Doctoral Thesis

Anti-Screening of the Galileon Force
Around a Disk Center Hole
ディスク中心の穴の周りにおける
ガリレオン力の反遮蔽効果

Hiromu Ogawa

Department of Physics, Graduate School of Science,
Rikkyo University

ii

Anti-Screening of the Galileon Force Around a
Disk Center Hole
Hiromu Ogawa

General relativity has passed a number of observational and experimental tests in the solar system and is now the most widely accepted andtheoretically simple theory of gravity. The standard cosmological model
based on general relativity is also highly compatible with cosmologicalobservations. General relativity is now considered to be one of the pillars
of modern physics.
Nevertheless, the origin of the current accelerated expansion of the
Universe remains one of the biggest mysteries in modern physics. In
order to explain this phenomenon in the context of general relativity, it isnecessary to introduce an unknown energy source, so-called dark energy.
The simplest possibility for dark energy is a cosmological constant.
However, the theoretically expected value of such a cosmological constantis much larger than the observed values of dark energy. The alternative
approach is that acceleration of universe is driven by modifications togeneral relativity. Modified gravity has been investigated intensively as
an alternative dark energy model.
Many modified gravity theories can be described effectively by addinga scalar degree of freedom into the two tensor modes of general relativity.
The simplest extensions to general relativity are generally called scalartensor theories wherein the presence of the scalar mediates a fifth force.
These models are strongly constrained by tests performed in the solar
system. If such a theory is equipped with a mechanism to suppress thisfifth force near concentrations of matter, the theory naturally evades theivconstraints coming from the solar system observations. This mechanism
is called a screening mechanism.
In this thesis, we investigate aspects of the screening mechanism inGalileon theory. Galileon theory arises as a four-dimensional effective
field theory among modified gravities. Despite the presence of thederivative self-interactions of the scalar field in the Galileon Lagrangian,the equation of motion remains a second-order equation. The presenceof the non-linear self-interaction term allows us to hide the fifth force:the Galileon theory is equipped with the Vainshtein mechanism, which
restores general relativity in the vicinity of concentrations of matter.
Therefore, the Galileon theory is well studied in the context of cosmology
and astrophysics as a model for dark energy. In this thesis, we focus on
cubic Galileon theory which is not excluded by the recent observation ofgravitational waves coming from a binary-neutron star merger. Cubic
Galileon also can be viewed as the simplest theory having the nonlinear
term. Therefore, we study the aspects of cubic Galileon theory and try
to reveal its nature.
The Vainshtein mechanism has been verified mainly in highly symmetric configurations of matter. We study the Vainshtein mechanism ina less symmetric setup in cubic Galileon theory. We numerically solve
the scalar field equation around a disk with a hole at its center in cubic
Galileon theory and find that the Galileon force is enhanced, rather
than suppressed, in the vicinity of the hole. This anti-screening effect islarger for a thinner, less massive disk with a smaller hole.
We investigate the cubic Galileon theory which can be viewed as thesimplest theory having the nonlinear term. It is therefore interesting tostudy whether or not the anti-screening effect occurs in the quartic andquintic Galileon models. This is a very important open problem which
we hope to study in the near future.

v

Acknowledgements
First and foremost, I would like to thank my supervisor, Tsutomu Kobayashi. He
always took a lot of his time to discuss with me. His well-directed comments and tips
helped me many times. My results and my thesis would not have been possible without
his help, advice about using techniques, ideas, and insight. I thank him for giving me
freedom and choices to study and find things my way. I especially appreciate him being
supportive of me to work with other researchers.
I should also thank two very important people Takashi Hiramatsu and Teruaki Suyama
who helped me many times with questions and works relating to my thesis. I am deeply
grateful to them for helping me with their ideas and contributing their analysis andcalculations so I could carry out my thesis.
I also thank Tomohiro Harada for his well-directed guidance and providing me withmuch theoretical knowledge, especially concerning general relativity and astrophysics.
He was always there for me whenever I had any questions or problems.
I had an excellent opportunity to join the University of Portsmouth as a visitor
student for four months. I thank Kazuya Koyama and David Wands for making thisvisit possible and supporting me during my stay at the University of Portsmouth. I
especially thank Kazuya Koyama for being my advisor and appreciate his hospitality at
the University of Portsmouth. I have had many great discussions with him. It was an
excellent time working with him.
I sincerely thank Shuichiro Yokoyama, Kazufumi Takahashi, Sanjay Jhingan, Emir
Gumrukcuoglu, and Takahisa Igata who guided and helped me many times with wonderful,interesting ideas and techniques concerning physics, especially general relativity, modified
gravity, and cosmology.
Kumiko Inagawa has always been helpful with any (troublesome) administrative
issues. I am grateful to her for providing the conditions that allowed me to carry out

vi

my research. It would have been hard to have finished this thesis without her helpful
support and effort.
I made a lot of friends outside my department who are essential to me and I would not
be able to imagine my life here without them. I am very grateful to Taishi Katsuragawa,Yuichiro Tada, Katsuki Aoki, Takahiro Hayashinaka, Mao Iwasa, Ippei Obata, Tomotaka
Kitamura, Tomoya Kinugawa, Sirachak Panpanich, Taisaku Mori, Seiga Sato, Keigo
Shimada, Shotaro Oka, Sakine Nishi, Tomo Tanaka, Kohji Yajima, Takafumi Kokubu,
Masataka Watanabe, and Susumu Hirakawa for encouraging comments and valuableadvice concerning my work, and also for making me a better person. I had so much fun
being able to spend a wonderful time together with you.
I am very grateful to the people in the Institute of Cosmology and Gravitation,
University of Portsmouth: Max Foxley-Marrable, Pierandrea Guarnieri, Robert Hardwick,
Natalie Hogg, Michael Kenna-Allison, Sam Lawrence, Tim Lingard, Ben Mawdsley, Taniya
Parikh, Christopher Pattison, Marco Surace, Elizabeth Swann, Andrius Tamosiunas,
Benjamin Thomas, Mike Wang, Matthew Withers, Bill Wright, and Paul Carter. Thankyou for your friendship, kindness, warmth, and support during my stay at the Universityof Portsmouth. Furthermore, I would like to express my gratitude to Matthew Hull andKanae Hull who helped and supported me many times during my stay at Portsmouth.
Further, I would like to express my gratitude to the members in the theoreticalphysics group at Rikkyo University. I especially thank Kota Ogasawara, Yasutaka Koga,
Shin’ichi Hirano, Shingo Akama, Aya Iyonaga, Keisuke Nakashi, Yosuke Mishima, Keitaro
Tomikawa, Yuko Mori, and Chiaki Nasu for answering all my (stupid) questions every
day and for having many conversations with me (both at university and the bar).
I would like to give a heartfelt thank you to my family, especially my Mum andDad, Michiko and Shigeru, for encouraging and loving me over the years. Without theirsupports and encouragements, this work would have not been possible.
Part of this work was supported by the Rikkyo University Special Fund for Research
and JSPS Overseas Challenge Program for Young Researchers.

Contents
1. Introduction

1

1.1. Dawn of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2. Beyond General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3. Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2. General Relativity

72.1. Newtonian Mechanics to General Relativity . . . . . . . . . . . . . . . .

72.2. General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1. Mathematical Preliminaries-tensor, covariant derivative and curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92.2.2. Formulation of Einstein’s Field Equations . . . . . . . . . . . . .

13

2.3. Cosmological Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.3.1. The FLRW Universe . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3.2. The

18-CDM Model . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.3. Problems in the-CDM Model . . . . . . . . . . . . . . . . . . .

20

2.3.4. Dark Energy Model . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.4. Black Holes and Stellar Solutions in General Relativity . . . . . . . . . .

24

2.4.1. Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.4.2. Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.5. Observational Results and Constraints on General Relativity . . . . . . .

37

2.5.1. Solar System Constraints . . . . . . . . . . . . . . . . . . . . . . .

37

2.5.2. Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . .

42

3. Modified Gravity

49

3.1. Beyond Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.2. Lovelock’s Theorem -Guiding Principles for Modified Gravity- . . . . . .

50

3.3. Theoretical Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.1. Ghost Instability . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.3.2. Gradient Instability . . . . . . . . . . . . . . . . . . . . . . . . . .

54

vii

viii

Contents

3.3.3. Tachyonic Instability . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.4. Ostrogradsky Ghosts . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.4. Scalar-Tensor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.4.1. Brans-Dicke Theory . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.4.2. Generalized Galileon Theory -Horndeski Theory- . . . . . . . . . .

62

3.4.3. Constraints on Horndeski Theory by the Observational Results ofGravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.5. Other Constraints on Modified Gravity . . . . . . . . . . . . . . . . . . .

72

3.6. Further Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.6.1. Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.6.2. Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4. Screening Mechanism

77

4.1. Fifth Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.2. Classification of Screening Mechanism . . . . . . . . . . . . . . . . . . . .

79

4.3. Vainshtein Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.4. Typical Value of the Vainshtein Radius and constraints . . . . . . . . . .

82

4.5. The Shape Dependence of the Vainshtein Mechanism . . . . . . . . . . .

84

5. Anti-Screening of the Galileon Force Around a Disk Center Hole89

5.1. Motivation for this work . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.2. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.2.1. The cubic Galileon . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.2.2. Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.3. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.4. Scalar-field profile in Schwarzschild geometry . . . . . . . . . . . . . . . .

97

6. Conclusions and Outlook
6.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99
99

6.2. Future Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A. Conformal Transformation Rules

103

A.1. Einstein and Jordan frames . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2. Transformation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.3. The Energy-Momentum Tensor and The Conservation Law . . . . . . . . 105

B. f (R) theories of gravity
107
B.1. Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
B.1.1. Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 108
B.1.2. Scalaron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B.2. f (R) theory and Brans-Dicke theory . . . . . . . . . . . . . . . . . . . . 110
B.2.1. f (R) Theory and Scalar-Tensor Theory . . . . . . . . . . . . . . . 111
B.2.2. Chameleon Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 112
B.3. Viable f (R) Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C. Numerical Scheme and Convergence of Results in Chapter 5117

Bibliography

121

List of figures

143

List of tables

145

x

Contents

Chapter 1.
Introduction
1.1. Dawn of Physics
Until the beginning of the 20th century, classical physics, such as classical mechanics,
electromagnetics, thermodynamics, and optics, had been extensively studied and developed. It was believed that all phenomena could be understood in the framework of thisclassical physics.
The 20th century was the starting point for a scientific revolution: general relativity
and quantum field theory began and were developed throughout these years. These
theories created a new paradigm in terms of how we describe nature: general relativity
unifies space and time, while quantum field theory unifies matter and fields. Generalrelativity describes gravity both at local scales such as the solar system and at the large
scale structure of the Universe. Quantum field theory describes phenomena at atomic
scales. Both theories describe physical phenomena well at different scales, and they haveagreed with a number of experiments and observations. Today, these two theories arefundamental to theoretical physics.
General relativity [1, 2] is now widely accepted as the standard theory of gravity. It
describes large-scale physical phenomena such as the evolution of the Universe. This
theory has passed all experimental and observational tests in the solar system [3].
Additionally, the robustness of general relativity has recently been reconfirmed by thedetection of gravitational waves [4, 5]; general relativity does not lose predictability evenin a very strong gravity regime. With this successful background, general relativity is
now considered to be one of the pillars of modern physics. However, there are several
problems that remain unsolved, which will now be described.
1

2

Introduction

The first problem occurs with quantum gravity theory. In the solar system, predictionsof general relativity are in good agreement with the observational data. However, at
the Planck scale, general relativity loses predictability. Significant research attempts
and efforts to unify these theories have failed so far; to our knowledge, the problem of
constructing a quantum gravity theory has not been solved. This is one of the most
severe and interesting problems in modern physics.
The second problem is dark matter. With the assumption of the cosmological principle(that the Universe is isotropic and homogeneous on a large scale and that there is no uniquepoint in the Universe), general relativity can describe the evolution of the Universe. The
precise measurements of the rotation curves of galaxies [6, 7], gravitational lensing [8, 9],
and numerical simulations of the formation of large-scale structures of the Universe [10]have provided evidence of invisible and mysterious matter. This matter does not stronglyinteract with standard matter or light and has not yet been directly observed, which is
why it is called dark matter. The discovery of dark matter is one of the main mysteriesin physics, and it could suggest that a new type of physics is required.
The third problem is an accelerated expansion of the Universe or dark energy. Measurements of type Ia supernovae [11,12], the cosmic microwave background (CMB) [13,14],and baryon acoustic oscillations [15, 16] provide evidence that our Universe is expandingat an accelerating rate. This evidence suggests the existence of an unknown energy
source, namely dark energy. The simplest candidate to account for dark energy would bea cosmological constant. The standard model of cosmology based on general relativity
is widely accepted and known as the -CDM model (where is the symbol for thecosmological constant and CDM is for cold dark matter). The -CDM model is highly
compatible with observations [17, 18]. However, this concordance model suffers from thefine-tuning problem of the cosmological constant; the theoretical value of this constant ismuch larger than its observed value [19].
Revealing the three above-mentioned challenging problems is at the frontier of modern
physics, especially cosmology and particle physics. However, the first and second problems
are beyond the scope of this thesis. In this thesis, general relativity and modifications
thereof are focused on.

Introduction

3

1.2. Beyond General Relativity
The mystery of the accelerated expansion of the Universe remains an unsolved andinteresting problem in our understanding of the Universe. To date, there are some ways
to solve the dark energy problem: i) an unknown mechanism or beyond standard physics
would tune or suppress the theoretical value of a cosmological constant, it is still achallenging problem. ii) one introduces a new exotic component of matter fields, suchas quintessence [20] or k-essence [21, 22]. The matter fields behave like a cosmologicalconstant today. iii) one tries to extend to and modify general relativity. The accelerated
expansion of the Universe is driven by modifications to general relativity. In this thesis,
the modifications to general relativity are focused on.
Numerous alternative theories of gravity have been proposed and intensively investigated as alternatives to the cosmological constant and dark energy. Most of them have a
new degree of freedom which could behave as dark energy. These theories are collectively
called modified gravity. By studying modified gravity, the two following points can beconsidered: i) Modified gravity can be regarded as quantum gravity at a low energy scale.
At this scale, general relativity loses predictability, it cannot be quantized, and it should
thus be modified. By studying modified gravity, we may obtain some new perspective
or guiding principle for constructing quantum gravity. ii) Studying modified gravity
offers us insight into and knowledge about gravitation. Furthermore, modifying andgeneralizing general relativity is one way in which to construct a new theory of gravity.
By investigating the differences between these modified gravity theories, we can approacha more well-behaved theory of gravity.
Most modified gravity has a new degree of freedom in addition to the two tensor
modes of general relativity. This degree of freedom mediates a new long-range force, i.e.,a fifth force, which could be dark energy; however, the presence of this force changes
gravitational law on a local scale. Since any deviation from established general relativity
is strongly constrained in the solar system, modified gravity theories are expected topossess a mechanism for screening the fifth force in the vicinity of concentrations ofmatter, such as in the solar system. A modified gravity which possesses a screening
mechanism can explain accelerated cosmic expansion without introducing dark energy,
also evading the constraints of current fifth-force searches [23–25].
Furthermore, in recent years, the first direct detection of gravitational waves from abinary-neutron star merger and a binary-black hole merger was confirmed by the LIGO

4Introduction(Laser Interferometer Gravitational-Wave Observatory) and Virgo collaboration [4, 5].
The result shows that the propagation speed of gravitational waves is close to that oflight. This propagation speed limit can be applied to a dark energy model. As a result,a large class of modified gravity theories have been constrained and ruled out by theobservational results on gravitational waves. Gravitational wave and multi-messenger
astronomy has opened a new window for testing theories of gravity in strong gravity
regimes.

1.3. Outline of This Thesis
This thesis describes the author’s research on aspects of the screening mechanism
employed in scalar-tensor theory, especially in Galileon theory [26]. Modified gravity canbe described effectively by adding a scalar degree of freedom to the gravitational action.
Thus, theories composed of a metric and a scalar field are ubiquitous and as such it isimportant to explore aspects of these scalar-tensor theories.
The Galileon action arises as an effective field theory and has been well studied in thecontext of cosmology to be used in a dark energy model. Even though the Galileon action
contains the derivative self-interaction terms, the theory has a second-order equationof motion for the metric and for the Galileon field. Furthermore, the self-interaction
restores general relativity on small scales, called the Vainshtein mechanism [27]. Galileon
theory is equipped with the Vainshtein mechanism and thus has been studied in thecontext of the solar system. Galileon theory is thought to be a viable dark energy modeland therefore a number of aspects of Galileon theory are studied, especially in relation
to the efficiency of the Vainshtein mechanism in a less symmetric system.
This thesis is organized as follows: the basic concept of general relativity is given in
Chapter 2 in which aspects of general relativity are reviewed: dark energy and the blackhole and stellar solution relating to the original work presented in this thesis.
In Chapter 3, a theoretical overview of models of modified gravity is given. Modified
gravity is firstly divided into a number of classes. In this thesis, gravitational theories that
include additional fields are focused on. Scalar-tensor gravity is especially focused on andBrans-Dicke theory [28], Galileon theory [26], and Horndeski theory [29] are introduced.
The Galileon and Horndeski theories are equipped with a screening mechanism, namely
the Vainshtein mechanism [27]. Current constraints on the theories are also reviewed,

Introduction

5especially those based on astrophysical tests. At the end of the chapter, further interesting
features of scalar-tensor theories are addressed.
In Chapter 4, an overview of the screening mechanisms is given. Firstly, the screening
mechanisms are classified into three types; weakly coupled, high mass, and kinetic. In
this thesis, the primary focus is on kinetic-type screening, especially the Vainshteinmechanism [27]. Then, the Vainshtein mechanism and its shape dependencies [30] arestudied.
In Chapter 5, how the Vainshtein mechanism works in a less symmetric setup isstudied. The scalar field equation around a disk with a hole at its center is numericallysolved in cubic Galileon theory and it is found that the Galileon force is enhanced,
rather than suppressed, in the vicinity of the hole. This chapter is entirely based on theauthor’s original work [31]. The author planned to explore and analyze the nature of theVainshtein mechanism around a disk with a hole at its center in cubic Galileon theory.
In Chapter 6, a summary of our results and future prospects are given.
List of Papers
The article included in this thesis is [31]:• H. Ogawa, T. Hiramatsu and T. Kobayashi, “Anti-screening of the Galileon force
around a disk center hole,” arXiv:1802.04969 [gr-qc]. (To be appeared in Modern
Physics Letters A, Mod. Phys. Lett. A 34, 1950013 (2019))
The author contributed to the research and participated in writing the article as a firstauthor.

6

Chapter 2.
General Relativity
Before we discuss modification to general relativity, we briefly summarize aspects ofgeneral relativity and its application to cosmology and astrophysics.

2.1. Newtonian Mechanics to General Relativity
In 1687 Isaac Newton published the book Philosophiae Naturalis Principia Mathematic[32]. In this book, Newton outlined his three laws of motion and was the first to propose
a mathematical description of the laws of gravity. He presented the well-known formulafor gravitational force
Fg = G

m1 m2
,
r2

(2.1)

where Fg is the gravitational force acting between two bodies, G is the Newtonian
gravitational constant, m1 and m2 are the masses of bodies, r is the distance between
the centers of the two bodies. The massive body of mass M produces the gravitational
potential(r) = ≠G

M
,
r

(2.2)

where (r) is the Newtonian potential. The Newtonian potential obeys Poisson’s equation
Ò2 (r) = 4ﬁGﬂ,
7

(2.3)

8

General Relativity

where ﬂ is the density of the object. Newton stated that gravity acts between massive
bodies and travels instantaneously throughout space. Newton’s theory of gravity wastested and validated in experiments up until the beginning of the 20th century.
In the 19th century, the advance of the perihelion of Mercury was measured to be 43
arc-seconds [33]. This experimental result was not predicted by Newtonian or celestialmechanics. Many hypotheses such as the existence of a dark planet which was not yet
observed were proposed to explain this result in the context of Newtonian gravity. This
mysterious observation would remain unexplained until the beginning of the 20th century.
Albert Einstein conceived and developed general relativity from 1907 to 1915 [1, 2].
General relativity dramatically changed the way of thinking about gravitation as well
as space and time. Einstein stated that gravity is not a force, but rather a distortion
or curvature of spacetime. Newton’s theory of gravity allowed for gravity to travel at
infinite speeds, whereas Einstein showed that nothing could travel faster than the speedof light, not even gravity. The predictions based on general relativity were in good
agreement with the observational data, such as the advance of the perihelion of Mercury
as well as other experimental tests [3, 34] (which will be discussed below). To date,general relativity is the well-established and accepted gravity theory and the basis for
the Standard Cosmological Model.

2.2. General Relativity
We first explain the principal of general relativity. Einstein’s field equations are formulatedusing Riemannian geometry. The geometric quantities are described by a fundamental
quantity called the metric. From the metric, we can compute the fundamental geometricquantities, such as distance in curved spacetime. Einstein constructed general relativity
under the assumptions of the general principle of relativity and general covariance. The
general principle of relativity states that the laws of physics should be invariant in all
frames, regardless of whether the frames are accelerated. General covariance statesthat the physical law should be written in a form which is invariant under a generalcoordinate transformation, namely a tensor. We give a brief review of general relativity
and associated mathematical preliminaries in this section based on [35–37].

General Relativity

92.2.1. Mathematical Preliminaries-tensor, covariant derivativeand curvature-

In general relativity, the presence of matter curves the geometry of spacetime. The
motion of the particles is affected by the curvature of spacetime. In order to study thecurvature of spacetime, we consider the infinitesimal distance between two points whichis called the line element. The line element is defined by a metric tensor gµ‹ , which is arank-two symmetric tensor, as follows:
ds2 = gµ‹ dxµ dx‹ .

(2.4)

Here we use Einstein’s summation convention:Aµ Bµ := gµ‹ Aµ B ‹ =:3
ÿgµ‹ Aµ B ‹ .

(2.5)

µ,‹=0

When the upper and lower indices are same, it means that we sum over the indices.
We assumed this notation throughout this thesis. We also write vectors and tensors incomponent form. Greek indices µ, ‹, ... stand for four dimensions of spacetime, while
Latin indices i, j, ... stand for three spatial dimensions.
For example, in the Euclidean space, the infinitesimal distance between two points isgiven byds2 = dx2 + dy 2 + dz 2 .

(2.6)

We also denote the flat-space or Minkowski metric by ÷µ‹ . In the Cartesian coordinate(t, x, y, z), the Minkowski spacetime is given by
ds2 = ÷µ‹ dxµ dx‹ = ≠dt2 + dx2 + dy 2 + dz 2 .

(2.7)

We use the metric signature {≠, +, +, +} throughout this thesis.
A metric tensor is used to define the inner product between two vectors. It is also thequantity that describe the gravitational field or geometry of the spacetime and motion ofparticles. That is why the metric tensor is one of the fundamental quantities of generalrelativity.

10

General Relativity

TensorNext, we explain a tensor. Using the tensor formalism, we can easily describe therelationship between the spacetime coordinates and the mathematical relations in a clearway.
Õ

Under a change of coordinates xµ æ xµ (xÕ ), tensors are transformed according to
the following ruleÕÕ

T µ1 ···µm ‹1Õ ···‹nÕ (xÕ )
Q

Õ

µ1

R

Q

Õ

µm

R

ˆx
ˆx
= a –1 b · · · a –m b ·
ˆx
ˆx

A

ˆx—1
ˆx

Õ

‹1

B

···

A

ˆx—n
ˆx

Õ

‹n

B

T –1 ···–m —1 ···—n (x).

(2.8)

A covariant tensor is denoted with lower indices while a contravariant tensor is denotedwith superscript indices. The metric or its inverse g µ‹ are used to rise or lower indices.
For example, A– = g–— A— and B – = g –— B— . The covariant and contravariant vectors
are related to each other by the metric, we call it the duality. When we consider asummation, these covariant and contravariant indices become important. The inverse of
the metric is defined by the relationshipg µ⁄ g⁄‹ = ” µ ‹ ,(2.9)

where ” µ ‹ is the Kronecker delta. A scalar is defined as a tensorial quantity with noindex and which is invariant under the coordinate transformations.
Covariant DerivativeÕ

Under a change of coordinates xµ æ xµ (xÕ ), the ordinary derivative of the vector istransformed as follows:
ˆAµ

Q

Õ

Õ

R

ˆx“ ˆ a ˆxµ µ b
=
A
Õ
Õ
“
ˆxµ
ˆx‹
ˆx‹ ˆxÕÕˆxµ ˆx“ ˆAµˆ 2 xµ ˆx“ µ
=
+
Õ
ÕA .
ˆxµ ˆx‹ ˆx“ˆx“ ˆxµ ˆx‹(2.10)

This is not a tensorial transformation, thus the derivative depends on the coordinatesin spacetime. Therefore, we need to introduce a new derivative operator that does notdepend on the coordinates. This derivative is called the covariant derivative, and theGeneral Relativity

11

covariant derivative is defined on a contravariant vector Aµ as follows:Ò‹ Aµ := ˆ‹ Aµ +µ(2.11)⁄‹⁄ A ,

where µ ‹⁄ is called the connection. For a covariant vector Aµ , the covariant derivativeis defined asÒ‹ Aµ = ˆ‹ Aµ ≠⁄(2.12)µ‹ A⁄ .

For a scalar, the covariant derivative is defined asÒµ „ = ˆµ „,(2.13)

and a tensor T µ‹ :
Ò“ T µ‹ = ˆ“ T µ‹ +

µ

“⁄ T

⁄‹

+

‹

“⁄ T

µ⁄

.

(2.14)

In general relativity, the connection is called as the Christoffel symbol which depends ongµ‹ and its derivative ˆ“ gµ‹ and defined by
µ

–— =

1 µ⁄
g (ˆ— g⁄– + ˆ– g⁄— ≠ ˆ⁄ g–— ),2which is symmetric in the two lower indices,

µ

–— =

µ

(2.15)—– .

Geodesic Equation
Here, we derive the equations of motion for a test particle in curved spacetime. Let us
consider the length of a curve between the points P and Q in spacetime. By integrating
the infinitesimal path length ds along the trajectory, we obtain the total length of the
trajectory:
s=

⁄ Q
P

ds =

⁄ QÒ
P

gµ‹ dxµ dx‹ .

(2.16)For a variation of (2.16) with respect to the coordinates which lie on this trajectory,
we can obtain the shortest distance between the points P and Q. A particle follows
this trajectory, called the geodesics of spacetime. In order to compute the integral, we12General Relativity

consider a parametrized worldline, x– = x– (⁄). We then consider an action
s=

⁄ Q
P

ds =

⁄ ⁄

Q

⁄P

A

B

⁄ ⁄
Q
ds
dxµ
d⁄ :=
L xµ ,,d⁄d⁄⁄P(2.17)

and find the curve which extremizes the action. Substituting the Lagrangian into theEuler-Lagrange equations, and we find that x– must satisfy the differential equation
ẍµ +

µ

–— ẋ

ẋ = 0,(2.18)– —where ẋ– := dx– /d⁄. This is called the geodesic equation 1 and its solutions are called
geodesic.

Curvature and Einstein’s Field Equations
Now we introduce the curvature of spacetime. Since spacetime is curved, covariant
derivatives do not commute with each other. The Riemann tensor R– —µ‹ measures thenoncommutativity (or curvature of the spacetime):[Ò– , Ò— ]Aµ = ≠Rµ ‹–— A‹ ,(2.19)

here we define the Riemann tensor as follows:
R– —µ‹ = ˆµ – —‹ ≠ ˆ‹ – —µ +

–

⁄µ

⁄

—‹ ≠

–

⁄‹

⁄

—µ .

(2.20)The Riemann tensor R–—µ‹ = g–ﬂ Rﬂ —µ‹ has the following properties:
R–—µ‹ = ≠R—–µ‹ ,(2.21)

R–—µ‹ = Rµ‹–— .

(2.23)

R–—µ‹ = ≠R–—‹µ ,(2.22)In addition, the Riemann tensor satisfies the Bianchi identities,
Ò“ R–—µ‹ + Ò‹ R–—“µ + Òµ R–—‹“ = 0.
1(2.24)

When one consider null paths, one cannot define the Lagrangian in the same manner, since these are zeroalong a null path, ds2 = 0. Therefore, we use the simpler Lagrangian L = (1/2)gµ‹ (dxµ /d⁄)(dx‹ /d⁄)instead of the Lagrangian in Eq. (2.17).

General Relativity

13

By contracting the Riemann tensor, the Ricci tensor R–— and the Ricci scalar R aredefined respectively as follows:Rµ‹ = R⁄ µ⁄‹ ,R = R⁄ ⁄ .

(2.25)The Einstein tensor is defined in terms of the Ricci tensor and the Ricci scalar:1Gµ‹ = Rµ‹ ≠ gµ‹ R.
2(2.26)

By virtue of the Bianchi identity (2.24), the Einstein tensor satisfies
Òµ Gµ‹ = 0.

(2.27)

2.2.2. Formulation of Einstein’s Field Equations
In this subsection, we will derive Einstein’s field equations from the variational principle.
The action for general relativity - the Einstein-Hilbert action - is defined as
SEH =

⁄

Ô
1
d4 x ≠g 2 R,
2Ÿ

(2.28)

Ô
here Ÿ is a constant, g denotes the determinant of the metric, and ≠gd4 x is theinvariant volume element. By varying the Einstein-Hilbert action with respect to themetric, we obtain Einstein’s field equations in a vacuum. Adding a matter action into
the Einstein-Hilbert action:
Stot = SEH + Sm [gµ‹ , ] =

⁄

6
Ô 5 1
d4 x ≠g
R
+
L
M ,
2Ÿ2

(2.29)

where denotes the matter fields and LM is the Lagrangian for the matter fields. By
varying the Stot with respect to the metric, it follows that
”Stot =

⁄

C

D

Ô
1
” Ô
dx
( ≠gLM )”g µ‹ .
2 ”(R ≠g) +
”g µ‹
2Ÿ
4

(2.30)

14

General Relativity

Using the following relations:Ô1Ô
” ≠g = ≠≠ggµ‹ ”g µ‹ ,2”R = Rµ‹ ”g µ‹ + g µ‹ ”Rµ‹ ,(2.31)(2.32)g µ‹ ”Rµ‹ = g µ‹ [Ò⁄ (” ⁄ µ‹ ) ≠ Ò‹ (” ⁄ ⁄µ )]= Òﬂ (g µ‹ ” ﬂ µ‹ ≠ g µﬂ ” ⁄ ⁄µ ),(2.33)

we obtain the total variations with respect to the metric as follows:
⁄

C

3

4

D

1
1
” Ô
µ‹
”Stot = d x ≠gµ‹ ( ≠gLM ) ”g
2 Rµ‹ ≠ gµ‹ R +
2
”g
2Ÿ
5
6
⁄
Ô
1
µ‹
ﬂ
µﬂ
⁄
+ d4 x ≠g
Ò
(g
”
≠
g
”
)
.
ﬂ
µ‹
⁄µ
2Ÿ2
4

Ô

(2.34)The last term of Eq. (2.34) is a total derivative, and this surface term does not contributeto the variation of the action since the metric dumps at infinity. Here we define theenergy-momentum tensor as follows:Ô2 ”( ≠gLM )Tµ‹ := ≠ Ô
.
≠g
”g µ‹

(2.35)

Since ”Stot = 0 for any variation ”g µ‹ , Equation (2.34) leads to the equations of motion
for the metric gµ‹ :2
1
1
Rµ‹ ≠ gµ‹ R = Ÿ2 Tµ‹ = 8ﬁGTµ‹ ,2(2.36)

which is known as Einstein’s field equations and here we choose Ÿ2 as 8ﬁG.
The left-hand side of the equation (2.36) describes the geometric part of the equation.
The right-hand side of the equation (2.36) describes the matter part of the equation and iscalled the energy-momentum tensor. Einstein’s field equations (2.36) can be interpreted
as how matter is related to the curvature of spacetime. After solving Einstein’s field
equations, one finds configurations for the gravitational field around matter (see Sec. 2.3
and 2.4).
Note that since the covariant derivative of the metric tensor is zero, we can also add
a factor gµ‹ into the left hand side of Eq. (2.36) without introducing a new degreeof freedom, whereis the cosmological constant. Adding the term ≠2 into theGeneral Relativity

15

Einstein-Hilbert action SEH , we arrive at
1Rµ‹ ≠ gµ‹ R + gµ‹ = 8ﬁGTµ‹ .
2(2.37)

We will explain more details of the cosmological constant later.
Energy-Momentum Tensor
The meaning of the components of the energy-momentum is expressed as follows
Q

R

Q

R

energy flux d
cT00 T0i dc energy density
Tµ‹ = ab=ab.
energy momentum stress tensor
Ti0 Tij(2.38)The energy-momentum tensor is a symmetric tensor. For example, the energy-momentum
tensor of a perfect fluid is given byT µ‹ = (ﬂ + P )uµ u‹ + P g µ‹ ,(2.39)

where ﬂ and P are the density and the pressure of the fluid respectively, and uµ = ẋµ is
the four-velocity of the fluid. The contracted Bianchi identity implies that the energymomentum tensor is conserved (locally)Ò‹ T µ‹ = 0.

