Supersymmetry, Supergravity, and String Theory Based Inflationary

Supersymmetry, Supergravity, and String Theory Based Inﬂationary Cosmology by Megan C. Kralj Submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020

Author...... Department of Physics May 8, 2020

Certiﬁed by...... Professor David I. Kaiser Professor of Physics, Germeshausen Professor of the History of Science, Associate Dean of the Social and Ethical Responsibilities of Computing Thesis Supervisor

Accepted by...... Professor Nergis Mavalvala Associate Department Head of Physics, Curtis and Kathleen Marble Professor of Astrophysics, Department of Physics Senior Thesis Coordinator 2 Supersymmetry, Supergravity, and String Theory Based Inﬂationary Cosmology by Megan C. Kralj

Submitted to the Department of Physics on May 8, 2020, in partial fulﬁllment of the requirements for the degree of Bachelor of Science in Physics

Abstract The aim of this thesis is to investigate and present an overview of inflationary theory. As part of this e↵ort, the main motivations for cosmic inflation, which are known as problems in the standard big bang scenario, are reviewed. These problems are the “flatness problem”, “horizon problem”, and the “monopole problem.” Analysis of inflation caused by a single scalar field rolling on a scalar potential is considered. This enables the introduction of the “slow-roll parameters” and their application. Since supersymmetry is considered a candidate beyond the standard model theory, we explore the connection between supersymmetry and the early universe. In order to incorporate gravity at energy scales of the early universe, supergravity is examined as well. Having considered inflation in the context of supersymmetry and supergravity, it is therefore natural to present the role of string theory in inflationary model building. As a result, axion monodromy is considered. Representative models are compared to analysis of precision measurements of the cosmic microwave background radiation. This comparison is achieved using observable derived constraints of the spectral index and tensor to scalar ratio.

Thesis Supervisor: Professor David I. Kaiser Title: Professor of Physics, Germeshausen Professor of the History of Science, Asso- ciate Dean of the Social and Ethical Responsibilities of Computing

3 4 Acknowledgments

First and most importantly, I would like to acknowledge Prof. D.I. Kaiser for being a great mentor and thesis advisor to me. I appreciate how he has always been extremely understanding of me as a student since the day I ﬁrst started working with him, as well as a great physics teacher when I needed help grasping challenging and new concepts. Prof. Kaiser gave me the perfect balance of research independence and academic guidance, which has helped me achieve the understanding required to put together this thesis. I feel like I owe a lot of my growth as a researcher to him, and that he has really helped encourage my passion for cosmology and particle physics. I plan to continue my work in these ﬁelds as a lifelong career.

Iwouldliketoacknowledgemyacademicadvisor,Prof.C.R.Canizares,whowas both an excellent mentor and provided academic guidance throughout my time at MIT. He helped me make key career decisions and supported me during dicult times. I would also like to thank Prof. J.A. Formaggio for submitting a recommendation to Prof. Kaiser when I was applying to work with him, as well as suggesting him as a potential advisor to me upon reading about my research interests. Additionally, I would like to thank Dr. S.P. Robinson, who was both a great boss and instructor to me in the MIT junior lab. He allowed me to express my personal research interests during my junior lab ﬁnal presentation to the physics faculty, which is what ultimately led me to working with Prof. Kaiser. I also want to acknowledge Prof. A. H. Guth, who was my instructor for electromagnetism 2, general relativity, and most importantly, an undergraduate cosmology course on the early universe. If it were not for these classes, I would not have discovered my interests in the topic of inﬂationary cosmology to begin with.

Moreover, I would like to acknowledge my parents, Mrs. Y.E. Kralj and Dr. D.R. Kralj. Being a student who commuted from home for nearly the entirety of my undergrad, they were present and supportive towards me at home on a daily basis. They are both great role models and always inspired me to pursue my interests and goals with resilience. I would also like to thank my grandparents, Mr. B. F. Santos

5 Perez and Mrs. Y.A. Gareca de Santos, for their economic and emotional support throughout my time at MIT. Finally, I would like to acknowledge my close friends, V. Liu, K. Keaney, and V. Phan, for their company and emotional support over the years during late nights of research and homework, as well as my loyal dog, Newton.

6 Contents

1 The Flatness and Horizons Problems 13 1.1 GeometryoftheUniverse ...... 13 1.1.1 IsotropyandHomogeneity ...... 14 1.1.2 Curvature ...... 19 1.2 TheFlatnessProblem ...... 24 1.2.1 FriedmannEquationsandtheFLRWMetric ...... 25 1.2.2 IntroductiontotheFlatnessProblem ...... 29 1.2.3 InﬂationasaSolutiontotheFlatnessProblem ...... 31 1.3 TheHorizonProblem...... 33 1.3.1 IntroductiontotheHorizonProblem ...... 33 1.3.2 InﬂationasaSolutiontotheHorizonProblem ...... 35

2 The Monopole Problem and the Inflationary Universe 41 2.1 The Monopole Problem ...... 41 2.1.1 GUTsandSpontaneousSymmetryBreaking ...... 42 2.1.2 Introduction to the Monopole Problem ...... 44 2.1.3 Inflation as a Solution to the Monopole Problem ...... 49 2.2 TheInflationaryUniverse ...... 51 2.2.1 TheEarlyUniverse...... 51 2.2.2 SlowRollApproximation...... 52 2.2.3 A Simple Toy Model: Quadratic Potential ...... 60

7 3 Supersymmetry and Supergravity Inflationary Models 63 3.1 Supersymmetry(SUSY) ...... 63 3.1.1 Conventions, Weyl Spinors, and the Dirac Wave Equation . . 63 3.1.2 GlobalSUSY ...... 70 3.1.3 SUSYAlgebraClosureandAuxilliaryFields ...... 72 3.2 SupersymmetryInflationaryModels...... 74 3.2.1 HybridInflation...... 75 3.2.2 SimpleF-TermSUSYInflation ...... 78 3.2.3 SUSYInflationaryModelwithFandD-Terms ...... 80 3.3 Supergravity(SUGRA)InflationaryModels ...... 83 3.3.1 Di↵erences Between SUGRA and SUSY in Context of Inflation 84 3.3.2 The ⌘ Problem ...... 86 3.3.3 SUGRAInflationaryModelwithFandD-terms ...... 88

4 Axion Monodromy Inflationary Models 95 4.1 Introduction to String Theory ...... 95 4.1.1 Concepts in String Theory and the Kaluza-Klein Model . . . . 95 4.1.2 Open Versus Closed Strings ...... 98 4.1.3 Open Versus Closed Superstrings ...... 101 4.2 Introduction to Compactification ...... 105 4.2.1 Compactification for the Kaluza-Klein Model ...... 105 4.2.2 Strings on Tori ...... 110 4.3 AxionMonodromyInflationaryModels ...... 115 4.3.1 Inflationary Model Using 6-Torus Compactification ...... 115 4.3.2 Inflationary Model Using Dp-Branes and Axions ...... 122

5 Observational Constraints for Di↵erent Inﬂationary Models 127 5.1 Tensor to Scalar Perturbation Ratio and the Spectral Index . . . . . 127 5.1.1 Overview of the Procedure for Comparison to Data ...... 127 5.1.2 Metric, Matter, and Curvature Perturbations ...... 129 5.1.3 Statistics of Perturbations ...... 133

8 5.1.4 Scalar and Tensor Perturbations ...... 134 5.1.5 Writing the Primordial Spectra in Terms of the Slow Roll Pa- rameters ...... 137 5.2 FinalResults ...... 140

5.2.1 Values of r and ns for Di↵erent Inﬂationary Model Potentials 140 5.2.2 Comparison to Data ...... 146 5.2.3 Conclusion ...... 148

9 10 List of Figures

1-1 Pseudo-Riemannian manifold M with metric gµ⌫.Thisﬁgureisin- spired by one from Knobel’s book [1]...... 14

1-2 ThreeTypesofSpatialHypersurfaces...... 19

1-3 Maximally Symmetric 3-Spaces for Increasing Comoving Distances . . 21

1-4 Determining if the Universe is Spherical Via Observation. This ﬁgure is based upon the course notes from Prof. Susskind’s Stanford OCW lecturesoncosmology[5]...... 24

1-5 Qualitative Plot of ⌦Versus Time...... 31

1-6 Horizon Distances of Causally Disconnected Regions of the Sky. . . . 34

1-7 Integrated Conformal Spacetime Diagram. The concept for this ﬁgure is inspired by Baumann’s lecture notes on Inﬂation [9]...... 35

1-8 Conformal Spacetime Diagram for an Inflationary Universe. The concept for this figure is inspired by Baumann’s lecture notes on Inflation [9]...... 37

1-9 Sketch of the Horizon Distance During Inflation. The concept for this figure is inspired by Baumann’s lecture notes on Inflation [9]...... 38

2-1 Running of the U(1), SU(2), and SU(3) interaction strength with increasing energy. In the left plot is the standard model, and in the right is the minimal supersymmetric model. This ﬁgure comes from one of Prof. Alan Guth’s presentation slides from when I took his cosmology course [3]...... 42

11 2-2 As the universe cools down and the Higgs field spontaneously breaks symmetry, neighboring regions tend to fall randomly into di↵erent states [46]...... 45 2-3 Timeline of the Universe. This figure is inspired by a timeline released bytheBICEP2team[14]...... 52 2-4 Numerical results that predict the expansion of the universe during the inflationary epoch...... 53

3-1 Particles and their superpartners. This figure is inspired by one in KamaluddinAhmed’spresentationslides[25]...... 65 3-2 Hybrid inflation potential. This figure is inspired by one from Jochen Baumann’s dissertation [43]...... 76 3-3 Hybrid inflation potential. This figure is inspired by one from Jochen Baumann’s dissertation [43]...... 77

4-1 A world line is replaced by a world sheet in string theory. This ﬁgure is inspired by one in Magdalena Larfors’s dissertation [41]...... 96 4-2 The brane-world concept, inspired by a diagram in Angel Uranga’s notes [32]...... 98

5-1 Quantum fluctuations as the scalar field rolls down the potential during inflation. This figure is inspired by one in Daniel Baumann’s lecture notes [38]...... 128 5-2 Anisotropies and perturbations in the CMB. This image is from Daniel Baumann’s lecture notes [8]...... 130

5-3 An overlay of the calculated values of ns and r for quadratic inflation, SUGRA D-Term inflation, and axion monodromy inflation on Planck collaboration 2018 results [45]. The square shaped points indicate the

analyzed constraint calculated in this thesis. Contours in the (ns,r) plane show the 68% and 95% conﬁdence levels...... 147

12 Chapter 1

The Flatness and Horizons Problems

The aim of this thesis is to conduct an in-depth analysis of various inflationary models. In the first portion of this thesis, we will review three of the central motivations for cosmic inflation- these problems are the “flatness problem,” “horizon problem,” and the “monopole problem.” In Chapter 1, we will place emphasis on examining the formalism for studying matter in curved spacetime, with special emphasis on the FLRW metric and the geometry of the universe, as well as examine how inflation resolves the flatness and horizon problems.

1.1 Geometry of the Universe

In section 1.1.1, which deals with fundamental geometric properties of the universe, the primary sources I used to learn the concepts include Christian Knobel’s book [1] and Chris Hirata’s cosmology course [2]. The goal of section 1.1.2 is to introduce di↵erent possible geometries and metrics the universe can have. The resources I used the most to write this subsection include my notes from Prof. Alan Guth’s cosmology course [3], Prof. David Kaiser’s informal cosmology primer [4], and Prof. Leonard Susskind’s online video lectures [5].

13 Figure 1-1: Pseudo-Riemannian manifold M with metric gµ⌫.Thisﬁgureisinspired by one from Knobel’s book [1].

1.1.1 Isotropy and Homogeneity

Let’s start by deﬁning a coordinate system of the form (t, x1,x2,x3). The spacetime metric turns the observer dependent vector xµ =(t, xi)intotheinvariantlineelement ds2.

3 2 µ ⌫ ds = gµ⌫dx dx (1.1) µ,⌫=0 X

In general relativity, the metric gµ⌫(t, x)inequation(1.1)dependsonposition in spacetime, which is determined by the distribution of matter and energy in the universe. In order to ﬁnd the metric of an Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, we write a general metric for curvilinear coordinates in terms of the spacetime interval.

ds2 = f(xk,t)dt2 +2g dtdxi + g dxidxj (1.2) 0i ij

14 Now, in accordance with the observed properties of the universe, we need to ensure that spacetime is isotropic and homogeneous, as discussed in [1]. Let spacetime be apseudo-RiemannianmanifoldM with metric gµ⌫, as shown in Figure [1-1]. The concept of homogeneity is that at an arbitrary time t,spacetimelooksthesameat any place. Consider the world line of an observer O that goes through spatial slices

⌃t1 ,⌃t2 ,⌃t3 of constant times t1, t2, t3.Inotherwords,M is sliced into 1-parameter family of spacelike hypersurfaces that are homogeneous. For any time t and any two points p, q ⌃ , there exists a di↵eomorphism (invertible function that maps one { }2 t di↵erentiable manifold to another, such that both the function and its inverse are smooth, e.g. coordinate transform) of spacetime that carries p into q and leaves gµ⌫ invariant.

The concept of isotropy is that if a particle is at rest and is isolated from any objects with a gravitational ﬁeld, it will not be accelerated in any spatial direction. Before discussing isotropy further, it is important to note that the universe will not look isotropic to any observer within that universe, which is why we need to use

µ µ world lines. For an observer at p,letu be tangent to O’s world line, and let v1 and µ µ v2 be unit vectors perpendicular to u .Theideabehindisotropyisthatthereisa µ µ µ di↵eomorphism with ﬁxed p and u that carries v1 into v2 and leaves gµ⌫ invariant. In other words, there is no preferred direction in space.

Let’s consider a comoving observer with velocity v↵.Becauseofisotropy,v↵ ⌃ . ? t i ↵ 1 Assuming we have ﬁxed x ,thevelocitycanbewrittenasv =(pf , 0, 0, 0), where the form of ﬁrst term is a result of normalization. So, we know that v↵ u for any vector ? 0 u tangent to ⌃t (such as u ). Further information on this process can be found in [2].

Taking the dot product of u and v yields

v u = g v↵u · ↵ 1 (1.3) = g vi 0i pf

Since v↵ u,weknowthatv u =0.Thus,wecanconcludethatg vi 1 =0. ? · 0i pf 15 g0i =0 (1.4) ) ij 1 g =(gij) (1.5)

Now, consider the acceleration of comoving observers:

aµ = v⌫ vµ (1.6) r⌫ By the deﬁnition of the covariant derivative,

µ ⌫ µ µ a = v (@⌫v +⌫v )(1.7)

For µ = i,

i ⌫ i i a = v (@⌫v +⌫v )

⌫ i = v ⌫v

0 i 0 = v 00v

1 1 2 i 2 = f 00f

1 i (1.8) = f 00

1 ij 1 @ = f g ( f) 2 @xj ij h1 1 @ i = g f ( f) 2 @xj ij⇣@ 1 ⌘ = g ln(f 2 ) @xj h i

Multiply both sides of the above equation by gij

@ g ai = g ln( f)(1.9) ij ij @xj p

@ a = ln( f)(1.10) ) j @xj p 16 aµ is measured by a comoving observer, which implies that it cannot have nonzero spatial components. Otherwise, it would suggest that there is a preferred direction in space, which violates isotropy. Thus, we can conclude that f is only dependent on t.

Deﬁning t0 as

1 t0 = f(t) 2 dt (1.11) Z

0i ij 1 Plugging the determined values of g , t0,andthefactthatg =(gij) into the expression for ds2 given by equation (1.2) yields

2 2 i j ds = dt0 + g dx dx (1.12) ij

Renaming t0 t,weget !

ds2 = dt2 + g dxidxj (1.13) ij

Now, note that vµ =(1, 0, 0, 0). We should consider a purely spatial vector u v ? µ that is also normalized, so uµu =1.

H(t, xi,v)=uµu v⌫ (1.14) ⌫rµ

0 Since u0 = u =0,plugthedeﬁnitionofthecovariantderivative.

µ ⌫ ⌫ H = u u⌫(@µv +µv )

i j j = u uj(@iv +i) (1.15) i j = u uji

i j = u uji0

17 j Now, we ﬁnd the value of i0

j µ ⌫ ⌫ i0 = u u⌫(@µv +µv ) 1 @g k @g k @g i = gjk i + 0 0 (1.16) 2 @t @xi @xj 1 ⇣@g k ⌘ = gjk i 2 @t

Substituting equation (1.17) into the Hamiltonian given in equation (1.15) yields

i jk 1 @gik H = u ujg 2 @t (1.17) 1 @g k = uiuk i 2 @t

i k @gik Recalling normalization v v gik =1,weknowthatthematrix @t must be a scalar multiple of gik

@g ik = Ig (1.18) ! @t ik

I By inspection, it is clear that H = 2 ,soweplugthisintoequation(1.18)

@g ik =2Hg (1.19) @t ik

The solution to this di↵erential equation is

j j 2 H(t)dt gik(t, x )=ik(x )e (1.20) R

Let’s deﬁne a(t) H(t)dt, which we use to re-write the above solution ⌘ R

j j 2 gik(t, x )=ik(x )[a(t)] (1.21)

ds2 = dt2 +[a(t)]2 (xk)dxidxj (1.22) ) ij 18 Figure 1-2: Three Types of Spatial Hypersurfaces.

1.1.2 Curvature

k i j Now, we want to ﬁnd the exact form of ij(x )dx dx . However, we must ﬁrst consider the di↵erent possible curvatures the universe could have, since gµ⌫ is extremely dicult to determine for an arbitrary matter distribution.

Recall that spatial homogeneity and isotropy allows for the universe to be represented by a time-ordered sequence of three-dimensional spatial slices ⌃t,eachof which is homogeneous and isotropic. The spacetime of the universe can be foliated into three di↵erent types of spatial hypersurfaces, as shown in Figure[1-2]. To get a sense of intuition for how the curvature of spacetime a↵ects observation, it is shown in the diagram that a triangle whose angles sum to 180 in a universe described by aﬂathypersurfacewouldhaveasumgreaterthan180 in a positively curved one, versus less than 180 in a negatively curved universe.

Now, let’s examine the properties of maximally symmetric 3-spaces. The ﬁrst hypersurface to consider is ﬂat space. The line element of the 3D Euclidean space is invariant under spatial translations and rotations, and is given by

2 2 i j dl = dx = ijdx dx (1.23)

From this element, we can obtain the Minkowski metric from special relativity, which

19 is the same everywhere in space and time.

10 0 0 0 0 10 01 g = (1.24) µ⌫ B C B 00 10C B C B C B 00 0 1 C B C @ A The second is positively curved space, whose 3-space is represented as a 3-sphere embedded in a four dimensional Euclidean space. The line element is given by

dl2 = dx2 + du2 (1.25) for x2 + u2 = a2 where a is the radius of curvature. The surface of the 3-sphere is homogeneous and isotopic as well due to the symmetry of the line element under 4D rotations. Finally, the third hypersurface is negatively curved space. The 3-space is represented as a 3-hyperboloid embedded in 4D Lorentzian space.

dl2 = dx2 du2 (1.26) for x2 u2 = a2

Homogeneity and isotropy are inherited from the symmetry of the line element under 4D pseudo-rotations.

We can see a visual representation of the corresponding 4 dimensional universes in Figure [1-3]. For example, in the positively curved case, we see a series of concentric 2-spheres, which we can see when we look out at distance .Theysweepthewhole 3-sphere.

For the speciﬁc cases of the 3-sphere and 3-hyperboloid, we have

dl2 = dx2 du2 ± (1.27) for x2 u2 = a2 ± 20 Figure 1-3: Maximally Symmetric 3-Spaces for Increasing Comoving Distances

Now, let x xa and u ua, as well as dx adx and du adu. ! ! ! !

dl2 = a2dx2 a2du2 ± (1.28) = a2(dx2 du2) ±

For

a2 = a2x2 a2u2 ± 1=x2 u2 ± d [1 = x2 u2] (1.29) dx ± du 2x 2u =0 ± dx udu = xdu ⌥

Putting equations (1.28) and (1.29) together, we can equate dl2 with the line

21 element for the spatial sections of the FLRW manifold.

dl2 = a2(dx2 du2) ± 2 2 2 2 x dx = a dx 2 ± u ! (1.30) x2dx2 = a2 dx2 ± 1 x2 ⌥ ! a2 dxidxj ⌘ ij

So, we have

x2dx2 a2 dx2 = a2 dxidxj ± 1 x2 ij ⌥ ! x2dx2 (1.31) dx2 = dxidxj ± 1 x2 ij ⌥ x x = i j ij ij ± 1 x xn ⌥ n

Now, let’s deﬁne curvature k for the di↵erent manifolds

k= 1 , Spherical 8 >0 , Euclidean > <> 1 Hyperbolic > > :> We can now re-write ij

kx x = + i j (1.32) ij ij 1 kx xn n Where dl2 > 0anda2 > 0. Converting to spherical coordinates as before using |x=0 the relation dx2 = dr2 + r2(d✓2 + sin2✓d2)forxdx = rdr as well as plugging in the

22 2 value of ij into dl yields

dr2 dl2 = a2 + d✓2 + r2(sin2✓d2) (1.33) 1 kr2 " # If we plug this into the spacetime di↵erential ds2 yields the FLRW metric.

ds2 = dt2 dl2 dr2 (1.34) = dt2 a2 + d✓2 + r2(sin2✓d2) 1 kr2 " #

Deﬁning d⌦2 r2(sin2✓d2) ⌘

dr2 ds2 = dt2 a2 + r2d⌦2 (1.35) 1 kr2 " # 2 dr2 2 2 Letting d 1 kr2 , r becomes a function Sk()for ⌘ Sk()= sinh ,k=1 8 >,k=0 > <> sin k= 1 > > Which yields :>

2 2 2 2 2 2 ds = dt a d + Sk()d⌦ (1.36) " # Where the comoving distance is given by

t0 dt z dz (z)= = (1.37) a(t) H(z) Zt1 Z0 for redshift z. Now, one might ask the question: How does an astronomer on Earth know if they are in an open, ﬂat, or closed universe? We can gain an intuition for what the apparent size of a galaxy would be as a function of comoving distance from the form

23 Figure 1-4: Determining if the Universe is Spherical Via Observation. This ﬁgure is based upon the course notes from Prof. Susskind’s Stanford OCW lectures on cosmology [5]. of the aforementioned equation.

