主論文

THESIS

Braneworld Cosmological Perturbations
ブレーン宇宙摂動論

Tsutomu Kobayashi
小林 努

Department of Physics, Kyoto University
January, 2006

Abstract
We explore aspects of cosmological perturbations in braneworld scenarios. In order to correctly evaluate perturbations on a four-dimensional brane, we need to solve higher dimensionalbulk perturbations, which reduces to the problem of solving partial diﬀerential equations withappropriate boundary conditions. Hence it is not as easy as in the four-dimensional standard cosmological scenario, and the lack of understanding about the behavior of the branecosmological perturbations has constrained so far the predictability of this revolutionary picture of the universe. In the Randall-Sundrum-type setup, the ﬁve-dimensional anti-de Sitterbulk with a maximally symmetric brane is the special case where gravitational wave (tensor)
perturbations are exactly solvable. Using this as a stepping stone, we further develop various
formulations toward understanding the generation and evolution of cosmological perturbations in the braneworld. We start with an analytic evaluation of leading order corrections
to the evolution of cosmological tensor perturbations at low energies. We explicitly show,
by a perturbative expansion scheme, that the corrections are indeed suppressed by 2 and2 ln , where  is the bulk curvature scale. Next, seeking for the possible high energy effects particular to the braneworld scenario, we consider the primordial spectrum of tensor
perturbations generated quantum-mechanically from inﬂation on the brane. To investigate
the eﬀect of nontrivial motion of the brane, ﬁrst we use the toy model called the junction
model and then develop a numerical scheme based on the Wronskian formulation, and showthat the primordial tensor spectrum in the braneworld can be reproduced with quite good
accuracy by a simple map from the result of the conventional four-dimensional calculation.
We ﬁnd that during inﬂation the vacuum ﬂuctuations in the initial Kaluza-Klein gravitonscontribute to the ﬁnal amplitude of the zero mode at a signiﬁcant level on small scales.
Then, using the same numerical formulation, we compute the late time spectrum of gravitational waves, evolving through the radiation-dominated stage after their generation frominﬂation. We show that in this case the eﬀect of the initial Kaluza-Klein vacuum ﬂuctuations
is subdominant, and therefore the damping due to the Kaluza-Klein mode generation andthe enhancement due to the modiﬁcation of the background Friedmann equation are in fact
the two dominant eﬀects, but they almost cancel each other, leading to the same spectral
tilt as the standard four-dimensional result. Finally, we introduce a (5 + m)-dimensional
vacuum description of ﬁve-dimensional bulk inﬂaton models with exponential potentials thatmakes an analysis of cosmological perturbations simple and transparent, and derive separable
master equations for all types of perturbations in this class of models.

Contents
1 Introduction

5

2 Overview of braneworld cosmology
2.1 Kaluza-Klein picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Randall-Sundrum braneworld . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Cosmology on the brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Friedmann-Robertson-Walker brane . . . . . . . . . . . . . . . . . . .
2.3.2 Dark radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8
8
8
11
11
15

3 Basic material on gravitational wave perturbations in braneworld

16

4 Leading order corrections to the evolution of tensor perturbations
4.1 Setup of the problem and background cosmological model . . . . . . . . . . .
4.2 Tensor perturbations on a FRW brane . . . . . . . . . . . . . . . . . . . . . .
4.3 New long wavelength approximation . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 General iteration scheme . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Long wavelength expansion of the zeroth order solution . . . . . . . .
4.3.3 Slow-roll inﬂation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Transition to the radiation stage . . . . . . . . . . . . . . . . . . . . .
4.4 Corrections in the radiation stage . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Corrections due to the unconventional expansion law . . . . . . . . . .
4.4.2 Suppression at sub-horizon scales . . . . . . . . . . . . . . . . . . . . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Appendix: Details of calculations . . . . . . . . . . . . . . . . . . . . . . . . .

23
24
26
27
27
28
29
30
34
35
36
38
39

5 The junction model
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Background metric, perturbations, and mode functions . . . . . . . .
5.1.2 Model with a jump in the Hubble parameter . . . . . . . . . . . . . .
5.1.3 Method to calculate Bogoliubov coeﬃcients . . . . . . . . . . . . . . .
5.2 Evaluation of Bogoliubov coeﬃcients . . . . . . . . . . . . . . . . . . . . . . .
5.3 Power spectrum of generated gravitational waves . . . . . . . . . . . . . . . .
5.3.1 Mapping formula: Pure de Sitter brane . . . . . . . . . . . . . . . . .
5.3.2 Model with variation of the Hubble parameter . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Appendix: Particle creation when connected to Minkowski brane . . . . . . .
5.6 Appendix: Details of calculations . . . . . . . . . . . . . . . . . . . . . . . . .

41
42
42
43
44
46
48
48
49
53
55
56

2

3
6 Quantum-mechanical generation of gravitational waves
6.1 Gravitational waves in inﬂationary braneworld . . . . . . . . . . . . . . . . .
6.1.1 Pure de Sitter brane . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 The “junction” model . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.3 A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59
60
60
62
63
63
66
67
69
74

7 The spectrum of gravitational waves
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 The background model . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Minkowski and de Sitter braneworlds . . . . . . . . . . . . . . . . . . .
7.2 Wronskian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Spectrum of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Generation of dark radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76
77
77
78
80
81
85
88

8 Cosmological perturbations in the bulk inﬂaton model
8.1 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Solutions in the Randall-Sundrum vacuum braneworld . . . . . . . . .
8.1.2 Bulk inﬂaton models from dimensional reduction . . . . . . . . . . . .
8.2 Single scalar ﬁeld in the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Examples of the background spacetime . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Kasner-type solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Kasner-type solutions with a cosmological constant . . . . . . . . . . .
8.4 Cosmological perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Vector perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.3 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Appendix: Perturbed Einstein equations and gauge transformations . . . . .
8.7 Appendix: Multiple scalar ﬁeld generalization . . . . . . . . . . . . . . . . . .

90
91
91
92
93
95
95
99
101
102
104
105
110
111
114

9 Conclusions

116

Appendix

118

A Review of cosmological perturbation theory in four dimensions
119
A.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.1.1 The unperturbed Universe . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.1.2 Perturbed metric and energy-momentum tensor . . . . . . . . . . . . . 120
A.2 Quantum theory of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 124
B Derivation of the eﬀective Einstein equations on the brane127

4
C Units

130

Chapter 1

Introduction
Cosmological inﬂation [39, 131, 96, 69, 94] explains how the large-scale structure observed
today emerges by stretching the quantum ﬂuctuations in inﬂaton and possibly other ﬁelds
in the universe. Small inhomogeneities in the energy density exit the horizon, and at alater stage they come back again inside the horizon, being responsible for the Cosmic Microwave Background (CMB) temperature anisotropies of the order of 1 part in 105 , whichwere indeed found by the Cosmic Background Explorer (COBE) in 1992 [7], and have been
observed with better resolution and sensitivity by the Wilkinson Microwave Anisotropy Probe(WMAP) [8, 70, 136, 121, 147]. Inﬂation also predicts the gravitational wave backgroundarising due to the quantum ﬂuctuations in the graviton ﬁeld. The detection of the gravitational waves of inﬂationary origin is a challenging future task for the Laser InterferometerSpace Antenna (LISA) [148] or the Deci-hertz Interferometer Gravitational Wave Observatory (DECIGO) [132]. These cosmological ﬂuctuations have rich information on the early
universe, and hence provide us with powerful tools to probe fundamental physics. Then, what
if there exist extra dimensions?
Motivated by string theory [122], there has been a growing attention to braneworld scenarios, in which our observable universe is realized as a “brane” embedded in a higher dimensional“bulk” spacetime (for pedagogical reviews, see e.g., Refs [129, 89, 102]). In this revolutionary picture, Standard Model particles are assumed to be trapped on the brane while gravity
is free to access the bulk. This is the key idea of braneworlds, allowing for rather “large”
extra dimensions as Newton’s law is conﬁrmed to hold down only to O(0.1 mm) by the current table-top experiments [99, 22]. In the naı̈ve construction proposed by Arkani-Hamed,
Dimopoulos, and Dvali (ADD) [2, 3], the braneworld idea is used to address the hierarchy
problem. If d extra dimensions are compactiﬁed with radius L, the gravitational Lagrangian
2+d (4+d)
2+d d (4)reads L ∼ M4+d·
R ∼ M4+d
L · R, and hence the Planck scale MPl is related to the(4 + d)-dimensional fundamental scale via the volume of extra dimensions as
2+d d
2
MPl
∼ M4+dL .

(1.1)

Taking, for example, d = 2 and L ∼ 0.1 mm, we have the low fundamental scale, M4+d ∼ 1
TeV, which is of the order of the electroweak scale.
Among various models, the Randall-Sundrum model [123, 124] is of particular interest
because it includes nontrivial gravitational dynamics despite rather a simple construction.
In the Randall-Sundrum type 2 model [124] with a single brane embedded in an anti-deSitter (AdS) bulk, although the ﬁfth dimension extends inﬁnitely, the warped structure of
the bulk geometry results in the recovery of four-dimensional general relativity on the brane5

6
at scales larger than the bulk curvature scale  or at low energies [31]. In order to reveal
ﬁve-dimensional eﬀects particular to the braneworld scenario, we have to focus on the scales
smaller than . It is natural to consider inﬂationary scenarios in the braneworld context, andcosmological perturbations from inﬂation will be quite useful for the purpose of probing suchsmall scales. Therefore, it is worthwhile clarifying the behavior of perturbations in the brane
universe.
Is it indeed observationally relevant to investigate cosmological dynamics on scales smaller
than the size of the extra dimension? To see this point, let us take a brief look at the possibilityof a direct detection of a gravitational wave background generated by some dynamics ofthe extra dimension with a typical length scale  [47]. Suppose that the extra dimensional
dynamics has left its trace when the Hubble radius was comparable to the typical size of theextra dimension: H∗−1 ∼ . The characteristic frequency of the gravitational waves redshiftedto the present day is f0 = H∗ a∗ /a0 . Using the relation a∗ T∗ = a0 T0 and the Friedmann
2 , we have the estimate
equation H∗2 ∼ T∗4 /MPl−4f0 ∼ 10


Hz ×


1 mm

−1/2
.

(1.2)

(We assumed here for simplicity that the standard Friedmann equation holds at energy scales
as high as ∼ T∗ .) Extra dimensions between  ∼ 1 and 10−6 mm can produce backgrounds
peaked in the LISA band (10−1 to 10−4 Hz); ground-based observations such as the Laser
Interferometer Gravitational-Wave Observatory (LIGO) [149] (up to ∼1000 Hz) can detect
activity from extra dimensions as small as  ∼ 10−14 mm [47].
While the cosmological perturbation theory in the conventional four-dimensional universe
has been rather established [67, 110, 4, 94], calculating cosmological perturbations in thebraneworld still remains to be a diﬃcult problem; we are far from taming it. In this thesis,
we explore aspects of cosmological perturbations in the braneworld. After an introductory
review of basic, well understood facts about cosmology and perturbations in the braneworld,
we ﬁrst evaluate leading order corrections to the cosmological evolution of gravitational waves
at low energies, conﬁrming that it is indeed small. Then, seeking for possible high energy
eﬀects, we move on to study the primordial and late time tensor spectra by invoking a
newly developed numerical method as well as the so called “junction model.” Finally, as anapplication of the simple braneworld setup where the cosmological perturbations are exactly
solvable, we discuss a braneworld model with bulk scalar ﬁelds and its perturbations.
The plan of this thesis is as follows:
Chapter 2 We overview homogeneous and isotropic cosmology on a brane.
Chapter 3 We summarize basic facts on cosmological (tensor) perturbations in the RandallSundrum braneworld.
Chapter 4 We analytically derive leading order corrections to the cosmological evolutionof tensor perturbations in the Randall-Sundrum braneworld at low energies. The originalcontribution is based on
T. Kobayashi and T. Tanaka, “Leading order corrections to the cosmological evolution of
tensor perturbations in braneworld,” JCAP 0410, 015 (2004) [63].

7
Chapter 5 We introduce the junction model to investigate the eﬀect of nontrivial motionof the brane on the primordial tensor spectrum. The original contribution is based on
T. Kobayashi, H. Kudoh and T. Tanaka, “Primordial gravitational waves in inﬂationary
braneworld,” Phys. Rev. D 68, 044025 (2003) [61].
Chapter 6 We formulate a numerical method to calculate the primordial spectrum ofgravitational waves, extending the result of the previous chapter. The original contribution
is based on
T. Kobayashi and T. Tanaka, “Quantum-mechanical generation of gravitational waves inbraneworld,” Phys. Rev. D 71, 124028 (2005) [65].
Chapter 7 Using the numerical formulation in the previous chapter, we clarify the late time
power spectrum of gravitational waves, evolving through the radiation-dominated epoch after
their generation during inﬂation. The original contribution is based on
T. Kobayashi and T. Tanaka, “The spectrum of gravitational waves in Randall-Sundrum
braneworld cosmology,” hep-th/0511186 [66].
Chapter 8 In the context of the bulk inﬂaton model, we present a new and powerful technic
to generate background cosmological solutions and to compute perturbations. The originalcontribution is based on
T. Kobayashi and T. Tanaka, “Bulk inﬂaton shadows of vacuum gravity,” Phys. Rev. D 69,
064037 (2004) [62].
Chapter 9 We draw the conclusions of this thesis.
We have tried to make each chapter to be organized in a self-contained manner so thatthe reader should be able to follow the content of one chapter independently from anotherwith prior reading of Chapters 2 and 3.

Chapter 2

Overview of braneworld cosmology
In this chapter, we outline the Randall-Sundrum braneworld scenario and homogeneous,isotropic cosmology on the brane.

2.1

Kaluza-Klein picture

To begin with, we will brieﬂy discuss the idea of the Kaluza-Klein scenario. Let us consider thesimplest setup with one extra spatial dimension y in addition to the usual four-dimensional(ﬂat) spacetime xµ . Let us assume that the extra dimension is a homogeneous circle ofradius L so that y runs from 0 to 2πL. A complete set of solutions to a ﬁve-dimensional
Klein-Gordon equation (a wave equation of a free massless particle), 2φ(xµ , y) = 0, is givenbyµ

φ(pµ , n) = eip xµ · einy/L ,(2.1)

where n = 0, ±1, ±2, · · · . The four-momentum pµ and the eigenvalue n of the extra momentum obeypµ pµ +

n2
= 0.
L2

(2.2)

Inhomogeneous modes with n = 0 are called Kaluza-Klein (KK) modes, and each mode canbe seen as a particle with mass
mn =

|n|
,
L

(2.3)

from the four-dimensional point of view. The KK modes can be excited at the energy scale∼ 1/L, and so in this scenario the size of the extra dimension must be microscopic (L  10−17cm) because no KK partners of ordinary particles such as photons have been observed.

2.2

Randall-Sundrum braneworld

The Randall-Sundrum (RS) braneworld models [123, 124] give us a completely diﬀerent mechanism to realize an eﬀectively four-dimensional world in a ﬁve-dimensional spacetime. A basic
ingredient of braneworld scenarios is that Standard Model particles and ﬁelds are localized on
89Figure 2.1: Schematic of the Randall-Sundrum braneworld.
a four-dimensional object (the “brane”) while gravity can propagate in the higher dimensional
spacetime (the “bulk”). In the RS braneworld gravity is localized at the four-dimensionalbrane not due to compactiﬁcation, but due to the curvature of the bulk spacetime, and forthis reason the mechanism is called “warped compactiﬁcation”.
The RS braneworld is described by the following action:


√
√
M3
S= 5
d5 x −g (R − 2Λ5 ) −(2.4)
d4 x −qi σi ,2iwhere M5 is the fundamental scale of gravity,6(2.5)
Λ5 = − 2is the bulk cosmological constant, and σi is the tension of the i-th brane. Since the cosmological constant Λ5 is negative, the bulk is basically given by an anti-de Sitter (AdS) spacetime.
There are two types of models of the Randall-Sundrum braneworld, and they are classiﬁed
as follows:
RS1

Let us assume the ﬁve-dimensional metric of the form
ds2 = a2 (y)ηµν dxµ dxν + dy 2 .

(2.6)

In the so called RS1 model [123], two branes are placed at y = 0 and y = L, and theZ2 -symmetry is imposed as y ↔ −y and y + L ↔ −y + L. The ﬁve-dimensional Einsteinequations read  2
1
a
= 2,
(2.7)
a

  
a1
3= − 3 [σ1 δ(y) + σ2 δ(y − L)] ,
(2.8)
a
M5

10
where a prime stands for a derivative with respect to y. From these two equations we cansee that the branes have equal and opposite tensions,
σ1 = −σ2 =

6M53
=: σ,


(2.9)

and the “warp factor” is given bya(y) = e−|y|/ .

(2.10)The eﬀective Planck scale MPl on the negative tension brane (second brane) is [123]2= M53 e2L/ − 1 ,(2.11)
MPland so if we set the separation of the branes to be L ∼ 35, we have M5 ∼ −1 ∼ O(TeV).
This implies that the RS1 model helps to solve the hierarchy problem, which is one of theoriginal motivation of the braneworld scenario.
One should note that the eﬀective Planck scale on the positive tension brane is


2
MPl
= M53 1 − e−2L/ .
(2.12)
RS2 The RS2 is a model with a single brane in an AdS bulk, and obtained by taking thenegative tension brane to inﬁnity, L → ∞ [124]. Then, from Eq. (2.12) we can see that the
Planck scale on the brane is2= M53 .
MPl(2.13)

Although the extra dimension extends inﬁnity, the warped structure of the bulk geometry
results in its ﬁnite contribution to the ﬁve-dimensional volume:



5 √
d4 x,
d x −g =2
which means that the eﬀective size of the extra dimension is . Gravitational perturbations
around the Minkowski brane in the RS2 model are extensively investigated in [31], and it isshown that for r   the Newtonian potential on the brane is given by


22
Gmm
(2.14)
1+ 2 ,
V (r)  −r
3r
where the second term is the small correction to four-dimensional gravity arising due to thecontribution of the massive KK modes of gravitons1 . At this moment, table-top experiments
conﬁrm that Newton’s law holds at scales down to O(0.1 mm) [99, 22]. Therefore we have  0.1 mm.

(2.15)

This in turn gives limits on the other parameters:σ  (1 TeV)4 ,M5  105 TeV.

(2.16)In what follows, we will mainly concentrate on the RS2 model.
For later purposes, we rewrite the metric (2.6) in a conformally ﬂat form by introducing
the Poincaré coordinates as
2ds2 = 2 (ηµν dxµ dxν + dz 2 ),(2.17)zwith z = ey/ . In these coordinates, the brane is located at z = .
1

The recovery of Einstein gravity beyond the linear order is conﬁrmed in Refs. [34, 82, 144].

11

2.3

Cosmology on the brane

2.3.1

Friedmann-Robertson-Walker brane

Our primary interests are in cosmological aspects of the braneworld, and in this section weshall consider a homogeneous and isotropic, Friedmann-Robertson-Walker (FRW) cosmological model. In the RS setup, cosmological expansion on the brane is mimicked by its motion
through the AdS bulk. To see this, let us consider the ﬁve-dimensional Schwarzschild-AdS
solution whose metric is given by [71, 51]
gAB dxA dxB = −f (r)dT 2 +1
dr2 + r2 γij dxi dxj ,f (r)(2.18)

C
r2
− 2,
2

r

(2.19)

where
f (r) := k +and γij is the metric of the maximally symmetric three-dimensional space. We allow for a
black hole in the bulk, and C is its mass parameter. If C vanishes, the bulk spacetime isexactly AdS. In that case, by a coordinate transformation z = /r and appropriate rescalingof the time coordinate we have the familiar form of the metric (2.17) when k = 0. As will beseen, nonvanishing C turns out to bring an interesting consequence on cosmology.
Suppose that the brane is moving in this static bulk and its trajectory is parameterized
by T = T (τ ) and r = r(τ ). The induced metric on the brane isqµν dxµ dxν = − f Ṫ 2 − f −1 ṙ2 dτ 2 + r2 (τ )γij dxi dxj ,(2.20)

where a dot denotes a derivative with respect to τ . Now it is clear that the function r(τ )plays a role of the scale factor: r(τ ) = a(τ ). The tangent to the brane isuA = Ṫ , ṙ ,uA = −f Ṫ , f −1 ṙ ,(2.21)

which is normalized as uA uA = −1 so that τ is the proper time on the brane. The unit
normal to the brane is(2.22)nA = ṙ, −Ṫ , nA = −f −1 ṙ, −f Ṫ .
From this we can calculate the extrinsic curvature Kµν := qµA qνB ∇A nB on the brane:
Kij = −f Ṫ r−1 gij ,Kτ τ = uA uB ∇A nB = ṙ−1

d 
dτ


f Ṫ .

(2.23)(2.24)

We assume that the energy-momentum tensor of the localized matter is given by that of aperfect ﬂuid,
T µν = diag(−ρ − σ, p − σ, p − σ, p − σ),(2.25)

12
where we include the tension of the brane σ. Then the junction conditions [52] imply that1(ρ + σ)δ i j ,6M53
1(2ρ + 3p − σ).
Kτ τ = −6M53

K ij = −From the spatial component Kij we have
 2
ṙ
1
1
k
C
=
(ρ + σ)2 − 2 − 2 + 4 .
H 2 :=r
r
r(6M53 )2

(2.26)(2.27)(2.28)

2 = M 3 and σ = 6M 3 /, we ﬁnally obtain the modified
After substituting the RS relations MPl
5
5Friedmann equation on the brane [9, 10, 111, 71, 51]
ρ 
C
ρ 
k
(2.29)
1+− 2 + 4.
H2 =
2
2σ
a
a
3MPl

When ρ
σ and C = 0, the conventional Friedmann equation is recovered on the brane.
The modiﬁcation to the standard cosmology arises from the quadratic term in ρ relevant atenergy scales higher than σ, and the so called “dark radiation” C/a4 which behaves as an
extra radiation component.
From the equation for Kτ τ , it can be seen that the standard conservation law holds on
the brane:ρ̇ + 3H(ρ + p) = 0.

(2.30)For an equation of state p = wρ with w constant, we have ρ ∝ a−3(1+w) . Then, assumingthat k = 0 and C = 0, the modiﬁed Friedmann equation (2.29) can be solved to givea = a∗

τ 2 + cττ∗2 + cτ∗

1/3(1+w)
,

22
ρ
1
=
,
σ
9(1 + w)2 τ 2 + cτ

(2.31)

where c := 2/3(1 + w). More speciﬁcally, in the case of a radiation ﬂuid (w = 1/3) we obtainσ) and a ∼ τ 1/4 at high energies (ρ  σ).
a ∼ τ 1/2 at low energies (ρAs an alternative approach, we may also use Gaussian normal coordinates [9, 10, 111]
and write the general line element that has the cosmological symmetry as
ds2 = −N 2 (τ, y)dτ 2 + A2 (τ, y)γij dxi dxj + dy 2 .

(2.32)In the Gaussian normal coordinates, the brane does not move but is located at a ﬁxed
coordinate position which can be set y = 0 without loss of generality. (The equivalence of thetwo description can be shown by ﬁnding the explicit coordinate transformation that links theGaussian normal and Schwarzschild-AdS coordinates [112].) The ﬁve-dimensional Einsteinequations together with the junction conditions give the explicit form of the metric functions
as [10],

ρ 2
2 C
1 1
+
1+
+
A(τ, y) = a(τ )2 2σ
a(τ )4

1/2

ρ
1 1
2 C
ρ 2
−
cosh(2y/)
−
1
+
+
−
,(2.33)1+sinh(2y/)2 2σa(τ )4
σ

13

T, t

ü

y

r

z

Figure 2.2: Gaussian normal coordinates, Poincaré and Schwarzschild static coordinates.
Solid curve represents the world surface of a cosmological brane.
andN (τ, y) =Ȧ(τ, y).
ȧ(τ )(2.34)In the above, a(τ ) is the scale factor on the brane and C is equivalent to the black hole massor the dark radiation parameter in the previous discussion. In deriving Eq. (2.34) we used
N (τ, 0) = 1. Of course, by invoking the junction conditions, one can show that the scale
factor obeys the modiﬁed Friedmann equation (2.29) in this coordinate system as well.
If the dark radiation vanishes, Eq. (2.33) can be simpliﬁed toρA(τ, y) = a(τ ) cosh(y/) − 1 +sinh(y/) ,(2.35)σand the explicit form of N (τ, y) also becomes simple:
N (τ, y) =A(τ, y) 3(ρ + p)+sinh(y/),a(τ )σ(2.36)

where we used the conservation law. These two expressions are frequently used in the literature. For a de Sitter brane (ρ = −p = const.), N depends only on y and A(τ, y) becomes aseparable function: A(τ, y) = a(τ )N (y), where
N (y) = cosh(y/) −1 + 2 H 2 sinh(y/).

(2.37)

14

Figure 2.3: Global structures for the Randall-Sundrum model. The Poincaré coordinates (t, z)
cover the shaded region in the left ﬁgure. For the de Sitter braneworld, (η, ξ) coordinates
cover the shaded region in the right ﬁgure. Typical trajectories of the FRW brane are shownby thick solid lines.
In this case, by a farther coordinate transformation−η = e−Hτ /H,(2.38)

sinh ξ = H/N (y),(2.39)

we obtain the following bulk metric:

1
2
2
ds =
(−dη 2 + δij dxi dxj ) + dξ 2 ,2sinh ξ η 2(2.40)

which will be used throughout this thesis. Note that η is the conformal time on the brane.
Denoting the brane position in these coordinates as ξ = ξb , we ﬁnd from Eq. (2.39) that
H = sinh ξb .
Let us mention the limitation of the Gaussian normal coordinates. The geodesics orthogonal to the brane eventually will cross, at which the Gaussian normal coordinates are ill
deﬁned. This coordinate singularity is at y = yc deﬁned byρ(2.41)
coth(yc /) = 1 + ,σwhere A(τ, y) vanishes [see Eq. (2.35)]. Thus, this coordinate system seems not so appropriate
for the high energy regime (ρ  σ) because the coordinate singularity is very close to thebrane. There is another potential singularity yh deﬁned byρ(2.42)
coth(yh /) = 1 − (2 + 3w) ,σ
where N (τ, y) vanishes [see Eq. (2.36)]. This singularity is relevant for w < −2/3, and for ade Sitter brane we have yc = yh .

15

2.3.2

Dark radiation

We are now in a position to give a short comment on the consequences of the “dark radiation”
term in the modiﬁed Friedmann equation (2.29). This term corresponds to the projected Weyltensor Eµν from the brane (four-dimensional) point of view (see Appendix B): C/a4 = Eτ τ /3.
The radiation-like behavior can be understood by seeing that Eµν is traceless.
The amount of the dark radiation can be constrained by nucleosynthesis [9]. At the timeof nucleosynthesis, the universe should be in the low energy regime (ρσ) in order not
to spoil the standard cosmology picture, and is dominated by the radiation energy density,
which is given byρr |N = g∗

π2 4
T ,
30 N

(2.43)

where g∗ is the eﬀective number of relativistic degrees of freedom at that time. In theStandard Model, g∗ = 10.75. The observed abundances of light elements strongly constrains
any deviation ∆g∗ from that value typically as ∆g∗ < 2. Thus, the energy density of the
2 C/a4 ) should satisfy the bounddark radiation ρDR (= 3MPl
ρDR
< 0.16.
ρr(2.44)

Note that this ratio is invariant because ρDR has the same dependence on the scale factor as
ρr . For a more detailed analysis on the observational constraints on the amount of the darkradiation, see Ref. [48].
It is likely that in the early universe gravitational waves escape into the bulk and theparameter C varies in time due to this process. For example, the high energy collision ofparticles on the brane results in the production of KK gravitons, which directly leads to thegeneration of dark radiation [42, 88, 90, 91]. Such a process was studied by using a toy model
based on the AdS-Vaidya solution [88, 91] and by resorting to a numerical calculation [90](see also Refs. [138, 33, 142, 53, 64]). In the most detailed study on this issue by Langlois
et al. [90], their estimate on the amount of dark radiation generated during the cosmologicalevolution of the brane universe points to a value close to the current observational bound.

Chapter 3

Basic material on gravitational
wave perturbations in braneworld
In the previous chapter we have brieﬂy overviewed homogeneous and isotropic cosmology on abrane. The rest of this thesis treats the generation and evolution of cosmological perturbationsin the braneworld [5, 6, 11, 14, 15, 17, 24, 25, 26, 27, 30, 36, 37, 38, 41, 45, 46, 49, 50, 61,
63, 65, 66, 68, 72, 73, 74, 76, 78, 79, 81, 83, 84, 85, 86, 92, 93, 100, 107, 113, 114, 115, 126,
127, 139, 141, 146]. Before going to the main discussions of the thesis, in this chapter we will
give some basic facts about cosmological perturbations in the Randall-Sundrum braneworld,
focusing on gravitational wave (tensor) perturbations [5, 6, 17, 27, 30, 36, 41, 45, 46, 49, 50,
61, 63, 65, 66, 78, 84, 139].
In the static Poincaré chart, tensor perturbations are given by
ds2 =


 i j

2 
2
i
2
+
δ
+
h
(t,
z,
x
)
dx
dx
+
dz
−dt
.
ij
ij
z2

(3.1)

(Here and from now on we assume a spatially ﬂat Friedmann-Robertson-Walker brane as a
background.) We decompose the perturbations into the spatial Fourier modes and consider
one Fourier mode at a time, so that
√


2
(A)
(A)
(3.2)
d3 kφk (t, z)eik·x eij ,
hij =
3/2
(2πM
)
5
A=+,×
(A)

where eij is a transverse and traceless tensor, and hereafter we suppress the index k. We will√
3/2
also omit the index A for the two polarization states. The meaning of the prefactor 2/M5
will become clear later when considering quantization of perturbations. The equation of
motion for φ is a Klein-Gordon-type equation:


3
2
2
2
(3.3)
∂t + k − ∂z + ∂z φ = 0,z
and it follows from the junction conditions that the boundary condition is given bynA ∂A φbrane = 0,

(3.4)

where nA is a unit normal to the brane. In Eq. (3.4) we neglected a possible anisotropic stressterm arising from perturbations in the brane energy-momentum tensor. Since Eq. (3.3) is16

17
the Bessel equation, its general solution can be easily obtained in an analytic form. This isthe bonus of working in the Poincaré coordinates. However, the cosmological brane moves inthe static bulk, and hence Eq. (3.4) reduces to a moving boundary condition:

 

2
2
H∂t − 1 +  H ∂z φ
= 0,
(3.5)
z=z(t)

which is diﬃcult to handle in general.
On the other hand, in the Gaussian normal coordinates the perturbed metric is given by
ds2 = −N 2 (τ, y)dτ 2 + A2 (τ, y) (δij + hij ) dxi dxj + dy 2 ,and the tensor perturbation obeys
 



 
k2
N
Ȧ Ṅ
A
1

φ̇ + 2 − φ − 3 +φ = 0,φ̈ + 3 −
N2
A N
A
A
N

(3.6)

(3.7)

where the explicit form of A(τ, y) and N (τ, y) is found in the previous chapter, and theoverdot (prime) denotes a derivative with respect to τ (y). Since the brane is located at aﬁxed coordinate position, the boundary condition takes a simple form∂y φ(τ, 0) = 0.

(3.8)

In the Gaussian normal coordinates, however, Eq. (3.7) is not separable for generic functions
A(τ, y) and N (τ, y). Thus, one will suﬀer from technical diﬃculties in any coordinate system.
Fortunately, in the exceptional cases of the maximally symmetric brane, one can choose
a coordinate system where the perturbation equation has a separable form and at the sametime a simple boundary condition is imposed at a ﬁxed coordinate position. This can beeasily seen by noting that we have A(τ, y) = a(τ )N (y) for a de Sitter brane, as is argued
in the previous chapter. Substituting this into Eq. (3.7), we obtain a manifestly separable
equation of motion, subject to the Neumann boundary condition at the brane. The de Sitter
braneworld includes the case of a Minkowski brane as a limiting case H → 0. The behaviorof gravitational wave perturbations are well understood in these two special cases; let us
summarize the basic known results.
Minkowski brane The Minkowski brane sits at z =  = const., and hence it is convenient
to use the Poincaré chart in this case. The zero mode solution is simply given by
ϕ0 (t) = Ae−ikt ,(3.9)

and the Kaluza-Klein mode solutions are written as

where ω =√ϕm (t, z) = Cz 2 [J2 (mz) − BY2 (mz)] e−iωt ,(3.10)

k 2 + m2 , and the boundary condition at z =  requiresB=

J1 (m).
Y1 (m)(3.11)At this stage the overall normalization is not so important and we leave it undetermined until
Chapter 7.

18

Figure 3.1: “Volcano” potential for gravitons around the Minkowski brane (left) and the deSitter brane (right).
Since the extra dimension is not compact in the RS2 model, the mass spectrum is continuous and m2 can take any non-negative value. This can be understood by introducing the
canonical variable ψ̂ := z −3/2 ϕm . In terms of ψ̂, the Klein-Gordon equation can be rewritten
in the form of Schrödinger equation,
∂2(3.12)− 2 + V (z) ψ̂ = m2 ψ̂,∂z
315(3.13)
V (z) = 2 − δ(z − ),4zwhere the boundary condition is incorporated as the delta-function in the potential. Fromthe left ﬁgure of Fig. 3.1, it is easy to ﬁnd that there is a single bound state (zero mode) as
well as a gapless continuum of KK modes.
The motion of the cosmological brane in the low energy regime is non-relativistic, inthe sense that |dz/dt|
1, and so such a brane may be approximated by a Minkowski
brane [5, 6, 27, 139, 63]. Having this fact in mind, in the next chapter we shall study theevolution of gravitational waves at low energies perturbatively taking into account small
deviation from a Minkowski brane [63].
De Sitter brane Even though realistic inﬂation should not be exactly de Sitter, a deSitter braneworld [84] is important in that it gives the simplest picture of how cosmologicalperturbations are generated and evolve during inﬂation in the early universe1 .
The perturbation equation in the Gaussian normal coordinates reduces to


k2
N 
1

φ̈
+
3H
φ̇
+
φ,
φ
=
φ
+
4
(3.14)
N2
a2
N
where
a(τ ) = a0 eH(τ −τ0 ) ,N (y) = cosh(y/) −1(3.15)
1 + 2 H 2 sinh(y/),(3.16)

Appendix A will be helpful to the readers who are not familer with the standard four-dimensional material
about the generation of cosmological perturbations from inﬂation.

