ABSTRACT Horava-Lifshitz Theory of Gravity and Its

ABSTRACT

Hoˇrava-Lifshitz Theory of Gravity and Its Applications to Cosmology

Yongqing Huang, Ph.D.

Chairperson: Anzhong Wang, Ph.D.

In this dissertation, I studied Hoˇrava-Lifshitz gravity and its applications to inﬂationary cosmology. After introducing the original proposal with the projectability and detailed balance conditions, I discussed its attracting features as well as the problems it faces. An extended model without the detailed balance condition was then studied and found to be stable in de Sitter spacetime, but still possess an extra scalar mode in the gravity sector. I then studied a model with an extra U(1) symmetry

dubbed as the projectable general covariant HL gravity and showed that it has the

same degree of freedom as general relativity, and is free of the stability problem. It

was found that the FLRW universe is necessarily ﬂat, given that the coupling with

matter takes a speciﬁc form. I also studied the scalar perturbations around the FLRW

metric and presented all the possible gauge choices.

Applications of the general covariant model in inﬂationary cosmology were stud-

ied in the second part. After deriving the slow-roll conditions in this model, I showed

that in the super-horizon regions the scalar perturbations become adiabatic, and the

comoving curvature perturbation is constant, because of this slow-roll condition. By

using the uniform approximation technique, power spectra and indices of primordial

scalar and tensor perturbations under the slow-roll approximations were expressed

explicitly in terms of the slow-roll parameters and the various coupling constants. I found that they are in general different from, but reducible to, the values in the class of simplest inflation models. Next I studied the non-Gaussianities of these perturbations. For scalar perturbations, by properly choosing the coupling constants, a large nonlinearity parameter fNL is possible. I also found that the bispectrum favors the equilateral shape as a result of the higher order spatial derivatives, and that folded shape is enhanced when the vacuum is from the Bunch-Davis vacuum. Both the squeezed and the equilateral shapes appear in the bispectrum for primordial gravitational waves. In addition, the polarization tensors of the tensor fields have strong effects on the shapes of the bispectrum. Hoˇrava-Lifshitz Theory of Gravity and Its Applications to Cosmology

Yongqing Huang, B.S.Phys.

A Dissertation

Approved by the Department of Physics

Gregory A. Benesh, Ph.D., Chairperson

Submitted to the Graduate Faculty of Baylor University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

Approved by the Dissertation Committee

Anzhong Wang, Ph.D., Chairperson

Gerald B. Cleaver, Ph.D.

Truell W. Hyde, Ph.D.

Jonatan C. Lenells, Ph.D.

Qin Sheng, Ph.D.

Accepted by the Graduate School May 2013

J. Larry Lyon, Ph.D., Dean

LIST OF FIGURES viii

ACKNOWLEDGMENTS ix

DEDICATION xi

1 Introduction to Hoˇrava-Lifshitz Theory of Gravity 1

1.1 Non-renormalizability of General Relativity ...... 1

1.2 Hoˇrava-Lifshitz Theory of Gravity ...... 2

1.2.1 The Lifshitz Scalar ...... 2

1.2.2 The Case for Gravity ...... 4

1.2.3 Attracting Features of HL Gravity ...... 8

1.2.4 Potential Problems of HL Gravity ...... 12

1.3 Organization of Dissertation ...... 15

2 Projetable Hoˇrava-Lifshitz Gravity without Detailed Balance Condition 18

2.1 Introduction to the Model ...... 18

2.2 The Instability Problem ...... 20

2.2.1 Scalar Perturbations in Flat FLRW Backgrounds ...... 20

2.2.2 Instability Around Minkowski Spacetime ...... 23

2.2.3 Stability Around de Sitter Spacetime ...... 24

2.3 New Features ...... 26

2.4 Persisting Problems ...... 28

2.5 Abandoning Both Conditions...... 29

iv 3 General Covariant Hoˇrava-Lifshitz Gravity with Variable λ 31

3.1 Introduction to the General Covariant Theory ...... 34

3.1.1 The Case with λ =1 ...... 34

3.1.2 A Minimum Substitution with Variable λ ...... 37

3.1.3 Coupling with Matter ...... 38

3.1.4 The Field Equations ...... 39

3.2 Scalar Cosmological Perturbations at Linear Order ...... 44

3.2.1 Flatness of FLRW Universe ...... 44

3.2.2 Linear Scalar Cosmological Perturbations ...... 46

3.2.3 Gauge Choices ...... 50

3.3 The Instability Problem ...... 53

3.3.1 Stability of the Minkowski Spacetime ...... 53

3.4 The Ghost and Strong Coupling Problems ...... 54

3.4.1 Ghost-free Condition ...... 55

3.4.2 Strong Coupling Problem ...... 56

3.5 Non-projectable General Covariant HL Gravity with Detailed Balance Condition Softly Breaking ...... 63

4 Slow-roll Single Field Inﬂation in General Covariant Hoˇrava-Lifshitz Gravity 66

4.1 The Inﬂation Paradigm ...... 68

4.1.1 The Horizon and Flatness Problems of Standard Model ...... 68

4.1.2 Accelerated Expansion and Slow-roll Approximation ...... 70

4.1.3 The Quantum Fluctuations ...... 72

4.1.4 Statistics of the Modes, Power Spectrum ...... 76

4.2 Slow-roll Inﬂation of a Scalar Field in HMT Model ...... 78

4.2.1 The Slow-roll Conditions ...... 78

4.2.2 Linear Scalar Perturbations Without Gauge-ﬁxing ...... 79

v 4.2.3 Uniform Density Perturbation ...... 81

4.2.4 Comoving Curvature Perturbation ...... 82

4.3 Scalar Perturbations in Sub- and Super-Horizon Scales ...... 83

4.3.1 Sub-Horizon Scales ...... 86

4.3.2 Super-Horizon Scales ...... 86

4.4 Power Spectra and Indices of Primordial Perturbations ...... 87

4.4.1 Scalar Perturbations ...... 88

4.4.2 Tensor Perturbations ...... 95

5 Primordial Non-Gaussianities in General Covariant Hoˇrava-Lifshitz Gravity 99

5.1 Non-Gaussianity ...... 102

5.1.1 The Three-point Correlator ...... 102

5.1.2 The In-in Formalism ...... 104

5.2 Primordial Non-Gaussianity of a Single Scalar Field ...... 106

5.2.1 Interaction Hamiltonian ...... 106

5.2.2 The Three-point Correlator ...... 112

5.2.3 Modiﬁed Dispersion Relation and Evolution of the Mode Functions ...... 113

5.2.4 Approximating the Mode Functionss ...... 115

5.2.5 The Bispectrum ...... 120

5.2.6 Shapes of the Bispectrum ...... 121

5.2.7 Projections onto Factorizable Templates ...... 128

5.2.8 Summary of the Assumptions ...... 130

5.3 Primordial Non-Gaussianity of Gravitational Waves ...... 131

5.3.1 The Interaction Hamiltonian ...... 132

5.3.2 The Mode Integration ...... 136

5.3.3 Shapes of the Bispectrum ...... 139

vi 6 Conclusions 144

6.1 Conclusions ...... 144

6.2 Future Work ...... 145

BIBLIOGRAPHY 146

vii LIST OF FIGURES

1.1 Causal structures of relativistic spacetime (left) and Galilean spacetime (right). In the relativistic case, events outside the light cone are causally disconnected from present, i.e. they have no physical contacts. In the Galilean case, the notion of time is unique up to a global shift and events that happen at time t2 are in principle aﬀected by all events at t < t2. . . 9 1.2 The introduction of an inﬂationary era makes all the observable parts of our universe causally connected at the beginning...... 10

5.1 The evolution of ωph ≡ ωk/a vs kph ≡ k/a in three diﬀerent regions, where Region I: t ∈ (ti, t1); Region II: t ∈ (t1, t2); and Region III: t ∈ (t2, t0)...... 114

5.2 Shape of the bispectrum (truncated). V3-term dominates, with the choice of the positive frequency...... 123

5.3 Shape of the bispectrum (truncated). V3-term dominates, with the choice of the negative frequency...... 124

5.4 Shape of the bispectrum (truncated). V5-term dominates, with the choice of the positive frequency...... 125

5.5 Shape of the bispectrum (truncated). V5-term dominates, with the choice of the negative frequency...... 126

−1 5.6 Shapes of (k1k2k3) G+++ (K) contributed by various terms. All are normalized to unity for equilateral limit...... 140

−1 5.7 Shapes of (k1k2k3) G++− (K) contributed by various terms. All are normalized to unity for equilateral limit...... 141

5.8 Shapes of the different configurations of the polarization tensors for s1 = s2 = s3 = 1. All are normalized to unity for equilateral limit. . . . . 142 5.9 Shapes of the different configurations of the polarization tensors for s1 = s2 = −s3 = 1. All are normalized to unity for equilateral limit. . . . 143

viii ACKNOWLEDGMENTS

I am deeply indebted to my advisor Dr. Anzhong Wang, who provided me with the opportunity to pursue my graduate study at Baylor. I am grateful for his constant and patient guidance, encouragements and valuable discussions on various topics. He has taught me ﬁeld theory, cosmology, how decent scientiﬁc research should be performed and much more than that. I wish to thank him for his guidance on the research that constitute this dissertation.

I want to thank Dr. Gerald B. Cleaver, Dr. Truell W. Hyde, Dr. Jonatan C.

Lenells and Dr. Qin Sheng for serving on my committee with their busy schedules and the suggestions on reﬁning this work.

Part of this dissertation has beneﬁted from the collaborations with Dr. Qiang

Wu and Dr. Tao Zhu. I learned numerical simulations of cosmological models under the directions of Dr. Qiang Wu. This experience broadened my knowledge and will no doubt beneﬁt me for my future studies. Discussions with Dr. Tao Zhu had cleared many confusions I had in cosmology, especially on the subject of spectrum indices and non-Gaussianities. It was a good experience working with them in the GCAP group.

Mrs. Chava Baker and Mrs. Marian Nunn-Graves have always provided as- sistance throughout my years at Baylor. I am grateful for Dr. Yumei Wu for her generous help in adjusting to life in the United States, which isn’t an easy task. I thank Karen Bland, Te Ha, Timothy Renner and my fellow graduate students for the discussions we had in the deep nights working on the hard physics problems, and for sharing the good times at Baylor. I learned a lot from Angela Douglass, whose guidance on serving as a teaching assistant not only made my life in graduate school much easier, but also beneﬁted the students I met in the laboratories.

ix I am deeply appreciative of the courses I had with Mr. Donghui Xu and Dr.

Haibin Li during the undergraduate years in China. My ﬁrst research projects on physics were under their guidance and those experience helped me a lot for my graduate studies. To the teachers under whom I studied: Ms. Chen, Ms. Sun, Ms. Sonyan

Dong, Ms. Liyun Guo, Mr. Xixian Liu, and all who guided me in my school life, thank you for your teachings. I could not have come this far without your guidance.

I would like to thank my friends, Long Li, Shuo Wang and Chenghao Yin back at my hometown, for their kind help oﬀered to my parents while I was half a globe way from them. Finally I would like to express my gratitude to my mother, SuHua

Zhang, and my father, MingChen Huang, for their enduring love and unconditional support of my pursuit of study in theoretical physics. I owe them the most.

x DEDICATION

To My Parents

And All My Teachers

xi CHAPTER ONE

Introduction to Hoˇrava-Lifshitz Theory of Gravity

1.1 Non-renormalizability of General Relativity

There is no wonder that general relativity (GR), the theory that describes the gravitation phenomenon, and the framework of relativistic quantum ﬁeld theory

(QFT), upon which the standard model of particle physics is built, are among the most successful achievements in the history of science. Yet despite the decades- long eﬀorts trying to incorporate these two into a broader theory, we are led to the conclusion that achieving such a theory would require us to change our view of the universe, fundamentally. The views proposed by string theory and loop quantum gravity are the vivid examples of such changes.

To achieve a quantum ﬁeld theoretic description of fundamental interactions, such as the eletroweak interaction, one can start with a classical action S. Inside the action, the ﬁelds are coupled through the interacting terms and coupling constants.

When we quantize the ﬁelds, these coupling constants are subject to quantum loop corrections, which can be calculated perturbatively with the help of Feynman diagrams. These perturbative series would later be summed over to give the correct physical value that physicists measure in experiments. The corrections normally are integrations in the momentum space and could be divergent. Devised to rescue the theory, the renormalization procedure is to bring extra terms into the action such that they can cancel these divergences and lead to ﬁnite, thus meaningful, values for the coupling constants.

The technical diﬃculty in applying this procedure to general relativity, or quantizing GR, is that the classical Einstein-Hilbert action ∫ 1 √ S = d4x −(4)g ((4)R − 2Λ) , (1.1) 16GN

1 where (4)R is the four-dimensional Ricci scalar, Λ is the cosmological constant and (4)g

(4) is the determinant of the four-dimensional metric gµν, would need inﬁnitely many extra terms to cancel the divergencies, hence is nonrenormalizable. The key reason

behind this is the fact that the coupling constant, GN , has a momentum dimension

of −2: [GN ] = −2 [1]. Studies of the one-loop divergence structure suggest the inclusion of (4)R2 terms for perturbative renormalization in 3 + 1 dimension. Stelle in 1977 [2] showed that this indeed makes the theory renormalizable. However, these higher order derivatives would include time derivative of order higher than two if one wants the theory to possess Lorentz symmetry, a symmetry that has been tested to a high accuracy.

Such dynamical systems are known to mathematicians to have problems. One can see the problems by looking at the propagator of an example of such models [3]

1 1 1 = − , (1.2) k2 − Gk4 k2 k2 − 1/G

(4) where we take gµν as the dynamical ﬁeld. The ﬁrst term corresponds to the usual massless graviton which also appears in GR, hence is considered benign. The trouble comes from the second one: it gives rise to a ghost excitation and leads to unitary issues, which is a fundamental property required in any successful quantum theory.

1.2 Hoˇrava-LifshitzTheory of Gravity

1.2.1 The Lifshitz Scalar

Among the various attempts to obtain a UV complete theory of gravitation, string theory attracts the most attention. The proposal by Hoˇrava [3] in 2009 took the view on string theory that it is such a rich theory considering the landscape of vacua permitted, that it can be viewed as framework, rather than a theory describing any phenomenon. Hence one may construct a theory of pure gravity that can later be imbedded in the “string framework”.

2 Hoˇrava’s theory is based on the basic assumption that the Lorentz symmetry,

instead of being exact, is fundamentally broken at high scales of energy and restores

only in the infrared (IR) limit. Below we shall refer to it as the Hoˇrava-Lifshitz

(HL) gravity. The construction began by observations on a class of models known

in the condensed matter physics and is referred to as the Lifshitz scalar ﬁeld theory.

Consider the simple free-action of some scalar ﬁeld ϕ in D + 1 dimension ∫ { } 1 S = dtdDx ϕ˙2 − ϕ (∆)z ϕ , (1.3) 2

where a dot represents diﬀerentiation with respect to time, ∆ = ∇⃗ 2 and z > 0. Notice how the time derivative is kept at the order of two while the spatial derivative can rise to an order higher than that. If one requires the action to be dimensionless, the appropriate dimensions of x and t would be

[x] = −1, [t] = −z, (1.4)

and the dimension of ϕ is

[ϕ] = (D − z)/2. (1.5)

Two things are immediately clear from this dimension analysis. One is that when

z > 1, this model does not respect Lorentz symmetry which requires space and time

to have the same dimension. z is called the dynamical critical exponent and measures

the degree of anisotropy between space and time. Second, when z = D, the scalar

ﬁeld itself is dimensionless. This indicates that the theory might be UV complete.

When one includes relevant deformation terms and the self-interaction terms,

∫ ∑z−1 ∫ ∑N D m D n SD = dtd x αmϕ(∆) ϕ, SI = dtd x gnϕ , (1.6) m=1 n=1 the dimensions of the coupling constants are

[αn] = 2(z − m), [gn] = D + z − n(D − z)/2. (1.7)

3 When z ≥ D, their dimensions are never negative. That is, D + z − n(D − z)/2 ≥ 0,

given max{n} = N < ∞, and 2(z − m) ≥ 2(z − (z − 1)) = 2. Such a theory is known

to be power counting renormalizable. One can also reach this conclusion using the

argument based on the superﬁcial degree of divergence [4].

1.2.2 The Case for Gravity

Motivated by this scalar ﬁeld theory, Hoˇrava constructs his theory of gravity

ﬁrst with the D-dimensional metric gij(i, j = 1, 2, ··· ,D). The kinetic part of the action consists of √ ijkl gg˙ijG g˙kl, (1.8)

where g is now the determinant of gij, and the generalized De Witt metric

Gijkl = gi(kgjl) − λgijgkl. (1.9)

( ) i(k jl) ≡ 1 ik jl il jk Here g g 2 g g + g g . If Lorentz symmetry is exact, the parameter λ would stay literally as 1 and is protected from receiving any quantum corrections.

In the present case however, λ loses this protection and becomes a true dynamical quantity by receiving the loop corrections.

In writing down the potential part of the action, one in principle needs to write down all terms of gij that respects the D-dimensional diﬀeomorphism. There are a great number of such terms. If one were to include all of them, the theory would lose its ability of making predictions. Hoˇrava reduces the choices by requiring the existence of an action W [gij] in D dimensions such that ∫ √ D ij kl SV = dtd x gE GijklE , (1.10) √ 1 δW [g ] gEij = ij , (1.11) 2 δgij

where Gijkl is the inverse of the generalized De Witt metric (1.9). This requirement is the gravitational analog of the detailed balance condition of the Lifshitz scalar.

4 A unique candidate for Eij that satisfy this condition while at the same time exhibits the UV anisotropic scaling with D = 3 = z leading to power-counting renormalizability in 3 + 1 dimensions is the Cotton-York tensor ( ) 1 Cij = εikl∇ Rj − Rδi , (1.12) k l 4 l which follows from a variation with respect to an action consists of the Chern-Simons

term ∫ 1 W [gij] = 2 ω3(Γ), (1.13) w ( ) 2 ω3(Γ) = Tr Γ ∧ Γ + Γ ∧ Γ ∧ Γ ( 3 ) ϵijk 2 = √ Γm∂ Γl + ΓnΓl Γm d3x, (1.14) g il j km 3 il jm kn

2 l where w is a coupling constants and Γkm is the Christoﬀel symbol of gij. Adding the relevant terms leads to the action ∫ ∫ 1 √ W = ω (Γ) + µ d3x g (R − Λ ) , (1.15) w2 3 W

where µ is another coupling constant and ΛW plays the role of cosmological constant. Although his reason for this detailed balance condition here is pragmatic: to

reduce the terms permitted in SV , the implementation Hoˇrava considered is nonethe- less very reminiscent of methods used in non-equilibrium critical phenomena and

quantum critical systems. This condition, in the context of condensed matter, makes

the quantization and renormalization procedure considerably easier [3]. Hence it

may have more meanings than just an assumption to reduce the number of coupling

constants [5].

The action that consists of (1.8) and (1.10) with W given in Eq.(1.15) is invari-

ant under time-independent spatial diﬀeomorphisms and global time translations, but

there is no Weyl invariance. To be able to be imbedded into, or to make connections

with, a broader framework, the model needs an extension of its gauge symmetry.

This in turn requires new gauge ﬁelds.

5 Considering the special role played by time, the extension Hoˇrava prescribed

is the spacetime diﬀeomorphisms that respects the preferred codimension-one foli-

ation F of spacetime M by the slices of ﬁxed time. A codimension-q foliation F

on a d-dimensional manifold M is deﬁned as M with an atlas of coordinate sys-

tems (ya, xi), a = 1, ..., q, i = 1, ..., d − q, such that the transition functions take the restricted form (˜ya, x˜i) = (˜ya(yb), x˜i(yb, xj)). Identifying ya as time and xi as the spatial coordinates, such “foliation-preserving diffeomorphisms” Diff(M, F) consists of spacetime-dependent spatial diffeomorphisms as well as time-dependent time reparametrizations

δt = f(t), δxi = ζi(t, x). (1.16)

Accompanying this extension are the new gauge fields N and Ni. The generators of Diff(M, F) (1.16) act on the fields via

k ˙ ˙ δN = ζ ∇kN + Nf + Nf,

k k ˙k ˙ ˙ δNi = Nk∇iζ + ζ ∇kNi + gikζ + Nif + Nif,

δgij = ∇iζj + ∇jζi + fg˙ij. (1.17)

There are two classes of functions that can be deﬁned on a foliation: functions that

depend on all coordinates, and those take on constant values on each leaf of the

foliation, i.e. the constant time slices. Functions that belong to the second class

are said to be projectable or to meet the projectability condition. Since N can be

considered as the gauge ﬁeld associated with the time reparametrization f(t), it

appears natural, but not mandated, to restrict it to be a projectable function on the

spacetime foliation F:

N = N(t). (1.18)

6 With this extension of symmetry, while keeping the projectability and detailed

balance condition, the action given by Hoˇrava is ∫ { √ 2 ( ) κ2 κ2µ S = dtdDx gN K Kij − λK2 − C Cij + ϵijkR ∇ Rl κ2 ij 2w4 ij 2w2 jl j k ( ) } κ2µ2 κ2µ2 1 − 4λ − R Rij + R2 + Λ R − 3Λ2 , (1.19) 8 ij 8(1 − 3λ) 4 W W

where the extrinsic curvature Kij is deﬁned as

−1 K = (g ˙ − ∇ N − ∇ N ) ,K = K gij, (1.20) ij 2N ij i j j i ij and ∇i is the covariant derivative with respect to gij. The arguments for its power counting renormalizability, similar to that of the Lifshitz scalar, can be found in [6].

It is important to emphasis the assumptions made in writing down (1.19):

• Space and time exhibit strong anisotropic scaling at short-distances (1.4);

• invariance under the foliation-preserving diﬀeomorphisms Diﬀ(M, F) (1.16);

• the detailed balance condition (1.15); and

• the ﬁeld N is projectable (1.18).

We shall see that except the ﬁrst one, at least one of the remaining three assumptions

will be abandoned in later developments of the theory in order to solve some problems

in the original HL model.

It is interesting to make connections of the ﬁelds gij, N, and Ni (i, j = 1, 2, 3) with ADM formalism in relativistic gravity [7]: ( )( ) 2 (4) µ ν − 2 2 i i j j ds = gµνdx dx = N dt + gij N dt + dx N dt + dx , (1.21)

where N and N i are lapse functions and shift vectors. It’s easy to see the relation

(4) ··· between the four-dimensional metric gµν (µ, ν = 0, , 3) and the ADM variables   2 i −N + NiN ,Ni (4)   gµν =   . (1.22) Ni , gij

7 (4) µλ (4) λ ij In a way similar to the four-dimensional case where g gµν = δν , g is deﬁned as the inverse of gij,

ik k g gij = δj . (1.23)

ij (4) And the shift vector is raised and lowered by g and gij. The inverse of gµν is expressed with these variables   −1/N 2 ,N i/N 2  (4)gµν =   . (1.24) N i/N 2 , gij − N iN j/N 2

i Hence from here below, we shall generally refer to gij,N and N as the metric, lapse function and shift vector.

1.2.3 Attracting Features of HL Gravity

Besides the key feature that it is power-counting renormalizable (and may well be renormalizable, though not proven strictly yet), HL theory has some other interesting aspects too.

The preferred foliation of spacetime deﬁnes a global causal structure at short distances. The causal structure in GR can be described by a light cone, see left of

Fig.1.1. Events outside the light cone are causally disconnected with the present. The causal structure of HL gravity at short-distances however, resembles that of Galilean spacetime, where a preferred “choice of coordinate” exists. This leads to an invariant notion of time, an important step towards understanding the “problems of time” in traditional means of quantizing gravity [8].

Because of this global causal structure, the notion of event horizon, which is a deﬁnitive character of black holes in GR, does not apply at short distances in HL gravity since all spacetime points are now causally connected. Therefore, although compact objects can still be described by HL gravity, the concept of black holes must be changed [9].

8 t t Future t = t2 Present t = t1 Future Past Present t = t0

x x Past

Figure 1.1: Causal structures of relativistic spacetime (left) and Galilean spacetime (right). In the relativistic case, events outside the light cone are causally disconnected from present, i.e. they have no physical contacts. In the Galilean case, the notion of time is unique up to a global shift and events that happen at time t2 are in principle aﬀected by all events at t < t2.

Another perspective to look at this problem is through the study of dispersion

relations. As will be shown in details in later chapters, because of the higher order

spatial derivatives, the dispersion relation in HL gravity can be written in a schematic

form as α α 2 2 2 1 4 ··· n 2z ω = cs(λ)k + 2 k + + 2z−2 k . (1.25) M1 Mn where the α’s are dimensionless coupling constants. This is fundamentally diﬀerent

2 ≤ from that of any relativistic theory, in which all the α’s are zero and cs 1. In particular, here the group speed, deﬁned as ω/k, would depend on the magnitude of the momentum and even on the value of λ, hence may well be unbounded from above.

This resembles the Galilean structure and is in direct contrast with the relativistic one where the speed of light is the limit.

This also has its implications in the study of cosmology, in particular the early universe when energy scale was high [10]. Our observed universe has one important property that challenges the standard model of cosmology before the proposal of the inﬂation paradigm: the diﬀerent patches of it share an astonishing degree of

9 Time Time

Present day

Matter domination Matter domination Last scattering surface (CMB) Matter−radiation equality Radiation Radiation Inflation domination domination era Distance Distance

Causally disconnected Causally connected

Figure 1.2: The introduction of an inflationary era makes all the observable parts of our universe causally connected at the beginning. homogeneity. According to the standard model, these patches must be causally disconnected in the distant past, given the constraint on the age of our universe from other observations, see Fig.1.2. Thus one must assume that they share the same state at very early time even though they are causally disconnected from each other. This is the horizon problem of the standard model of cosmology. The inflation paradigm, trying to tackle several puzzles of the standard model including the horizon problem, argues that the current cosmological observations of the these large-scale structures are sourced by the primordial quantum fluctuations. These fluctuations, initially connected, were stretched to super-horizon scales by a short period of “super-luminal” expansion of our universe and later re-entered the horizon and acted as seeds for the structures that we detect today. With the change of causal structure in the early universe in HL gravity, this inflationary period might not be necessary as far as the horizon problem is concerned because of the unbounded group speed. This was the argument in [11], which also found that it not only solves the horizon problem, but also capable of generating a scale-invariant power spectrum of the fluctuations, a well established observational fact. In addition, we shall see in Chapter Four that when the symmetry of the model is extended to U(1) ⋉ Diff(M, F), the universe is necessarily flat, thus solving the flatness problem.

10 An interesting implication speciﬁc to the projectable version of the model is

that dark matter appears naturally as an “integration constant” [12]. Since the lapse

function N(t) depends only on time under the projectability condition, a variation of

the action with respect to N leads to an Hamiltonian constraint in a integral form. In

the standard model, the Friedmann equation that relates the expansion and matter

contents of the smooth universe, described by the Friedmann-Lemaˆıtre-Robertson-

Walker (FLRW) spacetime, reads

8πG H2 = ρ, (1.26) 3

where H is the Hubble parameter describing the expansion rate of the universe and

ρ is the energy density of matter described by perfect ﬂuid. The matter also satisﬁes the conservation equation

ρ˙ + 3H (ρ + p) = 0, (1.27) where p is the pressure of the ﬂuid. These two together give rise to the dynamical equation ( ) 2H˙ + 3H2 = 8πp. (1.28)

One the other hand, in the projectable HL gravity we no longer have a Friedmann

equation, because the Friedmann equation follows from the Hamiltonian constraint.

While the constraint is satisﬁed for the whole universe, it is not necessarily so inside

our observable patch. Instead, we consider the dynamical equation ( ) 1 − 3λ 2H˙ + 3H2 = 8πp (1.29) 2 as an independent equation. Neither is the conservation equation satisﬁed at high energy

ρ˙ + 3H (ρ + p) = −q, (1.30)

where q ﬂows to 0 as the universe cools down. Integrating the dynamical equation

11 (1.29) with the use of (1.30), we obtain a ﬁrst integral [ ( ∫ )] − t 3(3λ 1) 2 1 ′ 3 ′ ′ H = 8πG ρ + 3 C0 + q(t )a (t )dt . (1.31) 2 a t0 We see that at later stages of the evolution when q → 0, the term on the right hand side inside the parentheses approaches a constant value. Interestingly enough, this constant behaves gravitationally like pressure-less dust (it scales as a−3), something like dark matter.

1.2.4 Potential Problems of HL Gravity

Though it has many attractive features, the original theory is plagued with several problems, most of which are related to the spin-0 “graviton”, a scalar field whose presence is due to the break from the full spacetime diffeomorphisms to the foliation preserving diffeomorphisms Diff(M, F).

It is well known that when a symmetry of a system is broken, a new species of particle might emerge and become a physical degree of freedom. In the present case, it is realized that this would be a scalar ﬁeld, in addition to the usual spin-2 ﬁeld in

GR. The mere presence of such ﬁeld is dangerous because it may appear at relative low energy scales and be within our current capability of detection. If so, the theory would have to explain why we haven’t observed such particles.

Since Minkowski spacetime is a solution of the theory when the detailed balance condition is broken (cf. Eq.(2.2)), consider for illustration purposes the linear scalar perturbations of the ADM variables (1.21) around vacuum Minkowski background:

¯ N = N(t) + ϵϕ, Ni = ϵB,i, gij = δij + ϵ (2E,ij − 2ψδij) , (1.32) where ϵ is the expansion parameter, and ϕ, B, E and ϕ are all scalars. In the projectable HL gravity, the quadratic action for the scalar perturbations in the low energy scales reads [13] ∫ [ ( )] 3λ − 1 S = dtd3x − ψ′2 − (∂ ψ)2 , (1.33) 1 − λ i

12 where we have dropped higher order spatial derivative terms since we consider the IR

limit. Apparently, the condition for a positive kinetic term is (3λ − 1)/(1 − λ) < 0, that is, the ghost-free condition on λ is 1 λ < , or λ > 1. (1.34) 3 This quadratic action also leads to a dynamical equation in momentum space 3λ − 1 ψ′′ + k2ψ = 0. (1.35) 1 − λ k k One immediately realizes that this is a wave equation, meaning that this scalar mode

is propagating with a λ-dependent speed even though no matter is present. This is

in direct contrast with the case in GR, where the equation is a Poisson equation so

that the only physical degree of freedom is the usual tensor ﬁeld.

A closer inspection of the dynamical equation (1.35) reveals that the condition

for stability of the system is (3λ − 1)/(1 − λ) > 0, or λ ∈ (1/3, 1). Unfortunately, this

inevitably violates the ghost-free condition (1.34). Thus the mode either becomes

a ghost or exhibits unstable behavior in the low energy limit. Though the scale of

instability can be pushed higher when considering the higher curvature terms, the

deviation of λ from 1 might still need to be ﬁne tuned to an extremely small value,

see Section 2.4. Such an issue is generally referred to as the instability and ghost

problem in the literature.1

The case is even worse in the non-projectable version of the model, where

the linear order evolution equation was found to be ﬁrst order in time diﬀerentials

around spatially inhomogeneous and time-dependent backgrounds [14]. The lapse

N(t, x) was also found to asymptotically tend to zero in the asymptotically ﬂat case,

thus preventing any interesting dynamics [15].

This extra scalar mode also exhibits another pathological behavior–the system

may become strongly coupled in the purely gravitation sector or when coupled with 1 Note that when λ = 1, the system is free of this problem. But λ in general cannot stay at this value as it expects quantum corrections.

13 matter in the low energy regime (IR limit), when the parameter λ ﬂows to its “rel-

ativistic value”, 1. To see this problem, consider again the perturbations in (1.32).

The action in quadratic order of ϵ written in a schematic form reads ∫ { } 2 3 −1 2 S2 ∝ ϵ dtd x (1 − λ) (∂iψ) + ··· , (1.36)

where the dots represents collectively other scalar perturbations including that for

matter, if available. But when we go to higher orders of perturbations, i.e. cubic

order of ϵ, the proportionality becomes (1 − λ)−n, where n > 1, ∫ ϵ3 { } ∝ 3 2 ··· S3 n dtd x ψ (∂iψ) + . (1.37) Mpl(1 − λ)

This suggests that when λ → 1 which is expected when the system under considera-

tion ﬂows to low energy, or when the process under consideration has a typical energy

scale higher than the strong coupling scale

∼ 1 ΛSC n , (1.38) Mpl(1 − λ) higher orders of perturbations may have an effect even bigger than that of lower order ones. Hence the perturbative description of the system breaks down. If λ cannot approach 1 at IR, no relativistic limit may be reached by the theory and this definitely contradicts with observations. Though strong coupling itself is not a problem, and one may expect that when nonlinear effects are taken into account the theory is still consistent with observations, this does put the above arguments on power-counting renormalizability invalid because those arguments are based on the assumption that the theory can be treated perturbatively.