(2.40)

This implies that particles move on geodesics. To see this, we consider the energymomentum tensor for P = 0 in Eq. (2.39), T µ‹ = ﬂuµ u‹ . We can rewrite the conservationof this energy-momentum tensor asuµ Òµ u‹ = uµ (ˆµ u‹ += ẍ‹ +

‹

µ— ẋ

‹

µ— u

—

)

ẋ = 0,µ —(2.41)

we obtain the geodesic equation again.
The Newtonian Limit
General relativity is generalization of Newtonian gravity, so it is required that generalrelativity restores Newtonian gravity in the limit of low velocities (ẋ π c) in weak

16

General Relativity

gravitational fields. A weak gravitational field means that the field is mostly flat and thecurvature is very small, such as a perturbation. We assume that the metric is given byg–— ≥ ÷–— + h–— ,|h–— | π 1,(2.42)

where hµ‹ is a perturbation. In the Newtonian limit, (ẋ π c), the geodesic equation(2.18) reduces to
d2 xi
i
i
2 = ≠ 00 = ≠Ò ,
d⁄
where

(2.43)

is the Newtonian potential, and we findh= ≠ 00 ,2(2.44)

ẍi = ≠Òi .

(2.45)and the particles move according to

2.3. Cosmological Solutions
The concept of cosmology is that there are no special places in the Universe. It is called
the Cosmological Principle. Measurements of the galaxy distribution and the CMB
show that the Universe is isotropic and homogeneous on a large scale (cosmologicalscale) [17, 18]. Any observer at a different position, such as in another galaxy, would
observe the same structure of the Universe as we do. Note that the Cosmologicalprinciple is not valid on small scales. Of course, several relatively small-scale systems,
such as our solar system and galaxy, do not obey the principle. We need to introduce
perturbations to the background at this scale; our Universe can be treated as isotropic
and homogeneous at zero-th order approximation. In this subsection, we observe that our
Universe seems to be well described by the solution of Einstein’s field equations, namely
Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions.

General Relativity

17

2.3.1. The FLRW Universe

According to the cosmological principle, the metric of the spacetime does not dependon position. The metric for an isotropic and homogeneous Universe is called the FLRW
metric which is given by222

ds = ≠dt + a(t)

C

D

dr2
2
2
2
2
2 + r (d◊ + sin ◊d„ ) ,1 ≠ Kr

(2.46)

where K is the spatial curvature of the Universe, a(t) is called the (expansion) scale
factor and denotes a parameter determined by the matter distribution of the Universe.
We compute the Ricci tensor and the Ricci scalar for the FLRW metric, and we obtain:
5

3

46

ä
ȧ K
+2
+
gij ,
a
a a2
ä
R00 = ≠3 ,a
6µ‹R = g Rµ‹ = 2 (äa + ȧ2 + K).
a
Rij =(2.47)(2.48)(2.49)

According to the cosmological principle, the energy-momentum tensor is written using
the perfect fluid,Tµ‹ = (ﬂ + P )uµ u‹ + P gµ‹ .

(2.50)

Substituting the FLRW metric (2.46) and the energy-momentum tensor (2.50) into
Einstein’s field equations (2.36), and using Eqs. (2.47–2.49), we obtain the two independentequations:
8ﬁGKﬂ ≠ 2,
3
a
ä
4ﬁG
=≠
(ﬂ + 3P ),
a
3

H2 =

(2.51)(2.52)

where we introduce the Hubble parameter as H := ȧ/a. These are referred to as theFriedmann equations which describe the evolution of the Universe. We can classify
the FLRW Universe into three types corresponding to the signature of K. i)K = 0:
corresponds to a flat (open) Universe. If a(t) = 1, the FLRW metric is equivalent to the
Minkowski metric, ii)K > 0: the Universe is closed and positively curved, iii)K < 0: theUniverse is open and negatively curved.

18

General Relativity

The system of the Friedmann equation and energy conservation equation for eachspecies:ﬂ̇i + 3H(ﬂi + Pi ) = 0,(2.53)

is not closed. Therefore, we must determine the equation of state of the fluid. Here, we
define the equation of state for each species as
Pi = wi ﬂi ,(2.54)

where w = 0, 1/3, and -1 corresponds to non-relativistic matter, radiation and thecosmological constant. Using Eq. (2.54), we can rewrite the Friedmann equation (2.49)
as follows
ä
4ﬁG ÿ
=≠
(1 + 3wi )ﬂi .
a
3 i(2.55)

If ﬂi > 0, the Friedmann equation describes the accelerated expansion of the Universe forwi < ≠1/3, whereas decelerated expansion for wi > ≠1/3. Note that we cannot describe
the accelerated expansion of the Universe in the case of ordinary matter w = 0 and 1/3
with ﬂ > 0.

2.3.2. The-CDM ModelIn this subsection, we consider a Universe model using the cold dark matter and thecosmological constant, namely the -CDM model. The -CDM model has been widely
accepted as the standard cosmological model because it agrees with observational results.
Several independent measurements and observations [11–16] have confirmed the latetime accelerated expansion of the Universe. In order to explain this phenomenon, we
need to modify Einstein’s field equations. As we noted above, we can add the term to
the Einstein-Hilbert action, and we obtain the (modified) Einstein’s field equations withthe cosmological constant:1
Rµ‹ ≠ gµ‹ R + gµ‹ = 8ﬁGTµ‹ .
2(2.56)

General Relativity

19

Hereafter, we assume that the cosmological constant drives the accelerated expansion of
the universe. From the FLRW metric, we obtain the Friedmann equations again:
8ﬁG
Kﬂ≠ 2 + ,
3
3
a
ä
4ﬁG
=≠
(ﬂ + 3P ) + .
a
3
3

H2 =

(2.57)(2.58)

Defining the energy density ﬂ and pressure P as follows:
ﬂ =

8ﬁG

, P =≠

8ﬁG(2.59),

and replacing ﬂ æ ﬂ + ﬂ , P æ P + P , we can omit the cosmological constant from theFriedmann equation. In other words, if one allow for the existence of matter which has anegative pressure, the cosmological constant can be regarded as the new matter. Thisnew matter/energy is called dark energy. From the definition of the dark-energy densityand pressure, its equation of states satisfies
ﬂ = ≠P , w = ≠1.

(2.60)

As we discussed in the previous subsection, dark energy drives the late-time accelerationof the Universe.
The density when

= 0, K = 0 at t = t0 is called the critical density:
ﬂc :=

3H02
,
8ﬁG

(2.61)

where we defined H0 which is the present-day value of the Hubble parameter. Using thecritical density (2.61), we can define the density parameters which are the ratio of thedensity of the Universe to the critical density for each different species,
ﬂi
,
ﬂc

i :=

(2.62)

here i represents the species, and i = , m, K, r corresponds to the cosmological constant,standard matter, curvature, and radiation. Using the density parameters, we can rewrite
the Friedmann equations as follows:
H

2

= H02

A

+

m
3 +

a

a

r
4 +

K
2

a

B

.

(2.63)

20

General Relativity

Equation (2.63) tells us that the evolution of the Universe depends on the amount ofeach energy density. At different times, a different energy source dominated the Universe,
and the evolution rate of the Universe changes. From measurements of the CMB [17, 18],
the curvature parameter is close to zero. Therefore, the curvature is very close to zero.

Components of the Universe
The -CDM model includes baryons, cold dark matter, radiation and a cosmologicalconstant. Combining measurements of the CMB by the Planck satellite [17, 18], of
Baryon Acoustic Oscillations [15, 16] , and of type Ia supernovae [11, 12] provides thedensity parameters: the approximate values are≥ 0.70 for the cosmological constant,c ≥ 0.25 for the cold dark matter, andb ≥ 0.05 for baryons. According to the results,
the cosmological constant, dark energy, and dark matter dominate the componentsof the current Universe. The origins of these dark components are still unknown. A
discussion for the origin of dark matter is beyond the scope of this thesis. Several dark
matter candidates have been proposed so far. These types are divided into two types;
baryonic and non-baryonic matter. Baryonic matter are astrophysical objects, such as
Machos [38]. Well studied non-baryonic types are axion [39, 40] and weakly interacting
massive particles [41]. The interested reader can find further details of dark matterin [42].

2.3.3. Problems in the-CDM Model

Despite the success of the -CDM model, several serious problems remain. The details ofthese problems are beyond the scope of this thesis; however, a brief summary is provided
here.
• The flatness problem:
Why is the curvature of the Universe today close to zero? The solution of theFriedmann equation with K = 0 is an unstable solution, and a small deviation fromflatness leads to either a closed or an open Universe. The order of today’s value K
is of order unity. This indicates that curvature of the Universe must have been oforder unity in the early Universe, but why and how?
• The horizon problem:
The measurement of the CMB [43] indicates that the CMB temperature is 2.7K,

General Relativity

21

anywhere in the sky. During the recombination period, when the Universe wasabout 380000 years old, the horizon subtended an angle of about 1.4¶ to the sky. In
other words, each matter at the last scattering was causally connected with eachmatter within the 1.4¶ on the CMB sky. Therefore, two points that were separated
by an angle of more than about 1.4¶ on the CMB surface before recombination werecausally disconnected at the time of the last scattering. Why are their temperatures
the same everywhere, even if they were causally disconnected?
• Magnetic-monopole problem:
Grand unification theory predicts that a stable magnetic monopole would be produced in the early Universe (at sufficiently high temperatures). However, it has
never been observed. Where have such monopoles gone?
One of the simplest solutions to these problems is the inflationary Universe. Inflation
is a period of rapid expansion, which occurs in the initial period of the Universe (roughly
from 10≠37 to 10≠33 sec after the Big-Bang). Such a rapid expansion quickly flattened
the Universe. Hence, the flatness problem can be solved by inflation. Two points whichwere causally disconnected before the recombination period were then causally connected
before the beginning of inflation, and these points later rapidly separated after inflation.
Inflation also solves the monopole problem. Monopoles existed before the start of inflation,and the rapid stretching of space greatly diluted their density.
The -CDM model is a viable cosmology model; however, many problems remain
unsolved. The most severe of these are the cosmological constant problem and thecoincidence problem.
The simplest candidate for dark energy is a cosmological constant. As noted above earlier, the cosmological constant can be written as an energy-momentum tensor. Therefore,
the cosmological constant can be interpreted as background vacuum energy. However,there is a large discrepancy between the theoretical and observational values of thecosmological constant; this is known as the fine-tuning problem. Why does this large
discrepancy between observation and theory occur? The discrepancy has been called theworst theoretical prediction in the history of physics. Indeed, the -CDM model cannot
explain the ratio of dark energy to cold dark matter for the current energy density, which
is known as a coincidence problem. The -CDM model cannot solve these problems.
These problems provide the motivation to construct other dark energy models.

22

General Relativity

2.3.4. Dark Energy Model
Dark energy plays an essential role of a repulsive force which drives the accelerated
expansion of the Universe. Numerous theoretical and experimental efforts have been
done to unveil the nature of dark energy, however, the origin is still unknown. This is themost mysterious problem in modern cosmology. One of the candidates for dark energy
is a dynamical scalar field with a potential which is called the Quintessence. Here, we
introduce a model for the Quintessence.

Quintessence

The idea of Quintessence is that dark energy behaves like a fluid with negative pressure
which drives accelerated expansion of the Universe up to and including today. The
simplest form of Quintessence is a canonical scalar field which is dynamical and slowly
rolling down its potential V („). We consider the Einstein-Hilbert action and add aminimally coupled scalar field with Lagrangian density
L„ =

Ô

3

4

1
≠g ≠ (ˆ„)2 ≠ V („) ,2(2.64)

into the Einstein-Hilbert action. Note that the field is minimally coupled to gravity
which means that field does not directly couple to any curvature tensors.
From the cosmological principle, the Universe is homogeneous; we can assume thatthe scalar field depends on time only on a cosmological scale. The equations of motionof a scalar field in FLRW metric is given bydV„¨ + 3H „˙ += 0.
d„(2.65)

This equation represents that the field „ rolls down a potential V („). The term 3H „˙isinterpreted as a friction term on the field. If the potential is too steep, H Æ m„ :=Òˆ 2 V („)/ˆ„2 where m„ is the mass of the scalar field, the field quickly rolls down. Hencethe equation of state does not behave like a cosmological constant.

General Relativity

23

The energy momentum tensor for the scalar field isTµ‹Ô
2 ”( ≠gL„ )
= ≠Ô
≠g
”gµ‹
3
4
µ
‹
µ‹ 1
µ
= Ò „Ò „ ≠ gÒµ „Ò „ + V („) .
2(2.66)(2.67)

From Eq. (2.67), we obtain the energy density1
ﬂ = „˙ 2 + V („),
2

(2.68)

1
P = „˙ 2 ≠ V („),
2

(2.69)and the pressureof the scalar field. The equation of state can be written as follows:
1 ˙2„ ≠ V („)
P
Ê=
= 21
.
ﬂ
2
˙
„ + V („)
2

(2.70)

We observed that when V („) ∫ „˙ 2 , the equation of state takes Ê ≥ ≠1, i.e., Quintessence
could drive the observed cosmic acceleration. Quintessence differs from the cosmologicalconstant explanation of dark energy in that Quintessence is dynamical, whereas, bydefinition, the cosmological constant does not change with time. Fromthe equations ofÒ2
motion for the scalar field, the slow-roll condition implies that H ≥ ˆ V („)/ˆ„2 ≥ m„ .
It means that the value of effective mass of Quintessence m„ should be that of the current
cosmological constant, roughly 10≠33 eV. If such a low mass of the scalar field directly
couples to matter, the scalar field gives rise to a long-range fifth force. One way to avoid
the fifth force is a screening mechanism. We will discuss the screening mechanism in
Chapter 4 for details. Furthermore, such a mass of the scalar field does not seem to benatural in the context of quantum field theory.
In fact, various Quintessence models face fine-tuning problems (namely, in the initialconditions and parameters in the potential) to reproduce the desired dynamics, but there
exist potential which gives rise to desire solutions. Two types of Quintessence have been
proposed depending on its evolution, namely thawing [44,45] and freezing models [46–48].
The details of Quintessence are beyond the scope of this thesis. See, [20] for example, fora review of Quintessence.

24

General Relativity

Other Possibilities
Quintessence is a dynamical model of dark energy in which the equation of state changes
with time. The extended type has been investigated, there are k-essence models [21, 22],
where the Lagrangian for the scalar field (2.64) is generalized to a function of „ and its
kinetic term X = ≠ˆ µ „ˆµ „/2. The kinetic energy is said to drive the current cosmic
expansion and this scenario is called k-essence.
An alternative approach is to modify general relativity to explain the current accelerated expansion of the Universe. In this framework, we do not need to introduce darkenergy to a cosmological model: modifications to general relativity drive accelerationof the Universe. As for now, numerous alternative theories of gravity have been proposed and intensively investigated as alternatives to the cosmological constant and darkenergy [49]. It has been suggested that a new particle produced from the alternative
theories of gravity can be a dark matter candidate. In this scenario, one might be able
to solve coincidence problem: one can predict the ratio of the dark energy density to the(dark) matter density today. We will discuss further details of this in Chapter 3.

2.4. Black Holes and Stellar Solutions in General
Relativity
In this subsection, we will explain stellar solutions, namely black hole and neutron starsolutions. These are solutions of Einstein’s field equations and tell us the interesting
effects of a strong gravity field. This subsection is based on [35, 37, 50].

2.4.1. Black Holes
Black holes are intriguing objects for many reasons. From a theoretical viewpoint,
black holes have a rich mathematical structure with regard to their uniqueness [51, 52]
in and beyond general relativity in four spacetime dimensions. Black holes behaveas thermodynamic objects and lead to some intriguing results: authors of [53–56]
showed that black holes radiate as black bodies, and lead to the notion of thermal
Hawking radiation. Hawking radiation has not been detected: the typical temperature
for astrophysical black holes is much lower than the CMB. The expression for thetemperature and entropy of black holes can be derived from different approaches toGeneral Relativity

25

quantum theory and string theory. This coincidence would suggest the existence ofuniversality related to the temperature and entropy. The second open question about
black holes is the information loss paradox [57]. Black holes radiate Hawking radiation,lose mass in the process and evaporate away. This evaporation violates a unitarity ofevolution which is forbidden in quantum mechanics. Studying the physics of black holes
will shed light on or provide a hint for the quantum theory of gravity and thermodynamics.
From an astrophysical viewpoint, the accretion of matter by black holes is accompaniedby electromagnetic radiation. The emission of X-rays during an accretion process from
Cygnus X-1 suggested that Cygnus X-1 is a black hole [58, 59], but this is the indirect
evidence of the existence of black hole. The recent detection of gravitational waves has
confirmed the existence of a (supermassive) black hole [4]. In the future, gravitationalwave astronomy should show interesting phenomena and reveal more of the nature of
black holes (in Sec. 2.5.2, gravitational-wave astronomy is discussed briefly). Amongthese various aspects, it is interesting to analyze black hole solutions in and beyond
general relativity.

Schwarzschild Solution
We first propose the static spherically-symmetric vacuum solution, namely the Schwarzschildsolution [60]. We will use spherical coordinates, so that the metric depends on the radial
component only. From this assumption, the metric does not contain a mixed term, suchas drd◊. By redefining the coordinates (t, r), the metric can be rewritten in the more
simple form:
ds2 = ≠e‹(r) dt2 + e⁄(r) dr2 + r2 (d◊2 + sin2 ◊dÏ2 ).

(2.71)

We can compute geometric quantities, such as the Ricci tensor and the Ricci scalar, fromthe metric (2.71). The Einstein tensors are given bye‹ (r)≠⁄(r)(1 ≠ r⁄Õ (r))],2 [1 ≠ er
e⁄(r)
G11 = ≠ 2 [1 ≠ e≠⁄(r) (1 + r‹ Õ (r))],r
ST
2
2 ≠⁄(r)
Õ
Õ
Õ
Õ
Õ
r e
‹
(r)
‹
(r)⁄
(r)
‹
(r)
≠
⁄
(r)
ÕÕ
U‹ (r) +
V,
G22 =
≠
+
2
2
2
r
G00 =

G33 = sin2 ◊G22 .

(2.72)(2.73)(2.74)(2.75)

26

General Relativity

We integrate Eq. (2.72)=0, and we obtain
⁄(r)

e

3

r
= 1≠ gr

4≠1,(2.76)

where rg is an integration constant. We will determine the integration constant later.
With G00 = G11 = 0, ‹ and ⁄ must satisfy the relation:d(‹(r) + ⁄(r)) = 0.
dr(2.77)

3(2.78)

We finally obtain
‹(r)

e

4

r
=C 1≠ g ,r

here C is an integration constant. Combining Eqs. (2.76) and (2.78), the metric (2.71)
can be rewritten as follows:
3

4

r
1
2
2
2
2
2 2
ds = ≠ 1 ≠ g dt2 + 1
rg dr + r (d◊ + sin ◊d„ ),r
1≠
2(2.79)rhere we rescale the time coordinate. This metric describes the Schwarzschild solution. We
derived the Schwarzschild solution with various assumptions, namely static, sphericallyand vacuum. According to Birkhoff’s theorem [61], the Schwarzschild solution is theunique spherically symmetric solution to the vacuum Einstein’s field equations, without
assuming a static background.
To determine the integration constant rg , we require that the Schwarzschild solution
is consistent with Newtonian gravity at large r. At large r, gtt satisfies (see above the
Newtonian limit in Sec. 2.2.2)
gtt æ ≠1 ≠ 2
2GM= ≠1 +.
r(2.80)

This yields
rg = 2GM,(2.81)

which is known as the Schwarzschild radius. The Schwarzschild metric represents
spacetime outside a static and spherically symmetric mass distribution in a vacuum. The

General Relativity

27

Schwarzschild metric diverges at r = 0 and 2GM . The divergence of r = rg = 2GM looks
like a singularity, but one can remove the divergence via a coordinate transformation. On
the other hand, the divergence of r = 0 cannot be removed. The surface at radius rg iscalled the event horizon, while the point r = 0 is called the singularity. If the singularity
is located outside of the event horizon, the singularity is called a naked singularity:
physical measurements diverge at the singularity. However, according to the cosmic
censorship conjecture [62], the naked singularity should be located inside of the event
horizon: As for now, we have never observed a naked singularity.
Next, we also introduce the other black hole solutions in general relativity. These
solutions are an interesting subject; however, a full treatment is beyond the scope of thisthesis. The interested reader can find a general description and details of black holesin [37, 63]. Here we will only focus on the well-known black hole solutions in generalrelativity.

Kerr Solution
The Kerr metric [64] is an exact axisymmetric solution of the vacuum Einstein’s field
equations and describes the spacetime of a rotating black hole without charge. The Kerr
metric in the Boyer-Lindquist coordinates (t, r, ◊, „) is given by23

ds = ≠ 1 ≠
+

A

rg r

4

2

dt ≠

(r2 + a2 )2 ≠2arg r sin2 ◊a2 sin2 ◊BdtdÏ +

dr2

sin2 ◊dÏ2 + d◊2 ,(2.82):= r2 + a2 ≠ rg r.

(2.83)

where
a=

J
,
M

:= r2 + a2 cos2 ◊,The metric represents a rotating black hole which has angular momentum J = aM . Notethat if the black hole is not rotating (a = 0), the metric reduces to the Schwarzschild
metric. By taking the limit r æ Œ, the Kerr metric reduces to the Minkowski metric;
thus, in this limit the Kerr spacetime is asymptotically flat.
The Kerr metric has singularities at = 0 and = 0. By transforming the coordinate
system, one finds that the metric is not singular at = 0, thus = 0 gives the coordinate
singularity. On the other hand, = 0 is a curvature singularity.

28

General Relativity

Here, we focus on the equation= 0. When 4a2 > rg2 , the equation has no real
solution. In this case, the curvature singularity is not within the horizon, a naked
singularity. From the cosmic censorship conjecture, this case is considered unphysical.
Therefore, we often focus on the case 4a2 5 rg2 . Solving = 0, we obtain+≠= (r ≠ rH)(r ≠ rH
) = 0,(2.84)

here we define the horizon of the Kerr black hole:Òrg ± rg2 ≠ 4a2
rH± :=,
2(2.85)+≠
rH
and rHare called the outer horizon and the inner horizon respectively. In the case of22
4a = rg , the Kerr metric describes the extremal Kerr solution.

Reissner-Nordström Solution
The Reissner-Nordström metric [65, 66] is an exact spherical symmetric vacuum solution
to Einstein’s field equations and describes the spacetime for a charged black hole. The
metric is given by
A

B

A

r
r
Q2
Q2
ds = ≠ 1 ≠ g + 2 dt2 + 1 ≠ g + 2
r
r
r
r
2

B≠1

dr2 + r2 (d◊2 + sin2 ◊d„2 ),(2.86)

where Q is the charge. If one takes Q = 0, the metric is equivalent to the Schwarzschild
metric. The Reissner-Nordström metric has divergence atÒ

rg ± rg2 ≠ 4Q2
±
rE =
.
2

(2.87)

If |2Q| > rg , the Reissner-Nordström metric has a naked singularity at r = 0. When
|2Q| = rg , the Reissner-Nordström metric describes the extremal Reissner-Nordström
solution.

The uniqueness theorem in General Relativity
The uniqueness theorem for black holes ia an interesting topic, and this gives rise to a
motivation for studying the no-hair theorem [51, 52]. In general relativity, black holes

General Relativity

29

are characterized by only three parameters: the mass, the angular momentum, and theelectric charge. Therefore, black holes cannot support other hairs in the frameworkof general relativity. On the other hand, several hairy black hole solutions have been
discovered in a beyond-general relativity context. We will discuss these briefly in Sec. 3.6.1.
From an astrophysical viewpoint, we do not know whether the astrophysical black hole isdescribed by Kerr spacetime or not. As we will discuss in Sec. 2.5.2, gravitational wave
astronomy could provide us with insights into dynamics of spacetime, and with a gravity
test in a strong gravity regime.

2.4.2. Neutron Stars
Neutron stars are highly dense stellar objects believed to be remnants of supernovas.
The existence of such objects was proposed by Badde and Zwicky in the 1930s [67, 68].
Subsequently, Tolman, Oppenheimer, and Volkoff theoretically formulated a model forsuch highly dense spherical stars [69, 70]. For decades, neutron stars were thought to be
hypothetical stars of only academic interest until the discovery of pulsars by Bell and
Hewish [71]. Pulsars are rapidly rotating and highly magnetized neutron stars. They
emit beams of radiation from their magnetic poles. The beams can be observed when
they are pointing towards the Earth. The rapid rotation usually causes a pulsing oflight to be observed. That is why we call them pulsars. Due to their similar behavior,pulsars are called the lighthouses of the Universe. Until now, thousands of neutron stars
have been discovered [72]. Most have properties of the order of a mass of 1 ≥ 2M§ anda radius of R ≥ 10km, i.e. a density of order 1016 g cm≠3 which significantly exceeds
the normal nuclear density O(1014 )g cm≠3 . Such extremely dense objects and mattercannot be created in a laboratory. Additionally, the structure of neutron stars is mostly
determined by their strong gravitational fields. Neutron stars provide us with a naturallaboratory to test gravity theories and for studying the poorly understood details andthe equation of state of extremely dense matter.

Structure
In order to investigate the structural parameters of a neutron star, such as its mass andradius, we need to solve the structure equations with the equation of state for nuclear
matter. However, there are several difficulties.

30

General Relativity

The internal structure of a neutron star still remains unknown. As noted above,the density of a neutron star is roughly 1015 g cm≠3 and may reach 1016 g cm≠3 . The
properties of such a high-density domain are still not well understood. The interior ofa neutron star can be divided into two regions according to the density [50]: (i) a corewith a density which could reach 1015 g cm≠3 . The structure and the composition of thecore are not well known. The inner core is expected to be composed of various exotic
particles, such as pions, hyperons, etc. (ii) a crust with a density ranging from 106 g
cm≠3 to 1014 g cm≠3 . The crust is further divided into an outer crust with a densityof (106 g cm≠3 to 1011 g cm≠3 ) and an inner crust of density (1011 g cm≠3 to 1014 g
cm≠3 ). The outer crust consists of neutron-rich atomic nuclei with free electrons. With
increasing density, the nuclei become neutron-rich and capture these electrons. They
then reach a critical density (O(1011 )g cm≠3 ), which is called the neutron drip line. At
this density of O(1011 ) g cm≠3 , neutrons begin to drip out of (or escape from) the nuclei
to become free neutrons. Explaining the inside of a neutron star is non-trivial. Future
observations, such as gravitational wave measurements and multi-messenger astronomy,
should reveal more about the nature of the neutron star.
The equation of state for dense matter above the neutron drip line density is still
unknown. Details of the equation of state for nuclear matter are beyond the scope of
this thesis. The reader is referred to [73, 74] for more information about the equationsof state. Numerous equations of state have been proposed to describe the structureof the neutron star thus far. The recent discovery of the pulsar PSR J1614-2230 [75]places tight constraints on equations of state for neutron stars. The results show theexistence of a massive neutron star: the estimated mass of the neutron star is M =(1.97 ± 0.04)M§ . The limits for the maximum mass of a neutron star excluded manyequations of state in the framework of general relativity (see, e.g., [73, 74] for a review
and references therein). Therefore, neutron stars provide us with a natural laboratory
for studying nuclear physics.

Lane-Emden Equation
The equation of state for densities below the neutron drip line is well studied. A
polytropic equation of state has been used for simulating a neutron star, which is given
as a power-law in density
P = Kﬂ ,(2.88)

General Relativity

31

where K and are constants. The ideal Fermi gas equation of states reduces to thisform with = 5/3 and 4/3. A hydrostatic star consists of a polytropic gas. The modelstar is called a polytrope. The Lane-Emden equation [76] describes polytrope properties.
In the non-relativistic limit, the conditions P π ﬂ, under the hydrostatic equilibrium
equations are given bydP (r)
GM (r)ﬂ(r) dM (r)=≠,= 4ﬁr2 ﬂ(r).
dr
dr
r2

(2.89)

Combining these equations, we obtain
d
dr

A 2

r dP
ﬂ dr

B

= ≠G

dM
dr

= ≠4ﬁGﬂr2 .

(2.90)

Substituting the polytropic equation of state (2.88) into Eq. (2.90), we obtainAddﬂr2 K ﬂ ≠2drdrB= ≠4ﬁGﬂr2 .

(2.91)

After defining quantities
1,n
ﬂ = ﬂc ◊ n ,=1+(2.92)(2.93)

where ﬂc = ﬂ(0) is the central density of the star, Equation (2.91) can be written as
follows
A(n + 1)Kﬂ1/n≠1 1 d
d◊
r2
2
4ﬁG
dr
r dr

B

= ≠◊n .

(2.94)

To make this equation dimensionless, we introduce a radial variable ›r,–Û(n + 1)Kﬂ1/n≠1
c
– :=
,
4ﬁG
› :=

(2.95)(2.96)

32

General Relativity

then finally we obtain the Lane-Emden equation for polytropes
A

1 d
d◊
›2
2
d›
› d›

B

= ≠◊n .

(2.97)The Lane-Emden equation is a second-order differential equation, it requires two boundary
conditions to solve the system. The two boundary conditions at the centre of a star are
given by-d◊ -◊(0) = 1,= 0.
d› -›=0(2.98)The boundary conditions come from Eq. (2.93) and dP/dr = 0 at the centre of a star(r = 0). In general, we solve the Lane-Emden equation numerically, but can analyticallysolve for n = 0, 1 and 5. For n 5 5, ◊ decrease monotonically and has zero-point at
the finite value of ›. For n > 5, the solution has no zero point, it means that gas isunbound. One solves the Lane-Emden equation from the centre of a star, obtaining
solutions decrease monotonically and eventually becomes zero at a value › = ›1 : thepoint is on the surface of the star, where P = ﬂ = 0. Therefore, one obtains the radius of
the star
R = –›1 =Û

(n + 1)Kﬂ1/n≠1
c
›1 .
4ﬁG

(2.99)

To obtain the total mass, we integrate the mass density over the region
M=

⁄ R
0

C

D

A

B

(n + 1)K (3≠n)/2n
2 d◊ -4ﬁr ﬂdr = 4ﬁ
ﬂc
≠›
.
4ﬁG
d› -›=›12(2.100)Using the Lane-Emden equation, we can estimate the mass of neutron stars and whitedwarfs. For the relativistic case, “ = 4/3, the mass is given by [50]A2
M = 1.457
µe

B2

M§ ,

(2.101)

where µe is a constant which is the number of nucleons per electron. For most whitedwarfs µe ≥ 2 and the maximum mass (2.101) is called the Chandrasekhar limit.

