University of Amsterdam

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1 University of Amsterdam MSc Physics Track: GRAPPA Master Thesis Going beyond the eective theory of ination by Lars Aalsma July EC September July 015 Supervisor: Dr. Jan Pieter van der Schaar Second reader: Dr. Ben Freivogel

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

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5 Acknowledgements First of all, I like to acknowledge Jan Pieter. Jan Pieter, thank you for supervising me the past year and introducing me to the various topics we have looked at. While I may have been a bit worried halfway through my project about the direction it would be heading, you ensured me that everything would fall into place, which it did. Furthermore, you have learned me a lot and not once have I left your oce without feeling inspired. Your enthusiasm is inspiring and I am looking very much forward to working with you in the coming years. Second of all, I would like to thank Ben for being my second reader. I have enjoyed the questions you asked during my presentation, which allowed me to show my audience some aspects of my thesis in a bit more detail. On top of that, I also thank Volkert van der Willigen. I am very grateful for your nancial support and I hope this thesis might add some knowledge to your understanding of cosmology. Of course, I also want to thank my fellow students. Adri, thank you for being my partner in crime as a student of Jan Pieter and the useful discussions we had. Vincent, Jonas and Jorrit, thank you for being friends and fellow students the past years. Physics would not be the same without you. Also, all other master students get a big thank you for creating the unique atmosphere C4.73A has. Besides closing the chapter of my life as a student, the past year has oered me much more. Thank you, my three sisters, for giving me the pleasure to see a new generation grow up. Even though you may not always hear much from me, I think of you. On the same note, I would like to thank my parents for supporting me in their own ways and being there for me when I need them. Finally, thank you Jolijn for making my life more interesting, exciting and enjoyable. I hope that the past year has set the stage for everything that will follow. Lars Aalsma July 015

6 Contents 1 Introduction 1 Physics of ination 4.1 Introduction to cosmology The Cosmic Microwave Background How natural are the initial conditions of our universe? Addressing the initial conditions of the universe with ination How to drive ination? Quantum uctuations during ination The energy scale and UV-sensitivity of ination The eective eld theory of ination Constructing an eective action Symmetries of ination The eective action of ination Why should we go beyond the eective eld theory of ination? From string theory to cosmology Aspects of string theory Eective N = 1 supergravity Axion ination Natural ination Saving natural ination Statistical generality of axion ination Non-minimal coupling Description of a non-minimal coupling in the Jordan and Einstein frame Supergravity formulation of a non-minimal coupling Cosmological attractors Radiative generation of a non-minimal coupling The Weak Gravity Conjecture and axion ination The Weak Gravity Conjecture for particles and gauge elds Generalization to p-forms Applying the Weak Gravity Conjecture to ination Contribution of gravitational instantons and a loophole iv

7 Contents 8 Conclusions and outlook 88 Bibliography 91 Appendix A The eective eld theory of ination 97 Appendix B Perturbative quantum gravity 101 v

8 Conventions Throughout this thesis, we will work in natural units, such that c = = 1. The metric convention we use is mostly plus, i.e. (-,+,+,+), unless stated otherwise. Furthermore we will use the reduced Planck mass which is dened as M p 8πG N = GeV. Mostly, we will be working in dimensions. If this is not the case we will denote the spacetime dimension by d. The Hubble slow-roll parameters are indicated with a tilde and are dened as ɛ Ḣ and η 1 Ḧ H HḢ. Here, the dot denotes a derivative with respect to time. The potential slow-roll parameters are dened as ɛ M p ( ) V (φ) and η M V (φ) p V (φ) V (φ), where the prime denotes a derivate with respect to φ. related to each other as These slow-roll parameters are ɛ ɛ and η η ɛ.

9 1 Introduction In the 0 th century, cosmological observations have led to a description of the origin and evolution of the universe, known as the hot Big Bang model. In this model, the origin of the universe is described as an extremely hot and dense state that expanded in a `Big Bang', after which it evolved under the inuence of gravity to today's observed universe. Whereas this view gave new insights into the history of the universe, it did not address its initial conditions. When we follow the evolution of an arbitrary initial state, the resulting universe will not look like our own. More specically, the isotropy of the Cosmic Microwave Background and the atness of the universe can not be explained. In order to obtain a universe with the observed properties, the initial state has to be extremely ne-tuned. This situation, in which a large amount of ne-tuning is needed, hints towards some underlying mechanism that creates more natural initial conditions. Alan Guth realized in 1981 that a period of exponential expansion, preceding the hot Big Bang phase, will drive an arbitrary state towards homogeneity, isotropy and atness [1]. This naturally sets the stage for the evolution of the universe during the hot Big Bang phase, as we can start from an arbitrary state and still end up with a universe with the correct properties. Unfortunately, this period of exponential expansion, known as ination (as space inates during this phase), could not be smoothly connected to the regular hot Big Bang phase of the universe, making this model not realistic. Nevertheless, it inspired the idea that the regular hot Big Bang phase of the universe was preceded by a period of exponential expansion. A step forward came when Andrei Linde suggested that a period of quasi-exponential expansion could be driven by a scalar eld with a potential energy that dominates over its kinetic energy []. This mechanism, known as slow-roll ination, does allow for a smooth transition to the regular hot Big Bang phase. However, it was shown that in this new inationary scenario there still would be a problem with its initial conditions, i.e. the universe was not likely to live long enough for ination to start [3]. At the time, it seemed that one problem with initial conditions was simply swapped for a dierent one. The resolution to this problem came with the introduction of chaotic ination [4], which in its essence states that there may exist dierent initial conditions in dierent parts of the universe or in dierent universes. This implies that there will always be some patch in which ination can occur, from which our universe originates. Of course, this notion was (and still is) quite controversial. Steinhardt and Vilenkin observed that this implies that our universe may be part of a multiverse where ination perhaps always takes place in some part (eternal ination) [5]. Despite its controversy, the inationary paradigm 1

10 1 Introduction is nowadays well-established, because all cosmological observations are consistent with it. Nevertheless, the origin of ination and the existence of a multiverse are still open questions. One of the predictions of ination that is conrmed by measurements is a nearly scaleinvariant spectrum of perturbations, originating from quantum uctuation in the scalar eld that drives ination (the inaton). These perturbations source small matter inhomogeneities that are visible in the Cosmic Microwave Background as small temperature uctuations [6]. These temperature uctuations have been measured with very high accuracy by satellites such as Planck (see gure.5) and are in agreement with a period of ination in the early universe. In addition, quantum uctuations of spacetime itself will lead to a nearly scale-invariant spectrum of tensor perturbations, also known as primordial gravitational waves. These gravitational waves should be visible in the polarization of the Cosmic Microwave Background [6]. A measurement of this polarization signal would establish the energy scale of ination. Currently, observations have only been able to put an upper bound on the amplitude of the gravitational waves, but upcoming measurements will push this bound further down or measure the amplitude of the gravitational waves. Despite the fact that ination is a compelling mechanism, it is theoretically not yet fully under control. It is well-known that when ination is treated in an eective way, which oers simplicity, the eect of new physics above the cuto of the theory can have a large eect on the eective theory, rendering ination dicult. This problem is known as the UV-sensitivity of ination. As we will motivate at the end of chapter, this UV-sensitivity can only be properly dealt with in a UV-complete theory, i.e. a theory that complements the eective theory at high energy. In the case of ination, this sensitivity reveals itself in terms of operators that are suppressed by the cuto, but can become active during ination. This implies that in order to have theoretical control over these operators, ination needs to be described in a theory of quantum gravity. Because the best candidate for such a theory is string theory, this motivates us to study ination in the context of string theory. Unfortunately, string theory is rather technical and only partially understood. Therefore, simplifying assumptions have to be made in order to have control over the theory. When considering string theory, such a simplication occurs when looking at its low-energy limit, i.e. supergravity. This opens up a theoretically accessible landscape, which also is non-trivial enough to allow for interesting dynamics. Above all, we know that supergravity is UV-completed into string theory, which makes it (in principal) possible to check if any low-energy assumptions are valid up to high energy. However, the situation is a bit more subtle. String theory is not yet fully understood, such that not all assumptions (e.g. the existence of de Sitter vacua) can be straightforwardly checked. In summary, a better understanding of string theory is needed, before we can truly embed ination in it. For now, the best we can do is pursue an understanding of ination in the parts of string theory that are well under control or make assumptions in a theory of supergravity.

11 The contents of this thesis are as follows. In chapter -4, we give a basic introduction to cosmology, an overview of the physics of ination, its description as an eective theory and the motivation for embedding ination into string theory. Continuing, chapters 5-7 give recent advances in particular well-motivated models, such as axion ination. In addition, chapter 6 also contains some original work. Finally, we conclude in chapter 8. Complementary, some appendices and a summary for non-physicists can be found at the end. 3

12 Physics of ination The goal of this chapter is to give a self-contained introduction to the basic concepts and formalism relevant for this thesis. Tools and techniques necessary to describe the evolution of the universe are presented and we will motivate how the initial conditions of the universe can be properly addressed. Continuing, we will show how theory and observations can be connected, paving the way for subsequent chapters..1 Introduction to cosmology The foundation of modern cosmology is the assumption that the universe is homogeneous and isotropic on large scales. From this assumption, the Einstein equations can be solved, which results in the Friedmann-Robertson-Walker (FRW) metric. In spherical coordinates, this metric is given by ( ) dr ds = dt + a(t) + 1 kr r dθ + r sin (θ)dφ. (.1) Here, a(t) is a scale factor that measures how much the universe has expanded in a certain amount of time and k is a parameter that determines the spatial curvature of the universe. Mostly, we will work in a at spacetime (k = 0) for which the FRW metric can be rewritten in Cartesian coordinates as ds = dt + a (t) [ dx + dy + dz ]. (.) Furthermore, we will also use comoving coordinates (τ, x), that x the expansion of the universe. Here, τ is conformal time, dened as dt dτ In terms of conformal time, (.) can be written as = a(t). (.3) ds = a (τ) [ dτ + dx + dy + dz ]. (.4) If we assume the energy content of the universe to have the stress-energy tensor of a perfect uid, which is given by T µν = diag (ρ, P, P, P ), (.5) 4

13 .1 Introduction to cosmology we can derive the Friedmann equations from the Einstein equations. H = ρ 3M p Here, H is the Hubble parameter which is dened as k a (.6) ä a = 1 (ρ + 3P ) (.7) 6Mp H ȧ a. (.8) As can be seen from (.8), the Hubble parameter describes the expansion rate of the universe at a certain time. Furthermore, ρ is the energy density of the uids in the universe, P the pressure and k the same parameter as in the FRW metric. The eect of the value of k can be most intuitively seen by introducing the density parameter Ω, which is dened as Ω ρ ρ crit = ρ 8πG N 3H. (.9) The critical energy density (ρ crit ) corresponds to the situation where the energy density of the universe is just right to sustain a at universe, while it evolves. Any small deviation of Ω > 1 results in a curved universe. To be more specic, Ω < 1 corresponds to negative spatial curvature and Ω > 1 to positive spatial curvature, see gure.1. In terms of the density parameter, (.6) can be written as Ω 1 = which allows us to make the following identication. > 1 k > 1 Closed universe Ω = 1 k = 0 Flat universe < 1 k < 1 Open universe k H a, (.10) (.11) Now that we know how the geometry of the universe depends on its energy content, we would also like to know how it evolves. For this, the relation between the energy density and the scale factor is needed. By using conservation of energy, the following conservation equation can be derived [7]. ρ ρ = 3(1 + w)ȧ a (.1) Here, w determines the equation of state of a particular uid, which is given by P = wρ. (.13) 5

14 Physics of ination Figure.1: The value of the density parameter Ω determines the geometry of the universe. Positive spatial curvature corresponds to a closed universe, negative spatial curvature to an open universe. Figure from wikipedia. The equation of state is bounded by energy conditions, which constrain w. For constant w, (.1) can be integrated to obtain ρ a 3(1+w). (.14) In the standard model of the universe (ΛCDM), which is well-established by observations, it is dominated at dierent epochs in time by three dierent uids; radiation, (nonrelativistic) matter and a component known as dark energy. For these uids, w takes the following values 1. 1 (radiation) 3 w = 0 (matter) (.15) 1 (dark energy) Given a particular uid, (.14) can be used to relate the energy density to the scale factor. For example, consider a universe that is spatially at (Ω = 1, k = 0) and dominated by radiation. By taking w = 1 and plugging it into (.14), we obtain 3 ρ a 4. (.16) Using this relation in the rst Friedmann equation (.6) and integrating it results in the expansion rate of the universe. a(t) t 1, (.17) 1 Observations tell us that dark energy has w 1, so values slightly larger than -1 are also allowed. 6