If we lived in a 2- sphere, a galaxy a billion lightyears away from us would look bigger than if we lived in a ﬂat universe.

1.2 The Flatness Problem

Subsection 1.2.1 focuses on finding the FLRW metric and deriving the Friedmann equations. The primary sources I found most helpful for putting this section together includes Prof. David Kaiser’s informal cosmology primer [10] and my notes from Prof. Alan Guth’s cosmology course [5]. 1.2.2 introduces the flatness problem, while 1.2.3 illustrates how an inflationary era resolves the flatness problem. These two sections closely follow steps in Stefani Orani’s dissertation [6], and references results from the 2018 Planck collaboration [44].

24 1.2.1 Friedmann Equations and the FLRW Metric

Continuing from the FLRW metric equation given by (1.34), by observation, it is easy to identify the nonzero components of the spacetime metric

a(t)2 g = 1 g = tt rr 1 kr2 2 2 2 2 2 g✓✓ = a(t) r g = a(t) r sin ✓

Now, we plug the metric components into the equation for the Christo↵el symbol to ﬁnd all of the nonzero symbols

aa˙ a˙ t = ✓ = t = r2aa˙ rr 1 kr2 ✓t a ✓✓ 1 ✓ = t = r2aa˙ sin2 ✓ ✓ = sin ✓ cos ✓ ✓r r a˙ a˙ kr r = = r = rt a t a rr 1 kr2 1 1 = r = (1 kr2) = r r ✓✓ ✓ tan ✓ r = r(1 kr2)sin2 ✓

Now, using the Christo↵el symbols above, it is possible to determine the components of the Reimann curvature tensor using the following equation

Rµ = @ µ @ µ +µ ⇢ µ ⇢ (1.38) ⌫ ⌫ ⌫ ⇢ ⌫ ⇢ ⌫

Plugging in the Christo↵el symbols yields the following nonzero Reimann tensor components

25 aa˙ a¨ Rt = R✓ = Rt = raa¨ rrt 1 kr2 tt✓ a ✓✓t k +˙a2 R✓ = Rt = r2aa¨ sin2 ✓R✓ = r2 sin2 ✓(k +˙a2) rr✓ 1 kr2 t ✓ a¨ a¨ Rr = R = Rr = r2(k +˙a2) ttr a tt a ✓✓r k +˙a2 R = Rr = r2 sin2 ✓(k +˙a2) R = r2(k +˙a2) rr 1 kr2 r ✓✓

It should be noted that some of the calculations for obtaining these components can be easily obtained using symmetries Rµ = Rµ . Next, using the Reimann ⌫ ⌫ tensor components above, it is possible to determine the components of the Ricci tensor using the following equation

t r ✓ R⌫ = R⌫t + R⌫r + R⌫✓ + R⌫ (1.39)

Plugging in the Reimann tensor components yields the following nonzero Ricci tensor components

a¨ 1 R = 3 R = (2k +2˙a2 + aa¨) tt a rr 1 kr2 2 2 2 2 2 R✓✓ = r (2k +2˙a + aa¨) R = r sin ✓(2k +2˙a + aa¨)

gij 2 It is worth noting that Rij = a2 (2k +2˙a + aa¨). Now, we use the Ricci tensor components and the FRW metric tensor components to get the Ricci scalar using the following equation

tt rr ✓✓ R = g Rtt + g Rrr + g R✓✓ + g R (1.40)

26 Thus, we compute the value of the Ricci scalar

a¨ 1 kr2 1 R =( 1)( 3 )+ (2k +2˙a2 + aa¨) a a2 1 kr2 1 1 + r2(2k +2˙a2 + aa¨)+ r2 sin2 ✓(2k +2˙a2 + aa¨) (1.41) a2r2 a2r2 sin2 ✓ a¨ a˙ 2 k =6 + + a a2 a2 ⇣ ⌘ The next step is to use the Ricci tensor components, Ricci scalar, and FRW metric components to obtain the components of the Einstein tensor using the equation

1 G = R g R (1.42) µ⌫ µ⌫ 2 µ⌫

The components of the Einstein tensor are

a˙ 2 k G =3 + tt a2 a2 ⇣ k ⌘ a¨ G = gij +˙a2a2 +2 ij a2 a ⇣ ⌘ We calculate is the energy-momentum tensor, which is given by

T µ⌫ =(⇢ + p)uµu⌫ + pgµ⌫ (1.43)

Now, we want to lower the indices

↵ ↵ Tµ⌫ =(⇢ + p)gµ↵g⌫u u + pgµ↵g⌫g

=(⇢ + p)uµu⌫ + pµg⌫ (1.44)

=(⇢ + p)uµu⌫ + pgµ⌫

So the nonzero components of the energy-momentum tensor are

Ttt = ⇢

Tij = pgij

27 Now, we can plug the Einstein tensor components, FRW metric components, and stress energy tensor components into Einstein’s equation

Gµ⌫ +⇤gµ⌫ =8⇡GTµ⌫ (1.45)

For µ = t and ⌫ = t,

Gtt +⇤gtt =8⇡gTtt (1.46)

a˙ 2 k 3 + ⇤=8⇡G⇢ (1.47) a2 a2 ⇣ ⌘ Rearranging, we obtain the ﬁrst Friedmann equation

a˙ 2 + k ⇤ 8⇡G = ⇢ (1.48) ) a2 3 3

Now, we take µ = i and ⌫ = j

Gij +⇤gij =8⇡gTij (1.49)

k a˙ 2 a¨ g + +2 +⇤g =8⇡Gpg (1.50) ij a2 a2 a ij ij ⇣ ⌘ Noting that g =0 ij 6

k a˙ 2 a¨ + +2 +⇤=8⇡Gp (1.51) a2 a2 a ⇣ ⌘

a˙ 2 + k a¨ = 2 ⇤+8⇡Gp (1.52) ! a2 a

Substituting (1.49) into the ﬁrst Friedmann equation

a¨ 8⇡G ⇤ 2 ⇢ +⇤=8⇡Gp (1.53) a 3 3 28 a¨ 2 8⇡G 2 ⇤=8⇡Gp(t)= ⇢ (1.54) a 3 3

Finally, we obtain the second Friedmann equation

a¨ 1 4⇡G ⇤= (⇢ +3p)(1.55) ) a 3 3

Deﬁne the Hubble parameter as H a˙ . Note that ⇤is the energy density of ⌘ a empty space, which we have set equal to zero for now.

Recalling the FLRW metric in the form given by equation (1.36), we re-write Sk as a function of r instead of ,wherewealsointroducetheuniverse’scurvatureradius

R0.

ds2 = dt2 + a2[dr2 + S2(r)(d✓2 + sin2✓d2)] (1.56) k

For

Sk(r)=

R sin r ,k =1 0 R0 8 ! > >,k=0 > <> R sinh r k = 1 0 R0 > ! > > :> 1.2.2 Introduction to the Flatness Problem

Next, we can re-write the ﬁrst Friedmann equation in terms of H, R0,and⇤=0.

2 8⇡G⇢ k H = 2 2 (1.57) 3 a R0

Setting R0 =1yields

8⇡G⇢ k H2 = (1.58) 3 a2 29 We can rearrange this as

k 8⇡G⇢ = H2 (1.59) a2 3

Now, let’s ﬁnd the critical density value ⇢cr for which k =0.

8⇡G⇢ 3H2 0= H2 ⇢ = (1.60) 3 ! cr 8⇡G

For clarity, let’s summarize what the value of ⇢cr tells us about the universe:

1. ⇢ = ⇢cr when k =0,theuniverseisﬂat,and⌦=1

2. ⇢>⇢cr when k>0, the universe is open, and ⌦ > 1

3. ⇢<⇢cr when k<0, the universe is closed, and ⌦ < 1 Continuing onwards from (1.58), and doing some mathematical manipulations.

2 k 8⇡G⇢ 3 2 k H + 2 = ⇢ = H 2 (1.61) a 3 8⇡G a ! Now, let’s deﬁne ⌦ ⇢ ⌘ ⇢cr

3 2 k ⌦= H + 2 8⇡G⇢cr a !

1 2 k (1.62) = 2 H + 2 H a ! k =1+ H2a2

Thus, we can deﬁne

k ⌦ 1 ⌦= (1.63) k ⌘ a2H2 Finally getting into the ﬂatness problem, we note that our universe is almost perfectly ﬂat (⌦ =0.0007 0.0037 according to the Planck collaboration) [44]. It is 0 ± averyspecialcasebecause⌦=1isapointofunstableequilibrium.Thequestion becomes why ⌦is so precisely tuned to 1?

30 Figure 1-5: Qualitative Plot of ⌦Versus Time.

1.2.3 Inﬂation as a Solution to the Flatness Problem

For a matter dominated universe with ⌦close to 1

2 (⌦ 1) t 3 (1.64) /

For a radiation dominated universe with ⌦close to 1

(⌦ 1) t (1.65) /

Assuming that matter-radiation equality occurs 50, 000 years after the big bang, we can calculate (⌦ 1) |t=1s

2 1s 50000yr 3 (⌦ 1) t=1s = (⌦0 1) (1.66) | 50000yr! 13.8Gyr!

Taking ⌦0 =0.1today,then

16 (⌦ 1) =1.4 10 (1.67) |t=1s ·

16 Then if we have ⌦ 0.1today,thatrequires(⌦ 1) =1.4 10 ,which 0 ' 1s ⇥ has a ﬁne-tuning of about 16 orders of magnitude. This result is unexplained by the

31 conventional hot big bang model.

The flatness problem is solved if an era of inflationary expansion precedes the Big Bang. During inflation, we have that

2N(t) (⌦ 1) e (1.68) /

We can calculate (⌦ 1) ,wheret is the initial time of inflationary expansion |t=ti i and teq is the final time of inflationary expansion.

2 3 2N(t=tf ) e tf teq (⌦ 1) t=ti = 2N(t=t ) (⌦ 1) t=tpresent (1.69) | e i teq ! tpresent ! |

Letting N = N(t ) N(t )andsolvingforN t f i t

2 1 t t 3 (⌦ 1) N = ln f eq |t=ti (1.70) t 2 t t (⌦ 1) " eq ! present ! |t=tpresent # Since our universe is almost ﬂat at present time, we have that

3 (⌦ 1) 10 (1.71) |t=tpresent ⇠

where we are using a current value of ⌦based upon the latest measurements from the Planck satellite [44]. At the beginning of inﬂation, we assume the following value of ⌦.

(⌦ 1) 1(1.72) |t=ti ⇠

The present age of the universe is t =4 1017s, while the time of equilibrium present · 12 tf is teq =1.5 10 s. The ratio is estimated using · teq

2 2 t T 1015GeV f = f = (1.73) teq Teq ! 0.8eV !

3 So if we use (⌦ 1) 1atthestartofinﬂation,and(⌦ 1) 10 ,we ti ⇠ today ⇠ 32 obtain

N 55 (1.74) t It is clear that inﬂation thus drives ⌦ 1. The value of ⌦ 1afterinﬂationis ! so small that it is still very small today.

1.3 The Horizon Problem

The objective of section 1.3.1 is to introduce the reader to the horizon problem, while section 1.3.2 presents inﬂation as a solution to the horizon problem. The references used to write these subsections include Andrew Liddle’s online cosmology notes [7], Daniel Baumann’s lecture notes on inﬂation [8], and Barbara Ryden’s cosmology textbook [9].

1.3.1 Introduction to the Horizon Problem

When we look at the cosmic microwave background (CMB), microwave photons emit- ted from opposite sides of the sky appear to be in thermal equilibrium. Although regions on the CMB sky are separated by more than 1 had no time to interact with

4 each other, their temperature is the same with an accuracy of less than 10 .The “problem” is that this remarkable uniformity cannot be explained by the conventional Hot Big Bang model unless it is assumed in the initial conditions. In order to show that this is true, we need to derive the comoving particle horizon. We start with the FLRW metric given by (1.34) and deﬁne conformal time ⌧ dt . ⌘ a(t) Assuming a ﬂat universe, so k =0,themetricbecomes

dr2 ds2 = dt2 + a2 + d✓2 + r2(sin2✓d2) 1 kr2 " # (1.75) = dt2 + a2[dr2 + d✓2 + r2(sin2✓d2)] = a2(⌧)[ d⌧ 2 + dr2 + d✓2 + r2(sin2✓d2)] 33 Figure 1-6: Horizon Distances of Causally Disconnected Regions of the Sky.

If we consider the radial propagation of light, the metric becomes

ds2 = a2(⌧)[ d⌧ 2 + dr2](1.76)

For light-like trajectories (photons), ds2 =0andr = ⌧ +constant. The maximal ± distance traveled by a photon, which is also the conformal time, is

t dt ⌧ ⌧ = 0 (1.77) i a(t ) Zti 0

Setting ti =0(originoftheuniverse)iswhatwerefertoasthecomovingparticle horizon. Now, we want to express the coordinate distance to a distant source in terms of directly observable quantities, particularly the redshift of the source in question.

t dt ⌧ ⌧ = 0 i a Zti t da = (1.78) aa˙ Zti t da = a2H Zti 34 Figure 1-7: Integrated Conformal Spacetime Diagram. The concept for this ﬁgure is inspired by Baumann’s lecture notes on Inﬂation [9].

a da dz Now, using 1 + z = and 2 = ,weget a0 a a0

z dz0 ⌧ ⌧i = a0H(z0) Z0 (1.79) z dz = 0 a H ⌦ (1 + z )3 +⌦ (1 + z )4 +⌦ Z0 0 o m,0 0 ,0 0 ⇤,0 p 5 Where ⌦ =0.3, ⌦ =8.8 10 ,and⌦ =0.3. Evaluating this integral m,0 ,0 · ⇤,0 numerically for z = 1090 (recombination), we obtain a graph which indicates that there is non-overlapping past light cones, and hence, regions of the sky that were never in causal contact.

1.3.2 Inﬂation as a Solution to the Horizon Problem

Nonetheless, inﬂation provides a way to explain the apparent homogeneity of the universe on scales much larger than the horizon size. The idea is that a small region of the universe in equilibrium experiences accelerated expansion on an enormous scale.

35 The basic strategy here is to ensure that

tdec dt t0 dt (1.80) a(t) ⇠ a(t) Z0 Ztdec This equation indicates that light can travel about the same amount before decoupling as after decoupling to the present.

In order to determine the behavior of the scale factor during inﬂation, we need to use the Friedmann ﬁeld equations

2 8⇡⇢ k H = 2 2 (1.81) 3mpl a

⇢˙ +3H(⇢ + p)=0 (1.82)

These equations can be combined into the acceleration equation

a¨ 4⇡ = 2 (⇢ +3p)(1.83) a 3mpl

For the case where p = ⇢,theﬂuidequationyields˙⇢ =0.Recognizingthat ⇢ = ⇢0,theaccelerationequationbecomes

a¨ 8⇡ 2 = 2 ⇢0 = H (1.84) a 3mpl

Solving this di↵erential equation gives

a(t) eHt for H =constant (1.85) /

During inﬂation, the conformal time is negative and evolves towards zero with increasing proper time

dt 1 H⌧ ⌧ = e (1.86) a(t) /H Z 36 Figure 1-8: Conformal Spacetime Diagram for an Inflationary Universe. The concept for this figure is inspired by Baumann’s lecture notes on Inflation [9].

Such that the scale factor evolves as

1 a(⌧)= (1.87) H⌧

Where ⌧ =0representst 0, thus implying that inﬂation continues forever. ! ⌧ =0isnottheBigBang,butrather,theendofinﬂation. Now, the horizon distance is given by

t dt d (t)=a(t)c (1.88) hor a(t) Z0

The horizon distance at a time ti prior to inﬂation is

ti dt dhor(ti)=a(ti)c 1 t 2 0 a(ti) Z ti (1.89)

=2cti

37 Figure 1-9: Sketch of the Horizon Distance During Inflation. The concept for this figure is inspired by Baumann’s lecture notes on Inflation [9].

The horizon distance at a time tf at the end of inﬂation is

ti tf N dt dt dhor(tf )=a(ti)e c 1 + [H (t t )] (1.90) t 2 a(ti)e i i 0 a(ti) ti ! Z ti Z If N is large, then

d (t ) eN 3ct (1.91) hor f ⇡ i Which indicates that inﬂation causes the horizon to grow exponentially. In order to estimate the amount of e-foldings needed to solve the horizon problem, let’s consider the comoving radius before inﬂation and impose the condition that it should be at least equal to the observable universe today.

1 1 (aiHi) =(a0H0) (1.92)

38 1 1 Now, we wish to write (a0H0) in terms of (af Hf ) ,thecomovingradiusafter inﬂation and at reheating. For simplicity, in this estimate, we will assume a radiation dominated universe. We can estimate the ratio as follows

2 3 a0H0 a0 af af T0 10 eV 28 = 15 =10 (1.93) af Hf ⇠ af a0 ! a0 ⇠ Tf ⇠ 10 GeV Using the two above equations, we obtain the desired relation before and after inﬂation

1 28 1 (a H ) 10 (a H ) (1.94) i i ⇠ f f Since H is assumed to be constant during inﬂation, this equation becomes

a a f =1028 = ln f =64 (1.95) ai ) ai ! This result indicates that the solution of the horizon problem requires approximately 64 e-foldings.

39 40 Chapter 2

The Monopole Problem and the Inﬂationary Universe

2.1 The Monopole Problem

The overall aim of this section is to provide the background for as well as to address the monopole problem. Section 2.1.1 essentially is a brief review of grand unified theories and spontaneous symmetry breaking. The references used to create this subsection include Barbara Hale’s lecture notes [13], one of Prof. Alan Guth’s slides from his cosmology course [3], Micheal Dine’s textbook on supersymmetry and string theory [22], Wim de Boer’s book on grand unified theories and supersymmetry [10], Owen Dando’s paper [12], and Prof. Alan Guth’s OCW lectures [17]. Section 2.1.2 introduces the audience to the monopole problem, which references Alan Guth’s OCW lectures [46], Owen Dando’s paper [12], John Preskill’s paper [11], and Prof. Alan Guth’s cosmology course lecture notes [3]. Finally, section 2.1.3 presents how inflation resolves the monopole problem, which is largely based upon one of Prof. Alan Guth’s cosmology course homework assignments [3].

41 Figure 2-1: Running of the U(1), SU(2), and SU(3) interaction strength with increasing energy. In the left plot is the standard model, and in the right is the minimal supersymmetric model. This ﬁgure comes from one of Prof. Alan Guth’s presentation slides from when I took his cosmology course [3].

2.1.1 GUTs and Spontaneous Symmetry Breaking

In the standard model, each type of gauge interaction has its own interaction strength, described by coupling constants. When the interaction strengths are extended to energies of the order of 1016 GeV, the three interaction strengths become about equal. As shown in Figure [2-1], the energies above 1016 GeV, the theory behaves like a fully unified SU(5) gauge theory [13]. The period after the big bang at which the temperature is higher than 1016 GeV can be divided into two epochs, called the Planck epoch and grand unified theory (GUT) epoch. During the Planck epoch, the four fundamental forces, electromagnetism U(1), gravitation, weak SU(2), and the strong SU(3) color, are assumed to all have the same strength, and be “unified” in one fundamental force. It should be pointed out that string theory is a good candidate for describing the Planck epoch. As the universe expands and cools down during the Planck epoch, gravitational interactions are no longer unified with electromagnetic U(1), weak SU(2), and the strong SU(3) color interactions. This period of time in the history of the universe is

42 called the GUT epoch. As stated above, during the GUT epoch the strong, weak, and electromagnetic fields are unified, and thus, a good candidate for a GUT is SU(5) symmetry. It should be noted that during the GUT epoch, the proton can decay, quarks are changed into leptons, and all the gauge particles (X,Y ,W ,Z,gluonsand photons), quarks and leptons are massless [22]. Assuming the symmetry groups SU(3) SU(2) U(1) are part of a larger group ⌦ ⌦ G, the smallest group G is the SU(5) group. So the minimal extension of the standard model towards a GUT is based on the SU(5) group [35]. SU(N) can be represented by matrices. Local gauge invariance requires the introduction of N 2 1gaugefields(themediators),whichcausetheinteractionsbetween the matter fields. The gauge fields transform under the adjoint representation of the SU(5) group which can be written in matrix form

G 2R G G XC Y C 11 p30 12 13 1 1 0 G G 2R G XC Y C 1 21 22 p30 23 2 2 24 = B 2R C C C (2.1) B G31 G32 G33 p X3 Y3 C B 30 C B W 3 3B + C B X1 X2 X3 + W C B p2 p30 C B W 3 3B C B Y1 Y2 Y3 W + C B p2 p30 C @ A In this matrix, G stands for the gluon ﬁelds. W and B are the gauge ﬁelds of the SU(2) symmetry group. X and Y represent the gauge bosons. The X and Y bosons represent interactions that transform leptons into quarks and quarks into leptons.