General Relativity

33

Tolman-Oppenheimer-Volkoff Equation
Here we explain the Tolman-Oppenheimer-Volkoff (TOV) equation [69, 70]. The TOV
equation can describe the interior structure of the static, spherically-symmetric perfect
fluid star in general relativity. When investigating the structure of the static, sphericallysymmetric perfect fluid star, the metric and the stress-energy tensor take the form ofEqs. (2.39) and (2.71) respectively. In the previous section, the Einstein tensor withspherically symmetric tensor is computed using Eqs. (2.72 - 2.74). Combining Eqs. (2.72- 2.74) and (2.39), we obtain Einstein’s field equations:1[r(1 ≠ e≠⁄(r) )]Õ e‹(r) = 8ﬁGe‹(r) ﬂ(r),r2‹ Õ (r) e⁄(r)≠ 2 (1 ≠ e≠⁄(r) ) = 8ﬁGe⁄(r) P (r),
rA
r
B
2
Õ
2
Õ
Õ
Õ
Õ
r
‹
(r)
‹
(r)⁄
(r)
‹
(r)
≠
⁄
(r)
‹ ÕÕ (r) +≠+e≠⁄(r) = 8ﬁGr2 P (r),222
r(2.102)(2.103)(2.104)

where the prime denotes the derivative with respect to r. The equations of motion formatter are given by the energy-momentum conservation relation. Since matter is staticand spherically symmetric, the conservation of the energy-momentum tensor implies
Òµ T 1µ = 0.

(2.105)The expression can be simplified to‹ Õ (r)(ﬂ(r) + P (r)) + P Õ (r) = 0.
2(2.106)

We start by evaluating Eq. (2.102). This equation can be simplified to1 ≠ (re≠⁄(r) )Õ = 8ﬁGﬂ(r).

(2.107)

Integrating both parts of Eq. (2.107) from 0 to r, we obtain the solution
⁄(r)

e

A

2GM (r)
= 1≠
r

B≠1

,

(2.108)here we define the mass of a shell with thickness dr as
dM (r) := 4ﬁﬂ(r)r2 dr,(2.109)

34

General Relativity

with the boundary condition M (0) = 0, which is needed to ensure that the metric isregular at r = 0. This equation determines ⁄, also can be written in the formM Õ (r) = 4ﬁr2 ﬂ(r).

(2.110)

We now consider Eq. (2.103). By using the solution Eq. (2.108), we can solve
Eq. (2.103) for ‹ Õ (r) as‹ (r) =ÕA

BA

2GM (r)+ 8ﬁGrP (r)r

2GM (r)1≠
rB≠1(2.111).

Combining Eqs. (2.111) and (2.106), we finally obtainA

G
2GM (r)P (r) = ≠ 2 (ﬂ(r) + P (r))(4ﬁr3 P (r) + M (r)) 1 ≠
r
r
Õ

B≠1

.

(2.112)This equation is known as the TOV equation. Taking the Newtonian limit, namelyc æ Œ, the TOV equation can be written in the form of the Newtonian theory ofhydrostatical equilibrium:P Õ (r) ≥ ≠Gﬂ(r)M (r).
r2(2.113)The Strategy for Solving the TOV Equation
Equations (2.106), (2.110), (2.112), and an equation of state P = P (ﬂ) can determine‹(r), ⁄(r), ﬂ(r) and P (r). M (r) is determined in terms of ‹(r) (2.110). Therefore, theseequations can be solved either analytically or numerically. We explain the strategy forsolving the equations;i) We set the initial value of ‹(r) and ⁄(r) at the centre of the star r = 0 as e‹(0) = ‹cand e⁄(0) = 1 (we should expand the metric around r = 0). We also set the initial valueof ﬂ(r) at r = 0 as ﬂ(0) = ﬂc . The central pressure Pc is determined by the equation ofstate.
ii) We can integrate the equations with these boundary conditions from r = 0.
Equation (2.106) shows that dP/dr < 0, therefore the pressure and the density decrease
gradually. By integrating the equations, the pressure and density take a value of zero ata radius R. R gives the surface of the star.

General Relativity

35iii) The value of ⁄(R) at the star surface is given by e⁄(R) = (1 ≠ 2GM (R)/R), wherethe value of mass function M (R) is the total mass of the star. When demanding thecontinuity of metric functions and their derivatives at the surface of star, the interior
solution is smoothly connected to the outer solution, namely the Schwarzschild metricfor r > R:e‹(r) = e⁄(r) = 1 ≠2GM (R).
r(2.114)

Therefore, we need to somehow find an initial condition ‹c which ensures the above
conditions (2.114). This two-point boundary value can be solved with the shooting
method to find the initial value of the ‹(r). Generally speaking, given equations of state,
we can solve the system numerically with a reasonable initial condition.

Solving the TOV equation with a constant density
Here, we will consider a simple model for a star with a constant density ﬂ = ﬂ0 . In thiscase, we can solve the TOV equation analytically. When the energy density is constant,the mass function M (r) (Eq. (2.110)) is given by:Y3_] 4 ﬁﬂ0 r 3 = M (R) r 3

M (r) = _ 3R[ 4 ﬁﬂ R3 = M (R)
3

0

rÆR

(2.115)r Ø R,

where R is the radius of the star and M (R) is the total mass of the star. Therefore, we
obtainY3__]e⁄(r) = _1

1 ≠ 2GMR(R)r3_[ 1 ≠ 2GM (R)
r

2

4≠1

2≠1

rÆR

(2.116)r Ø R.

Next, we derive the pressure of the star. Substituting Eq. (2.115) into the TOVequation (2.112), we obtainA

4ﬁGr
8ﬁGr2 ﬂ0
P (r) = ≠(ﬂ0 + P (r))(ﬂ0 + 3P (r)) 1 ≠
3
3
Õ

B≠1

.

(2.117)

36

General Relativity

This equation can be integrated:
ﬂ0 + 3P (r)=ﬂ0 + P (r)A

ﬂ0 + 3Pc
ﬂ0 + P cBA

2GM (R)r2
1≠
R3

B1/2

(2.118),where Pc = P (0) is the central pressure of the star. As we noted above, the pressure iszero at the surface, P (R) = 0, so Eq. (2.118) yields
1=A

ﬂ0 + 3Pc
ﬂ0 + Pc

BA

2GM (R)
1≠
R

B1/2(2.119).

Substituting Eq. (2.119) into Eq. (2.118), we obtain

P (r)
=
ﬂ0

A

2GM (R)r2
1≠
R3

A

2GM (R)
3 1≠
R

B1/2

B1/2

A

2GM (R)
≠ 1≠
R
A

B1/2

2GM (R)r2≠ 1≠
R3

B1/2 .

(2.120)Note that the central pressure is given by (r æ 0)
Pc
=
ﬂ0

A

2GM (R)1≠ 1≠
R
A

2GM (R)
3 1≠
R

B1/2

B1/2

(2.121).

≠1

Substituting Eq. (2.120) into Eq. (2.111) and integrating it, then we have
S A

2GM (R)e‹(r) = C Õ U3 1 ≠
R

B1/2

A

2GM (R)r2≠ 1≠
R3

B1/2 T2

V ,(2.122)

here C Õ is a constant of integration. We demand e‹(r) that is matched smoothly with theouter solution Eq. (2.114) at r = R, then we determine C Õ and obtainSA

3
2GM (R)e‹(r) = U
1≠
2
R

B1/2

A

1
2GM (R)r2
≠
1≠
2
R3

B1/2 T2

V .

(2.123)Equations (2.116) and (2.123) provide the description of a constant density star. According
to Eq. (2.121), the central pressure Pc diverges when 2GM/R æ 8/9. In other words, if

General Relativity

37

the star is supported by a finite pressure, the radius of the star should be taken the value
9
R > rg .
8(2.124)2.5. Observational Results and Constraints on
General Relativity
In this section, we will see that general relativity is widely accepted to be the correct
description of the physics of the gravitational field at the local scale and in a weak andstrong gravity regime.

2.5.1. Solar System Constraints
Newtonian gravity has been tested and can describe (not perfectly) the phenomena inthe solar system. Therefore, the corrections of Newtonian gravity can be treated as aperturbation in the solar system. Since any gravity theories should have the Newtonian
limit at the lowest order, the corrections of the Newtonian limit appear in the nextorder. We call these corrections post-Newtonian corrections. In order to specify the postNewtonian corrections for any gravity theory on solar system scales, a very powerful tool
has been developed and used. The tool is called the Parametrized Post-Newtonian (PPN)
formalism [3, 34]. The PPN formalism is a very general method for determining the postNewtonian metric; the PPN formalism provides a model-independent parametrization
for a given theory. That is why the PPN formalism is a very powerful tool for testing
gravity theory in the solar system. The details of the PPN formalism are beyond thescope of this thesis. The interested reader can find a general description for and detailsof the PPN formalism in [3, 34].
PPN Formalism
The PPN formalism was constructed for measuring and comparing observations withtheoretical predictions, and we now use the PPN formalism as the tool to constrain thedeviations from theoretical predictions. The critical feature of the PPN approximations
is the small book-keeping parameter ‘. In the PPN approximations, the matter in thesolar system can be idealized as a perfect fluid, and the velocity and field are measured

38

General Relativity

regarding the parameter ‘. The Newtonian potential U , planetary velocities v, theinternal energy per unit rest mass , pressure P , and density ﬂ are assigned an order ofmagnitude as
U ≥ v2 ≥P≥
ﬂ(2.125)≥ ‘.

As we noted above, the metric is expanded around the Newtonian background and canbe written as a series expansion in powers of ‘. We introduce PPN parameters in frontof each post-Newtonian terms in the metric. The metric is given by as follows
g00 = ≠ 1 + 2U ≠ 2—U 2 ≠ 2›

W + (2 + 2“ + –3 + ’1 ≠ 2›)+ 2(1 + 3“ ≠ 2— + ’2 + ›)2 + 2(1 + ’3 )1

3 + 2(3“ + 3’4 ≠ 2›)

4≠ (’1 ≠ 2›)A ≠ (–1 ≠ –2 ≠ –3 )Ê 2 U ≠ –2 Ê i Ê j Uij + (2–3 ≠ –1 )Ê i Vi + O(‘3 ),(2.126)1
1
g0i = ≠ (3 + 4“ + –1 ≠ –2 + ’1 ≠ 2›)Vi ≠ (1 + –2 ≠ ’1 + 2›)Wi
2
2
1
i
j
5/2
≠ (–1 ≠ 2–2 )Ê U –2 Ê Uij + O(‘ ),(2.127)2
gij =(1 + 2“U )”ij + O(‘2 ),(2.128)where “, —, ›, ’i and –i are the 10 post- Newtonian parameters and the Newtonian
potentialU=⁄ﬂ(xÕ ) 3 Õd x.
|x ≠ xÕ |(2.129)

General Relativity

39

The post Newtonian potentials

W,

i , Vi , Wi , A are listed as follows.

⁄ﬂÕ (x ≠ xÕ )i (x ≠ xÕ )j 3 Õ
d x,|x ≠ xÕ |3A ÕB⁄ Õ ÕÕ
ﬂ ﬂ (x ≠ xÕ )x ≠ xÕÕx ≠ xÕÕ
·d3 xÕ d3 xÕÕ ,
W =
ÕÕ ≠
Õ
ÕÕ
Õ 3
|x ≠ x | |x ≠ x ||x ≠ x |Uij =ﬂÕ [v Õ · (x ≠ xÕ )]2 3 Õd x,|x ≠ xÕ |3⁄
ﬂÕ v Õ23 Õ
1 =Õ d x,|x ≠ x |⁄ﬂÕ U Õ
=d3 xÕ ,
2|x ≠ xÕ |⁄ﬂÕ Õ=
d3 xÕ ,
3|x ≠ xÕ |
⁄
pÕ
3 Õ
4 =
Õ d x,|x ≠ x |⁄ﬂÕ viÕ
3 ÕVi =Õ d x,|x ≠ x |⁄ Õ Õ
ﬂ [v · (x ≠ xÕ )](x ≠ xÕ )i 3 Õ
Wi =
d x.
|x ≠ xÕ |3
A=⁄(2.130)(2.131)(2.132)(2.133)(2.134)(2.135)(2.136)(2.137)(2.138)The details of the ten PPN parameters are listed in table 2.1. General relativity
corresponds to the case “ = — = 1 and the rest of the terms ›, –i , and ’i are zero. The
PPN formalism has been developed and used to test the theory of gravity in a weak field.

40

Parameter

General Relativity

What it measures

Value ingeneral relativity“

Space-curvature gij produced by unit rest
mass

1

—

Non-linearity in the superposition law forgravity1›

Preferred-location effects

0–1

Preferred-frame effects

0

–2

0

–3

0

–3

Violation of conservation of total momentum(energy, angular momentum)

0

’1

0

’2

0

’3

0

’4

0

Table 2.1.: The PPN parameters. General relativity corresponds to the case “ = — = 1 [34].

General Relativity

41

Next, we will discuss the measurements of phenomena such as light deflection aroundthe Sun and the perihelion shift of Mercury.

Light Deflection around the Sun
The deflection angle (”◊) of a light ray which passes the Sun at a distance d is given by1
4M§ 1 + cos Ï”◊ = (1 + “),2
d2

(2.139)where M§ is the solar mass and Ï is the angle between the Earth-Sun line and the incoming
direction of the light ray. The observational results obtained by the very-long-baseline
radio interferometry yields a value for “ [77, 78]:
“ ≠ 1 = (≠1.6 ± 1.5) ◊ 10≠4 .

(2.140)The Shapiro Time Delay
The gravitational time delay experiment was proposed by Irwin Shapiro in 1964 [79].
Since a massive object, such as the Sun, curves spacetime, the radar signals (light rays)
which pass close to the massive object take a longer time to reach the observer than theradar signals travelling the same distance in vacuum. This time delay effect is called theShapiro time delay and given by as follows
SA(1 + “) U
d”t = ≠
240µs ≠ 20µs ln
2
R§

B2 A

BT

1AU V
,
rp

(2.141)

where d is the distance of closest approach of the radar signals, and rp is the distance
between the planet and the Sun. The bound on the value of “ is obtained from theDoppler tracking of the Cassini spacecraft on its way to Saturn [80]:“ ≠ 1 = (2.1 ± 2.3) ◊ 10≠5 .

(2.142)The Perihelion Shift of Mercury
As stated before, the advance of the perihelion of Mercury [33] was measured in the
19th century, and remained an unsolved problem in celestial mechanics for some time.

42

General Relativity

Several hypotheses had been proposed to explain it, for example, the existence of a planet
Vulcan near the Sun. However, the mysterious phenomenon remained unsolved until theadvent of general relativity. General relativity allows that the advance of the perihelionof Mercury value results from the curvature of spacetime around the Sun.
Taking into account the PPN contributions and the solar quadrupole moment J2(see [81] for a review), we obtain the advance of the perihelion of MercuryÊ̇ = 42.ÕÕ 98

3

4

1
J2
(2 + 2“ ≠ —) + 3 ◊ 10≠4 ≠7,
3
10(2.143)where J2 = (2.2 ± 0.1) ◊ 10≠7 . Combining the bound on “ [80], we obtain a bound on —— ≠ 1 = (≠4.1 ± 7.8) ◊ 10≠5 .

(2.144)

Other PPN parameters are well studied, see [34]. According to the above measurements,
the predictions based on general relativity are in good agreement with tests performedin the solar system.

2.5.2. Gravitational Waves
Einstein predicted the existence of gravitational waves 100 years ago [82, 83], but the firstobservation of gravitational waves was announced by LIGO in 2015 [4]. The discovery
confirmed a prediction of general relativity once more. In this section, we briefly review
gravitational waves.

Linearized General Relativity
First we study the linearization of Einstein’s field equations in the vacuum: We discuss
how the gravitational wave equation is derived from Einstein’s field equations.
Far away from compact objects, the gravitational field is weak. Therefore thespacetime geometry is nearly flat. Here, we consider the small perturbation hµ‹ on theflat background (Minkowski spacetime):gµ‹ = ÷µ‹ + hµ‹ , |hµ‹ | π 1.

(2.145)

General Relativity

43

The perturbation hµ‹ is defined on the background spacetime. We will work to linear orderin hµ‹ . Therefore the perturbation indices are raised and lowered using the background
Minkowski metric and its inverse.
To first order in hµ‹ , the Christoffel symbols are given by–1 –µ÷ (ˆ“ hµ— + ˆ— hµ“ ≠ ˆµ h—“ )2
1= (ˆ“ h– — + ˆ— h– “ ≠ ˆ – h—“ ).
2—“ ƒ(2.146)The Rich tensor (2.25) is reduced to,Rµ‹ = ˆ– – µ‹ ≠ ˆ‹ – µ– + – ⁄– ⁄ µ‹ ≠ – ⁄‹ ⁄ µ–1
ƒ (ˆµ ˆ– h– ‹ + ˆ – ˆ‹ hµ– ≠ ⇤hµ‹ ≠ ˆµ ˆ‹ h),2(2.147)

here h is the trace of the perturbation hµ‹ :h := h– – := ÷ –µ hµ– .

(2.148)Contracting Eq. (2.147) with ÷ µ‹ , we obtain the Ricci scalar:
R = R— — ƒ ˆ – ˆ — h–— ≠ ⇤h.

(2.149)Combining Eqs. (2.147) and (2.149), we obtain the linearized Einstein tensor
2Gµ‹ ƒ ˆ‹ ˆ – hµ– + ˆµ ˆ – h‹– ≠ ÷µ‹ ˆ – ˆ — h–— ≠ ⇤hµ‹ = 0,(2.150)where we introduce1hµ‹ := hµ‹ ≠ ÷µ‹ h.
2(2.151)

The reduced perturbation hµ‹ satisfies
h := h µ = ≠h,1hµ‹ = hµ‹ ≠ ÷µ‹ h.
2µ(2.152)(2.153)

We can further simplify the field equations by performing a gauge transformation.
One considers the following infinitesimal coordinate transformation xµ æ xµ + › µ , where44

General Relativity› is an arbitrary small vector field. Under this transformation, the perturbation hµ‹transforms as
hµ‹ æ hµ‹ ≠ ˆµ ›‹ ≠ ˆ‹ ›µ ,(2.154)hµ‹ æ hµ‹ ≠ ˆµ ›‹ ≠ ˆ‹ ›µ + ÷µ‹ ˆ – ›– .

(2.155)and hµ‹ transforms as

The transformed divergence of hµ‹ is given byˆ – h–— æ ˆ – h–— ≠ ⇤›— .

(2.156)One can find a gauge in which the divergence vanishes so that⇤›— = ˆ – h–— .

(2.157)

Therefore one can impose the following gauge condition on the field equations
ˆ – h–— = 0.

(2.158)In this gauge, the linearized Einstein field equations are given by⇤hµ‹ = 0.

(2.159)This equation represents the wave equation. The fluctuation of the spacetime propagatesas the wave in spacetime: hµ‹ represents gravitational waves. By performing gaugetransformations, we find that gravitational waves have two physical degrees of freedom.
This fact indicates that gravitational waves have only two polarization modes in theframework of general relativity. The existence of gravitational waves was first predicted
by Einstein in the framework of general relativity [82, 83].
Indirect Evidence for Gravitational Waves
The discovery of the Hulse-Taylor binary neutron star system [84], PSR B1913+16,
provided the first indirect evidence of gravitational waves. The Hulse-Taylor binary
neutron star system contains a neutron star and a pulsar, which emits a beam of radio
waves.

General Relativity

45

General relativity predicts gravitational radiation from binary stars, which carry theenergy and angular momentum out of the binary system (see [85], for a details). The
binary orbit shrinks due to the emission of gravitational radiation. The orbital period
decay Ṗb can be estimated using general relativity and is given by
192ﬁṖb = ≠
5

3

Pb
2ﬁ

4≠5/3

5/3
T§ m1 m2 M ≠4/3

3

4

73
37
1 + e2 + e4 (1 ≠ e2 )≠7/2 ,24
96(2.160)where e is the eccentricity, m1 and m2 are the masses of the compact stars, M = m1 + m2
is the total mass of the system and T§ = GM§ . See the details and calculations in [85].
This binary neutron star system allowed researchers to measure precise timing properties, such as an orbital period decay (2.160). Hulse and Taylor measured the rate ofchange of the orbital period for thirty years [86, 87] which showed a good agreement
between the observational and theoretical calculated value, according to general relativity.
The continued measurements of the binary pulsar system provide observational values
for Ṗb . The ratio of the observed value for Ṗb , Ṗbobs , to the predicted orbital period decayṖb computed from (2.160) is given byṖbobs= 1.0013 ± 0.0021,
ṖbGR(2.161)the agreement of the values is shown in Fig. 2.1. This represents indirect evidence forthe existence of gravitational waves which led to the 1993 Nobel Prize in Physics for
Hulse and Taylor.

Direct Detection of Gravitational Waves
Einstein predicted the existence of gravitational waves 100 years ago. In strong gravitational fields such as those found at neutron stars, the weak field approximation is not
valid. To describe this situation properly we need to solve the field equations numerically.
This field of study is known as numerical relativity (see [88–90] for examples). By using
this numerical method, we can evaluate gravitational wave forms for the binary blackhole and the neutron star. Many astronomers also tried to detect gravitational waves
directly. The first direct discovery of gravitational waves from a binary black hole merger
has been made by the LIGO (Fig. 2.2) [4]. This observation has directly confirmed
gravitational waves predicted by general relativity and the existence of (massive) black46

General Relativity

Figure 2.1.: Orbital decay of the Hulse-Taylor binary pulsar over 30 years of observation.
The points are experimental data and the solid line shows the cumulative shift
predicted by general relativity. Experimental results fall on the theoretical curve.
This is evidence for the existence of gravitational radiation from the binary
neutron stars. Figure from [87].

holes. A black hole merger does not generally emit electromagnetic radiation, therefore
we cannot observe them directly through electromagnetic telescopes.
The first detection of gravitational waves led to the 2017 Nobel Prize in physics forthe three physicists who led the LIGO experiment. This detection opened the door
to gravitational wave astronomy and also provided us the first glimpse into the strong
gravity regime.
Recently, the first direct detection of gravitational waves from a binary neutron starmerger, GW170817, has been made by the LIGO and Virgo collaborations [5]. The“-ray counterpart to GW170817, GRB 170817A, has been independently confirmed by
the Fermi Gamma-ray Burst Monitor and the International Gamma-Ray Astrophysics
Laboratory [91–93]. This detection opens the door to multi-messenger astronomy [94].
Using gravitational radiation from binary neutron stars, we can study the internal
structure of neutron stars, namely the equation of state. Needless to say, gravitational
wave measurements have become a new tool to test gravity and high energy physics
theories.

General Relativity

47

Figure 2.2.: The gravitational wave events GW150914 observed by LIGO Hanford (left column
panels) and Livingston (right column panels).Top row: Detector strain from thetwo sites Middle row: Solid line shows the numerical relativity model that is
consistent with GW150914, light grey line shows reconstructed strain signal, anddark grey line shows binary black hole merger template waveform Bottom line:
Residuals after subtracting the filtered numerical relativity waveform from thefiltered detector time series. Caption and figure from [4].

48

Chapter 3.
Modified Gravity
In recent decades, numerous gravity theories have been proposed. Exploring modified
gravity provides insight into aspects of general relativity, and may provide hints forunveiling the nature of gravitational theory and dark energy. This chapter is basedon [49, 95–98].

3.1. Beyond Einstein
As noted in the previous section, general relativity is a simple gravitational theory
and has been tested by several independent observations [3, 34]. However, the -CDM
model which is based on general relativity suffers from theoretical difficulties such as thecosmological constant problem. The accelerated expansion of the Universe cannot be
explained within general relativity without introducing dark energy, it leads to fine-tuning
problems. This may imply that general relativity needs to be modified on cosmologicalscales.
As we noted in Chapter 1, by studying modified gravity, the two following interests canbe considered: i) We may obtain a new perspective or guiding principle for constructing
quantum gravity. General relativity is not renormalizable in quantum field theory. This
indicates that general relativity should also be modified at a high energy scale. ii) We
can approach a more complete theory of gravity by investigating the differences between
these modified gravity theories. In fact, from a theoretical perspective, there is no reason
to consider the Einstein-Hilbert action as a fundamental action for gravity. By studying
modified gravity, we may understand why general relativity provides a good descriptionof gravity.
49

50

Modified Gravity

3.2. Lovelock’s Theorem -Guiding Principles for
Modified Gravity-

David Lovelock states that Einstein’s field equations with a cosmological constant are
unique second-order equations for the metric in four-dimensional space (Lovelock’s
theorem [99, 100]). Lovelock’s theorem indicates how we can obtain field equations forgravity which differ from Einstein’s field equations. This is why Lovelock’s theorem is a
guiding principle for modifying general relativity.
We demand that the action for a theory of gravity gives equations having up to second
order derivatives of the metric tensor, therefore we consider the action that depends onthe metric and its derivatives;
SLL =

⁄

d4 xL(gµ‹ , ˆ– gµ‹ , ˆ– ˆ— gµ‹ ).

(3.1)

Varying the action with respect to the metric”SLL =⁄

d4 x”g µ‹ Eµ‹ [L],

(3.2)

where
C

d
ˆL
d
E [L] = –≠ —dx ˆ(ˆ– gµ‹ ) dx
µ‹

A

ˆL
ˆ(ˆ– ˆ— gµ‹ )

BD

≠

ˆL
= 0.
ˆgµ‹

(3.3)

Lovelock’s theorem states that the only second-order field equations which can be obtained
from the action (3.1) in four dimensions are Einstein’s field equations
E [L] =µ‹

Ô

5 3≠g – Rµ‹

461≠ g µ‹ R + g µ‹ = 0,
2

(3.4)

where – is a constant. Note that, the Einstein-Hilbert term is not the only terms whichgive the expression Eq. (3.4). The action of the form
SGE =

⁄

Ô
d4 x ≠g(–R ≠ 2 + —G + “H),(3.5)

Modified Gravity

51

also gives the Einstein field equations, where “ and — are constants. Here we define H
and G as follows:
G := R2 ≠ 4Rµ‹ Rµ‹ + Rµ‹ﬂ‡ Rµ‹ﬂ‡ ,H := ‘µ‹ﬂ‡ R–— µ‹ R–—ﬂ‡ .

(3.6)
(3.7)

In four dimensions, both of them satisfy
E µ‹ [G] = 0,(3.8)

E µ‹ [H] = 0,

(3.9)

therefore, these terms do not contribute to the field equations in four-dimensional
spacetime. The former, called the Gauss-Bonnet term, does not contribute to the field
equations in four dimensions. The latter does not contribute to the field equations in any
number of dimensions. Lovelock’s theorem states that, in four dimensions, Einstein’s
field equations are unique second-order equations of the derivatives of the metric, andthe Einstein-Hilbert action is not the only action giving Einstein’s field equations.
From a mathematical perspective, we can consider other actions for gravity; we do
not need to consider the Einstein-Hilbert term as a fundamental action for gravity. As
we discussed above, Lovelock’s theorem states that, in four dimensions, in order to obtainthe field equations for a theory of gravity which differ from the Einstein field equations(or to modify the Einstein’s field equations), one should consider several possibilities asfollows:• Class I: Additional fields
The simplest modification to general relativity is to add an extra degree of freedom.
The metric tensor is directly coupled to the degrees of freedom and changes Einstein’s
gravity field equations. Theories with scalar fields (scalar-tensor theories, which
we will discuss below), vector fields (e.g., Proca theories [101–103]), and tensor
fields (for example TeVeS [104] and bimetric theory [105]) have been proposedas modifications to general relativity. However, one should avoid any instability
associated with the new degrees of freedom. One should pay attention to theoreticalconsistency: the existence the ghost instability, gradient instability, and Tachyonicinstability. (We will review this theoretical consistency below.)• Class II: Lorentz-violating
Lorentz invariance has been tested and verified by several experiments [106–108]

52

Modified Gravity

and is widely accepted in modern physics. However, one can construct a new classof gravity theories by breaking it. One of the classes of Lorentz-violating theories is
Horava-Lifshitz gravity [109, 110]. Due to the presence of a preferred time direction,
Lorentz invariance is broken, but the invariance is recovered at low energy levels.
Horava pointed out that the violation of Lorentz invariance leads to a modificationin graviton propagation in the ultraviolet regime. Horava gravity can be thought
to be a power-counting renormalizable theory of gravity [109]. Another type of
this class is Einstein-Ather theories [111]. The presence of a timelike vector breaks
Lorentz invariance. It is known that Class II theories fall within Class I theories.
• Class III: Higher dimensions
Lovelock’s theorem states that if one considers extra dimensions, the EinsteinHilbert action is not the only action giving Einstein’s field equations. It is possible
to construct a theory of gravity in higher spacetime dimensions, for example,
models based on the Gauss-Bonnet term and its generalizations. Exploring higherdimensional theories of gravity also raises theoretical interests in understanding
how the field equations depend on the dimensions of spacetime. Observational
measurements put constraints on the higher-dimensional effects, so the theory is
required to have a mechanism to hide them. There are multiple mechanisms, forexample, compactification [112] and braneworld [113, 114]. It is known that Class
IV theories, e.g., Kaluza-Klein and the Dvaili-Gabadadze-Porrati model, have some
limits: By reducing the dimensions D from D > 4 to four dimensions, an additional
scalar and gauge field appear. That is why higher-dimensional theories of gravity
can be effectively described by Class I theories.
• Class IV: Higher derivatives
If one considers some higher derivative terms in action, one can evade Lovelock’s
theorem. However, most such theories suffer the Ostrogradsky instability described
below in Ostrogradsky’s theorem [115, 116]. According to Ostrogradsky’s theorem,
if a nondegenerate action contains second and higher derivatives, the system isunstable. However, there is a loophole to evade this theorem: if the theory isdegenerate, the system is stable. We will describe the details below.
• Class V: Non-local gravity
One can construct the non-local theories by introducing inverse powers of the
Laplacian operator (e.g., R · f (⇤≠1 R)) [117]. Non-local term involving with the
Ricci tensor gives rise to cosmological instability [118]. Thus, most of the class areconstructed using the Ricci scalar such as R · f (⇤≠1 R)). Indeed, theories of non-local

Modified Gravity

53

gravity have been proposed to include quantum corrections in the Einstein-Hilbert
action [119].
Note that we can further consider some extension of general relativity: massive gravity [120, 121] etc. Most of these theories fall into the above classes within some limits,
mostly Class I.
As we review the tests of general relativity, general relativity predicts many physicalphenomena and agrees with all current experiments and observations in the solar system.
Therefore, a new or viable theory of gravity should restore general relativity on a small
scale such as the solar system. The theory should also explain the late-time accelerationof the Universe. In this thesis, we concentrate on and study theories which contain an
additional field. As we stated above, most modified gravity can be described by Class I
theories featuring additional fields.
We also require that the theory should satisfy several theoretical consistencies. In thenext section, we will briefly review these theoretical consistencies.

3.3. Theoretical Consistency
If we consider modified gravity, the system may appear to have some instabilities (orpathologies), namely ghost, gradient, and tachyonic instabilities. In constructing theories,
we must ensure that the theory is free from these instabilities in any frame. We describe
these instabilities in detail below.

3.3.1. Ghost Instability
The ghost instability is associated with ghosts (see [122] for a review). The ghost is afield entering an action and kinetic term with a wrong sign, for example
1
m2
Lghost = (ˆ )2 ≠
2
2

2

.

(3.10)Such a field has negative energy, the Hamiltonian is unbounded below, which indicatesan instability in an interacting system of the ghost field and another.

54

Modified Gravity

Let us consider the following Lagrangian with the field „ and the ghost field
m2
1
1
m2
L = ≠ (ˆ„)2 ≠ „ „2 + (ˆ )2 ≠
2
2
2
2

2

+ ⁄„2

2

.

:
(3.11)

The field „ has a kinetic term of the correct sign, while the field
has a wrong sign
kinetic term of the wrong sign. The wrong sign kinetic term leads to the the field
negative energy. When the interaction term (⁄„2 2 ) vanishes, the field „ does notexhibit instability. If the field is coupled to „ with the correct sign, the energy of theghost field is transferred to the other field „ via the interaction term. The particle
creation process of negative energy together with positive energy is permitted. However,the process costs zero energy and the production rate is infinite: it causes an unstablevacuum [123]. In constructing theories, the existence of a ghost field indicates that thetheory is ill-defined.

