

)

µν

ρσ

λδ

= Vµνρσλδ(p1, p2, p3) = iκ2fµνρσλδ(p1, p2, p3)

Figure 6.7: The relevant Feynman rules of perturbative quantum gravity coupled to scalarsup to O(κ2).

6.4.2 Feynman diagrams contributing to the non-minimal coupling

It is well-known that, even when a non-minimal coupling of matter to gravity is absent inthe classical action, it will be generated by quantum corrections, see [53] and referencestherein. In this section we will compute how the non-minimal coupling gµνRµνφ

aφa isgenerated. Therefore, we are only interested in the Feynman diagrams that generate thisterm. In order to identify which diagrams contribute to this operator, we need to knowits momentum structure. This structure can be found by expanding the gravitational partof the action. The terms linear in h, which could be omitted earlier when deriving theFeynman rules (since they were total derivatives) now give rise to the following operator.

ξgµνRµνφ2 = ξκφ2hµµ +O(κ2), (6.88)

where ξ is a dimensionless coecient. Because this operator contains one h and two φ's,it will be generated by loop corrections to the hφφ vertex. The one loop corrections tothe hφφ vertex are given in gure 6.8. All three diagrams contribute to the non-minimalcoupling, but the leading order terms are given by the divergent part of these diagrams

75


M(1)µν : hµν

φa

φb

M(2)µν : hµν

φa

φb

M(3)µν : hµν

φa

φb

Figure 6.8: One loop corrections to the hφφ vertex. These diagrams are responsiblefor generating the non-minimal coupling in addition to renormalizing already existingoperators.

that multiplies the non-minimal coupling. Only M(2)µν and M

(3)µν contain divergences that

multiply the generated non-minimal coupling. In Appendix B the details of the loopcalculation can be found. Here, we simply quote the result.

ξRφaφa ' Nm2

M2p

log

(q2

µ2

)Rφ2 + (subleading terms), (6.89)

up to some numerical factors. Here, q is the external momentum of hµν and µ is anarbitrary renormalization scale.

6.4.3 The renormalization of Mp

Of course, quantum corrections will not only have the eect of generating a non-minimalcoupling. Loop diagrams can also renormalize already existing operators. In particular,the renormalization of the graviton propagator will be of importance, as this is dependent

76


on the number of elds. We will show that this leads to a lowering of the cuto of oureective theory of semi-classical gravity. To be more precise, the cuto will be loweredby a factor of 1/

√N with respect to the naive cuto Mp. If the Feynman rules of the

previous section are used, we have to take the diagrams in gure 6.9 into account for therenormalization of the graviton propagator. However, only the diagrams with scalar loops

µν ρσ µν ρσ µν ρσ

µν ρσ µν ρσ

Figure 6.9: One-loop corrections to the graviton propagator. These diagrams contributeto the renormalization of the Planck mass.

will contribute a N -dependence to the graviton propagator. Comparing the quantumcorrections with the tree level result implies the following cuto of gravity in the large Nlimit [79].

Λ ' Mp√N, (6.90)

from which a relation between Mp and N follows.

M2p ' Λ2

UVN (6.91)

This implies that we can write the gravitational part plus the non-minimal correction as

√−g[NΛ2

UV

2R +

m2

Λ2UV

log

(q2

µ2

)φ2R

]. (6.92)

When we normalize gravity (absorb the N dependence), we obtain

√−g[

Λ2UV

2R +

m2

NΛ2UV

log

(q2

µ2

)φ2R

]. (6.93)

This result diers from the one obtained in [80], where the logarithmic and m2 dependencewas not present. However, on very general grounds it is expected that these factors shouldbe present. First of all, the logarithmic dependence should be present because the φaφaRterm is a marginal operator, which we expect to have a marginal (logarithmic) cutodependence. Furthermore, if we consider a theory which exhibits a weakly broken shiftsymmetry, m2 should be small. In the limit of m2 → 0 the shift-symmetry becomes exact.

77


This implies that terms violating this shift-symmetry (such as a non-minimal coupling)should also vanish. This can only be the case when the non-minimal coupling has anexplicit m2 dependence.

Hence, we conclude that a radiatively generated non-minimal coupling is suppressed byboth the number of elds due to the renormalization of Mp, as well as a parameter thatmeasures the breaking of the shift symmetry.

78

7 The Weak Gravity Conjecture and

axion ination

As was motivated in previous chapters, in order to properly address the UV-sensitivity ofination, an embedding into string theory is necessary. We saw in chapter 5 that axionsare particularly good candidates, although the single axion model seems to resist such anembedding [41]. One might wonder if there exists a general principle of quantum gravitythat tells you that such an embedding is not possible. On the other hand, we also sawthat models with multiple axions (with individual subplanckian axion decay constants) didnot seem incompatible with string theory from an eective point of view. Recently, somegeneral properties of quantum gravity have been used to try to nd out if such multi-axionmodels are compatible with string theory.

One rough argument that indicates that something peculiar is going on goes as follows.Using multiple axions, a superplanckian eld range can be obtained, that is enhancedby a factor of

√N or even more (see chapter 5). In the case of kinetic alignment, as

was considered in section 5.2.2, the maximum diameter was given by√Nfmax. However,

the eld corresponding to fmax is typically the lightest eld [43]. Therefore, it shouldbe possible to integrate out the heavier elds, leaving a single axion with an eectivesuperplanckian decay constant, which should not be allowed in string theory [41].