36 15 At about 10 seconds and an average thermal energy 10 GeV, a phase transition is believed to have taken place. In this phase transition, the vacuum state undergoes spontaneous symmetry breaking. When the phase transition takes place, the vacuum state transforms into a Higgs particle with mass and so-called “Goldstone bosons” with no mass. The Goldstone bosons “give up” their mass to the gauge particles, while the X and Y bosons gain masses 1015 GeV), The Higgs keeps its mass, which is on the order of magnitude ⇠ of the thermal energy of the universe, or kT 1015 GeV. ⇠ 43 In order to illustrate spontaneous symmetry breaking, we look at a simple model that Goldstone proposed, which is excellent for introductory purposes [12]. The Goldstone Lagrangian is

L = @ ¯@µ ( ¯ ⌘2)2 (2.2) µ 4 where is a complex ﬁeld and the term ( ¯ ⌘2)2 is the “Mexican-hat” potential. 4 The Goldstone Lagrangian is invariant under the group U(1) of phase transformations given by

(x) ei↵ (x)(2.3) ! However, if the potential is minimized at the values of the ground states for which is nonzero, the phase transformation under U(1) becomes ✓ ✓ + ↵. !

= ⌘ei✓ (2.4)

Consequently, this particular vacuum state is not invariant under the symmetry of the Lagrangian. Hence, the symmetry of the theory has been spontaneously broken.

2.1.2 Introduction to the Monopole Problem

As the universe cools below kT 1016 GeV, the matter goes through a phase transi- ⇠ tion, in which some of the components of the GUT Higgs field acquire nonzero values in the thermal equilibrium state, breaking the GUT symmetry. Higgs fields with nonzero values that form in one region may not align with those in another. Fields in di↵erent regions cannot be smoothly joined with patterns in neighboring regions, thus resulting in regions where the smoothing is imperfect, leaving defects, as illustrated using the Higgs field in Figure [2-2]. There are three main types of topological defects, two of which are domain walls and cosmic strings. For a domain wall, consider a system with a single scalar field as the temperature decreases and the system undergoes a phase transition. If there

44 Figure 2-2: As the universe cools down and the Higgs ﬁeld spontaneously breaks symmetry, neighboring regions tend to fall randomly into di↵erent states [46].

are two neighboring regions with di↵erent local minima, then Higgs ﬁelds acquire di↵erent nonzero expectation values. The domain wall is at the boundary between the two regions and consists of a sheet of high energy density [3].

For the cosmic string, consider a system with a complex field and a potential takes the Mexican hat form. The degenerate minima of this potential lies on a circle. Suppose we traverse some closed path in physical space. As we do so, the path could wind around the circle of degenerate minima with a change of 2⇡ in its phase space along the path, and hence, is not well-defined. The only way that the field can remain continuous is by rising to the local maximum of the Mexican hat potential, where the field is equal to zero. This departure from the vacuum manifold is accompanied by a nonzero energy density, which we associate with the string core. Unlike the domain wall, it is a continuous symmetry that is broken the zero field value occurs along a linear locus of points.

Monopoles form when a spherical symmetry is broken. To introduce the monopole, let’s consider the model ﬁrst studied by ’t Hooft and Polyakov. The Lagrangian for

45 the system is given by

1 1 L = (D i)2 (F i )2 (( i) ⌘2)2 2 µ 4 µ⌫ 2 D i = @ i e✏ijkAj k (2.5) µ µ µ F i = @ Ai @ Ai e✏ijkAj Ak µ⌫ µ ⌫ ⌫ µ µ ⌫

where Aµ is a gauge vector field, is the scalar field, and ⌘ is the mass of the particle. Consider the case of a real scalar field with three components. In this case, SU(2) symmetry is spontaneously broken to U(1), but ’t Hooft-Polyakov monopoles exist whenever a simple Lie group is broken to something with a U(1) factor, which is grand unification. Essentially, the field must leave the vacuum manifold to remain continuous, thus giving rise to a nonzero energy density because the vacuum manifold contains non-contractible 2-surfaces.

Far from the monopole core, the Higgs ﬁeld takes the “hedgehog” conﬁguration with r.Elementaryexcitationsaroundthesymmetrybreakingvacuumarea ⇠ massless photon, two charged vectors with mass m⌫ = e⌘, and a neutral massive

Higgs scalar with mass mH =2p⌘.

Now, we proceed to estimate the number of monopoles that are produced in the GUT phase transition [11], [3]. Using the estimate ﬁrst proposed by Kibble, the number density is given by

1 n (2.6) M ⇡ ⇠3

where ⇠ is the correlation length or the maximum distance over which the ﬁeld at one point in space is correlated with the ﬁeld of another point in space.

If the volume surrounding is set casually, then the correlation length is equal to the horizon distance at the time of the GUT phase transition. Assuming a radiation

46 dominated universe, then the horizon distance is

t c lhor = a(t) dt0 0 a(t0) Zt 1 c 2 (2.7) = t 1 dt0 Z0 t0 2 =2ct

The Friedmann equation for a radiation-dominated ﬂat universe is

8⇡ H2 = G⇢ (2.8) 3

where

u ⇢ = (2.9) c2

or equivalently

⇡2 (kT)4 ⇢ = g (2.10) 30 ~3c5

1 For a radiation dominated universe, a(t) t 2 .Thus,H becomes ⇠

a˙ 1 H = = (2.11) a 2t

which gives us the time at which the transition occurs.

1 t = (2.12) 2H

Finally, we can combine equations (2.12) and (2.10) to give us the time in terms of temperature.

3 5~3c5 1 t = 3 2 (2.13) 4s⇡ gGUTG (kTe)

47 3 5(1.055 10 34J s)3(2.998 108m s 1)5 1 t = ⇥ · ⇥ · 3 11 3 1 2 16 2 4s ⇡ 200 (6.674 10 m kg s ) (10 GeV) · · ⇥ · · 2 1GeV J (2.14) ⇥ 1.602 10 10J kg m2 s 2 ⇥ ! s · · 39 =1.713 10 s ⇥

Combining the results above, the number density of magnetic monopoles at the time of phase transition is

1 n M ⇡ ⇠3 1 3 ⇠ (2ct) (2.15) 1 = (2(2.998 108m s 1)(1/713 10 39s))3 ⇥ · ⇥ 89 3 =9.23 10 m ⇥

To judge whether this is big or small we need to calculate the present day quantity. Following Prof. Guth’s approach presented in his course, it is possible to estimate the present day monopole number density by considering the ratio of monopoles to photons [3]. In the conventional hot big bang model, the ratio of monopoles to photons would be about the same today as it was just after the phase transition.

nM nM,0 = n,0 (2.16) n

⇣(3) (kT )3 n =2 c ⇡2 (hc)3 (2.17) ⇣(3) (kT )3 n =2 0 ,0 ⇡2 (hc)3

In the context of the conventional, non-inﬂationary hot big bang model, the ratio

48 of monopoles to photons would be about the same today as it was just after the phase transition.

3 T0 nM,0 = nM Tc ! 3 89 3 2.73K (2.18) =9.23 10 m ⇥ 1.2 1029K ⇥ ! 3 =12031m

To put this in perspective, we would expect approximately 2 108 monopoles in ⇥ Lobby Seven at MIT (my school!). However, since MIT or other institutions have not detected a single monopole yet, it does not match the observation. This is one of the problems that motivated MIT’s Prof. Guth to discover inﬂation [3]. Inﬂation solves the monopole problem by diluting the number density of monopoles.

2.1.3 Inﬂation as a Solution to the Monopole Problem

During the inﬂationary period, we have

a˙ = Hia (2.19)

where Hi is the Hubble constant at the beginning of inﬂation. More speciﬁcally, it is equivalent to the reciprocal of GUT epoch time.

1 Hi = (2.20) tGUT Solving the di↵erential equation in (2.19), we can determine the scale factor a(t) during inﬂation.

Hi(t ti) a(t)=aie (2.21)

where ai is the initial value of the scale factor. Subsequently, we can deﬁne the

49 quantity in the exponential.

N H (t t )(2.22) ⌘ i f i

Equation (2.22) essentially allows us to calculate the factor by which a(t)increases by

a(t ) f = eN (2.23) a(ti)

Now, we can apply equation (2.23) to obtain the predicted monopole density at the end of inﬂation. In consequence, for N =100wehavethat

3N nM(tf ) = e nM(tGUT)

300 89 3 = e 9.23 10 m (2.24) · ⇥ 41 3 4.8 10 m ⇡ ⇥

In addition, from the end of inﬂation to the present, the universe experiences an

27 expansion given by the scaling a(t )=2 10 after inﬂation and a =1atthe f ⇥ 0 present time. Using this information, the estimated number density of monopoles at the present is given by

27 3 n =(0.5 10 ) n M(t0) ⇥ M(tf ) 81 41 3 =0.125 10 4.8 10 m (2.25) ⇥ · ⇥ 122 3 0.6 10 m ⇡ ⇥

As expected, inﬂation dilutes the monopoles. The probability of ﬁnding a monopole is very small.

50 2.2 The Inﬂationary Universe

This section provides an overview of the history, properties, and dynamics of an inflationary universe. Section 2.2.1 provides a brief summary of the history of an inflationary universe, which is largely based upon a figure released by the BICEP2 team [14] and Veneziano and Gasperini’s paper [15]. Section 2.2.2 derives the slow roll approximation of inflation. This section references a mix of Nicola Pintus’s paper [16], one of Prof. Phiala Shanahan’s quantum field theory homework assignments [17], Gary Watson’s online notes on inflationary cosmology [18]. Finally, section 2.2.3 deals with a simple toy model to provide an example of an inflationary potential. This section very closely follows Prof. David Kaiser’s informal cosmology primer [19].

2.2.1 The Early Universe

The standard cosmological model provides us with a description of the various stages in cosmological history, as illustrated in Figure [2.3]. These stages include the radiation era, the nucleosynthesis, the recombination era, the epoch of matter domination, and others. However, as shown in the diagram, at early times the model has an epoch of accelerated cosmic evolution, called “inflation.” As pointed out earlier, the inflationary epoch is needed to solve the flatness problem, the horizon problem, and the monopole problem. In addition, inflation provides with a mechanism by which quantum fluctuations are amplified which in turn produce the “seeds” for structure formation as well as the temperature fluctuations of the CMB radiation [15].

During inflation, evolution is dominated by the potential energy of the scalar field. For inflation to occur in a manner consistent with observations, it must meet the slow roll conditions. To provide context for the timeline, Figure [2.4] shows the predicted number of e-folds versus time during the inflationary epoch. Both, the slow roll conditions and a simple model with a scalar field will be discussed in the rest of this chapter.

51 Figure 2-3: Timeline of the Universe. This ﬁgure is inspired by a timeline released by the BICEP2 team [14].

2.2.2 Slow Roll Approximation

We start out by deriving the slow roll approximation for inflation [16]. We have an inflaton scalar field that is minimally coupled to gravity. Inflationary models assume there is a moment when the universe being dominated by starts and drives the universe into a De Sitter expansion with quasi-zero temperature.

We start with the action of the system

1 S = d4xp g gµ⌫@ @ + V () (2.26) 2 µ ⌫ Z h i More generally, µ S = dx L (A(x),@µA(x)) (2.27) Z µ µ µ µ Consider the inﬁnitesimal coordination transformation x x0 = x + x ! µ µ a for x = a ! ,where is a coecient, ! is the inﬁnitesimal transformation, and the index a =1, 2, 3,...,s is the set of s independent spacetime parameters.

µ µ µ⌫ This can include homogeneous Lorentz transforms x = ! + x⌫! noting that !µ⌫ = !⌫µ for N ﬁeld functions for which A = 1, 2, 3,...,N . A { } 52 Figure 2-4: Numerical results that predict the expansion of the universe during the inﬂationary epoch.

53 Letting

(x) 0 (x0)+ (x)(2.28) A ! A A

We can see that

a a (x)=0 (x) (x)=Y (x)! (2.29) A A A AB B

The Jacobian corresponding to inﬁnitesimal coordinate transformation is

@x µ J = det 0 = @ xµ (2.30) @x⌫ µ Thus, we write the change in action [ 17] as

tf S = dx0 d~xL (A(x),@µA(x))@x Zt0 Z (2.31) tf + dx0 d~xL (A(x),@µA(x)) Zt0 Z

tf S = dx0 d~xL (A(x),@µA(x))@x Zt0 Z tf + dx0 d~x@L (A(x),@µA(x))x (2.32) Zt0 Z tf + dx0 d~xL (A(x),@µA(x)) Zt0 Z

Noting that @x =0,

tf S = dx0 d~x@L (A(x),@µA(x))x Zt0 Z (2.33) tf + dx0 d~xL (A(x),@µA(x)) Zt0 Z 54 Combining the integrals, we obtain

tf S = dx0 d~x @L x + L (2.34) t Z 0 Z h i Renaming the index µ and rewriting L using the Euler Lagrange equation ! yields

tf µ L S = dx0 d~x @µL x + @µ A(x) (2.35) t (@µA(x)) Z 0 Z h ⇣ ⌘ i

Re-expressing the variation A(x)

tf µ L S = dx0 d~x @µL x + @µ A(x) (2.36) t (@µA(x)) Z 0 Z h ⇣ ⌘ i

tf µ L ⌫ S = dx0 d~x @µL x + @µ (A(x) @⌫A(x)x ) t0 " (@µA(x)) !# Z Z ⇣ ⌘ tf µ L ⌫ = dx0 d~x@µ ⌫ L (x) @⌫A(x) x t0 " (@µA(x)) ! # Z Z ⇣ ⌘ tf L + dx0 d~x@µ A(x) t0 " (@µA(x)) # Z Z ⇣ ⌘ (2.37)

µ Now, plug xµ @x !a for a = 1, 2,...,s and (x) (Y ) (x)!a ⌘ @!a { } A ⌘ a AB B for A = 1, 2,...,N ⇣ ⌘ { }

tf µ L µ a S = dx0 d~x@µ ⌫ L (x) @⌫A(x) a ! t0 " (@µA(x)) ! # Z Z ⇣ ⌘ (2.38) tf L a + dx0 d~x@µ (Ya)ABB(x)! t0 " (@µA(x)) # Z Z ⇣ ⌘ 55 Dividing through by a di↵erential of ! and rearranging

tf S µ L µ 0= a dx0 d~x@µ ⌫ L (x) @⌫A(x) a ! t0 " (@µA(x)) ! # Z Z ⇣ ⌘ (2.39) tf L dx0 d~x@µ (Ya)ABB(x) t0 " (@µA(x)) # Z Z ⇣ ⌘

Noting that S tf + dx d~x@ J µ =0 (2.40) !a 0 µ a Zt0 Z

We obtain the Noether current

µ L µ µ Ja (x)= @⌫A(x) L (x)⌫ a ) " (@µA(x)) # ⇣ ⌘ (2.41) L (Ya)ABB(x) (@µA(x)) ⇣ ⌘

Consider the following spacetime translations for a ⇢ = 0, 1, 2, 3 ⌘ { }

xµ = !µ

⌫ ⌫ a ⌘ ⇢

A(x)=0

There is no change under translations for classical relativistic wave ﬁelds, so we know that J µ(x) T µ(x)(2.42) a ! ⇢

µ⇢ @L (x) ⇢ µ⇢ T (x)= @ A(x) L g (2.43) ) @(@µA(x)) ⇣ ⌘ 56 Now, we compare the stress energy tensor to that of a perfect ﬂuid [18]

⇢ 000 0 0 p 001 T µ⌫ = (2.44) B C B 00p 0 C B C B C B 000p C B C @ A

Considering a Lagrangian of the form L = 1 @ @µ V () 2 µ

µ⇢ @ 1 2 ⇢ µ⇢ T (x)= (@µ) V () @ g L @(@µA(x)) 2 h i @ 1 2 ⇢ µ⇢ = (@µ) @ g L (2.45) @(@µA(x)) 2 = @µ@⇢ gµ⇢L

Renaming the index ⇢ ⌫ !

T µ⌫ = @µ@⌫ gµ⌫L (2.46) ) 57 So, we now want to ﬁnd the components of T µ⌫,startingwith⇢ = T 00

1 ⇢ = T 00 =(@0)2 g00 @ @ V () 2 1 h i =(@0)2 @ @ V () 2 1 h i = ˙2 g@ @ V () 2 h 1 i = ˙2 + V () ] g@ @ 2 2 1 h 00 i 11 22 33 = ˙ + V () g @0@0 + g @1@1 + g @2@2 + g @3@3 2 (2.47) 1h i = ˙2 + V () @ @ @ @ @ @ @ @ 2 0 0 1 1 2 2 3 3 1h i = ˙2 + V () (@ )2 ((@ )2 +(@ )2 +(@ )2) 2 0 1 2 3 1h i = ˙2 + V () ˙2 ( )2 2 r 1h 1 i = ˙2 + V () ˙2 + ( )2 2 2 r 1 1 = ˙2 + V ()+ ( )2 2 2 r

Finding T 11

1 T 11 = @1@1 g11 g@ @ V () 2 1 2 h 1 i =(@ ) V ()+ g @@ 2 (2.48) 1h i =(@1)2 V ()+ ˙2 ( )2 2 r 1h 1 i =(@1)2 V ()+ ˙2 ( )2 2 2 r

Likewise,

1 1 T 22 =(@2)2 V ()+ ˙2 ( )2 (2.49) 2 2 r 1 1 T 33 =(@3)2 V ()+ ˙2 ( )2 (2.50) 2 2 r

58 Now, we can use the components of T ij to obtain the value of p

1 p = (T 11 + T 22 + T 33) 3 1 3 3 = [(@1)2 +(@2)2 +(@3)2 3V ()+ ˙2 ( )2] 3 2 2 r (2.51) 1 3 3 = [( )2 3V ()+ ˙2 ( )2] 3 r 2 2 r 1 1 = ˙2 V () ( )2 2 6 r

Therefore, we obtain

1 1 ⇢ = ˙2 + V ()+ ( )2 (2.52) ) 2 2 r 1 1 p = ˙2 V () ( )2 (2.53) 2 6 r

Now, we assume the inﬂation scalar ﬁeld is spatially homogeneous. Physical gradi-

1 ents are related to comoving gradients by = a (t) .Thus,inhomogeneities rphys rcom in the ﬁeld are redshifted away during inﬂation since the scale factor increases by a large amount. Also, it is alright to neglect gradient contributions.

1 ⇢ = ˙2 + V ()(2.54) 2 1 p = ˙2 V ()(2.55) 2

We can take the time derivative of the expression for ⇢

⇢˙ = ˙¨ + V 0()˙ (2.56)

Now, we use the continuity equations to get a di↵erent expression for⇢ ˙,weplug

59 in the values of ⇢ and p obtained earlier

⇢˙ = 3H(⇢ + p) 1 1 = 3H( ˙2 + V ()+ (˙)2 V ()) (2.57) 2 2 = 3H˙2

Setting the two expressions for⇢ ˙ equal to one another

2 ˙¨ + V 0()˙ = 3H˙ (2.58)

¨ + V 0()= 3H˙ (2.59) )

Looking at the pressure and density, for p ⇢,wewant˙ V (), which is the ⇡ ⌧ slow roll approximation that gives a natural condition for inﬂation to occur.

2.2.3 A Simple Toy Model: Quadratic Potential

Let’s explore a single ﬁeld model by applying the slow roll conditions. Following Prof. Kaiser’s approach, we start by assuming a simple potential given in equation (2.60) [19].

1 V ()= m22 (2.60) 2

Using this potential and assuming a single scalar field, we wish to calculate the scaling function during inflation, and subsequently calculate the number of e-foldings at the end of inflation. Using the slow roll conditions, the Freidmann equation becomes

8⇡G H2 V () ⇡ 3 (2.61) 4⇡G = m2'2 3 60 The equation of motion thus becomes

3H'˙ V = m2' (2.62) ⇡ ,'

Using equation (2.61), we are able to write

a(t) da 4⇡G t = m dt0'(t0)(2.63) a 3 Zai r Zti

where ti is the starting time of inflation and ai is the initial value of the scale factor at the beginning of inflation. Also, we note that dt = d' because it a time dependent field. Substituting this ˙ relation into (2.62) yields

t '(t) d' dt0'(t0)= ''˙ Zti Z'i '(t) ' = d' m2 ' ' Z i 3H (2.64) '(t) 1 = d' p12⇡Gm' m2 Z'i ! p3⇡G = ['2 '2(t)] m i

Next, equation (2.64) can be used together with (2.62) to obtain

2 2 ' (t) a(t)=ai exp 2⇡G'i 1 2 (2.65) ( " 'i !#)

In Chapter 1, we predicted that the number of e-foldings before the end of inflation is supposed to be around 60. We denote the time of the beginning of inflation as t . ⇤ Just as we did in a prior section, we find the ratio between the scale factors at t and ⇤ tend using N =60.

2 2 a(t ) 60 2 ' (tend) ' (t ) ⇤ ⇤ = e =exp 2⇡G'i 2 2 (2.66) a(tend) " 'i 'i #!

61 From here, we would essentially use the slow roll parameter equations re-written for clarity in (2.67) and the results given by equation (2.66) in order to ﬁnd the desired slow roll parameter values at '(t ) for the quadratic potential. We will come back to ⇤ this calculation in Chapter 5.

2 H˙ 1 V,' ✏ = 2 H ⇡ 16⇡G V ! (2.67) H¨ 1 V ⌘ = ✏ , H'˙ ⇡ 8⇡G V !

62 Chapter 3

Supersymmetry and Supergravity Inﬂationary Models

3.1 Supersymmetry (SUSY)

The objective of this section is to familiarize the reader with the basics of supersymmetry. More specifically, the concept of auxilliary fields, which play a key role in supersymmetry based inflation models. Section 3.1.1 gives an overview of the notation, as well as reviews Weyl spinors and the Dirac equation. This sections references mostly a mix of Stephen Martin’s SUSY primer [24], Kamaluddin Ahmed’s presentation slides [25], and Aitchinson’s textbook on supersymmetry. Section 3.1.2 essentially introduces global SUSY, as well as provides the readers with some transformation relations to be used in the next subsection, 3.1.3, which introduces the concept of an auxilliary field. Both of these subsections closely follow Stephen Martin’s SUSY primer [24].