15 . The Cosmic Microwave Background This calculation can also be straightforwardly applied to other uids, which will give a dierent evolution, see gure.. Generally, when given an equation of state, we can derive a relation between the scale factor and the energy density. Plugging this into the Friedmann equation tells us the evolution of the universe. Of course, it is very well possible that multiple components contribute to the energy density of the universe, complicating the process of solving the Friedmann equation. There are two possible ways to proceed. Firstly, it is often reasonable to approximate a certain epoch as being dominated by only a single component. Secondly, if one does not want to make this assumption, the Friedmann equation can be solved numerically. Using these techniques, the ΛCDM model tells us that the universe is dominated in different periods of time by respectively radiation, matter and dark energy. On top of that, measurements have shown that the universe is very at, such that Ω = 1, k = 0 is a good approximation. a(t) Radiation : a(t) t Matter: a(t) t 3 Dark Energy: a(t) e t Figure.: The evolution of the scale factor for dierent uids. t. The Cosmic Microwave Background One of the greatest sources of cosmological information is the transition of the universe from being opaque to transparent. At a redshift of z 1100, the universe was in a hot and dense state, which consisted of photons, electrons and baryons. As photons were constantly Thomson scattering on electrons, their mean free path was small, causing the universe to be opaque. Only when the universe cooled suciently enough at z 1100, neutral atoms were able to form (an event known as recombination), which increased the mean free path of the photons dramatically (decoupling). As a consequence, photons were now able to travel freely, which made the universe transparent, see gure.3. After decoupling, the radiation emitted by the opaque universe could travel freely, carrying information about the universe at z This radiation, known as the Cosmic Microwave Background (CMB), is still measurable today as an isotropic radiation that is 7

16 Physics of ination Figure.3: When the universe cools down, neutral atoms are able to form, increasing the mean free path of the photons. Eventually, the photons are able to travel freely, making the universe transparent. visible when all foreground light is subtracted. Due to the expansion of the universe, it is signicantly redshifted with respect to its original wavelength, such that the CMB has the highest intensity at a temperature of T 0 =.7K. The accidental discovery of the CMB by Penzias and Wilson [8] conrmed this idea, which earned them the Nobel prize of Furthermore, because the early universe was in thermal equilibrium, it was also predicted that the CMB would have the spectrum of a black body. This spectrum was rst measured by the COBE satellite, see gure.4. Due to the large amount of information the CMB carries about the early universe, the initial discovery of the CMB initiated the launch of more surveys, such as WMAP and Planck that measured the CMB with an even higher precision. These surveys showed that, on top of black body spectrum, the CMB has tiny temperature uctuations of the order δt T 10 4, (.18) which are nearly scale-invariant, see gure.5. Later, we will comment on the important physical relevance of these uctuations. 8

17 .3 How natural are the initial conditions of our universe? Figure.4: Spectrum of the CMB as measured by the COBE satellite. The theoretical prediction of a black body with a temperature of T0 =.7K agrees excellently with the observations (the true error bars are even smaller). Figure from Wikipedia. Figure.5: All-sky map of the CMB taken by the Planck satellite. The colour di erence corresponds to small temperature uctuations. Figure from NASA..3 How natural are the initial conditions of our universe? As we mentioned in the introduction, a universe that is very homogeneous, isotropic and spatially at on large scales, is quite curious. Starting from an arbitrary state, and letting 9

18 Physics of ination it evolve according to the Friedmann equation, would result in a dierent universe than the one we are living in. Therefore, a logical question to ask ourselves is: how natural are the initial conditions of our universe? In this section, we will specify two problems concerning the initial conditions of the universe, known as the horizon and atness problem..3.1 The horizon problem As in any expanding geometry, our universe has a horizon. This cosmological horizon denes the maximal distance that is in causal contact with us. Any signal separated any further will not be able to reach us due to the expansion of the universe. Using comoving coordinates, this causal structure can be conveniently explored, because for ds = 0, dx = dτ. Thus, the maximum distance a light-like signal can have travelled between the singularity and a particular time t is given by the conformal time elapsed. τ = t 0 dt a(t ) = a(t) a(0) da(t ) a(t ) H, (.19) which can be rewritten in terms of the comoving Hubble radius R H (ah) 1. τ = log(a(t)) 0 d log(t ) a(t )H = log(a(t)) 0 d log(t )R H, (.0) The comoving Hubble radius is the radius of the Hubble sphere, the sphere that contains the observable universe. During radiation and matter domination (which was the dominant energy contribution during the largest part of the history of the universe) R H increases. Hence, we can see from (.0) that the biggest proper time contribution comes from late times. This observation is problematic if we want to explain the observed isotropy of the CMB. This can be seen as follows. The fact that we observe that every patch of the CMB has the same temperature (up to tiny uctuations) implies that they were in causal contact with each other before decoupling. If this was not the case, there is no reason why dierent patches would have the same temperature. However, the fact that the largest proper time contribution comes from late times, implies that there was not enough conformal time between the singularity and the surface of last scattering (the surface at redshift z = 1100 from which the CMB originates) for all dierent patches of the CMB to have been in causal contact with each other, see gure.6. 10

19 .3 How natural are the initial conditions of our universe? Figure.6: Not enough conformal time has elapsed since the singularity for dierent patches of the CMB on the surface of last scattering to have been in causal contact with each other. This can be seen from the fact that their light cones do not overlap. Because dierent patches of the CMB were space-like separated before decoupling, the fact that they all have the same temperature seems like a remarkable coincidence. Of course, we could ne-tune every patch of the CMB to have the same temperature, but such a large degree of ne-tuning is usually undesirable and not considered a solution. This issue is known as the horizon problem. To resolve this problem, we would like a mechanism that provides more conformal time between the singularity and the surface of last scattering, such that dierent patches of the CMB have had the time to be in causal contact with each other..3. The atness problem In addition to the homogeneity and isotropy of the universe, we also observe it to be exceptionally at. Following the previous discussion, we again examine how natural the initial conditions of such a universe are. Problematically, any initial deviation from atness, will grow under the inuence of gravity, such that an arbitrary initial state will evolve into a universe with large curvature, very dierent from our own. This can be seen from the rst Friedmann equation (.6). H = ρ 3M p k a (.1) 11

20 Physics of ination During the largest part of the history of the universe, the energy density was dominated by radiation and matter and the density respectively scaled as ρ rad a 4 and ρ mat a 3. Furthermore, if we interpret curvature as a uid, it scales as ρ curv a. Schematically, we can therefore write H = ρ i,rad a 4 + ρ i,mat a 3 + ρ i,curv a, (.) where the subscript i denotes an initial value. Since curvature only scales with a, it will dominate the energy density of the universe for large a(t), unless we ne-tune the initial curvature of the universe to be extremely close to zero. Thus, again we have to deal with a ne-tuning problem, which is known as the atness problem..4 Addressing the initial conditions of the universe with ination The issues raised in the previous section require a mechanism that explains the initial conditions of our universe, as we do not consider ne-tuning a solution. Of course, some people might take the point of view that the issues we treated are no problems at all, but simply a reection of our ignorance about the physics that is relevant at the energy scale just after the singularity. Indeed, to properly describe this period, a theory of quantum gravity is needed, of which we have limited knowledge. Nevertheless, we will argue that this is not a very reasonable point of view to take. Firstly, hoping that some unknown property of quantum gravity would take care of the initial conditions of the universe seems rather naive. Because we do not have a rm grasp on quantum gravity, it seems not very convincing to try to use it to address the raised issues. For this reason, any explicit mechanism that addresses the initial conditions is preferable. Secondly, the conventional mechanism used to resolve the horizon and atness problem (ination) not only addresses the initial conditions, it also makes predictions. By quantizing the theory of ination, a spectrum of perturbations from elds present in the early universe is obtained (see section.6). These perturbations were also present at the time that the photons in the early universe did not yet decouple. In particular density perturbations (originating from the inaton) will lead to small temperature anisotropies in the CMB. On top of that, the density perturbations also plant the seeds for the large scale structure. These predictions have been conrmed, which are very dicult to explain without ination. For these two reasons, we will take ination to be the best solution to address the initial conditions of the universe. As we mentioned, in order to solve the horizon problem, additional conformal time between the singularity and the surface of last scattering is needed. This can be achieved by introducing a period before the hot Big Bang phase of the universe where R H was decreasing, such that (.0) receives a dominant contribution from early times. In this way, enough conformal time will have elapsed for the light cones of dierent patches of the CMB to have overlapped in the past, see gure.7. 1

21 .4 Addressing the initial conditions of the universe with ination Figure.7: By introducing a period between the singularity and the hot big bang phase of the universe where R H was decreasing, additional conformal time is generated. Now, dierent patches in the CMB have been in causal contact with each other. A decreasing R H implies the following condition. Ṙ H = d ( ) [ ] 1 = Ḣ dt ah a H This can be captured by the Hubble slow-roll parameter ɛ. < 0 (.3) ɛ Ḣ H < 1 (.4) Rewriting the second Friedmann equation (.7) using the denition of Ḣ as Ḣ + H = 1 (ρ + 3P ), (.5) 6Mp allows us to combine this expression with the rst Friedmann equation (.6) in at space. ( Ḣ = 3H 1 + P ). (.6) ρ 13

22 Physics of ination Thus, a decreasing comoving Hubble sphere implies ɛ = Ḣ H = 3 ( 1 + P ) < 1. (.7) ρ This leads to the following equation of state. P < ρ 3 w < 1 3 (.8) If we now remember that curvature scales as ρ curv a, which corresponds to w = 1, 3 requiring w < 1 also solves the atness problem. That is, if a uid with an equation 3 of state w < 1 dominates the energy density, curvature eects are subleading and the 3 universe is driven towards atness. In the limit that Ḣ 0, we obtain an equation of state with w = 1, which leads to a constant energy density. The rst Friedmann equation (.6) for a spatially at universe (k = 0) is then given by ȧ ρ a =, (.9) 3M p which has the solution a(t) e ρ 3Mp t = e Ht. (.30) Thus, we see that ination corresponds to exponential expansion in the limit that Ḣ 0. It is necessary that H evolves slowly for ination to end, which corresponds to quasiexponential expansion. The amount of ination elapsed is measured by the number of efolds N, which is dened as the number of Hubble times ination lasts. N = t end t begin dt H(t ) (.31).5 How to drive ination? Now that it is clear how ination can properly address the initial conditions of the universe, we need to know how we can drive a period of ination. In the previous section, we observed that if a uid with an equation of state of w < 1 dominates the energy density, 3 R H will decrease, leading to a period of ination. An example of such a uid is a scalar eld with a potential energy that dominates over its kinetic energy. The action of a scalar eld, minimally coupled to gravity, is given by S = d 4 x [ M p g R + 1 ] µφ ν φg µν V (φ). (.3) Measurements of the CMB only give access to the last efolds of ination, so t begin has a value that is accessible to the CMB. 14

23 .5 How to drive ination? The equation of motion of this scalar eld in a FRW spacetime is given by Remember that we should satisfy φ + 3H φ + V (φ) = 0. (.33) ɛ = Ḣ < 1. (.34) H Combining (.33) with the rst Friedmann equation (.6) results in Using this expression, we can rewrite (.34) as which is satised when such that 3M p H = 1 φ + V (φ). (.35) ɛ = Ḣ 1 H = φ < 1, (.36) Mp H V (φ) 1 φ, (.37) 3M p H V (φ). (.38) Thus, a scalar eld with a potential energy that dominates over its kinetic energy can lead to a period of ination. Furthermore, to allow for a prolonged period of ination, the kinetic energy has to remain small during some time, which requires the acceleration φ to be small with respect to the Hubble friction 3H φ. By neglecting the Hubble friction in the equation of motion, ɛ can be written as ɛ = M p ( ) V 1. (.39) V We refer to ɛ as the potential slow-roll parameter (note that ɛ = ɛ). Neglecting the acceleration in the equation of motion, requiring a prolonged period of ination can be captured by a second Hubble slow-roll parameter η 1 Ḧ 1, (.40) ḢH which can also be equivalently captured by a potential slow-roll parameter ( η η ɛ). η M p V V 1 (.41) 15

24 Physics of ination Thus, when some potential V (φ) is specied, we can check if this potential can sustain ination by calculating the slow-roll parameters ɛ and η. Ination occurs as long as these parameters are small. Finally, at the end of ination, a smooth transition to the regular hot Big Bang phase of the universe must be made. At this phase, known as reheating, the energy density of the inaton needs to be transferred to standard model particles that are heated to a sucient temperature to start the hot Big Bang phase of the universe. Reheating is a rather model-dependent process and beyond the scope of this thesis. Therefore, will not treat reheating, but refer the interested reader to [9]..6 Quantum uctuations during ination The previous description of ination was executed completely classically. It is however necessary to take quantum eects into account, as ination couples the smallest to the largest scales. Furthermore, because we observe deviations from homogeneity and isotropy in the universe, we also want ination to provide the perturbations that are responsible for generating these deviations. Quantizing ination leads to spectra of perturbations. The average of the perturbations is zero, but the variance (the average of the squared amplitude) is non-zero. This leads to observational signatures of the primordial perturbations, generated by ination. Ination involves a metric and a scalar eld, so we need to consider perturbations to both of these. The metric has both scalar and tensor perturbations, the inaton only has scalar perturbations. In this section, we will show how these spectra can be obtained and what their observational consequences are. We will only describe the most important aspects and refer the reader to [6, 10] for a more in depth-discussion..6.1 Power spectrum of perturbations Here, we will consider tensor perturbations to the metric, as computing the scalar perturbations is a bit more involved. After that, we comment on how the computation can be performed for the scalar perturbations, and show its result. When expanding the metric in small uctuations two functions (h + and h ) that correspond to a dierent polarization are obtained. These functions describe the tensor perturbations and obey the following equation of motion. ḧ + Hḣ + k h = 0 (.4) When quantizing h, we obtain the following expansion in terms of creation and annihilation operators [10]. ĥ( [ ] k, τ) = v(k, τ)â k + v(k, τ)â am k (.43) p 16