19
with constant H. Equation (3.14) admits a zero mode satisfying
k2
φ0 = 0,a2(3.17)φm (τ, y) = ψm (τ ) · χm (y),(3.18)

φ̈0 + 3H φ̇0 +as well as massive KK modes,

where ψ(τ ) and χ(y) obey
 2

k
2
+ m ψm = 0,ψ̈m + 3H ψ̇m +a2
Nm2
χm + 4 χm + 2 χm = 0.
NN

(3.19)(3.20)

Equation (3.17) is nothing but the Klein-Gordon equation for a massless scalar ﬁeld in theFriedmann universe, while Eq. (3.19) is equivalent to the equation of motion for a massive
scalar ﬁeld.
Throughout this thesis we work mainly in the coordinate system deﬁned by Eqs. (2.38)
and (2.39) when discussing de Sitter inﬂation on the brane. In terms of the conformal timeη, the zero mode solution is given byi1
φ0 (η) = N · √e−ikη ,
η−
(3.21)k2k
where N is a normalization constant to be determined. Note that this coincides with thestandard mode function associated with the Bunch-Davies vacuum in de Sitter space [12, 16]
up to the normalization factor. In (η, ξ) coordinates the equation for the massive modes are
rewritten as
 2

∂
2 ∂
ν 2 + 9/4
2ψν (η) = 0,+k +−(3.22)∂η 2 η ∂η
η2


∂
9
1
∂
(3.23)+ ν2 +χν (ξ) = 0,
sinh3 ξ3
∂ξ sinh ξ ∂ξ
4
where


2

m =

9
ν +
4
2


H 2.

(3.24)The mode solutions are found for any non-negative values of ν 2 as
(1)ψν (η) = C3 (−η)3/2 Hiν (−kη),(3.25)−2(cosh ξ) − C2 Q−2χν (ξ) = C1 sinh2 ξ P−1/2+iν−1/2+iν (cosh ξ) ,(3.26)

where C3 and C1 are normalization constants and C2 is determined from the boundary
condition as
C2 =−1(cosh ξb )P−1/2+iνQ−1−1/2+iν (cosh ξb ),(3.27)

20
with ξb the position of the brane. The result m2 = (ν 2 + 9/4)H 2 ≥ 9H 2 /4 indicates that
there is a mass gap between the zero mode and the continuum of massive modes [32]2 . This
is in contrast to the case of the Minkowski brane. Since massive ﬁelds with m2 ≥ 9H 2 /4
decay at super-horizon scales during inﬂaton, it is expected that the zero mode will dominate
on the brane.
Again using the canonical variable ψ̂ := N 3/2 χm , the equation of motion can be rewritten
as

∂2
m2
− 2 + V (ξ) ψ̂ = 2 ψ̂,(3.28)∂ξ
H√
15
1 + 2 H 2
9δ(ξ − ξb ),−3(3.29)V (ξ) = +2
4 4 sinh ξHwhere the boundary condition is incorporated as the delta-function well. Noting that V (∞) =9/4, we clearly see that only massive modes with m2 /H 2 ≥ 9/4 can exist other than the zeromode (Fig. 3.1).
So far we have left undetermined the overall normalization of the mode functions. To give
appropriate normalization constants, we need to go to quantum theory. Now the reason for√
3/2
the presence of the factor 2/M5 in Eq. (3.2) is clear; with this choice the eﬀective action
for φ derived from the Einstein-Hilbert action is equivalent to the action of a canonicallynormalized scalar ﬁeld, whose kinetic term is given by (1/2)∂A φ∂ A φ. Therefore, we canquantize the graviton ﬁeld φ in a standard textbook manner [12]. Treating the ﬁeld φ as an
operator, it can be expanded in terms of the annihilation and creation operators as
(3.30)φ̂k (η, ξ) = â0 φ0 + â†0 φ∗0 + dν âν φν + â†ν φ∗ν ,where ân and â†n is respectively the annihilation and creation operators of a zero mode for
n = 0 and of a KK mode for n = ν. These operators satisfy the usual commutation relations.
In quantum ﬁeld theory, mode functions form an orthonormal system with respect to theWronskian (the scalar product) (φ1 · φ2 ). In the present case the Wronskian is naturally
deﬁned as [36, 61]
 ∞
3
∗
∗
dξ 2
(3.31)(φ1 · φ2 ) := −2i
3 (φ1 ∂η φ2 − φ2 ∂η φ1 ) ,
η
sinh
ξ
ξb
and the Wronskian conditions are given by(φ0 · φ0 ) = −(φ∗0 · φ∗0 ) = 1,

(φν · φν  ) = −(φ∗ν · φ∗ν  ) = δ(ν − ν  ),(φ0 · φν ) = (φ∗0 · φ∗ν ) = (φn · φ∗n ) = 0,

for n, n = 0, ν.

For the zero mode we have
N = C(H),
where



 ∞

−1/2
1
sinh3 ξξb
 −1/2


H
2 2
2
2
√
=
1 +  H +  H ln.
1 + 1 + 2 H 2C(H) =

2

2 sinh2 ξb(3.32)dξ(3.33)

This result has been generalized in the casep of de Sitter brane(s) with a bulk scalar ﬁeld, showing thatthe universal lower bound on the mass gap is 3/2H [30].

21
For x
1 we have C(x) ≈ 1, while we obtain C(x) ≈we separately impose

3x/2 for x  1. For the KK modesi3− 2 (ψν ∂η ψν∗ − ψν∗ ∂η ψν ) = 1,
η
 ∞
dξ
∗

2
3 χν  χν = δ(ν − ν ).
ξb sinh ξ(3.34)(3.35)We see from the condition (3.34) that the normalized mode function in the time direction is
given by√
π −3/2 −πν/2(1)e(−η)3/2 Hiν (−kη).
(3.36)ψν (η) =2
The condition (3.35) implies that the weighted mode function √(sinh ξ)−2/3 χν (ξ) behaves as(a linear combination of) the standard plane wave mode e±iνξ / 2π asymptotically far fromthe brane (ξ → ∞). More speciﬁcally, it is required that at inﬁnity
1 
A+ e−iνξ + A− eiνξ ,(3.37)(sinh ξ)−2/3 χν (ξ) ∼ √2πwith |A+ |2 + |A− |2 = 1. From this and the asymptotic behavior of the associated Legendre
functions,

Γ(iν)
1
Γ(−iν)−2e(−1/2+iν)ξ +e(−1/2−iν)ξ , (3.38)
P−1/2+iν (cosh ξ) ∼ √Γ(5/2 − iν)π Γ(5/2 + iν)√ Γ(iν − 3/2) (−1/2−iν)ξ
eπ,(3.39)
Q−2−1/2+iν (cosh ξ) ∼Γ(1 + iν)the normalization constant of the mode in the extra direction is determined as2 −1/2 Γ(iν) 2  Γ(−iν)
Γ(iν
−
3/2)
 +

− πC2
.
C1 = Γ(5/2 + iν)Γ(5/2 − iν)Γ(1 + iν) (3.40)

Now let us evaluate the amplitude of quantum ﬂuctuations. The amplitude of perturbations is often discussed in terms of the power spectrum deﬁned by0|φ̂k φ̂k |0 =(2π)3 (3) 

δ
k
+
k
Pφ (k).
4πk 3(3.41)In the present case φ̂k should be understood as the zero mode part of the graviton ﬁeld, andhence the power spectrum of four-dimensional gravitational waves in the braneworld is givenbyP(k) =

2k3· 3 |φ0 |2 ,2
2π M5

(3.42)

where the fundamental mass M5 appeared due to the normalization. Quantum ﬂuctuationsin the zero mode are stretched beyond the horizon radius by rapid expansion during inﬂation,and in the super-horizon regime we have |φ0 |2 ≈ C 2 (H)H 2 /2k 3 . Thus, 
2C 2 (H) H 2,(3.43)
P(k) =
2
2π
MPl

22
2 = M 3 . At low energies, we have C(H) ≈ 1, and we recover the standard
where we used MPl
5four-dimensional result. For high-energy inﬂation, we have

3H
P≈
2
MPl



H
2π

2
,

(3.44)

which is much greater than the standard result in four dimensions. The CMB observations
constrain the amplitude of the gravitational waves as P 1/2 < 10−5 , so that

H

MPl
M5

2
.

(3.45)

From this we see that even though the gravitational wave amplitude is enhanced, inﬂation atenergy scales as high as H  −1 is still allowed. In any case, exact de Sitter inﬂation on thebrane predicts the ﬂat spectrum of gravitational waves, in common with four-dimensional
general relativity.

Chapter 4

Leading order corrections to theevolution of tensor perturbations
The second Randall-Sundrum model (RS2) [124] with one brane in an anti de Sitter (AdS)
bulk is particularly interesting in the point that, despite an inﬁnite extra dimension, four
dimensional general relativity (4D GR) can be recovered at low energies/long distances on
the brane. What is leading order corrections to the conventional gravitational theory? It was
shown that the Newtonian potential in the RS2 braneworld, including the correction due tothe bulk gravitational eﬀects with a precise numerical factor [31, 35], is given byGmm
V (r)  −
r



22
1+ 2 .
3r(4.1)

Here  is the curvature length of the AdS space and is experimentally constrained to be   0.1
mm [99, 22]. It is natural to ask next what are leading order corrections to cosmologicalperturbations.
Great eﬀorts have been paid for the problem of calculating cosmological perturbationsin braneworld scenarios. In order to correctly evaluate perturbations on the brane, we needto solve bulk perturbations, which reduces to a problem of solving partial diﬀerential equations with appropriate boundary conditions. Hence it is not as easy as in the standard four
dimensional cosmology. A pure de Sitter braneworld [84] explained in the previous chapter,and its variations such as “junction” models [36, 61], which will be discussed in the next
chapter, and a special dilatonic braneworld [75, 77, 62], which will be the topic of Chapter 8,
are the rare examples where the bulk perturbations are analytically solved. In more genericcases, one has to resort to a numerical calculation [45, 46, 49, 50], or some approximate
methods [76, 27, 5, 78]. There is another diﬃculty concerned with a physical, fundamental aspect of the problem; we do not know how to specify appropriate initial conditions for
perturbations with bulk degrees of freedom.
In this chapter, we investigate cosmological tensor perturbations in the RS2 braneworld,and evaluate leading order corrections to the 4D GR result analytically. For this purpose,
we make use of the reduction scheme to a four dimensional eﬀective equation which iteratively takes into account the eﬀects of the bulk gravitational ﬁelds [139]. As a ﬁrst step, weconcentrate on perturbations which are initially on super-horizon scales.
The chapter is organized as follows. In Sec. 4.1 we deﬁne the setup of the problem
discussed in the present chapter more precisely. In Sec. 4.2 we brieﬂy review the derivation23

24
of the eﬀective equation of motion for tensor perturbations. In Sec. 4.3, using the longwavelength approximation we develop a method for evaluating leading order corrections, andevaluate the corrections considering slow-roll inﬂation and the radiation stage which follows
after inﬂation. The shortcoming of the long wavelength approximation is cured by introducing
another method suitable for analyzing the radiation phase in Sec. 4.4. Section 4.5 is devoted
to summary and discussion.

4.1

Setup of the problem and background cosmological model

In this chapter we investigate tensor perturbations in a Friedmann-Robertson-Walker (FRW)
braneworld model with an warped (inﬁnite) extra dimension [124, 9, 10, 111, 71, 51]. To
determine the spectrum of tensor perturbations observed at late time, we need to solve theperturbation equations with appropriate initial conditions. However, it is not an easy task to
specify what initial conditions we should use in the context of braneworld cosmology1 . At an
earlier stage of the evolution of the universe the physical wavelength of perturbation modes
is much shorter than the bulk curvature scale. Then, the correction to the four dimensional
standard evolution will not remain small. Therefore, in this regime it is not appropriate to
discuss the dynamics of perturbation modes under the assumption that a modiﬁcation to thefour dimensional standard one is small. Hence our strategy in this chapter is to isolate theissue of initial conditions from the rest. In this chapter we discuss a relatively easy part, i.e.,
the late time evolution after the eﬀect of the ﬁfth dimension becomes perturbative.
Our discussion will be restricted to the perturbation modes whose wavelength is initiallymuch longer than the Hubble horizon scale. We analytically derive a formula for small
corrections to the solution for the tensor perturbation equation due to the presence of theﬁfth dimension. We mainly focus on the behavior of a growing solution, since a growing
solution will be more important than a decaying one in general. Later we brieﬂy mention the
decaying solution in Sec. 4.5.
The background spacetime that we consider is composed of a ﬁve dimensional AdS bulk,
whose metric is given in Poincaré coordinates as
ds2 =2  2dz − dt2 + δij dxi dxj ,2zwith a FRW brane at z = z(t). Here  is the curvature length of AdS space. In the original
RS2 model [124], a brane is placed at a position of ﬁxed z, and Minkowski geometry is realizedon the brane. The induced metric on z = z(t) is
ds2 = a2 (t) −(1 − ż 2 )dt2 + δij dxi dxj ,(4.2)

a(t) = /z(t),where the overdot√denotes ∂t . From Eq. (4.2), we see that the conformal time on the brane
is given by dη = 1 − ż 2 dt. The brane motion is related to the energy density of matter
localized on the brane by the modiﬁed Friedmann equation [9, 10, 111, 71, 51] as
8πG
H =
3
2

1



ρ2
ρ+
2σ


,

The issue of initial conditions for perturbations will be addressed in a later chapter.

(4.3)

25
and
H :=

∂η a
−−1 ż
√=,a2
1 − ż 2(4.4)

where σ = 3/4πG2 is a tension of the brane. The quadratic term in ρ in the Friedmannequation modiﬁes the standard cosmological expansion law. At low energies (H1), thebrane motion is non-relativistic (ż1), and hence the conformal time on the brane η almost
coincides with the bulk conformal time t.
As a background FRW brane model, we consider slow-roll inﬂation at low energies followed
by a radiation dominant phase. During the slow-roll inﬂation the wavelength of perturbations
stays outside the Hubble scale, and it re-enters the Hubble horizon only during the radiationera.
We ﬁx a model of slow-roll inﬂation simply by providing a function
(η) := 1 −

∂η H
,
H2

(4.5)

where H := ∂η a/a = aH. During slow-roll inﬂation,  is supposed to be small. In the  → 0
limit, we recover the de Sitter case
a=

a0 η0
,
2η0 − ηH=

1,
2η0 − η(η < η0 ),(4.6)

where we have chosen the normalization of the scale factor and the origin of the η-coordinate
so that a = a0 and H = 1/η0 at η = η0 .
In the above alternatively we could specify the potential of the inﬂaton ﬁeld localized on
the brane. Then, for a given inﬂaton potential, the evolution of the background scale factor
would diﬀer from the conventional four dimensional one. However, as we will show below,
the correction to the tensor perturbations coming from the slow-roll inﬂation phase is always
small irrespective of the details of the evolution of the scale factor. Hence, we simply givetime-dependence of the slow-roll parameter to specify a model.
We assume that slow-roll inﬂation terminates at around η = η0 , and a radiation era
follows. During the radiation era the energy density decreases like ρ = ρ0 (a0 /a)4 as usual.
Then, solving Eq. (4.3) at low energies, we ﬁnd that the scale factor behaves like(0)a(η) =a (η) + δa(η) + O(4 ),withη
a (η) = a0 ,
η0

(0)


2

δa(η) = a0  H02(η > η0 ),

1
1
−
6 24



η0
η

3

1
−
8

(4.7)


η
η0


,

(4.8)

and H0 = (a0 η0 )−1 . Correspondingly, we have(0)H(η) = H (η) + δH(η) + O(4 ),with
1
H (η) = ,η(0)2
δH(η) = 2 3
6a η0



η0
η

3

(4.9)


−1 ,

(η > η0 ).

(4.10)

26
It is convenient to extend the deﬁnition of the slow-roll parameter  to the later epoch by
(0)

∂ηH

(η) := 1 − (0) .
H2

4.2(4.11)

Tensor perturbations on a FRW brane

We use the method to reduce the ﬁve-dimensional equation for tensor perturbations in a
Friedmann braneworld at low energies to a four-dimensional eﬀective equation of motion,
derived in Ref. [139]. Here we summarize the derivation.
Tensor perturbations on a Friedmann brane are given byds2 =2  2dz − dt2 + (δij + hij )dxi dxj .
2
z(4.12)We expand the perturbations by using Yijk (x), a transverse traceless tensor harmonics with
comoving wave number k, as hij = k Yijk (x)Φk (t, z). Then the equation of motion for thetensor perturbations in the bulk is given by


3
2
2
2
−∂z + ∂z + ∂t + k Φk = 0,
(4.13)z
Hereafter we will discuss each Fourier mode separately, and we will abbreviate the subscript
k. The general solution of this equation isΦ =dω Ψ̃(ω)e−iωt 2(pz)2 K2 (pz)
 pz 
(pz)2 (pz)4 3−iωt+− γ − ln
+ ··· ,
(4.14)1−=dω Ψ̃(ω)e416
42where p2 = −ω 2 + k 2 , and γ is Euler’s constant. We have chosen the branch cut of themodiﬁed Bessel function K2 so that there is no incoming wave from past null inﬁnity in thebulk. The coeﬃcients Ψ̃(ω) are to be determined by the boundary condition on the branenµ ∂µ Φ|z=z(t) = 0, where nµ is an unit normal to the brane, or, equivalently, by the eﬀective
Einstein equations on the brane [134],(4)Gµν = 8πG Tµν + (8πG5 )2 πµν − Eµν .

(4.15)(The derivation of this equation is summarized in Appendix. B.) Here πµν is quadratic in
the energy momentum tensor Tµν , and the ﬁrst two terms on the right hand side are totally
represented by the variables which reside on the brane. A projected Weyl tensor Eµν :=Cαµβν nα nβ represents the eﬀects of the bulk gravitational ﬁelds, giving rise to corrections tofour dimensional Einstein gravity in a fairly nontrivial way. In the present case, this can be
written explicitly, and the eﬀective four-dimensional equation reduces to 2∂η + 2H∂η + k 2 φ = −2E,
(4.16)

1−2E = (H)2 (∂t2 + ∂z2 ) − 2H 1 + (H)2 ∂t ∂z + ∂z2 − ∂z Φ,(4.17)zz=z(t)

27
where φ(t) := Φ(t, z(t)) is the perturbation evaluated on the brane. At low energies (H1),
we can use the approximation like


1
1
2
2
2
2
2
∂t Φ  ∂η Φ, ∂t ∂z Φ  − ∂η (∂η + k )Φ, ∂z Φ  −dω Ψ̃e−iωη p2  − (∂η2 + k 2 )Φ,2a2
2
neglecting higher order corrections of O(4 ). Then, using the lower order equation to eliminate
the higher derivative terms in −2E, we ﬁnally obtain our basic equation:


(0)
2
2
2
(4.18)∂η + 2 H ∂η + k φ  (0) (S0 [φ] + S1 [φ] + S2 [φ]) ,
a2
where
(0)

S0 [φ] := −2 a2
δHφ ,
2(0)(0) (0)(4.19)(0)S1 [φ] := (3 H 3 − 2 H H  )φ + k 2 H 2 φ,
  

p
1
+γ ,
S2 [φ] := −dωp4 φ̃e−iωη ln
22a(4.20)(4.21)and the prime denotes ∂η . The ﬁrst term arises due to the non-standard cosmological expansion included in H, while the last two terms are the corrections from the bulk eﬀects Eµν . We
can see that all terms are suppressed by 2 (or 2 ln ) 2 . S2 is essentially nonlocal because of
the presence of the log term. A time-domain expression for the quantity p4 φ which appearsin S2 can be rewritten, by eliminating the higher derivative terms in a similar way as before,
as




(0)
(0) (0)
(0)
(0)
(0)
4


3
2
2

φ,
(4.22)p̂ φ  −2 H − 6 H H + 4 H ∂η + 2k H − H
where we have introduced a derivative operator p̂2 = ∂η2 + k 2 .

4.3

New long wavelength approximation

Let us begin with the case that the wavelength of perturbations is much longer than the
Hubble scale. The long wavelength approximation not only allows us an easy treatment of
perturbations, but also turns out to be suﬃcient to derive all the major corrections except
for that coming from S0 . We would like to stress that what we call the long wavelength
approximation here is diﬀerent from a low energy expansion scheme or a gradient expansion
scheme in the literature [57, 58, 135].

4.3.1

General iteration scheme

The equation we are considering takes the form of


(0)
2
2
2
∂η + 2 H ∂η + k φ = (0) S[φ].
a22Hereafter, we will refer to both terms suppressed by 2 and by 2 ln  as O(2 ).

(4.23)

28
This can be solved iteratively taking 2 as a small parameter. For this purpose, we write
(0)

F (η)

φ(η) =φ (η)e

 η

(0)

f (η  )dη  .

=φ (η) exp(4.24)(0)

A zeroth order solution φ by deﬁnition satisﬁes the equation obtained by setting the righthand side of Eq. (4.23) to be zero. Then f obeys⎛∂η f + 2⎞(0)⎝ ∂η φ
(0)

2

(0)

+H⎠ f =

(0) (0)(0)S[φ ],(4.25)a2 φ

φwhich can be integrated immediately to givef (η) =2 η(0) (0)(0)(0)

dη  φ S[φ ].

(4.26)

a2 φ 2Integrating this expression, we obtain the ﬁrst order correction F . In the present case, S
consists of three parts, S0 , S1 and S2 . We hereafter denote the corrections f and F comingfrom Si as fi and Fi , respectively.

4.3.2

Long wavelength expansion of the zeroth order solution

In the long wavelength approximation, we can write the zeroth order solution explicitly as
(0)

φ
(0)

∂η φ


 η
 Ak 1 − k 2dη  I(η  ) ,(4.27)

 −Ak k 2 I(η),(4.28)

where Ak is an amplitude of the perturbation which can depend on k, and(0)I(η) := a −2 (η)

 η

(0)

a 2 (η  )dη  .

(4.29)−∞

Here we simply keep the terms up to O(k 2 ).
We have already introduced a parameter (η) in Eq. (4.11), which parameterizes thecosmological expansion. In the inﬂationary universe, it reduces to a slow-roll parameter andis assumed to be small. Slow-roll inﬂation assumes that time diﬀerentiation of  is O(2 ).
Hence, we treat  as a constant during slow-roll inﬂation. The transition from inﬂation to theradiation stage occurs at around η = η0 , and the behavior of  there depends on the detailsof the reheating process. During the radiation stage,  keeps again a constant value,  = 2,(0)because the scale factor behaves like a ∝ η. In general  reduces to a constant for the scale(0)

factor a proportional to a power of the conformal time.
Using  and the zeroth order solution in the long wavelength approximation, we can write

29
(0)

(0)

S1 [φ ] and p̂4 φ as
(0)

(0)

S1 [φ ] = H

(0)

p̂4 φ



(0)

(0)

2

(0)

(1 + 2) H ∂η φ + k 2 φ

(0)
(0)
(0)
 Ak k 2 H 2 1− H I − 2 H I ,

(0)
(0)
(0)
(0)
2
 − 2(1 + ) H ∂η φ − 2k 2 φ
 2H


(0)
(0)
2
2

 −2Ak k H  I − 2 (1 + ) H I − 1 ,(4.30)(4.31)

which are generic expressions not relying on any speciﬁc time dependence of the scale factor.
Equation (4.31) can be further rewritten into a more suggestive form. First, integrating
(0)

(0)

(0)

the identity ∂η (a 2 / H ) = (1 + ) a 2 , it can be shown that
(0)

H

(0)

1− H I = (0)
a2

 η

(0)

 a 2 dη  .

−∞Then, integration by parts leads(0)H(1 + ) H I − 1 = (0)a2(0)

and thus we have

⎛

 η

(0)

(4.32)−∞(0)H
p̂4 φ  −2Ak k 2 H 2 ⎝ I − 2(0)
a2
(0)

(0)

 a 2 Idη  ,

 η

⎞
(0)

 a 2 Idη  ⎠ .

(4.33)−∞

Since  is of second order in the slow-roll parameters during slow-roll inﬂation, we can safely
neglect this term until the end of inﬂation. Moreover, for suﬃciently smooth  this term is
expected to be small, and  = 0 for the scale factor proportional to a power of the conformal
time. Only at the time when the equation of state of the universe changes abruptly, i.e., ata sudden transition from inﬂation to the radiation stage,  possibly becomes signiﬁcantly
large like a delta function. After such a violent transition, the integral in Eq. (4.33) leaves
a constant value and thus this second term decreases in proportion to the damping factor
(0)

(0)

H / a 2.

4.3.3

Slow-roll inﬂation

Now let us consider slow-roll inﬂation on the brane. In this case S0 vanishes, since δH = 0 by
(0)construction. In the previous section we have also shown that p̂4 φ vanishes at ﬁrst order inthe slow-roll parameters during inﬂation. Consequently, the only relevant term in the present
situation is S1 .
Equation (4.30) together with Eq. (4.32) tells us that(0)

S1  −Ak k 2 H 2 ,

(4.34)

30
and thus η

k 2 2f (η)  − (0)
a2
(0)

(0)

(0)

(0)

dη   H 2 .

(4.35)(0)

Using the relations H a /a0 η0 and dη = a0 η0 d a / a 2 , and neglecting the time variation of
the slow-roll parameter, we obtain
k 2 2aia,(4.36)f (η)  (0) −a0 a0
η0 a 2where ai = a(ηi ) and ηi is the lower boundary of integration. Integrating this expression, we
have
⎡  
 2 ⎤
3
1 ai ⎦
k 2 2 ⎣ 1 ai−
F (η)  − 2 .
(4.37)
3 (0)
2 (0)
ai
a
a
We chose the integration constant so that F vanishes in the limit of a(η) → ∞. This meansthat we have renormalized Ak so that it represents the amplitude of ﬂuctuations that we seeat late times if slow-roll inﬂation lasts forever.
Although our present long wavelength approximation is not valid when the wavelength
ai k −1 becomes comparable or shorter than the Hubble scale H, we can still arrange ai to be
suﬃciently small so that we can neglect the ﬁrst term in Eq. (4.37). For such ai , we have an
expression independent of ai ,F (η) ≈

k 2 2(0),(4.38)

2 a2

and we do not need to care about the choice of the “initial time” ηi . We see that the correction
arising during inﬂation is very tiny, suppressed by the slow-roll parameter  in addition to(0)
the factor k 2 2 / a 2 . For pure de Sitter inﬂation there is no correction, as is expected.

4.3.4

Transition to the radiation stage

In this section, we investigate the eﬀects of the transition from the inﬂation stage to theradiation stage. First, just for the illustrative purpose, we plot the behavior of S1 and S2 forthe modes well outside the horizon in the neighborhood of the transition time η0 in Fig. 4.1
and Fig 4.2. These plots are for  given by(η) = tanh[(η − η0 )/s] + 1,

(4.39)

where the parameter s controls the smoothness of the transition. From the ﬁgures it can beseen that the corrections from Eµν become signiﬁcant only around the transition time.
From now on we shall consider the limiting case where  is given by (η) = 2θ(η − η0 ).
Here we neglected the tiny eﬀect of the non vanishing slow-roll parameter during inﬂation.
For this instantaneous transition model, the scale factor and the comoving Hubble parameter
(0)

(0)

for η < η0 are given by Eq. (4.6), while for η > η0 they are given by a and H in Eqs. (4.8)
and (4.10).

31

Figure 4.1: Behavior of S1 around the transition time with the vertical axis in an arbitrary
unit. Red solid line shows the case of a sharp transition with s = 0.02η0 , while blue dashedline represents the case of a smooth transition with s = 0.5η0 . The wavelength of the modeis chosen to be kη0 = 0.01.

Figure 4.2: Behavior of S2 around the transition time with the vertical axis in an arbitrary
unit. Red solid line shows the case of a sharp transition with s = 0.02η0 , while blue dashedline represents the case of a smooth transition with s = 0.5η0 . The choice of the parameteris the same as in Fig. 4.1, kη0 = 0.01.

32
Assuming such a step function-like transition, we can evaluate the corrections from S0 at
η > η0 as




η03
η03
k 2 2 η η  dη 
1
−
1
+
2
,
(4.40)f0 (η)  − (0)3η3η3
9 a 2 η0 η0 2  5 
k 2 2η0
η0η
−
5 −9+5.
(4.41)F0 (η)  −
2
η0
η
η
90a0
Here the integration constant is chosen so that the correction vanishes before the transition.
From this expression, we ﬁnd that F0 diverges for a large η, and hence the correction to theﬁnal amplitude of ﬂuctuations from S0 looks inﬁnitely large. However, this is an artifact of
the long wavelength approximation. At a late time during the radiation era the zeroth ordersolution becomes oscillatory. This oscillation suppresses the late time contribution. However,in the long wavelength approximation, this oscillation is not taken into account. In the
succeeding section, we will develop another method which takes into account this oscillation.
There we will ﬁnd a ﬁnite correction.
As for S1 , we have



η03
2 k 2 2 η dη 
1 + 5 3 ,(4.42)
f1 (η)  − (0)3 a 2 η0 η 2
η

 
 
  
1 η0 2 1 η0 5
2 k 2 2 3 9 η0−+.
(4.43)+
F1 (η)  −3 a20 2 4 η
2 η
4 η
Long time after the transition (but of course before horizon re-entry), the correction becomestime independent,
lim F1 (η) = −η→∞

k 2 2.
a20(4.44)The next step is to evaluate the correction coming from S2 . This can be done as follows.
(0)First, for the instantaneous transition p̂4 φ becomes


(0)
η03
η3
4
2 1
2
δ(η − η0 ) − 4θ(η − η0 ) 5 = −4Ak k ∂η θ(η − η0 ) 04 .
p̂ φ  −4Ak kη0ηη(4.45)(0)Then, integrating by parts, the Fourier transform of p̂4 φ is obtained as∼(0)
2Ak k 2
η3

4
p φ =
iω dη  θ(η − η0 ) 04 eiωη
π
η
2
2Ak k
iωeiωη0 q(ωη0 ),
=
π
with
 ∞
eiωη0 (y−1)
dy.
q(ωη0 ) :=y4
1
We separate S2 into two parts,



 
 2

∼
(0)
(0)
k
1
η0
1
p
4
−iωη
4
γ
+
ln
p̂
−
+
ln
S2 = −lnφ,dωp φ e
4
k2
2
2a0
η

(4.46)(4.47)

33
and, by substituting the expressions obtained above, we have 2
Ak k 2p−iωTdω q(ωη0 )∂η e
ln
S2 =
2π
k2



 

k
η0
η3
2
+2Ak k γ + ln
+ ln∂η θ(T ) 04 ,2a0
ηη

(4.48)

where T := η − η0 . Then, it can be integrated to give
 

 2
k 2 2 1p−iωTdω q(ωη0 ) eln
f2 (η) = (0)k2a 2 2π 

 
 η
η0
η03
k
2 η3
+ ln
θ(T ) 4 ++2 γ + lndη  θ(η  − η0 )  04 . (4.49)
2a0
η
η
η η
−∞
The branch cuts of log function and the path of ω integration in the ﬁrst term should bechosen so that the retarded boundary condition is ensured, which are presented in Fig. 4.3.
The integration is dominated by ω 2 ≈ k 2 , where we may approximate q(ωη0 ) ≈ q(kη0 ) ≈ 1/3
because kη0 is assumed to be small. Then, the ﬁrst term can be written by using the formula
presented in Sec. 4.6, and can be integrated to give a part of F2 (η) as
2cos[k(η  − η0 )] − 1
2k 2 2 η
 η0

θ(η
dη
−
η
)
−
0
η 2
η  − η0
3a20 −∞+∂η [θ(η  − η0 ) ln(k(η  − η0 ))] + γδ(η  − η0 )



η02
2k 2 2 η02
ln(k(η − η0 )) + ln(kη0 ) 1 − 2−η
3a20 η 2η − η0ln(η/η0 − 1)
η0
+γ
+ ln
−
−1 +
η
η
(η/η0 )2
2k 2 2[γ − 1 + ln(kη0 )] ,
−→ −
η→∞
3a20
where we have neglected the term proportional to cos[k(η  − η0 )] − 1 in the integrand because
this term brings the higher order contributions in k. The other part of F2 (η) is obtained byintegrating the remaining parts in Eq. (4.49) as






η05
k
1
1 η05
ln(η/η0 )2k 2 2
1− 5 +
γ + ln
−1+5
2
5
2a0
5
η
25 η
(η/η0 )5
a0




η0
1
η5
1
1−
−
1 − 05
+
4
η
20
η

 

2
2
k
1
4
2k 
γ + ln
.
+
−→
2
η→∞ a
5
2a0
25
0
Combining the above two, we ﬁnally obtain the correction from S2 ,






1
2k 2 2 37
2
k
H0
+
−
γ
+
ln
ln
.
lim F2 (η) =
η→∞
75 15
a0 H0
5
2
a20

(4.50)

34

Figure 4.3: Branch cuts and the contour of the integration in Eq. (4.49) for T > 0. For T < 0
we close the contour on the upper half plane.

