Doctoral thesis

Toward Tests of Degenerate Higher-OrderScalar-Tensor Theories on Small and Large
Scales
小スケール・大スケールにおける縮退重力理論の検証に向けて

Department of Physics, Graduate School of
Science, Rikkyo University
Shin’ichi Hirano
Supervisor: Tsutomu Kobayashi

Abstract

We explore the viability of Degenerate Higher-Order Scalar-Tensor (DHOST) theories as alternatives to darkenergy. We study the solution on a static spherically symmetric spacetime in DHOST theories in which gravitational waves propagate at the speed of light and do not decay into scalar fluctuations. We also study thestatistical property of the density fluctuations at linear and non-linear orders.
In generic DHOST theories, the standard inverse square law of the gravitational potentials is partially broken
inside matter as Sun. The screening mechanism in DHOST theories evading gravitational wave constraints
operates very diﬀerently from that in generic DHOST theories. We derived a spherically symmetric solution in
the presence of non-relativistic matter. General relativity is recovered in the vacuum exterior region provided
that functions in the Lagrangian satisfy a certain condition, implying that fine-tuning is required. Gravity in thematter interior exhibits novel features: although the gravitational potentials still obey the inverse square law, theeﬀective gravitational constant is diﬀerent from its exterior value, and the two metric potentials do not coincide.
We discuss possible observational constraints on this subclass of DHOST theories, and argue that the tightest
bound comes from the Hulse-Taylor pulsar.
We investigate the potential of cosmological observations, such as galaxy surveys, for constraining DHOST
theories, focusing in particular on the linear growth of the matter density fluctuations. We develop a formalism
to describe the evolution of the matter density fluctuations during the matter dominated era and in the earlystage of the dark energy dominated era in DHOST theories, and give an approximate expression for the gravitational growth index in terms of several parameters characterizing the theory and the background solution underconsideration. By employing the current observational constraints on the growth index, we obtain a new constraint on a parameter space of DHOST theories. Combining our result with other constraints obtained from the
Newtonian stellar structure, we show that the degeneracy between the eﬀective parameters of DHOST theories
can be broken without using the Hulse-Taylor pulsar constraint.
The Horndeski scalar-tensor theory and its recent extensions allow nonlinear derivative interactions of thescalar degree of freedom. We study the matter bispectrum of large scale structure as a probe of these modified
gravity theories, focusing in particular on the eﬀect of the terms that newly appear in the so-called “beyond
Horndeski” theories. We derive the second-order solution for the matter density perturbations and find that theinteractions beyond Horndeski lead to a new time-dependent coeﬃcient in the second-order kernel which diﬀers
in general from the standard value of general relativity and the Horndeski theory. This coeﬃcient can deform thematter bispectrum at the folded triangle configurations, while it is never possible within the Horndeski theory.

iii

Contents
Chapter 1

Introduction

1

Chapter 2

Modified gravity

4

2.1

Standard cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Modified gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Degenerate Higher-Order Scalar-Tensor (DHOST) theories

20

Chapter 3
3.1

Ostrogradsky ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2

Degenerate theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3

Quadratic DHOST theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4

Partial breaking of Vainshtein screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5

GWs constraints

Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

On the screening mechanism in DHOST theories evading gravitational wave constraints

25

4.1

Screening mechanism in DHOST theories without graviton decay . . . . . . . . . . . . . . . 25

4.2

Observational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3

Summary

Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Constraining DHOST theories with linear growth of density fluctuations

31

5.1

Cosmological perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2

DHOST theories: background and perturbation equations . . . . . . . . . . . . . . . . . . . 39

5.3

Modeling DHOST cosmology in the matter dominated era . . . . . . . . . . . . . . . . . . . 43

5.4

Constraining DHOST cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5

Summary

Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Matter bispectrum beyond Horndeski theories

49

6.1

Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2

Matter density perturbations in GLPV theory . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.3

Matter bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.4

GWs constatints

6.5

Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 7

Conclusions

61

Appendix A

Explicit expressions for some coeﬃcients in Chapter 5

64

Appendix B

The coeﬃcients of Eqs. (6.22)–(6.24) and Eqs. (6.33)–(6.35) in Chapter 6

66

Bibliography

68

1

Chapter 1

Introduction
The elucidation of the evolution of the universe, so-called cosmology, is one of the most interesting topics inhumanity. With the development of technology of observations, we can test cosmology precisely. We are in the
golden ages of cosmology. The standard cosmology is based on two well-established theories in modern physics.
The first is General Relativity (GR) suggested by A. Einstein. The other is the standard model of particle
physics. Using these foundations, we can predict phenomena in the universe. Precise observations have proved
most predictions. However, there are some mysteries in modern cosmology.
Fig. 1.1 is the rate of energy components in the present universe [1]. The standard matter is about 5%, andthe remnants 95% are dark components in the universe, and we do not know these origins. Dark energy is thecomponent that plays a role of the present late-time acceleration [2, 3]. Based on GR and standard model, themost straightforward origin of the acceleration is the cosmological constant. However, there is the cosmologicalconstant problem [4]. That is the significant discordance between the observed value and the prediction basedon quantum field theory of the cosmological constant. An interesting alternative to the cosmological constantis modif ied gravity (see Ref. [5, 6, 7, 8] for reviews). On cosmological scales, modified gravity explains thelate-time acceleration while it needs to recover the result of tests of gravity on the Solar system. Its modification
produces an accelerating expansion without parameter fine-tunings and is screened on the Solar system scales.
Modified gravity is the modification of GR. The way of the modification typically is to add some new degreeof freedom (DoF) in addition to metric tensor. The simple example is scalar-tensor theories to add a new scalarFig. 1.1 The components of the present universe [1].

Chapter 1 Introduction

2

field to GR. For example, f (R) gravity [10] (see Refs. [11, 12, 13, 14] for detailed discussion) has some higherorder corrections to Einstein-Hilbert (EH) action. By the redefinition of variables, f (R) gravity is equivalent
to the EH action and canonical scalar field conformally coupled to matter. DGP braneworld [15] has the EH
action and massless scalar field with non-linear kinetic self-interactions. Comprehensively treating these variable
models, scalar-tensor theories have been generalized in the direction based on removing ghost modes due to higher
derivatives, Ostrogradsky ghost [16, 17]. It is called the Horndeski theory [18, 19, 20] which has the most general
theory with second-order equations of motion for metric tensor and scalar field. As for further developments,
Degenerate Higher-Order Scalar-Tensor (DHOST) theories have been built [21, 22, 23, 24, 25, 26, 27]. Despite
higher-order Euler-Lagrangian equations, the system can reduce second-order equations thanks to its degeneracy.
Thus, there is no problematic ghost mode. However, due to the presence of its degeneracy, DHOST theories havehigher derivative operators. In this thesis, we would like to study the properties of DHOST theories on small
and large scales toward its tests.
This thesis is organized as follows:
Chapter 2

We overview the late-time acceleration based on GR. We introduce modified gravity as aninteresting one of the possibilities to explain the late-time acceleration which is consistent with our universe.
Chapter 3

we overview DHOST theories and introduce its viable classes evading gravitational wave con-straints.
Chapter 4 We study the screening mechanism in a subclass of DHSOT theories in which gravitational waves
propagate at the speed of light and do not decay into scalar fluctuations. We derive a spherically symmetricsolution in the presence of a non-relativistic matter. GR is recovered in the vacuum exterior region provided
that functions in the Lagrangian satisfy a certain condition, implying that fine-tuning is required. Gravity inthe matter interior exhibits novel features: although the gravitational potentials still obey the standard inverse
power law, the eﬀective gravitational constant is diﬀerent from its exterior value, and the two metric potentials
do not coincide. We discuss possible observational constraints on this subclass of DHOST theories and argue
that the tightest bound comes from the Hulse-Taylor pulsar. This chapter is based on S. Hirano, T. Kobayashi,
and D. Yamauchi, “Screening mechanism in degenerate higher-order scalar-tensor theories evading gravitational
wave constraints,” Phys. Rev. D 99 (2019) no.10, 104073 [arXiv:1903.08399 [gr-qc]] [28].
Chapter 5 We investigate the potential of cosmological observations, such as galaxy surveys, for constraining
DHOST theories, focusing in particular on the linear growth of the matter density fluctuations. We develop aformalism to describe the evolution of the matter density fluctuations during the matter-dominated era and in
the early stage of the dark energy dominated era in DHOST theories, and give an approximate expression for thegravitational growth index in terms of several parameters characterizing the theory and the background solution
under consideration. By employing the current observational constraints on the growth index, we obtain a new
constraint on a parameter space of DHOST theories. Combining our result with other constraints obtained fromthe Newtonian stellar structure, we show that the degeneracy between the eﬀective parameters of DHOST theories
can be broken without using the Hulse-Taylor pulsar constraint. This chapter is based on S. Hirano, T. Kobayashi,
D. Yamauchi, and S. Yokoyama, “Constraining degenerate higher-order scalar-tensor theories with linear growthof matter density fluctuations,” Phys. Rev. D 99 (2019) no.10, 104051 [arXiv:1902.02946 [astro-ph.CO]] [29].
Chapter 6 The Horndeski scalar-tensor theory and its recent extensions allow nonlinear derivative interactions

Chapter 1 Introduction

3of the scalar degree of freedom. We study the matter bispectrum of large scale structure as a probe of thesemodified gravity theories, focusing in particular on the eﬀect of the terms that newly appear in the so-called“beyond Horndeski” theories. We derive the second-order solution for the matter density perturbations and findthat the interactions beyond Horndeski lead to a new time-dependent coeﬃcient in the second-order kernel which
diﬀers in general from the standard value of GR and the Horndeski theory. This coeﬃcient can deform thematter bispectrum at the folded triangle configurations, while it is never possible within the Horndeski theory.
This chapter is based on S. Hirano, T. Kobayashi, H. Tashiro, and S. Yokoyama, “Matter bispectrum beyond
Horndeski theories,” Phys. Rev. D 97 (2018) no.10, 103517 [arXiv:1801.07885 [astro-ph.CO]] [30].
Chapter 7

We summarize the conclusions in this thesis.

Through this thesis, we use the natural units, ℏ = c = 1.

4

Chapter 2

Modified gravity
In this section, we discuss the introduction of modified gravity. In Sec. 2.1, we show that the late-time acceleration
is explained by the cosmological constant in standard cosmology. In Sec. 2.2, we introduce modified gravity andexplain its property, screening mechanism, and self-accelerating solution.

2.1 Standard cosmology
In 1915, Einstein suggested General Relativity (GR) based on general coordinate invariance (general relativistic
principle) and equivalence principle which gravity universally couples to matter. In GR, the dynamics of spacetime(i.e., metric tensor gµν ) is determined by Einstein equations,Gµν = 8πG Tµν .

(2.1)

Gµν is the Einstein tensor which is expressed by the metric tensor and its derivatives, and Tµν is an energymomentum tensor of matter contents. In general, these equations are non-linear, so it is diﬃcult to solve.
Assuming the symmetry to a spacetime and matter distribution, we can solve. For example, static sphericallysymmetric case or homogeneous and isotropic case are those.
Our universe would be spatially homogeneous and isotropic at a cosmological scale (which is larger than about
Mpc scales) from current cosmological observations. Assuming the homogeneity and isotropy to spacetime, themetric tensor is determined by the Einstein equations
ds2 = − dt2 + a2 (t)γij dxi dxj ,
(
)
1
2
2
γij = diag
, r , r sin θ .
1 − Kr2

(2.2)
(2.3)

This is called Friedmann-Lemaitre-Robertson-Walker (RW) metric. K is the spatial constant curvature. In thefollowing discussion, we set K = 0 because the eﬀect of the spatial curvature is negligible to other eﬀects in our
universe [1]. We also use the Cartesian coordinates as spatial coordinates. The resultant metric is given by
ds2 = −dt2 + a2 (t)δij dxi dxj .

(2.4)

On the matter sector, we consider the universe filled with the perfect fluid with the energy-momentum describedbyTµν = (ρ + p)uµ uν + pgµν ,(2.5)

where ρ and p are the energy density and pressure of the fluid, and uµ = (−1, 0, 0, 0) is the 4-velocity co-moving
with this coordinate.

Chapter 2 Modified gravity

5

Let us calculate the geometrical quantities in this spacetime. The components of the Christoﬀel symbol are
Γ000 = Γ00i = Γ0i0 = Γ0i
00 = 0,Γ0ij = a2 Hγij ,Γi0j = Γij0 = Hδ ij ,1
Γijk = (3) Γijk = γ il (γjl,k + γkl,j − γjk,l ).
2

(2.6)

The components of the Riemann tensor and the Ricci scalar are
R00 = −3(Ḣ + H 2 ),R0i = Ri0 = 0,()
Rij = Ḣ + 3H 2 γij ,()
R = 6 Ḣ + 2H 2 .

(2.7)

Then, the components of the Einstein tensor are
G00 = 3H 2 ,G0i = Gi0 = 0,()
Gij = − 2Ḣ + 3H 2 γij .

(2.8)

Thus, the (0, 0) and (i, j) components of the Einstein equations are given by

2
− MPl

(

2
3MPl
H 2 = ρ,)3H 2 + 2Ḣ = p.