3.3.2. Gradient Instability
Wrong spatial derivatives also give rise to another instabilities, called the gradientinstability. To see this pathology of the field theory clearly, we consider the simple
example of the Lagrangian for a scalar field with spatial gradients of the wrong sign1
1
L = „˙ 2 + (ˆ„)2 .
2
2(3.12)In Fourier space, the solutions to the equation are„k (t) ≥ e ± kt ,(3.13)

where k := |k|. The growing solution ekt grows exponentially without bound, the timescale of the growing mode istinst ≥ k ≠1 .
Therefore, the high energy mode leads to an instability.

(3.14)

Modified Gravity

55

3.3.3. Tachyonic Instability
Another instability that appears in modified gravity is the presence of a tachyon. A
tachyonic instability [124] appears as a field with a negative mass term. We consider atypical Lagrangian: a scalar field with a negative mass1
m2 2
L = ≠ (ˆ„)2 +„.
2
2(3.15)Taking the long-wavelength k æ 0 limit, the solution is given by„(t) ≥ e ± mt .

(3.16)The solution indicates an instability. However, unlike the gradient instabilities, thetimescale of the instability is bounded and is given by
tinst ≥ m≠1 .

(3.17)The timescale is independent of k. Unlike the gradient instability, the tachyonic instability
is present only for long wavelengths. Therefore, gradient instabilities are more dangerous
than tachyonic one.

3.3.4. Ostrogradsky Ghosts
One could construct higher-derivative theories, in most cases, these theories are pathological: they suffer from the Ostrogradsky instability [115, 116, 125]. It is known from Ostrogradsky’s theorem [115]: if a nondegenerate Lagrangian (the matrix Mij = ˆ 2 L/ˆ q̇ i ˆ q̇ j
is not invertible, where q̇ denotes the velocity) contains higher derivatives, the system isunstable.1
In this section, we will show the theorem and why such a higher derivative term
leads to instability in the context of classical mechanics. For simplicity, we consider the
Lagrangian which is assumed to be L = (q, q̇, q̈). It depends on the position q of a point
1

In other words, a nondegenerate Lagrangian depends on higher derivatives, the Hamiltonian of thesystem contains linear momenta, thus, is necessarily unbounded.

56

Modified Gravity

particle as a function of time. The Euler-Lagrange equation is given by
ˆL
d ˆL
d2 ˆL
≠
+ 2
= 0.
ˆq
dt ˆ q̇dt ˆ q̈(3.18)For a nondegenerate Lagrangian, this equation contains fourth derivatives of q, thus
solutions depend on four independent initial data. In the Hamiltonian formulation, we
need four independent canonical variables. Ostrogradsky’s choices are
ˆL
d ˆL
≠
,
ˆ q̇
dt ˆ q̈ˆL
Q2 = q̇, P2 =.
ˆ q̈
Q1 = q, P1 =(3.19)(3.20)The Lagrangian is nondegenerate, so we can solve for q̈ in terms of the canonical variables
Q1 , Q2 and P2 ; that is to say there exists a function F (Q1 , Q2 , P2 ) which is given by-ˆL -= P2 .
ˆ q̈ -q=Q1 ,q̇=Q2 ,q̈=F(3.21)The Hamiltonian is obtained by the Legendre transformation,
H(Q1 , Q2 , P1 , P2 ) = P1 q̇ + P2 q̈ ≠ L= P1 Q2 + P2 F (Q1 , Q2 , P2 ) ≠ L(Q1 , Q2 , F (Q1 , Q2 , P2 )).

(3.22)

Here we observed that the Hamiltonian is linear in the canonical momentum P1 ; theHamiltonian is unbounded below, which means that this system is unstable.
The above discussion is general, but it is also useful to consider an example. We
consider a higher derivative theory:L=≠g 2 1 2 Ê2 2q̈ + q̇ ≠ q ,
2
2
2Ê 2

(3.23)

where g and Ê are constants. The dimensionless parameter g quantifies its deviationfrom a simple harmonic oscillator. The Euler-Lagrange equation and solution for (3.23)
areg ....
q + q̈ + Ê 2 q = 0,2
Êq(t) = C+ cos (k+ t) + S+ sin (k+ t) + C≠ cos (k≠ t) + S≠ sin (k≠ t),(3.24)(3.25)

Modified Gravity

57

where
k±

Ô
ı̂
ı 1 ± 1 ≠ 4g
Ù
=Ê
,
2g

2
2
k≠
q0 + q¨0k+
q + q¨
C+ = 2C≠ = 2 0 2 0 ,2 ,k≠ ≠ k+k+ ≠ k≠...
...
22
k≠ q̇0 + q0
k+
q̇0 + q0
S+ =
,
S
=
≠
2
2
2
2 ,
k+ (k≠≠ k+)k≠ (k+≠ k≠)(3.26)(3.27)(3.28)...
where q0 , q˙0 , q¨0 and q0 are the initial data. For this theory, Ostrogradsky’s canonicalmomenta are given by
g
q̈,
Ê2
g
P2 = ≠ 2 q̈.
Ê
P1 = q̇ +(3.29)(3.30)The Hamiltonian can be recast in terms of the canonical variables
Ê2 2 1 2 Ê2 2
P2 ≠ Q2 + Q1
2g
2
2
1Ô
1Ô
2
2
=
1 ≠ 4gk+(C+2 + S+2 ) ≠1 ≠ 4gk≠(C≠2 + S≠2 ).
2
2

H = P1 Q2 ≠(3.31)The momentum P1 appears linearly in the Hamiltonian: the Hamiltonian shows that the+ mode has positive energy and the ≠ mode has negative energy: namely there exists a
ghost state. As we mentioned above, such a negative energy mode allows for a state with
arbitrarily high energy, which excites other states with the same amplitude and opposite
sign. This propagating mode with negative energy is called the Ostrogradsky ghost.
Ostrogradsky’s theorem states that any nondegenerate higher derivative theories
exhibit ghost instabilities. In order to avoid the presence of the Ostrogradsky ghost we
require that the matrix Mij = ˆ 2 L/ˆ q̈ i ˆ q̈ j is not invertible. This is called the degeneracy
condition.2 Imposing the condition, the unstable mode can be eliminated (throughintegration by parts), and the equations of motion reduces to a second-order one. We
will discuss an example of this theory later.

2

In this example theory, the degeneracy condition is ˆ 2 L/ˆ q̈ 2 = 0, which means that g = 0

58Modified Gravity

3.4. Scalar-Tensor Theory
In the framework of general relativity, the gravitational field is mediated by a metric tensor
gµ‹ , or a massless spin-2 particle, the graviton. It is possible that gravity is mediated byother degrees of freedom. The simplest model that one could consider in this context
would be other scalar fields, vector fields, and tensors. In this thesis, we consider thescalar-tensor theory, in which the gravitational field is mediated by scalar fields together
with gravitons. Scalar-tensor theory is one of the well-studied alternative theories ofgravity as a simple modification of general relativity. The scalar-tensor theory arises
naturally as the effective theory of these higher-dimensional theories (e.g., Kaluza-Klein
theory [112]). That is why the scalar-tensor theory can be viewed as the low energy limitof string theory; many modified gravity theories can be described effectively by adding ascalar degree of freedom. The simplicity of scalar-tensor theory allows us to solve theequations for many interesting physical phenomena in cosmology and astrophysics. That
is why the scalar-tensor theory has played an important role in providing the framework
for testing gravity. Therefore, it is important to explore the aspects of scalar-tensor
theory.

3.4.1. Brans-Dicke Theory
The prototype of scalar-tensor theory has been proposed by Fierz [126], Jordan [127],
Brans and Dicke [28], which is now referred to as FJBD theory, or Brans-Dicke theory(hereinafter, this will be called the Brans-Dicke theory.) This theory contains an extra
scalar field „, one free parameter Ê and the Einstein-Hilbert action. In the limit Ê æ Œ,
general relativity is recovered. The Brans-Dicke theory is originally motivated by Mach’s
Principle (which states that inertial forces experienced by a body are determined by
the environment, such as the distribution of matter in the Universe.) The scalar field
non-minimally couples to the metric, namely gravity, and modifies the gravitational
constant G; the gravitational constant depends on the scalar distribution in the Universe.
The Brans-Dicke theory is known as one of the simplest scalar-tensor theories.
The action of Brans-Dicke theory is defined as follows:
J
SBD
=

⁄

C

A

BD

1
Ê
d x ≠g„R ≠ 0 Òµ „Òµ „
16ﬁG
„
4

Ô

+ Sm [gµ‹ ,m ],(3.32)

Modified Gravity

59

where Ê0 is a constant parameter which controls the strength of interactions between thescalar field and matter fields, and m represents the matter field. From the action (3.32),
one can redefine an effective gravitational constant,Geff =

G.
„(3.33)

That is why, Brans-Dicke theory was proposed as a theory of gravity based on Mach’s
principle.
The matter field couples to the metric and does not couple to the scalar field. The
first term of Eq. (3.32) is the gravitational action. The scalar field non-minimallycouples to the Ricci scalar as well as the metric tensor. The scalar field contributes
to the field equations via the metric tensor. Given the action in this form (3.32), theenergy-momentum tensor of the matter fields is conserved, as we proved in Chapter 2.
This frame is referred to as the Jordan frame, and the metric tensor gµ‹ is called the
Jordan-frame metric. Indeed, test particles follow the geodesics of that metric.
Nordtvedt, Bergmann, and Wagoner have generalized Brans-Dicke theory by replacing
the parameter Ê with a potential function of „ as follows [128–130]:
J
SGBD
=

⁄

C

A

BD

1
Ê(„)
d x ≠g„R ≠Òµ „Òµ „ ≠ U („)
16ﬁG
„
4

Ô

+ Sm [gµ‹ ,m ],(3.34)

where U („) is the potential function of the scalar field. When Ê(„) = ÊBD is constantand U („) = 0, the theory reduces to Brans-Dicke theory.
It is well known that f (R) gravity [131] is equivalent to a specific class of scalar-tensor
theory, including Brans-Dicke theory with Ê = 0 in the Jordan-frame. F (R) gravity is afunction of the scalar curvature R, and one of the simplest extensions of general relativity.
For the interested reader, a brief review of f (R) theory can be found in Appendix B.

Conformal Transformation
In the Jordan frame, a scalar field is directly coupled to the Ricci scalar. By transforming
the frame, we can eliminate the non-minimal coupling between the scalar field and the
Ricci scalar. This transformation is referred to as the conformal transformation and60

Modified Gravity

defined as follows:gµ‹ æ ĝµ‹ := (x)2 gµ‹ ,(3.35)

where (x) is an arbitrary function. Under the conformal transformation, the Jordanframe is transformed into another frame. This other frame is called the Einstein frame
which is described by the Einstein-frame metric ĝµ‹ . Hereafter, we will denote thequantities that are expressed in the Einstein frame with a hat. As we will see below,
the Ricci scalar which enters into the action is minimally coupled to the scalar. The
transformation changes the line element as
ds2 æ dŝ2 = ĝµ‹ dx̂µ dx̂‹ = (x)2 ds2 .

(3.36)The transformation changes the distance between any two points on the manifold but
does not change the angle of any two vectors. Therefore, this transformation is called theconformal transformation. Hereafter, we define the conformal transformation as follows:gµ‹ æ ĝµ‹ = f („)gµ‹ ,(3.37)

where f („) is a function of the scalar field „. Under the conformal transformation, thevolume element transforms as (see Appendix A for the conformal transformation rules ofgeometric quantities)ÔÒ≠g = f („)≠2 ≠ĝ.

(3.38)

We input the geometrical quantities, thus the action is transformed as:Ô≠g„R =

Ò≠ĝ(„f („)≠1 R̂ . . .).

(3.39)In order to remove the scalar term from the coefficient of the Ricci scalar, we choosef („) = „. The kinetic term of the scalar field should be canonical, so we redefine thescalar field as
ln „ =

Û

16ﬁG.
3 + 2Ê0(3.40)

Modified Gravity

61

Finally, we obtain the Brans-Dicke action in the Einstein-frame as follows:
E
SBD
=

⁄

A

2
MPl
1
d x ≠ĝR̂ ≠ Ò̂µ Ò̂µ
2
2
4

Ò

B

+ Sm (e2—„/MPl gµ‹ ,m ),(3.41)
Ò2
where MPl:= (8ﬁG)≠1 is the Planck mass and we have introduced 2— := 1/ 3/2 + Ê0 forJE
convenience. The two actions SBD
and SBDare connected via the conformal transformation, thus the corresponding field equations are related by the transformation. Therefore,
the Jordan-frame and Einstein-frame are mathematically equivalent.

Field Equations
We will derive the field equations in the Jordan-frame (3.32) and the Einstein-frame(3.41).
Substituting F (R) = „R into Eq. (B.7), we obtain the tensor field equations asfollows:„Gµ‹ ≠

56

Ê01
ˆµ „ˆ‹ „ ≠ gµ‹ (Ò„)2 + (gµ‹ ⇤ + Òµ Ò‹ )„ = 8ﬁGTµ‹ .
„2(3.42)

Varying the action (3.32) with respect to „ yields the equations of motion for a scalar
field in the Jordan-frame„R + 2Ê0 ⇤„ ≠Ê0(Ò„)2 = 0.
„(3.43)

Using Eq. (3.42) and Eq. (3.43), we eliminate the Ricci scalar and obtain the following
equation:
⇤„ =

8ﬁG
T.
2Ê0 + 3(3.44)

Varying the action (3.41) we obtain the tensor field equations in the Einstein-frame(3.41)
Ĝµ‹ = ˆµ ˆ‹1≠ ĝµ‹ ˆ⁄ ˆ ⁄ + 8ﬁGT̂µ‹ ,2(3.45)

where the energy-momentum tensor T̂µ‹ is defined in the Einstein-frame metric ĝµ‹ .
Varying Eq. (3.41) with respect to the „, we obtain the equations of motion for the scalar

62

Modified Gravity

field
ˆ =
⇤

Û

4ﬁG
—
T̂ =
T̂ ,
2Ê + 32MPl(3.46)

where —, which is defined in (3.41), is the coupling function which controls the strengthof the coupling between the scalar field and matter. In the limit Ê æ Œ, the coupling
function becomes zero, and Brans-Dicke theory restores general relativity.
The PPN parameters of Brans-Dicke theory with an arbitrary function Ê(„) are:“=1 + Ê(„0 ),2 + Ê(„0 )Ê Õ („0 )— =1+,(3 + 2Ê(„0 ))2 (4 + 2Ê(„0 ))(3.47)(3.48)

where „0 is the present value of the scalar. The bound on the PPN parameter “ in theBrans-Dicke theory, as given in [3, 34], leads to a bound on the Brans-Dicke parameter ÊÊ(„0 ) > 40000.

(3.49)

Although Ê suffers from the constraint, Brans-Dicke theory can explain the current
accelerated expansion of the Universe without introducing exotic matter or a cosmologicalconstant [132].
Typically, dimensionless coupling parameters such as Ê are expected to be of orderunity. From this, we can consider that the models of modified gravity satisfy the following
condition: a new degree of freedom should be screened in the solar system, while drives
the acceleration of the Universe today. This mechanism is called the screening mechanism.
We will explain the mechanism further in Chapter 4. When modified gravity is equippedwith the screening mechanism, the theory naturally evades the constraints coming fromsolar system observations (see Chapter 4).

3.4.2. Generalized Galileon Theory -Horndeski TheoryOver the decades, several studies have been devoted to finding and constructing a moregeneral scalar-tensor theory having the second-order field equations. Generalized Galileon
theory [26, 133, 134] includes all possible terms which satisfy the Galilean symmetry, andits equations of motion are second order equations in an arbitrary number of dimensions.

Modified Gravity

63

In flat spacetime, the Galileon theory is invariant under the generalization of Galilean
symmetry in the scalar field ﬁ(x):ﬁ(x) æ ﬁ(x) + bµ xµ + c,

(3.50)

which is called the Galileon shift symmetry. Scalar fields that respects Galilean shift
symmetry are called Galileon. This symmetry initially was found in the decoupling limitof the Dvali-Gabadadze-Porrati (DGP) braneworld model [113, 114, 135]. The non-linear
derivative coupling term such as ⇤„(Ò„)2 arise from the DGP model. The term recovers
general relativity on small scales3 .

Galileon theory in flat spacetime
Flat-spacetime Galileon theory was inspired by the DGP model and developed in [26],and later generalized to curved spacetime [133, 134]. Flat-spacetime Galileon theory has
five Lagrangians that leads to second-order equations of motion for the Galileon field,
and describes a Galileon field propagating on a flat spacetime (Minkowski spacetime).
The generalized action which is invariant under the Galilean shift (3.50) and gives thesecond-order equations of motion is given by:S=⁄Ô
d4 x ≠g[LGR + Lﬁ ],(3.51)

where LGR is the Lagrangian for a linearized general relativity and Lﬁ = L(ﬁ, ˆﬁ, ˆˆﬁ)
is the generalization of the Galileon Lagrangian. The Lagrangian Lﬁ is given by [26]L(ﬁ, ˆﬁ, ˆˆﬁ) =

3

5
ÿ
i=1

ci Li (ﬁ, ˆﬁ, ˆˆﬁ),(3.52)

It is known that in the DGP braneworld model the self-accelerating solution is unstable due to thepresence of a ghost [114, 136, 137]. Historically, Galileon theory was applied to the four-dimensional
effective field theory of the DGP model [26]. It showed that, in Galileon theory, one could constructa self-accelerated solution without the presence of ghosts. The Galileon theory also arises in othercontexts [120, 138].

64

Modified Gravity

where ci are constants, and
L1 = ﬁ,(3.53)

1
L2 = ≠ (ˆﬁ)2 ,2
1
L3 = ≠ ˆ 2 ﬁ(ˆﬁ)2 ,25
6
1 1 2 22
L4 = ≠
ˆ ﬁ ≠ (ˆˆﬁ)2 (ˆﬁ)2 ,25
6
12
1 1 2 232
2
3
L5 = ≠
ˆ ﬁ ≠ 3 ˆ ﬁ (ˆˆﬁ) + 2(ˆˆﬁ) (ˆﬁ)2 ,2(3.54)(3.55)(3.56)(3.57)

where ˆ 2 = ˆ– ˆ – , (ˆﬁ)2 = ˆ– ﬁˆ – ﬁ and (ˆˆﬁ)n = (ˆ–1 ˆ –1 ﬁ)(ˆ–2 ˆ –2 ﬁ) · · · (ˆ–n ˆ –n ﬁ).
Even though the Galileon actions contain derivative self-interactions terms, the theory
has second-order equations of motion for the metric and the Galileon field. Indeed,
the scalar self-interaction recovers general relativity on small scales. This mechanism
is described further in Chapter 4. We do not present the equations of motion for the
Galileon field here. The interested reader can find them in [26].

Covariant Galileon theory

It is interesting to consider covariant Galileon theory, but the property that the Galileon
theory has second-order equations of motion does not hold in curved spacetime. One cannaively covariantize the flat-spacetime Galileon Lagrangian,÷µ‹ æ gµ‹ , ˆµ æ Òµ .

(3.58)

One obtains the equations of motion containing higher derivatives, which leads to a ghost
mode. To eliminate the higher derivatives terms, one should introduce non-minimal
gravitational coupling to „, for example Gµ‹ Òµ „Ò‹ „ where Gµ‹ is the Einstein tensor.
The action for the covariant Galileon in four dimensions is given by [133, 134]
S=

⁄

5

6

1
d x ≠gR + Lcov
gal + Smatter ,
16ﬁG
4

Ô

(3.59)

Modified Gravity

where Lcov
gal =L1 = „,

65qcov
is given byi ci Li(3.60)

1
L2 = ≠ (Ò„)2 ,(3.61)
2
1
L3 = ≠ ⇤„(Ò„)2 ,(3.62)
2
5
6
1
22
1
1
L4 = ≠ (Ò„)2 (⇤„)2 ≠ Òµ Ò‹ „ ≠ (Ò„)2 R ,(3.63)
2
4
5
6
1
23
1
22
1
2
3
µ ‹
ﬂ
L5 = ≠ (Ò„) (⇤„) + 2 Òµ Ò‹ „ ≠ 3⇤„ Òµ Ò‹ „ ≠ 6G‹ﬂ Òµ „Ò Ò „Ò „ ,2(3.64)
where R is the Ricci scalar and(Òµ Ò‹ „)2 = Òµ Ò‹ „Òµ Ò‹ „, (Òµ Ò‹ „)3 = Òµ Ò‹ „Ò‹ Ò⁄ „Ò⁄ Òµ „.

(3.65)

Although the equations of motion are second order, the Galilean symmetry is generally
broken in curved spacetime due to the presence of the self-interacting terms. Note thatthese terms are not unique which yields second-order equations of motion for the metricand the scalar field. The interested reader can find the equations of motion for the
Galileon field in [133, 134].

Generalized Galileon
The generalized Galileon Lagrangian in four dimensions gives second-order equations of
motion for the metric and the scalar field. The action is given as follows [139, 140]:
SH =

⁄

4

Ô

d x ≠g

C 5
ÿ
i=2

D

Li + Lm ,(3.66)

where
L2 = G2 („, X),

L3 = ≠G3 („, X)⇤„,(3.67)
ËÈ

(3.68)

L4 = G4 („, X)R + G4X („, X) (⇤„)2 ≠ (Òµ Ò‹ „)2 ,(3.69)
ËÈ
1
L5 = G5 („, X)Gµ‹ Òµ Ò‹ „ ≠ G5X („, X) (⇤„)3 + 2(Òµ Ò‹ „)3 ≠ 3⇤„(Òµ Ò‹ „)2 ,6(3.70)

66

Modified Gravity

where G2 , G3 , G4 and G5 are arbitrary functions of „ and the canonical kinetic term X,
2
and GiX stands for ˆGi /ˆX. We use units so that MPl= 1. After the authors of [139]derived the action, it was shown in [140] that the generalized Galileon theory is equivalent
to the Horndeski theory in four dimensions. After that, the Galileon theory has been
extensively studied in the context of cosmology [135, 141–145].

Horndeski Theory
After Lovelock’s theorem was established, Horndeski found, when he was a student ofLovelock, the most general theories with second-order field equations for the metric andthe scalar field [29]. He considered that the theory depends on the metric, a single scalar
field, and an arbitrary number of their derivatives,
LH = LH (gµ‹ , gµ‹,ii , · · · , gµ‹,i1 ,...,ip , „, „,i1 , · · · , „,i1 ··· ,iq ),(3.71)

where p, q Ø 2. Horndeski required that the equations of motion of the theory are limited
to second order. If the theory has the derivative terms higher than two in its equationsof motion, the extra degree of freedom appears and propagates in the system, this is an
Ostrogradsky ghost as we discussed in the previous section4 . The propagating modes
are associated with some instabilities in the system. Such an extra degree of freedom isrestricted by Ostrogradsky’s theorem. That is why many researchers took for granted
that the theory of gravity should have second-order equations.
The original Horndeski Lagrangian is given by [29]–—“
LH =”µ‹ﬂ

52
Ÿ1 Òµ Ò– „R—“ ‹‡ + Ÿ1X Òµ Ò– „Ò‹ Ò— „Ò‡ Ò“ „ + Ÿ3 Ò– „Òµ „R—“ ‹‡3
6–—+ 2Ÿ3X Ò– „Òµ „Ò‹ Ò— „Ò‡ Ò“ „ +”µ‹[(F + 2W )R–— µ‹ + 2FX „Òµ Ò– „Ò‹ Ò— „+ 2Ÿ8 Ò– „Òµ „Ò‹ Ò— „] ≠ 6(F„ + 2W„ ≠ XŸ8 )⇤„ + Ÿ9 ,[–(3.72)– ]here ”—11—22...—nn = ”—11 ”—22 . . . ”—nn . Ÿ1 , Ÿ3 , Ÿ8 , Ÿ9 and F are functions of „ and X which isdefined as X := ≠ 12 Òµ „Òµ „. The original Horndeski Lagrangian has not been recognizedor forgotten since Deffayet independently rediscover the action of the generalized Galileon
theory [139, 140].
– – ...–

4–

Historically, Horndeski did not suppose the Ostrogradsky ghost. He assumed such a constraint because
most of the theories that had been built up were described by second-order differential equations.

Modified Gravity

67

Note that the constructions of generalized Galileon [133] and Horndeski theory [29]are based on a different guiding principle. A starting point of the generalized Gaileon is toconstruct the most general scalar theory in an arbitrary-dimensional flat spacetime withsecond-order field equations [26], and later covariantize [133]. Horndeski theory [29], on
the other hand, determines the most general scalar-tensor theories in four dimensions withsecond-order field equations for the metric and the scalar field. The different approachand construction determines most general scalar-tensor theories in four dimensions whichleads to second-order equations of motion for the metric and the scalar field [140].
Recently, many studies have tried to construct more general scalar-tensor theories infour dimensions which yield equations of motion of order two. We will discuss the details
below later.
Subclass of Galileon/Horndeski theory
Galileon/Horndeski theory includes a wide range of theories of gravity with a single
scalar degree of freedom, as well as broad dark energy models such as quintessence andk-essence [21, 146]. We list the examples below.
• General Relativity [1, 2]:General relativity is obtained by choosing
1
G2 = 0, G3 = 0, G4 = , G5 = 0.
2(3.73)• Quintessence [20] and k-essence [21, 22]:
Quintessence is obtained by choosing
1
G2 = X ≠ V („), G3 = 0, G4 = , G5 = 0,2(3.74)

where V („) is the potential of scalar field. K-essence is given by the functions
1
G2 = G2 („, X), G3 = 0, G4 = , G5 = 0.
2(3.75)• Brans-Dicke theory [28, 126, 127] (F (R) theory [131]):
The action of Brans-Dicke theory with the scalar potential V („) is given by
G2 = ≠ÊBD„X ≠ V („), G3 = 0, G4 = , G5 = 0.
2„2(3.76)

68

Modified Gravity

f (R) gravity is equivalent to the Brans-Dicke theory with ÊBD = 0 and the potential
V („) = (Rf (R),R ≠ f (R))/2. The interested reader can find the relation between
the f (R) theory and the Brans-Dicke theory in Appendix B.
• Covariant Galileon theory [133]:
Covariant Galileon theory is obtained by choosing
G2 = ≠c2 X, G3 = ≠c3

X
1
X2
X2
,
G
=
≠
c
,
G
=
3c
,
4
4
5
5
2
M3
M6
M9

(3.77)

where the ci (i = 2, ..., 5) are constants and M is a constant with dimensions of mass.
• Kinetic Gravity Braiding [147–149]:
When we set G4 = G4 („) and G4X = G5 = 0, and we obtain the actionS=⁄Ô
d4 x ≠g [G4 („)R + K(„, X) ≠ G3 („, X)⇤„] .

(3.78)

This model is called the kinetic gravity braiding model. This model contains the
cubic Galileon and DGP models. Indeed, we will mention later, the action is themost general Horndeski theory for dark energy models after the GW170817/GRB
170817A [5, 91–93].
• Non-minimally coupled with the Gauss-Bonnet term:
When we set [140]d4 ›(„) 2d3 ›(„)
G2 = X + 8
X (3 ≠ log X), G3 = 4
X(7 ≠ 3 log X),
d„4
d„3
1
d2 ›(„)
d›(„)
G4 = + 4
log X,
2 X(2 ≠ log X), G5 = ≠42d„d„(3.79)

where ›(„) is a function of the field, Horndeski theory reproduces the non-minimal
coupling of the field with the Gauss-Bonnet term (see for example Einstein-dilatonGauss-Bonnet gravity [150], the theory emerges naturally from string theory):
S=

⁄

Ô

5

6

R
d x ≠g+ X + ›(„)G .
2
4(3.80)• Fab-Four theory [151]:
Fab-Four theory is that the most general subclass of Horndeski theory in which it
has a flat spacetime solution with the (arbitrary) cosmological constant term. The
theory has a flat-spacetime solution with any values of the cosmological constant,Modified Gravity

69

namely the self-tuning cosmological constant (see [152] for the details of the selftuning mechanism). The presence of the scalar field tunes the value of the barecosmological constant: we can get rid of the large cosmological constant or thecosmological constant problem. If one demands that Horndeski theory has theself-tuning mechanism, one obtains the simple four Lagrangians as:
LJohn = VJohn („)Gµ‹ Òµ „Ò‹ „,1
LPaul = ≠ VPaul („)Áµ‹⁄‡ Á–—“” R⁄‡“” Òµ „Ò– „Ò‹ „Ò— „,4
LGeorge = VGeorge („)R,
1

2

LRingo = VRingo („) Rµ‹–— Rµ‹–— ≠ 4Rµ‹ Rµ‹ + R2 .

(3.81)

The correspondence between the Fab-Four Lagrangian (3.81) and the Horndeski
Lagrangian is presented in [152].
Beyond Horndeski Theory
Recently, many studies have tried to construct more general scalar-tensor theories. The
starting point of the studies was to construct a theory which would lead to the secondorder equations of motion for the metric and a scalar field. The point of requiring
second-order field equations is to avoid the Ostrogradsky ghost (see Sec. 3.3.4). GleyzesLanglois-Piazza-Vernizzi (GLPV) theory [153] was proposed as the first example of abeyond-Horndeski theory. There are two additional Lagrangians in the Horndeski case.
The theory contains the third order field equations of motion, but in a particular gauge,
the third-order equations of motion reduce to the second-order equations of motion.
Many researchers have thought that the requirement for the equations of motion for thefields to be second-order equations (in a particular gauge) is the necessary condition to
avoid the Ostrogradsky instability.
However, the second-order equations of motion are indeed a not-necessary but sufficientcondition for evading the Ostrogradsky instability. As mentioned in Sec 3.3.4, theOstrogradsky theorem states that any nondegenerate higher derivative theories exhibit
ghost instabilities [125]. Recent works [125, 154, 155] have shown that higher derivative
theories do not always lead to the Ostrogradsky instability: if the theory is degenerate,
the theory can evade the Ostrogradsky instability. The author of [154] named the theory
as degenerate higher order scalar-tensor (DHOST) theory. As noted in Sec. 3.3, to avoid
the Ostrogradsky instability, the kinetic matrix ˆ 2 L/ˆ q̈ i ˆ q̈ j should be non-invertible.

70

Modified Gravity

After imposing the degeneracy conditions, one can reduce the system to another withsecond-order equations of motion. The reader is referred to [156] for more information
about the DHOST theories.

3.4.3. Constraints on Horndeski Theory by the Observational
Results of Gravitational Waves
Many modified gravity theories have been studied and constrained in the cosmologicalcontext, namely via the CMB and large-scale structure of the Universe (see [49] for areview and references therein). The recent observation of the gravitational wave event
GW170817 from a neutron star binary merger by LIGO and VIRGO [5] and of its
electromagnetic counterpart, the gamma-ray burst GRB 170817A [91–93], which places
severe constraints on the speed of gravitational waves [157–160].
The result shows that the propagation speed of gravitational waves is close to that oflight, more precisely (c2T ≠ c2 )/c2 Æ 6 ◊ 10≠15 , where cT is the speed of the gravitational
waves and c is the speed of light. The gravitational waves originated from a binaryneutron star merger which is located in the host galaxy NGC4993, with a low-redshiftz ≥ 0.008 [91–93]. This propagation speed limit can be applied to a dark energy model,
namely modified gravity, in which cT is modified from the prediction by general relativity:
cT = 1 within z ≥ 0.008 [91–93]. In the natural unit where the speed of light is 1, thedeviation from cT = 1 occurs in several modified gravity theories.
Next, we will explain the constraints on Horndeski theory by the observational
results on gravitational waves. In a cosmological background, the evolution of tensor
perturbations is given by [157, 161]ḧij + [2 + –M (a)]H ḣij + (1 + –T )k 2 hij = 0,(3.82)

where –T is the tensor speed excess that represents how much the gravitational waves
speed deviates from light speed
c2T = 1 + –T .