4.4

Corrections in the radiation stage

So far we have worked in the long wavelength approximation, and found k-dependent small
corrections to the spectrum of tensor perturbations due to S1 and S2 , which correspond to Eµνterm. It might be possible, however, that further corrections arise after the mode re-enters
the horizon. We will show that such corrections are highly suppressed. A basic observation
supporting this conclusion is that the contributions from Eµν term in the right hand sideof Eq. (4.18) become signiﬁcantly large only at the transition time. It decreases in powersof a(η) after the transition, and will be negligible when the long wavelength approximation
brakes down.
In the long wavelength approximation, we could not obtain a meaningful estimate forthe leading order correction caused by unconventional cosmic expansion, S0 . Here we will
give a more rigorous treatment, and will resolve this drawback of the approach taken in thepreceding section.
In the radiation stage,u(η) := a(η)φ(η),(4.51)
is a convenient variable. In terms of this new variable, we can rewrite the equation of
motion (4.18) asu + k 2 u =2 S̄0 + S1 + S2 ,awith

(4.52)au.
(4.53)
2
If we neglect the deviation of the expansion law from the standard one, we have a ∝ η in theradiation stage and so a = 0. This means that the ﬁrst term on the right hand side gives
a contribution of O(2 ) from the modiﬁcation of the expansion law as S0 in the preceding
S̄0 =

35
section. Thus, all the terms collected on the right hand side are of O(2 ), and at zeroth orderu behaves just like a harmonic oscillator. The zeroth order solution is expressed by a linear
combination of the standard modes:
 −ikη(0)eeikη
u (η) = Ak α √+ β√,(4.54)
2k
2kwhere the coeﬃcients α and β are determined by matching the solutions at the onset of theradiation stage.
We deﬁne “energy” of the harmonic oscillator asE :=1   2 k 2 2
u + |u| .
22(4.55)

Then, this energy is conserved at the zeroth order, i.e.,
(0)

E =

k
(|α|2 + |β|2 )|Ak |2 = const.
2

Taking into account the corrections of O(2 ), the total variation of the energy can be estimated
as
2
2  ∞


∆E =
∆Ej =
Ej dη,(4.56)
j=0

j=0

η0

with
E0 =
Ej

=

2  ∗(u ) S̄0 + c.c.,2a2  ∗(u ) Sj + c.c.,2afor j = 1, 2.
(0)(4.57)(0)

The amplitude of the oscillator |Ak |eRe[F ] is proportional to E 1/2 a /a. The factor a /a here
is to be attributed to unconventional cosmic expansion. Hence we have
#
"
∆E0
δa(η)
=
,
lim Re[F0 (η)] + (0)
(0)
η→∞
a (η)
2E
∆Ej
,
for j = 1, 2.
(4.58)
lim Re[Fj (η)] =
(0)
η→∞
2E

4.4.1

Corrections due to the unconventional expansion law

We start with computing the correction due to the unconventional cosmic expansion, F0 ,which we could not determine in the long wavelength approximation. To do so, we ﬁrst
need to determine the coeﬃcients in Eq. (4.54) by requiring that u(0) and its derivative arecontinuous across the transition from the de Sitter stage to the radiation stage. We assume
for a moment that the perturbation in the initial phase is given by the de Sitter growingmode,k 2 (2η0 − η)2
Ak a0 η01+.
(4.59)v(η) 
2η0 − η2

36
Solving(0)v(η0 ) = u (η0 ),(0)∂η v(η0 ) = ∂η u (η0 ),we obtain the coeﬃcients as

ia0
2i
α √
1 − (kη0 )3 ,3η0 2k

β = α∗ .

(4.60)The energy of the harmonic oscillator at zeroth order is approximately given by
(0)

E 

|Ak |2 a20.
2η02(4.61)

Next we integrate
E0 =
=

i a
(|α|2 − |β|2 − αβ ∗ e−2ikη + α∗ βe2ikη )|Ak |2 + c.c.
4 aaIm[αβ ∗ e−2ikη ]|Ak |2 .
a(4.62)The scale factor correct up to O(2 ) is given in Eq. (4.7). Using this expression, we have
2 H02 η04
a
−
.
a
2 η6

(4.63)

Then, integrating Eq. (4.62), we have
∆E0
(0)

2E


 ∞
2 H02 η02
η04 −2ikη
∗
 −
Im
αβ
dη
e
η6
2a20
η0


2


1
2 H02
i
2i
2
2i
3
2
3
4

− kη0 − (kη0 ) + (kη0 ) + O (kη0 )Im 1 − (kη0 )
4kη0
3
5 2
3
3
2
2
1 k 
1
.
(4.64)
 − 2 H02 +8
10 a20The ﬁrst term in the last line is not suppressed for the k → 0 limit. This is because this termarises due to the asymptotic behavior of δa(η). In fact one can see from Eq. (4.58) that theexpression for the physical amplitude of the perturbations eF0 does not have this term, andﬁnally we have1 k 2 2.
(4.65)
lim Re[F0 (η)] =η→∞
10 a204.4.2

Suppression at sub-horizon scales

Here we shall show that corrections from the Eµν term are suppressed at sub-horizon scales
and hence the result obtained in the preceding section by using the long wavelength approximation suﬃces for our purpose.

37
(0)

(0)

(0)

(0)

Substituting φ =u / a into S1 [φ ], energy gain or loss due to this term is obtained as



2
5
k 2 (0)  ∗ (0)5 (0)  2uE1 = (0)||+−+(u ) u + c.c..
(4.66)
3
4
2
η
η
η
2
a
2
(0)

(0)

(0)

Note that | u  |2 and (u  )∗ u consist of just constant parts and oscillating parts with constant
amplitudes. Therefore, it is manifest that when we integrate E1 the dominant contribution
comes from around the lower boundary of the integral, η ≈ η0 , and so corrections arisingin the sub-horizon regime are suppressed compared to those imprinted beforehand in the
super-horizon regime.
A similar expression for E2 can be obtained as
 2

∼
(0)
2 (0)  ∗
p

4
−iωη
ln
E2 = − (0)(u )dωp φ e
2
k
8a

  


3 (0)  2k3k 2 (0)  ∗ (0)
22
u
+
γ
(u ) u + c.c.. (4.67)
|
|
+
−
+
+ (0) ln
(0)
η3
η4 η2
a2
2a
The second term is local and suppressed for large η for the same reason above. To examine
the late time behavior of the ﬁrst term, we substitute the expression(4.54) without limitation$) 4−iω(η−ηη > η0 . This approximation is justiﬁed because the kernel dωep ln(p2 /k 2 ) decays3
at least as fast as 1/(η − η ) for a large time separation, as will be shown in Sec. 4.6. Under
this approximation for φ(η), it is easy to calculate its Fourier transform3⎧∼⎨ −(α + β) for ω ≤ −k,
(0)
iη0 1
√ ×
(α − β)for |ω| ≤ k,φ (ω) =
(4.68)
2a0 2k ⎩(α + β)for ω ≥ k.
The contour of the ω-integration in the ﬁrst term of Eq. (4.67) is such shown in Fig. 4.3, andhence it can be separated into three parts as(0)
k 2 k 2 2
−k−i∞
−k
k
I
(4.69)
The ﬁrst term in (4.67) = −(u  )∗
+
I
+
I
kη
0
→k−i∞ ,→−k→k
16 a (0)a0
where−k−i∞ ∗
kI→k−i∞= (I→−k
)

−k
I→k



e−ikη ∞πi,
= (α + β) √dy(y 2 + 2iy)2 e−kη·y ln(2y − iy 2 ) −
2
2k 0
 1
i
= (α − β) √dy(1 − y 2 )2 e−ikη·y ln(1 − y 2 ).
2k −1

(4.70)(4.71)

We can see that all Is are suppressed for large values of kη, since their integrand becomes a
product of a sooth function and a rapidly oscillating function. The dominant contribution in
the integration of E  therefore comes from around the lower boundary of the integral.
Based on the above analysis, we conclude that all the dominant corrections arise in the
super-horizon regime, and thus it is suﬃcient to evaluate the corrections in the long wavelength approximation.
3

Depending on the choice of the integration path near η = 0, the Fourier transform of φ(η) changes by aconstant independent of ω. Here we took the principal values.

38

4.5

Summary

We have investigated leading order corrections to the tensor perturbations in the RS2 braneworld
cosmology by using the perturbative expansion scheme of Ref. [139]. We have studied a modelcomposed of slow-roll inﬂation on the brane, followed by a radiation dominant era. The unperturbed ﬁve dimensional bulk is AdS space with curvature radius . In our expansion
scheme the asymptotic boundary conditions in the bulk are imposed by choosing outgoingsolutions of bulk perturbations, whose general expression is known in the Poincaré coordinate system. Hence, the issue of bulk boundary conditions is handled without introducing
an artiﬁcial regulator brane. This is one of the notable advantages of the present scheme.
We set the initial condition when the wave length ak −1 is already longer than the Hubble
radius during slow-roll inﬂation. We will not lose much by neglecting the modes which are
already inside the Hubble scale at the time of transition to the radiation era, since they are notcosmologically so interesting. We compared the resulting amplitudes of ﬂuctuations after thehorizon re-entry with and without the eﬀect of an extra dimension. To do so, we normalized
the amplitude so that the late time amplitude of ﬂuctuations becomes identical in two cases if
the slow-roll inﬂation lasts forever. As the reference without the eﬀect of an extra dimension,
we took a growing mode solution. The corrections due to gravity propagation through theﬁfth dimension dominantly come from the contribution around the transition time. Therefore
they can be estimated by using the long wavelength approximation with a suﬃcient accuracy.
Since the correction due to the modiﬁed cosmic expansion comes from relatively late epoch,
it is necessary to take into account the oscillatory behavior of the solution after the re-entry
to the Hubble horizon.
Combining the results obtained in Eqs. (4.44), (4.50) and (4.65), we ﬁnd that the amplitude of a ﬂuctuation with comoving wave number k is modiﬁed by a factor eRe[F ] due to theeﬀect of an extra dimension with






2
4
k
H0
k 2 2 13+ ln−
γ + ln,(4.72)
lim Re[F (η)] = 2
η→∞150 15a0 H0
52a0
where a0 and H0 are the scale factor and the Hubble parameter at the transition time, andγ is the Euler’s constant. Leading corrections are proportional to k 2 2 /a20 or k 2 2 /a20 log k,
as is expected from the dimensional analysis. However, our calculation here determined theprecise numerical factors analytically.
Throughout this chapter, we have dropped the contribution from the decaying mode forsimplicity. Since the decaying mode during the initial slow-roll inﬂation phase is usually
suppressed when the wavelength is longer than the Hubble scale, it does not give a signiﬁcanteﬀect. However, in the context of braneworld, we have not yet understood how to ﬁx theinitial conditions for ﬂuctuations. Therefore, in principle, there is a possibility that thecontamination of the decaying mode can be extremely large, although it seems quite unlikely.
It is quite easy to estimate the leading order correction due to the decaying mode. The
eﬀect is not due to the non-trivial evolution of a solution but totally due to unconventional
initial condition. Adding a decaying mode will modify v given in Eq. (4.59) toAk a0 η0k 2 (2η0 − η)2+ Cd k 3 (2η0 − η)3 ,v(η) 
1+
2η0 − η2where Cd is a complex number which parameterizes the amplitude of the decaying component.
Repeating a similar calculation leading to Eq. (4.65), we ﬁnd that the relative change in the

39

Figure 4.4: Branch cut and the contour of the integration in Eq. (4.76).
amplitude due to the decaying mode is given by
Re[Cd ]

2 H02(kη0 )3 .
2(4.73)

Thus the correction due to the decaying component is more suppressed in the sense of thepower of k, but it might be more important than the other corrections for kη0 > Re[Cd ]−1 if
Re[Cd ] is extremely large.
Lastly we would like to mention the limitation of the present work. The perturbative
expansion scheme that we have adopted in this chapter is valid for scales larger than the
AdS curvature length (k/a1) at low energies (H1); besides our study is restricted
to initially super-horizon perturbations. We gave an initial condition for a solution by hand,but of course such an approach is not satisfactory. To determine the initial condition, we
need to investigate the evolution of perturbations at the initial phase where the wavelength ofthe mode is too short and reduction to an eﬀective four dimensional problem is not possible.
Furthermore, to seek for interesting eﬀects that might arise at high energies (H  1), we
have to develop a new formalism, which is a next challenge and will be addressed in the restof this thesis.

4.6

Appendix: Details of calculations

In this appendix, we will show the following formula:

dωe

−iωT


 2
cos(kT ) − 1p+ ∂T [θ(T ) ln(kT )] + γδ(T ) ,= −4π θ(T )
ln
2
k
T

(4.74)

where p2 = k 2 − ω 2 and γ is Euler’s constant. The branch cuts are all running on the lower
half complex plane of ω (Fig. 4.3), which ensures the retarded boundary conditions.

40
We divide the above integration into two parts:
 2 
 2 



p
−p
−ω 2
−iωT
−iωT
−iωT
ln
ln
ln
dωe
= dωe
+ dωe
.
k2
ω2
k2

(4.75)

The ﬁrst term of the right hand side can be rewritten as 2 

−p
−iωT
dωe
= dωe−iωT [ln(ω − k) + ln(ω + k) − 2 ln ω] .
ln
ω2
The integration can be performed as
 0

−iωT
dωe
ln(ω − k) = θ(T )(−i)dre−iT (k−ir) ln(re3πi/2 )∞ ∞(−i)dre−iT (k−ir) ln(re−πi/2 )
+
0

e−ikT
,
= −2πθ(T )T
and the other two integrals are obtained by setting k → −k and k → 0. Thus we have 2cos(kT ) − 1−p−iωT= −4πθ(T )
.
ln
dωe
2
ω
T
This expression is regular at T = 0. To evaluate the second term in the right hand side ofEq. (4.75), we need a trick because the integrand suﬀers from a slow fall-oﬀ at large valuesof |ω|. It should be interpreted as





−iω
e−iωT
e−λ
ln
= 2i∂T θ(T )(ln(−λ) − ln(kT )) ,(4.76)
2i∂T
dω
dλ
ω
k
λ
where we put λ = iωT . Now the branch cut and the contour of the integration on λ plane
are such shown in Fig. 4.4. Thanks to this trick, the expression for the second term becomesa well-deﬁned one, though the result does not change as long as convergence is guaranteed.
Then, performing the integration ase−λ(ln(−λ) − ln(kT )) = lim ∂ν dλe−λ (−λ)ν−1 + 2πi ln(kT )
dλ
ν→0
λ


= lim ∂ν (e−πiν − eπiν )Γ(ν) + 2πi ln(kT )ν→0= 2πi [ln(kT ) + γ] ,we ﬁnally obtain the formula (4.74).
$
For T > 0 the formula (4.74) reduces to dωe−iωT ln p2 /k 2 = −4π cos(kT )/T and hence
the kernel that appears in S2 becomes
 2

 2

k
3k
p
3
−iωT 4= 32πcos(kT ) − 4 sin(kT ) ,p ln−
for T > 0. (4.77)
dωe
k2
T3 T5
T
From this expression we ﬁnd that the non-local source S2 does not keep the past history fora long time.

Chapter 5

The junction model
In this chapter we study gravitational waves from an inﬂating brane. If the Hubble parameteron the brane is constant, the power spectrum becomes scale-invariant [84], as was explained in
Chapter 3. However, the Hubble parameter usually changes even during inﬂation. The changeof the Hubble parameter, i.e., the nontrivial motion of the brane in the ﬁve-dimensional bulk,“disturbs” the graviton wave function. As a result, zero mode gravitons, which correspondto the four-dimensional gravitational waves, are created from vacuum ﬂuctuations in the
Kaluza-Klein modes as well as in the zero mode1 . It is also possible that gravitons initially inthe zero mode escape into the extra dimension as the “dark radiation” [111, 42, 88, 90, 91].
Therefore we expect that the nontrivial motion of the brane may leave characteristic featuresof braneworld inﬂation. If so, it is interesting to search for a signature of the extra dimension
left on the primordial spectrum. However, there is a technical diﬃculty. When the Hubble
parameter is time dependent, the bulk equations are no longer separable. Then we have tosolve a complicated partial diﬀerential equation. To cope with this diﬃculty, we consider asimple model in which two de Sitter branes are joined at a certain time; namely, we assume
that the Hubble parameter changes discontinuously. In this model we can calculate the power
spectrum almost analytically. This is a milder version of the transition described in the workby Gorbunov et al. [36], in which they considered a simpliﬁed inﬂation model in which deSitter stage of inﬂation is instantaneously connected to Minkowski space and hence it ispossible to solve the perturbation equations including the bulk to some extent. Thus we willclosely follow the same technic as Ref. [36].
This chapter is organized as follows. In the next section we describe the setup of our
ﬁve-dimensional model, and explain the formalism introduced in Ref. [36] to solve the mode
functions for gravitational wave perturbations. Using this formalism, we explicitly evaluate
the Bogoliubov coeﬃcients in Sec. 5.2. In Sec. 5.3 we translate the results for the Bogoliubov
coeﬃcients into the power spectrum of gravitational waves, and its properties are discussed.
We show that the power spectrum for our ﬁve-dimensional model can be reproduced withgood accuracy from that for the corresponding four-dimensional model by applying a simple
mapping. Section 5.4 is devoted to conclusion.

1

The situation here is quite similar to particle production by a moving mirror [12, 36]

41

42

5.1

Preliminaries

5.1.1

Background metric, perturbations, and mode functions

Let us start with the simple case in which the background is given by a pure de Sitter
brane in AdS5 bulk spacetime [84]. The situation here is exactly the same as what was
introduced in Chapter 3. We solve the ﬁve-dimensional Einstein equations for gravitational
wave perturbations. For this purpose, it is convenient to use a coordinate system in which
the position of the brane becomes a constant coordinate surface. In such a coordinate system
the background metric is written as

1
2
ds =
(−dη 2 + δij dxi dxj ) + dξ 2 ,2sinh ξ η 2
2

(5.1)

where  is the bulk curvature radius, and the de Sitter brane is placed at ξ = const. = ξb .
Note that here η is supposed to be negative. On the brane, the scale factor is given bya(η) = 1/(−ηH) and the Hubble parameter is
H = −1 sinh ξb .

(5.2)

Note that under the coordinate transformations
t = η cosh ξ − η0 cosh ξb ,z = −η sinh ξ,(5.3)

with a constant η0 , the metric (5.1) becomes the AdS5 metric in the Poincaré coordinates.
The metric with gravitational wave perturbations is written as2
ds =sinh2 ξ
2




1 
2
i
j
−dη
+ dξ 2 .
+
(δ
+
h
)
dx
dx
ij
ij
η2

(5.4)

We decompose the transverse-traceless tensor hij into the spatial Fourier modes as√2
hij (η, x, ξ) =(2πM5 )3/2


d3 p φ(η, ξ; k)eik·x eij ,(5.5)

where eij is the polarization tensor, and the summation over diﬀerent polarization was supis related to the fourpressed. M5 represents the ﬁve-dimensional Planck mass,√ and it3/2
2 . The factor
dimensional Planck mass MPl by M53 = MPl
2/(M5 )
is chosen so that theeﬀective action for φ corresponds to the action for the canonically normalized scalar ﬁeld.
Then, the Einstein equations for the gravitational wave perturbations reduce to the KleinGordon equation for a massless scalar ﬁeld, 2φ = 0, in AdS5 . We assume Z2 -symmetry acrossthe brane. Assuming that anisotropic stress is zero on the brane, Israel’s junction condition
gives the boundary condition for the perturbations as∂ξ φ|ξ=ξb = 0.

(5.6)

Since the equation is separable, the mode functions are given in the form of φν (η, ξ) =ψν (η)χν (ξ), and they are found in Chapter 3.

43
As will be seen, we need to evaluate the value of the wave function at the location of
the brane χν (ξb ), and in some special cases χν (ξb ) reduces to a rather simple form. For
sinh ξb
1 and ν sinh ξb
1, we have(χν (ξb ) ≈)ν tanh πν
2ν 2 + 1/4
(sinh ξb )2 ,
ν 2 + 9/4

(5.7)

while, for sinh ξb  1 or ν sinh ξb  1, we have1χν (ξb ) ≈ √ (sinh ξb )3/2πν
ν 2 + 9/4

.

(5.8)

For the derivation of these two expressions, see Ref. [36].

5.1.2

Model with a jump in the Hubble parameter

We consider a model in which the Hubble parameter changes during inﬂation. As we have
explained, in the case of constant Hubble parameter the brane can be placed at a constantcoordinate plane. When the Hubble parameter varies, we need to consider a moving brane in
the same coordinates. For simplicity, we consider the situation in which the Hubble parameterchanges discontinuously at η = η0 from H1 to
H2 = H1 − δH.

(5.9)

Here δH/H1 is assumed to be small. For later convenience, we deﬁne a small quantity H by
H==

1 + (H2 )2 − H2 1 + (H1 )2
H2




δH 2
1
2 + 3(H1 )2
δH 3
δH
+
+O
.
H1
2[1 + (H1 )2 ]3/2 H1
1 + (H1 )2 H1

H1(5.10)

To describe the motion of the de Sitter brane after transition, it is natural to introduce a
˜ deﬁned by
new coordinate system (η̃, ξ)
t = η̃ cosh ξ˜ − η̃0 cosh ξ˜b ,˜z = −η̃ sinh ξ.

(5.11)

Then, the brane expanding with Hubble parameter H2 is placed at ξ˜ = ξ˜b by choosing two
constants ξ˜b and η̃0 so as to satisfy H2 = −1 sinh ξ˜b and η0 sinh ξb = η̃0 sinh ξ˜b . The trajectoryof the brane is shown in Fig. 5.1. Apparently, mode functions in this coordinate system takethe same form as those in Chapter 3, but the arguments (ξ, η) and the Hubble parameter˜ η̃) and H2 . We refer to these second set of modes as φ̃0 and φ̃ν . The
H1 are replaced by (ξ,˜ isrelation between (η, ξ) and (η̃, ξ)*η̃ = − η 2 + 2H η0 η cosh ξ + 2H η02 ,tanh ξ˜ = (η cosh ξ + H η0 )−1 η sinh ξ.

(5.12)

44

Figure 5.1: Trajectory of the brane (thick solid line) in static coordinates. Dashed (respectively dotted) lines represent surfaces of ξ = constant (respectively ξ˜ = constant).
As explained above, the variation of the Hubble parameter is assumed to be small. For atechnical reason, we further impose a weak restriction that the wavelength of the perturbations concerned is larger than δH/H 2 . These conditions are summarized as follows:δH
H
δH
k|η0 |
H

5.1.3

1,

(5.13)

1.

(5.14)

Method to calculate Bogoliubov coeﬃcients

We consider the graviton wave function ϕ that becomes the zero mode φ̃0 at the inﬁnite
future η̃ = 0. We write the wave function ϕ as
ϕ = φ̃0 + δϕ,(5.15)

where the second term δϕ arises because φ̃0 does not satisfy the boundary condition for t < 0.
Writing down φ̃0 and δϕ at the inﬁnite past, η = −∞, as a linear combination of φ0 , φν , andtheir complex conjugates, we can read the Bogoliubov coeﬃcients.
At η → −∞ the ﬁrst term φ̃0 is expanded asφ̃0 −→(U0M φM + V0M φ∗M ) ,(5.16)
η→−∞M =0,ν

45
where the summation is taken over the zero mode and the KK modes. The coeﬃcients U0ν
and V0ν are written by using the Wronskian [Eq. (3.31)] aslim φ̃0 · φM ,
U0M =
η→−∞


(5.17)
V0M = − lim φ̃0 · φ∗M .
η→−∞Evaluation of the Wronskian at η = −∞ leads to V00 = V0ν = 0 [36], while
H2 C(H2 ) −i H kη0 cosh ξbe
1 − iH kη0 C 2 (H1 ) − cosh ξb
U00 ≈H1 C(H1 )1− (H kη0 )2 sinh2 ξb ,2
U0ν

≈ H kη0−2ieiπ/4 H2
C(H2 )χν (ξb ).
ν 2 + 1/4 (H1 )2(5.18)(5.19)

1 is satisﬁed. The
These expressions are approximately correct as long as k|η0 |δH/H1
derivation of these equations is explained in Sec. 5.6.
The second term δϕ in Eq. (5.15) is obtained as follows. Since both ϕ and φ̃0 satisfy
the Klein-Gordon equation in the bulk, δϕ also obeys the same equation. The boundary
condition for δϕ is derived from Eq. (5.6) as ∂ξ δϕ = −∂ξ φ̃0 . Therefore, the solution δϕ isgiven by η0dη  Gadv (η, ξ; η  , ξb ) ∂ξ φ̃0 (η  , ξ  ) ,(5.20)
δϕ = 2ξ =ξb−∞with the aid of the advanced Green’s function that satisﬁes

1
∂
∂
∂2
∂
+ k 2 η 2 − sinh3 ξGadv (η, ξ; η  , ξ  ) = δ(η − η  )δ(ξ − ξ  ). (5.21)
η 2 2 − 2η3∂η∂η∂ξ sinh ξ ∂ξThe explicit form of the Green’s function is [36]Gadv (η, ξ; η  , ξ  ) =



 ∗

i3

 
∗
 
3   4 θ(η − η) φM (η, ξ)φM (η , ξ ) − φM (η, ξ)φM (η , ξ ) . (5.22)
sinh ξ η
M

Taking the limit η → −∞, we can expand δϕ in terms of in-vacuum mode functions,δϕ −→[u0M φM + v0M φ∗M ] ,(5.23)
η→−∞M =0,ν

where the coeﬃcients are given by
 η0
dη
3
φ∗ (η, ξb ) ∂ξ φ̃0 (η, ξ),(5.24)
u0M = −2i
3
4 M
ξ=ξb
sinh
ξ
η
−∞
b
 η0
dη
3
φ (η, ξb ) ∂ξ φ̃0 (η, ξ).
(5.25)
v0M = 2i
3
4 M
ξ=ξb
−∞ sinh ξb ηwritten in terms of the
To evaluate these coeﬃcients, we need the source term ∂ξ φ̃0 
ξ=ξbcoordinates (η, ξ), which is√ 2
C(H2 )2 2= −−3/2 sinh ξ˜b √ikH η0 η sinh ξb eik η +2 H η0 η cosh ξb + H η0 .
∂ξ φ̃0 
ξ=ξb
2k(5.26)

46
From Eqs. (5.16) and (5.23), we ﬁnally obtain the Bogoliubov coeﬃcients relating theinitial zero mode or KK modes to the ﬁnal zero mode,ϕ=(α0M φM + β0M φ∗M ) ,(5.27)

M

where
α0M

= U0M + u0M ,β0M= v0M .

(5.28)

From these coeﬃcients we can evaluate the number and the power spectrum of the generated
gravitons.

5.2

Evaluation of Bogoliubov coeﬃcients

Now let us evaluate the expressions for the Bogoliubov coeﬃcients obtained in the preceding
1) andsection. We concentrate on the two limiting cases: the low energy regime (H1high energy regime (H1  1). We ﬁrst evaluate the coeﬃcients α00 and β00 , which relate
the initial zero mode to the ﬁnal zero mode. We keep the terms up to second order in H (orequivalently in δH/H1 ).
Substituting Eq. (5.26) into Eq. (5.25), we haveH2
β00 = C(H1 )C(H2 ) H η0
H1



 η0
dη
−∞

1
i
− 3
2
η
kη



“ √
”
−ik η− η 2 +2 H η0 η cosh ξb + 2H η02e. (5.29)

Because there is a factor H in front of the integral, we can neglect the correction of O(2H )in the integrand. Then we can carry out the integration to obtain
β00 ≈ C(H1 )C(H2 )H2i −2ikη0 −ikη0 H cosh ξb
He.
H1 2kη0(5.30)

Similarly, we get
H2
α00 ≈ U00 + C(H1 )C(H2 ) H
H1


1+

i
2kη0



e−ikη0 H cosh ξb ,(5.31)

where U00 is given by Eq. (5.18).
At low energies (H11), the Bogoliubov coeﬃcients α00 and β00 become

α00 ≈
β00 ≈

i δH −ikη0 δH/H2e,2kη0 H1i δH −2ikη0 −ikη0 δH/H2e.
2kη0 H1
1+(5.32)(5.33)

It is worth noting that these expressions are correct up to the second order in δH/H1 . This
result agrees with the result of the four-dimensional calculation α(4D) and β (4D) .

47
At high energies (H1  1), the coeﬃcients are





i δH
3 δH 2 ikη0 δH
3+−
α00 ≈ 1 +2 2kη0 H1
8 H1
2 H1



(kη0 )2 δH 2 −ikη0 δH/H1 −ikη0 (3/2)(δH/H1 )2
e
−
,
2
H1


2
3
i δH
β00 ≈
e−2ikη0 −ikη0 (δH/H1 )−ikη0 (3/2)(δH/H1 ) .
2 2kη0 H1(5.34)(5.35)

Here we stress that the last two terms in the square brackets of Eq. (5.34), both coming from
U00 , are enhanced at k|η0 |  1.
Next, we calculate the Bogoliubov coeﬃcients α0ν and β0ν , which relate the initial KK
modes to the ﬁnal zero mode, up to the leading ﬁrst order in δH/H1 . Although we willcalculate the power spectrum up to second order in δH/H1 in the succeeding section, theexpressions up to ﬁrst order are suﬃcient for α0ν and β0ν , in contrast to the case for α00 and
β00 . Again, substituting ∂ξ φ̃0 (η, ξb ) into Eq. (5.25), we have
β0ν =

√

3/2

2k 

˜

sinh ξb
C(H2 )χ∗ν (ξb )H η0
sinh2 ξb

 η0

√
dη
ik η 2 +2 H η0 η cosh ξb + 2H η02
ψ
(η)e
.
ν
3
−∞ η

(5.36)

Setting H in the integrand to zero, the coeﬃcient reduces to
(
β0ν ≈

H2
π
C(H2 )χ∗ν (ξb )H kη0 e−πν/22(H1 )2

 k|η0 |∞dxx−3/2 Hiν (x)eix ,(1)(5.37)

where we have introduced the integration variable x = −kη. Similarly we have
(
∗
∗
α0ν
≈ U0ν
−

π
H2
C(H2 )χ∗ν (ξb )H kη0 e−πν/22(H1 )2

 k|η0 |∞dxx−3/2 Hiν (x)e−ix ,(1)(5.38)

where U0ν is given by Eq. (5.19) and is O(H )2 .
Now let us discuss the dependence of α0ν and β0ν on H1 and δH/H1 in the limitingcases, H1
1 and H1  1. At low energies, we ﬁnd, using Eq. (5.7),|β0ν | , |α0ν | ∝ (H1 )
2

2

2

δH
H1

2
(H1

1) ,

(5.39)

where we have omitted the dependence on ν and k|η0 |. These coeﬃcients are suppressedby the factors of H1 and δH/H1 . Recall that α00 and β00 agree with the standard fourdimensional result at low energies. Thus, because of the suppression of α0ν and β0ν at low
energies, the four-dimensional result is recovered only by the contribution from the initial
zero mode.
At high energies, we obtain from Eq. (5.8)
|β0ν | , |α0ν | ∝
2

2

2

δH
H1

2

(H1  1) ,(5.40)The integral including the Hankel function is written in terms of generalized hypergeometric functions.

48
where we have again omitted the dependence on ν and k|η0 |. In contrast to the result in the
low energy regime (5.39), this high energy behavior is not associated with any suppression
factor.
Integrating |β0ν |2 over the KK continuum, we obtain the total number of zero mode
gravitons created from the initial KK vacuum. It can be shown that the coeﬃcients behave∗ ∼ (k|η |)1/2 at k|η |as β0ν ≈ −α0ν
1, and we have00∗β0ν + α0ν∼ O(k|η0 |)3/2 .

(5.41)

Thus the number of created gravitons is proportional to k outside the horizon and is evaluated
as
⎧

2
⎪
2 δH
⎪
⎪
(H1
1),
 ∞
⎨ 0.5 × k|η0 |(H1 )H1
2(5.42)|β0ν | dν ≈


⎪
δH 2
0
⎪
⎪
0.3 × k|η0 |(H1  1).
⎩
H1
(1)

On the other hand, making use of the asymptotic form of the Hankel function Hν (x) ∼√ei[x−(2ν+1)π/4] / x for x → ∞, we can evaluate the integral in Eq. (5.37) in the k|η0 | → ∞limit as k|η0 |e2ix
1(k|η0 | → ∞),dx 2 ∼(5.43)
β0ν ∝ k|η0 |xk|η0 |∞
where we have carried out the integration by parts and kept the most dominant term. Thisshows that the creation of gravitons is suppressed well inside the horizon. Since the assumption of the instantaneous transition tends to overestimate particle production at large k [1],the number of particles created from the initial zero mode and KK modes is expected to be
more suppressed inside the horizon than Eq. (5.43) if we consider a realistic situation in which$ k|η |
the Hubble parameter changes smoothly. Note that u0ν ∝ k|η0 | ∞ 0 x−2 dx is constant fork|η0 | → ∞. Therefore, the α0ν coeﬃcient is dominated by U0ν at large k, which behaves like
|α0ν |2 ∼ |U0ν |2 ∝ (kη0 )2 .
An example of numerical calculation is shown in Fig. 5.2. The ﬁgure shows that the
spectrum has a peak around the Hubble scale and then decreases inside the horizon, and we
conﬁrmed that the behavior of the number density outside the horizon is well described by
Eq. (5.42).