In addition, the tension between the eective theory of axion ination and the absenceof superplanckian eld ranges in string theory becomes even more apparent when oneconsiders the KNP alignment mechanism. Here, a change of basis relates two subplanckianaxions to a theory in which there arises a superplanckian direction. The discomfort thatis felt by this approach is best captured by the following quote.

If God does not like large axion decay constants in his theory of quantumgravity, it seems strange that we could fool him by a mere basis choice.- Tom Rudelius [81]

Indeed, it is unexpected that a dierent physical conclusion is obtained, just by performinga change of basis.

One of the properties of quantum gravity that has been applied to resolve this confusionwas discovered by Arkani-Hamed, Motl, Nicolis and Vafa [82] in a dierent context. Thisproperty is known as the Weak Gravity Conjecture (WGC). Recently, the WGC wasgeneralized to many instantons coupling to axions to show that the WGC in its strongestform forbids axion ination [83]. On the other hand, some authors have suggested that inparticular cases, axion ination is still allowed [84, 85]. Nevertheless, a reaction to these

79

7 The Weak Gravity Conjecture and axion ination

arguments showed that evading the constraints by the WGC is rather cumbersome andonly possible in very non-generic circumstances [86].

In this chapter, we will rst review the WGC and its motivation in its original formulation.After that, we continue to show the generalization of the WGC to axions. Finally, weconclude with the consequences the WGC has for axion ination and comment on apossible loophole.

7.1 The Weak Gravity Conjecture for particles and gauge

elds

With the discovery of the string theory landscape1, it was discovered that string theorygives rise to an overwhelmingly large number of vacua. Explicitly probing the landscapecan learn us about the properties of string theory, but is a dicult task. Therefore, aneective description is preferable, but a priori it is not known if an eective theory admitsa UV-completion in the landscape.

As it turns out, not all consistent low-energy eective theories have this property, the setof which is named the swampland [87]. The WGC is one of the tools that can help todistinguish the landscape from the swampland. Generalizing the WGC has proven to befruitful in order to nd out if eective theories of axion ination have a UV-completion inthe landscape or belong to the swampland.

Qualitatively, the WGC can be described as the notion that in any theory containinggravity, gravity should be the weakest force. More precisely, in a four-dimensional theorycontaining gravity and a U(1) gauge eld with mass m and coupling g, there should exista particle that satises

m

g≤ 1, (7.1)

in appropriate units. A natural question that arises from this conjecture is; why shouldwe believe (7.1) to be true? On top of the fact that we observe this bound to be satisedin any theory that we know of, the WGC can also be motivated from dierent arguments,such as the absence of global symmetries in (quantum) gravity [17]. Here, we present theargument by Arkani-Hamed, Motl, Nicolis and Vafa [82]. If we take the gauge couplingg → 0, the gauge symmetry becomes indistinguishable from a global symmetry. Therefore,something should prevent us from taking this limit. The WGC resolves this, as it tells usthat g cannot be taken arbitrarily small. To see what happens more in detail, consider thatwe have a very tiny gauge coupling g ∼ 10−100 and a black hole with a mass M ∼ 10Mp.The black hole should satisfy the bound

M

Q≥ 1. (7.2)

1The set of all stable vacua of string theory is usually referred to as the landscape. While we saw inchapter 4 that the current status of dS vacua is unclear, a landscape of AdS vacua is well-established.

80

7.2 Generalization to p-forms

Now, we let the black hole evaporate until it reaches its extremal limit (M = Q). If thereexists no light charged state, the black hole will have a possible charge ranging from 0to 10100 which it is not able to radiate away. This leads to 10100 remnants [88], whichare associated with the same problems that global symmetries in gravity have [89]. Wetherefore conclude that there must exist a state that satises (7.1), in agreement with theWGC.

When accepting that such a light charged state must exist, a further distinction betweenthe properties of this state can be made, leading to two versions of the WGC.

• The strong WGC: mq≤ 1 for the lightest state in the theory.

• The mild WGC: mq≤ 1 for any state in the theory.

While this distinction is sound, it is important to mention that no violations of the strongWGC have ever been found.

7.2 Generalization to p-forms

In the previous section, we formulated the WGC for a four-dimensional theory of gravitywith particles coupling to a U(1) gauge eld. To apply the WGC conjecture to axions,a generalization to p-forms is required (The U(1) gauge eld we considered before is a1-form). This generalization was already made in [82] and translates into a bound on thetension of the object electrically charged under the p-form gauge eld.

T .

(g2

GN

)1/2

(7.3)

For an axion (0-form) this bound describes the tension of instantons charged under anaxion, which is given by the instanton action

Sinst .Mp

f(7.4)

Here, f is the axion decay constant. This is consistent with the result that led to theconclusion that ination with a single axion (which requires f > Mp) is not possible.Banks, Dine, Fox and Gorbatov found that whenever f > Mp, an instanton with ananomalous small action contributes higher harmonics to the potential (the instanton con-tribution appears as e−Sinst in the potential), spoiling ination [41]. Indeed, (7.4) indicatesthat whenever f > Mp, one would expect unsuppressed instantons to contribute to thepotential to satisfy the WGC.