(3.83)

Recent observations of GW170817/GRB 170817A placed constraints on bound≠6 ◊ 10≠15 Æ –T Æ 1.4 ◊ 10≠15 .

(3.84)

Modified Gravity

71

In Horndeski theory, –T is given by5
–T =

È
2X Ë
¨ ≠ „H)G
˙
2G
≠
2G
≠
(
„
.
4X
5„
5X
Mú2

(3.85)The Planck mass run rate is given by –M and is defined as:
d ln Mú2,–M =d ln a(3.86)

where Mú2 is the effective Planck mass. For Horndeski theory Mú2 = 2(G4 ≠ 2XG4X +˙
XG5„ ≠ „HXG
5X ). We see that –T and –M depend on the theory parameter. It was
shown in [157] that the speed limit imposes Horndeski parameters
G4X ≥ 0, G5 ≥ const,(3.87)

in which case L5 vanishes by virtue of the Bianchi identities. Therefore, the resulting
constrained Horndeski action is given by:S=⁄Ô
d4 x ≠g[G4 („)R + G2 („, X) ≠ G3 („, X)⇤„].

(3.88)

The action is known as Kinetic Gravity Braiding [147–149] and the most general Horndeski
theory for dark energy models after the GW170817/GRB 170817A. As a result, a large
class of Horndeski theories have been constrained and ruled out by the observational
results on gravitational waves. Indeed, other theories of gravity are also constrained [158–160, 160, 164]. However, DHOST theories satisfy cT =1 and survive the observational
constraints from gravitational waves [157–160]. After all, gravitational wave and multimessenger astronomy has opened a new window for testing theories of gravity in strong
gravity regimes.
As we observed, the difference between the speed of graviton and photon has been
constrained by the detection of gravitational waves originating from the binary-neutronstar GW170817 and its electromagnetic counterpart GRB 170817A [5, 91–93]. Note thatone should keep two important facts in mind: i) the gravitational waves originated from a
Neutron star binary merger which is located with z ≥ 0.01, or 40 Mpc, while cosmological
data, such as Planck data, comes from higher redshifts. Therefore, alternative inflation
models are of course free from this constraint. ii) the authors of [165] pointed out that,for Horndeski theory (not only the Horndeski theory, see [165] for details), one can
5

The explicit expressions are given for beyond Horndeski in [162], for DHOST theories in [163].

72

Modified Gravity

have the speed of gravitational waves differing from photons at low energy scales (onthe cosmological scale) while keeping the speed of light at a high energy scale. If thisstatement is true, modified gravity ruled out by the recent gravitational waves constraints
could survive as a dark energy model at this time.
The constraints on the speed of the gravitational waves also place constraints on
graviton mass, the dispersion relation, and the equivalence principle. In the future, withincreasing number of multi-messenger events and detector improvements (sensitivity,
the number of observatories), one can further test and place constraints on the theoryof gravity: additional polarization of gravitational waves(already measured [166] by
three-detectors), damping of gravitational waves etc. (see [167, 168] for a review of
constraints on theories of gravity and dark energy models). Multi-messenger astronomy
should give us the insight to unveil the nature of gravity and dark energy.

3.5. Other Constraints on Modified Gravity
Any theory of gravity should pass tight observational constraints from several experiments.
For example, in the solar system, the fifth force is tested by the PPN formalism (asdiscussed in Sec. 2.5.1) [80, 169, 170] which provides a constraint on space-time curvature.
The modification of general relativity contributes to a modification of Newton’s lawof gravity with a Yukawa potential parametrization [23–25]. Current tests of gravity
showed no implications of deviations from general relativity. Therefore, these tests ofgravity provide the tight constraints on modified gravity, but, thanks to the screening
mechanism, most modified gravity theories can pass these tests.
Many observational tests could also be considered, these experiments focus on exploring
the screening scale of the fifth force. All objects in the Universe are clustered, forexample, galaxy clusters. The clusters can be divided into the two regions: linear andnon-linear scales for the matter density perturbation. Therefore, they have non-screened
and screened regions, then one can consider the following situation; while the fifth
force may be screened in the inner regions of the clusters, the fifth force may not becompletely screened in the outer regions of the clusters. In this sense, cosmologicalobservations, such as the CMB and the large-scale structure [171–173], give constraints
on modified gravity. Higher order cosmological perturbations [174–176], redshift-spacedistortions [177,178], and the gravitational weak-lensing effect of galaxy clusters [179,180]
also provide constraints on modified gravity. Combining these observations with others,Modified Gravity

73

such as from gravitational wave astronomy would improve the the knowledge of thenecessary constraints on modified gravity.

3.6. Further Physics
As discussed above, astrophysical measurements provide powerful constraints for Horndeski theory as well as other models of dark energy. Cosmological measurements also
provide constraints for Horndeski theory. Recently, [181] pointed out that Horndeski
theory with arbitrary functions G4 and G5 needs fine-tuning to explain the accelerating
Universe. The PPN formalism in Horndeski theory is also studied in [169]. [170] studied
the Nordtvedt effect and the Shapiro time delay in Horndeski theory using the measurements of the lunar laser ranging experiments [182] and the Shapiro time delay [79]. These
results constraint the surviving theory after the gravitational wave tests [5, 91–93].
In the following, several aspects and features of the scalar-tensor theory will be
discussed. A physical phenomenon in scalar-tensor theory is not the same as in generalrelativity; when one investigates physical phenomena in scalar-tensor theory, one can findseveral interesting physical aspects which do not occur in general relativity. Scalar-tensor
theory has been extensively studied in the context of cosmology. The details are beyond
the scope of this thesis. The reader is referred to [95, 183, 184] for more information
about scalar-tensor theory in the context of cosmology. In this subsection, aspects ofcompact objects in scalar-tensor theory are focused on.6

3.6.1. Black Holes
Aspects of scalar-tensor theory in a vacuum spacetime, namely a black hole spacetime,
have been extensively studied. Many researchers pointed out and suggested that theno-hair theorem as it is in general relativity holds in scalar-tensor theory: stationary
black hole solutions cannot support the scalar-hair. This was first pointed out in thecontext of the Brans-Dicke theory [185]. Later the theorem was extended to moregeneral scalar-tensor theories [186] including k≠essence [187, 188]. It was also proventhat a Galileon cannot develop a nontrivial configuration around a static and sphericallysymmetric black hole [189]. The theorem makes the following assumptions: i) spacetime
is static and spherically symmetric ; II) spacetime is asymptotically flat ; iii) scalar field
6

In this thesis, we focus on scalar-tensor theory, so we refer the reader to [96] for further information.

74

Modified Gravity

respects the symmetries of the metric („ = „(r)) ; iv) the theory is a shift-symmetrictheory (the theory is invariant under the transformation „ æ „ + c) ; V) Noether’scurrent J 2 is finite. Under these hypotheses, the scalar field is constant everywhere andthus the black hole solution is identically a Schwarzschild solution. Therefore, the sameconclusion would hold true in shift-symmetric Horndeski theory (See [190, 191] for areview and references therein).
Recently, there have been many attempts to relax the assumptions of and to findloopholes in the no-hair theorem [189]. The authors of [192, 193] pointed out that when
the scalar field is coupled to the Gauss-Bonnet term, the black hole can support thenon-trivial scalar hair. The scalar charge depends on the black hole mass; the black holehair is called a secondary-hair. [194] pointed out that when one relaxes assumption iii),
the scalar field is linearly time-dependent „(t, r) = qt + (r), one finds the black holesolution with a non-trivial scalar hair, while the metric is a Schwarzschild black holemetric. It is called a stealth solution: the scalar field configuration does not back-react
on the spacetime. Soon after [194] proposed it, it was generalized in the context of the
Horndeski theory [195]. If one assumes that one or more assumptions are broken, one
can find several hairy black holes in Horndeski theory and in beyond Horndeski theories(see [196] for a review and references therein).

3.6.2. Neutron Stars
The physical phenomena of neutron stars in scalar-tensor theory are also interesting.
Here we consider a scalar tensor theory where „ is directly coupled to the Ricci scalar R.
The relationship between the Einstein-frame metric gµ‹ and the Jordan-frame metric ĝµ‹can be parametrized by ĝµ‹ = A2 („)gµ‹ , where A(„) is a function of „. Here we further
define the quantities –(„) := d(ln A(„))/d„. We can expand the function –(„) aroundthe asymptotic value „0–(„) = –0 + —0 („ ≠ „0 ) + O(„20 ).

(3.89)

The choice of –(„) = –0 = constant, corresponds to the Brans-Dicke theory. The Cassini
measurements of the Shapiro time delay [80] placed the bound on –0 , as –0 < 3.5 ◊ 10≠3 .
Even though –0 is quite small (–0 π 1, therefore, this theory agrees with Solar System
experiments), the physical phenomena which do not occur in general relativity occur
inside neutron stars. Inside the star, the presence of a non-minimal scalar-matter couplingModified Gravity

75

can cause a tachyonic instability of the scalar field; this effect is known as spontaneous
scalarization [197–199]. The scalarization occurs inside the star, the structure of the staris significantly modified; the gravitational constant decreases and gravity becomes weakin the scalarization phase. Therefore, this phenomenon allows for more massive neutronstars than general relativity. For spherically symmetric neutron stars, the scalarization
occurs for —0 < ≠4.35 due to the existence of a scalar field. The binary neutron stars
lose energy faster by emitting gravitational waves. The binary-pulsar observations put
stringent bounds on —0 , as —0 > ≠4.5. Recently, [200, 201] pointed out that spontaneous
scalarization can occur for —0 > 0. The presence of the scalarization effect —0 < 0 occurs.
Many works have focused on this region of the parameter. Many observations [199, 202]
placed a constraint on the parameter in the case of —0 > 0; the details in the case of—0 > 0 should be explored.
The structure of neutron stars in scalar-tensor theory differs from that in generalrelativity, so we can distinguish these theories by measurements of the neutron stars,
in principle. However, there is a degeneracy between uncertainties in the equation ofstate of dense nuclear matter and strong-field gravity. An equation-of-state-independent
relation does exist between the moment of inertia (I), the gravitational Love number
and the quadrupole moment (Q) called the I-Love-Q relation [203, 204]. It resolves
the degeneracy between uncertainties in the equation of state of dense nuclear matterand strong-field gravity, that is why the equation-of-state-independent -relation is key
to explore the nature of strong gravity. The measurements of the moment of inertiaof the pulsar/neutron star should be confirmed by Radio observation [205], while themeasurements of the pulsar/neutron star tidal Love number should be detected bygravitational-wave observations [206]. The I-Love-Q relation has been intensively studied
and will play an important role for strong-field tests of gravity (see [207] for a review
and references therein). Indeed, dipole radiation is predicted by scalar-tensor theory. As
the number of gravitational waves events increase, future gravitational wave observations
can places constraints on scalar-tensor theory. Therefore, the strong gravitational fieldaround neutron stars provides us a laboratory to test a theories of gravity.
The authors of [208] pointed out that the no-hair theorem for spherically symmetricand static stars holds in shift-symmetric Horndeski theories with minimal matter coupling.
The theorem was proved under the same assumptions which were previously discussed
before for the no-hair theorem of black holes in shift-symmetric Horndeski theories [189].
The authors [208] assumed that the metric functions and the scalar field are regular at
the centre of stars. By violating one or more assumptions, for example, as discussed

76

Modified Gravity

previously, assuming the scalar field is linearly time-dependent „(t, r) = qt + (r), one
can find relativistic star solutions (see, e.g., [209] neutron stars in fab four theory).
Beyond Horndeski and DHOST theories exhibit a partial breaking of the Vainshteinmechanism inside a (non) relativistic star, thus the structure of a star is significantly
modified [210, 211]. Recently, relativistic stars have been studied in the GLPV [212, 213]
and in DHOST [214]. They reported that a significant modification of the structure fromthat of general relativity, depending on its parameters, occurs.

Chapter 4.
Screening Mechanism
In the previous chapter, we reviewed modified gravity, especially scalar-tensor theory.
Scalar-tensor theory introduces a new degree of freedom and couples strongly to matter;it mediates the fifth force. Observational constraints in the solar-system do not allow
for the existence of the fifth force or a new degree of freedom in the vicinity of the solar
system, but a new degree of freedom for the dark energy model is required.
As mentioned in Sec. 3.4.1, Brans-Dicke theory [28, 126, 127] can explain the current
accelerated cosmic expansion by introducing a large coupling parameter Ê. On the other
hand, one would like to consider modifications to general relativity which do not introduce
such a coupling parameter. The theory should be equipped with a screening mechanism:
we can suppress the fifth force in the environment-dependent way. Consequently, thetheory can evade the solar system constraints and can explain the observed cosmic
expansion. In this chapter, we summarize and introduce the screening mechanism to
avoid the solar system constraint. This chapter is based on [49, 95–98].

4.1. Fifth ForceAs we mentioned in Sec. 3.4, in the Einstein frame the energy-momentum tensor definedin the Einstein frame metric is not conserved in this frame:Òµ T̂ µ‹ =d logT̂ Ò‹ „,
d„(4.1)

where T̂m = ĝµ‹ T̂mµ‹ is the trace of the energy-momentum tensor. According to Eq. (4.1),
the matter does not move on the geodesic of ĝµ‹ . Taking a non-relativistic limit, we can77

78

Screening Mechanism

find an additional terms in geodesic equation (2.18). This additional term represents thefifth force.

Explicit expression for Fifth force
We will show the explicit expression for the fifth force. In the Jordan-frame, a test
particle follows the geodesic:
ẍµ +

µ

⁄ —
⁄— ẋ ẋ = 0,

(4.2)

where gµ‹ is the Jordan-frame metric. Under the conformal transformation:gµ‹ æ ĝµ‹ = e2—„/MP1 gµ‹ ,(4.3)

where ĝµ‹ is the Einstein-frame metric, we can rewrite (4.2) regarding the Einstein-frame
metric:ẍµ + ˆ µ⁄— ẋ⁄ ẋ— +2— 1
2ˆﬂ „ẋﬂ ẋµ + ĝ µ– ˆ– „ = 0.
MPl

(4.4)

Taking the Newtonian limit and the first and second terms of Eq. (4.4) givesẍ + Ò .
The third term of Eq. (4.4) gives a fifth force. Finally, we find that a test particle
feels the fifth force which is given by:F„ = ≠

—
Ò„.
MPl

(4.5)

If a theory (Lagrangian) is given in the form:
L ∏ A(„)T̂m ,(4.6)

where A(„) is an arbitrary function of scalar field, the scalar field is non-minimallycoupled to matter. Then, the energy-momentum tensor is not conserved:
Òµ T̂ µ‹ =—(„)
T̂m Ò‹ „,
MPl

(4.7)

Screening Mechanism

79

where
—(„) := MPl

dA(„)
,
d„

(4.8)

it tunes the coupling strength between the scalar field and matter. As we showed above,taking the non-relativistic limit, the left-hand side of Eq. (4.7) is the geodesic equation.
The right hand side of Eq. (4.7) is the fifth force (4.5). Therefore, when one assumes thatthe scalar field (or new degree of freedom) is coupled to matter in the Einstein-frame,
the coupling between the scalar field and matter mediates the fifth force. However,
the existence of a fifth force is tightly constrained by experimental tests in the solar
system [24, 34]. Therefore, for modified gravity, it is necessary to include a screening
mechanism which hides effects at the local scale without introducing large coupling
function.

4.2. Classification of Screening Mechanism
We can classify the screening mechanism into three types. Let us consider a general
Lagrangian expanded around the background solution of the scalar, „ = „¯ + ”„
1
¯ µ ”„Ò‹ ”„ ≠ m2eff („)”„¯ 2 + —(„)¯ ”„ Tm + · · · ,L = ≠ Z µ‹ („)Ò
2
MPl

(4.9)

where Z µ‹ represents derivative self-interactions of the field. Z µ‹ , the effective mass meffand the coupling function —(„), coupled to the trace of the energy-momentum tensor are
field-dependent and can vary according to the environment. One should tune the valueof the field or hide the fifth force, so that fifth force becomes negligible compared withthe usual gravitational force on local scales. There are three ways to hide the fifth force.
• i)Weakly coupled type:
When the coupling constant —(„) depends on the environment, the coupling function
can be small enough in regions of high density (such as the solar system) for the fifth
force to become negligible compared with the usual gravitational force. By contrast,
in regions of low density, such as the void of space, where —(„) can be of order unity;
the fifth force remains as large as the usual gravitational force. Symmetron [215]and dilaton [216] screening mechanisms are based on this assumption.

80

Screening Mechanism• ii)High Mass type:The effective mass of the perturbation of a scalar field meff („) depends on the local
matter density. In regions of high density, such as in the Earth, the scalar acquires
a mass and meff is massive; its force range ⁄ = m≠1
eff becomes very short. Therefore,
the fifth force becomes an unobservable force at present. By contrast, in low-density
regions, such as in space, the force range becomes long-range and the fifth force
remains as large as the usual gravitational force. Chameleon screening [217, 218] isbased on this assumption.
• iii)Kinetic type:
When the kinetic function Z µ‹ is large in dense regions, the scalar is effectively¯¯ π 1, near the source; the fifth force isweakly coupled to matter, —(„)/Z(„)
negligible compared with the usual gravitational force1 . This is called kinetic
screening. This class can be divided into two subclasses, depending on whether the
first or second derivatives of the scalar field play a crucial role. The former includes
various models [219–221], and the latter includes the Galileons [26, 133, 134]. The
last class of these models is called the Vainshtein mechanism [27] and has beenstudied extensively (see [222] for a review).
The reader is referred to [95,223] for more information about the screening mechanism.
This thesis is primarily concerned with the class of theories which possess the Vainshteinmechanism. We describe their properties in detail below.

4.3. Vainshtein Mechanism
The idea of the Vainshtein mechanism [27] was originally discovered in the context ofmassive gravity. The Vainshtein mechanism relies on the nonlinear derivative interactionsof an additional field without tuning the potential of a scalar field. When the Vainshteinmechanism is in effect, the nonlinear terms dominate within the so-called Vainshtein
radius from the source and near the source, it is hence effectively weakly coupled tomatter where its gradient is large. Well outside the Vainshtein radius, the linear term
dominates, and linearization can be applied. Such a nonlinear interaction giving secondorder equations of motion for a scalar field has been investigated in the context of Galileon
theory [180, 224–226].
1

Here, we assume that Zµ‹ takes on the form Zµ‹ = ÷µ‹ Z for simplicity.

Screening Mechanism81

Let us consider the cubic Galileon which is the simplest theory employing the Vianshtein mechanism:
1c—
L = ≠ c2 (ˆ„)2 ≠ 33 (ˆ„)2 ⇤„ +„Tµµ ,
2MPl
M(4.10)

where c2 and c3 are dimensionless parameters, — ≥ O(1) is the coupling factor, MPl is
the Planck mass, M is another mass scale, and Tµµ is the trace of the matter energymomentum tensor, which is defined in the Einstein-frame. The first and second terms of
Eq. (4.10) are invariant under Galilean symmetry „ æ „ + c + bµ xµ . The third term is
not invariant with the Galilean shift, however, by virtue of the cut-off scale MPl ∫ M ,the breaking of symmetry can be ignored.
The coupling between matter and the metric fluctuations around the Minkowski
spacetime, hµ‹ , is expressed as (1/2)hµ‹ T µ‹ . As we discussed in Sec. 4.1, this implies
that the Jordan frame metric is given by hJµ‹ = hµ‹ + (2—/MPl )„÷µ‹ , and thus a test
particle of mass m feels the fifth force as—˛F˛„ = ≠
mÒ„,MPl

(4.11)

˛ 00 .
in addition to the usual gravitational force F˛G = (m/2)Òh
We assume that matter is non-relativistic so that Tµµ ƒ ≠ﬂ. Varying the action withrespect to „, we obtain the field equation,c2 „ +

È
c3 Ë
—
2
i j
(
„)
≠
Ò
Ò
„Ò
Ò
„
=
ﬂ,
i
j
3
MPl
M

(4.12)

where we assumed that „ is static. For a given configuration of matter, one can integrate
Eq. (4.12) to obtain the profile of „. Due to the non-linear interaction, it is difficult tosolve the equation of motion for the scalar field in many cases. However, thanks to thesymmetry of spacetime, we can solve the equation. Here, let us consider the profile of „around a spherical matter configuration. Using the spherical coordinates we obtainQ3
˛ = c2 M r a≠1 +
Ò„
4c3

Û

3

r
1+ v
r

43

R

b˛
er ,

(4.13)

82

Screening Mechanism

where ˛er is the radial unit vector and M is the mass of the spherical body and we haveintroduced the Vainshtein radiusrv =A

2c3 —M
c2 ﬁM 3 MPl

B1/3

.

(4.14)

We can consider two regimes: located outside of the Vainshtein radius (r ∫ rv ) and deep
inside of the Vainshtein radius (r π rv ).
Outside of the Vainshtein radius (r ∫ rv ): At long distances from the source (r ∫ rv ),the linear term in Eq. (4.12) dominates, and the scalar profile (4.13) is
d„
M
≥—
.
dr
4ﬁMPl r2

(4.15)In this case, the ratio of the fifth force to the gravitational force F„ /FG is given byF„≥ 2— 2 .
FG(4.16)

This implies that the fifth force is as large as the usual gravitational force if — = O(1).
Deep inside the Vainshtein radius (r π rv ): At short distances from the source(r π rv ), the non-linear term in Eq. (4.12) dominates, and the scalar profile (4.13) tends
Ô
to as ≥ 1/ r. In this case, the ratio of the fifth force to the gravitational force F„ /FG isgiven byAF„r
≥ 4— 2
FG
rv

B3/2

,

(4.17)and thus the fifth force is screened in the vicinity of the body.

4.4. Typical Value of the Vainshtein Radius andconstraints
According to measurements performed in the solar system, a fifth force should be screenedby virtue of the screening mechanism. Modified gravity theories, such as Galileon theory,
possessing the Vainshtein mechanism are constrained by solar system tests. We would
like to find a typical value of (in other words constraints on) the Vainshtein radius.

Screening Mechanism

83

For simplicity, our starting point is the cubic Galileon in the Einstein-frame (Eq. (4.10)).
Deep inside the Vainshtein radius, the ratio of the fifth force to the gravitational forceF„ /FG is proportional to (r/rv )3/2 , the fifth force is suppressed in the region. The
2
Vainshtein radius can be written in terms of the Schwarzschild radius rg = M/4ﬁMPl,
we obtain
rv = (4ﬁ—rg L2 )1/3 ,(4.18)

where L = (2c3 MPl /c2 M 3 )1/2 . For — ≥ O(1), the Vainshtein radius for the mass of solar
mass is rv ≥ 100pc ≥ 3.5 ◊ 1018 m, which is significantly larger than the radius of the Sunr§ ≥ 10≠8 pc ≥ 6.7 ◊ 108 m.2 Other examples for the Vainshtein radius can be obtained(see, e.g. [227]). According to the result, the typical value of the Vainshtein radius issignificantly larger than astrophysical objects; in the vicinity of those objects, the fifth
force is significantly suppressed.
The current constraints on L come from continuous observations of the Earth-Moon
distance by lunar laser ranging (see [182] for a review of lunar laser ranging). The
measurements provide the constraints on a correction to the Newtonian potential:
”

2

—
≥
=
2

A

r
rv

B3/2

Æ 2.4 ◊ 10≠11 .

(4.19)

Using Eq. (4.18), we obtain a bound on L [228–230]
L Ø 150— ≠3/2 Mpc.

(4.20)For — ≥ O(1), the constraints on the Vainshtein radius for the solar mass is
rv§ Ø O(102 )pc.

(4.21)

Laboratory test
The Vainshtein mechanism has also been tested in the laboratory [231]. The authorof [231] suggested Casimir force experiments. The Casimir force is measured the forces
between a sphere and a plate. The Galileon force mediates between two parallel plates,
which are completely flat and parallel: the force does not screen by the Vainshtein2Note that Proxima Centauri is the closest star to the Solar System and located about 1.3pc from thesolar-system. Therefore, in the vicinity of the solar system, the fifth force is almost suppressed.

84

Screening Mechanism

mechanism as discussed in Sec. 4.5. The un-screened Galileon force could be observedin this situation, however, it is difficult to produce the laboratory setup, until now theexperiments could only place weak constraints on Galileon theory.

4.5. The Shape Dependence of the Vainshtein
Mechanism
Previous works mostly focused on the Vainshtein mechanism around spherical distributions of matter, as a star can be well approximated by a sphere. The authors of [30] haveinvestigated the systems analytically with cylindrical and planar symmetries, and foundthat screening is weaker in the cylindrical symmetric case and does not occur in a systemwith planar symmetry. Let us summarize their work with the planar and cylindricalsymmetry case, in this subsection.
Our starting point is the flat space Galileon [26] in the Einstein frame:
S=

⁄

C

D

1
1
⁄4
⁄5
—„ µ
d x ≠g ≠ L2 ≠T µ ,3 L3 ≠
6 L4 ≠
9 L5 +
2
MPl
2M
2M
2M
4

Ô

(4.22)

where M is a constant with dimensions of mass andL2 = (Ò„)2 ,(4.23)

L3 = ⇤„(Ò„)2 ,Ë

(4.24)

È

L4 = (Ò„)2 (⇤„)2 ≠ Òµ Ò‹ „Òµ Ò‹ „ ,Ë

(4.25)

L5 = (Ò„)2 (⇤„)3 ≠ 3(⇤„)Òµ Ò‹ „Òµ Ò‹ „
È+ 2Òµ Ò‹ „Ò‹ Òﬂ „Òﬂ Òµ „ .

(4.26)

Here we assume that matter is non-relativistic, so that Tµµ ƒ ≠ﬂ with constant density.
We solve the field equations under the assumption of static configurations. By varying
the above action under these assumptions, we obtain the field equations:È—1 Ë
ﬂ = ⇤„ + 3 (⇤„)2 ≠ (Òµ Ò‹ „)(Òµ Ò‹ „)
MPl
M
Ë
È
⁄
+ 46 (⇤„)3 ≠ 3⇤„(Òµ Ò‹ „)(Òµ Ò‹ „) + 2(Òµ Ò‹ „)(Ò‹ Ò“ „)(Ò“ Òµ „) . (4.27)
MScreening Mechanism

85

Next, we investigate the efficiency of Vainshtein screening in a planar configuration.
Now let us consider solving Eq. (4.27) in planar symmetry using the metric:
ds2 = ≠dt2 + dx2 + dy 2 + dz 2 .

(4.28)We choose the scalar profile as „ = „(z), and also assume that ﬂ = ﬂ(z). The cubic andquadratic terms do not contribute to the field equations, and we obtain—ˆ 2 „(z)ﬂ(z) =.
MPlˆz 2(4.29)For the simplicity, we consider a planar source with constant density ﬂ = ﬂ0 between± z0 . By integrating this profile, we obtain
Y
_
_
_
]

and

—ﬂ0
z
ˆ„(z)
M
Pl
= _ —ﬂ
0
ˆz
_
_
z0
[
MPl
Y
_
_
_
]

—ﬂ0 2
z
2MPl 34„(z) =—ﬂ0 z0
z0
_
_
_
z
≠
[
MPl
2

|z| < z0(4.30)|z| Ø z0|z| < z0

(4.31)|z| Ø z0

where we demand ˆz „ = 0 at the origin by symmetry. The non-linear terms, second
derivatives of the scalar, do not contribute to the field equations. Therefore, theVainshtein mechanism does not occur around a planar source. In fact, we evaluate the2usual gravitational force outside the plane FG = 2ﬂ0 z0 m/MPl. The ratio of the scalar
force F„ (4.5) to the usual gravitational force F„ /FG is given byF„= 2— 2 .
FG(4.32)

Next, we investigate the efficiency of Vainshtein screening in a cylindrical configuration.
Now let us consider solving Eq. (4.27) in cylindrical symmetry using the metric:
ds2 = ≠dt2 + dr2 + r2 d◊2 + dz 2 .

(4.33)

86

Screening Mechanism

We choose the scalar profile as „ = „(r), and also assume that ﬂ = ﬂ(r). The cubic andquadratic terms contribute to the field equations, and we obtain—ˆ 2 „(r) 1 ˆ„(r)
2 ˆ„(r) ˆ 2 „(r)ﬂ(r) =
+
+
.
MPl
r ˆr
ˆr2
rM 3 ˆrˆr2(4.34)For the simplicity, we consider a planar source with constant density ﬂ = ﬂ0 for r < r0 .
We choose boundary conditions as „(0) = 0. Indeed, we demand ˆr „(0) = 0 by symmetry.
By integrating the profile with these boundary conditions, we obtain solutions.
If the cubic term does not contribute to the field equations, we obtain
Y
_
_
_
]

—ﬂ0
r
ˆ„(r)
2MPl
=
_
—ﬂ0 r02ˆr__[
2MPl r|r| < r0

(4.35)|r| Ø r02
We evaluate the usual gravitational force outside the cylinder source FG = mﬂ0 r02 /4MPlr.
The ratio of the scalar force F„ to the usual gravitational force F„ /FG is given byF„= 2— 2 .
FG(4.36)

If the non-linear terms do not contribute to the field equations, the Vainshtein mechanism
does not occur around a cylindrical source.
If the cubic term contributes the field equations, we obtain
Y
Q
R
ı̂
_
3
2
_
ı
_
M r cÙ
rvd___
a 1 + 2 ≠ 1b
_
]
2
r0
ˆ„(r)
=
QÛ
R
_
3
2
ˆr
_
_
M
r
r
_
_
a 1 + v ≠ 1b
_
_
[
2
r2

|r| < r0

(4.37)|r| Ø r0

where we define the Vainshtein radiusrvÛı̂ı 2—ﬂ0 r02
=Ù
:=

MPl M

3

2—mc
,
ﬁMPl M 3(4.38)

Screening Mechanism

87

where mc is the linear mass density. Deep inside the Vainshtein radius, the ratio of thescalar force F„ to the usual gravitational force F„ /FG is given byF„r= 4— 2 .
FG
rv(4.39)

Within the Vainshtein radius, the Vainshtein mechanism occurs around a cylindricalsource, then scalar force is suppressed compared to the usual gravitational force. Comparing Eq.(4.39) with Eq.(4.17), the screening effect in the cylindrical case is weaker
than in the spherical case.
The authors of [30] have also investigated Vainshtein screening in k-essence theory.
They conclude that Vainshtein screening might be sensitive to the shape of the matter
distribution. This result implies that the Vainshtein screening mechanism around lesssymmetric matter configurations is quite nontrivial. Therefore, we have studied how the
Vainshtein mechanism works in a less symmetric setup in Chapter 5.

88Chapter 5.
Anti-Screening of the Galileon Force
Around a Disk Center HoleIn this chapter, we consider a disk with a hole at its center as a source and solve theGalileon field equation fully numerically in order to address the consequence of nonlinear
derivative interactions in a less symmetric system. We only study the cubic Galileons for
simplicity. A similar system in a different theory of modified gravity has been consideredin [232, 233], where scalar field profiles around a black hole accretion disk have been
investigated in the context of the chameleon theory.

5.1. Motivation for this work
As we described in Chapter 4, several types of screening mechanisms have been known sofar. The first one relies on the potential term of the scalar degree of freedom. The shapeof the potential is designed so that the scalar becomes effectively massive in a high density
region. This class of models include chameleon [217], symmetron [215], and dilaton [234]mechanisms. The second one relies on nonlinear derivative interactions of the scalar field,
by which the kinetic term of the scalar becomes effectively large and hence it is effectively
weakly coupled to matter near the source where its gradient is large. This class can bedivided into two subclasses depending on whether first or second derivatives of the scalar
field play a crucial role. The former includes models of [219–221] and the latter includes
the Galileons [26, 133, 134]. The screening mechanism in this last class of models is called
the Vainshtein mechanism [27], and has been studied extensively (see [222] for a review).
The Vainshtein mechanism has been investigated [180, 224–226] even in the context of89

90

Anti-Screening of the Galileon Force Around a Disk Center Holethe most general scalar-tensor theory with second-order field equations [29] because it
can be obtained by generalizing the Galileons [139, 140] and the mechanism can thus
be implemented naturally. See [235–238] for the Vainshtein mechanism (and its partial
breaking) in more general scalar-tensor theories which have been developed recently.
As noted in Chapter 4, mostly focused on the Vainshtein mechanism around spherical
distributions of matter, as a star can be well approximated by a sphere. The authorsof [30] pointed out that Vainshtein screening might be sensitive to the shape of the matter
distribution. It is, however, difficult in general to study the shape dependence of theVainshtein mechanism because one has to treat derivative nonlinearities in less symmetricsystems. In [239], a two-body system was investigated numerically and it was shownthat the equivalence principle can be violated apparently in such systems. Approximate
solutions for slowly rotating stars in the cubic Galileon theory were obtained in [240]. As
for a dynamical aspect of the Vainshtein mechanism, the emission of scalar modes froma binary system was evaluated in [241]. Indeed, there are several works which investigate
the Vainshtein mechanism in the context of cosmology (see for example [242,243] and thereferences therein). Very recently the shape dependence of screening in the chameleon
theory was addressed numerically in [244].