5.3

Power spectrum of generated gravitational waves

So far, we have discussed the Bogoliubov coeﬃcients to see the number of created gravitons.
However, our main interest is in the power spectrum of gravitational waves because themeaning of “particle” is obscure at the super Hubble scale.

5.3.1

Mapping formula: Pure de Sitter brane

Gravitational waves generated from pure de Sitter inﬂation on a brane have the scale invariant
spectrum 
2C 2 (H) H 2,(5.44)
P5D =
2
2π
MPl

49

Figure 5.2: Spectra of number density of zero mode gravitons created from vacuum ﬂuctuations in the initial KK modes. Integration over ν is performed numerically. Solid line
represents the low energy case with H1 = 0.1, while dashed line shows the high energy case.
The latter should be understood as the limiting case H1 → ∞ since at high energies thenumber does not depend on H1 ; see Eq. (5.40).
which is deﬁned by the expectation value of the squared amplitude of the vacuum ﬂuctuation,
8πk 3 |φ0 |2 /(2πM5 )3 , evaluated at a late time. Since C 2 ∼ 1 at H
1, this power spectrum
agrees at low energies with the standard four-dimensional spectrum [94, 95]
2
P4D = 2
MPl



H
2π

2
.

(5.45)

At high energies, however, the power spectrum (5.44) is enhanced due to the factor C(H),
and is much greater than the four-dimensional counterpart. This ampliﬁcation eﬀect was
found in Ref. [84]. These results say that the diﬀerence between Eq. (5.44) and Eq. (5.45) isabsorbed by the transformation
H → HC(H).

5.3.2(5.46)

Model with variation of the Hubble parameter

Now let us turn to the case in which the Hubble parameter is not constant. Time variation of
the Hubble parameter during inﬂation brings a small modiﬁcation to the spectrum, and theresulting spectrum depends on the wavelength. Here we consider the amplitude of vacuum
ﬂuctuation of the zero mode in the ﬁnal state. It will be a relevant observable for the observers
on the brane at a late epoch because the KK mode ﬂuctuations at super Hubble scale rapidly
decay in the expanding universe due to its four-dimensional eﬀective mass. Since the behaviorof the zero mode at the inﬁnite future η̃ → 0 is known from the explicit form of the mode
function [see Eq. (3.21)], we ﬁnd that the vacuum ﬂuctuation for zero mode at a late epoch

50
is given by

2 
2 





∗
∗
∗
∗
lim α00 φ̃0 − β00 φ̃0  + dν α0ν φ̃0 − β0ν φ̃0 
η̃→0



C 2 (H2 ) H22
∗ 2
∗ 2
|α
.
+
β
|
+
dν
|α
+
β
|
=
00
0ν
00
0ν

2k 3

(5.47)

2 , we obtain theMultiplying this by (2/M53 )(k 3 /2π 2 ) and recalling the relation M53 = MPl
power spectrum
zero
KK
P5D (k) = P5D(k) + P5D(k),

(5.48)

with
zero
P5D
(k)

=

KK
(k) =
P5D

 
2C 2 (H2 ) H2 2
∗ 2
|α00 + β00
| ,
2
2π
MPl



2C 2 (H2 ) H2 2 ∞∗ 2dν|α0ν + β0ν
| .
2
2π
MPl
0

(5.49)The power spectrum in the four-dimensional theory, computed in the same way, is given by
2
P4D (k) = 2
MPl



H2
2π

2 
2
 (4D)
+ β ∗(4D)  .
α(5.50)

The appearance of the coeﬃcient α in the power spectrum may look unusual. This is due
to our setup in which the ﬁnal state of the universe is still inﬂating. In such a case, theﬂuctuations that have left the Hubble horizon never reenter it. Outside the horizon, thenumber of particle created, does not correspond to the power spectrum.
There are two apparent diﬀerences between P5D and P4D ; the normalization factor C(H)
KK . In the low energy regime, however, the twoand the contribution from the KK modes P5D
spectra agree with each other:
P5D ≈ P4D

(H1

1).

(5.51)

This is because, as is seen from the discussion about the Bogoliubov coeﬃcients in the prezero is exactly the same as P
ceding section, the zero mode contribution P5D
4D (up to the
KKnormalization factor), and the Kaluza-Klein contribution P5D is suppressed by the factor(H1 )2 . On the other hand, when H1 is large, the amplitude of gravitational waves deviatesfrom the four-dimensional one owing to the ampliﬁcation of the factor C(H).
We have observed for pure de Sitter inﬂation that the correspondence between the ﬁvedimensional power spectrum and the four-dimensional one is realized by the map (5.46). It isinteresting to investigate whether the correspondence can be generalized to the present case.
It seems natural to give the transformation in this case by

whereh → hC(h),(5.52)

 (4D)∗(4D) h(k) = H2 α+β;(5.53)

51
namely, the rescaled power spectrum Pres (k) is deﬁned as 
2C 2 (h) h 2
Pres (k) =
.
2
2π
MPl

(5.54)

We will see that this transformation works well and mostly absorbs the diﬀerence between
the ﬁve- and four-dimensional amplitudes.
We examine the diﬀerences between the ﬁve-dimensional spectrum and the rescaled fourdimensional spectrum by expanding them with respect to δH/H1 as


δH 3
(0)
(1)
(2)
KK
+ P5D
,
(5.55)
P5D (k) = P5D + P5D + P5D + O
H


δH 3
(0)
(1)
(2)
.
(5.56)
Pres (k) = Pres + Pres + Pres + O
HHere the quantity associated with the superscript (n) represents the collection of the terms ofO ((δH/H)n ). On the right hand side of Eq. (5.55), all the terms except for the last one comefrom the initial zero mode. The direct expansion shows that the leading terms in Eqs. (5.55)
and (5.56) exactly agree with each other up to the ﬁrst order in δH/H1 ,
(0)

(1)

P5D + P5D

(0)
(1)
= Pres+ Pres  "
2C 2 (H1 ) H1 2
sin(2kη0 )
=
−2
1
+
2
2π
kη0
MPl

C 2 (H1 )

δH
2
1 + (H1 ) H1#
. (5.57)

These terms contain only the contribution from the initial zero mode.
KK is of order (δH/H )2 because both α
The contribution from the initial KK modes P5D10ν
and β0ν are O(δH/H1 ). Thus, to examine whether the agreement of the spectrum continues
to hold even after including the KK modes, we investigate the second order part of the(2)(2)
spectrum. The second order terms P5D and Pres are given by


  "
−2 (H ) + C 2 (H )C
2C 2 (H1 ) H1 2
(2)
1
1
(kη0 )2−2
P5D (k) =22π
MPl
1 + (H1 )2
C 2 (H1 )sin2 (kη0 ) C 2 (H1 )+1 ++ cos(kη0 )(kη0 )2
1 + (H1 )2
1 + (H1 )2
sin(2kη0 ) 2 + 3(H1 )2
6C 2 (H1 )+−
2kη0
1 + (H1 )2
1 + (H1 )2
#


δH 2
C 2 (H1 )3 + 4(H1 )2
4C 2 (H1 )−,(5.58)+
1 + (H1 )2
1 + (H1 )2
1 + (H1 )2 H1
  "2C 2 (H1 ) H1 2
sin2 (kη0 )(2)
2cos(kη
)+Pres (k) =022π(kη0 )2MPlsin(2kη0 ) 2 + 3(H1 )2
4C 2 (H1 )+−kη0
1 + (H1 )2
1 + (H1 )2
sin2 (2kη0 ) 4 + 5(H1 )2
4C 2 (H1 )−−(2kη0 )2
1 + (H1 )2
1 + (H1 )2

52+

4C 2 (H1 )3 + 4(H1 )2−
1 + (H1 )2
1 + (H1 )2
(2)

#



C 2 (H1 )
1 + (H1 )2

δH
H1

2
.

(5.59)(2)

From these equations, we notice that Pres and P5D do not agree with each other. We seethat the diﬀerence is enhanced in particular at k|η0 |  1. However, as we have mentioned
earlier, the KK mode contribution also gives a correction of the same order, and in fact weshow that approximate agreement is recovered even in this order by adding the KK mode
contribution.
(2)(2)
1. Expanding P5D and Pres with respectFirst we observe the power spectrum at k|η0 |to k|η0 |, we have(2)

(2)
P5D (k) Pres(k),∼ (kη0 )2
(0)
(0)
P
P



δH
H1

2
.

(5.60)On the other hand, Eq. (5.41) leads to
KK (k)
P5D∼ (k|η0 |)3
P (0)



δH
H1

2
.

Therefore the diﬀerence is small outside the horizon as2 P (2) + P KK − P (2) 
res 
 5D
2 δH
5D
)
∼
(kη


0


H1
P (0)
(2)

(5.61)(k|η0 |

1) ,(5.62)(2)

although the cancellation between P5D (k) and Pres (k) does not happen. By a similar argument, at k|η0 |  1, we have
 P (2) + P KK − P (2)   δH 2
res 
 5D
5D
(k|η0 |  1) .
(5.63)
∼



H1
P (0)
The situation is more interesting when we consider the spectrum inside the Hubble hori(2)zon. There is a term proportional to (kη0 )2 in P5D , which is dominant at k|η0 |  1, while
(2)
(2)
(2)
there is no corresponding term in Pres . Hence, the diﬀerence between P5D and Pres is




(2)
(2)
−2 (H ) + C 2 (H )P5D − PresδH 2
C
C 2 (H1 )1
1
2≈ −(kη0 ) 2 −.
(5.64)P (0)
1 + (H1 )2
1 + (H1 )2 H1
Our approximation is valid for k|η0 |δH/H1  1 [Eq. (5.14)]. Hence, within the region ofvalidity, this diﬀerence can be as large as P (0) . We also note that the terms proportional
to (k|η0 |)2 come from α00 , while the contribution from β00 is suppressed at k|η0 |  1. On
KK (k), is dominated by α
the other hand, the$ contribution from$ initial KK modes, P5D
0ν at
KK
∗
2
2
k|η0 |  1: P5D ∝ dν|α0ν + β0ν | ∼ dν|α0ν | . This means that, although the creation ofzero mode gravitons from the initial KK modes is negligible inside the horizon, a part of theamplitude of the ﬁnal zero mode comes from the initial KK modes losing their KK momenta.
KK behaves as
Since the coeﬃcient α0ν is proportional to k|η0 | at large k|η0 |, P5D
KK (k)
P5D∼ +(kη0 )2
P (0)



δH
H1

2
.

(5.65)

53(2)

This KK mode contribution cancels P5D . The cancellation can be proved by looking at theproperty of the Bogoliubov coeﬃcients2
2(5.66)|α00 | − |β00 | + dν(|α0ν |2 − |β0ν |2 ) = 1.
This relation together with Eq. (5.47) implies that the power spectrum cannot signiﬁcantly
deviate from C 2 (H2 )H22 /2k 3 in the region where β00 and β0ν are negligibly small. We canalso demonstrate the cancellation by explicit calculation in the low and high energy limits.
For k|η0 |  1, we have α0ν ≈ U0ν . Then from Eqs. (5.7), (5.8), and (5.19), we obtain
⎧ ∞

2
2ν tanh(πν)dν
⎪
2
2 δH
⎪
⎪
× (kη0 ) (H1 )
(H1
1),

⎨
(ν 2 + 1/4)(ν 2 + 9/4)
H1
02
(5.67)
dν|α0ν | ≈

2

⎪ 6 ∞
ν 2 dν
⎪2 δH⎪× (kη0 )(H1  1),⎩
π 0 (ν 2 + 1/4)2 (ν 2 + 9/4)
H1
which gives (kη0 )2 (H1 )2 (δH/H1 )2 in the low energy regime and (3/4)(kη0 )2 (δH/H1 )2 in the
KK (k) cancels P (2) (k)3 .
high energy regime. Comparing these with Eq. (5.64), we see that P5D
5D
To summarize, we have observed that the agreement between the rescaled spectrum
Pres (k) and the ﬁve-dimensional spectrum P5D (k) is exact up to ﬁrst order in δH/H1 . The
agreement is not exact at second order, but we found that the correction is not enhanced at
any wavelength irrespective of the value of H1 . Just for illustrative purpose we show theresults of numerical calculations in Fig. 5.3 and Fig. 5.4.

5.4

Summary

In this chapter we have investigated the generation of primordial gravitational waves and its
power spectrum in the inﬂationary braneworld model, focusing on the eﬀects of the variationof the Hubble parameter during inﬂation. For this purpose, we considered a model in which
the Hubble parameter changes discontinuously.
In the case of de Sitter inﬂation with constant Hubble parameter H, the spectrum isknown to be given by Eq. (5.44) [84]. It agrees with the standard four-dimensional one[Eq. (5.45)] at low energies H1, but at high energies H  1 it signiﬁcantly deviatesfrom Eq. (5.45) due to the ampliﬁcation eﬀect of the zero mode normalization factor C(H).
One can say, however, the ﬁve-dimensional spectrum is obtained from the four-dimensional
one by the map H → HC(H).
In a model with variable Hubble parameter, gravitational wave perturbations are expected
to be generated not only from the “in-vacuum” of the zero mode but also from that of the
Kaluza-Klein modes. Hence, it is not clear whether there is a simple relation between theﬁve-dimensional spectrum and the four-dimensional counterpart. Analyzing the model witha discontinuous jump in the Hubble parameter, we have shown that this is indeed approximately the case. More precisely, if the squared amplitude of four-dimensional ﬂuctuations isgiven by (h/2πMPl )2 , we transform (h/2πMPl )2 into C 2 (h)(h/2πMPl )2 , then the resulting
rescaled spectrum exactly agrees with the ﬁve-dimensional spectrum P5D (k) up to ﬁrst orderin δH/H1 . At second order O(δH/H)2 the agreement is not exact, but the diﬀerence is not
3

The small H expansion C 2 (H) ≈ 1 − (H)2 [1/2 + ln(H/2)] is used here to investigate the low energy
case.

54

Figure 5.3: Five-dimensional power spectrum of gravitational waves P5D (k) and the fourdimensional one P4D (k) at low energies (H1 = 10−2 ) with δH/H1 = 10−3 . These two agreewith each other. In this case initial KK modes give negligible contribution.

Figure 5.4: Power spectra of gravitational waves at high energies (H1 = 103 ) with δH/H1 =zero (k) + P KK (k) and the rescaled four-dimensional one
10−3 . Five-dimensional spectrum P5D
5D
zero (k)Pres (k) agree well with each other (solid lines), while only the zero mode contribution P5D
gives the reduced ﬂuctuation amplitude inside the horizon (dotted line).

55
enhanced at any wavelength irrespective of the value of H1 . Hence, in total, the agreement
is not signiﬁcantly disturbed by the mismatch at second order. As a non-trivial point, we
also found that the initial KK mode vacuum ﬂuctuations can give non-negligible contribution
to the ﬁnal zero mode states at second order.
One may expect that the power spectrum of gravitational waves in the braneworld model
would reﬂect the characteristic length scale corresponding to the curvature (or “compactiﬁcation”) scale of the extra dimension . However, our analysis showed that the resultant
power spectrum does not depend on the ratio of the wavelength of gravitational waves to thebulk curvature scale, k|η0 |H.
Here we should mention the result of Ref. [36] that is summarized in Sec. 5.5. Their
setup is the most violent version of the transition H1 → H2 = 0. If the wavelength of
the gravitational waves is much longer than both the Hubble scale and the bulk curvaturescale, the power spectrum is given by Eq. (5.71) and obviously it is obtained from the fourdimensional counterpart by the map H → HC(H). However, if the wavelength is longer thanthe Hubble scale but much smaller than the bulk curvature radius, the amplitude is highly
damped as is seen from Eq. (5.74) and the map H → HC(H) does not work at all. This
damping of the amplitude can be understood in the following way. In the high energy regimeH  1, the motion of the brane with respect to the static bulk is ultrarelativistic. At the
moment of transition to the Minkowski phase, the brane abruptly stops. Zero mode gravitons
with wavelength smaller than the bulk curvature scale can be interpreted as “particles”
traveling in the ﬁve dimensions. These gravitons make a “hard hit” with the brane at the
moment of this transition, and get large momenta in the ﬁfth direction relative to the static
brane. As a result, these gravitons escape into the bulk as KK gravitons, and thus theamplitude (5.74) is damped. On the other hand, in our model the change of the Hubble
parameter is assumed to be small, and hence such violent emission of KK gravitons does not
happen. If this interpretation is correct, the mapping rule h → hC(h) will generally give agood estimate for the prediction of inﬂationary braneworld models as far as time variation of
the Hubble parameter is smooth. If we can conﬁrm the validity of this prescription in moregeneral cases, the analysis of gravitational wave perturbations will be simpliﬁed a lot, and infact it will be conﬁrmed by a numerical study in the next chapter.

5.5

Appendix: Particle creation when connected to Minkowski
brane

Here for comparison with our results we brieﬂy summarize the results obtained by Gorbunov
et al. [36] focusing on the power spectrum of gravitational waves. Their method is basicallythe same that we have already explained in the main text. They considered the situationthat de Sitter inﬂation on the brane with constant Hubble parameter H suddenly terminates
at a conformal time η = η0 , and is followed by a Minkowski phase. The power spectrum ofgravitational waves is expressed in terms of the Bogoliubov coeﬃcients as
 2



H
2
2
2
2
P5D (k) = 2(5.68)(kη0 ) 1 + 2|β00 | + 2 dν|β0ν | ,MPl 2πwhere |β00 |2 and |β0ν |2 are the number of zero mode gravitons created from initial zero modeand KK modes, respectively. At super Hubble scale (k|η0 |1) we can neglect the ﬁrst termin the parentheses, which corresponds to the vacuum ﬂuctuations in Minkowski space.

56
According to Ref. [36], when k|η0 |H
1 [i.e., the wavelength of gravitational wave
−1(k/a) is much larger than the bulk curvature scale ] and k|η0 |1, the coeﬃcients are
given byC 2 (H),4(kη0 )2

k|η0 |(H)2∼k|η0 |H|β00 |2 ≈dν|β0ν |2(5.69)(H1),(H  1).

(5.70)

One can see that the contribution from initial KK modes is suppressed irrespective of the
expansion rate H. Therefore the power spectrum is evaluated as 
C 2 (H) H 2.
(5.71)
P5D ≈
2
2π
MPl
Equation (5.71) is half of the power spectrum on the de Sitter brane [Eq. (5.44)], and thisresult can be understood as follows. The amplitude of ﬂuctuations at the super Hubble scale
stays constant. After the sudden transition from the de Sitter phase to the Minkowski phase,the Hubble scale becomes inﬁnite. Therefore, those ﬂuctuation modes are now inside the
Hubble horizon, and they begin to oscillate. As a result, the mean-square vacuum ﬂuctuation
becomes half of the initial value.
1 (these conditions require H  1),
On the other hand, when k|η0 |H  1 and k|η0 |the Bogoliubov coeﬃcients are given by|β00 |2 ≈dν|β0ν |2 ∼C 2 (H)1,
2(kη0 ) (kη0 H)21.
(kη0 H)2(5.72)(5.73)As before, the contribution from initial KK modes is negligible, and that from |β00 |2 dominates
the power spectrum, 
C 2 (H) H 2
4.
(5.74)
P5D ≈22π(kη0 H)2MPl
One can see that the spectrum is suppressed by the factor 4/(kη0 H)2 .

5.6

Appendix: Details of calculations

We derive Eqs. (5.18) and (5.19) by calculating the Wronskian (5.17). Expanding Eq. (5.12)
in terms of H , we haveη̃ = η + H η0 cosh ξ − 2H η02 (sinh ξ)2 /(2η) + · · · ,ξ˜ = ξ − H η0 sinh ξ/η + 2H η02 cosh ξ sinh ξ/η 2 + · · · ,(5.75)(5.76)

which reduce to the following forms at the inﬁnite past (η → −∞),
η̃ − η = H η0 cosh ξ,ξ˜ − ξ = 0.

(5.77)(5.78)

57
Then, U00 is evaluated at η = −∞ as
U00



dξ
∗
∗
φ̃0 ∂η φ0 − φ0 ∂η φ̃0 = −2i
22
ξb η sinh ξ
η→−∞
 ∞
−i
kη
0 cosh ξH
H2
e=C(H1 )C(H2 ) · 2(H1 )2dξ
H1
sinh3 ξξb
H2 C(H2 ) −i H kη0 cosh ξbe· 2(H1 )2 C 2 (H1 ) ×≈
H1 C(H1 )

 ∞
1
1
i H kη0 cosh ξb
1 − iH kη0 cosh ξ − (H kη0 )2 cosh2 ξ , (5.79)
dξ
×e
3
2
sinh ξ
ξb3

 ∞where we expanded the integrand with respect to H in the last line. Integrating each term,
we ﬁnally obtain
U00 ≈H2 C(H2 ) −i H kη0 cosh ξbe
H1 C(H1 ) 1
× 1 − iH kη0 C 2 (H1 ) − cosh ξb − (H kη0 )2 sinh2 ξb .
2

(5.80)

Note that the condition H k|η0 | cosh ξb (≈ k|η0 |δH/H)
1 is required in order to justify the
expansion of the exponent. Because the integral is saturated at ξ ≈ ξb , we do not have toworry about the validity of the expansion for large cosh ξ.
The explicit form of U0ν is needed up to the ﬁrst order in H . A similar calculation leads
to
U0ν



dξ
∗
∗
φ̃0 ∂η φν − φν ∂η φ̃0 = −2i
2 sinh3 ξ
η
ξb
η→−∞
 ∞
−3/2
H2 
dξ
= −2i3 · −1/2 C(H2 ) √ · √χν (ξ)
22k
2k ξb η sinh3 ξ




i
−ikη̃
ikη+iπ/4ikη+iπ/4−ikη̃ e· (−1 − ikη)e− (−η̃) e· (−ikη)e
×
η̃ −

k
η→−∞
 ∞
−i
kη
cosh
ξ
0
H
b
e
= −2eiπ/4 H2 C(H2 )dξ
(5.81)χν (ξ).
sinh3 ξξb3

 ∞The integral in the last line, which we call I, can be calculated as follows. Again, expanding
the integrand in terms of H , we have
I≈

 ∞
dξ
ξb

χν (ξ)(1 − iH kη0 cosh ξ) .
sinh3 ξ(5.82)Let us consider the ﬁrst term in the parentheses. The spatial wave function χν satisﬁes(sinh ξ)−3 χν = −(ν 2 + 9/4)−1 ∂ξ [(sinh ξ)−3 ∂ξ χν ] with the boundary condition ∂ξ χν (ξb ) = 0.
Therefore, together with the behavior at inﬁnity, (sinh ξ)−3 ∂ξ χν ∼ (sinh ξ)−3 ∂ξ (sinh ξ)3/2 →0, we ﬁnd that the integral of the ﬁrst term vanishes. Then, using the integration by parts

58
twice, we have


9
ν +
4
2


I ≈
=
=
≈


1
∂
∂
iH kη0χν (ξ)dξ cosh ξ3∂ξ sinh ξ ∂ξ
ξb
 ∞
1
∂
dξ
−iH kη0
2 ∂ξ χν (ξ)
sinh ξξb ∞−2 cosh ξχν (ξb )χν (ξ)+ iH kη0dξiH kη02
sinh ξbsinh3 ξξbχν (ξb )+ 2I,iH kη0(H1 )2
 ∞(5.83)

from which we can evaluate U0ν . Note that the approximation is valid when k|η0 |δH/H1

1.

Chapter 6

Quantum-mechanical generation ofgravitational waves
The generation and evolution of perturbations are among the most important issues in cosmology because of their direct link to cosmological observations such as the stochastic gravitational wave background and the temperature anisotropy of the cosmic microwave background(CMB), by which we can probe the early universe. While the cosmological perturbation theory in the conventional four-dimensional universe is rather established [67, 110, 4, 94], calculating cosmological perturbations in the braneworld still remains to be a diﬃcult problem.
Although some attempts have been made concerning scalar perturbations [74, 76, 126, 78,
79, 146, 81], further progress is awaited to give a clear prediction about the CMB anisotropy
in the Randall-Sundrum braneworld. Almost the same is true for gravitational wave (tensor)
perturbations [41, 84, 36, 61, 45, 46, 49, 50, 27, 5, 6, 139, 63]. However, since their generationand evolution depend basically only on the background geometry, they are slightly easier tohandle.
During inﬂation super-horizon gravitational wave perturbations are generated from vacuum ﬂuctuations of gravitons. The pure de Sitter braneworld is the special case that allows
deﬁnite analytical computation of quantum ﬂuctuations. Thanks to the symmetry of the deSitter group, the perturbation equation becomes separable and can be solved exactly [84].
The time variation of the Hubble parameter generally causes mixing of a massless zero modeand massive Kaluza-Klein modes, and this eﬀect was investigated based on the “junction”
models, which assume an instantaneous transition from a de Sitter to a Minkowski brane [36]or to another de Sitter brane [61], as was explained in the previous chapter. There we discussed the quantum-mechanical generation of gravitational waves within such limited andsimpliﬁed models.
As for the classical evolution of gravitational waves during the radiation dominated epoch,several studies have been done [45, 46, 49, 50], assuming that a given single initial mode foreach comoving wave number dominates. The focus of these works is mainly on the evolutionof modes which re-enter the horizon in the high-energy regime. While the late time evolution
is worked out analytically in Refs. [63, 78] by resorting to low-energy approximation methods.
In this chapter we consider the generation of primordial gravitational waves during inﬂation in more general models of the Randall-Sundrum type. For deﬁniteness, we adopt a
simple setup in which both initial and ﬁnal phases are described by de Sitter braneworlds.
Two de Sitter phases with diﬀerent values of the Hubble expansion rate are smoothly inter59

60
polated. This work is an extension of the work on the “junction” models in Chapter 5. In
the previous analysis the transition of the Hubble rate was abrupt and the gap was assumed
to be inﬁnitesimal. Here we extend the previous results to more general models with smoothtransition by using numerical calculations with a reﬁned formulation.
This chapter is organized as follows. In the next section we brieﬂy summarize paststudies [84, 36, 61] on the generation of gravitational waves via quantum ﬂuctuations during
inﬂation in the Randall-Sundrum braneworld, emphasizing the mapping formula introducedin the previous chapter [61]. In Sec. 6.2 we describe our numerical scheme to investigate thegeneration of gravitational waves, and then in Sec. 6.3 we present results of our calculations.
Section 6.4 is devoted to discussion.

6.1

Gravitational waves in inﬂationary braneworld

6.1.1

Pure de Sitter brane

The background spacetime that we consider is composed of a ﬁve-dimensional AdS bulk,
whose metric is given in the Poincaré coordinates by
ds2 =


2 
2
i
j
2
−dt
,
+
δ
dx
dx
+
dz
ij
z2

(6.1)

and a Friedmann brane at z = z(t).
First let us consider the generation of gravitational waves from pure de Sitter inﬂation on
the brane [84]. The coordinate system appropriate for the present situation is


1 
2
2
2
i
j
(6.2)
+
δ
dx
dx
ds =
−dη
+ dξ 2 ,ij
22ηsinh ξ
which is obtained from Eq. (6.1) by a coordinate transformationt = η cosh ξ + t0 ,(6.3)

z = −η sinh ξ,(6.4)

where t0 is an arbitrary constant and η is the conformal time, which is negative. The deSitter brane is located atξ = ξb = constant,(6.5)

and the Hubble parameter on the brane is given by
H = −1 sinh ξb .
The gravitational wave perturbations are described by the metric


2
1 
2
2
i
j
−dη
+ dξ 2 ,
+
(δ
+
h
)dx
dx
ds =
ij
ij
sinh2 ξ η 2
and we decompose the perturbations into the spatial Fourier modes as
√

2
hij =
d3 k φk (η, ξ)eik·x eij ,(2πM5 )3/2

(6.6)

(6.7)

(6.8)

61
where eij is the transverse-traceless polarization tensor and M5 is the fundamental mass scale
2 . Hereafter wewhich is related to the four-dimensional Planck mass MPl by (M5 )3 = MPlwill suppress the subscript k. The linearized Einstein equations give the Klein-Gordon-type
equation for φ:


∂2
1
2 ∂
sinh3 ξ ∂∂2+k
φ = 0.
−
−
3
2
2
∂η
η ∂η
η
∂ξ sinh ξ ∂ξ

(6.9)

Assuming the Z2 -symmetry across the brane, the boundary condition on the brane is givenby ∂ξ φ|ξ=ξb = 0. The perturbation equation (6.9) admits one discrete zero mode φ0 (η) as
well as massive Kaluza-Klein (KK) modes φν = ψν (η) · χν (ξ), which were already given in
Chapter 3.
In inﬂationary cosmology, ﬂuctuations in the graviton ﬁeld (and other ﬁelds such as theinﬂaton) are considered to be generated quantum-mechanically. As explained in Chapter 3,
we can quantize the graviton ﬁeld following the standard canonical quantization scheme.
Then the expectation value of the squared amplitude of the vacuum ﬂuctuation in the zeromode is given by|φ0 |2H2(1 + k 2 η 2 )2k 3
H2→ −1 C 2 (H) 3 ,2k=

−1 C 2 (H)

(6.10)

where the expression in the second line is obtained by evaluating the perturbation in the
super-horizon regime/at a late time, and hence this is the amplitude of the growing mode. In
terms of the power spectrum deﬁned by
P :=

4πk 3
2
·
|φ0 |2 ,3(2π) (M5 )3(6.11)

we have
2C 2 (H)
P=
2
MPl



H
2π

2
,

(6.12)

2 . This is the standard ﬂat spectrum up to the overall factor
where we have used (M5 )3 = MPl
2
C (H). In conventional four-dimensional cosmology the power spectrum of the primordial
gravitational waves from de Sitter inﬂation is given by [94, 95]

P4D =

2
2
MPl



H
2π

2
.

(6.13)

Thus just by rescaling the amplitude in four-dimensional cosmology as
H → HC(H),
the braneworld result (6.12) is exactly obtained.

(6.14)

62

6.1.2

The “junction” model

Pure de Sitter inﬂation on the brane described in the previous subsection is a special casewhere the amplitude of the growing zero mode can be obtained completely analytically. Asa next step to understand gravitational waves from more general inﬂation with H = const, a
discontinuous change in the Hubble parameter was considered in the previous chapter [61].
In such a “junction” model the Hubble parameter is given byHi ,η < η0 ,H(η) =(6.15)
Hf = Hi − δH, η > η0 ,
and
δH
Hi

1,

(6.16)

is assumed.
The evolution of gravitational wave perturbations can be analyzed by solving the equationof motion backward using the advanced Green’s function [36]. A set of mode functions can beconstructed for the initial de Sitter stage, and another for the ﬁnal de Sitter stage, either ofwhich forms a complete orthonormal basis for the graviton wave function. The ﬁnal (growing)zero mode may be written as a linear combination of the initial zero and KK modes. The
Bogoliubov coeﬃcients give the creation rate of ﬁnal zero mode gravitons from the initial
vacuum ﬂuctuations in the Kaluza-Klein modes as well as in the zero mode. Thus the power
spectrum may be written as a sum of two separate contributions,
P = P0 + PKK ,(6.17)

where P0 and PKK are the parts coming from the initial zero and Kaluza-Klein modes,
respectively.
An important result brought by analyzing this junction model is that P is well approximated by the rescaled spectrum Pres obtained by using a simple map speculated by theexact result of pure de Sitter inﬂation [Eq. (6.14)]. More precisely, let P4D (k) be the gravitational wave spectrum evaluated in the standard four-dimensional inﬂationary universe withthe same time dependence of the Hubble parameter (6.15), and then the rescaled spectrum
is deﬁned by
 2
 
hk
2
2C 2 (hk ) hk 2
→ Pres =.
(6.18)
P4D = 22
2π
MPl 2πMPl
Comparing the rescaled spectrum with P obtained from a ﬁve-dimensional calculation, one
ﬁnds that the diﬀerence between these two is suppressed to be second order like [61] 2 P − Pres   δH1.
(6.19)
P
Hi
It might be worth noting here that the KK contribution PKK is necessary for realizing thisinteresting agreement between the braneworld result and the (basically) four-dimensional
result especially at high energies H  1. The above agreement (6.19) raised a speculationthat the mapping formula hk → C(hk ) may work with good accuracy in more general
inﬂation models with a smoothly changing expansion rate. The central purpose of the present
chapter is to check whether this speculation is correct or not.

63

6.2

Formulation

6.2.1

Basic equations

Now let us explain the formulation that we use to study the generation of gravitational waves
without assuming a pure de Sitter brane. Our formulation is based on double null coordinates,
which are presumably the most convenient for numerical calculations.
The metric (6.1) can be rewritten by using double null coordinates
u = t − z,

(6.20)

v = t + z,

(6.21)

42 −dudv + δij dxi dxj .
2(v − u)(6.22)

in the form ofds2 =The trajectory of the brane can be speciﬁed arbitrarily by
v = q(u).

(6.23)

By a further coordinate transformation
U = u,(6.24)

q(V ) = v,

(6.25) 
42−q (V )dU dV + δij dxi dxj ,2[q(V ) − U ](6.26)

we obtain
ds2 =where a prime denotes diﬀerentiation with respect to the argument. Now in the new coordinates the position of the brane is simply given by
U = V.

(6.27)

We will use this coordinate system for actual numerical calculations.
The induced metric on the brane isds2b = 
42−q (V )dV 2 + δij dxi dxj ,2[q(V ) − V ](6.28)

from which we can read oﬀ the conformal time η and the scale factor a, respectively, asq  (V ) dV,2,a=q(V ) − Vdη =(6.29)(6.30)and hence the Hubble parameter on the brane is written asH =1
2 q  (V )[1 − q  (V )],(6.31)

64
or equivalentlyq  (V ) =

1 + 2 H 2 − H

2.