Even so, this result was already known. How does the WGC constrain theories withmultiple axions? The extension to multiple U(1) gauge elds was given in [90], where theset of vectors

~zi =~qimi

(7.5)

81


was introduced. Here, ~qi is the vector that contains the charges of the particles (labeledwith the index i) under the gauge groups. In order for an extremal black hole (which has|~z| = 1) to be able to decay, it is necessary for the WGC to be satised in every directionof charge space. Satisfying the WGC is thus equivalent to the statement that the convexhull of the ~zi's contains the unit ball [90], see gure 7.1. This condition will be referredto as the convex hull condition (CHC). The last step necessary to be able to apply the

|~z|=

1

|~z|=

1

Figure 7.1: In the left gure, four particles coupling to two U(1) gauge groups fail tosatisfy the CHC, as the convex hull of the charge vectors does not contain the unit ball.In the right gure, two additional particles ensure the CHC to be satised.

WGC to axions is the translation of the CHC in the case of multiple U(1) gauge groups tomultiple axions. This complicated procedure was executed in the context of string theoryin [83]. In the case of axions, the following set of vectors is introduced.

~zk =~QkSk

(7.6)

Here, ~Qk is the vector that contains the charges of the instantons (labeled with the index k)under the axions (the charges are roughly proportional to the inverse decay constants). TheCHC then states that the convex hull of the charge vectors contains a ball of a particularradius, where the precise number depends on the type of axion we are considering. Forexample, in type IIB string theory where the axion originates from a C2 form, the convexhull should contain the ball of radius 2/

√3 [83].

7.3 Applying the Weak Gravity Conjecture to ination

Now that we know how the WGC manifests itself for axions, we are ready to calculatehow it constrains axion ination. Specically, we will see that the eld range of the axionsis limited to be subplanckian when the strong WGC is obeyed. While this was clear inthe single-axion case, there exist an ambiguity in dening the eld range in the multi-axion case. In the single-axion case, the axion decay constant completely determines the

82


maximum eld range. In contrast, in the multi-axion case we have to dene the eld rangewith respect to the `physical' (canonical) elds that do not solely depend on the decayconstants, but also on basis transformations, as we saw in chapter 5.

To prevent criticism regarding the denition of the eld range, we will use two dierentdenitions and obtain the same conclusions. First, we use the denition of the eld rangein terms of the mass matrix, as was done in [83] and secondly, we will dene the eldrange by looking at the periodicity of the axions, as was done in [86].

7.3.1 Constraints on the eld range from the mass matrix

Consider the potential arising from instantons (labeled with k) coupling to a set of axions~θ.

V =∑k

Λ4e−Sk[1− cos

(~Qk · ~Φ

)](7.7)

Here, the kinetic term is already canonically normalized. Expanding (7.7) around itsminimum and taking all Sk ∼ S gives

V = Λ4e−S~ΦT (QTQ)~Φ. (7.8)

We see that QTQ is proportional to the mass matrix. As small masses are required forination, the eigenvalues of QTQ are a good proxy for the square of the inverse decayconstants.

fi =Mp

mi

, (7.9)

where m2i are the eigenvalues of QTQ. Thus, if we are able to constrain mi, we automati-

cally have a constraint on fi. In order to be in a regime of perturbative control (in whichthe instanton contributions are under control), we require

Sk ≥ r−1, (7.10)

where r is the radius of the ball necessary to satisfy the CHC. Furthermore, we onlyconsider the minimum allowed value of Sk, to show that even in the most optimal case,we will not be able to obtain a superplanckian eld range. The CHC now simplies tothe statement that the convex hull of the ~Qk's should contain the unit ball. Finally, wewill use a mathematical reformulation of the CHC which tells us that the following twostatements are equivalent.

1. The convex hull of the ~Qk's contains the unit ball.

2. For any unit vector ψ, there exists a unique set of numbers αj and r such that∑j

αj = 1, r > 1 and∑j

αj ~Qk = rψ.

83


For the proof of this equivalence, we refer the reader to [83]. Now, consider the eigenvalueequation

QijQjkψ

k(n) = m(n)ψ

i(n). (7.11)

Here, ψ(n) is the nth eigenvector with eigenvalue m(n). Note that the eigenvectors are

orthonormal, since QTQ is symmetric. Multiplying (7.11) with a particular eigenvector ~φwith eigenvalue m gives

m = (Qijφi)(Qjkφ

k)

m =∑j

(Qijφi)2

m =∑j

α2j ,

(7.12)

where we dened αj2 ≡ (Qijφi)2. We also dene

αj ≡αj∑k

αk. (7.13)

By dividing the eigenvalue equation of ~φ by∑l

αl and rewriting a bit, we obtain the

following relation.