(2.9)
(2.10)

A dot denotes time derivative of the coordinate time, and H is the Hubble parameter, H := ȧ/a. MPl is the2
reduced Planck mass, MPl:= 1/(8πG). The first equation is the so-called Friedmann equation. The second

equation is the so-called evolution equation for the scale factor a. Also, the Bianchi identity of the Einsteintensor and the Einstein equation induce the energy-momentum conservation ∇µ Tµν = 0. The ν = 0 componentof this conservation gives the continuity equationρ̇ + 3H(ρ + p) = 0.

(2.11)The evolution of the universe is determined by Eqs. (2.9)–(2.11), and the equation of state (EoS).
We also have the metricity, ∇ρ gµν = 0. So, we can add the cosmological constant term as matter content to
the Einstein equations
Gµν = 8πG Tµν − Λgµν .

(2.12)

This Λ is the cosmological constant. This term is known as the simplest origin of the late-time acceleration of
the universe.
Let us consider the matter contents. As matter contents, we can consider non-relativistic matter, radiation, andcosmological constant. We wrote these components by the indices m, r, λ, respectively. In the case of radiation,
the EoS parameter w (:= p/ρ) is 1/3. Then, from Eq. (2.11), ρr ∝ a−4 . In the case of matter, the EoS parameter22
is 0. Then, from Eq. (2.11), ρm ∝ a−3 . In the case of cosmological constant, ρΛ = MPlΛ, pΛ = −MPlΛ, and thus

w = −1.
Next, let us derive the evolution of the universe in each era. The Friedmann equation (2.9) determines the
expansion law of the universe.

Chapter 2 Modified gravity

6

ρm0
ρr0
+
+ ρΛ
a3 ( a4
)
Ωm0
Ωr0
2
2
2
⇔ ȧ = H0+ 2 + a ΩΛ .
a
a

2
3MPl
H2 =

(2.13)

H0 is the present-day Hubble parameter. ρi0 (i = r, m, Λ) is the energy density of each matter content at the2present time. Ωi0 (:= ρi0 /(3MPl
H02 ))(i = r, m, Λ) are the energy fractions to the total energy density at thepresent time. From the Planck observation [1] on CMB (Cosmic Microwave Background radiation), we candetermine Ωr0 ≈ 10−5 , Ωm0 ≈ 0.3, and ΩΛ0 ≈ 0.7. Note that non-relativistic matter contains dark matter andbaryons, and the most component of it is dark matter. Thus, dark matter plays an essential role in structureformation.
Usually, the scale factor is normalized to unity at present. In the early stage of the universe (a ≪ 1),
the relativistic matter is dominated. After that, the non-relativistic matter becomes dominant. Finally, thecosmological constant becomes dominant, and this situation would be that of our universe at present.
Let us study the expansion law of each stage of the universe. Radiation Dominance (RD): If the universe
becomes dominated by radiation,ȧ2 ≈H02 Ωr0.
a2(2.14)

Then, the solution is given bya(t) = (2H0√
Ωr0 t)1/2 .

(2.15)

So, the RD universe is expanding, but the rate of the expansion ȧ is ∼ 1/t. The expansion in the RD era is thedecelerate expansion. Matter Dominance (MD): If the universe becomes dominated by non-relativistic matter,ȧ2 ≈H02 Ωm0.
a(2.16)

Then, the solution is given by(a(t) =⇔ H=

3 √H0 Ωm0 t2)2/3,(2.17)

2.
3t(2.18)

So, the MD universe is expanding, but the rate of the expansion ȧ is ∼ 1/t. The expansion in the MD era is
the decelerate expansion. Cosmological Constant Dominance: If the universe become dominated by cosmologicalconstant,ȧ2 ≈ H02 ΩΛ .

(2.19)

Then, the solution is given bya(t) = a0 eH0

√

ΩΛ t

.

The cosmological constant dominance is expanding, and the acceleration of the expansion ä is ∼ eH0(2.20)√ΩΛ t. Thisexpansion in the cosmological constant dominance is the accelerated expansion. This solution is known as the deSitter solution. Therefore, the cosmological constant term is the most straightforward candidate for the origin of
the late-time acceleration within GR and the standard model.

Chapter 2 Modified gravity

7

From the joint analysis of CMB and Ia Supernova observations [1], EoS parameter for dark energy at the“present” time is constrained to be
wde = −1.03 ± 0.03.

(2.21)

This is very close to that of the cosmological constant. Does the cosmological constant drive the late-timeacceleration? However, there is the cosmological constant problem [4]. That is the significant discordance
between the observed value and the prediction based on quantum field theory of the cosmological constant. An
interesting alternative to the cosmological constant is modif ied gravity (see Ref. [5, 6, 7, 8] for reviews).

2.2 Modified gravity
In this subsection, we introduce modified gravity models which can explain the late-time acceleration. First, we
explain why we can modify Einstein’s general relativity. Then, we introduce the prototypes of modified gravity
models. Using prototypes, we explain why these are consistent with the tests of gravity on small scales andquantum eﬀects. Finally, we show the concrete cosmological dynamics of modified gravity models by using shift
symmetric Horndeski theories.

2.2.1 Why we can modify the gravity theory on cosmological scales
On scales of the Earth, Newtonian gravity is known as the appropriate description of gravity. On scales of the
Solar system, we need general relativistic corrections to Newtonian gravity. Recognizing Newtonian gravity to
be the limit of GR with slow-motion approximation, GR is our gravity law on these scales. From observationsof GWs, the gravity on strong gravitational field regime also could be GR. However, the tests of gravity are not
performed on all scales. Figure. 2.1 and 2.2 summarize where we have tested gravity. Gravity is schematicallyparametrized by the amplitudes of gravitational potential GM/r and curvature GM/r3 where M is the mass ofa point source, and r is the distance from the source. General relativity is well tested in the Solar system and byusing binary pulsars. These are large curvature regions in these figures. Gravitational wave detectors will test
gravity on strong gravity regimes. Gravity on low curvature regions only partially was tested. This region is on
cosmological scales. The gravity theory can be modified on cosmological scales, and also we need to challenge
tests of gravity on cosmological scales.

2.2.2 Lovelock theorem and its breakIn this section, we introduce the direction for modifications of gravity. To do so, let us consider the Lovelock
theorem [44]. In 4 dimensional spacetime, we assume that the gravity theory is constructed by the metric tensor
and its derivatives up to second order with general coordinate invariance, L = L(gµν , ∂ρ gµν , ∂ρ ∂σ gµν ). Permitting
the equations of motion based on this action to be second order, the equations of motion are uniquely determined
to be the Einstein equations with the cosmological constant,
(
)
1 µν
µν
µν
E = α R − g R + Λg µν ,2(2.22)

where α is coupling constant. This statement is Lovelock theorem.
Treating the violation of the Lovelock theorem as a no-go theorem to go beyond the Einstein equations, ways
how to violate that is the direction to modify GR. Possibilities to violate the Lovelock theorem are as follows• To add a scalar field,• Higher-order curvatures,Chapter 2 Modified gravity

Fig. 2.1 The parameter space for gravitational
fields in experiments. See Ref. [45] for details.

8Fig. 2.2 The parameter space for gravitational
fields in predictions of GR. See Ref. [45] for details• Extra dimensions.
These are eﬀectively to add extra degrees of freedom to GR. The scalar-tensor theories are its simplest models
in the case which have a scalar field in addition to the metric tensor.

2.2.3 Prototypes of modified gravity: Scalar-Tensor theories
In this section, we introduce the modified gravity models. For simplicity, we treat typical models in scalar-tensor
theories.
The simplest model is the Brans-Dicke theory [9]. The action is given by
[
]
∫
√
wBD
S = d4 x −g ψR +(∂ψ)2 ,
ψ

(2.23)

where ψ is the scalar field, and wBD is a constant. The first term is the non-minimal coupling between the scalar
and tensor fields, and ψ means a substitute for the gravitational constant when we compare to the Einstein-Hilbert
action. This theory was considered as a model that can change the gravitational constant by the dynamics of thescalar field.
Second, let us consider f (R) gravity [10] (see [11, 12, 13, 14] for reviews). f (R) gravity is the generalization of
the Einstein-Hilbert action. The action is given by
∫
∫
√
√
M2
d4 x −g [R + f (R)] + d4 x −g Lm (gµν , Ψm ),S = Pl2(2.24)

Chapter 2 Modified gravity

9where f is the arbitrary function in Ricci tensors, and Lm is the matter Lagrangian and Ψm is its field. Naively
speaking, there exist two additional DoFs in this action because its equation of motion is fourth order due to
Rn ⊃ (∂ 2 g)n . This is not correct. To see it, let us consider changes in the variables. First, we introduce theLagrangian multiplier λ to replace the Ricci scalar to the scalar field,
M2
Sg = Pl2∫√d4 x −g [χ + f (χ) + λ(R − χ)] .

(2.25)The variation for χ induces λ = 1 + fχ . Substituting this equation into the action and varying with respect to χ
again, we obtainfχχ (R − χ) = 0.

(2.26)

This requires that fχχ ̸= 0 for recovering the original action (2.24). If we define φ = 1 + fχ (χ), the action can be
rewritten by∫S=
U (φ) =√
d4 x −g[]2
MPlφR − U (φ) ,2(2.27)

2MPl[χ(φ)φ − f (χ(φ))] .
2(2.28)

Egµν → gµν
= φ gµν(2.29)

Performing the conformal transformationand the field redefinition
ϕ
=
MPl

√

3
ln φ,
2

we further rewrite the action as
] ∫
[ 2
∫
√
√
MPl E 1 µν
4E
R − gE ∂µ ϕ∂ν ϕ − U (ϕ) + d4 x −g Lm (gµν , Ψm ),S = d x −g
22
1
U (ϕ) = 2 [χ(φ)φ − f (χ(φ))],
φ
[ √
]
2 ϕ
gµν = exp −gE .
3 MPl µν(2.30)(2.31)(2.32)(2.33)

So, f (R) gravity is equivalent to the system with GR and the canonical scalar field with conformal couplings tomatter. Note that the frame with standard kinetic terms for metric, EH action, is called Einstein frame while
the frame with the minimal coupling to matter is called Jordon frame. Eq. (2.24) is the action in Jordon frame
while Eq. (2.33) is that in Einstein frame. Using the slow-rolling phase in the potential such as inflation, theaccelerating expansion can be realized. However, with not only the scalar field as an extra energy component
in the universe but also its non-minimal couplings to matter, the scalar field would propagate as an extra force.
There exists a suppression mechanism of the couplings to matter, and then the scalar field cannot propagate (wewill see this topic in Sec. 2.2.4). For example, there exists a viable model in f (R) gravity, so-called Hu-Sawicki
model [46],f (R) = −H02

c1 (R/H02 )n.
1 + c2 (R/H02 )n(2.34)

As you see, the f (R) term is sub-dominant on a higher curvature regime, R ≫ H02 (i.e., early universe), and thenit drives the accelerating expansion on a late-time universe.

Chapter 2 Modified gravity

10

Another example is the DGP braneworld [15]. The idea of braneworld is inspired by the D-brane based on
string theory. In this model, our world (brane) is embedded in 5-dimensional spacetime (bulk). The standard
particles are confined to the brane while gravity can propagate through the whole spacetime. The action is given
by
M3
S= 5
2

∫
5

d x

√

−(5) g (5) R +

M42
2

∫
d4 x

√

−g (R + Lm ),(2.35)

where (5) g and (5) R are the 5D metric tensor and Ricci scalar respectively. M4 and M5 are Planck mass in eachdimension respectively. The Friedmann equation in this model is given by
H2 =

H
ρm
+
,
rc
3M42

(2.36)

where rc = M42 /2M53 is the cross-over scale. The origin of the model parameter is explained by the geometryof spacetime. At early stages, H ≫ 1/rc , we recover the usual Friedmann equation. At late times, we canobtain the solution H = 1/rc , and then the de Sitter expansion can occur. This solution is called self-acceleration
brunch, the late-time acceleration can be realized if we choose rc = H0−1 . However, this solution is unstable under
perturbations [47, 48], so we need to generalize this model to evade this instability. However, we can capture
the property of its generalization from this model. The eﬀective action projected in 4D flat spacetime from theoriginal action (2.35) in the decoupling limit (Λ is fixed, M4 → 0, and M5 → ∞) is given by [49][]∫(∂π)2 21124
∂ π + √ 2 πT ,S = d x (metric perturbations) − (∂π) − √
2
6 6Λ3
2 6M4