3.1.1 Conventions, Weyl Spinors, and the Dirac Wave Equa- tion

Supersymmetry (SUSY) is a type of spacetime symmetry that maps particles and ﬁelds of integer spin (bosons) into ﬁelds of half-integer spin (fermions). The minimal

63 supersymmetric standard model successfully predicts many phenomena that GUTs cannot explain. Some of these include the unification of the coupling constants, renormalization of the Higgs mass, a possible explanation of dark matter, electroweak symmetry breaking at scale lower than that of unification, and proton decay [39]. As apotentialcandidateforbeyondthestandardmodelphysics,supersymmetryalso has useful applications in inflationary cosmology.

Before jumping into supersymmetry using Martin’s approach [24], it is useful to make a few statements on notation and conventions. Recall the 4-component Dirac fermion D with mass M has a Lagrangian of the form

L = i µ@ M (3.1) Dirac D µ D D D

where µ is given by

0 µ 10 µ = 5 = 0 µ 0 1 0 011 @ A @ A

in terms of the Pauli matrices

10 01 0 = 1 = 0 011 0 101 @ A @ A 0 i 10 2 = 3 = 0 i 0 1 0 0 1 1 @ A @ A The aim of this section is to introduce and demonstrate the necessity of auxilliary fields in supersymmetry. Now, consider a 4D field theory with minimum fermion content. Let be a single left-handed 2-component Weyl fermion, and be a complex superpartner scalar field, as shown in Figure [3-1].

64 Figure 3-1: Particles and their superpartners. This ﬁgure is inspired by one in Ka- maluddin Ahmed’s presentation slides [25].

The corresponding action is

4 S = d x(Lscalar + Lfermion)(3.2) Z

Substituting the general forms of the Lagrangian yields

4 µ µ S = d x( @ ⇤@ + i †¯ @ µ † )(3.3) µ µ Z The ﬁrst and third terms are easily recognizable as general scalar and fermion potential terms. However, where does the second term come from? We can ﬁgure this out by using Weyl fermions! For the rest of this subsection, we will be using Aitchison’s approach [20].

In order to understand equation (3.3), we start by deﬁning = .Each 0 1 of its components and are the two-component Weyl spinors. @mustA obey the

65 Dirac equation in momentum space for c = ~ =1givenbyequation(3.4).

E =(↵ p~ + m) (3.4) ·

~ 0 01 where ↵ = and = 0 0 ~ 1 0 101 @ A @ A

0 i Note that using this convention, we have that = and = i 0 1 5 i 0 10 @ A .Wealsohavetheanticommutationrelations 5, = 5, =0. 0 0 1 1 { } { } @ A We can plug the deﬁnition of into the Dirac equation given by (3.4).

~ p~ E = · + m (3.5) 0 1 0 ~ p~ 1 0 1 · @ A @ A @ A where ~ ( , , ) ⌘ 1 2 3

Equation (3.5) can be re-written in terms of a system of two equations.

(E ~ p~ ) = m · 8 (3.6) >(E + ~ p~ ) = m < · :> Note that and are eigenstates of 5.

10 85 0 1 = 0 1 0 1 =10 1 > 0 0 1 0 0 > > B C B C B C B C (3.7) > @ A @ A @ A @ A <> 0 10 0 0 = = 1 5 0 1 0 1 0 1 0 1 > 0 1 > B C B C B C B C > > @ A @ A @ A @ A : 66 Now, let’s deﬁne two projection operators to construct the 5 eigenstates.

1 1 2 0 2 0 10 P = I+5 = + = 8 R 2 0 1 0 1 0 1 > 0 1 0 1 00 > 2 2 > B C B C B C (3.8) > @ 1 A @ 1 A @ A <> 2 0 2 0 00 I 5 PL = 2 = 0 1 + 0 1 = 0 1 > 0 1 0 1 01 > 2 2 > B C B C B C > @ A @ A @ A :> 10 8PR =0 1 0 1 = 0 1 > 00 0 > > B C B C B C (3.9) > @ A @ A @ A <> 00 0 PL =0 1 0 1 = 0 1 > 01 > > B C B C B C > @ A @ A @ A :> 2 2 Note that also, PRPL =0,PR = PR,andPL = PL.Theeigenvaluesof5 found in equations (3.7) represent the chirality of each spinor: is a right handed fermion with chirality +1 and is a left handed fermion with chirality 1. Thus, we denote the spinors as R and L respectively. Now, consider two types of inﬁnitesimal transformations on (E,p~ ):

1. 3D rotations with inﬁnitesimal parameter ~✏ =(✏1,✏2,✏3):

E E0 = E ! (3.10) p~ p~ ~✏ p~ ! 0 ⇥

2. Velocity transformations with inﬁnitesimal parameter ~⌘ =(⌘1,⌘2,⌘3):

E E0 = E ~⌘ p~ ! · (3.11) p~ p~ ~⌘ E ! 0

Thus, and transforms as

i 1 0 =(1+ ~✏ ~ ~⌘ ~ ) ! 2 · 2 · (3.12) i 1 0 =(1+ ~✏ ~ + ~⌘ ~ ) ! 2 · 2 · 67 1 Note that and both have spin 2 and behave the same under rotations, but di↵erently under boosts. and also satisfy

(E0 ~ p~ ) 0 = m0 · 0 8 (3.13) >(E0 + ~ p~ )0 = m 0 < · 0 In SUSY, we work with 2-component:> Weyl spinors, and .Deﬁningtheterms inside of the parenthesis in equation (3.12), we get

i 1 V =(1+ ~✏ ~ ~⌘ ~ ) 2 · 2 · 1 i 1 V =(1 ~✏ ~ + ~⌘ ~ ) (3.14) 2 · 2 · i 1 V † =(1 ~✏ ~ ~⌘ ~ ) 2 · 2 ·

where the last expression is a result of the Hermiticity of ~ . Note that

1 1 i 1 (V †) =(V )† =1+ ~✏ ~ ~⌘ ~ (3.15) 2 · 2 · By inspection of equation (3.12) and utilizing the deﬁnitions from equations (3.14), we can see that from the transformation relations

0 = V (3.16) 1 1 0 =(V †) =(V )†

At this point, we have the tools to deal with the four component Dirac spinor in order to construct a Lorentz invariant and a 4-vector as shown in equations (3.17) and (3.18) respectively.

¯ = † (3.17)

µ ¯ = †(,↵) = †(I,↵) (3.18)

68 Where in equation (3.18), we use the relation that 2 = I,whereI is the identity matrix.

Now, let’s re-write equation (3.17) in terms of Weyl spinors

01 † = † † 0 101 0 1 ⇣ ⌘ @ A @ A (3.19) = † † 0 1 ⇣ ⌘ @ A = † + †

In addition, we re-write equation (3.18) in terms of Weyl spinors.

01 ~ 0 †(1,↵) = † † , 0 101 0 0 ~ 1 ! 0 1 ⇣ ⌘ @ A @ A @ A ~ = † † , (3.20) 0 1 0 ~ 1 ! ⇣ ⌘ @ A @ A = † + † †~ †~ ⇣ µ µ ⌘ † + †¯ ⌘

(3.20) is also written in terms of the µ,andthegammamatricesbecomeµ = 0¯µ . 0 µ 0 1 @ The DiracA equation (3.6) can be re-written as follows.

µ pµ = 8 (3.21) µ <>¯ pµ =

> µ Note that the Lorentz transform: character for pµ is the same as ,andthe µ Lorentz transform character for ¯ pµ is the same as . Here, we start to see the ﬁrst

69 signs of supersymmetry! We used the deﬁnition of to re-write the Dirac Lagrangian.

µ µ µ ¯ (i @ m) = †i @ + †i¯ @ m( † + † )(3.22) µ µ µ

If we assume that m =0,weessentiallyreduceourLagrangiantothatofthe massless, non-interacting Wess-Zumino model.

µ µ LW-Z = †i @µ + †i¯ @µ (3.23)

Note that the Wess-Zumino model consists of a single left-handed massless fermion, given by the second term

µ Lfermion = †i¯ @µ (3.24)

Renaming to in equation (3.24), we get

µ Lfermion = †i¯ @µ (3.25)

which gives us the second mystery term in (3.3).

3.1.2 Global SUSY

The next step to understanding the need for auxilliary fields is to impose the global supersymmetry condition on this system. Continuing with Martin’s approach [24], let’s start by considering the supersymmetric transform with an infinitesimal parameter ✏.Thistransformationoperatesonthescalarfieldasfollows.

= ✏ (3.26) ⇤ = ✏† †

↵ ↵ For global supersymmetry, impose the condition that @µ✏ =0since✏ is equal to a constant.

70 Given our scalar Lagrangian

µ µ L = @ ✏ @ ⇤ @ ✏† †@ (3.27) scalar µ µ

Applying the transformation given equations (3.26) yields

µ µ L = ✏@ @ ⇤ ✏†@ †@ (3.28) scalar µ µ

We want the action to be invariant under supersymmetry, so that the terms in

µ Lfermion and Lscalar cancel each other out. Since Lfermion = i †¯ @µ ,weknowthat

should be linear in ✏† and ,anditshouldalsohaveonespacetimederivative.

We can show that the variation in ↵ and its conjugate transpose is given by

µ ↵ = i( ✏†)↵@µ (3.29) µ † = i(✏ ) @ ⇤ ↵ ↵ µ

This can be used to obtain the variation in the fermionic Lagrangian.

µ ⌫ ⌫ µ L = ✏ ¯ @ @ ⇤ + †¯ ✏†@ @ (3.30) fermion ⌫ µ µ ⌫

We can substitute the following Pauli matrix identities into equation (3.29).

µ n ⌫ µ µ⌫ [ ¯ u + ¯ ]↵ = 2⌘ ↵ (3.31) ˙ ˙ [¯µ⌫ +¯⌫µ] = 2⌘µ⌫ ↵˙ ↵˙

Thus, we can re-write Lfermion.

µ µ ⌫ µ µ µ L = ✏@ @ ⇤ + ✏†@ †@ @ (✏ ¯ @ ⇤ + ✏ @ ⇤ + ✏† †@ )(3.32) fermion µ µ µ ⌫ 71 We can use equations (3.30) and (3.32) to write the change in action

4 S = d x(Lscalar + Lfermion) Z 4 ⌫ µ µ µ = d x @ (✏ ¯ @ ⇤ + ✏ @ ⇤ + ✏† †@ ) (3.33) µ ⌫ Z ! =0

Where in the last step, we note that we are integrating over a total derivative, which causes the integral to go to zero.

3.1.3 SUSY Algebra Closure and Auxilliary Fields

Continuing to follow Martin’s SUSY primer [24], the next step is to show that the supersymmetry algebra closes. In order to prove this, it must be shown that the commutator of two supersymmetry transformations parameterized by two di↵erent spinors ✏1 and ✏2 is another symmetry of the system. We recall the transformations for the bosonic and fermionic ﬁelds from earlier in equations (3.26) and (3.29) respectively. We start by looking at the scalar ﬁeld.

( ) = (✏ ) ( ) ✏2 ✏1 ✏1 ✏2 ✏2 1 ✏1 ✏2 µ µ = i✏ ✏†@ + i✏ ✏†@ (3.34) 1 2 µ 2 1 µ µ µ = i(✏ ✏† i✏ ✏†)@ 1 2 2 1 µ

Note that the final line of (3.33) has a coecient of @µ,whichisaderivativeof the field. In the Heisenberg picture, i@µ corresponds to the generator of spacetime translation Pµ.Werepeattheprocessforthefermionfield.

µ µ (✏2 ✏1 ✏1 ✏2 ) = ✏2 ( i ✏1†@µ) ✏2 ( i ✏2†@µ) (3.35) µ µ = i ✏†✏ @ + i ✏†✏ @ 1 2 µ 2 1 µ 72 Now, let’s use the Fierz identity.

µ ✏1† = ✏2 @µ (3.36) µ ✏† = ✏ @ 2 1 µ

Substituting this into equation (3.34) yields

µ µ (✏2 ✏1 ✏1 ✏2 ) = i ✏1†@µ✏2 + i ✏2†@µ✏1 (3.37) µ µ µ µ = i( ✏ ✏† + ✏ ✏†)@ + i(✏ ✏†¯ ✏ ✏†¯ )@ 1 2 2 1 µ 1 2 2 1 µ

The ﬁrst term is clearly the same as the translation for the scalar ﬁeld found in equation (3.34). The second term vanishes “on-shell,” which means that if the

µ classical equation of motion is satisfied, ¯ @µ =0. In order to make sure supersymmetry holds quantum mechanically, we now introduce a new scalar field F ,calledan“auxilliaryfield.”Thisfieldshouldhaveno kinetic term and will allow the algebra to close “o↵-shell.” Let the Lagrangian density for F and its conjugate be

Lauxilliary = F ⇤F (3.38)

where

F = F ⇤ =0 (3.39)

Now, we want to make F transform into a multiple of the equation of motion of

µ ,¯ @µ =0.Let’sstartbyﬁndingthevariationinF .

µ F = i✏†¯ @µ (3.40) µ F ⇤ = i@µ †¯ ✏

73 We can use equations (3.38) and (3.40) to ﬁnd the variation in the auxilliary Lagrangian.

µ µ L = i✏†¯ @ F ⇤ + i@ †¯ ✏F (3.41) auxilliary µ µ This vanishes on-shell, but not o↵-shell. We can deal with this by adding extra terms to the transform of .

µ = i( ✏†)@µ + ✏F (3.42) µ † = i( ✏)@µ⇤ + ✏†F ⇤

With some simple algebra, it can actually be shown that [24]

L = Lscalar + Lfermion + Lauxilliary (3.43)

And that for a general X = ,⇤, , †,F,F⇤ [24] { }

µ µ ( )X = i( ✏ ✏† + ✏ ✏†)@ X (3.44) ✏2 ✏1 ✏1 ✏2 1 2 2 1 µ

This shows that supersymmetry would be valid o↵-shell. Therefore, we have introduced the auxilliary ﬁeld F in order for SUSY to work o↵shell and ensure the algebra closes.

3.2 Supersymmetry Inﬂationary Models

Section 3.2.1 deals with presenting the concept of hybrid inflation, which is a class of inflationary models that include both SUSY and string theory base inflation. The primary reference for this chapter is Julien Lesgourges’s lecture notes on inflationary cosmology [40]. Other sources include Jochen Baumann’s dissertation [43] and Copeland et al’s paper [26]. Section 3.2.2 presents a simple SUSY inflationary F-term model. It closely follows the work done in Copeland et al’s paper [26], but also in-

74 cludes information from Micheal Dine’s textbook [22]. Finally, section 3.2.3 goes over a more complex SUSY model involving both F-term and D-term inﬂation. It closely the follows Julien Lesgourges’s lecture notes on inﬂationary cosmology [40].

3.2.1 Hybrid Inﬂation

Before jumping into SUSY based inflationary models, it is important to consider a new class of inflationary models known as “hybrid inflation.” Hybrid inflation describes models exiting the inflationary stage with a phase transition triggered by an auxilliary field. Since the secondary auxiliary field does not actually play a role during inflation, this model belongs to the single field category [40], [26]. Atypicalhybridpotentialhastheform

2 2 2 2 2 Vtotal(,)=(M ) + V ()+˜ (3.45) = M 4 +(˜ 2 M 2)2 + 4 + V ()

where and are two real scalar fields. M is a mass term, while and are constants. In this case, plays the role of the inflaton field, while is the trigger e field. One might wonder why the potential has this specific form? Essentially, any potential for which low values of leads to a phase transition is a valid choice [40]. In this specific case, when is big, the mass term of the field is positive. We assume the initial condition ˜ 2 >M2,sinceatthebeginningofinflation,wewant apotentialoftheform+C2,whereC is a real and positive constant. A diagram of the desired potential is shown in Figure [3-2]. Up until reaches a critical value

critical, will essentially ensure a small e↵ective mass of the system. Due to the positive squared mass, rolls quickly to zero. When =0,wegetthee↵ective potential of the form

4 Ve↵()=M + V ()(3.46)

If the potential is flat enough in the direction, inflation can happen. In this case, the slow roll conditions can be satisfied when >critical.Thisisthepointatwhich

75 Figure 3-2: Hybrid inﬂation potential. This ﬁgure is inspired by one from Jochen Baumann’s dissertation [43].

the e↵ective mass of the ﬁeld vanishes. We can see from equation (3.44) that this

2 2 occurs when ˜ = M . After = critical,the field leaves the equilibrium point. = 0 and rolls towards one of the absolute minima at (,)=( M,0). During ± the phase transition, the e↵ective mass of the field is on the order of magnitude Mp. ⇠ When the e↵ective mass of the field of the system is much greater than H,the phase transition is fast. This occurrence is also known as a “waterfall.” In this case, inflation ends at = critical exactly. If the slow roll conditions break before the inflaton reaches its critical value, inflation ends. In this case, the phase transition occurs when critical is finally reached.

The phase transition at the end of inﬂation can represent the breaking of larger symmetries as the ﬁelds reach di↵erent minima at di↵erent locations. In the most simple case, one of these topological defects could be a domain wall, as shown in Figure [3-3]. It could also lead to many other types of topological defects, such as cosmic strings and monopoles.

76 Figure 3-3: Hybrid inﬂation potential. This ﬁgure is inspired by one from Jochen Baumann’s dissertation [43].

In general, hybrid inﬂationary models have an e↵ective potential of the form

Ve↵()=V0(1 + f()) (3.47)

where f() > 0.

As we learned in Chapter 2, the slow roll approximation gives the conditions ✏ 1 ⌧ and ⌘ 1, where | |⌧

2 m2 V () ✏ pl 0 ⌘ 16⇡ V () ! (3.48) m2 V () ⌘ pl 00 ⌘ 8⇡ V () !

Substituting Ve↵ into equations (3.47) yields the slow roll parameters for a general

77 hybrid inﬂationary potential.

2 m2 @f ✏ = pl @ 16⇡ 1+f ! 2 (3.49) 2 @ f m 2 ⌘ = pl @ 8⇡ 1+f !

Before continuing onwards, let’s look at the simplest possible example of what an F -term only inﬂationary model using SUSY is.

3.2.2 Simple F-Term SUSY Inﬂation

In order to investigate F-term inﬂation, we will start with the simplest superpotential that breaks U(1) symmetry, published by Copeland et al [26]. The superpotential consists of three chiral superﬁelds 1, 2,andandisgivenby

2 W = ( 1 2 +⇤ )(3.50)

where ⇤is a mass term, and is a coupling constant. In this case, U(1) symmetry is given by

i✓ 1 e 1 ! (3.51) i✓ e 2 ! 2

We obtain the scalar potential from the superpotential as follows

2 2 2 @W @W @W V = + + @ @ 1 @ 2 (3.52) = 2 + ⇤2 2+ 2( 2 + 2) 2 | 1 2 | | 1| | 2| | |

Following the model described in reference [26], a term with a supersymmetry

78 breaking mass m on the order of 1TeV is added for the chiral superﬁeld .The ⇠ resulting potential is given by

V = 2 +⇤2 2 + 2( 2 + 2) 2 + m2 2 (3.53) | 1 2 | | 1| | 2| | | | |

Identifying as the inﬂaton, it is desired to obtain the e↵ective potential during inﬂation.

The potential is minimized under 2 conditions. The ﬁrst condition is that arg( 1)+ arg( 2)=⇡.Thesecondconditionisthatthecanonicallynormalizedﬁeldassociated with the phase of that has a smaller magnitude has an e↵ective mass ⇤atthe  potential minimum.

It is important to note that V is independent of 2 angular degrees of freedom. The first is the axion field from Peccei-Quinn theory, which is given by arg( ) arg( ), 1 2 while the second degree of freedom is the argument of . The Peccei-Quinn theory refers to a type of symmetry that occurs in QCD. Essentially, in QCD, the vacuum energy can be written as a function of a specific parameter. When the parameter goes to zero, the vacuum energy is minimized, which leads to an unwanted axion. In the standard Lagrangian for an axion, there is a term that is dependent on that parameter. In order to minimize the vacuum energy and allow the term to drop out, we apply what is called a “Peccei-Quinn” symmetry transformation. QCD systems should generally be invariant under these transformations [22].

Going back to Copeland et al’s example [26], writing the potential in terms of the three canonically normalized ﬁeld yields

= p2 | | = p2 (3.54) 1 | 1| = p2 2 | 2| 79 We thus obtain

2 2 1 V (, , )= ( 2⇤2)2 + ( 2 + 2)2 + m22 (3.55) 1 2 4 1 2 4 1 2 2

For >0, which corresponds to the normalized saxino field, or the axion superpartner field, it is possible to prove that the field is fixed at its minimum value during inflation [26]. This allows us to write the constraint on the critical value

>criticalp2⇤.

Also, in terms of the canonically normalized ﬁeld = p2 1 2 and ,thepotential is

2 2 1 V (, )= ( 2 4⇤2)2 + 2 2 + m22 (3.56) 16 4 2

Now, it is easy to see that if =0,whichisalocalminimumoftheﬁeld,then

1 V (, )=2⇤4 + m22 (3.57) 2

And inﬂation occurs.

3.2.3 SUSY Inﬂationary Model with F and D-Terms

In global SUSY, the scalar potential is divided into 2 contributions, the F-term and the D-term.

V = VF + VD (3.58)

Let the superpotential W be a holomorphic function of at most order 3 in complex scalar ﬁelds i.