Some of the these problems are more challenging in the version of the model with projectability abandoned, while others persist regardless of whether we keep the projectability condition. Hence when possible, one may wish to eliminate this spin-0 particle all together.

14 Several other issues speciﬁc to the versions of models that satisfy the detailed

balance condition include parity violation and wrong-sign cosmological constant prob-

lem. The parity violation problem is in fact obvious when looking at the action (1.19).

∇ l Speciﬁcally, the term Rjl jRk involves spatial derivatives of order ﬁve on the metric gij. To see the wrong-sign cosmological problem, we consider the last two terms in (1.19) κ2µ2 Λ (R − 3Λ ) . (1.39) 8 (1 − 3λ) W W For the coupling constant in front of R to be positive so that we obtain a reasonable

Newtonian constant GN ,ΛW must stay negative when λ > 1/3, comparing with the classical Einstein-Hilbert action written in ADM variables ∫ √ ( ) 1 3 ij 2 S = dtd xN g KijK − K + R − 2Λ . (1.40) 16πGN

Since we can identify R − 3ΛW as the term R − 2Λ, ΛW plays the role of the cosmological constant. This ΛW is of the wrong sign to be compatible with cosmological observations. In addition, a scalar ﬁeld with detailed balance condition is not UV stable [16] and models with this condition also generates cosmological perturbations that are not scale-invariant [17].

1.3 Organization of Dissertation

For any new theory to be considered successful, it must be self-consistent and able to pass the known experimental tests and make new predictions. The following chapters will give details on the origin of the aforementioned problems, efforts in trying to resolve these issues and the modifications to the original model along with these efforts, and on the cosmological tests.

Chapter Two introduces a generalization of the original HL model (1.19) keeping the projectability condition but without the detailed balance condition, and the problems with this generalization. In particular, we show that although Minkowski

15 spacetime is not stable, de Sitter spacetime is and hence can be taken as the legitimate

background. This chapter is based on the work published in [18].

Chapter Three gives details on a further modiﬁcation co-developed by Hoˇrava

[19]. Since the above mentioned problems all seem to relate to the extra scalar

mode, whose existence stems from the breaking of the local Lorentz symmetry, and

the deviation of λ from 1, an extension of the symmetry from Diﬀ(M, F) might eliminate this particle and restore λ back to a constant. The extension considered in

[19] is an inclusion of a local U(1) symmetry with accompanying new ﬁelds A and φ.

This extension eliminates the scalar field and, acclaimed in [19], fixes λ = 1 at the same time. However, it was soon realized that the newly introduced φ might be too powerful and an action more general than the one given in [19] is constructed [20] which again allows a variable λ. We refer to this model with variable λ as the HMT model. We study the flatness of the FLRW universe in this chapter, and discuss all possible gauge-choices for the linear scalar perturbations under the new symmetry.

We also show that the strong coupling problem re-appears when gravity is coupled with matter, but note that it can be cured with an introduction of a cutoﬀ scale M∗.

This chapter is based on the work published in [21].

Chapters Four and Five are devoted to the applications of the HMT model to the

studies on statistics of primordial cosmological perturbations. In particular, Chapter

Four investigates the application of the HMT model to the slowly-rolling inﬂationary

universe. The class of simplest inﬂation models predicts that the statistics of these

primordial quantum ﬂuctuations are largely Gaussian, i.e. the power spectrum (or

the 2-point correlation function of the ﬁelds) characterizes almost all the information

of them. In addition, the power spectrum should be almost (energy-)scale-invariant.

All these predictions are veriﬁed by observations. Hence we try to test the HMT

model with such observations. Although the statistics are essentiality described by

a Gaussian distribution, the non-Gaussian signatures are nevertheless an important

16 discriminator in model selections. It also gives richer information about the high energy physics than the ones provided with the power spectrum. This will be the topic of Chapter Five, where we evaluate the magnitude and study the k-dependence of the bispectrum–the 3-point correlation function. These two chapters are based on the work published in [22, 23, 24].Conclusions will be presented in Chapter Six.

Throughout this dissertation, the following conventions will be adopted:

• We study only the minimum case in three dimensional space, that is D = 3 = z.

• Indices that are marked with Greek letters run from 0 to 3.

• Indices marked with Latin letters run from 1 to 3.

• We adopt the Einstein summation convention.

(4) − • The signature of gµν is ( + ++). • The three-dimensional Christoﬀel symbol is deﬁned as

gil Γi ≡ (g + g − g ); (1.41) jk 2 jl,k kl,j jk,l

the three-dimensional Riemann and Ricci tensors are

i ≡ i − i e i − e i R jkl ∂kΓjl ∂lΓjk + ΓjlΓek ΓjkΓel, ≡ k Rij R ikj. (1.42)

17 CHAPTER TWO

Projetable Hoˇrava-Lifshitz Gravity without Detailed Balance Condition

This chapter published as: Y. Huang, A. Wang and Q. Wu, “Stability of the de Sitter spacetime in Hoˇrava-Lifshitz theory”, Mod. Phys. Lett. A 25, 2267-2279 (2010).

In this chapter, the projectable HL theory of gravity without detailed balance condition [25, 26] will be discussed. This generalization keeps the projectability condition but abandons the detailed balance condition. While the parity violation and

Newtonian limit problems are solved, it still displays the instability and strong coupling problem of the original model. It can also reproduce some of the results in cosmology that were originally derived under general relativity, but with possibly assumptions in deriving them. This chapter was published in citemine-1.

2.1 Introduction to the Model

Realizing that parity violation, the wrong-sign cosmological problem and the dynamical inconsistence are the inevitable consequences of the detailed balance condition, one might be tempted to break the condition softly or discard it completely.

The proposal in [25, 26] adopts the second choice: to abandon the detailed balance condition while keeping the projectability condition. The price with this discrimi- nation between the two conditions, from the pragmatic point of view, is that extra terms must be brought into the action: the number of coupling constants actually rise from 5 in Hoˇrava’s original model to 11 in the this generalization, see Eq. (2.2) below.

Abandoning the detailed balance condition has no eﬀect on the kinetic part of the action, as is obvious from discussions in Chapter One. The potential part however, must reﬂect this change. For the action to be (power-counting) renormalizable and

18 preserve parity, it must contain spatial derivatives on gij of even order up to the sixth

order. That is, SV must have the most general combination of the Ricci tensors and scalars and their derivatives ( ) 2 ij 3 ij i j k ∇ ∇i jk ··· R,R ,RijR ,R ,R∆R,RRijR ,RjRkRi , ( iRjk) R , (2.1)

where ... represents terms that are dependent of the ﬁrst nine terms due to [25]

• commutator identities;

• Bianchi identities;

• special relations appropriate to three dimensions; and

• discarding surface terms in integration by parts.

The Riemann tensors are explicitly excluded because in three dimensions they are

dependent of the above Ricci tensors and the metric gij. Terms of spatial derivatives

of the extrinsic curvature Kij are not included either since it is of ﬁrst order in time derivative (hence equivalent to third order spatial derivative for z = 3). Hence we

are lead to the most general form ∫ √ ( ) 2 3 −2 S = ζ dtd xN g LK − LV + ζ LM , (2.2)

ij 2 LK = KijK − (1 − ξ) K , (2.3) 1 ( ) L = ζ2g + g R + g R2 + g R Rij V 0 1 ζ2 2 3 ij 1 ( ) + g R3 + g RR Rij + g Ri Rj Rk ζ4 4 5 ij 6 j k i 1 [ ( )] + g R∆R + g (∇ R ) ∇iRjk , (2.4) ζ4 7 8 i jk

where LM represents the Lagrangian for matter, the g’s are dimensionless coupling constants and 1 Λ = ζ2g (2.5) 2 0 is the cosmological constant. The coupling constant ξ is related to λ through

ξ = 1 − λ. (2.6)

19 The relativistic limit in the IR, on the other hand, requires,

2 2 1 Mpl g1 = −1, ζ = = . (2.7) 16πGN 2

When ξ → 0 and g2,··· ,8 → 0, the action goes back to the Einstein-Hilbert one. The Hamiltonian constraint, super-momentum constraint and the dynamical equations can then be derived by variation of the action (2.2) with respect to N,Ni and gij. For detail, we refer readers to the articles [13, 25].

2.2 The Instability Problem

2.2.1 Scalar Perturbations in Flat FLRW Backgrounds

To study the instability problem of the this generalization, we give a very brief introduction to the scalar perturbations of ﬂat FLRW background. We delay the discussions on the gauge invariance of these perturbations to Section 3.2. The detailed ﬁeld equations and the gauge-invariant variables in this generalization are

ﬁrst given in [13, 27]. The vector and tensor perturbations are throughly studied in the article [28]. In particular, the vector perturbations are found to be identical to those given in GR and non-propagating. And the tensor perturbations are found to be generating a scale-invariant power spectrum.

The homogeneous and isotropic ﬂat universe is described by the FLRW metric,

( ) 2 2 2 i j ds = a (η) −dη + δijdx dx . (2.8) where a is the scale factor whose time dependence describes the expansion behavior of the universe and η is the conformal time (or sometimes called the comoving time) deﬁned as

a(η) dη = dt. (2.9)

¯ ¯ ′ For this metric, Kij = −aHδij and Rij = 0, where H = a /a and an overbar denotes

20 a background quantity. Then, the Hamiltonian constraint yields, ( ) 3 H2 8πG Λ 1 − ξ = ρ¯ + , (2.10) 2 a2 3 3 while the dynamical equations give rise to ( ) 3 H′ G 1 1 − ξ = −4π (¯ρ + 3¯p) + Λ, (2.11) 2 a2 3 3 whereρ ¯ andp ¯ denote the energy density and pressure of matter of the FLRW background. Similarly to GR, the super-momentum constraint is then satisﬁed identically, while the conservation laws of energy and momentum yield

ρ¯′ + 3H (¯ρ +p ¯) = 0. (2.12)

This is not an independent equation as it can be derived from Eqs. (2.10) and (2.11).

Clearly, replacing G and Λ, respectively, by G/(1−3ξ/2) and Λ/(1−3ξ/2), Eqs. (2.10) and (2.11) reduce exactly to the ones given in GR.

Linear scalar perturbation of the FLRW metric are given by1

2 δgij = a (η)(−2ψδij + 2E,ij) ,

2 δNi = a (η)B,i δN = a(η)ϕ(η). (2.13)

Choosing the quasi-longitudinal gauge [13],

ϕ = 0 = E, (2.14) we ﬁnd that the two gauge-invariant quantities deﬁned in [13] reduce to,

Φ = HB + B′, Ψ = ψ − HB. (2.15)

1 Note that we can decompose the perturbations into scalars, vectors and tensor and study separately their evolution equations at linear order only because of the O(3) symmetry possessed by the FLRW metric. At higher orders however, these perturbations are coupled to each other and no such simpliﬁcations are available. See [29] for details on this issue.

21 To second order the kinetic and potential parts of the action (2.2) take the forms, ∫ { [ ] ( ) 9 (2) 2 3 2 − ′2 H ′ ′ 2 H2 2 SK = ζ dηd xa 3ξ 2 3ψ + 6 ψψ + 2ψ ∂ B + ψ } 2 + ξB∂4B , (2.16) ∫ { } ( ) 2α ( ) 2α S(2) = ζ2 dηd3xa2 2 ∂ψ 2 − 3Λa2ψ2 − 1 ∂2ψ 2 + 2 ψ∂6ψ , (2.17) V a2 a4 ( ) ( ) 2 4 where α1 ≡ 8g2 + 3g3 /ζ and α2 ≡ 3g8 − 8g7 /ζ . The matter perturbations are written as

δJ t = −2δµ, δJ i = a−2q,i, [ ] δτ ij = a−2 (δP + 2¯pψ) γij + Π,⟨ij⟩ . (2.18)

The angled brackets on indices deﬁne the trace-free part:

1 ,k f ⟨ ⟩ ≡ f − δ f . (2.19) , ij ,ij 3 ij ,k

In GR, δµ reduces to the density perturbation δρ, and q, δP, Π to, respectively, ( ) −a(¯ρ+p ¯) v+B , the pressure perturbation δp, and the scalar mode of the anisotropic pressure.

To ﬁrst-order the Hamiltonian constraint is ∫ [ ( ) ] 3 ( ) d3x ∂2ψ − 1 − ξ H ∂2B + 3ψ′ − 4πGa2δµ = 0. (2.20) 2

The integrand is a generalization of the Poisson equation in GR [29]. The supermomentum constraint, on the other hand, reads

(2 − 3ξ) ψ′ = ξ∂2B + 8πGa q, (2.21) which generalizes the general relativity 0i constraint [29]. The trace and trace-free

22 parts of the perturbed dynamical equations yield, respectively, ( ) ξ α α ψ′′ + 2Hψ′ − 1 + 1 ∂2 + 2 ∂4 ∂2ψ 2 − 3ξ a2 a4 [ ] 8πGa2 P − 2 = − 3δ + (2 3ξ)∂ Π , (2.22) ( 3(2 3ξ) ) ( )′ α a2B = a2 + α ∂2 + 2 ∂4 ψ − 8πGa4Π. (2.23) 1 a2

The conservation laws give ∫ [ ] d3x δµ′ + 3H (δP + δµ) − 3 (¯ρ +p ¯) ψ′ = 0, (2.24) 2 q′ + 3Hq = aδP + a∂2Π. (2.25) 3

2.2.2 Instability Around Minkowski Spacetime

To see how the ghost and instability problems of scalar perturbations are

avoided in the de Sitter background, it is instructive ﬁrst to recall how they raise

in the Minkowski background. Since in this section we are mainly concerned with

IR limit, the terms proportional to α1 and α2 are highly suppressed by the Planck

2 4 scales Mpl and Mpl, respectively. Then, in the following discussions it is quite safe to neglect all these terms.

In the Minkowski background, without matter perturbations, Eq. (2.21) in the

momentum space gives, 3ξ − 2 k2B = ψ′ , (2.26) k ξ k for ξ ≠ 0. Then, Eq.(2.22) becomes

1 ′′ 2 2 ψk + k ψk = 0, (2.27) cψ

2 ≡ − 2 ≥ ≤ where cψ ξ/(2 3ξ). Clearly, it is stable only when cψ 0, that is, 0 < ξ 2/3. However, submitting Eq. (2.26) into Eqs. (2.16) and (2.17), we ﬁnd that ( ) ( ) L ≡ L − L − 1 ′2 − 2 K V = 2 ψ ∂ψ . (2.28) cψ

23 2 Therefore, unless cψ < 0 (or ξ < 0), the spin-0 graviton is a ghost. But, when ξ < 0 the scalar ﬁeld becomes unstable. Note that the spin-0 graviton becomes stable when

ξ = 0 [13].

2.2.3 Stability Around de Sitter Spacetime

The de Sitter spacetime is described by a(η) = −1/(Hη), where η ≤ 0 and the

Hubble parameter H stays as constant. In particular, η = −∞ corresponds to the

initial (t = 0) of the universe, while η = 0− to the future inﬁnity (t = ∞). When matter is not present, we have

q = δP = δµ = Π = 0, (2.29)

and the momentum constraint (2.21) yields the same equation (2.26) for ξ ≠ 0. Then, from Eqs. (2.16) and (2.17), we ﬁnd that { [ ] } 2(3ξ − 2) 9 L = α2 ψˆ′2 + (2 − ξ)(3ξ − 2)H2 − 2k2 ψˆ2 , (2.30) ξ k 2 k

where α ψ = αψˆ , α = 0 , (2.31) k k a3ξ/2

with α0 being an arbitrary constant. Thus, to have the kinetic part non-negative, we must assume either ξ ≤ 0 or ξ ≥ 2/3. However, the GR limit requires ξ = 0.

Therefore, one needs to restrict to the range ξ ≤ 0. But, in the following we shall

leave this possibility open, and show that the de Sitter spacetime is stable against

gravitational scalar perturbations for any given ξ in the IR limit. To this purpose, we ﬁrst notice that Eq. (2.22) can be cast in the form, ( ) ξk2 2 χ′′ + − χ = 0, (2.32) k 2 − 3ξ η2 k

where χk = aψk. Depending on the values of ξ, the above equation has diﬀerent solutions. In the following we consider them separately.

24 Case 1) ξ/(2 − 3ξ) < 0: In this case, Eq. (2.32) has the general solution, ( ) ( ) 1 1 χ = c 1 − ez + c 1 + e−z, (2.33) k 1 z 2 z where c1 and c2 are two integration constant, and

2 1/2 ξk z ≡ η = z0η. (2.34) 2 − 3ξ

Then, ψk and Bk are given by ( ) ( ) z −z ψk =c ˜1 z − 1 e +c ˜2 z + 1 e , ( ) (3ξ − 2)z B = c˜ ez − c˜ e−z , (2.35) k ξk2 1 2

− which are all ﬁnite as kη → 0 or t → ∞, wherec ˜i ≡ −Hci/z0. Inserting the above into Eq.(2.15), we ﬁnd that the gauge-invaraint quantities Φ and Ψ are given by − ( ) z0z(3ξ 2) z −z Φk = 2 c˜1e +c ˜2e , [ ξk ] z (3ξ − 2) − ξk2 Ψ =c ˜ z + 0 ez k 1 ξk2 [ ] z (3ξ − 2) − ξk2 +˜c z − 0 e−z, (2.36) 2 ξk2

which are also ﬁnite in the IR limit kη → 0−.

Case 2) ξ/(2 − 3ξ) > 0: In this case, Eq. (2.32) has the general solution,

( ) c ( ) χ = c sin z + c + 1 cos z + c , (2.37) k 1 2 z 2

while ψk,Bk and the gauge-invaraint quantities Φ and Ψ are ﬁnite in the IR limit kη → 0− are ﬁnite too: [ ( ) ( )] ψk =c ˜1 z sin z + c2 + cos z + c2 , c˜ z (3ξ − 2) ( ) B = 1 0 z cos z + c , k ξk2 2 2 − ( ) −c˜1z0(3ξ 2) Φk = 2 z sin z + c2 , [ ξk ] c˜ ( ) ( ) Ψ = 1 ξk2 + z2(3ξ − 2) cos z + c +c ˜ z sin z + c . (2.38) k ξk2 0 2 1 2

25 0 0 Case 3) ξ = 0: In this case, Eq. (2.21) yields ψk = ψk, where ψk is a constant, while Eq. (2.23) gives

0 2 2 Bk = ψkη + c0H η , (2.39) where c0 is another integration constant. Clearly, both of these two terms represent decaying modes (kη → 0−). Then, the corresponding Φ and Ψ are given by

2 Φk = Ψk = c0H η ≃ 0, (2.40) as η → 0−.

Therefore, it is concluded that for any given ξ the de Sitter spacetime is stable against the gravitational scalar perturbations.

2.3 New Features

By keeping the projectability condition, this generalization preserves most of the attracting features of the original model discussed in Section 1.2.3. On the other hand, the abandonment of the detailed balance condition solves some of the problems faced by the original HL model.

First, by keeping the projectability condition the system does not have the dynamical inconsistence shown in [15].

Second, by construction the action (2.2) does not possess any parity violation terms in the pure gravitation sector. In the meantime, the coupling constant g0 and g1 are independent, hence can be adjusted separately to avoid the wrong-sign cosmological constant problem, see Eqs. (2.5) and (2.7).

In the same spirit, the Lorentz violation scales are now separated from the

2 Planck scale by the coupling constants g2,··· ,8. Take for example the R term. Com- paring to the R term, it is suppressed by a mass scale √ √ ζ2 M 2 M = = pl . (2.41) g2 2g2

26 And the various Lorentz-violating terms are suppressed by various mass scales. They begin to manifest themselves only when the physical momenta are of the order of these mass scales, though the detailed mechanism for such suppression is still an open question.

The Newtonian limit, which does not exist if the detailed balance condition is kept, is now obtainable but needs a diﬀerent interpretation [10]. In GR, the New- tonian limit is obtained when the Newtonian potential–whose divergence gives the

Newtonian gravitational force–is taken as the space-dependent part of the lapse N

(4) (or the g00 component of the 4-metric). In this generalization however, the projectability condition prohibits this interpretation. Nevertheless, it can be interpreted in such a way that in the low energy limit, the Newtonian potential is rather encoded in Ni and gij while the motion of matter still doesn’t exhibit any diﬀerence from the GR case.

Applications of this model to cosmology were studied in [13, 27, 30]. In particular, it was found that the higher curvature terms act as eﬀective gravitational anisotropic stress on small scales. The Klein-Gordon equation for scalar matter ﬁeld in cosmology was derived and was shown to reduce to the standard GR form at the

FLRW background. The gravitational field equations have the Friedmann equation form after identifying G/(1 − 3ξ/2) with the Newton constant GN . Two quantities that are important in the study of early cosmology–the comving curvature perturbation R and the curvature perturbation on uniform density hyper- surface ζ (their definitions will be given later in Eqs.(4.24) and (4.25))–are gauge- invariant even under the symmetry Diff(M, F). A particular interesting result is that similar to the case in GR, they become adiabatic and stay constant on super-horizon scales but for a very different reason. In GR, these two gauge-invariant quantities conserve on large scales because of the local Hamiltonian constraint and energy conservation [31]. In the projectable HL gravity however, the Hamiltonian constraint and

27 energy conservation are global and the constancy is due to the slow-roll condition on

the time-evolution of the de Sitter background [27].

2.4 Persisting Problems

As was shown in Section 2.2.1, this generalization still faces the instability

problem of the scalar ﬁeld in the IR limit around the Minkowski metric. Moreover,

the system is still strongly coupled at a rather low energy limit.

Recall that in the analysis of the instability of the Minkowski metric, we ignored

the α1 and α2 terms (see the paragraph above Eq.(2.26)). An inclusion of these higher curvature terms would push the scale of inability to k ∼ 1/(|cψ|M), where M is the scale above which the higher curvature terms must be considered, similar to that

considered in Eq.(2.41). If the instability were not to develop within the age of our

universe, that is 1 1 > , (2.42) |cψ|M H0 then this would inevitably require either a rather low scale for M or a ﬁne tuning of

λ. Considering the observational constraints on H0 and M, the value of λ must be ﬁne-tuned to a value |1−λ| < 10−61 [32, 33]. But this would push the strong coupling scale (1.38) too low to be acceptable, and as was pointed out in [32], λ cannot be too close to 1 considering the Cherenkov process. Besides, quantum corrections will in no doubt drive λ away from this range. Hence we need a better solution for this issue.

The strong coupling problem, on the other hand, is investigated following the eﬀective ﬁeld theory approach in [33]. There it was shown that the mechanism proposed in [34] does not really solve the strong coupling problem in the projectable generalization as it inevitably leads to the requirements on the Lorentz-violating scales

3/2 M ≲ Mpl|cψ| , (2.43)

28 a too much too low scale. Note that when Vainshtein mechanism is applied, that

is, considering the nonlinear eﬀects, then the spherically symmetric, static vacuum

spacetimes are free from the strong coupling problem, as was shown in [35]. In

addition, Wang and Wu [33] found a class of non-perturbative cosmological solutions

and showed that it reduces smoothly to the de Sitter spacetime in the relativistic

time. This, together with the stability of the de Sitter spacetime, hints that one

may take the de Sitter spacetime as a legitimate background and the scalar graviton

decouples in the IR limit once the nonlinear eﬀects are taken into account.

Because these problem are all related to the scalar graviton, whose presence

is the result of breaking the general diﬀeomorphism, a possible cure for these two

persisting problems is an extension of the symmetry of the theory and hence elimi-

nates the scalar particle all together and forces λ = 1. This is the model that will be discussed in details in the following chapters.

2.5 Abandoning Both Conditions

At this point, we should point out that another line of development parallel to the generalization discussed above and to the later HMT model is the model proposed by Blas, Pujol`asand S. Sibiryakov (BPS) [36], which is usually referred to as the “healthy extension” of the the HL model.

The BPS model abandons both the projectability and the detailed balance condition. As expected, the number of coupling constants of their model grows to a much higher value (in fact, more than 60) due to the mandated inclusion of the terms involving the spatial derivatives of the lapse

−1 ai ≡ N ∂iN. (2.44)

With this new constituent the IR behavior can be signiﬁcantly improved, particularly

i with the term aia which is of second order in spatial derivative. (Note that the only second order spatial derivative in the original HL model (1.19) and the the projectable

29 generalization (2.2) is the three-dimensional Ricci scalar R which also appears in

GR.) This removes the instability problem of the models that kept the projectability

condition. The solution to the strong coupling problem is through the control of the

UV cutoﬀ scale M∗ above which the linear perturbations become invalid and making

the strong coupling scale lower than it [34]

1 ∗ ∼ M > ΛSC n . (2.45) Mpl(1 − λ)

The model is also free of the dynamical inconsistence of other models [15] that abandons the projectability condition [37]. On the other hand, the model does not posses

GR as its low energy limit, but a Lorentz-violating scalar-tensor theory. The scalar mode still exists, though its eﬀects can be made week and compatible with observations [36] by adjusting those coupling constants.

30 CHAPTER THREE

General Covariant Hoˇrava-Lifshitz Gravity with Variable λ

This chapter published as: Y. Huang and A. Wang, “Stability, ghost, and strong coupling in nonrelativistic general covariant theory of gravity with λ ≠ 1”, Phys. Rev. D 83,104012 (2011).

With all the problems caused by the extra spin-0 gravitons and the deviation

of λ from 1, it would be a great improvement if some mechanism can eliminate them

and at the same time ﬁxes λ.

The solution proposed by Hoˇrava and C.M. Melby-Thompson [19] achieved this goal with an extension of the symmetry respected by the theory to

U(1) ⋉ Diﬀ(M, F). (3.1)

Two new auxiliary ﬁelds with this extension–a U(1) gauge ﬁeld A and the Newtonian prepotential φ–are required for invariance with respect to (3.1). Under this new symmetry, the special status of time maintains, so that the anisotropic scaling (1.4) with z > 1 is still realized, whereby the UV behavior of the theory can be considerably improved.

As shown explicitly in [19], the U(1) symmetry pertains speciﬁcally to the case

λ = 1. And it was conﬁrmed that the extra scalar degree of freedom in the gravity sector is indeed eliminated [38]. However, this claim on λ was soon challenged by da Silva [20], who argued that the introduction of the Newtonian prepotential is so powerful that an action with λ diﬀerent from 1 respects the enlarged symmetry

(3.1) as well. Once λ deviates from 1, the old issues may all rise again and deserve solutions, if possible.

In this chapter, we ﬁrst illustrates how the action must be modiﬁed in cope with the extended symmetry [19], followed by the generalization of [20]. We then

31 investigate the same challenging problems in the new setup (HMT model) in detail.

In Section 2, we study the FLRW universe with any given spatial curvature, and derive

the generalized Friedmann equation and conservation law of energy. An immediate

by-product of the setup is that, the FLRW universe is necessarily ﬂat, when it is

ﬁlled with (multi-) scalar, vector or fermionic ﬁelds, provided that the coupling of the

gravitational sector to matter is described by the recipe given in [20]. We develop

the general formulae for the linear scalar perturbations of the FLRW universe. We

also discuss gauge choices in the HMT setup and found that some choices that were

not available before the symmetry extension are now available.

Applying these formulae to the Minkowski background in Section 3, we study

the stability problem, and show explicitly that it is stable and the spin-0 graviton is

eliminated even for λ ≠ 1. This conclusion is the same as that obtained by da Silva

for the maximal symmetric spacetimes with detailed balance condition, in which

the Minkowski spacetime is not a solution of the theory [20]. The gauge ﬁeld A

and the Newtonian pre-potentail φ have no contributions to the vector and tensor

perturbations, so the results presented in [28] in the SVW setup can be equally

applied to the HMT setup even with λ ≠ 1. In particular, it was shown that the vector perturbations vanish identically in the Minkowski background. Thus, similar to general relativity, a free gravitational ﬁeld in this setup is completely described by a spin-2 massless graviton even with λ ≠ 1.

We study the ghost and strong coupling problems in Section 4, and derive the ghost-free conditions in terms of λ. To study the strong coupling problem, we consider two diﬀerent kinds of spacetimes all ﬁlled with matter 1 : (a) spacetimes in which

1 Note that, to count the number of the degrees of the propagating gravitational modes, one needs to consider free gravitational fields. Another way to count the degrees of the freedom is to study the structure of the Hamiltonian constraints [19, 39]. On the other hand, to study the ghost and strong coupling problems, one needs to consider the cases in which the gravitational modes are different from zero. In this chapter, this is realized by the presence of matter fields. That is, it is the matter that produces the gravitational modes. Clearly, this does not contradict to the conclusion that the spin-0 mode is eliminated in such a setup. A similar situation also happens in GR, in which

32 the ﬂat FLRW universe can be considered as their zero-order approximations; and

(b) spherical statics spacetimes in which the Minkowski spacetime can be considered

as their zero-order approximations. We ﬁnd that the strong coupling problem indeed

exists in both spacetimes. To avoid this, one may follow [34] to introduce an energy

scale M∗ that satisﬁes the condition [40],

M∗ < Λω, (3.2)

where M∗ is the suppression energy of the sixth-order derivative terms, and Λω is the would-be strong coupling energy scale, given by, ( ) 3/2 ζ 5/4 Λω ≃ |λ − 1| Mpl, (3.3) c1

where ζ is related to the Planck mass Mpl through Eq.(2.7), and c1, defined in (3.31), represents the coupling of a scalar field with the gauge field A. In the case without

the projectability condition, the observed alignment of the rotation axis of the Sun

15 with the ecliptic requires M∗ ≲ 10 GeV [34]. Similar considerations have not been

carried out in the current version of the HL theory, and the upper bound of M∗ is unknown. From the above expression, it is clear that λ cannot be precisely equal to one, in order for the BPS mechanism to work either without [34] or with [40] the projectability condition. This chapter was published in [21].

We would like to note that the consistence of this model with solar system tests was investigated recently [41], and found that it is consistent with all such tests, provided that the gauge ﬁeld and Newtonian prepotential are part of the metric.

Finally in Section 5, we brieﬂy review a model proposed in [42] that endows the extended symmetry but abandons the projectability condition and breaks the detailed balance condition softly.

scalar perturbations of the FLRW universe in general do not vanish, although the only degrees of the freedom of the gravitational sector are the spin-2 massless gravitons.

33 3.1 Introduction to the General Covariant Theory

3.1.1 The Case with λ = 1

Though the original HL model was expected to resemble GR in the low energy

limit, the similarity has its limits, with the diﬀerence in the propagating degree of

freedom the most challenging one. In order to get closer to general relativity, this

extra scalar mode is better to be eliminated as a gauge artifact. The extension of the

gauge symmetry of the theory is a means to achieve such a goal.

The construction in [19] had its inspiration from the ultralocal theory of gravity

which drops the spatial scalar curvature term R in the classic Einstein-Hilbert action

(cf. Eq.(1.40)). This step selects a preferred foliation F of spacetime, a similarity

with the symmetry of the original HL model. The spatial diﬀeomorphism symmetries

have been kept (the commutator of the generators of supermomentum constraint is

the same as that of GR [43]), but the time reparametrization symmetry has been

linearized, and its algebra contracted to a local U(1) gauge symmetry [19, 44]–the

generator of the superhamiltonian constraint commute with itself, whereas the com-

mutator in GR gives a ﬁeld-dependent result [43]. In terms of the Lagrangian symme-

tries, the ultralocal theory’s action possess a symmetry group which takes the form

of a semi-direct product

U(1) ⋉ Diﬀ(M, F). (3.4)

This maintains the special status of time and as a result is friendlier with the way time is treated in quantum mechanics, a virtue shared with the ADM formalism.

On the other hand, it has an interesting extra beneﬁt–it gives the theory the same number of local gauge symmetries as GR, i.e. the theory is generally covariant. This

is the symmetry that the proposal [19] tried to implement.

34 As noted in [3], the linearized HL model around ﬂat spacetime with λ = 1 enjoys an (extra) global symmetry, whose generator α acts on the ﬁelds via

δαN = δαgij = 0, δαNi = ∂iα, (3.5) where α is a function of spatial coordinates only, i.e.α ˙ = 0. Hence the linearized theory, when λ = 1, does not have that extra scalar ﬁeld. This global symmetry can be gauged (promoting α to a function of x and t) in the linearized theory with

λ = 1 easily with an introduction of an auxiliary ﬁeld A that has the transformation property

δαA =α, ˙ (3.6) and a proper coupling of it with the other ﬁelds (or rather the linearized ﬁelds).

