Doctoral thesis

Observational Predictionsof Generalized Galilean Genesis

Sakine Nishi

Department of Physics, Rikkyo University
15RA002B

Abstract
Inflation is a very successful scenario for the early universe. However, even an inflationary
universe has a singularity in the past, and the alternative scenarios motivated by thisproblem have been discussed. In this thesis, we focus on one of the alternative scenarios
called Galilean genesis. This scenario has the fascinating feature that the universe starts
expanding from approximately Minkowski space-time. If this model gives the same observational prediction as that of inflation and solves the problems that inflation solves, wehave to discuss how one can distinguish the two models. Thus, in this thesis, we discuss
two aspects of primordial gravitational waves generated in the genesis phase. One is thegravitation and solution of GWs during the genesis-reheating transformation and, after
that, the radiation-dominated universe. The other is the possibility of the various spectrum of GWs by extending the existing genesis models. The genesis models we discussin this thesis are described in the Horndeski theory, which is the generalized scalar-tensor
theory having up to second derivative terms.
For the phase transition and the reheating, we figure out the spectrum in some casesand discuss the range of the frequency of the detector, which can catch the gravitational
waves. As a result, we can find the spectrum for genesis by using the detector, which
can detect at a higher frequency than we project now. For the low-frequency region, we
can not find the gravitational waves because the amplitude cannot grow in Minkowski
space-time.
For the various spectrum of gravitational waves, we study the extension of the existingmodel. The generalized Galilean genesis model has one constant parameter, and thenwe additionally introduce a parameter. In the previous study, we can find the various
spectrum only for scalar perturbation by introducing one parameter. However, by extending the model, we can find the spectrum of gravitational waves and scalar perturbation
are varying by two parameters. This result implies that whatever observational result wefind, it can be explained by this Galilean genesis model. In this model, we can distinguishbetween the models with the consistency relation.

Contents
Introduction

3

Acknowledgements

5

I

7

Review of Cosmology

1 Standard Big Bang Cosmology
1.1 Expanding Universe and the Hubble law . . . . . . . . . . . . . . . . . . . .
1.2 Big bang model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9
10
11
12
15

2 Inflation
2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Slow-roll inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 K-inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 G-inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
18
20
24
25

3 Modified Gravity
3.1 f (R) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Brans-Dicke theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Kinetic gravity braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Galileon theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Horndeski theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 GLPV theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusions in this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
28
29
31
33
35
36

4 Cosmological perturbations
37
4.1 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1

2

CONTENTS
4.2
4.3
4.4
4.5
4.6

Matter perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cosmological perturbations in General Relativity . . . . . . . . . . . . . . .
Cosmological perturbations in Horndeski Theory . . . . . . . . . . . . . . .
Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40
41
42
43
44

5 Some topics in Modern Cosmology
5.1 Curvaton field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Reheating of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Alternative scenarios to inflation . . . . . . . . . . . . . . . . . . . . . . . .

53
53
54
58

II

63

Galilean Genesis

6 Generalized Galilean genesis
6.1 Generalized genesis solutions . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Background evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Primoridal perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Curvaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Conclusions in this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
66
68
74
78
79

7 Reheating and primordial gravitational waves in GGG
7.1 Gravitational particle production . . . . . . . . . . . . . . . . . . . . . . . .
7.2 The spectrum of primordial gravitational waves . . . . . . . . . . . . . . . .
7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Conclusions in this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . .

81
82
87
91
96

8 Scale-invariant perturbations
99
8.1 A new Lagrangian for Galilean Genesis . . . . . . . . . . . . . . . . . . . . . 100
8.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.3 Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.5 Conclusions in this chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Conclusions

113

Introduction
How our Universe started may be one of the interesting subjects not only for researchers
but also for everyone. The Big Bang cosmology is the well-known scenario of the universe, and then the inflationary scenario was proposed to solve the problems of the big
bang. Nowadays the standard cosmological scenario of the primordial universe is inflation,
however, this scenario has the problem that the universe has a singularity at the initial
time.
Motivated by this problem, there are many alternative scenarios. In this thesis, we
focus on the specific model called Galilean genesis [1]. Though the scenario which has
accelerating expansion is called inflation, we call inflation for the model that the expansion
law is the approximately de-Sitter expansion to distinguish between the inflation and its
alternatives.
In this way, there exist many models of the early universe, and our interest is to findwhich scenario was caused. To consider this, what we often do is to check the instabilities
in the model, to compare to the observation and so on. There is the possibility the
problems appeared by this resarch can be avoided by extending the model.
For the scalar perturbation, WMAP and Planck satellites are the famous detector.
They observe the cosmic microwave background (CMB) and the temperature of Universe,
and we can find many cosmological parameters. For the gravitational waves, there are
famous detectors such as LIGO, Virgo, DECIGO, and KAGRA. The event may be freshin our memory that the direct detection was achieved and reported in February 2016. The
gravitational waves generated from binary black hole can be detected in present days, andwe expect the detection of primordial gravitational waves in the future.
This thesis is based on my work during doctoral course [2] and [3]. The main topic
of these study is gravitational waves generated during genesis phase. The information wealready have to compare the models are the flat spectrum in inflation and blue spectrum
in alternative scenarios. The contents of this thesis are as follows.
In part 1, we introduce some basic topics of cosmology. First, we introduce the discovery of expanding universe such as Hubble law and how to explain the physical values incosmology. As our goal is to discuss the alternative scenario to inflation by using modified
gravity theory, we will introduce the models of inflation and some modified theories. Then
finally we introduce some cosmological topics we will discuss in Galilean genesis in part 2.
3

4

CONTENTS

In part 2, we discuss some topics for Galilean genesis. First, we introduce the generalized model of the Galilean genesis [4]. Then we will discuss the gravitational reheating in
generalized Galilean genesis, and compare the shape of the power spectrum [2]. Moreover,then we discuss farther modification of generalized Galilean genesis to get the various
spectrum of the scalar perturbation and gravitational waves [3]. Then finally, we conclude
the discussion of this thesis.

Conventions
Indices for spacea, b, c, ... = 1, 2, 3

Indices for spece-time

α, β, γ, ... = 0, 1, 2, 3

Unit

c = 1, ℏ = 1

Signeture(−, +, +, +)Christoﬀel symbol1Γρµν = g ρσ (gµρ,ν + gρν,µ − gµν,ρ )2

Riemann tensorRρµνσ = −∂µ Γρνσ + ∂ν Γρµσ − Γκνσ Γρµκ + Γκµσ Γρνκ

Ricci tensor

Rµν = g ρσ Rρµσν

Ricci scalar

R = g µν Rµν

Einstein tensor
Gravitational constant1Gµν = Rµν − gµν R
2
G = GNewton =

−2
MPl
8π

Acknowledgements
First of all, I would like to thank my supervisor, Tsutomu Kobayashi. I have learned manythings from him since I was an undergraduate student. And I would like to thank all themembers of the group of theoretical physics at Rikkyo University and all the people whohave discussed with me. It is a very fortunate for me to be able to study and discuss
together in a very active group. I also would like to thank my family and friends. The
works I wrote in this thesis were supported in part by the JSPS Research Fellowships for
Young Scientists No. 15J04044.

5

Part I

Review of Cosmology

7Chapter 1

Standard Big Bang Cosmology
Inflation is the most standard scenario for explaining the evolution of the early universe.
In this thesis, our goal is to discuss the alternative scenario to inflation and to compare
the scenarios. Thus let us introduce the basic materials of cosmology in this section, andreview the history of cosmology from discovering the expanding universe to developing the
inflationary scenario.

1.1

Expanding Universe and the Hubble law

The fact that the Universe is expanding is known very well. The man who found thisis Edwin Hubble [5]. Vesto M. Slipher found that the spectrum of most galaxies was
redshifted, and this suggests the galaxies are moving far away. Meanwhile, many models
for the universe were discussed as a solution of general relativity (GR). Then in 1929,
Hubble found the law from the observation shown as fig.1.1, and the law is described asv = Hr,(1.1.1)

where v is the velocity of the galaxy, H is called the Hubble parameter explained in thefollowing section, and r is the distance from here to the galaxy. Defining a(t) as therepresentation of length between observers, the Hubble parameter is defined by the scale
factor a(t),H(t) ≡ȧ.
a(1.1.2)

According to the observational results, many galaxies go away, and those far from us moves
faster than that is near here. Eq.(1.1.1) shows the velocity of the galaxy, and we can findthe fact that the universe expands.
9

10

CHAPTER 1. STANDARD BIG BANG COSMOLOGY

Figure 1.1: The relation of velocity and distance for nebulae. The black points have therelation of proportion shown as the line. After combining these points into groups as thecircles, the relation is shown as the dashed line. This figure is cited from [5].

1.2

Big bang model

As the universe is expanding, we can consider easily the universe begins from the hot andhigh-density state. Such a scenario the universe started from the hot universe is called
the big bang. However, before we obtain the evidence of big bang theory, the influential
theories were explaining the evolution of the universe such as the steady-state theory or bigbang theory. In Steady-state theory, the energy density is unvarying although universe
expands. The temperature at present is calculated, and the observation founded it forradio astronomy. Arno A. Penzias and Robert W. Wilson tried to remove the noise byimproving observation equipment, and they found big band theory explains the noise.
The noise has the spectrum of the black body at the temperature of 3K, and this number
is accord to what calculated by researchers who studied big bang theory. This is called
cosmic microwave background (CMB). The equipment for CMB is currently developed,
and now we have the temperature of the universe by the result of COBE and Planck(fig.1.2)
T0 ∼ 2.7 K.

(1.2.1)

Also, there was the problem that the total amount of light elements in the universe wasmuch larger than that can be generated with stars. However, calculating the amount oflight elements produced by Big Bang nucleosynthesis, we found it is same to the obser-

1.3. METRIC

11

Figure 1.2: The CMB map of Planck [6]. The noise of temperature characterized by Kcmb
is mapped with color coding.
vation. In this way, big bang theory became standard. However, there is some unsolved
point such as flatness and horizon problem and so on. We will review these problems in
the later section.

1.3

MetricFrom now let us introduce many expressions to discuss some cosmological topic. As wetreat the evolution of space-time, we would like to introduce the metric at first. Generally
we describe the line element as
ds2 = gµν dxµ dxν ,(1.3.1)

which can be divided by time and space as
ds2 = g00 dt2 + g0i dx0 dxi + gij dxi dxj .

(1.3.2)Seeing the large scale from the fundamental observer with the proper time x0 = ct, we
can assume the universe is isotropic and homogeneous and we can set
g00 = 1,

(1.3.3)g0i = 0.

(1.3.4)

12

CHAPTER 1. STANDARD BIG BANG COSMOLOGY

The distance between two observer dl(t0 ) changes with a(t) as
dl(t) = a(t)dl(t0 ),(1.3.5)where a(t) is called scale factor. Similarly, gij is given as
gij = a2 (t)γij ,(1.3.6)

where γij shows the metric of 3 dimensional space. Therefore we haveds2 = −dt2 + a2 (t)γij dxi dxj .

(1.3.7)This is called Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) metric. Note that we often
use the conformal time defined by
dη =

dt
.
a(t)

In polar coordinate system, it can be written as[]
ds2 = −dt2 + a2 (t) dr2 + r2 (dθ2 + sin2 θdϕ2 ) .
Then, introducing curvature K, the metric is given by[]
dr2
ds2 = −dt2 + a2 (t)+ r2 (dθ2 + sin2 θdϕ2 ) .
1 − Kr

1.4(1.3.8)(1.3.9)(1.3.10)

Field equations

It is known that our universe evolves throughout some eras such as radiation dominant
phase or matter dominant phase. The description of the evolution depends on how theequation of state is shown. Let us we discuss the evolution of the universe in each epochwith solving the field equations. First, we introduce the FLRW metric with curvature
[
]
dr2
2
2
2
2
2
2
2
ds = −dt + a (t)+ r (dθ + sin θdϕ ) .
(1.4.1)
1 − KrApplying this metric to the Einstein equation with cosmological constant Λ, we obtain1
Rµν − gµν R + gµν Λ = 8πGTµν ,2where Tµν is the energy momentum tensor which is given asρ 0 0 0
 0 p 0 0 Tµν =  0 0 p 0 .
0 0 0 p(1.4.2)(1.4.3)

1.4. FIELD EQUATIONS

13

Then we can find
( )2
K
ȧ
8πG
Λ
ρ− 2 + ,
=
a
3
a
3
ä
4πG
Λ
= −
(ρ + 3p) + ,a3
3(1.4.4)(1.4.5)where ρ is the energy density, and p is the pressure. These equations are derived fromEinstein equations and called Friedmann equation eq.(1.4.4) and acceleration equationeq.(1.4.5). The energy momentum tensor Tµν is subject to the conservation law asTµν;ν = 0,

(1.4.6)ȧρ̇ + 3 (ρ + p) = 0.
a(1.4.7)from which we obtain

If the equation of state is written as
p = wρ,(1.4.8)where w is a constant parameter depending on what is dominant in the era we are considering, we can solve eq.(1.4.7) asρ = a−3(w+1) .

(1.4.9)To consider the rate of the energy, we introduce the density parameter Ωa for energy
density whose component is characterized by a
ρa0
,
ρc0

(1.4.10)

3H02
.
8πG

(1.4.11)Ωa0 =where ρc0 is the critical energy density,
ρc0 =The critical energy density is given by eq.(1.4.4) with taking K = 0, Λ = 0 and present
time t = t0 . For K and Λ the density parameter is given as
K
,
H02
Λ
ΩΛ0 =
,
3H02
ΩK0 =

(1.4.12)(1.4.13)

14

CHAPTER 1. STANDARD BIG BANG COSMOLOGY

where we set the scale factor at present time a0 = 1. The total energy density ρtot iswritten as∑ρtot =
ρa .
(1.4.14)
Although various components is supposed, we assume ρtot has radiation ρr , matter ρm ,and dark energy ρd from now on. The general equation for scale factor is given as[{[ ∫]}]1/2
da
Ωm0
2 Ωr0
2
−1
,
= H0
+ Ωd0 a exp 3 (1 + w)a da+(1.4.15)dt
a2
aPractically, the evolution of scale factor for present era can be find by fixing these parameters by observation. Let us solve the equations by assuming the case that the components
except one can be negligible.

1.4.1

Radiation dominant phase

In the radiation dominant phase, coeﬃcient w in the equation of state is1
w= ,(1.4.16)3
Thus, the equation for the scale factor a(t) in radiation dominant phase is given asȧ(t) =

H02 Ωr0.
a2(1.4.17)

The solution of this equation is
(
)1/2
1/2
a(t) = a(ti ) 2H0 Ωr0 (t − ti ),(1.4.18)

where ti is the end of the inflation and we assume the integral constant can be negligible.

1.4.2

Matter dominant phase

During the matter dominant phase w is given as
w = 0.

(1.4.19)

Therefore, we can find the equation for the scale factor a(t) in matter dominant phase asȧ(t) =
Thus the solution is(a(t) = a(tr )H02 Ωm0
.
a

)3/2
3
1/2
H0 Ωm0 (t − tr ),2

(1.4.20)(1.4.21)

where tr is the start of the matter dominant phase and we assume the integral constantcan be negligible.

1.5. HORIZONS

1.4.3

15

Dark energy

In the case that Ωd is dominant, we do not have information for w because we do notknow what the dark energy is. The equation for scale factor is given as
[ ∫
]
2
2
−1
ȧ(t) = H0 Ωd a exp 3 (1 + w)a da .
(1.4.22)
If this dark energy is cosmological constant, the parameter for the equation of state is
w = −1.

(1.4.23)

To make the accelerating expansion of the universe, the condition for w is
1
w<− .
3

1.5

(1.4.24)

Horizons

To discuss the following topics, we have to introduce the definitions of length and horizonin cosmology. Let us consider the length that the light runs from t = t0 to t = t1 . Light
goes on the null geodesic line ds = 0 and from the isotropy we have
∫ t1
1
le =
(1.5.1)dt.
a(t)t0This length is measured in comoving coordinate, and le extend with scale factor a(t). We
introduce the length lh as
lh = a(t)le ,(1.5.2)

Considering the path of light for the past, the region which has causal relation characterized by∫ t01
lp = a(t)dt,(1.5.3)a(t)tiniwhere we take t = tini as the birth of the universe and t = t0 as the present time. We callthis lp particle horizon. Conversely, the causal relation for future is shown by∫ t∞
1
le = a(t)(1.5.4)dt,a(t)t0
and it is called event horizon. These expressions so far are for the distance for space. Then
let us consider the distance for time especially the age of the universe. Taking t0 as theinitial time of big bang and t as the present time, we obtain
∫ t0
∫ 1
da
tage = to − t =dt =,(1.5.5)
t
a aH

16

CHAPTER 1. STANDARD BIG BANG COSMOLOGY

where we take the scale factor at the present time as a(t) = 1. This can be calculatedroughly and described as inverse of the Hubble parametertage ∼ H0−1 .

(1.5.6)

Precisely, applying the solution of eq.(1.4.15) with determining Ωa from observation, we
can find the age of universe.
tage ∼ 13.8 Gyr.

(1.5.7)

Chapter 2

Inflation
In this section, we introduce some models of inflation. Inflation needs accelerating expansion, so we haveä > 0.

(2.0.1)

Let us review some typical models of inflation. From now on we assume K = 0 and Λ = 0
in eq.(1.4.4) and eq.(1.4.5), and we use
( )2
ȧ
8πG
=
ρ,
a
3
4πG
ä
= −
(ρ + 3p).
a
3(2.0.2)(2.0.3)

Inflation is known as the scenario which can solve these problems of the big bang.
There are many models of inflation, and we will discuss the specific models in the later
section. Basically, in inflationary phase the expansion law is approximated by de Sitter
expansion asa(t) = a(ti )eH(t−ti ) ,ȧ
H ≡ = const,a(2.0.4)(2.0.5)where t = ti is the initial time of inflation. How long the inflationary phase continued ischaracterized by e-folding number shown as∫af= Hdt.
N = log(2.0.6)ai
Let us review how inflation could solve the problems and derive the e-folding number N
which is needed to do that.
17

18

CHAPTER 2. INFLATION

Obs

t

t=t0

t=tL
p1

p2

pn

t=t*

Figure 2.1: The schematic picture of the horizon problem. t0 is the present time, tL is thelast scattering time and t∗ is the start of radiation dominant phase. Obs is the observer
seeing the CMB, he or she observe particle horizon scale. The event horizon is smaller
than the particle horizon.

2.1

Problems

Before the introduction of many models of inflationary scenario, we have to know what
are the problems for Big Bang scenario. In this section, let us discuss the problems by
inflationary scenario. Following discussion is the review of [7, 8].

2.1.1

Horizon problem

In introduction we referred to CMB observation. Figure 1.2 is the observational resultof CMB from Planck2015 [6]. As we can see in this figure, the structure of CMB has
isotropy and small fluctuation. It is natural to consider that this structure was initiallythe quantum fluctuation and it was expanded by the evolution of the universe. However,in big bang scenario, we can not explain the figure of CMB. The event horizon from the
beginning of big bang to the last scattering is smaller than the particle horizon shownin fig.2.1. In big bang scenario, CMB structure has to show some regions which havediﬀerent temperature because the regions do not have causality each other. This problem
is called horizon problem. The structure shown as CMB is explained by the expansion of2.1. PROBLEMS

19

Inflationary phase, and thus let us consider this problem. The event horizon is given as
∫ t∗
1
dt,
le = a(tL )(2.1.1)a(t)tL
where we take tL is the time of last scattering, and t∗ is the end of inflation. As the
expansion in inflation is given in eq.(2.0.4), the particle horizon is
∫ t∞
1
dt.
lp = a(t)(2.1.2)a(t)t0
To obtain the CMB structure we need le > lp , after some calculations we can find thee-folding number needed to solve this problem. For the event horizon, we have∫ t∗
1
le = a(tL )dttL a(t∗ )exp[Hinf (t − t∗)]a(tL )(eN − 1)=a( t∗)Hinf≃a(tL )eN .
a( t∗)Hinf(2.1.3)For particle horizon, by using eq.(1.4.15) we obtaina(tL )
a0 H0(2.1.4)a∗ Hinf.
a0 H0(2.1.5)le ≃Thus we findeN >Then, applying the observational results, we can estimate the number of N as
N > 62.

2.1.2(2.1.6)

Flatness problem

For the number of curvature K, some observation suggests |ΩK | < 1. In addition, CMB
suggests the most optimum number is ΩK = 0. However, this is at present. If we seek
the exact value, K becomes too high precision. Considering inflationary scenario solves
this problem because ΩK becomes negligible during the inflationary era in any K. Let us
focus on the density parameter Ωρ and ΩK . From the Friedmann equation, we need theconditionΩρ > ΩK .

(2.1.7)

20

CHAPTER 2. INFLATIONV( )Slow-roll●

Reheating●Figure 2.2: Skematic picture of inflationary scenario. The inflaton rolls down on its
potential, and after that the inflaton oscillates at the bottom of the potential.
In inflation, recalling af = ai eN and H = const., we can anticipate easily that this problem
can be solved because ΩK is written asΩK =K
a2 H 2.

(2.1.8)After some calculations we can find the e-folding number needed to solve[this problem.] If
4the energy density at the beginning of radiation dominant phase is ρr = 2 × 101 6GeV ,we haveN > 62.

2.2(2.1.9)Slow-roll inflation

Constructing the model of inflationary scenario, we define the Lagrangian as
Ltot = LGR + Linf .

(2.2.1)

The models of inflationary scenario is shown by the gravitational field gµν and a scalar
field ϕ called inflaton, and its Lagrangian is shown as
R
,
16πG
1
Linf = − (∂ϕ)2 − V (ϕ),
2LGR =(2.2.2)(2.2.3)

2.2. SLOW-ROLL INFLATION

21

where V (ϕ) is the potential of the inflaton. From this we obtain energy density ρ, pressurep and the field equation for ϕ as1
ρ = ϕ̇2 + V (ϕ),(2.2.4)2
1
p = ϕ̇2 − V (ϕ),(2.2.5)2
ϕ̈ + 3H ϕ̇ + Vϕ = 0,(2.2.6)
where “ϕ ” is the derivative of ϕ. For the slow-roll inflation, we consider the scenario thatthe inflaton rolls down on its potential as fig.2.2. Then at the end of inflation, the inflaton
oscillates at the bottom of the potential and decay into the other field. The oscillationand decay make high temperature, and it called reheating we will discuss in sec.5.2 . To
construct the slow-roll inflation we assumeϕ̇2 ≪ V,(2.2.7)|ϕ̈| ≪ 3H|ϕ̇|.

(2.2.8)Thus we have
3H ϕ̇ + Vϕ = 0.

(2.2.9)From these equations, we can find the conditions for the potential( )2Vϕ≪ 24πG(2.2.10)V
Vϕϕ≪ 24πG(2.2.11)V
When we discuss the model of inflation, we often use slow-roll parameters [9] as follows.
The slow-roll parameters are explained by derivative of potential V (ϕ)
( )2
Vϕ
1
ϵV =
,
(2.2.12)
8πG V
1 Vϕϕ
ηV =
,
(2.2.13)8πG Vor written by Hubble parameter H(t)ϵH = −

Ḣ
3ϕ̇
=
,
H2
2V + ϕ̇2(2.2.14)

ηH = −

1 Ḧ
ϕ̈
.
=−
2 H Ḣ
H ϕ̇

(2.2.15)

To consider the slow-roll approximation, we need the absolute numbers of these parameters
are much smaller than 1. In this inflationary scenario, the end of inflation quits when theslow-roll parameter becomes larger than 1.Then let us consider some specific models ofslow-roll inflation.

22

2.2.1

CHAPTER 2. INFLATION

Large-field inflation

The Large-field inflation (chaotic inflation) is proposed by A. D. Linde [10]. In this model
the potential of the inflaton is given as
V (ϕ) = V0 ϕn ,(2.2.16)

where n is the number and V0 is the constant. In this model, the slow-roll parameter isgiven as
ϵV =

n2
.
8πGϕ2

(2.2.17)

Thus the inflation in this model quit when the parameter becomes ϵ > 1 after the scalar
field rolling down its potential. For example, now we consider the case of1
V (ϕ) = m2 ϕ2 ,(2.2.18)
2
where m is the mass of inflaton. As the second derivative of time for inflaton ϕ written as
ϕ̈ =

dϕ̇
dϕ̇
= ϕ̇ ,dt
dϕ(2.2.19)

the field equations eq.(2.0.2) and eq.(2.2.6) without the approximations derive the equation√dϕ̇
12πG(ϕ̇2 + m2 ϕ2 )1/2 + m2 ϕ2=−(2.2.20).
dϕ
ϕ̇
Therefore we can figure out this relation in fig.2.3. As this dϕ̇/dϕ attracts to the state of
dϕ̇
≃ 0,
dϕ

(2.2.21)

we obtain from eq.(2.2.20)
ϕ̇ = − √m.
12πG(2.2.22)

Thus we find the solutions asm
ϕ(t) = ϕi − √(t − ti ),
12πG
[√
]
4πG
m2
a(t) = ai exp
mϕi (t − ti ) −(t − ti )2 ,3
6

(2.2.23)(2.2.24)

where t = ti is the initial time of inflation. Therefore, to make the e-folding number larger
than N = 60, we takeϕ i ≳ G− 2 .
1(2.2.25)

2.2. SLOW-ROLL INFLATION

23

Figure 2.3: The phase space in large-scale inflation. The state attracts to the thick line.
This figure is cited from [11] with the unit G = 1.