(2.37)

where π is scalar field which is the perturbations of the position of the brane in the 5th-direction. There exist the
non-linear self-interactions of the scalar field. Due to this interaction, a suppression mechanism of the couplings
to matter can occur, and then the inverse power law can be kept (we will see this topic in Sec. 2.2.4).
We expect that Modified gravity can realize the late-time acceleration by using such as self-accelerating deSitter solution. De Sitter spacetime is a maximally symmetric solution and has conformal symmetry. In Ref. [50],the scalar field interactions with respect to conformal invariance have been constructed at the short distance limit.
By definition, the constructed theory has a self-accelerating solution. The authors obtain the scalar interactions
by building geometric quantities with respect to conformal invariance and using the conformal transformationgµν = e2π ηµν . In 4D, these are given by1
L2 = − (∂π)2 ,21
L3 = − (∂π)2 ∂ 2 π,
21
L4 = − (∂π)2 [(∂ 2 π)2 − (∂µ ∂ν π)2 ],2
1
L5 = − (∂π)2 [(∂ 2 π)3 − 3(∂µ ∂ν π)2 ∂ 2 π + 2(∂µ ∂ν π)3 ].
4(2.38)(2.39)(2.40)(2.41)The indices of L mean the numbers of π in the interactions. The second one is similar to that of the non-linear
interaction in the DGP model. These interactions have the following symmetry in field space∂µ π → ∂µ π + bµ ,(2.42)

up to total derivatives and constant terms. bµ is a constant vector. This symmetry is similar to Galilean shift
symmetry in Newtonian dynamics. So, this symmetry is called Galiean shift symmetry, and the scalar field with
Galilean symmetry in field space is called the galileon. The above interactions include higher derivatives. In
general, equations of motion are fourth order. The degrees of freedom are two against the presence of a single

Chapter 2 Modified gravity

11

scalar field. This extra DoF is known as Ostrogradsky ghost [16, 17]. However, the galileon has no Ostrogradsky
ghost thanks to the special combination in interactions with respect to Galilean shift symmetry.
The covariantized theory of the galileon is not derived from the galileon in flat space. This is because Ostrogradsky ghost appears in terms of higher derivatives of metric tensor when we replace the partial derivatives tocovariant derivatives. The covariantization of the galileon has been accomplished by introducing its counterterms
to eliminate higher derivatives of the metric tensor. The covariant action which leads to second-order EoMs for
metric tensor and scalar field is given by [51]1L2 = − (∇ϕ)2 ,2
1
L3 = − (∇ϕ)2 □ϕ,
2
1
1
L4 = (∇ϕ)4 R − (∇ϕ)2 [(□ϕ)2 − (∇µ ∇ν ϕ)2 ],
8
2
3
1
4 µν
L5 = − (∇ϕ) G ∇µ ∇ν ϕ − (∇ϕ)2 [(□ϕ)3 − 3(∇µ ∇ν ϕ)2 □ϕ + 2(∇µ ∇ν ϕ)3 ].
8
4(2.43)(2.44)(2.45)(2.46)The kinetic mixing couplings between metric and scalar field for L4 and L5 are the counter terms to remove
higher derivative terms of the metric tensor.
Based on the context of eliminating higher derivatives for the metric tensor and scalar field, this action can befurther generalized to the generalized Galileon[19]. As shown in Ref. [20], its action is equivalent to the Horndeski
theory [18] and the following action is given by
SH =

5 ∫
∑

d4 x

√

−g Li

(2.47)

i=2

L2 = G2 (ϕ, X),(2.48)

L3 = −G3 (ϕ, X)□ϕ,(2.49)

L4 = G4 (ϕ, X)R + G4X [(□ϕ) − (∇µ ∇ν ϕ) ],1
L5 = G5 (ϕ, X)Gµν ∇µ ∇ν ϕ − G5X (ϕ, X)[(□ϕ)3 − 3(∇µ ∇ν ϕ)2 □ϕ + 2(∇µ ∇ν ϕ)3 ].
6
2

2(2.50)(2.51)

G2 , G3 , G4 , and G5 are the arbitrary functions of ϕ and its kinetic term X := −(1/2)g µν ∇µ ϕ∇ν ϕ. GiX denotes
X derivative of the arbitrary functions. Horndeski does not give this action. However, this action has recently
been called the Horndeski theory. This theory is the most general action which leads to second-order EoMs forthe metric and scalar field. There does not exist the Galilean symmetry explicitly, so the Horndeski theory does
not necessarily have a self-accelerating solution. We will see viable models with self acceleration in Sec. 2.2.6.
In the presence of non-minimal couplings to curvatures, the scalar field couples to matter in the Einstein frame.
Sourced by matter, the scalar field can propagate on small scales so that the inverse power law could be modified.
In the Horndeski theory, there exists a suppression mechanism of the couplings to matter, and then the inverse
power law can be kept (we will see this topic in Sec. 2.2.4).
On 17 Aug. 2017, the gravitational waves (GWs) from the neutron star (NS)-neutron star merger have been
detected. This event is called GW170817 [52]. At the end of a NS-NS merger, a gamma-ray burst would occur.
In this event, the Fermi satellite detected the gamma-ray burst, GRB 170817 [53]. Assuming the mechanism of
the gamma-ray burst, we can constrain the speed of GWs from the diﬀerence of arrival time between GWs andgamma-ray. The bound roughly is given by|c2GW − 1| ≲ 10−15 .

(2.52)The upper bound is derived from the first arrival of GWs due to its superluminal propagation when GWs andgamma-ray emitted simultaneously (Fig. 2.3). The lower bound is derived from the time lag of the beginning ofChapter 2 Modified gravity

12Fig. 2.3 The image of the diﬀerence of the arrival
time between GWs and gamma-ray with superluminal
propagation, cGW > c. τobs is the diﬀerence of thearrival time observed in the experiments, and Dmin is
the minimal distance from source object to the Earth.

Fig. 2.4 The image of the diﬀerence of the arrival
time between GWs and gamma-ray with subluminal
propagation, cGW < c. τint is the time lag of the beginning of emitting gamma-ray against that of GWs.
The maximum value of τobs is roughy 10s in typicalmodels of gamma-ray burst.

emitting gamma-ray against that of GWs (Fig. 2.4). A lot of modified gravity models have been ruled out, only
modified gravity models which can survive this event have minimal or conformal couplings to matter.
In the Horndeski theory, the propagation speed of GWs on a FLRW spacetime is given by [20]c2GW =

G4 − X(ϕ̈G5X + G5ϕ ).
G4 − 2XG4X − X(H ϕ̇G5X − G5ϕ )(2.53)

Maybe, this should correspond to unity. If the fine-tuning for the dynamics of the scalar field exists for somereason, the denominator and numerator in Eq. (2.53) can become the same. However, if GWs go through local
structures on the propagation, the speed of GWs can locally change due to the variation of the local value of thescalar field [55]. This implies G4X = G5 = 0. The resultant Horndeski theory after GW180817 is given by
LH = G2 (ϕ, X) − G3 (ϕ, X)□ϕ + G4 (ϕ)R.

(2.54)

Assuming we have minimal couplings to matter in this Jordon frame, this theory can be transformed to the
Einstein frame where there are no kinetic couplings between the scalar field and metric tensor as the conformaltransformation,Egµν → gµν
= G4 (ϕ)gµν .

Then, the action in the Einstein frame is
[
] ∫
∫
√
√
M2
S = d4 x −g E G2 (ϕ, X) − G3 (ϕ, X)□E ϕ + Pl RE + d4 x −g Lm (gµν , Ψm ),2(2.55)(2.56)

where Lm is the Lagrangian for matter and Ψm is its field. This theory has the conformal couplings to matter.
Is the Horndeski theory the most general scalar-tensor theories to explain the late-time acceleration? Thisanswer is No! We will discuss the further developments of scalar-tensor theories next chapter, so-called Degenerate
Higher-Order Scalar-Tensor (DHOST) theories.

Chapter 2 Modified gravity

13

2.2.4 Recovering General relativity: Screening mechanism
In the Solar system, GR is tested with high precision. Let us consider a static spherically symmetric spacetimearound matter, ds2 = −(1 + 2Φ(r))dt2 + (1 − 2Ψ(r))(dr2 + r2 dΩ). For example, the ratio between gravitational
potentials Φ and Ψ is strongly constrained to unity,Ψ/Φ − 1 ≤ O(10−5 ),(2.57)

from the observations of the deflection angle and time delation due to the gravitational field of the Sun [56, 57].
In GR, this diﬀerence is exactly zero, Φ = Ψ. This fact implies that gravity theory at a small scale like theSolar system is GR. Modified gravity has additional degrees of freedom (DoFs) in addition to the DoFs of themetric tensor. These additional DoFs generically propagate at all scale sourced by the trace part of the energymomentum tensor. Then, the relation Φ = Ψ would be violated.
We assume scalar-tensor theories for simplicity. In order to be satisfied with the relation between the gravitational potentials, scalar-tensor theories should be required to suppress the propagation of additional DoFs on
small scales. This mechanism is called Screening mechanism. This mechanism is mainly induced by the eﬀect
of nonlinear self-interactions of additional DoFs. There are two types.
The first is non-linear potential terms (this type screening is so-called Chameleon mechanism [58, 59]). In thiscase, from couplings to energy-momentum tensor, the eﬀective potential for the scalar field depends on the energy
density of matter. On a small scale, i.e., a high-dense region, the eﬀective mass of the scalar increases drastically.
The solution of the scalar field is roughly described by the Yukawa potential, ∼ e−meff r /r, where r is the distancefrom a source. Thus, the propagation of the scalar can be suppressed on a small scale, and the relation Φ = Ψ
is kept. However, the Chameleon mechanism works by the variation of the field value at local and cosmologicalscales. In the transition from the RD era to the MD era, the conformal couplings to matter suddenly appear. In
Refs. [60, 61], the authors claim that this sudden appearance of matter field catastrophically kicks the value ofthe field on cosmological scales to the very high energy scale near the Planck scale quantum-mechanically. The
classical background evolution would be spoiled. Due to this obstacle, modified gravity models with Chameleon
mechanism may not produce viable cosmology to explain the late-time acceleration.
The second is non-linear kinetic terms. This type of screening has kinetic screening [62] with first-orderderivatives and Vainshtein screening [63] with second-order derivatives. Because both screenings use the same
principle without diﬀerent order of derivative, we focus on the Vainshtein screening and discuss it in detail. Thismechanism works typically in the Horndeski theory. Let us consider the cubic Gaileon [50] as a typical model
within Horndeski theories,
)
(
2
MPl
1
□φ
2
LcG =
+ (conformal couplings to matter),R − (∇φ) 1 +2
2
2Λ3(2.58)

where Λ is the energy scale related to the late-time acceleration, Λ = (MPl H02 )1/3 . We would like to study a static
spherically symmetric spacetime sourced by non-relativistic point source like a star. Expanding the Lagrangianaround Minkowski space, we obtain the eﬀective Lagrangian,
)
(
1
∂2φ
1
2
+
φT.
Leﬀ = (metric perturbations) − (∂φ) 1 +
2
2Λ3
MPl

(2.59)

The first term is the schematic description of the kinetic term for h. T is the trace prat of the energy momentum
tensor for matter. The variations for h and φ yield the equations of motion. The equation of motion for h canreproduce the usual laws, that is, Φ = Ψ and ∂r Φ = GM/r2 . Focusing on the scalar part, the coupling to thetrace of the energy-momentum tensor sources the propagation of the scalar field. Roughly speaking, the geodesic

Chapter 2 Modified gravity

14Fig. 2.5 The image of the Vainshtein screening.

equation could be influenced by this scalar field as an external force. Note that this implies the violation of Φ = Ψin the Jordon frame, but now we analyzed the behavior of the gravitational fields in the Einstein frame. Thus,
the violation does not appear explicitly. However, thanks to the presence of the non-linear interaction (∂φ)2 ∂ 2 φ,
the violation can be avoided. In the following discussion, we solve the equation of motion directly and show thatthe propagation of the scalar field is suppressed in the sight of field-theoretic insights.
We analyze the dynamics of φ. Let us assume Tνµ = −M δ(x)δ0µ δν0 and φ = φ(r). Integrating the equation of
motion for ϖ with the regularity at the center of a matter, we obtain the following equation
∂r φ
1
+ 2
r
H0

(

∂r φ
r

)2
=

GM
.
r3

This is the schematic description, so we omitted numerical factors. The solution is given by( ) GM r 3/2(r ≪ rv )2r
rv∂r φ(r) ≈ GM(r ≪ r)r2(2.60)(2.61)v, where rv := Λ−1 (M/MPl )1/3 is called Vainshtein radius. Inside the Vainshtein radius, the gradient of the scalar
field can be suppressed by the second term in LHS, non-linear kinetic interactions. Thus, the scalar field does not
aﬀect geodesic motions around matter as an external force. The value of rv is O(100) pc for Sun. For a galaxy
cluster, rv is O(1) Mpc. At least for Sun, the value of the Vainshtein radius is much larger than the size of the
Solar system. The cubic Galileon can pass the most tests of gravity on the Solar system scale.
We focus on the coupling to matter to interpret the physical meaning. Recalling the scalar part of the Lagrangian, it is given by
)
(
1
∂2φ
1
+
φT.
Lφ = − (∂φ)2 1 +
3
2
2Λ
MPl

(2.62)

Performing the field redefinition as
√
∂φ

1+

∂2φ
→ ∂π,
2Λ3

(2.63)

Chapter 2 Modified gravity

15

the resultant Lagrangian is naively written by1
Lπ ≈ − (∂π)2 +
2

√
MPl

1
2

πT.