3 2 W (1,...,n)=µ + µi i + µijij + µijkijk (3.59) i ij X X Xijk Recalling the derivation in the previous section, we know that by defnition, the

80 F-term is given by

2 @W VF = (3.60) @n n X On the other hand, the D-term is given by

2 1 V = g2 q 2 (3.61) D 2 n| | n ! X

where qn is the charge of n under U(1). In addition, it is possible to add a constant term ⇠ to the sum.

2 1 V = g2 q 2 + ⇠ (3.62) D 2 n| | n ! X ⇠ is called the Fayet-Illiopoulos term, which can be freely inserted into the equation by hand and still respects the system’s covariance.

Now, let’s evaluate an inﬂationary model that contains both F and D terms. Let’s start with a superpotential of the form

+ 2 W = ↵ S µ S (3.63)

+ where ,,andS are scalar fields with charges 1, 1, and 0 respectively. In the F-term inflationary limit, we make the assumption that ⇠ vanishes. We now proceed to find VF and VD by using equations (3.60), (3.61), and (3.63). The result for the F-term is given by

2 @W VF = @n n X 2 2 2 @W @W @W (3.64) = + + + @ @ @S + 2 2 2 2 + 2 2 = ↵ µ + ↵ S ( + ) | | | | | | | | 81 Proceeding to calculate the D-term yields

1 2 + 2 2 2 = g ( + ) 2 | | | |

2 In F-term inflation, when S is above some critical value S 2 µ ,thecharged | | | | 2 + fields remain in their equilibrium configuration, which is = =0.Thiscondition on S keeps the e↵ective mass term positive, so that constantly is driven towards | | n the equilibrium configuration. Also note that the tree level potential is held constant at V = µ2.Thiscorresponds to the flat direction, which is lifted only by loop corrections. The Coleman-Weinberg formula can subsequently be applied to obtain the e↵ective potential of the system as documented in reference [26].

2 4 2 2 2 ↵ µ 3 ↵ S Ve↵ = µ + 2 +ln | 2 | (3.66) 16⇡ 2 ⇤ ! The critical value is S = µ . | |critical p↵ When the modulus of S goes below the value of S ,thesystemundergoesa | |critical “waterfall” transition to the true minimum, which is given by

µ + ei✓ ! p↵ µ i✓ e (3.67) ! p↵ S 0 !

where V = 0. Thus, SUSY is restored for F-term inﬂation! Now, on the other hand, in the D-term inﬂationary limit, we now assume that ⇠

82 in VD does not vanish. So for VD,wenowhave

1 2 + 2 2 2 = g ( + + ⇠) 2 | | | |

1 2 2 The constant tree level potential is now V = 2 g ⇠ .Thisisaﬂatdirection,sowe can once again use the one-loop correction given by the Coleman-Weinberg formula to get the e↵ective potential [26].

g2⇠2 g4⇠2 3 ↵2 S 2 Ve↵ = + 2 +ln | 2 | (3.69) 2 10⇡ 2 ⇤ !

The critical value for this model is S = gp⇠ . | |critical ↵ When the modulus of the S ﬁeld falls below the critical value, the system undergoes a “waterfall” transition to the true minimum, given by

+ 0 ! i↵ ⇠e (3.70) ! S 0p !

where V = 0. Thus, SUSY is restored for D-term inﬂation as well!

3.3 Supergravity (SUGRA) Inﬂationary Models

The aim of this section is to use what we have learned so far about SUSY to work through a few examples of SUGRA inﬂation. Section 3.3.1 is more conceptual in nature, and addresses the di↵erences between SUGRA and SUSY. This section is heavily inspired by Masahide Yamaguchi’s paper

83 [39] and Julien Lesgourges’s lecture notes on inflationary cosmology [40]. Section 3.3.2 introduces the ⌘ problem, which is an extremely relevant issue that arises when de- veloping both supergravity and string theory based inflationary models. The primary references here are Pierre Binetruy’s textbook [21] and Julien Lesgourges’s lecture notes on inflationary cosmology [40]. Finally, section 3.3.3 gives an example of both F-term and D-term SUGRA inflation, which extremely closely follows two examples originally given in Masahide Yamaguchi’s paper [39].

3.3.1 Di↵erences Between SUGRA and SUSY in Context of Inﬂation

The scalar part of the SUGRA Lagrangian is determined by three functions that are all functions of chiral superﬁelds. The ﬁrst function is called the Kahler potential,

K(i, i⇤). It consists of real functions of scalar fields and their conjugates, but it is not holomorphic. The second function is the superpotential W (i), and the third function is the gauge kinetic function f(i). Both the superpotential and the gauge kinetic function are holomorphic functions of complex scalar fields. As expected, the action of complex scalar fields minimally coupled to gravity has both kinetic and potential terms.

4 1 S = d xp g L V ( , ⇤) (3.71) p g kinetic i i Z " # The kinetic terms of the scalar ﬁelds are determined in part by the Kahler potential K and are given by

1 µ⌫ L = K D D ⇤g (3.72) p g kinetic ij⇤ µ i ⌫ j @2K where Kij⇤ = and Dµ is the gauge covariant derivative. @ij⇤ Just as in the previous section on SUSY inﬂationary models, we have a potential that comprises of an F-term and a D-term V = VF + VD. However, this time, the F-term is determined by the superpotential W ,aswellastheKahlerpotentialK.

84 The F-term VF is given by

K 1 2 VF = e [D WK D W ⇤ 3 W ](3.73) i ij⇤ j⇤ | |

@W @K where D W = + W . i @i @i The D-term is related to the gauge symmetry of the system and is given by both the gauge kinetic function fa and the Kahler potential K.

1 1 V = [f ( )] g2D2 (3.74) D 2 < a i a a a " # X i @K where Da =i(Ta) + ⇠a. Note that the subscript a represents the gauge j @j symmetry and Ta represents the associated generator and ga is a gauge coupling constant. It should also be pointed out the Fayet-Illiopoulos term ⇠a is back, since it takes on a nonzero value for Abelian gauge symmetry U(1).

The kinetic and potential terms are invariant under the following Kahler transformations.

K(i, i⇤) K(i, i⇤) U(i) U ⇤(i⇤) ! (3.75) W ( ) eU(i)W ( ) i ! i

for U(i) is any holomorphic function of ﬁelds i.

Investigating equation (3.78), it could be determined that in order to get the positive energy density needed for inﬂation, at least one the Di W terms must be nonzero. Since these terms are order parameters of SUSY, inﬂation is always accompanied by SUSY breaking.

Using the near-canonical Kahler potential

2 K( , ⇤)= + ... (3.76) i i | i| i X 85 The equation for VF can be approximated as

2 @W @W⇤ 2 VF exp i + ... +(i⇤ + ...)W (ij + ...) +(j + ...)W ⇤ 3 W ⇡ | | @ @⇤ | | i !(" i # ij " j # ) X X = V + V 2 + ... global global | | i X (3.77)

where Vglobal is the e↵ective potential in the global SUSY limit given by

2 @W Vglobal = (3.78) @i i X

3.3.2 The ⌘ Problem

The main issue with incorporating inﬂation into supergravity is known as the ⌘ problem.

In supergravity, we have that

8⇡K VF =exp [...](3.79) mpl !

where the higher order terms in equation (3.83) correspond to fields related to W , K, and a mix of first and second order derivative terms of these functions. Using a Taylor expansion of the Kahler potential about the origin of the field space, we can cancel the lowest-order terms with transformations to obtain

K = Kmnmn⇤ + ... (3.80) mn X

where Kmn is a matrix coecient of order 1.

Now, assume p2 is the inflaton field. The inflationary potential will take ⌘ | 1| 86 the form

VF =exp 4⇡K11 + ... [...](3.81) mpl ! !

with derivatives

@VF 8⇡K11 = 2 VF + ... @ mpl 2 (3.82) @ VF 8⇡K11 2 = 2 VF + ... @ mpl

Substituting the equations in (3.85) and (3.86) into the slow roll parameter ⌘ yields

2 mpl 8⇡K11 ⌘ = 2 + ... 8⇡ mpl (3.83)

= K11 + ...

However, K11 is order of magnitude 1... so there is no way ⌘ can possibly be small! Thus, the SUGRA VF violates the slow roll conditions by construction unless the extra terms in W and K cancel the terms contributing to the result in (3.87) exactly. In other words, for instance, choosing a superpotential and convenient Kahler potential that allows terms to cancel perfectly in such a way that allows for the condition ⌘ K to be met. Other options include having models where the ⌧| 11| F-term potential is negligible or models where the inﬂaton does not appear in the F-term potential. These options do not often show up, since the F-term potential plays an important role in current models. Note that choosing a convenient Kahler potential is only a temporary solution to the ⌘ problem, and that the issue may arise again in constructing inﬂationary models that incorporate string theory [40], [21].

87 3.3.3 SUGRA Inﬂationary Model with F and D-terms

F-Term Inﬂation

In order to avoid the ⌘ problem, we must use a special form of the superpotential and near canonical Kahler potential. As we will see, if we choose a superpotential that is linear in the inflaton field, the inflaton e↵ective mass becomes negligible, which allows us to avoid the ⌘ problem in turn. We execute this plan of action by imposing R-symmetry with the R charge of the inflaton to be 2 and the others to be 0. Let’s look at an example right now! Start with a superpotential of the form

W = S ¯ µ2S (3.84)

where S is a gauge-singlet superﬁeld. and ¯ areaconjugatepairofsuperﬁelds that transform as nontrivial representations of gauge group G. and µ are positive parameters less than 1.

W is linear in S with R-symmetry. The inﬂaton S as well as and ¯ transform as

S(✓) e2i↵S(✓ei↵) ! (✓) e2i↵ (✓ei↵) (3.85) ! 2i↵ i↵ ¯ (✓) e ¯ (✓e ) !

For this analysis, we will use the canonical Kahler potential as documented in reference [40].

K = S 2 + 2 + ¯ 2 (3.86) | | | | | | 88 Using equation (3.60) to compute VF

V (S, , ¯ ) = exp S 2 + 2 + ¯ 2 (1 S 2 + S 4) ¯ µ2 2 | | | | | | !( | | | | | | (3.87) 2 2 2 2 2 2 2 + S (1 + ) ¯ µ ⇤ + (1 + ¯ ) µ ¯ + VD | | "| | | | | | | | #)

Since (3.87) depends only on the modulus of complex scalar field S,wecanwrite the real part as p2[ (S)] with the inflaton field. ⌘ < 1 1 Assuming 1andletting ( ¯ ⇤), and thus ¯ ( + ¯ ⇤), the ⌧ ⌘ p2 ⌘ p2 mass terms are for and ¯ are

2 1 2 4 2 2 2 V µ ( ¯ + ¯ ⇤ ⇤)+ ( + µ ) ( ¯ + )(3.88) mass ⇡ 2 | | | |

Deﬁning = 1 ( ¯ ) and ¯ = 1 ( + ¯ ) yields sqrt2 sqrt2

V =[(2 + µ4) S 2 + µ2] 2 +[(2 + µ4) S 2 µ2] ¯ 2 (3.89) mass | | | | | | | |

Next, we need to ﬁnd the eigenvalues M for the eigenstates and ¯. ±

M =(2 + µ2) S 2 µ2 ± | | ± 1 (3.90) = (2 + µ4)2 µ2 2 ±

where = ¯ . Note that M 2 is always positive, so it allows us to set = 0, so ⌥ ⇤ + that = ¯ ⇤.

Now, let’s consider the dynamics of S and . When gets smaller than a critical µp2 2 ¯ value critical p for M < 0, rolls down to a global minimum, thus ending ⌘ inﬂation.

Note that for F-term inﬂation, ¯ isaﬂatdirection,sosetting = ⇤ and

89 assuming µ , we obtain the e↵ective potential ⌧

1 V =( 2 µ2)2 + 22 2 + µ44 + ... (3.91) | | | | 8

with tree level potential V = µ4.

As a result of SUSY breaking (and the resulting positive energy density) during inﬂation, we get a separation of mass between and ¯ witheigenvaluesM 2 ± ⇡ 2 2 µ2 and superpartner fermions with mass M = . Now, we proceed to use 2 ± p2 the formula to obtain one loop corrections as we did in SUSY:

2 2 2 2 2 N 2 2 2 +2µ 2 2 2 2µ V1L = 2 ( +2µ ) ln 2 +( 2µ ) ln 2 128⇡ " ⇤ ⇤ (3.92) 2 2 4 2 ln 2 ⇤ #

where ⇤is a renormalization scale and N is the dimensionality of G.

Letting yields critical

˜2 4 1 4 Ve↵ µ 1+ 2 ln + (3.93) ⇡ 8⇡ critical 8 !

where ˜ pN . ⌘ The number of e-foldings using the potential in (3.93) is estimated [39] as

N V N = d V Zc 0 N 8⇡2 d (3.94) ⇡ 2 Zc N 2 4⇡ 2 N ⇡ e2 Zc

602 e where for N =60,d = 4⇡2 .Substitutingthee↵ectivepotentialVe↵ into the slow e 90 roll parameters yields

˜4 ✏ ⇡ 128⇡42 N (3.95) ˜2 ⌘ 2 2 ⇡8⇡ N

D-Term Inﬂation

As stated earlier, the potential for D-term inﬂation depends on the Kahler potential and gauge kinetic function, but it does not contain the exponential factor that leads to the ⌘ problem. In this section, we provide an example D-term inﬂation developed by Yamaguchi [39].

This time, let’s start out with the superpotential

+ W = S (3.96)

+ where S, ,and are 3 chiral superﬁelds with charges 1, 1, and 0 respectively, and is a coupling constant.

The superpotential W is invariant under U(1) and has R-symmetry. Hence, the chiral superﬁelds have the transformation properties given by

S(✓) e2i↵S(✓ei↵) ! (3.97) + i↵ (✓) + (✓e ) !

In addition, the Kahler potential is given by

2 2 2 K = S + + + (3.98) | | | | | |

which is invariant under gauge and R-symmetry.

Now we proceed to ﬁnd V = VF + VD, this time including the Fayet-Iliopoulos

91 term and using the equations for VF and VD,weobtain

2 2 2 2 2 2 2 V (S, +, )= exp( S + + + ) + + S + S+ | | | | | | "| | | | | | 2 2 2 2 2 g 2 2 2 +(S + + + +3)S+ + ( + + + ⇠) | | | | | | | | # 2 | | | | (3.99)

In the last term, please note that g is the gauge coupling constant and ⇠ is the Fayet-Illiopoulos term. The potential in (3.99) has a unique global minimum at

S =+ =0 (3.100) = ⇠ p

For large values of S ,thepotentialhasalocalstableminimawithpositive | | energy density at + = = 0. Assuming that inﬂation occurs at the trajectory where + =0and =0,wecanproceedtocalculatethemasstermsfor+ and .Thesetermsaregivenby

2 2 2 2 Vmass = m+ + + m (3.101) | | | |

2 2 gp⇠ Using the last equation for m ,wecancalculatethatform 0, S . | | This is identiﬁed as the critical value Scritical.Recallingthattheminimumisstable, g2⇠2 inﬂation is driven by positive potential energy density 2 . Additionally, the mass split between these two terms is associated with radiative corrections. The one loop

92 correction is given by

2 2 S 2 2 1 2 2 S 2 2 2 S e| | + g ⇠ | | V1L = 2 ( S e + g ⇠) ln | | 32⇡ " | | ⇤ ! 2 2 S 2 2 2 2 S 2 2 2 S 2 2 2 S e| | g ⇠ 4 4 2 S 2 S e| | +( S e| | g ⇠) ln | | 2 S e | | ln | | | | ⇤ ! | | ⇤ !# (3.102)

Assuming that S S ,theoneloopcorrectionreducesto | || critical|

4 2 2 2 S 2 g ⇠ S e| | 3 V1L 2 ln | | 2 + (3.103) ⇡ 16⇡ " ⇤ ! 2# As a result, we get the e↵ective potential that is valid during inﬂation.

2 2 2 2 2 S 2 g ⇠ g S e| | Ve↵ 1+ 2 ln | | 2 (3.104) ⇡ 2 " 8⇡ ⇤ !# Note that the potential in (3.106) only depends on the modulus of complex scalar ﬁeld S,sowecanidentifytherealpartas p2[ (S)]. ⌘ < Thus, for 1, the e↵ective potential becomes critical ⌧ 

g2⇠2 g2 22 Ve↵ 1+ 2 ln 2 (3.105) ⇡ 2 " 8⇡ 2⇤ !#

where is the inﬂaton and ⇤is a renormalization scale. Using equation (3.107), the slow roll parameters for D-term SUGRA inﬂation [39] are given by

g4 ✏ 4 2 ⇡ 32⇡ (3.106) g2 ⌘ ⇡4⇡22

93 94 Chapter 4

Axion Monodromy Inﬂationary Models

4.1 Introduction to String Theory

The aim of this section is to present a conceptual overview of the basics of string theory on an advanced undergraduate level. A very useful reference that I used in learning the concepts for the ﬁrst time is Daniel Baumann’s notes [8]. The ideas for section 4.1.1 come from a mix of Magdalena Larfors’s dissertation [40] and Angel Uranga’s notes [32]. Sections 4.1.2 and 4.1.3 both follow Micheal Dine’s textbook on supersymmetry and string theory extremely closely [22]. I found that out of all of the texts I have read, Dine’s approach to string and superstring theory was the most mathematically accessible to undergraduates, especially with a background in supersymmetry or quantum ﬁeld theory.

4.1.1 Concepts in String Theory and the Kaluza-Klein Model

Start by assuming that the fundamental building blocks are one-dimensional objects called “strings.” It is called an “open” string if it has end points, and a “closed” string if it is a loop. A particle moving through space sweeps out a path called a “world line,” whereas a string sweeps out a 2D surface called a “world sheet.” As

95 Figure 4-1: A world line is replaced by a world sheet in string theory. This ﬁgure is inspired by one in Magdalena Larfors’s dissertation [41].

shown in Figure [4-1], in string theory, point particles are replaced with strings, and world lines are replaced with world sheets. We can parametrize a path in terms of timelike and spacelike coordinates (⌧,).

View the world sheet as a fundamental geometric object, and xn as ﬁelds living on the world sheet, which can be used to describe embedding and motion of strings in spacetime for 2D theory. By quantizing the action of 2D theory, we can get the quantum mechanical description of vibrating string dynamics and the associated spectrum. The fundamental particles in the standard model correspond to the oscillation modes of the string (more speciﬁcally, each oscillation mode is an eigenstate of the energy). Thus, the oscillation modes of the fundamental string can give rise to the particles of that standard model. In the low energy limit, one can only reach fundamental modes.

AgoodintroductiontotheconceptsbehindstringtheoryisexaminingtheKaluza Klein concept. It is good to think of the appearance of 4D gauge bosons as components of a metric tensor in a higher dimension spacetime. Consider a 5D spacetime M S1 4 ⇥ with G where M,N = 0, 1,...,4 .Weareinthelowerenergylimitofthe4D M,N { } { } 1 theory, as in a lower energy than the compactiﬁcation scale MC = R .

96 For µ, ⌫ = 0, 1, 2, 3 , { } { }

G G ,4Dgraviton MN ! µ⌫ G A ,4Dmasslessgaugeboson (4.1) µ4 ! µ G ,4Dmasslessscalar 44 !

The di↵eomorphism invariance in the ﬁfth dimension is what allows for the gauge invariance of the 4D vector boson Aµ,whichisaU(1) gauge boson.

If we want to generalize this system to d dimensions, we start by taking the dim(4 + d)spacetimeoftheformM X .Themetricin(4+d)dimensionsgives 4 ⇥ d rise to the 4D metric and gauge bosons associated with the isometry group of Xd,or µ the set of bijective isometries from Xd to itself. In this case, if we let ka be a set of a N Killing vectors in Xd, the 4D gauge bosons would be obtained using Aµ = GMNka . The pitfall with the Kaluza Klein approach is that it is extremely dicult to construct manifolds that have an isometry group that would yield the particles in the standard model.

We now shift to a new idea called the Brane-World concept, in which macro- scopically large extra dimensions exist, but the standard model does not propagate through them with the exception of gravity. The standard model essentially lives on a “brane,” the generalization of a membrane in higher dimensional spacetime, while gravity propagates through the bulk of spacetime. This concept is illustrated in Figure [4-2].

In this case, we take the string scale to be M 1018GeV[32]. s ⇡ Now, let’s delve into the di↵erences between a string versus a superstring. In the case of spacetime coordinates being viewed as ﬁelds that live on the world sheet, there are two main issues. First of all, the ground states for open and closed strings have imaginary mass, which are tachyons, thus rendering the theory unstable. The second issue is that the states in the spectra are bosonic, and we clearly need fermions to produce the standard model. The solution to these issues is using superstring theory,

97 Figure 4-2: The brane-world concept, inspired by a diagram in Angel Uranga’s notes [32]. since it introduces fermionic ﬁelds on the world sheet. In this thesis, the speciﬁc type of string theory we will be using is Type IIB.

For basic string theory, the idea is that the elementary particles are 1D extended objects, or strings. The oscillation modes of unique types of strings correspond to di↵erent particles (e.g. di↵erent quantum numbers).

Vacuum state: 0 ,scalar | i! First excited state: aµ 0 A ,vector (4.2) | i! µ Second excited state: aµa⌫ 0 G ,tensor | i! µ⌫

The mass of the corresponding particles goes up with the number of excited oscillator modes.

4.1.2 Open Versus Closed Strings

In the conformal gauge, selecting X+ = ⌧ and a momentum density P + that is constant along the string. For a D dimensional system, the independent degrees of freedom are the set of coordinates Xl(,⌧ )wherel = 1, 2,...,D 2 . { } 98 For an open string, the action of a free 2D ﬁeld is

T S = d2((@ Xl)2 (@ Xl)2)(4.3) 2 ⌧ Z 1 1 Using the Regge slope ↵0 = 2⇡T = 2 , S becomes

1 S = d2((@ Xl)2 (@ Xl)2)(4.4) 2⇡ ⌧ Z To obtain the equations of motion for the string, we deﬁne the boundary conditions in .Weknowthatopenstringshave2freeends,sowehavethefollowingboundary conditions:

l l Neumann boundary condition: @X (⌧,0) = @X (⌧,⇡) (4.5) Dirichlet boundary condition: Xl(⌧,0) = Xl(⌧,⇡)=d

where in Dirichlet boundary condition, d is a constant. Consider the Neumann boundary condition only for now. Let’s start by Fourier expanding Xl.