(6.32)

Given the Hubble parameter as a function of η, one can integrate Eqs. (6.29) and (6.32) toobtain q as a function of V .
When the brane undergoes pure de Sitter inﬂation (and thus q  (V ) = constant), thefollowing relation between (U, V ) and (η, ξ) will be useful:t0 − U1ξ = ξb + ln,
(6.33)
2t0 − Vη = −e−ξb [(t0 − U )(t0 − V )]1/2 ,(6.34)andq(V ) = e−2ξb (V − t0 ) + t0 .

(6.35)The Klein-Gordon-type equation for a gravitational wave perturbation φ in the (U, V )coordinates reduces to6(∂V − q  (V ) ∂U ) + q  (V ) k 2 φ = 0,
(6.36)
4∂U ∂V +q(V ) − Usupplemented by the boundary condition[∂U − ∂V ]φU =V = 0.

(6.37)The expression for the Wronskian evaluated on a constant V hypersurface is given by

 V
3
2
(X · Y ) = 2idU
(X∂U Y ∗ − Y ∗ ∂U X),(6.38)
q(V ) − U−∞which is independent of the choice of the hypersurface.
When inﬂation on the brane deviates from pure de Sitter one, the decomposition into
the zero mode and KK modes becomes rather ambiguous. For this reason we require theinitial and ﬁnal phases of inﬂation to be pure de Sitter, though arbitrary cosmic expansion
is allowed in the intermediate stage. In both de Sitter phases, q(V ) can be ﬁt by Eq. (6.35),
and ξb and t0 are determined, respectively. Hence we have two sets of de Sitter coordinates
˜ We distinguish the coordinates in the ﬁnal phase by associating them with(η, ξ) and (η̃, ξ).
tilde. In the ﬁnal de Sitter phase the mode will be well outside the horizon, and hence we
expand the graviton ﬁeld in terms of the growing and decaying zero mode solutions φ̃g andφ̃d as(6.39)
φ = Âg φ̃g + Âd φ̃d + dν Âν φ̃ν + Â†ν φ̃∗ν ,whereφ̃g := Im φ̃0 ,(6.40)

φ̃d := Re φ̃0 ,(6.41)

65
and the mode functions with tilde are deﬁned in the same way as φ0 and φν with the substi˜ η̃) for (ξ, η). It can be easily seen that the growing and the decaying modes aretution of (ξ,normalized as






1
φ̃g · φ̃d = ,φ̃g · φ̃g = φ̃d · φ̃d = 0,
(6.42)
2f
f
f
where subscript f means that the expression is to be evaluated in the ﬁnal de Sitter phase.
Notice that de Sitter mode functions thus deﬁned as functions of (U, V ) through (ξ, η) [or˜ η̃)] do not satisfy the equation of motion outside the initial (or ﬁnal) de Sitter phase. Back(ξ,in the initial de Sitter phase the graviton ﬁeld can be expanded as
(6.43)
φ = â0 φ0 + â†0 φ∗0 + dν âν φν + â†ν φ∗ν .
We assume that initially the gravitons are in the de Sitter invariant vacuum state annihilated
by â0 and âν ,â0 |0 = âν |0 = 0.

(6.44)

We would like to evaluate the expectation value of the squared amplitude of the vacuum
ﬂuctuation in the growing mode at a late time, 2 φ̃g  0|Â2g |0,or equivalently, the power spectrum,
P(k)

=
→

 2
4πk 3 2 ·
φ̃g  0|Â2g |0(2π)3 (M5 )3

2C 2 (Hf ) Hf 2
0|Â2g |0,
2
2π
MPl

(6.45)

where Hf is the Hubble parameter in the ﬁnal de Sitter phase. In order to obtain the ﬁnal
amplitude, it is not necessary to solve the evolution of all the (inﬁnite number of) degrees
of freedom with their initial conditions set by Eq. (6.43). In fact, we have only to solve thebackward evolution of the ﬁnal decaying mode, as explained below.
Suppose that a solution Φ(U, V ) is chosen so as to satisfy Φ = φ̃d in the ﬁnal de Sitter
phase. From the Wronskian condition (6.42) we see that


1
Âg =
φ · φ̃d = (φ · Φ)2f
= (φ0 · Φ)i â0 +dν(φν · Φ)i âν + h.c.,where subscript i means that the expression is to be evaluated in the initial de Sitter phase.
In the above we used the fact that the Wronskian is constant in time. Thus we obtain
2
2(6.46)
0|Âg |0 = 4 |(φ0 · Φ)i | + dν|(φν · Φ)i |2 ,

66

Figure 6.1: Numerical (backward) evolution scheme.
from which we can calculate the power spectrum of the primordial gravitational waves. It is
obvious that the spectrum is written in the form of Eq. (6.17) with

6.2.2

P0 :=

8C 2 (Hf )2
MPl

PKK :=

8C 2 (Hf )
2
MPl




Hf
2π
Hf
2π

2

|(φ0 · Φ)i |2 ,

2 dν|(φν · Φ)i |2 .

(6.47)(6.48)

Numerical scheme

Here we describe the algorithm that we employ to solve the backward evolution of the gravitational perturbations numerically. We set the boundary conditions for the mode Φ so as tobe identical to a decaying zero mode φ̃d at a time V = Vf in the ﬁnal de Sitter phase. We
decompose Φ asΦ(U, V ) = φ̃d (U, V ) + δΦ(U, V ),(6.49)

67
and solve the equation for δΦ(U, V ) instead of Φ(U, V ). The equation of motion for δΦ is
obtained as
64∂U ∂V +(∂V − q  (V )∂U ) + q  (V )k 2 δΦq(V ) − U6
= − 4∂U ∂V +(∂V − q  (V ) ∂U ) + q  (V ) k 2 φ̃d .
(6.50)
q(V ) − U
Since both Φ and φ̃d satisfy the boundary condition of the form of Eq. (6.37), the boundary
condition for δΦ is also written as(6.51)[∂U − ∂V ]δΦU =V = 0.
We immediately ﬁnd that we do not have to solve the backward evolution in the ﬁnal deSitter phase, since δΦ identically vanishes there.
In some cases Φ is not disturbed so much from its ﬁnal conﬁguration φ̃d (U, Vf ), especiallywhen we discuss a small deviation from the pure de Sitter case. It is advantageous then to use
a small quantity δΦ as a variable in numerical calculations. Of course, there is no problem
even when δΦ does not stay small.
Our scheme to obtain the values of δΦ at the initial surface V = Vi is as follows. We givethe boundary conditions at V = Vf as f1,1 , f2,1 , · · · , fN,1 = 0, wherefn,m := δΦ(U = Vf − ε(n − 1), V = Vf − ε(m − 1)).
A sketch of the numerical grids is shown in Fig. 6.1. Here N is taken to be suﬃciently
large, and we use the same grid spacing ε both in the U and V directions. At a virtual site
(U, V ) = (Vf , Vf − ε) we set
f1,2 = f2,1 ,(6.52)

so that the boundary condition (6.51) is satisﬁed at (Vf − ε/2, Vf − ε/2). From f1,1 , f2,1 ,and f1,2 we can determine the value of f2,2 by using the equation of motion (6.50) at (Vf −ε/2, Vf − ε/2), and then from f2,1 , f3,1 , and f2,2 we can determine f3,2 , and so on. We repeat
the same procedure in the subsequent time steps until we obtain fM,M , fM +1,M , · · · , fN,M at
Vi = Vf − ε(M − 1).

6.2.3

A toy model

As stated above, to make the problem well posed, we consider models which have initial andﬁnal de Sitter phases. To perform numerical calculations, as a concrete example, we adopt a
simple toy model in which the Hubble parameter is given by an analytic formη − η0H(η) = H̄ − ∆ tanh,
(6.53)
s
where
H̄ :=

Hi + Hf,2∆ :=Hi − Hf,2(6.54)

with Hi and Hf the initial and ﬁnal values of the Hubble parameter, s is a parameter that
controls the smoothness of the transition, and η0 indicates a transition time. Taking the

68

Figure 6.2: The Hubble parameter and the slow-roll parameter  in our model. This is a plot
for the model of Fig. 6.9.
initial time ηi and the ﬁnal time ηf so that η0 − ηi  s and ηf − η0 ≈ −η0  s, we haveH ≈ Hi for η → ηi and H ≈ Hf for η → ηf . The scale factor for this inﬂation model is given
by




−1
η − η0a(η) = s∆ ln 2 cosh,(6.55)− H̄η + ∆η0swhere the integration constant is determined by imposing that a ≈ (−Hf η)−1 at η ≈ ηf ≈ 0.
The slow-roll parameter,  := −∂η H/(aH 2 ), is obtained as=∆{s∆ ln[2 cosh((η − η0 )/s)] − H̄η + ∆η0 },s[H̄ cosh((η − η0 )/s) − ∆ sinh((η − η0 )/s)]2(6.56)

which takes maximum at η = η0 . The maximum value is(η0 ) = max =∆sH̄ 2 a(η0 ).

(6.57)The behavior of the Hubble parameter and the slow-roll parameter is shown in Fig. 6.2.

Power spectrum

69

0.95
0.9
0.85
0.8

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)H
0.0008î

0.00060.00040.0002
0
-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)HFigure 6.3:
Top panel:
power spectra of gravitational waves normalized by
2 )(H /2π)2 . Red diamonds (upper ones) indicate the result including the con(2C 2 (Hi )/MPlitributions from the initial Kaluza-Klein modes, while green diamonds (lower ones) represent
the contribution from the initial zero mode. Although the rescaled four-dimensional spectrum
is shown by blue diamonds, they are almost hidden by the red ones since the result of the
ﬁve-dimensional calculation is well approximated by the rescaled four-dimensional spectrum.
Bottom panel: diﬀerence between the ﬁve-dimensional and rescaled four-dimensional spectra.
Here we should mention the limitation of this simple toy model. For ﬁxed values of Hi ,Hf , and max one can prolong the period of the transition ∼ s measured in conformal timeby taking large s. However, using the deﬁnition of max (6.57), the transition time scale inproper time is found to be given by ∼ a(η0 )s = H̄ 2 max /∆. Therefore we cannot change thetransition time scale independently of the other parameters, Hi , Hf , and max .

6.3

Numerical results

Using various parameters shown in Table. 6.1 we performed numerical calculations, the resultsof which are presented in Figs. 6.3–6.10. Numbers of grids are N × M = 50000 × 1000, andthe grid separation ε is chosen to be (0.11−2.5)× depending on the energy scale of inﬂation.
Note here that the step width in conformal time η is given by ∆η = q  (V ) ε [Eq. (6.29)],
and q  (V ) becomes smaller for a larger value of H [Eq. (6.32)]. Our choice of ε makes ∆ηabout the same size, ∆η ≈ 0.1, in all the calculations. Integration over ν is performed upPower spectrum

70

0.9
0.8
0.7
0.6
0.5
-0.25

0

0.25

0.5

0.75

1

0.75

1

î

ö]
log[k=a(ñ 0)H
0.001
0.00080.00060.00040.00020-0.0002-0.0004
-0.25

0

0.25

0.5

ö]
log[k=a(ñ 0)HPower spectrum

Figure 6.4: Same as Fig. 6.3, but the parameters are diﬀerent.

0.9
0.8
0.7
0.6

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)H
0.004

î

0.003
0.002
0.001
0

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)HFigure 6.5: Same as Fig. 6.3, but the parameters are diﬀerent.

Power spectrum

71

0.9

0.7

0.5

0.3
-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)H
0.004
0.003

î

0.002
0.001
0
-0.001-0.002-0.003

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)HPower spectrum

Figure 6.6: Same as Fig. 6.3, but the parameters are diﬀerent.
0.8
0.6
0.4
0.2
0
-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.8

1

ö]
log[k=a(ñ 0)H
0.01

î

0.005
0
-0.005-0.01-0.015-0.4 -0.2

0

0.2

0.4

0.6

ö]
log[k=a(ñ 0)HFigure 6.7: Same as Fig. 6.3, but the parameters are diﬀerent.

Power spectrum

72

0.9
0.8
0.7
0.6
0.5
0.4
-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)H
0.004

î

0.003
0.002
0.001
0
-0.001

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)HPower spectrum

Figure 6.8: Same as Fig. 6.3, but the parameters are diﬀerent.

0.9
0.8
0.7
0.6
0.5

-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)H
0.004
0.003

î

0.002
0.001
0
-0.25

0

0.25

0.5

0.75

1

ö]
log[k=a(ñ 0)HFigure 6.9: Same as Fig. 6.3, but the parameters are diﬀerent.

Power spectrum

73

0.8
0.6
0.4
0.2
0
-0.5

-0.25

0

0.25

0.5

0.75

1

0.75

1

ö]
log[k=a(ñ 0)H
0.01

î

0
-0.01
-0.02
-0.5

-0.25

0

0.25

0.5

ö]
log[k=a(ñ 0)HFigure 6.10: Same as Fig. 6.3, but the parameters are diﬀerent.

Figure 6.11: Wronskian |(φν · Φ)i |2 as a function of ν. The parameters are given by those of
Fig. 6.9 and k/a(η0 )H̄ = 10.

74

Table 6.1: Parameters used for the numerical calculations presented in the ﬁgures.

Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10

Hi
0.11
0.11
1.1
1.25
1.5
3.5
11
15

Hf
0.10
0.08
1.0
1.0
1.0
3.0
10
10

max
0.21
0.69
0.21
0.45
0.75
0.34
0.19
0.75

/s
0.09
0.1
0.09
0.09
0.09
0.095
0.085
0.09

η0 /
−50.45−50.46−50.21−50.21−50.21−50.08−50.02−50.02to ν = 20 – 40 with an equal grid spacing of 0.15. The typical behavior of the integrand|(φν · Φ)i |2 is shown in Fig. 6.11.
In order to compare the ﬁve-dimensional power spectrum with a four-dimensional counterpart, we solve the conventional evolution equation for gravitational waves in four-dimensions, 2(6.58)∂η + 2aH∂η + k 2 Φ = 0,
where H(η) is given by the same function as used for the corresponding ﬁve-dimensional
computation, and then we calculate the rescaled power spectrum obtained from the fourdimensional ‘bare’ spectrum by using the mapping formula (6.18).
We introduce a parameter that represents a diﬀerence between the ﬁve-dimensional power
spectrum and the rescaled four-dimensional spectrum:
δ(k) :=

Pres − P.
P(6.59)In all the cases of our calculations we clearly see that |δ|
1 and thus we conclude that theprimordial spectrum of the gravitational waves in the braneworld is quite well approximatedby the rescaled four-dimensional spectrum. We also ﬁnd that the diﬀerence becomes larger forlarger values of the slow-roll parameter , but |δ| is no greater than O(10−2 ) even for models
with max  0.75. Notice that the universe is not inﬂating any more if  is greater than unity.
At high energies H  1 it can be seen that dependence of δ on H is weak, while at low
energies like H ∼ 0.1, δ is further suppressed compared to the high energy cases.
Contributions from the initial KK modes can be signiﬁcant for small wavelength modes.
For example, for the modes with k/a(η0 )H̄  10 we see that PKK /P ∼ O(10−2 ) at low
energies, but the KK contribution can become as large as PKK /P ∼ 0.6 in our most ‘violent’model with H ∼ 10 and max  0.75.

6.4

Summary

In this chapter we have investigated the generation of gravitational waves during inﬂation
on a brane and computed the primordial spectrum. Extending the previous work [61], wehave considered a model composed of the initial and ﬁnal de Sitter stages, and the transition
region connecting them smoothly. We have numerically solved the backward evolution of

75
the ﬁnal decaying mode to obtain the amplitude of the growing zero mode at a late time bymaking use of the Wronskian method,
We found that the power spectrum P is well approximated by the rescaled spectrum Presbasically calculated in standard four-dimensional inﬂationary cosmology. Here the rescaling
formula for the amplitude is given by a simple map hk → hk C(hk ), where C is the normalization factor of the zero mode. Although the diﬀerence of the two spectra, δ := (Pres −P)/P,
depends on the energy scale of inﬂation and the slow-roll parameter, our numerical analysis
clearly shows that in any case the mapping formula works with quite good accuracy, yielding|δ|  O(10−2 ). This implies that the mapping relation between the two spectra holds quite
generally in the Randall-Sundrum braneworld, which we believe is a useful formula.
We have taken into account the vacuum ﬂuctuations in initial Kaluza-Klein modes aswell as the zero mode, and found that both of them contribute to the ﬁnal amplitude of thezero mode. The amount of the initial KK contribution can be large on small scales, andit increases as the energy scale of inﬂation H becomes higher. This gives rise to a quite
interesting picture. When the expansion rate changes during inﬂation, zero mode gravitons
escape into the bulk as KK gravitons, but at the same time bulk gravitons come onto thebrane to compensate for the loss, and these two eﬀects almost cancel each other. This seems
to happen, irrespective of the energy scale, in a wide class of the inﬂation models even witha not-so-small slow-roll parameter. It is suggested that this is not the case in the radiation
dominated decelerating universe [45, 46, 49, 50], where the decay into KK gravitons reduces
the amplitude of the gravitational waves on the brane. The junction model of Ref. [36] joining
de Sitter and Minkowski branes also show the suppression of the gravitational wave amplitude
at high energies.
What is the reason for the remarkable agreement of the braneworld spectrum and therescaled four-dimensional spectrum? Extremely long wavelength modes, which leave thehorizon during the initial de Sitter stage much before the Hubble parameter changes, have a2 )(H /2π)2 , and the amplitude of the perturbations stays
squared amplitude of (2C 2 (Hi )/MPliconstant during the subsequent stages 1 . The same is true for four-dimensional inﬂationary
cosmology, and so the mapping formula is applicable to these long wavelength modes. On
the other hand, a mode whose wavelength is much shorter than the Hubble horizon scale atthe transition time will feel the transition as adiabatic and hence the particle production is
2 )(H /2π)2 . The
exponentially suppressed, 0|Â†g Âg |0 ≈ 1, leading to P ≈ (2C 2 (Hf )/MPlfsame argument can be applied to the conventional cosmology. Thus it is not surprising thatthe mapping formula works for such short wavelength modes. However, at present there
seems no simple reason for the amplitude of the modes with k ∼ O(a(η0 )H̄) to coincide withthe rescaled one.

1

In the k2 → 0 limit, φ=const is a growing solution of Eq. (6.36).

Chapter 7

The spectrum of gravitational
waves
Cosmological inﬂation predicts the gravitational wave background arising due to quantum
ﬂuctuations in the graviton ﬁeld. Gravitational wave ﬂuctuations are stretched beyond thehorizon radius by rapid expansion during inﬂation, and at a later stage they come back inside
the horizon possibly with rich information on the early universe and hence on high energy
physics. Though yet undetected, gravitational waves will provide us with a powerful tool toprobe fundamental physics in near future [106].
Gravitational waves from inﬂation on the brane was ﬁrst studied by Langlois et al. [84],under an assumption that inﬂation is exactly de Sitter (Chapter 3). In this special case,the perturbation equation is separable and analytically solvable. A toy model called the“junction model” [36, 61] is an extended version of the pure de Sitter braneworld, whichallows a sudden change of the Hubble parameter H by joining two maximally symmetric(i.e., de Sitter or Minkowski) branes at some time (Chapter 5). Later, the junction modelis extended to a more general inﬂation model with a smooth expansion rate [65] (Chapter 6). To make the cosmological model more realistic, one should take into account theradiation-dominated phase that follows after inﬂation, and, at least in the low energy regime(H1), corrections to the evolution of gravitational waves are shown to be small [139, 63](Chapter 4). In a much more general and interesting case, i.e., in the high energy (H  1)
radiation-dominated phase, the perturbation equation no longer has a separable form andhence one cannot even deﬁne a “zero mode” and “Kaluza-Klein modes” without ambiguity.
To understand the evolution of gravitational waves in that regime, numerical studies havebeen done by Hiramatsu et al. [45, 46] and by Ichiki and Nakamura [49, 50]. Their results
give us a lot of implications, for example, on the damping nature of the gravitational wave
amplitude due to the Kaluza-Klein mode generation, but the initial condition they adopt isnaive, neglecting initial quantum ﬂuctuations in the Kaluza-Klein modes. Hence, its validity
is open to question.
The goal of the present chapter is to clarify the late time power spectrum of gravitational
waves in the Randall-Sundrum brane cosmology, evolving through the radiation-dominatedstage after their generation during inflation. We closely follow the same line in the previous
chapter [65], in which, using the Wronskian formulation, we have formulated a numerical
scheme for the braneworld cosmological perturbations. The initial condition in our analysis
is imposed quantum-mechanically, and therefore we will be able to obtain a true picture of76

77
the generation and evolution of gravitational perturbations in the braneworld.
This chapter is organized as follows. In Section 7.1, we start with giving the backgroundcosmological model and summarize basic known results concerning the gravitational wave
mode functions in the de Sitter and Minkowski braneworlds. In Section 7.2, we describe
the Wronskian formulation to obtain the power spectrum of gravitational waves, and thenwe show our numerical results in Sec. 7.3. In Section 7.4 we discuss an amount of the darkradiation generated due to excitation of Kaluza-Klein modes. Finally we conclude in Sec. 7.5.

7.1

Preliminaries

7.1.1

The background modelNow we describe a model for the background. We shall work in the cosmological setting of the
Randall-Sundrum braneworld, and so the bulk is given by a ﬁve-dimensional AdS spacetime.
The AdS metric in the Poincoré coordinates is
ds2 =

2
(−dt2 + δij dxi dxj + dz 2 ),
z2

(7.1)

where  is the bulk curvature scale and constrained by table-top experiments as   0.1
mm [99, 22]. A cosmological brane moves in this static bulk, the trajectory of which is
given by z = z(t). The scale factor of the universe is related to the position of the brane asa(t) = /z(t).
We consider the following cosmological model on the brane. The initial stage of the model
is given by de Sitter inﬂation with a constant Hubble parameter H = Hi , which is smoothly
connected to the radiation-dominated phase. (In order to join the two phases smoothly, thebrane is not exactly de Sitter at the very last stage of inﬂation.) In the radiation stage thescale factor evolves subject to the modiﬁed Friedmann equation [9, 10, 111, 71, 51]
H2 =

ρr 
ρr 
1
+
,
2
2σ
3MPl

(7.2)

2 /2 is the tension of the brane. Since
where ρr is the radiation energy density and σ = 6MPl
the conventional conservation law holds on the brane, we have ρr ∝ a−4 . Thus, in terms of
the proper time τ on the brane we obtaina(τ ) = a(τ1 )

τ
τ1

2


+ 2c

τ
τ1

1/4


− 2c

,

(7.3)

where τ1 is a ﬁducial time, c := 1 + [ρr (τ1 )/2σ] − 1, and ρr (τ1 )/σ = 2 /8τ12 . After a periodof time the energy scale of the universe becomes suﬃciently low, and the radiation-dominatedphase is then smoothly connected to the Minkowski phase. This artiﬁcial connection will notcause any unexpected problems on our ﬁnal result (i.e., power spectra of gravitational waves)because at the end of the radiation stage the brane universe is already in the low energy
regime. Just for simplicity we assume that the Minkowski brane is located at z = . Namely,
the scale factor is normalized so that a = a0 = 1 when the universe ceases expanding. The
motion of the brane is shown in Fig. 7.1.

78

t

U

V

Minkowski

lh

na
Fi
yp
er
ce

fa

r
su

Radiation-dominated

z
al

iti

In
ce

fa

ur

rs

pe

hy

de Sitter

Figure 7.1: Brane trajectory in static coordinates.

7.1.2

Minkowski and de Sitter braneworlds

Let us consider tensor perturbations in AdS spacetime bounded by a brane. We can decompose the graviton ﬁeld into a zero mode and Kaluza-Klein (KK) modes without ambiguity
when the brane is maximally symmetric. This is the reason why the initial and ﬁnal stagesof the background model are given by the de Sitter and Minkowski phases, respectively.
We write the perturbed metric as
ds2 =


2 
−dt2 + (δij + hij )dxi dxj + dz 2 ,
2
z

(7.4)

where hij is the transverse-traceless metric perturbation. We decompose it into the spatial
Fourier modes as usual.
For the analysis of perturbations from the Minkowski brane, the above Poincaré coordi-

79
nate system will be best suited, and the perturbation equation is
 2

∂
∂2
3 ∂
2
φ = 0,+k − 2 +
∂t2
∂z
z ∂z

(7.5)

subject to the boundary condition∂z φz= = 0.

(7.6)

The mode solutions of Eq. (7.5) were already found in Chapter 3. Going to quantum theory,
the graviton ﬁeld can be expanded in terms of the zero mode and KK modes as
 ∞ 

† ∗
(7.7)
dm Âm ϕm + Â†m ϕ∗m ,φ = Â0 ϕ0 + Â0 ϕ0 +0where Ân and Â†n (n = 0, m) are the annihilation and creation operators, respectively, of
their corresponding modes. The normalized zero mode function is given by
ϕ0 (t) = √1 −ikt
,
e
2k

(7.8)

while the normalized KK mode function isϕm (t, z) = √12ω3

e−iωt um (z),

(7.9)

with
(
um (z) := z2

m Y1 (m)J2 (mz) − J1 (m)Y2 (mz),2
[Y1 (m)]2 + [J1 (m)]2(7.10)andω=

k 2 + m2 .

(7.11)The normalization here is determined by the Wronskian conditions(ϕ0 · ϕ0 ) = −(ϕ∗0 · ϕ∗0 ) = 1,

(7.12)(ϕm · ϕm ) = −(ϕ∗m · ϕ∗m ) = δ(m − m ),(ϕ0 · ϕm ) = (ϕ∗0 · ϕ∗m ) = 0,
(ϕn · ϕ∗n ) = 0,
for n, n = 0, m,

where the Wronskian is deﬁned by [36](X · Y ) := −2i

 ∞


 3

dz
(X∂t Y ∗ − Y ∗ ∂t X) .
z

(7.13)In the de Sitter braneworld we introduce another set of coordinates (η, ξ), which is related
to (t, z) ast = η cosh ξ + t0 ,z = −η sinh ξ,(7.14)

80
where t0 is an arbitrary constant. In (η, ξ) frame the de Sitter brane is located at a ﬁxed
coordinate position ξ = ξb = constant, and the Hubble parameter on the brane is given
by Hi = −1 sinh ξb . The perturbation equation again has a separable form subject to the
Neumann boundary condition at the brane. Treating φ as an operator, the graviton ﬁeld canbe expanded as
φ = â0 φ0 + â†0 φ∗0 + ∞ dν âν φν + â†ν φ∗ν ,(7.15)

0where ân and â†n (n = 0, ν) are the annihilation and creation operators of each mode. The
explicit form of the normalized mode functions was already given before in Chapter 3.

7.2

Wronskian formulation

Due to the presence of an inﬁnite tower of Kaluza-Klein modes, cosmological perturbations in
the braneworld have inﬁnite degrees of freedom. Instead of solving an initial value problem
for such a system, it would be better to use the Wronskian formulation in order to takenecessary degrees of freedom out of inﬁnite information. In the present case, we would liketo know the ﬁnal amplitude of the zero mode, and therefore in fact what we need to do issolving the (backward) evolution of a single degree of freedom [36, 61, 65]. Following thesame line in the previous chapter [65], we shall compute the amplitude of gravitational waves
in the ﬁnal Minkowski phase using the double null coordinates and the Wronskian.
As explained in Section 7.1, our cosmological model is composed of the de Sitter inﬂationary phase followed by the radiation-dominated epoch, which is connected smoothly tothe ﬁnal Minkowski phase. In the initial de Sitter phase, the graviton ﬁeld can be expanded
as Eq. (7.15), while in the ﬁnal Minkowski phase it can be expanded as Eq. (7.7). We assume
that initially the gravitons are in the de Sitter invariant vacuum state annihilated by â0 andâν ,â0 |0 = âν |0 = 0.

(7.16)The expectation value of the squared amplitude of the zero mode in the ﬁnal stage is20| ϕ0 Â0 + ϕ∗0 Â†0 |0 = |ϕ0 |2 0| 1 + 2Â†0 Â0 |0 + oscillating part


1
Nf ,
k

where Nf := 0|Â†0 Â0 |0 is the number of created zero mode gravitons. Here we used thecommutation relation Â0 , Â†0 = 1 and assumed that Nf  1. The ﬁnal power spectrum isthen given by
P(k) :==

4πk 3 21· Nf
33(2π) (M5 ) kk2
2 Nf .
π 2 MPl

(7.17)

81
The operator Â0 can be projected out by making use of the Wronskian relations. Noting that
the Wronskian is constant in time, we haveÂ0 = (φ · ϕ0 )f = (φ · Φ)
= (φ0 · Φ)i â0 + dν(φν · Φ)i âν + h.c.,(7.18)

where Φ is a solution of the Klein-Gordon equation (6.36) whose ﬁnal conﬁguration is thezero mode function ϕ0 in the Minkowski phase, and subscript f and i denote the quantities
evaluated on the ﬁnal and initial hypersurfaces, respectively. Thus we clearly see that ﬁnal
zero mode gravitons are created from the vacuum ﬂuctuations both in the initial zero mode
and in the KK modes:
∗2(7.19)
Nf = |(φ0 · Φ)i | + dν|(φ∗ν · Φ)i |2 .
Correspondingly, the power spectrum (7.17) can be written as a sum of the two contributions:
P = P0 + PKK ,(7.20)

where
P0 :=PKK :=

7.3

k2∗
2
2 |(φ0 · Φ)i | ,π 2 MPlk2dν|(φ∗ν · Φ)i |2 .
2
π 2 MPl(7.21)(7.22)

Spectrum of gravitational waves

De Sitter inﬂation on the brane predicts the ﬂat primordial spectrum [84] 
2C 2 (Hi ) Hi 2.
δT2 :=
22π
MPl(7.23)

During inﬂation the gravitational wave perturbations are stretched to super-horizon scales,
and then they stays constant until horizon reentry, with their amplitude given by Eq. (7.23).
(φ = constant is a growing solution of Eq. (6.36) in the limit k 2 → 0.) The primordial
spectrum of gravitational waves from non-de Sitter inﬂation is studied extensively in Chapter 6 [65], and so in this chapter we concentrate on the simple case where inﬂation is givenby the exact de Sitter model.
For long wavelength modes with kk∗ , wherek∗ := a∗ H∗ = a∗ /

(7.24)

labels the mode that reenters the horizon when H = 1, the amplitude will decay as hµν ∝ a−1
after horizon reentry, because gravity on the brane is basically described by four-dimensional
general relativity in the low energy regime (H
1). In fact, it is explicitly shown thatleading order corrections to the cosmological evolution of gravitational waves are suppressedby 2 and 2 ln  at low energies [139, 63]. In particular, modes with k < k0 , wherek0 := a0 H0 = H0 ,(7.25)

82

(a)
-0.5
-1
-1.5
-2
-2.5
-0.4

-0.2

0

0.2

0.4

0.6

0.2

0.4

0.6

0.8

1

0.6

0.8

log( spectrum )

(b)
-1.5
-2
-2.5
-3
-3.5
0

(c)
-1.5
-2
-2.5
-3
-3.5
-4
-4.5

0

0.2

0.4

1

1.2

1.4

log(k/k )Figure 7.2: Power spectra of gravitational waves from inﬂation with Hi = 10 (a), 42 (b),and 100 (c), normalized by δT2 /2. The total power spectrum is shown by red circles, whileblue crosses represent the contribution only from the initial zero mode. Green squires indicate results from basically four-dimensional calculations, only including the eﬀect of themodiﬁcation of the background expansion rate.

83
and H0 is the Hubble parameter evaluated at the end of the radiation-dominated phase (i.e.,just before the Minkowski phase)1 , reenter the “horizon” in the ﬁnal Minkowski phase, andthen they begin to oscillate (but their amplitude will not decay). As a result, the mean-squarevacuum ﬂuctuations of such modes become half of the initial value, leading to
P=

δT2
,
2

for k < k0 .

(7.26)In Ref. [36], the same spectrum is obtained for scales larger than the AdS and horizon scales
by using the “junction model”, in which an instantaneous transition from a de Sitter to a
Minkowski brane is assumed. For the reason mentioned above, modes with k0 < kk∗ havethe standard spectrum,


δ2
P= T
2

k
k0

−2
.

(7.27)

Something nontrivial may happen to gravitational waves with k  k∗ . If the eﬀects ofmode mixing are neglected, only the modiﬁcation of the background expansion rate alters
the spectrum for these short wavelength modes toδ2P̃  T2kk∗

−2/3 

k∗
k0

−2
.

(7.28)

However, this evaluation will not be correct because mode mixing is expected to be eﬃcientat high energies.
Our procedure to obtain a correct power spectrum is as follows. We solve the perturbation
equation (6.36) with its boundary condition set to be Φ(U, Vf ) = ϕ0 on the ﬁnal hypersurface.
The numerical backward evolution scheme we use here is the same as that used in the previous
chapter [65], and the detailed description of the scheme is found there. After obtaining
the conﬁguration on the initial hypersurface, we evaluate the Wronskian to get the power
spectrum.
We performed numerical calculations for three diﬀerent values of the inﬂationary Hubble
parameter, Hi = 10, 42, and 100. The radiation-dominated phase is terminated when the
Hubble parameter decreases down to H  0.03/ =: H0 , and then it is connected smoothly
to the Minkowski phase, so that we can see the amplitude of the well-deﬁned zero mode. The
numbers of grids are 90000 in the U -direction and 12000 in the V -direction, and the grid
separation is chosen to be ∼ 0.005 × . Integration over the KK index ν is performed up toν = 25 with an equal grid spacing of 0.05.
The power spectra of gravitational waves are shown in Fig. 7.2. Assuming that the power
spectrum is of the form

P=A

k
k0

n
,

(7.29)

1
To be more precise, the scale factor in Eq. (7.25) should be replaced by a bit smaller one than a0 = 1,
because a0 is deﬁned as the scale factor in the Minkowski phase (where the Hubble parameter vanishes).
However, as the smooth period connecting the radiation and Minkowski phases is taken to be very short, thispoint is not important.