25 .6 Quantum uctuations during ination Here, the hat indicates that we are dealing with operators and the bar indicates complex conjugation. When we use the fact that H evolves slowly, the v's satisfy the following dierential equation. ( v + k ) v = 0 (.44) τ Now, the variance is given by ĥ ( k, τ)ĥ( k, τ) = v( a M k, τ) (π) 3 δ (3) ( k k ) p (π) 3 P h (k)δ (3) ( k k ). (.45) Here, P h (k) is referred to as the power spectrum of tensor perturbations. To determine the power spectrum, we need to know v( k, τ), for which we need to solve (.44). We will solve this equation for two cases. First, we solve it for the case when the wavelength of the perturbation is inside the horizon. After that, we consider the case when the wavelength of the perturbation is longer than the horizon. The general solution of (.44) is [ v = e ikτ 1 i k kτ ]. (.46) Inside the horizon, k τ 1, such that we can neglect the i/kτ term. If we directly took this limit in (.44), we would have obtained the equation of motion of a simple harmonic oscillator. The solution becomes v = e ikτ k. (.47) Comparing this with (.45), we see that the amplitude of the power spectrum decreases. Therefore, if this would remain the case during ination, we would not be able to see any eect of the perturbations. However, because the comoving Hubble sphere shrinks, something remarkable occurs. When the perturbation exits the horizon, k τ < 1, such that the solution becomes v = e ikτ k i kτ. (.48) Comparing this with (.45) and using τ 1/(aH), we see that outside of the horizon, the power spectrum becomes constant! Taking both polarizations into account, this xes the tensor power spectrum. P h (k) = 4H M p k 3 (.49) Here, the indicates that a quantity should be evaluated at horizon crossing. It is convenient to also introduce a dimensionless power spectrum, dened as h k3 π P h(k) = H. (.50) π Mp 17

26 Physics of ination Deriving the power spectrum for scalar perturbations is a bit more involved, as we need to account for scalar perturbations of the metric, as well as the inaton. As it turns out, it is useful to work in dierent gauges to compute perturbations and dene the scalar power spectrum in terms of gauge-invariant quantities. Often, the comoving curvature perturbation R is used to parametrize the scalar perturbations. In the comoving gauge, this quantity appears in the spatial part of the metric as follows. g ij = a e R δ ij (.51) Hence, we see that R generates small spatial curvature perturbations. Apart from this subtlety, the same logic as for the tensor power spectrum applies. Again, it is observed that the scalar perturbations vanish outside of the horizon. The variance of the scalar perturbations is given by and the scalar power spectrum is given by R( k, τ)r( k, τ) = (π) 3 P R (k)δ( k k ), (.5) P R (k) = H 4ɛ M p 1 k 3. (.53) We can also dene a dimensionless power spectrum. R(k) k3 π P R(k) = H 8π ɛ M p (.54).6. Observational signatures of cosmological perturbations Importantly, we saw that the expressions for the dimensionless power spectrum did not depend explicitly on the wavelength of a perturbation, but only weakly via H and ɛ on a scale (both parameters should vary slowly). Therefore, a robust prediction of ination is the fact that it generates a nearly scale-invariant spectrum of perturbations. Deviations from scale invariance can be captured by the spectral indices, which are dened as d log( R ) d log(k) n s 1 d log( h ) d log(k) n t, (.55) where n s and n t are the spectral indices of respectively the scalar and tensor power spectra. The limit n s 1 and n t 0 corresponds to scale invariance. The power spectra are related to the theory by the slow-roll parameters. By using the expression of the dimensionless power spectra we obtain n s 1 = η 6ɛ (.56) n t = ɛ (.57) r = 16ɛ, (.58) 18

27 .6 Quantum uctuations during ination where r is the tensor-to-scalar ratio that is dened as r = R (k). (.59) h (k) While it was nice that we saw that perturbations freeze outside of the horizon, this does not yet have observational consequences. Only when the comoving Hubble sphere grows during the regular hot Big Bang phase, perturbations can re-enter the horizon again, see gure.8. Figure.8: Perturbations exit R H during ination after which they freeze until they reenter 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 perturbations, these resulted in small matter inhomogeneities in the early universe, which have a profound eect. The small matter inhomogeneities have grown under the inuence of gravity after the photons decoupled, which has led to the large scale structure we observe today. Additionally, photons in the early universe could also feel the presence of these matter inhomogeneities. This resulted in temperature anisotropies in the CMB that, on large scales, can be directly attributed to primordial perturbations. Complementary, the polarization of the CMB also gives primordial information. Local quadrupole anisotropies present before the CMB decoupled, can result in a polarization of the CMB. Scalar perturbations result in a curl-free polarization signal (E-modes), while the tensor perturbations can result in a divergence-free polarization signal (B-modes), see gure.9. 19

28 Physics of ination Figure.9: Schematic illustration of possible polarization signals of the CMB. E-modes correspond to scalar perturbations and B-modes to tensor perturbations. Figure from [1]. Therefore, in order to put ination to the test, measurements of the CMB should reveal that the temperature anisotropies of the CMB are nearly scale-invariant. Furthermore, polarization measurements of the CMB should determine the size of the primordial tensor perturbations. By measurements of the CMB, we only have access to the last efolds of ination. Typically, a reference scale k (called the pivot scale) which is accessible to the CMB is used at which all observable quantities are evaluated. In eld space, this scale φ corresponds to the point after which ination lasted of efolds given by N = φ φ end dφ 1. (.60) M p ɛ While the scalar power spectrum has been measured via the temperature uctuations of the CMB, there has not yet been a measurement of the tensor power spectrum. Consistent with ination, the spectrum of scalar perturbations is indeed nearly scale-invariant. Indeed, the most recent measurements [13] have measured n s and put a bound on r 3. n s = ± (.61) r < 0.11, (.6) in agreement with ination. Now, in order to compare theoretical predictions with observations, we have to calculate n s and r for particular potentials. 3 The pivot scale at which Planck measured these quantities is k = 0.05Mpc 1 0

29 .6 Quantum uctuations during ination.6.3 A simple example: V (φ) = 1 m φ We will illustrate how inationary predictions are made by considering the following simple potential. The slow-roll parameters for (.63) are given by and ination is possible as long as φ > M p, see gure.10. V (φ) = 1 m φ (.63) ɛ = η = M p φ, (.64) V(ϕ) ϵ = 1 ϕ Figure.10: A scalar eld rolls slowly down its potential V (φ) = 1 m φ while driving ination as long as ɛ, η < 1. The shaded area corresponds to the region where ination is possible. A reasonable estimate of the number of efolds required for ination to address the initial conditions of the universe is N = 60 [14]. From (.60), we then nd that φ = 16M p. As the end of ination (which corresponds to ɛ = 1) occurs at φ end = M p, we observe an interesting property of this model; the eld displacement φ during ination is superplanckian. Models which exhibit this property are known as large-eld ination models and have some interesting properties. Most importantly, they are intimately related to fundamental physics, a point which we will come back to in section.7. The spectral index and the tensor-to-scalar ratio of this model are given by (.56) and (.58). n s = 0.97 r = 0.13, (.65) where we used φ = 16M p. Unfortunately the bound on r by the Planck 015 data is in disagreement with this particular model, but it shows how computing observables from an inationary model can be performed. An overview of some popular ination models compared with Planck 015 data is given in gure.11. 1

30 Physics of ination Figure.11: Planck 015 constraints in the n s -r plane on various inationary models. Figure from [15]..7 The energy scale and UV-sensitivity of ination Using the tools developed in the previous sections, it is now possible to connect theoretical predictions of inationary models to observations. However, this leaves a large parameter space with models consistent with observations. In particular, the energy scale of ination, which is given by E inflation = V 1/4 = (3H M p ) 1/4, (.66) is not yet well-constrained by measurements, because in the scalar power spectrum the value of H during ination is obscured by ɛ, see (.54). In contrast, a measurement of the tensor power spectrum does establish the scale of ination, as the tensor power spectrum directly depends on H, see (.50). In terms of r, the energy scale is given by [14] ( r ) 1/4 E inflation = (3H Mp ) 1/4 = Mp. (.67) 0.1 The upper bound on r (.6) therefore gives an upper bound on the scale of ination E max = 10 M p. (.68) As was noted by Lyth [16], a detection of primordial gravitational waves would not only pin down the scale of ination to be high (E inflation E max ), it also implies that ination was superplanckian. This can be seen from the bound ( ) φ r 0.01, (.69) M p

31 .7 The energy scale and UV-sensitivity of ination which is known as the Lyth bound. This motivates us to consider models of ination that predict an observable gravitational wave signal for two reasons. Firstly, (.67) implies that such a model is tied to interesting fundamental physics, as ination occurred at a scale where new physics is expected to be relevant. Secondly, upcoming polarization measurements of the CMB will constrain r further or measure its value, allowing such models to be falsiable, which promotes a healthy way of doing physics. Problematically, (.69) implies that such a model should be superplanckian, which makes model building rather dicult, as we will see. Nevertheless, quantum corrections are not exclusively an issue for large-eld ination. All models of slow-roll ination are sensitive to high-energy physics, but large-eld eld ination models suer in a more dramatic way 4. This sensitivity to high-energy physics is known as the UV-sensitivity of ination, which makes constructing large-eld ination models a challenge. For example, consider quantum corrections to an eective (only valid up to a cuto scale Λ) theory of ination. In the absence of any symmetry, the mass of the inaton will receive a correction of the order [14] δm Λ. (.70) Because Λ > H, the slow-roll parameter η will receive a large correction. δη = M p = δm 3H δv V Λ H > 1. (.71) This is known as the η-problem and needs to be addressed in every model of slow-roll ination. One way of preventing large corrections to the inaton mass, is by appealing to a symmetry. Specically, imposing a shift symmetry φ φ + constant (.7) forbids any term other than the kinetic term in the action. The potential may only weakly break this shift-symmetry, such that the inaton obtains a small mass. Still, one should be careful as physics above the cuto scale may not respect this symmetry. In particular, it is well known that any consistent theory of quantum gravity should not contain global symmetries (see for example [17]), while the low energy eective theory may exhibit this symmetry. Therefore, high-energy physics might still induce corrections to the inaton mass that will lead to the η-problem. Determining whether such corrections will be large is a subtle question that can only be properly answered by knowing the theory which completes the eective theory above the cuto scale (the UV-completion). 4 From the perspective of a model builder, this pessimistic view is understandable. In contrast, a more optimistic theorist may view this as an opportunity to use ination to probe a regime of unknown physics. 3

32 Physics of ination The situation is even worse in large-eld ination. When the eld displacement during ination ( φ) is superplanckian, all operators that were previously suppressed by M p are now no longer suppressed and contribute to the potential. An innite amount of operators should be ne-tuned to keep the atness of the potential, which is clearly not natural. From an eective eld theory perspective the atness of the potential is not sustainable. Therefore, to properly construct a large-eld model, it is necessary to know if an eective theory admits a UV-completion in which the size of the corrections can be checked. These issues have inspired theorists to work in high-energy theories such as string theory and theories that are known to admit a UV-completion (such as supergravity) to construct models of ination. Summarizing, the UV-sensitivity of ination is both a blessing and a curse. While it makes explicit constructions dicult, it also brings about the hope that one day ination can be used to probe physics at an energy scale where we expect new physics to be relevant. 4

33 3 The eective eld theory of ination The eective eld theory of ination is a powerful principle, that allows for the systematic study of ination in an eective and model independent way. This means that, in a certain energy regime, only the relevant degrees of freedom are studied, while degrees of freedom that are important at a dierent scale will decouple. An EFT description has many benets. For example, an eective eld theory only incorporates degrees of freedom up to a cuto scale, which oers a simplication with respect to the full theory. Moreover, an EFT description that assumes a particular set of symmetries encompasses all possible theories with the same symmetry structure. 3.1 Constructing an eective action There are two ways of obtaining an eective action of a theory that describes the physics at a certain energy regime. When the full theory (S UV ), which is valid up to high energy, is known, heavy degrees of freedom 1 above the cuto scale Λ can be integrated out. Integrating out the heavy degrees of freedom results in an eective action that is only valid up to the cuto and consists of the low-energy action (S IR ) and higher dimensional corrections. The eective action in d spacetime dimensions can then be schematically written as S eff = S IR + d d O ni x c i Λ. (3.1) n i d i Here, O ni denotes an operator of dimension [mass] n i. This gives an ordering principle of the dierent operators in the eective action. Operators with n i > d are called irrelevant; they are important in the UV (high energy) and can be neglected in the IR (low energy). Operators with n i < d are called relevant; they are unimportant in the UV and become important in the IR. Operators with n i = d are called marginal. 1 Heavy degrees of freedom are degrees of freedom that are only probed at a high energy scale 5