(2.64)

1 + ∂2Λφ3

2

Using the solution for φ, Eq. (2.61), ∂2Λφ3 ∼ (rv /r)3/2 . Then, on r ≪ rv , the coupling to matter is strongly
suppressed. So, we interpret the Vainshtein mechanism as the suppression of the propagation of the scalar field
due to non-linear kinetic interactions.
We expect that the Horndeski theory can pass the tests of gravity on the Solar system. However, the general
Horndeski theory does not always have successful Vainshtein screening thanks to self-interactions. In Ref. [64, 65],(∂ 2 ϕ)3 terms enhance the propagation of the scalar field, that is, the coupling to matter becomes enhanced rather
than suppressed. In the same situation, the perturbations around the Vainshtein solution have instabilities [66].
In order to have successful Vainshtein screening in Horndeski theories, we should set G5 to zero.

2.2.5 Quantum field-theoretic topics for modified gravity
In the context of the quantum field theory, the non-linear operators such as∂2φ
1− (∂φ)2 3
4
Λ

(2.65)are irrelevant operators which have negative mass dimension couplings. These operators cannot be renormalizable.
The non-renormalizability means the violation of the predictability as quantum theories because couplings intheories should be observables which renormalized due to loop corrections. Thus, theories with irrelevant operators
must be regarded as Eﬀective Field Theories (EFT) below a certain scale of negative mass dimension couplings.
This scale is called the cut-oﬀ scale. In the above example, the cut-oﬀ scale is Λ. In the cubic Galileon, there is
no problem as quantum field theory as long as we treat the phenomenology in the energy scale below the cut-oﬀ
scale Λ. Comparing this scale to the corresponding scale to the Vainshtein radius,
(
Λ≫Λ

MPl
M

)1/3
(= 1/rv ).

(2.66)

So the discussion for the Vainshtein screening at a classical level can be valid even if we take into account of
quantum eﬀects [67, 68]. However, on the energy scale, Λ, irrelevant operators cannot be suppressed by the
cut-oﬀ scale, and then the discussion at a classical level is invalid due to back-reactions of non-perturbative
quantum eﬀects. Theories lose their predictability. Most of these situations are regarded as the appearanceof new physics at this scale, and other sophisticated theories must appear thanks to a certain successful UVcompleted mechanism. For example, we expect the quantum gravity beyond GR exists below the Planck massscale.
In the context of EFT, EFT below cut-oﬀ scale should be constructed by adding possible terms based onsymmetry or integrating out heavy DoFs above the cut-oﬀ scale on an UV-complete theory. As a result, EFT has
infinite irrelevant operators suppressed by the cut-oﬀ scale, and it follows that EFT generally has Ostrogradsky
ghost due to higher derivatives. To see it, let us consider a below simple toy model in a flat space,
1
1
1
g
1
L = − (∂H)2 − M 2 H 2 − (∂π)2 − m2 π 2 − π 2 H 2 ,
2
2
2
2
4

(2.67)

where H and π are the heavy field and light field respectively (M ≫ m), and g is a coupling constant. Integratingout the heavy field, the Lagrangian is schematically described by]∞ [1 2 2 ∑ ci (g) 4+2i di (g)
1
2 2i
2
π
+
(∂π) π + · · · .
L = − (∂π) − mR π −
2
2
M 2i
M 2i
i=1

(2.68)

Chapter 2 Modified gravity

16

Fig. 2.6 The regions for some eﬀects: classical, quantum, linear, and non-linear regions in scalar field.

The summation is understood as that of irrelevant operators and M is the cut-oﬀ scale. As you see, the summation
is not the combinations to eliminate the Ostrogradsky ghost. This picture seems to be compatible with ghost-freescalar-tensor theories such as the Horndeski theory. In Ref. [69], the authors study the equation of motion in EFT
in a perturbative way. They show that the ghost-free combinations of operators can only aﬀect physical solutions,and the others are higher-order eﬀects. This could imply that we should consider only ghost-free operators in
EFT.
In the previous sections, the propagation speed of GWs should be close to that of light in GW170817 observedin advanced LIGO and its optical counterpart event. The sensitivity of the detector in LIGO is eﬃcient on 100
Hz (the corresponding wavelength to it is O(1000) km). This scale is roughly equivalent to the cut-oﬀ scale basedon EFT of dark energy,Λ = (MPl H02 )1/3 ∼ 100 Hz.

(2.69)In Ref. [70], the authors point out this point and mention the uncertainty of the constraint on the speed of GWs in
the modified gravity model as an EFT. Based on their discussion, it will be clear whether their opinion is correct
or not with the detection of GWs with lower energy (longer wavelength) observed by spacecraft interferometer
LISA than that detected by LIGO. This discussion is based on the calculation in a flat space. On the other hand,taking into account of the eﬀect of curved backgrounds, the cut-oﬀ scale could be shifted. In Ref. [49], in the
background sourced by a point source, the cut-oﬀ scale can be renormalized by the ratio of the Vainshtein radius
to radial coordinate asΛ → Λeﬀ ≈ Λ( r )3/4v

r.

(2.70)In the case of Earth, Λeﬀ (rE ) ∼ 10 GHz. Thus, the stringent constraint on the speed of GWs could be valid in
modified gravity models. But, the validity of this discussion is beyond the scope of this doctoral thesis.

2.2.6 Cosmological dynamics
In this section, we would like to discuss the concrete cosmological dynamics of modified gravity. Let us consider
shift-symmetric Horndeski theories with cGW = 1. These models are viably applied to alternatives to dark energyChapter 2 Modified gravity

17and can capture the typical behavior of theories with Vainshtein screening. The Lagrangian is given by
L=

2
MPl
R + G2 (X) − G3 (X)□ϕ.
2(2.71)

23MPl
H 2 = ρm + ρϕ ,(2.72)The background equations are given by

2
MPl(3H 2 + 2Ḣ) = −pm − pϕ ,(2.73)(a3 J )· = 0,(2.74)whereρϕ = −G2 + J ϕ̇,(2.75)

pϕ = G2 − 2X ϕ̈G3X ,(2.76)J ϕ̇ = 2XG2X + 6H ϕ̇XG3X .

(2.77)The matter sector contains non-relativistic matter and radiation. From the field equation (2.74), J ∝ a−3 .
Going through the early universe, inflation, J vanishes regardless of the initial value. Thus, these models havethe attractor, J = 0. On the attractor J = 0, we haveρϕ = −G2 ,(2.78)

d ln X 2 2
pϕ = G2 +M H αB ,d ln ad ln X
12αB d ln H=.
J˙ ϕ̇ = 0 ↔
d ln aαK d ln a(2.79)(2.80)The α parameters are given by2MPl
H 2 αK = 2X(G2X + 2XG2XX ) + 12HX ϕ̇(G3X + XG3XX ),2MPl
HαB = −ϕ̇XG3X .

(2.81)(2.82)

These can characterize the linear perturbations of this model. The EoS parameter is
wϕ :=
α :=

pϕ
2α
d ln H= −1 +,ρϕ3(1 − Ωm ) d ln a

(2.83)

26αB
αK(2.84)

Substituting above equations into BG equations,ρϕ= 1 − Ωm ,3M 2 H 2d ln H3
1= − Ωm,d ln a2
1+αd ln Ωmα= −3(1 − Ωm ) − 3Ωm.
d ln a1+α(2.85)(2.86)(2.87)

Using above equation, the EoS parameter for the scalar field further reduces towϕ = −1 −Ωmα.
1 − Ωm 1 + α(2.88)

We would like to study the time evolution directly. Most of all models are not solved analytically, but one canobtain it in the following situation as an example. Let us consider analytic models with the tracker condition (for

Chapter 2 Modified gravity

18

detailed discussion, see [71, 72]). We assume that the arbitrary functions depend only on a power-law functionof X as(2
G2 = −c2 MPl
H02

X
2 H2
MPl
0

)p

(
, G3 = c3 MPl

X
2 H2
MPl
0

) p3
,

(2.89)

where p, q, c2 , c3 are constants. Then, we also assume the tracker condition ϕ̇2q H = const. and the present Hubble)−q(
H0 if we set the present value for theparameter H0 . This means that we obtain the equation H = M 2XH 2Pl02
kinetic energy of the scalar field X0 = MPl
H02 , and q = −αK /(12αB ) = const. by using Eq. (2.80). Substituting

these ansatzes to the attracter J = 0, we obtain the condition for the coeﬃcients and the relation between eachpower of X
p3 = p + q − 1/2,

(2.90)− pc2 + 3 · 21/2−q c3 p3 = 0.

(2.91)

These imply that parameters are)2+p/q
(
c2
H0
Ωϕ =
,
3 · 2p/q H
3p3 c3αB = − 1/2 Ωϕ =: cB Ωϕ ,2 c2
αK = −12qαB =: cK Ωϕ .

(2.92)(2.93)(2.94)

From the above equations, the energy density of the scalar field becomes suppressed during an early universe if
the models are satisfied with 2 + p/q > 0(it is called cosmological V ainshtein mechanism [74]) . Most of thescalar-tensor theories cannot be satisfied with this simple description because we restrict the form of arbitrary
functions to one power-law term of X. Generally, arbitrary functions are polynomials with respect to X, and α
parameters are not described by Ωϕ directly (for instance, see [73]).
Closing the late-time universe, the component of the scalar field becomes dominated. In this case, the EoS
parameter wϕ is
wϕ = −1 −
c :=

c Ωm
,
1 + c (1 − Ωm )

6c2B
cK

At the asymptotic limit, i.e., early time a ≪ 1 and DE dominance (a ≈ 1), it reduces to{−(1 + c) (a ≪ 1),wϕ ≈−1(a ≈ 1).

(2.95)(2.96)(2.97)

Thus, in the dominant stage of the scalar field, the de Sitter expansion can be realized. This behavior is the sameas that of the self-accelerating solution in Sec. 2.2.3. This behavior is not general in Horndeski theories. Thereexist models that cannot be analytically solved have this typical attractor behavior during the evolution of theuniverse.
In this section, we study very viable models that have the so-called tracker solution as an alternative to the CC.
There are other models such as the early dark energy model [75] and the model which has the scaling solution [76].
In general, the dark energy constituent can become a sub-dominant component before the late-time acceleration
phase while in the tracker scenario, the dark energy is much more suppressed than other components. If the darkenergy becomes dominated before late times, the expansion history of the universe is modified. For example,

Chapter 2 Modified gravity

19

CMB observables are very sensitive to this fact. In Ref. [77], the authors analyze the angular power spectrum
for temperature fluctuations in the theory,
ω(ϕ)(∂ϕ)2 ,
ϕ[]−1
2ω(ϕ) + 3 = α02 − β ln (ϕ/ϕ0 ),L = ϕR −(2.98)

where ϕ0 , α0 , and β are the present value of ϕ and constants. The pattern of the amplitude oscillation ischaracterized by the BAO (Baryon Acoustic Oscillation), and this pattern is shifted due to the modification of
the expansion history on early stages of the universe (Fig. 2.7). It could be no consistent scenario for dark energy
without the tracker scenario.

Fig. 2.7 The angular power spectrum for CMB temperature fluctuations in each value of the constants:
the pattern of the amplitude oscillation is shifted due to the modification of the expansion history on earlystages of the universe. For detailed discussion, see Ref. [77].

20

Chapter 3

Degenerate Higher-Order Scalar-Tensor (DHOST)
theories
In this chapter, I introduce the recent developments of ghost-free scalar-tensor theories, in particular Degenerate
Higher-Order Scalar-Tensor (DHOST) theories [21, 22, 23, 24, 25, 26, 27].

3.1 Ostrogradsky ghost
We will show that there is a ghost mode, so-called Ostrogradsky ghost [16, 17], due to higher derivatives in theanalytical mechanics. Let us consider the Lagrangian L = L(q, q̇, q̈), where q(t) is the position of a particle.
Varying with respect to q, we obtain Euler-Lagrange equation,
d dL
d2 dL
dL
−
+ 2
= 0.
dq
dt dq̇
dt dq̈

(3.1)

Because of d2 /dt2 (dL/dq̈) ̸= 0 in general, this equation is the 4th-order system. Changing the variables q and q̇
to Q1 and Q2 respectively and defining its canonical momentum as follows,
dL
d dL
−
,
dq̇
dt dq̈
dL
P2 :=
,
dq̈

P1 :=

(3.2)
(3.3)

the Hamiltonian is given by
H = P1 Q2 + P2 f (Q1 , Q2 , P2 ) − L,

(3.4)

where f is the function in terms of Q1 , Q2 , and P2 . P1 and Q2 have arbitrary signs with motion. Thus, theHamiltonian is unbound below. Returning to the Lagrangian picture, the system has a mode with the positive
kinetic term and the other with the negative kinetic term. (Here, I do not show this fact directly. For example,
see Ref. [17]) The appearance of the additional DoF would be equivalent to the system with a mode with thenegative kinetic term. This additional DoF is called ”Ostrogradsky ghost.”
In order to construct ghost-free theories with higher derivatives, we must eliminate this Ostrogradsky ghost
under the construction of theories. In this section, we consider the system written by the single variable q with its
second derivatives. Avoiding the Ostrogradsky ghost, we must eliminate the dependence of q̈ in the Lagrangian.
Moving to the multi-variables system, this condition changes.