QijQjkφ

k∑l

αl=

mφi∑l

αl

QijQjkφ

kαl

αl=∑l

(α2l

αl

)φi

∑l

αl ~Ql =∑l

(α2l

αl

)~φ

(7.14)

In the second line, we used equation (7.13) on the left hand side and (7.12) on the righthand side. In the third line, we used the denition of αl. From the denition of the CHC,we obtain ∑

l

(α2l

αl

)= r > 1. (7.15)

It follows that at least one of the αl's must be bigger than one, in order to satisfy (7.15).By looking at (7.12), we see that this implies m > 1. Because m is an arbitrary eigenvalue,we can repeat this argument and obtain the same bound for the other eigenvalues. Thisleads via (7.9) to

fi ≤Mp. (7.16)

Because we needed fi > Mp for ination, this forbids ination using this proxy of the eldrange.

84


7.3.2 Constraints on the eld range from periodicity

Here, we use the same potential (7.7), but use the periodicity of the cosines to dene theeld range, as was also done in chapter 5. Now, every instanton gives a constraint.

| ~Qk · ~Φ| ≤ π ∀k (7.17)

It will be convenient to use the following parametrization.

~Qk · ~Φ = c ~Qk · Φ, (7.18)

where Φ is a unit vector in an arbitrary direction. The diameter of the fundamentaldomain D is then most strongly constrained by

D(Φ) =2π

maxk( ~Qk · Φ)(7.19)

and the largest possible diameter is given by the direction Φ that maximizes D. We willnow calculate how this notion of the eld range is limited by the CHC. Remember thatthe statement that the convex hull of the ~zk's contains the unit ball is equivalent to∑

k

αk~zk = rΦ, (7.20)

for a unique set αk and where r > 1. Now, we assume that it is possible to obtain

D(Φ) > 2π. (7.21)

From (7.19) it then follows that

~Qk · Φ < 1 ∀k. (7.22)

Using the denition of ~zk in (7.20), we get∑k

αk~QkSk

= rΦ (7.23)

and taking the dot product with ~Qj results in∑k

αkSkQkj = r Φ · ~Qj︸︷︷︸

<1

, (7.24)

where we dened

Qij ≡ ~Qi · ~Qj. (7.25)

From (7.24) we now obtain the bound

r >∑k

αkSkQkj. (7.26)

85


Taking the norm of (7.23) results in∑kj

αkαjSkSj

Qkj = r2, (7.27)

and plugging it into (7.26) leads to

r > r2∑j

Sjαj

r > r2

r < 1,

(7.28)

where in the rst line we used∑j

αj = 1 and required Sk > 1, to be in a region of

perturbative control. This is however in contradiction with the CHC, which requiresr > 1. Therefore, we conclude that D ≤ 2π along any direction, such that the eld rangeof the axions is limited to be subplanckian.

7.4 Contribution of gravitational instantons and a

loophole

In the previous section, we saw that it was not possible to obtain a superplanckian eldrange, while remaining in a regime of perturbative control. Nevertheless, in an eectivetheory it is straightforward to obtain a superplanckian eld range. Moreover, a super-planckian eld range was even found in explicit string compactications [47]. How can wereconcile these seemingly contradicting observations? The point of view motivated by theWGC is that when one obtains a transplanckian eld range in string theory, perturbativecontrol is lost and additional contributions are expected that will ensure the WGC to besatised. This also strongly constrains the eld range. In constructions where a trans-planckian eld range is obtained, these additional contributions are not captured by theeective theory [84].

As was calculated in [84], these additional contributions come from gravitational instan-tons. When the CHC is not satised, contributions of gravitational instantons to the po-tential will be non-negligible and these extra contributions ensure the CHC to be satised,see gure 7.2. At the same time, the gravitational instantons contribute higher harmonicsto the potential, because they have a potential that is proportional to cos(nφ/f), wheren is an integer, which spoils an initial superplanckian eld range [84].

The only way in which the CHC can be satised while at the same time gravitational in-stantons do not contribute higher harmonics, is in a case where the gravitational instantonsare subdominant with respect to the instantons responsible for generating a transplanck-ian eld range. To illustrate this, consider a potential with two contributions. The rst

86

7.4 Contribution of gravitational instantons and a loophole

|~z|=

1

|~z|=

1

Figure 7.2: In the left gure, four instantons (blue dots) with orthogonal ~Qk's (as in N -ation) fail to satisfy the CHC. We expect an additional contribution from gravitationalinstantons (green dots, right gure) to satisfy the CHC. However, these instantons alsocontribute higher harmonics to the potential.

contribution is responsible for generating a large at direction and the second one is thecontribution from a gravitational instanton, which ensures the CHC to be satised.

V = Λ41e−S1

[1− cos

(φ

f

)]+ Λ4

2e−S2

[1− cos

(nφ

f

)](7.29)

Here, f > Mp and f/n < Mp. In order for the second term to be negligible with respectto the rst, S2 S1, if we assume Λ1 ' Λ2. Such a setup is however forbidden bythe strong WGC, as in this case it is not the particle with the smallest Sk that satisesthe WGC. Therefore, if we believe that the strong WGC should not be violated, wearrive at the same conclusion that obtaining a superplanckian eld range with axionsis not possible. Contrary, if we believe that it is reasonable for only the mild WGC tobe satised, this loophole may be exploited. There has been some eort in constructingmodels that exploit this loophole, although it is not clear whether these constructions arefully realizable in explicit string compactications, as they depend on subtle details of themoduli stabilization scheme employed [91].