2.2.2

Small-field inflation

In this model the potential of the inflaton is typically written as[
V (ϕ) = V0

( )n ]
ϕ1−,µ(2.2.26)

where µ has the mass scale. Now we consider the example as
V (ϕ) =λ 2
(ϕ − v 2 )2 ,4

(2.2.27)

where the vacuum state is at ϕ = ±v. Solving the equations, we can find the solution fora(t)[√a(t) = ai exp

]
4πG
m2
2
mϕi (t − ti ) −(t − ti ) ,36(2.2.28)

where t = ti is the initial time of inflation. In this model, the slow-roll parameter is givenas[( )n ]−2n2 ϕ−2
µ
ϵV =
1−
.
8πG
ϕ

(2.2.29)

To satisfy ϵV ≪ 1, we obtain ϕ < µ and G− 2 < ϕ. Thus we find the inflation occurs if
1µ > G− 2 .
1

24

2.2.3

CHAPTER 2. INFLATION

Starobinsky inflation

The inflation models we reviewed up to now have the kinetic energy and the various
potential of the inflaton in GR. However, there are many models inflation generated inmodified gravity. Starobinsky inflation is a kind of these models [12]. In this model, theLagrangian of GR is extended as
L = R + αR2 ,(2.2.30)

where α is the parameter. Such model is included in f (R) gravity in sec.3.1. In this model,
the field equations are−(1 + 2αR)Ḣ = αR̈ − αṘH,
13(1 + 2αR)H 2 = αR2 − 6αH Ṙ.
2(2.2.31)(2.2.32)

As the slow-roll parameters during inflation are much smaller than 1, we can find thesolutions asH≃H−1(t − ti ),12αa ≃ ai exp[Hi (t − ti ) −R ≃ 12H 2 −(2.2.33)
1(t − ti )2 ],
72α

1
.
6α2

(2.2.34)(2.2.35)

In this model, the slow-roll parameter is given as
ϵH = 02.3(2.2.36)

K-inflation

In the inflation models we reviewed so far, inflation is caused by the inflaton rolling down
its potential. In K-inflation, there are no potential and inflaton driven by kinetic energy.
This is studied in [13] and [14]. The Lagrangian and detail are shown in sec.2.3. Thismodel is not a modified gravity but the subclass of Galileon theory we will review in thelater section. In this case, we have
L = K(ϕ, X),(2.3.1)

where K(ϕ, X) is arbitrary function of ϕ and X. X is the kinetic term of scalar field ϕ as
1
X = − g µν ∂µ ϕ∂ν ϕ.
2(2.3.2)

2.4. G-INFLATION

25

The models which have such like this Lagrangian are called K-essence [13] or K-inflation[14]. In this model, the energy momentum tensor Tµν isTµν = KX ∂µ ϕ∂ν ϕ − gµν K,(2.3.3)and thus the energy density ρ and the pressure p areρ = 2XKX − K,(2.3.4)

p = K.

(2.3.5)Thus we can find the parameter of the equation of state w
w=

2.4

K
p
=
.
ρ
2XKX − K(2.3.6)G-inflation

G-inflation is first discussed in [15] and generalized in [16]. In this inflation model, we
discuss general model of inflation which is described in Horndeki theory [17, 18] which
we will review in sec.3.5. In that paper, we calculate under the assumption that the stateof universe attracts to the de-Sitter expansion. The inflationary models written by one
scalar field and gravitational field are included in this model.

Chapter 3

Modified Gravity
In this section, we introduce some modified gravity theories. General relativity is very
beautiful and successful theory. However, there are some problems we can not explain thephenomena in cosmology. We have not understood what dark energy and dark matter are
and how these come from yet. To find the answer to the questions we have two methods,
introducing some new fields or modifying the gravitational theory. Now we focus on thelatter method. Nowadays many modified gravity theories are discussed. The later sectionin this thesis, we use generalized Galilean theory. This contains some theories of modified
gravity. Thus, now we introduce some theories and understand the features of them.
There are many kinds of modified gravity theory, and now we will show some of them.

3.1

f (R) theoryf (R) theory is investigated by Hans A. Buchdahl [19] and the action is written as∫√
1
Sein =
d4 x −gf (R) + Smatter .
(3.1.1)
16πG
In this theory, we consider the extended Lagrangian
Lf (R) = f (R),

(3.1.2)where f (R) is the function of R. The field equations are given by1f ′ (R)Rµν − f (R)gµν − ∇µ ∇ν f ′ (R) + gµν f ′ (R) = 8πGTµν .
2(3.1.3)

In FLRW metric, we obtain1
3f ′ (R)H 2 = (f ′ (R)R − f (R)) − 3H f˙ ′ (R)2−2f ′ (R)Ḣ = f¨′ (R) − H f˙ ′ (R)27(3.1.4)(3.1.5)

28

CHAPTER 3. MODIFIED GRAVITY

Now let us consider the case off (R) = R + αRn .

(3.1.6)

If we take n = 2, we can find that Starobinski inflation we discussed in sec.(2.2.3) is
included in this theory. In this case, eq.(3.1.4) becomes1
3(1 + nαRn−1 )H 2 = (n − 1)αRn − 3n(n − 1)αHRn−2 Ṙ,
2(3.1.7)and we need 1 + nαRn−1 ≫ 1 to make the accelerating expansion of the universe. Thus
we obtain
(
)
n
−
1
Ṙ
H2 ≃
R − 6nH.
(3.1.8)6n
R

3.2

Brans-Dicke theory

Brans-Dicke theory (BD theory) is given by C. Brans and R. H. Dicke [20]. The action is
given as[]∫1
ωBD (ϕ)SBD =d4 x ϕR −∇µ ϕ∇µ ϕ − U (ϕ) ,(3.2.1)16πGϕwhere ωBD is the constant in the original version. However, we discuss ωBD as the functionof the scalar field in many of the BD theories.
It is known that this theory has the relation to f (R) theory [19, 21]. Let us introduce
new scalar field and consider the Lagrangian
L = f (ψ) + fψ (R − ψ),(3.2.2)∂fwhere fψ = ∂ψ. Taking the variation with ψ, we obtainfψψ (R − ψ) = 0.

(3.2.3)

ψ = R.

(3.2.4)

Because of fψψ ̸= 0, we find

If we take ϕ = fψ , the Lagrangian eq.(3.2.2) is equal to the Lagrangian of the f (R) theory
eq.(3.1.2). Moreover, eq.(3.2.2) becomes
L = fψ R − fψ ψ + f (ψ)= ϕR − U (ϕ),
U (ϕ) = ϕψ(ϕ) − f (ψ(ϕ)).

(3.2.5)(3.2.6)

3.3. KINETIC GRAVITY BRAIDING

29

The Lagrangian eq.(3.2.5) is eq.(3.2.1) with ωBD = 0.
Moreover, by using the conformal transformationgµν → ĝµν = Ω2 gµν ,(3.2.7)

let us see that f (R) theory and BD theory can be described in Einstein frame. Under thistransformation, the action of f (R) theory eq.(3.2.5) changes as
[
]
∫
√
1
S = dx4 −g
fR R − U
16πG
[
]
∫
√
1
4
−2
µν
−4
→ dx −g̃fR Ω (R̃ + 62̃(ln Ω) − 6g̃ ∂µ (ln Ω)∂ν (ln Ω)) − Ω U
16πG(3.2.8)where
fR R − f
.
16πG

U=

(3.2.9)If we choose Ω2 = fR (> 0) and define
√
κϕ ≡

3
ln fR ,
2

(3.2.10)

we obtain∫S=√dx −g̃

[4

V (ϕ) =

1
1
R̃ − g̃ µν ∂µ ϕ∂ν ϕ − V (ϕ)
16πG
2

]

fR R − f.
2κ2 fR2(3.2.11)(3.2.12)

Therefore, we can find these theories are included in scalar-tensor theory.

3.3

Kinetic gravity braiding

In this case, the action is written as
L = K(ϕ, X) − G(ϕ, X)2ϕ,(3.3.1)

where K(ϕ, X), G(ϕ, X) are arbitrary functions. We call this model Kinetic gravity braiding [22]．We assume the metricds2 = −dt2 + a2 (t)dx2 .

(3.3.2)

30

CHAPTER 3. MODIFIED GRAVITY

The field equation for scalar field is
Pϕ − ∇µ J µ = 0,

(3.3.3)where we havePϕ = Kϕ − [(∇µ ϕ) ∇µ ] Gϕ ,Jµ= (LX − 2Gϕ ) ∇µ ϕ − GX ∇µ X,(3.3.4)(3.3.5)where Pϕ = ∂ϕ P and the pressure P
P = K − [(∇µ ϕ) ∇µ ] G.

(3.3.6)The energy momentum tensor is given as
2 δSϕTµν ≡ √
= LX ∇µ ϕ∇ν ϕ − gµν P − ∇µ G∇ν ϕ − ∇ν G∇µ ϕ.
−g δg µν(3.3.7)

Thus the energy density ρ and the pressure p is
ρ = ϕ̇J + 2XGϕ − K,(3.3.8)p = K − 2XGϕ − 2XGX ϕ̈.

(3.3.9)Assuming the scalar field depends on time ϕ = ϕ(t), only the component for J 0 in eq.(3.3.5)
is remaining. Thus the equation for scalar field is reduced toJ˙ + 3HJ = Pϕ ,(3.3.10)J ≡ J = (KX − 2Gϕ + 3H ϕ̇GX )ϕ̇,(3.3.11)0

Focusing on energy density in eq.(3.3.8), we have
ρ = 2X(KX − Gϕ ) − K + 6ϕ̇HXGX ,(3.3.12)and we can find ρ includes the Hubble parameter H. As we may already find fromeq.(2.3.4), the energy density in k-essence does not have the Hubble parameter H. In suchcase, we can find Hubble parameter from energy density, however, in this case, energy
density ρ and Hubble parameter H are corresponding to each other. This feature is called
as kinetic gravity braiding. Introducing the parameter κ as the rate of kinetic braiding
κ = XGX ,(3.3.13)

we can rewrite ρ with replacing the term which do not have κ to E as
ρ = E + 6ϕ̇Hκ.

(3.3.14)

3.4. GALILEON THEORY

31

Thus the Hubble parameter H is1
H 2 = 2ϕ̇Hκ + E,
3

(3.3.15)

and we can solve the equation√
H = κϕ̇ + σ1(κϕ̇)2 + E,
3

(3.3.16)

where we take σ = ±1. This suggests us one energy state has two ways of the evolutionof universe.Then let us consider the evolution of the universe. Assuming this model isinvariant under the transformation for ϕ
ϕ → ϕ + c,

(3.3.17)J˙ + 3HJ = 0.

(3.3.18)

we can find Pϕ = 0 from eq.(3.3.10)

Solving this equation, we obtain
J=

const.
= (KX − 2Gϕ + 3H ϕ̇GX )ϕ̇.
a3

(3.3.19)

This suggests that the state becomes J → 0 as the universe expands. The state such as
J = 0, in this case, is called an attractor.

3.4

Galileon theory

The Galileon theory was constructed by A. Nicolis, R. Rattazzi and E. Trincherini [23].
This section is a review of [24, 25, 26]. As its name is ”Galileon”, this model have Galilean
shift symmetry∂µ ϕ → ∂µ ϕ + bµ ,(3.4.1)

where ϕ is the scalar field. First, let us consider the Lagrangian of the original Galileon
theory in flat space-time. Now we assume the Lagrangian as L(ϕ, ∂µ ϕ, ∂µ ∂ν ϕ), and in thiscase the Euler Lagrange equation is written as∂L∂L∂L+ ∂µ ∂ν
= 0,
− ∂µ
∂ϕ
∂ϕµ
∂ϕµν

(3.4.2)where we define,ϕµ ≡ ∂µ ϕ,

ϕµν ≡ ∂µ ∂ν ϕ.

(3.4.3)

32

CHAPTER 3. MODIFIED GRAVITY

The Lagrangian in D dimension is given asµ1 ...µn ν1 ...νnϕµ1 ν1 ϕµn νn ,Ln = T(2n)(3.4.4)

where (2n) is the number of indices which T has, and we defineµ ...µν ...νn+1µ1 ...µn ν1 ...νn
1
n+1 1
T(2n)
ϕµ1 ν1 ≡ A2(n+1)
1 ...µm ν1 ...νm
Aµ(2m)≡

ϕµn+1 ϕνn+1 ,(3.4.5)1ϵµ1 ...µm ρ1 ...ρm ϵν1 ...νmρ1 ...ρm .
(D − m)!

(3.4.6)

The Levi-Civita tensor is given as1[µϵµ1 ...µm ≡ − √ δ1 1 δ2µ2 ...δmµm ] .
−g(3.4.7)

Deriving the field equation from this Lagrangian, we have no higher derivative terms of
ϕ because these terms are canceled. For example, the Lagrangian in 4 dimensional flatspace-time has four Lagrangians as
L2 = −ϕµ ϕµ ,(3.4.8)

L3 = ϕ ϕ ϕµν − ϕ ϕµ 2ϕ,µ νµ(3.4.9)

L4 = −(2ϕ) (ϕµ ϕ ) + 2(2ϕ)ϕµ ϕ ϕν + (ϕµν ϕ ) − 2ϕµ ϕ ϕνρ ϕ ,
2

µ

µν

µν

µν

L5 = −(2ϕ) ϕµ ϕ + 3(2ϕ) ϕµ ϕ ϕν + 3(2ϕ)ϕµν ϕ ϕρ ϕ
3

µ

2

µν

µν

ρ

(3.4.10)ρ−6(2ϕ)ϕµ ϕµν ϕνρ ϕν − 2(ϕµν ϕνρ ϕρµ )(ϕλ ϕλ )−3(ϕµν ϕµν )(ϕρ ϕρλ ϕλ ) + 6(ϕµ ϕµν ϕνρ ϕρλ ϕλ ).

(3.4.11)

So far we have considered the Lagrangian in the flat space-time. In the curved spacetime, we use
ϕµ ≡ ∇µ ϕ,

ϕµν ≡ ∇µ ∇ν ϕ.

(3.4.12)Thus the Galilean shift symmetry is broken in curved space-time. This alteration generates
the third or∑higher derivatives terms for the field equation of L4 and L5 . For the equationof motion
Ei = 0, we obtain E4 for L4 as
1
1
E4 = − (ϕµ ϕµ )(ϕµµρρ − ϕµρµρ ) + ϕµ ϕν (2ϕµρνρ − ϕµνρρ − ϕρρµν )
2
2
5
µ
ν
µ
+ (2ϕ)ϕ (ϕµν − ϕν µ ) − 3ϕµ ϕµν (ϕρρν − ϕνρρ )2−2ϕµ ϕνρ (ϕνρµ − ϕµνρ ) + (2ϕ)3 + 2(ϕµν ϕνρ ϕρµ ) − 3(2ϕ)(ϕµν ϕµν ). (3.4.13)
By integrating by parts, we can remove the higher derivative terms of ϕ. Thus E4 has
1
1
5
E4 ⊃ + ϕµ ϕµ ϕν ∇ν R − ϕµ ϕν ϕρ ∇ρ Rµν − 2ϕϕµ Rµν ϕν
4
2
2
1
µν
ρ
µ
νρ
+2ϕµ ϕ Rνρ ϕ + ϕµ ϕ ϕνρ R + 2ϕµ ϕν ϕρσ Rµρνσ ,2(3.4.14)

3.5. HORNDESKI THEORY

33

and we find the higher derivative terms of metric. To remove these terms, we introduce
Lnonmin4= (ϕµ ϕµ )(ϕν Gνλ ϕλ ).
Lnonmin4(3.4.15)

[
]
1
µ
2
µν
µ
L4 + Lnonmin
=
(ϕ
ϕ
)
2(2ϕ)
−
2(ϕ
ϕ
)
−
(ϕ
ϕ
)R
.
µ
µν
µ
4
2

(3.4.16)

Thus we have

In the same way, we introduce Lnonminfor L5
5
Lnonmin= (ϕµ ϕµ )(ϕν Gνλ ϕλ ).
5(3.4.17)and we have the total Lagrangian for L5 as
[
]
µ
3
µν
ν λ µ
µν
λ
L5 + Lnonmin
=
(ϕ
ϕ
)
(2ϕ)
−
32ϕ(ϕ
ϕ
)
+
2(ϕ
ϕ
ϕ
)
−
6(ϕ
ϕ
G
ϕ
)
.
µ
µν
µ
νλ
5
µ ν λ
(3.4.18)

3.5

Horndeski theory

As we discussed so far, many modified gravity theories exist. Later we discuss some topics
of galilean genesis, and we use Horndeski theory. Horndeski theory is known as the mostgeneralized scalar tensor theory in which the field equations do not have more than thirdtime derivative terms. This theory is investigated by Horndeski at 1972 [17] and widely
known by recent research about generalization of the galilean theory [18] and applying thecosmological context [16]. The Lagrangian is written as∫√
SHor =
d4 x −g (L2 + L3 + L4 + L5 ) ,L2 = G2 (ϕ, X),
L3 = −G3 (ϕ, X)2ϕ,[]
L4 = G4 (ϕ, X)R + G4X (2ϕ)2 − (∇µ ∇ν ϕ)2 ,[]1
L5 = G5 (ϕ, X)Gµν ∇µ ∇ν ϕ − G5X (2ϕ)3 − 32ϕ(∇µ ∇ν ϕ)2 + 2(∇µ ∇ν ϕ)3 ,6(3.5.1)
where (∇µ ∇ν ϕ)2 = ∇µ ∇ν ϕ∇ν ∇µ ϕ , (∇µ ∇ν ϕ)3 = ∇µ ∇ν ϕ∇ν ∇λ ϕ∇λ ∇µ ϕ , and Gi (ϕ, X) (i =2, 3, 4, 5) is the arbitrary function of scalar field ϕ and its kinetic term X =: − 21 (∂ϕ)2 . We
can obtain the Lagrangian we have already discussed. For example, if we take
G2 (ϕ, X) = G3 (ϕ, X) = G5 (ϕ, X) = 0,
G4 (ϕ, X) = const.,(3.5.2)

34

CHAPTER 3. MODIFIED GRAVITY

we can find the Lagrangian of general relativity, or we take
G2 (ϕ, X) = K(ϕ, X),
G3 (ϕ, X) = G(ϕ, X),
G4 (ϕ, X) = G5 (ϕ, X) = 0,

(3.5.3)

we can find the Lagrangian of kinetic gravity braiding in sec(3.3). Although this theory
includes Galilean theory, we do not assume the Galilean shift symmetry eq.(3.4.1) in Horndeski theory. The field equations in Horndeski theory is derived in [16]. The Friedmannequation is5∑

Ei = 0,(3.5.4)i=2E2 = 2XG2X − G2 ,(3.5.5)E3 = 6X ϕ̇HG3X − 2XG3ϕ ,(3.5.6)

E4 = −6H G4 + 24H X (G4X + XG4XX ) − 12HX ϕ̇G4ϕX − 6H ϕ̇G4ϕ ,2

2(3.5.7)E5 = 2H X ϕ̇ (5G5X + 2XG5XX ) − 6H X (3G5ϕ + 2XG5ϕX ) ,3

2(3.5.8)The evolution equation is given as
5∑

Pi = 0,(3.5.9)i=2P2 = G2 ,()(3.5.10)

P3 = −2X G3ϕ + ϕ̈G3X ,(3.5.11)()
P4 = 2 3H 2 + 2Ḣ G4 − 12H 2 XG4X − 4H ẊG4X − 8ḢXG4X − 8HX ẊG4XX()()(3.5.12)+2 ϕ̈ + 2H ϕ̇ G4ϕ + 4XG4ϕϕ + 4X ϕ̈ − 2H ϕ̇ G4ϕX ,()
P5 = −2X 2H 3 ϕ̇ + 2H Ḣ ϕ̇ + 3H 2 ϕ̈ G5X − 4H 2 X 2 G5XX)([]
+4HX Ẋ − HX G5ϕX + 2 2 (HX)· + 3H 2 X G5ϕ + 4HX ϕ̇G5ϕϕ ,(3.5.13)
Finally, the field equation for ϕ is given by
1 d 3(a J) = Pϕ ,a3 dt

(3.5.14)

3.6. GLPV THEORY

35

where
J

= ϕ̇G2X + 6HXG3X − 2ϕ̇G3ϕ + 6H 2 ϕ̇(G4X + 2XG4XX ) − 12HXG4ϕX+2H 3 X(3G5X + 2XG5XX ) − 6H 2 ϕ̇(G5ϕ + XG5ϕX ),(3.5.15)Pϕ = Kϕ − 2X(G3ϕϕ + ϕ̈G3ϕX ) + 6(2H + Ḣ)G4ϕ + 6H(Ẋ + 2HX)G4ϕX
2−6H 2 XG5ϕϕ + 2H 3 X ϕ̇G5ϕX ,3.6(3.5.16)

GLPV theory

We can construct the more general class of theory. The more extended theory is discussed
by Gleyzes et al. [27]. This theory can have higher time derivative terms, in spite of the
Horndeski theory have no more than second order time derivative terms.
LGLP V = L2 + L3 + L4 + L5 ,L2 = G2 (ϕ, X),
L3 = −G3 (ϕ, X)2ϕ,

[
]
′ ′ ′
L4 = G4 (ϕ, X)R + G4X (2ϕ)2 − (∇µ ∇ν ϕ)2 + F4 (ϕ, X)ϵµνρσ ϵµ ν ρ σ ϕµ ϕµ′ ϕνν ′ ϕρρ′ ,[]1
L5 = G5 (ϕ, X)Gµν ∇µ ∇ν ϕ − G5X (2ϕ)3 − 32ϕ(∇µ ∇ν ϕ)2 + 2(∇µ ∇ν ϕ)36µνρσ µ′ ν ′ ρ′ σ ′+F5 (ϕ, X)ϵϵϕµ ϕµ′ ϕνν ′ ϕρρ′ ϕσσ′ ,(3.6.1)

where F4 and F5 are arbitrary functions and ϵµνρσ is the total antisymmetric Levi-Civita
tensor. For this to reduce to the Horndeski theory we take
F4 (ϕ, X) = 0,
F5 (ϕ, X) = 0.

(3.6.2)

Now we use ADM decomposition in which the metric is given as
ds2 = −N 2 dt2 + γij (dxi + N i dt)(dxj + N j dt),

(3.6.3)where N is lapse function, N i is shift vector and γij is the spatial metric. The Lagrangianof the GLPV theory is written as
L2 = A2 (t, N ),L3 = A3 (t, N )K,
2
L4 = A4 (t, N )(K 2 − Kij) + B4 (t, N )R(3) ,L5 = A5 (K323− 3KKij+ 2Kij) + B5 (t, N )K ij

(
)
1
(3)
Rij − gij R,
2(3.6.4)

36

CHAPTER 3. MODIFIED GRAVITY

where Kij is the extrinsic curvature
Kij =1(g˙ij − ∇i Nj − ∇j Ni ) ,2N(3.6.5)Ai (t, N ) and Bi (t, N ) are arbitrary functions which have the following relations
∫
√
G3ϕ
√
A2 = G2 + 2X
dX,
2 2X
∫
√
√
A3 = − G3X 2XdX − 2 2XG4ϕ ,A4 = −G4 + 2XG4X − XG5ϕ − 4X 2 F4 ,(2X)3/2
G5X + (2X)5/2 F5 ,
6
∫
√
G5ϕ
√
dX,
B4 = G4 − 2X
4 2X∫√
B5 = − G5X 2XdX.
A5 =(3.6.6)

For this expression, In order to reduce to the Horndeski theory, we take the conditions
A4 = −B4 + 2XB4X ,1
A5 = − XB5X .
3

3.7(3.6.7)

Conclusions in this chapter

In this section, we have discussed many modified theories. As we have reviewed in thischapter, many kinds of modified theories are discussed. We have introduced GLPV theory
as the beyond theory of Horndeski, but there is the more expanded theory called XG3 [28].
First, we have studied relatively simple models such as f (R) theory, Brans-Dicke theory
or K-essence, and after those theories, we have studied more complicated and generalized
theory. In such a generalized theory, we assume some conditions or ansatz. What wehave reviewed is for one scalar field and one gravitational field, but there are many models
which have multi-scalar fields or gravitational fields. In part 2, although there are many
generalized theories of scalar-tensor theory, we use Horndeski theory as the simple example
to see the typical features of each term.