3.2 Degenerate theories
Next, let us consider below multi-variables system with second-order derivatives
L=

1 2 1
1
aϕ̈ + k0 ϕ̇2 + kij q̇ i q̇ j + bi ϕ̈q̇ i + ci ϕ̇q̇ i − V (ϕ, q i ),
2
2
2

(3.5)

Chapter 3 Degenerate Higher-Order Scalar-Tensor (DHOST) theories

21

where there are 4 DoFs, ϕ(t), and q i (t) (i = 1, 2, 3). a, k0 are constants, and bi , ci are constant vectors, andkij is a constant tensor. As treated in the previous section, we define the new variables by using a Lagrangian
multiplier. Instead of ϕ̇, we use Q as
L=

1 2 1
1
aQ̇ + kij q̇ i q̇ j + bi Q̇q̇ i + ci Qq̇ i + k0 Q2 − V (ϕ, q i ) + λ(Q − ϕ̇),222

where λ is the Lagrange multiplier. The Euler-Lagrange equations are given by)()( ) ( ici q̇ + k0 Q − λ
a bi
Q̈
,
=
∂V
−cj Q̇ − ∂qbj kijq̈ ij(The matrix

a

bi

(3.6)

(3.7)

ϕ̇ = Q,(3.8)∂V
λ̇ = −
.
∂ϕ

(3.9)

)

is called kinetic matrix. If this matrix is invertible, we need ten initial conditions to
bj kijsolve the system. Then, there exists an additional DoF, that is, Ostrogradsky ghost. If the kinetic matrix is
non-invertible, Dofs are degenerate. Then, the additional cannot appear on the system. The invertibility of the
kinetic matrix reads oﬀ

(
det

a
bj

bi
kij

)

= 0 ↔ det kij · [a − bi bj (k −1 )ij ] = 0.

(3.10)

In general det kij ̸= 0, then the condition which the kinetic matrix is not invertible is a − bi bj (k −1 )ij = 0.
This condition is called degeneracy condition. Applying this trick to eliminate the Ostrogradsky ghost, one canconstruct multi-variables systems with second-order derivatives, such as scalar-tensor theories.

3.3 Quadratic DHOST theories
In this section, let us consider scalar-tensor theories with second-order derivatives up to quadratic order (as fordetailed discussions, see Refs. [21, 22]). The general Lagrangian is given by∫√
S[ϕ, g] = d4 x −g [G2 (ϕ, X) − G3 (ϕ, X)□ϕ + f (ϕ, X)R + C µνρσ (ϕ, X)ϕµν ϕρσ ] ,(3.11)

whereC µνρσ (ϕ, X) =11
a1 (ϕ, X)(g µρ g νσ + g µσ g νρ ) + a2 (ϕ, X)g µν g ρσ + a3 (ϕ, X)(ϕµ ϕν g ρσ + ϕρ ϕσ g µν )2
2
1
+ a4 (ϕ, X)(ϕµ ϕρ g νσ + ϕν ϕρ g µσ + ϕµ ϕσ g νρ + ϕν ϕσ g µρ ) + a5 (ϕ, X)ϕµ ϕν ϕρ ϕσ ,4

(3.12)

with X = −∇µ ϕ∇µ ϕ/2, ϕµ = ∇µ ϕ, and ϕµν = ∇ν ∇µ ϕ. f, ai (i = 1, · · · , 5) are arbitrary functions in terms of ϕand X.
First, let us define the new variables by using a Lagrangian multiplier. Instead of ϕµ , we use Aµ . The action
related to the kinetic structure is given by∫√
Skin = d4 x −g [f (ϕ, X)R + C µνρσ (ϕ, X)∇µ Aν ∇ρ Aσ + λµ (ϕµ − Aµ )] ,(3.13)

where λµ is the Lagrange multiplier.
To analyze the time evolution, we do the ADM decomposition. We assume the existence of a 3-dimensional
spacelike hypersurface. We introduce the normal vector nµ which is time-like. Then, the induced metric is given
by hµν = gµν + nµ nν . Using these geometrical quantities, we can express the projection of Aµ to the hypersurface
Âµ = hνµ Aµ ,(3.14)

Chapter 3 Degenerate Higher-Order Scalar-Tensor (DHOST) theories

22

and its normal component of Aµ
A = n µ Aν .

(3.15)We also introduce the time direction vector tµ which can be written bytµ = N nµ + N µ ,(3.16)

where N µ is orthogonal to nµ . N is the lapse function, and N µ is the shift vector. Using this vector, the time
derivative is determined as the Lie derivative with respect to it. We haveȦ := tµ ∇µ A.

(3.17)

Also, the dynamics of the hypersurface is described by the extrinsic curvatureKµν =1(ḣµν − Dµ Nν − Dν Nµ ),2N(3.18)

where Dµ denotes the covariant derivative associated with hµν .
Doing the ADM decomposition of Eq. (3.13), we obtain below kinetic part of the theories [21, 22]
Lkin = AȦ2 + 2B µν ȦKµν + Kµνρσ Kµν Kρσ ,(3.19)where1[a1 + a2 − (a3 + a4 )A + a5 A2 ],N2AA
B µν =(2a2 − a3 A2 + 4fX )hµν −(a3 + 2a4 − 2a5 A2 )Âµ Âν
2N
2NKµνρσ = (a1 A2 + f )hµ(ρ hν)σ + (a2 A2 − f )hµν hρσ + · · · .
A=(3.20)(3.21)(3.22)The structure of this Lagrangian is same as that in previous section, Eq. (3.6). The correspondence is
A ↔ Q, Kµν ↔ q̇ i , A ↔ a, B µν ↔ bi , Kµνρσ ↔ kij .

(3.23)

So, the degeneracy condition is given by−1
A − B µν B ρσ Kµνρσ= 0.

(3.24)

The case A = 0 and B = 0 corresponds to Horndeski theories. More general case A ̸= 0 and B ̸= 0 corresponds
to DHOST theories. The explicit form of this condition and its classification are given by [21, 22].
Applying to cosmology, DHOST theories must be stable under perturbations around the FLRW background.
In Ref. [78, 79], the authors study the stability of tensor perturbation in the quadratic DHOST theories. Theyfind that the stable class in the theory is conformally/ disformally related to the Horndeski theories. This classis called class I DHOST theory [21, 22]. The degeneracy conditions read oﬀ
a2 = −a1 ̸= −f /X,
1{4f [3(a1 − 2fX )2 − 2a3 f ] − a3 X 2 (16a1 fX + a3 f )a4 =8(f − a1 X)2(3.25)

2+ 4X(3a1 a3 f + 16a21 fX − 16a1 fX− 4a31 + 2a3 f fX )}1(2a1 − a3 X − 4fX )[a1 (2a1 + 3a3 X − 4fX ) − 4a3 f ].
a5 =8(f − a1 X)2(3.26)

Thus, there are 5 free functions G2 , G3 , f , a1 , and a3 .

(3.27)

Chapter 3 Degenerate Higher-Order Scalar-Tensor (DHOST) theories

23

3.4 Partial breaking of Vainshtein screening
As we see in Sec. 2.2.4, modified gravity models recover the standard result in GR in the Solar system. In
the class I DHOST theory, however, the standard behavior of the gravitational potentials is recovered aroundmatter while it is violated inside matter [80, 81, 82, 83, 84]. The gradients of gravitational potentials inside theVainshtein radius are given bydΦ
GM (r)
GM ′′ (r)
=
+ Υ1
,
2
dr
r
4
GM (r) 5Υ2 GM ′ (r)dΨ=+ Υ3 GM ′′ (r),
−
dr
r2
4
r

(3.28)(3.29)

where(8πG)−1 = 2f − 2XfX − 6X 2 a3 ,(3.30)

2(fX + Xa3 )
,
a3 f
8XfX
Υ2 =
,
5f
f 2 − X 2 a23.
Υ3 = X
4a3 f
Υ1 = −(3.31)(3.32)(3.33)

M (r) is enclosed mass inside a radius r and G is the gravitational constant in DHOST theories. Thus, the strengthof the partial breaking inside matter depends on Υ parameters, and in particular, Υ1 is used for constraints on
DHOST theories. Existing constraints on DHOST theories mainly come from the Newtonian stellar structure
modified due to the partial breaking of the Vainshtein mechanism, which is characterized by a single parameterΥ1 (the definition here is for theories with c2GW = 1) [80, 82, 83]. The lower bound on Υ1 has been obtained
from the requirement that gravity is attractive at the stellar center: Υ1 > −2/3 [85]. The upper bound is given
by comparing the minimum mass of stars with the hydrogen burning with the minimum mass of observed red
dwarfs: Υ1 < 1.6 [86]. There are several attempts for improving the above bounds [87, 88, 89, 90, 91], including
the one concerning the speed of sound in the atmosphere of the Earth [90].

3.5 GWs constraints
As we see in Sec. 2.2.3, we can constrain the speed of GWs close to that of light. In the class I DHOST theories,
the propagation speed of GWs on a FLRW spacetime is given by [78, 79]
c2GW =

f.
f − Xa1(3.34)

Thus, we must choose a1 = 0 for c2GW = 1.
The dark energy field spontaneously breaks Lorentz symmetry. So graviton can decay into the dark energy
field. Usually, this channel is not more eﬃcient than other scatterings. In Ref. [92, 93], the authors study eﬃcient
decay channels of graviton into the dark energy field in the context of eﬀective field theory of dark energy. The
most eﬃcient interaction is given by
Sγππ =

2Mpl
m̃24
2 + 2m̃2
Mpl
4∫d4 x γ̈ij ∂i π∂j π,(3.35)

where γij is tensor perturbations and π is dark energy field. m̃4 is the time-dependent mass scale related tobackground spacetime. (see Ref. [92] for details) In this calculation, they assume that the propagation speed ofChapter 3 Degenerate Higher-Order Scalar-Tensor (DHOST) theories

24

GWs is unity. The ratio of the decay rate to the present Hubble is given by
Γγππ= 1020
H0

(

Λ3
Λ⋆

)6

(1 − c2s )2,480πc7s(3.36)

where cs is the propagation speed of dark energy field which in general is not unity. Λ⋆ is the model-dependent
parameter. Estimating the ratio, we chose the energy of graviton to the scale which is observed in advanced
LIGO, Λ3 . This is because GWs propagating on cosmological distance has been observed in advanced LIGO.
Thus, this huge ratio must be much smaller than unity. In the class I DHOST theory with cGW = 1,
Λ3
∝ a3 .
Λ⋆(3.37)

So a3 should vanish.
Under the above two GWs constraints, the resultant class I DHOST theory which is viable to explain the originof the late-time acceleration is given by
L = G2 (ϕ, X) − G3 (ϕ, X)□ϕ + f (ϕ, X)R +2
3fXϕµ ϕµσ ϕσν ϕν .
2f

(3.38)The disformal coupling to matter can change the propagation speed of graviton. Of course, this theory is related
to the viable Horndeski theory through the conformal transformation without matter [92]. The explicit form of
this theory in the Einstein frame is given by
[
] ∫
∫
2
√
MPl
4 √
R + d4 x −g̃ Lm (g̃µν , Ψm ),S = d x −g G2 (ϕ, X) − G3 (ϕ, X)□ϕ +
2
1
gµν ,
g̃µν =f (ϕ, X)

(3.39)(3.40)

where Lm is the Lagrangian of matter and Ψm is its field. Note that the screening mechanism in this surviving
theory is diﬀerent from that of generic quadratic DHOST theories. We will discuss this topic in the next section.

25

Chapter 4

On the screening mechanism in DHOST theories
evading gravitational wave constraints
In this chapter, we study the screening mechanism in a subclass of DHSOT theories in which GWs propagate
at the speed of light and do not decay into scalar fluctuations. This topic is based on S. Hirano, T. Kobayashi
and D. Yamauchi, “Screening mechanism in degenerate higher-order scalar-tensor theories evading gravitational
wave constraints,” Phys. Rev. D 99 (2019) no.10, 104073 [arXiv:1903.08399 [gr-qc]] [28].
As we saw in the previous section, the Lagrangian for DHOST theories in which gravitons propagate at the
speed of light and do not decay into dark energy is described by
L = G2 (ϕ, X) − G3 (ϕ, X)□ϕ + f (ϕ, X)R +2
3fXϕµ ϕµσ ϕσν ϕν ,
2f

(4.1)

where R is the Ricci scalar, ϕµ = ∇µ ϕ, ϕµν = ∇µ ∇ν ϕ, X := −ϕµ ϕµ /2, and fX = ∂f /∂X. Cosmology derived
from the Lagrangian (4.1) is explored in Ref. [117]. It turns out that in this particular subclass of DHOST theories
the screening mechanism operates in a diﬀerent way from that in generic DHOST theories, as already inferredin Ref. [92]. The purpose of the present chapter is to clarify how the (breaking of the) Vainshtein screening
mechanism occurs in the above theory.

4.1 Screening mechanism in DHOST theories without graviton decay
A weak gravitational field is described by the line element
ds2 = −[1 + 2Φ(t, x)]dt2 + [1 − 2Ψ(t, x)]dx2 ,(4.2)

with the scalar-field configuration
ϕ = ϕ0 (t) + π(t, ⃗x).