This leads us to the conclusion that the strong WGC forbids any construction of axionination. While there might be possibilities to use axions for ination by violating thestrong WGC, no good motivation exists to do so. Even though, it would be interesting tond out if such constructions are consistent, as they would lead to new insights, not onlyof ination, but also about the nature of quantum gravity.

87

8 Conclusions and outlook

Now that we have reached the end of this thesis, it is time to recap and conclude to seewhat we have learned. In chapter 2, we saw that in order to address the initial conditionsof the universe, a period of ination that precedes the hot Big Bang phase of the universeis required. Without this period, the initial state of the universe that leads to today'sobserved homogeneity, isotropy and atness would have to be extremely ne-tuned. Thisperiod of ination leads to natural initial conditions of the hot Big Bang phase of theuniverse.

Furthermore, when treating ination quantum mechanically, a nearly scale-invariant spec-trum of scalar and tensor perturbations is predicted. The scalar perturbations lead tosmall temperature anisotropies in the CMB and have been measured with a very highprecision. From these measurements, information about the potential of the scalar eld(s)driving ination can be inferred.

These measurements however, do not yet establish the energy scale at which inationoccurred, because there is a degeneracy between the Hubble scale H and the slow-roll pa-rameter ε in the scalar power spectrum. This degeneracy would be lifted by a measurementof the tensor power spectrum, which is directly proportional to H. However, the amplitudeof the tensor perturbations is not necessarily observable and depends on the inationarymodel one is considering. If ination does produce observable tensor perturbations, thisimplies that ination occurred at a relatively high energy scale: E ' 10−2Mp.

Moreover, the Lyth bound tells us that a model producing an observable signal of tensorperturbations has to be a large-eld model. On the one hand, these models are fascinating,since they are intimately related to interesting physics due to their extreme UV-sensitivity.On the other hand, this extreme sensitivity renders ination impossible without symme-tries that prohibit large corrections.

If we assume for a moment that the symmetries during ination forbid large correctionsand are furthermore UV-completed, we can build an eective eld theory of ination,as we have done in chapter 3. Instead of starting with a scalar eld and computingthe slow-roll parameters to see if the potential can allow for ination, we assume aninating background and describe perturbations around it. This has many advantages;the eective action encompasses all single-eld slow-roll models, loop corrections can easilybe incorporated and degrees of freedom that are irrelevant decouple from the theory.

Nevertheless, the low-energy symmetries are not necessarily UV-completed. A well-knownexample of this is the absence of global symmetries in quantum gravity. Therefore, pur-suing an eective description of ination without a proper justication of the low-energy

88

symmetries might be dangerous. This urges us to go beyond the eective theory of inationand check symmetry assumptions in a UV-complete theory. Because ination incorporatesgravity and the UV-completion of gravity is a theory of quantum gravity, ination needsto be embedded in quantum gravity.

String theory is the best candidate for a theory of quantum gravity, which motivates usto embed ination into string theory. Aspects of string theory were treated in chapter 4,where we also saw that building models of ination in string theory is a dicult endeavour.Using inspiration from string theory, we can construct novel eective theories (for examplein supergravity) which we know are UV-completed in string theory. An example of sucha model is axion ination, which we presented in-depth in chapter 5. However, in orderto construct a large-eld model we needed to use many axions, because the axion decayconstants are limited to be subplanckian in string theory.

In chapter 6, we digressed for a bit and considered a non-minimal coupling to gravity. Inparticular, we discussed an interesting class of models which exhibit attractor behaviour(the α and ξ-attractors). Here, we also took a completely bottom-up perspective andcalculated how a non-minimal coupling is always generated by quantum corrections andshowed that it is suppressed by the number of elds. This result complements alreadyexisting results in the literature, showing that from an eective point of view, models witha large number of scalars are theoretically under control.

Coming back to the main issues of this thesis in chapter 7, we argued that recent workconcerning the Weak Gravity Conjecture strongly constrains superplanckian eld ranges,in contrast with the eective results of chapter 5. Only in very constrained set-ups,one can still obtain a superplanckian direction in eld space. Currently, it is unclear ifan explicit construction of ination using axions is possible, while satisfying the WeakGravity Conjecture.

Concluding, the general lesson we can learn from the issues we discussed in this thesis isthat the eective description of ination is not sucient to capture all the subtleties thatarise from its UV-sensitivity. In order to theoretically understand ination, one needs to gobeyond the eective theory of ination. Embedding ination in string theory is thereforenecessary to have a better understanding of the high-energy origin of our universe.

Unfortunately, there are many obstacles to overcome before this goal can be reached.Foremost, it is unknown if de Sitter vacua are present in string theory, as it is unclear ifthe uplifting procedure used by KKLT to construct de Sitter vacua is consistent. Becauseination is described by a quasi-de Sitter phase, this question rst needs to be answered.Alternatively, an interesting direction of future research is to nd other ways of construct-ing de Sitter vacua. Some research in this direction has already been performed [31, 32],but there do not yet exist conclusive answers.

A second interesting direction of research would be to investigate holography in the con-text of de Sitter space. Holography in the form of the AdS/CFT correspondence hasrevolutionized the way we think about (quantum) gravity. Whereas this correspondencehas led to many new insights, it only is applicable to AdS and not dS space. Therefore,extending holography to a dS/CFT correspondence (if it exists) could lead to new insights

89

8 Conclusions and outlook

about quantum gravity, the nature of dS space and therefore ination. This direction wasrst pursued by Strominger in 2001 [92]. Since then, there has been some progress, but itis not nearly as well-established as the AdS/CFT correspondence.