Chapter 4

Cosmological perturbations
In this section, we introduce cosmological perturbations. First, we discuss how the perturbation behaves in general relativity. As we use Horndeski theory in sec.3.5, we finallyderive the quadratic action or equation for perturbations in Horndeski theory. This sectionis the review of [11].

4.1

Perturbations

Let us introduce gauge transformation for point xγ as
xγ → x̃γ = xγ + ξ γ ,(4.1.1)and discuss some tensor value Aµν under this transformation. Under the transformation,Aµν changes to Ãµν as
Aµν (xγ ) =(0) Aµν (xγ ) + δAµν ,γÃµν (x̃ ) =(0)γAµν (x̃ ) + δ Ãµν ,(4.1.2)(4.1.3)and this can be written in another expression under the transformationAµν (xγ ) → Ãµν (xγ ) =dxµ dxνAαβ (xγ ).
dxα dxβ37(4.1.4)

38

CHAPTER 4. COSMOLOGICAL PERTURBATIONSThis equation is derived from the transformation of tensor field. Applying the transformation for scalar field q and vector field vµ , we obtainq̃(x̃γ ) = q(xγ )=(0)q(xγ ) + δq,dxαṽµ (x̃γ ) =vα (xγ )dx̃µd(x̃α − ξ α )=vα (xγ )dx̃µ≃ vµ (xγ ) −(0) vα (xγ )ξµα=

(0)(4.1.5)vµ (xγ ) + δvµ −(0) vα (xγ )ξ α,µ ,(4.1.6)

and
(0)
(0)

q(x̃γ ) =

(0)

γ

(0)

vµ (x̃ ) =q(xγ ) +(0) q,α ξ α ,γvµ (x ) +(0)

(4.1.7)αvµ,α ξ .

(4.1.8)Therefore, we find for δq and δvµ as
δq → δ q̃ = δq −(0) q,α ξ α ,δvµ → δṽµ = δvµ −(0) vµ,α ξ α −(0) vα ξ α,µ .

(4.1.9)(4.1.10)

Let us apply this transformation for metric gµν . Decomposing the metric into g00 , g0i , gij
and referring to the result for Aµν , we find{}
1 ( 0 )′− aξ,aδg̃00 = δg00 − 2a}{ ′δg̃0i = δg0i + a2 (ξs − ξ 0 ),i − ξ⊥i,0 ,
2

δg̃ij

= δgij + a2 (−ξ⊥j,i − ξ⊥i,j − 2ξs,ij − 2Hδij ξ 0 ),(4.1.11)(4.1.12)(4.1.13)

If we set the metric including only scalar perturbations as[]
ds2 = a(η)2 −(1 + 2ϕ)dη 2 + 2B,i dηdxi + {(1 − 2ψ)δij + 2E,ij } dxi dxj ,(4.1.14)the conpornents of gµν is written as
δg00 = −2a2 ϕ,(4.1.16)2(4.1.17)

δg0i = a B,i ,δgij(4.1.15)2= a (−2ψδij + 2E,ij ),4.1. PERTURBATIONS

39

and thus the transformations for ϕ, B, E, ψ are
1 ( 0 )′aξ ,a
B → B̃ = B + ξs ′ − ξ 0 ,ϕ → ϕ̃ = ϕ −E → Ẽ = E + ξs ,(4.1.18)(4.1.19)(4.1.20)

ψ → ψ̃ = ψ + Hξ .
0

(4.1.21)

Now we have 4 degrees of freedom shown by ξ µ , and we can fix a gauge freely. Aseq.(4.1.18)-eq.(4.1.21) have 2 of 4 compornents, however, we can obtain the gauge invariant
values without fixing the gauge. Introducing the gauge invariant value Φ and Ψ, we can
find
1
′ ′
Φ = ϕ − [a(B − E )] ,(4.1.22)a′Ψ = ψ + H(B − E ).
(4.1.23)
In the other case, if we take the metric with vector perturbations[]
ds2 = a(t)2 −dη 2 + 2Si dηdxi + (δij − Fi,j − Fj,i ) dxi dxj ,(4.1.24)the metric is descrbed asδg00 = 0,(4.1.25)2

δg0i = a Si ,(4.1.26)= −a (Fi,j + Fj,i ).
2δgij(4.1.27)

We can find the transformations for Si and Fi′

Si → S̃i = Si − ξ⊥i ,(4.1.28)

Fi → F̃i = Fi + ξ⊥i .

(4.1.29)

In this case, we can find the gauge invariant value Vi as′

Vi = Si + Fi ,For the tensor perturbations, we take this metric[]ds2 = a(t)2 −dη 2 + (δij + hij ) dxi dxj ,(4.1.30)(4.1.31)

and the metric is given as
δg00 = δg0i = 0,

(4.1.32)2= a hij .

(4.1.33)→ h̃ij = hij ,(4.1.34)

δgij
For the tansformation of hij , we can findhij
and this shows hij is gauge invariant.

40

4.2

CHAPTER 4. COSMOLOGICAL PERTURBATIONS

Matter perturbation

In this subsection, we discuss matter perturbations which we ignored in pre sections. Now
we assume the perfect fluid and discuss the perturbations for the matter.
Tµν = (ρ + p)uµ uν + gµν p,(4.2.1)Introducing δTµν as the perturbation for matter, we haveTµν = (0) Tµν + δTµν ,(4.2.2)

Taking the transformation and we define T̃µν as the tensor after transformation for Tµνδ T̃00 = δT00 − 2(0) T00 ξ 0′ − (0) T00 ′ ξ 0= δT00 − 2a2 ρ0 ξ 0′ − (a2 ρ0 )′ ξ 0{}= δT00 − a2 ρ0 (B − E ′ ) − (B̃ − Ẽ ′ ) ,(4.2.3)

δ T̃0i = δT0i − (0) T00 ξ ′ i − (0) Tij ξ j ′{}= δT0i − a2 (ρ0 + p0 ) (B − E ′ ) − (B̃ − Ẽ ′ ) ,(4.2.4)= δTij − (0) Tki ξs,k,j − (0) Tkj ξs,k,i − (0) Tij ′ ξ 0{}= δTij − a2 δij p′ 0 (B − E ′ ) − (B̃ − Ẽ ′ ) ,(4.2.5),i

δ T̃ijThese equations suggest usδ T̄00 = δT00 − a2 ρ0 ′ (B − E ′ ),(4.2.6)′

δ T̄0i = δT0i − a (ρ0 + p0 )(B − E ),i ,2

δ T̄ij′′= δTij − a δij p0 (B − E ).
2

(4.2.7)(4.2.8)

As the components of the energy momentum tensor Tµν have the energy density ρ, thepressure p and the 4-velocity uµρ = ρ0 + δρ,(4.2.9)p = p0 + δp,(4.2.10)uµ = u0µ + δuµ ,(4.2.11)

we can obtain from above equationsδ ρ̄ = δρ − ρ′ 0 (B − E ′ ),′(4.2.12)′δ p̄ = δp − p 0 (B − E ),′(4.2.13)′

δ ū0 = δu0 − {a(B − E )} ,′

δ ūi = δui − a(B − E ),i ,(4.2.14)(4.2.15)

4.3. GAUGE

41

δTµν = (δρ + δp)uµ uν + (ρ + p)(δuµ uν + uµ δuν ) + gµν δp + δgµν p,(4.2.16)

Thus we obtain the components of perturbations of the energy momentum tensor asδ T̄00 = a2 δ ρ̄,(4.2.17)

δ T̄0i = a(ρ0 + p0 )δ ūi ,(4.2.18)2(4.2.19)

δ T̄ij4.3= a δ p̄.

Gauge

We have discussed the perturbations, and we find there are some degrees of freedom. Thus
we can fix the gauge, and there are some ways to fix them. In this subsection, we introduce
some gauges.

4.3.1

Conformal Newtonian gauge

First, we introduce conformal Newtonian gauge (or longitudinal gauge, conformal gauge).
In this gauge, we take the condition as
B = E = 0,

(4.3.1)[]ds2 = a2 (η) −(1 + 2Φ)dη 2 + (1 − 2Ψ)δij dxi dxj .

(4.3.2)and thus we obtain

4.3.2

Comoving gauge

For the comoving gauge, we take the rule for perturbation of energy momentum tensor as
δTi0 = 0.

(4.3.3)In this case, contravariant vector for velocity becomes
11
δui = − Bi + 2 δui .
aa(4.3.4)As the gauge transformation for δui and B is
δ ũi = δui − aξ 0,i ,′B̃ = B + ζs − ξ 0 ,we can find we have to fix ζ ′ = 0.

(4.3.5)(4.3.6)

42

4.4

CHAPTER 4. COSMOLOGICAL PERTURBATIONS

Cosmological perturbations in General Relativity

Applying the components of metric, we discussed before into the Einstein equation
δGµν = 8πGδTµν ,(4.4.1)

we can obtain the equations for perturbations.

4.4.1

Scalar perturbation

For scalar perturbation, we obtain the equations with scalar part of the energy momentum(S)
tensor δTµν as follows{∆ψ − 3H(ψ ′ + Hϕ) = 4πGδT00 ,(S)(4.4.2)(S)
4πGδT0i ,(4.4.3)(S)(4.4.4)(ψ + Hϕ),i =′′′δij 2ψ + ∆(ϕ − ψ) + 2H(2ψ + ϕ) + 2H (ϕ − 2ψ)}+2H′ (ϕ + ψ) + (ψ − ϕ),ij
2= 8πGδTij .

Focusing on eq.(4.4.4), if we assume δTij ∝ δij , we fan find ψ − ϕ = 0. Therefore eq.(4.4.4)
becomesϕ′′ + 3Hϕ + (2H′ − H2 )ϕ = 4πGδTij .
(S)

4.4.2

(4.4.5)Vector perturbation

For vector perturbation, the equations with vector part of the energy momentum tensor(V )δTµν is′∆Vi = −16πGδT0i ,(V )(4.4.6)(V )−16πGδTij ,(4.4.7)(Vi,j + Vj,i ) + 2H(Vi,j + Vj,i ) =From eq.(4.4.7) we find the vector mode is always decaying.

4.4.3

Tensor perturbation

For tensor perturbations, the equation with tensor part of the energy momentum tensor(T )δTij is
′′

′

(T )

hij + 2Hhij − ∆hij = 16πGδTij .

(4.4.8)

4.5. COSMOLOGICAL PERTURBATIONS IN HORNDESKI THEORY

4.5

43

Cosmological perturbations in Horndeski Theory

In this way, we derived the equations for the cosmological perturbations from the equationof motion. However, there is another approach where we derive the equations from thevariation of quadratic action. In this thesis, we consider the model written by the scalar
field and the gravitational field. Let us consider the equations for Horndeski theory inthis section. This is the review of [15]. The action and background equations have been
already shown in sec.3.5. Then in this section, let us see the perturbations in FRLW
metric and unitary gauge in ADM formalism in the unitary gauge.
ds2 = −N 2 dt2 + γij (dxi + N i dt)(dxj + N j dt).

4.5.1

(4.5.1)

Unitary gauge

When we consider the case that the Lagrangian is written by the scalar field and thegravitational field, we can use the unitary gauge defined as follows. In the unitary gauge,
we set that the scalar field depends on only timeϕ = ϕ(t).

(4.5.2)

From eq.(2.3.2), we find
X=−

ϕ̇2
.
2N 2

(4.5.3)Thus the kinetic term of scalar field ϕ is the function of t and N , and the arbitrary
functions in eq.(3.6.4) are written as the function of t and N .

4.5.2

The quadratic action in Horndeski theory

The Friedmann equation eq.(3.5.8) and the evolution equation eq.(3.5.13) are derived fromthe variation of δa and δN . The quadratic Lagrangian for tensor perturbations is
[
]
∫
1
FT
3
3
2
2
ST =
dtd xa GT hij − 2 (∇hij ) ,(4.5.4)
8
a
where
GT
FT

[
(
)]
:= 2 G4 − 2XG4X − X H ϕ̇G5X − G5ϕ ,[()]:= 2 G4 − X ϕ̈G5X + G5ϕ .

The quadratic Lagrangian for scalar perturbation in unitary gauge is
[
]
∫
FS
3
3
2
2
SS = dtd xa GS ζ̇ − 2 (∇ζ) ,a(4.5.5)(4.5.6)(4.5.7)

44
where

CHAPTER 4. COSMOLOGICAL PERTURBATIONS
1 d ( a 2)
G − FT ,a dt Θ T
Σ 2
GS :=G + 3GT ,Θ2 TΣ := XG2X + 2X 2 G2XX + 12H ϕ̇XG3X

FS :=(4.5.8)(4.5.9)+6H ϕ̇X 2 G3XX − 2XG3ϕ − 2X 2 G3ϕX − 6H 2 G4[()
6 H 2 7XG4X + 16X 2 G4XX + 4X 3 G4XXX()]−H ϕ̇ G4ϕ + 5XG4ϕX + 2X 2 G4ϕXX+30H 3 ϕ̇XG5X + 26H 3 ϕ̇X 2 G5XX)(
+4H 3 ϕ̇X 3 G5XXX − 6H 2 X 6G5ϕ + 9XG5ϕX + 2X 2 G5ϕXX
=

X

5
∑
∂Ei
i=2

∂X

1 ∑ ∂Ei
,
2
∂H
5

+

(4.5.10)i=2

Θ := −ϕ̇XG3X + 2HG4 − 8HXG4X − 8HX 2 G4XX + ϕ̇G4ϕ + 2X ϕ̇G4ϕX()−H 2 ϕ̇ 5XG5X + 2X 2 G5XX + 2HX (3G5ϕ + 2XG5ϕX )1 ∑ ∂Ei
.
6
∂H
5

=

−

(4.5.11)i=2

The definition of Ei (i = 2, 3, 4, 5) is written in eq.(3.5.8). The propagation speed fortensor perturbations cT and the scalar perturbation cS are described as
FT
,
c2T =
(4.5.12)
GT
FS
c2S =
(4.5.13).
GS

4.6

Power spectrum

When we want to distinguish between the models of early universe, we discuss the observation of the cosmological perturbations. In this subsection, we discuss how we can see
the perturbation from observation. First what we review is the power spectrum, which isvery useful value for distinguishing between the models. When we calculate this, we needto solve the equation for perturbation and take the initial condition. Thus we will review
the Bunch-Davies vacuum which is usually used [29].

4.6.1

Initial state

We can choose how define the initial state. Now we introduce Bunch-Davies vacuum Thereis an argument how we take the vacuum depends on the physical value. Now we assume

4.6. POWER SPECTRUM

45

the action for the inflaton ϕ as
Sϕ =
=

∫
[
]
1 √
−gd4 x −g µν ϕα ϕβ − m2 ϕ2
2
∫
]
[
1 √
−gd4 x ϕ̇2 + (∇ϕ)2 − m2 ϕ2 ,2

(4.6.1)for simplification, we introduce χ as
χ = a(η)ϕ,(4.6.2)and thus the action eq.(4.6.1) can be rewritten for the action of χ(x, η) as
[
(
) ]
∫
1
a′′
Sχ =
d3 xdη χ′2 − (∇χ)2 − m2 a2 −
χ2 .
2
a

(4.6.3)Using the Fourier transformation for χ(x, η)∫χ(x, η) =

d3 k
(2π)

3
2

χk (η)eik·x ,(4.6.4)

we can obtain the equation for χk (η)χk ′′ + ωk2 (η)χk = 0,

(4.6.5)where ωk2 (η) is defined as
ωk2 (η) ≡ k 2 + m2 a2 −

a′′
.
a

(4.6.6)Introducing the canonically conjugate momentum π∂L= v′,(4.6.7)∂v ′()′′
from eq.(4.6.3) by defining m2ef f = m2 a2 − aa we can find the Hamiltonian for χπ≡Hχ (η) =

1
2

∫

[
]
d3 x π 2 + (∇χ)2 + m2ef f χ2 ,(4.6.8)and the commutation relation between π and χ are[π(x, η), π(y, η)] = [χ(x, η), χ(y, η)] = 0,

(4.6.9)[χ(x, η), π(y, η)] = iδ(x − y).

(4.6.10)

46

CHAPTER 4. COSMOLOGICAL PERTURBATIONSThe solution of χ explaining by mode function vk (η)
]
1 [
+
∗
χk (η) = √ a−k vk (η) + a−k vk (η) .
2(4.6.11)+a−k and ak are the operator,a−k =√ W [vk , χk ]2,
W [vk , vk∗ ]−a+k = (ak ),where W is the Wronskian. Thus we have∫]1
d3 k [ ik·x − ∗χ(x, η) = √e
ak vk + e−ik·x a+
v
.
k
3
−k
2
(2π) 2
Applying the solution for eq.(4.6.8), we obtain
∫
[
(
)∗
(
)
1
−
+ +
′2
2 2
′2
2 2
Hχ (η) =d3 k a−
a
v
+
ω
v
+
a
a
v
+
ω
v
k
k
k k
k k
k −k
k −k
4
]
( + −
)
(
)
+ 2ak ak + δ 3 (0) |vk ′ |2 + ωk2 |vk |2 .
Taking account of a−k |0 >= 0, the expectation is given as∫()
1< 0|Hχ |0 >=d3 kδ 3 (0) |vk ′ |2 + ωk2 |vk |2 ,4

(4.6.12)(4.6.13)(4.6.14)(4.6.15)

and thus
E = |vk ′ |2 + ωk2 |vk |2 .
where vk ′ , vk and vk∗ ′ , vk∗ have the relation[ ′ ]vk , vk = vk ′ vk∗ − vk vk∗ ′ = 2i,(4.6.16)(4.6.17)

Assuming vk as
vk = rk eiα ,(4.6.18)

rk2 αk ′ = 1.

(4.6.19)

we have

Therefore, eq.(4.6.16) is explained as
(
)
1 ′2
1
2 2
E=
r k + ωk rk + 2 .
4
rk(4.6.20)

4.6. POWER SPECTRUM

47

If the energy is minimal, we obtainrk ′ = 0,(4.6.21)−1
rk = ωk 2 ,(4.6.22)and thus vk , vk ′ and αk as the initial state of the perturbation1
vk = √ e−iαk ,ωk√vk ′ = i ωk e−iαk ,αk ′ = ωk .

(4.6.23)

In the following subsection, we use this initial state for scalar and tensor perturbations.

4.6.2

Definition of power spectrum

Let us introduce the definition of the power spectrum, which is defined by the two-point
correlation for the function at the vacuum state. Now we consider the wave function f (k)and the definition of the power spectrum is< 0|f (k)f (k ′ )|0 >≡▷2π 2 (3)δ (k + k ′ )P(k).
k3(4.6.24)

Scalar perturbation

We have reviewed the quadratic action for scalar and tensor perturbations and the initialcondition for the perturbations. To calculate the power spectrum, we have to solve theequation for the perturbation. Now we consider the quadratic action for scalar perturbation which can be written as
∫
[
]
(2)
⃗ 2 ,
Sζ = dtd3 x C1 (−t)2p ζ̇ 2 − C2 (−t)2q (∇ζ)
(4.6.25)
where C1 and C2 are constant, p and q are the constant parameters. Then we take as avariable
1/2

−y :=

(−t)1−p+q,
1/2 1 − p + q
CC2(4.6.26)

1

and we introduce u related to ζ as√u := 2(C1 C2 )1/4 (−t)(p+q)/2 ζ,(4.6.27)

48

CHAPTER 4. COSMOLOGICAL PERTURBATIONS
log(physical scale)aksuperhorizon

H-1

log(t)

subhorizon
Inflation

Radiation dom.

Figure 4.1: Schematic picture of the relation for the evolution of H −1 and a/k. We
call H −1 ≪ a/k superhorizon and a/k ≪ H −1 subhorizon. On superhorizon scale, theperturbation do not perturb.
Thus we can find1
S (2) =
2[(∫
dyd3 x

∂u
∂y

)2

]
2 − 1/4
ν⃗ 2+− (∇u)
u2 ,y2(4.6.28)

where
ν :=

1 − 2p.
2(1 − p + q)

By using the Fourier transformation, the solution u is given as
√
π√
uk =
−yHν(1) (−ky).
2(4.6.29)(4.6.30)

As we can see from this formula or in fig.4.1, the behavior of wave depends on |ky|. Once
the wavelength becomes larger than Hubble scale, the wave quit perturbation. Now we
discuss the waves for early universe, thus we take the sub horizon limit |ky| ≪ 1. The
solution of ζ is[]ν−1/2√1/2
C12|ν|−3/2π(1 − p + q) 1/2 (−y)|ζk | ≃k −|ν| (−y)1/2−|ν|π(C1 C2 )1/4
C
2

∝k

−|ν|

|y|

ν−|ν|

.

(4.6.31)

4.6. POWER SPECTRUM

49

Therefore we obtain[]22|ν|−3/2 Γ(|ν|) k 3−2|ν| C1ν−1Pζ =(1 − p + q)2ν−1 |y|2(ν−|ν|) .
Γ(3/2)
8π 2 C2ν
▷(4.6.32)

Tensor perturbations

For the tensor perturbations, we can calculate in the same way to the scalar perturbation.
We assume the quadratic action for the tensor perturbations as
∫
[
]
1
(2)
⃗ ij )2 ,Sh =dtd3 x C1 (−t)2p ḣ2ij − C2 (−t)2q (∇h(4.6.33)
8
and we obtain[2|ν|−3/2 Γ(|ν|)
Ph = 16Γ(3/2)▷]2k 3−2|ν| C1ν−1(1 − p + q)2ν−1 |y|2(ν−|ν|) .
8π 2 C2ν

(4.6.34)

G-inflation

The power spectrum for scalar perturbation in generalized G-inflation [16] is given as
follows. First, in the quadratic action of scalar perturbation eq.(4.5.7), we takecS
dyS = dt,√azS = 2(FS GS )1/4 ,u = zS ζ.

(4.6.35)(4.6.36)(4.6.37)

The action becomes
1
(2)
SS =
2

∫

[

]
zS′′ 2
2⃗
dyS d x (u ) − (∇u) + u ,
zS
3

′ 2

where“′ ” is the derivative of yS . On superhorizon scales, we have the solution∫dt′.
ζ = C1 + C2
a3 GS

(4.6.38)(4.6.39)

where C1 and C2 are constant and GS is the varying function. Now we define and assume
ḞS≃ const,
HFS
ĠS
gS ≡
≃ const,HGS
fS ≡(4.6.40)(4.6.41)

50

CHAPTER 4. COSMOLOGICAL PERTURBATIONS

and we have
1/2

Ps =

γ S Gs H 2
2 Fs3/2 4π 2,

(4.6.42)kyS =1

whereγS = 2νT ≡2νS −3

Γ(νS )Γ( 32 )

2(fS
gS
1 − ϵH −
+
2
2

)
,

3 − ϵH + gT.
2 − 2ϵH − fT + gT

(4.6.43)(4.6.44)In the similar way to the scalar perturbation, we have the power spectrum for the tensor
perturbations. From the quadratic action of tensor perturbations eq.(4.5.4), we obtain
1/2

Ph = 8γTGT

H2
3/2 4π 2
F
T(4.6.45),
kyT =1where
γT = 22νT −3

Γ(νT )Γ( 32 )

2(

gT
fT
+
1 − ϵH −22),

3 − ϵH + gT,
2 − 2ϵH − fT + gT
ḞT
ĠT
fT ≡
, gT ≡
,
HFT
HGT
νT ≡

(4.6.46)(4.6.47)(4.6.48)

γS and γT are constant in inflationary phase, and they take γS , γT ≃ O(1).
▷Inflation

In the slow-roll inflation, whose action is described by eq.(2.2.2) and eq.(2.2.3), the arbitrary functions in Horndeski theory are written as
1G2 (ϕ, X) = − g µν ∂µ ϕ∂ν ϕ + V (ϕ) = X + V (ϕ),
2G3 (ϕ, X) = 0,
1
G4 (ϕ, X) =,16πG
G5 (ϕ, X) = 0.

(4.6.49)(4.6.50)(4.6.51)(4.6.52)

4.6. POWER SPECTRUM

51

Substituting these functions into the definitions in Horndeski theory, we obtain
1
,
8πG
ϵH
.
FS =
8πG

1
,
8πG
ϕ̇2
GS =
,
2H 2
GT =

FT =

(4.6.53)(4.6.54)

and thusgT = 0,fT = 0,gS = 2(ϵH − ηH ),(4.6.55)
fS = 2(ϵH − ηH ),(4.6.56)

As the slow-roll parameters are much smaller than 1 during inflation, we find gS , fS ≃ 0and νS , νT ≃ 23 . Therefore, in inflation, we can find the power spectrum for scalar andtensor perturbations as
Ps =
Ph =

2
16GHinfπ2
4Hinf

4π 2 ϕ̇2.

,(4.6.57)(4.6.58)

Chapter 5Some topics in Modern Cosmology
Above all, we have discussed the standard inflationary scenario and modified gravity theories. After this section, we will discuss the alternative to inflation scenario called Galilean
genesis. To compare the scenarios, we discuss some relevant topics in inflation.