5.2. Basic equations
5.2.1. The cubic Galileon
In this study, we employ the cubic Galileon theory1 [26, 133] as an example of themodel endowed with the Vainshtein mechanism. This theory has been well studied incosmology [142, 143, 245, 246] including weak gravitational lensing [247], and black holesolutions were found in [248] and the quantum aspect of this theory has been discussed in[249]. In the Einstein frame, the Galileon field „ and its coupling to matter are describedby the action
S=

1

⁄

C

D

1
c
—
d x ≠ (ˆ„)2 ≠ 33 (ˆ„)2 ⇤„ +„Tµµ ,
2
M
M
Pl
4

(5.1)

As stated in the introduction, we only consider the Galileon field living in a flat background. We also
investigate how our result depends on the background curvature in Appendix C.

Anti-Screening of the Galileon Force Around a Disk Center Hole91

where c3 and — are dimensionless parameters, MPl is the Planck mass, M is another massscale, and Tµµ is the trace of the matter energy-momentum tensor, which is defined in the
Einstein frame. We assume that matter is non-relativistic, so that Tµµ ƒ ≠ﬂ. Varying
the action with respect to „, we obtain the field equation,
„+

È
c3 Ë
—
2
i j
(
„)
≠
Ò
Ò
„Ò
Ò
„
=
ﬂ,
i j
3
MPl
M

(5.2)

where we assumed that „ is static. For a given configuration of matter, one can integrate
Eq. (5.2) to obtain the profile of „.
So far, successful Vainshtein screening has been confirmed mainly for sphericallysymmetric configurations. The Vainshtein mechanism in the systems with planar andcylindrical symmetry has been investigated in [30] and in Sec. 4.5, and it was foundthat the screening of the fifth force is sensitive to the shape of the matter distribution.
Only in such highly symmetric cases the non-linear equation (5.2) can be integrated
analytically, and one has to employ numerical methods in general cases. In [239] theGalileon field equation is integrated numerically for a two-body system. In [30], theprofile of the Galileon field around a matter distribution that has not been investigatedpreviously, i.e., a disk with a hole.

5.2.2. Numerical setup
Specifically, we model the system by the following uniform density profile
ﬂ(r, ◊) = ﬂ0 U (r ≠ r1 )U (r2 ≠ r)U (◊0 ≠ ◊)U (◊0 + ◊),ﬂ0 = const,

(5.3)
(5.4)

where U is the Heaviside function, with r1 , r2 , and ◊0 being the inner radius, the outerradius, and the (half of the) opening angle of the disk, respectively (Fig. 5.1). Note that
here we are using the spherical coordinates whose definition is slightly different from theusual one, x = r cos ◊ cos Ï, y = r cos ◊ sin Ï, z = r sin ◊, with ≠ﬁ/2 Æ ◊ Æ ﬁ/2.
To implement numerical integration, we introduce the following dimensionless quantities:
„¯ :=

„
,
M 3 r02

r̄ :=

r
,
r0

µ :=

—ﬂ0
,
M 3 MPl

(5.5)

92

Anti-Screening of the Galileon Force Around a Disk Center Hole

Figure 5.1.: A disk object with a hole in spherical coordinates.

where r0 is some arbitrary length scale and µ is the parameter that corresponds to thecoupling between matter and the Galileon for fixed ﬂ0 . At a sufficiently large distancefrom the disk object, it can be regarded as a point particle and hence we have „¯ ≥ µ/r̄2 .
Therefore, it can be said that µ controls the nonlinearity of the scalar field. We rewrite
Eq. (5.2) in terms of the above variables assuming that „ is axisymmetric.
The boundary conditions we impose are given by-ˆ „¯ -= 0,
ˆr̄ -r̄=0ˆ „¯ -ˆ „¯ -== 0,ˆ◊ -◊=0ˆ◊ -◊=ﬁ/2¯ max , ◊) = 0,
„(r̄

(5.6)
(5.7)
(5.8)

where r̄max := rmax /r0 corresponds to the boundary of the computational domain. The
boundary condition (5.6) amounts to the regularity at the center, while the condition (5.7)
reflects the symmetry of the system. Since the field equation is invariant under theconstant shift of the scalar field, „ æ „ + c, we may impose the boundary condition (5.8)
without loss of generality.
One may naively expect that derivative nonlinearity of the Galileon field is large forr . (c3 —ﬂ0 V /M 2 MPl )1/3 , where V is the size of a massive object. If we roughly taker1 ≥ r2 , we can estimate V as V ≥ r23 ◊0 . Thus, in terms of the dimensionless variables,
we see that the nonlinear effect is large for r̄ . (c3 µ◊0 )1/3 r̄2 .

Anti-Screening of the Galileon Force Around a Disk Center Hole

93

Figure 5.2.: The vectors represent that the dimensionless force fields for r̄1 = 8, ◊0 = 0.05,
and µ = 36.8, with c3 = 1 (left) and c3 = 0 (right). The pink regions represents
the disk.

5.3. Numerical Results
We now present our numerical solutions to Eq. (5.2). We fix r̄2 and r̄max as r̄2 = 30 andr̄max = 80, respectively, and performed numerical calculations for different values of r1 ,◊0 , and µ. The number of data points is 200 in the r direction and 100 in the ◊ direction.
Since the nonlinear effect would be significant in the vicinity of the hole, we solve thefield equation (5.2) with inhomogeneous grid spacing (see the details of the numericalcomputation in Appendix C).
˛ for
In Fig. 5.2 we show a vector plot of the dimensionless force field ≠(M 3 r0 )≠1 Ò„
c3 = 1, r̄1 = 8, ◊0 = 0.05, and µ = 36.8. In order to clarify the effect of the nonlinear
terms in Eq. (5.2), we also calculated the force field with the same parameters, but with
c3 = 0. The result is also presented in Fig. 5.2 for comparison. It can be seen that in
the c3 = 1 case the fifth force is suppressed compared to the c3 = 0 case in almost every
region, as expected. This is clear in particular for r̄ & 20 around the disk. However,surprisingly enough, the nonlinear effect enhances, rather than suppresses, the fifth force
in the vicinity of the hole.
To quantify this anti-screening effect, we introduce the following scalar quantity,R=˛ c =1|Ò„|
3.
˛|Ò„|c =0

(5.9)

3

We may say that screening is successful if R < 1. Figure 5.3 shows R for the above case,which clearly indicates that the fifth force is enhanced near the hole.

94

Anti-Screening of the Galileon Force Around a Disk Center Hole

Figure 5.3.: 2D plot of the degree of (anti-)screening R for the case shown in Fig. 5.2 (left).
The pink line represents the disk. R along ◊ = ﬁ/10 as a function of r̄ is also
shown (right).

To see how the enhancement of the fifth force depends on the parameters, we provide
numerical results for different values of r1 , ◊0 and µ in Figs. 5.4–5.6. Figure 5.4 shows R
for different sizes of the hole, r̄1 = 4 and 20, with c3 , r1 , ◊0 being fixed to the previous
values and µ being given such that the total mass of the disk is unchanged from theprevious case. It is found that the fifth force around the hole is stronger for a smallerhole size, as is most clearly seen in the bottom panel of Fig. 5.4. Figure 5.5 represents
the dependence of the enhancement effect on the thickness of the disk. We see that forsmaller ◊0 , i.e., for a thinner disk, the fifth force around the hole is stronger. We also
see that the enhancement effect occurs in the disk interior for larger ◊0 . This is because
the x̄ component of Galileon force both in the case of c3 = 0 and c3 = 1 cross zero atdifferent points, so that the ratio R becomes large when the denominator crosses the zero.
Finally, we see from Fig. 5.6 how increasing µ changes the result with other parameters
fixed. For larger µ, the enhancement of the Galileon force is less evident. This is because
larger µ implies that the disk is (effectively) more dense or more massive, and thus thescreening effect from the disk itself is more efficient. To sum up, the anti-screening effectis larger for a thinner, less massive disk with a smaller hole.

Anti-Screening of the Galileon Force Around a Disk Center Hole95

Figure 5.4.: R for r1 /r0 = 4 (top left) and r1 /r0 = 20 (top right). The other parameters are
the same as in the previous plots. The white areas represent regions where R isgreater than 5. R along ◊ = ﬁ/10 as a function of r̄ is also shown (bottom).

Figure 5.5.: R for ◊0 = 0.1 (top left) and ◊0 = 0.2 (top right). R along ◊ = ﬁ/10

as a function of r̄ is also shown (bottom).

96

Anti-Screening of the Galileon Force Around a Disk Center Hole

Figure 5.6.: R for µ = 369 (left) and µ = 3690 (right). R along ◊ = ﬁ/10

as a function of r̄ is also shown (bottom).

Anti-Screening of the Galileon Force Around a Disk Center Hole

97

5.4. Scalar-field profile in Schwarzschild geometry
We have solved the Galileon field equation in the flat background. In order to see thescalar-field profile in the curved background, let us consider the covariant version of
Eq. (5.2) in a fixed background. The cubic Galileon action is given by
S=

⁄

C

D

2
MPl
1
c
d x ≠gR ≠ g µ‹ Òµ „Ò‹ „ ≠ 33 ⇤„g µ‹ Òµ „Ò‹ „ + Sm [e2—„ gµ‹ ; ].
2
2
M
(5.10)
4Ô

where Òµ is the covariant derivative with respect to gµ‹ , and Sm represents the matteraction for matter fields . The Einstein frame metric gµ‹ is related to the Jordan frame
JJ
metric gµ‹
via the conformal transformation, gµ‹= e2—„ gµ‹ . The energy momentum
tensor is defined byÔ≠gTµ‹ = 2

”Sm
.
”g µ‹

(5.11)

We assume that matter is non-relativistic, so that Tµµ ƒ ≠ﬂ. Varying the action withrespect to the field, we obtain the field equation
⇤„ +

È
c3 Ë
—
2
µ ‹
µ‹
(⇤„)
≠
Ò
Ò
„Ò
Ò
„
≠
R
Ò
„Ò
„
=
ﬂ,
µ
‹
µ
‹
3
MPl
M

(5.12)

where Rµ‹ is the Ricci tensor, and we take3gµ‹ dxµ dx‹ = ≠ 1 ≠

4

3

rg
r
dt2 + 1 ≠ g
r
r

4≠1

dr2 + r2 d2
2,

(5.13)

with rg = r0 . We solve Eq. (5.12) numerically for the matter configuration with r̄1 = 8,
◊0 = 0.05, and µ = 36.8. The resultant profile of „ is shown in Fig. 5.7 and should becompared with that in the flat background in Fig. 5.3. It can be seen that the profiles
are not so different. We thus conclude that the effect of the background curvature does
not change the result of anti-screening.

98

Anti-Screening of the Galileon Force Around a Disk Center Hole

Figure 5.7.: R in the Schwarzschild background. This should be compared with the left panelof Fig. 5.3.

Chapter 6.
Conclusions and Outlook
6.1. Summary
As we reviewed in Chapter 2, general relativity has been widely accepted as the standard
model of gravity and cosmology. However, the concordance cosmological model basedon general relativity, the -CDM model, is consistent with the observational results,
but the origin of dark energy and dark matter remain unsolved. The discovery of suchdark components may point to the existence of new physics and give us the motivation
to study the modification of the theory of gravity. Modified gravity can explain theaccelerating expansion without dark energy, and yield a dark matter candidate.
In Chapter 3, among modified gravity theories, the scalar-tensor theory is the mostwell studied and understood gravity models. A scalar field appears often in the context
of cosmology, such as in inflation and quintessence models. Modified gravity can beeffectively described, at least, by adding a scalar degree of freedom to the gravitational
action. Thus, theories composed of a metric and a scalar field are ubiquitous. In thisthesis, the scalar-tensor theory has been focused on, especially Galileon theory which was
proposed as a four-dimensional higher-derivative theory without introducing higher orderderivatives in the equations of motion. The Galileon has been studied in the context ofcosmology as a dark energy model.
In contrast to general relativity, in scalar tensor theory, the interactions between
the scalar field and matter mediates a new long-rang fifth force. Experimental tests
performed in the solar system show that all results are in agreement with generalrelativity. Therefore many viable modified gravity theories possess a screening mechanism
to suppress the fifth force in the vicinity of concentrations of matter which is discussed
99

100

Conclusions and Outlook

in Chapter 4. The Galileon theory is equipped with the Vainshtein mechanism which
relies on the nonlinear derivative interactions of the scalar field. It was confirmed that
the Vainshtein mechanism occurs around spherical distributions of matter. The efficiencyof the Vainshtein mechanism in less symmetric systems was not clear due to the presenceof derivative nonlinearities. Therefore, in this thesis, the aspects of the Vainshteinmechanism in Galileon theory was explored.
In Chapter 5, we have studied numerically the fifth force around a disk with a hole atits center in the cubic Galileon theory. Since our source is thin but finite, we have seenthat screening still occurs in almost every region around the disk. However, we havefound that the hole at the centre causes an unexpected consequence: the Galileon force is
not suppressed but enhanced in the vicinity of the hole, namely, anti-screening operates.
Anti-screening we have seen in this paper occurs in the region where nonlinearity of theGalileon field is dominant and the configuration of matter is less symmetric. Due to thiscomplexity, so far we have not arrived at analytic understanding of our result.

6.2. Future Issues
There are issues which we left for future study. The cubic Galileon theory can evade thetight constraints on the propagation speed of gravitational waves [5, 91, 94], while someparameter region has already been strongly constrained by ISW measurements [171] asdark energy model. Note, however, that by generalizing the present model slightly one
can easily find models that are consistent with the ISW data [157, 250]. We have foundthe anti-screening effect in a particular cubic Galileon theory, and we believe that it
occurs in such generalized Galileon models.
We investigate the cubic Galileon theory which can be viewed as the simplest theory
having the nonlinear term. It is therefore interesting to study whether or not theanti-screening effect occurs in the quartic and quintic Galileon models. This is a very
important open problem which we hope to study in the near future.
Furthermore, some of the parameters we have used in our numerical calculations
might not be realistic. In particular, we have seen that we need µ . 103 in order for theforce to be enhanced. For larger µ, the effect of anti-screening is washed away by thescreening effect from the disk in the present setup. If the Galileon field is responsible
for the current cosmic acceleration, one would expect M 3 ≥ MPl H02 ≥ ﬂ̄/MPl , where H0

Conclusions and Outlook

101

is the present Hubble parameter and ﬂ̄ is the average energy density of the Universe.
The energy density of our disk is thus given by ﬂ0 ≥ µﬂ̄, assuming that — = O(1). Our
numerical calculations correspond to such very low density matter distribution. Therefore,
at this stage it is difficult to derive direct implications of our results for astrophysics
and experiments. Nevertheless, we believe that it is interesting to further explore how
the (anti-)screening mechanism operates for nontrivial configurations of matter and thepresent work provides a first step toward understanding this complicated problem.

102

Appendix A.
Conformal Transformation Rules
In this appendix, we present the conformal transformation properties of geometric tensors
that we used to derive several results in Chapter 3. We then show how the energy
conservation law for the Einstein frame can be obtained from the Jordan frame.

A.1. Einstein and Jordan frames
Consider a scalar-tensor theory; the action is given by
S=

⁄

C

A

B

1
Ê
d x ≠g„R ≠ 0 Òµ „Òµ „ + Lm (gµ‹ ,
16ﬁG
„
4

Ô

D

m)

.

(A.1)

When the action is given in this form, the energy-momentum tensor of the matter fields isconserved. Here, we use the Jordan-frame metric gµ‹ . We define the energy-momentum
tensor of the matter fields in Jordan frame as covariantly conserved,
Òµ T µ‹ = 0.

(A.2)

In the Jordan frame, test particles follow the geodesics of the spacetime or of the metric.
Indeed, the scalar is non-minimally (or directly) coupled to the gravity and the matter iscoupled to the Jordan frame metric. We now consider the conformal transformationgµ‹ æ ĝµ‹ =Here we defined the Einstein frame metric ĝµ‹ .
103

2

gµ‹ .

(A.3)

104

Conformal Transformation Rules

A.2. Transformation Rules

We are interested in several quantities such as the Riemann tensor, which is constructedusing the metric and Christoffel symbols. First, one can easily derive the transformationof the Christoffel symbols,
⁄

µ‹

1 ⁄ﬂ
g (ˆµ g‹ﬂ + ˆ‹ gµ⁄ ≠ ˆﬂ gµ‹ )2= ˆ ⁄µ‹ + (”µ⁄ ˆ‹ + ”‹⁄ ˆµ ≠ ĝµ‹ ĝ ⁄ﬂ ˆﬂ ) log= ˆ ⁄µ‹ + A⁄ µ‹ .
=

(A.4)

Here we define the quantities,A⁄ µ‹ =⁄µ‹ ≠ˆ ⁄µ‹ ,

(A.5)

for the convenience of the following applications. Next, to find the transformation of the
Riemann tensor, we use the definition of the Riemann tensor,
R– —µ‹ = ˆµ – —‹ ≠ ˆ‹ – —µ +

–

⁄µ

⁄

—‹ ≠

–

⁄‹

⁄

—µ ,

(A.6)

and combine with the above result, then we have–R– —µ‹ = R̂—µ‹+ 2A– ﬂ[µ Aﬂ ‹]— + 2Ò[µ A– ‹]— .

(A.7)

Contracting equation (A.7) and combining equation (A.5) gives the transformation of
the Ricci tensor,
Rµ‹ = R̂µ‹ ≠ ĝµ‹ ⇤ log≠ 4Òµ Ò‹ log≠ 4ĝµ‹ Ò‡ log Ò‡ log(A.8)+ 2Òµ log Ò‹ log ,and contracting with g µ‹ gives the transformation of the Ricci scalar,R=

1

2 (R̂ ≠ 6⇤ log≠ 6Ò⁄ log Ò⁄ log ).

(A.9)

Note that the term proportional to ⇤ log is a total derivative, therefore this term does
not contribute to the equations of motion.

Conformal Transformation Rules

105

A.3. The Energy-Momentum Tensor and The
Conservation Law
As mentioned in the previous section, the energy-momentum is conserved in the Jordan
frame. From now on, we derive the quantities that are expressed in the Einstein frame.
In other words, we derive the relation of the energy-momentum tensors between the twoframes. Using the definition of the energy-momentum tensor,
2 ”SM [g–— , ]T µ‹ = ≠ Ô
,
≠g
”gµ‹

(A.10)

we obtainTµ‹A2 ”SM [ĝ–— , ]= 4 ≠Ô
”ĝ⁄ﬂ
≠ĝ
”ĝ
1
:= 4 T̂ ⁄ﬂ ⁄ﬂ”gµ‹1= 6 T̂ µ‹ .
1

B

”ĝ⁄ﬂ
”gµ‹

(A.11)

We can calculate the conservation law in the Einstein frame from the Jordan frame
relationÒµ T µ‹ = Òµ (= Òµ (≠6T̂ µ‹ )

≠6T̂ µ‹ ) +≠6

(Aµ µ⁄ + A‹ µ⁄ )T̂ ⁄µ = 0.

(A.12)

Substituting definition A⁄ µ‹ (A.5) we obtainÒµ T̂ µ‹ =d logT̂ Ò‹ „,d„(A.13)

where T̂m = ĝµ‹ T̂mµ‹ is the trace of the energy-momentum tensor. The energy-momentum
tensor is not conserved in the Einstein-frame; a test particle does not follow the geodesicsof the metric.

106Appendix B.
f (R) theories of gravity
In this appendix, we review f (R) gravity theory and its properties. f (R) gravity is one of
the natural extensions of general relativity, which was first proposed in [131]. Replacing
the Ricci scalar in the Einstein-Hilbert action with an arbitrary function of the Ricci
scalar, we obtain the action of f (R) theory as:
J
Sf(R)
=

1 ⁄ 4 Ôd x ≠gf (R) + Sm [gµ‹ , ],
2Ÿ

(B.1)

here Ÿ is a constant. In the context of inflation, the Straobinsky model L ≥ R + R2
is one of the viable inflation models [251]. The Straobinsky model may also drive thelate-time acceleration of the Universe. That is why, the cosmological and astrophysicalimplications of f (R) theory have been vastly investigated (see [252] for a review).

B.1. Field Equation
Varying Eq. (B.1) with respect to the metric, we obtain
”S =
=

⁄
⁄

⁄
Ô
Ô
d4 x”( ≠g)f (R) + d4 x ≠g”(f (R))

4
⁄
Ô 3 1
Ô
d4 x ≠g ≠ gµ‹ f (R) ”g µ‹ + d4 x ≠g”(f (R)).
2

(B.2)

The variation of the Ricci scalar is given by”R = Rµ‹ ”g µ‹ + g µ‹ {Ò– ” – µ‹ ≠ (Ò‹ ” – µ– )},
107

(B.3)

108

f (R) theories of gravity

then we evaluate the second term of Eq. (B.3):” – µ‹ = ”(g –— ” —µ‹ )=+ g –— ” —µ‹1= —µ‹ ”g –— + g –— (Òµ ”g—‹ + Ò‹ ”g—µ ≠ Ò— ”gµ‹ ) +2
1 –—= g (Òµ ”g—‹ + Ò‹ ”g—µ ≠ Ò— ”gµ‹ ),2—µ‹ ”g–—⁄µ‹ ”g—⁄(B.4)

and we also obtain1” – µ– = g –— ”g–—;µ .
2

(B.5)

Combining Eqs. (B.4) and (B.5), we obtainÓÔg µ‹ (” – µ‹ );– ≠ (” – µ– );‹ = gµ‹ ⇤”g µ‹ ≠ Òµ Ò‹ ”g µ‹ .

(B.6)

Finally we obtain the field equation for f (R) gravity
ˆf (R)1
ˆf (R)
Rµ‹ ≠ gµ‹ f (R) + (gµ‹ ⇤ ≠ Òµ Ò‹ )= ŸTµ‹ .
ˆR
2
ˆR

(B.7)

B.1.1. Energy Conservation
Next, we prove that the energy-momentum tensor in f (R) theory is covariantly conserved.
Hereafter, for simplicity we use the notation:
ˆf (R):= f,R ,ˆR

Dµ‹ := gµ‹ ⇤ ≠ Òµ Ò‹ .

Taking the covariant derivative Òµ of Eq. (B.7), we obtain1Òµ (Rµ‹ f,R ) ≠ Ò‹ f + Òµ Dµ‹ f,R = ŸÒµ Tµ‹ .
2

(B.8)

Ò‹ f = (Ò‹ R)f,R ,(B.9)

Using the chain rule:f (R) theories of gravity

109

and the formulae:
Òµ Dµ‹ f,R = ≠(⇤Ò‹ ≠ Ò‹ ⇤)f,R= [Ò‹ , ⇤]f,R= ≠Rµ‹ (Òµ f,R ),1
Òµ Rµ‹ = Ò‹ R,
2(B.10)(B.11)

we rewrite Eq. (B.8) as
1ŸÒµ Tµ‹ = (Òµ Rµ‹ )f,R + Rµ‹ (Òµ f,R ) ≠ (Ò‹ R)f,R ≠ Rµ‹ (Òµ f,R )2
1µ= (Ò Rµ‹ )f,R ≠ (Ò‹ R)f,R2= 0.

(B.12)

We conclude that the energy momentum tensor is covariantly conserved.

B.1.2. Scalaron
In f (R) theory, a new scalar degree of freedom propagates, it is called the scalaron. By
taking trace of Eq. (B.7) with (Tµ‹ = 0), we obtainRµ‹ f,R ≠ 2f (R) + 3⇤f,R = 0.

(B.13)

We can rewrite the equation as2
11
2f ≠ Rf,R
3
dV eff:= ≠.
dR⇤f,R =(B.14)

Thus, this equation implies that the Ricci scalar, which corresponds to an arbitrary
function f (R), is dynamical.

110

f (R) theories of gravity

B.2. f (R) theory and Brans-Dicke theory

As we noted above, f (R) theory has a new scalar degree of freedom. This means thatf (R) theory can be seen as one example of a scalar-tensor theory. Here, we prove thatf (R) theory is equivalent to Brans-Dicke theory with ÊBD = 0. We introduce an auxiliary
filed ‰ and we consider the following action:S=

1 ⁄ 4 Ôd x ≠g[f (‰) + f,‰ (‰)(R ≠ ‰)] + Sm (gµ‹ , Œi ).
2Ÿ

(B.15)

Varying Eq.(B.15) with respect to ‰ we obtain⁄ÈÔ Ë
d4 x ≠g f,‰ ”‰ + f,‰‰ (R ≠ ‰)”‰ ≠ f,‰ ”‰⁄Ô= d4 x ≠g[f,‰‰ (R ≠ ‰)]”‰.

0=(B.16)

Assuming f,‰‰ ”= 0 for all R, (B.16) must holds for any variation ”‰. Therefore, we find
‰ = R.

(B.17)In this case, the action (B.15) is equivalent to f (R) theory. If we define„ := f,‰ (‰),

(B.18)

we can rewrite Eq. (B.15) as follows
S=

⁄

5

6

1
d x ≠g„R ≠ U („) + Sm [gµ‹ , ],2Ÿ

(B.19)‰(„)„ ≠ f (‰(„)).
2Ÿ

(B.20)

4Ô

where we define U („) asU („) =

Comparing Eqs. (B.19) and (3.34), we find that f (R) theory is equivalent to Brans-Dicke
theory with ÊBD = 0.

f (R) theories of gravity

111

B.2.1. f (R) Theory and Scalar-Tensor Theory
We can rewrite f (R) gravity as a scalar-tensor theory by using the conformal transformation. As we stated, we can recast the action of f (R) theory (B.1) into
Sf (R) =6
Ô 51
d4 x ≠g
Rf,R ≠ U + Sm [gµ‹ , ],2Ÿ⁄(B.21)

where
U :=

1
(Rf,R ≠ f ).
2Ÿ(B.22)We define the conformal transformation2gµ‹ æ ĝµ‹ =(B.23)gµ‹ ,here ĝµ‹ is the Einstein-frame metric. Under this transformation, the Christoffel symboltransforms according to (see Appendix A)ˆ –—“ =

–

—“ +

≠1

(”—– Ò“+ ”“– Ò—≠ g—“ Ò– ),(B.24)and the Ricci scalar also transforms according to2

R=(R̃ + 6⇤ú Ê ≠ 6g̃ µ‹ ˆµ Êˆ‹ Ê),(B.25)

where we have definedÊ := ln .

(B.26)

Under the conformal transformation, we obtain the action in the Einstein frame
E
Sf(R)
=

⁄

Ò

d4 x ≠ĝ

5

1
f,R
2Ÿ
≠

≠2

≠4

R̂ ≠

1
f,R
2Ÿ

1
U + (6f,R
2Ÿ

≠2
≠2

(6ĝ µ‹ ˆµ Êˆ‹ Ê)
6⇤ú Ê) + Sm [≠2ĝµ‹ , ],(B.27)The underline in Eq. (B.27) is the surface term and vanishes. To eliminate the nonminimal coupling in the action, we define the conformal transformation as
f,R

≠2

= 1,

(B.28)

112

f (R) theories of gravity

here we assume f,R > 0. We also define a new field (scalaron)
Ê :=

Ú

Ÿ
1
„ = ln , „ =
6
2

Û

6
ln f,RŸ(B.29)

so that the second term of Eq. (B.27) becomes the kinetic term. Substituting Eqs. (B.28)
and (B.29) into Eq. (B.27), we obtain the action
S=

⁄

5

6

1
1
d x ≠ĝR̂ ≠ Ò̂µ „Ò̂µ „ ≠ V („) + Sm [
2Ÿ
2
4

Ò

≠2

ĝµ‹ , ],(B.30)

where we define
V („) :=

Rf,R ≠ f
U
.
2 =
2
f,R
2Ÿf,R

(B.31)

The action (B.30) is the Einstein-Hilbert term with an additional scalar „. Since thescalar field couples to the matter field, the energy-momentum tensor associated with thematter field is no longer conserved.

B.2.2. Chameleon Mechanism
In f (R) gravity, the scalaron modifies the gravitational interaction, which is tightly
constrained by the observational results. Most models of f (R) gravity have a screening
mechanism to suppress the fifth force in the vicinity of concentrations of matter, whichis called the chameleon mechanism [217].
Here, we consider the non-relativistic perfect fluid in the Einstein-frame:Tµ‹ = diag [ﬂ, 0, 0, 0], T = g µ‹ Tµ‹ = ≠ﬂ.

(B.32)We redefine „ by
e2

Ô

Ÿ/6„

:= e≠2—„/MPl ,(B.33)

and rewrite Eq. (B.30) in terms of ĝµ‹ and „:
SST =

⁄

A

B

2
MPl
1
d x ≠ĝR̂ ≠ ĝ µ‹ Òµ „Ò‹ „ ≠ V („) + Sm [e2—„/MPl ĝµ‹ ,
2
2
4

Ò

i ].

(B.34)

f (R) theories of gravity

113

Varying Eq. (B.34) with respect to „, we obtain the equation of motion of the scalaronin the Einstein-frame:ˆ = dVeff („) ,⇤„d„(B.35)

where Veff is defined as
Veff := V („) +—ﬂe—„/MPl .
MPl(B.36)The effective potential depends on the matter density, the scalaron also depends on thematter density. When the effective potential reaches a minimum with „ = „min , the masssquared of the scalaron m„ is defined by the second derivative of the effective potential
with respect to „m2„ :=

d2 Veff („)
d2 V („min )— 2 —„min /MPl
=
+
.
2 ﬂe
d„2
d„2
MPl

(B.37)The Compton wave length of the scalaron is inversely proportional to the mass, therefore
one finds that, in a high density region, the Compton wavelength becomes short: thescalaron is screened in the solar system. This is the chameleon mechanism [217].
We live in a very dense region, the earth, so the scalaron (chameleon field) mass islarge enough to evade the current equivalence principle violation and fifth force search.
Thin-shell region near the surface of the earth only contribute to the field equation
outside the earth which is called thin-shell mechanism. If a body has thin-shell, the fifth
force is completely screened. The chameleon and thin-shell mechanism will be tested byupcoming satellite experiments [253, 254].

B.3. Viable f (R) Theories
In the previous section, we reviewed the general properties of the f (R) theory. Now, we
review viable models for the f (R) theory.
Inflation Model
We explain that f (R) theory is one of the strongest candidates to describe inflation. As
we reviewed, inflation is an epoch of accelerated expansion in the early Universe. An

114

f (R) theories of gravity

inflation epoch should end, so there is a new energy source, namely the inflaton. On theother hand, f (R) theory can drive inflation without introducing the inflaton: the degreeof freedom comes from the higher-order curvature.
Starobinsky has investigated the inflation models which apply quantum corrections
to Einstein’s field equations to drive inflation [251]. The original Starobinsky model
contains all quadratic curvature terms such as Rµ‹ Rµ‹ and R–—µ‹ R–—µ‹ . In isotropic andhomogeneous flat spacetime in four dimensions, the curvature corrections are described
only by the scalar curvature term
34

1 ⁄ 4 Ô1SSt =d x ≠g R +
R2 ,
2Ÿ
6M 2

(B.38)

where M is the mass scale parameter. In the case of the Starobinsky inflation model, theconformal transformation is given byÔf,R = e 2Ÿ/3„ .