84

0.1

rKK

0.01
0.001
0.0001
0

0.2

0.4

0.6

0.8

1

1.2

1.4

log(k/k )Figure 7.3: Contribution of initial Kaluza-Klein ﬂuctuations, rKK , for the model with Hi =100.
we ﬁnd that irrespective of the inﬂationary energy scale, the parameters are approximately
given byδT2,2n  −2.

A (7.30)(7.31)

Namely, we have the same spectrum as the standard one [Eq. (7.27)] even for short wavelength
modes with k  k∗ . As is shown in Fig. 7.3, the contribution of the vacuum ﬂuctuations in
the initial KK modes to the ﬁnal spectrum,rKK (k) :=PKK,
P0 + PKK(7.32)

never exceeds 10% so far as the present calculations are concerned, and hence it gives a
subdominant eﬀect. On the other hand, the excitation of KK modes suppresses the amplitudeof the gravitational waves relative to Eq. (7.28), and our result implies that the eﬀect of themodiﬁcation of the background Friedmann equation compensate this suppression, leading
to approximately the same spectral tilt as that in conventional four-dimensional cosmology.
This is consistent with the numerical study by Hiramatsu et al. [46], in which they assume
the initial conﬁguration in the bulk to be a de Sitter zero mode and obtain P0 ∝ k −2 .
Unfortunately, due to the limited number of grids in the U -direction, it is diﬃcult toevaluate accurate values of rKK ; the convergence is not so good (Fig. 7.4). However, sincerKK decreases with an increasing number of grids, it is strongly expected that the eﬀect of
the initial KK ﬂuctuations is negligibly small. To obtain a more accurate evaluation of thecontribution of the initial KK modes, we need an improved numerical formulation, though
it seems quite unlikely that rKK turns to increase at a much larger number of grids2 . (Weconﬁrmed that the convergence of the zero mode part P0 is suﬃciently good.)2

After completing the original contribution (hep-th/0511186 [66]), we have improved the numerical code

85

0.07

rKK

0.06
0.05
0.04

50000

60000

70000

80000

90000

Number of grids
Figure 7.4: rKK for Hi = 100 and log(k/k∗ ) = 1.12, versus the number of grids in the
U -direction.

7.4

Generation of dark radiation

So far we have concentrated on ﬁnal zero mode gravitons created from initial vacuum ﬂuctuations. In this section we shall discuss the generation of KK gravitons. In particular, we
are interested in the KK mode gravitons created from initial ﬂuctuations in the zero mode,
because from the ﬁve-dimensional point of view they are interpreted as gravitons that escapefrom the brane. At suﬃciently late times, all the emitted gravitons fall deep into the bulk,
and then the bulk spacetime is described as an AdS-Schwarzschild black hole, the mass ofwhich divided by a4 is viewed as the “dark radiation” from a brane observer [42, 88, 90, 91].
Before going to the estimation of the energy density of ﬁnal KK mode gravitons, ﬁrst
let us take a look at the energy density of zero mode gravitons ρGW . Since at low energies
gravitational waves evolve in a standard manner, their energy density behaves as ρGW ∝ a−4 .
Therefore, the ratio ρGW /ρr is an invariant quantity in the low energy regime, irrespective of
the cosmic expansion. Evaluating it at the end of the radiation stage, we have
  2
ρGW,0
δ
ki
ρGW
· T ∼ δT2 .
= ln(7.33)
ρr
ρr,0k0
6Note that δT < 10−5 . In deriving the estimate (7.33), we used the formula ki
2d ln k k 2 P(k),ρGW,0 = MPl(7.34)

k0

and substituted the numerical result obtained in the previous section, P  (δT2 /2)(k/k0 )−2 ,where k0 can be eliminated in favor of ρr,0 by using the Friedmann equation at low energies
and obtained more accurate values of rKK . The result is not so diﬀerent from the above. In the case of
Hi = 100, for example, we have rKK  0.022 for log(k/k∗ ) = 1.12 and rKK  2.4 × 10−4 for log(k/k∗ ) = 0.40.
Thus we safely conclude that the eﬀect of the initial KK ﬂuctuations is negligibly small.

86
2 ). The upper limit of the integral may be given by the inverse horizon
k02 = H02  ρr,0 /(3MPlscale at the end of inﬂation,

ki := ai Hi ,(7.35)

because the particle production is exponentially suppressed on sub-horizon scales.
Let ρ̃GW be the energy density of gravitational waves obtained by neglecting the mode
mixing eﬀect. More precisely, ρ̃GW is the energy density of gravitational waves hµν wherehµν is a solution of the conventional perturbation equation (∂τ2 + 3H∂τ + k 2 /a2 )hµν = 0 withthe cosmic expansion given by a solution of the modified Friedmann equation. This would bemuch greater than ρGW . Then, ∆ρ := ρ̃GW − ρGW  ρ̃GW is the energy density that leaks
from the brane, and by deﬁnition ∆ρ is proportional to a−4 as long as it is evaluated in the
low energy regime. Thus, ∆ρ/ρr is an invariant quantity. Now ∆ρ,0 can be calculated fromthe spectrum of the form (7.28), and we have an estimate
δ2
∆ρ,0
∆ρ
=
 T × Hi ,
ρr
ρr,0
8

(7.36)

where we used a4 ρr = a4∗ σ = ρr,0 and the Friedmann equation at high energies, H 2 
2 σ. The estimate (7.36) implies that a large amount of energy (compared to ρρ2 /6MPl
GW ) islost from the brane. Is the escaped energy ∆ρ directly transferred to the ﬁnal bulk gravitons?
To discuss this point, we compare it with the energy density of the generated dark radiation.
Since KK modes are excited dominantly at high energies but not at low energies, the darkradiation, as is deduced from its name, behaves like a radiation component, ρDR ∝ a−4 , atlate times. Hence, we shall see the ratio ρDR /ρr but it may be evaluated at the end of theradiation stage.
The total number of the created bulk gravitons is given again by the Wronskian as




†
∗
2
dm0|Âm Âm |0 = dm |(φ0 · Φm )i | + dν|(φ∗ν · Φm )i |2 ,where Φm is a solution of the Klein-Gordon equation (6.36) whose ﬁnal conﬁguration is a KK
mode function ϕm in the Minkowski phase. Concentrating on the ﬁrst part, |(φ∗0 · Φm )|2 dm
is identiﬁed as the number of KK gravitons coming from the initial zero mode ﬂuctuations,with their mass between m and m + dm. Thus the energy density is expressed asρDR,0 :=

=

d3 k
(2π)3
d ln kdm ω|(φ∗0 · Φm )i |2
d ln mmk 3 ω ∗|(φ0 · Φm )i |2 .
2π 2

(7.37)As a side remark, in the junction model of Ref. [36], the second part, i.e., the number of KK
gravitons created from the initial ﬂuctuations in KK modes, is suppressed compared to theﬁrst one.
We numerically calculated the Wronskian |(φ∗0 · Φm )|2 in a similar manner to the previous
section. The Hubble parameter during inﬂation is chosen to be Hi = 42. In this case, thenumber of grids are 150000 in the U -direction and 20000 in the V -direction, and the grid
separation is about 0.0036 × .

87Figure 7.5: Energy of bulk gravitons multiplied by the phase space factor.
The integrandd ln mmk 3 ω ∗|(φ0 · Φm )i |2
2π 2

is plotted in Fig. 7.5. For each k, integration over m is performed up to m ∼ 2/ with a grid
spacing of ∆ ln m  0.17. Performing the integration over k, we obtain
ρDR,0 ≈ 0.04 × −4 ,(7.38)

where one should note that contributions from modes with k < k∗ are suppressed. The
radiation energy density can be written as2
ρr,0 = a4i ρr,i  a4i (6MPlσ)1/2 Hi 

9 1
(ai Hi )4 ,2π 2 δT2(7.39)

which can be obtained by using the modiﬁed Friedmann equation and noting that δT2 
2 for H  1. From Eqs. (7.38) and (7.39) we have an estimate
3Hi3 /4π 2 MPliρDR< O(1) × δT2 .
ρr(7.40)

This result indicates that the energy density of the generated dark radiation is not larger thanthat of zero mode gravitons [Eq. (7.33)]. Of course, this is a completely harmless amountof an extra radiation component [48]. The scattering of particles on the brane in the early
universe, discussed in [42, 88, 90, 91], can be a more eﬃcient way to produce bulk gravitons.
Although a large amount of energy is lost from the brane via excitation of KK modes in
the high energy regime, the ﬁnal energy density of the dark radiation is much smaller thanthat, without an enhancement factor like Hi in Eq. (7.36). This discrepancy is explained asfollows [42, 90]. In the high energy regime, the motion of the brane is so relativistic (in the

88frame deﬁned by the static bulk coordinates) that emitted gravitons run almost parallel tothe brane trajectory. These gravitons stay in the vicinity of the brane and bounce oﬀ it manytimes during the high energy stage, until eventually they are reﬂected by the non-relativisticbrane to fall oﬀ into the bulk. During this process, the gravitons lose a large portion of their
momentum transverse to the brane because they repeatedly hit the retreating brane. Thisqualitatively accounts for the smallness of the ﬁnal energy density of the dark radiation. To
justify the above interpretation quantitatively, a more rigorous analysis will be needed in the
direction of Refs. [90, 109], which includes calculating the pressure to the brane due to theeﬀective energy-momentum tensor of the bulk gravitational waves.
Here we should comment on the result of the junction model obtained by Gorbunov etal. [36]. In terms of the power spectrum, their result is summarized as
⎧
δT2
⎪
⎪
,
(k
k∗ ),
⎨
2
P≈
(7.41)
δT2
4
⎪
⎪
, (k∗
k
ki ),
⎩
2 (k/k∗ )2where k∗ = a∗ /, ki = ai Hi , and a∗ = ai (= 1) because a de Sitter inﬂationary stage is directly
joined to a Minkowski phase in the junction model (see also Appendix A of Ref. [61]). From
this we can estimate the energy density that leaks from the brane as2
Hi2 δT2 ≈ ρe.i. δT2 × Hi ,∆ρ ≈ MPl(7.42)

where ρe.i. is the energy density at the “end of inﬂation”. On the other hand, according toAppendix D of Ref. [36], the energy density of created KK gravitons is given byρDR ≈ Hi4 ≈ ρe.i. δT2 .

(7.43)

Thus, we ﬁnd that ∆ρ ∼ ρDR × Hi , which is consistent with our present result. [Notethat in the junction model the energy density of ﬁnal zero mode gravitons is estimated as
2 /2 )δ 2 and hence ρρe.i. δT2 ∼ ρDR .]ρGW ∼ (MPl
GW
T

7.5

Summary

We have examined the power spectrum of the gravitational wave background in the cosmological scenario of the Randall-Sundrum braneworld. There are three possible ingredients which
may lead the power spectrum to a non-standard one: the unconventional background expansion rate due to the ρ2 term in the Friedmann equation, the excitation of KK modes during theradiation-dominated stage at high energies, and the eﬀect of initial vacuum ﬂuctuations in KK
modes. Previous estimates are based on a rather simple toy model [36] or numerical studies
about the classical evolution of perturbations, neglecting the initial KK ﬂuctuations [45, 46].
In the present analysis, initial conditions are set in a quantum-mechanical manner and hence
the eﬀect of the initial KK ﬂuctuations is included. Along the same line in Ref. [65], we makeuse of the Wronskian formulation to obtain the ﬁnal amplitude of the zero mode gravitational
waves numerically. We have found that the eﬀect of initial KK vacuum ﬂuctuations are subdominant: rKK < 0.1. Our result conﬁrms that the damping of the amplitude due to the
KK mode excitation and the enhancement due to the modiﬁcation of the background expansion rate mainly work, but almost cancel each other. Consequently, the power spectrum is

89basically the same as the standard one obtained in conventional four-dimensional cosmology.
We believe that the cancellation between the two eﬀects is a phenomenon peculiar to theradiation-dominated phase. To make the particularity of the radiation stage clear, it would
be interesting to investigate consequences of a diﬀerent equation of state parameter w(= p/ρ)
after the inﬂationary stage. This is the next issue we plan to report in a future publication.
We have also estimated the energy density of the generated dark radiation numerically,and shown that only a tiny amount is generated. It is smaller than the energy density of zeromode gravitational waves.

Chapter 8

Cosmological perturbations in thebulk inﬂaton model
So far we have elaborated on cosmological tensor perturbations in the Randall-Sundrum
braneworld scenario [123, 124], where the bulk is empty except for a negative cosmologicalconstant. In this chapter, we shall consider a diﬀerent class of a braneworld model so called
the bulk inﬂaton model. A conservative construction of inﬂation models in the context of the
RS model assumes that slow-roll inﬂation is driven by a scalar ﬁeld conﬁned to the brane,
and such a model is considered, e.g., in Ref. [100]. An empty bulk, however, seems less likely
from the point of view of uniﬁed theories, which often require various ﬁelds in addition to
gravity. Considering a bulk scalar ﬁeld, Himemoto et al. [43, 44, 130, 108] have shown that,interestingly, a bulk scalar ﬁeld can mimic the standard slow-roll inﬂation on the brane undera certain condition (see also Ref. [60]).
Also in the context of heterotic M theory, cosmological solutions has been studied [97,125, 98, 133]. In the model discussed in Refs. [97, 125], the scalar ﬁeld has an exponential
potential in the bulk and the tensions of the two branes are also exponential functions of
the scalar ﬁeld. In this model the power-law expansion (but not inﬂation) is realized on thebrane. A single-brane model with such exponential-type potentials is also interesting, andit has been investigated for a static brane case [54, 23, 13] and a dynamical (cosmological)
case [118, 29, 87, 19]. Recently an inﬂationary solution was found in a similar setup by
Koyama and Takahashi [75, 77], extending the results of Refs. [23, 13, 118, 29]. A striking
feature of their model is that cosmological perturbations can be solved analytically.
As we have stressed through this thesis, for the purpose of giving a prediction in braneworldmodels it is essential to take into account perturbations in the bulk. To do so, generally wecannot avoid solving partial diﬀerential equations in the bulk with discouragingly complicated boundary conditions. Only a few cases are known where perturbation equations can beanalytically solved [84, 36, 61]. One of them is the special class of bulk inﬂaton models mentioned above [75, 77]. In this chapter we clarify the reason why the perturbation equations
are soluble in this special case. Based on this notion, we present a new systematic method
to ﬁnd a wider class of background cosmological solutions and to analyze perturbations fromthem. The essence as to why our method works is deeply connected to the fact that tensor
perturbations in the vacuum RS braneworld with a de Sitter brane are exactly solvable.
This chapter is organized as follows. In the next section, we explain our basic ideas ofconstructing background solutions and of analyzing cosmological perturbations. In Sec. 8.2
90

91
we consider a model with a single scalar ﬁeld in the bulk, which is the main interest of
this work, and derive an eﬀective theory on the brane. Then, in Sec. 8.3 we present someexamples of exact solutions for the background cosmology obtained by making use of theideas explained in Sec. 8.1. Section 8.4 deals with cosmological perturbations. Section 8.5 isdevoted to discussion.

8.1

Basic ideas

8.1.1

Solutions in the Randall-Sundrum vacuum braneworld

We begin with a model whose action is given by
where
Sg =



1
2κ2D+1S = Sg + Sb ,

(8.1)

√
dD+1 X −G (R[G] − 2ΛD+1 ) ,(8.2)

is the action of (D + 1)-dimensional Einstein gravity with a negative cosmological constantΛD+1 = −D(D − 1)/22 ,

√
(8.3)
Sb = − dD X −g σ,is the action of a vacuum brane with a tension σ, and g is the determinant of the induced
metric on the brane.
We assume Z2 -symmetry across the brane, so that the tension of the brane is determinedby the junction condition asκ4D+1 σ 2
1
− 2,
(8.4)
2
4(D − 1)
where H0 is related to the D-dimensional cosmological constant induced on the brane Λb by1
Λb = (D − 1)(D − 2)H02 ,(8.5)
2
and it represents the deviation of σ from the ﬁne-tuned Randall-Sundrum value 2(D −1)/κ2D+1 .
One of the key ideas in the present chapter is to make use of the following well-known
fact. If a metric ds2(D) is a solution of the D-dimensional vacuum Einstein equations with acosmological constant Λb , thends2(D+1) = e2ω(z) dz 2 + ds2(D) ,
(8.6)
H02 =

is a solution of the (D + 1)-dimensional model deﬁned by Eq. (8.1), and the warp factor eω(z)is given byH0eω(z) =.
(8.7)
sinh(H0 z)(For a Ricci-ﬂat brane, we have Λb = 0. In this case the warp factor reduces to eω(z) = /z.)Namely, we can construct a (D +1)-dimensional solution in the Randall-Sundrum braneworldfrom a vacuum solution of the D-dimensional Einstein equations. A well-known example isthe ﬁve-dimensional black string solution obtained from the four-dimensional Schwarzschildsolution [18].

92

8.1.2

Bulk inﬂaton models from dimensional reduction

We now explain howto obtain an (n + 2)-dimensional braneworld model with bulk scalarﬁelds from (n + 2 + ji )[= D + 1]-dimensional spacetime by dimensional reduction. We use
n to represent the number of uncompactiﬁed spatial
dimensions on the brane, which is three
in realistic models. Let us consider (n + 2 + ji )-dimensional spacetime whose metric isgiven byds2(D+1) = GAB dX A dX B = Gab (x)dxa dxb +e2φi (x) dσi2 ,(8.8)
where dσi2 is the line element of a ji -dimensional constant curvature space with the volume
Vi . Here the indices a and b run from 0 to (n + 1), and φi is assumed to depend only on the(n + 2)-dimensional coordinates xa .
Then dimensional reduction to (n + 2) dimensions yields



√
1
(n+2)
n+2
Q
ji G ab ∂a φi ∂b φi + G ab ∂a Q∂b Q=
x −G e R[G] −
Sg
d
2
2κn+2


−2φi
,
(8.9)
Ki ji (ji − 1)e−2ΛD+1 +
where
Q :=



ji φi ,

(8.10),
κ2n+2 := κ2D+1 / Vi and Ki represents the signature of the curvature of the metric dσi2 : −1
(open), 0 (ﬂat), or 1 (closed). Making a conformal transformation to the “Einstein frame,”G̃ab = e2Q/n Gab ,we have
Sg(n+2)* 
1=
ji G̃ ab ∂a φi ∂b φi − G̃ ab ∂a Q∂b Q
dn+2 x −G̃ R[G̃] −nKi ji (ji − 1)e−2φi −2Q/n .
−2ΛD+1 e−2Q/n +12κ2n+2(8.11)

(8.12)

Notice that the kinetic term of each scalar ﬁeld has an appropriate signature since ji > 0. A
parallel calculation gives

√
(n+2)
= − dn+1 x −q σ̃e−Q/n ,(8.13)
Sb,
where σ̃ = σ Vi , so that κ2n+2 σ̃ = κ2D+1 σ, and q is the determinant of the induced metricon the brane, qâb̂ := G̃âb̂ |z=zb = e2Q/n Gâb̂ |z=zb . The caret upon indices represents restriction
to the subspace parallel to the brane. Hence â and b̂ run from 0 to n. Here zb represents thelocation of the brane. In this manner, we can derive (n + 2)-dimensional braneworld models
with bulk scalar ﬁelds which have exponential-type potentials both in the bulk and on thebrane.

93

Figure 8.1: Schematic of the higher dimensional vacuum description of an (n+2)-dimensionalbulk inﬂaton model and the derivation of the (n + 1)-dimensional eﬀective theory on thebrane. The top left picture represents the bulk inﬂaton model that we are interested in.
To analyze cosmological background solutions (and “zero mode” perturbations), we use the(n + 1)-dimensional description shown in the bottom left corner. On the other hand, we makeuse of the (D + 1)-dimensional description presented in the top right corner to simplify the
perturbation analysis.

8.2

Single scalar ﬁeld in the bulk

From now on, for simplicity, we focus on models with a single bulk scalar ﬁeld. The mdimensional space represented by e2φ dσ 2 is compactiﬁed on either a torus (K = 0), a sphere(K = 1), or a compact hyperboloid (K = −1). Using a canonically normalized ﬁeld ϕ :=κ−1n+2 m(m + n)/n φ, the (n + 2)-dimensional reduced action is written as

S

(n+2)


*
1
1
d
x −G̃
R[G̃] − G̃ ab ∂a ϕ∂b ϕ − V (ϕ)
22κ2n+2√− dn+1 x −q U (ϕ),
n+2=

(8.14)

where the potentials are
V (ϕ) = −(m + n)(m + n + 1) −2√2bκn+2 ϕ Km(m − 1) −√2κn+2 ϕ/nbe−e,2κ2n+2 22κ2n+2(8.15)√

U (ϕ) = σ̃e− 2bκn+2 ϕ ,(8.16)

94
with

(
b :=

m
.
2n(m + n)(8.17)

If we assume that the metric ds2(D+1) is given in the form of (8.6), the action can further
reduced to the (n + 1)-dimensional eﬀective one on the brane. We write the D-dimensionalpart of the metric in the form of(n+1)(x)dxâ dxb̂ + e2α(t) dσ 2 .
âb̂ds2(D) = gÂB̂ dxÂ dxB̂ = g(8.18)

Then, comparing the coeﬃcient in front of dσ 2 , it follows thatφ(t, z) := α(t) + ω(z).

(8.19)

Also, the (n + 2)-dimensional metric G̃ab is written asG̃ab dxa dxb = e2mφ/n Gab dxa dxb = e2(m+n)ω/n qâb̂ dxâ dxb̂ + e2mα/n dz 2 ,(8.20)

where we have set eω(zb ) = 1. Substituting the above expression into (8.14), we perform theintegration over z to obtain
1
m(m + n) mα/n âb̂
(n+1)
n+1 √
S
d
= 2
x −q emα/n R[q] −q ∂â α∂b̂ α
e
n
2κn+1
−2Λb e−mα/n + Km(m − 1)e−2α−mα/n ,where κ2n+1 = κ2D /V with
κ2D := κ2D+1(8.21)  ∞−12e(m+n)ω dz,
zb

and V is the volume of the m-dimensional compactiﬁed space.
The above reduction to (n + 1) dimensions can be done more easily, starting with the(D + 1)[= n + 2 + m]-dimensional action. First we perform the integration over z and obtaina D-dimensional eﬀective action,
√1
dD x −g(R[g] − 2Λb ),(8.22)
S (D) = 22κD
where Λb is, as before, the one deﬁned by Eq. (8.5). The obtained eﬀective action is that for
D-dimensional pure gravity with a cosmological constant Λb . Compactifying m dimensions
further, and taking into account
(n+1)
,
âb̂

qâb̂ = e2mα/n g(8.23)

the same expression for the (n + 1)-dimensional eﬀective action (8.21) can be recovered in aparallel way as we did for the reduction from (D + 1) dimensions to (n + 2) dimensions. Using
ϕBD := emα/n and ωBD := n(m + n)/m, we can rewrite the action in a familiar form:


1
ωBD
(n+1)
n+1 √
S
= 2
x −q ϕBD R −(∂ϕBD )2
d
ϕBD
2κn+1
−(m+2n)/m−2Λb ϕ−1
BD + Km(m − 1)ϕBD,(8.24)

95
which implies that the eﬀective theory on the brane is described by a scalar-tensor theory.
The action (8.21), or equivalently (8.24), describes not only the background unperturbed
cosmology but also the zero mode perturbation, both of which are independent of the extradimensional coordinate, z, apart from the overall factor e2ω .
The induced metric on the brane qab is the metric in the Jordan frame. If we use themetric in the Einstein frame,2mαq̃âb̂ = e n(n−1) qâb̂ ,the eﬀective action becomes


(n+1)
n+1
= d
x −q̃
S
where ϕ̃ := κ−1
n+1(8.25)

1
1
R[q̃] − q̃ âb̂ ∂â ϕ̃∂b̂ ϕ̃ − Ṽ (ϕ̃) ,
2
2
2κn+1

(8.26)

m(m + n)/n α and the potential is√

2 2nKm(m − 1) −e
κ2n+1 Ṽ (ϕ̃) = Λb e− n−1 bκn+1 ϕ̃ −2√2n(m+n−1)bκn+1 ϕ̃m(n−1)

2.

(8.27)

Obviously, the system deﬁned by the above action is equivalent to Einstein gravity with ascalar ﬁeld. A discussion of this type of potential can be found in Ref. [117].

8.3

Examples of the background spacetime

In this section, we give some examples of D-dimensional vacuum solutions, which generate(D + 1)-dimensional braneworld solutions by making use of the prescription described inSec. 8.1.1. Here we discuss models with a single scalar ﬁeld and investigate their cosmologicalevolution in detail. Generalization to the case of multiple scalar ﬁelds is given in Sec. 8.7.

8.3.1

Kasner-type solutions

We ﬁrst consider the following Kasner-type solution as an example of the Ricci-ﬂat case,Λb = 0,gÂB̂ dxÂ dxB̂ = e2α(η) −dη 2 + γµν dy µ dy ν + e2β(η) δij dxi dxj ,where γµν is the metric of m-dimensional constant curvature space, and i and j run from 1 ton. Under the assumption of the above metric form, we solve the vacuum Einstein equations,e2α Rηη = n β  − α β  + β  + mα = 0,2(8.28)

e2α Rµν = δµν K(m − 1) + nα β  + (m − 1)α + α = 0,(8.29)

e2α Ri j = δi j nβ  + (m − 1)α β  + β  = 0,(8.30)

22

where the prime denotes diﬀerentiation with respect to η. Eliminating α and α from theabove three equations, we have(n(m + n − 1)  4
β =∓β − K(m − 1)2 β  2 .
m

96
We can easily integrate this equation. For example, when K = 1 (compactiﬁed on them-sphere S m ), the solution of this equation becomesnβ  =±(m − 1)q,sin[(m − 1)η]

where
(
q :=

mn
,
m+n−1

(8.31)and the integration constant was used to shift the origin of time. Thus we ﬁnd ±q/nm−1η
.
eβ = tan2
Substituting this result into Eq. (8.30), we haveα = cot[(m − 1)η] ∓q.
sin[(m − 1)η]This can be integrated as(m−1)αe= sin[(m − 1)η] cot

m−1
η
2

 ±q
.

The solution for K = −1 is easily obtained by replacing sin, tan, and cot in the above
expressions by sinh, tanh, and coth, respectively. The solution for K = 0 behaves as eβ ∝
η ±q/n and e(m−1)α ∝ η 1∓q .
Let us further investigate cosmology of the above example. Setting n = 3 and K = 1, theinduced four-dimensional metric becomesqâb̂ dxâ dxb̂ = e2mα/3 −e2α dη 2 + e2β δij dxi dxj ,(8.32)where1qeα = {sin[(m − 1)η]} m−1 {cot[(m − 1)η/2]} m−1 ,eβ = {tan[(m − 1)η/2]}q/3 ,(8.33)and q = 3m/(m + 2) 1 . We set q to be positive without any loss of generality since thesignature of q is ﬂipped by a shift of the origin of time, η → η +π. Here one remark is in order.
In the original (D+1)-dimensional model m represents the number of compactiﬁed dimensions
and therefore is supposed to be an integer. However, m is just a number parameterizing
the form of the scalar ﬁeld potential when we start with the action (8.14) obtained after
dimensional reduction. We therefore ﬁnd that m can be any real positive number in this1

We should remark that the dynamical solutions in Ref. [29] can be obtained if we compactify the ndimensional section e2β δij dxi dxj and regard the m-dimensional section e2α γµν dy µ dy ν as our three-space instead. The ﬂat case (K = 0) corresponds to the solution in Ref. [118]. We should also mention that recently
there has been a lot of discussion about the solutions with K = −1 in an attempt to explain accelerated
expansion of the universe in the context of M or string theory [117, 140, 119, 120, 128, 28, 20, 40, 21]. Note,however, that these arguments are not in the braneworld context.

97
context. The positivity needs to be assumed to keep the appropriate signature of the kinetic
term for the scalar ﬁeld φ, or equivalently to keep the relation between φ and ϕ real. (Strictly
speaking, the case with m < −n = −3 is √also allowed.) Then q can be regarded as acontinuous parameter with its range 0 < q < 3.
The cosmological time τ is related to η viadτ = e(m+3)α/3 dη.

(8.34)

Recall that qâb̂ , the metric induced on the brane in the ﬁve[= n+2]-dimensional model (8.14),
is related to gâb̂ by Eq. (8.23). Hence the scale factor associated with the metric qâb̂ is given
by a = emα/3+β , and therefore the Hubble parameter on the brane H := a−1 da/dτ is given
by H = (mα /3 + β  )e−mα/3−α . Substituting the above solution (8.33) into these expressions,
we obtain
q(q 2 −3)

2q 2

a = {sin[(m − 1)η]} 9(q2 −1) {tan[(m − 1)η/2]} 9(q2 −1) ,2
q(q 2 −9)q q 2 − 3 + 2q cos[(m − 1)η]− 8q−
9(q 2 −1) {tan[(m − 1)η/2]} 9(q 2 −1) .
H ={sin[(m−1)η]}
3(q 2 − 3)
The relation between the coordinate time η and the cosmological time τ (8.34) is not
so obvious, but the asymptotic behavior can be easily studied. When η → 0, we haveq(q+9)τ ∝ η 9(q+1) → 0 andq(q+3)a ∝ η 9(q+1) ∝ τ p− ,H ∝ ηeα ∝ η

q(q+9)− 9(q+1)
3−q 2− 3(q+1)

q+9
− q+3

∝a

,

→ +∞,

where the exponent p− is deﬁned below in Eq. (8.35). For η̄ := η − π/(m − 1) → 0, the−q(q−9)

the propercosmological time is (locally) expressed as τ ∝ η̄ 9(q−1) . Therefore the range of √time τ is inﬁnite for the parameter region 0 < q < 1, while it is ﬁnite for 1 < q < 3. In thislimit η̄ → 0, the scale factor, the Hubble parameter, and the scalar ﬁeld behave asq(q−3)(0 < q < 1),τ p+ ,− 9(q−1)√∝a ∝ η̄
p+(τend − τ ) , (1 < q < 3),
q(q−9)q−9− q−3

H ∝ η̄ 9(q−1) ∝ aeα ∝ η̄

3−q 2− 3(1−q)→,+∞, (0 < q < 1),√
0, (1 < q < 3).

In the above expressions, we have used
m+3.
(8.35)
4m + 9 ± 3m(m + 2)√√
The ranges of p+ and p− are 1/(4 + 3) < p+ < 1/3 and 1/3 < p− < 1/(4 − 3).
The behavior of this solution is easily understood from the viewpoint of the four-dimensionaleﬀective theory described by the action (8.26), as was discussed in Ref. [28]. The potential (8.27) with Λb = 0 and K = 1 is shown in Fig. 8.2. For 0 < q < 1 (0 < m < 1) the
p± :=

98

Figure 8.2: The motion of the scalar ﬁeld in the potential. The behavior of the scale factor
in the Jordan frame is also presented.
The potential is positive for 0 < q < 1 (left ﬁgure),√
whereas it is negative for 1 < q < 3 (right ﬁgure).

Figure 8.3: Sketch of the behavior of the scale factor a (dashed lines) and the Hubble parameter H (solid lines) in the Jordan frame as functions of the proper time on the brane τ . The
left ﬁgure shows the solution of (8.33) with q > 1. The right ﬁgure describes the solutionof (8.40) with the plus sign in the exponent.

potential is positive, while for q > 1 (m > 1) the potential is negative. In the former case, thescalar ﬁeld α starts at α = ∞, climbs up the slope of the potential, turns around somewhere,
and ﬁnally goes back to α = ∞. In the latter case, α starts to roll down from α = ∞. The
universe expands for a period of time and eventually it starts to contract. Finally α falls into
the bottomless pit within a ﬁnite time, where the universe ends up with a singularity.
Here we note that the above picture based on the four-dimensional eﬀective theory describes the dynamics in the conformally transformed frame in which the metric is given byq̃âb̂ , whereas we suppose that the “physical” metric is given by the induced metric on thebrane qâb̂ . In principle, the cosmic expansion law can look very diﬀerent depending on theframe we choose. The dynamics in the “physical” frame therefore can be apparently very
diﬀerent. However, the above discussion is still useful since the conformal rescaling does notchange the causal structure of the spacetime.