5.1

Curvaton field

In some cases, we consider the power spectrum is explained by curvaton field χ [30]. In
this section, we review basic information for the curvaton. As the mass of curvaton islighter than that of inflaton, Inflation does not contribute to making the power spectrum
due to the curvaton field lives longer than the inflaton. Let us consider this mechanism.
Definition of the curvature perturbation is written asζ ≡ −ψ −Hδρ.
ρ̇

(5.1.1)

If there exists some kind of fields, the total curvature perturbation ζtot is written asζtot ≡

∑ ρ̇i

ζi ,

(5.1.2)

H
δρi .
ρ̇i

(5.1.3)ρ̇

where ζi is given as
ζi ≡ −ψ −Note that the definition of ζ in this section is in the gauge of δρ = 0. In part 2, we use
δϕ = 0. Now we consider inflaton and curvaton. As inflaton changes to radiation before
curvaton decay, ζtot can be written asζtot =

4ρr ζr + 3ρχ ζχ
,4ρr + 3ρχ
53

(5.1.4)

54

CHAPTER 5. SOME TOPICS IN MODERN COSMOLOGY

where χ behaves as the matter, thus the energy density of each field areρr ∝ a−4 ,(5.1.5)−3(5.1.6)ρχ ∝ a,where we assume that the scalar field behaves as a matter, and thus we take w = 0
for χ. Eq.(5.1.4) suggests that the rate of the energy density determines ζtot , and theenergy density of radiation decays faster than that of curvaton. Therefore, we can findthe curvature perturbation is made from curvaton, and ζtot can be described asζtot ≃ ζχ .

5.2

(5.1.7)

Reheating of the universe

It is considered the temperature of the universe once grows at the phase transition frominflation to radiation dominant phase. As we usually consider the universe was started
from the high energy state, we call this reheating. (The scenario of Galilean Genesis we
discuss in chap.2 suggests the low energy initial state.) Various mechanisms are explaining
how reheating was caused. We introduce two ways to reheat the universe, from oscillatingof the inflaton and energy transformation by gravitation.
We assume the matter generated during reheating phase as χ. The Lagrangian of thematter is written as11
L = − g µν ∂µ χ∂ν χ − m2 X 2 ,2
2(5.2.1)

where m is the mass of the matter. By deriving the equation for χ, using the Fouriertransformation, and define X asχk (η) = e− 2
3

∫

Hdη

X(η),

(5.2.2)

we obtainẌk + ω 2 Xk = 0,

(5.2.3)k2
3
9+ m2 − Ḣ − H 2 .
a2
2
4(5.2.4)

where ω is defined as
ω2 =

5.2.1

Reheating from oscillating inflaton

Let us consider the decay of inflaton. We usually assume the scenario that inflaton oscillates on the bottom of its potential and decays with the rate Γϕ . In this case, the equations

5.2. REHEATING OF THE UNIVERSE

55

for the energy density are written asdρϕ+ 3Hρϕ = −Γϕ ρϕ ,dtdρr
+ 4Hρr = Γϕ ρϕ .
dt

(5.2.5)(5.2.6)

Solving these equations gives us( a )3∗ρϕ (t) = ρϕ (t∗ )e−Γϕ (t∗ −t) ,
a
∫ t∗
Γϕ
ρr (t) = 4
a4 (t′ )ρϕ dt,
a (t) t(5.2.7)(5.2.8)where t = t∗ is the end of inflation. From the relation between energy density ρ andreheating temperature TR ,ρ=π2g∗ TR4 ,30(5.2.9)where g∗ is the eﬀective number of relativistic species of particles, we can find the reheating
temperature. Moreover, by using Friedmann equation, the temperature can be shown by
Hubble parameter.

5.2.2

Bogolybov coeﬃcients

Recalling the action eq.(4.6.1), we have chosen the mode function vk and the operators â±k.
However, we can select the diﬀerent function and operators. Let us consider the function
uk which satisfies eq.(5.2.3) given as
uk = αk vk (η) + βk vk∗ (η).

(5.2.10)

This mode function uk also satisfy the normalization condition eq.(4.6.17), and thus therelation between α and β is|α|2 − |β|2 = 1.

(5.2.11)

Introducing the operator b̂±k corresponding to the mode function uk which is described as
]
1 [
+
∗
χk (η) = √ a−k vk (η) + a−k vk (η)
2]1 [ − ∗,
= √ bk uk (η) + b+
−k uk (η)
2

(5.2.12)

56

CHAPTER 5. SOME TOPICS IN MODERN COSMOLOGY±
we find the relations between the operators â±k and b̂k+∗ −â+k = αk b̂k + βk b̂−k ,
+
∗ −
â−
k = αk b̂k + βk b̂−k ,
+
∗ −
b̂+
k = αk âk − βk â−k ,
+
∗ −
b̂−
k = αk âk − βk â−k .

(5.2.13)

These are called Bogolybov transformation and α and β are called the Bogolybov coeﬃcients, and we can obtain the coeﬃcients is given from the relations as
W (uk , vk∗ ),2i
W (vk ), uk
,
βk =
2i

αk =

(5.2.14)(5.2.15)

where the function W is the Wronskian. Using this transformation gives us the informationof the density for the state of b. Taking < 0| and |0 > as the vacuum state for b and−Nk = â+k âk as the number operator. The particle number is−< 0|Nk |0 >=< 0|â+k âk |0 >
+
∗ −
∗ −
=< 0|(αk b̂+
k + βk b̂−k )(αk b̂k + βk b̂−k )|0 >+=< 0|(βk∗ b̂−−k )(βk b̂−k )|0 >= |βk |2 δ 3 (0).

(5.2.16)

Therefore we obtain the particle number for wave number k as
nk = |βk |2 .

(5.2.17)

Then we can obtain the energy density
1
ρ=
2πa4

∫
nk k 3 dk.

(5.2.18)

and the temperature from eq.(5.2.9).

5.2.3

Gravitational reheating

For the phase transition from inflation to the next phase, we can consider another scenario.
In one case, the oscillation of inflaton on the bottom of its potential generates the matter
we discussed the last subsection. In the other case, the energy transition generates thematter. In genesis model, we discuss reheating phase caused in the latter case. Let us
consider the gravitational reheating in inflation discussed in [31]. This subsection is the5.2. REHEATING OF THE UNIVERSE

57

review of [29]. Assuming the Bogolybov coeﬃcients as the function of η, we can show thesolution of eq.(4.6.11)
α(η)
β(η)
Xk (η) = √ e−ikη + √ eikη ,2k
2k(5.2.19)

and applying this into eq.(5.2.3) the equations for α, β is shown asa′′
β = 0,aa′′′′′
α − 2ikα − α = 0.
a′′′

β + 2ikβ −Then we can find its solutioniβ(η) = −2k∫ ∞e2ikη V (η)dη.

(5.2.20)(5.2.21)(5.2.22)−∞

The particle number nk and the energy density ρk for generated matter χ is written by
the coeﬃcients β as∫ ∞1nk = 2 3|βk |2 k 3 dk,(5.2.23)
2π a −∞∫ ∞1
ρk = 2 4|βk |2 k 3 dk,(5.2.24)
2π a −∞and thus the total energy density is given as
∫ ∞
∫ ∞
1
′
′
ρ=−
dη
dη2 ln(m|η1 − η2 |)V (η1 )V (η2 ),124128π a −∞−∞(5.2.25)

and we inserted some arbitrary mass scale m in ln for a dimensional reason, though ρχwill be dependent only logarithmically on m. The definition of V (η) in eq.(5.2.25) is′′′

f f − (f )2
V (x) =,
f2

f (η) ≡ a2 (η),(5.2.26)

where x = Hinf η. This shows us the time derivative of V (η) generated the created matter.
Now we consider the phase transition from inflation to the radiation dominant phase as
an example. The function V (x) is related to the time derivative of Hubble parameter.
Whenever f (x) in (5.2.26) does not have the formula, we can calculate ρ with assuming
the function as follows
 1(x < x0 ) x2 ,
c0 + c1 x + c2 x2 + c3 x3 , (x0 < x < x1 )f (x) =
(5.2.27)d0 (x + d1 )(x1 < x),

58

CHAPTER 5. SOME TOPICS IN MODERN COSMOLOGY

V( )

η

Δη

Figure 5.1: The schematic picture of the evolution of Hubble parameter during the phasetransition from inflationary phase to radiation dominant phase. ∆η which we assume very
short is the phase transition era.
where x0 is the end of inflation and x1 is the beginning of the radiation dominant phase.
Finding each coeﬃcients in eq.(5.2.27), we can find the function V (η). This method isavailable when ∆η is very small, and V during ∆η is approximately a line like fig.(5.1).
Therefore in eq.(5.2.25), we can find ρ by calculating during only ∆η, and we obtain thereheating temperature from eq.(5.2.9).

5.3

Alternative scenarios to inflation

In the later section, we discuss one of the alternative scenarios to inflation. There aremany models which explain the behave of early universe, thus in this section, we review
the feature of alternative scenarios and some scenarios such as bouncing cosmology andstring gas cosmology.

5.3.1

Null energy condition

In the study of the model of the early universe, we have to discuss some stability conditions.
The null energy condition (NEC) is one of that stability. This condition appears in thebasic field equation, and let us derive it. The review of this topic is written in [32]. The5.3. ALTERNATIVE SCENARIOS TO INFLATION59

NEC is given byTµν k α k β > 0,(5.3.1)

where Tµν is the energy momentum tensor and k α is the null vector. The component of
the energy momentum tensor is shown in eq.(1.4.3), and thus
ρ + p > 0,(5.3.2)

Let us applying this condition into the cosmological context. The Friedmann equationand evolution equation givesḢ = −4πG(ρ + p).

(5.3.3)Thus when NEC is violated, Hubble parameter satisfy
Ḣ > 0.

(5.3.4)In the cosmological context, we call the null energy condition as this equation. Let us see
some examples of this condition. In standard inflationary scenario, we can obtainρ + p = ϕ̇2 > 0,(5.3.5)and this says NEC is satisfied. In the K-inflation scenario, NEC is violated if
ρ + p = 2XKX < 0.

(5.3.6)In alternative scenarios of early universe, it is known that NEC is violated without generating instabilities.

5.3.2

Bouncing cosmology

In inflationary model, we have reviewed the scale factor always grows, and it may have thesingularity. To avoid this problem, we can consider the scale factor grows after contraction
as we can see in fig.5.2. Such scenarios are called bouncing scenario, and there are manymodels of the bouncing universe (see the review [33, 34]). The feature of this scenario
is the Hubble parameter takes H = 0 at the turning point in bouncing phase. As an
example. let us see the models of the two field matter bouncing universe [35]. In thismodel, we consider two scalar fields, and the action is given like
L = K(ϕ, X) − G(ϕ, X)2ϕ + P (ψ, Y ),1
Y = − g µν ∂µ ψ∂ν ψ,
2(5.3.7)(5.3.8)

60

CHAPTER 5. SOME TOPICS IN MODERN COSMOLOGYlog(physical scale)ak|H 1 |
log(t)

BounceFigure 5.2: The schematic picture in the bouncing scenario.
where
K(ϕ, X) = MPl [1 − g( ϕ)]X + βX 2 − V (ϕ)(5.3.9)G(ϕ, X) = γX,(5.3.10)where P (ψ, Y ) is the function of the scalar field ψ and its kinetic term, V (ϕ) is the potentialof the scalar field ϕ, and beta, γ are the positive constant parameters.
e−

√

2g0

√
,
(5.3.11)+ ebg 2/qϕ
2V0√
V (ϕ) = − √(5.3.12),e− 2/pϕ + ebv 2/qϕ√√
where bg , bV are the constants. In − p/2 ln(2g0 ) < ϕ < p/2 ln(2g0 )/bg , we can obtainthe solutions of the bouncing phase asg(ϕ) =

2/pϕH ∼ Υt,a(t) ∼ aB e

(5.3.13)
1
Υt2
2

,

(5.3.14)

where aB is the scale factor at the beginning of the bouncing phase.

5.3.3

Galilean genesis

The Galilean genesis is proposed by P. Creminelli, A. Nicolis and E. Trincherini [1]. The
schematic picture of this scenario is described in fig.5.3. Now let us review the original5.3. ALTERNATIVE SCENARIOS TO INFLATION

61

log(physical scale)H-1aklog(t)

GenesisRadiation dom.

Figure 5.3: The schematic picture in the genesis scenario.
model of Galilean genesis constructed by using the Lagrangian of the form [1, 36]
L=1
2f 3 λ4 2 2f 3 λ3
R + 2f 2 λ2 e2λϕ X +
X +
X2ϕ,
16πG
Λ3
Λ3

(5.3.15)

where f , λ and Λ are constants. For this Lagrangian at t → −∞, we consider the Hubble
parameter takes H → 0 and eλϕ behaves like the de-Sitter solution as
1
,
H0 t
2Λ3
H0 =
.
3f
eλϕ = −

Then, we obtain the energy density and the pressure from the Lagrangian,
[
]
1
2 2
2λϕ 2
2 4
3
ρ = −f λ e ϕ̇ − 2 (λ ϕ̇ + 4λH ϕ̇ )
H0
]
[
1 2 4
2λϕ 2
2 2
≃ −f λ e ϕ̇ − 2 λ ϕ̇ ,
H0
[
]
1
4λ
2 2
2λϕ 2
3
2 4
p = −f λ e ϕ̇ −∂t ϕ̇ ) .
(λ ϕ̇ −
3
3H02

(5.3.16)(5.3.17)(5.3.18)(5.3.19)

We can confirm NEC is violated in genesis solution by applying eq.(5.3.16) to ρ and p.
Recalling the Friedmann equation eq.(2.0.2), we find neglecting the terms of H in ρ leads

62

CHAPTER 5. SOME TOPICS IN MODERN COSMOLOGYp ≫ ρ. Thus, Ḣ is described asḢ ≃ −4πGp,(5.3.20)and thus we obtain
8πGf 2
1(−∞ < t < 0),
3 H02 (−t)3[]4πGf 2
1a(t) ≃ a0 1 +,3 H02 (−t)2H(t) ≃(5.3.21)(5.3.22)

where we take a(t)|t→−∞ = a0 . The solution of the scale factor describes the universethat starts expanding from singularity-free Minkowski in the asymptotic past. The samegenesis solution can also be obtained from the DBI conformal galileons [37, 38]. Thereis the other scenario in which the evolution of scale factor starts from a(t) ≃ const..
The string gas cosmology motivated by superstring theory was investigated by Robert
Brandenberger [39] also suggests the universe started from Minkowski space-time.
In this way, there are some alternative scenarios which can avoid the singularity. If
these scenarios do not have the other theoretical problems, how can we distinguish between
the models of the early universe? As the tool for distinction, we can see the diﬀerencein the spectrum of perturbations. Especially, in chapter 2, we focus on the gravitational
waves. We have discussed the spectrum of gravitational waves in inflation is approximately
flat, however, it is known that the spectrum of alternative scenarios is blue.
The discovery of detect of gravitational waves may be fresh in our mind, however,it takes more time to obtain the information of primordial universe. In chapter 2, let
us discuss the distinction of the spectrum of gravitational waves with looking forward
the success of detecting the information in the future. The main topic of chapter 2 isthe growth of the gravitational waves and extending the model for generating various
spectrum, so we will conclude these discussions at the conclusions.

Part II

Galilean Genesis

63

Chapter 6

Generalized Galilean genesis
It is fair to say that inflation [40, 7, 41] followed by a hot Big Bang is a standard scenario of modern cosmology. Inflation is attractive because the period of quasi-de Sitter
expansion in the early universe resolves several problems that would otherwise indicate
the need for fine-tuning. Moreover, curvature perturbations are naturally generated fromquantum fluctuations during inflation, which seed large-scale structure of the universe [42].
The basic prediction of inflation is that the primordial curvature perturbations are nearly
scale-invariant, adiabatic, and Gaussian. This is in agreement with observations of CMB
anisotropies [43, 44, 45]. Inflationary models also predict the quantum mechanical production of gravitational waves [40], the detection of which would be the evidence for inflation.
Despite the success of inflation, it would be reasonable to ask whether only inflation
can be a consistent scenario compatible with observations. It should also be noted thatan inflationary universe is past geodesically incomplete [46] and so the problem of an
initial singularity still persists. From this viewpoint, various alternative scenarios havebeen proposed so far, such as bouncing models. Although such models can eliminate
the initial singularity, many of them are unfortunately plagued by instabilities originatedfrom the violation of the null energy condition (NEC), the growth of shear, and primordial
perturbations incompatible with observations [47].
In the context of cosmology, the violation of the NEC implies that
dH
> 0,
dt

(6.0.1)

where H is the Hubble rate and t is cosmic time. This signals ghost instabilities ingeneral relativity. Recently, however, it was noticed that in noncanonical galileon-type
scalar-field theories the NEC can be violated stably,1 and based on this idea, Creminelli etal. proposed a novel, stable alternative to inflation named galilean genesis [1]. (See also
1

The NEC can be violated stability at least within linear perturbation analysis. However, at nonlinear
order, it is not clear whether there are no instabilities [48].

65

66

CHAPTER 6. GENERALIZED GALILEAN GENESIS

Ref. [49].) In the galilean genesis scenario, the universe is asymptotically Minkowski inthe past and starts expanding from this low energy state. As such, this scenario is devoidof the horizon and flatness problems. Aspects of galilean genesis have been studied inRefs. [50, 51, 52, 53, 54] and the original model has been extended in Refs. [36, 37, 38]
to possess improved properties. See also Refs. [22, 15, 55, 56, 57, 58, 59, 60, 61] for otherinteresting NEC violating cosmologies in galileon-type theories and Ref. [32] for a related
review.
In this chapter, we introduce a unified treatment of the galilean genesis models andgive a generic Lagrangian admitting the genesis solutions. This is done by using the Horndeski theory [17], which is the most general scalar-tensor theory with second-order field
equations. Our generalized galilean genesis (GGG) Lagrangian contains four functional
degrees of freedom and a constant parameter denoted α. This parameter determines the
behavior of the Hubble rate. For specific choices of those functions and α = 1, our Lagrangian reproduces the previous models explored in Refs. [1, 36, 37, 38]. As is often thecase with inflation alternatives, it turns out that the galilean genesis models in general failto produce nearly scale-invariant curvature perturbations. We show, however, that with
an appropriate tuning of α it is possible to have a slightly tilted spectrum consistent withobservations.

6.1

Generalized genesis solutions

In Ref. [62] it was noticed that the genesis solution (5.3.21) is obtained generically in the
subclass of the Horndeski theory with
G2 = e4λϕ g2 (Y ), G3 = e2λϕ g3 (Y ),M2
G4 = Pl + e2λϕ g4 (Y ), G5 = e−2λϕ g5 (Y ),2(6.1.1)

where each gi (i = 2, 3, 4, 5) is an arbitrary function of
Y := e−2λϕ X.

(6.1.2)This extends the Lagrangian given in Ref. [52] to include the Horndeski functions G4 andG5 . The Lagrangian (5.3.15) and the DBI conformal galileon theory are included in thegeneral framework defined by (6.1.1) as specific cases.
In this chapter, we further generalize (6.1.1) and consider
G2 = e2(α+1)λϕ g2 (Y ), G3 = e2αλϕ g3 (Y ),M2
G4 = Pl + e2αλϕ g4 (Y ), G5 = e−2λϕ g5 (Y ),2(6.1.3)where α (> 0) is a new dimensionless parameter. The four functions, g2 , g3 , g4 , and g5 ,are arbitrary as long as several conditions presented in this section and in Sec. 6.3 are6.1. GENERALIZED GENESIS SOLUTIONS

67

2 /2 as Y → 0. The
satisfied. We assume, however, that g4 (0) = 0, so that G4 → MPl
Horndeski theory with (6.1.3) admits the following generalized galilean genesis solution:

1
1
eλϕ ≃ √,
λ 2Y0 (−t)H≃h0(−t)2α+1(−∞ < t < 0),(6.1.4)for large |t|, where Y0 and h0 are positive constants. We see that Y ≃ Y0 for this background. The parameter α in the Lagrangian results in controlling the evolution of the
Hubble rate. The scale factor is given by
a≃1+

1 h0
,
2α (−t)2α(6.1.5)and hence the solution describes the universe that starts expanding from Minkowski inthe asymptotic past, similarly to the original galilean genesis solution which corresponds
to the case of α = 1. The “slow-expansion” model considered in Ref. [63] is reproducedby taking the particular functions gi with α = 2. We thus obtain a one-parameter familyof the generalized genesis solutions as an alternative to inflation. Note that, although
the evolution of the scale factor is very diﬀerent from quasi-de Sitter, the universe inthis scenario is also accelerating: ∂t (aH) > 0, and hence fluctuation modes will leave thehorizon during the genesis phase.
Substituting Eq. (8.1.10) to the background equations (3.5.4)–(3.5.13) and picking up
the dominant terms at large |t|, we haveE ≃ e2(α+1)λϕ ρ̂(Y0 ) ≃ 0,
P ≃ 2G(Y0 )Ḣ + e(6.1.6)p̂(Y0 ) ≃ 0,

(6.1.7)()
ρ̂(Y ) := 2Y g2′ − g2 − 4λY αg3 − Y g3′ ,(6.1.8)2(α+1)λϕ

wherep̂(Y ) := g2 − 4αλY g3+8(2α + 1)λ2 Y (αg4 − Y g4′ ),()
2G(Y ) := MPl− 4λY g5 + Y g5′ ,(6.1.9)(6.1.10)an overdot stands for diﬀerentiation with respect to t, and a prime for diﬀerentiation withrespect to Y . The constant Y0 is determined as a root ofρ̂(Y0 ) = 0,

(6.1.11)and then h0 is determined from Eq. (8.1.13) as
h0 = −1p̂(Y0 ).
2α+12(2α + 1)(2λ Y0 )G(Y0 )(6.1.12)

68

CHAPTER 6. GENERALIZED GALILEAN GENESIS

As will be seen shortly, this background is stable for G(Y0 ) > 0. Therefore, the above NEC
violating solution is possible provided thatp̂(Y0 ) < 0.

(6.1.13)

As will be demonstrated in the next section, the generalized genesis solution will develop a singularity H → ∞ at some t = tsing , as in the original genesis model. We therefore
assume that the genesis phase is matched onto the standard radiation-dominated universe
before t = tsing , ignoring for the moment the detail of the reheating process. In conventional general relativity, matching two diﬀerent phases can be done by imposing that the
Hubble parameter is continuous across the two phases. However, the matching conditions
are modified in general scalar-tensor theories as second-derivatives of the metric and thescalar field are mixed in the field equations. The modified matching condition [62] reads∫e(2α+1)λϕ Y0 √2
MPl Hrad = G(Y0 )H −2y g3′ (y)dy20()+2λϕ̇e2αλϕ αg4 − Y0 g4′ ,(6.1.14)
and we require that the subsequent radiation-dominated universe is expanding: Hrad > 0.
This condition translates to√ ∫ Y0 g3−g2 − 2λY0 g3 + (2α + 1)λ Y0(6.1.15)√ dy > 0.
y0
It is easy to see that in the case of α = 1 all the expressions presented above reproduce
the previous results [62].
Before closing this section, let us emphasize that (generalized) galilean genesis has the
Minkowski phase only in the asymptotic past. The true Minkowski spacetime solution
corresponds to the special case of Y = 0, i.e., ϕ = const. The Y = 0 solution is found only
if g2 (0) = 0. One may wonder if the true Minkowski vacuum (Y = 0) in our neighborhood
begins to expand to form a genesis universe (Y = Y0 > 0). This is forbidden because thetwo diﬀerent stable solutions cannot be interpolated, as argued in Ref. [52]. (See, however,Ref. [53].)

6.2

Background evolution

To see whether or not the generalized genesis solution presented in the previous section is
an attractor, we trace the background evolution starting from generic initial conditions.

6.2.1

Analytic argument

Let us begin with a simplified discussion neglecting gravity, i.e., the eﬀect of the cosmic
expansion [52]. It is convenient to introduce a new variableψ := e−λϕ (> 0).

(6.2.1)

6.2. BACKGROUND EVOLUTION

69

Figure 6.1: Examples of the curves defined by Eq. (6.2.3). Horizontal dashed lines
correspond to the genesis solutions.
In terms of ψ we have Y = ψ̇ 2 /(2λ2 ). For any homogeneous solutions the scalar-field
equation of motion (3.5.14) with the functions (6.1.3) can be written as]
d [ −2(α+1)ψρ̂(Y ) = 0.
(6.2.2)dt
Integrating this, we obtainρ̂(Y ) = Cψ 2(α+1) ,(6.2.3)where C is an integration constant. Equation (6.2.3) defines a curve in the (ψ, ψ̇) spacefor each C, as shown in Fig. 6.1. With an initial condition (ψi , ψ̇i ) away from the genesis−2(α+1)solution, the integration constant is determined as C = ψiρ̂(ψ̇i2 /2λ2 ). If ψ̇ < 0initially, the scalar field rolls along the curve toward ψ → 0, i.e., ρ̂ → 0. Hence, thissolution approach to one of the genesis solutions which are denoted as horizontal lines(ψ̇ = const) in the (ψ, ψ̇) plane. If ψ̇ > 0 initially, the scalar field rolls the opposite way
along the curve and goes further away from the genesis solutions. This is the time reversalof the ψ̇ < 0 solutions.
The above analytic argument implies that the genesis solution is the attractor for initialconditions such that ψ̇ < 0 (⇔ (eλϕ )˙ > 0). In the next subsection we perform numericalcalculations to show that this is basically true even if one takes into account of the eﬀect ofgravity. The numerical analysis also allows us to see the final fate of the genesis solutions
for which the eﬀect of the cosmic expansion cannot be ignored.