(4.3)

Here, ϕ0 (t) is a slowly evolving background determined from the cosmological boundary condition and π(t, x) isa fluctuation. Since we are interested in gravity on scales well inside the horizon, we ignore the cosmic expansion.
Following Refs. [66, 80], we expand the action in terms of the metric perturbations and π, keeping the higherderivative terms relevant to the screening mechanism in the quasi-static regime. The resultant eﬀective Lagrangianis given by
[
(
)
η
3β
4ξ
2(α − ξ) 2
Leﬀ = f −2Ψ∂ 2 Ψ + 4(1 − 2β)Ψ∂ 2 Φ −(∂π)2 + 4β 1 −Φ∂ 2 Φ + 1/2 Ψ∂ 2 π +
Φ∂ π
2f
2
f
f 1/2
4β
6β 2
α
2β (1 − 3β)
2 2
2 2
(∂π)
∂
Φ
−
(∂π)
∂
Ψ
+
+
(∂π)2 ∂ 2 π +∂i π∂j π∂i ∂k π∂k ∂j π
3
fΛ
f Λ6
f 1/2 Λ3
f 1/2 Λ3]4β(1 − 3β)ϕ̇0 2
8β ϕ̇06β 2 ϕ̇0
6β 2
2
2
2 2
Φ∂ π̇ + 1/2 3 Ψ∂ π̇ +(∂π) ∂ π̇ − Φρ,(4.4)+ 1/2 3 (∂ π̇) −f Λ6f Λf 1/2 Λ3f ΛChapter 4 On the screening mechanism in DHOST theories evading gravitational wave constraints

26

where we introduced dimensionless quantities
α :=ϕ̇20 G3X,
2f 1/2β :=ϕ̇20 fX
,
2f

ξ :=

fϕ
,
f 1/2

(4.5)

and defined an energy scale Λ := (ϕ̇20 /f 1/2 )1/3 . The dot denotes the diﬀerentiation with respect to t. The explicit
expression for the coeﬃcient η is not important here. In deriving the Lagrangian (4.4) we ignored ϕ̈0 since ϕ0
is a slowly varying field. We assume that matter is minimally coupled to gravity, so that we add the term −Φρwhere ρ = ρ(t, x) is the density of a nonrelativistic matter source. The Lagrangian (4.4) is a particular case of
the general eﬀective Lagrangian for the Vainshtein mechanism in DHOST theories [81, 82, 83]. However, thescreening mechanism in this particular subclass operates in a very diﬀerent way than in generic cases, as we willsee below.
Let us consider a spherically symmetric matter distribution, ρ = ρ(t, r), where r is the radial coordinate.
Varying the action with respect to Ψ, Φ, and π, we obtain the following equations:2ϕ̇0
β ẋ = 0,Λ3[α − ξ + (1 − 3β)βξ]x + 2β(2 − 3β)y + 2(1 − 2β)z
(1 − β)ξx + (1 − 2β)y − z − 2βx(rx)′ ++ 2β(1 − 3β)x(rx)′ −2ϕ̇0β(1 − 3β)ẋ = A,
Λ3

(4.6)

(4.7)

and
F(x, ẋ, x′ , ẍ, ẋ′ , x′′ , y, ẏ, y ′ , z, ż, z ′ ) = 0,

(4.8)

where the prime denotes diﬀerentiation with respect to r and we defined the dimensionless variables asπ′
f 1/2 Φ′f 1/2 Ψ′, y :=, z :=
,
3
3
Λ r
Λ r
Λ3 r
1 M (t, r)
1
M (t, r)A :==,8πf 1/2 Λ3 r3
8π ϕ̇20 r3
x :=

(4.9)
(4.10)

with∫ rρ(t, r̄)r̄2 dr̄M (t, r) := 4π(4.11)

0

being the mass contained within r. In deriving Eqs. (4.6)–(4.8) we integrated the field equations once and fixed
the integration constants so that x, y, and z are regular at r = 0. The explicit form of F is complicated.
From Eqs. (4.6) and (4.7) we have
y=

ϕ̇0 β
A + 2β(1 − β)x(rx)′+ c1 x − 3ẋ,2(1 − β)2
Λ 1−β(4.12)z=(1 − 2β)A − 2β(1 − β)x(rx)′ϕ̇0 β+ c2 x + 3
ẋ,22(1 − β)Λ 1−β(4.13)

where c1 and c2 are written in terms of α, β, and ξ. Then, substituting Eqs. (4.12) and (4.13) to Eq. (4.8), we
obtain[](r3 A)′
2ϕ̇04(α − 3βξ)(1 − β)x2 + c3 − 2β(1 − β)x = [α + (1 − 2β)ξ − 2ζ]A − 3 (1 − β)β Ȧ,r2
Λ

(4.14)

where we definedζ :=ϕ̇20 fϕX,
2f 1/2(4.15)

Chapter 4 On the screening mechanism in DHOST theories evading gravitational wave constraints

27

and the explicit expression for c3 (which is written in terms of α, β, etc. and their time derivatives) is notimportant. As expected from the degeneracy of the theory, the final result (4.14) is just an algebraic equation
for x, with no derivatives acting on x. In generic quadratic DHOST theories, however, one would obtain at thisfinal stage a cubic equation for x. The present theory is special in the sense that the coeﬃcient of the cubic term
vanishes identically.
From now on, let us consider the case where the source is static, ρ = ρ(r). Then, since we are assuming thatϕ̇0 is approximately constant, A is also independent of time. Thus, Ȧ in Eq. (4.14) can be neglected.
One may define the typical radius rV below which nonlinearities are large by A(rV ) = 1. We are mainlyinterested in the solutions to Eq. (4.14) for A ≫ 1 both inside and outside the matter source. Outside the matter
distribution we have A ∝ r−3 , whereas we have (r3 A)′ ̸= 0 inside.
Let us first consider the exterior region. For A ≫ 1 we have
x≃±

[
]1/2
1 α + (1 − 2β)ξ − 2ζA.
2 (α − 3βξ)(1 − β)(4.16)

From this it can be seen that the terms linear in x in Eqs. (4.12) and (4.13) are suppressed relative to the otherterms. We thus find, irrespective of the sign of Eq. (4.16), thatα(4 − β) − β(13 − 2β)ξ + 2βζA,8(α − 3βξ)(1 − β)2α(4 − 7β) − 11β(1 − 2β)ξ − 2βζz≃A,8(α − 3βξ)(1 − β)2y≃(4.17)(4.18)

This shows that Φ ̸= Ψ in general, implying that the present subclass of DHOST theories does not evade thesolar-system constraints. However, if the parameters satisfy*1
3α − ξ(1 + 10β) + 2ζ = 0,(4.19)

GR is recovered, yieldingA,
2(1 − β)M1.
Φ′ = Ψ ′ =16πf (1 − β) r2
y=z=⇔(4.20)The eﬀective gravitational constant is given by
GN,out =1.
16πf (1 − β)(4.21)

Thus, fine-tuning is needed in order for the screening mechanism to work successfully in the vicinity of a source.
This is in contrast to generic DHOST theories [80, 81, 82, 83].
Next, let us look at the interior region. We have two branches, one of which is given by(I) :β(r3 A)′≫ 1,2(α − 3βξ) r2(4.22)

α + (1 − 2β)ξ − 2ζ r2 A
= O(1).
2β(1 − β)(r3 A)′(4.23)x≃and the other by(II) :

x≃−*1 More precisely, the condition for successful screening is β[3α − ξ(1 + 10β) + 2ζ] = 0. Clearly, the case with β = 0 corresponds

to the subclass of the Horndeski theory. This is the trivial case exhibiting the Vainshtein mechanism [64, 65, 66].

Chapter 4 On the screening mechanism in DHOST theories evading gravitational wave constraints
In Branch I, the behavior of gravity is far away from the normal one:[]
9β 3
(r3 A)′(r3 A)′
3′′y=(r A) −+ O(A),(1 − β)3 ξ 2 r3
r
[
]
9β 3
(r3 A)′(r3 A)′3′′z=−(r A) −+ O(A),(1 − β)3 ξ 2 r3r

28

(4.24)(4.25)

where Eq. (4.19) was assumed. It then follows thatΦ′ ≃ −Ψ′ ∝M ′ M ′′
(M ′ )2−.
2
rr3(4.26)We therefore conclude that this branch would not describe the stellar structure appropriately, and hence must
be excluded.
Branch II is phenomenologically more interesting. In this branch, all x’s in Eqs. (4.12) and (4.13) can beneglected, leading toA(1 − 2β)A, z=.
22(1 − β)2(1 − β)21
M⇔ Φ′ =, Ψ = (1 − 2β)Φ.
216πf (1 − β) r2y=(4.27)

From this we see that the eﬀective gravitational constant inside the matter distribution is diﬀerent from theexterior value by a factor of (1 − β)−1 :GN,in =

GN,out.
1−β(4.28)

This must be contrasted with the way of breaking the screening mechanism in generic DHOST thoeries, where
M ′ and M ′′ appear in Φ′ and Ψ′ as corrections to the standard gravitational law with the same gravitational
constant as the exterior one [80, 81, 82, 83]. We also see that Φ and Ψ do not coincide in the matter interior.
One should note that Eq. (4.19) is not used to derive Eq. (4.27).
Let us finally comment on the solution for A ≪ 1. We have two branches, namely, x ∼ y ∼ z ∼ A andx ∼ y ∼ z ∼ 1. By inspecting the explicit solutions to Eq. (4.14), we find that the former branch, which is
phenomenologically more acceptable, is matched onto Branch II ifβ(1 − β)c3 < 0(4.29)

is satisfied.
As an example, we show in Fig. 4.1 the Branch II profiles of x, y, and z for A(r) = B(r)/B(1000) (namely,rV = 1000) with B(r) = (r3 + 1)−1 . The density profile mimics a star with the radius r ∼ 1. The parameters
are given by ξ = α = 1, β = ζ = 1/4, and c3 = 1. (For x we plot an exact solution to Eq. (4.14), but for y and z
the terms linear in x are ignored because they are subdominant for r ≪ rV .)We also present in Fig. 4.2 the Branch II solution for the NFW density profile, ρ(r) = ρ0 /[(r/rs )(1 + r/rs )2 ]with rs = 1 and ρ0 chosen so that rV = 1000. The parameters are again given by ξ = α = 1, β = ζ = 1/4, andc3 = 1. Since there is no definite surface in this case, we see deviations from GR everywhere.

4.2 Observational constraints
We have seen that though the particular subclass of DHOST theories (4.1) could evade solar-system tests byrequiring the fine-tuned relation (4.19), (i) Φ and Ψ do not coincide inside the matter distribution, and (ii) thegravitational constant in the matter interior is diﬀerent from its exterior value. Let us discuss briefly possible
observational constraints on such modifications of gravity.

Chapter 4 On the screening mechanism in DHOST theories evading gravitational wave constraints
���

�

���

�

29

����
��
�����

-�

�����
����

����

�

��

���

����

�
Fig. 4.1 An example of a Branch II solution for rV = 1000 and the stellar radius ∼ 1. The dashed line
corresponds to the potentials in GR with the gravitational constant GN,out .

���

�

���

�

����
��
�����

-�

�����
����

����

�

��

���

����

�
Fig. 4.2 The Branch II solution for the NFW density profile. The dashed line corresponds to the potentials
in GR with the gravitational constant GN,out .

The diﬀerence between the two potentials in the nonvacuum region, Ψ/Φ − 1 = −2β, can be measured bycomparing the X-ray and lensing profiles of galaxy clusters, as has been investigated for diﬀerent types of modifications in Refs. [113, 118, 119]. In particular, the constraints obtained for beyond Horndeski theories in Ref. [113]read |Φ/ΦGR − 1| < O(10−1 ) and |Ψ/ΨGR − 1| < O(10−1 ). Thus, we would expect constraints of the same orderof magnitude, |β| < O(10−1 ), from galaxy clusters.
A diﬀerent value of the gravitational constant inside the Sun would lead to changes in the solar structure, andthereby modify the sound speed and solar neutrino fluxes. Based on the solar standard model, it has been argued
that a relative diﬀerence of O(10−2 ) is still allowed by observations [120]. Thus, the Sun could potentially be
used to test a diﬀerent value of the gravitational constant inside extended objects.
Note, however, that currently the most stringent bound comes from the diﬀerence between the measured valueof the gravitational constant, GN (= GN,out or GN,in ), and the gravitational coupling for GWs, GGW , which isconstrained from the orbital decay of the Hulse-Taylor pulsar: −7.5×10−3 < GGW /GN −1 < 2.5×10−3 [83, 121].
In the present case, we have GGW = (16πf )−1 [78, 79], so the constraint is given by|β| < O(10−3 ),which is orders of magnitude tighter than the possible constraint from galaxy clusters.

(4.30)

Chapter 4 On the screening mechanism in DHOST theories evading gravitational wave constraints

30

4.3 Summary
In this chapter, we have studied the screening mechanism in a particular subclass of degenerate higher-orderscalar-tensor (DHOST) theories in which the speed of GWs is equal to the speed of light and gravitons do notdecay into scalar fluctuations. By inspecting a spherically symmetric gravitational field, we have found thatthe screening mechanism operates in a very diﬀerent way from that in generic DHOST theories [80, 81, 82, 83].
First, the fine-tuning is required so that solar-system tests are evaded in the vacuum exterior region. This is
in contrast to generic DHOST theories, in which the implementation of the Vainshtein screening mechanism
outside the matter distribution is rather automatic. Second, the way of the Vainshtein breaking inside extended
objects is also diﬀerent from that in generic DHOST theories. We have shown that in the interior region themetric potentials obey the standard inverse power law, but the two do not coincide. Moreover, the eﬀective
gravitational constant diﬀers from its exterior value. However, the current most stringent bound comes from thefact that the eﬀective gravitational coupling for GWs is diﬀerent from the Newtonian constant [83, 121], rather
than from the above interesting phenomenology. The obtained constraint is as tight as
XfX< O(10−3 ).
f(4.31)

Thus, we conclude that the allowed parameter space is small for DHOST theories as alternatives to dark energy
evading gravitational wave constraints.