(B.39)In the Einstein frame, the action can be seen with the Einstein-Hilbert term having aminimally coupled scalar field where the degree of freedom behaves as inflaton.
Next, we consider the equation of motion for the scalar field. In our case, we find
1
2
2R ,
6M
1
f,R = 1 +
R.
3M 2
f =R+

(B.40)(B.41)

We can rewrite the equations of motion for the scalaron (B.14)(⇤ ≠ M 2 )R = ŸM 2 T.

(B.42)

From the above Eqs. (B.40-B.41), we can solve these equations with respect to R, andwe obtainÔR = 3M 2 (e 2Ÿ/3„ ≠ 1).

(B.43)

Finally, we compute the potential of the scalar filed which is given by
V („) =

Ô
3M 2
(1 ≠ e≠ 2Ÿ/3„ )2 .
4Ÿ

(B.44)

f (R) theories of gravity

Taking the limitÔ115Ÿ„ ∫ 1, we obtainV („) ≥

3M 2
= const.
4Ÿ

(B.45)

This limit is equal to the slow-roll inflation, M 2 is interpreted as the inflaton mass. OnÔthe other hand, in the limit Ÿ„ π 1 one obtains
1
V („) ≥ M 2 „2 .
2

(B.46)In these limits, the field oscillates around the origin, which is called reheating. Thus, theStarobinsky inflation models can explain slow-roll inflation. The model predicts the valueof the tensor-to-scalar ratio; the value is at the centre of the Planck constraint [255].
f (R) theory can explain the accelerated expansion of the early Universe. Therefore,
one expects that the theory also can explain the present accelerated expansion of the
Universe by the scalar degree of freedom.
Dark Energy models
f (R) theory has been studied over the decades with the general form f (R). Even thoughf (R) theory might explain the accelerated expansion of the Universe, the function f (R)is not a free function. There are several constraints coming from experimental andobservational results and theoretical consistency requirements [252]:• In order to avoid a ghost state, we must have f (R) > 0 everywhere. If f (R) < 0,
the coefficient of the Einstein-Hilbert term is negative. Thus, the graviton becomesthe ghost modes, and the gravitational constant becomes negative.
• In order to avoid the tachyonic instability in regions of high curvature, we musthave f,RR > 0. Perturbing the trace of equation of motion around a backgroundcurvature R = Rb , we obtain
1
1
Ÿ”T = ⇤”f,R (Rb ) ≠
3
3

A

B

f,R (Rb )≠ Rb ”f,R (Rb )f,RR (Rb ):= ⇤”f,R (Rb ) ≠ m2s ”f,R (Rb ),(B.47)

here we define the effective mass ms for a scalar degree of freedom. If the masssquared is negative, the scalar mode becomes a tachyonic mode. Therefore, we musthave f,RR in this regime.

116

f (R) theories of gravity• In order to satisfy solar system constraints on a fifth force, f (R) takes the formR ≠ 2 for R ∫ R0 ≥ H02 ≥ . f (R)theory should restore the -CDM model at thescale [256].
(R)• 0 < f ,RR(R) < 1 when Rf,R = 2f (R) for the stability of the present and late-time,Rde Sitter limit of the Universe [257]. The condition comes from Eq. (B.47)
Rf

1
3

A

B

f,R (R)≠ R > 0.
f,RR (R)(B.48)• f (R) æ 0 when R æ 0 which ensure that there is flat spacetime solution in f (R)theory.
A number of theories satisfying the above constraints have been proposed so far. Several
choices for the form of f (R) have been well studied:• The Starobinsky model [258]:
SAT(B.49)

c1 (R/Rc )2n,c2 (R/Rc )2n + 1(B.50)

R2
U
f (R) = R + —Rc 1 + 2
RcB≠nwhere n, — and Rc > 0 are constant.

≠ 1V ,• The Hu and Sawicki model [259]:
f (R) = R ≠ —Rc

where n, —, Rc > 0. c1 and c2 are constant.
• The Tsujikawa (or hyperbolic) model [260]:
AB

R
f (R) = R ≠ —Rc tanh,
Rc

(B.51)

where —, Rc > 0 are constant.
All the above models satisfy the above conditions, both cosmological and local gravity
constraints.

Appendix C.
Numerical Scheme and Convergenceof Results in Chapter 5
In this appendix, we explain the details of the numerical computation used in Chapter 5.
Throughout Chapter 5, we employed the numerical scheme developed in Ref. [239] tosolve the field equation (5.2).
To achieve the high resolution around the hole of disk, we first transform the radial
coordinate
r =‰+

– 3
‰,
3r02

(C.1)

where – < 1 is a constant. Through out in Chapter 5, we use –=0.2. According to thiscoordinate transformation, the spatial interval in r coordinate is given byr(‰) = A(‰) ‰,‰2
A(‰) := 1 + – 2 ,
r0

(C.2)

where ‰ = ‰max /(Nr ≠ 1) and ‰max is given by solving Eq. (C.1) with r = rmax , andin general ‰max < rmax if – < 1. Then we can achieve high resolution near ‰ ≥ 0 incomparison with the original spatial interval, r(‰) < rmax /(Nr ≠ 1), with fixed Nr ,though the resolution at large distance gets worse where „ becomes almost constant and
we do not need the resolution.
In the new coordinate (‰, ◊), we discretize the computational domain as ‰i = (i +1/2) ‰ and ◊j = (j + 1/2) ◊ with i = 0, 1, . . . , Nr ≠ 1 and j = 0, 1, . . . , N◊ ≠ 1 where◊ = ﬁ/(2N◊ ). The derivatives with respect to r are given in terms of that with respect
117

118

Numerical Scheme and Convergence of Results in Chapter 5

to ‰,
ˆ„
1 ˆ„
=
,
ˆr
A(‰) ˆ‰ˆ2„
1 ˆ2„
AÕ (‰) ˆ„=≠,ˆr2
A(‰)2 ˆ‰2 A(‰)3 ˆ‰(C.3)

The derivatives with respect to ‰ and ◊ are approximated by the central finite differenceson the grid as„≠ „i≠1,jˆ„¥ i+1,j,ˆ‰
2 ‰„≠ „i,j≠1ˆ„¥ i,j+1,ˆ◊
2 ◊„i+1,j ≠ 2„i,j + „i≠1,j
ˆ2„
,
2 ¥
ˆ‰
‰2
„i,j+1 ≠ 2„i,j + „i,j≠1
ˆ2„
,
2 ¥
ˆ◊
◊2

(C.4)
(C.5)

where „i,j := „(‰i , ◊j ). Here we shift the ‰ and ◊ coordinates by half spatial intervals‰ and ◊, respectively, since it is to be able to easily impose the Neumann boundary
condition at r = 0 (r = rh in the case of a black hole) and ◊ = 0, ﬁ/2. All the derivatives
in Eq. (5.2) are replaced by these differences.
We regard the non-linear terms of Eq. (5.2), the terms proportional to c3 , as the extra
source term such that
—„ =

—
c
ﬂ ≠ 33 N [„].
MPl
M

(C.6)

At the first step, we solve the linear equation with setting N [„] = 0 and obtain thesolution „ú . Then we update „ in the following manner:„new (r, ◊) = (1 ≠ Ê)„old (r, ◊) + Ê„ú (r, ◊),(C.7)

with a mixing parameter Ê = O(0.01). Note that, unless the parameter Ê is small, thisiteration scheme does not work since the non-linear term N [„] induces quite a largechange of the field configuration. At the next step, evaluating N [„new ], we solve the field
equation again, and „ is further updated. This iteration procedure is terminated when
the update of „ is well suppressed, namely,||„new ≠ „old ||< ‘,||„new ||
Òq

(C.8)

2
≠8
where the norm ||„|| is defined as ||„|| :=ij „(ri , ◊j ) and we set ‘ = 10 . We
compare the results obtained for ‘ = 10≠7 and ‘ = 10≠8 , and confirmed that the relative

Numerical Scheme and Convergence of Results in Chapter 5

119

error between them is less than 1%. Therefore,it is sufficient to solve the field equation(5.2) with ‘ = 10≠8 . For details, see Ref. [239].
The field equation solved in Chapter 5, given in Eq. (5.2), is highly non-linear, andthus it should be confirmed whether our numerical results are reliable in the sense thatthey are well converged with the iteration scheme mentioned above. To see this, we solve
the field equation with changing the number of grid points in the coordinate of (r, ◊),Nr and N◊ , and the position of the boundary in the radial direction, r̄max . The fiducial
values of Nr , N◊ and r̄max in Chapter 5 are Nr = 200, N◊ = 100 and r̄max = 80. Notethat we do not focus on the other numerical parameters since they control solely theconvergence speed and the precision, and thus do not affect the final results.
Figure C.1 shows R evaluated at ◊ = ﬁ/10. From the left panel, we find that our
result is insensitive to the size of the computational box, which means that artificial
effects from the boundary at r = rmax do not affect the feature at all. The remaining
panels show the dependences of R on the number of grids, N◊ (center panel) and Nr(right panel.) While the detailed structure of the peak around r̄ ≥ 1 is sensitive to thespatial resolution, the fact that R can be larger than the unity at r̄ . 8 is confirmed to
be robust.

Figure C.1.: The convergence check of numerical results; the dependence on r̄max (left), onN◊ (middle) and on Nr (right).

120

Bibliography[1] A. Einstein, “The Field Equations of Gravitation,” Sitzungsber. Preuss. Akad.
Wiss. Berlin (Math. Phys.) 1915 (1915) 844–847.
[2] A. Einstein, “The Foundation of the General Theory of Relativity,” Annalen Phys.
49 no. 7, (1916) 769–822. [,65(1916)].
[3] C. M. Will, Theory and experiment in gravitational physics. Cambridge University
Press, 1993.
[4] Virgo, LIGO Scientific Collaboration, B. P. Abbott et al., “Observation ofGravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116
no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc].
[5] Virgo, LIGO Scientific Collaboration, B. Abbott et al., “GW170817:
Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys.
Rev. Lett. 119 no. 16, (2017) 161101, arXiv:1710.05832 [gr-qc].
[6] A. Borriello and P. Salucci, “The Dark matter distribution in disk galaxies,” Mon.
Not. Roy. Astron. Soc. 323 (2001) 285, arXiv:astro-ph/0001082 [astro-ph].
[7] V. C. Rubin, N. Thonnard, and W. K. Ford, Jr., “Rotational properties of 21 SC
galaxies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/
to UGC 2885 /R = 122 kpc/,” Astrophys. J. 238 (1980) 471.
[8] D. Clowe, A. Gonzalez, and M. Markevitch, “Weak lensing mass reconstruction of
the interacting cluster 1E0657-558: Direct evidence for the existence of dark
matter,” Astrophys. J. 604 (2004) 596–603, arXiv:astro-ph/0312273[astro-ph].
[9] M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman,
C. Jones, S. Murray, and W. Tucker, “Direct constraints on the dark matter
self-interaction cross-section from the merging galaxy cluster 1E0657-56,”
121

122

BIBLIOGRAPHY

Astrophys. J. 606 (2004) 819–824, arXiv:astro-ph/0309303 [astro-ph].
[10] C. Conroy, R. H. Wechsler, and A. V. Kravtsov, “Modeling luminosity-dependent
galaxy clustering through cosmic time,” Astrophys. J. 647 (2006) 201–214,
arXiv:astro-ph/0512234 [astro-ph].
[11] Supernova Cosmology Project Collaboration, S. Perlmutter et al.,“Measurements of Omega and Lambda from 42 high redshift supernovae,”
Astrophys. J. 517 (1999) 565–586, arXiv:astro-ph/9812133 [astro-ph].
[12] Supernova Search Team Collaboration, A. G. Riess et al., “Observational
evidence from supernovae for an accelerating universe and a cosmological constant,”
Astron. J. 116 (1998) 1009–1038, arXiv:astro-ph/9805201 [astro-ph].
[13] WMAP Collaboration, D. N. Spergel et al., “First year Wilkinson Microwave
Anisotropy Probe (WMAP) observations: Determination of cosmologicalparameters,” Astrophys. J. Suppl. 148 (2003) 175–194, arXiv:astro-ph/0302209[astro-ph].
[14] WMAP Collaboration, D. N. Spergel et al., “Wilkinson Microwave Anisotropy
Probe (WMAP) three year results: implications for cosmology,” Astrophys. J.
Suppl. 170 (2007) 377, arXiv:astro-ph/0603449 [astro-ph].
[15] SDSS Collaboration, D. J. Eisenstein et al., “Detection of the Baryon Acoustic
Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies,”
Astrophys. J. 633 (2005) 560–574, arXiv:astro-ph/0501171 [astro-ph].
[16] SDSS Collaboration, K. N. Abazajian et al., “The Seventh Data Release of the
Sloan Digital Sky Survey,” Astrophys. J. Suppl. 182 (2009) 543–558,
arXiv:0812.0649 [astro-ph].
[17] Planck Collaboration, Y. Akrami et al., “Planck 2018 results. I. Overview andthe cosmological legacy of Planck,” arXiv:1807.06205 [astro-ph.CO].
[18] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmologicalparameters,” arXiv:1807.06209 [astro-ph.CO].
[19] S. Weinberg, “The Cosmological Constant Problem,” Rev. Mod. Phys. 61 (1989)
1–23. [,569(1988)].
[20] S. Tsujikawa, “Quintessence: A Review,” Class. Quant. Grav. 30 (2013) 214003,
arXiv:1304.1961 [gr-qc].

BIBLIOGRAPHY

123[21] C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, “Essentials of kessence,” Phys. Rev. D63 (2001) 103510, arXiv:astro-ph/0006373 [astro-ph].
[22] T. Chiba, T. Okabe, and M. Yamaguchi, “Kinetically driven quintessence,” Phys.
Rev. D62 (2000) 023511, arXiv:astro-ph/9912463 [astro-ph].
[23] E. Fischbach and C. Talmadge, “Six years of the fifth force,” Nature 356 (1992)
207–214.
[24] E. G. Adelberger, B. R. Heckel, and A. E. Nelson, “Tests of the gravitational
inverse square law,” Ann. Rev. Nucl. Part. Sci. 53 (2003) 77–121,
arXiv:hep-ph/0307284 [hep-ph].
[25] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D.
Hoyle, and H. E. Swanson, “Tests of the gravitational inverse-square law below thedark-energy length scale,” Phys. Rev. Lett. 98 (2007) 021101,
arXiv:hep-ph/0611184 [hep-ph].
[26] A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modificationof gravity,” Phys. Rev. D79 (2009) 064036, arXiv:0811.2197 [hep-th].
[27] A. I. Vainshtein, “To the problem of nonvanishing gravitation mass,” Phys. Lett.
39B (1972) 393–394.
[28] C. Brans and R. H. Dicke, “Mach’s principle and a relativistic theory ofgravitation,” Physical review 124 no. 3, (1961) 925.
[29] G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional
space,” Int. J. Theor. Phys. 10 (1974) 363–384.
[30] J. K. Bloomfield, C. Burrage, and A.-C. Davis, “Shape dependence of Vainshtein
screening,” Phys. Rev. D91 no. 8, (2015) 083510, arXiv:1408.4759 [gr-qc].
[31] H. Ogawa, T. Hiramatsu, and T. Kobayashi, “Anti-screening of the Galileon force
around a disk center hole,” arXiv:1802.04969 [gr-qc].
[32] I. Newton, Philosophiae naturalis principia mathematica. William Dawson & Sons
Ltd., London, 1687.
[33] U. Le Verrier, “Lettre de m, le verrier à m: Faye sur la théorie de mercure et sur lemouvement du périhélie de cette planète,” Comptes rendus hebdomadaires desséances de l’Académie des sciences 49 (1859) 379–383.

124

BIBLIOGRAPHY[34] C. M. Will, “The Confrontation between General Relativity and Experiment,”
Living Rev. Rel. 17 (2014) 4, arXiv:1403.7377 [gr-qc].
[35] S. Misao, General Relativity. Sangyou Tosyo, 1996.
[36] L. D. Landau and E. M. Lifshitz, The classical theory of fields; 3rd ed. Course oftheoretical physics. Pergamon, Oxford, 1971.
http://cds.cern.ch/record/101808. Trans. from the Russian.
[37] E. Poisson, A relativist’s toolkit: the mathematics of black-hole mechanics.
Cambridge university press, 2004.
[38] MACHO Collaboration, C. Alcock et al., “The MACHO project: Microlensing
results from 5.7 years of LMC observations,” Astrophys. J. 542 (2000) 281–307,
arXiv:astro-ph/0001272 [astro-ph].
[39] R. D. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons,”
Phys. Rev. Lett. 38 (1977) 1440–1443. [,328(1977)].
[40] R. D. Peccei and H. R. Quinn, “Constraints Imposed by CP Conservation in the
Presence of Instantons,” Phys. Rev. D16 (1977) 1791–1797.
[41] G. Jungman, M. Kamionkowski, and K. Griest, “Supersymmetric dark matter,”Phys. Rept. 267 (1996) 195–373, arXiv:hep-ph/9506380 [hep-ph].
[42] G. Bertone and D. Hooper, “A History of Dark Matter,” Submitted to: Rev. Mod.
Phys. (2016) , arXiv:1605.04909 [astro-ph.CO].
[43] Planck Collaboration, P. A. R. Ade et al., “Planck 2015 results. XIII.
Cosmological parameters,” Astron. Astrophys. 594 (2016) A13,
arXiv:1502.01589 [astro-ph.CO].
[44] T. Chiba, “Slow-Roll Thawing Quintessence,” Phys. Rev. D79 (2009) 083517,
arXiv:0902.4037 [astro-ph.CO]. [Erratum: Phys. Rev.D80,109902(2009)].
[45] R. J. Scherrer and A. A. Sen, “Thawing quintessence with a nearly flat potential,”
Phys. Rev. D77 (2008) 083515, arXiv:0712.3450 [astro-ph].
[46] R. R. Caldwell and E. V. Linder, “The Limits of quintessence,” Phys. Rev. Lett.
95 (2005) 141301, arXiv:astro-ph/0505494 [astro-ph].
[47] T. Chiba, “W and w’ of scalar field models of dark energy,” Phys. Rev. D73(2006) 063501, arXiv:astro-ph/0510598 [astro-ph]. [Erratum: Phys.

BIBLIOGRAPHY

125

Rev.D80,129901(2009)].
[48] M. Sahlen, A. R. Liddle, and D. Parkinson, “Quintessence reconstructed: Newconstraints and tracker viability,” Phys. Rev. D75 (2007) 023502,
arXiv:astro-ph/0610812 [astro-ph].
[49] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, “Modified Gravity andCosmology,” Phys. Rept. 513 (2012) 1–189, arXiv:1106.2476 [astro-ph.CO].
[50] S. L. Shapiro and S. A. Teukolsky, Black holes, white dwarfs, and neutron stars:
The physics of compact objects. John Wiley & Sons, 2008.
[51] W. Israel, “Event horizons in static vacuum space-times,” Phys. Rev. 164 (1967)
1776–1779.
[52] W. Israel, “Event horizons in static electrovac space-times,” Commun. Math. Phys.
8 (1968) 245–260.
[53] J. D. Bekenstein, “Black holes and the second law,” Lett. Nuovo Cim. 4 (1972)
737–740.
[54] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D7 (1973) 2333–2346.
[55] S. W. Hawking, “Black hole explosions,” Nature 248 (1974) 30–31.
[56] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43(1975) 199–220. [,167(1975)].
[57] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black Holes: Complementarity
or Firewalls?,” JHEP 02 (2013) 062, arXiv:1207.3123 [hep-th].
[58] B. L. Webster and P. Murdin, “Cygnus x-1-a spectroscopic binary with a heavy
companion?,” Nature 235 no. 5332, (1972) 37.
[59] J. Kormendy and D. Richstone, “Inward bound: The Search for supermassive
black holes in galactic nuclei,” Ann. Rev. Astron. Astrophys. 33 (1995) 581.
[60] K. Schwarzschild, “On the gravitational field of a mass point according to
Einstein’s theory,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916
(1916) 189–196, arXiv:physics/9905030 [physics].
[61] G. D. Birkhoff and R. E. Langer, Relativity and modern physics, vol. 1. Harvard
University Press Cambridge, 1923.

126

BIBLIOGRAPHY[62] R. Penrose, “Gravitational collapse: The role of general relativity,” Riv. Nuovo
Cim. 1 (1969) 252–276. [Gen. Rel. Grav.34,1141(2002)].
[63] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Princeton University
Press, 2017.
[64] R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraicallyspecial metrics,” Phys. Rev. Lett. 11 (1963) 237–238.
[65] H. Reissner, “Über die eigengravitation des elektrischen feldes nach dereinsteinschen theorie,” Annalen der Physik 355 no. 9, (1916) 106–120.
[66] G. Nordström, “On the energy of the gravitation field in einstein’s theory,”
Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B
Physical Sciences 20 (1918) 1238–1245.
[67] W. Baade and F. Zwicky, “On super-novae,” Proceedings of the National Academyof Sciences 20 no. 5, (1934) 254–259.
[68] W. Baade and F. Zwicky, “Remarks on super-novae and cosmic rays,” PhysicalReview 46 no. 1, (1934) 76.
[69] J. R. Oppenheimer and G. M. Volkoff, “On massive neutron cores,” PhysicalReview 55 no. 4, (1939) 374.
[70] R. C. Tolman, “Static solutions of einstein’s field equations for spheres of fluid,”
Physical Review 55 no. 4, (1939) 364.
[71] A. Hewish, S. J. Bell, J. Pilkington, P. F. Scott, and R. A. Collins, “Observationof a rapidly pulsating radio source,” Nature 217 no. 5130, (1968) 709.
[72] J. Antoniadis et al., “A Massive Pulsar in a Compact Relativistic Binary,” Science
340 (2013) 6131, arXiv:1304.6875 [astro-ph.HE].
[73] F. Ã zel and P. Freire, “Masses, Radii, and the Equation of State of NeutronStars,” Ann. Rev. Astron. Astrophys. 54 (2016) 401–440, arXiv:1603.02698[astro-ph.HE].
[74] R. R. Silbar and S. Reddy, “Neutron stars for undergraduates,” Am. J. Phys. 72(2004) 892–905, arXiv:nucl-th/0309041 [nucl-th]. [Erratum: Am. J.
Phys.73,286(2005)].
[75] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, “Shapiro DelayBIBLIOGRAPHY

127

Measurement of A Two Solar Mass Neutron Star,” Nature 467 (2010) 1081–1083,
arXiv:1010.5788 [astro-ph.HE].
[76] H. J. Lane, “On the theoretical temperature of the sun, under the hypothesis of agaseous mass maintaining its volume by its internal heat, and depending on thelaws of gases as known to terrestrial experiment,” American Journal of Scienceno. 148, (1870) 57–74.
[77] E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson, “Progress in measurements ofthe gravitational bending of radio waves using the vlba,” The AstrophysicalJournal 699 no. 2, (2009) 1395.
[78] S. B. Lambert and C. Le Poncin-Lafitte, “Determination of the relativistic
parameter gamma using very long baseline interferometry,” Astron. Astrophys.
499 (2009) 331, arXiv:0903.1615 [gr-qc].
[79] I. I. Shapiro, “Fourth test of general relativity,” Physical Review Letters 13 no. 26,(1964) 789.
[80] B. Bertotti, L. Iess, and P. Tortora, “A test of general relativity using radio links
with the Cassini spacecraft,” Nature 425 (2003) 374–376.
[81] J.-P. Rozelot and C. Damiani, “History of solar oblateness measurements andinterpretation,” The European Physical Journal H 36 no. 3, (2011) 407–436.
[82] A. Einstein, “Approximative Integration of the Field Equations of Gravitation,”
Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916) 688–696.
[83] A. Einstein, “Über Gravitationswellen,” Sitzungsber. Preuss. Akad. Wiss. Berlin(Math. Phys.) 1918 (1918) 154–167.
[84] R. A. Hulse and J. H. Taylor, “Discovery of a pulsar in a binary system,” The
Astrophysical Journal 195 (1975) L51–L53.
[85] P. C. Peters and J. Mathews, “Gravitational radiation from point masses in a
Keplerian orbit,” Phys. Rev. 131 (1963) 435–439.
[86] J. H. Taylor and J. M. Weisberg, “Further experimental tests of relativistic gravity
using the binary pulsar psr 1913+ 16,” The Astrophysical Journal 345 (1989)
434–450.
[87] J. M. Weisberg and J. H. Taylor, “Relativistic binary pulsar B1913+16: Thirty

128

BIBLIOGRAPHY

years of observations and analysis,” ASP Conf. Ser. 328 (2005) 25,
arXiv:astro-ph/0407149 [astro-ph].
[88] T. Nakamura, K. Oohara, and Y. Kojima, “General relativistic collapse to black
holes and gravitational waves from black holes,” Progress of Theoretical Physics
Supplement 90 (1987) 1–218.
[89] M. Shibata and T. Nakamura, “Evolution of three-dimensional gravitational waves:
Harmonic slicing case,” Physical Review D 52 no. 10, (1995) 5428.
[90] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, “Accurate
evolutions of orbiting black-hole binaries without excision,” Physical Review
Letters 96 no. 11, (2006) 111101.
[91] Virgo, Fermi-GBM, INTEGRAL, LIGO Scientific Collaboration, B. P.
Abbott et al., “Gravitational Waves and Gamma-rays from a Binary Neutron Star
Merger: GW170817 and GRB 170817A,” Astrophys. J. 848 no. 2, (2017) L13,
arXiv:1710.05834 [astro-ph.HE].
[92] V. Savchenko et al., “INTEGRAL Detection of the First Prompt Gamma-Ray
Signal Coincident with the Gravitational-wave Event GW170817,” Astrophys. J.
848 no. 2, (2017) L15, arXiv:1710.05449 [astro-ph.HE].
[93] A. Goldstein et al., “An Ordinary Short Gamma-Ray Burst with Extraordinary
Implications: Fermi-GBM Detection of GRB 170817A,” Astrophys. J. 848 no. 2,(2017) L14, arXiv:1710.05446 [astro-ph.HE].
[94] B. P. Abbott et al., “Multi-messenger Observations of a Binary Neutron Star
Merger,” Astrophys. J. 848 no. 2, (2017) L12, arXiv:1710.05833[astro-ph.HE].
[95] A. Joyce, B. Jain, J. Khoury, and M. Trodden, “Beyond the CosmologicalStandard Model,” Phys. Rept. 568 (2015) 1–98, arXiv:1407.0059[astro-ph.CO].
[96] E. Berti et al., “Testing General Relativity with Present and Future AstrophysicalObservations,” Class. Quant. Grav. 32 (2015) 243001, arXiv:1501.07274[gr-qc].
[97] T. P. Sotiriou, Modified Actions for Gravity: Theory and Phenomenology. PhD
thesis, SISSA, Trieste, 2007. arXiv:0710.4438 [gr-qc].

BIBLIOGRAPHY

129

http://hdl.handle.net/1963/5273.
[98] J. Sakstein, Astrophysical Tests of Modified Gravity. PhD thesis, Cambridge U.,
DAMTP, 2014. arXiv:1502.04503 [astro-ph.CO].
https://www.repository.cam.ac.uk/handle/1810/246265.
[99] D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12(1971) 498–501.
[100] D. Lovelock, “The four-dimensionality of space and the einstein tensor,” J. Math.
Phys. 13 (1972) 874–876.
[101] A. Proca, “Sur la theorie ondulatoire des electrons positifs et negatifs,” J. Phys.
Radium 7 (1936) 347–353.
[102] G. Tasinato, “Cosmic Acceleration from Abelian Symmetry Breaking,” JHEP 04(2014) 067, arXiv:1402.6450 [hep-th].
[103] L. Heisenberg, “Generalization of the Proca Action,” JCAP 1405 (2014) 015,
arXiv:1402.7026 [hep-th].
[104] J. D. Bekenstein, “Relativistic gravitation theory for the MOND paradigm,” Phys.
Rev. D70 (2004) 083509, arXiv:astro-ph/0403694 [astro-ph]. [Erratum:
Phys. Rev.D71,069901(2005)].
[105] S. F. Hassan and R. A. Rosen, “Bimetric Gravity from Ghost-free Massive
Gravity,” JHEP 02 (2012) 126, arXiv:1109.3515 [hep-th].
[106] T. Jacobson, S. Liberati, and D. Mattingly, “Lorentz violation at high energy:
Concepts, phenomena and astrophysical constraints,” Annals Phys. 321 (2006)
150–196, arXiv:astro-ph/0505267 [astro-ph].
[107] V. A. Kostelecky, “Gravity, Lorentz violation, and the standard model,” Phys. Rev.
D69 (2004) 105009, arXiv:hep-th/0312310 [hep-th].
[108] V. A. Kostelecky and N. Russell, “Data Tables for Lorentz and CPT Violation,”
Rev. Mod. Phys. 83 (2011) 11–31, arXiv:0801.0287 [hep-ph].
[109] P. Horava, “Quantum Gravity at a Lifshitz Point,” Phys. Rev. D79 (2009) 084008,
arXiv:0901.3775 [hep-th].
[110] D. Blas, O. Pujolas, and S. Sibiryakov, “Consistent Extension of Horava Gravity,”
Phys. Rev. Lett. 104 (2010) 181302, arXiv:0909.3525 [hep-th].

130

BIBLIOGRAPHY[111] T. Jacobson, “Einstein-aether gravity: A Status report,” PoS QG-PH (2007) 020,
arXiv:0801.1547 [gr-qc].
[112] J. M. Overduin and P. S. Wesson, “Kaluza-Klein gravity,” Phys. Rept. 283 (1997)
303–380, arXiv:gr-qc/9805018 [gr-qc].
[113] G. R. Dvali, G. Gabadadze, and M. Porrati, “4-D gravity on a brane in 5-D
Minkowski space,” Phys. Lett. B485 (2000) 208–214, arXiv:hep-th/0005016[hep-th].
[114] A. Nicolis and R. Rattazzi, “Classical and quantum consistency of the DGP
model,” JHEP 06 (2004) 059, arXiv:hep-th/0404159 [hep-th].
[115] M. Ostrogradsky, “Mémoires sur les équations différentielles, relatives au problèmedes isopérimètres,” Mem. Acad. St. Petersbourg 6 (1850) 385–517.
[116] R. P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,”
Scholarpedia 10 no. 8, (2015) 32243, arXiv:1506.02210 [hep-th].
[117] S. Deser and R. P. Woodard, “Nonlocal Cosmology,” Phys. Rev. Lett. 99 (2007)
111301, arXiv:0706.2151 [astro-ph].
[118] H. Nersisyan, Y. Akrami, L. Amendola, T. S. Koivisto, J. Rubio, and A. R.
Solomon, “Instabilities in tensorial nonlocal gravity,” Phys. Rev. D95 no. 4, (2017)043539, arXiv:1610.01799 [gr-qc].
[119] T. Biswas, E. Gerwick, T. Koivisto, and A. Mazumdar, “Towards singularity andghost free theories of gravity,” Phys. Rev. Lett. 108 (2012) 031101,
arXiv:1110.5249 [gr-qc].
[120] C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,”
Phys. Rev. Lett. 106 (2011) 231101, arXiv:1011.1232 [hep-th].
[121] K. Hinterbichler, “Theoretical Aspects of Massive Gravity,” Rev. Mod. Phys. 84(2012) 671–710, arXiv:1105.3735 [hep-th].
[122] F. Sbisá, “Classical and quantum ghosts,” Eur. J. Phys. 36 (2015) 015009,
arXiv:1406.4550 [hep-th].
[123] J. M. Cline, S. Jeon, and G. D. Moore, “The Phantom menaced: Constraints on
low-energy effective ghosts,” Phys. Rev. D70 (2004) 043543,
arXiv:hep-ph/0311312 [hep-ph].