99

8.3.2

Kasner-type solutions with a cosmological constant

The next example is a generalization of the Kasner-type spacetime including a cosmologicalconstant Λb . Let us assume that the metric is in the form ofgÂB̂ dxÂ dxB̂ = −dt2 + e2α(t) δµν dy µ dy ν + e2β(t) δij dxi dxj ,(8.36)

where i and j again run from 1 to n, but here the metric of m-dimensional space is chosen to
be ﬂat (K = 0) because otherwise the solution with Λb = 0 is not obtained analytically. The
D[= n + 1 + m]-dimensional vacuum Einstein equations with a cosmological constant reduce
to




(8.37)m α̈ + α̇2 + n β̈ + β̇ 2 = (m + n)H02 ,α̈ + α̇ mα̇ + nβ̇ = (m + n)H02 ,(8.38)β̈ + β̇ mα̇ + nβ̇ = (m + n)H02 .
(8.39)
From this we obtain two types of solutions (see Sec. 8.7). One is a trivial solution, namely,
D-dimensional de Sitter spacetime,
α = β = H0 t.
Of the other type are the following two solutions,emα+nβ = sinh[(m + n)H0 t], ±1/q
m+n
α−β
e
H0 t
= tanh,2(8.40)

where q is the one that has been introduced in Eq. (8.31). The range of the time coordinate
t is (−∞, ∞) for the former de Sitter solution and [0, ∞) for the latter non-trivial solutions.
Applying the method discussed in Sec. 8.1.1 to these solutions, one can construct backgroundsolutions for a (D + 1)-dimensional braneworld model.
First we brieﬂy mention the relation to the bulk inﬂaton model proposed by Koyama and
Takahashi [75, 77]. Identifying their model parameters ∆ and δ as
∆ = 4b2 − 8/3 = −2(m + 4)
,
m+3

δ=

m+4
2 H02,4(m + 3) 1 + 2 H02(8.41)

we will ﬁnd that our model is equivalent to theirs. The parameter m is supposed to takeany positive number. Thus it follows that ∆ varies in the same region −8/3 ≤ ∆ ≤ −2
considered in [75, 77]. The background metric obtained by substituting the simplest solution
α = β = H0 t is indeed the case discussed in their paper.
Next we consider the cosmic expansion law. We start with the simplest case α = β = H0 t.
The dimensionally reduced metric (on the brane) isqâb̂ dxâ dxb̂ = e2mH0 t/3 −dt2 + e2H0 t δij dxi dxj .
Introducing the cosmological time τ and the conformal time η deﬁned bydτ = adη = emH0 t/3 dt,

100

Figure 8.4: The motion of the scalar ﬁeld in the potential and the behavior of the scale factor
in the Jordan frame. The solution found by Koyama and Takahashi [75, 77] is described in
the left ﬁgure. One of the non-trivial solutions with its behavior at the starting point a ∝ τ p+is also shown in the left ﬁgure, while the other one with the exponent p− behaves as shownin the right ﬁgure.
the scale factor on the brane a = e(m+3)H0 t/3 is written in terms of τ or η asa ∝ τ (m+3)/m ∝ η −(m+3)/3 .

(8.42)

Since 1 ≤ (m + 3)/m < ∞, power-law inﬂation with any exponent can be realized.
Furthermore, we have non-trivial solutions (8.40). The behavior of the solutions is asfollows. At early times (t ∼ 0), the scale
factorbehaves
like a = emα/3+β ∼ t1/3 , and”“√the cosmological time is given by dτ ∼ t
Therefore, we havem±

3m(m+2) /3(m+3)

a ∼ τ p± ,dt (and so τ → 0 as t → 0).

(8.43)

with p± introduced previously, which implies that the universe is not accelerated at earlytimes. At late times (t → ∞), we see that α → H0 t and β → H0 t, and the solution shows
power-law expansion as is given in Eq. (8.42).
A rather intuitive interpretation of the behavior of these three solutions can be madefrom the four-dimensional point of view again. The situation is summarized in Fig. 8.4. This
time the potential is always positive. For one of the non-trivial solutions with the exponent
p+ in (8.43), the scalar ﬁeld starts to roll down from α = −∞. For the other non-trivial
solution with the exponent p− , the ﬁeld starts to climb up the slope of the potential from
α = +∞, turns around somewhere, and rolls down back to α = +∞. Suppose that the ﬁeld
α is increasing. Let us trace the evolution of α backward in time. If the kinetic energy islarger than a certain critical value, α will not have a turning point in the past. In this caseα continues to decrease, reaching −∞. This corresponds to the case with the exponent p+ .
If the kinetic energy is lower, α will have a turning point. Then, we will have α → +∞ at
t → −∞. This corresponds to the case with the exponent p− . The case of the power-law
inﬂation (8.42) is, in fact, the marginal case between these two. In this case, α does not turnaround. Therefore the evolution of α is similar to the case with p+ . In any case, information
about the initial velocity is lost as the universe expands. Therefore the late time behavior of
the solutions is unique, and is given by Eq. (8.42).

101

8.4

Cosmological perturbations

We consider cosmological perturbations in the (n + 2)-dimensional bulk inﬂaton models deﬁned by Eqs. (8.12) and (8.13). The analysis of perturbations is complicated if we work in theoriginal (n + 2)-dimensional models with a bulk scalar ﬁeld. Our (n + 2)-dimensional system,
however, is equivalent to the (D + 1)[= n + 2 + m]-dimensional one deﬁned by Eqs. (8.2)
and (8.3). We will show that the perturbation analysis becomes very simple and transparent
in the (D + 1)-dimensional picture, in which we just need to consider pure gravity without
any matter ﬁelds.
We begin with the following form of background metric:
GAB dX A dX B = e2ω(z) dz 2 − dt2 + e2α(t) γµν dy µ dy ν + e2β(t) δij dxi dxj ,(8.44)
where the latin and roman indices in the lower case, respectively, run m- and n-dimensional
subspaces, and the warp factor is given by eω(z) = H0 / sinh(H0 z). We assume that α(t) andβ(t) are chosen so that gÂB̂ is a solution of the (n + 1 + m)-dimensional vacuum Einsteinequations with a cosmological constant,
(8.45)m α̈ + α̇2 + n β̈ + β̇ 2 = (m + n)H02 ,(8.46)α̈ + α̇ mα̇ + nβ̇ + K(m − 1)e−2α = (m + n)H02 ,β̈ + β̇ mα̇ + nβ̇ = (m + n)H02 .
(8.47)The background solutions with γµν = δµν (K = 0), H0 = 0 and with K = ±1, H0 = 0
were discussed in the preceding section. In the following discussions, we include more generalcases with K = ±1 and H0 = 0. Although the background solution cannot be obtained in an
explicit form for such non-ﬂat compactiﬁcations with a cosmological constant, we will ﬁnd
that general properties of perturbations can be explored to a great extent.
We write the perturbed metric as(GAB + δGAB )dX A dX B= e2ω (1 + 2N )dz 2 + 2Adtdz − (1 + 2Φ)dt2 + e2α (1 + 2S)γµν dy µ dy ν+e2β (1 + 2Ψ)δij dxi dxj + 2Eij dxi dxj + 2Bi dxi dt + 2Ci dxi dz . (8.48)
These perturbations are assumed to be homogeneous and isotropic in the directions of them-dimensional compactiﬁed space spanned by the coordinates y µ . From the assumption of
isotropy, mixed components such as δGµt dy µ dt and δGµi dy µ dxi are set to zero. Concerning
the metric perturbations of the compactiﬁed space, therefore, only the overall volume perturbation S is considered. After reduction to (n + 2) dimensions, S is to be interpreted as thescalar ﬁeld perturbation. Here, we also assume that the dependence on the n-dimensionalicoordinates xi is given by eiki x .
Metric perturbations are decomposed into scalar, vector, and tensor components basedon the behavior under the transformation of the n-dimensional spatial coordinates xi in thefollowing manner:ki
Bi = B S + BiV , k i BiV = 0,
ik


ki kj
δij
1
1 T
V
Eij = − 2 ++ Eij,
E S + k(i Ej)
k
n
ik
2
T
T
= 0, δ ij Eij= 0.
k i EiV = k i Eij(8.49)

102
The quantities with a superscript S, V , and T represent scalar, vector and tensor perturbations, respectively. The perturbations δGAB obey the linearized Einstein equations supplemented by boundary conditions at the position of the brane, δKAB |z=zb = 0 where KAB is theextrinsic curvature of the brane. From the (D + 1)-dimensional point of view matter sources
are absent on the D-dimensional brane, and this makes boundary conditions considerably
simple. Each component of the Einstein equations is presented in Sec. 8.6.
Here we would like to discuss the number of physical degrees of freedom in scalar, vector,
T has (n + 1)(n − 2)/2 indeand tensor perturbations. The transverse traceless tensor Eij
pendent components, each of which obeys a second order diﬀerential equation. For vector
perturbations there are three variables BiV , EjV , and CiV . The coordinate transformation has
one vector mode, and correspondingly there is one vector constraint equation. Therefore wehave only 1[= 3 − 1 × 2] vector remaining as a physical mode. Since a transverse vector has(n − 1) independent components, we ﬁnd that there are 1 × (n − 1) degrees of freedom in
vector perturbations, corresponding to the “graviphoton.” For scalar perturbations there are
8 variables, and the coordinate transformation has 3 independent modes. Since there are thesame number of constraint equations, the number of physical modes is 2[= 8 − 3 × 2]. Oneof them corresponds to the bulk scalar ﬁeld and the other corresponds to the “graviscalar.”
In total, there are 1 + (n + 2)(n − 1)/2 = 2 + (n − 1) + (n + 1)(n − 2)/2 physical degrees
of freedom. The ﬁrst “1” on the left hand side corresponds to the bulk scalar and the other(n + 2)(n − 1)/2 degrees of freedom to those of (n + 2)-dimensional gravitational waves.

8.4.1

Tensor perturbations

Since tensor perturbations are gauge invariant from the beginning, they are in general easy
to analyze. The equations for tensor perturbations are read from the {i, j}-component of theEinstein equations (8.114) as
T
= 0,
LEij

where we have deﬁned a diﬀerential operator
L := ∂t2 + mα̇ + nβ̇ ∂t + e−2β k 2 − ∂z2 − (m + n)(∂z ω)∂z .

(8.50)(8.51)The perturbed junction condition implies that boundary conditions are Neumann on thebrane,
T
= 0.
(8.52)∂z Eijz=zb

T = χ(t)ψ(z)Y T (x )Since the perturbation equations are manifestly separable, we write Eij
ij
where YijT is a transverse, traceless tensor harmonics. Then χ(t) and ψ(z) obey
(8.53)
χ̈ + mα̇ + nβ̇ χ̇ + e−2β k 2 + M 2 χ = 0,∂z2 ψ + (m + n)(∂z ω)∂z ψ + M 2 ψ = 0.

(8.54)

Here M 2 is a separation constant and represents the squared Kaluza-Klein mass for observers
on the D-dimensional brane.
Now we discuss the mode function in the z-direction, ψ(z). Using a canonical variableψ̂ := eµω ψ, with
µ :=

3
m+n
≥ ,
2
2

(8.55)

103
Eq. (8.54) is rewritten into a Schrödinger-type equation−∂z2 ψ̂ + V (z)ψ̂ = M 2 ψ̂,(8.56)

κ2D+1 σH022 2
δ(z − zb ).
+
µ
H
−
0
2
sinh2 (H0 z)(8.57)

where the potential is
V (z) = µ(µ + 1)

The delta-function term is introduced so that ψ automatically satisﬁes the boundary conditionψ  (zb ) = 0. The presence of the zero mode, for which ψ is constant in z, is obvious from
Eq. (8.54). From the asymptotic value of the potential V (∞) = µ2 H02 , we can say that there
is a mass gap δM = µH0 between the zero mode and the KK continuum.
The z-dependence of the massive modes is given in terms of the associated Legendre
functions by−1/2−µ−1/2−µψM (z) = cT (sinh(H0 z))1/2+µ Q1/2+iν (cosh(H0 zb ))P−1/2+iν (cosh(H0 z))−1/2−µ−1/2−µ−P1/2+iν (cosh(H0 zb ))Q−1/2+iν (cosh(H0 z)) ,where cT is a normalization constant and
)
ν :=

M2
− µ2 .
H02(8.58)(8.59)

These general properties of the mass spectrum and the mode functions in the z-direction holdirrespective of the speciﬁc form of the background solution α and β.
Let us move on to the time dependenceof tensor perturbations. Using the cosmological$time on the brane deﬁned by τ = emα/n dt, Eq. (8.53) is rewritten as
 2


d
k2
m dα d(8.60)++nH++ e−2mα/n M 2 χ = 0,dτ 2n dτ dτa2where one must recall that a = emα/n+β and H = a−1 da/dτ . For the zero mode (M 2 = 0),
this reduces to the equation for the tensor perturbations in the scalar-tensor theory deﬁned
by the action (8.21). An apparent diﬀerence from Einstein gravity is the presence of the term(m/n)(dα/dτ ). The Kaluza-Klein mass with respect to observers on the brane is expressed
asm2KK (t) = e−2mα(t)/n M 2 ,(8.61)and so mKK = e−mα/n µH0 for the lightest one, whereas the Hubble parameter at that time
is given bymα̇ + β̇ .
H = e−mα/nnFor α = β = H0 t, this implies H = 2n−1 e−mα/n µH0 and therefore the mass gap and H areof the same order. On the other hand, when the background is given by Eq. (8.40), we have104
H = 2n−1 e−mα/n µH0 coth[(m + n)H0 t] and the mass gap can be very small compared to H,
but only for a short period near t = 0.
Despite its rather simple form, Eq. (8.53) cannot be solved analytically in general. One
exception is the case α = β = H0 t discussed in [75]. In this case, Eq. (8.53) readsχ̈ + (m + n)H0 χ̇ + e−2H0 t k 2 + M 2 χ = 0,
which, using the conformal time η = −e−H0 t /H0 , can be rewritten as 2
1−m−n dM2
d2
χ = 0.
+
k
+
+
dη 2
η
dη
H02 η 2(8.62)

This indeed has analytic solutions,χ0 ∝ (−η)µ Hµ(1) (−kη),(8.63)(1)

χM ∝ (−η)µ Hiν (−kη).

(8.64)

This is not a surprise because the background of the current model is just an AdSm+2+n bulk
with a de Sitter brane.

8.4.2

Vector perturbations

Next we consider vector perturbations. From the perturbed junction conditions δKi j |z=zb = 0
and δKi t |z=zb = 0, we have

(8.65)∂z EiV z=z = 0, CiV z=z = 0, ∂z BiV z=z = 0.
bbbUnder a vector gauge transformation xi → x̄i = xi + ξ iV , the metric variables transform as
ĒiV = EiV + kξiV ,B̄iV = BiV − ξ˙iV ,C̄iV = CiV − ∂z ξiV .

(8.66)

Thus we are allowed to set EiV = 0 by choosing an appropriate gauge. We expand theremaining variables by using the transverse vector harmonics YiV as
BiV = BYiV ,CiV = CYiV .

(8.67)For convenience, we introduceΩ := k −2 emα+(n+2)β e(m+n)ω ∂z B − C˙ .

(8.68)

Then Eqs. (8.117) and (8.120) are written as
B = e−mα−nβ e−(m+n)ω ∂z Ω,−mα+nβ −(m+n)ω

C=e

e

Ω̇.

(8.69)(8.70)

It is easy to see that the remaining third equation is automatically satisﬁed if the above two
equations hold. Substituting these two into Eq. (8.68), we obtain a master equation
(8.71)Ω̈ − mα̇ + nβ̇ Ω̇ + e−2β k 2 Ω − ∂z2 Ω − (m + n)(∂z ω)∂z Ω = 0.

105This equation looks similar to the equation for tensor perturbations (8.50). The diﬀerenceis that the signatures of the terms containing ﬁrst derivatives such as Ω̇ and Ω are reversed.
From Eqs. (8.65), the boundary condition for Ω on the brane turns out to be Dirichlet,Ω|z=zb = 0.

(8.72)

Since the master equation (8.71) is separable, we write Ω(t, z) = χ(t)Ω(z). The canonicalvariable Ω̂(z) := e−µω Ω again obeys a Schrödinger-type equation−∂z2 Ω̂ + V (z)Ω̂ = M 2 Ω̂,(8.73)

with the potential
V (z) = µ(µ − 1)

H02
+ µ2 H02 .
sinh2 (H0 z)(8.74)

The crucial diﬀerence from tensor perturbations is the absence of the delta-function potential
well. Because of this, there is no zero mode and only the massive modes with M 2 ≥ V (∞) =µ2 H02 exist. The z-dependence of the mode functions is given byΩM (z) = cV (sinh(H0 z))1/2−µ−1/2+µ−1/2+µ

Q−1/2+iν (cosh(H0 zb ))P−1/2+iν (cosh(H0 z))−1/2+µ−1/2+µ,−P−1/2+iν (cosh(H0 zb ))Q−1/2+iν (cosh(H0 z)) (8.75)
where cV is a normalization constant. When α = β = H0 t, we can ﬁnd an analytic solution
for the time-dependence of the mode functions, which, using the conformal time, is given byχM (η) ∝ (−η)−µ Hiν (−kη).
(1)

8.4.3

Scalar perturbations

Gauge choice, the boundary condition, and the mode decomposition
Since scalar perturbations are more complicated, we begin with ﬁxing the gauge appropriately
in order to simplify the perturbed Einstein equations. We impose the Gaussian-normal gauge
conditions
N = A = C = 0.

(8.76)

Diﬀerent from the case of vector perturbations, these conditions do not ﬁx the gauge completely. In the case of scalar perturbations we need to take care of perturbations of the brane
location. Here we make use of the remaining gauge degrees of freedom to keep the brane
location unperturbed at z = zb . In the Gaussian-normal gauge, boundary conditions on thebrane for all remaining variables become Neumann:∂z Ψ|z=zb = ∂z E|z=zb = ∂z S|z=zb = ∂z Φ|z=zb = ∂z B|z=zb = 0.

(8.77)

Three of eight scalar perturbation equations are the constraint equations, and the otherﬁve are the evolution equations. First, let us examine the constraint equations (8.116), (8.122),

106
and (8.123). Eq. (8.122) reduces to ∂z2 (Φ + nΨ + mS) + ∂z ω∂z (Φ + nΨ + mS) = 0. Taking
into account the boundary conditions, this equation is once integrated to give∂z (Φ + nΨ + mS) = 0.
Eqs. (8.116) and (8.123) reduce to∂z kB + 2Φ̇ + 2 mα̇ + nβ̇ Φ − 2mα̇S − 2nβ̇Ψ = 0,






1
−2β
∂z Ḃ + mα̇ + nβ̇ + 2β̇ B + 2e
−1 E
k Ψ+
n

(8.78)(8.79)= 0.

(8.80)

With the aid of Eq. (8.78), we ﬁnd that all the perturbation equations are separable. Furthermore, the z-dependent parts of these equations are the same as those of tensor perturbations
with the same type of boundary conditions. Therefore we can expand all variables by using
the same mode functions in the z-direction as those for tensor perturbations:EM (t)ψM (z),Φ = Φ0 (t)ψ0 (z) +ΦM (t)ψM (z),(8.81)··· ,E = E0 (t)ψ0 (z) +where ψ0 is constant, ψM is given by Eq. (8.58), and M 2 ≥ µ2 H02 2 . Consequently, Eqs.
(8.78), (8.79), and (8.80) are automatically satisﬁed for the zero mode. For the massivemodes these constraint equations giveΦ + nΨ + mS = 0,kB + 2Φ̇ + 2 mα̇ + nβ̇ Φ − 2mα̇S − 2nβ̇Ψ = 0,





1
−2β
− 1 E = 0,k Ψ+Ḃ + mα̇ + nβ̇ + 2β̇ B + 2en(8.82)(8.83)(8.84)

where the subscript M was abbreviated. Note that these three equations are nothing but thecomponents of the divergence of the metric perturbations,∇A δGzA = ∇A δGtA = ∇A δGiA = 0.
In other words, the transverse traceless conditions are automatically satisﬁed if one imposes
the Gaussian-normal gauge conditions except for the contribution coming from the zero mode.
(∇A δGzA gives the traceless condition.) Below we discuss the KK modes and the zero mode
separately.
KK modesBy using the constraint equations (8.82)-(8.84), the Einstein equations (8.111), (8.112), (8.119),(8.121), and (8.124) are simpliﬁed to give
2
− 1 k β̇B + 2β̇ Φ̇ + 2(m + n)H02 Φ,(8.85)
LΨ = −nLE = 2k β̇B,

(8.86)−2αK(m − 1)(S − Φ) + 2(m + n)H02 Φ,

LS = k α̇B + 2α̇Φ̇ + 2eLΦ = −2 mα̈ + nβ̈ Φ + 2k β̇B + 2mα̈S + 2nβ̈Ψ + 2(m + n)H02 Φ,
LB = − mα̈ + nβ̈ B − 4β̇ Ḃ − 4β̇ 2 B − 2(m + n)H02 B − 4e−2β k β̇Φ,22(8.87)(8.88)(8.89)In Ref. [145], it was shown that the non-normalizable graviscalar mode with KK mass M = (2µ − 1)H02
excites a small black hole in the bulk in the long wavelength limit, which corresponds to the dark radiation.

107
where L is deﬁned in Eq. (8.51). Two of them give independent master equations for themassive modes, and the remaining three equations do not give any new conditions. With theaid of the constraint equations (8.82)-(8.84), Eqs. (8.85) and (8.88) can be rewritten as4LΨ = −2(n − 2)β̇ α̇ − β̇ Ψ + β̇ Φ̇nn−2
β̇ (m + 1)α̇ + nβ̇ − (m + n)H02 Φ,(8.90)−2n.
LΦ = −4β̇ Φ̇ − 2 (m + 1)α̈ + nβ̈ + 2β̇ (m + 1)α̇ + nβ̇ − (m + n)H02 Φ−2n α̈ − β̈ + 2β̇ α̇ − β̇ Ψ.
(8.91)
Unfortunately, except for the simplest case (to be discussed later) we do not know how todisentangle these two equations, although there is no problem in solving these equations
numerically. Once we solve these coupled equations, the other variables S, B, E are easily
determined just by using the constraint equations.
Zero mode
To discuss the zero mode, it is useful to look at the cosmological perturbations in the corresponding (n + 1)-dimensional theory deﬁned by the action (8.26). In the case of the (n + 1)dimensional Friedmann universe with a single scalar ﬁeld, there is only one physical degreeof freedom in scalar perturbations. One can derive a second order diﬀerential equation for
one master variable [110]. Back in the braneworld context, the background metric and its
zero-mode perturbations are also described by the same eﬀective action (8.26). Therefore,
the analysis of the zero mode is no diﬀerent from the conventional (n + 1)-dimensional cosmological perturbation theory. Below we will explain this fact more explicitly.
To begin with, we consider (n + 1 + m)-dimensional spacetime whose metric is given by
ds2 = −(1 + 2Φ)dt2 + e2α (1 + 2S)γµν dy µ dy ν+e2β (1 + 2Ψ)δij dxi dxj + 2Eij dxi dxj + 2Bi dxi dt ,(8.92)

where only scalar perturbations are imposed and they are again assumed to be homogeneous and isotropic with respect to the m-dimensional compactiﬁed space spanned byy µ . Then, the perturbed Einstein equations δRÂB̂ = (m + n)H02 δgÂB̂ become identical toEqs. (8.111), (8.112), (8.119), (8.121), and (8.124) with N , A, C, and the terms diﬀerentiated by z dropped. Hence, it is manifest that the analysis of zero-mode perturbations in our(n + 2 + m)-dimensional spacetime is equivalent to that of the above system.
As for perturbations in (n + 2 + m)-dimensional spacetime, we have already ﬁxed thegauge by imposing three gauge conditions (8.76). However, these gauge conditions do notﬁx the gauge completely. As is manifest from Eqs. (8.126), gauge transformations satisfyingξ z = 0 and ξ S = ξ t = 0 do not disturb the conditions (8.76). On the other hand, on the(n + 1)-dimensional side there are two scalar gauge transformationst → t̄ = t + ξ t ,xi → x̄i = xi + k i ξ S /ik.
The transformation of metric variables under these gauge transformations is the same as thatobtained by setting ξ z = 0 and ξ S = ξ t = 0 in the last ﬁve equations in (8.126).

108
If we think of the size of the compactiﬁed dimension S as a scalar ﬁeld in (n + 1)dimensional spacetime, the system reduces to a conventional (n+1)-dimensional model with ascalar ﬁeld. In the conventional cosmological perturbation theory, Φ and Ψ in the longitudinalgauge (B = E = 0) are known to be convenient variables. Here one remark is that we needto take account of a conformal transformation to map the theory to the conventional (n + 1)dimensional one,2m

ds̃2 = e n(n−1)

(α+S)

2m

2m

· e n (α+S) ds2 = e n−1 (α+S) ds2 ,(8.93)

which follows from the discussion in Sec. 8.2. Then the variables corresponding to the socalled Sasaki-Mukhanov variables are
m
S,
n−1
m
Ψ̂ := Ψ +S,n−1

Φ̂ := Φ +(8.94)(8.95)

in the longitudinal gauge.
Eliminating N , A, B, C, E and the terms diﬀerentiated by z, Eq. (8.112) becomesΦ̂ + (n − 2)Ψ̂ = 0.

(8.96)

Similarly, from Eqs. (8.119), (8.111), and (8.124), we have
m+n−1
˙
mα̇S,
(n − 1)Ψ̂ + (n − 2) mα̇ + (n − 1)β̇ Ψ̂ = −n−1
˙L0 Ψ = −2(n − 1)β̇ Ψ̂ + 2(m + n)H02 Φ,˙
˙ − 1)αΨ̂
L0 S = −2(n+ 2(m + n)H 2 Φ − 2K(m − 1)e−2α (Φ − S),

0(8.97)

where L0 f := f¨ + mα̇ + nβ̇ f˙ + e−2β k 2 f . Combining all these, we obtain the equation of
motion for Ψ̂:





α̈ ˙
α̈
Ψ̂ + 2(n − 2) β̈ − β̇Ψ̂ = 0.
(8.98)
L0 Ψ̂ − 2 β̇ +α̇α̇$Using the conformal time η = e−β dt, we can rewrite this into a more familiar form as



α
α


2

(8.99)
Ψ̂ + (n − 1)H − 2  Ψ̂ + k Ψ̂ + 2(n − 2) H − H  Ψ̂ = 0,ααwhere we have deﬁned
H := (ln ã) =1[mα + (n − 1)β] ,n−1(8.100)and ã = emα/(n−1)+β is the scale factor in the Einstein frame.
Since there is a mass gap between the zero mode and the massive modes in general in
our models except for a short period in the cases of K = 0, the massive modes would not be
excited easily. Hence, the behavior of the zero mode is especially important. Since we foundthat the zero mode can be described by the corresponding (n + 1)-dimensional conventionalcosmology, it can be easily analyzed in general.

109
Exactly solvable caseLet us consider the simplest background given by α = β = H0 t with K = 0. In this specialcase, scalar perturbations including the KK modes are solved exactly. The most remarkableadvantage of our approach may be that z-dependence of the modes can be derived for ageneral background as we did in the earlier part of this section. The time-dependent part,which is usually non-trivial especially for the KK modes, is also solved easily as shown below
when α = β = H0 t.
Substituting α = β = H0 t into Eq. (8.91), the equation for Φ is decoupled ﬁrst,
LΦ = −4H0 Φ̇ − 2(m + n + 2)H02 Φ.

(8.101)

By assuming the z-dependence given in Eq. (8.58), we expand as Φ = ΦM ψM . Then, using
the conformal time, the above equation is rewritten as 2
m+n+3 d1 M2
d2
+k + 2−+ 2(m + n + 2) ΦM = 0.
(8.102)
dη 2
η
dη
η
H02
The solution is given in terms of the Hankel function byΦM = cS1 (−η)2 ρ(η)(8.103)with(1)

ρ = (−η)µ Hiν (−kη),
where cS1 is a constant and µ and ν were deﬁned in Eqs. (8.55) and (8.59). Then, substituting
this into Eq. (8.84) with the aid of Eq. (8.83), BM is immediately obtained as(8.104)
BM = −2cS1 k −1 H0 (−η)3 ρ + (2µ − 1)(−η)2 ρ .
The result is consistent with the evolution equation for B [Eq. (8.89)].
Eqs. (8.86), (8.87) and (8.89) are combined to give a simple equation,
L[ΨM + EM /n − SM ] = 0.
The operator L is the one that appeared in tensor perturbations, and so the mode solutions
are already known:ΨM + EM /n − SM = cS2 ρ,(8.105)where cS2 is another constant. Substituting ΦM and BM , the constraints (8.82) and (8.84)
reduce to two algebraic equations for ΨM , EM , and SM asnΨM + mSM
ΨM + (1/n − 1)EM= −cS1 (−η)2 ρ,=−cS1 (−η)2 ρ − cS1 k −2× (2µ − 1)ηρ + (ν 2 − 3µ2 + 2µ)ρ .

(8.106)(8.107)

Solving Eqs. (8.105), (8.106), and (8.107), we obtain the expressions for ΨM , EM , and SM .
Thus, all the metric variables can be analytically solved. Note that one of the above two
independent solutions was already obtained in Ref. [75, 77].

110
The zero-mode solution is also easily obtained. In this background, the master equation
becomesΨ̂ −

m+n−3 Ψ̂ + k 2 Ψ̂ = 0,η(8.108)and the solution is(1)Ψ̂ = cS (−η)µ−1 Hµ−1 (−kη).

8.5(8.109)

Summary

We have shown that a wide class of braneworld models with bulk scalar ﬁelds can beconstructed by dimensional reduction from a higher dimensional extension of the RandallSundrum model with an empty bulk. The sizes of compactiﬁed dimensions translate into
scalar ﬁelds with exponential potentials both in the bulk and on the brane. We have mainly
concentrated on models with a single scalar ﬁeld, which include the power-law inﬂation solution of Ref. [75, 77].
First we have investigated the evolution of ﬁve[= n + 2]-dimensional background cosmologies, giving an intuitive interpretation based on the four[= n + 1]-dimensional eﬀective
description. Then we have studied cosmological perturbations in such braneworld models.
Lifting the models to (5 + m)[= n + 2 + m]-dimensions is a powerful technique for this purpose. The degrees of freedom of a bulk scalar ﬁeld in (n + 2)-dimensions are deduced froma purely gravitational theory in the (n + 2 + m)-dimensional Randall-Sundrum braneworld,which consists of a vacuum brane and an empty bulk. We would like to emphasize that theanalysis is greatly simpliﬁed thanks to the absence of matter ﬁelds. From the (n + 2 + m)dimensional perspective, we have derived master equations for all types of perturbations. We
have shown that mode decomposition is possible for all models which are constructed byusing this dimensional reduction technique. Moreover, the dependence in the direction of theextra dimension perpendicular to the brane can always be solved analytically.
As for scalar perturbations, there are two physical degrees of freedom for the massivemodes and the equations are not decoupled in general. For the zero mode, however, thesituation is equivalent to the standard four-dimensional inﬂation driven by a single scalarﬁeld. Hence, only one degree of freedom is physical. Therefore we end up with a single
master equation. To sum up, our “embedding and reduction” approach enables a systematic
study of cosmological perturbations in a class of braneworld models with bulk scalar ﬁelds.
In this chapter, we have not discussed quantum mechanical aspects. In order to evaluate
the amplitude of the quantum ﬂuctuations, the overall normalization factor of the perturbations must be determined. For this purpose, one needs to write down the perturbed action
up to the second order written solely in terms of physical degrees of freedom as is done in
the standard cosmological perturbation theory.
In this chapter our investigation is restricted to the parameter region m > 0, where m isthe number of compactiﬁed dimensions. If m > 0, a singularity could exist at z = ∞, but
it is null. For m < −3, we have the right sign for the kinetic term of the scalar ﬁeld and so
it is possible to consider such models. In this parameter region, however, there is a timelike

111singularity at z = 0 and therefore we need a regulator brane to hide it. This case includes
the cosmological solution of heterotic M theory [97] (which corresponds to m = −18/5),
and the analysis of cosmological perturbations in such two-brane models [133] would also bemeaningful. This issue is left for future work.

8.6

Appendix: Perturbed Einstein equations and gauge transformations

In this section, we write down the components of the perturbed Einstein equations
1
1
1+m+n
δRAB = ∇C ∇(A δGB) C − 2δGAB − ∇A ∇B δG = −δGAB .
22
2
The perturbed quantities are decomposed into scalar, vector, and tensor components whosebasic deﬁnitions are given by Eq. (8.49). Note that in the following expressions no gauge
conditions have been imposed yet.
• {i, j}-component
1ḧij + (mα̇ + nβ̇)ḣij − hij − (m + n)ω  hij − e−2β ∂ k ∂k hij + 2e−2β ∂ k ∂(i hj)k2− ∂(i Ḃj) + (mα̇ + nβ̇)∂(i Bj) − ∂(i Cj)− (m + n)ω  ∂(i Cj)−δij β̇ ∂ k Bk − ω  ∂ k Ck + δij 2(m + n)H02 (N − Φ) + 2(1 + m + n)ω  N + ω  N − mα̇ + (m + 2n)β̇ ω  A − β̇A − ω  Ȧ − ω  (Φ + mS + nΨ).
+β̇ (N − Φ + mS + nΨ)˙ − e−2β ∂i ∂j (N + Φ + mS + nΨ) = 0,(8.110)where hij = 2Ψδij + 2Eij and the dot (prime) denotes ∂/∂t (∂/∂z). (We use the prime
to denote diﬀerentiation with respect to z only in Sec. 8.6.)
Trace Part:Ψ̈ + (mα̇ + nβ̇)Ψ̇ + e−2β k 2 Ψ − Ψ − (m + n)ω  Ψ





1
1
−2β 2 2Ψ+
− 1 E − k Ḃ + mα̇ + 2nβ̇ B − C  − (m + 2n)ω  C
−e
k
n
n
n
2

 
+ 2(m + n)H0 (N − Φ) + 2(1 + m + n)ω N + ω N− mα̇ + (m + 2n)β̇ ω  A − β̇A − ω  Ȧ − ω  (Φ + mS + nΨ).
+β̇ (N − Φ + mS + nΨ)˙+e−2β k 2 (N + Φ + mS + nΨ)

1
= 0.
n

(8.111)

Trace-free Part:−2β 2Ë + (mα̇ + nβ̇)Ė + e







k E − E − (m + n)ω E + 2e
+k Ḃ + (mα̇ + nβ̇)B − k C  + (m + n)ω  C−e−2β k 2 (N + Φ + mS + nΨ) = 0.