34 3 The eective eld theory of ination By ordering the eective action according to this criteria, the dominant contributions can be identied. Often, S UV is not known and the procedure of integrating out heavy modes cannot be executed. This brings us to the second way of obtaining the eective action. As we saw, ignorance about UV-physics will be parametrized by irrelevant operators. By making assumptions about the symmetries of the UV-theory and allowing all operators that are consistent with these symmetries, the same eective action can be obtained. This allows us to construct an eective action, even though we do not have information about the UV-theory. An eective theory is said to be UV-completed when there exists a S UV that complements S IR above Λ. An example that shows that these dierent ways of obtaining the eective action lead to the same result is given in chapter of [14]. 3. Symmetries of ination In order to nd the eective action that describes ination, we have to nd the symmetries of ination. In chapter, we saw that ination corresponds to a period that satises ɛ = Ḣ < 1. (3.) H In the limit that Ḣ 0, the inationary background can be described by a de Sitter space; a maximally symmetric vacuum solution of the Einstein equations with a positive cosmological constant. However, ination cannot be described by a pure de Sitter space, as ination does not end in a de Sitter space. Said dierently, the isometries of de Sitter space contain time-translations, which do not allow ination to end because this would introduce an explicit time dependence. This implies that we can describe ination as a quasi-de Sitter phase where time dieomorphisms are spontaneously broken by a time-dependent scalar φ(t) (the inaton). This scalar acts as a clock that measures the amount of ination elapsed. Perturbations of the inaton around the inating background transform the following way under time-dieomorphisms (given by x 0 x 0 + ξ 0 (x 0, x)). δφ(x 0, x) δφ(x 0, x) + φ(x 0 )ξ 0 (x 0, x) (3.3) Because the inaton spontaneously breaks time dieomorphisms, the theory will contain an associated Goldstone boson [18] that we will denote by π, which parametrizes δφ. We can use the gauge freedom of choosing our coordinates in a way we prefer to let x 0 coincide with the slicing of spacetime introduced by φ(t). This is known as unitary gauge and has the feature that the scalar perturbations (and thus the Goldstone boson) vanish. δφ = π = 0 (3.4) Of course, the Goldstone boson degree of freedom does not disappear, but can be described in terms of an additional degree of freedom of the graviton. Hence, the choice of unitary gauge is convenient, because the theory of cosmological perturbations can be described 6

35 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. are allowed. x i x i + ξ i (t, x) (3.5) 3.3 The eective action of ination Working in unitary gauge, we now want to write down the most general action consistent with spatial dieomorphism invariance. This was rst done by Cheung et al. in [19]. The eective action is given by S = d 4 x [ 1 g M p R c(t)g 00 Λ(t) + 1 M (t) 4 (δg 00 ) M 3(t) 4 (δg 00 ) 3 M 1 (t) 3 (δg 00 )δk µ µ M (t) δ(k µ µ) M ] 3 (t) δk ν µ δkµ ν (3.6) Here, K µν is the extrinsic curvature: the curvature of constant time slices. Furthermore, we expanded the eective action in perturbations of the metric and derivatives. All quantities are written in terms of perturbations, while the unperturbed quantities are absorbed in g [ c(t)g 00 Λ(t) ]. (3.7) For example g 00 = g 00 F RW + δg 00 = 1 + δg 00, (3.8) where the factor 1 can be absorbed in Λ(t). In Appendix A, we show that allowing all operators consistent with spatial dieomorphism invariance will lead to (3.6). The coecients c(t), Λ(t), M (t), M 3 (t), M1 (t), M (t) and M 3 (t) all have a generic time dependence. Since ination only breaks time dieomorphism invariance weakly (the slowroll parameters that break the time dieomorphism invariance have to be small), we expect that it is also natural for all other time-dependent parameters to not vary signicantly per Hubble time Restoring full dieomorphism invariance Going to unitary gauge allowed us to write down the most general action compatible with spatial dieomorphism invariance in an expansion in perturbations and derivatives. It is however convenient to explicitly reintroduce the Goldstone boson, which will non-linearly 7

36 3 The eective eld theory of ination restore the full dieomorphism invariance. This reintroduction can be realized by using the so-called Stückelberg trick, see Appendix A. The advantage of this method is that, at suciently high energy, the Goldstone boson decouples from the metric perturbations. This allows us to describe the EFT of ination only in terms of the Goldstone boson, which simplies the action. This statement is formalized in Goldstone's equivalence theorem [18]. The Goldstone boson can be made explicit again by performing a broken time dieomorphism. We then substitute t t = t + ξ 0 (x) (3.9) ξ 0 (x( x)) π( x). (3.10) Under the time dieomorphism, the 00 component of the metric transforms as g 00 (x) g 00 ( x(x)) = x0 (x) x 0 (x) g µν (x) (3.11) x µ x ν and the the determinant of the metric transforms as ( ) x(x) g g( x(x)). (3.1) x Using these transformation rules one can see that an action of the form d 4 x g [ A(t) + B(t)g 00 (x) ], (3.13) transforms as d 4 x [ ] g( x(x)) x x 0 x 0 x A(t) + B(t) x µ (x) x ν (x) gµν ( x(x)), (3.14) which can be simplied by changing the integration variable to x, such that the Jacobian exactly cancels the x/ x term. d 4 x [ g( x) A( t ξ 0 (x( x))) + B( t ξ 0 (x( x))) ( t ] ξ 0 (x( x))) ( t ξ 0 (x( x))) g µν ( x) x µ (x) x ν (x) (3.15) Next, introduce π, by using (3.10) and drop all tildes to obtain d 4 x [ ] (t + π(x)) (t + π(x)) g(x) A(t + π(x)) + B(t + π(x)) g µν (x). (3.16) x µ (x) x ν (x) By evaluating the second term (t + π(x)) x µ (x) (t + π(x)) g µν (x) =(1 + π(x)) g 00 + (1 + π(x))( x ν i π(x))g 0i (x) + ( i π(x))( j π(x))g ij, (3.17) 8

37 3.3 The eective action of ination we can rewrite (3.6) in terms of the Goldstone boson as S = d 4 x [ 1 g M p R Λ(t + π) c(t + π) [ (1 + π) g 00 + (1 + π)( i π)g 0i + ( i π)( j π)g ij] + 1! M (t + π) 4 [ (1 + π) g 00 + (1 + π)( i π)g 0i + ( i π)( j π)g ij + 1 ] + 1 3! M 3(t + π) 4 [ (1 + π) g 00 + (1 + π)( i π)g 0i + ( i π)( j π)g ij + 1 ] ], (3.18) where the x dependence of π was omitted. Summarizing, full dieomorphism invariance is now restored, because π transforms as π(x) π(x) ξ 0 (x) (3.19) under t t + ξ 0 (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 Equations of motion The leading terms in (3.6) are given by S = d 4 x [ ] 1 g M p R Λ(t) c(t)g 00, (3.0) which determine the unperturbed background evolution. Because we know that the unperturbed evolution is described by a FRW metric, we are able to x the coecients Λ and c by comparing the equations of motion of (3.0) to the Friedmann equation. Additionally, this cancels any tadpoles (terms linear in π that result in a shift of the vacuum expectation value), a procedure known as tadpole cancellation [18]. We can nd the equations of motion of (3.0) by deriving the stress-energy tensor and plugging it into the Einstein equations. The stress energy tensor is given by T µν g δs δg µν (3.1) The variation of (3.0) is given by δs = d 4 x [ δ g Λ δ g cg 00 g c δµδ 0 νδg 0 µν], (3.) which can be rewritten using the identity δ g = 1 ggµν δg µν. (3.3) By denition, only the matter terms contribute to T µν, so the Ricci scalar gives no contribution. 9

38 3 The eective eld theory of ination We then obtain δs = d 4 x [ 1 g g µνλ + 1 ] g µνc g 00 c δµδ 0 ν 0 δg µν, (3.4) so the stress-energy tensor is given by The Einstein equations can be written as T µν = g µν ( Λ + c g 00 ) + c δ 0 µδ 0 ν. (3.5) G µν = 1 T Mp µν. (3.6) We can now determine Λ and c by calculating the components of the Einstein tensor of a at FRW metric and plugging it into the left hand side of the Einstein equations and the derived stress-energy tensor in the right hand side. The relevant components of the Einstein tensor are { G 00 = 3H G µν = G µ µ = 1H 6Ḣ. (3.7) Comparing this with the stress-energy tensor results in 3H (t) = 1 [Λ + c] Mp (3.8) 1H Ḣ = 1 [Λ c], 3Mp (3.9) where we used (3.8) to derive (3.9). Solving these equations for Λ and c results in c = M p Ḣ and Λ = M p [ ] 3H + Ḣ (3.30) Decoupling limit Now that we have xed the background evolution of the eective action, we continue to see how it simplies in a certain energy regime. In a general gauge theory with a spontaneously broken symmetry, Goldstones equivalence theorem tells us that, at high energy, the interaction of a longitudinally polarized gauge boson can be described in terms of the interaction of the Goldstone boson, see Appendix A. In our case, the gauge boson is analogous to the graviton. Similar to the gauge theory example, we can neglect metric uctuations at suciently high energy. Essentially, the metric uctuations decouple from the theory, which allows us to describe the entire dynamics of the theory in terms of the Goldstone boson. Following the gauge theory example, 30

39 3.3 The eective action of ination we can estimate the energy at which this simplication occurs by canonically normalizing π. From (3.18) we see that the kinetic term of π is given by M p Ḣ µπ ν πg µν. (3.31) Thus, the canonically normalized Goldstone boson is dened as The kinetic term for δg 00 is such that If, in analogy with the gauge theory, we then identify π c M p Ḣ 1/ π. (3.3) M p ( µ δg 00 )( ν δg 00 )g µν, (3.33) δg 00 c M p g 00. (3.34) M p Ḣ 1/ = m g, (3.35) where m = Ḣ1/ is the `mass' of the Goldstone boson and g = 1/M p the coupling to gravity. By this identication, the Goldstone self-interaction gets strongly coupled at a scale Λ M p Ḣ 1/. (3.36) Thus, the energy scale at which metric uctuations decouple is given by E m = Ḣ1/. (3.37) Hence, the energy scale at which metric uctuations decouple and we can still trust the theory is given by Ḣ 1/ E Mp 1/ Ḣ 1/4. (3.38) As a consistency check, we can look at the leading order mixing term between δg 00 and π for simple slow-roll ination (M = M 3 = 0). M p Ḣ πδg00 = Ḣ1/ π c δg 00 c, (3.39) from which we see that metric uctuations indeed decouple at an energy E Ḣ1/. 31

40 3 The eective eld theory of ination The simplest slow-roll eective action When taking the decoupling limit, the eective action simplies greatly. Take for example the eective action describing the simplest 3 scenario of slow-roll ination (M = M 3 = 0). By expanding (3.18) and taking the decoupling limit, we obtain S = d 4 x [ 1 g (c + ċπ) M p R Λ Λπ ( (1 + π) + iπ i π a By expanding (3.40) up to two π's or derivatives we get S = d 4 x [ ( 1 g M p Λ Λπ + c π ) iπ i π a )]. (3.40) ] + c π + ċπ. (3.41) Now, we apply integration by parts to the the term containing π (only with respect to time). d 4 x g c π = d 4 x g [3Hcπ + ċπ] (3.4) Plugging this into (3.41) gives S = d 4 x g [ 1 M p Λ Λπ + c ( π ) ] iπ i π ċπ 6Hcπ. (3.43) a Inserting the derived expressions for Λ and c into (3.43) cancels the terms linear in π, as we promised, and yields the following eective action for the Goldstone boson. S π = d 4 x [ ( g Mp Ḣ π + )] iπ i π (3.44) a This simple action describes all possible simple slow-roll models of ination to leading order Moving beyond the simple slow-roll scenario The true power of the EFT of ination becomes visible when we go beyond the simplest slow-roll scenario by turning on other operators (setting M and M 3 non-zero). This allows us to describe deviations from the simple slow-roll scenario. For example, if M is large, the leading order kinetic term of π is given by M 4 µ π ν πg µν. (3.45) 3 With simple we mean a minimal coupling to gravity, canonical kinetic term and a potential that satises the slow-roll conditions. 3

41 3.3 The eective action of ination Thus, the canonically normalized Goldstone boson is dened as π c M π, (3.46) from which we can identify m = M /M p. The energy scale at which the theory becomes strongly coupled becomes Λ M. (3.47) Thus, in the same manner as for the simple slow-roll scenario, we see that the energy scale at which metric uctuations decouple while the theory is still valid is Because the leading order mixing term is given by M M p E M. (3.48) M 4 πδg 00 = M M p π c δg 00 c, (3.49) this is indeed consistent. This results (up to cubic order in π and derivatives) in S π = d 4 x [ ( ) g Mp Ḣ +M 4 π iπ i π a ( π + π 3 π iπ i π a ) 43 M 43 π 3 ]. (3.50) The fact that M is turned on has some interesting consequences. For example, it results in a dierent coecient for the time and spatial kinetic term of π (M 4 M PlḢ) π and M PlḢ iπ i π a, (3.51) which leads to a non-trivial speed of sound (c s 1) of π. c s 1 M 4 M Pl Ḣ. (3.5) Rewriting (3.50) in terms of c s yields [ S π = d 4 x MPl Ḣ ( π + c i π i π c s ) + M s a p 4 3 M 4 3 π 3 ]. Ḣ(1 c s )( π 3 π iπ i π ) a (3.53) Interestingly, a small speed of sound can enhance non-gaussianties (three point correlation functions, known in cosmology as the bispectrum) due to the factor c s in front of the terms cubic in π (which contribute to the bispectrum). 33