1 Xl = xl + pl⌧ + i ↵l ein⌧ cos n (4.6) n n n=0 X6 l l l l where ↵n = pnaˆn and ↵ n = pnaˆn† . Here, x and p are normalized zero modes corresponding to the position and momentum of the string’s center of mass. Now, let’s impose the condition that

l J i ij [@ X (,⌧ ),X (0,⌧)] = ( 0)(4.7) ⌧ ⇡

This condition is satisﬁed when

[xl,pJ ]=ilJ (4.8) [↵l ,↵J ]=n lJ n n0 n+n0,0

99 where p is the transverse momenta. The commutation relation in equation (4.7) enabled us to quantize 2D free fields in a finite volume. The integers denote the occupation numbers for the infinite set of oscillator modes. The Hamiltonian for this system has the form

H = p~ 2 + N + c (4.9)

l l where the number operator is given by N = n1=1 ↵ n↵n and c is an ordering l constant. Thus, the states are labeled as p , Nni P. | { }i + + + Deﬁning the gauge choice as X = p ⌧, p is conjugate to the light cone time x

H + 2 and is given by p = p+ .Thisdeﬁnitionalsogivestherelationsp p = p~ + N + c 2 l and M = N + c.Thus,wecanwritetheﬁrstexcitedstateasA = ↵0 1 p~ with | i | i 2 mass mA =1+c.Itisclearfromthislastrelationthattheorderingconstantisgiven by c = 1. Since we found the value of c, we know that the lowest state is a tachyon.

T (p~ ) = p~ , 0 p~ (4.10) | i | { }i ⌘ It should be noted A~ is a vector field with D 2components.Forasystemwith D dimensions, D 1degreesoffreedomcorrespondstoamassivefield,whileD 2 degrees of freedom corresponds to a massless field. Thus, A~ must be massless and c =1ifwewanttomaintainLorentzinvariance.Inordertoensurethisinvariance, we need to construct a full set of Lorentz generators, as described in reference [22]. This calculation yields D =26andc = 1. For a closed string, we utilize the Dirichlet boundary condition

Xl( + ⇡,⌧)=Xl(,⌧ )(4.11)

Performing a Fourier expansion, we obtain

l l l i 1 l 2in(⌧ ) l 2in(⌧+) X = x + p ⌧ + (↵ e +˜↵ e )(4.12) 2 n n n n=0 X6 100 The corresponding commutators for the closed string are

[xl,pJ ]=ilJ

[↵l ,↵J ]=n lJ (4.13) n n0 n+n0 [˜↵l , ↵˜J ]=n lJ n n0 n+n0

The Hamiltonian has the form

H = p~ 2 + N + N + b (4.14)

e l l where the number operators N and N are given by N = n1=1 ↵ n↵n and N = l l n1=1 ↵˜ n↵˜n. P e e P Since translations should not a↵ect states, we use the Noether procedure to obtain the generator of constant shifts in .

l l P = d@⌧ X @X Z (4.15) = N N e Equation (4.15) tells us that we need to impose the condition that N = N.Thus, we obtain that the lowest level state is given by T = p~ with mass m2 = b,and | i | i T e l l that the ﬁrst excited state is given by lJ =˜↵ 1↵ 1 p~ ,asindicatedinreference | i | i [22].

4.1.3 Open Versus Closed Superstrings

Consider an open superstring. We have a fermion l corresponding to coordinate Xl. This yields the fermionic action

1 S = d2i ¯l(@ ↵) l (4.16) 2⇡ ↵ Z 0 1 In two dimensions, the gamma matrices are given by = 2 and = i1.

101 Since in this basis, Dirac equation is imaginary, consider the fermions to be real

Majorana spinors. We now utilize the eigenfunctions of 3. It should be noted that in 4D, 3 = 3

l l = (4.17) 0 l 1 + @ A Where we are dealing with light cone coordinates on the world sheet ± = ⌧ = . ± Equation (4.16) becomes

1 2 l l l l S = d ( +@ + + +@+ )(4.18) 2⇡ Z Once again, we need boundary conditions at the ends of the string. We use the Euler Lagrange equation to ﬁnd them [22]. We know that surface terms have the form + + ,sotheboundarytermsmustgotozerowhen + = by ± inspection. Thus, ignoring the overall sign, at =0,wehave

l l +(0,⌧)= (0,⌧)(4.19)

And at = ⇡,

l (⇡,⌧)= l (⇡,⌧)(4.20) ± ± As indicated in reference [22], the fermions denoted with a “+” sign are referred to as being in the Ramond sector, while the “-” fermions are in the Neveu-Schwarz sector. For the Ramond sector, the mode expansions are given in equations (4.21). More- over, the Neveu-Schwarz sector mode expansions are given in (4.22)

l 1 l in(⌧ ) = dne p2 n Z X2 (4.21) l 1 l in(⌧+) = d e + p2 n n Z X2 102 l 1 l ir(⌧ ) = bre p2 1 r Z+ 2X2 (4.22) l 1 l ir(⌧+) + = bre p2 1 r Z+ 2X2

We employ the following anticommutation relation to quantize the system

l J lJ (,⌧ ) , (0,⌧) = ⇡( 0) (4.23) { ± ±} ±±

Which for the modes, yields the anticommutation relations

l J lJ br,bs = r+s { } (4.24) dl ,dJ = lJ { m n} m+n

For the Ramond sector, the Hamiltonian in the conformal gauge is given by

H = p~ + N↵ + Nd (4.25)

l l l l Where the number operators are given by N↵ = m1=1 ↵ m↵m and Nd = m1=1 d mdm. The zero modes in the Ramond sector are what giveP rise to spacetime fermions.P

For the Neveu-Schwarz sector, let’s start by replacing Nd with Nb = r1= 1 mb rbr. 2

These states are actually the eigenstates of the fermion number operatorsPbn† bn,dn† dn,..., for n =0witheigenvalues0and1. 6 In summary, when working with finite volume field theory (0 <<⇡), the mechanics are a bit di↵erent. Our superstring theory so far describes a bosonic field and a fermionic field. The fermionic field has two sector with their own independent Hilbert spaces. For a (D 2)-dimensional system, we have the anticommutation relation dl ,dJ = lJ obtained from equation (4.24) with m = n =0associatedwith { 0 0 } the orthogonal group O(D 2) [22]. In preparation for the work to developed over the next few paragraphs, let’s assume

103 D =10.WewilluseDiracmatricesintheO(8) group. O(8) would be comprised of

8-dimensional representations of two spinor representations, 8s and 8s0 and one vector representation, 8v. We start by labeling states as f˙ and f. Consider the expectation of the ladder

l operator d0

˙ l 1 l f d f = ˙ (4.26) h | 0| i p2 ff How exactly do we go about constructing states in terms of the creation and annhilation operators? Let’s start by thinking of O(8) as acting upon 8 coordinates xl,re-writingthemasfollows

z1 = x1 + ix2

z2 = x3 + ix4 (4.27) z3 = x5 + ix6

z4 = x7 + ix8

1 2 which corresponds the embedding of U(4) in O(8). We can deﬁne f 0 d +id for ⌘ 0 0 i j ij f ,f † = ,whichisthecommonly-knownanticommutationrelationforfermionic { } ladder operators, even though the Neveu-Schwarz sector deals with bosons. It should be noted that there are 2 out of 8 representations that can be formed using our new creation operator f † by acting on the ground state. The ﬁrst representation is one in which an even number of f †’s act on the ground state, whereas the other one is when there is an odd number.

1 2 1 2 3 4 ”Even” Representation: 0 ,f †f † 0 ,f †f †f †f † 0 ,... | i | i | i (4.28) 1 1 2 3 ”Odd” Representation: f † 0 ,f †f †f † 0 ,... | i | i

Finally, consider the closed superstring. Once again, our boundary conditions

l J now require periodic currents of the form + + [22]. The Neveu-Schwarz boundary

104 conditions are applied to right moving particles, and the Ramond boundary conditions are applied to left moving particles, which is allowed since the Lagrangian can be broken up into left and right moving fermion ﬁelds.

l l 2in(⌧ ) = dne n Z X2 (4.29) ˜l ˜l 2in(⌧+) = dne n Z X2

l l ir(⌧ ) = bre 1 r Z+ 2X2 (4.30) ˜l ˜l ir(⌧+) = bre 1 r Z+ 2X2

4.2 Introduction to Compactiﬁcation

The goal of this section is to introduce the reader to the mathematics behind the compactiﬁcation of higher dimensions. 4.2.1 follows a mix of Micheal Dine’s textbook on supersymmetry and string theory [22] and Anze Zaloznik’s seminar notes on Kaluza Klein theory [42]. 4.2.2 once again mostly follows Dine’s methods on string tori compacitiﬁcation, since I found it to be the most friendly and accessible [22], but also includes references to Florian Wolf’s dissertation [29], Sebastiaan Schotten’s dissertation [34], and Pierre Binetruy’s textbook on supersymmetry [21].

4.2.1 Compactiﬁcation for the Kaluza-Klein Model

This subsection aims to demonstrate that using Klein’s compactification of a fifth dimension into a small circle of radius R,aworldoffixeddimensionsexists,even though we observe four. Kaluza and Klein intended to show that five dimensional coordinate invariance gave rise to a four dimensional coordinate invariance and U(1) gauge invariance. Their goal was to unify electromagnetism and gravity.

105 We start by assuming ﬁve dimensional spacetime M 4 S1.Considera5dimen- ⇥ µ= 0,1,2,3 sional scalar ﬁeld with coordinates (x { },y), where 0 y<2⇡R,whereR is  the internal radius of the 1-sphere. Note that for section 4.2.1 only, I am distinguish- ing the Ricci scalar R from the internal radius R so as not to confuse the reader. Since y is periodic along the 1-sphere manifold, it can be expanded with Fourier modes. e

1 (x, y)= (x)eipny (4.31) p n n 2⇡R X

n where pn = R .Thus,weobtaintheaction

1 d4xdyL = d4xdy [()2 + M 22] 2 Z Z 1 (4.32) = d4x [@ 2 +(M 2 + p2 )2] 2 µ n n Z X

n Since R is very small, R is large, which represents the massive states. The argument here is that these massive fields can be integrated out, so that only the massless field is obtained. Therefore, the e↵ects of massive fields can be seen only in tiny, higher dimensional operators.

Now we are going to consider the Kaluza Klein theory. As stated previously, we examine five-dimensional Einstein gravity with the fifth dimension compactified. The Kaluza-Klein Lagrangian is given by

1 L = pgR (4.33) 22 e where 2 =16⇡G.

The inﬁnite number of massive states representing the modes of the 5D metric,

106 the components of the metric are identiﬁed as

Tensor: gµ⌫

Vector: gµ4 (4.34)

Scalar: g44

The massless states form modes independent of y.Wecanre-writethe5Dfieldas a4Doneusingagaugefield.Thetransformationofthefifthdimensionisoftheform x4 = x4 + ✏4(x). So, the coordinate invariance associated with this transformation is

gµ4 = gµ4 + @µ✏4(x)(4.35)

which looks like a gauge ﬁeld! Now, let

gµ⌫ = gµ⌫

gµ4 = Aµ (4.36)

2(x) g44(x)=e

We can proceed to substitute the relations in (4.37) back into the 5D action by working out the Christo↵el symbols and Reimann curvature tensor components, which is not shown for clarity. This lengthy procedure was demonstrated in Chapter 1. The result obtained is

2⇡R 1 2 L = pge R + e F (4.37) 22 4 µ⌫ e where R is the radius of the compact manifold, R is the four-dimensional curvature scalar, and F = @ A @ A . As a consequence, we have obtained a theory of µ⌫ µ ⌫ ⌫ µ e gravity in four dimensions and an Abelian gauge ﬁeld, which were derived from the same initial 5D Einstein gravity. As a quick aside, we should write out Fµ⌫ explicitly.

107 We start by considering the form of the ﬁve dimensional metricg ˆµ⌫.

gµ⌫ gµ4 gˆ = (4.38) µ⌫ 0 1 g⌫4 g44 @ A where the ﬁfth dimension is compactiﬁed.

Now, consider the set of basis vectors e and e for µ = 0, 1, 2, 3 ,forwhiche µ 4 { } 4 corresponds to the ﬁfth direction. It is important to note that e = g =0,which µ· µ4 6 means that we can break apart eµ into terms that are orthogonal to e4 and terms that are not. Thus, we have eµ = eµ + eµ and eµ e4 =0.Sincee4 is parallel to ? || ? · gµ4 eµ ,werecognizethateµ = e4.Itisnowpossibletousethislastrelationtowrite || || g44 the components of gµ⌫ as follows

gµ4g⌫4 gµ⌫ = eµ e⌫ = ⌘µ⌫ + (4.39) · g44

where ⌘µ⌫ is our regular 4D Minkowski metric.

We now deﬁne

gµ4 Bµ ⌘ g44 (4.40) g ⌘ 44

Substituting equations (4.41) and (4.42) into equation (4.39) yields the metric tensor.

⌘mn +BmBn Bm gˆ = (4.41) mn 0 1 Bn @ A Now, let’s consider a constraint on the system. We utilize what is called the “cylinder condition,” which accounts for the fact that the ﬁfth dimension is compact- iﬁed and unobservable, and thus does not interact with the other metric components.

108 This can be written as

@4gˆ↵ =0 (4.42)

where ↵ = 0, 1, 2, 3 and = 0, 1, 2, 3 . { } { } Under the cylinder condition, consider an inﬁnitesimal translation in the direction of the ﬁfth dimension.

µ µ x0 x ! (4.43) 4 4 x0 x + ✏(x) !

Applying this transformation to the metric along with the cylinder condition yields

@x⌫ @x4 @x4 @x4 0 gµ4 = µ 4 g⌫4 + µ 4 g44 (4.44) @x0 @x0 @x0 @x0

Thus, we can identify that

Bµ0 = Bµ + @µ✏ (4.45)

If we postulate that the extra dimension is geometrically circular, we can identify Bµ as the electromagnetic potential because the form of gauge transformations in electrodynamics are locally U(1). Essentially, any shift in x4 corresponds to an

Abelian gauge transformation of Bµ. From Chapter 1, we have the general equation of the Ricci scalar, re-written in equation (4.47) for clarity. We proceed to use the Christo↵el symbols to obtain the 5-dimensional Ricci scalar in equation (4.48).

R = g↵(@ @ + )(4.46) ↵ ↵ ↵ ↵ e

1 µ⌫ 2 µ R5D = R Fµ⌫F @µ@ p (4.47) 4 p e e 109 where we have our explicit value of Fµ⌫

F = @ B @ B (4.48) µ⌫ ⌫ µ µ ⌫

It is worth noting that in our original Lagrangian for the Kaluza Klein given

2(x) in equation (4.38), we have set g44 = e ,whereasinequation(4.48),g44 is an arbitrary scalar field . This result in equation (4.47) is very encouraging because it contains components corresponding to the electromagnetic and gravitational fields. Going back to equation (4.38), if we perform a Weyl rescaling of the metric to obtain the conformal transform gµ⌫, one will find can be treated as a modulus and nothing sets its expectation value. Quantum e↵ects actually generate a potential for even at one loop correction, but vanishes as the radius becomes large [22].

4.2.2 Strings on Tori

We start by compactifying one dimension X9 on a circle of radius 2⇡R.Thiscondition requires states to be invariant for translations of 2⇡R,thusallowingustoquantize

9 n momenta p = R . Now, we employ a new feature from our field theory based procedure. Due to identification of points, string fields X9 does not have to be strictly periodic, which allows us to write X9 using a mode expansion.

9 9 9 i 1 9 in(⌧ ) 9 in(⌧+) X = x + p ⌧ +2mR + (↵ e +˜↵ e )(4.49) 2 n n n n=0 X6 where m Z. The states for which m = 0 are referred to as ”winding modes,” 2 6 which correspond to the string wrapping extra dimensions. The mass operator gives

9 n2 2 2 acontribution(p )= R2 and another contribution m R from windings if there is no momentum. These two contributions respond di↵erently to value of R.IfR is small,

n2 the term R2 will be large and becomes heavy. If R is large, then the winding term m2R2 becomes large and therefore very heavy. This relation produces a symmetry between large and small radii with respect to when the R is below or above the string scale.

110 Now we proceed to integrate the winding term in the superstring theory developed in the previous sections. We separate X9 by left versus right moving ﬁelds, which is described by the following equations. By looking at equation

9 9 x n i 1 9 in(⌧ ) X = + + mR (⌧ )+ ↵ e L 2 2R 2 n n ! n=0 X6 (4.50) 9 9 x n i 1 9 in(⌧+) X = + mR (⌧ + )+ ↵˜ e R 2 2R 2 n n ! n=0 X6

As we can see in equation (4.50), the momentum now will include the winding term and because we are dealing with a closed string, we will have a left and right moving momenta, which then we proceed to deﬁne as

n pL + mR ⌘ 2R (4.51) n p mR R ⌘ 2R

As a result, the mass operators become

1 2 L0 = pL + N 2 (4.52) 1 L˜ = p2 + N˜ 0 2 R

Thus, the process is completed as we compactify on the product of circles with coordinate Xl.Themomentainequations(4.51)become

l l n l l pL l + m R ⌘ 2R (4.53) nl pl mlRl R ⌘ 2Rl

For informational purpose and completeness, Dine provides the student with a

111 ﬂavor of the richness of the theory by providing the spectrum of 10D gauge bosons for a heterotic string with O(32) symmetry unbroken [22].

AB A B AM = 1 1 M 1 p (4.54) | i 2 2 2 | i

To my amazement, Dine mentions that this decomposes into a set of 4D gauge bosons in the conformal gauge to the subscript value M = 2, 3 and 6 scalars to { } M = L.Thegraviton,scalar,andantisymmetrictensorfieldsdecomposeasaset of fields glJ , BlJ , gµi, Bµl, gµ⌫, bµ⌫,and. Note that gµi and Bµl are vectors, and that gµ⌫, bµ⌫,and correspond to the 4D graviton, antisymmetric tensor, and scalar field respectively [22]. The reason I included these specific details is to highlight that the theory includes a set of particles and force carriers that unifies gravity with other tensor and scalar fields, which is what we would wish to obtain from a high energy unified theory to be used in the early universe.

For a model including fermions, we use the conformal gauge and O(8) spinors. Instead of using 4 ladder operators to create representations as we did in the previous section, we group our creation operators as ai=1,2,3 and b, and annhilation operators

i as a † and b†.

In the previous section, for a ladder operator a,8s would have been given by i j 1 2 3 4 0 ,a†a † 0 ,a †a †a †a † 0 ...,aswritteninequation(4.28)usingaslightlydi↵erent | i | i | i notation (operators are labelled as f instead of a). However, now that we are deal-

i j ing with strings on a torus, the representation would now be given by 0 ,a†a † 0 , | i | i j 1 2 3 b†a † 0 , b†a †a †a † 0 ,.. We now have 4 states with no b’s, and 4 states with one | i | i b,whichindicatesthatthesetwogroupshaveoppositehelicityandcorrespondto transformations under O(6).

Since O(6) is isomorphic to SU(4), we have that 8s =4+4.¯ Furthermore, we can decompose this further by noting that SU(3) transformations are a subgroup of

SU(4), thus giving us the representation 8s =3+3+1+¯ 1.¯ This fact is signiﬁcant because we can clearly see the embedding of SU(4), SU(3), and U(1) in our theory, which is what we would need to produce a theory that yields the symmetry of the

112 standard model.

There is a moduli term that is a result of the torus compactiﬁcation. This term comes from the diagonal metric components and are related to the internal radii. Conceptually, in the string solutions for Rl, there can be o↵-diagonal matrix entries, which are not be dealt with in the context of this thesis. This is a result of non-ﬁxed internal radii and the geometric fact that the torus is not strictly required to be a product of circles. One can learn more about the subject of moduli stabilization in Florian Wolf’s dissertation [29] and Sebastiaan Schotten’s dissertation [34].

In the previous paragraphs, we have been describing compactiﬁcation in one direction. In this section, we consider compactiﬁcation using an N-dimension torus. Starting the basis vectors el where a = 1, 2,...,N ,wehave a { }

l l a l X = X +2⇡n ea (4.55)

l where ea must statisfy

l l e˜aeb = a,b (4.56)

which corresponds to the dual space of the original lattice in the previous section

l with unit vectore ˜a.

The momenta for the N-dimensional torus is given by

l a l p = n e˜a (4.57)

Likewise, the windings for an N-dimensional torus are given by

l a l w = m ea (4.58)

Both of these quantities depend on the basis vectors that characterize the general torus. For completeness, the momenta pl can be decomposed into left-moving and

113 right-moving momenta.

l l p l pL = + w 2 (4.59) pl pl = wl R 2

It should be noted that throughout our use of string theory for building inflationary models, we will be dealing with Type II theories, which encompasses general compactifications with N = 8 SUSY. On the other hand, the Type II heterotic case encompasses N = 4 for four dimensions, which implies a torus with di↵erent dimensions. Additionally, we have moduli contribution from Wilson lines. When dealing a gauge spacetime lattice with lattice spacing a,wewanta to be small, since this enables Lorentz invariance. A small a corresponds to weak QCD gauge coupling. For small lattice spacing calculations, Green’s functions can be approximated using perturbation theory. Moreover, as the lattice gets smaller, more computational power is needed to obtain large scale quantities. In order to preserve gauge invariance and reduce the required computational resources, we work with a set of variables called “Wilson lines,” which contribute the additional moduli in our case. Wilson essentially used these objects to discretize path integrals [21]. Recalling the introductory Kaluza Klein model with a circle compactification, we obtained a Lagrangian that included amodulus in equation (4.37). Similarly, in the toroidal case, these moduli are constant gauge fields, which is approximately the gauge transformation given by

l ix Al l ix Al A = ie l @ e l (4.60) l = ig@ g†

For clarity, Dine proposes redeﬁning charged ﬁelds = g0.

l 2⇡iR Al 0(X )=e l 0 (4.61)

114 The conceptual point of this last equation is that the lattice can be thought of using background ﬁelds, which we will see in the following section.