One is thus tempted to generalize this success to the nonlinear case. Such an extension however, is nontrivial. First, in generalizing the global symmetry to the nonlinear case, the transformations of the ADM variables take the form

δαN = δαgij = 0, δαNi = N∇iα. (3.7)

This requires the scaling dimension

[α] = z − 2. (3.8)

The time independence of α must now be expressed with the covariant time derivative

i α˙ − N ∇iα = 0, (3.9) and the scaling dimension of A would be

[A] = 2z − 2. (3.10)

The auxiliary gauge ﬁeld now transforms under the foliation-preserving diﬀeomor- phisms (1.16) and the gauge symmetry as

˙ ˙ i i δA = fA + fA + ζ ∂iA, δαA =α ˙ − N ∇iα = 0. (3.11)

35 The obstruction lies in gauging the symmetry is that, in 3 + 1 dimension with a sole introduction of a coupling between A and gij, the original HL model (and hence the projectable generalization without the detailed balance condition) is not invariant under the transformation α(x, t) even at λ = 1.

The solution found in [19] is an introduction of another auxiliary ﬁeld φ, referred to as the Newtonian prepotential, which transforms as

i δφ = fφ˙ + ζ ∂iφ, δαφ = −α, (3.12) and has scaling dimension same as α

[φ] = z − 2. (3.13)

Note the diﬀerence between the notations used here and the ones used in [19, 20].2

Decorating the action with couplings between the ADM variables and these two new ﬁelds, the action with λ = 1 ∫ { 2 √ [ ] 3 ij − 2 − L Gij ∇ ∇ S = 2 dtd x g N KijK K V + φ (2Kij + i jφ) κ }

− A (R − 2Λg) , 1 Gij = Rij − gijR + Λ gij, (3.14) 2 g is now invariant under the symmetry (3.1). This action, when varied with respect to

A, yields a constraint equation

R − 2Λg = 0 (3.15) rather than a dynamical one. It is exactly this constraint that eliminated the extra scalar mode in the gravity sector. Interestingly, the ﬁeld A can be interpreted as the

Newtonian potential in an expansion in powers of 1/c as was argued in [19], where c is a formal expansion parameter. This was later conﬁrmed in [41].

2 − HMT HMT − HMT G HMT In particular, we shall have Kij = Kij , Λg = Ω , φ = ν , ij = Θij , where quantities with the super-index “HMT” are those used in [19, 20].

36 Note that a composite quantity

N A = −φ˙ − N i∇ φ − ∇i∇ φ (3.16) i 2 i transforms in the same way as A,

˙ ˙ i i δA = fA + fA + ζ ∂iA, δαA =α ˙ − N ∇iα. (3.17)

Hence A in (3.14) can be replaced with a linear combination of A and A.

3.1.2 A Minimum Substitution with Variable λ

If the model in (3.14) is truly the most general one that respects the symmetric

(3.1), it would be a major success. But the claim was soon challenged in [20] which showed that the extended symmetry actually allows an action that is more general than the one given in (3.14) while at the same time permits a variable λ.

The minimum substitution considered in [20] is actually simple,

Ni → Ni − ∇iφ. (3.18)

With such substitution the action ∫ √ ( ) 2 3 −2 S = ζ dtd xN g LK − LV + Lφ + LA + Lλ + ζ LM , (3.19)

is also invariant under the extended symmetry, where LV is given as before in Eq.(2.4) and the coupling with A is kept the same ( ) A L = 2Λ − R , (3.20) A N g

but the kinetic part now allows a variable λ

ij 2 LK = KijK − λK , (3.21) and the coupling with φ is extended with a λ dependent term ( ) ij Lφ = φG 2Kij + ∇i∇jφ , ( )[( ) ] 2 2 2 Lλ = 1 − λ ∇ φ + 2K∇ φ . (3.22)

37 3.1.3 Coupling with Matter

How the Hoˇrava-Lifshitz theory of gravity couples with matter is an important

topic. Couplings with scalar and vector matter ﬁelds in the original HL model were

first studied in [16, 26, 45, 46, 47]. Specifically, when a scalar matter field couples

i only with N,N , gij, the most general action takes the form [26, 27], ∫ ( ) √ (0) i 3 L(0) Sχ N,N , gij; χ = dtd xN g χ , (3.23) where ( ) f(λ) 2 L(0) − i∇ − V χ = χ˙ N iχ , 2N 2 ( ) 1 V = V (χ) + + V (χ) (∇χ)2 + V (χ) P2 + V (χ) P3 2 1 2 1 3 1 2 +V4 (χ) P2 + V5 (χ)(∇χ) P2 + V6(χ)P1P2, (3.24)

with V (χ) and Vn(χ) being arbitrary functions of χ, and

n Pn ≡ ∆ χ. (3.25)

Note that in the kinetic term we added a factor f(λ), which is an arbitrary function

of λ, subjected to the requirements: (i) The scalar ﬁeld must be ghost-free in all the

energy scales, including the UV and IR. (ii) In the IR limit, the scalar ﬁeld has a

well-deﬁned velocity, which should be equal or very closed to its relativistic value.

(iii) The stability condition in the IR requires [48],

f(λ) > 0. (3.26)

In addition to proposing the minimum substitution above, da Silva [20] also

proposed a way to couple with matter in the HMT setup. By requiring the matter

sector to respect the symmetry (3.1) as well, the action for the matter sector was

prescribed to have a schematic form ∫ √ 3 SM = dtd x gZ(ψn, gij, ∇k)(A − A), (3.27)

38 where Z is the most general scalar operator under (3.1). It is built with the matter

ﬁelds and gij and has the scaling dimension

[Z] = 2. (3.28)

This is necessary as the Lagrangian density must have dimension 2z = 6.

To couple a scalar matter ﬁeld with the gauge ﬁeld A and the Newtonian prepotential φ, we make the replacement in Eq. (3.23) ( ) ( ) (0) i → i Sχ N,N , gij; χ Sχ N,N , gij, A, φ; χ , (3.29) where ( ) ( ( ) ) i ≡ A (0) i ∇i Sχ N,N , gij, A, φ; χ Sχ (χ, A) + Sχ N, N + N φ , gij; χ , (3.30) with ∫ √ [ ( ) ]( ) A ≡ 3 ∇ 2 − A Sχ dtd x g c1(χ)∆χ + c2(χ) χ A . (3.31)

Thus, the action can be cast in the form, ∫ √ { } 3 L(0) L(A,φ) Sχ = dtd xN g χ + χ , (3.32)

(0) where Lχ is given as before in Eq. (3.24) and [ ] A − A ( ) L(A,φ) = c ∆χ + c ∇χ 2 χ N 1 2 ( )( )( ) f i k − χ˙ − N ∇iχ ∇ φ ∇kχ N[ ] f ( )( ) 2 + ∇kφ ∇ χ , (3.33) 2 k

3.1.4 The Field Equations

With the action (3.19), it can be shown that the Hamiltonian and super- momentum constraints are given respectively by, ∫ ∫ √ [ ( )( ) ] √ 3 ij 2 2 3 t d x g LK + LV − φG ∇i∇jφ − 1 − λ ∇ φ = 8πG d x g J , (3.34) [ ( ) ] j 2 ∇ πij − φGij − 1 − λ gij∇ φ = 8πGJi, (3.35)

39 where

δ (NL ) δL J t ≡ 2 M ,J ≡ −N M , π ≡ −K + λKg . (3.36) δN i δN i ij ij ij

Variations with respect to φ and A yield, respectively,

( ) ( ) ( ) ij 2 2 G Kij + ∇i∇jφ + 1 − λ ∇ K + ∇ φ = 8πGJφ, (3.37)

R − 2Λg = 8πGJA, (3.38) where δL δ (NL ) J ≡ − M ,J ≡ 2 M . (3.39) φ δφ A δA On the other hand, the dynamical equations now read, { } [ ] 1 √ ( ) √ g πij − φGij − 1 − λ gij∇2φ N g ,t ( ) ( )[( ) ] = −2 K2 ij + 2λKKij − 2 1 − λ K + ∇2φ ∇i∇jφ + Kij∇2φ [ ] 1 [ ] ( ) ∇ k ij − k(i j) − ∇(i j) − ij∇ k + k N π 2π N + 1 λ 2 Fφ g kFφ N ( ) 1 + gij L + L + L + L + F ij + F ij + F ij + 8πGτ ij, (3.40) 2 K φ A λ φ A

2 ij ≡ il j ≡ where (K ) K Kl , f(ij) (fij + fji) /2, and ( √ ) 1 δ − gL ∑8 ij V ns ij F ≡ √ = gsζ (Fs) , g δgij [ ( s=0 ) ] 1 F ij = ARij − ∇i∇j − gij∇2 A , A N ( ) N i F i = K + ∇2φ ∇iφ + ∇2φ, φ N ∑3 ij ij Fφ = F(φ,n), (3.41) n=1

ij with ns = (2, 0, −2, −2, −4, −4, −4, −4, −4). The stress 3-tensor τ is deﬁned as (√ ) 2 δ gL τ ij = √ M , (3.42) g δgij

40 ij and the geometric 3-tensors (Fs)ij and F(φ,n) are given originally in [13]

1 (F ) = − g , 0 ij 2 ij 1 (F ) = − g R + R , 1 ij 2 ij ij 1 (F ) = − g R2 + 2RR − 2∇ ∇ R + 2g ∇2R, 2 ij 2 ij ij (i j) ij 1 (F ) = − g R Rmn + 2R Rk − 2∇k∇ R + ∇2R + g ∇ ∇ Rmn, 3 ij 2 ij mn ik j (i j)k ij ij m n 1 (F ) = − g R3 + 3R2R − 3∇ ∇ R2 + 3g ∇2R2, 4 ij 2 ij ij (i j) ij 1 (F ) = − g RRmnR + R RmnR + 2RR Rk − ∇ ∇ (RmnR ) 5 ij 2 ij mn ij mn ki j (i j) mn n 2 mn 2 mn − 2∇ ∇(iRRj)n + gij∇ (R Rmn) + ∇ (RRij) + gij∇m∇n (RR ) , 1 3 ( ) (F ) = − g RmRnRp + 3RmnR R + ∇2 R Rn 6 ij 2 ij n p m ni mj 2 in j 3 ( ) ( ) + g ∇ ∇ Rk Rln − 3∇ ∇ R Rnk , 2 ij k l n k (i j)n 1 (F ) = − g (∇R)2 + (∇ R)(∇ R) − 2R ∇2R + 2∇ ∇ ∇2R − 2g ∇4R 7 ij 2 ij i j ij (i j) ij 1 (F ) = − g (∇ R )(∇pRmn) − ∇4R + (∇ R )(∇ Rmn) 8 ij 2 ij p mn ij i mn j ( ) ( ) ∇ ∇p n ∇n∇ ∇2 ∇ n ∇ m + 2 ( pRin) Rj + 2 (i Rj)n + 2 n Rm (iRj) ( ) ( ) − ∇ ∇ mn − ∇ ∇n m − ∇n∇m∇2 2 n Rm(j i)R 2 n Rm(i Rj) gij Rmn, (3.43) and {( ) ( ) 1 ij ∇2 ij − j ∇j∇ ik F(φ,1) = φ 2K + φ R 2 2Kk + kφ R 2 ( ) ( )( )} − i ∇i∇ jk − − ij ∇i∇j 2 2Kk + kφ R 2Λg R 2K + φ , [ ( ) ] 1 ( ) 2N k 2φ F ij = ∇ 2 φGk(i∇j)φ − φGij + ∇kφ + Gk(i∇ N j), (φ,2) 2 k N N k 1 { ( ) } F ij = 2∇ ∇(if j)k − ∇2f ij − ∇ ∇ f kl gij , (3.44) (φ,3) 2 k φ φ k l φ where {( ) ( ) } 1 f ij = φ 2Kij + ∇i∇jφ − 2K + ∇2φ gij . (3.45) φ 2

41 t i ij The matter components (J ,J ,Jφ,JA, τ ) satisfy the conservation laws, ∫ [ √ 1 (√ ) 2N (√ ) d3x g g˙ τ kl − √ gJ t + √k gJ k kl g ,t N g ,t ] A √ −2φJ ˙ − √ ( gJ ) = 0, (3.46) φ N g A ,t 1 √ J k N ∇kτ − √ ( gJ ) − (∇ N − ∇ N ) − i ∇ J k ik N g i ,t N k i i k N k J +J ∇ φ − A ∇ A = 0. (3.47) φ i 2N i

When the matter is a scalar ﬁeld and couple with gravity in the form prescribed in the Eq. (3.32), the matter currents take the form [ ( ) ] [ ] f 2 ( ) ( ) J t = −2 χ˙ − N k∇ χ + V − c △χ + c ∇χ 2 ∇φ 2 2N 2 k 1 2 [ ] ( )( ) 2 k +f ∇ φ ∇kχ , (3.48) [ ( ) ] [ ] f ( ) ( ) J i = χ˙ − N k + N∇kφ ∇ χ ∇iχ + c △χ + c ∇χ 2 ∇iφ, (3.49) N k 1 2 { [ ( ) ]} 1 √ 2 Jφ = √ g c1△χ + c2 ∇χ N g ,t { [ ] 1 ( )( ) − ∇ f χ˙ − N k + N∇kφ ∇ χ ∇iχ N i k [ ( ) ] ( )} 2 i i + c1△χ + c2 ∇χ × N + N∇ φ , (3.50) [ ( ) ] 2 JA = 2 c1△χ + c2 ∇χ . (3.51)

The stress 3-tensor reads

(0) φ τij = τij + τij , (3.52)

(0) L(0) with τij corresponds to χ in Eq.(3.24) and is given as [27] { [( ) ]} (0) L(0) ∇ V V ∇k V ∇k τij = gij χ + k ,1 + ∆ ,2 χ + ,2 ∆χ ( ) + 1 + 2V1 + 2V5P2 (∇iχ)(∇jχ) ( )( ) ( )( ) −2 ∇(i∆V,1 ∇j)χ − 2 ∇(iV,2 ∇j)∆χ , (3.53)

42 φ L(A,φ) and τij stems from χ in Eq.(3.33) { } 1 [ ( ) ] φ L(A,φ) − ∇ − A ∇k τij = gij χ k c1 A χ [N ] 2(A − A) ( )( ) + c ∇ ∇ χ + c ∇ χ ∇ χ N 1 i j 2 i j [ ( ) ]( )( ) 2 + c1∆χ + c2 ∇χ ∇iφ ∇jφ [ ( )( )]( )( ) 2f k k + χ˙ − N + N∇ φ ∇kχ ∇(iχ ∇j)φ N [ ] 2 ( ) + ∇ c A − A ∇ χ , (3.54) N (i 1 j) where

∂V V ≡ P P2 P ,1 = 2V2 1 + 3V3 1 + V6 2, ∂P1 V ∂ 2 V,2 ≡ = V4 + V5(∇χ) + V6P1. (3.55) ∂P2

On the other hand, the variation of the action (3.32) with respect to χ yields the following generalized Klein-Gordon equation, { } √ [ ] f g ( ) √ χ˙ − N k + N∇kφ ∇ χ N g N k { ,t } [ ] f ( ) ( ) = ∇ χ˙ − N k + N∇kφ ∇ χ N i + N∇iφ N 2 i k { } [ ] 2gij + ∇ ∇ (A − A)c − (A − A)c ∇ χ N (i j) 1 2 j) [ ] [ ] A − A ( ) ( ) + c′ ∆χ + c′ ∇χ 2 + ∇i 1 + 2V + 2V P ∇ χ N 1 2 1 5 2 i 2 −V,χ − ∆ (V,1) − ∆ (V,2) , (3.56)

′ ≡ where c1 ∂[c1(χ)]/∂χ, and

∂V V ≡ = V ′ + V ′(∇χ)2 + V ′P2 + V ′P3 ,χ ∂χ 1 2 1 3 1 ′P ′ ∇ 2P ′P P + V4 2 + V5 ( χ) 2 + V6 1 2, (3.57)

′ ≡ with the convention Vi ∂[Vi(χ)]/∂χ.

43 3.2 Scalar Cosmological Perturbations at Linear Order

3.2.1 Flatness of FLRW Universe

The homogeneous and isotropic FLRW universe is described by,

¯ ¯ 2 N = 1, Ni = 0, g¯ij = a (t)γij, (3.58)

( )− 1 2 2 2 ≡ 2 2 2 where γij = δij 1 + 4 κr , with r x + y + z , κ = 0, 1. To be consistent, we would use symbols with bars to denote the quantities of background from here below.

Using the U(1) gauge freedom, on the other hand, we can always set

φ¯ = 0. (3.59)

Then we ﬁnd

¯ 2 ¯ Kij = −a Hγij, Rij = 2κγij,

¯ij −4 ¯ ij ¯ij ¯i FA = 2κa Aγ , Fφ = 0, Fφ = 0,

¯ij −2 ij −2 2 −4 3 −6 F = a γ (−Λ + κa + 2∆1κ a + 12∆2κ a ) , (3.60)

2 where H =a/a, ˙ Λ ≡ ζ g0/2, and

3g + g 9g + 3g + g ∆ ≡ 2 3 , ∆ ≡ 4 5 6 . (3.61) 1 ζ2 2 ζ4

Hence we obtain

( ) 6κ 12∆ κ2 24∆ κ3 ¯ 2 ¯ 1 2 LK = 3 1 − 3λ H , LV = 2Λ − + + , ( ) a2 a4 a6 ¯ ¯ −2 ¯ ¯ LA = 2A Λg − 3κa , Lφ = 0 = Lλ. (3.62)

It can be shown that the super-momentum constraint (3.35) is satisﬁed identically for J¯i = 0, while the Hamiltonian constraint (3.34) yields,

1( ) κ 8πG Λ 2∆ κ2 4∆ κ3 3λ − 1 H2 + = ρ¯ + g + 1 + 2 , (3.63) 2 a2 3 3 a4 a6

44 where J¯t ≡ −2¯ρ. On the other hand, Eqs.(3.37) and (3.38) give, respectively, ( ) κ 8πG H Λ − = − J¯ , (3.64) g a2 3 φ 3κ − Λ = 4πGJ¯ , (3.65) a2 g A while the dynamical equation (3.40) reduces to ( ) 1( )a¨ 4πG 1 2∆ κ2 8∆ κ3 1 κ 3λ − 1 = − (¯ρ + 3¯p) + Λ − 1 − 2 + A¯ − Λ , (3.66) 2 a 3 3 g a4 a6 2 a2 g whereτ ¯ij =p ¯g¯ij. The conservation law of momentum (3.47) is satisﬁed identically, while the one of energy (3.46) reduces to,

¯ ¯ ρ¯˙ + 3H (¯ρ +p ¯) = AJφ. (3.67)

It is interesting to note that the energy of matter is not conserved in general, due to its interaction with the gauge ﬁeld A¯ and the Newtonian pre-potentialφ ¯. This might have profound implications in cosmology.

From Eqs.(3.64) and (3.65), one can see that when

¯ ¯ JA = 0 = Jφ, (3.68)

the universe is necessarily flat, κ = 0 = Λg. This is true for the case where the ¯ source is a scalar field, as can be seen from Eqs.(3.50) and (3.51), where both JA and ¯ Jφ are proportional to the spatial gradients of the scalar field χ. This can be easily generalized to the case with multi-scalar fields.

For a vector ﬁeld (A0,Ai), we have [A0] = 2, [Ai] = 0 [26]. Then, we ﬁnd

i Z(A0,Ai, gij, ∇k) = KBiB , (3.69)

i where K is an arbitrary function of A Ai, and

1 ε jk B = √i F , ∇iB = 0, (3.70) i 2 g jk i

45 with Fij ≡ ∂jAi − ∂iAj. This can be easily generalized to several vector ﬁelds,

(n) (n) (A0 ,Ai ), for which we have ∑ ⃗ ⃗ ∇ K (m) (n)i Z(A0, Ai, gij, k) = mnBi B , (3.71) m,n

K (k)i (l) where m,n is an arbitrary function of A Ai . Then, it is easy to show that in ¯ ¯(m) the FLRW background, we have JA = 0, because Bi = 0 [47], as can be seen from ¯ Eq.(3.70). With the gauge choice (3.59), one can also show that Jφ = 0. Therefore, an early universe dominated by vector fields is also necessarily flat. This can be further generalized to the case of Yang-Mills fields [45]. For fermions, on the other hand, their dimensions are [ψn] = 3/2 [46]. Then, Z(ψn, gij, ∇k) cannot be a functional of ¯ ¯ ψn. Therefore, in this case JA and Jφ vanish identically. Although we cannot exhaust all the matter fields, with the special form of the coupling given by Eq.(3.27), it is quite reasonable to argue that the universe is necessarily flat for all cosmologically viable models in the HMT setup. Therefore, from now on, we shall consider only the flat FLRW universe, i.e.,

κ = 0 = Λg, (3.72) for which Eq.(3.68) holds.

3.2.2 Linear Scalar Cosmological Perturbations

When we consider perturbations, it’s most convenient to turn to the conformal ∫ time η, where η = dt/a(t). Under this coordinate transformation, the gravitational and gauge ﬁelds transform as,

˜ i ˜ i N = aN,N = aN , gij =g ˜ij,

A = aA,˜ φ =φ, ˜ (3.73) where the quantities with tildes are the ones deﬁned in the coordinates (t, xi). With these in mind, we write the linear scalar perturbations of the metric in terms of the

46 conformal time η as,

( ) 2 2 δN = aϕ, δNi = a B,i, δgij = −2a ψγij − E,ij ,

A = Aˆ + δA, φ =φ ˆ + δφ, (3.74) where

Aˆ = aA,¯ φˆ =φ, ¯ (3.75) and A¯ andφ ¯ are the gauge ﬁelds of the background in the (t, xi) coordinates. Quan- tities with hats denote the ones of the background in the coordinates (η, xi). Note also that we have gone from the covariant derivatives with respect tog ¯ij in Eq.(3.58) to the partial derivatives, since we have proven that under the matter-couplings considered here, we have spatially ﬂat FLRW universe. Under the gauge transformations

(1.16), we ﬁnd that

′ ϕ˜ = ϕ − Hξ0 − ξ0 , ψ˜ = ψ + Hξ0,

B˜ = B + ξ0 − ξ′, E˜ = E − ξ,

′ δφ˜ = δφ − ξ0φˆ′, δA˜ = δA − ξ0Aˆ′ − ξ0 A,ˆ (3.76) where f = −ξ0, ζi = −ξ,i, H ≡ a′/a, and a prime denotes the ordinary derivative with respect to η. Under the U(1) gauge transformations, on the other hand, we ﬁnd that

ϵ ϕ˜ = ϕ, ψ˜ = ψ, E˜ = E, B˜ = B − , a δφ˜ = δφ + ϵ, δA˜ = δA − ϵ′, (3.77) where ϵ = −α. Then the gauge transformations of the whole group U(1)⋉Diﬀ(M, F) will be the linear combination of the above two. Since we have six unknown and three arbitrary functions, the total number of the gauge-invariants of U(1) ⋉ Diﬀ(M, F)

47 is N = 6 − 3 = 3. These gauge-invariants can be constructed as, ( ) ( )( ) − 1 − ′ − 1 ′′ − H ′ − Φ = ϕ ′ aσ δφ ( ) φˆ φˆ aσ δφ , a − φˆ a − φˆ′ 2 H ( ) Ψ = ψ + aσ − δφ , a − φˆ′ [ ( ) ( )]′ a δφ − φˆ′σ − Aˆ aσ − δφ Γ = δA + , (3.78) a − φˆ′ where σ ≡ E′ − B.

Then, considering Eq.(3.59), we ﬁnd that the above expressions reduce to

1 ′ Φ = ϕ − (aσ − δφ) , a H Ψ = ψ − (δφ − aσ) , a [ ]′ Aˆ Γ = δA + δφ − (aσ − δφ) , (φ ˆ = 0). (3.79) a

The expressions for Φ and Ψ now take precisely the same forms as those deﬁned in

[13], which are also identical to those given in GR [29]. Using the U(1) gauge freedom

(3.77), we shall set

δφ = 0. (3.80)

This choice completely ﬁxes the U(1) gauge.

For the general perturbations (3.74), we have { [ ]} ′ δKij = a ψ δij − σ,ij + H (2ψ + ϕ) δij − 2E,ij ,

2 δRij = ψ,ij + ∂ ψδij. (3.81)

Thus, to ﬁrst-order the Hamiltonian and momentum constraints become, respectively, { } ∫ [ ] 1( ) d3x ∂2ψ − 3λ − 1 H 3(ψ′ + Hϕ) − ∂2σ − 4πGa2δµ = 0, (3.82) 2 ( ) ( ) 1 (3λ − 1) ψ′ + Hϕ + (1 − λ)∂2 σ − δφ = 8πGaq + ∆(η), (3.83) a where 1 1 δµ ≡ − δJ t, δJ i ≡ q,i, (3.84) 2 a2

48 ,i ij q = δ q,j, and ∆(η) is an integration function. In GR, it is usually set to zero [29]. However, in the present case, since ϕ = ϕ(η), another interesting choice is

∆(η) = (3λ − 1)Hϕ, which will cancel the second term in the left-hand side of

Eq.(3.83).

On the other hand, the linearized equations (3.37) and (3.38) reduce, respectively, to [ ( )] ( ) ( ) 1 2H∂2ψ + 1 − λ ∂2 3 ψ′ + Hϕ − ∂2 aσ − δφ = 8πGa3δJ , (3.85) a φ 2 2 ∂ ψ = 2πGa δJA. (3.86)

The linearly perturbed dynamical equations can be divided into the trace and traceless parts. The trace part reads, ( ) 1 ( ) ψ′′ + 2Hψ′ + Hϕ′ + 2H′ + H2 ϕ − ∂2 σ′ + 2Hσ ( )3 [ ] 2 α α Λ a ( ) − 1 + 1 ∂2 + 2 ∂4 ∂2ψ − g −Aϕˆ + δφ′ + δA 3(3λ − 1) a2 a4 (3λ − 1) 2 ( ) λ − 1 ( ) + ∂2 Aψˆ − δA + Hδφ + ∂2 δφ′ + Hδφ 3(3λ − 1)a (3λ − 1)a 8πGa2 = δp, (3.87) 3λ − 1 where 8g + 3g 8g − 3g α ≡ 2 3 , α ≡ 7 8 , 1 ζ2 2 ζ4 [ ] 2 1 ( ) δp ≡ δP + p∂¯ 2E, δτ ij = δP + 2¯pψ δij + Π, , 3 a2 1 ΠGR ≡ Π + 2¯pE, Π, = Π,ij − δij∂2Π. (3.88) 3 The traceless part is given by ( ) [ ] 1 α 1 ( ) ψ − ϕ + σ′ + 2Hσ + α + 2 ∂2 ∂2ψ − Aψˆ − δA − Hδφ a2 1 a2 a = 8πGa2ΠGR + G(η), (3.89) where G(η) is another integration function. Again, in GR it is set to zero [29]. But, similar to the momentum constraint (3.83), one can also choose G(η) = −ϕ so that the second term in the left-hand side of the above equation is canceled.

49 The conservation laws (3.46) and (3.47) to ﬁrst order are given, respectively, by, ∫ { [ ( ) ] 3 ′ 2 2 ′ ¯ ′ d x 2a δµ + 3H (δP + δµ) + 2¯pH∂ E + (¯ρ +p ¯) ∂ E − 3ψ − Jφδφ ( ) ( ) ( ) } ˆ ′ ¯ ˆ ˆ ¯ 2 ′ − A δJA + 3HδJA − 3HJA δA − Aϕ + AJA 3ψ − ∂ E = 0, (3.90) 2a q′ + 3Hq − aδp − ∂2ΠGR = I(η), (3.91) 3 where I(η) is another integration function of η only. In GR, it is usually chosen to be zero [29].

This completes the general description of linear scalar perturbations in the ﬂat

FLRW background in the framework of the HMT setup with any given λ [20], without choosing any speciﬁc gauge for the linear perturbations. However, before closing this section, let us consider some possible gauges.

3.2.3 Gauge Choices

Though the field equations listed above give a complete description of the dynamical behaviors of the linear perturbations, they are nevertheless cumbersome to work with and may hinder the physical implications of the equations. On the other hand, due to the symmetries of the theory, one is in general free to make gauge transformations (“coordinate-system changes”) so that under a particular choice, some of the perturbations can be set zero. Historically, early analysis of cosmological perturbations were performed in different gauges and this often lead to different or even contradicting results. Bardeen clarified this issue with his seminal paper in 1980 [49] by the introduction of two gauge-invariant quantities that are linear combinations of the metric perturbations. These two perturbations, when choosing a particular gauge, take a simplified form yet the physical conclusions based on them are gauge- independent. Hence by constructing the gauge-invariant perturbations one is free from the problem of gauge artifacts and at the same time can enjoy the simplicity of some particular gauge.

50 To consider the gauge choices, we ﬁrst note that

ξ0 = ξ0(η), ξ = ξ(η, x).

Then, from Eqs.(3.76) and (3.77) one immediately ﬁnds that the spatially ﬂat gauge

ψ˜ = 0 = E˜ [29] is impossible in the HMT setup. Since ϕ = ϕ(η), a natural gauge for

the time sector is

ϕ˜ = 0, (3.92)

for which ξ0 is uniquely ﬁxed up to a constant C, ∫ 1 η C ξ0(η) = a(η′)ϕ(η′)dη′ + . (3.93) a(η) a(η)

Then, depending on the choices of ξ and ϵ, we can have various diﬀerent gauges.

3.2.3.1 Longitudinal Gauge. The longitudinal gauge in GR is deﬁned as [29],

E˜ = 0 = B,˜ (3.94) which is impossible in the HL theory without the U(1) symmetry [13]. However, with the U(1) gauge freedom, Eqs.(3.76) and (3.77) show that now this gauge becomes possible with the choice,

ξ = E, ϵ = a(ξ0 − σ), (3.95) where ξ0 is given by Eq.(3.93). It should be noted that this gauge is fundamentally diﬀerent from that given in GR [29], because now we also have ϕ˜ = 0.

3.2.3.2 Synchronous Gauge. In GR, the synchronous gauge is deﬁned as [29],

ϕ˜ = 0 = B.˜ (3.96)

However, this is already implied in the above longitudinal gauge. With the extra

U(1) gauge freedom ϵ, we can further require,

(i) δφf = 0, or (ii) δAf = 0. (3.97)

51 The former will be referred to as the Newtonian synchronous gauge, while the latter the Maxwell synchronous gauge. For the Newtonian synchronous gauge, ϵ and ξ are given by ∫ [ ] η 1 ( ) ξ(η, x) = B + ξ0 + δφ − ξ0φˆ′ dη′ + D(x), a ϵ(η, x) = ξ0φˆ′ − δφ, (3.98)

where D(x) is an arbitrary function of xi only. For the Maxwell synchronous gauge,

they are given by ∫ [ ] η ( )′ 0 ˆ ′ ϵ(η, x) = δA − ξ A dη + D1(x), ∫ ( ) η ϵ ξ(η, x) = B + ξ0 − dη′ + D (x), (3.99) a 2

i where D1(x) and D2(x) are other two arbitrary functions of x only. From the above one can see that none of them can ﬁx the gauge uniquely.

3.2.3.3 Quasi-longitudinal Gauge. In [38], the gauge,

ϕ˜ = E˜ = δφf = 0, (3.100) was used. With this gauge, we ﬁnd that

ξ(η, x) = E(η, x), ϵ(η, x) = ξ0φˆ′ − δφ(η, x), (3.101)

and ξ0 is given by Eq.(3.93). Thus, in this case the gauge freedom of Eqs.(3.76) and

(3.77) are also uniquely determined up to the constant C, similar to the longitudinal

gauge (3.94).

Note that instead of choosing the above gauge, one can also choose

ϕ˜ = E˜ = δAf = 0, (3.102)

for which we have

ξ(η, x) = E(η, x), ∫ [ ] η ( )′ 0 ˆ ′ ϵ(η, x) = δA − ξ A dη + D3(x), (3.103)

52 i where D3(x) is another integration function of x only. Thus, unlike the gauge (3.100),

now the gauge is ﬁxed only up to a constant C and an arbitrary function D3(x). To be distinguished from the one deﬁned in the case without the U(1) symmetry

[13], we shall refer the gauge (3.100) to as the Newtonian quasi-longitudinal gauge,

and Eq.(3.102) the Maxwellian quasi-longitudinal gauge.

3.3 The Instability Problem

The stability of the maximal symmetric spacetimes in the HMT setup with

λ ≠ 1 was considered in [20] with detailed balance condition. Since the Minkowski

is not a solution of the theory when detailed balance condition is imposed, so the

analysis given in [20] does not include the case where the Minkowski spacetime is

the background. However, for the potential given by Eq.(2.4), the detailed balance

condition is broken, and the Minkowski spacetime now is a solution of the theory.

Therefore, in this section we study the stability of the Minkowski spacetime with any

given λ. The case with λ = 1 was considered in [38], so in this section we consider

only the case with λ ≠ 1.

We adopt the Newtonian quasi-longitudinal gauge (3.100) in this section.