42 3 The eective eld theory of ination Observables in the power spectrum A careful reader might be worried that the energy scale at which the simplication of the eective action occurred (the decoupling limit) is not the energy scale relevant for us, because the observations we do today are at much lower energies than the scale at which decoupling takes place. Luckily, we saw in chapter that perturbations that exit the comoving Hubble sphere are conserved outside of the horizon and we therefore have to evaluate all inationary observables at the scale at which horizon crossing took place, which we denoted with a star ( ). Therefore, the relevant scale for ination is V 1/4, which is safely above the decoupling scale, see (.66). Because the Goldstone boson is directly related to the comoving curvature pertubation R (see chapter ) via the relation [0] R = Hπ, (3.54) we can relate correlation functions of the Goldstone boson to correlation functions of the comoving curvature perturbation. For the simple slow-roll scenario with the Goldstone action S π = d 4 x [ ( g Mp Ḣ π + )] iπ i π, (3.55) a this of course results in the same power spectrum we saw earlier, as it should. R(k) k3 π P R(k) = H. (3.56) 8π ɛmp Moving away from simple slow-roll ination by turning on the parameter M in (3.18), results in a non-trivial speed of sound (c s 1) of π. In addition to the non-zero threepoint correlation functions that are generated when M 0, the power spectrum of R also undergoes a change. By only looking at the quadratic terms in π (which are the ones needed to calculate the power spectrum), we obtain from (3.50) the following action S π = d 4 x [ MPl Ḣ ( π + c c s s ) ] i π i π (3.57) a Calculating the power spectrum of R from this action shows that it depends on the speed of sound [0]. R = H. (3.58) 8π c s ɛmp Using the denitions of the spectral tilt (.55), one can see that this parameter also undergoes a change with respect to (.56). n s 1 = 6ɛ + η s, (3.59) 34

43 3.4 Why should we go beyond the eective eld theory of ination? where we dened a new parameter s which measures the change of the speed of sound. s c s (3.60) Hc s The Goldstone actions (3.55) and (3.57) do not generate a non-trivial speed of sound for the tensor perturbations, thus the tensor power spectrum is the same as our original result (.50). h k3 π P h(k) = H π M p (3.61) Hence, the expression for the tensor spectral index (.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 ination in terms of φ. It can incorporate all models of single-eld ination in a simple and systematic way and turning on dierent operators controls deviations from the simplest slow-roll scenario. Moreover, the size of these operators can be constrained by measurements. On top of that, one-loop corrections to correlation functions of cosmological perturbations can be straightforwardly taken into account, as the symmetry structure of the action only allows for a nite number of terms. This implies that renormalization will not generate new operators. Conversely, the normal action of φ contains an innite number of operators. Taking loop corrections into account is therefore possible. Despite the benets of the EFT of ination, it also has shortcomings. As we discussed in section.7, ination and in particular large-eld ination is sensitive to UV-physics. specically, quantum gravity need not to respect the symmetries of the eective theory. This would brutally violate the simplicity of the Goldstone action that we derived, because previously forbidden corrections now have to be taken into account. This urges us to go beyond the eective theory and take a more direct approach in considering quantum gravity eects. Because the best candidate for a theory of quantum gravity that we know of is string theory, it is of a particular interest to construct inationary models in string theory. In practice, this is technically very dicult, although there has been progress over the years. In the next chapter, we will treat aspects of string theory relevant for ination and show how realistic models of ination might come about. 35

44 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 be interesting to construct a model of ination in string theory, for a number of reasons. First of 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 seen if low-energy symmetries are completed in the UV. More specically, quantum gravity does 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 physics is 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 eective actions. For example, models of axion ination originate from string theory, but also admit a phenomenological low-energy description. In this chapter, we will treat some aspects of string theory and supergravity, without making any pretences of being thorough or complete, as this is beyond the scope of this thesis. Instead, we skip technical details and focus on conceptual aspects relevant for ination. For a more in-depth introduction to string theory in the context of ination, we refer 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 can be described as vibrating strings. This string can be dened by a two dimensional worldsheet action which leads to a target space that describes the spacetime. In order for this description to be consistent, the target space has to have more dimensions than our ordinary four-dimensional spacetime. Specically, superstring theory (string theory with supersymmetry on the target space) gives rise to a ten-dimensional spacetime [1]. In the 90's, it was discovered that the ve existing superstring theories 1 were are all related 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, these superstring theories need to be compactied. This process reduces the ten dimensions of the target space to four, by making six dimensions compact. Only at very high energy, the compact dimensions are probed [14]. 1 Type I, Type IIA, Type IIB, heterotic E 8 E 8 and heterotic SO(3) string theory. 36

45 4.1 Aspects of string theory Obtaining a four-dimensional eective action from string theory In order to connect string theory to our four-dimensional world by compactifying the theory, 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 the superstring) results in a ten-dimensional supergravity theory. The next step is the compactication of the supergravity theory. While the ten-dimensional theory enjoys supersymmetry, the four-dimensional eective theory that one obtains after compactifying does not necessarily. However, it was shown that compactifying on Calabi- Yau manifolds can preserve supersymmetry, such that we obtain a four-dimensional effective supergravity action with supersymmetry preserved. Calabi-Yau manifolds have a very 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 fourdimensional theory, moduli appear as massless scalar elds. An example of a modulus is the volume of the compact space. The moduli need to be stabilized (given a mass), such that they do not destabilize the four-dimensional theory. For example, if the modulus corresponding to the volume of the compact space would not be stabilized, the compact space could decompactify. Typically, Calabi-Yau compactications contain many moduli which 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, they decouple from the low-energy theory. The process of generating a potential for the moduli is known as moduli stabilization. Summarizing, the process of obtaining a four-dimensional eective 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 arrow illustrates compactication and moduli stabilization. All compactication data (such as moduli etc.) can be stored by two functions, which specify the supergravity theory. These are the real Kähler potential K and the holomorphic superpotential W, which we will come back to in section De Sitter vacua from string theory Readers with little prior knowledge of string theory, may get the impression that obtaining a stable four-dimensional vacuum from string theory is rather straightforward. This is far from the truth. The fact that we have swept all technical details under the rug obscures the diculty of these constructions. Obtaining stable vacua from string theory, that can be used for cosmology (de Sitter vacua), is an extremely complicated task that is only executed by the bravest of physicists. As a matter of fact, it is currently not even clear if string theorists have really succeeded in this task. Because we think that ination and dark energy domination can be described by an approximate de Sitter vacuum, this is a crucial question to answer. In addition, we would like the minimum of the potential 37

46 4 From string theory to cosmology Figure 4.1: All ten-dimensional superstring theories are related to each other by dualities and to eleven-dimensional M-theory. The low energy limit of a superstring theory is a ten-dimensional supergravity theory. By performing a compactication (on a Calabi-Yau manifold), a four-dimensional eective theory of supergravity is obtained. of the inaton to be only slightly positive, to explain the smallness of the cosmological constant (as V min (φ) = Λ), see gure 4.. As we mentioned before, in order to obtain a stable vacuum we have to employ some moduli stabilization scheme. The moduli appear as at directions in the potential of the supergravity theory and can be stabilized in type IIB string theory by perturbative and non-perturbative corrections to the Kähler and superpotential [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 places 3. As is described in section of [14], competition between at least two corrections is needed, The Kähler potential receives perturbative corrections while the rst correction to the superpotential is non-perturbative, due to a famous non-renormalizable theorem []. 3 This is known as the Dine-Seiberg problem. For a more precise formulation of this statement, we refer the reader to [3]. 38

47 4.1 Aspects of string theory V(ϕ) Λ ds ϕ Λ AdS Figure 4.: Two vacua from a compactication of string theory. The minimum of the potential determines the size of the cosmological constant. to prevent an instability from destroying the stability of the vacuum. Moreover, even when 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 [4]. A possible evasion of this no-go theorem was published in 003, 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 [5]. It was claimed that in this way, a landscape of stable ds vacua could be constructed. In contrast, this mechanism has received a large amount of critique as it was observed that the anti-branes used for the uplifting procedure create a singularity [6, 7, 8]. Dierent attempts have been made to resolve this singularity [9, 30], but to no avail. Concluding, it is currently unclear if it is possible to resolve the singularity in order to obtain a metastable ds vacuum from string theory, despite the fact that a lot of hard work has been done to proof or disproof their existence 4. More philosophically, one might wonder why it seems so dicult to obtain a stable ds vacuum from string theory. After all, we think that string theory should describe nature and that nature can be approximately described by a ds vacuum. How can we resolve this tension? On the one hand, if we believe that it should be possible to obtain ds vacua from string theory, the current absence of convincing constructions might simply be due to our limited creativity and calculational power. On the other hand, perhaps the framework of string theory cannot really be applied to cosmology in the way that we have 4 Recently, dierent approaches of obtaining ds vacua have been presented, such as perturbing supersymmetric Minkowski vacua [31, 3]. 39

48 4 From string theory to cosmology tried. Currently, we are only guided by our theoretical intuition and have no observational signatures of string theory. Nevertheless, here we take the optimistic point of view that these issues will be settled in the future and embedding ination in string theory is an interesting direction to explore. 4. Eective N = 1 supergravity While the eective four-dimensional supergravity theory does not have the complete information of string theory, it is a whole lot simpler. Its simplicity makes constructing models of ination more easily realizable than in string theory. Furthermore, we are equipped with the knowledge that supergravity admits a UV-completion, in contrast with an ordinary eective theory. In this section, we give a brief introduction to supersymmetry, following the 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 interested reader to Martins solid introduction into the subject. The foundation of supersymmetry lies in the idea that bosons and fermions should be related by a symmetry (supersymmetry) to each other. An operator Q that realizes such a 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.) {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 a spacetime symmetry. A supersymmetric theory can have more supersymmetry generators than 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 to work with supermultiplets; states that contain an equal number of bosonic and fermionic degrees of freedom. The components of the supermultiplet are related to each other by the supersymmetry generators. Satisfying the demand that a supermultiplet contains an equal amount of bosonic and fermionic degrees of freedom allows for dierent occupations of the supermultiplet. In the simplest case, a supermultiplet contains one Weyl fermion with two spin helicity states and two real scalars that can be combined in a single complex eld. Such a supermultiplet is known as a chiral supermultiplet. Moving on, we can also take a massless spin 1 vector boson (with two helicity states) accompanied by a spin 1/ Weyl fermion. This combination is known as a vector multiplet. Finally, we can also incorporate the graviton in a supermultiplet together with a spin 3/ Weyl fermion. 40

49 4. Eective N = 1 supergravity A supersymmetric theory can be conveniently reformulated in so-called superspace instead of ordinary four-dimensional spacetime. Superspace has four (bosonic) coordinates x µ (the usual spacetime coordinates) and four (fermionic) coordinates θ α, θ α with dimension [mass] 1/, 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 the coordinates 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 x µ, such that it is invariant under supersymmetry. We can now make a distinction between two dierent types of elds. If a supereld depends on all fermionic 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 term is known as a F-term. Typically, chiral superelds only depend on half of the fermionic coordinates and vector superelds depend on all fermionic coordinates. To obtain an action in four dimensions, the dierent contributions to the action are integrated over the fermionic coordinates. As the inaton is a scalar eld, for a description of ination it is sucient 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]. As we mentioned before, all data of a string compactication can be captured by these two functions. Therefore, one can explicitly construct a model of ination that is inspired by string theory. For N = 1 supergravity, this results in the following general action for a set of chiral superelds [14]. S = d 4 x ] g [K A B µ Φ A µ B Φ V F (4.5) Here, K A B is the Kähler metric and V F is the F-term potential. The F-term potential is given by [14] V F = e K/M p [K A BD A W D BW 3 where D A denotes the Kähler covariant derivative ] W, (4.6) Mp D A W = A W + 1 ( Mp A K)W (4.7) and a subscript indicates a derivative with respect to the superelds. A W = W Φ A (4.8) 41

50 4 From string theory to cosmology 4..1 An example of ination in supergravity We will now construct a simple inationary model in supergravity to illustrate some issues one encounters in trying to do so. The most straightforward choice of Kähler potential one can consider is This makes the Kähler metric rather simple K = Φ Φ. (4.9) K A B = δ A B, (4.10) such that the kinetic term in (4.5) is canonical when the superelds are normalized. It was however quickly realized that this is a particular unfortunate choice, as this induces an exponential steepening of the potential by e K/M p = e Φ Φ/M p, (4.11) in the potential (4.6). A resolution to this problem was proposed in [35]. The authors argued that instead of using (4.10), another type of Kähler potential can be used that also leads to a canonical kinetic term. When one considers K = 1 ( Φ + Φ), (4.1) 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 Φ that 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 not appear 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. Consider a Kähler potential and superpotential of the form K = 1 ( Φ + Φ) + S S (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 holomorphic function. If we assume that we can stabilize the elds S and Re(Φ) at zero [36], the potential becomes V = e K/M p [K A BD ] W A W D B W 3 = S W S W = m g(φ) M p Re(Φ)=S=0 (4.16) 4

51 4. Eective N = 1 supergravity This gives us a large freedom of obtaining almost any potential we like, because g(φ) is an arbitrary holomorphic function. For example, if we take the simple choice we obtain g(φ) = Φ, (4.17) V = m Im(Φ) (4.18) which, as promised, is the familiar quadratic ination potential when we parametrize Im(Φ) = φ/. Although we saw in chapter that this model is in disagreement with observations, it illustrates how a model of ination in supergravity might come about. In chapter 5 and 6, we will see other well-motivated models of ination in supergravity. Because we know that supergravity is UV-completed by string theory, such models are a step in the right direction to embed ination in string theory. 43