BIBLIOGRAPHY

131[124] A. D. Dolgov and M. Kawasaki, “Can modified gravity explain accelerated cosmic
expansion?,” Phys. Lett. B573 (2003) 1–4, arXiv:astro-ph/0307285[astro-ph].
[125] R. P. Woodard, “Avoiding dark energy with 1/r modifications of gravity,” Lect.
Notes Phys. 720 (2007) 403–433, arXiv:astro-ph/0601672 [astro-ph].
[126] M. Fierz, “On the physical interpretation of P.Jordan’s extended theory ofgravitation,” Helv. Phys. Acta 29 (1956) 128–134.
[127] P. Jordan, Schwerkraft und Weltall: Grundlagen der theoretischen Kosmologie,
vol. 107. Friedr. Vieweg & Sohn, 1955.
[128] K. Nordtvedt, Jr., “Post-Newtonian Metric for a General Class of Scalar-Tensor
Gravitational Theories and Observational Consequences.,” 161 (Sept., 1970) 1059.
[129] R. V. Wagoner, “Scalar tensor theory and gravitational waves,” Phys. Rev. D1(1970) 3209–3216.
[130] P. G. Bergmann, “Comments on the scalar tensor theory,” Int. J. Theor. Phys. 1(1968) 25–36.
[131] H. A. Buchdahl, “Non-linear Lagrangians and cosmological theory,” Mon. Not.
Roy. Astron. Soc. 150 (1970) 1.
[132] N. Banerjee and D. Pavon, “Cosmic acceleration without quintessence,” Phys. Rev.
D63 (2001) 043504, arXiv:gr-qc/0012048 [gr-qc].
[133] C. Deffayet, G. Esposito-Farese, and A. Vikman, “Covariant Galileon,” Phys. Rev.
D79 (2009) 084003, arXiv:0901.1314 [hep-th].
[134] C. Deffayet, S. Deser, and G. Esposito-Farese, “Generalized Galileons: All scalarmodels whose curved background extensions maintain second-order field equations
and stress-tensors,” Phys. Rev. D80 (2009) 064015, arXiv:0906.1967 [gr-qc].
[135] C. de Rham and A. J. Tolley, “DBI and the Galileon reunited,” JCAP 1005
(2010) 015, arXiv:1003.5917 [hep-th].
[136] K. Koyama, “Are there ghosts in the self-accelerating brane universe?,” Phys. Rev.
D72 (2005) 123511, arXiv:hep-th/0503191 [hep-th].
[137] M. A. Luty, M. Porrati, and R. Rattazzi, “Strong interactions and stability in the
DGP model,” JHEP 09 (2003) 029, arXiv:hep-th/0303116 [hep-th].

132

BIBLIOGRAPHY[138] C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,” Phys.
Rev. D82 (2010) 044020, arXiv:1007.0443 [hep-th].
[139] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade, “From k-essence to generalised
Galileons,” Phys. Rev. D84 (2011) 064039, arXiv:1103.3260 [hep-th].
[140] T. Kobayashi, M. Yamaguchi, and J. Yokoyama, “Generalized G-inflation:
Inflation with the most general second-order field equations,” Prog. Theor. Phys.
126 (2011) 511–529, arXiv:1105.5723 [hep-th].
[141] T. Kobayashi, “Cosmic expansion and growth histories in Galileon scalar-tensor
models of dark energy,” Phys. Rev. D81 (2010) 103533, arXiv:1003.3281[astro-ph.CO].
[142] N. Chow and J. Khoury, “Galileon Cosmology,” Phys. Rev. D80 (2009) 024037,
arXiv:0905.1325 [hep-th].
[143] T. Kobayashi, H. Tashiro, and D. Suzuki, “Evolution of linear cosmologicalperturbations and its observational implications in Galileon-type modified gravity,”
Phys. Rev. D81 (2010) 063513, arXiv:0912.4641 [astro-ph.CO].
[144] R. Gannouji and M. Sami, “Galileon gravity and its relevance to late time cosmic
acceleration,” Phys. Rev. D82 (2010) 024011, arXiv:1004.2808 [gr-qc].
[145] A. De Felice and S. Tsujikawa, “Generalized Galileon cosmology,” Phys. Rev. D84(2011) 124029, arXiv:1008.4236 [hep-th].
[146] C. Armendariz-Picon, V. F. Mukhanov, and P. J. Steinhardt, “A Dynamicalsolution to the problem of a small cosmological constant and late time cosmic
acceleration,” Phys. Rev. Lett. 85 (2000) 4438–4441, arXiv:astro-ph/0004134[astro-ph].
[147] C. Deffayet, O. Pujolas, I. Sawicki, and A. Vikman, “Imperfect Dark Energy fromKinetic Gravity Braiding,” JCAP 1010 (2010) 026, arXiv:1008.0048 [hep-th].
[148] A. De Felice, R. Kase, and S. Tsujikawa, “Vainshtein mechanism in second-orderscalar-tensor theories,” Phys. Rev. D85 (2012) 044059, arXiv:1111.5090[gr-qc].
[149] O. Pujolas, I. Sawicki, and A. Vikman, “The Imperfect Fluid behind Kinetic
Gravity Braiding,” JHEP 11 (2011) 156, arXiv:1103.5360 [hep-th].

BIBLIOGRAPHY

133[150] R. R. Metsaev and A. A. Tseytlin, “Order alpha-prime (Two Loop) Equivalence of
the String Equations of Motion and the Sigma Model Weyl Invariance Conditions:
Dependence on the Dilaton and the Antisymmetric Tensor,” Nucl. Phys. B293
(1987) 385–419.
[151] C. Charmousis, E. J. Copeland, A. Padilla, and P. M. Saffin, “General second
order scalar-tensor theory, self tuning, and the Fab Four,” Phys. Rev. Lett. 108(2012) 051101, arXiv:1106.2000 [hep-th].
[152] C. Charmousis, E. J. Copeland, A. Padilla, and P. M. Saffin, “Self-tuning and thederivation of a class of scalar-tensor theories,” Phys. Rev. D85 (2012) 104040,
arXiv:1112.4866 [hep-th].
[153] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Healthy theories beyond
Horndeski,” Phys. Rev. Lett. 114 no. 21, (2015) 211101, arXiv:1404.6495[hep-th].
[154] D. Langlois and K. Noui, “Degenerate higher derivative theories beyond Horndeski:
evading the Ostrogradski instability,” JCAP 1602 no. 02, (2016) 034,
arXiv:1510.06930 [gr-qc].
[155] D. Langlois and K. Noui, “Hamiltonian analysis of higher derivative scalar-tensor
theories,” JCAP 1607 no. 07, (2016) 016, arXiv:1512.06820 [gr-qc].
[156] D. Langlois, “Dark Energy and Modified Gravity in Degenerate Higher-OrderScalar-Tensor (DHOST) theories: a review,” arXiv:1811.06271 [gr-qc].
[157] J. M. Ezquiaga and M. ZumalacÃπrregui, “Dark Energy After GW170817: Dead
Ends and the Road Ahead,” Phys. Rev. Lett. 119 no. 25, (2017) 251304,
arXiv:1710.05901 [astro-ph.CO].
[158] P. Creminelli and F. Vernizzi, “Dark Energy after GW170817 and GRB170817A,”
Phys. Rev. Lett. 119 no. 25, (2017) 251302, arXiv:1710.05877 [astro-ph.CO].
[159] J. Sakstein and B. Jain, “Implications of the Neutron Star Merger GW170817 for
Cosmological Scalar-Tensor Theories,” Phys. Rev. Lett. 119 no. 25, (2017) 251303,
arXiv:1710.05893 [astro-ph.CO].
[160] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, and I. Sawicki, “Strong
constraints on cosmological gravity from GW170817 and GRB 170817A,” Phys.
Rev. Lett. 119 no. 25, (2017) 251301, arXiv:1710.06394 [astro-ph.CO].

134

BIBLIOGRAPHY[161] E. Bellini and I. Sawicki, “Maximal freedom at minimum cost: linear large-scale
structure in general modifications of gravity,” JCAP 1407 (2014) 050,
arXiv:1404.3713 [astro-ph.CO].
[162] J. Gleyzes, D. Langlois, F. Piazza, and F. Vernizzi, “Exploring gravitational
theories beyond Horndeski,” JCAP 1502 (2015) 018, arXiv:1408.1952[astro-ph.CO].
[163] D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi, “Effective Description ofHigher-Order Scalar-Tensor Theories,” JCAP 1705 no. 05, (2017) 033,
arXiv:1703.03797 [hep-th].
[164] J. Oost, S. Mukohyama, and A. Wang, “Constraints on Einstein-aether theory
after GW170817,” Phys. Rev. D97 no. 12, (2018) 124023, arXiv:1802.04303[gr-qc].
[165] C. de Rham and S. Melville, “Gravitational Rainbows: LIGO and Dark Energy at
its Cutoff,” arXiv:1806.09417 [hep-th].
[166] Virgo, LIGO Scientific Collaboration, B. P. Abbott et al., “GW170814: A
Three-Detector Observation of Gravitational Waves from a Binary Black Hole
Coalescence,” Phys. Rev. Lett. 119 no. 14, (2017) 141101, arXiv:1709.09660[gr-qc].
[167] J. M. Ezquiaga and M. ZumalacÃπrregui, “Dark Energy in light ofMulti-Messenger Gravitational-Wave astronomy,” arXiv:1807.09241[astro-ph.CO].
[168] R. Kase and S. Tsujikawa, “Dark energy in Horndeski theories after GW170817: A
review,” arXiv:1809.08735 [gr-qc].
[169] M. Hohmann, “Parametrized post-Newtonian limit of Horndeski’s gravity theory,”
Phys. Rev. D92 no. 6, (2015) 064019, arXiv:1506.04253 [gr-qc].
[170] S. Hou and Y. Gong, “Constraints on Horndeski Theory Using the Observations of
Nordtvedt Effect, Shapiro Time Delay and Binary Pulsars,” Eur. Phys. J. C78
no. 3, (2018) 247, arXiv:1711.05034 [gr-qc].
[171] J. Renk, M. ZumalacÃπrregui, F. Montanari, and A. Barreira, “Galileon gravity inlight of ISW, CMB, BAO and H0 data,” JCAP 1710 no. 10, (2017) 020,
arXiv:1707.02263 [astro-ph.CO].

BIBLIOGRAPHY

135[172] K. Koyama, “Cosmological Tests of Modified Gravity,” Rept. Prog. Phys. 79 no. 4,(2016) 046902, arXiv:1504.04623 [astro-ph.CO].
[173] M. Ishak, “Testing General Relativity in Cosmology,” arXiv:1806.10122[astro-ph.CO].
[174] Y. Takushima, A. Terukina, and K. Yamamoto, “Bispectrum of cosmologicaldensity perturbations in the most general second-order scalar-tensor theory,” Phys.
Rev. D89 no. 10, (2014) 104007, arXiv:1311.0281 [astro-ph.CO].
[175] S. Hirano, T. Kobayashi, H. Tashiro, and S. Yokoyama, “Matter bispectrum
beyond Horndeski theories,” Phys. Rev. D97 no. 10, (2018) 103517,
arXiv:1801.07885 [astro-ph.CO].
[176] D. Yamauchi, S. Yokoyama, and H. Tashiro, “Constraining modified theories ofgravity with the galaxy bispectrum,” Phys. Rev. D96 no. 12, (2017) 123516,
arXiv:1709.03243 [astro-ph.CO].
[177] A. Taruya, K. Koyama, T. Hiramatsu, and A. Oka, “Beyond consistency test ofgravity with redshift-space distortions at quasilinear scales,” Phys. Rev. D89 no. 4,(2014) 043509, arXiv:1309.6783 [astro-ph.CO].
[178] B. Bose and K. Koyama, “A Perturbative Approach to the Redshift Space PowerSpectrum: Beyond the Standard Model,” JCAP 1608 no. 08, (2016) 032,
arXiv:1606.02520 [astro-ph.CO].
[179] T. Narikawa and K. Yamamoto, “Testing gravity with halo density profiles
observed through gravitational lensing,” JCAP 1205 (2012) 016,
arXiv:1201.4037 [astro-ph.CO].
[180] T. Narikawa, T. Kobayashi, D. Yamauchi, and R. Saito, “Testing generalscalar-tensor gravity and massive gravity with cluster lensing,” Phys. Rev. D87(2013) 124006, arXiv:1302.2311 [astro-ph.CO].
[181] S. Arai and A. Nishizawa, “Generalized framework for testing gravity withgravitational-wave propagation. II. Constraints on Horndeski theory,” Phys. Rev.
D97 no. 10, (2018) 104038, arXiv:1711.03776 [gr-qc].
[182] K. Nordtvedt, “Lunar laser ranging: A Comprehensive probe of postNewtonian
gravity,” in Recent Developments in Gravitational Physics: Rome, Italy, September
9-12, 2002, pp. 97–113. 2003. arXiv:gr-qc/0301024 [gr-qc]. [,97(2003)].

136

BIBLIOGRAPHY[183] C. D. Kreisch and E. Komatsu, “Cosmological Constraints on Horndeski Gravity
in Light of GW170817,” arXiv:1712.02710 [astro-ph.CO].
[184] V. Salvatelli, F. Piazza, and C. Marinoni, “Constraints on modified gravity from
Planck 2015: when the health of your theory makes the difference,” JCAP 1609
no. 09, (2016) 027, arXiv:1602.08283 [astro-ph.CO].
[185] S. W. Hawking, “Black holes in the Brans-Dicke theory of gravitation,” Commun.
Math. Phys. 25 (1972) 167–171.
[186] T. P. Sotiriou and V. Faraoni, “Black holes in scalar-tensor gravity,” Phys. Rev.
Lett. 108 (2012) 081103, arXiv:1109.6324 [gr-qc].
[187] A. A. H. Graham and R. Jha, “Nonexistence of black holes with noncanonicalscalar fields,” Phys. Rev. D89 no. 8, (2014) 084056, arXiv:1401.8203 [gr-qc].
[Erratum: Phys. Rev.D92,no.6,069901(2015)].
[188] A. A. H. Graham and R. Jha, “Stationary Black Holes with Time-Dependent
Scalar Fields,” Phys. Rev. D90 no. 4, (2014) 041501, arXiv:1407.6573 [gr-qc].
[189] L. Hui and A. Nicolis, “No-Hair Theorem for the Galileon,” Phys. Rev. Lett. 110(2013) 241104, arXiv:1202.1296 [hep-th].
[190] C. A. R. Herdeiro and E. Radu, “Asymptotically flat black holes with scalar hair:a review,” Int. J. Mod. Phys. D24 no. 09, (2015) 1542014, arXiv:1504.08209[gr-qc].
[191] E. Babichev, C. Charmousis, and A. Lehébel, “Black holes and stars in Horndeski
theory,” Class. Quant. Grav. 33 no. 15, (2016) 154002, arXiv:1604.06402[gr-qc].
[192] T. P. Sotiriou and S.-Y. Zhou, “Black hole hair in generalized scalar-tensor
gravity,” Phys. Rev. Lett. 112 (2014) 251102, arXiv:1312.3622 [gr-qc].
[193] T. P. Sotiriou and S.-Y. Zhou, “Black hole hair in generalized scalar-tensor gravity:
An explicit example,” Phys. Rev. D90 (2014) 124063, arXiv:1408.1698 [gr-qc].
[194] E. Babichev and C. Charmousis, “Dressing a black hole with a time-dependent
Galileon,” JHEP 08 (2014) 106, arXiv:1312.3204 [gr-qc].
[195] T. Kobayashi and N. Tanahashi, “Exact black hole solutions in shift symmetricscalar-tensor theories,” PTEP 2014 (2014) 073E02, arXiv:1403.4364 [gr-qc].

BIBLIOGRAPHY

137[196] E. Babichev, C. Charmousis, and A. Lehébel, “Asymptotically flat black holes in
Horndeski theory and beyond,” JCAP 1704 no. 04, (2017) 027,
arXiv:1702.01938 [gr-qc].
[197] T. Damour and G. Esposito-Farese, “Nonperturbative strong field effects in tensor- scalar theories of gravitation,” Phys. Rev. Lett. 70 (1993) 2220–2223.
[198] M. Salgado, D. Sudarsky, and U. Nucamendi, “On spontaneous scalarization,”
Phys. Rev. D58 (1998) 124003, arXiv:gr-qc/9806070 [gr-qc].
[199] T. Damour and G. Esposito-Farese, “Tensor - scalar gravity and binary pulsar
experiments,” Phys. Rev. D54 (1996) 1474–1491, arXiv:gr-qc/9602056[gr-qc].
[200] R. F. P. Mendes, “Possibility of setting a new constraint to scalar-tensor theories,”
Phys. Rev. D91 no. 6, (2015) 064024, arXiv:1412.6789 [gr-qc].
[201] R. F. P. Mendes and N. Ortiz, “Highly compact neutron stars in scalar-tensor
theories of gravity: Spontaneous scalarization versus gravitational collapse,” Phys.
Rev. D93 no. 12, (2016) 124035, arXiv:1604.04175 [gr-qc].
[202] J. Novak, “Neutron star transition to strong scalar field state in tensor scalar
gravity,” Phys. Rev. D58 (1998) 064019, arXiv:gr-qc/9806022 [gr-qc].
[203] K. Yagi and N. Yunes, “I-Love-Q,” Science 341 (2013) 365–368,
arXiv:1302.4499 [gr-qc].
[204] K. Yagi and N. Yunes, “I-Love-Q Relations in Neutron Stars and their
Applications to Astrophysics, Gravitational Waves and Fundamental Physics,”
Phys. Rev. D88 no. 2, (2013) 023009, arXiv:1303.1528 [gr-qc].
[205] J. M. Lattimer and B. F. Schutz, “Constraining the equation of state with momentof inertia measurements,” Astrophys. J. 629 (2005) 979–984,
arXiv:astro-ph/0411470 [astro-ph].
[206] J. S. Read, C. Markakis, M. Shibata, K. Uryu, J. D. E. Creighton, and J. L.
Friedman, “Measuring the neutron star equation of state with gravitational wave
observations,” Phys. Rev. D79 (2009) 124033, arXiv:0901.3258 [gr-qc].
[207] K. Yagi and N. Yunes, “Approximate Universal Relations for Neutron Stars andQuark Stars,” Phys. Rept. 681 (2017) 1–72, arXiv:1608.02582 [gr-qc].

138

BIBLIOGRAPHY[208] A. Lehébel, E. Babichev, and C. Charmousis, “A no-hair theorem for stars in
Horndeski theories,” JCAP 1707 no. 07, (2017) 037, arXiv:1706.04989 [gr-qc].
[209] A. Maselli, H. O. Silva, M. Minamitsuji, and E. Berti, “Neutron stars in Horndeski
gravity,” Phys. Rev. D93 no. 12, (2016) 124056, arXiv:1603.04876 [gr-qc].
[210] K. Koyama and J. Sakstein, “Astrophysical Probes of the Vainshtein Mechanism:
Stars and Galaxies,” Phys. Rev. D91 (2015) 124066, arXiv:1502.06872[astro-ph.CO].
[211] R. Saito, D. Yamauchi, S. Mizuno, J. Gleyzes, and D. Langlois, “Modified gravity
inside astrophysical bodies,” JCAP 1506 (2015) 008, arXiv:1503.01448[gr-qc].
[212] E. Babichev, K. Koyama, D. Langlois, R. Saito, and J. Sakstein, “Relativistic
Stars in Beyond Horndeski Theories,” Class. Quant. Grav. 33 no. 23, (2016)
235014, arXiv:1606.06627 [gr-qc].
[213] J. Sakstein, E. Babichev, K. Koyama, D. Langlois, and R. Saito, “Towards Strong
Field Tests of Beyond Horndeski Gravity Theories,” Phys. Rev. D95 no. 6, (2017)064013, arXiv:1612.04263 [gr-qc].
[214] T. Kobayashi and T. Hiramatsu, “Relativistic stars in degenerate higher-orderscalar-tensor theories after GW170817,” Phys. Rev. D97 no. 10, (2018) 104012,
arXiv:1803.10510 [gr-qc].
[215] K. Hinterbichler and J. Khoury, “Symmetron Fields: Screening Long-Range Forces
Through Local Symmetry Restoration,” Phys. Rev. Lett. 104 (2010) 231301,
arXiv:1001.4525 [hep-th].
[216] P. Brax, C. van de Bruck, A.-C. Davis, and D. Shaw, “Dilaton and modified
gravity,” Physical Review D 82 no. 6, (2010) 063519.
[217] J. Khoury and A. Weltman, “Chameleon cosmology,” Phys. Rev. D69 (2004)
044026, arXiv:astro-ph/0309411 [astro-ph].
[218] D. F. Mota and D. J. Shaw, “Evading Equivalence Principle Violations,Cosmological and other Experimental Constraints in Scalar Field Theories with aStrong Coupling to Matter,” Phys. Rev. D75 (2007) 063501,
arXiv:hep-ph/0608078 [hep-ph].
[219] E. Babichev, C. Deffayet, and R. Ziour, “k-Mouflage gravity,” Int. J. Mod. Phys.

BIBLIOGRAPHY

139

D18 (2009) 2147–2154, arXiv:0905.2943 [hep-th].
[220] P. Brax, C. Burrage, and A.-C. Davis, “Screening fifth forces in k-essence and DBI
models,” JCAP 1301 (2013) 020, arXiv:1209.1293 [hep-th].
[221] C. Burrage and J. Khoury, “Screening of scalar fields in Dirac-Born-Infeld theory,”
Phys. Rev. D90 no. 2, (2014) 024001, arXiv:1403.6120 [hep-th].
[222] E. Babichev and C. Deffayet, “An introduction to the Vainshtein mechanism,”
Class. Quant. Grav. 30 (2013) 184001, arXiv:1304.7240 [gr-qc].
[223] P. Brax, “Screened modified gravity,” Acta Phys. Polon. B43 (2012) 2307–2329,
arXiv:1211.5237 [hep-th].
[224] R. Kimura, T. Kobayashi, and K. Yamamoto, “Vainshtein screening in acosmological background in the most general second-order scalar-tensor theory,”
Phys. Rev. D85 (2012) 024023, arXiv:1111.6749 [astro-ph.CO].
[225] K. Koyama, G. Niz, and G. Tasinato, “Effective theory for the Vainshteinmechanism from the Horndeski action,” Phys. Rev. D88 (2013) 021502,
arXiv:1305.0279 [hep-th].
[226] R. Kase and S. Tsujikawa, “Screening the fifth force in the Horndeski’s mostgeneral scalar-tensor theories,” JCAP 1308 (2013) 054, arXiv:1306.6401[gr-qc].
[227] B. Li, G.-B. Zhao, and K. Koyama, “Exploring Vainshtein mechanism on
adaptively refined meshes,” JCAP 1305 (2013) 023, arXiv:1303.0008[astro-ph.CO].
[228] G. Dvali, S. Hofmann, and J. Khoury, “Degravitation of the cosmological constantand graviton width,” Phys. Rev. D76 (2007) 084006, arXiv:hep-th/0703027[HEP-TH].
[229] T. W. Murphy, “Lunar laser ranging: the millimeter challenge,” Rept. Prog. Phys.
76 (2013) 076901, arXiv:1309.6294 [gr-qc].
[230] G. Dvali, A. Gruzinov, and M. Zaldarriaga, “The Accelerated universe and themoon,” Phys. Rev. D68 (2003) 024012, arXiv:hep-ph/0212069 [hep-ph].
[231] P. Brax, C. Burrage, and A.-C. Davis, “Laboratory Tests of the Galileon,” JCAP1109 (2011) 020, arXiv:1106.1573 [hep-ph].

140

BIBLIOGRAPHY[232] A.-C. Davis, R. Gregory, R. Jha, and J. Muir, “Astrophysical black holes inscreened modified gravity,” JCAP 1408 (2014) 033, arXiv:1402.4737[astro-ph.CO].
[233] A.-C. Davis, R. Gregory, and R. Jha, “Black hole accretion discs and screened
scalar hair,” JCAP 1610 no. 10, (2016) 024, arXiv:1607.08607 [gr-qc].
[234] P. Brax, C. van de Bruck, A.-C. Davis, and D. Shaw, “The Dilaton and Modified
Gravity,” Phys. Rev. D82 (2010) 063519, arXiv:1005.3735 [astro-ph.CO].
[235] T. Kobayashi, Y. Watanabe, and D. Yamauchi, “Breaking of Vainshtein screeningin scalar-tensor theories beyond Horndeski,” Phys. Rev. D91 no. 6, (2015) 064013,
arXiv:1411.4130 [gr-qc].
[236] M. Crisostomi and K. Koyama, “Vainshtein mechanism after GW170817,” Phys.
Rev. D97 no. 2, (2018) 021301, arXiv:1711.06661 [astro-ph.CO].
[237] D. Langlois, R. Saito, D. Yamauchi, and K. Noui, “Scalar-tensor theories andmodified gravity in the wake of GW170817,” Phys. Rev. D97 no. 6, (2018) 061501,
arXiv:1711.07403 [gr-qc].
[238] A. Dima and F. Vernizzi, “Vainshtein Screening in Scalar-Tensor Theories before
and after GW170817: Constraints on Theories beyond Horndeski,” Phys. Rev.
D97 no. 10, (2018) 101302, arXiv:1712.04731 [gr-qc].
[239] T. Hiramatsu, W. Hu, K. Koyama, and F. Schmidt, “Equivalence Principle
Violation in Vainshtein Screened Two-Body Systems,” Phys. Rev. D87 no. 6,(2013) 063525, arXiv:1209.3364 [hep-th].
[240] J. Chagoya, K. Koyama, G. Niz, and G. Tasinato, “Galileons and strong gravity,”
JCAP 1410 no. 10, (2014) 055, arXiv:1407.7744 [hep-th].
[241] C. de Rham, A. Matas, and A. J. Tolley, “Galileon Radiation from Binary
Systems,” Phys. Rev. D87 no. 6, (2013) 064024, arXiv:1212.5212 [hep-th].
[242] B. Falck, K. Koyama, and G.-B. Zhao, “Cosmic Web and Environmental
Dependence of Screening: Vainshtein vs. Chameleon,” JCAP 1507 no. 07, (2015)
049, arXiv:1503.06673 [astro-ph.CO].
[243] B. Falck, K. Koyama, G.-B. Zhao, and M. Cautun, “Using Voids to Unscreen
Modified Gravity,” Mon. Not. Roy. Astron. Soc. 475 no. 3, (2018) 3262–3272,
arXiv:1704.08942 [astro-ph.CO].

BIBLIOGRAPHY

141[244] C. Burrage, E. J. Copeland, A. Moss, and J. A. Stevenson, “The shapedependence of chameleon screening,” JCAP 1801 no. 01, (2018) 056,
arXiv:1711.02065 [astro-ph.CO].
[245] F. P. Silva and K. Koyama, “Self-Accelerating Universe in Galileon Cosmology,”
Phys. Rev. D80 (2009) 121301, arXiv:0909.4538 [astro-ph.CO].
[246] A. Barreira, B. Li, W. A. Hellwing, C. M. Baugh, and S. Pascoli, “Nonlinear
structure formation in the Cubic Galileon gravity model,” JCAP 1310 (2013) 027,
arXiv:1306.3219 [astro-ph.CO].
[247] B. R. Dinda, “Weak lensing probe of cubic Galileon model,” JCAP 1806 no. 06,(2018) 017, arXiv:1801.01741 [astro-ph.CO].
[248] E. Babichev, C. Charmousis, A. Lehébel, and T. Moskalets, “Black holes in a cubic
Galileon universe,” JCAP 1609 no. 09, (2016) 011, arXiv:1605.07438 [gr-qc].
[249] I. D. Saltas and V. Vitagliano, “Quantum corrections for the cubic Galileon in thecovariant language,” JCAP 1705 no. 05, (2017) 020, arXiv:1612.08953[hep-th].
[250] R. Kimura, T. Kobayashi, and K. Yamamoto, “Observational Constraints on
Kinetic Gravity Braiding from the Integrated Sachs-Wolfe Effect,” Phys. Rev. D85(2012) 123503, arXiv:1110.3598 [astro-ph.CO].
[251] A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without
Singularity,” Phys. Lett. B91 (1980) 99–102. [,771(1980)].
[252] T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. 82(2010) 451–497, arXiv:0805.1726 [gr-qc].
[253] A. Sanders, A. Alexeev, S. Allison, K. Bronnikov, J. Campbell, M. Cates,
T. Corcovilos, D. Earl, T. Gadfort, G. Gillies, et al., “Project see (satellite energy
exchange): proposal for space-based gravitational measurements,” Measurement
Science and Technology 10 no. 6, (1999) 514.
[254] S. G. Turyshev, “Experimental tests of general relativity: recent progress andfuture directions,” Physics-Uspekhi 52 no. 1, (2009) 1–27.
[255] Planck Collaboration, Y. Akrami et al., “Planck 2018 results. X. Constraints on
inflation,” arXiv:1807.06211 [astro-ph.CO].

142

BIBLIOGRAPHY[256] T. Faulkner, M. Tegmark, E. F. Bunn, and Y. Mao, “Constraining f(R) Gravity asa Scalar Tensor Theory,” Phys. Rev. D76 (2007) 063505,
arXiv:astro-ph/0612569 [astro-ph].
[257] V. Muller, H. J. Schmidt, and A. A. Starobinsky, “The Stability of the De Sitter
Space-time in Fourth Order Gravity,” Phys. Lett. B202 (1988) 198–200.
[258] A. A. Starobinsky, “Disappearing cosmological constant in f(R) gravity,” JETP
Lett. 86 (2007) 157–163, arXiv:0706.2041 [astro-ph].
[259] W. Hu and I. Sawicki, “Models of f(R) Cosmic Acceleration that Evade
Solar-System Tests,” Phys. Rev. D76 (2007) 064004, arXiv:0705.1158[astro-ph].
[260] S. Tsujikawa, “Observational signatures of f(R) dark energy models that satisfy
cosmological and local gravity constraints,” Phys. Rev. D77 (2008) 023507,
arXiv:0709.1391 [astro-ph].

List of figures
2.1. Orbital decay of the Hulse-Taylor binary pulsar over 30 years of observation.
The points are experimental data and the solid line shows the cumulative
shift predicted by general relativity. Experimental results fall on thetheoretical curve. This is evidence for the existence of gravitationalradiation from the binary neutron stars. Figure from [87]. . . . . . . . .

46

2.2. The gravitational wave events GW150914 observed by LIGO Hanford (leftcolumn panels) and Livingston (right column panels).Top row: Detector
strain from the two sites Middle row: Solid line shows the numerical
relativity model that is consistent with GW150914, light grey line showsreconstructed strain signal, and dark grey line shows binary black holemerger template waveform Bottom line: Residuals after subtracting thefiltered numerical relativity waveform from the filtered detector time series.
Caption and figure from [4]. . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.1. A disk object with a hole in spherical coordinates. . . . . . . . . . . . . .

92

5.2. The vectors represent that the dimensionless force fields for r̄1 = 8,
◊0 = 0.05, and µ = 36.8, with c3 = 1 (left) and c3 = 0 (right). The pinkregions represents the disk. . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.3. 2D plot of the degree of (anti-)screening R for the case shown in Fig. 5.2(left). The pink line represents the disk. R along ◊ = ﬁ/10 as a functionof r̄ is also shown (right). . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.4. R for r1 /r0 = 4 (top left) and r1 /r0 = 20 (top right). The other parameters
are the same as in the previous plots. The white areas represent regions
where R is greater than 5. R along ◊ = ﬁ/10 as a function of r̄ is also
shown (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

143

144

LIST OF FIGURES

5.5. R for ◊0 = 0.1 (top left) and ◊0 = 0.2 (top right). R along ◊ = ﬁ/10 . . .

95

5.6. R for µ = 369 (left) and µ = 3690 (right). R along ◊ = ﬁ/10 . . . . . . .

96

5.7. R in the Schwarzschild background. This should be compared with theleft panel of Fig. 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

C.1. The convergence check of numerical results; the dependence on r̄max (left),on N◊ (middle) and on Nr (right). . . . . . . . . . . . . . . . . . . . . . . 119

List of tables
2.1. The PPN parameters. General relativity corresponds to the case “ = — =1 [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14540