Concluding, this thesis shows that ination is well established in an eective way. How-ever, to describe its theoretical origin, a better understanding of high-energy physics in acosmological context is needed. In the end, our hope is that cosmology and high-energyphysics are two subjects that will be joined together, to answer the biggest questions aboutour universe.

90

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A The eective eld theory of ination

The most general action

The following quantities invariant under spatial dieomorphisms can be present in theeective action [19].

• Quantities invariant under all (spatial + time) dieomorphisms, since this leaves theaction invariant, e.g. the Ricci scalar R.

• General functions depending on t. Since ∂µt = δ0µ (in unitary gauge t = t, so t loses

its space dependence), is a general function depending on t we can always use δ0µ to

contract an index. Thus we can also write quantities with an upper index 0.

• Quantities that live on surfaces of constant t. In order to construct these quantities,we introduce a unit vector perpendicular to this surface.

nµ =∂µt√

−gµν∂µt∂ν t=

δ0µ√−g00

(A.1)

One can easily check this denes a unit vector perpendicular to t. Take for examplethe unperturbed FRW metric, in this case nµ = (1, 0, 0, 0). We also dene theinduced metric hµν ≡ gµν + nµnν . This metric projects vectors onto the surface ofconstant t. One can show this by contracting hµν with an arbitrary vector.

• With the induced metric, we can construct three-dimensional quantities such as(3)Rµνρσ.

• Terms with the covariant derivative acting on ∂µt. Projected on the surface ofconstant t, this denes the extrinsic curvature.

Kµν ≡ hσµ∇σnν (A.2)

Where we only need to project the σ, since if we also project ν

97


hσµhνλ∇σnλ =hσµ (gνλ + nνnλ)∇σn

λ (A.3)

=hσµ∇σnν + hσµnνnλ∇σnλ (by metric compatibility) (A.4)

(A.5)

and nλ∇σnλ = 1

2∇σ(nνnν) = 0.

• Furthermore, we can also write

nσ∇σ = −1

2(−g00)−1hµν∂µ(−g00) (A.6)

so this can be rewritten in terms we already allowed.

• Finally, it is important to note that the 3-dimensional Riemann tensor can be writ-ten in terms of quantities we already noted, using the Gauss-Codazzi relation [19].Also, any three-dimensional quantity can be written as a four-dimensional quantityprojected using hµν and nµ.

Stückelbergs trick and the decoupling limit in a gauge

theory

Consider a general non-abelian gauge theory with gauge eld Aµ which we want to beinvariant under elements U of a gauge group G. The lagrangian is given by

L = −1

4FµνF

µν − 1

2m2AµA

µ (A.7)

Performing a gauge transformation

Aµ → UAµU† +

i

gU∂µU

† ≡ i

gUDµU

†, (A.8)

results in

L =− 1

4FµνF

µν − 1

2m2(

i

gUDµU

†)(i

gUDµU †) (A.9)

=− 1

4FµνF

µν − 1

2

m2

g2DµU

†DµU (A.10)

98

where we have used the identity DµU† = −U †(DµU)U †. This can be proven by observing

that U is unitary.

Dµ(U †U) = (DµU†)U + U †DµU = 0 (A.11)

→ DµU† = −U †(DµU)U †

This lagrangian is not gauge invariant. We can restore gauge invariance by dening U ≡eiπ

aTa , where T a are the generators of G and demand transforms non-linearly as

eiπaTa → Λ(t, ~x)eiπ

aTa (A.12)

under the gauge transformation Λ(t, ~x) [19]. This is called the Stückelberg trick.

From (A.8) we derive

DµU† = −igAµU † + ∂µU

† (A.13)

DµU = igUAµ − U(∂µU†)U

Now the lagrangian can be expanded as

L = −1

2

m2

g2DµU

†DµU (A.14)

= −1

2

m2

g2[−igAµ(1− πaT a) + ∂µ(1− πaT a)]

× [ig(1 + πaT a)Aµ − (1 + πaT a)∂µ(1− πaT a)(1 + πaT a)]

= −1

2

m2

g2

[g2AµA

µ − 2igAµ∂µ(πaT a)− ∂µ(πaT a)∂µ(πaT a) + . . .]

= −1

2m2AµA

µ + im2

gAµ∂µ(πaT a) +

1

2

m2

g2∂µ(πaT a)∂µ(πaT a) + . . . ,

where the dots denote higher order terms and we dropped the FµνFµν term.

The leading order mixing term between the gauge eld and π in (A.15) is proportionalto

im2

gAµ∂

µπa = imAµ∂µπac , (A.15)

where we canonically normalized πac ≡ m/g · πa. We can ignore such mixing terms in thelimit m→ 0 and g → 0, while keeping m/g xed.

99


In this limit, we get

L =1

2

m2

g2∂µ(πaT a)∂µ(πaT a) (A.16)

= −1

2

m2

g2∂µU

†∂µU,

This is equivalent to saying that Aµ decouples from π when E m.