31

Chapter 5Constraining DHOST theories with linear growthof density fluctuations
In this chapter, we investigate the potential of cosmological observations, such as galaxy surveys, for constraining
DHOST theories, focusing in particular on the linear growth of the matter density fluctuations. This topic
is based on S. Hirano, T. Kobayashi, D. Yamauchi and S. Yokoyama, “Constraining degenerate higher-orderscalar-tensor theories with linear growth of matter density fluctuations,” Phys. Rev. D 99 (2019) no.10, 104051[arXiv:1902.02946 [astro-ph.CO]] [29].
One of the most stringent constraints on gravity theories is obtained from the gravitational wave event
GW170817 [52] and its optical counterpart GRB 170817A [53], which gave the constraint on the speed of GWs,
cGW , as |cGW − 1| ≲ 10−15 . This observation can be used to rule out scalar-tensor theories which predict a variable gravitational-wave speed at low redshifts [95, 96, 98, 99, 100, 101, 82]. One finds that there still is a broadclass of viable scalar-tensor theories. In particular, a certain subclass of quadratic DHOST theories [21, 22, 23]
survived after this event.
Of course, even before GW170817 lots of stringent constraints on local gravity had been obtained, implyingthat gravity must be consistent with GR at least on small scales and in the weak gravity regime. Therefore,
viable scalar-tensor theories are required to have a mechanism that suppresses the fifth force mediated by thescalar field on small scales, and Vainshtein screening is a typical one of such mechanisms in the Horndeski andrelated theories. Interestingly, DHOST theories generically exhibit Vainshtein screening outside matter, whereas
its partial breaking occurs inside [80, 82, 81, 83]. As the gravitational laws inside an astrophysical body diﬀer
from the standard ones, this phenomenon leads to a modification of its internal structure, which can be used
to constrain DHOST theories [85, 86, 87, 88, 89, 90]. The authors of Ref. [83] applied this idea to the DHOST
theories satisfying c2GW = 1 and obtained constraints on the parameters which characterize the theories.
In this chapter, in addition to the above constraints, we investigate the possibility of constraining DHOST
theories from the current/future precise cosmological observations. In particular, we focus on the linear evolutionof the matter density fluctuations, which can be measured by observations of large scale structure. Measuring thelinear growth rate of large-scale structure, f (a), is known to be a powerful tool to test modifications of gravity
responsible for the present cosmic acceleration. To compare the observational data with theoretical predictions,the simplest approach is to introduce an additional parameter called gravitational growth index, γ, defined interms of the linear growth rate and the fraction parameter of non-relativistic matter Ωm as [122]γ :=d ln f.
d ln Ωm(5.1)

The purpose of this chapter is to obtain a novel constraint on DHOST theories with c2GW = 1 from the observationsof the linear growth rate. To do so, we develop a formalism to describe DHOST cosmology during the matter
dominated era and the early stage of the dark energy dominated era, and evaluate the growth index at highChapter 5 Constraining DHOST theories with linear growth of density fluctuations

32

redshifts. We expect that the current observations of the growth index yield new constraints on DHOST theories
which are complementary to the existing bounds.
The chapter is organized as follows. In Sec. 5.1, We introduce cosmological perturbations and overview theevolution of density fluctuations. In Sec. 5.2, we derive cosmological background equations in class I quadratic
DHOST theories. Then we consider linear cosmological perturbations and derive the evolution equation of thedensity fluctuations. In Sec. 5.3, we introduce our formalism to model DHOST cosmology and evaluate thegrowth index as a probe of modifications gravity. We thereby give novel constraints on DHOST theories fromcurrent observations in Sec. 5.4. Finally, we discuss our results and future prospects in Sec. 5.5.

5.1 Cosmological perturbations
Inflation and Big-Bang cosmology pass many cosmological observations. Inflation is the accelerating expansionof the universe and can solve the problems of Big-Bang cosmology. Also inflation generate the seed of the density
fluctuations of large scale structure. In this section, we see the evolution of density fluctuations based on theinitial conditions generated by inflation.
Let us consider so-called cosmological perturbation around a background spacetime. The metric is given bygµν = ḡµν + δgµν , where ḡµν is the background metric and δgµν is the metric pertrubation. From the Einsteinequations, the equations of motion in terms of the linear perturbation is given byδGµν (δgµν ) = 8πG δTµν .

(5.2)

This is the perturbed Einstein equations. Determining the form of the metric perturbations δgµν , we derive thedynamics of these from above equations.

5.1.1 Metric perturbations and gauge freedom
The metric perturbation δgµν depends on gauge transformations because the theory has 4D-diﬀeomorphism. Dueto an infinitesimal transformation xµ → x̃µ := xµ + ξ µ , the transformation of δgµν is given by the Lie derivative
along ξ µ , that is,δg̃µν = δgµν − ḡµν,ρ ξ ρ − ḡµρ ξ ρ,ν − ḡρν ξ ρ,µ .

(5.3)

For convention, we decompose δgµν to the scalar, vector, and tensor-type variables under the symmetry in FLRW
spacetime, O(3) . The concrete form of the metric perturbation is given by
ds2 = −(1 + 2A)dt2 − 2a(B,i + Si )dtdxi + a2 [(1 − 2ψ)δ + 2E,ij + 2F(i,j) + hij ]dxi dxj ,(5.4)

where A, B , ψ, and E are the scalar-type variables, and Si , Fi are the vector-type variables, and hij is thetensor-type variable. The vector type is transverse, and the tensor type is transverse and traceless. Thanks tolinearity, the equations of motion on each types are independent of the other types. In the following discussion,
we focus on the scalar-type perturbations at a linear level to mention the dynamics of density fluctuations. The
scalar perturbations are given by
ds2 = −(1 + 2A)dt2 − 2aB,i dtdxi + a2 [(1 − 2ψ)δ + 2E,ij ]dxi dxj .

(5.5)

For example, ψ is related to a geometric quantity. Let us consider a t = const. hypersurface. Then, the intrinsic
curvature is given by
(3)

R=

4 2
∂ ψ,
a2

(5.6)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

33

where ∂ 2 := δ ij ∂i ∂j is the 3-dimensional spatial Laplacian. ψ is the so-called curvature perturbation. The eachcomponents of the perturbed Einstein tensor are given by
[
(
)]
2
B
δG00 = 6H(ψ̇ + HA) − 2 ∂ 2 ψ + a2 H Ė +,aa
δG0i = −2∂i (ψ̇ + HA),

(5.7)
(5.8)

δGij = 2[ψ̈ + 3H ψ̇ + H Ȧ + (3H 2 + 2Ḣ)A]δji(

1+ ∂ ∂j − δji ∂ 2
3

)[

i

(
)]
1
B
(ψ − A) + (1 + 3H) Ė +
.
a2
a

(5.9)

A dot denotes a derivative with respect to the cosmic time.
Moving to the right hand side of the Einstein equation, we set the energy-momentum tensor of fluids. The
4-velocity of fluids is defined by uµ = (1 − A, a−1 ∂ i (v + B)). v is the velocity potential. The components of theenergy-momentum tensor are given byδT00 = −δρ,

(5.10)

δTi0 = ∂i [a(ρ + p)v] =: ∂i δq,(5.11)

(
)
1 i 2
i
i
i
δTj = δpδj + ∂ ∂j − δj ∂ δΠ,3(5.12)

where δρ, δp are the fluctuations of the energy density and pressure respectively, and δΠ is the anisotropic stress.
These scalar-type metric perturbations depend on a gauge transformation xµ → xµ + ξ µ (ξ µ = (δt, ∂ i δx))below
A → A − δ̇t,
1
B → B − δt + aδ̇x,
a
E → E − δx,(5.13)

ψ → ψ + Hδt.

(5.16)(5.14)(5.15)

Also, the energy-momentum tensor transforms according to the law of the transformation for tensors. Then, thefluid fluctuations are changed byδρ → δρ − ρ̇δt,(5.17)

δp → δp − ṗδt,(5.18)

δq → δq − (ρ + p)δt,(5.19)

δΠ → δΠ.

(5.20)Let us consider the relationships between these perturbative quantities and observables. The perturbative
quantities depend on a gauge transformation (5.3), but observables are not independent of gauge freedom. First,
we discuss the gauge invariant way. Eliminating the spatial gauge dependence δxi , we consider below quantity,
σ := Ė +B.
a(5.21)

Then, σ → σ − δt/a under a gauge transformation. So, rewriting the Einstein equations by this quantity, we
can eliminate the spatial gauge dependence. In the matter sectors, the perturbative variables depend only on atime-dependent gauge transformation without δΠ. Thus, in order to describe the gauge-invariant equations of
motion in terms of scalar-type variables, we can kill the remaining the time-direction gauge shift δt with σ. For
example, the gauge-independent values areδ G̃00 := δG00 − a2 Ġ00 σ,(5.22)

δ G̃0i := δG0i − a2 (G00 − Gjj /3)∂i σ,(5.23)

δ G̃ij := δGij − a2 Ġij σ.

(5.24)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

34

As for matter sector, we can obtain the gauge-invariant variables by using T instead of G. These treatment is
the so-called gauge-invariant perturbation theory [123, 124, 125].
For convention, let us consider the gauge-invariant quantities related to A and ψ. It is convenient to use thebelow combinations,Φ := A − (a2 σ)· ,(5.25)

Ψ := ψ + a2 Hσ.

(5.26)

These are the so-called Bardeen variables and are broadly used to describe the evolution of density fluctuations.
Going to Fourier space and rewriting the variables by above gauge-inv. variables, the components of Einsteinequations in (0, 0)(0, i) are given byk2
Ψ = −4πGδρ,a2Ψ̇ + HΦ − a2 Ḣσ = −4πGδq,

3H(Ψ̇ + HΦ − a2 Ḣσ) +(5.27)(5.28)

where k is the length of a comoving wave-vector. In matter sector, we have not defined gauge-inv. values. We
define the gauge-inv. variable δρ(GI) = δρ − 3Hδq. Combining the Eqs. (5.27) and (5.28), we can obtain the
Poisson equation,k2− 2 Ψ = 4πGδρ(GI) .
a(5.29)In the context of a gauge fixing which we will discuss after , δρ(GI) is equivalent to that at a comoving gauge
which this coordinate is the comoving frame with fluids, v = 0.
The trace and traceless parts of the Einstein equations in (i, j) areΨ − Φ = 8πGδΠ,()
2ψ̈ + 2H ψ̇ − H Ȧ − (3H 2 + 2Ḣ)A = 4πG δp + ∇2 δΠ .
3

(5.30)(5.31)

From (5.30), we can obtain Ψ = Φ without an anisotropic stress δΠ. In the case without δΠ, combining Eq. (5.31)
rewritten by Φ and Ψ and Ψ = Φ, we obtain the evolution equation of the gravitational potential Φ,Φ̈ − 3(1 + c2s )H Φ̇ + [3H 2 (1 + c2s ) + 2Ḣ]Φ +c2s k 2
Φ = 4πGδpnad ,a2(5.32)

where c2s is the propagation speed of fluids, c2s := (∂p/∂ρ)|s = ṗ/ρ̇. The source term in the right hand side ofabove equation, δpnad , means the fluctuation of entropy,
δpnad = δp(GI) − c2s δρ(GI) ,(5.33)

δp(GI) := δp − a2 ṗσ.

(5.34)

Assuming the component of fluid, we can obtain the evolution of the gravitational potential.
The conservation laws of the energy-momentum tensor δ(∇µ Tνµ ) = 0 are given by2

˙ + 3H(δρ + δp) − 3ψ̇(ρ + p) = k [δq + a2 (ρ + p)σ],δρa2
˙ + 3Hδq + (ρ + p)A + δp = 2 k 2 δΠ.
δq3(5.35)(5.36)

The first is the relativistic version of the energy equation, and the second is called the Euler equation. Because wehave Bianchi identity of the Einstein tensor Gij and the Einstein equations, above conservation equations (5.35)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

35

and (5.36) are not independent. However, it is convenient to calculate the dynamics of fluids, that is, the density
fluctuation.
It is convenient to fix the gauge freedom instead of using gauge-invariant variables. In this section we will use
a Newtonian gauge. In this gauge, we fix B = E = 0 (⇒ σ = 0). This means that A = Φ, ψ = Ψ, δρ(GI) =δρ − 3Hδq, δp(GI) = δp. These dynamical variables have no gauge freedom.