52 5 Axion ination In chapter, we saw that large-eld ination is extremely UV-sensitive. This requires that we impose some symmetry to suppress dangerous corrections. In chapter 4, we motivated why we need to account for this symmetry in string theory. In large-eld models, only a shift symmetry is able to suppress all dangerous corrections. Thus, we would 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 are shift-symmetric particles that arise naturally in string theory [37]. The shift symmetry of the axions is weakly broken to a discrete symmetry by non-perturbative corrections (instantons) [38]. This makes axions radiatively stable, such that they can accommodate large-eld ination. 5.1 Natural ination Phenomenologically, we can describe ination as driven by a pseudo nambu-goldstone boson from the breaking of a shift symmetry and this observation has led to a model of ination known as natural ination [39]. We can think of the pseudo-nambu goldstone boson as an axion with the following potential. V (φ) = Λ 4 [1 cos ( )] φ f (5.1) Here, f is the axion decay constant, which determines the periodicity of the eld (and therefore the maximal eld range available for ination) and Λ is some generated energy scale. 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 eective N = 1 supergravity. We consider two complex elds; the chiral supereld Φ (which can arise as a Kähler modulus) and a supereld S (for example a goldstino, which is associated with supersymmetry breaking). If we then take the following Kähler and superpotential K = Φ + Φ + S S g(s S) W = Λ S(1 e at ), (5.) 44

53 5. Saving natural ination the role of the inaton is played by the linear combination of Φ and Φ that does not appear in the Kähler potential. It will be useful to parametrize Φ and S in terms of real elds. Φ = 1 (β + iφ) S = 1 (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 ( )] aφ V = Λ [1 4 cos. (5.4) Modifying the Kähler and superpotential can lead to similar natural ination-like potentials. Whereas natural ination is a well-motivated model from string theory, it unfortunately has a fatal aw. In order to be consistent with observations, the axion decay constant f has to be larger than the Planck mass. This is problematic, because string theory does not allow for such large axion decay constants. To be more precise, whenever one nds a superplanckian axion decay constant in string theory, there are also unsuppressed corrections that destroy the superplanckian eld displacement, by generating additional maxima along the axion potential [41]. For a more elaborate discussion of this point, see section 7.. We conclude that natural ination, in its original form, is not compatible with observations, as it is not possible to obtain superplanckian decay constants in string theory. 5. Saving natural ination Even though natural ination is not a good model for obtaining large-eld ination in string theory, there have been attempts at modifying natural ination, such that it can be realized in string theory and also is compatible with observations. The leading proposals for modifying natural ination such that it keeps its features (well-motivated from string theory, radiatively stable and having a large eld range) and also is consistent with observations, can be divided into three dierent categories: N-ation, alignment mechanisms and axion monodromy ination. We will only focus on the rst two proposals and refer the interested reader to [38] for a review of axion monodromy ination (and other models of axion ination). The idea of N-ation is based on the following observation: while individual axions must have a subplanckian axion decay constant, it might be possible to use many subplanckian axions 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 the axions is not canonical. In order to make the kinetic term canonical, a non-trivial rotation of the axions is required, which brings the axions into a new basis. In this new basis, 45

54 5 Axion ination a large direction in eld space may be present, while it was not manifest in the original basis. We will see some examples of both of these approaches and comment on the issues and features these models have N-ation Dimopoulos, Kachru, McGreevy and Wacker were the rst ones to present the idea of N-ation; using multiple axions to obtain a superplanckian eld range [4]. If we consider N copies of the potential (5.1), each of which breaks a dierent shift symmetry, we obtain the following action. S = d 4 x g N [ 1 ( ( ))] φ µφ i µ φ i Λ 4 i i 1 cos i=1 f i (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 if its possible to obtain a large at direction in eld space, we introduce the notion of the fundamental domain, which is a N-dimensional space constrained by the periodicity of the axions. θ i π i (5.6) In the case of (5.5), the fundamental domain is a hypercube with N sides of π. For large- eld ination to occur, we have to nd a large direction in eld space, while satisfying the constraints (5.6). In the language of the fundamental domain, this means nding a superplanckian distance in the fundamental domain. If we now dene the inaton Φ to be the linear combination of the individual axions that is parallel to a diagonal of the fundamental domain, we can obtain an enhancement of the eld displacement Φ. If the dierent axion decay constants are comparable, the eld displacement is enhanced by a factor of N, with respect to the single-eld case. The diameter of the fundamental domain is given by 1 D = π f fn πf N. (5.7) This situation is illustrated for two axions in gure 5.1. Summarizing, even when we start with all f i < M p, a superplanckian eld displacement can arise in a particular direction in eld space. Nevertheless, it can be questioned how general this is, as it requires picking out a special direction in the fundamental domain along which ination occurs. 1 Note that only part of the total diameter of the fundamental domain can be used for ination. 46

55 5. Saving natural ination θ 1 = π φ = fπ ˆφ Φ = π f 1 + f φ 1 = f 1 π ˆφ 1 θ = π Figure 5.1: Picking out a direction in eld space parallel to the diagonal of the fundamental domain 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 two axions, for which the fundamental domain is a square. 5.. 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 = d 4 x [ g 1 N K ( ( ij µ θ i µ θ j ))] 1 cos θ i, (5.8) where K ij is the metric on eld space. In the case that K ij is diagonal, it is given by and we can rescale i=1 Λ 4 i K = diag(f i ), (5.9) φ = diag(f i ) θ, (5.10) to obtain the same action as in (5.5). In the case that K ij is not diagonal, we have to perform a basis transformation that diagonalizes K ij and then rescale Here, R is a rotation matrix that diagonalises K. i=1 φ = diag(f i )R T θ. (5.11) R T KR = diag(f i ) (5.1) Performing the rotation and rescaling, we obtain S = d 4 x [ ( ( ))] N g 1 µφ i µ φ i Λ 4 (R 1 φ)i i 1 cos. (5.13) f i 47

56 5 Axion ination Due to the rotation R, necessary to bring the kinetic term into a canonical form, there now is a non-trivial relation between the axions that dene the fundamental domain and the canonical axions. If one of the canonical axions φ i is parallel to a diagonal of the fundamental domain, we can again obtain an enhancement of the eld displacement by a factor N, but now only one axion decay constant needs to be relatively large, see gure 5.. The maximal diameter is now obtained when the largest axion decay constant f max corresponds to an axion that is parallel to a diagonal in the fundamental domain. This is known as kinetic alignment [43]. The diameter is now given by D = πf max N. (5.14) θ 1 = π φ = f π ˆφ φ1 = f 1 π ˆφ 1 Φ = πf 1 θ = π Figure 5.: The canonical axions φ are related by a non-trivial rotation to the axions θ that dene the fundamental domain. If a canonical axion with a relatively large axion decay constant is parallel to a diagonal in the fundamental domain, an enhancement of the eld displacement by a factor N is obtained Lattice alignment Another type of alignment that leads to a large axion decay constant is known as lattice alignment. This mechanism was introduced by Kim, Nilles and Peloso [44]. They considered two axions θ 1 and θ appearing in the action as follows. S = d 4 x [ g 1 ( ( µθ i µ θ i Λ 4 θ1 1 1 cos + θ )) ( ( + Λ 4 θ1 1 cos + θ )) ] f i=1 1 g 1 f g (5.15) For simplicity, take f = f 1 = f. If in addition we take g 1 g, the same linear combination of θ 1 and θ appears in the cosines. Thus, a at direction appears for the orthogonal 48

57 5.3 Statistical generality of axion ination combination θ 1 g 1 θ f. (5.16) When changing to the normalized elds Ψ = Φ = fg 1 f + g 1 ( θ1 f + θ ) g 1 ( fg 1 θ f + g1 f θ 1 g 1 ), (5.17) it can be seen that the eld Φ has an axion decay constant of If we dene f Φ = g g 1 g f + g 1. (5.18) g 1 g = ε, (5.19) a parametrically large axion decay constant can be obtained in the limit ε 0, whereas the original decay constants f, g are all subplanckian. Again, the question arises how general the situation is where alignment occurs. 5.3 Statistical generality of axion ination In the previous section, we found that obtaining a superplanckian direction in eld space is in principle no problem, when multiple axions are used. Nevertheless this does not proof that such a situation is very generic in string theory. It can be the case that the situations we considered above are extremely rare and, in realistic setups, not realizable. Due to the technical nature of explicit constructions in string theory, checking this point is not an easy task. Therefore, a more accessible approach of nding the occurrence of superplanckian directions is to make use of statistical arguments in the large N limit (when many axions are present). Using statistical arguments in the string landscape is not new [45] and has also been applied to axion ination by McAllister et al., which led to some interesting observations [43, 46, 47] Emergent properties of N-ation The statistical properties of models of N-ation with a large number of axions were rst derived by Easther and McAllister [46]. They showed that in the large N limit, the See chapter 7 for another issue with lattice alignment. 49

58 5 Axion ination spectrum of masses of the axions is an emergent property, which does not depend on the exact 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 = (π) M ij θ i θ j + O(θ 3 ), (5.0) where M ij is the mass matrix. Now, the action is given by S = d 4 x g [ 1 ] K ij µ θ i µ θ j (π) M ij θ i θ j (5.1) As before, we can perform a change of basis to render the kinetic term canonical and absorb the eigenvalues of K ij (the axion decay constants squared) with a eld redenition. S = d 4 x [ g 1 µφ i µ φ i (π) M ] ij φ i φ j (5.) f i f j To characterize the behaviour of the axions, information about the mass matrix and in particular its eigenvalues (the squared masses of the axions) is needed. However, a full microscopic consideration is rather complicated. Luckily, when M ij is large enough, it is reasonable to approximate it by a random matrix. In particular, Easther and McAllister have shown that is is reasonable to approximate M ij by a Wishart matrix: a matrix of the form M = A T A, (5.3) where A is a (N + P ) N random matrix with entries that can be (but are not limited to be) Gaussian random variables. It is well-known that the spectrum of eigenvalues of a large Wishart matrix is described by the Mar cenko-pastur probability distribution [48]. P (m ) = (b m )(m a) πm βσ, (5.4) where a = σ (1 β 1 ) (5.5) b = σ (1 + β 1 ) (5.6) β = N N + P. (5.7) Since the variance σ has to be tuned such that the average mass scale matches observations, β is the only free parameter. Even this parameter is not entirely free, because N can 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

59 5.3 Statistical generality of axion ination In gure 5.3, 5.4 and 5.5, we show the eigenvalue distribution of a set of mass matrices M = A T A with dierent values for β, where the entries of A were randomly picked from a normal distribution. It can be seen that this spectrum is well-tted by a Mar cenko-pastur distribution. Entries m Figure 5.3: Spectrum of eigenvalues of matrices with β = 1, tted with a 3 Mar cenko-pastur distribution. Entries m Figure 5.4: Spectrum of eigenvalues of matrices with β = 1, tted with a Mar cenko-pastur distribution. 51

60 5 Axion ination Entries m Figure 5.5: Spectrum of eigenvalues of matrices with β = 1, tted with a Mar cenko-pastur distribution. From these gures, we can make some interesting observations. The Mar cenko-pastur distribution provides an excellent t to the eigenvalue spectrum of the mass matrix. Since the 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 occur with N-ation is still a rather dicult question to answer. As the spectrum of masses is typically broad 3, the axions will not all evolve in the same manner. Therefore, to obtain the full dynamics, we have to follow the evolution of each of the axions, which is a very laborious task with a large number of axions. Thus, from this analysis, it is not conclusive if it is possible to construct a model of N-ation in which a superplanckian eld range occurs naturally Eigenvector delocalization in models with kinetic mixing At rst sight, models with kinetic alignment seem less ne-tuned than N-ation, due to the fact that the large direction in the fundamental domain is generated by only a single axion (with a relatively large axion decay constant). To nd a more thorough answer how general kinetic alignment is, we will determine how likely it is for a canonical axion to be parallel to a diagonal in the fundamental domain. This can be found out by looking at the eigenvectors of the kinetic matrix. As was remarked by Bachlechner, Dias, Frazer and McAllister in [43], kinetic alignment is a rather generic occurrence. This can be intuitively seen from the fact that at large N, the fundamental domain has only N sides, but N diagonals. It is therefore statistically more likely for an eigenvector to be parallel to a diagonal than not, if we expect the eigenvectors to be randomly distributed. This can be complemented by mathematical statements about random matrix theory. If the kinetic matrix K can be approximated by a Wishart matrix, 3 Easther and McAllister argued that, by considering the renormalization of the Planck mass in the presence of a large number of species, β = 1 is favoured. 5