100

B Perturbative quantum gravity

Expansion of√−g

We expand gµν = gµν√−g in small uctuations around a Minkowski background.

gµν = ηµν + κhµν

gµν = ηµν − κhµν +O(κ2)(B.1)

The expansion of√−g can be derived by observing that

det(gµν) = det(√−ggµν) = (−1)

d2det(gρσ)

d2det(gµν). (B.2)

Here, d is the spacetime dimension. This can be rewritten by multiplying with det(gµν).

√−g = (−1)−

1d−2det(gµν)

1d−2 (B.3)

Now, this expression can be expanded to quadratic order in κ.

√−g = (−1)

−1d−2 exp(

1

d− 2Tr log(ηµν + κhµν))

= (−1)−1d−2 exp(

1

d− 2Tr[log(ηµν) + κh− κ2

2h2])

= exp(1

d− 2Tr[κh− κ2

2h2])

= 1 +1

d− 2Tr(κh− κ2

2h2) +

1

2(d− 2)2Tr(κh− κ2

2h2)2 +O(κ3)

= 1 +κ

d− 2hµµ −

κ2

2(d− 2)hµνh

νµ +

κ2

2(d− 2)2(hµµ)2 +O(κ3)

(B.4)

101


Feynman rule of the three-graviton vertex

The Feynman rule of the three-graviton vertex is

Vµνρσλδ(p, q, r) = −iκ2

[q(µrν)

(2ηρ(ληδ)σ −

2

d− 2ηρ(σηλ)δ

)+p(ρrσ)

(2ηµ(ληδ)ν −

2

d− 2ηµ(νηλ)δ

)+ p(λqδ)

(2ηµ(ρησ)ν −

2

d− 2ηµ(νηρ)σ

)+ 2r(ρησ)(µην)(λqδ) + 2p(ληδ)(ρησ)(µqν)

+ 2q(µην)(ληδ)(ρqσ) + q · r(

2

d− 2ηµ(ρησ)νηλδ

+2

d− 2ηµ(ληδ)νηρσ − 2ηµ(ρησ)(ληδ)ν

)+ p · r

(2

d− 2ηρ(µην)σηλδ +

2

d− 2ηρ(ληδ)σηµν

−2ηρ(µην)(ληδ)σ)

+ p · q(

2

d− 2ηλ(µην)δηρσ

+2

d− 2ηλ(ρησ)δηµν − 2ηλ(µην)(ρησ)δ

)],

(B.5)

as was derived in [93].

Calculation of loop integrals

Here, we show the details of the calculation of the loop integrals that will generate thenon-minimal coupling that appears in section 6.4.2. In order to calculate the leadingcontribution of the integrals to the non-minimal coupling, we need to carefully identifythese terms by their momentum structure.

ξ(a)gµνRφaφa = ξ(a)κφaφahµµ +O(κ2)

= ξ(a)κφaφaq2ηµνh

µν (in momentum space)(B.6)

Here, q is the momentum associated with the graviton. The rst diagram in gure 6.8 isgiven by

M (1)µν =

∫d4k

(2π)4V ρσ(p1, k)Dρσλδ(p1 − k)V λδ(q + k, p2)D(k)D(q + k)Vµν(q + k, k) (B.7)

102

Note that the indices corresponding to dierent scalars are already contracted. By onlylooking at the momentum structure that corresponds to the non-minimal coupling, thisresults in the following integral.

I(M1)µν = κ3

∫d4k

(2π)4

[m4aq

2ηµν4(p1 − k)2((q + k)2 −m2

a)(k2 −m2

a)

](B.8)

The second diagram in gure 6.8 is given by

M (2)µν =

∫d4k

(2π)4Vρσ(p1, p1 − k)D(p1 − k)Vλδ(q + k, p2)

×Dρσαβ(k)Dγελδ(q + k)Vµνγεαβ(−q, q + k,−k).

(B.9)

The relevant terms are

I(M2)µν = κ3

∫d4k

(2π)4

q2ηµν(q + k)2k2((k − p1)2 −m2

a)

×[

1

4m4a +

1

4m2ak

2 − 3

8k2p2

1

].

(B.10)

Putting the external momenta of the scalars on shell, this becomes

I(M2)µν = κ3

∫d4k

(2π)4

q2ηµν(q + k)2k2((k − p1)2 −m2

a)

×[

1

4m4a −

1

8m2ak

2

].

(B.11)

Finally, the last diagram in gure 6.8 is

M (3)µν =

∫d4k

(2π)4VλδγεD

ρσλδ(k)Dαβγε(q − k)Vµνρσαβ(−q, k, q − k). (B.12)

The only relevant term is

I(M3)µν = κ3

∫d4k

(2π)4

−17

8

m2aq

2ηµνk2(k − q)2

. (B.13)

The leading contribution to the non-minimal coupling is given by the divergent part ofthe above integrals. The integral (B.8) is UV-nite and can therefore be ignored. Therst term in (B.11) is UV-nite and the second term is logarithmically divergent. Lastly,the integral (B.13) is also logarithmically divergent. Summarizing, the divergent integralsare

I(1)µν = κ3

∫d4k

(2π)4

−1

8

m2ak

2q2ηµν(q + k)2k2((k − p1)2 −m2

a)(B.14)

I(2)µν = κ3

∫d4k

(2π)4

−17

8

m2aq

2ηµνk2(k − q)2

. (B.15)

103


Since these integrals are divergent, we regulate them by introducing a Euclidean momen-tum cut-o |k| < Λ.