3.3.1 Stability of the Minkowski Spacetime

It is easy to show that the vacuum Minkowski spacetime,

a = 1, A¯ =φ ¯ = κ = 0, (3.104)

is a solution of the HMT model even with λ ≠ 1, provided that

¯ ¯ Λg = Λ = JA = Jφ =ρ ¯ =p ¯ = 0. (3.105)

Then, the linearized Hamiltonian constraint (3.82) is satisﬁed identically, while the

super-momentum constraint (3.83) yields,

3λ − 1 ∂2B = ψ,˙ (3.106) 1 − λ

53 2 ij where ∂ = δ ∂i∂j. Eqs.(3.85) and (3.86) reduce to, ( ) ∂2 ∂2B + 3ψ˙ = 0, (3.107)

∂2ψ = 0. (3.108)

The trace and traceless parts of the dynamical equations reduce, respectively, to

2 1 ψ¨ − ∂2δA + ∂2B˙ = 0, (3.109) 3(3λ − 1) 3 B˙ = δA − ψ. (3.110)

It can be shown that Eqs.(3.107) and (3.109) are not independent, and can be ob-

tained from Eqs.(3.106), (3.108) and (3.110). Eq.(3.108) shows that ψ is not propa-

gating, and with proper boundary conditions, we can set ψ = 0. Then, Eqs.(3.106)

and (3.110) show that B and δA are also not propagating, and shall vanish with proper boundary conditions. Therefore, we ﬁnally obtain

ψ = B = δA = 0. (3.111)

Thus, the scalar perturbations even with λ ≠ 1 vanish identically in the Minkowski

background. Hence, the spin-0 graviton is eliminated in the HMT setup even for any

given coupling constant λ.

3.4 The Ghost and Strong Coupling Problems

To consider the ghost and strong coupling problems, we ﬁrst note that they are

closely related to the fact that λ ≠ 1. The parts that depend on λ are the kinetic

part, LK , and the interaction part Lλ(Kij, φ) between the extrinsic curvature Kij and the Newtonian pre-potenital φ. With the gauge choice φ = 0, we can see that

the latter vanishes identically. Then, it is suﬃcient to consider only the kinetic part

SK , the IR terms R and Λ, and the source term SM , ∫ √ ( ) 3 SIR = dtd xN g LK + R − 2Λ + LM . (3.112)

54 Second, the presence of matter is to produce non-zero perturbations. Otherwise,

the spacetimes, to zero-order, are the maximally symmetric spacetimes. In these

backgrounds, when matter is not present, the corresponding metric and gauge ﬁeld

perturbations, ψ, B and δA, vanish identically, as shown in the last section for the Minkowski spacetime, and in [20] for the (anti-) de Sitter one. On the other hand, LM depends on λ only through f(λ), and this dependence is regulated to have nice behaviors, so it does not contribute to the strong coupling and ghost problems.

Therefore, the only role that LM plays here is to produce non-vanishing ψ, B and δA. It is interesting to note that to study the strong coupling problem, in [36] the authors assumed that the background metric has non-vanishing extrinsic and spatial

¯ 2 ¯ curvatures in the scale L: Rij ∼ 1/L and Kij ∼ 1/L, instead of non-vanishing ψ and B assumed here as well as in [50, 51]. But, the purposes are the same: to provide an environment so that the strong coupling problem can manifest itself properly, if it exists. In the following, we shall consider two diﬀerent kinds of gravitational

fields: one represents spacetimes in which the flat FLRW universe with Λ = 0 can be considered as their zero-order approximations; and the other represents static weak gravitational fields, in which the Minkowski spacetime can be considered as their zero-order approximations.

3.4.1 Ghost-free Condition

In the ﬂat FLRW background, the quadratic part of SIR is given by, { ∫ [ ] ( ) 9 S(2) = ζ2 dηd3xa2 1 − 3λ 3ψ′2 + 6Hψψ′ + 2ψ′∂2B + H2ψ2 IR 2 } ( ) ( ) + 2 ∂ψ 2 + (1 − λ) ∂2B 2 . (3.113)

Note that in writing the above expression, we had ignored the term LM since it has no contributions to both the ghost and the strong problems, as mentioned above.

55 Then, from the super-momentum constraint (3.83), we ﬁnd that

3λ − 1 8πGaq ∂2B = ψ′ − . (3.114) 1 − λ 1 − λ

Substituting it into Eq.(3.113), we obtain ∫ { } 2 2 ′ ( ) 9λ(3λ − 1) q˜ (2) 2 3 2(1+δ) − ˜ 2 ˜ 2 − H2 ˜2 SIR = ζ dηd xa 2 ψ + 2 ∂ψ ψ + 2 , (3.115) cψ 2 cψ

where √ 1 − λ 3λ − 1˜q c2 = , ψ = aδψ,˜ q = , (3.116) ψ 3λ − 1 8πGa1−δ ≡ − − 2 and δ 3(1 λ)/2. Thus, the ghost-free condition requires cψ < 0, or equivalently,

1 i) λ > 1, or ii) λ < , (3.117) 3

which are precisely the conditions obtained in the SVW setup [13].

It should be noted that the conditions (3.117) also hold in the non-ﬂat FLRW

backgrounds, as one can easily show by following the above arguments.

3.4.2 Strong Coupling Problem

3.4.2.1 Flat FLRW Background In this case we adopt the gauge,

2 B 2 −2ψ N = a, Ni = a e ∂iB, gij = a e δij, (3.118)

which reduces to the linear perturbations studied in the previous sections to the ﬁrst

order of ψ and B. This gauge is slightly diﬀerent from the one used in [33, 50, 51].

Then, we ﬁnd ( ) 2e2ψ ( ) R = 2∂2ψ − ∂ψ 2 , (3.119) a2

and the kinetic action SK is given by ∫ { [ ] ( ) 27 (2) 2 3 2 − ′2 H ′ ′ 2 H2 2 SK = ζ dηd xa 1 3λ 3ψ + 18 ψψ + 2ψ ∂ B + ψ } 2 + (1 − λ)B∂4B . (3.120)

56 Note that this is diﬀerent from the expression given by Eq.(3.113). The reason is that,

in the calculations of Eq.(3.113), the 3-metric gij is approximated to the ﬁrst-orders of ψ and B, as one can see from Eq.(3.74), while gij to their second orders (so does √ g). For detail, we refer readers to [52]. On the other hand, in the derivation of

ij Eq.(3.120), we expanded both gij and g to second orders. It is interesting to note that this diﬀerence does not aﬀect the super-momentum constraint (3.114), which

(2) can be also obtained by the variation of SIR with respect to B. Since the B-terms in both expressions of Eqs.(3.113) and (3.120) are the same, so is the resulting equation

(2) obtained by the variation of SIR with respect to B. Substituting Eq.(3.114) into Eq.(3.120), we find that ∫ { } 2 −2 ′ ( ) 27(3λ − 2)(3λ − 1) q˜ (2) 2 3 2(1+δ) ˜ 2 ˜ 2 − H2 ˜2 SIR = ζ dηd xa 2 ψ + 2 ∂ψ ψ + 2 , cψ 2 cψ (3.121) (2) (2) (2) ˜ where SIR = SK + SR , and cψ, ψ andq ˜ are defined by Eq.(3.116) but now with δ = 9(λ − 1)/2. Then, we find that the ghost-free conditions are the same as that

given by Eq.(3.117). One can show that the conclusions regarding to the strong

coupling problem are also independent of the use of either the expression (3.121) or

(2) (3.115) for SIR . To the third-order of ψ and B, we ﬁnd that ∫ { [ ( ) ] (3) 2 3 2 2 2 SIR = ζ dηd xa 2ψ ψ∂ ψ + ∂ψ ( ) 9( ) + 3λ − 1 2ψψ′2 + 6Hψ2ψ′ + 3H2ψ3 2 [ ( ) ] ′ ,k ′ 2 +(3λ − 1) 2(ψ + Hψ) ψ B,k + ψ(2ψ + Hψ)∂ B [ ]( ) 2 ,k −2 (3λ − 1)HB − (λ − 1)∂ B ψ B,k ( )( )[ ( ) ] ′ 2 2 ,kl −2 3λ − 1 ψ + Hψ B∂ B + ∂B + 4ψ,kB,lB [ ( ) ] ) ,kl 2 2 2 2 +(ψ + 2B) B B,kl − λ ∂ B − 2λ(∂B ∂ B } [ ( ) ] 2 2 ,kl +(3λ − 1)HB B∂ B + 2 ∂B + 2B B,kB,l . (3.122)

57 Following [33], we ﬁrst write the quadratic action (3.115) in its canonical form

with order-one coupling constants, by using the coordinate transformations,

i i η = αη,ˆ x = α|cψ|xˆ , (3.123)

and redeﬁnitions of the canonical variables, √ ˆ ˜ ψ 2ˆq ψ = 1/2 , q˜ = 1/2 2 . (3.124) Mpl|cψ| α Mpl|cψ| α

It must not be confused with the constant α used here and the one used in the previous sections for the U(1) gauge generator. Then, from Eq.(3.114) we ﬁnd that ( ) 1 8πGaq Bˆ B = − ψ′ − = , |c |2∂2 3λ − 1 M |c |1/2 ψ( √ )pl ψ aδ 2 Bˆ = − ψˆ∗ + δHˆψˆ − qˆ , (3.125) ∂ˆ2 3λ − 1

where ψˆ∗ = ∂ψ/∂ˆ η,ˆ Hˆ = a∗/a. Inserting Eqs.(3.123)-(3.125) into Eq.(3.122), we

obtain { ∫ 3 ( ) 1 |c | 2 1 1 (3) 3 2 ψ Lˆ(3) Lˆ(3) Lˆ(3) Lˆ(3) Lˆ(3) SIR = dηdˆ xaˆ 1 + 1 2 + 3 + 4 + 5 5 2Mpl α |c | 2 α |c | 2 α ψ }ψ ( ) ( ) 1 Lˆ(3) Lˆ(3) 1 Lˆ(3) Lˆ(3) α Lˆ(3) + 1 6 + 7 + 5 8 + 9 + 1 10 , (3.126) |cψ| 2 |cψ| 2 |cψ| 2

L(3) where i ’s are given below [ ] 9 ( ) ( ) Lˆ(3) = (3λ − 1)a3δ 2ψˆ ψˆ∗ + δHˆψˆ 2 + 6Hˆψˆ2 ψˆ∗ + δHˆψˆ + 3Hˆ2ψˆ3 , 1 2 [ ( ) ] Lˆ(3) = 2a3δ ψˆ2∂ˆ2ψˆ + ψˆ ∂ˆψˆ 2 , 2 { ( ) [ ]( )( ) Lˆ(3) − 2δ ˆ∗ Hˆ ˆ ˆ ˆ ˆi ˆ 3 = 3λ 1 a 2 ψ + (1 + δ) ψ ∂iψ ∂ B } [ ]( ) + ψˆ 2ψˆ∗ + (1 + 2δ)Hˆψˆ ∂ˆ2Bˆ , ( )( )( ) Lˆ(3) − δ ˆ2 ˆ ˆ ˆ ˆi ˆ 4 = 2(3λ 1)a ∂ B ∂iψ ∂ B , [ ( )( )( ) ( ) ( ) ] Lˆ(3) δ ˆk ˆl ˆ ˆ ˆ ˆ ˆ ˆk ˆl ˆ 2 − ˆ2 ˆ 2 5 = a 4 ∂ ∂ B ∂kψ ∂lB + ∂ ∂ B λ ∂ B , (3.127)

58 [ ][ ( ) ( ) ] Lˆ(3) − δ ˆ∗ Hˆ ˆ ˆ ˆ2 ˆ ˆ ˆ 2 6 = 2(1 3λ)a ψ + (1 + δ) ψ B ∂ B + ∂B , ( )( ) Lˆ(3) − δHˆ ˆ ˆ ˆ ˆi ˆ 7 = 2(1 3λ)a B ∂iψ ∂ B , [( ) ( ) ] Lˆ(3) ˆ ˆk ˆl ˆ 2 − ˆ2 ˆ 2 8 = 2B ∂ ∂ B λ ∂ B , [( )( )( ) ( ) ( )] Lˆ(3) ˆk ˆl ˆ ˆ ˆ ˆ ˆ − ˆ ˆ 2 ˆ2 ˆ 9 = 2 ∂ ∂ B ∂kB ∂lB λ ∂B ∂ B , [ ( ) ( ) ] Lˆ(3) − Hˆ ˆ ˆ ˆ2 ˆ ˆ ˆ 2 10 = (3λ 1) B B ∂ B + 2 ∂B . (3.128)

L(3) Clearly, for any chosen α some of the coeﬃcients of i ’s always become unbounded as cψ → 0, that is, the corresponding theory is indeed plagued with the strong coupling problem.

To study it further, let us consider the rescaling,

− − ηˆ → s γ1 η,ˆ xˆi → s γ2 xˆi,

ψˆ → sγ3 ψ,ˆ qˆ → sγ4 q.ˆ (3.129)

(2) Then, SIR given by Eq.(3.115) is invariant for γ1 = γ2 = γ3 = γ4/2 = γ. Without loss of generality, one can set γ = 1. For such a choice of γ, it can be shown that Bˆ is scale-invariant,

Bˆ → B.ˆ (3.130)

(3) 1 Then, in SIR of Eq.(3.126) the ﬁrst ﬁve terms are scaling as s , and the sixth to ninth terms all scaling as s0, while the last term is scaling as s−1. Thus, all the

first five terms are irrelevant in the low energy limit, and diverge in the UV, so they are all not renormalizable [53]. The sixth to ninth terms are marginal, and are strictly renormalizable, while the last term is relevant and superrenormalizable. This indicates that the perturbations break down when the coupling coefficients greatly exceed units. To calculate these coefficients, let us consider a process at the energy

(3) scale E. Then, we ﬁnd that the ten terms in the cubic action SIR have, respectively, the magnitudes, (E,E,E,E,E,E0,E0,E0,E0,E−1), for example, ∫ ( )( ) 3 ˆ∗ ˆ ˆ ˆi ˆ dηdˆ xˆψ ∂iψ ∂ B ≃ E. (3.131)

59 Since the action is dimensionless, all the coeﬃcients in (3.126) must have the dimen-

−ns sions E , where ns = (1, 1, 1, 1, 1, 0, 0, 0, 0, −1), (s = 1, 2, 3, ..., 10). Writing them in the form, ( ) ˆ ns λs λs = , (3.132) Λs ˆ where λs is a dimensionless parameter of order one, one ﬁnds that Λs for s = 1, 2, 3, 4, 5, 10 are given by

4Mplα 1 | | 2 Λ1 = 3 , Λ2 = Mpl cψ α, 9(3λ − 1)|cψ| 2 1 1 2M |c | 2 α M |c | 2 α Λ = pl ψ , Λ = pl ψ , 3 3λ − 1 4 3λ − 1 5 (3λ − 1)α | | 2 Λ5 = 2Mpl cψ α, Λ10 = 1 . (3.133) 2|cψ| 2 Mpl

Translating it back to the coordinates η and xi, the energy and momentum scales are given by

ω Λs k Λs Λs = , Λs = . (3.134) α α|cψ| For s = 6, 7, 8, 9, the coupling coeﬃcients are given by

3λ − 1 λ6 = λ7 = 1/2 , |cψ| Mpl 1 λ8 = λ9 = 5/2 . (3.135) |cψ| Mpl

ω k From these expressions, one can see that the lowest scale of Λs and Λs ’s is given by

ω ≃ | |5/2 Λmin = Λ5 cψ Mpl, (3.136)

as cψ → 0. For any process with energy higher than it, the corresponding coupling constants become larger than unit, and then the strong coupling problem rises.

Thus to be consistent with observations in the IR, on one hand, λ is required to be closed to its relativistic value λIR = 1, on the other hand, to avoid the strong coupling problem, the above shows that it cannot be too closed to it.

60 3.4.2.2 Static Weak Gravitational Fields When a static gravitational field produced by a source is weak, such as the solar system, one can treat the problem as perturbations of the Minkowski spacetime. Since the Minkowski background is a particular case of the flat FLRW spacetime, one can consider its perturbations still given by Eq.(3.118) but now with a = 1. Since the presence of matter, ψ now is in general different from zero. Then, we find that ∫ ( ) ( ) (8πGq)2 S(2) = ζ2 dtd3x 2 ∂ψ 2 − , (3.137) IR λ − 1 where 8πG B = q. (3.138) (λ − 1)∂2 Setting

t = αt,ˆ xi = αxˆi, √ ψˆ 3λ − 1|c |qˆ ψ = √ , q = ψ , (3.139) 2ζα 8πGζα2 we ﬁnd that S(2) given by Eq.(3.137) takes its canonical form, ∫ (( ) ) (2) ˆ 3 ˆ ˆ 2 − 2 SIR = dtd xˆ ∂ψ qˆ . (3.140)

On the other hand, we have ∫ { [ ( ) ] ( ) (3) 2 3 2 2 − 2 ,k ,kl SIR = ζ dtd x 2ψ ψ∂ ψ + ∂ψ + 2(λ 1)∂ B ψ B,k + 4ψ,kB,lB [ ( ) ] ,kl 2 2 + (ψ + 2B) B B,kl − λ ∂ B } ) 2 2 ,kl − 2λ(∂B ∂ B + 2B B,kB,l ∫ { ( ) } 1 L(3) 2L(3) L(3) 2 3/2 L(3) ˆ 3 1 2 3 4 = dtd xˆ + + 2 + 3 , Mpl α α (3λ − 1)|cψ| α 3λ − 1 |cψ| (3.141)

61 where

[ ( ) ( ) ] (3) ˆ ˆ ˆ2 ˆ ˆ ˆ 2 L1 = ψ ψ ∂ ψ + ∂ψ , ( )( )( ) (3) ˆ2 ˆ ˆk ˆ ˆ ˆ L2 = ∂ B ∂ ψ ∂kB , [( ) ( ) ] ( )( )( ) (3) ˆ ˆ ˆ ˆ 2 − ˆ2 ˆ 2 ˆk ˆ ˆl ˆ ˆ ˆ ˆ L3 = ψ ∂k∂lB λ ∂ B + 4 ∂ ψ ∂ B ∂k∂lB , [( ) ( ) ] ( )( ) ( )( )( ) (3) ˆ ˆ ˆ ˆ 2 − ˆ2 ˆ 2 − ˆ2 ˆ ˆ ˆ 2 ˆk ˆ ˆl ˆ ˆ ˆ ˆ L4 = B ∂k∂lB λ ∂ B λ ∂ B ∂B + ∂ B ∂ B ∂k∂lB ,

(3.142)

but now with ( ) 1 1 Bˆ B = √ qˆ ≡ √ . (3.143) 2 ζ|cψ| 3λ − 1 ∂ˆ ζ|cψ| 3λ − 1 ˆ (2) Considering the rescaling (3.129) with t =η ˆ, we ﬁnd that SIR given by (3.140)

is invariant, provided that γ3 = (γ1 + γ2)/2 and γ4 = (γ1 + 3γ2)/2. Without loss ˆ of generality, we can set γ1 = γ2 = 1, and then B scales exactly as that given by

(3) 1 1 1 Eq.(3.130), while the four terms in SIR of Eq.(3.141) scale, respectively, as s , s , s and s0. Then, following the analysis given between Eqs.(3.132) and (3.136), we ﬁnd

k ω that Λs = Λs for s = 1, 2, 3, where

ω ω Λ1 = 2Λ2 = Mpl, ω − | |2 Λ3 = (3λ 1)Mpl cψ , (3.144)

and ( ) 2 3/2 1 λ4 = 3 . (3.145) 3λ − 1 Mpl|cψ| Clearly, as λ → 1, the coupling also becomes strong. In particular, since the fourth

term scales as s0, its amplitude remain the same, as the energy scale of the system

changes. That is, this term is equally important at all energy scales. The strength of

this term gives the lowest energy scale, as cψ → 0. Therefore, now we have

3 Λmin ≃ Mpl|cψ| . (3.146)

62 It should be noted that, in the above we studied the strong coupling problem

only in terms of ψ. Then, one may argue that our above conclusions may be gauge- dependent. In the following, we shall show that this is not true. Let us ﬁrst note that in the static case the gauge invariant quantity Ψ is precisely equal to ψ, as one can see from Eq.(3.79). Therefore, in this case the coupling indeed becomes strong when

3 E > Mpl|cψ| , even in terms of the gauge-invariant quantity. On the other hand, in the cosmological case, from Eqs.(3.79) and (3.125) we

ﬁnd that the gauge-invariant quantity Ψ can be written as

aδΨˆ Ψ = 1/2 , (3.147) Mpl|cψ| α

where [ √ ] αH ( ) 2 Ψˆ ≡ ψˆ + α ψˆ′ + δHψˆ − qˆ . (3.148) ∂ˆ2 3λ − 1 Since the lowest energy scale (3.136) is independent of α (as it should be), we can

d ˆ ˆ always choose α ∝ |cψ| , (d > 0), so that Ψ ≃ ψ and Ψ ≃ ψ as |cψ| → 0. Then, one can repeat the analysis in terms of Ψ and Ψˆ and ﬁnds that the same conclusions are

resulted.

3.5 Non-projectable General Covariant HL Gravity with Detailed Balance Condition Softly Breaking

While we’ve mainly studied the projectable HMT setup without the detailed

balance condition , it is worthwhile noticing that a non-projectable version with the

extended symmetry (3.1) exists [42]. Similar to the BPS model, this version includes

a new ingredient ai deﬁned in (2.44). The diﬀerence with the BPS model is that the ZWWS model breaks the detailed balance condition only softly and has hence only

15 coupling constants.

63 The kinetic part and the coupling with A is unaﬀected by this breaking, yet

the coupling with φ takes modiﬁcations ∫ { √ 3 ij Sφ = dtd x gN φG (2Kij + ∇i∇jφ + ai∇jφ) , [( ) ( ) ] 2 i 2 2 i + (1 − λ) ∇ φ + ai∇ φ + 2 ∇ φ + ai∇ φ K [ 1 ( ) + Gˆijkl 4 (∇ ∇ φ) a ∇ φ + 5 a ∇ φ a ∇ φ 3 i j (k l) (i j (k l) ( ) ] + 2 ∇(iφ aj)(k∇l)φ + 6Kija(l∇k)φ , (3.149)

where Gˆijkl is inverse of the general De Witt metric (1.9) with λ = 1. The poten-

tial part, on the other hand, is required to satisfy a “generalized” detailed balance

condition ˆ ij kl i j LV = E GijklE − gijA A . (3.150)

The ﬁrst term on the right hand side is given in Eqs.(1.11) and (1.13), while Ai and

Wa are given as ∫ 1 δW 1 √ ∑n=1 Ai = √ a ,W = d3x gai b ∇na , (3.151) g δa a 2 n i i n=0 where bn are arbitrary constants. Adding lower dimension relevant terms, which breaks the above detailed balance condition softly, leads to the potential

( ) ( ) 2 i −2 2 ij LV = γ0ζ − β0aia − γiR + ζ γ2R + γ3RijR [ ( ) ( ) −2 i 2 i 2 i j +ζ β1 aia + β2 (() ai) + β3 aia aj ( ) ] ij i ij i + β4a aij + β5 aia R + β6aiajR + β7Rai [ ( ) ] −4 ij i 2 +ζ γ5CijC + β8 ∆a , (3.152)

where the β’s and γ’s are dimensionless constants and Cij is the Cotton-York tensor

deﬁned in Eq.(1.12). Note that in writing down the above potential term, only parity

preserving terms are considered. The parity violating terms are in principle allowed

by the generalized detailed balance condition.

64 It was shown that the scalar graviton is indeed eliminated even with variable

λ. Hence the model is free of the ghost, instability and strong coupling problems in the gravitational sector. When coupled with matter however, the model does display the strong coupling problem where the strong coupling scale becomes [42] ( ) 3/2 Mpl 5/2 Λω ≃ Mpl|cψ| . (3.153) c1

A solution is possible if one follows the idea of the BPS mechanism introduced at the end of Chapter 2 [42].

Studies of cosmological perturbations at linear order was carried out in [54], in which the same ﬂatness conclusion was drawn. The vector perturbations, interestingly, does not receive any contributions from ai because the lapse N does not posses any vector perturbations. Hence the vector perturbations have the same behavior as the version studied in Chapter 2 [28].

Single-field slow-roll inflation was also studied in [55]. There it was found that the power spectra and spectrum index of the scalar perturbations under the slow-roll approximations, acquire tiny corrections from the theory, as the adiabatic condition holds in the case considered here. Remarkably, part of corrections are of the consequence of the non-projectability condition. In the relativistic limit, the spectrum and index reduce to the standard results given in GR. It was also found in [56] that this model is capable of producing large polarizations in the primordial gravitational waves if the parity violating terms are added to the potential or if the modes of the primordial quantum fluctuations underwent a non-adiabatic period.

65 CHAPTER FOUR

Slow-roll Single Field Inﬂation in General Covariant Hoˇrava-Lifshitz Gravity

This chapter published as: Y. Huang, A. Wang and Q. Wu, “Inﬂation in general covariant theory of gravity”, J. Cosmol. Astropart. Phys. 10 (2012) 010.

Having studied the consistency of the Hoˇrava-Lifshitz theory of gravity, we now turn to the phenomenological test of the theory. Because the theory has its most significant differences from general relativity in high energies, the study of early universe in the current theory naturally serves as one of the best probe for its consistency with observations. Since the model considered in Chapter 3 is free of the ghost and stability problems, while the strong coupling problem is curable, we shall consider its applications to inflation in this Chapter. This chapter was published in [22]

Proposed to address several challenges faced by the standard hot big bang model of cosmology [57], the inflation paradigm argues that our universe, at its extremely early age (10−36 ∼ 10−32 seconds after the big bang), underwent a stage of violent expansion, and the primordial quantum fluctuations that were initially within causal connections were stretched so much that their Compton wavelength became much larger than the Hubble radius–the limit of classical causal connection in relativistic physics. As the universe expands almost exponentially, its temperature dropped down to almost zero and later restored to the high energy in the reheating phase. This is then followed by radiation- and matter-dominated stages. The observationally interesting super-horizon modes re-entered the Hubble horizon during the matter- dominated era (the later it re-entered, the longer its wavelength) and acted as seeds of the large-scale structures and the cosmic microwave background radiation (CMB) that we observe today [58]. Hence the initial state, and physics that happens during and after the inflationary stage all leave their footprints in the structures we detect.

66 We shall brieﬂy introduce the technical details of the simplest inﬂation models in

Section 1.

In Section 2, we ﬁrst consider the ﬂat FLRW background, and argue that the

slow-roll conditions imposed in GR are also needed here, in order to obtain enough

e-fold to solve the problems such as horizon, monopole, domain walls, and so on [59].1

In addition, we show that in the super-horizon regions the perturbations become adiabatic, and the comoving curvature perturbation is constant. But the reason for the adiabaticity and constancy, similar as the one in [27], is diﬀerent from that in GR.

In Section 3, we show explicitly that in the sub-horizon regions, the metric and scalar

ﬁeld are tightly coupled and have the same oscillating frequencies, while in the super- horizon regions, the dynamical evolution leads to vanishing metric perturbations at late times. It is remarkable that a master equation for the scalar perturbations exists, in contrast to the case without the U(1) symmetry [27].

We calculate the power spectra and indices of both scalar and tensor perturbations in Section 4 with the slow-roll approximations, by using the uniform approximation [60]. We express them explicitly in terms of the slow-roll parameters and the coupling constants of the high-order spatial derivative terms. We ﬁnd that the power spectra of both scalar and tensor perturbations can reduce to the expressions obtained in the minimum inﬂation scenario in GR, under some conditions on the coupling constants. Moreover, they are nearly scale-invariant. The spectrum tilt (index) of tensor perturbations is the same with the one given in the minimum scenario.

The index of the scalar perturbations on the other hand, diﬀers from the one in the minimum scenario and receives corrections that depends on λ. This is expected as

the main diﬀerence of HL gravity from GR in perturbations is in the scalar case. The

scalar-to-tensor ratio was found to depend on higher-order spatial derivative terms,

1 Note that the horizon problem is solvable and scale-invariant spectra of primordial perturbations can be obtained in the HL theory with out inﬂation, as ﬁrst noted in [11]. But to solve other problems, such as the relics of topological defects, including monopoles, it may still be needed.

67 but can reduce to the value in the minimum case. Therefore, as far as slow-roll inﬂation is concerned, the HL theory is consistent with observations.

4.1 The Inﬂation Paradigm

Since the inﬂation paradigm is a very rich ﬁeld of study, we give only brief introductions to the general ideas behind the class of simplest models (or minimum scenario) which assumes the following [61]

• gravity is described by general relativity;

• inﬂation is driven by a single scalar ﬁeld;

• with canonical kinetic term for the mater;

• slow-roll condition is always satisﬁed during inﬂation; and

• the Bunch-Davies vacuum.

Violations of any of these assumptions could lead to dramatically diﬀerent conclusions. For detailed reviews of inﬂation dynamics, interested readers can refer to

[59, 62].

4.1.1 The Horizon and Flatness Problems of Standard Model

According to the standard hot big bang theory of cosmology, our universe started at about 13.7 billion years ago from a very hot (energetic) and dense state.

This initial universe then expanded as it cooled down and light elements, clusters of galaxies and stars formed along this expansion history. The various predictions of this standard model has been veriﬁed by experiments during the past several decades

[63]. However, it faces several severe challenges such as the horizon, ﬂatness and monopole problems.

The cosmological principle, one of the fundamental assumptions (the other two being GR and Weyl’s postulate [63]) of the standard model, states that our universe, on large scales, is homogeneous (translationally smooth) and isotropic (no preferred

68 direction). Mathematically, this can be expressed as that the maximally symmetric metric–the FLRW metric (3.58)–describes our universe at the background level. To understand the horizon and ﬂatness problems, it suﬃces to look at the dynamics of the universe at the background level.

The Friedmann equation and the acceleration equation, ﬁrst considered in

Chapter 3 in Eqs.(3.63) and (3.66) take the following form in the standard model

κ 8πG H2 + = ρ,¯ (4.1) a2 3 a¨ 4πG = − (¯ρ + 3¯p), (4.2) a 3 where we’ve dropped the cosmological constant. For ordinary constituents of the universe described by perfect ﬂuids, the energy density ρi and its corroding pressure pi satisfy the equation of state

ρ¯i = wip¯i, (4.3) where wi are non-negative constants. For matter moving slowly, w = 0, i.e. the matter is pressure-less. For relativistic matter such as photons, w = 1/3. Hence the acceleration equation tells us that the expansion rate of our universe, since its birth, is always decreasing.

However, if one traces back this expansion history, the observed homogenous patches of CMB would be seemingly disconnected at the beginning (cf. Fig.1.2 in

Chapter 1, where the slope (time/distance) during the radiation dominated era was taken to be 1 and for radiation dominated era was higher than 1): the particle horizon–the maximum distance a signal can be transmitted during a period in relativistic physics– ∫ ∫ t ′ a ′ ≡ dt d ln a Dp(t) a(t) ′ = a(t) ′ ′ (4.4) 0 a(t ) 0 a H(a ) is much smaller than (in fact only 10−5 of) the size of the last scattering surface

≪ 1 Dp(tLS) . (4.5) H(tLS)

69 Thus the standard model would be forced to impose a rather strange initial

data: although the diﬀerent parts of the universe were causally disconnected, they

share the same state. This is the horizon problem–parts that were out of each others’

horizon look the same.

The ﬂatness problem can be seen by manipulating the Friedmann equation into

the following form κ ≡ ρ 1 + 2 2 = Ω , (4.6) a H ρcritical

2 where the critical density ρcritical is deﬁned as 3H /8πG. It’s obvious that if the expansion is always slowing down, a2H2 =a ˙ 2 keeps decreasing, and Ω would move

away from 1. But current observations limit Ω to be extremely close to unity [62].

4.1.2 Accelerated Expansion and Slow-roll Approximation

The inﬂation paradigm, in solving these two problems, introduced a short period

of time before the radiation dominated era. During this short period, the expansion

was actually speeding up. (See Fig.1.2 in Chapter 1, where the inﬂation era has a

less-than-unity slope of time/distance.)

Clearly, if the universe underwent an accelerating expansion, and if the stage

lasted long enough, then a2H2 in (4.6) would grow large enough during this period

that Ω was driven to a value close enough to unity and later decelerated expansion

barely moves Ω. Hence the ﬂatness problem could be solved.

For the horizon problem, this indicates that during the acceleration, the rate

of change of comoving particle horizon is higher than that of the comoving Hubble

radius (in fact, the comoving Hubble radius shrinks) ( ) ( ) d D (t) 1 d 1 p = > 0 > . (4.7) dt a(t) a(t) dt aH

Again, if the period lasted long enough, the comoving particle horizon might grow to a size much larger than the comoving Hubble radius.