4.3 Axion Monodromy Inﬂationary Models

This section essentially introduces the string theory based inflationary models, otherwise known as axion monodromy. Section 4.3.1 covers torus string compactification with and without flux and closely follows the work done in Sebastiaan Schotten’s dissertation [34]. 4.3.2 introduces axion monodromy and Dp-branes. This section utilizes amyriadofreferences,includingSebastiaanSchotten’sdissertation[34],PierreBi- netruy’s textbook on supersymmetry [21], Yi-Fu Cai’s et al.’s paper [30], and Takahiro Terada’s paper [37]. Other references that I used to gain an understanding of axion monodromy include Bailin and Love’s textbook on cosmology and gauge field theory [28], Jose Edelstein’s notes [31], Aitor Landete’s paper [33], Florian Wolf’s dissertation [29], and Tim Wrase’s video lecture [35]

4.3.1 Inﬂationary Model Using 6-Torus Compactiﬁcation

Compactiﬁcation Without Flux

To further illustrate the concept of compactiﬁcation, we will go over the next few paragraphs consider the e↵ect of the gauge ﬁelds on the action. We consider an extremely simple toy model heavily based upon reference [34]. Start with qualitative analysis of the following six dimensional action

SEHM = SEH SM (4.62) = dx6p g (M 4R M 2 F 2) 6 6 6 6 | | Z

where R6 is the six-dimensional curvature scalar, M6 is the six-dimensional reduced

Planck mass, F is the six-dimensional Maxwell ﬁeld, and g6 is the determinant of the six-dimensional spacetime metric. We label the action with the subscript EHM, which

115 stands for Einstein-Hilbert-Maxwell. The action is divided into two components, the Einstein-Hilbert term and the Maxwell term.

The ﬁrst case we want to look at is when there is no ﬂux. F =0,sotheportion of the action with the Maxwell term drops out. so for SEH.Weareleftwith

S = M 4 d6xp g R (4.63) EH 6 6 6 Z We are interested in obtaining a four-dimensional theory. As a result, two dimensions must be compactiﬁed. This factorizes the manifold into M 4 M ,whereg is ⇥ g the genus, or the number of holes in a manifold. For clarity, Schotten provides an

2 example where he states that “M0 corresponds to the two-sphere S , M1 corresponds to the two-torus T 2,etc”[34].

2 Consider a compactiﬁcation on manifold Mg with “volume” R ,whereR is a modulus scalar ﬁeld dependent upon coordinates xµ.Thismodulusentertheanalysis through the metric. Assuming the metric can be separated into a 4-dimensional spacetime metric, we have

2 M N ds = gMNdx dx

µ ⌫ 2 i j = gµ⌫dx dx + R (x)˜gijdy dy (4.64)

µ ⌫ i j = gµ⌫dx dx +ˆgijdy dy

Examination of equation (4.64) gives us with two additional deﬁnitions– the “physical” metricg ˆ R2(x)˜g and the “unit” metricg ˜ on M . With respect to the unit ij ⌘ ij ij g metric, we have the condition

d2y g˜ =1 (4.65) Mg Z p

where Mg is our manifold of genus g. Regarding the “physical” metric, it should

116 be noted that

gˆ(x) det(ˆg (x)) ⌘ ij 2 =det(R (x)˜gij) (4.66) = R4(x)˜g

Substituting the result of equation (4.66) into d2y gˆ(x)yields Mg R p

d2y gˆ(x)= d2y R4(x)˜g Mg Mg Z p Z p = R2(x) d2y g˜ (4.67) Mg Z p = R4(x)

where we substituted equation (4.65) in the second line of (4.67).

At this point, we can separate the integrand in the for SEH into an internal compact part and non-compact part. The reason we do this is in preparation for integrating out the compact portion. In order to separate the six-dimensional spacetime, we start with factorizing the g6 term as follows

p g = g gˆ(x) 6 4 = p g gˆ(x) (4.68) 4 = p g Rp2(x) g˜ 4 p 117 The R6 term in equation SEH becomes

MN R6 = RMNg

P MN = RMPNg =(@ P @ P +P Q +P Q )gMN P MN N PM PQ NM NQ PM 2 1 µ 2 µ = R + R (x)R 4R (x)g R(x) 2R (x)g ( R(x))( R(x)) 4 2 rµr rµ r (4.69)

It is encouraging to see the four-dimensional curvature scalar R4 terms. As pointed out by Schotten in [34], the last two terms in (4.69) can be written as total derivatives, which means that they will vanish upon integration. Now, let’s substitute equations (4.68) and (4.69) into (4.63), while dropping the terms that vanish. The desired action becomes

4 4 2 2 2 S = M d xd yp g R (x) g˜[R + R R ] EH 6 4 4 2 (4.70) Z p = M 4 d4xp g [R2(x)R + M ] 6 4 4 g Z

where Mg is the so-called “Euler characteristic.” This term is a topological invariant of the manifold Mg.

Note that V2 is the volume of compact space, so that the Planck mass in four dimensions is given by

2 4 MP = M6 V2 (4.71) 4 2 = M6 R (x)

Since in our exercise, we are going from six to four dimensions.

We want to pull the x independent terms out of the integral. Let’s redeﬁne

118 M 2 M 2 = M 4R2.Thus, P ! P 6 0

2 4 2 4 R (x) 1 SEH = M6 R0 d xp g4 2 R4 + 2 Mg " R0 R0 # Z (4.72) R2(x) 1 = M 2 d4xp g R + M P 4 R2 4 R2 g Z " 0 0 #

Since the R4 term is multiplied by a function of a scalar, it is desired to bring this action into the Einstein frame. To achieve this, we use the transformation

g h = R2(x)g (4.73) µ⌫ ! µ⌫ µ⌫

This transformation yields the relations in equations (4.75) and (4.76).

4 p g = R (x)p h (4.74) 4

R = R2(x)R(h) +6hµ⌫R(x) (h) (h)R(x) 12hµ⌫( (h)R(x))( (h)R(x)) (4.75) 4 4 rµ r⌫ rµ r⌫

After performing the integral, we obtain

2 M S = P d4xp h[R(h) V (r)] (4.76) EH R 4 0 ! Z Where

(M ) V (R)= g R4(x) (4.77) 2g 2 = R4(x)

Where (M )=2g 2. Equation (4.78) is our inﬂationary potential for this g model! The highlighting characteristic of this potential is its dependence on the genus

119 of the manifold.

Compactiﬁcation With Flux

Now, consider the same model as before, but with a nonzero value of FMN in the

Maxwell term of equation (4.62). FMN is called the “Maxwell field,” and is responsible for a flux permeating the compact manifold Mg. An additional comment on the nonzero flux is that the flux is quantized in part due to the Dirac condition.

1 F = F dyi dyj = n (4.78) 2 ij ^ ZMg ZMg Clearly, n is an integer that corresponds to the number of units of magnetic ﬂux. We now focus only on the Maxwell term of the six-dimensional Einstein-Hilbert- Maxwell action given in equation (4.62). The factorized Maxwell ﬁeld can be written in the following form

F 2 = F F MN | | MN µ⌫ ai bj = Fµ⌫F + Fijgˆ (x)ˆg (x)Fij (4.79)

µ⌫ 4 ai bj = Fµ⌫F + R (x)˜g g˜ FijFab

µ⌫ where Fµ⌫F is the four-dimensional term. Using the Dirac quantization condition in equation (4.78), we obtain an expression that can be used to work on the two-dimensional component of equation (4.79). This equation is

1 2 ij n = d y gF˜ ij✏ (4.80) 2 Mg Z p It should be noted that since F = F and ✏ ✏ij =2,thenF = n✏ . ij ji ij ij ij 120 Substituting this back into SM yields

4 2 µ⌫ 4 ai bj S = M d xd yp g [F F + R (x)˜g g˜ F F ] M 6 6 µ⌫ ij ab Z 2 (4.81) R(x) 2n2 = M R d4xp g F F µ⌫ + P 0 4 R µ⌫ R4(x) Z 0 ! " #

2 MP where M6 = R2(x) . Just as before,q we employ the transformation g h = R2(x)g used to get µ⌫ ! µ⌫ µ⌫ from equation (4.73) to (4.77), which yields

1 2n2 S = M R d4xp h F F µ⌫ + M P 0 R2(x)R2 µ⌫ R4(x) Z 0 !" # 2 (4.82) M R 1 2n2 = P d4xp h 0 F F µ⌫ + R M R2(x) µ⌫ R6(x) 0 ! Z P !" #

Now, using the same result for SEH as given by equation (4.77) in the previous section, we substitute the values of SM and SEH into SEHM.

S = S S EHM EH M 2 M R 2g 2 n2R = P d4xp h R(h) 0 F F µ⌫ 0 R 4 M R2(x) µ⌫ R2(x) M R6(x) 0 ! Z " P P # (4.83)

Using simple inspection, we can identify the last two terms in equation (4.84) as the inﬂationary potential V (r).

2 2g 2 n R0 V (r)= 2 + 6 (4.84) R (x) MPR (x) which is clearly dependent upon ﬂux as well as genus. This important result potentially relates compact dimensions to cosmological inﬂation!

121 4.3.2 Inﬂationary Model Using Dp-Branes and Axions

As mentioned before in Section 4.1, when one solves the equation of motion for a relativistic string, we either have the Neumann condition or the Dirichlet condition. In the case of an open string, it must satisfy the Neumann boundary condition in p spatial directions. In the case of a closed string, it must satisfy the Dirichlet boundary condition, so that the endpoints become restricted to move in a p dimensional hyperplane. This hyperplane is called a “Dp-brane.” Dp-branes ensure momentum conservation for the string-brane system [34].

In this section, in order to find our inflationary potential, we want to show T- duality by compactifying bosonic string theory in one periodic direction of size R.In the one dimensional case, consider the momenta from the previous section given by equations (4.53). They are symmetric under the R 1 transformation, which is ! 2R T-duality [21]. Essentially, it means that we can consider the internal radius to be ar- bitrarily small. However, in the case of the torus, applying the duality transformation to a compactified direction X yields

XL XL ! (4.85) X X R ! R

For an open string, the Neumann boundary condition for left and right moving bosons is originally given by equation (4.87), but becomes the relation in (4.88) after a T-duality transformation. It should be noted that essentially, the Neumann boundary condition becomes the Dirichlet boundary condition.

@⌧ X =(@+ + @ )X =0 (4.86)

@X =(@+ @ )X =0 (4.87)

Under T-duality, the spectrum states are invariant. A similar phenomenon occurs

122 under simultaneous changes as compactiﬁcation occurs on large and small circles. The associated spectra for each case are equivalent. In this section, we present the action of a Dp-brane. The intent is not to derive it, but rather, provide a conceptual overview based upon the work on the previous section on compactiﬁcation. The action of the Dp-brain is generally given by

1 p+1 SDp = p+1 d ⇠e det(G↵ + B↵ +2⇡↵0Fab) (2⇡)p(↵ ) 2 0 Z q (4.88) = dp+1⇠ det(G + B +2⇡↵ F )T ↵ ↵ 0 ab Dp Z q where g e is a modulus where is a dilaton field to be discussed shortly, F s ⇠ ab is the world-volume (higher-dimensional equivalent of a world sheet) of the familiar gauge field, ⇠ represents the coordinates, and TDp is defined to be the brane tension. ↵ 0, 1, 2,...,p+1 and 0, 1, 2,...,p+1 are the world-volume indices. G is 2{ } 2{ } ↵ the induced world volume metric. It is actually a (p +1)by(p +1)matrix,andis given by equation (4.89).

M N G↵ = GMN@↵X @X (4.89)

On the topic of dilaton fields, compactification introduces the so-called “moduli” into the theory. A moduli is a scalar field with a potential that is flat over a large region of field space. The role of moduli in string theory is that they govern the shape and size of the compact manifold and the strength of the 4-dimensional coupling constant via the dilaton scalar e mentioned in the previous paragraph.

Going back to equation (4.88), the last unidentiﬁed term is B↵,whichhasanin- duced Neveu-Schwarz–Neveu-Schwarz (NS-NS) 2-form. This is given by the equation (4.90) as documented in Schotten [34].

M N B↵ = BMN@↵X @X (4.90)

Which ﬁnally allows us to introduce the deﬁnition of an axion. Of interest to the

123 work in this thesis, axions arise by integrating the NS-NS 2-form ﬁeld B2,suchas the one identiﬁed in equation (9.40). Of interest to the reader, axions can also arise by integrating other quantities, such as the R-R p-form potential, which will not be addressed in this document. For more detail, see references [28], [31], and [33].

For example, when writing B2 as

l B2 = bl(x)!2 (4.91)

J J J where bl is the axion and !2 is 2-form such that (2) !2 = ↵0l ,wecanobtain ⌃1 the axion bl by integrating the two-form ﬁeld over theR two-cycle, so that it yields the condition given by (4.92)

B2 = ↵0bl(x)(4.92) ⌃(2) Z 1 In turn, equation (4.92) is used to evaluate the integral

S = dp+1⇠ det(G + B )T (4.93) Dp ↵ ↵ Dp Z q Now, let’s assume B2 only has nonzero components in the direction of the 2-cycle. Working through the intermediate steps as documented in Schotten’s appendix A.2.3 [34], we have

1 SD5 = 5 3 e det(G↵ + B↵) (2⇡) (↵0) M4 ⌃2 Z ⇥ q 1 p = 5 3 g4e det(G5 + B5) (2⇡) (↵0) M ⌃ Z 4 Z 2 q (4.94) 1 G44 B45 + G45 p = 5 3 g4e vdet (2⇡) (↵0) M ⌃ u 0 1 Z 4 Z 2 u B45 + G45 G55 u t @ A 1 4 2 2 2 2 2 p = 5 3 d x g4e (↵0l ) +(↵0b) +(↵0l(45)) (2⇡) (↵0) Z q where 4, 5 and 4, 5 are both indices in the 2-cycle direction. As 2{ } 2{ } Schotten points out, in order to obtain the desired results, he uses the relation

124 2 2 G = G = ↵0l as length squared for both sides of a 2-cycle. G = ↵0l ⌃2 44 ⌃2 55 ⌃2 45 (45)

RrepresentsR any possible o↵-diagonal terms in the matrix G↵ [34]. R

If we assume G45 =0,weobtain

1 4 2 2 2 p SD5 = 5 3 d x g4e (↵0l ) +(↵0b) (4.95) (2⇡) (↵0) Z p Which after some mathematical manipulation, yields a term that can be once again identiﬁed as the inﬂationary potential by inspection.

1 3 p 2 2 V4D, b-axion = e 5 2 l + b (4.96) (2⇡) (↵0) It should be noted that for b l2,thee↵ectivepotentialislinearinb.Inorder ⌧ to summarize what we have just learned, let’s consider the form of the corresponding Lagrangian of the canonically normalized inﬂaton ﬁeld '.

1 L = (@')2 c ' (4.97) 2 1

where c1 is a constant given by

1 1 c1 = 5 2 (4.98) f (2⇡) (↵0) for f is a constant that corresponds to the axion decay, as documented by Cai et al [30]. In this speciﬁc exercise, the e↵ective potential is simply a linear term! A literature search shows many variants of the potential that can arise from axion monodromy, as stated in [30]. The general form is given in terms of exponent p and mass term m.

1 2 4 p p L = (@') m ' (4.99) 2 where the last term in the Lagragian is the potential, which can be written in its most general form [37] as

n V = cn (4.100)

125 We now use the equations for a modiﬁed version of the slow roll parameters given by Terada [37]. Note that they have the same form as our usual slow roll parameters, but slightly di↵erent constant coecients.

2 1 V ✏ = 0 2 V ! (4.101) V ⌘ = 00 V

Substituting equation (4.104) into equations (4.105) and doing some simple algebra yields the our inﬂationary model’s slow roll parameters.

n2 ✏ = 2 (4.102) n(n 1) ⌘ = 2

In summary, we have discussed how axions in string theory arise by using wrapped branes and, in this case, integrating B2-ﬂuxes over a nontrivial cycle on the compact manifold. Monodromy combined with suitably wrapped branes are used to develop an unbounded ﬁeld range, which leads to a potential that tends towards a pure power law.

126 Chapter 5

Observational Constraints for Di↵erent Inﬂationary Models

Overall, the objective of section is to provide the final details required to conduct a quantitative comparison between quadratic inflation, SUGRA D-Term inflation, and axion monodromy inflation. The subsections essentially walk the reader through the concepts and derivation of the scalar to tensor perturbation ratio, as well as the spectral index, both of which are required to compare our inflationary models to publish CMB data. A large number of references were used to write this section, including Daniel Baumann’s notes on inflation [8] and [48], Prof. David Kaiser’s informal primers [19] and [47], Sebastiaan Schotten’s dissertation [34], Takahiro Terada’s paper [37], Alberto Vasquez’s paper [38], and Masahide Yamaguchi’s paper [39].

5.1 Tensor to Scalar Perturbation Ratio and the Spectral Index

5.1.1 Overview of the Procedure for Comparison to Data

In this chapter, the theoretical predictions for di↵erent inﬂationary models parameterized by a single scalar ﬁeld potential V ()arecomparedwithcurrentobserva- tional data on the ns versus r plane, where ns is the scalar spectral index and r is

127 Figure 5-1: Quantum fluctuations as the scalar field rolls down the potential during inflation. This figure is inspired by one in Daniel Baumann’s lecture notes [38]. the tensor-to-scalar ratio of the amplitudes of the power spectra. This connection between theoretical and observational cosmology makes it possible to evaluate the viability or at least to constrain the inflationary model under investigation [38].

Consider a scalar field rolling down a potential as shown in Figure [5-1]. During inflation, as the scalar field rolls down the potential, the classical background evolution is slightly perturbed by the e↵ects of quantum fluctuations. These fluctuations may introduce local time delays and a↵ect the end of inflation, which is marked as end in the figure. It should be noted that for di↵erent parts of the universe, inflation will end at slightly di↵erent times, as stated in reference [38]. As a result, di↵erent parts of the universe may not reach end at the same time. This fact implies slightly di↵erent evolutions, which in turn induces the relative density fluctuations ⇢(t, x).

It should be pointed out that the statistics of quantum fluctuations of the inflaton are approximately Gaussian and scale invariant [37]. During inflation, these fluctuations are expanded, which causes a local variation that ultimately results in a density

128 perturbation. These fluctuations are actually viewed as the seed of density perturbations. These density perturbations are divided into two types- tensor perturbations and curvature perturbations. Curvature perturbations occur when the inflaton fluctuations are converted to the scalar part of the metric. On the other hand, tensor perturbations occur when inflaton fluctuations are converted to fluctuations of the transverse traceless part of the metric.

5.1.2 Metric, Matter, and Curvature Perturbations

This section is heavily based upon Daniel Baumann’s notes [28] and two of Prof. David Kaiser’s informal primers [47] and [19]. So far, in this thesis, we have been considering the homogeneous portion of the inflaton. However, as stated in the previous paragraph, during inflation, the inflaton experiences quantum fluctuations. Since during inflation, the inflaton dominates energy density, when the universe becomes matter-dominated well after inflation, the perturbations induce matter perturbations. These perturbations are seen as temperature fluctuations in the CMB, as shown in Figure [5-2]. The aim of this section is to provide a brief overview on how to identify the correlation functions of the primordial perturbations [19]. First, we want to consider a density perturbation and separate terms that are purely time dependent from terms that have both spatial and time dependence. X(t, ~x)representsvariousquantities,suchasthespacetimemetric,stress-energyten- sor, and any particle fields. We write the variation of X(t, ~x)inequation(5.1).

X(t, ~x) X(t, ~x) X(t)(5.1) ⌘ where X¯(t)isthehomogeneousbackground. e Using Einstein’s equation and assuming that the perturbations are small, we can

ﬁnd the perturbation of the metric Gµ⌫ in terms of small changes in the stress energy tensor Tµ⌫.

Gµ⌫ =8⇡GTµ⌫ (5.2)

129 Figure 5-2: Anisotropies and perturbations in the CMB. This image is from Daniel Baumann’s lecture notes [8].

In Chapter 1, we showed that the universe after inﬂation is spatially ﬂat, isotropic, and remarkably homogeneous. This fact can be used to symmetrically justify decom- posing the metric and stress energy tensor perturbations into scalar and tensor parts [34].

This scalar and tensor decomposition can be described in Fourier space.

3 i~k ~x X~k(t)= d ~x X (t, ~x)e · (5.3) Z where X ,g ,... and ~k is a Fourier wavevector. ⌘{ µ⌫ } In order to study metric perturbations, let

(t, ~x)=¯(t)+(t, ~x) (5.4) gµ⌫(t, ~x)=¯gµ⌫(t)+gµ⌫(t, ~x)

where gµ⌫ is the metric and is the inﬂaton. The overbar indicates the homoge- nous background solution of a speciﬁc quantity. As a result, for an FLRW universe,

130 we get the inﬁnitesimal spacetime interval.