70

6.2.2

CHAPTER 6. GENERALIZED GALILEAN GENESIS

Full numerical analysis

In the Horndeski theory with (6.1.3) the Friedmann equation can be written asE= e2(α+1)λϕ ρ̂(Y ) + 6H ϕ̇e2αλϕ c1 (Y )[]−3H 2 c2 (Y ) + e2αλϕ d2 (Y ) + 2H 3 ϕ̇e−2λϕ c3 (Y )(6.2.4)= 0,
where
c1 = Y g3′ − 2αλg4 + 2(3 − 2α)λY g4′ + 4λY 2 g4′′ ,
c2 =
c3 =
d2 =

2
MPl
− 12λY g5 − 28λY 2 g5′ − 8λY 3 g5′′ ,5Y g5′ + 2Y 2 g5′′ ,2g4 − 8Y g4′ − 8Y 2 g4′′ .

(6.2.5)(6.2.6)(6.2.7)(6.2.8)Equation (6.2.4) is exact and hence can be used even if the background evolution isaway from the genesis solution. Similarly, one can substitute Eq. (6.1.3) to the evolution
equation P = 0 and the scalar-field equation of motion to write straightforwardly theexact equations for the background. The resultant equations are integrated numerically,giving the background evolution starting from generic initial conditions.
Given the initial conditions (ϕ(t0 ), ϕ̇(t0 )), the initial value for H is determined fromthe Friedmann equation (6.2.4). Therefore, the initial values (ϕ(t0 ), ϕ̇(t0 )) must be chosen
in such a way that Eq. (6.2.4) admits a real root H. Equation (6.2.4) is quadratic in H if
g5 = 0 and cubic if g5 ̸= 0. In both cases, the discriminant D for eλϕ ≪ 1 is given by
D = e2(α+1)λϕ c2 (Y )ρ̂(Y ) + O(e2(2α+1)λϕ ).

(6.2.9)In the g5 = 0 case, the initial data (ϕ(t0 ), ϕ̇(t0 )) must lie in the region where D ≥ 0 is
satisfied. In the g5 ̸= 0 case, the Friedmann equation has at least one real root for any(ϕ(t0 ), ϕ̇(t0 )).
Concrete numerical examples are presented in Figs. 6.2–6.4. In Figs. 6.2 and 6.3 weshow the cases where the Friedmann equation is quadratic in H. The shaded regions(D < 0) cannot be accessed because H would be imaginary there. In Fig. 6.2 we have one
genesis solution, while we have two in Fig. 6.3. In both cases, generalized galilean genesis
is the attractor for ψ̇ < 0. At late times where ψ ≪ 1, the numerical solutions are nolonger approximated by Eq. (8.1.10), and within a finite time the Hubble rate H diverges.
As the Friedmann equation is quadratic in H, we have two branches of the solutions, oneof which may be contracting initially (H < 0). The cosmological evolution neverthelessapproaches the same genesis solution and the trajectories in the (ψ, ψ̇) space are almost
indistinguishable.
The behavior of the models with g5 ̸= 0 is more complicated, as illustrated in Fig. 6.4.
In the white region, we have D > 0 and so there are three possible choices for the initial6.2. BACKGROUND EVOLUTION

71

3

2

Ψ

1

0

-1

Genesis
-2

-3
0

200

400

600

800

Ψ

Figure 6.2: Numerical results of the background evolution for the model with g2 = −Y +Y 2 , g3 = Y , and g4 = g5 = 0. The parameters are given by MPl = 1, λ = 1, and α = 1.

1.5

1.0

Ψ

0.5

Genesis

0.0

-0.5

-1.0

-1.5

0

200

400

600

800

Ψ

Figure 6.3: Numerical results of the background evolution for the model g2 = −Y + 3Y 2 −
Y 3 , g3 = Y , and g4 = g5 = 0. The parameters are given by MPl = 1, λ = 1, and α = 2.

72

CHAPTER 6. GENERALIZED GALILEAN GENESIS
3

2

Ψ

1

0

Genesis
-1

-2

-3
0

200

400

600

800

Ψ

Figure 6.4: Numerical results of the background evolution for g2 = −Y + Y 2 , g3 = Y ,g4 = 0, and g5 = −Y . The parameters are given by MPl = 1, λ = 1, and α = 1.
value of H. Two of the three branches converge to the genesis solution similarly to theg5 = 0 case, as shown as the black lines in Fig. 6.4. Also in this case we find H → ∞within a finite time. However, the remaining one branch never converges to the genesis
solution. The corresponding examples are shown as the gray lines in Fig. 6.4. In theshaded region, we have D < 0 and there is only one possible initial value for H at eachpoint, which corresponds to the latter branch. Therefore, the generalized galilean genesis
solution can be a dynamical attractor for the initial data in the white (D > 0) region.
We thus conclude that the galilean genesis solution is the attractor provided thatψ̇ < 0 (⇔ (eλϕ )˙ > 0) initially, though the situation in the presence of g5 is involved. In
the inflationary scenario, usually it does not matter which direction the scalar field rolls
initially, but the universe must be expanding initially. In contrast to the case of inflation,
the galilean genesis scenario allows both for expanding and contracting universes at theinitial moment, while the scalar field must roll in a particular direction initially. As far as
we have investigated numerically, all the solutions develop a singularity H → ∞ at sometime t = tsing in the future. In passing, we have checked that the numerical examples in
Figs. 6.2 and 6.4 satisfy the stability conditions presented in the next section.

6.2.3

Spatial curvature

We have so far neglected the spatial curvature. In this subsection, let us justify thisassumption by showing that the spatial curvature does not interfere with the evolution

6.2. BACKGROUND EVOLUTION

73

of the genesis background. We will use the cosmological background equations with thespatial curvature K in the Horndeski theory.
Let us take an( initial) condition such that H is suﬃciently small in the equation of
motion for ϕ and eλϕ ˙ > 0. Then, in a universe with K ̸= 0 the equation of motion for
ϕ can be written as[]d 2(α+1)λϕ
K5 (Y )
2αλϕ K4 (Y )eρ̂(Y ) − e−= 0,(6.2.10)
dt
a2
a2
where
(
)
K4 (Y ) := 6 g4 − 2Y g4′ K,()
K5 (Y ) := −12λY g5 + Y g5′ K.

(6.2.11)(6.2.12)

Even if e2(α+1)λϕ ρ̂ ∼ e2αλϕ K4 , K5 at the initial moment, the curvature terms become
smaller relative to the ρ̂ term as the scalar field rolls. Thus, we have the same attractor
solution Y = Y0 satisfying ρ̂(Y0 ) = 0, i.e., eλϕ ∼ (−t)−1 . Along this attractor, theevolution equation reads[ 2]K
2G(Y0 )Ḣ + e2(α+1)λϕ p̂(Y0 ) + MPl+ 4λY0 g5 (Y0 ) 2 ≃ 0,a
(6.2.13)
where we assumed that Ḣ ≫ H 2 . Equation (6.2.13) implies that the curvature term
becomes subdominant as the scalar field rolls, and as a result Ḣ is determined by the p̂
term, recovering the evolution of the genesis background. Thus, the flatness problem isresolved in the genesis model.

6.2.4

Anisotropy

In conventional cosmology, an initial anisotropy is wiped out during inflation [64]. However, in alternative scenarios such as bouncing cosmology, it is often problematic that theinitial anisotropy grows in a contracting phase [65, 66, 67] (see however [58]). In this subsection we will show that adding the initial anisotropy on the generalized Galilean genesis
solution does not destabilize the background evolution.
We consider the Kasner metric[]ds2 = −dt2 + a2 e2θ1 (t) dx2 + e2θ2 (t) dy 2 + e2θ3 (t) dz 2 ,(6.2.14)
where it is convenient to write√
θ1 = β+ + 3β− ,θ 2 = β+ −

√

3β− ,

θ3 = −2β+ .

(6.2.15)

74

CHAPTER 6. GENERALIZED GALILEAN GENESISIn Horndeski theory, the equations for anisotropy is given as)]}(d { 3[
2
2
− β̇−
= 0,
a GT β̇+ − 2X ϕ̇G5X β̇+
dt
[
{
]}
d
a3 GT β̇− + 4X ϕ̇G5X β̇+ β̇−= 0.
dt(6.2.16)(6.2.17)

If the deviations from the genesis background are not large, it follows from Eqs. (6.2.16)
and (6.2.17) that
)]
(
d [
2
2
G β̇+ − 2e−2λϕ ϕ̇Y0 g5′ β̇+
− β̇−
= 0,
dt
]
d [
G β̇− + 4e−2λϕ ϕ̇Y0 g5′ β̇+ β̇− = 0.
dt

(6.2.18)(6.2.19)

In the models with g5′ = 0, this simply gives
β̇+ , β̇− ∼ const,(6.2.20)so that the initial anisotropy dilutes as θi ∼ (−t). In the models with g5′ ̸= 0, we have thefollowing possibilities:
√
√
1
3
1
3
(β̇+ , β̇− ) = (0, 0), (b, 0), (− b,b), (− b, −
b),
2
2
2
2

(6.2.21)

where
b :=

G
2e−2λϕ ϕ̇Y0 g5′∼ (−t)−1 .

(6.2.22)In this case, for nonzero β̇± the initial anisotropy can grow logarithmically: θi ∼ ln(−t).
However, this should be compared with ln a ∼ (−t)−2α ; we see that the logarithmic growthof θi does not spoil the genesis background.

6.3

Primoridal perturbations

Let us now study the behavior of primordial tensor and scalar perturbations around thegeneralized genesis background to obtain predictions of our scenario as well as to impose
stability conditions. To do so, we utilize the general quadratic action for cosmologicalperturbations in the Horndeski theory derived in Ref. [16].

6.3. PRIMORIDAL PERTURBATIONS

6.3.1

75

Tensor perturbations

The quadratic action for tensor perturbations hij in the genesis phase is given by
1
(2)
Sh =
8

[

∫
dtd x a G(Y0 )33c2ḣ2ij − t2 (∇hij )2a],(6.3.1)

where
c2t =

2 + 4λY g (Y )
MPl
0 5 0
G(Y0 )(6.3.2)and note that a ≃ 1. It can be seen that stability against tensor perturbations is assured
if
G(Y0 ) > 0,

(6.3.3)

2+ 4λY0 g5 (Y0 ) > 0,
MPl

(6.3.4)are satisfied.
Since both G(Y0 ) and c2t are constant during the genesis phase, the tensor perturbations
are eﬀectively living in Minkowski without regard to α and the concrete form of gi (Y ),and consequently amplification of quantum fluctuations does not occur as opposed to thecase of quasi-de Sitter inflation. This means that no detectable primordial gravitational
waves are generated from our generic class of the genesis models.

6.3.2

Scalar perturbations

The quadratic action for the curvature perturbation ζ in the unitary gauge is given by
(2)
Sζ =

[

∫
dtd x a GS
3

3

]
c2s
2
ζ̇ − 2 (∇ζ) ,a2(6.3.5)where with some manipulation GS and c2s in the genesis phase are written as[(2α + 1)λξ 2 (Y0 )Y0 ξ ′ (Y0 )ξ ′ (Y0 )p̂(Y0 ),ξ(Y0 )ρ̂′ (Y0 )]2

GS = 2c2s =ρ̂′ (Y0 )e−2αλϕ ,(6.3.6)(6.3.7)with
ξ(Y ) := −Y G(Y ).
p̂(Y )(6.3.8)

76

CHAPTER 6. GENERALIZED GALILEAN GENESIS

Equations (6.3.6) and (6.3.7) show that GS ∝ (−t)2α and c2s = const. It follows from
Eqs. (6.1.13) and (6.3.3) that ξ(Y0 ) > 0. We thus find that stability against scalar perturbations is guaranteed if
ρ̂′ (Y0 ) > 0,

(6.3.9)′(6.3.10)

ξ (Y0 ) < 0,are fulfilled. We can choose the functional degrees of freedom so that this is possible.
Let us evaluate the power spectrum of ζ. To simplify the notation, it is convenient towrite GS = A(−t)2α , where A is a constant deduced from Eq. (6.3.6), the value of which
depends on the model, i.e., α and the concrete form of gi (Y ). The equation of motion
derived from the action (6.3.5) is given by
ζ̈k +

2α
ζ̇k + c2s k 2 ζk = 0,twhere we moved to the Fourier space. This equation can be solved to give
√
1
π
1
ζk =
(−t)ν Hν(1) (−cs kt), ν := − α,
2 2A(Y0 )2(6.3.11)(6.3.12)(1)

where Hν is the Hankel function of the first kind and the positive frequency modes havebeen chosen. On large scales, |cs kt| ≪ 1, we haveζk ≃ Ak + Bk (−t)1−2α ,(6.3.13)where√π Γ(ν)(cs k)−ν ,
2A π
√
[
]
π
1
i cos(πν)Γ(−ν)
−ν−1
:= 2
−
2A Γ(ν + 1)πν×(cs k) .

Ak := −i2

ν−1

Bk

(6.3.14)(6.3.15)

If 0 < α < 1/2, the second term in Eq. (6.3.13) decays as is common to usual cosmologies, leaving the constant mode at late times. Thus, in this case the power spectrum isgiven byPζ (k) =Γ2 (ν) 3−2ν22ν−4 c−2ν
s
k
,
π3A

(6.3.16)

and the spectral index is found to bens = 2α + 3,

(6.3.17)

6.3. PRIMORIDAL PERTURBATIONS

77

yielding a blue spectrum incompatible with observations.
The case of α > 1/2 is more subtle, because the second term in Eq. (6.3.13) grows
and dominates on large scales. This is what happens in the original galilean genesis model(α = 1) [1]. To extract the late-time amplitude of ζk , let us consider the following situation.
Suppose that the genesis phase terminates at t = tend and is matched onto some otherphase. We assume that the scalar field is homogeneous on the t = tend hypersurface. In
the subsequent phase, the curvature perturbation on large scales may be written asζk = Ck − Dk

∫ ∞
t

dt′
,
a3 (t′ )GS (t′ )(6.3.18)

where we do not specify GS (t) for t > tend , but assume that Eq. (6.3.18) gives theconstant and decaying modes and hence the integral converges. The late-time amplitude is given by Ck . The matching conditions [62] imply that ζk and GS ζ̇k are continuous across the two phases (cf. [68, 69, 70]). It is then straightforward to obtain
Ck = Ak + Bk (−tend )1−2α (1 + I) ≃ Bk (−tend )1−2α (1 + I), where
I := (2α − 1)∫ ∞a3 (tend )A(−tend )2α dt′
a3 (t′ )GS (t′ )|tend |
tend

(6.3.19)is independent of k. We may thus use the estimatePζ (k) ∼ C × Pζ (k)|t=tend ,(6.3.20)with C being some k-independent factor. The power spectrum evaluated at the end of thegenesis phase is given byPζ (k)|t=tend =22−2ν−4 c2ν
s Γ (−ν)|tend |4ν k 3+2ν ,π3A(6.3.21)so thatns = 5 − 2α.

(6.3.22)

Although the overall amplitude depends on the details of the model construction, thespectral index depends only on α and not on the concrete form of gi (Y ). We have anexactly scale-invariant spectrum for α = 2, and this is in sharp contrast to the original
galilean genesis model having α = 1, which produces a blue-tilted spectrum of curvature
perturbations. A particular realization of α = 2 is found in Ref. [63, 71], where the sameconclusion is reached. Taking α = 2.02, one can obtain the nearly scale-invariant, butslightly red-tilted, spectrum with ns ≃ 0.96.

78

6.4

CHAPTER 6. GENERALIZED GALILEAN GENESISCurvatonIn the previous section we have seen that the nearly scale-invariant spectrum for curvature
perturbations is possible only in the case of α ≃ 2. In the other cases we need to consider analternative mechanism such as the curvaton in order to obtain a scale-invariant spectrum.
In this section, we study slightly in more detail the curvaton coupled to a conformal metric,
the basic idea of which was proposed earlier in Ref. [1]. A similar mechanism was proposed
in Ref. [51].
To make a scale-invariant power spectrum, we introduce a curvaton field σ coupled tothe conformal metric,ĝµν = e2βλϕ gµν ,(6.4.1)

where β is a constant parameter which is assumed to be close to unity, β ≃ 1. Assuming
the simplest potential, we consider the following action for σ:
[
]
∫
√
1
1
Sσ = d4 x −ĝ − ĝ µν ∂µ σ∂ν σ − m2 σ 2 .
(6.4.2)2
2
The conformal metric (6.4.1) implies that the eﬀective scale factor for the curvaton iseβλϕ ∼ (−t)−β with β ≃ 1, so that σ lives eﬀectively in a quasi-de Sitter spacetime.
The equations of motion for the homogeneous part σ = σ0 (t) is given byσ̈0 + (2βλϕ̇ + 3H)σ̇0 + e2βλϕ m2 σ0 = 0.

(6.4.3)On the genesis background, one can ignore H ∼ (−t)−(2α+1) relative to λϕ̇ ∼ (−t)−1 ,leading to2βm2
σ0 = 0.
σ̇0 + √
(6.4.4)t[λ 2Y0 (−t)]2β√
The eﬀective Hubble rate for the curvaton is ∼ λ 2Y0 . For the “light” curvaton withσ̈0 −m2 ≪ λ2 Y0 ,(6.4.5)we thus have σ0 ≃ const and the other independent solution decays quickly.
The energy density and pressure of σ are given by
ρσ =
pσ =

1 2βλϕ 2 1 4βλϕ 2 2e
σ̇0 + em σ0 ∼ (−t)−4β ,221 2βλϕ 2 1 4βλϕ 2 2e
σ̇0 − em σ0 ∼ (−t)−4β .
22(6.4.6)(6.4.7)Equations (8.1.12) and (8.1.13) imply that the dominant part of the cosmological background equations grows as ∼ (−t)−2(α+1) . Thus, in order for the (initially subdominant)curvaton not to spoil the genesis background as time proceeds, we require thatα + 1 ≥ 2β.

(6.4.8)

6.5. CONCLUSIONS IN THIS CHAPTER

79

The fluctuation of the curvaton, δσ(t, x), obeys2¨ − 2β δσ˙ − ∇2 δσ + √ m
δσ
δσ = 0.
t[λ 2Y0 (−t)]2β

Neglecting the mass term, this can be solved in the Fourier space to give
√ ( √
)β
π
(1)
δσk =
λ 2Y0 (−t)β+1/2 Hβ+1/2 (−kt),2(6.4.9)(6.4.10)where the positive frequency modes have been chosen. Thus, the power spectrum of thecurvaton fluctuations isPδσ (k) =23β−2 λ2β Y0β Γ2 (β + 1/2) 2−2βk,π3(6.4.11)and we findns = 3 − 2β.

(6.4.12)In the case of β = 1, the eﬀective scale factor for the curvaton is that of exact de Sitter,and hence the power spectrum is exactly scale-invariant, as is expected. Taking β = 1.04
we obtain ns = 0.96. The curvaton fluctuations can be converted into adiabatic ones
after the genesis phase, where σ behaves as a conventional scalar field in a true expanding
universe, in the same way as the usual curvaton field in the inflationary scenarios. Note,however, that due to the restriction (6.4.8) the present curvaton mechanism works only
for the models with α ≥ 2 − ns (> 1).

6.5

Conclusions in this chapter

In this chapter, we have extended the galilean genesis models [1, 36, 37, 38] and constructeda generic Lagrangian from the Horndeski theory that admits the generalized galilean genesis solution. In generalized galilean genesis, the universe starts expanding from Minkowski
in a singularity free manner with the increasing Hubble rate H ∼ (−t)−(2α+1) , whereα (> 0) is a new constant parameter in the Lagrangian. We have investigated the background evolution and shown that the generalized galilean genesis solution is the attractor
for a wide range of initial conditions. In particular, we have seen that the spatial curvature
and an initial anisotropy do not hinder the evolution of the genesis phase.
We have then studied the primordial perturbations from the generalized galilean genesis models. From the quadratic actions for cosmological perturbations we have imposedseveral stability conditions on the functions in our generic Lagrangian. In contrast to thecase of quasi-de Sitter inflation, tensor fluctuations are not amplified in the genesis phasein all the galilean genesis models we have constructed, and hence no detectable primordial

80

CHAPTER 6. GENERALIZED GALILEAN GENESIS

gravitational waves are expected. The evolution of the curvature perturbation ζ depends
on the parameter α and has turned out to be more interesting. In the case of α > 1/2,
ζ grows on large scales, as in the original galilean genesis model (α = 1) [1]. The tilt of
the power spectrum at the end of the genesis phase is given by ns = 5 − 2α, irrespectiveof the other details of the model. Thus, we have a slightly red-tilted spectrum for α ≳ 2.
In the case of α < 1/2, the constant mode dominates on large scales as in conventionalcosmology. In this case, the power spectrum has been shown to be always blue-tilted. We
have also discussed the possibility of the curvaton mechanism in the generalized galilean
genesis scenario.
We have ignored the reheating process in our scenario. It would be interesting to
explore how the universe reheats and how matter is created at the end of generalized
galilean genesis. Since the Lagrangian defined by (6.1.3) excludes a cosmological constant,
it is not clear how the genesis phase is connected finally to the late-time universe describedby the ΛCDM model. These are the open questions.

Chapter 7

Reheating and primordial
gravitational waves in GGG
Inflation [7, 41] is now the standard model of the early universe, solving the problems
that Big Bang cosmology faces and explaining the origin of large scale structure in theway consistent with observations. However, the inflationary universe still suﬀers from the
problem of initial singularity [72] as long as the null energy condition (NEC) is satisfied.
To circumvent the singularity, alternative scenarios such as the bouncing models [73, 74,
75, 76] have been proposed so far, but a majority of the alternative models are unstable dueto the violation of the NEC. Recently, an interesting class of scalar field theories called theGalileon has been constructed [23] and various aspects of the Galileon theory have been
explored extensively in the literature. One of the most intriguing nature of this theory isthat the null energy condition can be violated stably1 . Thus, the Galileon opens up a new
possibility of stable, singularity-free models alternative to inflation [55, 56, 57, 58, 59, 60],
as well as stable NEC violating models of dark energy [22] and inflation [15].
In this chapter, we consider an initial phase of NEC violating quasi-Minkowski expansion driven by the Galileon field as an alternative to inflation. The scenario is called theGalilean genesis [1], and diﬀerent versions of genesis models can be found in Refs. [63, 36,
37, 38]. The purpose of this chapter is to address how the genesis phase is connected to thesubsequent hot universe through the reheating stage. In the case of inflation, reheating
usually proceeds with coherent oscillation of the scalar field at the minimum of the potential and its decay. However, the scalar field that drives Galilean genesis does not have thepotential. The situation here is similar to that in kinetically driven inflation models suchas k- [14] and G-inflation [15]. We therefore consider reheating through gravitational particle production [77]. Reheating after Galilean genesis has also been discussed in Ref. [50].
See also Refs. [54, 61, 78, 79] for the other aspects of genesis models.
The exit from k- and G-inflation is accompanied by the kination phase, i.e., the phase1

See Ref. [48] for a subtle point beyond linear perturbations.

81

82CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
where the kinetic energy of the scalar field is dominant. Similarly, the kination phase
can also be incorporated at the end of Galilean genesis as we will do in this chapter.
The primordial gravitational waves that re-enter the horizon during this kination era havea blue spectrum, giving enhanced amplitudes at high frequencies [80]. In addition tothis, the primordial spectrum of gravitational waves is expected to be blue due to NEC
violating quasi-Minkowski expansion in the earliest stage of the universe. The combined
eﬀect will therefore give rise to strongly blue gravitational waves. In the second part of thischapter we evaluate the spectrum of gravitational waves generated from Galilean genesis
and explore the possibility of testing this alternative scenario e.g., with the advanced
LIGO detector.
In the previous chapter [4], we have developed a general framework unifying the original
genesis model [1] and its extensions [63, 36, 37, 38], using which we have derived thestability conditions and primordial power spectra of scalar and tensor perturbations. The
framework is based on the Horndeski theory [17, 18, 16], the most general second-orderscalar-tensor theory in four dimensions. We use this general framework also in this chapter.
The plan of this chapter is as follows. In the next section, we review the Galilean genesis
solution and its generalization. In Sec. III, we investigate the creation of massless scalarparticles after Galilean genesis and compute the reheating temperature. Using the result ofSec. III, we then evaluate the power spectrum of gravitational waves in Sec. IV. We present
some concrete examples and discuss the detectability of the primordial gravitational waves
in Sec. V. Finally, we conclude in Sec. VI.