70 To quantify the acceleration, note that ( ) ( ) d 1 d 1 a¨ = = − dt aH dt a˙ (aH)2 aH2 + aH˙ = , (aH)2

introduce a¨ = H2 + H˙ ≡ H2 (1 − ϵ ) (4.8) a H where H˙ d ln H ϵ ≡ − = − . (4.9) H H2 Hdt

When this slow-roll parameter ϵH approaches 1 from below, the acceleration phase ends. To ensure a long-enough acceleration, a second small parameter must be introduced

ϵ˙H ηH ≡ . (4.10) ϵH H On the other hand, the desired acceleration phase requires an equation of state that 1 ρ¯ + 3¯p < 0, or w < − . (4.11) 3 The simplest inflation model assumes the canonical coupling of a scalar field χ (often called the inflaton) with gravity ∫ [ ] √ 1 S = d4x −(4)g g ∂ χ∂ χ − V (χ) , (4.12) 2 µν µ ν

which gives energy density and pressure

1 1 ρ¯ = χ¯˙ 2 + V,¯ p¯ = χ¯˙ 2 − V.¯ (4.13) ϕ 2 ϕ 2

If the potential dominates over the kinetic part, then Eq.(4.11) is satisﬁed

p¯χ wϕ = ≃ −1. (4.14) ρ¯χ

71 The Friedman equation, acceleration equation and the dynamical equation of the

scalar ﬁeld then read ( ) ˙ 2 2 8πG ¯ χ¯ H = V 1 + ¯ , (4.15) 3 ( 2V) a¨ 8πG χ¯˙ 2 = V¯ 1 − , (4.16) a 3 V¯ χ¯¨ + 3Hχ¯˙ + ∂V¯ /∂χ¯ = 0. (4.17)

The dominance of potential over kinetic energy indicates, together with (4.8),

( )( )− χ¯˙ 2 χ¯˙ 2 1 3χ¯˙ 2 1 − ϵ = 1 − 1 + ≃ 1 − , (4.18) H V¯ 2V¯ 2V¯ or 1 χ¯˙ 2 ϵ ≃ . (4.19) H 8πG 2H2 A second set of slow-roll parameters describing the shape of the potential can

be introduced ( ) M 2 V¯ ′ 2 V¯ ′′ ϵ ≡ pl , η ≡ M 2 (4.20) V 2 V¯ V pl V¯ where V¯ ′ ≡ dV¯ /dχ¯. They are related to the previous set through [61]

1 ϵ = ϵ , η = − η + 2ϵ . (4.21) V H V 2 H V

Under the slow-roll approximation ϵV , |ηV | ≪ 1, the background is described by quasi-de Sitter spacetime with the scale factor

1 a ≃ − , (4.22) Hη(1 − ϵH )

where η is the conformal time.

4.1.3 The Quantum Fluctuations

Soon after inflation was proposed, it was realized that the quantum fluctuations during that time could be responsible for the generation of the later formed large scale structures or inhomogeneities. For simplicity, we consider only the quantization of the inflaton in the quasi-de Sitter spacetime.

72 The quantization of a ﬁeld in curved spacetime is actually complicated because

no unique vacuum choice can be used to deﬁne a ground state [64]. Observers in

diﬀerent coordinate systems may disagree with each other on whose vacuum is the

true ground state–the classical background may be “pumping energy” to the one’s

vacuum, viewed from another observer. Luckily, quantization in the ﬂat de Sitter

background is simple enough because there exists a conformal transformation between

the ﬂat FLRW metric–to which the de Sitter spacetime belongs–and the Minkowski

metric [ ] 2 2 2 2 2 µ ν ds = a (η) −dη + dr = a (η)ηµνdx dx . (4.23)

As discussed in Chapter 3, the cosmological perturbations, because of the gauge-

invariance of the underlying gauge symmetry of the gravitational theory, is better

to be performed with the gauge invariant quantities. Two gauge-invariant quantities

that are particularly important in relativistic cosmology are perturbation on uniform-

density hypersurfaces ζ and the comoving curvature perturbation R, deﬁned as [29]

H −ζ ≡ ψ + δρ, (4.24) ρ¯′ H R = ψ + δχ. (4.25) χ¯′

Note that this ζ is not to be confused with the ζ related to Mpl. The uniform-density hypersurfaces are deﬁned to the hypersurfaces on which δρ = 0 and hence ζ = −ψ,

while the comoving curvature perturbation R = ψ in the comoving gauge δχ = 0.

On large scales, when the spatial derivative terms are ignored, they are conserved

ζ˙ = 0 = R˙ (4.26) for adiabatic perturbations for which we have

p¯˙ δp = δρ . (4.27) ρ¯˙

Hence we perform the quantization of R below.

73 Assuming general relativity and canonical coupling with a single scalar ﬁeld χ,

it can be shown that the free (quadratic) action in terms of R under the comoving

gauge reads [59] ∫ [ ] 1 χ¯˙ 2 S(2) = dtd3xa3 R˙ 2 − a−2 (∂ R)2 . (4.28) 2 H2 i To proceed, deﬁne the normalized Mukhanov variable χ¯˙ 2 v = zR, z2 = a2 = 2a2M 2 ϵ. (4.29) H2 pl

This leads to ∫ [ ] 1 z′′ S(2) = dηd3x v′2 − (∂ v)2 + v2 . (4.30) 2 i z When we promote the variable v to a quantized field ∫ 3 [ ] d k † v → vˆ(η, x) = v (η)â eikx + v∗ (η)â e−ikx , (4.31) (2π)3 k k −k k where vk is called the mode function, then canonical commutation relation between

′ quantum ﬁeldv ˆk and its conjugate momentumv ˆk is,

⟨ | ′ | ⟩ ℏ 0 [ˆvk, vˆk] 0 = i . (4.32)

If we further assume that the mode function is dependent only on the magnitude of the momentum k, then vk satisﬁes the second order classical equation of motion

(EoM) ( ) z′′ v′′ + k2 − v = 0. (4.33) k z k If we want to have † 3 3 − ′ [ˆak, aˆk′ ] = (2π) δ (k k ), (4.34)

the norm (Wronskian) has to be

∗ ′ − ∗′ − ℏ vkvk vk vk = i . (4.35)

Besides the normalization condition, we need another boundary/initial condition to determine the mode functions completely as it satisfies a second order differential equation. The standard choice of the initial condition is such that at sufficiently

74 early times (η → −∞), the mode was deep inside the horizon and didn’t feel the expansion of the universe. In this case, the EoM reduces to an equation for a simple harmonic oscillator

′′ 2 vk + k vk = 0. (4.36)

Requiring that the vacuum state to be the ground state of the Hamiltonian

ˆ H |0⟩ = E0 |0⟩ , (4.37)

where the vacuum is deﬁned asa ˆk |0⟩ = 0, vk takes the asymptotic form in the far past

1 −ikη vk(η → −∞) ⇒ √ e . (4.38) 2k The diﬃculty in solving the EoM (4.33) is that the eﬀective mass term z′′/z is in general a complicated function of time. In the de Sitter limit,

z′′ 2 = , (4.39) z η2

and the solution, determined by the initial condition (4.38), is ( ) 1 1 −ikη vk = √ 1 + e . (4.40) 2k ikη

If we make the slow-roll assumption, then [62]

z′′ ν2 − 1/4 2 − 3η + 9ϵ = ≃ V V , (4.41) z η2 η2 3 ν ≃ + 3ϵ − η , (4.42) 2 V V

and the exact solution to this approximated EoM is the Hankel function √ π|η| [ ] v = ei(1+2ν)π/4 c H(1)(k|η|) + c H(2)(k|η|) , (4.43) k 2 1 ν 2 ν

where c2 = 0 after we impose the initial condition. Note that the case ν = 3/2 corresponds to the de Sitter limit where the slow-roll parameters vanish. And we

75 shall see later that it is the deviation of this eﬀective term z′′/z from the de Sitter

value 2/η that generates the slight tilt of the spectrum [65].

In the super-horizon regions, that is k2 ≪ z′′/z, the mode is frozen ∫ dη v ≃ C + C . (4.44) k 1 2 a2

Hence the behavior of the mode during the horizon-crossing time is the most impor-

tant.

The tensor perturbations can be treated in a similar way, and we shall delay

this to Section 4.4.

4.1.4 Statistics of the Modes, Power Spectrum

The comoving curvature perturbation would later re-enter the horizon after the

inﬂation terminates and acts as the gravitational potential that aﬀects the structure

formation [63]. The tensor perturbations aﬀect the CMB in its polarization eﬀects.

Hence the statistics of these primordial perturbations must be studied. Again we

demonstrate the case for scalar perturbations here but delay the case for tensors to

Section 4.4.

Apparently, by our assumption of the homogeneity and isotropy of the uni-

verse, the spatial average of these perturbations must be zero. Hence the two-point

correlator is the ﬁrst interesting quantity for study

⟨R(x)R(x + r)⟩ . (4.45)

If we deﬁne the power spectrum as Fourier transform of the two-point correlation

function [59] ∫ 3 −ik·r PR(k) = d re ⟨R(x)R(x + r)⟩ , (4.46) then the correlation of the Fourier images of R(x) with diﬀerent momenta is given

76 by ∫ 3 3 ′ 3 −ik·r ⟨RkRk′ ⟩ = (2π) δ (k + k ) d re ⟨R(x)R(x + r)⟩

3 3 ′ = (2π) δ (k + k ) PR(k). (4.47)

On the other hand, the covariance is given as ∫ ⟨ ⟩ 1 2 R2 3 σR = (x) = 3 d kPR(k) ∫(2π) 2 ≡ d ln k∆R(k). (4.48)

Hence we have 3 2 k ∆ (k) = PR(k). (4.49) R 2π2 The scale-dependence is deﬁned by the spectral index (tilt) d ln ∆2 (k) n − 1 = R . (4.50) s d ln k Using the asymptotic form of the Hankel function, it is then easy to verify that the spectrum is nearly scale-invariant [65] ( ) [ ]( ) − H2 2 Γ(ν) k 3 2ν ∆2 (k) ≃ 22ν−3 , (4.51) R 2πχ¯˙ Γ(3/2) aH − ns = 2ηV 6ϵV , (4.52)

where all quantities are evaluated at the horizon-crossing time. If we make the slow-

roll parameters explicitly zero (or ν = 3/2), then the spectrum takes the simple

form ( ) H2 2 ∆2 (k) = . (4.53) R 2πχ¯˙ The next correlation function in line would be the three-point function. How-

ever, due to the smallness of the second slow-roll parameter ηV , the self-interaction of the inﬂaton is minimal. This indicates that the harmonic oscillators in the ﬁeld

space should be independent from, or only weakly coupled with, each other. Hence

the statistics should be almost all encoded in the two-point function. We shall deal

with the three-point function in the next chapter.

77 4.2 Slow-roll Inﬂation of a Scalar Field in HMT Model

In Section 3.1.3, we constructed the action for a single scalar field. In this section, we apply the perturbations developed in Section 3.2 to study inflationary models of such a scalar field. To this goal, let us first consider the slow-roll conditions.

4.2.1 The Slow-roll Conditions

For the ﬂat FLRW background, we ﬁnd that ( ) 1 J¯t = −2f χ¯˙ 2 + V˜ (¯χ) ≡ −2¯ρ, 2 J¯i = J¯ = J¯ = 0, φ ( A ) 1 τ¯ = fa2 χ¯˙ 2 − V˜ (¯χ) δ ≡ a2pδ¯ , (4.54) ij 2 ij ij

˜ where V (¯χ) ≡ V (¯χ)/f. Then, Eqs.(3.63)-(3.65) and (3.67) yield Λg = 0 and ( ) 8πG˜ 1 Λ˜ H2 = χ¯˙ 2 + V˜ (¯χ) + , (4.55) 3 2 3 where 2fG 2Λ G˜ ≡ , Λ˜ ≡ . (4.56) 3λ − 1 3λ − 1 On the other hand, Eq.(3.56) reduces to,

χ¯¨ + 3Hχ¯˙ + V˜ ′ = 0. (4.57)

Eqs.(4.55) and (4.57) are identical to these given in GR [29], if one identiﬁes G˜ and Λ˜ as the Newtonian and cosmological constants, respectively. As a result, all the conditions for inﬂationary models obtained in GR are equally applicable to the current case, as long as the background is concerned. In particular, the slow-roll conditions,

ϵ˜V , |η˜V | ≪ 1, (4.58)

78 need to be imposed in order to get enough e-fold, where ( ) 2 ˜ 2 ˜ ′ Mpl V 3λ − 1 ϵ˜V ≡ = ϵV , 2 V˜ 2f ( ) ˜ ′′ − ˜ 2 V 3λ 1 η˜V ≡ M = ηV , (4.59) pl V˜ 2f

˜ 2 ≡ ˜ with Mpl 1/(8πG), and ϵV and ηV are the ones deﬁned in (4.20). However, due to the presence of high-order spatial derivatives, the perturbations will be dramatically diﬀerent, as to be shown below.

4.2.2 Linear Scalar Perturbations Without Gauge-ﬁxing

In this section, in order for the formulae developed below to be applicable to as many cases as possible, we shall not restrict ourselves to any speciﬁc gauge. To

ﬁrst-order we ﬁnd that fχ¯′ ( ) V δρ ≡ δµ = δχ′ − χ¯′ϕ + 4 ∂4δχ + V ′δχ, a2 a4 fχ¯′ δp = (δχ′ − χ¯′ϕ) − V ′δχ, a2 fχ¯′ q = δχ, a 2c − GR 1 2 Π = 2¯pE, Π = 0, δJA = 2 ∂ δχ, [ a ] 1 ( ) δJ = c′ χ¯′ + c H − fχ¯′ ∂2δχ + c ∂2δχ′ . (4.60) φ a3 1 1 1 Hence Eqs.(3.82), (3.83), (3.85) and (3.86) read, respectively, ∫ { } 1( ) [ ] d3x ∂2ψ − 3λ − 1 H 3(ψ′ + Hϕ) − ∂2σ ∫ 2 { } V = d3x 4πG fχ¯′ (δχ′ − χ¯′ϕ) + 4 ∂4δχ + a2V ′δχ , (4.61) a2 ( ) 1 (3λ − 1)ψ′ + (1 − λ)∂2 σ − δφ = 8πGfχ¯′δχ, (4.62) [ ( a )] ( ) 1 2Hψ + 1 − λ 3ψ′ − ∂2 aσ − δφ a [( ) ] ′ ′ H − ′ ′ = 8πG c1χ¯ + c1 fχ¯ δχ + c1δχ , (4.63)

ψ = 4πGc1δχ. (4.64)

79 Note that in writing Eq.(4.62), we had chosen ∆(η) = (3λ − 1)Hϕ. It is also interesting to note that, unlike the case without the U(1) symmetry [27], now the metric perturbation ψ is proportional to δχ. It is this diﬀerence that leads to a master equation, as to be shown below. Without the U(1) symmetry, this is in general impossible

[27].

The trace and traceless parts of dynamical equation read, respectively,

( ) 1 ( ) ψ′′ + 2Hψ′ + Hϕ′ + 2H′ + H2 ϕ − ∂2 σ′ + 2Hσ ( ) 3 2 α α − 1 + 1 ∂2 + 2 ∂4 ∂2ψ 3(3λ − 1) a2 a4 2 ( ) λ − 1 ( ) 2 ˆ − H 2 ′ H + − ∂ Aψ δA + δφ + − ∂ δφ + δφ 3(3λ [1)a ] (3λ 1)a 8πG = fχ¯′(δχ′ − χ¯′ϕ) − a2V ′δχ , (4.65) 3λ − 1 ( ) [ ] 1 α 1 ( ) ψ + σ′ + 2Hσ + α + 2 ∂2 ∂2ψ − Aψˆ − δA − Hδφ = 0, (4.66) a2 1 a2 a where in writing Eq.(4.66) we had set G(η) = −ϕ. The energy conservation law now takes the form, ∫ { d3xa2χ¯′ fδχ′′ + 2Hfδχ′ + a2V ′′δχ + 2a2V¯ ′ϕ [ ] ′ } ( ) ′ A¯ (ac δχ) − fχ¯′ ϕ − ∂2E − 3ψ − 1 χ¯′ a ∫ { ( ) } − 3 4 ′ ′ ′ − H = d x ∂ V4δχ + V4 χ¯ V4 δχ . (4.67)

The momentum conservation is identically satisﬁed, while the Klein-Gordon equation becomes

{ [ ] } ( ) f δχ′′ + 2Hδχ′ − χ¯′ 3ψ′ + ϕ′ − ∂2σ + 2a2V ′ϕ + a2V ′′ − ∂2 δχ 2 [ ] ∂ ˆ ′ − − ′ ′ = 2A (c1 c2) δχ c1δφ + fχ¯ δφ + c1δA a ( ) V + V ′ V + 2 V − 2 4 ∂2 − 6 ∂4 ∂2δχ, (4.68) 1 a2 a4

80 which can be rewritten as a perturbed energy balance equation, ( ) 1 δρ′ + 3H(δρ + δp) − (¯ρ +p ¯) 3ψ′ − ∂2σ − ∂2(v + B) f 1 = (¯ρ +p ¯)δQHMT, (4.69) f where [ ( ) ] V 1 V 1 V H δQHMT = 4 ∂4δχ′ + 2V − 2 6 ∂4 − 2V + V ′ + 4 ∂2 ∂2δχ a2χ¯′2 χ′ 1 a4 a2 2 4 χ¯′ [ ] 1 + ∂2 2Aˆ (c′ − c ) δχ − c δφ′ + fχ¯′δφ + c δA , aχ¯′ 1 2 1 1 q ≡ −a(¯ρ +p ¯)(v + B). (4.70)

4.2.3 Uniform Density Perturbation

Under the gauge transformations (3.76) and (3.77), δχ and δρ transform, re-

spectively, as

δχf = δχ − ξ0χ¯′, δρe = δρ − ξ0ρ¯′. (4.71)

Therefore, the quantity ζ deﬁned earlier H −ζ ≡ ψ + δρ, (4.72) ρ¯′ is gauge-invariant in the HL theory as well. It can be shown that it obeys the evolution

equation [ ] Hδp 1 ∂2 ζ′ = − nad + δQHMT − ∂2σ − (v + B) , (4.73) ρ¯ +p ¯ 3 f where the non-adiabatic pressure perturbation is deﬁned as p¯′ δp ≡ δp − δρ = δpGR + δpHMT, (4.74) nad ρ¯′ nad nad with, ( ) [ ] 2 χ¯′′ δpGR ≡ 2 + χ¯′ (δχ′ − χ¯′ϕ) − (¯χ′′ − Hχ¯′)δχ nad 3a2 Hχ¯′ [( ) ( )′ ] 2aV˜ ′ χ¯′ χ¯′ = − (δχ′ − χ¯′ϕ) − δχ , H ′ ( 3 χ¯ )a a 2¯χ′′ V δpHMT ≡ 1 + 4 ∂4δχ. (4.75) nad Hχ¯′ 3a4

81 Note that Eq.(4.73) is quite similar to that of the case without the U(1) symmetry

[27], and the only diﬀerence is the inclusion of the U(1) gauge ﬁeld A and Newtonian

prepotential φ in δQHMT, as one can see from Eq.(4.70) given above and Eq.(4.3) given in [27]. But these terms vanish in the super-horizon region. As a result, all the conclusions obtained in [27] in this region are equally applicable to the present case. In particular, the perturbations in this region are adiabatic during the slow-roll inﬂation, as in GR.

4.2.4 Comoving Curvature Perturbation

On the other hand, the comoving curvature perturbation H R = ψ + δχ, (4.76) χ¯′ is also gauge-invariant in the HL theory. From its deﬁnition, it can be shown that R satisﬁes the equation,

H′ − H2 R′ = HS + δχ + ψ′ + Hϕ, (4.77) χ¯′ where the dimensionless intrinsic entropy perturbation S is deﬁned as δχ′ − χ¯′ϕ χ¯′′ − Hχ¯′ 3H S ≡ − δχ = − δpGR , (4.78) χ¯′ χ¯′2 2V˜ ′χ¯′ nad where to get the last step Eq.(4.75) was used. In terms of R the super-momentum

constraint (4.62) can be written in the form, ( ) λ − 1 δφ R′ = HS + ∂2 σ − , (4.79) 3λ − 1 a which reduces to R′ = HS on all scales in the relativistic limit λ → 1. Thus, in

the slow-roll approximations and neglecting the spatial gradients on large scales, we

obtain the same conclusion as that given in [27], namely, the comoving curvature

perturbation has two modes on large scales, a constant mode and a rapidly decaying

mode, given by ∫ dη R ≃ C + C . (4.80) 1 2 a2

82 In addition, unlike that in GR where the local Hamiltonian constraint enforces adi-

abaticity on large scales, in the HMT setup it is the extreme slow-roll evolution

(χ¯¨ = 0, or,χ ¯′′ = Hχ¯′) that leads to rapidly decaying entropy perturbations at late

times. Hence more generally if the expansion of the universe during the inﬂationary

epoch deviates from de Sitter expansion severely (hence slow-roll approximation is

no longer valid), the constancy of ζ and R and the adiabaticity are not guaranteed.

Note that we could also ﬁnd the ﬁrst-order equation for S by using the Klein-

Gordon equation, which can be written in the form, ( ) { [ ] χ¯′′ 1 2fχ¯′ δφ S′ + 2 + H S = [fχ¯′3Hϕ] + ∂2 (1 + 2V ) δχ − (σ − ) χ¯′ fχ¯′ 1 3λ − 1 a 2 [ ] ∂ ˆ ′ ′ + 2A (c1 − c2) δχ + c1δA − c1δφ a [ ] } ∂4 V − 2 (V + V ′) + 6 ∂2 δχ . (4.81) a2 2 4 a2

4.3 Scalar Perturbations in Sub- and Super-Horizon Scales

In this section, for convenience of calculations, we shall restrict ourselves to the

Newtonian quasi-longitudinal gauge deﬁned by Eq.(3.100) in Section 3.2.3, i.e.,

ϕ = E = δφ = 0. (4.82)

Then, Eqs.(4.61)-(4.68) can be cast in the forms ∫ { [ ]} 1( ) d3x ∂2ψ − 3λ − 1 H 3ψ′ + ∂2B 2 ∫ { ( ) } 3 ′ ′ 2 ′ V4 4 = 4πG d x fχ¯ δχ + a V + 2 ∂ δχ , (4.83) { a } ∫ ′ A¯ (ac δχ) d3xa2χ¯′ fδχ′′ + 2Hfδχ′ + a2V ′′δχ − 3fχ¯′ψ′ − 1 χ¯′ a { } ∫ ( ) − 3 4 ′ ′ ′ − H = d x ∂ V4δχ + V4 χ¯ V4 δχ , (4.84)

83 (3λ − 1)ψ′ − (1 − λ)∂2B = 8πGfχ¯′δχ, (4.85) ( )( ) [( ) ] H − ′ 2 ′ ′ H − ′ ′ 2 ψ + 1 λ 3ψ + ∂ B = 8πG c1χ¯ + c1 fχ¯ δχ + c1δχ , (4.86)

ψ = 4πGc1δχ, (4.87) 1 ( ) ψ′′ + 2Hψ′ + ∂2 B′ + 2HB 3 ( ) 2 α α 2 ( ) − 1 + 1 ∂2 + 2 ∂4 ∂2ψ + ∂2 Aψˆ − δA 3(3λ − 1) a2 a4 3(3λ − 1)a ( ) 8πG = fχ¯′δχ′ − a2V ′δχ , (4.88) 3λ − 1 ( ) ( ) ( ) 1 α 1 ψ − B′ + 2HB + α + 2 ∂2 ∂2ψ − Aψˆ − δA = 0, (4.89) a2 1 a2 a { [ ] } f δχ′′ + 2Hδχ′ − χ¯′ 3ψ′ + ∂2B + a2V ′′δχ ( ) [ ] 1 V + V ′ V 1 = 2 + V − 2 4 ∂2 − 6 ∂4 ∂2δχ + ∂2 2Aˆ (c′ − c ) δχ + c δA . 2 1 a2 a4 a 1 2 1 (4.90)

Recall Aˆ = aA¯. It can be shown that Eqs.(4.86) and (4.88) are not independent, and can be obtained from the others. Therefore, in the present case there are four independent diﬀerential equations, (4.85), (4.87), (4.89), and (4.90), for the four unknowns, ψ, B, δA and δχ. Hence we can express ψ, B and δA in terms of δχ,

and then submit them into Eq.(4.90), we obtain a master equation for δχ, which can

be written as

δχ′′ + Pδχ′ + Qδχ = F∂2δχ, (4.91)

where 4πGc 2 β ≡ f + 1 , 0 | 2 | cψ P ≡ 1 ′ H (β0 + 2 β0) , β0 { ( ) 1 4πGc c′′χ¯′2 8πG Q ≡ a2V ′′ + 1 1 − fχ¯′2 f − c ′ β |c2 | λ − 1 1 0 (ψ )} c′ 1 − 4πGc a2V ′ 3 + 1 − , (4.92) 1 | 2 | | 2 | f cψ cψ

84 and { 1 ( ) F ≡ 1 + 2V + 2A¯(c ′ − c ) − 4πGc 2 1 − A¯ β 1 1 2 1 0 } ( ) ( ) 2 2 − V + V ′ + 2πGα c 2 ∂2 − V + 2πGα c 2 ∂4 , (4.93) a2 2 4 1 1 a4 6 2 1

with 1 − λ c2 ≡ . (4.94) ψ 3λ − 1

Setting ( ∫ ) 1 δχ = exp − Pdη u, (4.95) 2 one can write Eq.(4.91) in the momentum space in the form,

′′ 2 uk + ωkuk = 0, (4.96)

where

1 ( ) 2 2F − P′ P2 − Q ωk = k k 2 + 4 , { 4 1 ( ) F ≡ 1 + 2V + 2A¯(c ′ − c ) − 4πGc 2 1 − A¯ k β 1 1 2 1 0 } 2k2 ( ) 2k4 ( ) + V + V ′ + 2πGα c 2 − V + 2πGα c 2 . (4.97) a2 2 4 1 1 a4 6 2 1

Note that the above hold only for λ ≠ 1. When λ = 1, we have a ﬁrst-order equation for δχ ′ ′ χ¯ ′ − δχ + (c1 f) δχ = 0, (4.98) c1 which has the general solution, {∫ } ′ ( ) χ¯ − ′ δχ = exp f c1 dη δχ1(x), (λ = 1), (4.99) c1

where δχ1(x) is an arbitrary function of x only. Since in this dissertation we are mainly interested in the case λ ≠ 1, in the following we shall not consider this case

further.

85 Also, for the ﬁeld to be stable in the UV regime, the condition

2 V6 + 2πGα2c1 < 0, (4.100) has to be satisﬁed. To study Eq.(4.96) further, we consider the sub- and super-horizon scales, separately.

4.3.1 Sub-Horizon Scales

In this region, we have k ≫ H, and the dispersion relation reduces to,

2k6 ( ) 2 ≃ − 2 ωk 4 V6 + 2πGα2c1 . (4.101) β0a

With the extreme slow-roll condition, we have a ≃ −1/Hη and H,V6, c1 ≃ constants. Then, from Eq. (4.96) we ﬁnd that

iωkη uk ∝ e . (4.102)

Unlike the case without the U(1) symmetry [27], the metric perturbations ψ and B now oscillate with the same frequency as δχ, as one can see from Eqs.(4.85) and (4.87). Therefore, they are always coupled to the scalar ﬁeld modes.

4.3.2 Super-Horizon Scales

In this region, we have k ≪ H, and to the order of k2, we ﬁnd that [ ] k2 ( ) 2P′ + P2 2 ≃ ¯ ′ − − 2 − ¯ Q − ωk 1 + 2V1 + 2A(c1 c2) 4πGc1 1 A + . (4.103) β0 4

In the extreme slow-roll and massless limit (¯χ′ ≃ 0 ≃ V ′,V ′′ ≃ 0), we obtain the following solution { } [ ] D k2η2 ( ) u = − 1 1 + 1 + 2V + 2A¯(c ′ − c ) − 4πGc 2 1 − A¯ k Hη 2β 1 1 2 1 { 0 } 2 2 [ ( ) ] 2 k η ¯ ′ 2 ¯ +D2η 1 − 1 + 2V1 + 2A(c1 − c2) − 4πGc1 1 − A 10β0 2 ∼ D1a + D2η , (4.104)

86 where the ﬁrst term represents a constant perturbation, while the second term rep-

resents a decaying mode. Then, we ﬁnd that 12πGc ≃ − 3 ≃ 2 ≃ − 1 2 δχ D1 D2Hη , ψ 4πGc1δχ, k B 2 D2η . (4.105) |cψ| In terms of the gauge-invariant quantities (3.79), we obtain

Ψk = ψk − HBk, H ′ Φk = Bk + Bk, ( ) ( ) 2 2 2 2 2 2 ˆ Φk − Ψk = H k η H k η α2 − α1 ψk + Hη Aψk − δAk . (4.106)

Thus, like in the case without the U(1) symmetry [27], the dynamical evolution now

leads to Φ = Ψ → 0 at late times (η → 0).

4.4 Power Spectra and Indices of Primordial Perturbations

To calculate the spectra and indices of primordial scalar and tensor perturbations with the slow-roll approximations, we shall use the uniform approximation, proposed in [60], and applied to the studies of tensor perturbations in the HL theory without the U(1) symmetry in [28, 66]. As illustrated in the Section 4.1, the

EoM for the mode functions lacks an exact solution in generic inﬂationary background. In the leading order slow-roll approximation, the EoM for the mode function around horizon crossing time (ω2 ≃ z′′/z) enjoys an exact solution built of Hankel

functions. The aim of the uniform approximation is to obtain a single approximate

solution to the EoM with z′′/z = C(η)/η2 where C(η) is varying only slowly with

time. The method at leading order is able to reproduce the result originally derived

under slow-roll approximations. Better, the method has deﬁnite error bounds and

is systematically improvable. This latter virtue is of more importance to us for the

current case since we’ve made the slow-roll approximation. We shall closely follow

the treatment presented in [28]. In particular, for perturbations given by,

′′ vk = [g(k, η) + q(η)]vk, (4.107)

87 2 where q(η) = −1/4η , and vk is the canonically normalized ﬁeld, the corresponding power spectrum and index at leading order of the uniform approximation are given as [28],

k3 2 ≡ | |2 ∆v(k) kη→0− 2 vk 2π − kη→{0 } k3 exp 2D(k, η) = lim √ , (4.108) kη→0− 2 2 4π a g(k, η) 2 d ln ∆v nv − 1 ≡ , (4.109) d ln k kη→0− where ∫ η √ D(k, η) ≡ g(k, η′)dη′, (4.110) η¯(k) andη ¯(k) denotes the turning point g(k, η¯) = 0. Note that in writing the above expressions, we assumed that there is only one turning point, that is, we consider only the case where g(k, η) = 0 has only one real root. For detail, see [28]. In the following, we shall apply the above to the cases of scalar and tensor perturbations.

4.4.1 Scalar Perturbations

With the help of the master equation (4.91) and the deﬁnition of gauge-invariant

R in (4.76), the second order action reads, ∫ [ ] 1 S(2) = dηd3xa2h2 β R′2 − β R2 − β (∂ R)2 − β (∂2R)2 − β (∂ ∂2R)2 ,(4.111) 2 0 4 1 i 2 3 i where

2 | 2 | β0 = f + 4πGc1/ cψ , [ ] ≡ ¯ ′ − − 2 − ¯ β1 1 + 2V1 + 2A(c1 c2) 4πGc1(1 A) , 2 ( ) β ≡ V + V ′ + 2πGc2α , 2 a2 2 4 1 1 2 ( ) β ≡ − V + 2πGc2α , 3 a4 6 1 2 ′ h′2 (a2β hh′) β ≡ β Q − β + 0 4 0 0 h2 a2h2 ( )− H 1 δχ h ≡ 4πGc + = . (4.112) 1 χ¯˙ R

88 After introducing the variable

2 2 2 v ≡ zR, z ≡ a h β0, (4.113) the action is normalized to ∫ [ ] 1 β β β (2) 3 ′ 2 − 1 2 − 2 2 − 2 2 2 − 3 2 2 S = dηd x (v ) (∂iv) meﬀv (∂ v) (∂i∂ v) . (4.114) 2 β0 β0 β0

2 Here meff is defined to be z′′ β − 2 ≡ − 4 meff . (4.115) z β0

Going through the quantization procedure as described in Section 4.1, the classical equation of motion (EoM) for mode functions vk are ( ) ′′ 2 2 vk + ωk + meff vk = 0, (4.116) where 2 ( ) 2 k 2 4 ωk = β1 + β2k + β3k . (4.117) β0 Looking at the expressions of the β coefficients, we see that they contain terms ¯ of c1, c2,V1,V2,V4,V6 and A. Now go back to the Lagrangian describing the inflaton

χ, Eqs.(3.24) and (3.33), one can see that V1,V2,V4,V6 all stem from the potential term V, while c1 and c2 appear through the first line of (3.33), which can also be taken as a “potential term”. (Note that the second and the third line of (3.33) correspond to modifications of dynamical coupling terms due to the presence of φ.). Therefore, we could assign their respective “slow-roll” parameters describing their time evolution during inflation in a manner similar to V . However, unlike V , which appears in the background equation (4.55), these “potential terms” are not constrained by the background equations. As an approximation, we assume here that the time depen- ¯ dence of c1, c2,V1,V2,V4,V6 and A are at least second order in terms of the slow-roll parameters. Since we only consider the first order approximations in this paper, they can be taken as constant throughout inflation. This also leaves dβ0/dη ∝ dc1/dη = 0.