61 5.3 Statistical generality of axion ination 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 high probability. This is known as eigenvector delocalization. Furthermore, eigenvector delocalization is not very sensitive to the type of distribution that is chosen. Therefore, we are not only limited to the case when K is a Wishart matrix. This was also cross-checked with explicit string constructions by Bachlechner, Long and McAllister in [47], who showed that in Calabi-Yau compactications of string theory the kinetic matrix is well-approximated by a random matrix from an ensemble that exhibits eigenvector delocalization. In particular, they compared their results to an explicit compactication [50] and found agreement with their statistical results Diameter of the fundamental domain of the most general axion action In order to nd more realistic examples of kinetic alignment, we consider a generalization of (5.8),which is inspired by explicit string constructions [47]. This generalization takes into 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 = d 4 x [ g 1 P K ( ij µ θ i µ θ j 1 cos(q i j θ j ) )]. (5.8) Here, Q is a matrix with integer entries. It will proof useful to introduce the decomposition ( ) Q Q =, (5.9) where Q is a square N N matrix and Q R a (P N) N matrix. This allows us to rewrite (5.8) as S = d 4 x [ g 1 N K ( ij µ θ i µ θ j 1 cos(q i j θ j ) ) P N ( 1 cos((qr ) i jθ j ) )]. i=1 Λ 4 i Q R i=1 Λ 4 i i=1 Λ 4 i (5.30) In order to dene the fundamental domain, we perform a basis transformation that renders the fundamental domain hypercubic, as we did before. Q θ = φ with (Q 1 ) T KQ 1 Ξ (5.31) Now, (5.30) becomes S = d 4 x [ g 1 Ξ ij φ i φ j N i=1 Λ 4 i ( 1 cos(φ i ) ) P N i=1 Λ 4 i ( 1 cos((q R Q 1 φ) i )) ]. (5.3) 53

62 5 Axion ination In this basis, it becomes manifest that the fundamental domain is a hypercube with sides π, cut by (P N) additional constraints. Summarizing, the fundamental domain is dened by φ i π i and (Q R Q 1 φ) i π i. (5.33) Lastly, in order to determine the eld range of the canonical eld Φ in this fundamental domain, a last basis transformation that diagonalizes Ξ is needed. The canonical elds are then dened as Φ = diag(ξ i )S T φ with S T ΞS = diag(ξ 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 = d 4 x g [ 1 Ξ ij φ i φ j N Λ 4 i i=1 ( 1 cos(φ i ) )], (5.35) such that the fundamental domain is a hypercube with sides π. This is similar to the situation we encountered for the action (5.8), with the important dierence that K and Ξ are non-trivially related. The diameter is now given by D = Φ T Φ = φ T Sdiag(ξ i )S T φ = φ T Ξφ (5.36) If we denote the maximal possible diameter squared by r max, we see from (5.36) that the diameter is dened by an ellipsoid in the fundamental domain, where the principal axes are given by the normalized eigenvectors of Ξ and with a length determined by 1/ξ i. Finding the maximal diameter then amounts to nding the largest r for which the entire ellipsoid intersects the fundamental domain. The maximal possible displacement is obtained when the eigenvector Ψ max of Ξ with the largest eigenvalue ξ max (corresponding to the shortest axis of the ellipse) is parallel to a diagonal in the fundamental domain, see gure

63 5.3 Statistical generality of axion ination φ 1 = π Ψ min Ψmax φ = π Figure 5.6: Finding the maximal eld displacement amounts to nding the maximal r for which the ellipse intersects the fundamental domain. The maximal diameter is then given by the shortest axis. Summarizing, to obtain the largest possible diameter, we have to rotate the ellipse in such a way that the eigenvector Ψ max corresponding to the largest eigenvalue ξ max is parallel to a diagonal and then nd the largest possible r, such that the ellipsoid intersects the fundamental domain. The maximal possible diameter is then given by D = πξ max N, (5.37) which is similar to (5.14) with the dierence that there is a non-trivial relation between the eigenvalues of Ξ and K. Case #: More instantons than axions Secondly, there can also be more instantons than axions (P > N), such that the fundamental domain is a hypercube, cut by P N additional constraints, see (5.33). It is now no longer possible to obtain a closed expression for the maximal eld displacement as we were able to in the previous case. Nevertheless, Bachlechner, Long and McAllister showed in [47] that it is still possible to compute the eld displacement in an arbitrary direction when the matrices Q and K are known. Furthermore, even when these matrices are not known we can use the fact that, when they are large, they reach a universal limit in which their properties are statistically determined. This makes it still possible to obtain bounds on the maximal eld range. 55

64 5 Axion ination The strongest constraint on the fundamental domain is given by the P N constraints in (5.3), that cut the hypercube. Therefore, we can obtain the eld displacement in a particular direction ˆv by introducing an operator that rescales the ellipse on which ˆv ends, such that it intersects the fundamental domain. W (ˆv) = π ˆv (5.38) Max i ((Q R Q 1 φ) i ) It will be convenient to parametrize ˆΨ as a linear combination of some of the eigenvectors multiplied by the square root of its corresponding eigenvalue. v = i ξ i Ψi (5.39) The maximal diameter along the direction ˆv is then given by (using (5.34) and (5.38)) D = diag(ξ i )S T W (ˆv) = W (ˆv) diag(ξ i )S T j ξ j Ψj = W (ˆv) diag(ξ i )S T ξ ( Ψj 's are orthonormal) = W (ˆv) ξi 4, i (5.40) where ξ is a vector that contains the ξ i 's that correspond to the eigenvectors we used to construct v, the remaining entries are zero. This gives us an analytical expression to compute the diameter in an arbitrary direction, if Q and the eigenvalues of Ξ are known Statistical bound on the eld range In the case that Q and K are not known, but can be approximated by a random matrix from some ensemble, it is possible to obtain a lower bound on the eld displacement. To calculate this, an estimate of W (ˆv) is needed. In the case that Q is square (P = N), the operator W (ˆv) simply evaluates to W (ˆv) = π ˆv, (5.41) Max i (ˆv i ) as the fundamental domain is now a hypercube with sides π. Now, if it is reasonable to approximate Ξ by a random matrix that is rotationally invariant, its eigenvectors will have entries that are normally distributed, such that they will exhibit eigenvector delocalization. The median size of the largest entry can then be approximated by [47] Max i (ˆv i ) = erf 1 ( 1/N ) N, (5.4) 56

65 5.3 Statistical generality of axion ination such that W (ˆv) = π N erf 1 ( 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 Max i ((Q R Q 1 φ) i ) erf 1 ( 1 P N ) log(p N), (5.44) such that W (ˆv) = π log(p N) (P > N). (5.45) In the case that P = N, the enhancement of W (ˆv) grows with N, see gure 5.7, but when P > N a large enhancement only occurs when P is slightly bigger than N and decreases at large N, see gure 5.8. It is now possible to obtain a bound on the maximal W(v ) N Figure 5.7: W (ˆv) for the case P = N. The enhancement grows with N. diameter when we consider dierent ensembles of random matrices. For example, if we consider the kinetic matrix to be diagonal (K = diag(f i )), Ξ becomes By diagonalizing Ξ with S, we obtain the relation Ξ = f (QQ T ) 1. (5.46) diag(ξi ) = f diag( 1 ), (5.47) Q i from which it follows that the largest eigenvalue of Ξ is given by ξ max = f Min(Q i ). (5.48) 57

66 5 Axion ination W(v ) P N =1.1 P N =1.4 P N =1.7 P N = N Figure 5.8: W (ˆv) for dierent values of P/N. A large enhancement is possible for a small P/N, which decreases with N. By using some properties of random matrix theory, (5.43) and (5.45), it was shown that the maximal diameter is bounded by [47] { fn 3/ (P = N) D (5.49) fn (P > N). Interestingly, this exceeds the naive enhancement of N we derived earlier. This shows that for generic congurations, obtaining superplanckian diameters is no problem. In contrast, its realization in string theory is something that is currently under discussion, see chapter 7. 58

67 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 = d 4 x g M p R. (6.1) In contrast, one might add some function f(φ) multiplying the Ricci scalar, such that the action deviates from the canonical Einstein-Hilbert form. S = d 4 x gf(φ)r. (6.) Adding a non-minimal coupling can be motivated in dierent ways. For example, it is well-known that there exists interesting models of ination that require a non-minimal coupling, such as Higgs ination [51] and cosmological attractor models [5]. Whereas the non-minimal coupling is sometimes considered a pathology in Higgs ination, it is a feature in the attractor models. Furthermore, even when a non-minimal coupling is absent in 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 spoil the atness of the potential [54]. We therefore have to make sure that the non-minimal coupling is theoretically under control. In this chapter, we will treat some important aspects that one encounters when including a non-minimal coupling and show how a nonminimal coupling can be introduced in supergravity. Continuing, we treat some interesting models and show how a non-minimal coupling is radiatively generated in the presence of N 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 the non-minimal coupling in the Jordan frame, it appears as a function that multiplies the Einstein-Hilbert term in the action. Alternatively, we can perform a conformal transformation on the metric that renders the Einstein-Hilbert term canonical, but changes the action for the scalar eld. This description is known as the Einstein frame. These two different descriptions are physically equivalent, since they are simply related by a conformal transformation. Nevertheless, the non-trivial relation between the two frames can hide 59

68 6 Non-minimal coupling 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 term in d spacetime dimensions. S Jordan = d d x [ g f(φ)r 1 ] gµν µ φ ν φ V (φ) (6.3) Here, M d is the d-dimensional reduced Planck mass. We can perform a conformal transformation on the metric g µν. The Ricci scalar ˆR constructed from ĝ µν is then given by [55] ˆR = 1 Ω [ R ĝ µν = Ω g µν (6.4) (d 1) Ω Ω ] (d 1)(d 4), (6.5) Ω with Ω = g µν µ ν Ω = 1 g µ ( gg µν ν Ω ). (6.6) Furthermore, the determinant of the metrics are related as ĝ = Ω d g. (6.7) Using these identities, the gravitational part of (6.3) can be written as d d x gf(φ)r = [ ĝ d d x Ω f(φ) Ω (d 1) ˆR + Ω d Ω ] (d 1)(d 4) + g µν Ω µ Ω ν Ω. (6.8) From the rst term in (6.8), it is clear that in order to render the gravitational part of the action canonical, we have to make the identication f(φ)ω d M ˆR d = d ˆR. (6.9) Now, the second term in (6.8) can be integrated by parts, which results in 1 d d x ĝ (d 1) f(φ) Ω = Ω d+1 d d x ĝm d (d 1)(d 3) d ĝ µν Ω ˆ µ Ω ˆ ν Ω. (6.10) 1 Note that there is a dierence between ˆ and. However, we are only acting on scalars, such that Ω = ˆ Ω. 60

69 6.1 Description of a non-minimal coupling in the Jordan and Einstein frame The last term in (6.8) can be rewritten as. d d x ĝ M d d (d 1)(d 4) ĝ µν Ω ˆ µ Ω ˆ ν Ω (6.11) Adding these contributions, the gravitational part of (6.3) can be written in quantities constructed from ĝ µν as S = d d x ĝ M d [ ] d (d 1)(d ) ˆR ĝ µν Ω ˆ µ Ω ˆ ν Ω. (6.1) The scalar part of (6.3) can be written as follows. d d x [ ĝ 1 Ω d ˆ µ φ ˆ ν φ V (φ) ] Ω d (6.13) Finally, we can write the complete action in terms of quantities constructed from ĝ µν as S = d d x [ M d d ĝ ˆR M d d (d 1)(d ) ĝ µν Ω ˆ µ Ω ˆ ν Ω 1 ] (6.14) Ω d ĝ µν ˆ µ φ ˆ ν φ ˆV (φ) where we dened ˆV (φ) V (φ)/ω d. This action can be conveniently rewritten in terms of f(φ) by using (6.9). The Einstein frame action is then given by S Einstein = d d x [ M d d ĝ ˆR 1 ] K(φ)ĝµν ˆ µ φ ˆ ν φ ˆV (φ). (6.15) We dened the kinetic function K(φ) as where K(φ) M d 4 d ( d 1 d ) f φ f(φ) + M d d f(φ), (6.16) f φ = φ f. (6.17) In four dimensions, (6.15) can be written as S Einstein = d 4 x [ M p ĝ ˆR 1 ] K(φ)ĝµν µ φ ν φ ˆV (φ). (6.18) We conclude that a scalar non-minimally coupled to gravity with a canonical kinetic term in the Jordan frame is equivalent to a minimally coupled scalar with a non-canonical kinetic term in the Einstein frame. We can straightforwardly generalize the above derivation to the case of multiple scalars. If we take the Jordan frame action for multiple scalars S Jordan = d d x [ g f(φ A )R 1 ] δ AB g µν µ φ A ν φ B V (φ A ), (6.19) 61

70 6 Non-minimal coupling the corresponding Einstein frame action in four dimensions is S Einstein = d 4 x [ M p ĝ ˆR 1 ] K AB(φ A )ĝ µν µ φ A ν φ B ˆV (φ A ), (6.0) with and K AB (φ) = M p [ 3 f A f B f + δ ] AB f (6.1) f A = A f. (6.) Canonical normalization of the kinetic term We just saw that we could describe a non-minimal coupling in the Einstein frame by a canonical Einstein-Hilbert term, at the expense of having a non-canonical kinetic term for the scalar eld(s). However, in the case of a single scalar eld, we can always perform a eld redenition that renders the kinetic term canonical. In other words, we want to dene a eld ˆφ such that Dividing by µ φ µ φ gives 1 ĝµν µ ˆφ ν ˆφ = 1 K(φ)ĝµν µ φ ν φ. (6.3) ( d ˆφ ) = K(φ). (6.4) dφ Thus, the canonically normalized eld can be obtained by integrating [ d ˆφ 3 = ± We conclude that (6.18) can be simplied to S Einstein = f φ f + 1 M p f ] 1/ dφ. (6.5) d 4 x [ M p ĝ ˆR ĝµν 1 ] µ ˆφ ν ˆφ ˆV ( ˆφ). (6.6) One should be careful in thinking of this action as being equivalent to the simple single scalar eld action (.3) rst used to describe ination. Due to the conformal transformation and eld redenition to bring it into this form, (6.6) can show a very dierent behaviour than when we start with the simple action (.3). In the multield case, the situation is a bit more subtle. Because the kinetic term K AB can be interpreted as a metric on eld space, the statement that there exists a set of canonical 6