7.1

Gravitational particle production

In this chapter, we consider the generalized Galilean genesis shown in chapter 6 and now
we redefine the scale factor[]
1 h0
a ≃ aG 1 +,(7.1.1)2α (−t)2αwhere aG is the constant value. This means the scale factor approaches to aG as t → −∞.
The genesis solution is the approximate one valid forh0≪ 1.
(−t)2α(7.1.2)The background evolution begins to deviate from the above solution when h0 /(−t)2α ∼ 1,
and one need to invoke a numerical calculation for a later epoch. To make the calculations
tractable analytically, in this chapter we impose the technical assumption that the genesis
phase terminates before Eq. (7.1.2) is violated. Denoting the Hubble parameter at theend of genesis by H∗ , the condition (7.1.2) can be written as−1/2αH∗ ≪ h0,(7.1.3)

7.1. GRAVITATIONAL PARTICLE PRODUCTION

83

or, equivalently,H∗ (−t∗ ) ≪ 1,

(7.1.4)

where t∗ is the proper time at the end of the genesis phase. This inequality defines therange of validity of our computation.
The universe must eventually be connected to a radiation dominated phase. In standard inflationary cosmology, the inflaton field oscillates about the minimum of the potential and reheat the universe. The radiation dominated universe follows after this reheatingstage. However, the Lagrangian for generalized Galilean genesis does not have the potential term and hence conventional reheating would not be suitable for the genesis scenario.
This is also the case for k-inflation [14]. In this chapter, we therefore explore the possibility of gravitational reheating after the genesis phase, i.e., reheating via gravitational
particle production at the transition to the intermediate phase where the kinetic energyof the scalar field is dominant (the kination phase). See [50] for preheating after Galilean
genesis via a direct coupling between the scalar field and the other fields. Note in passingthat we have another possibility that the genesis phase is followed by inflation [61, 78, 79].
The inflationary stage is then naturally described by k- or G-inflation [15], so that also
in this case we employ gravitational reheating. In this chapter, we do not consider thislatter possibility and focus on the scenario in which the genesis phase is followed by
the intermediate kination phase and then by the radiation dominated phase. To realize
the background evolution suitable to this reheating mechanism, we assume that the Lagrangian of the form (6.1.3) is only valid until ϕ (which is increasing during the genesis
phase) reaches some value, ϕ∗ , and for ϕ > ϕ∗ the Lagrangian is such that the kinetic
energy of ϕ decreases quickly to make only the standard kinetic term X relevant in the2 /2)R + X. The genesis phase thus terminates and is followed by
Lagrangian: L ≃ (MPl2 H 2 = ρ ∝ a−6 .2 The basic scenario here is essentially thethe kination phase where 3MPlϕsame as those at the end of k- and G-inflation [14, 15] described in some detail in [31].
We do not provide a concrete model realizing this because it is reasonable to presume
that this is indeed possible by using the functional degrees of freedom in the Horndeski
theory,3 and also because particle creation is only sensitive to the evolution of the scale
factor but not to an underlying concrete model of generalized Galilean genesis. Note thatin many cases it is likely that perturbations become unstable at the transition between
the two phases. Even if this occurs we can solve the issue by using the idea in Ref. [79],though one must go beyond the Horndeski theory to do so.
Let us consider the creation of massless, minimally coupled scalar particles at thetransition from the genesis phase to kination. The created particle is denoted by χ, whose2

The necessary requirement here is that the energy density of ϕ dilutes more rapidly than that ofradiation. We assume the kination phase just for simplicity.
3
We assume the particular form of the Lagrangian (6.1.3) only during the genesis phase. At the very
end of genesis, the Lagrangian may be away from the form (6.1.3).

84CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
the Lagrangian is given by1
Lχ = − g µν ∂µ χ∂ν χ.
2(7.1.5)Its Fourier component, χk , obeys
(
)
1
a′′
′′
2
(aχk ) + k −χk = 0,aa(7.1.6)

where a dash stands for diﬀerentiation with respect to the conformal time η. Since a ≃aG = const in the genesis phase, we have η ≃ t/aG and hence η = −∞ corresponds to theasymptotic past. We write the solution using the Bogoliubov coeﬃcients as
αk (η)βk (η)aχk = √ e−ikη + √ eikη ,2k
2k(7.1.7)and impose the boundary conditions that αk → 1 and βk → 0 as η → −∞. For k 2 ≫|a′′ /a|, we have [81, 82, 83, 84]
∫ η
i
a′′
βk (η) = −e−2iks ds.
(7.1.8)2k −∞aThe energy density of the created particles can be computed as∫ ∞1ρχ = 2 4
k 3 |βk (∞)|2 dk,
2π a 0and this can be described as eq.(5.2.25) and eq.(5.2.26).
In the genesis phase, we have
[
]
h0
1
2
f (η) ≃ aG 1 +(η < η∗ ),α (−aG η)2α

(7.1.9)(7.1.10)with η∗ := t∗ /aG , while in the subsequent kination phase a ∝ (η + const)1/2 and hence f
is of the formf (η) = c0

η
+ c1 ,−η∗(7.1.11)

where c0 and c1 are to be determined by requiring that f and f ′ are continuous at η = η∗ .
Under this “sudden transition” approximation, f ′′ is discontinuous and consequently V
is also discontinuous at η = η∗ . Due to this discontinuity, the integral (5.2.25) diverges,
which is unphysical. To avoid divergence, we insert a short stage having a time scale ∆η in
between the genesis and kination phases, and join the two phases smoothly. The idea here
is also employed in connecting inflation to the radiation/kination phase smoothly [77, 31].

7.1. GRAVITATIONAL PARTICLE PRODUCTION

85

Thus, we make the ansatz{
b0 + b1 η + b2 η 2 + b3 η 3 (η∗ < η < η∗ + ∆η)f (η) =,
c0 η/(−η∗ ) + c1
(η∗ + ∆η < η)

(7.1.12)after the end of genesis, and determine the six coeﬃcients by requiring that f , f ′ , andf ′′ are continuous at η = η∗ and η = η∗ + ∆η. This allows us to obtain continuous V
and hence finite ρχ . For the current purpose we do not have to write b0 , b1 , b2 , and b3
explicitly, but we only need[()]
2a2G h0
∆η
c0 =
1+O
,
(7.1.13)(−t∗ )2α
−η∗
{
[
(
)]}
(2α + 1)h0∆η
c1 = a2G 1 +1+O.
(7.1.14)α(−t∗ )2α−η∗
Now we have
V (η) ≃ 2a2G (2α + 1)h0 (−aG η)−2(α+1)(η < η∗ ),(7.1.15)

and |V | ≪ V (η∗ ) for η > η∗ + ∆η. To evaluate the integral (5.2.25), it is suﬃcient toapproximate V as a straight line for η∗ < η < η∗ + ∆η. Thus,
V ≃ a2G2(2α + 1)h0(−η + η∗ + ∆η)(−t∗ )2(α+1) ∆η(η∗ < η < η∗ + ∆η).

The main contribution to the integral (5.2.25) comes from the region η∗ < η < η∗ + ∆η
where V ′ gets much larger than in the genesis and kination phases. Performing the integral
in this domain, we obtain()( a )4(2α + 1)2
1
h20
G
ρχ =
ln
(7.1.16).
24(α+1)
32π
aG H∗ ∆η (−t∗ )aA logarithmic divergence is now manifest in the sudden transition limit, ∆η → 0. In thecase of the smooth transition, however, we may take ln(1/aG H∗ ∆η) = O(1), leading to a
finite energy density of relativistic particles. Now since h0 has the dimension of [mass]−2α ,it is convenient to introduce the dimensionless quantity eh0 defined as2αeh0 := MPlh0 .

(7.1.17)

Then, to make Eq. (7.1.16) more suggestive, we rewrite the equation in diﬀerent ways by
1/(2α+1) −1/(2α+1)
using −t∗ = h0
H∗
and Eq. (7.1.17). First, we haveρχ =2
A e − 2α+1
h
0
32π 2

(

H∗
MPl

)− 4α

2α+1

H∗4

( a )4Ga,(7.1.18)

86CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
where

(
2

A := (2α + 1) ln

1
aG H∗ ∆η)(7.1.19)

is a number of O(1). This result is in contrast to that in the inflationary scenario, where
4 /a4 , with H
the energy density of the created particles is simply given by ρχ ∼ Hinf
inf
being the inflationary energy scale. We may also write ρχ asA
H∗4 ( aG )4ρχ =,(7.1.20)128π 2 α2 δ∗2awhere δ∗ is defined asa(η∗ )− 1 (≪ 1).
(7.1.21)aG
This second expression shows that actual ρχ is much larger than the naive estimate deduced
from the case of inflation, ρχ ∼ H∗4 (aG /a)4 .
The reheating temperature TR is determined from ρχ = ρϕ , where ρϕ is the energy
density of the scalar field after the end of genesis,( a )6G
2
H∗2
ρϕ = 3MPl(7.1.22).
a
The scale factor at the time when ρχ = ρϕ occurs, aR , is given by
√
(
) 1
1
aR
96π 2 H∗ − 2α+1 e 2α+1
(7.1.23)
=
h0 .
aG
A
MPl
δ∗ :=

Since aR > aG , we have the condition()(2α+1)/2
96π 2
H∗
e
<
h0 .
MPl
A

(7.1.24)Equating the radiation energy density at a = aR to (π 2 g∗ /30)TR4 , where g∗ is the eﬀective
number of relativistic species of particles, we obtain
(
) α+2
)
(
3
2α+1
− 2(2α+1)
30 1/4
H
TR
A3/4
∗
e
√
=
.
(7.1.25)
h0
2
MPl
π g∗
MPl
3(32π 2 )3/4
2 /M .
This result is again in contrast to that in the inflationary scenario, where TR ∼ Hinf
Pl
One can also write TR as)(H∗2
30 1/4
A3/4
√
TR =
.
(7.1.26)
π 2 g∗3(32π 2 )3/4 (2α)3/2 MPl δ∗3/2
From this one can clearly see that the reheating temperature is much higher than thenaive estimate deduced from inflation, TR ∼ H∗2 /MPl . In other words, for a fixed reheating
temperature, the Hubble parameter at the end of genesis is smaller than the corresponding
inflationary value.

7.2. THE SPECTRUM OF PRIMORDIAL GRAVITATIONAL WAVES

7.2

87

The spectrum of primordial gravitational waves

Having obtained the background evolution in the previous section, let us discuss the spectrum of primordial gravitational waves from generalized Galilean genesis. As expectedfrom the null energy condition violating nature of the scenario, generated gravitational
waves exhibit a blue spectrum and hence are relevant only at high frequencies.
The quadratic action for the gravitational waves in generalized Galilean genesis is [4]
[
]
∫
c2t
1
(2)
2
3
3
2
Sh =
dtd x a G(Y0 ) ḣij − 2 (∂k hij ) ,(7.2.1)8a
2 +4λY g (Y )]/G(Y ). Note that the coeﬃcients G(Y ) and c2 are constantwhere c2t := [MPl
0 5 0
0
0
t
during the genesis phase. The equation of motion for each Fourier mode of two polarization
states, hλk (λ = +, ×), reads
(
)
a′′
1
(ahk )′′ + c2t k 2 −hk = 0,(7.2.2)
a
a

where
a′′
h0
2α + 1≃·2αa(−aG η)η2(7.2.3)and λ is omitted here and hereafter. From Eq. (7.2.3) it can√ be seen that the gravitational
waves freeze not at |ct kη| ∼ 1 but at later times, |ct kη| ∼ h0 /(−aG η)2α ≪ 1. The WKB
solution for the subhorizon modes is given by√
1
2 −ict kη.
hk =e(7.2.4)a Gct kSince a ≃ aG , the power spectrum is already constant at early times,
k3
2k 2
2
|h
|
=
.
k
π2
π 2 Gct a2G(7.2.5)After each mode exits the “horizon” and ceases to oscillate, this amplitude is retained,
as illustrated by the numerical example in Fig. 7.1. Thus, the power spectrum of theprimordial gravitational waves evaluated at the end of the genesis phase is given by(p)

Ph (k) =2k 2.
π 2 Gct a2G(7.2.6)

In contrast to the gravitational waves from inflation, the primordial spectrum is blue, andhence gravitational waves could be relevant observationally only at high frequencies.

88CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG

1.0Èhk HΗLÈ
0.5
0.0
-0.5
Re@hk HΗLD
-1.0
-8

-6

-4

0

-2
Ηc
log
Η

2

4

Figure 7.1: The numerical evolution of gravitational waves during generalized Galilean
genesis with α = 1. ηc denotes the horizon-crossing time.
The density parameter for the gravitational waves per log frequency interval, Ωgw ,evaluated at the present epoch is given by
Ωgw =

k2
Ph (k),
12H02(7.2.7)

where H0 is the present value of the Hubble parameter and the power spectrum Ph characterizes the present amplitude of the gravitational waves. Since hk = const on superhorizon
scales and hk ∝ a−1 after horizon re-entry, we have((p)
Ph (k) = Ph (k)

ak
a0

)2
,

(7.2.8)where ak is the scale factor at horizon re-entry and a0 (= 1) is the scale factor at present.
Suppose that a mode with wavenumber k re-enters the horizon at the epoch characterized by the equation-of-state parameter w. Since k = ak Hk , where the Hubble parameter−3(1+w)/2
at horizon re-entry obeys Hk ∝ ak
, we findak ∝ k −2/(1+3w) .

(7.2.9)In our scenario we have the kination phase (w = 1) after the end of genesis and subse-

7.2. THE SPECTRUM OF PRIMORDIAL GRAVITATIONAL WAVES

89quently the conventional radiation and matter dominated phases. Thus, we obtain2kR keq
k04



2


k k k4 2 R4 eq
keq k0Ωgw = Ω(p)gw (k) ×
4
 k 2 keq


4


 k0
k4

(kR < k < k∗ )(keq < k < kR )(7.2.10)(k0 < k < keq ),where we writeΩ(p)gw (k) =k2
k4(p)
Ph = 2 2.
2
12H0
6π H0 Gct a2G(7.2.11)Here, the wavenumbers k∗ , kR , keq , and k0 correspond to the modes that re-enter thehorizon at the end of genesis, at the reheating time, at the epoch of matter-radiation
equality, and at the present epoch. Explicitly, we have()
h−4
k0 = a0 H0 = 2.235 × 10
Mpc−1 ,
(7.2.12)0.67()
2−2 Ωm hkeq = aeq Heq = 1.028 × 10
Mpc−1 .
(7.2.13)0.141
It is also useful to write
( )2
k∗
aR
=
kR
aG
1
) 1+2α ()− 2(2+α)
(
(
) 2
1
1+α
32π 2 2(2+α) π 2 g∗
TR − 2+α
2+α
e
2+α
h0
,
=3
A
30
MPl

(7.2.14)

where we used the fact that the Hubble parameter at the reheating time, HR , follows fromHR = H∗ (aG /aR )3 . In terms of the frequency, kR corresponds to(g∗ )1/6
fR ≃ 0.026
106.75(TR
6
10 GeV)Hz.

(7.2.15)

Using (7.2.14) one can also write f∗ . A schematic picture of Ωgw as a function of thefrequency f is shown in Fig. 7.2. This should be compared with the standard prediction
from inflation, where one has a flat spectrum for feq < f < fR [85].
To see the possibility of detecting the gravitational waves from generalized Galilean
2
genesis, let us evaluate Ωgw at k = kR and k = k∗ . For simplicity, we assume that G = MPland ct = 1. This is exact for g5 = 0 models, and extending the following estimate to the90CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG

logIWgw M

f3

f2
f0

fR

feq

f*

logH f L
Figure 7.2: Schematic shape of the power spectrum of gravitational waves in generalized
Galilean genesis.
g5 ̸= 0 case is straightforward. Then, it can be seen from Eq. (7.2.10) that
2 k4
kR
0
2 H 2 a2 k 2
6π 2 MPleq
0 G()2 ( )−4H∗
aR≃ 10−5 ×
,
MPl
aG
k3
Ωgw (k∗ ) = Ωgw (kR ) × 3∗
kR
(
) ( )
H∗ 2 aR 2≃ 10−5 ×,MPlaGΩgw (kR ) =1(7.2.16)(7.2.17)

where we used k0 = H0 . The ratio aR /aG (> 1) itself depends on H∗ and model parameters
through Eq. (7.1.23). At this stage it is instructive to compare those results with the flat
spectrum in the inflationary scenario,
−5
Ωinf
×
gw ≃ 10

(

Hinf
MPl

)2
.

(7.2.18)

For fixed TR , it follows from Eq. (7.1.26) that H∗ < Hinf . This shows that Ωgw (kR )in generalized Galilean genesis is smaller than the inflationary prediction for fixed TR .
However, this does not hold true for Ωgw (k∗ ).

7.3. EXAMPLES

91

Now, by the use of Eqs. (7.1.23) and (7.1.25) we obtainΩgw (kR ) = 10

−5

·3

1
− 2+α

(

32π 2
A

1+2α () 2(2+α)π 2 g∗30

3+2α) 2(2+α)

1

e
h02+α

(

TR
MPl

) 2(3+2α)2+α,(7.2.19)andΩgw (k∗ ) = 10

−5

·3

2+3α
2+α

(

32π 2
A

) 2(1+2α)(2+απ 2 g∗
30

α
) 2+α

4

e
h02+α

(

TR
MPl

) 4α

2+α

.

(7.2.20)

) α (

)3
f
.
100 Hz(7.2.21)In the frequency range fR < f < f∗ , we have
Ωgw (f ) = 10

−31

·3

1
− 2+α

(

32π 2
A

1+2α () 2(2+α)π 2 g∗30

1+α) 2(2+α)

1
2+α

e
h0

(

TR
MPl

2+α

One could enhance Ωgw by taking large eh0 , but too large e
h0 would violate Eq. (7.1.3),
indicating the breaking of the approximation a ≃ aG [1 + h0 /2α(−t)2α ]. Using Eqs. (7.1.3)
and (7.1.25) we have, for fR < f < f∗ ,−30

Ωgw (f ) ≲ 10

A

−1/4

(

g∗ )1/4
106.75(f
100 Hz)3.

(7.2.22)

Note that the right hand side is independent of α and the reheating temperature. To
generate a higher amplitude of the gravitational waves, one would relax the limitation
imposed by Eq. (7.1.3).
In the above argument we have assumed ct = 1 for simplicity, but it is worth notingthat we can enhance the amplitude of the gravitational waves by assuming that ct < 1.

7.3

ExamplesTo be more specific, let us move to the discussion of concrete examples in the this section.

7.3.1

The original model of Galilean genesis

First, we apply the above results to the original model of Galilean genesis [1], whichcorresponds to the α = 1 case. The original model is given by choosing the arbitrary
functions and parameters in (6.1.3) as
g2 = −2µ2 Y +g4 = g5 = 0,

2µ3 2
2µ3
Y
,
g
=
Y,
3
Λ3
Λ3
λ = 1, α = 1,

(7.3.1)

92CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
where µ and Λ are the parameters having the dimension of mass. Note that ϕ is taken
to be dimensionless and so [Y ] = mass2 . It is easy to verify that the model satisfies thestability conditions for scalar and tensor perturbations. From Eqs. (8.1.12) and (6.1.12)
we find
Y0 =

Λ3
,
3µ

h0 =

µ3
2 Λ3 .
2MPl(7.3.2)

Assuming that A = O(1), the reheating temperature can be written as−3

TR ≃ 10(g∗ )−1/4
106.75( )3/2ΛH∗ .
µ(7.3.3)The condition that the analytic approximation for the genesis background solution is valid,
Eq. (7.1.3), translates to
µ
< 102
Λ

(

TR
10
10 GeV)−1/3 (g∗ )−1/12,
106.75(7.3.4)

while aR /aG > 1 yieldsµ> 10−7Λ(TR
10
10 GeV)2/3 (g∗ )1/6.
106.75(7.3.5)The density parameters at k = kR and k = k∗ are given respectively by)10/3( g )5/6 ( µ ) ( T
∗
R
−31
,
Ωgw (kR ) ≃ 10
106.75
Λ
1010 GeV(7.3.6)and−10Ωgw (k∗ ) ≃ 10(g∗ )1/3 ( µ )4
106.75Λ(TR
10
10 GeV)4/3,(7.3.7)

where the corresponding frequency isf∗ = 10

9

(µ)(
Λ

TR
1010 GeV)1/3
Hz.

(7.3.8)In the frequency range fR < f < f∗ we have−32

Ωgw (f ) ≃ 10(g∗ )1/3 ( µ )106.75
Λ(TR
1010 GeV)1/3 (

f
100 Hz)3.

(7.3.9)Figure 7.3 shows the examples of the power spectra compared with the anticipatedsensitivity of the advanced LIGO. We consider two diﬀerent reheating temperatures. One

7.3. EXAMPLES

93

Adv LIGO

10-10

Wgw

10-18
10-26
10-34
10-42
fR

fR

-50

10

1000

105

107

109

f HHzL

Figure 7.3: The density parameters Ωgw (f ) for diﬀerent reheating temperatures in theoriginal model of Galilean genesis. The two plots correspond to TR ∼ 1010 GeV (fR =100 Hz) with µ/Λ = 102 and TR ∼ 1016 GeV (fR = 108 Hz) with µ/Λ = 1. Here µ/Λ
is taken to be the largest value allowed by Eq. (7.3.4). The anticipated sensitivity of theadvanced LIGO is also marked.

94CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
is TR ∼ 1010 GeV and it follows from Eq. (7.2.15) that the corresponding frequency is given
by fR = 100 Hz. In this case, the amplitude is too small to be detected by the advanced
LIGO. The other is as high as TR ∼ 1016 GeV, giving fR = 100 MHz. Also in this case it
is unlikely to be able to detect the primordial gravitational waves at f = 100 Hz. In bothcases we have Ωgw ∼ 10−12 at f = 100 MHz, as can be seen directly from Eq. (7.2.22).
The gravitational reheating stage after Galilean genesis could therefore be probed by thegravitational waves at very high frequencies.

7.3.2

A model generating the scale invariant curvature perturbation

As a second example, let us consider the case of α = 2. This class of models is ofparticular interest because it gives rise to a scale-invariant spectrum of the curvature
perturbation [63, 4]. The Lagrangian is given in a similar form to the previous example
byg2 = −2µ2 Y +g4 = g5 = 0,

2µ3 2
Y ,
Λ3
λ = 1,

g3 =

µ3
Y,
Λ3
(7.3.10)

but now α = 2. It follows from Eqs. (8.1.12) and (6.1.12) that
Y0 =

Λ3
,
µ

h0 =

µ4
2 Λ6 .
10MPl(7.3.11)

The reheating temperature is obtained as−2TR ≃ 10(g∗ )−1/4 Λ9/5 H∗.
2/5
106.75
M µ6/54/5(7.3.12)Pl

Equation (7.1.3) in this case reduces to
MPl µ2
< 106
Λ3

(

TR
10
10 GeV)−1 (g∗ )−1/4,
106.75(7.3.13)

while aR /aG > 1 gives
MPl µ2> 10−11
Λ3

(

TR
10
10 GeV)(g∗ )1/4.
106.75(7.3.14)The density parameters at k = kR and k = k∗ are computed respectively asΩgw (kR ) ≃ 10−32(g∗ )7/8
106.75(MPl µ2
Λ3)1/2 (

TR
1010 GeV)7/2,(7.3.15)

7.3. EXAMPLES

95

Adv LIGO

10-10

Wgw

10-18
10-26
10-34
10-42
fR

fR

-50

10

1000

105

107

109

f HHzL

Figure 7.4: The density parameters Ωgw (f ) for diﬀerent reheating temperatures in the
α = 2 model, in comparison with the anticipated sensitivity of the advanced LIGO. The
parameter MPl µ2 /Λ3 is taken to be the possible largest values: MPl µ2 /Λ3 = 106 forTR ∼ 1010 GeV (fR = 100 Hz) and MPl µ2 /Λ3 = 1 for TR ∼ 1016 GeV (fR = 108 Hz).
and−15

Ωgw (k∗ ) ≃ 10(g∗ )1/2
106.75(MPl µ2
Λ3

)2 (

TR
10
10 GeV)2,(7.3.16)Hz.

(7.3.17)

with the corresponding frequencyf∗ = 108(g∗ )1/24
106.75(MPl µ2
Λ3)1/2 (

TR
10
10 GeV)1/2

In the range fR < f < f∗ , we findΩgw (f ) ≃ 10−33(g∗ )3/8
106.75(MPl µ2
Λ3)1/2 (

TR
10
10 GeV)1/2 (

f
100 Hz)3.