89 With the above assumptions, it can be shown that h2 relating δχ and R is of

order O(ϵV ). In fact, from its deﬁnition, [ ] ( )− ( ) −2 H 2 χ¯˙ 2 c χ¯˙ h2 = 4πGc + = 1 + 1 1 ˙ 2 χ¯ H 2MplH [ ] −2 ˙ ˜ 2 c1χ¯ = 2MplϵV 1 + 2 , (4.118) 2MplH

and ˙ ˙ √ c1χ¯ √ c1 × √ 1 χ¯ √ c1 ≪ 2 = = ϵ˜V 1, 2MplH 2Mpl 2Mpl H 2Mpl

where |c1| ≃ M∗ ≪ Mpl [40], we have, to ﬁrst order of the slow-roll parameters,

2 ≃ ˜ 2 h 2MplϵV . (4.119)

On the other hand, a(η) ≃ −(1 + ϵV )/(Hη), which leads to

2 2 − 3ηV + 9ϵV ∆m −m2 ≃ + . (4.120) eﬀ η2 η2

Here the ﬁrst term comes from z′′/z and is the same as that from GR under the above assumptions, whereas the second term introduces new eﬀects, [ ( ) ( ) ] 2 √ 2 1 2f 2 c1 ∆m ≃ 3 (β − 1) ηV + − 6β ϵ − √ ϵ . (4.121) 0 − 0 V − V β0 λ 1 λ 1 2Mpl

We see that these eﬀects depend in general on λ. Since it’s the deviation of meﬀ from the de Sitter value 2/η2 that breaks the exact scale-invariance, we would expect that

observations on the power index, which will be derived below, place constraints on

the value of λ.

The function g(k, η) deﬁned through Eqs.(4.107) and (4.116) is now given by

k2 [ ( )] g(k, η) = a2 − a y2 + a y4 + a y6 , (4.122) y2 0 2 4 6

90 where

y ≡ kη,

2 1 2 2 9 2 a ≡ − m η = − 3ηV + 9ϵV + ∆m , 0 4 eﬀ 4 1 [ ] ≡ ¯ ′ − 2 − ¯ a2 1 + 2V1 + 2A(c1 + c2) 4πGc1(1 A) , β0 1 [ ( )] ≡ 2 − ′ 2 a4 2H (1 2ϵV ) V2 + V4 + 2πGc1α1 , β0 1 [ ( )] ≡ − 4 − 2 a6 2H (1 4ϵV ) V6 + 2πGc1α2 . (4.123) β0

Thus the power spectrum of R is given by,

( ) [ ] ( ) − 2 − | |3 2a0 4 6 3 2a0 2 H (1 2ϵV ) y0 2 a4y0 + a6y0 y ∆R(k) = exp lim , (4.124) 2 2 → 4π β0a0h e 3a0 y 0 y0

where y0 is deﬁned to be the turning point of g(k, η¯), namely,

6 4 2 − 2 a6y0 + a4y0 + a2y0 a0 = 0. (4.125)

Clearly, y0 is independent of k, as an’s are. Then, we ﬁnd

2 2 nR − 1 = 2η − 6ϵ − ∆m . (4.126) V V 3

Due to the term ∆m2, the spectrum index of the scalar perturbation is in general

diﬀerent from that given GR. In particular, we see that it depends on the value

1/(λ − 1) (Note that β0 is also a function of λ − 1). One may worry that in the relativistic limit at low energy λ → 1 these terms will diverge, hence breaking the

near-scale-invariance of the spectrum. However, during inﬂation, we are in a region

where UV physics dominates, thus the value of λ is expected to be far away from its

relativistic ﬁxed point at that time.

We note that even when the index can restore the standard value in GR,2 it is

a consequence of our assumption that all the potential terms c1, c2,V1,V2,V4 and V6

2 2 Mathematically, in the limit c1 = 0, β0 → 1 and f /3β0(λ − 1) → 1, the standard result can be restored.

91 are time-independent. Comparing the deﬁnition of z given here in Eq.(4.113) with

the one given in GR, we can see some extra terms of c1 appear. By our assumption,

2 dc1/dη = 0, this makes dβ0/dη = 0 and dh /dη ∝ d(χ/H¯˙ )/dη, which leaves the term z′′/z exactly the same as that of a single ﬁeld in GR. Further more, in the modiﬁed

dispersion relation (4.117), the term β1/β0 corresponds to the relativistic case, and

the ones β2/β0 and β3/β0 are induced by Lorentz-symmetry-breaking eﬀects, which are assumed to be time-independent. If, however, these potential terms are evolving

with time during inﬂation, one needs to take into account of the time-dependence of

the dispersion relation and of the varying eﬀective mass [67]. The same arguments

also apply to the studies of tensor spectrum and index.

We would also like to note that, the exact form of scale-dependence of the scalar

spectrum depends on the instant when it’s evaluated [68], and can receive further

corrections when we incorporate a second order uniform approximation [60]. What’s

more, from an observational point of view, as long as the scale-invariance is not broken

severely, the connection between tensor-scalar-ratio and slow-roll parameters is more

important than the tilt itself.

Setting the slow roll parameters to zero exactly, the power spectrum given above

can be put in the simple form, [ ] 4H4|y |3 2 ( ) ∆2 (k) = 0 exp a y4 + a y6 . (4.127) R 2 3 2 4 0 6 0 3π e β0χ¯˙ 9

In the relativistic limit (a2 = β0 = 1, a4 = a6 = c1 = 0, and gs = 0, (s = 2, ..., 8), this yields the well-known result obtained in GR (4.53), ( ) 18 H2 2 ∆2GR = , (4.128) R e3 2πχ¯˙

except for the factor 18/e3 ∼ 0.896. This diﬀerence in magnitude is due to the way

we normalize the power spectrum in the uniform approximations. As will be shown

later, the same factor also appears in the expression for the power spectrum of tensor

perturbations, so that the ratio of them does not depend on this factor.

92 To estimate the eﬀect from higher curvature terms on the power spectrum, let us ﬁrst write the dispersion relation (4.117) in the form,

k4 k6 2 ≡ 2 ωk b1k + b2 2 2 + b3 4 4 , (4.129) a MA a MB where [40]

−1/2 −1/4 MA ≡ |g3| Mpl,MB ≡ |g8| Mpl, (4.130) and

1 [ ] ≡ ¯ ′ − 2 − ¯ b1 1 + 2V1 + 2A(c1 + c2) 4πGc1(1 A) , β0 2 b2 ≡ [λ2 + λ4 + λ23] , β0 −2 b3 ≡ [λ6 + λ78] , (4.131) β0 with

≡ 2 ≡ ′ 2 ≡ 4 λ2 V2MA, λ4 V4 MA, λ6 V6MB, c2M 2 c2M 4 ≡ 1 A ≡ 1 B − λ23 4 (8g2 + 3g3), λ78 6 (8g7 3g8). (4.132) 2Mpl Mpl

Since g2 and g3 all both the coeﬃcients of the fourth-order derivative terms, as one can see from Eq.(2.4), it is quite reasonable to assume that g2 and g3 are in the same order, g3/g2 ≃ O(1). Similarly, one can argue that gs/g4 ≃ O(1) for s = 5, 6, 7, 8, as all of these terms are the coeﬃcients of the sixth-order derivative terms. For the sake of simplicity, we further assume MA ≃ MB = M∗. Taking c1 ≃ c2 ≃ M∗, based on

1/2 considerations on the strong coupling problem from Eq.(3.2) we ﬁnd M∗ ≤ Mpl|cψ| [40]. Then, from Eq.(4.112) we obtain ( ) 4/5 1 M∗ β0 ≃ f(λ) + ≃ O(1), (4.133) 2 Λω as f(λ) ≃ O(1). To determine the scales of Vn, we assume that bn deﬁned in Eq.(4.131) are all of order 1, i.e.,

bn ≃ O(1), (n = 1, 2, 3), (4.134)

93 which is a reasonable assumption, considering the physical meanings of the energy

scales MA and MB. In fact, one can deﬁne MA and MB so that b2 = b3 = 1 precisely, as originally deﬁned in [40]. To have b1 = 1, one can properly choose V1. On the other hand, since A¯ is undetermined, and for the sake of simplicity, we further set

A¯ = 0.

With all the above assumptions, we ﬁnd that the function g(k, η) now reads, [ ] k2 9 ( ) g(k, η) = − y2 1 + ϵ y2 + ϵ2 y4 , y2 4 HL HL 2 ≡ H ϵHL 2 . (4.135) M∗

Thus, depending on the energy scale H when inﬂation occurs, one can have diﬀerent ≪ ≫ turning point y0. In the following we consider only two limits, ϵHL 1 and ϵHL 1. In

addition, in writing Eq.(4.135) we have set bn = 1 = β0 precisely. General expressions

without setting bn = 1 can be found at the end of this section.

4.4.1.1 Case 1) ϵHL ≪ 1. When ϵHL ≪ 1, to its second order, we ﬁnd that ( ) 9 9 81 y2 ≃ 1 − ϵ + ϵ2 , (4.136) 0 4 4 HL 16 HL

for which the power spectrum is given by, ( ) 9 729 ∆2 (k) ≃ ∆2GR 1 − ϵ + ϵ2 . (4.137) R R 4 HL 128 HL

It is interesting to note that the condition ϵHL ≪ 1 is equivalent to ( ) 3 Λ 2 ≪ − ω 4 ≪ V (¯χ) (3λ 1) Mpl, (ϵHL 1). (4.138) 2 Mpl

≫ ≫ 4.4.1.2 Case 2) ϵHL 1. When ϵHL 1, to ﬁnd the turning point y0, we ﬁrst write g(k, η) given by Eq.(4.135) in the form, [ ] k2 9 ( ) g(k, η) = ϵ2 η2 − y2 η2 + η y2 + y4 , (4.139) HL y2 4 HL HL HL

94 where ηHL ≡ 1/ϵHL ≪ 1. Then we ﬁnd the perturbative solution ( ) [ ( ) ( ) ] 3η 2/3 1 4η 1/3 2 4η 2/3 y2 ≃ HL 1 − HL − HL , (4.140) 0 2 3 9 9 9 for which the power spectrum takes the form, √ [ ( ) ] 4η e 1 4η 1/3 ∆2 (k) ≃ ∆2GR(k) HL 1 − HL . (4.141) R R 9 2 9

Thus, if the inﬂation happened way above the scale M∗, the spectrum will be sup-

2 2 pressed by the factor M∗ /H , comparing with that of GR.

4.4.2 Tensor Perturbations

The tensor perturbations can be written in the form [13, 28]

2 δgij = a (δij + hij) , (4.142)

i j where hij is traceless and transverse, i.e., h i = 0 = ∂ hij. For a single scalar inﬂaton, the anisotropic stress is zero, so the tensor perturbations are source-free. In the ADM formalism, with the results of constraint equations derived in Section 3.1.4, it can be shown that the second order action is given by ∫ { 1 ζ2a2 S(2) = dηd3x (∂ h )2 − (1 − A¯)(∂ h )2 2 2 η ij c ij } g ( ) g ( ) − 3 ∂2h 2 − 8 ∂ ∂2h 2 . (4.143) ζ2a2 ij ζ4a4 c ij

Deﬁning the following expansion in the momentum space [59], ∫ d3k ∑ h = ϵs (k)hs (η)eikx, (4.144) ij (2π)3 ij k s=+,×

i s s′ where ϵii = k ϵij = 0 and ϵij(k)ϵij(k) = 2δss′ , the above action becomes ∫ { } ∑ 2 4 6 a ′ g k g k S(2) = dηd3k (h s)2 −(1−A¯)k2(hs )2 − 3 (hs )2 − 8 (hs )2 . (4.145) 2ζ2 k k ζ2a2 k ζ4a4 k s=+,×

s To make the action canonically normalized, we introduce vk by

s ≡ s vk aζhk. (4.146)

95 Then, the action (4.145) becomes [ ] ∑ ∫ 1 ′ S = dηd3k (v s)2 − (ω2 + m2 )(vs )2 , (4.147) (2) 2 k k eﬀ k s=+,× but now with

g k4 g k6 ω2 = (1 − A¯)k2 + 3 + 8 , k ζ2a2 ζ4a4 a′′ m2 = − . (4.148) eﬀ a

One can see that each spin state of the tensor perturbation acts like a scalar. The classical equation of motion for the mode functions again read

′′ 2 2 vk + (ωk + meﬀ)vk = 0, (4.149)

2 where in writing the above equation, we had dropped the super indices “s”, and ωk

2 and meff are now defined by Eq.(4.148). From the above, we can directly read off g(k, η) for tensor perturbations, { [ k2 9 g(k, η) = (1 + 2ϵ ) − (1 − A¯)y2 y2 4 V } ] g H2 g H4 + 3 (1 − 2ϵ )y4 + 8 (1 − 4ϵ )y6 , ζ2 V ζ4 V y ≡ kη. (4.150)

2 ≡ 2 Thus, its turning point y0 (kη¯) satisﬁes the cubic equation

9 g H2 g H4 (1 + 2ϵ ) = (1 − A¯)y2 + 3 (1 − 2ϵ )y4 + 8 (1 − 4ϵ )y6. (4.151) 4 V 0 ζ2 V 0 ζ4 V 0

Then the dimensionless spectrum and index for the tensor perturbations can be de-

ﬁned as [59],

k3 |v |2 ∆2 (k) ≡ 4 × k , (4.152) T kη→0− 2π2 ζ2a2 kη→0− d ln ∆2 ≡ T nT . (4.153) d ln k kη→0−

96 Here the factor of 4 accounts for the two spin states and the normalization of the

polarization tensors to 2.

Again, assuming that the gauge field A¯ is constant during inflation, we find [ ( )] 16H2|y |3 2H2y4 H2 ∆2 (k) = 0 exp 0 g + g y2 , (4.154) T 3π2e3ζ2 9ζ2 3 8 0 ζ2

nT = −2ϵV . (4.155)

In the relativistic limit, Eq.(4.154) yields the well-known results obtained in

GR [59]

2 2GR 18 2H ∆T (k) = 3 2 2 . (4.156) e π Mpl

Because of the normalization of the power spectrum in the uniform approximation,

as mentioned in the last section, a diﬀerence of a factor 18/e3 also appears in the tensor perturbations.

To study the eﬀect of high order curvature terms, following what we did for the

scalar perturbations, we consider the two cases ϵHL ≪ 1 and ϵHL ≫ 1, separately.

4.4.2.1 Case 1) ϵHL ≪ 1. In this case, the power spectrum (4.154) takes the form ( ) 9 729 ∆2 (k) ≃ ∆2GR(k) 1 − ϵ + ϵ2 . (4.157) T T 2 HL 32 HL

Then, from Eqs.(4.137) and (4.157), we ﬁnd that the scalar-tensor ratio is given by ( ) ∆2 (k) 9 2187 ≡ T ≃ − 2 r 2 16ϵV 1 ϵHL + ϵHL . (4.158) ∆R(k) 4 128

For the general case, see Eq.(4.163).

≫ ≫ 2 4.4.2.2 Case 2) ϵHL 1. When ϵHL 1, from Eq.(4.150) we ﬁnd that y0 is given by, ( ) [ ( ) ( ) ] 3η 2/3 1 16η 1/3 1 16η 2/3 y2 ≃ HL 1 − HL − HL , (4.159) 0 4 6 9 18 9

97 and the power spectrum takes the form, [ ( ) ] 2e1/2η 1 16η 1/3 ∆2 (k) ≃ ∆2GR(k) HL 1 − HL . (4.160) T T 9 4 9

Then, the combination of it with Eq.(4.141) yields [ ( ) ] 2 − 22/3 4η 1/3 r ≃ 8ϵ 1 + HL . (4.161) V 4 9

For arbitrary bn, see Eq.(4.165).

4.4.2.3 General Expressions for Power Spectra. Here we give the more general expressions of spectra without setting b1 = b2 = b3 = 1 = β0. MA = MB = M∗ (and 2 ¯ thus by (4.130) g3 = g8) and A = 0 is still assumed. In the limiting case when

2 ϵHL ≡ (H/M∗) ≪ 1, we ﬁnd ( ) 2 1 c χ¯˙ ∆2 (k) ≃ ∆2GR 1 + 1 R R (b )2/3β 2M 2 H [ 1 0 pl ] 9b 81(17b2 − 8b b ) × − 2 2 1 3 2 1 2 ϵHL + 4 ϵHL , (4.162) 4b1 128b1 ( ) −2 c χ¯˙ r ≃ 16ϵ (b )2/3β 1 + 1 V 1 0 2M 2 H [ pl ] 9(b − 2b2) 81(36b4 − 17b2 + 8b b ) × 2 1 1 2 1 3 2 1 + 2 ϵHL + 4 ϵHL . (4.163) 4b1 128b1

In the limit ϵHL ≫ 1, we obtain ( ) [ ] 1 2 ( ) 1 4e 2 η c χ¯˙ b 4b 3 2 ≃ 2GR √ HL 1 − 2 3 ∆R(k) ∆R 1 + 2 1 ηHL , (4.164) 9 b3β0 2MplH 2b3 9 ( ) √ −2 b β c χ¯˙ r ≃ 16ϵ 3 0 1 + 1 V 2 2M 2 H { [ pl ] } ( ) ( ) 1 1/3 3 1 b2 b3 16 × 1 − − ηHL . (4.165) 4 2b3 4 9

Recall ηHL = 1/ϵHL. Clearly, the magnitude of the ratio r are dependent on the values of b1 and b3.

98 CHAPTER FIVE

Primordial Non-Gaussianities in General Covariant Hoˇrava-Lifshitz Gravity

This chapter published as: Y. Huang and A. Wang, “Non-Gaussianity of a single scalar ﬁeld in general covariant Hoˇrava-Lifshitz gravity”, Phys. Rev. D 86,103523 (2012).

Though the minimum scenario of the inflation models predicts that the statistics of the primordial perturbations is highly Gaussian, it is nevertheless important to study the non-Gaussian characteristics, since it may provide further information in the early universe and physics beyond the standard model. While a change in high energy physics during or before inflation gives only mild modifications to the power spectrum and index, non-Gaussianities, measuring the nonlinearity of the system and evaluated by higher order correlation functions of the fluctuations (higher than two, the power spectrum), is much more sensitive to the new physics in the ultraviolet

[69]. The strength of such signals could be well within the range of detection of the current Planck satellite [70] and the forthcoming experiments, such as the CMBPol mission [71]. Hence non-Gaussianities have attracted lot of attention within the past decade and been studied extensively in various models [61, 72]. Since the HL theory diﬀers from GR signiﬁcantly in high energies, in this chapter we study the primordial non-Gaussianity in the HMT setup.

Note that the current and near future experiments mainly measure the non-

Gaussian signals in the CMB, as the statistics of large scale structures is more prone to the contamination of nonlinear local gravitational effects and hence it’s more difficult to extract information from it. On the other hand, the detection of non-Gaussianities in CMB are through the statistics of the temperature fluctuations ∆T/T . Though it is sourced by the primordial perturbations ζ and R, it is not free from effects from low-energy physics such as the Sachs-Wolfe effect [65, 73]. Hence we consider only

99 the primordial non-Gaussianities in the perturbations generated during the slow-roll inﬂation era.

To study non-Gaussianity, various techniques have been developed. In particular, the in-in formalism, developed initially by Schwinger some decades ago [74], becomes the standard after it was ﬁrst explored by Maldacena in his pioneer work

[75] in calculating the high-order correlators for cosmological perturbations. This is further developed by Weinberg in his seminal paper [76]. Therefore we ﬁrst describe the in-in formalism in Section 1.

Primordial non-Gaussianities in HL theory were studied previously, one without the projectability condition [77], and the other with it [78] but in the curvaton scenario

[79], and many interesting results were obtained. However, the study presented in this chapter are different from those in at least two aspects: (i) we shall study the problem in the HMT setup, in which the degree of gauge freedom is the same as that in GR; and (ii) continuing with the work in Chapter 4, we shall investigate the problem in the inflationary scenario for a single scalar field with slow-roll conditions.

Because of these differences, the results presented in this chapter are significantly different from theirs.

Because non-Gaussianities measures the nonlinearity of a system, we must go to higher orders of the primordial perturbations. As first noted in Chapter 2, unlike the much easier case in linear order, scalar, vector and tensor perturbations beyond the linear order couple to each other. Just like the background quantities may affect the evolutions of linear perturbations, perturbations beyond linear order are generally affected by all the lower-order quantities (in this sense the background is the zeroth order). Hence one can no longer study the evolutions of scalar, vector and tensor perturbations separately as the “background” now includes lower order perturbations and does not possess the O(3) symmetry of the FLRW metric [29]. Thus the non-

Gaussian structure of these primordial perturbations is much richer than that of the

100 power spectrum. In particular, the scalar perturbations may now be correlated with

tensor perturbations, which is not possible in the power spectrum. We study only

the correlations of the scalar-scalar-scalar and tensor-tensor-tensor forms.

For the correlator of scalar-scalar-scalar perturbations in Section 2, we ﬁrst give

the interaction Hamiltonian and analyze the leading order terms. With some reason-

able assumptions, we have found that the leading order terms in self-interaction of

3/2 the HMT model are of order ϵ αˆn (n = 3, 5), where ϵ is the slow-roll parameter ≡ 4 andα ˆn are dimensionless constants, defined as Vn αˆn/Mpl, and Vn are the coupling coefficients of sixth order derivative operators of the scalar field, and have no contributions to the power spectra and indices, as shown explicitly in Chapter 4. Clearly, by properly choosing those coefficients, a large non-Gaussianity is possible. This is different from the standard result of minimum scenario in general relativity, where the interaction terms are of order ϵ2 [75, 76]. We then discusses the modified dispersion relation and its effect on the mode function and the shapes of bispectrum using the matching method proposed in [80]. By dividing the history of inflation into three regions, in which the dispersion relation takes different asymptotic forms, the gauge- invariant comoving curvature perturbation R has different asymptotic solutions. In particular, we have found that the mode function is in general a superposition of oscillatory functions. This is different from the standard choice of the Bunch-Davis vacuum, where only one positive frequency branch of the plane wave is selected, and results in an enhancement of the folded shape in the bispectrum. The bispectrum is enhanced, and gives rise to a large nonlinearity parameter fNL [cf. Eq.(5.70)], as long as M∗, above which the non-linear terms in the dispersion relation dominate, isn’t much lower than the Planck scale [See (5.71) and (5.72)]. These results are also compared with those presented in [78]. At the end of this section we summarize the assumptions made along the way of analysis. This section was published in [23, 24].

101 The 3-point correlator for tensor perturbations is studied in the last section.

There we ﬁnd that the interaction Hamiltonian, at leading order, receives contribu-

ij i j k tion from four terms built of the 3-dimensional curvature: R,RijR ,RjRkRi and ( ) i jk ∇ R (∇iRjk). The Ricci scalar R yields the same k-dependence as that in general relativity, i.e. its signal peaks at the squeezed limit regardless of the spins of the tensor ﬁelds, but with diﬀerent magnitude due to coupling with the U(1) gauge

field A and a UV history when the dispersion relation is significantly different from the relativistic form. This magnitude could lead to potentially large non-linearity parameter, if the new mass scale M∗ is not too much lower than the Planck scale. ( ) ij i jk The two terms RijR and ∇ R (∇iRjk) generate shapes similarly to the R term.

i j k The term RjRkRi favors the equilateral shape when spins of the three tensor fields are the same but peaks in between the equilateral and squeezed limits when spins are mixed. We find that this is due to the effect of the polarization tensors: when spins are mixed, the product of the three polarization tensors strongly favors the squeezed shape. Thus an absence of the equilateral shape in the bispectrum of the tensor perturbations cannot rule out gravity theories of higher order derivatives.

5.1 Non-Gaussianity

5.1.1 The Three-point Correlator

Take for example the case for scalar perturbations, the quantity we want to compute is the three-point correlator ⟨Rˆ 3(t)⟩, 1 which has a deep connection to non-Gaussian statistics. We know from statistics that for a (standard) Gaussian distribution, any moment of higher order ⟨x2n⟩ can be expressed in terms of the variance ⟨x2⟩ and odd order moments vanish due to symmetry. This is known as the

Isserlis theorem in probability theory.

1 Note that in this chapter the quantities with a hat are quantized ﬁelds.

102 Similarly, for quantum mechanics (QM), free harmonic oscillators’ ground state

wave-function is in the form e−x2 . In the Copenhagen interpretation of QM, the mod-

ules square of the wave-function is the probability distribution function (PDF). Hence

any expectations of operators evaluated with this PDF has the same conclusions as

above: ⟨xˆ2n⟩ can be expressed as parings of ⟨xˆ2⟩ while ⟨xˆ2n+1⟩ vanish. The case for

QFT is similar, since we can interpret it as to each spacetime point we attach a harmonic oscillator. In other words, if there were no non-linear (self-)interactions power spectrum gives us all the information.

Thus to have a higher order correlator, the mode function/vacuum has to be perturbed by non-linear (self-)interaction. The in-in formalism can be viewed as the perturbative approach to this problem. This also have a connection with how statisticians treat a problem of ﬁnding higher moment by perturbing the Gaussian measure [81]: For the simple case of a single variable (a single ﬁeld in the case of

QM), the expectation value is ∫ ⟨xk⟩ = dx xkP (x), (5.1)

where P (x) is the PDF. If P (x) is weakly non-Gaussian

∑ l 2 2 λ x P (x) ∝ e−x /2σ −V (x),V (x) = l , (5.2) l! l≥3 then ⟨xk⟩ no longer vanishes for k = 2n + 1 and the calculation can be treated

perturbatively. For QFT, the expectation value in the path integral formulation is ∫ ⟨Q(ϕˆ)⟩ = D[ϕ]Q(ϕˆ)eiSR , (5.3)

where SR is the action with Minkowskian signature. After a Wick rotation it → t,

− − eiSR → e SE0 SEI , (5.4) where SE0 is the quadratic (or free) action with Euclidean signature and ∑ λ ϕˆl S ∝ l . (5.5) EI l! l≥3

103 5.1.2 The In-in Formalism

Recall the treatment of the quantities in the past chapters, such as the metric

and the scalar ﬁeld. In studying their behavior we assumed a time-dependent-only

¯ ¯ i classical background value (c-number) for each of them (N, N , g¯ij andχ ¯), and at- { Sch} tached perturbations δφa to them respectively. The classical background values’ evolution is determined by the background equations (4.55) and (4.57). For the per-

turbations of the action, we can break them into two parts, the linear order that can

be shown to vanish by utilizing the background equations, and higher order ones. The

higher order perturbations of the action can be further broken into two sub-categories,

the second (quadratic) order S(2) which we wrote down explicitly in Section 4.4, and

higher orders SI that we shall compute below. We then quantized the perturbations with the quadratic action S(2) and solved the EoM for the mode function vk of the quantized fields. In the language of Hamilto- nian formalism, these quantized fields and their conjugate momentum {δφÎ, δπÎ} are

the interaction picture ﬁelds, whose evolution is generated by the Heisenberg equa-

ˆ ˆ 2 tions with the quadratic Hamiltonian H0(t), and thus act as free ﬁelds under H0(t). { Sch} On the other hand, the evolution of perturbations δφa is not just determined by ˆ ˆ H0(t), but also higher orders HI (t). If the higher order perturbations do not exist, then the vacuum deﬁned in the

free theory, which we use |0⟩ to denote, is the true quantum vacuum of the theory

(recall that classical vacuum is “literally nothing”). And 2n + 1-point correlators

evaluated on it will simply vanish, just like ⟨x2n+1⟩ = 0 on a Gaussian measure.

So our task under this interaction picture is two-fold: a) to see the eﬀect of HI on ˆ |0⟩ and project these free vacuum onto the interaction vacuum |Ω⟩, and b) how ζSch ⟨ | ˆ3 | ⟩ evolves under HI. Thus the expectation value we are calculating is Ω ζSch(t) Ω . The expectation value of a product of operators QˆSch in the in-in formalism is given

2 Note that now the quadratic Hamiltonian is time dependent explicitly as the background c-numbers are dynamical.

104 by [61]

⟨ | ˆSch | ⟩ ⟨ | ˆ−1 ˆ −1 ˆSch ˆ ˆ | ⟩ Ω Q (t) Ω = Ω F U0 Q (t0) U0 F Ω

= ⟨Ω| Fˆ−1QˆI(t)Fˆ |Ω⟩ , (5.6)

in which the “unitary time-evolution operators” are ( ∫ ) t ˆ ˆ ′ ′ F (t, t0) = T exp − i HI(t )dt , (5.7) t0 ( ∫ ) t ˆ ˆ ′ ′ U0(t, t0) = T exp − i H0(t )dt , (5.8) t0 where T is the time-ordering operator.

We can further replace |Ω⟩ by the free vacuum |0⟩, in most cases the Bunch-

Davis vacuum, as the in state. As explained in Weinberg’s seminal paper [76], to evaluate the expectation value of operators in the interaction vacuum, one could first perform the calculation in the free vacuum, and later multiply this with a numerical factor contributed by the vacuum fluctuations. This numerical factor, in the in-in formalism, is simply ⟨[ ∫ ][ ∫ ]⟩ ( t ) ( t ) ¯ ˆ ˆ T exp i HI(t)dt T exp − i HI(t)dt = 1. t0 t0 “Hence in the ‘in-in’ formalism all vacuum fluctuation diagrams automatically cancel.”

Thus the 3-point correlator becomes ( ) 3 ⟨Ω| Rˆ 3(t) |Ω⟩ = ⟨0| Fˆ−1 Rˆ I(t) Fˆ |0⟩ , ( ∫ ) t ˆ ˆ ′ ′ F (t, t0) = T exp − i HI(t )dt . (5.9) t0

To the leading order in Hˆ (t), this becomes, in the momentum space ⟨ ⟩ Rˆ Rˆ Rˆ k1 (t) k2 (t) k3 (t) ∫ t ⟨[ ]⟩ ≃ ′ ˆ | ′ Rˆ Rˆ Rˆ i dt H (3)(t ), k1 (t) k2 (t) k3 (t) . (5.10)

105 Theoretical studies of primordial non-Gaussianities usually concentrate on two

characteristics of the three-point correlator: its magnitude and its k-dependence.

While the magnitude tells whether the signal can survive the later contaminations

from low-energy phenomenon and be detected by observations, the k-dependence

gives us the detailed information, if detectable, about the dynamics of inﬂation [61]:

for example, if the 3-point function peaks where all three modes share the similar

momentum, this usually indicates higher derivate coupling between matter and grav-

ity; if the mode was not in its ground state in the beginning, the signal peaks when

k2 + k3 ≃ k1; if other light fields other than the inflaton were present during inflation and responsible for the curvature perturbations, the 3-point function would display

a local shape. Below we study both the magnitude and the k-dependence using the

in-in formalism.

5.2 Primordial Non-Gaussianity of a Single Scalar Field

5.2.1 Interaction Hamiltonian

To generalize the linear scalar perturbations considered in previous chapters to the nonlinear case, we consider in this chapter the perturbations given by

2 2ζ 2 ˆ gij = a e δij, Ni = a B,i,N = N,

A = Aˆ + aδA, φ = 0, χ =χ ˆ + δχ. (5.11)

Clearly, to ﬁrst order, they reduce to the ones given by Eq.(3.74) if one identiﬁes

ζ as ζ = −ψ (note that this ζ is not to be confused with the Planck mass ζ in

previous chapters). To simplify calculations, we need to specify a gauge. There

are two popular choices in GR, either put the scalar metric perturbation ζ to zero

or the matter perturbations δχ to zero [75]. None of these, however, is possible in

HL gravity because of the constraint (4.64) which relates ζ with δχ. On the other

hand, we can take advantage of this relation and the analysis can be performed with

106 the quantity R. Substituting Eq.(5.11) into the total action (3.19) with the matter

coupling prescribed in Eq.(3.32), we ﬁnd that its cubic part is given by Eq.(5.12).