The result of the integrals is then (up to some numerical factor)

−κ3q2m2aηµν log

(Λ2

q2

). (B.16)

Comparing with (B.6), we see that the following operator is generated.

−m2a

M2p

logΛ2

q2Rφaφa (B.17)

If we take all masses to be approximately equal (ma ' m), we obtain

−N m2

M2p

logΛ2

q2Rφ2. (B.18)

Renormalization

In order to renormalize the non-minimal coupling, we have to cancel the logarithmicdivergence. We can do this by adding a bare non-minimal coupling to the action of theform ξ0Rφ

aφa, where

ξ0 = ξ + δξ. (B.19)

The counter-term δξ is chosen such that it only cancels the divergent part of the logaritm.

δξ = Nm2

M2p

log(Λ2)φ2 (B.20)

This results in the following renormalized operator.

ξRRφaφa = N

m2

M2p

log

(q2

µ2

)Rφ2 (B.21)

Here, µ is an arbitrary mass scale, that determines the strength of the non-minimal cou-pling. We choose µ such that it is equal to the energy at which we dene our theory, suchthat the non-minimal coupling vanishes at that scale. When we move from this scale tohigher energies, the non-minimal coupling is `turned on'.

Note that we only captured the leading order contribution to the renormalization of thenon-minimal coupling. Furthermore, considering other loop diagrams will renormalizeother terms in the action. Nevertheless, we are only interested in the non-minimal couplingand will not consider renormalization of other operators.

104

Summary for non-physicists

Observations of the cosmos, using both telescopes and satellites, have allowed cosmolo-gists to deduce the history of the universe. To do this, cosmologists had to make threereasonable assumptions. Firstly, it was assumed that the universe is homogeneous on largescales, which means that it has the same matter density at dierent places. Secondly, itwas assumed that the universe is isotropic, which means that it looks the same in everydirection. Thirdly, the universe is approximated to be spatially at. With these assump-tions, a standard model of cosmology was developed. This model, known as the hot BigBang model, describes the history of the universe as follows.

The universe started 14 billion years ago in a very tiny, hot and dense state and expandedin a `Big Bang', causing it to cool dramatically. While cooling down, structures such asatoms, molecules and later galaxies were able to form. At the same time, the expansionslowed down until it reached a point in our not-to-far past where the expansion acceleratedagain. While observations strongly support this model, it still leaves some open questions.Foremost, this model does not answer the question what caused the universe to bang inthe rst place. Furthermore, it also does not specify the exact properties that the initialstate should have in order to result in a universe like our own.

In particular, when considering the second question, this leads to a peculiarity. When wefollow the evolution of an arbitrary initial state, we can show that the resulting universewould not look like our own. More specically, the shape and matter distribution of thisuniverse would not correspond to our universe. Therefore, the initial state must have beenspecial (it should have very specic properties) in order to lead to our universe.

However, because we do not have any reason to believe that our universe is special in thisway, the hot Big Bang model in this formulation is incomplete. It was realised in the 1980sthat these issues could be solved if we allow a period of exponential expansion to precedethe phase of the universe when the expansion decelerated. This period, known as inationbecause space inates an enormous amount during this phase, solves both questions aboutthe hot Big Bang model simultaneously. Foremost, it gives rise to natural properties of theinitial state of the universe (any deviations from atness, inhomogeneity and anisotropyare `diluted' by ination). On top of that, it also predicts small matter clumps to bepresent after ination, which evolve to large structures during the decelerating phase ofthe universe. Schematically speaking, the laws of quantum physics predict extremely tinymatter uctuations, that are `blown up' during ination, which results in matter clumps.This leads to the remarkable conclusion that structures of the universe that we observetoday (galaxies etc.), originate from extremely tiny `quantum uctuations'.

Interestingly, all modern observations of the universe are in agreement with the predictionsthat ination makes. However, ination is theoretically not yet well-understood. Thereason for this is that describing ination requires the combination of two fundamentaltheories of modern physics that are understood separately, but not together. On the onehand, the behaviour of particles at the smallest scales is described by quantum physics.On the other hand, a large amount of matter at large scales is described by Einsteinstheory of gravity. Because ination tells us that our entire universe (large amount ofmatter) originates from a very small patch (where quantum physics is relevant) that isinated, it is necessary to combine quantum physics with gravity to describe this process.Unfortunately, combining these two pillars of modern physics in a theory of `quantumgravity' is a very dicult problem and not yet solved.

In contrast, there exists a candidate for a theory of quantum gravity; string theory. Inthis thesis, I describe how ination can be described in the context of string theory andtreat promising models of ination that are inspired by string theory. In particular, Ihave shown that models with many particles that drive a period of ination are natural instring theory and also plausible when we take corrections expected from quantum physicsinto account. This result complements already existing results in the literature.

However, because string theory is only partially understood, we conclude that more workis needed to understand the foundations of quantum gravity, before a full model of inationin quantum gravity can be constructed.

university of amsterdam - uva · 2020. 7. 9. · university of amsterdam msc physics track: grappa...

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