1
−1 E
k Ψ+
n

−2β 2

(8.112)

112
Vector :ËiV + (mα̇ + nβ̇)ĖiV − EiV  − (m + n)ω  EiV +k ḂiV + (mα̇ + nβ̇)BiV − k CiV  + (m + n)ω  CiV = 0.

(8.113)Tensor :T∂t2 + (mα̇ + nβ̇)∂t + e−2β k 2 − ∂z2 − (m + n)ω  ∂z Eij= 0.

(8.114)• {z, i}-component j 
∂ hij − (∂i A)˙− mα̇ + nβ̇ − 2β̇ ∂i A + ∂ j ∂i Cj − ∂ j ∂j Ci−2∂i Φ + mS  + nΨ + 2(m + n)ω  ∂i N+e2β C̈i − Ḃi + mα̇ + nβ̇ + 2β̇ Ċi − Bi  = 0.
Scalar :

(8.115)








1
k 2Ψ + 2− 1 E  − Ȧ − mα̇ + nβ̇ − 2β̇ A − 2 Φ + mS  + nΨn+2(m + n)ω  N − e2β C̈ − Ḃ  + mα̇ + nβ̇ + 2β̇ Ċ − B  = 0. (8.116)
Vector :
kEiV  + k 2 CiV + e2β C̈iV − ḂiV  + mα̇ + nβ̇ + 2β̇ ĊiV − BiV  = 0.

(8.117)• {t, i}-component j ∂ hij ˙+ (∂i A) + (m + n)ω  ∂i A + ∂ j ∂i Bj − ∂ j ∂j Bi






+2 mα̇ + nβ̇ − β̇ ∂i Φ − 2∂i Ṅ + mṠ + nΨ̇ + 2∂i β̇N + m β̇ − α̇ S+e2β −Bi  + Ċi + (m + n)ω  Ċi − Bi  = 0.
(8.118)
Scalar :Vector :








1
k 2Ψ̇ + 2
− 1 Ė + A + (m + n)ω  A + 2 mα̇ + nβ̇ − β̇ Φ
n




−2 Ṅ + mṠ + nΨ̇ + 2β̇N + 2m β̇ − α̇ S−e2β −B  + Ċ  + (m + n)ω  Ċ − B  = 0.

k ĖiV + k 2 Bi + e2β −BiV  + ĊiV  + (m + n)ω  ĊiV − BiV  = 0.

(8.119)(8.120)

113• {t, t}-componentk Ḃ + 2β̇B − ω  C − (nΨ + mS)¨+ ω  (Φ + nΨ + mS) + Ȧ + (1 + m + n)ω  Ȧ+Φ + (m + n)ω  Φ − e−2β k 2 Φ + mα̇ + nβ̇ Φ̇ − 2nβ̇ Ψ̇ − 2mα̇Ṡ
−N̈ − ω  N  − 2(1 + m + n)ω  N − 2(m + n)H02 (N − Φ) = 0.

(8.121)• {z, z}-componentN̈ + e−2β k 2 N + k C  + ω  C − (Φ + nΨ + mS) − ω  (Φ + nΨ + mS)


 

− Ȧ + ω  Ȧ − mα̇ + nβ̇ A + ω  A − Ṅ+(1 + m + n) ω  N  + 2ω  N = 0.
(8.122)• {t, z}-componentk Ċ + 2β̇C + B  + e−2β k 2 A + 2 mα̇ + nβ̇ Φ + 2ω  (m + n)Ṅ
(8.123)−2 nΨ̇ + mṠ  − 2nβ̇Ψ − 2mα̇S  − 2(m + n)H02 A = 0.
• {µ, ν}-componentS̈ + 2mα̇ + nβ̇ Ṡ − S  − (2m + n)ω  S  + e−2β k 2 − 2e−2α K(m − 1) S+nα̇Ψ̇ − α̇Φ̇ − ω  Φ + nΨ + 2e−2α K(m − 1)Φ + k ω  C − α̇B+α̇Ṅ + ω  N  + 2(1 + m + n)ω  N − ω  Ȧ − α̇A − [(2m + n)α̇ + nβ] ω  A+2(m + n)H02 (N − Φ) = 0.

(8.124)

Lastly we summarize the gauge transformations of the metric variables. Under a scalargauge transformation,t →t̄ = t + ξ t ,z →z̄ = z + ξ z ,kix̄i = xi + ξ S ,
ik

xi →

(8.125)the metric variables transform as
N →
A →
C →
B →

N̄ = N − ξ z  − ω  ξ z ,Ā = A + ξ t − ξ˙z ,C̄ = C + e−2β kξ z − ξ S
B̄ = B − e−2β kξ t − ξ˙S ,E →Φ̄ = Φ − ξ˙t − ω  ξ z ,1Ψ̄ = Ψ − kξ S − ω  ξ z − β̇ξ t ,nĒ = E + kξ S ,S →S̄ = S − ω  ξ z − α̇ξ t .

Φ →
Ψ →

(8.126)

114

8.7

Appendix: Multiple scalar ﬁeld generalization

Let us give the generalization of the Kasner-type metric discussed in Sec. 8.3. First we
generalize the case without Λb but including the curvature for one of the spatial sections:M

gM N dx dx

N

2α(η)

=e

N

 
(i)
2
µ ν
e2γi (η) δM N dxM dxN ,−dη + γµν dy dy +
i=1

(i)

where δM N is the metric of a ji -dimensional ﬂat space and γµν is the metric of a m̄-dimensional
maximally symmetric space. As before we identiﬁed j1 with n. If we compactify (m̄+ N
i=2 ji )dimensions leaving (n+1) dimensions, the compactiﬁed space is divided into N sectors having
diﬀerent scale factors. Then the resulting cosmology after dimensional reduction will possess
N scalar ﬁelds.
The set of vacuum Einstein equations becomes2
ji γi − α u + u + m̄α = 0,(8.127)e2α Rηη =e2α Rµν = δµν K(m̄ − 1) + α u + (m̄ − 1)α + α = 0,

(i)
(i) 
e2α RMN δN L = δM L γi u + (m̄ − 1)α γi + γi = 0,2where we have introducedu :=

ji γi .

(8.128)(8.129)(8.130)From Eq. (8.129), we obtainu2 + (m̄ − 1)α u + u = 0.

(8.131)

Then it is easy to see that Eqs. (8.128) and (8.130) are equivalent to Eqs. (8.29) and (8.30)
in the example of Sec. 8.3.1 by identifying u with nβ  . Therefore the solution of Eqs. (8.128)
and (8.130) for K = 1 is written asu=±(m̄ − 1)q̄,sin[(m̄ − 1)η]α = cot[(m̄ − 1)η] ∓q̄
.
sin[(m̄ − 1)η](8.132)Since these two equations (8.128) and (8.130) do not have dependence on the number ofdimensions, q̄ has not been ﬁxed yet. Substituting this solution into Eq. (8.129), we obtain(m̄ − 1)γi cot[(m̄ − 1)η] + γi = 0,
which is integrated to giveγi =(m̄ − 1)ci,sin[(m̄ − 1)η]where ci is an integration constant. Substituting this into the remaining equation (8.127)
and the deﬁnition of u (8.130), we ﬁnd that the solution is given by
 ±q̄
 
m̄ − 1(m̄−1)αη= sin[(m̄ − 1)η] cot
,
e
2
 ci


m̄ − 1γiη,e = tan
2

115
with


m̄ − q̄ 2,
ji ci = ±q̄.
m̄ − 1
The next is a generalization of the Kasner-type spacetime including a cosmological constant Λb . Let us assume that the metric is in the form of

ji c2i =gM N dxM dxN = −dt2 +

N


(i)

e2γi (t) δM N dxa dxb .

(8.133)i=0Here all the spatial sections are taken to be ﬂat (Ki = 0), because otherwise an analytic
2solution with Λb = 0 cannot be found. The Einstein equations RM N [g] = N H0 gM N , with
N := D − 1 = ni reduce to (8.134)ji γ̈i + γ̇i2 = N H02 ,
(8.135)ji γ̇i = N H02 ,γ̈i + γ̇iwhere the overdot denotes diﬀerentiation with respect to t. These equations admit a trivial
solution of D-dimensional de Sitter spacetime,
γi = H0 t.

(8.136)
There is another type of non-trivial solution. From Eq. (8.135) we ﬁnd that u :=ji γ̇iobeysu̇ + u2 = N 2 H02 .
The solution for this equation isu = N H0 coth(N H0 t).
Substituting this into Eq. (8.135), we obtain[sinh(N H0 t)γ̇i ]˙ = N H02 sinh(N H0 t).
This can be easily integrated and the integration constants are determined from Eq. (8.134).
Then we have
H0 [cosh(N H0 t) + ci ]γ̇i =,sinh(N H0 t)
ji c2i = N (N − 1).
ji ci = 0,(8.137)(8.138)

Finally, integrating Eq. (8.137), we obtain
N γi

e

 ci


N H0
t
= sinh(N H0 t) tanh.
2

(8.139)Integration constants were removed by rescaling the spatial coordinates. There are only these
two types of solutions (8.136) and (8.139) for Eqs. (8.134) and (8.135).

Chapter 9

Conclusions
In this thesis we have discussed aspects of cosmological perturbations in the Randall-Sundrumtype braneworlds. In Chapter 4 we have calculated leading order corrections to the evolutionof tensor perturbations in the RS2 braneworld cosmology at low energies by using the perturbative expansion scheme of Ref. [139], and analytically determined the precise numericalfactors of the correction terms of O(2 ) and O(2 ln ). Thus we have conﬁrmed that leading order corrections are indeed small and conventional four-dimensional cosmology can be
reproduced at low energies on the brane.
In Chapter 5 we have introduced the so called “junction model,” in which two maximallysymmetric branes are joined in order to investigate the eﬀect of nontrivial motion of thebrane, and calculated the primordial spectrum of gravitational waves generated quantummechanically from inﬂation. Through the analysis of this toy model, we have conjectured
that the primordial tensor spectrum in the RS braneworld is quite well approximated by
the “rescaled” spectrum obtained by using a simple map speculated by the exact result ofpure de Sitter inﬂation. To check this, in Chapter 6 we have developed a new numericalmethod based on the Wronskian formulation. Our numerical results strongly indicate thatthe above speculation is correct in general inﬂationary models. The vacuum ﬂuctuations of
the initial KK gravitons contribute to the ﬁnal amplitude of the zero mode at a signiﬁcantlevel for high-energy inﬂation, and thus we have obtained an interesting picture: when the
expansion rate changes during inﬂation, zero mode gravitons escape into the bulk as KK
gravitons, but at the same time bulk gravitons come onto the brane to compensate for theloss, and these two eﬀects almost cancel each other. In Chapter 7, we have followed the sameline as the previous chapter and numerically studied the late time power spectrum of tensor
perturbations, focusing on the high-energy eﬀects in the radiation-dominated stage. In our
analysis, initial conditions of perturbations are imposed quantum-mechanically. The issue of
initial conditions has been less considered in previous studies. We have found that the eﬀect
of initial KK vacuum ﬂuctuations are subdominant, contributing not more than 10% of thetotal power spectrum so far as the present calculation is concerned, and thus the damping
due to the generation of KK modes and the enhancement due to the modiﬁcation of the
background Friedmann equation mainly work. The combination of these two eﬀects leads
to the same spectral tilt as the standard four-dimensional result. We have also estimatedthe energy density of the “dark radiation,” which is generated via KK mode excitation, andshowed that only a tiny amount is generated due to that process.
The topic of Chapter 8 was on more complicated braneworld models with scalar ﬁelds
in the bulk. However, concerning the bulk inﬂaton models in Chapter 8 the analysis of per116

117
turbations is essentially the same as that of the simplest setup: tensor perturbations in the
Randall-Sundrum-type de Sitter braneworld. The key idea is that the ﬁve-dimensional bulk
inﬂaton models can be described by (5 + m)-dimensional vacuum gravity via dimensional
reduction. We have shown that various background solutions including the power-law inﬂation model [75, 77] are generated from known (5 + m)-dimensional vacuum solutions of puregravity, and derived master equations supplemented by simple boundary conditions for all
types of perturbations.
As a ﬁnal remark, a number of issues still remain to be solved. In this thesis we haveelaborated mainly on tensor perturbations in the Randall-Sundrum-type braneworld, but ofcourse it is important to study scalar cosmological perturbations, aiming to give a clear prediction about the CMB temperature anisotropy on the brane. The generation and evolutionof curvature perturbations are much more complicated than that of tensor perturbations,because the problem reduces to solving the system of coupled brane-bulk ﬁelds [80]. We
believe that the approaches reviewed in this thesis will be helpful as well for understanding
the behavior of scalar perturbations in the braneworld. From the point of view of generalrelativistic cosmology, the analysis of the Randall-Sundrum-type models is accessible thanks
to their simple construction. It would be interesting to explore general relativistic and cosmological consequences of more “stringy” models than those relying on just ﬁve-dimensional
AdS geometry, going in the direction of, e.g., Refs. [59, 116] inspired by the recent seminal
proposal of KKLT [55] and KKLMMT [56].

Acknowledgments
First of all I would like to thank Takahiro Tanaka for valuable discussions and collaborations. He has guided me through the ﬁelds of general relativity and cosmology. I havelearned a lot from him during these ﬁve years. It is my pleasure to thank Takashi Nakamura,head of the theoretical astrophysics group, who provides an encouraging atmosphere around
me. If I have succeeded in making my work scientiﬁcally interesting and important, I owe it
to his critical comments. I am grateful to all the colleagues in Department of Physics at Kyoto University and Yukawa Institute for Theoretical Physics, especially to Hiroyuki Tashiro,
Teruaki Suyama, Shuichiro Yokoyama, and Makoto Yoshikawa, for everyday discussions andconversations (and for an occasional drink). Many thanks to my parents, my friends, and myfavorite rock music! T.K. is supported by the JSPS.

Appendix

Appendix A

Review of cosmologicalperturbation theory in four
dimensions
Here we give a brief review of cosmological perturbation theory in the conventional fourdimensional Universe. For comprehensive reviews, see Refs. [67, 110, 4, 94].

A.1

Governing equations

A.1.1

The unperturbed Universe

The unperturbed Universe is described by the Friedmann-Robertson-Walker spacetime, whose
metric is given by
ds2 = −dt2 + a2 (t)δij dxi dxj .

(A.1)

Just for simplicity we assume that the geometry of a homogeneous and isotropic 3-space is
ﬂat. The energy-momentum tensor which is compatible with homogeneity and isotropy of
3-space is that of a perfect ﬂuid:
Tab = (ρ + p)ua ub + pgab .

(A.2)

The components of the Einstein equations are
G00 = −3H 2 = −8πGρ,
Gi j = − 3H 2 + 2Ḣ δ i j = 8πGpδ i j ,

(A.3)
(A.4)

where H := ȧ/a = (da/dt)/a is the Hubble parameter. The ﬁrst equation is called theFriedmann equation. The energy-momentum conservation equation ∇µ T µν = 0 readsρ̇ + 3H(ρ + p) = 0.
119

(A.5)

120

A.1.2

Perturbed metric and energy-momentum tensor

Perturbed quantities can be decomposed into scalar, vector, and tensor parts according totheir transformation properties with respect to the three-dimensional space, where scalarparts are related to a scalar potential, vector parts to transverse vectors and tensor parts totransverse, traceless tensors. Such a splitting is found to be quite useful to study cosmologicalperturbation theory at least in linear order, because the diﬀerent modes are decoupled fromeach other in the ﬁeld equations.
Scalar perturbations
We write the perturbed metric as
ds2 = −(1 + 2A)dt2 + 2a∂i Bdtdxi + a2 [(1 − 2ψ)δij + 2∂i ∂j E] dxi dxj .

(A.6)

The components of the Christoﬀel symbol and the perturbed Einstein tensor are, respectively,
Γ000 = Ȧ,Γi00 = ∂ i A/a2 + Ḃ + HB /a ,

Γ0ij = a2 Hδij − 2a2 HAδij − a∂i ∂j B +Γ00i = ∂i (A + aHB) ,d  2a (−ψδij + ∂i ∂j E) ,dt

Γi0j = Hδ i j − ψ̇δ i j + ∂ i ∂j Ė,Γkij = −aH∂ k Bδij − ∂i ψδ kj − ∂j ψδ ki − ∂ k ψδij + ∂ k ∂i ∂j E,
and




δG00 = 6H ψ̇ + HA − 2a−2 ∂ k ∂k ψ + H a2 Ė − aB ,
δG0i = −2∂i ψ̇ + HA ,






δGi j =∂ i ∂j − δ i j ∂ k ∂k a−2 (ψ − A) + Ė − B/a ˙+ 3H Ė − B/a+2 ψ̈ + 3H ψ̇ + H Ȧ + 3H 2 + 2Ḣ A δ i j .

(A.7)
(A.8)

(A.9)

The ﬂuid four-velocity is given byuµ = 1 − A, a−1 ∂ i v ,uµ := gµν uν = (−1 − A, a∂i (v + B)) ,(A.10)and then the perturbed energy-momentum tensor is found to beδT 00 = −δρ,

(A.11)

δT 0i(A.12)

δT ij= ∂i [a(ρ + p)(v + B)] =: ∂i δq,


1 i k
i
i
= δpδ j + ∂ ∂j − δ j ∂ ∂k δΠ.
3(A.13)The perturbed quantities are generally not invariant under a scalar gauge transformationt → t + δt,

xi → xi + ∂ i δx,(A.14)

121
but they transform as
˙
A → A − δt,−1

B → B+a(A.15)˙
δt − aδx,(A.16)

ψ → ψ + Hδt,

(A.17)

E → E − δx,(A.18)

δρ → δρ − ρ̇δt,(A.19)

δq → δq + (ρ + p)δt,(A.20)

δΠ → δΠ.

(A.21)

andThe following gauge-invariant combinations have been widely used in the literature:d  2Φ := A −a Ė − aB ,(A.22)dt(A.23)
Ψ := ψ + H a2 Ė − aB .
In terms of these variables we can write the gauge-invariant Poisson equationk2
Ψ = −4πGδ,
a2(A.24)

where we have deﬁned the gauge-invariant comoving density perturbationδ := δρ − 3Hδq,(A.25)and the spatial Laplacian has been replaced by their eigenvalue: ∂ i ∂i → −k 2 . The tracelesspart of the Einstein equation δGi j = 8πGT ij gives the constraint equationΨ − Φ = 8πGa2 δΠ.

(A.26)

Thus we have Ψ = Φ when anisotropic stresses vanish, which is the case for a cosmic ﬂuid
in which there is negligible diﬀusion or freestreaming. Assuming that δΠ is negligible, we
obtain the evolution equation for the potentialk2Φ̈ + (4 + 3c2s )H Φ̇ + 2Ḣ + 3H 2 (1 + c2s ) Φ + c2s 2 Φ = 4πGδpnad ,a(A.27)

where c2s := ṗ/ρ̇ is the sound velocity and the source term in the right hand side,δpnad := δp − c2s δρ,(A.28)

is the non-adiabatic pressure perturbation or entropy perturbation.
The perturbation of energy-momentum conservation δ (∇µ T µν ) = 0 gives
˙ + 3H(δρ + δp) − 3ψ̇(ρ + p) =δρ
˙ + 3Hδq + (ρ + p)A + δp =
δq



k2
2
Ė
−
aB
,
δq
+
(ρ
+
p)
a
a2
2 2
k δΠ.
3

(A.29)(A.30)

122
Using the curvature perturbation on uniform density hypersurfaces,ζ := −ψ −

H
δρ,
ρ̇

(A.31)

the ﬁrst equation can be rewritten into a quite important expression:ζ̇ = −

H
δpnad + O(k 2 ).
ρ+p(A.32)

This implies that in the absence of entropy perturbations ζ is conserved on super-horizon
a2 H 2 ). What should be stressed here is that the above result is obtained
scales (k 2without invoking the Einstein equations; so long as energy-momentum conservation holds,
we have Eq. (A.32) in a braneworld or in alternative theories of gravity [143].
Another gauge-invariant variable which is commonly used is the comoving curvature perturbation
R := ψ −

H
δq.
ρ+p

(A.33)

Since with the aid of the Poisson equation (A.24) we have
R+ζ =−

H k2
H
δ =
Ψ,
ρ̇
4πGρ̇ a2

(A.34)

R and −ζ coincide and thus R is constant for adiabatic perturbations on super-horizon scales.
Now let us consider a universe ﬁlled with a scalar ﬁeld ϕ, for which the perturbed energymomentum tensor is given by
˙ + V,ϕ δϕ − ϕ̇2 A ,δT 00 = −δρ = − ϕ̇δϕ
δT 0i = ∂i δq = ∂i (−ϕ̇δϕ) ,˙ − V,ϕ δϕ − ϕ̇2 A δ i .
δT ij = δpδ i j = ϕ̇δϕj

(A.35)(A.36)(A.37)

Note that anisotropic stress δΠ vanishes for a scalar ﬁeld (and for multiple scalar ﬁelds). The
non-adiabatic pressure can be calculated asδpnad = −

2V,ϕ
δ,
3H ϕ̇

(A.38)

from which we ﬁnd that in the case of a single scalar ﬁeld, perturbations become adiabaticon large scales.
The evolution equation for the gauge-invariant potential is obtained by using the Einsteinequations as




ϕ̈
k2
ϕ̈
Φ̇ + 2 Ḣ − H
Φ + 2 Φ = 0.
Φ̈ + H − 2ϕ̇ϕ̇a(A.39)

123
Vector perturbations
The perturbed metric is
ds2 = −dt2 − 2aBi dtdxi + a2 δij + 2∂(i Ej) dxi dxj ,(A.40)

where Bi and Ei are transverse vectors: ∂ i Bi = 0 and ∂ i Ei = 0.
The components of the perturbed Einstein tensor are
1
δG0i = − ∂ k ∂k σi ,2a−2 δGij= ∂(i σ̇j) + 3H∂(i σj) − 2 3H 2 + 2Ḣ ∂(i Ej) ,(A.41)(A.42)

whereσi := Ėi + Bi /a,(A.43)

while the perturbed energy-momentum tensor is given by
δT 0i = δqi ,δTij(A.44)= 2p∂(i Ej) + ∂(i δΠj) .

(A.45)

Under a vector gauge transformation xi → xi + δxi , the metric perturbations transformas
˙ i,
Bi → Bi + aδx

Ei → Ei − δxi ,(A.46)and so σi is a gauge-invariant quantity. Both δqi and δΠi are gauge-invariant. Thus, we
obtain the following equations that govern vector perturbations:σ̇i + 3Hσi = 8πGδΠi
2k σi = 16πGδqi .

(A.47)(A.48)

Equation (A.47) shows that in the absence of anisotropic stress sources, vector metric perturbations decay away due to Hubble friction.
Tensor perturbations
Tensor perturbations correspond to gravitational waves. We write the perturbed metric as
ds2 = −dt2 + a2 (δij + hij ) dxi dxj ,(A.49)

where hij is transverse and traceless: ∂ i hij = 0 and δ ij hij = 0. The equation of motion is
given byḧij + 3H ḣij +k2
hij = 16πGδΠij ,a2(A.50)

where δΠij stands for the tensor component of the matter anisotropic stress, and two possible
polarization states, “+” and “×”, are suppressed.

124

A.2

Quantum theory of perturbations

Quantization of ﬁelds in a nontrivial background generally gives rise to particle production,
which is responsible for structure formation in the inﬂationary universe and thus provides
the initial conditions for the classical evolution of perturbations described in the previous
section. We now explain how we quantize cosmological perturbations.
Scalar perturbations
Quantization of scalar perturbations in single-ﬁeld inﬂation is based on the comoving curvature perturbation,
R=ψ+

H
δϕ,
ϕ̇

(A.51)and in terms of this we deﬁne the new variable [110]ϕ̇u := zR = a δϕ + ψ ,H(A.52)

where
z :=

aϕ̇
.
H

(A.53)

Then, the Einstein-Hilbert action truncated to quadratic order in a small ﬂuctuation u isgiven by



1
z 
1
3
3
(A.54)dηd xL =dηd x (∂η u)2 − δ ij ∂i u∂j u + u2 ,S=22zwhere η is the conformal time and a prime denotes partial diﬀerentiation with respect toη. This is equivalent to the action for a scalar ﬁeld in ﬂat spacetime with a time-dependenteﬀective mass m2 = −z  /z, and here its origin is attributed to the variation of the backgroundspacetime.
The theory is now quantized by promoting u and its conjugate momentum,π=∂L= ∂η u,∂(∂η u)(A.55)

to operators satisfying the following equal time commutation relations:[û(η, x), π̂(η, y)] = iδ (3) (x − y),(A.56)[û(η, x), û(η, y)] = [π̂(η, x), π̂(η, y)] = 0.

(A.57)The operator û can be expanded asd3 kû(η, x) =uk (η)âk eik·x + u∗k (η)â†k e−ik·x ,
3/2
(2π)

(A.58)

where the mode uk and its complex conjugate u∗k form a complete orthonormal basis withrespect to the Wronskian:(uk · uk ) = − (u∗k · u∗k ) = 1,

(uk · u∗k ) = 0,

(A.59)

125
where(φ · ϕ) := −i (φ∂η ϕ∗ − ϕ∗ ∂η φ) .

(A.60)The equal time commutation relations for û and π̂ imply the usual commutation relations forthe annihilation and creation operators âk and â†k :[âk , â†k ] = δ (3) k − k ,[âk , âk ] = [â†k , â†k ] = 0.

Then the vacuum is deﬁned as the state annihilated by âk , so that âk |0 = 0.
The ﬁeld equation for uk is given by


z 

2
uk + k −uk = 0,z(A.61)(A.62)(A.63)and the correct form of the mode is determined so that ordinary quantum ﬁeld theory in ﬂatspacetime is reproduced at short distances. Namely, we impose, in the limit of k/aH → ∞,1
uk → √ e−ikη .
2k(A.64)The spectrum of the comoving curvature perturbation is deﬁned by
Rk Rk  =(2π)3 (3)δ (k + k )PR (k),4πk 3(A.65)

where the statistical average · · ·  can now be replaced by the expectation value 0| · · · |0,
leading to the formula
PR (k) =k 3  uk 2  .
2π 2 z(A.66)

Tensor perturbations
For tensor perturbations, the Einstein-Hilbert action truncated to quadratic order reduces to
2 MPl(A.67)
S=dηd3 x a2 (η)∂µ hij ∂ µ hij ,8and deﬁning v (A) (A = +, ×) by
MPl(×) (×)√ a(η)hij = v (+) e(+)
eij ,ij + v2
we obtain the following eﬀective action

2
 1
a  (A) 2
3.
S=dηd x ∂η v (A) − δ ij ∂i v (A) ∂j v (A) +v
2a(A.68)(A.69)

A=+,×

For each polarization state, this gives the action equivalent to that for a scalar ﬁeld in ﬂatspacetime with a time-dependent mass m2 = −a /a. Thus, along the same path as scalar

126
perturbations, we can quantize tensor perturbations and compute the power spectrum justby replacing z by a. Since a depends only on the background geometry, the situation is mucheasier now.
Unlike the case of scalar perturbations, where metric and matter perturbations are simultaneously quantized and hence a nonvanishing matter component is required, it is possible to quantize gravitational waves in a de Sitter background. For de Sitter inﬂation,
√ we

2
−ikη
have a /a = 1/η . The mode solution vk that satisﬁes the condition vk → e/ 2k ask/aH = −kη → ∞ is given by


i
1
√
1−
vk =
e−ikη .
(A.70)kη
2k
Thus, the spectrum well after horizon exit is 2
H2
k 3  vk 2
2.
PGW = 2 · 2   = 2
MPl 2π aMPl 2π

(A.71)The growing mode of tensor perturbations hij stays constant well outside the horizon andhence the amplitude is basically determined by the Hubble parameter at horizon exit. As aresult, slow-roll (i.e., non de Sitter) inﬂation predicts a slightly tilted tensor spectrum: 2 H 
2
PGW (k) =.
(A.72)
2
MPl 2π 
k=aH

For a detailed calculation including scalar power spectra, see Ref. [137].

Appendix B

Derivation of the eﬀective Einsteinequations on the brane
In this appendix we replicate the derivation of the well-known projected Einstein equations
on the brane [134] (see also [101]). The extension to more complicated braneworld models isfound in Refs. [103, 104, 105].
Our starting point is the ﬁve-dimensional Einstein equations that determine the bulk
geometry:(5)

GAB = −Λ5 gAB + κ25 (5) TAB .

(B.1)

The Gauss and Codazzi equations relate the four-dimensional Riemann tensor Rµνλσ , theextrinsic curvature Kµν on the brane, and the ﬁve-dimensional curvature tensor as follows:Rµνλσ = (5) RABCD qµA qνB qλC qσD + Kµλ Kνσ − Kµσ Kνλ ,(B.2)∇ν K νµ − ∇µ K = (5) RAB qµA nB ,(B.3)

where nA is the unit normal to the brane, qµν := gµν − nµ nν is the induced metric on thebrane, and ∇µ is the covariant derivative associated with qµν . Contracting µ and λ in theGauss equation (B.2) we obtain the four-dimensional Ricci tensor:
Rµν = (5) RAB qµA qνB − (5) RABCD nA nC qµB qνD + KKµν − Kµλ Kλν .
It follows from Eqs. (B.1) and (B.4) that
12 2 (5)
1 (5)
A B
(5)
A B
TAB qµ qν +TAB n n −T qµνGµν = − Λ5 + κ52
34

1  λσ+KKµν − Kµλ Kλν +K Kλσ − K 2 qµν − Eµν ,
2

(B.4)

(B.5)

where Eµν is the projected Weyl tensor1 :Eµν = (5) CABCD nA nC qµB qνD .
1

(B.7)

The d-dimensional Riemann tensor can be decomposed into Ricci and traceless parts:
Rµανβ =´22 `gµ[ν Rβ]α − gα[ν Rβ]µ −Rgµ[ν gβ]α + Cµανβ .
d−2(d − 1)(d − 2)This is the deﬁnition of the Weyl tensor Cµνλσ .

127

(B.6)

128
Let y be a Gaussian normal coordinate orthogonal to the brane so that nA dxA = dy. Let
the energy-momentum tensor be of the form
(5)

TAB = TAB + SAB δ(y),(B.8)

SAB = −σqAB + TAB ,(B.9)

where σ is the tension of the brane and TAB is the contribution from brane matter. (Thebrane is located at y = 0 without loss of generality.) To make the ﬁnal result as general as
possible, we allow for the energy-momentum tensor TAB of any bulk matter (e.g., dilaton andmoduli ﬁelds). The junction conditions2 imply that


κ25
1
(B.10)
Sµν − Sqµν ,Kµν |y=0+ = −23
where we assumed the Z2 -symmetry across the brane. Substituting this into Eq. (B.5), we
obtainGµν = −Λqµν + κ2 Tµν + 6

κ2κ2
πµν − Eµν + 4 Fµν ,σσ(B.11)

where
κ2 =
Λ =

1 4
κ σ,
6 5

1
Λ5 + κ2 σ ,2(B.12)(B.13)

andπµν
κ25 Fµν11
1
1:= − Tµα Tν α + T Tµν + qµν Tαβ T αβ − qµν T 2 ,
4
12
8
24


1
:= TAB qµA qνB + TAB nA nB − T qµν .
4(B.14)(B.15)

This is the ﬁeld equations projected on the brane. The eﬀect of Kaluza-Klein modes is
encoded into Eµν . If the bulk is empty except for the cosmological constant, Fµν vanishes,
as is the case in the Randall-Sundrum model. These two terms carry bulk information andtherefore in general the geometry on the brane cannot be determined solely from the above
2

The junction conditions will be most easily derived as follows. Let us consider a (d − 1)-dimensional
discontinuous (timelike) hypersurface in d-dimensional spacetime, and let the energy-momentum tensor be of
the form (d) TAB = · · · + Sµν δ(y), as in the main text. In the Gaussian normal coordinates we haveKµν = −Γyµν =1∂y gµν ,2

which is discontinuous across the hypersurface. It is straightforward to show that the Ricci tensor is expressedin terms of the extrinsic curvature as Rµν = −∂y Kµν + · · · . Then we have
„
«
Z 
1
lim
Rµν dy = κ2d Sµν −Sqµν→0 −
d−2=−Kµν |y=0+ + Kµν |y=0− ,where we used the d-dimensional Einstein equations in the ﬁrst line. Here κ2d is the d-dimensional gravitational
constant. This together with the Z2 -symmetry Kµν |y=0+ = −Kµν |y=0− yields Eq. (B.10).

129
eﬀective equations; one needs information on the bulk geometry which is determined fromthe full ﬁve-dimensional ﬁeld equations. Nevertheless, the eﬀective Einstein equations areuseful in that they provide us some insight into what is the modiﬁcation to the standard
four-dimensional general relativity.
Using the ﬁve-dimensional Einstein equations (B.1), the Codazzi equation (B.3), and thejunction condition (B.10), we can derive the (non-)conservation equation:∇ν T νµ = −2TAB nA g Bµ ,(B.16)

which means that generally there is exchange of energy-momentum between the brane andthe bulk. The standard conservation equation ∇ν T νµ = 0 holds in the absence of the bulk
matter.

Appendix C

UnitsNatural units• MPl = (c/G)1/2 = 2.177 × 10−5 g• lPl = 1.616 × 10−33 cm• tPl = 5.391 × 10−44 sec• TPl = 1.416 × 1032 K = 1.221 × 1019 GeV
Conversion from natural units• 1 cm = 5.068 × 1013 GeV−1 • 1 sec = 1.519 × 1024 GeV−1 /c• 1 g = 5.608 × 1025 GeV/c2• 1 erg = 6.242 × 102 GeV• 1 K = 8.618 × 10−14 GeV/kB

130

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