Under the Newtonian quasi-longitudinal gauge (4.82), with perturbations given

by (5.11), after tedious calculations, we ﬁnd that the action can be written in the

form, | |GR |GR |HC | |HL S (3) = Sg (3) + Sχ (3) + Sg (3) + SA (3) + Sχ (3) , (5.12) where the “GR parts” are given by3 { ∫ [ ] [ ] 9ζ3 f(λ) 9ζ2 S |GR = dηdxa2 χ¯′2 − a2V + f(λ)¯χ′δχ′ − a2V ′δχ χ (3) 2 2 2 [ ] ( ) 3ζ ′ 2 2 ′′ 2 ′ k + f(λ)(δχ ) − a V δχ − 3ζf(λ)¯χ (∂kB) ∂ δχ 2 } ( ) V ′′′ − f(λ)δχ′ (∂ B) ∂kδχ − a2 (δχ)3 , (5.13) k 6

∫ { 1 − 3λ 27 ( ) ( ) S |GR = dηdxa2 H2ζ3 + 9Hζ2 3ζ′ − ∂2B − 2ζ′ ∂ B∂kζ g (3) 16πG 2 k [ ( ) ] ′ 2 k ′ 2 + 3 3ζ (ζ ) − 2Hζ ∂kB∂ ζ − 2ζζ ∂ B ( ) 3 2 3λ 2 2 + ζ (∂ijB) − ζ ∂ B 1 − 3λ 1 − 3λ } 1 + λ ( )( ) 8 ( ) − 2 ∂2B ∂ B∂kζ + ∂ B∂iB∂jζ 1 − 3λ k 1 − 3λ ij ∫ { } 1 − dηdxa2 9Λa2ζ3 + ζ2∂2ζ . (5.14) 16πG

And the “HL gravitational part” is given by, { ∫ [ ] 1 2 ( ) S |HC = dηdxa2 − (3ζ) (16g + 5g ) ∂2ζ 2 + g (∂ ζ)2 g (3) 16πG a2M 2 2 3 3 ij pl } ( ) [ ] 2 2 ( )( ) ( ) − 2 4 2 2 2 (3ζ) 2 2 16g7 ∂ ζ ∂ ζ + 5g8 ∂k∂ ζ + g8 (∂ijkζ) , a Mpl − 2L |HC a V (3) , (5.15)

3 Note that though this is labeled as the “GR part”, it cannot reproduce the exact expression of GR, as a result of the diﬀerence in symmetry.

107 where [ 2 ( ) ( ) 2L |HC − − 2 2 2 2 a V (3) = 2 2 ( 64g2 20g3) ζ ∂ ζ + (16g2 + 6g3) ∂ ζ (∂kζ) a Mpl ( ) ] ij 2 − 2g3 ∂iζ∂jζ∂ ζ − 4g3ζ (∂ijζ) ( ) [ 2 2 ( ) ( ) + − 12g ∂ ζ∂ ζ∂ijkζ + 16g ∂ ζ∂ ∂2ζ∂ijζ a2M 2 8 ij k 8 i j pl ( ) ( ) 2 k 2 2 2 − (32g7 + 20g8) ∂ ζ∂kζ∂ ∂ ζ − 30g8ζ ∂k∂ ζ ( ) 2 ij k − 6g8ζ (∂ijkζ) − g6 ∂ ζ∂jkζ∂i∂ ζ ( ) 2 3 − (64g4 + 20g5 + 6g6 + 32g7) ∂ ζ ( ) 2 2 + (16g7 − 4g5 − 3g6) ∂ ζ (∂ijζ) ] 2 4 4 2 − 96g7ζ∂ ζ∂ ζ + 8g7∂ ζ (∂kζ) . (5.16)

The “gauge part” is given by, ∫ { } 1 ( ) ( ) S | = dηdxa2 2Aζ¯ 2 ∂2ζ + 2Aζ¯ (∂ ζ)2 + 4ζ ∂2ζ δA + 2 (∂ ζ)2 δA . A (3) 16πG k k (5.17)

Finally, the “HL matter part” is given by { ∫ [ ] 9ζ2 V ( ) ( ) S |HL = dηdxa2 − 4 ∂4δχ + Ac¯ ∂2δχ χ (3) 2 a2 1 } ( ) [ ] [ ] 2 L |HL 2 L |HL + 3a ζ χ (2) + a χ (3) , (5.18) with ( ) 1 L |HL = − + V a−2 (∂ δχ)2 χ (2) 2 1 k [ ] V V ( ) V ′ ( ) V ( )( ) − 4 a4P | − 2 ∂2δχ 2 − 4 δχ ∂4δχ − 6 ∂2δχ ∂4δχ a4 2 (2) a4 a4 a6 [ [ ] ] A¯ ( ) c δA ( ) + c′ δχ ∂2δχ + c a2P | + c (∂ δχ)2 + 1 ∂2δχ , (5.19) a2 1 1 1 (2) 2 k a2 (5.20)

108 ( ) [ ] 1 V ′ V ( ) L |HL = 2a−2 + V ζ (∂ δχ)2 − 1 δχ (∂ δχ)2 − 2 2 ∂2δχ a2P | χ (3) 2 1 k a2 k a4 1 (2) [ ] V ′ ( ) V ′ V ′′ ( ) − 2 2 2 − 4 4P | − 4 2 4 4 (δχ) ∂ δχ 4 (δχ) a 2 (2) 4 (δχ) ∂ δχ a [ a ] 2a V ( ) V V ( ) − 3 2 3 − 4 4P | − 5 4 2 6 ∂ δχ 4 a 2 (3) 6 ∂ δχ (∂kδχ) a a [ a ] V ′ ( )( ) V ( ) ( ) − 6 (δχ) ∂2δχ ∂4δχ − 6 (∂2δχ) a4P | + (∂4δχ) a2P | a6 a6 2 (2) 1 (2) ( [ ] [ ]) A¯ 2P | ′ 2P | + c1 a 1 (3) + c1 (δχ) a 1 (2) a2 [ ] A¯ c′′ ( ) + 1 (δχ)2 ∂2χ + (−2c ζ + c′ δχ)(∂ δχ)2 a2 2 2 2 k [ ] δA ( ) + c′ δχ ∂2δχ + c a2P | + c (∂ δχ)2 , (5.21) a2 1 1 1 (2) 2 k where ( ) ( ) 2 k 2 a P1|(2) = (∂kζ) ∂ δχ − 2ζ ∂ δχ , [ ( ) ( ) ] 4 k 2 k 2 a P2|(2) = (∂kδχ) ∂ ∂ ζ + 2 (∂kζ) ∂ ∂ δχ ( ) ( ) ij 4 +2 (∂ijζ) ∂ δχ − 4ζ ∂ δχ ,

2 2 2 a P1|(3) = 3ζ ∂ δχ, ( ) 4 2 4 ij a P2|(3) = 8ζ ∂ δχ − 8ζ (∂ijζ) ∂ δχ [ ( ) ( ) ] k 2 k 2 −4ζ (∂kδχ) ∂ ∂ ζ + 2 (∂kζ) ∂ ∂ δχ ( ) [ ( ) ( ) ] i j j + ∂ ζ ∂ ζ (∂ijδχ) + ∂ δχ (∂ijζ) . (5.22)

One can further substitute B, δA in terms of ζ ≃ −ψ by solving the linearized

ﬁeld equations 1 ( ) 2fχ¯′ ( ) B = ∂−2ζ′ − ∂−2ζ , (5.23) | 2 | − cψ (1 λ)c1 [ ] δAˆ α α 1 ( ) = 1 + 1 ∂2 + 2 ∂4 − A¯ ζ + ∂−2ζ′′ a z2 a4 |c2 | [ ]ψ 2fχ¯′2 c′ 2V ′ ( ) + 1 + ∂−2ζ − 2 − [(1 λ)c1 c1 a ](1 λ)c1 2fχ¯′ 2H ( ) − − ∂−2ζ′ , (5.24) − | 2 | (1 λ)c1 cψ −1 δχ = − (4πGc1) ζ, (5.25)

109 where (∂−2∂2) ζ ≡ ζ. This lengthy procedure, in the case of GR, simpliﬁes the

interaction terms condierably and lets one easily identify the leading order (in slow-

roll parameters) terms. See [82] for a very good and detailed review of the subject.

Then we ﬁnd that the interaction can be cast in the schematic form,

∫ ∑3 { } S = ∂2kζm · ∂2lδχ(3−m) , (5.26) m=0 where k and l are non-negative integers.

To ﬁnd the leading order terms in the self-interaction, let us ﬁrst note that the matter perturbation δχ is related to the gauge-invariant quantity R through

1/2 δχ = hR ∝ ϵ MplR, (5.27)

where ϵ is the slow-roll parameter, and R is the comoving curvature perturbations,

deﬁned explicitly in Eq.(4.76). From Eq.(5.25), we also have

− − c1δχ ζ = 4πGc1δχ = 2 . (5.28) 2Mpl

Assuming that c1 ≃ M∗ ≪ Mpl [40], we find that ζ ≪ δχ ≪ R. This implies that to find the leading order terms, it suffices to look for terms which are of cubic order of δχ [m = 0 in (5.26)], since all terms in lower orders of δχ are of higher orders of

ζ, hence further suppressed by factors of M∗/Mpl. With this as our guideline, it can |GR be shown that only six terms are left for considerations. One from the part Sχ (3) in Eq.(5.13), identiﬁed as

V ′′′(χ)δχ3. (5.29)

However, since it is the third-order derivative term of the potential, one can immedi-

ately ignore it, as the slow-roll conditions require |V ′′′| ≪ 1. The other ﬁve are from

|HL Sχ (3) and are identiﬁed as, V ′ V ′ ( ) V ′′ ( ) 1 δχ (∂ δχ)2 , 2 δχ ∂2δχ 2 , 4 (δχ)2 ∂4δχ , a2 k a4 a4 V ( ) V ( ) 3 ∂2δχ 3 , 5 ∂4δχ (∂ δχ)2 . (5.30) a6 a6 k

110 Out of the ﬁve, three are proportional to derivatives of the coupling functions V1,V2,

and V4. They all appear in the linear perturbations, and it was assumed in Chapter 4 that their derivatives with respect to χ vanishes. To be consistent, in this chapter

we keep this assumption. Hence, we are ﬁnally left only with two terms, that are

proportional to V3 and V5, ∫ { } dηd3x ( ) ( ) h3 V ∂2R 3 ,V ∂4R (∂ R)2 , (5.31) a2 3 5 k √ where h is estimated in Eq.(4.119) to be of order ∼ O( ϵ). In contrast to the

minimum scenario in GR, which predicts that the self-interaction should be of the

2 3 3/2 order of ϵ , in the current case, the leading order is ofα ˆnh ∼ αˆnϵ , whereα ˆn are

4 dimensionless parameters, deﬁned by Vn =α ˆn/M∗ , (n = 3, 5).

Despite the fact that these two terms are similar to the α2 and α3 terms given in [78], a key diﬀerence, however, exists. In [78], the authors worked in the framework of curvatons [79], and inﬂation was not necessary to produce the scale-invariant power spectrum. As a result, the time of interest was assumed to be the period in which

H ≫ M∗. Thus, the quantization of their Lifshitz scalar ϕ, which is responsible for generating the primordial curvature perturbations, can be carried out as [78], ϕ ζ ∝ , µ ∫ [ ] 1 3 −ikx ∗ † ϕ (x, t) = 3 d ke uk(t)ˆak + uk(t)ˆa−k , (2π) [ ∫ ] 3 ∝ M∗ − k dt uk(t) 3/2 exp i 2 2 , (5.32) k M∗ a where µ is an energy scale. The mode freezes after it leaves the sound horizon

3 −1/4 (HM∗ ) , which is much smaller than the Hubble horizon in the inﬂation scenario, and gives rise to a power spectrum, 2 M∗ Pζ = . (5.33) (2π)2 µ2 This is quite diﬀerent from the expression H4 P GR = , (5.34) ζ (2π)2 χ¯˙ 2

111 obtained with inﬂation in GR. In contrast, slow-roll inﬂation is required in our current

model and the power spectrum was found in Chapter 4 to resemble the GR expression,

2 2 given by Eq.(4.137) when H ≪ M∗, but suppressed by orders of M∗ /H when H ≫

M∗.

5.2.2 The Three-point Correlator ∫ With the self-interaction at hand, and the relation S|(3) = − dtH|(3) for the case of a single scalar ﬁeld, we perform the calculations of the bispectrum by using

the in-in formalism and ﬁnd that to the leading order in HI we have ⟨ ⟩ Rˆ Rˆ Rˆ k1 (t) k2 (t) k3 (t) ∫ t ⟨[ ]⟩ ≃ ′ ˆ | ′ Rˆ Rˆ Rˆ i dt H (3)(t ), k1 (t) k2 (t) k3 (t)

− 3 3 3 ′ − = ih{ (2π) δ (k1 + k2 + k3)[U(t ; t) c.c.] } [ ] [ ( ) ( ) ] × 2 2 2 6 6 6 − 4 2 2 4 6V3 k1k2k3 + V5 k1 + k2 + k3 k1k2 + k1k2 + cyclic , (5.35)

where ∫ t dt′ U(t′; t) ≡ r∗ (t)r∗ (t)r∗ (t) r (t′)r (t′)r (t′). (5.36) k1 k2 k3 a3(t′) k1 k2 k3

4 rk(t) is the mode function for R. In writing down (5.35), we assumed that

d(V ) 3,5 = V ′ χ¯˙ = 0. (5.37) dt 3,5

We’ll see that, once this assumption is relaxed, it will generate more interesting

features in the shapes of the bispectrum.

4 Note that quantization of the gauge-invariant perturbation R was performed in Chapter 4 2 2 2 through the canonically normalized ﬁeld v = zR, z ≡ a h β0 in Eq.(4.113). For the simplicity of calculations, here we introduce the mode functions of R, rk(η), which relates to vk(η) through the relations,

v (η) v z(η) = k = . rk(η) R

112 5.2.3 Modiﬁed Dispersion Relation and Evolution of the Mode Functions

To study the shapes of the bispectrum, in this section we ﬁrst consider the time

evolution of the mode function in the (quasi-)de Sitter background.

In the (quasi-)de Sitter background a ≃ −1/(Hη), the equation of motion

(EoM) of the mode function takes the form, ( ) 2 ′′ 2 − vk (η) + ω 2 vk(η) = 0, (5.38) ( η ) 2 4 2 2 2 H 2 2 H 4 4 ω = k cs + b2 2 k η + b3 4 k η , (5.39) M∗ M∗

2 where cs = b1 and b2, b3 are deﬁned in Eq.(4.131). Due to the time-dependence of the dispersion relation, the mode function vk(η) (or equivalently rk(t)) will not take a simple plane wave form as in GR. Rather, its form will evolve with time. This

complicates the calculations of the bispectrum considerably.

One may worry that, like in the case of [78], relevant scales may have left the

horizon at a time when the k6 term dominated the dispersion relation. However, this

seems not reasonable in the slow-roll inﬂation scenario. For the mode to leave the

horizon at that time, two conditions have to be met,

2 4 4 ≫ 2 ∼ b3ϵHLk η cs 1, (UV regime) a′′ 2 b ϵ2 k6η4 ≪ = , (Super-horizon region). (5.40) 3 HL a η2

2 2 ≪ ≫ 2 2 2 ≫ This would indicate that k η 1 and at the same time b3 1/(ϵHLk η ) 1.

However, it was argued in Chapter 4 (cf. Eq. (4.134)) that b3 in general is of order one. Therefore, we consider this case as physically not realistic.

Then the evolution of the mode function during inﬂation has to be taken into

account. To deal with this problem, Brandenberger and Martin (BM) proposed a

matching procedure [80]. Three regions were identiﬁed for the evolution history of

the mode function (See Fig. 5.1. More divisions are possible given a speciﬁc model).

Region I is the region in which the UV eﬀects dominate, alias ω ∼ k3 in the present

113 ωph

t Η 2

0 k 2 k1 k i kph

Figure 5.1: The evolution of ωph ≡ ωk/a vs kph ≡ k/a in three diﬀerent regions, where Region I: t ∈ (ti, t1); Region II: t ∈ (t1, t2); and Region III: t ∈ (t2, t0). case. Then, the solution of the EoM in this region can be approximated with the

Bessell functions of the ﬁrst kind, √ √ I ≃ | | | | vk(η) A1 η Jν [z(η)] + A2 η J−ν [z(η)] . (5.41)

In Region II, the dispersion relation restores its relativistic form ω ∼ k, so that the mode function can be safely approximated with plane wave solutions,

II ≃ − vk (η) B1 exp [ icskη] + B2 exp [icskη] . (5.42)

Note that unlike in the case of the Bunch-Davies vacuum, both positive and negative frequencies appear in the mode function in this region. Region III is the super horizon region when the mode freezes, i.e. R˙ = 0 (cf. Eq.(4.80)). The initial conditions in

Region I are the ones that minimize the energy of the ground state of the ﬁeld [80], given, respectively, by Eqs.(5.51) and (5.52), from which the two constants A1 and

A2 are fixed. The undetermined coefficients of the solutions in Regions II and III are fixed by matching conditions across each boundary of these regions, by requiring that the mode function and its first order time derivative be continuous.

114 On the other hand, we’ve seen that the evaluation of the 3-point correlator

(5.35) all boils down to the following integration of the mode function U(t′; t), ∫ t dt′ r∗ (t)r∗ (t)r∗ (t) r (t′)r (t′)r (t′). (5.43) k1 k2 k3 a3(t′) k1 k2 k3

A technical diﬃculty arises when the dispersion relation is of the form (5.39) as no exact solutions exist. Thus, the matching procedure presented in [80] seems a natural choice in approximating the solution of the mode function to the EoM. Below we evaluate the mode integration (5.36) and calculate the bispectrum and its shapes more explicitly using the BM matching procedure.

5.2.4 Approximating the Mode Functionss

Dividing the integration into three regions, as mentioned previously, one can see that only that over Regions I and II need to be considered, as Region III is the super-horizon region, and the mode function is frozen out. Then, we ﬁnd that ∫ t dt′ r∗ (t)r∗ (t)r∗ (t) r (t′)r (t′)r (t′) k1 k2 k3 a3(t′) k1 k2 k3 ∫ ∫ t1 dt′ t dt′ = Ψ∗(t) × rI (t′)rI (t′)rI (t′) + Ψ∗(t) × Ψ(t′) 3 ′ k1 k2 k3 3 ′ ti a (t ) t1 a (t ) I ′ II ′ ≡ U (t ∈ (ti, t1); t) + U (t ∈ (t1, t); t) , (5.44) where

II II II Ψ(t) = rk1 (t)rk2 (t)rk3 (t). (5.45)

In writing the mode-integration in this form, we are assuming that the all three modes

(k1, k2 and k3) are making the transitions from Region I to Region II and from Region II to Region III at approximately the same time. If this is not the case, then the integrand should include case of rII (t′)rI (t′)rI (t′). This indicates that one of the k1 k2 k3 modes is significantly longer (the “squeezed” limit) than the other two. However, the correlations between modes with vastly different momenta should be minimum in single-field slow-roll inflation models, as first pointed out in [75]: if one mode left the

115 horizon and froze far earlier than the other two, it’s only eﬀect would be rescaling the time, as is obvious from (5.11). Hence we do not consider this case in this section.

To carry out the above integrations, we ﬁrst note that in Region I, t ∈ (ti, t1), in ∫ ′ contrast to the case considered in [78] where the modes all take the form e−i dt ω/a3 ∼

′ e−ik3η 3 , only those mode that are in the kernel of the integration take this asymptotic shape, while rII (t) takes the form of plane waves. ki Below we re-derive the coeﬃcients following the BM guidelines of matching, but within our concrete model. From (4.111), the mode function rk(t) satisﬁes the EoM ( ) ω2 r¨ (t) + 3H + m2 + k r (t) = 0, (5.46) k a2 k

2 2 where m = β4/β0, ωk is the same as defined in Eq. (5.39) and a dot represents differential w.r.t time t. In Region I where the nonlinear effects in the dispersion dominates, the solution takes the asymptotic form (below we use t and conformal time η interchangeably, noting that adη = dt.), [ ( )] [ ( )] H2k3 H2k3 I ≃ − 3 − 3 3 − 3 rk(t) A1 exp i 2 η ηi + A2 exp +i 2 η ηi [ 3M ] [ ]3M H2k3 H2k3 ˜ − 3 ˜ 3 = A1 exp i 2 η + A2 exp +i 2 η , [ 3M ] 3M H2k3 ˜ ≡ 3 A1 A1 exp +i 2 ηi , [ 3M ] H2k3 A˜ ≡ A exp −i η3 , (5.47) 2 2 3M 2 i β a4 2 |V + 2πGc2α | M −4 ≡ 3 = 6 1 2 . (5.48) 2 | 2 | β0 f + 4πGc1/ cψ

It can be shown that M here is related to M∗ deﬁned in Chapter 4 (cf. the paragraph

4 4 2 2 above Eq.(4.133)) through M∗ = b3M . Since b3 ∼ O(1), the condition H /M∗ ≪ 1 implies H2/M 2 ≪ 1.

116 In Region II, the mode function is a superposition of plane waves,

II ≃ − − − rk (t) B1 exp [ icsk (η η1)] + B2 exp [icsk (η η1)] ˜ ˜ = B1 exp [−icskη] + B2 exp [+icskη] , ˜ B1 ≡ B1 exp [+icskη1] , ˜ B2 ≡ B2 exp [−icskη1] , (5.49) β 1 + 2V + 2A¯(c′ − c ) − 4πGc2(1 − A¯) c2 ≡ 1 = 1 1 2 1 . (5.50) s 2 | 2 | β0 f + 4πGc1/ cψ The initial conditions are chosen such that [80]

vk(ηi) 1 rk(ti) = = √ , (5.51) z(ηi) z(ηi) 2ω(ηi) where z(η) is deﬁned in Eq.(4.113). Its initial time-derivative takes value such that the energy density is minimized at the initial time (not necessarily inﬁnite past ηi → −∞) ( √ ) ( )′ ′ ′ vk(ηi) 1 ω(ηi) − z(ηi) √ 1 rk(ti) = = i . (5.52) z(ηi) z(ηi) 2 z(ηi) 2ω(ηi)

At the time of matching between region I and II, t1,

I II ′I ′II rk(t1) = rk (t1), rk (t1) = rk (t1). (5.53)

These four conditions ﬁx the four undetermined constants. We found that [ ] H2k3 3 ( ) exp +i 2 η 3M∗ i 2i A˜ = √ 1 + , 1+ η ω (η ) z (η[i) 2ω (ηi)] i i − H2k3 3 ( ) exp i 2 η ˜ 3M∗ i 2i A2+ = √ 0 − , z (ηi) 2ω (ηi) ηiω (ηi) 3 2 2 ≃ k H ηi ω (ηi) 2 , (5.54) M∗ √ ′ 1 ω(ηi) for r (ti) = +i (positive frequency choice), or k z(ηi) 2 [ ] H2k3 3 ( ) exp +i 2 η 3M∗ i 2i A˜ − = √ 0 + , 1 η ω (η ) z (η[i) 2ω (ηi)] i i − H2k3 3 ( ) exp i 2 η ˜ 3M∗ i 2i A2− = √ − 1 − . (5.55) z (ηi) 2ω (ηi) ηiω (ηi)

117 √ ′ 1 ω(ηi) for r (ti) = −i (negative frequency choice). And k z(ηi) 2 [ ( ) ( ) e+icskη1 k2H2η2 H2k3 ˜ 1 ˜ − 3 B1 = 1 + 2 A1 exp i 2 η1 2 csM∗ 3M∗ ( ) ( ) ] k2H2η2 H2k3 − 1 ˜ 3 + 1 2 A2 exp +i 2 η1 , csM∗ 3M∗ [ ( ) ( ) − e icskη1 k2H2η2 H2k3 ˜ − 1 ˜ − 3 B2 = 1 2 A1 exp i 2 η1 2 csM∗ 3M∗ ( ) ( ) ] 2 2 2 2 3 k H η1 ˜ H k 3 + 1 + 2 A2 exp +i 2 η1 . (5.56) csM∗ 3M∗

The time of matching η1 (or equivalently, t1) is a critical quantity. Here we choose it to be [83]

2 − 2 2 2 2 ωk(η1) csk = csk , (5.57) or from Eq.(5.39)

4 2 β3 (η1) k + β2 (η1) k = β1. (5.58)

Now recall that by dividing the evolution history into three regions, we implicitly assumed that the β2 term was never dominant during the history of inﬂation, as a result, the above condition can be approximated with

4 β3 (η1) k = β1, (5.59) which leads to

2 4 4 4 2 V6 + 2πGc1α2 k H η1 = β1. (5.60)

Hence

(kHη )2 = c M 2 (1 + 2δ ) , 1 s √∗ B c M∗ η ≃ − s (1 + δ ) , (5.61) 1 k H B where we have used the deﬁnition of cs and M∗, and introduced a quantity δB to denote any deviations from (5.59), for example, when we consider the k4 term’s

118 minimal eﬀect on η1. (In [83], it was found that a diﬀerence in this matching time could result in an unnecessary oscillatory component for the power spectrum, and we shall see below that this indeed happens through an extra oscillatory phase for the

B coeﬃcients.)

Looking closer at the equation (5.56), this choice of η1 indicates that the inﬂu- ˜ ˜ ˜ ˜ ∼ 2 2 3 3 ence of A2 (A1) on B1 (B2) is minimal. Also note that ηiω (ηi) (H /M∗ )k ηi is in general a very large quantity, this allows us to make further approximations on the

expressions of the coeﬃcients A˜ and B˜ and arrive at

M∗ ˜ √ +iϕA 1 A1 = √ e ( + iδA) , 3 0 h β0 2k −M∗ − ˜ √ iϕA 0 A2 = √ e ( + iδA) , (5.62) 3 1 h β0 2k

and

M∗ ˜ √ iϕB1 B1+ = √ e (1 + iδA + δB ) , 3 h β0 2k ( ) −M∗ − ˜ √ iϕB1 iϕB2 B2+ = √ iδAe + δB e , 3 h β0 2k ( ) M∗ − ˜ √ iϕB1 iϕB2 B1− = √ iδAe + δB e , 3 h β0 2k −M∗ − ˜ √ iϕB1 B2− = √ e (1 + iδA + δB ) , (5.63) 3 h β0 2k

where

2 ≡ 2 2M∗ δA = 3 3 2 , (5.64) ηiω (ηi) k ηi H H2k3 2 ≡ 3 ϕA 2 ηi = , 3M∗ 3δA [ ( ) ] 3 cskη1 ηi ϕB1 ≡ (1 + 2δB ) + 2 (1 − δB ) , 3 η1 [ ( ) ] 3 cskη1 ηi ϕB2 ≡ (1 + 2δB ) − 2 (2 + δB ) . (5.65) 3 η1

119 5.2.5 The Bispectrum

Having derived the expressions for the mode functions, we can ﬁnally write down the integral (5.44) explicitly. In Region I we have,

M 2 ∑ ∑ σI I ′ ∈ B∗ lmn U (t (ti, t1); t) = xyz , i Klmn x,y,z l,m,n =1,2 (=1,2 ) ∗ ˜∗ ˜∗ ˜∗ B ≡ B (k1)B (k2)B (k3) exp − icsKxyzη xyz x y z [ ] iH2 ( ) σI ≡ A˜ (k )A˜ (k )A˜ (k ) exp K η3 − η3 , (5.66) lmn l 1 m 2 n 3 3M 2 lmn 1 i where the coeﬃcients A˜ and B˜ are given above, and

K ≡ − l 3 − m 3 − n 3 lmn ( 1) k1 + ( 1) k2 + ( 1) k3,

x y z Kxyz ≡ (−1) k1 + (−1) k2 + (−1) k3. (5.67)

On the other hand, in Region II we ﬁnd that { } ∑ ∑ η2 − η2 2 (η − η ) 2 II ′ ∈ 2 B∗ II × 1 1 − U (t (t1, t2); t) = H xyz σlmn + 2 3 , iKlmn K iK x,y,z l,m,n lmn lmn =1,2 =1,2 [ ] II ≡ ˜ ˜ ˜ K − σlmn Bl(k1)Bm(k2)Bn(k3) exp ics lmn (η η1) . (5.68)

Putting all the above together, we ﬁnd that the bispectrum can be cast in the form,

⟨R R R ⟩ k1 k2 k3 { } 3 3 3 I II = 2h (2π) δ (k1 + k2 + k3) × Im U + U { [ ] [( ) ( )]} × 2 2 2 6 6 6 − 4 2 2 4 6V3 k1k2k3 + V5 k1 + k2 + k3 k1k2 + k1k2 + cyclic . (5.69)

Using the expression of the mode function rk(t) given above, we ﬁnd that the nonlinearity parameter fNL can be estimated as, ( ) ⟨RRR⟩ 3 6 6 2 2 2 3 (2π) M∗ k M∗ H η1 (k + k + k) fNL ≡ ≃ [ ] √ [ + ] 2 3 2 2 3 3 3 3 ⟨RR⟩ (2π) 2π2 h β0 PR k + k + k (k + k + k) [ ( ) ( )] × h3 6V k2k2k2 + V 3k6 − 6k6 ( ) 3 5 ( ) 6 −5 2 3 M∗ 10 M∗ ≃ O h V3,5 , (5.70) h (4.9 × 10−5)4

120 where we have made the assumption k1 = k2 = k3 = k, β0, cs ∼ 1 and used the ﬁducial COBE normalization [84] of the power spectrum, from which we are able to reproduce the spectrum presented in GR under the assumption H << M∗. Writing

−n −2 M∗ = 10 Mpl and assigning for the slow-roll parameter ϵ ∼ 10 [85], when n = 2, we obtain 10 ( ) ( ) f ∼ O h3V ∼ 107 × O h3V ; (5.71) NL ϵ3 3,5 3,5 and when n = 3, we ﬁnd that

10−7 ( ) ( ) f ∼ O h3V ∼ 10−1 O h3V . (5.72) NL ϵ3 3,5 3,5

Therefore, a large non-Gaussianity can be produced with a relative small V3 and V5,

given that the new scale M∗ isn’t much lower than Mpl.

5.2.6 Shapes of the Bispectrum

Let us ﬁrst note that, from (5.69), the k-dependence of the spectrum, a.k.a

the shape and running, depends not only on the action, but also on the form of the

mode function. To study the k-dependence of the bispectrum in more detail, let us

ﬁrst look at the expressions of U(t′; t) in Eqs. (5.66) and (5.68). The corresponding

conformal time η is usually taken to be at very late of the inﬂation era, i.e., kη → 0.

To be simple, here we take η = 0.

The bispectrum is plotted in Figures 5.2 - 5.5 for various choices of the parame- ˜ ˜ ters Ai’s and Bi’s. From these ﬁgures, we can see that when δA and δB are taken to be

zero, our result resembles the shape of α2 and α3 terms in [78], despite the fact that our integration (5.44) is actually diﬀerent from theirs. This is quite understandable

from the following considerations: when we make δA = 0 = δB , only the positive (or negative, depending on the choice of sign for the initial condition) frequency modes

exist in both regions. This makes the product of the six mode functions in Eq.(5.43),

which in general has 26 terms since each of them has two branches, collapses into one

121 single term. Substituting this into Eqs. (5.66) and (5.68), we obtain

1 1 I ′ ∈ ∝ U (t (ti, t1); t) 3 3 3 3 , (k1k2k3) k1 + k2 + k3 1 1 II ′ ∈ ∝ U (t (t1, t); t) 3 3 . (5.73) (k1k2k3) (k1 + k2 + k3)

We see that the contribution from Region I has the same k-dependence as the α2 and

α3 terms in Eqs.(4.5) and (4.6) of [78]. The contribution from Region II does not exist there since the modes were assumed to exist the sound horizon in Region I, nor

does it have the same k-dependence as in the relativistic cases.

The real diﬀerence comes in when we make either δA or δB nonzero. The impact of these on the shape of the bispectrum is the enhancement of the folded shape [61] (or sometimes called the ﬂattened shape [69]), namely, that the bispectrum peaks at the

− → 3 3 − 3 → limit k2 + k3 k1 0 and k2 + k3 k1 0. A non-zero δA for the positive frequency choice of the initial condition results in the appearance of “negative-frequency” modes in both rI (t) and rII (t), whereas a non-zero δ leads to a mixture only in rII (t). It’s k1 k1 B k1 this mixture that makes the assumption which is usually kept in the standard choice

of the Bunch-Davis (BD) vacuum,that is, only positive frequency modes appear in

the mode function, invalid. In fact, when we take ηi to be inﬁnite past like in the BD

vacuum, δA will be zero automatically as can be seen from its deﬁnition in (5.64); at ˜ −∞ the same time, ϕA that appears in Ai’s is inversely related to δA, goes to (1 + iε).

When we take δA = 0 but δB ≠ 0, this is similar to the case studied in [69] and our result is consistent with theirs.