71 6. Supergravity formulation of a non-minimal coupling elds {φ A } is equivalent to the statement that there exists a conformal transformation that renders K AB ˆK AB = δ AB. (6.7) Thus, in order to obtain a canonical kinetic matrix, K AB must be conformally at [55]. In order to be conformally at, all components of the Riemann tensor constructed from the metric must vanish [56]. R a bcd = 0 (6.8) The number of independent components of the Riemann tensor is given by [56] 1 1 N (N 1), (6.9) where N is the number of elds. We see that it becomes increasingly more dicult for K AB to be conformally at with an increasing number of elds. We can deal with this diculty in the following way. On a curved manifold, it is always possible to dene local inertial coordinates, for which the metric is at, but only in some nite region. Therefore, this method can only be applied when ination does not take the elds to far from their original value. When this condition is satised, new slow-roll parameters can be dened and we can continue in the same manner as usual, see Appendix A of [57]. 6. Supergravity formulation of a non-minimal coupling A non-minimal coupling can also be formulated in supergravity. Some years ago, a new formulation of supergravity was discovered, which makes it possible to write the N = 1 supergravity action in the Jordan frame [58, 59]. This inspired new models of ination in supergravity [60]. To show the relation between the Einstein and Jordan frame, we start from the Einstein frame supergravity action (in the absence of gauge interactions) with a set of superelds { z A, zā}. S Einstein = d 4 x [ 1 ĝ M p ˆR 1 ] K A Bĝ µν µ z A ν z B ˆV (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 K A B is the Kähler metric. If we then dene the conformal transformation between ĝ µν and g µν as ĝ µν = Ωg µν, (6.31) For N =, N = 3, it is sucient that the components of respectively the Ricci scalar and Ricci tensor vanish. 63

72 6 Non-minimal coupling we can relate quantities constructed from both metrics in the following way. ĝ = Ω g (6.3) ˆR = 1 [ R log Ω + 3 ] g µν Ω Ω µω ν Ω (6.33) Using these identities, the action can be written in the Jordan frame. S Jordan = d 4 x [ M ] p g ΩR + 3M p Ω A Bg µν µ z A ν z B V (z), (6.34) where V (z) = ˆV (z)/ω 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]. ( ) Ω pol = 1 1 B za z δ 3 A B Mp + J(z) + J( z), (6.36) where J(z) is an arbitrary holomorphic function. Alternatively, the authors of [61] also have considered an exponential Ω, given by [ ( )] Ω exp = exp 1 B za z δ 3 A B + J(z) + M J( z). (6.37) p This makes it clear how a non-minimal coupling can be introduced in supergravity Examples of non-minimal supergravity models The models considered in [61] have two superelds (z A = (S, Φ)) and the function J(z) was specied as J(z) = 3χ 4 Φ Mp, (6.38) where χ is some real number. The strength of the non-minimal coupling can be measured by ξ, which is dened as 3M p ξ = χ 4. (6.39) Using this parameter and (6.38) we can rewrite (6.36) and (6.37) as Ω pol = 1 1 [ S 3M S 3ξ p (Φ + Φ) 1 ] (3ξ + 1)(Φ Φ) (6.40) [ ( 1 Ω exp = exp S S 3ξ (Φ + Φ) 1 )] (3ξ + 1)(Φ Φ). (6.41) 64

73 6.3 Cosmological attractors If we assume that it is possible to stabilize S = Im(Φ) = 0, we can describe ination by only a single eld Re(Φ) = φ/. Ω pol S=Im(Φ)=0 = 1 + ξ φ M p Thus, the action in the Jordan frame is given by S pol Jordan = S exp Jordan = (6.4) Ω exp S=Im(Φ)=0 = e ξ φ Mp (6.43) d 4 x [ 1 g (M p + ξφ )R 1 ] gµν µ φ ν φ V (φ) (6.44) d 4 x [ M ] φ p 1 g eξ Mp R gµν µ φ ν φ V (φ). (6.45) We see that ξ indeed determines the strength of the non-minimal coupling of the inaton to gravity. Specically for S pol Jordan, χ = 0 corresponds to a canonical Kähler potential which leads to a value of ξ = 1. When ξ takes this value, the inaton is conformally 6 coupled, which means that it does not get renormalized by quantum corrections [6]. Now that we know the form of the action of these models in the Jordan frame, we can construct models of ination with a non-minimal coupling in supergravity. Remember from section 4..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 ξ breaks this symmetry. When choosing the superpotential as W = Sg(Φ), (6.46) this leads to the Einstein frame potential V (Φ) = e K/M p g(φ). Thus for (6.40) and (6.41) we get ) 3 V pol S=Im(Φ)=0 = (1 + ξ φ g(φ/ ) V exp S=Im(Φ)=0 = e 3ξ φ Mp M p g(φ/ ) which for an appropriate choice of parameters, can sustain ination. (6.47), (6.48) 6.3 Cosmological attractors The most interesting behaviour of a non-minimal coupling becomes explicit in the context of cosmological attractors. These attractors were discovered when it was observed that 65

74 6 Non-minimal coupling di erent types of models had the same cosmological predictions to leading order [63]. The attractor models known as ξ -attractors have predictions that converge to [64, 65] ns = 1 N ns = 1 N (6.49) 1 r =, N for large values of ξ. Here, N is the number of e-folds. Similarly, there exist a class of α-attractors which has the predictions [66] (6.50) 1α r=, N for small values of α. These models encompass already existing models of in ation (such as Starobinsky [67] and Higgs [68, 51] in ation). Furthermore, it was shown that the ξ and α-attractors overlap for speci c values of the relevant parameters [69] (see also gure 6.1) and that the α-attractors can also be embedded in supergravity [66, 70]. Figure 6.1: Classi cation of the di erent attractor models. Figure from [69]. In this section, we will present the di erent attractors models and show how these can be embedded in supergravity ξ -attractors The ξ -attractors have the following action in the Jordan frame. Z Mp 1 4 µν SJordan = d x g Ω(φ)R KJ (φ)g µ φ ν φ V (φ) 66 (6.51)

75 6.3 Cosmological attractors By taking K J = 1 and using the conformal transformation (6.31) this action can be brought into the Einstein frame. S Einstein = d 4 x [ M p ĝ ˆR 1 ] K(φ)ĝµν µ φ ν φ ˆV (φ), (6.5) with K = 3 ( ) Ωφ + 1 Ω(φ) Ω(φ). (6.53) If we now make the choice Ω = 1+ξg(φ), this model exhibits attractor behaviour for large ξ. We can see this as follows. For large ξ, the rst term in K dominates, such that (6.5) becomes d ˆφ 3 Ω φ = ± dφ. (6.54) Ω(φ) Integrating this expression yields 3 ˆφ = ± log (1 + ξg(φ)). (6.55) One has to be careful in specifying the correct sign, as we want ˆφ to be dierentiable at φ = 0 (at which ˆφ vanishes when we take g(φ = 0) = 0). If g(φ) is odd, ˆφ is well-dened for any value of φ. In contrast, when g(φ) is even, we should pick the opposite solutions for φ < 0 and φ > 0 [71]. Without loss of generality, we can take the Jordan frame potential to be V (φ) = λ ξ (Ω(φ) 1). (6.56) If g(φ) is odd, we can pick the positive solution of (6.55), such that the Einstein frame potential becomes ˆV odd ( ˆφ) = λ [ 1 e ˆφ] ξ 3. (6.57) This potential is plotted in gure 6.. We see that, independent of the choice of g(φ), we obtain the potential (6.57), which corresponds to Starobinsky ination with the predictions given by (6.49). Conversely, when g(φ) is even, the Einstein frame potential becomes ˆV even ( ˆφ) = λ [ 1 e ξ φ ˆ 3 ], (6.58) which is the potential corresponding to Higgs ination. Because these two potentials dier only for negative ˆφ, but have the same plateau that can be used for ination, they have the same predictions. In the previous section, we saw how this model with Ω(φ) = 1 + ξφ could be embedded in supergravity. 67

76 6 Non-minimal coupling V odd(ϕ ) ϕ Figure 6.: Potential corresponding to (6.57). The model corresponding to this potential is 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. 3 ˆφ = ± log(ξg(φ)) (6.59) Again, the choice of sign is subtle. In this case, φ = 0 corresponds to innite ˆφ. It was argued in [71] that it is sucient to pick the positive solution with 0 < φ <, because it maps to all possible values of ˆφ. By taking the same Jordan frame potential, this results in the same potential as for Starobinsky ination. Furthermore, it was also shown in [71, 7] that there exists another attractor at weak coupling that leads to the same predictions as for quadratic ination (V (φ) m φ ). At weak coupling, the canonical eld is dened by the expression d ˆφ = 1 dφ. (6.60) Ω(φ) Because there does not exist a simple expression of the canonical eld, independent of the form of g(φ), it requires a bit more work to check that at weak coupling, there indeed exists an attractor. By expanding Ω(φ), it can be seen that the predictions of models with a weak coupling are equivalent to those of simple quadratic ination [7]. An example of a supergravity embedding of induced ination was given in the previous section. 68

77 6.3 Cosmological attractors 6.3. α-attractors In addition to the discussed ξ-attractors, there also exist models which exhibit the same double 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 a T-model, which has the action [73] S T = d 4 x [ 1 g R 1 g µν µ φ ν φ (1 φ /(6α)) 1 ] m φ. (6.61) When we integrate we obtain the canonical eld d ˆφ 1 = ± dφ (6.6) 1 φ /(6α) ˆφ = ± ( ) φ 6α arctanh. (6.63) 6α Thus, the potential in terms of the canonical eld is given by ( ) ˆφ V T ( ˆφ) = 3αm tanh. (6.64) 6α Interestingly, for large α this model returns to quadratic ination with its corresponding predictions. lim α V T ( ˆφ) = 1 m ˆφ (6.65) In contrast, for α N (where N is the number of efolds) the potential is more similar to the Higgs and Starobinsky type potentials with the predictions (6.50), see gure 6.3. V T (ϕ ) α=1 α= α= ϕ Figure 6.3: Potential corresponding to (6.64) plotted for dierent values of α. The model corresponding to this potential is known as a T-model. 69

78 6 Non-minimal coupling Similarly, one can also specify a somewhat dierent potential with the same kinetic term. S = d 4 x [ ] 1 g R 1 g µν µ φ ν φ (1 φ /(6α)) 1 φ m (6.66) (1 + φ 6α ) Because the kinetic term is the same as in the T-models, the canonical eld is again given by (6.63). This results after rewriting the tanh's a bit in the folowing potential. V E ( ˆφ) = 3α [ 1 e ˆφ] 3 α (6.67) These models are known as E-models [74]. Again, in the limit of large α this model converges 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. V E (ϕ ) α= α=3 α= ϕ Figure 6.4: Potential corresponding to (6.67) plotted for dierent values of α. The model corresponding to this potential is known as an E-model 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 Z Z + α 1 (Z Z) ] S S (6.68) 1 Z Z 3 W = 3αmSZ ( 1 Z ). (6.69) We then need to stabilize the eld S = 0 (as we saw before) or, alternatively, take it to be a nilpotent eld (S = 0). Additionally, we stabilize Im(Z) = 0 and Re(Z) = z plays the role of the inaton. Thus, for the Kähler metric we obtain K zz = K Z Z S=Im(Z)=0 = 3α (z 1) (6.70) 70

79 6.3 Cosmological attractors The canonical eld is dened by d ˆφ 3α = ± dz, (6.71) (z 1) from which we obtain ˆφ = 3α arctanh(z). (6.7) The potential is given by the standard supergravity F-term potential V T (z) = e K/M p [K A BD ] W A W D B W 3 1 ( ) = K zz (1 z ) 3 S W S W = 3αm z. M p S=Im(Z)=0 (6.73) Thus, in terms of the canonical eld we obtain V T ( ˆφ) = 3αm tanh ( ) ˆφ, (6.74) 3α which is equivalent to (6.64) up to a factor of in the tanh, which can easily be introduced 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, but also supersymmetry breaking and dark energy [76]. The attractor models of ination have many features. They oer a somewhat modelindependent approach to ination, while being well-motivated from the perspective of supergravity. Furthermore, by modifying α, one has a large theoretical control over the inationary predictions, such that these models t comfortably inside the n s r contour, independent of future measurements. The T-models have the same n s for any α but only dier in r, see gure In contrast, the E-models show a somewhat dierent trajectory for α, see gure

80 6 Non-minimal coupling 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 interesting to see where this behaviour originates from. In particular, we would like to know if the α-attractors are also realizable in string theory, which is not completely clear. Whereas we will not answer the last question, there has been research towards the interpretation 7