(7.3.18)

The basic conclusions for the α = 2 model is the same as those for α = 1, as presented inFig. 7.4. Though detectable gravitational waves are not expected in the sensitive bands of,say, the LIGO detector, Ωgw is enhanced at high frequencies up to the model independent
value as shown in Eq. (7.2.22) and Ωgw ∼ 10−12 at f = 100 MHz under the optimistic
choice of the parameters.
Since the α = 2 model can generate scale-invariant curvature perturbations, let us
now consider consistency between the above optimistic estimate of Ωgw and the prediction

96CHAPTER 7. REHEATING AND PRIMORDIAL GRAVITATIONAL WAVES IN GGG
for the curvature perturbations. The power spectrum of the curvature perturbation ζ
evaluated at the end of Galilean genesis is given by [4]Γ2 (−3/2)(−t∗ )−6 ,(7.3.19)
2π 3 c3s A√
where in the present model it is found that cs = 3 and A = (100/9)(MPl µ2 /Λ3 )2 . For
simplicity we assume that the evolution of ζ in the post-genesis phase is negligible. We
can write t∗ in terms of TR to obtainPζ |t=t∗ =−11Pζ ≃ 10(g∗ )3/8
106.75(MPl µ2
Λ3)1/2 (

TR
10
10 GeV)3/2.

(7.3.20)

Using Pζ ∼ 10−9 , the parameters are fixed as
MPl µ2
∼ 104
Λ3

(

TR
1010 GeV)−3.

(7.3.21)

This gives smaller values of MPl µ2 /Λ3 than the most optimistic ones used in Fig. 7.4. For
TR ∼ 1010 GeV we have Ωgw ∼ 10−13 at f = 100 MHz. If one would use the curvaton
mechanism to produce the observed spectrum of the curvature perturbation even in thecase of α = 2, one has Pζ < 10−9 but then MPl µ2 /Λ3 < 104 (TR /100 GeV)−3 .

7.4

Conclusions in this chapter

In this chapter, we have studied the reheating stage after the initial quasi-Minkowski
expanding phase and its consequences on the primordial gravitational waves. The initialstage was described in a unified manner as generalized Galilean genesis [4] in terms of the
Horndeski scalar-tensor theory. Since the scalar field does not have a potential in which it
oscillates, we have considered reheating through the gravitational production of masslessscalar particles at the transition from the genesis phase to the kination phase. To avoid
an unphysical diverging result which would come from a sudden transition approximation,
we have followed the previous works [77, 31] and considered a smooth matching of the twophases. We have computed in that way the energy density of created particles and thereheating temperature. In the case of gravitational reheating after inflation, it is known
4 , where H
that the created energy density is given by ∼ Hinf
inf is the inflationary Hubble
parameter [77]. We have shown that the energy density of massless scalar particles createdafter Galilean genesis is not simply given by the naive replacement, ∼ H∗4 , where H∗ is
the Hubble parameter at the end of the genesis phase, but rather by a more involved form
which depends on the model parameters as well as H∗ . In particular, it has been foundthat for the same reheating temperature the Hubble parameter at the end of genesis issmaller compared with the corresponding value in the inflation scenario.

7.4. CONCLUSIONS IN THIS CHAPTER97

We have then discussed the spectrum of the primordial gravitational waves from generalized Galilean genesis. The combined eﬀects of the quasi-Minkowski expanding background and the kination phase give rise to the blue gravitational waves, Ωgw ∝ f 3 , at highfrequencies, while their amplitude is highly suppressed at low frequencies in contrast with
the inflationary gravitational waves. Unfortunately, the expected amplitude is too small tobe detected in the sensitive bands of the advanced LIGO detector. However, it is possible
to have Ωgw ∼ 10−12 at f ∼ 100 MHz. Thus, the primordial gravitational waves having aspectrum Ωgw ∝ f 3 at f ≳ 100 MHz and the lack thereof at low frequencies would oﬀer
an interesting test of Galilean genesis in future experiments.
Finally, let us comment that the equations of motion for a massless scalar field andgravitational waves are practically the same, though we have discussed the particle production of the former on subhorizon scales while the latter cross the horizon. This implies
that the gravitons can also be generated on subhorizon scales at the transition between
the two phases in the same way as the massless scalar field, and this concern must be
taken care of in the gravitational reheating scenario.

Chapter 8

Scale-invariant perturbations
It is no exaggeration to say that inflation [40, 7, 41] is now a part of the “standard model”
of the Universe. Not only homogeneity, isotropy, and flatness of space, but also theinhomogeneous structure of the Universe originated from tiny primordial fluctuations [42],can be elegantly explained by a phase of quasi-de Sitter expansion in the early Universe.
However, even the inflationary scenario cannot resolve the initial singularity problem [46],which raises the motivation for debating the possibilities of alternatives to inflation (for areview, see, e.g., Refs. [73, 34]). In order to be convinced that the epoch of quasi-de Sitter
expansion did exist in the early Universe, one must rule out such alternatives.
A typical feature of singularity-free alternative scenarios is that the Hubble parameter
H is an increasing function of time in the early universe. The null energy condition
requires that for all null vectors k µ the energy-momentum tensor satisfies Tµν k µ k ν ≥ 0,
which, upon using the Einstein equations, translates to the condition for the Ricci tensor,
Rµν k µ k ν ≥ 0. In a cosmological setup this reads Ḣ ≤ 0, and hence the NEC1 is violated insuch alternative scenarios. Unfortunately, in many cases the violation of the NEC implies
that the system under consideration is unstable. Earlier NEC-violating models are indeed
precluded by this instability issue [47]. Recently, however, it was noticed that scalar-field
theories with second-derivative Lagrangians admit stable NEC-violating solutions [1, 22,15], which revitalizes singularity-free alternatives to inflation [55, 56, 57, 58, 59, 60, 78, 32].
One can avoid the initial singularity also in emergent universe cosmology [86, 87, 88, 89]
and in string gas cosmology [39, 90].
The future detection of primordial gravitational waves (tensor perturbations) is supposed to give us valuable information of the early Universe. It is folklore that a nearly
scale-invariant red spectrum of primordial gravitational waves is the “smoking gun” of
inflation. The reason that this is believed to be so is the following. The amplitude ofeach gravitational wave mode is determined solely by the value of the Hubble parameterIn this chapter, we use the terminology NEC when referring to Rµν kµ kν ≥ 0, which is, more properly,
the null convergence condition.
1

99

100

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

evaluated at horizon crossing. During inflation H is a slowly decreasing function of time,
while in alternative scenarios the time evolution of H is very diﬀerent. This folklore is
not true, however, even in the context of inflation, because some extended models of inflation can violate the NEC stably and thereby the Hubble parameter slowly increases,
giving rise to nearly scale-invariant blue tensor spectra [15]. Then, does the detection ofnearly scale-invariant tensor perturbations indicate a phase of quasi-de Sitter expansion?
Naively, the gross violation of the NEC in alternative models implies strongly blue tensor
spectra, and by this feature one would be able to discriminate inflation from alternatives.
In this chapter, we show that this expectation is not true: nearly scale-invariant scalar
and tensor perturbations can be generated from quantum fluctuations on a NEC-violatingbackground.2 Thus, it is possible that the individual spectrum has no diﬀerence from thatof inflation, though the consistency relation turns out to be diﬀerent.
The model we present in this chapter is a variant of Galilean Genesis [1], in which
the universe starts expanding from Minkowski by violating the NEC stably. The earlier proposal of Galilean Genesis [1, 36, 37, 38] fails to produce scale-invariant curvature
perturbations (without invoking the curvaton), but it was shown in [4, 63, 71] that it ispossible if one generalizes the original models. In all those models, the tensor perturbations have strongly blue spectra and hence the amplitudes are too small to be detected at
low frequencies [2]. In our new models of Galilean Genesis, however, the primordial tensor
spectrum can be red, blue, or scale invariant, depending on the parameters of the model,
and the curvature perturbation can have a nearly scale-invariant spectrum. We work in the
Horndeski theory [17, 18, 16], the most general scalar-tensor theory with second-order field
equations, to construct a general Lagrangian admitting the new Genesis solution with theabove-mentioned properties. As a specific case our Lagrangian includes the Genesis model
recently obtained by Cai and Piao [93], which yields scale-invariant tensor perturbations
and strongly red scalar perturbations.
The plan of this chapter is as follows. In Sec. II, we introduce the general Lagrangianfor our new variant of Galilean Genesis, and study the background evolution to discuss
whether homogeneity, isotropy, and flatness of space can be explained in the present
scenario. Then, in Sec. III, we calculate primordial scalar and tensor spectra. We givea concrete example yielding scale-invariant scalar and tensor perturbations in Sec. IV. In
Sec. V we draw our conclusions.

8.1

A new Lagrangian for Galilean Genesis

Now let us present a new variant of generalized Galilean Genesis that enjoys a similar
background evolution but exhibits a novel behavior of perturbations compared to the2It has been known that in string gas cosmology scale-invariant scalar and tensor perturbations are
generated from thermal string fluctuations [73, 34]. Nearly scale-invariant tensor perturbations can also
be sourced by gauge fields in bouncing [91] and ekpyrotic [92] scenarios.

8.1. A NEW LAGRANGIAN FOR GALILEAN GENESIS

101

existing Genesis models. As the arbitrary functions Gi (ϕ, X) in the Horndeski theory wechooseG2 = e2(α+1)λϕ g2 (Y ) + e−2(β−1)λϕ a2 (Y ) + e−2(α+2β−1) b2 (Y ),G3 = e2αλϕ g3 (Y ) + e−2βλϕ a3 (Y ) + e−2(α+2β) b3 (Y ),G4 = e−2βλϕ a4 (Y ) + e−2(α+2β)λϕ b4 (Y ),G5 = e−2(α+2β+1)λϕ b5 (Y ),(8.1.1)

where g2 and g3 are arbitrary functions of Y , but ai (Y ) and bi (Y ) are such thata2 (Y ) = 8λ2 Y (Y ∂Y + β)2 A(Y ),(8.1.2)a3 (Y ) = −2λ(2Y ∂Y + 1)(Y ∂Y + β)A(Y ),(8.1.3)a4 (Y ) = Y ∂Y A(Y ),(8.1.4)3

2

3

b2 (Y ) = 16λ Y (Y ∂Y + α + 2β + 1) B(Y ),(8.1.5)b3 (Y ) = −4λ Y (2Y ∂Y + 3)(Y ∂Y + α + 2β + 1) B(Y ),(8.1.6)b4 (Y ) = 2λY (Y ∂Y + 1)(Y ∂Y + α + 2β + 1)B(Y ),(8.1.7)b5 (Y ) = −(2Y ∂Y + 1)(Y ∂Y + 1)B(Y ),(8.1.8)2

2

with arbitrary functions A(Y ) and B(Y ). We thus have four functional degrees of freedom,
as well as two constant parameters α and β in this setup. We assume thatα+β >0(8.1.9)in order to obtain the background evolution which we will present shortly. However, at
this stage we do not impose that α > 0 and β > 0.
We assume the ansatz,
Y ≃ Y0 = const,

H≃

h0
,
2α+2β+1(−t)(8.1.10)and substitute this into the field equations to see that eq. (8.1.10) indeed gives a consistentsolution for a large |t|. (The range of t is −∞ < t < 0.) The scale factor for a large |t| isgiven by
a≃1+

1
h0
.
2(α + β) (−t)2(α+β)(8.1.11)The (00) and (ij) components of the gravitational field equations read, respectively,
ρ̂(Y0 ) + O(|t|−2(α+β) ) = 0,

(8.1.12)2GT Ḣ + e2(α+1)λϕ p̂(Y0 ) + O(|t|−2(2α+β+1) ) = 0,

(8.1.13)

102

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

where
GT ≃ −2e−2βλϕ Y0 (A′ + 2Y A′′ )+ 2e−2(α+2β+1)λϕ H ϕ̇Y0 (6B ′ + 9Y B ′′ + 2Y 2 B ′′′ ).

(8.1.14)

Note thatGT ∝ (−t)2β ,(8.1.15)and hence GT Ḣ = O(|t|−2(α+1) ). Equation (8.1.12) fixes Y0 as a root ofρ̂(Y0 ) = 0,

(8.1.16)

and then eq. (8.1.13) is used to determine h0 . Since there is H in GT , eq. (8.1.13) reduces
to a quadratic equation in h0 in general. We have sensible NEC-violating cosmology only
for h0 > 0. Since it will turn out that the condition
GT > 0(8.1.17)is required from the stability of tensor perturbations, one must imposep̂(Y0 ) < 0,(8.1.18)

though this is not a suﬃcient condition for h0 > 0.
A particular case of this class of Genesis models can be found in [93], which corresponds
to A ∝ Y −2 , B = 0 with α = β = 2.
We have thus found that the Horndeski theory with (8.1.1) admits the Genesis solution (8.1.10), which is similar to previous ones [4]. However, we will show in the next
section that the evolution of tensor perturbations is quite diﬀerent: they can even growon superhorizon scales and can give rise to a variety of values of the spectral index nt .
Before seeing this, let us address more about the background evolution.

8.2

Problems

8.2.1

Flatness Problem

Now let us move on to discuss the problems which inflation solves. In the inflationary
universe, the curvature term in the Friedmann equation is diluted exponentially relative
to the other terms, and thus the flatness problem in standard Big Bang cosmology is
resolved. Since cosmic expansion is very slow in Galilean Genesis, a ≃ 1, one may wonder
if the flatness problem is solved as well in this scenario. We have shown in the previous
chapter [4] that the curvature term is eventually diluted away in all existing Galilean
Genesis models. We now check this point in our new variant of Galilean Genesis.

8.2. PROBLEMS

103

The background equations in the presence of the spatial curvature K are given by [4]
3GT K
≃ 0,
a2
FT K
2GT Ḣ + e2(α+1)λϕ p̂(Y0 ) +≃ 0,a2e2(α+1)λϕ ρ̂(Y0 ) −(8.2.1)(8.2.2)

where
FT ≃ 2e−2βλϕ Y0 A′− 4e−2(α+2β)λϕ (1 + 2α + 4β)λY02 (2B ′ + Y B ′′ ).

(8.2.3)We have{(−t)2βFT ∝(−t)2(α+2β)(2B ′ + Y0 B ′′ = 0).
(2B ′ + Y0 B ′′ ̸= 0)(8.2.4)In order for the flatness problem to be resolved, the curvature term has to be negligible
compared to the other terms. In eq. (8.2.1) the ratio of the curvature term to the firstterm is ∼ (−t)2(α+β)+1 , and due to the condition α + β > 0 the curvature term becomes
negligible as time proceeds. It can be seen using eq. (8.2.4) that the same is true ineq. (8.2.2). We have thus confirmed that the flatness problem can be solved as well in our
new variant of Galilean Genesis.

8.2.2

Anisotropy

Given the large-scale isotropy of the Universe, let us consider the evolution of anisotropies
in the Genesis phase and check whether the universe can safely be isotropized. In standard
cosmology, the shear term in the Friedmann equation is diluted rapidly as ∝ a−6 . However,the situation is subtle in Galilean Genesis.
We describe an anisotropic universe using the Kasner-type metric. From the equationsof motion for β+ and β− with a ≃ 1, we obtain [4]
(
)]
d [
2
2
GT β̇+ − 2X ϕ̇G5X β̇+
− β̇−
= 0,
dt
]
d [
GT β̇− + 4X ϕ̇G5X β̇+ β̇− = 0,dt

(8.2.5)(8.2.6)

where GT ∝ (−t)2β andX ϕ̇G5X = ϕ̇e−2(α+2β+1)λϕ Y0 b′5 (Y0 )∝ (−t)2α+4β+1 .

(8.2.7)

104

CHAPTER 8. SCALE-INVARIANT PERTURBATIONSIn the b′5 (Y0 ) = 0 case, it is easy to see that β̇± ∝ (−t)−2β , and henceβ̇±∝ (−t)2α+1 .
H(8.2.8)This implies that if α > −1/2, the universe is isotropized as it expands.
To see what happens in the general case of b′5 (Y0 ) ̸= 0, it is convenient to defineb :=GT∼ H ∝ (−t)−2(α+β)−1 .
2X ϕ̇G5X(8.2.9)One can integrate eqs. (8.2.5) and (8.2.6) to obtain
(

β̇+
b
(

)

(

)2 ( )2β̇+β̇−−+= const × (−t)2α+1 ,bb)( )( )
β̇+
β̇−
β̇−
+2
= const × (−t)2α+1 .
bbb(8.2.10)(8.2.11)

If α < −1/2, the right hand sides grow as the universe expands, leading to the growthof β̇± /b, i.e., the growth of β̇± /H. Therefore, this case is not acceptable. If α > −1/2
and the initial anisotropies are suﬃciently smaller than b (∼ H), the quadratic terms in
eqs. (8.2.10) and (8.2.11) can be ignored and we have β̇± /b ∝ (−t)2α+1 , i.e.,β̇±∝ (−t)2α+1 ,H(8.2.12)

implying that the universe is isotropized. However, if α > −1/2 and the initial anisotropies
are as large as β̇± = O(b), it is possible that the solution approaches one of the attractors
shown in eq.(6.2.21). In this case, the anisotropies remain,β̇±= const,H(8.2.13)which is not acceptable. In light of the above result, it is required that
α>−

1
2

(8.2.14)

to avoid a highly anisotropic universe. We also assume in the general case of b5 (Y0 ) ̸= 0
that the initial anisotropies β̇± are not as large as b (∼ H).

8.3. POWER SPECTRA

8.3

105

Power Spectra

In the previous section we have seen that the Horndeski theory with (8.1.1) oﬀers a variantof generalized Galilean Genesis, which has a similar background solution to the previous
Genesis models. A significant point of this variant is found in the dynamics of scalar
and tensor perturbations. In this section, let us discuss the evolution of the cosmologicalperturbations and their power spectra. As we will see, the quadratic Lagrangian for thecurvature perturbation ζ is of the form
⃗ 2,
L ∼ C1 (−t)2p ζ̇ 2 − C2 (−t)2q (∇ζ)

(8.3.1)

where C1 , C2 , p, and q are constants satisfying C1 , C2 > 0 and 1 − p + q > 0. The
Lagrangian for the tensor perturbations hij is of the same form.

8.3.1

Tensor Perturbations

The quadratic action for tensor perturbations in the Horndeski theory is given in eq.(4.5.4),
where GT and FT were already defined in eqs. (8.1.14) and (8.2.3). In the previous models
of generalized Galilean Genesis, as well as in conventional models of inflation (in Einstein
gravity), we have GT , FT ≃ const, giving
hij ∼ const and decaying solution,(8.3.2)on superhorizon scales. The amplitude of the dominant constant mode is proportional to
H at horizon crossing, leading to a slightly red spectrum in the case of inflation and astrongly blue spectrum in NEC-violating cosmologies such as Galilean Genesis.
In our new models of Galilean Genesis, we still have a ≃ 1. However, now GT and
FT are strongly time-dependent, as shown in eqs. (8.1.15) and (8.2.4). Noting that GT ∝(−t)2β , we obtain two independent solutions on superhorizon scales,∫ tdt′hij ∼ const and∼ (−t)1−2β .
(8.3.3)a3 GTThis indicates that, while we have constant and decaying solutions as usual for β < 1/2,
for β > 1/2 the would-be decaying mode grows on superhorizon scales. This is in sharp
contrast to the previous Genesis models.
This peculiar evolution of the tensor perturbations for β > 1/2 can be explained in atransparent manner by moving to the “Einstein frame” for the gravitons. Performing adisformal (and conformal) transformation,3−2ea = MPlFT GT a,(8.3.4)−2 3/4 −1/4
det = MPlFT GT dt,

(8.3.5)

1/4 1/4

3

It was shown in Ref. [94] that in cosmology a pure disformal transformation is equivalent to rescaling
the time coordinate.

106

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

the action (7.2.1) can be recast into the standard form [95],
[
]
2 ∫
MPl
(2)
⃗ ij )2 ,Sh E =
detd3 x ea3 (∂et hij )2 − ea−2 (∇h8(8.3.6)

where the scale factor in this “Einstein frame” readsea ∼ (−et )(n+1)/3(−∞ < et < 0)(8.3.7)

with n > 0 for β > 1/2. Clearly, this is the scale factor of a contracting universe where thetensor perturbations are eﬀectively living. This is the reason for the superhorizon growthof hij . It is worth emphasizing that the frame we have moved to is the Einstein frame
only for the gravitons. This frame is not convenient for studying and interpreting thebackground dynamics and the evolution of the curvature perturbation; it was introduced
just to understand the evolution of the tensor perturbations.
Growing tensor perturbations on superhorizon scales imply that the anisotropic shear
also grows. Indeed, it can be seen that β̇± in Sec. 8.2.2 and ḣij share the same timedependence, ∝ (−t)−2β . Nevertheless, this does not spoil the Genesis scenario because the
Hubble rate grows faster provided that α > −1/2, as discussed in Sec. 8.2.2.
The power spectrum of the tensor perturbations is dependent not only on GT but also
on FT , and from eq. (8.2.4) one finds two distinct cases depending on whether 2B ′ (Y0 ) +Y0 B ′′ (Y0 ) vanishes or not. Both cases yield the power spectrum of the form
Ph = AT k nt .

(8.3.8)Since the explicit expression for AT is messy, we do not give it here. Based on a concrete
example we will evaluate AT in the next section. To see the spectral index, let us firstconsider the case of 2B ′ (Y0 )+Y0 B ′′ (Y0 ) = 0. In this case, we have FT = 2e−2βλϕ Y0 A′ (Y0 ) ∝(−t)2β . The spectral index is dependent only on the parameter β and is given by
nt = 3 − 2|ν|with ν :=

1
− β,
2

(8.3.9)where the constant mode is dominant for ν > 0, while the would-be decaying mode growsfor ν < 0. The flat spectrum is obtained for β = −1, 2. In the case of 2B ′ (Y0 )+Y0 B ′′ (Y0 ) ̸=
0 we have FT ∝ (−t)2(α+2β) , so that nt is determined from the two parameters α and β asnt = 3 − 2|ν|with ν :=1 − 2β.
2(α + β + 1)

(8.3.10)

The flat spectrum is obtained for 3α + 5β + 2 = 0 and 3α + β + 4 = 0, though the lattercase is not allowed under the conditions α + β > 0 and α > −1/2.
In the previous study, we typically have blue spectra for tensor perturbations in NECviolating alternatives to inflation. However, we have confirmed that the spectra can also
be flat and red, depending on the parameters, in our variants of Galilean Genesis.

8.3. POWER SPECTRA

8.3.2

107

Curvature Perturbation

The quadratic action for the curvature perturbation is eq.(4.5.7), whereΣGT2+ 3GT ,
Θ2 ()
1 d aGT2FS :=− FT ,
a dt
Θ
GS :=

(8.3.11)(8.3.12)and in the present class of Genesis models Σ and Θ are given byΣ ≃ e2(1+α)λϕ Y0 ρ̂′ (Y0 ) ∝ (−t)−2(α+1) ,Θ ≃ −e(8.3.13)

Y0 ϕ̇g3′
−2βλϕ

2αλϕ

− 2e

HY0 (3A′ + 12Y0 A′′ + 4Y02 A′′′ )+ e−2(1+α+2β)λϕ H 2 Y0 ϕ̇(30B ′ + 75Y0 B ′′ + 36Y02 B ′′′ + 4Y03 B (4) )∝ (−t)−(2α+1) .

(8.3.14)Thus, we have again two distinct cases and if ρ̂′ (Y0 ) = 0 we find thatGS ≃ 3GT ∝ (−t)2β ,(8.3.15)

while if ρ̂′ (Y0 ) ̸= 0 we obtain
GS ≃

ΣGT2
∝ (−t)2(α+2β) .
Θ2Irrespective of whether FT ∝ (−t)2β or ∝ (−t)2(α+2β) , we have( 2)GTF S ≃ ∂t
− FT ∝ (−t)2(α+2β) .
Θ(8.3.16)(8.3.17)

It is easy to evaluate the power spectrum of the curvature perturbation,Pζ = AS k ns −1 .

(8.3.18)

Again, the explicit expression for AS turns out to be messy. Therefore, AS will be evaluatedthrough a concrete example in the next section and here we focus only on the spectral
index. In the special case of ρ̂′ (Y0 ) = 0, we obtain the spectral index
ns − 1 = 3 − 2|ν|with ν :=1 − 2β,2(α + β + 1)

(8.3.19)

which shares the same expression as eq. (8.3.10). The spectrum is therefore scale invariant
for 3α + 5β + 2 = 0. In the general case of ρ̂′ (Y0 ) ̸= 0 we havens − 1 = 3 − 2|ν| with ν :=1
− α − 2β,2(8.3.20)

108

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

Figure 8.1: Tensor and scalar tilts as functions of α and β in models with 2B ′ (Y0 ) +Y0 B ′′ (Y0 ) = 0 and ρ̂′ (Y0 ) = 0, plotted in a viable parameter range, α + β > 0 and
α > −1/2. The thick solid line shows the parameters giving a scale-invariant spectrum of
the curvature perturbation, ns = 1. The dashed lines correspond to scale-invariant tensor
perturbations, nt = 0, and the red (blue) region represents the parameters for which thetensor spectrum is red (blue). For a nearly scale-invariant scalar spectrum, ns ≈ 1, only
a red tensor spectrum is obtained in this class of models.