In addition, the choice of negative frequency A˜− essentially changes the sign of

the bispectrum, and this is expected by comparing (5.62) with (5.63). 5

5 Note that our self-interaction terms (5.31) have opposite sign w.r.t [78]. This explains why our shape reproduces theirs only when we take the negative frequency.

122 0 0.00004 -1.´10-6 0.00003 1.0 1.0 -2.´10-6 0.00002 - 0.00001 -3.´10 6 0 -4.´10-6 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

(a) δA = 0 = δB (b) δA = −0.1, δB = 0

0.00000 0.00002

-0.00005 1.0 0 1.0

-0.00010 -0.00002

-0.00015 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

Figure 5.2: Shape of the bispectrum (truncated). V3-term dominates, with the choice of the positive frequency.

123 4.´10-6 0.00002 3.´10-6 1.0 0 1.0 2.´10-6 -0.00002 1.´10-6 -0.00004 0 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

(a) δA = 0 = δB (b) δA = −0.1, δB = 0

0.0001 0.0001 1.0 1.0

0.00005 0.00005 0.0000 0.0000 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

Figure 5.3: Shape of the bispectrum (truncated). V3-term dominates, with the choice of the negative frequency.

124 0 0.00002

- 1.0 1.0 -1.´10 6 0.00001 0 -2.´10-6 -0.00001 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

(a) δA = 0 = δB (b) δA = −0.1, δB = 0

0.00005 0.00001

0 1.0 0 1.0

-0.00005 -0.00001 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

Figure 5.4: Shape of the bispectrum (truncated). V5-term dominates, with the choice of the positive frequency.

125 2.´10-6 0.00001 1.0 0 1.0 -6 1.´10 -0.00001 -0.00002 0 0.8 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

(a) δA = 0 = δB (b) δA = −0.1, δB = 0

0.00005 0.00005 1.0 1.0 0 0

-0.00005 0.8 -0.00005 0.8 0.0 0.0

0.5 0.6 0.5 0.6

1.0 1.0

Figure 5.5: Shape of the bispectrum (truncated). V5-term dominates, with the choice of the negative frequency.

126 We do not have an enhanced “squeezed” triangle signal as in the minimum

scenario (though the overall magnitude there is very small), nor do we have a large

local form, which is a typical result of multi-ﬁeld models. The lack of an enhanced

“squeezed” triangle signal is partially because of the assumptions made in evaluating

the mode-function-integral (5.44) where we assumed that the modes all have the same

functional form. However, as argued there, the “squeezed” signal should be minimal

as the correlation between modes with too diﬀerent momentum should be small. The

reason for suppression of the local form is that the interaction terms which could

generate the local form are all suppressed by factors of either c1/Mpl or ϵ and are of sub-leading order [See discussions between (5.28) and (5.31)].

′ ′ Now recall that in writing down (5.35), we have assumed that V3 and V5 are

zero (cf.Eq.(5.37)). However, this is not physically necessary since V3 and V5 appears neither in the background equations, nor in the linear perturbations thus not constraint by the slow-roll conditions, and a strongly varying shape of V3 and V5 (not to be too strong to invalidate the perturbative expansion) would actually give both a higher non-linearity and new features in the bispectrum, such as the sinusoidal running and resonant running, as shown in [61]. Hence we have a natural mechanism for generating such kinds of shapes in the bispectrum yet only mildly restrained by power spectrum.

Though the ﬁnal integration was separated into two distinct periods, we would like to point out that this is a result of the matching procedure employed due to the lack of exact solutions for (5.38) in our model. One should, in principle, evaluate the integration as a whole and study the shapes of the bispectrum. A possible solution is a development of the uniform approximation [60] with some numerical integration techniques involved.

127 5.2.7 Projections onto Factorizable Templates

To study the quantitative behavior of the shapes of the bispectrum, in general, factorable ansatz or template functions are utilized. Among them, three are of the most importance to us [61], ( ) ≡ − 1 1 1 − 48 TOrth. (k1, x, y) 18 3 + 3 + 3 3 2 2 (x y x y ) x y 1 +18 2 3 + 5 perms. , ( x y ) ≡ 1 1 1 18 TFold. (k1, x, y) 6 3 + 3 + 3 3 + 2 2 (x y x y ) x y − 1 6 2 3 + 5 perms. , (x y ) ≡ − 1 1 1 − 12 TEqui. (k1, x, y) 6 3 + 3 + 3 3 2 2 (x y x y ) x y 1 +6 + 5 perms. , (5.74) x2y3 where x ≡ k2/k1, y ≡ k3/k1. Then the bispectrum (5.69) can be cast in the form, ( ) 6 M∗ ⟨R R R ⟩ ≡ 3 3 √ k1 k2 k3 (2π) δ (k1 + k2 + k3) THMT (k1, k2, k3) , (5.75) h β0 where

THMT = cOTOrth. + cETFold. + cFTEqui., (5.76) with

THMT · TOrth. THMT · TFold. THMT · TEqui. cO = , cF = , cE = , (5.77) TOrth. · TOrth. TFold. · TFold. TEqui. · TEqui. where the inner product is deﬁned as [78] ∑ T1 (k1, k2, k3) T2 (k1, k2, k3) T1 · T2 ≡ Pk1 Pk2 Pk3 ∫ki ∫ 1 1 4 4 ∝ x dx y dyT1(1, x, y)T2(1, x, y). (5.78) 0 1−x

128 When δA = 0 = δB , the projection gives the following result ( ) H2 2 − 3 cO = M 1 + 2 (0.0033V3 0.0018V5) h , ( M∗ ) H2 2 − 3 cF = M 1 + 2 (0.0092V3 0.0004V5) h , ( M∗ ) H2 2 − 3 cE = M 1 + 2 (0.0063V3 0.0036V5) h , (5.79) M∗ for the positive frequency, and ( ) H2 2 − 3 cO = M 1 + 2 ( 0.0004V3 + 0.0002V5) h , ( M∗ ) H2 2 − 3 cF = M 1 + 2 ( 0.0001V3 + 0.0000V5) h , ( M∗ ) H2 2 − 3 cE = M 1 + 2 ( 0.0008V3 + 0.0005V5) h . (5.80) M∗ for the negative frequency. We see that the projection onto the equilateral template has the biggest magnitude for most of the cases. This is consistent with the observations in [86] that derivative coupling favors the correlation between modes with similar momenta.

However, these analyses are only very rough approximations. This is because these three bases do not form a complete basis set, as pointed out in [78], nor are they un-correlated, since we can compute correlations between them and get

Corr (Equi., Orth.) ∼ 0.204,

Corr (Fold., Orth.) ∼ −0.748,

Corr (Fold., Equi.) ∼ 0.489, (5.81) which indicate that there are strong correlations between the three, where

T · T Corr (S1, S2) ≡ √ S1 √ S2 . (5.82) TS1 · TS1 TS2 · TS2

When δA and δB are not zero, the bispectrum, if taken naively, diverges at the − → 3 3 − 3 → folded limit k2 + k3 k1 0 or k2 + k3 k1 0, and the above inner product fails

129 to converge. To understand the divergence, let us look at the integration (5.66) in

Region I more closely. The divergence occurs because, in addition to Eq.(5.73), we

still have (26 − 2)-terms of the form,

1 (δ )m (δ )n . (5.83) A B 3 3 − 3 kx + ky kz

The denominator appears through the dη′ integration in (5.44). When the denomina-

tor approaches zero in the folded limit, if δA and δB are not exactly zero, these terms blow up. The divergences must be regulated away if we want to have a quantitative estimate of the projection. A possible solution is to introduce a cut-oﬀ so that the integration yields,

1 (δ )m (δ )n (5.84) A B 3 3 − 3 3 k2 + k3 k1 + kc

where

kc = kc (δA, δB ) , (5.85)

in order to keep (5.84) ﬁnite. As noted in [61], this regularization is highly model-

dependent and requires some systematic study. Putting this divergence aside, its is

reasonable to expect that the projection onto the folded base would be enhanced

signiﬁcantly, in comparing with (5.79) and (5.80).

5.2.8 Summary of the Assumptions

We would like to summarize the assumptions made along the analysis, as the

invalidation of any of these assumptions will certainly change some of the conclusions.

In particular, in obtaining the leading order term in the self-interaction, we made the

assumption that c1 ∼ M∗ ≪ Mpl. This eliminates a great number of terms which are not of order δχ3 in the third order expansion, and makes all terms that will

give non-local eﬀect (or local shape bispectrum) of sub-leading order. However, the

phenomenological upper bound on M∗ for the projectable version of the HL theory

130 is not established yet [41]. Dropping this assumption will bring back a lot of terms

and enhance the local shape signal in the bispectrum. ′ ≃ We’ve also kept the assumption made in Chapter 4 that Vn 0. For V1,V2 and

V4, this assumption further eliminate some possible contributions to the leading order eﬀect. For V3 and V5, as noted in the discussion at the end of Part.C in Section IV, if this assumption is dropped, new features (sharp and periodic shapes) will appear.

The condition H/M∗ << 1 has been also assumed to obtain an expression of the power spectrum similar to that of GR. However, if this is dropped out, the power spectrum will receive large corrections, as noted in Eqs. (4.141) and (4.160).

By dividing the time of interest into three regions, we have assumed that the period of dominance of the k4 term in the dispersion relation is so short that its eﬀect

can be incorporated into the small parameter δB . This is a strong assumption, as

2 2 noted in [56, 87]. During this period of time, ωk may actually go below H and then the mode function will be no longer oscillating, but grow with the scale factor a. This

is an important problem and deserves further investigations.

5.3 Primordial Non-Gaussianity of Gravitational Waves

Having studied the characteristics of the scalar-scalar-scalar correlation func-

tion, let us now turn to the tensor-tensor-tensor correlator. We assume that no tensor

perturbations exists in the matter sector, and the tensor perturbations of the metric

around a spatially ﬂat FLRW metric is [88] ( ) i 2 h N = a, N = 0, gij = a e ij , (5.86)

where ( ) 1 1 eh = δ + h + h hk + h hjkh + ··· . (5.87) ij ij ij 2 ik j 6 ij ki The metric perturbation is deﬁned in this way such that there is no cubic term

involving two time derivatives in GR [75]. The small dimensionless quantity hij is the same as deﬁned in Eq.(4.142) and hence still satisﬁes the transverse and traceless

131 ij condition. Moreover, its indices are lowered and raised by δij and δ . Thus for simplicity of notations, in this section we will not distinguish super-indices and sub-

indices for it, assuming it is understood that when an index appears twice, summation

over that index is performed. We further introduce a short hand notation

n ≡ ··· (h )ij hik1 hk1k2 hkn−1j. (5.88)

5.3.1 The Interaction Hamiltonian

With these perturbations, the interaction Hamiltonian at leading order is found

to receive contributions from only four terms

( ) ij ∇i jk ∇ i j k R,RijR , R ( iRjk) ,RjRkRi . (5.89)

The kinetic part of the action SK , like the case in GR, does not contribute to HI even though λ now could diﬀer from 1. We ﬁnd that,

−2 a ij,ab R = h (2hiahjb − hijhab) , (5.90) 4 [ ] ( ) 1 R Rij = a−4 ∂2hij h h ij 2 ia,b jb,a ( ) [ ] −4 2 ij ab 1 1 +a ∂ h h hia,jb − hab,ij − hij,ab , (5.91) [ 4 2 ] ( ) a−6 ( ) ( ) ( ) ∇iRjk (∇ R ) = ∂2h hab ∂2h − 4hai,b ∂2h i jk 4 ij ij ,ab bj ,a −6 ( ) [ ] a 4 ab + ∂ hij h − 4hia,bj + 2hij,ab + hab,ij 4 [ ] a−6 ( ) − ∂4h h h , (5.92) 2 ij ia,b jb,a a−6 [ ][ ][ ] Ri Rj Rk = − ∂2 (h ) ∂2 (h ) ∂2 (h ) . (5.93) j k i 8 ij jk ki

The contribution from R is the same as that obtained in [88]. The contribution

i j k from RjRkRi differs from theirs in an essential way. The model considered in [88] describes the class of single-field inflation models where the coupling between the

inﬂaton and gravity could be diﬀerent from the canonical form while Lorentz sym-

metry is kept. This explains why their interaction term posses time-derivatives. On

132 the other hand, coupling of the gravitational sector with scalar matter in our model is in the canonical form (cf. Eq. (3.24)). The higher spatial derivative terms exist because of the Lorentz symmetry breaking.

We deﬁne the Fourier image of hij here similarly to Eq.(4.144) ∫ d3k ∑ h = εs (k)hs (t)eikx, (5.94) ij (2π)3 ij k s=+,−

s i s s s′ but now εii(k) = k εij(k) = 0 and εij(k)εij(k) = δss′ . This set of polarization tensors is related to those introduced in (4.144) through

1 ( ) ε = ϵ+ iϵ× , (5.95) ij 2 ij ij

By properly choosing the phase of this new set of polarization tensors, they satisfy the relation [88] [ ] s ∗ −s s − εij(k) = εij (k) = εij( k). (5.96)

Then in momentum space, we ﬁnd { [ ] q q a2R ; 1a 1b εs1 (q ) εs2 (q ) εs3 (q ) + εs1 (q ) εs2 (q ) εs3 (q ) 24 ij 1 ab 2 ij 3 ij 1 ij 2 ab 3 } q q − 1a 1b εs1 (q ) εs2 (q ) εs3 (q ) + cyclic of (1, 2, 3) , (5.97) 6 ij 1 ia 2 jb 3 { [ ] q2q q q q (q2 + q2) a4RijR ; 1 1a 1b + 1a 1b 2 3 εs1 (q ) εs2 (q ) εs3 (q ) ij 6 6 ij 1 ia 2 jb 3 [ q q − 1a 1b 2 s1 s2 s3 q3 εij (q1) εab (q2) εij (q3) 12 ] 2 s1 s2 s3 + q ε (q1) ε (q2) ε (q3) 2 ij ij ab } [ ] q2 − 1 εs1 (q ) εs2 (q ) εs3 (q ) q q + q q + cyclic , (5.98) 24 ij 1 ab 2 ab 3 2i 2j 3i 3j (q q q )2 a6Ri Rj Rk ; 1 2 3 εs1 (q ) εs2 (q ) εs3 (q ) + cyclic, (5.99) j k i 24 ij 1 jk 2 ki 3

133 { ( ) q q (q4 + q4 + q4 + q2q2 + q2q2) a6 ∇iRjk (∇ R ) ; 1a 1b 1 2 3 1 2 1 3 εs1 (q ) εs2 (q ) εs3 (q ) i jk 6 ij 1 ia 2 jb 3 [ q q ( ) − 1a 1b 2q4 + q2q2 εs1 (q ) εs2 (q ) εs3 (q ) 24 3 1 3 ij 1 ab 2 ij 3 ( ) ] 4 2 2 s1 s2 s3 + 2q + q q ε (q1) ε (q2) ε (q3) 2 1 2 ij ij ab } [ ] q4 − 1 εs1 (q ) εs2 (q ) εs3 (q ) q q + q q + cyclic , 24 ij 1 ab 2 ab 3 2i 2j 3i 3j (5.100)

where the symbol ; is deﬁned to have meaning of ∫ · ∑ d3q d3q d3q ei(q1+q2+q3) x 1 2 3 s1 ′ s2 ′ s3 ′ = × h (t ) h (t ) h (t ) . (5.101) (2π)3 3 q1 q2 q3 s1,s2,s3 Hence the Hamiltonian reads ∫ (2π)3δ(q + q + q ) d3q d3q d3q ∑ ′ 1 2 3 1 2 3 s1 ′ s2 ′ s3 ′ HI(t ) = × h (t ) h (t ) h (t ) a−3ζ2 (2π)3 3 q1 q2 q3 [ s1,s2,s3 ] ( ) g ( ) g ( ) g × A¯ − 1 R + 3 RijR + 8 ∇iRjk (∇ R ) + 6 Ri Rj Rk , ζ2 ij ζ4 i jk ζ4 j k i (5.102)

ij ∇i jk ∇ i j k where expressions of R, (R Rij) , ( R )( iRjk) and RjRkRi in momentum space are given in Eqs.(5.97-5.100). For the simplicity of notation, deﬁne the following shorthand notations [ ] ≡ s1 s2 s3 C1 (123) k1ak1bεij (q1) εia (q2) εjb (q3) , [ ]q ≡ s1 s2 s3 C2 (123) k1ak1bεij (q1) εab (q2) εij (q3) , [ ]q ≡ s1 s2 s3 C3 (123) k1ak1bεij (q1) εij (q2) εab (q3) , [ ]q ≡ s1 s2 s3 C4 (123) k1ak1bεij (q1) εab (q2) εab (q3) , [ ]q ≡ s1 s2 s3 C5 (123) εij (q1) εjk (q2) εki (q3) . (5.103) q

s Promoting the scalar variable hk(t) to a quantized ﬁeld

ˆs ∗ † − hk(t) = hk(t)ˆas(k) + h−k(t)ˆas( k), (5.104)

134 where the creation and annihilation operators satisfy the commutation relation [ ] † ′ 3 ′ − ′ aˆs(k), aˆs′ (k ) = (2π) δ(k k )δss , (5.105) we are now in the position to employ the in-in formalism. The 3-point correlator we seek can be calculated at leading order

⟨hˆs1 (t) hˆs2 (t) hˆs3 (t)⟩ ∫k1 k2 k3 t [ ] ≃ i dt′⟨ Hˆ (t′), hˆs1 (t) hˆs2 (t) hˆs3 (t) ⟩ I k1 k2 k3 ti ∫ t 3 ′ { [ ]} 3 ′ a (t ) − ′ ′ − ∗ = i (2π) δ (k1 + k2 + k3) dt 2 Fs1s2s3 ( K, t ) W (K, t ; t) W ti ζ

+ 5 permutations of (k1, k2, k3), (5.106) where we have deﬁned the product of mode functions

′ ≡ ′ ∗ ′ ∗ ′ ∗ W (K, t ; t) hk1 (t )hk1 (t)hk2 (t )hk2 (t)hk3 (t )hk3 (t), (5.107) and for the contraction pairing {(q1, k1), (q2, k2), (q3, k3)} ( ) ¯ − [ ]∗ − ′ ≡ A 1 1 − 1 − 1 Fs1s2s3 ( K, t ) ′ C1 C2 C3 (123) a2(t ) 6 4 4 k 2 2 2 [ ]∗ − g3 k1 + k2 + k3 − 1 − 1 ′ C1 C2 C3 (123) a4(t )ζ2 6 4 4 k 4 4 4 [ ]∗ − g8 k1 + k2 + k3 − 1 − 1 ′ C1 C2 C3 (123) a6(t )ζ4 6 4 4 k 2 2 [ ]∗ − g8 k1k3 − 1 ′ C1 C2 (123) a6(t )ζ4 6 4 k 2 2 [ ]∗ − g8 k1k2 − 1 ′ C1 C3 (123) a6(t )ζ4 6 4 k 2 2 2 [ ]∗ g6 k1k2k3 + ′ C5 (123) + cyclic, (5.108) a6(t )ζ4 24 k with

[ ]∗ ≡ s1 − s2 − s3 − C1 (123) k1ak1bεij ( k1) εia ( k2) εjb ( k3) k

−s1 −s2 −s3 = k1ak1bεij (k1) εia (k2) εjb (k3) [ ]∗ s1 s2 s3 = k1ak1b εij (k1) εia (k2) εjb (k3) . (5.109)

135 ij We see that the ﬁrst line of (5.108) is contributed by R, the second line by R Rij, ( ) ∇i jk ∇ i j k the third, fourth and ﬁfth line by R ( iRjk) and last line by RjRkRi .

5.3.2 The Mode Integration

As was pointed out in Section 2, the k-dependence of the bispectrum receives contributions from both the interaction Hamiltonian and the mode function, whose behavior is determined by the EoM it satisﬁes and its initial state (mathematically the initial condition). In principle, one should perform the integration in (5.106) tracing the behavior of the mode function through out its history from the initial time (ti) until several e-folds after horizon exit when the mode freezes (t). However, during the early times the modes are highly oscillatory and the integration during that time should be mostly canceled out by itself. This, of course, requires some conditions on the integrands.

Looking at the integrands in (5.106), we see that the non-oscillating time-

3 ′ −2n ′ dependent parts come from the common factor a (t ), the factor of a (t ) in Fs1s2s3 (n = 1, 2, 3) and from the three mode functions. The non-oscillating time-dependent part in the mode function is modeled as

′ m l ′ l h ∝ (η ) exp[iλlk (η ) ], (5.110)

where m and l are dynamical, i.e. they change when diﬀerent terms dominate the dis-

persion relation. Since we are working with quasi-de Sitter space, then a ≃ −1/(Hη),

hence the integration can be written in a schematic form ∫ ( ) ′ l l l ′l ′3m+(2n−4) dη exp[iλl k1 + k2 + k3 η ]η . (5.111)

When 3m + (2n − 4) < 0, the integration must be small in value as η′3m+(2n−4)

changes very slowly during the early times (η′ → −∞) while the oscillating part has

high frequencies; similarly, when 0 < 3m + (2n − 4) < l, the oscillation smoothes

out the changes in η′3m+(2n−4) as one can always perform a change of variables τ =

136 η′3m+(2n−4)+1 and integrate over τ with a purely oscillating integrand. When 3m +

(2n − 4) > l on the other hand, one must be careful. However, if 3m + (2n − 4) is not

higher than l by too much, the errors introduced by ignoring this history should not

be too large. In fact for the current model, when k6 term dominates the dispersion

relation, m = 0, l = 3 and nMAX = 3, hence 3m + (2n − 4) ≤ 2 < l. Therefore we shall work under this assumption below, and consider only the period when the k2 term

dominates the dispersion.

The oscillating mode function should take the form during this period ( ) ( ) C (kη ) 1 C−(kη ) 1 + i −icT kη i icT kη vk(t) = √ 1 − e + √ 1 + e ,, (5.112) 2k icT kη 2k icT kη 2 ≡ − ¯ 6 where cT (1 A), the canonically normalized ﬁeld vk(t) is related to hk(t) through

aM v (t) = pl h (t), (5.113) k 2 k

and the constants C+ and C− are in general functions of H, Mpl, the new mass scale

M∗ and transition time ηi. The reason for this dependence on (kηi) is the UV stage when the dispersion diﬀers from the relativistic form signiﬁcantly. The detailed form

however, depends on the procedure of matching of this “relativistic solution” with

the solution in the UV region. One example is the matching we considered in Section

5.2.4.

Summing up the discussions, we see that the UV history has at least three

eﬀects on the integration we are considering: one is that ηi can no longer be extended

to Euclidean space −∞(1 + iϵ); second, the dependence of the normalization on ηi; third, if the mode underwent a non-adiabatic period (ω2 < 2/η2), then a “negative

frequency” branch exists (C− ≠ 0). Since we’ve seen in Section 2 that a mixture of

“negative frequency” and “positive frequency” modes gives an enhanced folded shape

in the bispectrum, in this section we focus on the case when C− = 0.

6 Note that the relation here is diﬀerent from the relation in Eq.(4.146) due to the diﬀerent normalization we choose for the polarization tensors. The EoM’s however, are the same for both.

137 The bispectrum is then given as

⟨hˆs1 (t) hˆs2 (t) hˆs3 (t)⟩ k1 k2 k3 √ 6 2HC 2G∗ (K) = (2π)3 δ (k + k + k ) ζ−2 + s1s2s3 1 2 3 3 3 3 cT Mpl k1k2k3

+E(UV) + E(Finite ηi) + 5 perm.’s, (5.114) where we have introduced two error terms signifying that we considered only the relativistic region in the integration, and deﬁned, for the contraction pairing

{(q1, k1), (q2, k2), (q3, k3)}, ( ) [ ]∗ ∗ ≡ ¯ − I 1 − 1 − 1 Gs s s (K) A 1 C1 C2 C3 (123) 1 2 3 6 4 4 k 2 2 2 [ ]∗ g3 k1 + k2 + k3 1 1 − II C1 − C2 − C3 (123) ζ2 6 4 4 k 4 4 4 [ ]∗ g8 k1 + k2 + k3 1 1 − III C1 − C2 − C3 (123) ζ4 6 4 4 k 2 2 [ ]∗ g8 k1k3 1 − III C1 − C2 (123) ζ4 6 4 k 2 2 [ ]∗ g8 k1k2 1 − III C1 − C3 (123) ζ4 6 4 k 2 2 2 [ ]∗ g6 k1k2k3 + III C5 (123) + cyclic, (5.115) ζ4 24 k and − 3 I (k1 + k2 + k3) + (k1 + k2 + k3)(k1k2 + k2k3 + k3k1) + k1k2k3 = cT 2 , (k1 + k2 + k3) 3 II − (k1 + k2 + k3) + (k1 + k2 + k3)(k1k2 + k2k3 + k3k1) + 3k1k2k3 = 2 4 , cT (k1 + k2 + k3) (k + k + k )3 + 3 (k + k + k )(k k + k k + k k ) + 15k k k III = 8 1 2 3 1 2 3 1 2 2 3 3 1 1 2(5.116)3 . 3 6 cT (k1 + k2 + k3)

We see that the magnitude of the bispectrum depends on C+ which in turn depends on the new mass scale M∗. Since we’ve found in Chapter 4 that the general relativistic value of the power spectrum for tensor perturbations is obtainable in the HMT framework when H ≪ M∗, we can have the same conclusion as that in Section 2: a large bispectrum is possible, provided that M∗ is not too much lower than the Planck scale.

138 5.3.3 Shapes of the Bispectrum

We are now ready to plot the shapes of the bispectrum. For s1 = s2 = s3 = 1

and s1 = s2 = −s3 = 1, we plot the shapes contributed by the various terms separately in Fig.’s 5.6 and 5.7. These two conﬁgurations represent all possible conﬁgu-

rations as we do not have parity violating terms.

When spins of all 3 tensor ﬁelds are the same (s1 = s2 = s3 = 1), the signal

generated by the relativistic term R peaks at the squeezed limit (k3/k1 → 0) while i j k ≃ ≃ the term RjRkRi favors the equilateral shape (k2/k1 k3/k1 1). It’s interesting ( ) ij i jk to note that the other two terms–R Rij and ∇ R (∇iRjk)–also generate higher signal in the squeezed limit, though they are of higher-order derivatives. This is

indeed expected, if one realizes that the k-dependence of them are similar to that

of the GR term (cf. Eq.(5.115)). Thus an absence of the equilateral shape in the

bispectrum of the tensor perturbations cannot rule out gravity theories of higher

order derivatives, at least for the R2 type of theories.

Note that although the shape of the R term is similar to that in GR and in

[88], magnitude of the signal in the current model could be signiﬁcantly diﬀerent

from those, due to coupling with the U(1) gauge ﬁeld A and a UV history when the

dispersion relation is signiﬁcantly diﬀerent from the relativistic form.

ij When the spins are mixed (s1 = s2 = −s3 = 1), the terms R,R Rij and ( ) i jk ∇ R (∇iRjk) generate shapes similar to the previous case. A particular interesting

i j k result is that the signal generated by RjRkRi no longer favors the equilateral shape but peaks in between the equilateral and squeezed limits. This can be understood as that for the mixed spin the product of the polarization tensors gives a strong favor of the squeezed shape. To illustrate this, we plot the k-dependence of them in both

139 cases ((+ + +) and (+ + −)) in Fig.’s 5.8 and 5.9, where we’ve defined [ ] 1 1 ∗ Configuration1 = C1 − C2 − C3 (123) + cyclic, [ 4 4] k 1 ∗ 2 2 − Configuration2 = k1k3 C1 C2 (123) + cyclic, [ 4 ]k 1 ∗ 2 2 − Configuration3 = k1k2 C1 C3 (123) + cyclic, 4 k [ ]∗ Configuration4 = C5 (123) + cyclic. (5.117) k

ij R R Rij

2 2 1.0 1.0

1 1

0 0

0.5 k3 0.5 k3 0.6 0.6 k1 k1

0.8 0.8 k2 k2

k1 0.0 k1 0.0 1.0 1.0 (a) (b)

ij k R Rij k i k j Rj Ri Rk

1.0

2 1.0 1.0 0.5 1

0 0.0 k 0.5 3 0.5 k3 0.6 k 0.6 1 k1

0.8 0.8 k2 k2

k1 0.0 k1 0.0 1.0 1.0 (c) (d)

−1 Figure 5.6: Shapes of (k1k2k3) G+++ (K) contributed by various terms. All are normalized to unity for equilateral limit.

140 ij R R Rij

20 20 15 1.0 15 1.0 10 10 5 5 0 0

0.5 k3 0.5 k3 0.6 0.6 k1 k1

0.8 0.8 k2 k2

k1 0.0 k1 0.0 1.0 1.0 (a) (b)

Rij k R i k j ij k Rj Ri Rk

0 2.0 -5 1.0 1.5 1.0 -10 1.0

-15 0.5 0.0

0.5 k3 0.5 k3 0.6 0.6 k1 k1

0.8 0.8 k2 k2

k1 0.0 k1 0.0 1.0 1.0 (c) (d)

−1 Figure 5.7: Shapes of (k1k2k3) G++− (K) contributed by various terms. All are normalized to unity for equilateral limit.

141 k 2 k2

k1 k 0.6 1 0.6 0.8 0.8 1.0 1.0 1.0 1.0

0.5 0.5

0.0 0.0 1.0 1.0 0.5 0.5 k 0.0 k3 0.0 3

k1 k1 (a) Conﬁguration 1 (b) Conﬁguration 2

k1 0.6 0.8 1.0 1.0 1.0

0.5 0.5

0.0 0.0 1.0 1.0 0.6

0.5 0.8 0.5 k k3 2 k3 0.0 1.00.0 k k1 1 k1 (c) Conﬁguration 3 (d) Conﬁguration 4

Figure 5.8: Shapes of the diﬀerent conﬁgurations of the polarization tensors for s1 = s2 = s3 = 1. All are normalized to unity for equilateral limit.

142 k2

k1 0.6 0.8 1.0 20 2

15 0 10 -2 5 1.0 0 0.6 1.0 0.8 0.5 0.5 k 2 k3 k 3 1.00.0 0.0 k 1 k1 k1 (a) Conﬁguration 1 (b) Conﬁguration 2

0.6 k1

0.8

1.0

2 4

0 2

-2 0 1.0 1.0 0.6

0.8 0.5 0.5

k2 k3 k3 1.00.0 0.0 k 1 k1 k1 (c) Conﬁguration 3 (d) Conﬁguration 4

Figure 5.9: Shapes of the diﬀerent conﬁgurations of the polarization tensors for s1 = s2 = −s3 = 1. All are normalized to unity for equilateral limit.

143 CHAPTER SIX

Conclusions

6.1 Conclusions

In this dissertation, I studied two models of Hoˇrava-Lifshitz theory of gravity: one without the U(1) symmetry and the other with it, but both assumed the projectability condition N = N(t). Then I investigated applications of the general covariant model with projectability condition (the model with the U(1) symmetry) to inﬂationary cosmology.

The ﬁrst part is devoted to studies of and solutions to the several problems of the theory faced by the original proposal, such as the infrared instability [18] and strong coupling problems [21]. Some modiﬁcations to the original model are proposed to make the theory healthy so that such pathologies can be cured. I found that, among the two models mentioned above, the projectable general covariant HL gravity (the

HMT model) is the most successful one [21]. It has the same degree of freedom as general relativity does, hence is free of the infrared instability issue and the strong coupling problem in the gravity sector. Though the strong coupling problem still appears when the theory couples with matter, it was noted that introduction of a cutoﬀ scale M∗, above which the Lorentz-symmetry-breaking eﬀects dominate, could solve such a problem.

Applications of the HMT model to slow-roll inﬂation driven by a single scalar

ﬁeld is studied next [22, 23]. Three main conclusions are obtained in this part. First,

I found that the FLRW universe is necessarily ﬂat, given that coupling with matter takes a speciﬁc form [22]. Second, I found that under some reasonable conditions, the power spectra of the primordial scalar and tensor perturbations are nearly scale- invariant [22]. Corrections to the power spectra due to the higher spatial curvature

144 2 2 terms are found to be in the order of H /M∗ where H is the Hubble parameter during the inﬂation era [22]. Third, I found that large bispectrum of both scalar and tensor perturbations are possible, given that M∗ isn’t much lower than the Planck scale [23].

The bispectrum of scalar perturbations was found to show equilateral shape at leading order [23], while the bispectrum of the tensor perturbations shows both equilateral and squeezed shape [24]. Therefore this model is consistent with these cosmological tests and can make predictions that could be tested in future experiments.

6.2 Future Work

In obtaining the above conclusions, various assumptions are made on the coupling constants for the Lorentz-violating terms. It is important to study the eﬀects and possible new features when these assumptions are relaxed. Further, the integration over mode functions with the modiﬁed dispersion in the study of primordial non-Gaussianity deserves systematic and deeper analysis. It would also be very interesting to apply the technique to the model developed in [42].

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