2 µ ⌫ ds = gµ⌫dx dx (5.5) = (1 + 2)dt2 +2aB dxidt + a2[(1 2 ) + E ]dxidxj i ij ij

where for this metric, B @ B S , @iS =0,E 2@ E +2@ F +h , @iF =0, i ⌘ i i i ij ⌘ ij i j ij i i i and hi = @ hij =0.Inthiscase,Si and Fi corresponds to vector perturbations. As Baumann argues in [8], these two vector perturbations are neglected since they are not created by inflation and decay away. E and B are purely gravitational vector quantities, which are labelled similarly to the electric and magnetic fields because they transform in a similar way, but have no actual connection to electromagnetism. andaregaugeinvariantBardeenpotentials[47]. Next, let’s examine how scalar fluctuations change under a change of coordinates, as given by

t t + ↵ ! (5.6) xi xi + ij ! ,j

Using the coordinate transformations in equation (5.6), the scalar metric perturbations should transform as follows [8].

↵˙ ! 1 B B + a ↵ a˙ ! (5.7) E E ! +H↵ !

Now we introduce some perturbed ﬂuid quantities. ⇢ is the perturbation in density, p is the perturbation in pressure, and q is the perturbation in momentum density q (¯⇢ +¯p)v . Using these quantities, we ﬁnd the variation in di↵erent sets i ⌘ i 131 of components of the stress energy tensor.

T 0 = (¯⇢ + ⇢) 0 0 Ti =(¯⇢ +¯p)avi i i (5.8) i v B T0 = (¯⇢ +¯p) a !

i i i Ti = j(¯p + p)+⌃j

i where ⌃j is the gauge invariant anisotropic stress. We can show that tensor fluctuations are gauge invariant, while scalar fluctuations change under a coordinate transformation. Thus, once again, applying the transformations given by equations (5.6) to the fluid perturbations themselves yields

⇢ ⇢ ⇢↵¯˙ ! p p p¯˙↵ (5.9) ! q q +(¯⇢ +¯p)↵ !

Additionally, we can deﬁne one other gauge invariant scalar, the comoving curvature perturbation R [8] as

H R = q (5.10) ⇢¯ +¯p

0 where q is the scalar part of Ti = @iq,whichisthe3-momentumdensitycom- ponents of the stress energy tensor. Since T 0 = @¯˙ during inﬂation, we can re-write (5.10) as i i

H R = + (5.11) ¯˙ From a geometry standpoint, R measures the spatial curvature of constant comoving hypersurfaces.

132 5.1.3 Statistics of Perturbations

The power spectrum of comoving curvature perturbation R is given by equation (5.12) [8].

3 R~ ,R~ =(2⇡) (~k + ~k0)PR(k)(5.12) h k k0 i

where R~ ,R~ is the mean value fluctuations of the ensemble and PR(k)denotes h k k0 i a power spectrum. We also now define a scalar value in equation (5.13), with the intention of using this definition to write the spectrum in power law form later on.

k3 2 2 = P (k)(5.13) s ⌘ R 2⇡2 R

Next, we deﬁne the scalar spectral index ns,whichgivesusthefollowingscale dependence.

d ln 2 n 1 s (5.14) s ⌘ d ln k

We also deﬁne the running of the spectral index, ↵.

dn ↵ s (5.15) s ⌘ d ln k

Finally, using some straightforward algebra and the relations given by equations (5.12), (5.13), (5,14), and (5.15), we can obtain an approximate power law that describes the power spectrum.

1 k ns(k ) 1+ ↵s(k )ln( ) ⇤ 2 ⇤ k ⇤ 2 k s(k)=As(k ) (5.16) ⇤ k ⇤ ! where k is a scale factor. It should be noted that quantum ﬂuctuations of the ⇤ inﬂaton are approximately Gaussian and scale invariant to the extent of the work presented in this thesis.

133 As for the tensor perturbations, the correlation function is given by

3 h~ h~ =(2⇡) (~k + ~k0)Ph(k)(5.17) h k k0 i where once again, we deﬁne a scalar value that will enable us to write a power law later on.

k3 2 = P (k)(5.18) h 2⇡2 h The power spectrum of the tensor perturbations is deﬁned as the sum of the power spectra for the 2 polarizations of the priomordial gravitational waves. In this chapter, we are interested in determining the tensor to scalar ratio, however, the quantum generation of tensor perturbations could lead to special signatures in the polarization of the CMB. This topic is beyond the scope of this work, but there are multiple exper- iments searching for polarization information associated with tensor perturbations, and therefore, primordial gravitational waves [48]. The tensor perturbation power spectrum is given by

2 22 (5.19) t ⌘ h

Now, we deﬁne the tensor index nt.

d ln 2 n t (5.20) t ⌘ d ln k Finally, we write our ﬁnal approximate law using equations (5.17),(5.18),(5.19) and (5.20).

nt(k ) ⇤ 2 k t (k)=At(k ) (5.21) ⇤ k ⇤ !

5.1.4 Scalar and Tensor Perturbations

It should be noted that this section is heavily based upon Daniel Baumann’s notes [8] and [48]. Perturbations in the comoving curvature perturbation R are related to

134 perturbations in the inﬂaton ﬁeld via the relation given by equation (5.11). Setting =0yields

R = H ˙ (5.22) Ht ⌘

The power spectrum of R can be rewritten in terms of is thus derived using equation (5.22).

2 H R~kR~k = ~k,~k (5.23) h 0 i ˙ ! h 0 i

where ~ ,~ is given by h k k0 i

2 2 3 2⇡ H ~ ~ 0 ~k,~k =(2⇡) (k + k ) 3 (5.24) h 0 i k 2⇡ ! As derived in [8]. Thus, the power spectrum for R is given by

2 2 2 H H ⇤ ⇤ R(k)= 2 2 (5.25) (2⇡) ˙ ⇤ where H denotes the horizon crossing a H = k. ⇤ ⇤ ⇤ Expanding the Einstein-Hilbert action and after quite some e↵ort, a second order action S(2) for the tensor perturbation can be obtained [8].

2 mpl 3 2 2 2 S = d⌧dx a [(h0 ) (@ h ) ](5.26) (2) 8 ij l ij Z We deﬁne the Fourier expansion h in the action as

3 d k ~˙ h = ✏s (k)hs (⌧)eik~x (5.27) ij (2⇡)3 ij ~k s=+, Z X⇥ i s s where ✏ii = k ✏ij =0and✏ij(k)✏ ij0 (k)=ss0 . Substituting the Fourier expansion in equation (5.27) into the Hilbert-Einstein

135 action in equation (5.26) yields

4 a 2 s s 2 s s S = d⌧d~k m [h 0h 0 k h 0h 0](5.28) (2) 4 pl ~k ~k ~k ~k s X Z Let’s now deﬁne a canonically normalized ﬁeld.

a vs m hs (5.29) ~k ⌘ 2 pl ~k

Substituting equation (5.29) into (5.28), one obtains the following form for the action.

s 2 2 a00 s 2 S = d⌧d~k (v 0) k (v 0) (5.30) (2) ~k a ~k s " ! # X Z

a00 2 where a = ⌧ 2 .

Baumann recognizes the above equation as the equivalent of the action of a har- monic oscillator, which after a signiﬁcant amount of work, allows him to calculate the corresponding power spectrum. Each polarization of the gravitational wave is a massless ﬁeld that has been normalized, is given in equations (5.31).

2 hs = s ~k m ~k pl (5.31) v s ~k ~k ⌘ a

Using the results for the power spectrum of s, which is derived by Baumann [48], ~k the tensor perturbation evaluated at the horizon crossing is given by

2 2 4 H ⇤ h(k)= 2 (5.32) mpl 2⇡ !

136 As a result, the power spectrum of tensor ﬂuctuations becomes

2 2 t =2h(k) 2 H2 (5.33) ⇤ = 2 2 ⇡ mpl

Finally, we obtain the tensor to scalar perturbation ratio by dividing the amplitudes of the power spectra of each.

2 t (k) r 2 (5.34) ⌘ s(k)

5.1.5 Writing the Primordial Spectra in Terms of the Slow Roll Parameters

Once again, it should be noted that this section is heavily based upon Daniel Bau- mann’s notes [8]. The calculated power spectra of the scalar and tensor ﬂuctuations evaluated at the horizon crossing condition k = aH are given by equations (5.35) and (5.36).

2(k) 2 (k) s ⌘ R 1 H2 1 (5.35) = 2 2 8⇡ mpl ✏ k=aH

2(k) 22 (k) t ⌘ h 2 H2 (5.36) = 2 2 ⇡ mpl k=aH where

d ln H ✏ = (5.37) dN 137 The corresponding tensor to scalar ratio using the results above are given by the following equation.

2 t r 2 =16✏ (5.38) ⌘ s ⇤ Now, we turn to the scale dependence of the spectra. The scalar spectral index is given by equation (5.39).

d ln 2 n 1 s (5.39) s ⌘ d ln k

Applying the chain rule to the value of n 1yields s

d ln 2 d ln 2 dN s = s (5.40) d ln k dN ⇥ d ln k

With some mathematical manipulation, we can obtain the derivative of the scalar power spectrum with respect to the number of e-foldings.

d ln 2 d ln H d ln ✏ s =2 (5.41) dN dN dN

By setting ⌘ = d ln H, ,wecanuseequations(5.37)and(5.41)toobtainthe dN following equation.

d ln ✏ =2(✏ ⌘)(5.42) dN

In order to avoid confusion, at this point, I would like to note that Baumann introduces the deﬁnition of the Hubble slow roll parameters, given by

@ ln H 2H ⌘ = | ,| = , @N H 2 (5.43) @H H ✏ = =2 , @N H !

These deﬁnitions evaluate the rate of change of H and H,.Inaddition,Baumann

138 provides the relation between the Hubble slow roll parameters and the potential slow roll parameters as

⌘ ✏v ⇡ (5.44) ✏ ⌘ ✏ ⇡ v v

Before we continue on, we note that when evaluated at k = Ha,lnk = N +lnH. The second term in equation (5.40) becomes

1 dN d ln k = d ln k " dN # 1 d ln H (5.45) = 1+ " dN # 1+✏ ⇡

Finally, using equations (5.39), (5.41), and (5.45) gives us to ﬁrst order the spectral index in terms of the Hubble slow roll parameters.

ns 1=2⌘ 4✏ (5.46) ⇤ ⇤

Now, we turn to calculating the spectral index and tensor to scalar ratio in terms of the desired slow roll results, which Baumann deﬁnes as the potential slow roll parameters. The spectral index is

n 1=2⌘⇤ 6✏⇤ (5.47) s v v

Finally, the tensor to scalar ratio is

r =16✏v⇤ (5.48)

139 5.2 Final Results

This section essentially provides a qualitative comparison between quadratic inflation, SUGRA D-Term inflation, and axion monodromy inflation. Subsection 5.2.1 essentially uses the inflationary potentials derived in previous chapters to derive the slow roll parameters for each model. Subsequently, the slow roll parameters are used to find exact values for the spectral index and scalar to tensor perturbation ratio. Subsection 5.2.2 includes the final plotted result, which are plotted against observable constraints provided by the 2018 Planck Collaboration in order to determine the viability of each model [45]. Finally, subsection 5.2.3 includes the final conclusion and closing remarks.

5.2.1 Values of r and ns for Di↵erent Inﬂationary Model Po- tentials

An excellent description of the dynamics with a quadratic potential is documented in Prof. Kaiser’s primer [19] and Alberto Vasquez’s notes [38]. For the SUGRA D-term potential, Yamaguchi provides an in-depth review in his paper, which is published in [39]. For the axion monodromy subsubsection, the general results for monomial axion monodromy are best presented by Sebastiaan Schotten’s dissertation [34] and Takahiro Terada’s dissertation [37].

Quadratic Potential

In Chapter 2, we examined a simple single scalar ﬁeld inﬂationary model with a quadratic potential, re-written in equation (5.49).

1 V ()= m22 (5.49) 2

Continuing the computation in Section 2.2.3, we start where we left o↵with

140 equation (2.66) which is once again re-written for clarity as equation (5.50).

a(t ) 60 ⇤ = e a(tend) 2 2 (5.50) 2 ' (tend) ' (t ) ⇤ =exp 2⇡G'i 2 2 ( " 'i 'i #)

We now proceed to solve for '(t ), the value of the inﬂaton at the time of the ﬁrst ⇤ Hubble crossing.

30 ' = ⇤ r ⇡ (5.51) =3.1mpl

where we have re-labeled '(t )as' ⇤ ⇤ We now proceed to calculate the slow roll parameters using equations (5.52) and (5.53).

H˙ ✏ = H2 2 (5.52) 1 V,' ⇡ 16⇡G V !

'¨ ⌘ = ✏ H'˙ 2 (5.53) 1 V,'' ⇡ 8⇡G V !

Utilizing the slow roll parameters as well as equation (5.51), we calculate the

141 numerical values of the slow roll parameters in equation (5.54).

2 1 1 mpl ⇡ 1 3 ✏ = = = =8.3 10 ⇤ 2 2 4⇡G ' 4⇡ 30mpl 120 ⇥ ⇤ 2 (5.54) 1 1 mpl ⇡ 1 3 ⌘ = = = =8.3 10 ⇤ 2 2 4⇡G ' 4⇡ 30mpl 120 ⇥ ⇤

Finally, we use our slow roll parameters to ﬁnd the spectral index and scalar to tensor perturbation ratio.

1 ns =1 6✏ +2⌘ =1 =0.967 ⇤ ⇤ 30 (5.55) r =16✏ =0.128 ⇤

SUGRA D-Term Potential

In section 3.3.3, we examined two SUGRA potentials, an F-term model and a D-term model. Let’s focus on the D-term superpotential, re-written for clarity in equation (5.57).

W = S+ (5.56)

This superpotential led us to an e↵ective inﬂationary potential of the form

2 2 2 2 2 S 2 g ⇠ g S e| | Ve↵ 1+ 2 ln | | 2 (5.57) ⇡ 2 " 8⇡ ⇤ !#

Which can be re-written in terms of inﬂaton ﬁeld .

g2⇠2 g2 22 Ve↵ 1+ 2 ln 2 (5.58) ⇡ 2 " 8⇡ 2⇤ !#

142 From this potential, we obtain the following values of the slow roll parameters.

g4 ✏ 4 2 ⇡ 32⇡ (5.59) g2 ⌘ ⇡4⇡22

Using our conclusions throughout Chapters 1 and 2, we assume the value of the number of e-foldings at the end of inﬂation N to be equal to 60. Using reference [39],

2 the upper bound for the value of g is given by g 2 10 .Weproceedtousethe  ⇥ equation for number of e-foldings to obtain an approximation for the ﬁeld values at the beginning and end of inﬂation.

N 4⇡2 N d ⇡ g2 Ze (5.60) 2⇡2 = (2 2) g2 N e

If hybrid inflation ends when the inflaton reaches the critical value c or the slow roll conditions are not satisfied using ⌘ =1,then = g .Yamaguchiassumes | | f 2T the coupling is not too small, so that inflation ends before reaching the critical value

2 . Assuming N =60and ,usingequation(5.60),weobtain2 = g N . c N e N 2⇡2 2 Since = = g and = gN ,thenourassumption is well satisﬁed! e f 2⇡ N p2⇡ N e Substituting the value of N in equations (5.59) yields

g2 ✏ ⇡ 16⇡N (5.61) 1 ⌘ ⇡2N

Finally, we calculate the spectral index and scalar to tensor perturbation ratio in

143 terms of N.

1 ns 1= 6✏ +2⌘ 2⌘ ⇡ ⇡N (5.62) g2 r =16✏ ⇡ ⇡2N

Substituting N =60intotheseequationsyieldsthevaluesinequation(5.63).

ns =0.9833 (5.63) 7 r =6.7 10 ⇥

Axion Monodromy Potential

Towards the end of Section 4.3.2, we obtained the general axion monodromy inﬂa- tionary potential re-written in equation (5.64).

n V = cn (5.64)

with the corresponding slow roll parameters assuming a slow roll approximation of the form given by equation (2.59) in Chapter 2.

n2 ✏ = 22 (5.65) n(n 1) ⌘ = 2

At the end of inflation, ✏ =1,whichallowsustocalculatethefieldattheendof inflation using equation (5.65).

n2 ✏end = 2 =1 (5.66) 2end

144 so

n end = (5.67) p2

The number of e-foldings can subsequently be calculated.

tend N = dtH ⇤ t Z ⇤ (5.68) ⇤ 1 = d p2✏ Zend

where N indicates the number of e-foldings from the horizon crossing to the end ⇤ of inﬂation.

Substituting ✏,weobtain

⇤ N = d ⇤ n Zend (5.69) 2 n = ⇤ 2n 4

Using our usual relations between the slow roll parameters and 1 n and r,we s obtain the following relations in terms of N,thenumberofe-foldings,andpolynomial order n.

n(n +2) n +2 1 n = = s 2 2(N + n ) 4 (5.70) 8n2 4n r = 2 = n N + 4

145 Assuming n =1inequations(5.70),weobtain

3 ns =1 2 N + 1 4 (5.71) 4 r = 1 N + 4 Finally, we compute the spectral index and scalar to tensor perturbation ratio by assuming the number of e-foldings at the end of inﬂation to be Ne =60.

ns =0.9751 (5.72) r =0.0664

5.2.2 Comparison to Data

Figure [5-3] displays the observational CMB constraints on ns and r in the (ns,r) plane published by the 2018 Planck Collaboration [45]. The contours show the 68% and 95% CL derived recent published data by the Planck collaboration in 2018 [45]. The data represents Planck alone (grey region), Planck+BK15 data (red region), Planck+BK15+BAO data (blue region), as indicated in the legend of the ﬁgure. It shows the 68% and 95% CL contours in the

(ns,r) plane. The experimental constraints are compared with predictions of the three inﬂationary models discussed in the previous section at 60 e-folds.

1 2 2 For inflation produced by a massive scalar field V ()= 2 m ,thepredicted data is outside the boundary of the 68% CL of the Planck alone data. Therefore, this model is excluded with a high statistical significance. In the case of inflation resulting from the SUGRA D-term model, the predicted data is also outside of the boundary of the 68% CL contour. Therefore, this model is strongly disfavored by the Planck 2018 data. Inflation from axion monodromy is represented by the yellow square. It is at the boundary of 68% CL contour. It should be pointed out that in this specific case the potential is approximately linear, V () . As pointed out in ⇠ 146 Figure 5-3: An overlay of the calculated values of ns and r for quadratic inflation, SUGRA D-Term inflation, and axion monodromy inflation on Planck collaboration 2018 results [45]. The square shaped points indicate the analyzed constraint calculated in this thesis. Contours in the (ns,r)planeshowthe68%and95%confidence levels.

147 [39], in this speciﬁc case, the axion arises from a D5 brane wrapped around a two- cycle with B2 ﬂux. Out of all of the models, axion monodromy is the most favored in accordance with the Planck data. However, it could be excluded if it is compared with the Planck+BK15 or Planck+BK15+BAO data.

5.2.3 Conclusion

In the first portion of this thesis, we have reviewed three of the central motivations for cosmic inflation, which are known as problems in the standard big bang scenario. These problems are the “flatness problem,” “horizon problem,” and the “monopole problem.” As part of the investigation, we have placed emphasis on examining the formalism for studying matter in curved spacetime. Moreover, we have placed special emphasis in the FLRW metric and its dynamical properties. We have also reviewed the theory of cosmological inflation as a solution to the flatness and horizon problem of the standard cosmological model. As an example, we have analyzed inflation caused by a single scalar field with a quadratic potential. This simple model allowed us to introduce the “slow-roll parameters” and their application in inflationary theory. In the second half of this thesis, we have explored the connection between supersymmetric theories and the evolution of the early universe. Of particular relevance for the topics treated in this thesis is the potential for the scalar components of the chiral multiplet, which can be identified from the Lagrangian. To illustrate the role of supersymmetry in inflationary model-building, hybrid inflation was considered as an example. In supersymmetry, unlike in the standard model, the scalar potential is evidently strongly constrained by gauge symmetries and supersymmetry. Supersym- metry can be realized either as a global or a local symmetry. Incorporating gravity into supersymmetry allows us to introduce supergravity. Since the early universe corresponds to high energy scales, supersymmetry in this context implies supergravity, and possibly also string theory. Having considered inflation in the context of supersymmetry and supergravity, we explained the key concepts required for understanding inflation in context of Type IIB superstring theory. We introduced key concepts in string theory, such as D-branes and compactification. To illustrate the

148 role of string theory in inflationary model-building, we investigated the concept of axion monodromy. The final chapter considers three representative models of inflation, which lead to observational predictions of the spectral index and tensor to scalar perturbation ratio. In order to better understand these concepts, we have provided a brief overview of the production and amplification of perturbations during cosmic inflation. Inflation produced by a massive scalar field with a quadratic potential, inflation resulting from a supergravity D-term model, and inflation from an axion monodromy monomial potential have all been compared to observed data of the CMB. It has been determined that the predicted values of the spectral index and tensor to scalar perturbation ratio are only within the 68% CL contour of the observational constraints for axion monodromy. This implies that out of all of the viable inflationary models, axion monodromy best supports the data. As stated by my thesis advisor, Prof. Kaiser, “The close agreement between the spectrum of primordial perturbations predicted by inflation and the latest precision measurements of the cosmic microwave background radiation (CMB) is one of the crowning achievements of recent work in cosmology” [19]. Even though the predicted results seem to support axion monodromy as the most viable inflationary model, the additional data sets further constraints on the allowed values of ns and r.Thisfactisprovidingfurtherguidanceonlookingfor model parameters that better fit the data. Beyond the scope of this thesis, other models have been devised that fit the CMB data better than the axion monodromy with n = 1 case [45]. Future analysis of observational data is bounded to improve the constraint, which in turn will provide better guidance as to which models should be further investigated.

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