Figure 8.2: Same as Fig. 8.1, but for models with 2B ′ (Y0 ) + Y0 B ′′ (Y0 ) ≠ 0 and ρ̂′ (Y0 ) = 0.
In this case, nt = ns − 1, and hence the lines giving scale-invariant tensor and scalar
spectra coincide. For ns = 0.96, the tensor tilt is given by nt = 0.96 − 1 = −0.04.

8.3. POWER SPECTRA

109

Figure 8.3: Same as Fig. 8.1, but for models with 2B ′ (Y0 ) + Y0 B ′′ (Y0 ) = 0 and ρ̂′ (Y0 ) ̸= 0.
Both tensor and scalar spectra are scale invariant for (α, β) = (1, −1), (4, −1), though theformer is located at the edge of the viable parameter range. For a nearly scale-invariant
scalar spectrum, ns ≈ 1, both red and blue tensor spectra are possible in this class ofmodels.

Figure 8.4: Same as Fig. 8.1, but for models with 2B ′ (Y0 ) + Y0 B ′′ (Y0 ) ≠ 0 and ρ̂′ (Y0 ) ̸= 0.
For a nearly scale-invariant scalar spectrum, ns ≈ 1, the tensor spectrum is always bluein this class of models.

110

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

and the spectrum is scale invariant for the parameters satisfying α + 2β + 1 = 0 or
α + 2β − 2 = 0.
Having thus obtained the spectral indices nt and ns , we summarize the results in
Figs. 8.1–8.4. Of particular interest are the cases presented in Figs. 8.2 and 8.3. In theformer case both scalar and tensor perturbations have nearly scale-invariant spectra for
3α + 5β + 2 ≃ 0, while in the latter case this is possible for α ≃ 4 and β ≃ −1. In theother two cases, i.e., the cases given in Figs. 8.1 and 8.4, the parameters leading to scaleinvariant scalar and tensor perturbations are on the boundaries of the viable parameterregions.
Before closing this section, let us comment on the stability of the Genesis solutions. By
requiring that GT , FT , GS , FS > 0, one can obtain a stable Genesis phase. However, as
shown in [96, 97], non-singular cosmological solutions in the Horndeski theory are plagued
with gradient instabilities which occur at some moment in the entire expansion history,
provided that the integrals
∫ t
∫ ∞
′
aFT dt andaFT dt′(8.3.21)−∞t

do not converge. The Genesis models with FT ∼ (−t)n (n ≥ −1) satisfies the postulatesof this no-go theorem, and hence, even though a single genesis phase itself is stable, gradient instability occurs eventually after the Genesis phase. The models with FT ∼ (−t)n(n < −1) can evade the no-go theorem [97], but then the universe would be geodesically incomplete for gravitons [98]. If one would prefer a geodesically complete universe
for gravitons, some new terms beyond Horndeski must be introduced to avoid gradientinstabilities [98, 61, 79, 99].

8.4

An example

As a concrete example, let us focus on the case with B(Y ) = 0, α = 4, and β = −1,
which gives rise to exactly scale-invariant spectra for scalar and tensor perturbations. Our
example is given by
g2 = −Y +

Y2
,
µ4

[

(

and
A=M

2

Y
−
µ4

Y
µ4

g3 =

)2

2
+
5

Y
,
8λµ4
(

Y
µ4

(8.4.1))3 ],(8.4.2)where µ and M are parameters having dimension of mass, and it follows from eq. (8.4.1)
that Y0 = 2µ4 /3.

8.5. CONCLUSIONS IN THIS CHAPTER111

One can solve the background equations to obtain√
5 3 µ2 7λϕH=e∝ (−t)−7 .
56 λM 2(8.4.3)

It is straightforward to compute
4
GT = M 2 e2λϕ ,94FT = M 2 e2λϕ ,1549 2 4 −4λϕGS = λ M e,7270
FS = λ2 M 4 e−4λϕ ,27(8.4.4)

which shows that this model is stable. The primordial power spectra of tensor and curvature perturbations are given, respectively, by√
λ 2 µ45 10λ2 µ4
Ph =
≃
1.3
×
,
(8.4.5)
3 π2M 2M2
andPζ ≃ 0.017 × (λH∗ )6/7

( µ )16/7
M,(8.4.6)

where H∗ is the Hubble parameter at the end of the genesis phase. Note that the curvature
perturbation grows on superhorizon scales and hence Pζ depends on the time when the
Genesis phase ends, while tensor perturbations do not. The tensor-to-scalar ratio has a
non-standard expression (i.e., it does not depend on nt or the slow-roll parameter) andreadsr ∼ 10−2 × (λH∗ )−6/7 (λµ)12/7 (λM )2/7 ,(8.4.7)

which can be made suﬃciently small by choosing the parameters.
One can improve the above model by introducing slight deviations from α = 4 andβ = −1 to have ns ≃ 0.96. The lesson we learn from this example is that it is rather
easy to construct a stable model of Galilean Genesis generating primordial curvature
perturbations that are consistent with observations and tensor perturbations that can be
hopefully detected by future observations.

8.5

Conclusions in this chapter

In this chapter, we have proposed a variant of generalized Galilean Genesis as a possible alternative to inflation. A general Lagrangian for this new class of models has been

112

CHAPTER 8. SCALE-INVARIANT PERTURBATIONS

constructed within the Horndeski theory. The Lagrangian has four functional degrees of
freedom in addition to two constant parameters, and includes the model studied in Ref. [93]as a specific case. We have confirmed that under certain conditions the background evolution of our Genesis models leads to a stable, homogeneous and isotropic universe with flat
spatial sections. We have then calculated power spectra of primordial perturbations andshown that a variety of tensor and scalar spectral tilts can be obtained, as summarized in
Figs. 8.1–8.4. In some cases, curvature/tensor perturbations grow on superhorizon scales
and for this reason the primordial amplitudes depend not only on the functions in the
Lagrangian but also on the time when the Genesis phase ends. It should be emphasizedthat in spite of the gross violation of the null energy condition the tensor spectrum canbe (nearly) scale invariant, though the consistency relation is still non-standard.
We have thus seen that in the Galilean Genesis scenario both scalar and tensor power
spectra can be nearly scale invariant as in the standard inflationary scenario. It is therefore
crucial to evaluate the amount of non-Gaussianities in the primordial curvature perturbations produced during the Genesis phase. This point will be reported elsewhere.

Conclusions
We have reviewed the history and the fundamental information of cosmology in part 1
and discuss many topics of the generalized Galilean genesis as an alternative scenario toinflation in part 2. These studies aimed to distinguish the models of the early universe,
as we have seen in fig.4.1, fig.5.2 and fig.5.3, the evolution of the universe in the model
is diﬀerent each other. However, as a matter of course, we can not see this evolution
directly, and thus we have discussed how can we see the diﬀerence between the models
from observations.
We know the gravitational waves in alternative scenarios shows the blue spectrum sofar, however, we have found what kind of genesis models can generate the flat spectrum andother various spectrums. This result suggests we can not reject the alternative scenarios
if we detect the flat spectrum in the future, but the consistency relation can distinguishbetween the models.
The other way to distinguish between the models, we should consider the other cosmological prediction such as non-gaussianity or find the instability in specifically modified
gravity theories. For example, in the class of Horndeski theory, the scenario without initial
singularity have the instability [97, 100]. This study suggests the genesis model has theinstability, and thus we have to consider the genesis scenario in beyond Horndeski theory
to avoid this instability. The other remarkable point except for the distinction of the models is the anisotropy from genesis. In part 2, we have discussed the anisotropy and derived
the condition not to evolve the anisotropy. This suggests the possibility to construct themodel of the anisotropic inflation in the Horndeski theory without vector field.

113

Bibliography[1] Paolo Creminelli, Alberto Nicolis, and Enrico Trincherini. Galilean Genesis: An
Alternative to inflation. JCAP, 1011:021, 2010.
[2] Sakine Nishi and Tsutomu Kobayashi. Reheating and Primordial Gravitational
Waves in Generalized Galilean Genesis. JCAP, 1604(04):018, 2016.
[3] Sakine Nishi and Tsutomu Kobayashi. Scale-invariant perturbations from NEC
violation: A new variant of Galilean Genesis. Phys. Rev., D95(6):064001, 2017.
[4] Sakine Nishi and Tsutomu Kobayashi.
1503(03):057, 2015.

Generalized Galilean Genesis.

JCAP,[5] Edwin Hubble. A relation between distance and radial velocity among extra-galactic
nebulae. Proc. Nat. Acad. Sci., 15:168–173, 1929.
[6] R. Adam et al. Planck 2015 results. I. Overview of products and scientific results.
Astron. Astrophys., 594:A1, 2016.
[7] Alan H. Guth. The Inflationary Universe: A Possible Solution to the Horizon and
Flatness Problems. Phys. Rev., D23:347–356, 1981.
[8] Steven Weinberg. Cosmology. 2008.
[9] Andrew R. Liddle, Paul Parsons, and John D. Barrow. Formalizing the slow rollapproximation in inflation. Phys. Rev., D50:7222–7232, 1994.
[10] Andrei D. Linde. Chaotic Inflation. Phys. Lett., 129B:177–181, 1983.
[11] V. Mukhanov. Physical Foundations of Cosmology. Cambridge University Press,
Oxford, 2005.
[12] Alexei A. Starobinsky. A New Type of Isotropic Cosmological Models Without
Singularity. Phys. Lett., B91:99–102, 1980.
[13] Takeshi Chiba, Takahiro Okabe, and Masahide Yamaguchi.
quintessence. Phys. Rev., D62:023511, 2000.
115

Kinetically driven

116

BIBLIOGRAPHY[14] C. Armendariz-Picon, T. Damour, and Viatcheslav F. Mukhanov. k - inflation. Phys.
Lett., B458:209–218, 1999.
[15] Tsutomu Kobayashi, Masahide Yamaguchi, and Jun’ichi Yokoyama. G-inflation:
Inflation driven by the Galileon field. Phys. Rev. Lett., 105:231302, 2010.
[16] Tsutomu Kobayashi, Masahide Yamaguchi, and Jun’ichi Yokoyama. Generalized Ginflation: Inflation with the most general second-order field equations. Prog. Theor.
Phys., 126:511–529, 2011.
[17] Gregory Walter Horndeski. Second-order scalar-tensor field equations in a fourdimensional space. Int. J. Theor. Phys., 10:363–384, 1974.
[18] C. Deﬀayet, Xian Gao, D. A. Steer, and G. Zahariade. From k-essence to generalised
Galileons. Phys. Rev., D84:064039, 2011.
[19] Hans A. Buchdahl. Non-linear Lagrangians and cosmological theory. Mon. Not.
Roy. Astron. Soc., 150:1, 1970.
[20] C. Brans and R. H. Dicke. Mach’s principle and a relativistic theory of gravitation.
Phys. Rev., 124:925–935, 1961.
[21] Antonio De Felice and Shinji Tsujikawa. f(R) theories. Living Rev. Rel., 13:3, 2010.
[22] Cedric Deﬀayet, Oriol Pujolas, Ignacy Sawicki, and Alexander Vikman. Imperfect
Dark Energy from Kinetic Gravity Braiding. JCAP, 1010:026, 2010.
[23] Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini. The Galileon as a local
modification of gravity. Phys. Rev., D79:064036, 2009.
[24] Nathan Chow and Justin Khoury. Galileon Cosmology. Phys. Rev., D80:024037,
2009.
[25] C. Deﬀayet, Gilles Esposito-Farese, and A. Vikman. Covariant Galileon. Phys. Rev.,
D79:084003, 2009.
[26] C é dric Deﬀayet and Dani è le A. Steer. A formal introduction to Horndeski andGalileon theories and their generalizations. Class. Quant. Grav., 30:214006, 2013.
[27] Jérôme Gleyzes, David Langlois, Federico Piazza, and Filippo Vernizzi. Healthy
theories beyond Horndeski. Phys. Rev. Lett., 114(21):211101, 2015.
[28] Xian Gao. Unifying framework for scalar-tensor theories of gravity. Phys. Rev.,
D90:081501, 2014.
[29] Viatcheslav Mukhanov and Sergei Winitzki. Introduction to quantum eﬀects in gravity. Cambridge University Press, 2007.

BIBLIOGRAPHY117[30] David H. Lyth, Carlo Ungarelli, and David Wands. The Primordial density perturbation in the curvaton scenario. Phys. Rev., D67:023503, 2003.
[31] Taro Kunimitsu and Jun’ichi Yokoyama. Higgs condensation as an unwanted curvaton. Phys. Rev., D86:083541, 2012.
[32] V. A. Rubakov. The Null Energy Condition and its violation. Phys. Usp., 57:128–
142, 2014. [Usp. Fiz. Nauk184,no.2,137(2014)].
[33] M. Novello and S. E. Perez Bergliaﬀa. Bouncing Cosmologies. Phys. Rept., 463:127–
213, 2008.
[34] Robert Brandenberger and Patrick Peter. Bouncing Cosmologies: Progress andProblems. 2016.
[35] Yi-Fu Cai, Evan McDonough, Francis Duplessis, and Robert H. Brandenberger. Two
Field Matter Bounce Cosmology. JCAP, 1310:024, 2013.
[36] Paolo Creminelli, Kurt Hinterbichler, Justin Khoury, Alberto Nicolis, and Enrico
Trincherini. Subluminal Galilean Genesis. JHEP, 02:006, 2013.
[37] Kurt Hinterbichler, Austin Joyce, Justin Khoury, and Godfrey E. J. Miller. DBI Realizations of the Pseudo-Conformal Universe and Galilean Genesis Scenarios. JCAP,
1212:030, 2012.
[38] Kurt Hinterbichler, Austin Joyce, Justin Khoury, and Godfrey E. J. Miller. DiracBorn-Infeld Genesis: An Improved Violation of the Null Energy Condition. Phys.
Rev. Lett., 110(24):241303, 2013.
[39] Robert H. Brandenberger and C. Vafa. Superstrings in the Early Universe. Nucl.
Phys., B316:391–410, 1989.
[40] Alexei A. Starobinsky. Spectrum of relict gravitational radiation and the earlystate of the universe. JETP Lett., 30:682–685, 1979. [Pisma Zh. Eksp. Teor.
Fiz.30,719(1979)].
[41] K. Sato. First Order Phase Transition of a Vacuum and Expansion of the Universe.
Mon. Not. Roy. Astron. Soc., 195:467–479, 1981.
[42] Viatcheslav F. Mukhanov and G. V. Chibisov. Quantum Fluctuations and a
Nonsingular Universe. JETP Lett., 33:532–535, 1981. [Pisma Zh. Eksp. Teor.
Fiz.33,549(1981)].
[43] D. Larson et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Power Spectra and WMAP-Derived Parameters. Astrophys. J. Suppl.,
192:16, 2011.

118

BIBLIOGRAPHY[44] P. A. R. Ade et al. Planck 2013 results. XXII. Constraints on inflation. Astron.
Astrophys., 571:A22, 2014.
[45] P. A. R. Ade et al. Planck 2013 results. XVI. Cosmological parameters. Astron.
Astrophys., 571:A16, 2014.
[46] Arvind Borde and Alexander Vilenkin. Singularities in inflationary cosmology: A
Review. Int. J. Mod. Phys., D5:813–824, 1996.
[47] D. Battefeld and Patrick Peter. A Critical Review of Classical Bouncing Cosmologies.
Phys. Rept., 571:1–66, 2015.
[48] Ignacy Sawicki and Alexander Vikman. Hidden Negative Energies in Strongly Accelerated Universes. Phys. Rev., D87(6):067301, 2013.
[49] Paolo Creminelli, Markus A. Luty, Alberto Nicolis, and Leonardo Senatore. Starting
the Universe: Stable Violation of the Null Energy Condition and Non-standard
Cosmologies. JHEP, 12:080, 2006.
[50] Laurence Perreault Levasseur, Robert Brandenberger, and Anne-Christine Davis.
Defrosting in an Emergent Galileon Cosmology. Phys. Rev., D84:103512, 2011.
[51] Yi Wang and Robert Brandenberger. Scale-Invariant Fluctuations from Galilean
Genesis. JCAP, 1210:021, 2012.
[52] V. A. Rubakov. Consistent NEC-violation: towards creating a universe in the laboratory. Phys. Rev., D88:044015, 2013.
[53] Benjamin Elder, Austin Joyce, and Justin Khoury. From Satisfying to Violating theNull Energy Condition. Phys. Rev., D89(4):044027, 2014.
[54] Damien A. Easson, Ignacy Sawicki, and Alexander Vikman. When Matter Matters.
JCAP, 1307:014, 2013.
[55] Taotao Qiu, Jarah Evslin, Yi-Fu Cai, Mingzhe Li, and Xinmin Zhang. Bouncing
Galileon Cosmologies. JCAP, 1110:036, 2011.
[56] Damien A. Easson, Ignacy Sawicki, and Alexander Vikman. G-Bounce. JCAP,
1111:021, 2011.
[57] Yi-Fu Cai, Damien A. Easson, and Robert Brandenberger. Towards a Nonsingular
Bouncing Cosmology. JCAP, 1208:020, 2012.
[58] Yi-Fu Cai, Robert Brandenberger, and Patrick Peter. Anisotropy in a Nonsingular
Bounce. Class. Quant. Grav., 30:075019, 2013.

BIBLIOGRAPHY

119[59] M. Osipov and V. Rubakov. Galileon bounce after ekpyrotic contraction. JCAP,
1311:031, 2013.
[60] Taotao Qiu, Xian Gao, and Emmanuel N. Saridakis. Towards anisotropy-freeand nonsingular bounce cosmology with scale-invariant perturbations. Phys. Rev.,
D88(4):043525, 2013.
[61] David Pirtskhalava, Luca Santoni, Enrico Trincherini, and Patipan Uttayarat. Inflation from Minkowski Space. JHEP, 12:151, 2014.
[62] Sakine Nishi, Tsutomu Kobayashi, Norihiro Tanahashi, and Masahide Yamaguchi.
Cosmological matching conditionsand galilean genesis in Horndeski’s theory. JCAP,
1403:008, 2014.
[63] Zhi-Guo Liu, Jun Zhang, and Yun-Song Piao. A Galileon Design of Slow Expansion.
Phys. Rev., D84:063508, 2011.
[64] Robert M. Wald. Asymptotic behavior of homogeneous cosmological models in thepresence of a positive cosmological constant. Phys. Rev., D28:2118–2120, 1983.
[65] Kerstin E. Kunze and Ruth Durrer. Anisotropic ’hairs’ in string cosmology. Class.
Quant. Grav., 17:2597–2604, 2000.
[66] Joel K. Erickson, Daniel H. Wesley, Paul J. Steinhardt, and Neil Turok. Kasnerand mixmaster behavior in universes with equation of state w ¿= 1. Phys. Rev.,
D69:063514, 2004.
[67] BingKan Xue and Paul J. Steinhardt. Evolution of curvature and anisotropy near anonsingular bounce. Phys. Rev., D84:083520, 2011.
[68] Jai-chan Hwang and Ethan T. Vishniac. Gauge-invariant joining conditions forcosmological perturbations. Astrophys. J., 382:363–368, 1991.
[69] Nathalie Deruelle and Viatcheslav F. Mukhanov. On matching conditions for cosmological perturbations. Phys. Rev., D52:5549–5555, 1995.
[70] Ruth Durrer and Filippo Vernizzi. Adiabatic perturbations in pre - big bang models:
Matching conditions and scale invariance. Phys. Rev., D66:083503, 2002.
[71] Yun-Song Piao. Adiabatic Spectra During Slowly Evolving. Phys. Lett., B701:526–
529, 2011.
[72] Arvind Borde and Alexander Vilenkin. Eternal inflation and the initial singularity.
Phys. Rev. Lett., 72:3305–3309, 1994.

120

BIBLIOGRAPHY[73] Robert H. Brandenberger. Alternatives to the inflationary paradigm of structureformation. Int. J. Mod. Phys. Conf. Ser., 01:67–79, 2011.
[74] Robert H. Brandenberger. Cosmology of the Very Early Universe. AIP Conf. Proc.,
1268:3–70, 2010.
[75] Robert H. Brandenberger.
ICFI2010:001, 2010.

Introduction to Early Universe Cosmology.

PoS,[76] Jaume de Haro and Yi-Fu Cai. An Extended Matter Bounce Scenario: current status
and challenges. Gen. Rel. Grav., 47(8):95, 2015.
[77] L. H. Ford. Gravitational Particle Creation and Inflation. Phys. Rev., D35:2955,
1987.
[78] Zhi-Guo Liu, Hong Li, and Yun-Song Piao. Preinflationary genesis with CMB Bmode polarization. Phys. Rev., D90(8):083521, 2014.
[79] Tsutomu Kobayashi, Masahide Yamaguchi, and Jun’ichi Yokoyama. Galilean Creation of the Inflationary Universe. JCAP, 1507(07):017, 2015.
[80] Hiroyuki Tashiro, Takeshi Chiba, and Misao Sasaki. Reheating after quintessential
inflation and gravitational waves. Class. Quant. Grav., 21:1761–1772, 2004.
[81] N. D. Birrell and P. C. W. Davies. Quantum Fields in Curved Space. Cambridge
Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK,
1984.
[82] Leonard Parker. Quantized fields and particle creation in expanding universes. 1.
Phys. Rev., 183:1057–1068, 1969.
[83] Ya. B. Zeldovich and Alexei A. Starobinsky. Particle production and vacuum polarization in an anisotropic gravitational field. Sov. Phys. JETP, 34:1159–1166, 1972.
[Zh. Eksp. Teor. Fiz.61,2161(1971)].
[84] Ya. B. Zeldovich and Alexei A. Starobinsky. O skorosti rozhdeniya chastits v
gravitatsionnykh polyakh,. Pisma Zh. Eksp. Toer. Fiz., 26:373, 1977. [On the rateof particle production in gravitational fields, JETP Lett. 26 (1977) 252].
[85] Michele Maggiore. Gravitational wave experiments and early universe cosmology.
Phys. Rept., 331:283–367, 2000.
[86] George F. R. Ellis and Roy Maartens. The emergent universe: Inflationary cosmology
with no singularity. Class. Quant. Grav., 21:223–232, 2004.

BIBLIOGRAPHY

121[87] George F. R. Ellis, Jeﬀ Murugan, and Christos G. Tsagas. The Emergent universe:
An Explicit construction. Class. Quant. Grav., 21(1):233–250, 2004.
[88] Yi-Fu Cai, Mingzhe Li, and Xinmin Zhang. Emergent Universe Scenario via Quintom
Matter. Phys. Lett., B718:248–254, 2012.
[89] Yi-Fu Cai, Youping Wan, and Xinmin Zhang. Cosmology of the Spinor Emergent
Universe and Scale-invariant Perturbations. Phys. Lett., B731:217–226, 2014.
[90] Thorsten Battefeld and Scott Watson. String gas cosmology. Rev. Mod. Phys.,
78:435–454, 2006.
[91] Ido Ben-Dayan. Gravitational Waves in Bouncing Cosmologies from Gauge Field
Production. JCAP, 1609(09):017, 2016.
[92] Asuka Ito and Jiro Soda. Primordial Gravitational Waves Induced by Magnetic
Fields in Ekpyrotic Scenario. 2016.
[93] Yong Cai and Yun-Song Piao. The slow expansion with nonminimal derivative
coupling and its conformal dual. JHEP, 03:134, 2016.
[94] Guillem Domènech, Atsushi Naruko, and Misao Sasaki. Cosmological disformal
invariance. JCAP, 1510(10):067, 2015.
[95] Paolo Creminelli, Jérôme Gleyzes, Jorge Noreña, and Filippo Vernizzi. Resilience of the standard predictions for primordial tensor modes. Phys. Rev. Lett.,
113(23):231301, 2014.
[96] M. Libanov, S. Mironov, and V. Rubakov. Generalized Galileons: instabilities ofbouncing and Genesis cosmologies and modified Genesis. JCAP, 1608(08):037, 2016.
[97] Tsutomu Kobayashi. Generic instabilities of nonsingular cosmologies in Horndeski
theory: A no-go theorem. Phys. Rev., D94(4):043511, 2016.
[98] Paolo Creminelli, David Pirtskhalava, Luca Santoni, and Enrico Trincherini. Stability of Geodesically Complete Cosmologies. JCAP, 1611(11):047, 2016.
[99] Yong Cai, Youping Wan, Hai-Guang Li, Taotao Qiu, and Yun-Song Piao. The
Eﬀective Field Theory of nonsingular cosmology. JHEP, 01:090, 2017.
[100] Shingo Akama and Tsutomu Kobayashi. Generalized multi-Galileons, covariantized
new terms, and the no-go theorem for nonsingular cosmologies. Phys. Rev.,
D95(6):064011, 2017.