5.1.2 Evolution of density fluctuations
Our universe is filled with rich structures, stars, galaxies, clusters, and super clusters. However our universe isclose to homogeneous and isotropic universe at 0th order. If our universe is perfectly homogeneous and isotropic
universe at an initial state, these rich structures cannot be born. Fortunately, inflation can generate classicalfluctuations of spacetime from quantum fluctuations. Transferring these fluctuations to density fluctuations
during the evolution of the universe, the rich structures can be realized.
The gravity plays an important role of making structures. The density fluctuations grow due to the gravity
and the evolution of the universe. When the density of matter increases beyond certain threshold value, thestructure such as stars can be born. This is called gravitational instability. Cosmological observations canobserve the statistical property of density fluctuations based on the perturbative calculations. We would like to
study the evolution of density fluctuations in terms of non-relativistic matter deeply inside horizon scales which
we can observe at a linear level. In order to the gravitational evolution of the density, the density fluctuations
need to grow at early times. The higher-order corrections can be negligible. The non-linear property of density
fluctuations will be discussed in Ch. 6.
Deeply inside sub-horizon scale (k ≪ aH), rewriting Eqs. (5.35)(5.36) by Φ, δ := δρ(GI) /ρ̄, δp, v instead ofδρ(GI) and δp(GI) at a Newtonian gauge, we obtain equations for matter,
v
= 0,
a
[
]
k2
δp
v̇ + Hv =Φ−.
aρ̄(1 + δ)δ̇ +(5.37)(5.38)

δ is the so-called density fluctuation. In the RD era, δp can aﬀect the dynamics of the system. Also, we have the
Poisson equation (5.29),
1 2∂ Φ = 4πGρ̄δtot .
a2(5.39)

Note that the cosmological constant does not appear on the Poisson equation. Here, we do not restrict the single
fluid component. The dynamics of gravitational potentials is determined by the total fluid. So, we wrote thesub-scription tot to emphasize it. In the above system, there are the four unknown variables under the threeequations. So, we need to add the extra information for matter content, equation of state parameter w.
In order to the gravitational evolution of the density, the density fluctuations need to grow at the early time
when these are much smaller than unity. We study the evolution of density fluctuations at linear level where thehigher-order eﬀect can be negligible.
For simplicity, let us consider only non-relativistic matter universe, that is, Einstein de Sitter universe. Then,
we can neglect the entropy fluctuation δpnad (recall δp = c2s ρ̄δ + δpnad ). Combining Eqs. (5.37) and (5.38) toeliminate the velocity potential v, we can obtain the evolution equation of the density fluctuation δ,)(
c2s k 2
δ = 0.
δ̈ + 2H δ̇ − 4πGρ̄ − 2a(5.40)

Thanks to linearity, Fourier modes do not mix with each other. This picture is not used to go to non-linear order.

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

36

We can understand the behavior of the solution in Eq. (5.40) without solving concretely when we regard δas the position of a particle. The first term is the acceleration of a particle, the second one is the friction, andthe third one is the potential term which can change its sign. Thus, this equation is equivalent to that of 1
dimensional motion with the friction and the potential V (δ) = −(4πGρ̄ − c2s k 2 /a2 )δ 2 /2. Note that in this casethe friction term and potential term are time-dependent.
If the coeﬃcient of the potential 4πGρ̄ − c2s k 2 /a2 is negative, the potential is convex downwardly. Then, δcannot grow, and dampedly oscilate. This situation happen in the case with the large sound speed. Because
fluids with the large sound speed have large pressure, the gravitational force cannot grow the density of matter
against it. Also, at a small scale, there is no suﬃcient mass to grow gravitationally. This oscillation due to thepressure is called the acoustic oscillations.
If the coeﬃcient of the potential 4πGρ̄ − c2s k 2 /a2 is positive, the potential is convex upwardly. The density
fluctuation can gravitationally grow even if there exist the friction term and the pressure of fluids. The friction
term is induced by the cosmic expansion, it prevents the grow of the density fluctuation from the gravitational
force.
These two diﬀerent situations are separated by kJ . It is determined by the condition that the coeﬃcient of thepotential vanishes, that is,kJ =√a 4πGρ̄.
cs(5.41)

The corresponding wavelength in the real space is
λJ :=

4πa
= cs
kJ

√

π
.
Gρ̄

(5.42)

It is called Jeans length. This quantity give the criteria of the structure formation. Therefore, the modes smaller
than it cannot grow against the pressure while the modes larger than it can do. Considering the mass includedin the volume whose sides are Jeans length,c3
MJ := ρ̄λ3J = √ s ,G3 ρ̄(5.43)

it is the threshold of minimum mass which can form the structure. This MJ is called Jeans mass.
The speed of non-relativistic matter (includes dark matter) is suﬃciently smaller than the speed of light. In
the MD era, the Jeans length is much smaller than the horizon size lH := c/H ∼ c/(Gρ̄)1/2 (λJ ∼ (cs /c)lH ≪ lH ).
Thus, the structure formation occur broadly inside the horizon due to the gravitational instability.

5.1.3 Growth of density fluctuations thanks to gravitational instability
We focus on the modes with wavelengths larger than the Jeans length, and we derive the growing solution. This
growing solution describes the linear growth of density fluctuations for dark matter during the MD era and after
that. This is the important fact on the structure formation.
At a scale much larger than the Jeans scale, k ≪ kJ , the evolution equation for density fluctuation (5.40) is
given byδ̈ + 2H δ̇ − 4πGρ̄δ = 0.

(5.44)

As you see above, this equation does not depend on the wave-number k. This equation holds in Fourier space.
Thus, the evolution of the density fluctuation is independent of that at other positions.
At first, let us consider the Einstein de Sitter universe which is filled with non-relativistic matter and K = Λ = 0.
The background dynamics is given by Eq. (2.18), H = 2/3t. Then, the background energy density is given byChapter 5 Constraining DHOST theories with linear growth of density fluctuations37ρ̄ = (6πGt2 )−1 . The above equation reduces to
δ̈ +

4
2
δ̇ − 2 δ = 0.
3t3t(5.45)

Assuming the power law form of solutions, δ ∝ tn , we obtain the general solution
δ = At2/3 + Bt−1 ,(5.46)

where A and B are integration constants. The first term is the growing mode solution which increases in time,
and the second one is the decaying mode solution which decreases in time. The decaying mode is irrelevant in
the structure formation while the growing mode plays an important role of the growth of density fluctuations.
The time-dependence of the growing mode is propotional to t2/3 . This dependence is same as that of the scale
factor in the Einstein de Sitter universe. Therefore, the growth of density fluctuation is given by δ ∝ a in the
MD era.
In general situation, for example, after the matter dominace, we can obtain the solution for δ if we give theexpansion law of the universe. To eliminate the first-order derivative we introduce the variable y = aδ. Then,the equations (5.44) is rewritten by this new variable as
(
)
ä
ÿ −
+ 4πGρ̄ y = 0.
a(5.47)

By the way, the evolution equation for the scale factor (2.9) and the conservation law of matter at the backgroundlevel (2.11) have been given by
ä
4πG
Λ
=−
ρ̄ + ,a33ρ̄˙ = −3H ρ̄.
Diﬀerentiating the evolution equation in time and using the conservation law, we obtain
(
)
ä
...
a−
+ 4πGρ̄ ȧ = 0.
a(5.48)(5.49)(5.50)

Comparing this equation to Eq. (5.47), y = ȧ is the particular solution in this ordinary diﬀerential equation. Let
us study the general solution. Assuming the form of the solution, y = ȧw(t), Eq. (5.47) further reduces toȧẅ + 2äẇ = 0.
We can solve this equation in terms of ẇ imediately. The solution is ẇ ∝ ȧ−2 . Integrating once in time
∫
∫
dt
da
w∝
=
.
ȧ2
ȧ3
Summarizing above result, the independent solutions in Eq. (5.44) are{ ∫a
H 0 a3da
H 3 =: D+ ,
δ∝HWe recall the Hubble parameter in general era (5.58) has been given by√Ωm0
1 − Ωm0 − ΩΛ0+ ΩΛ0 +.
H = H03aa2(5.51)(5.52)(5.53)(5.54)

This is the monotonous decreasing function in time. Thus, the above solution defined by D+ in Eq. (5.53) isthe growing mode while the below one is the decaying mode. This function D+ (t) is the so-called linear growth
factor.

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

38

It is known that the linear growth factor D+ is described by the integral form
D+ =

5
aΩm
2

∫ 1
0

dx
(Ωm /x + ΩΛ

x2 + 1 − Ω

3/2
m − ΩΛ ),(5.55)

where we fix the integration constant as D+ → a at the limit a → 0. In the limit a → ∞, the cosmologicalconstant becomes dominant. Then, H = const. From Eq. (5.53), the linear growth factor is
D+ = const.

(5.56)

Thus, the matter density become sparse due to the accelerating expansion, and the density fluctuation cannot
grow. The decaying mode can be negligible after suﬃcient time. The density fluctuations at the present time are
given byδ(k, t) = D+ (t)δL (k),(5.57)

where δL (k) is the initial distribution of the density fluctuations. *1
Note that above discussion we have derived the evolution of density fluctuations of “dark matter”, so notbaryon. As you know, structures are constructed by baryon. In the theoretic prediction, baryon grows undergravitational potentials made from dark matter due to gravitational couplings. The density fluctuations of baryon
are sourced by that of dark matter. In the MD, baryon’s density fluctuations becomes same dynamics as that ofdark matter (so-called catch-up) The relation between baryon and dark matter fluctuations is known as biased
relation. At a linear level, baryon’s density fluctuations is related to that of dark matter as shifted with constantfactor, δb = b δ. In fact, we can observe statistical property of δb with red-shift space distortion in cosmologicalobservations.

5.1.4 Gravitational potential and velocity field
The time evolution of gravitational potentials is determined by the solution of the density fluctuations (5.57) and
the Poisson equation (5.39),
1 2∂ Φ = 4πGρ̄δ.
a2(5.58)In the MD era, ρ̄ ∝ a−3 . Then, the time evolution of gravitational potential is given by Φ ∝ D+ (t)/a(t)|MD ∼const. In the dark energy dominance, ρ̄ = const. Then, Φ ∝ a2 (t)D+ (t)|DED ∼ const.
Going to Fourier space, the solution in the Poisson equation isΦ(k, t) = −4πGa2 ρ̄=−δ(k, t)k2

a2 H 2δ(k, t).
k2(5.59)

From the first line to the second one, we used the Friedman equation (2.9).
Let us study the behavior of the velocity field at linear order. In the previous section, we decomposed thetensor quantities to pure scalar-type, vector-type, and tensor-type variables. The 4-velocity has the vector-type
component. In the fluid equations which will be derived later, we linearize this equation and take the rotation,()∂(5.60)+ H ϵijk ∂ j v k = 0 ⇔ ϵijk ∂ j v k ∝ a−1∂tThis means that the rotational modes of the velocity (so-called vorticity) decays proportional to a−1 due to thecosmic expansion.
*1 For convention, the linear growth factor sometimes is normalized by that at the initial time.

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

39

Neglecting this decaying component, there is the single scalar-type variable v which is the divergence componentof the velocity field v i . For convention, we define the velocity-divergence field θ as∂i v i (
v )
=
.
aH
aH

θ=

(5.61)

Using the continuity equation (5.37) and the solution of the density fluctuation (5.57), we can obtain the relation
between δ and θθ = −f (t)δ,(5.62)

d ln D+Ḋ+=.
d ln aHD+(5.63)

where
f (t) :=

This function means the time evolution rate of the growth factor, and is called linear growth rate.
Roughly speaking, f is given by the time derivative of the growth factor. Diﬀerentiating Eq. (5.55) in time,
we obtain the below integral
f = −1 −Ωm+ ΩΛ +
2

[∫ 1
0

dx
(Ωm0 /x + ΩΛ0 x2 + 1 − Ωm0 − ΩΛ0 )3/2]−1.

(5.64)In the Einstein-de Sitter universe, f = 1 identically. The rate f is the criteria of the distinguishment of the theoryof gravity discussed in this thesis. In the era after the matter dominance but suﬃcient near (i. e, 1 − Ωm ≪ 1),
we obtain the approximate expression of f
6
(1 − Ωm )11≈ Ω6/11m .

f ≈1−(5.65)

This value, 6/11, is the typical value of GR, and is the landmark of the test of gravity on large scale structure.
In general, this power law index is called growth index which we introduced Eq. (5.1). In Sec. 5.4, we estimate
the growth index in DHOST theories.

5.2 DHOST theories: background and perturbation equations
5.2.1 Action
We recall the action of the generic quadratic DHOST theories [21, 22] which is given by
∫
S=

√

[

d x −g G2 (ϕ, X) − G3 (ϕ, X)□ϕ + G4 (ϕ, X)R +
4

5
∑

]
ai (ϕ, X)Li ,i=1

where we have several functions of the scalar field ϕ and its kinetic term X := (−1/2)ϕµ ϕµ . The Lagrangians Li
are quadratic in the second derivatives of ϕ and are given by
L1 = ϕµν ϕµν ,L2 = (□ϕ)2 ,L3 = (□ϕ)ϕµ ϕµν ϕν ,L4 = ϕµ ϕµρ ϕρν ϕν ,L5 = (ϕµ ϕµν ϕν )2 ,where ϕµ := ∇µ ϕ and ϕνρ := ∇ρ ∇ν ϕ.
In order for this higher-derivative theory to be free of Ostrogradsky ghosts, we must impose the degeneracy
conditions that relate G4 and ai . The quadratic DHOST theories are classified in several subclasses [21, 22],
among which we are interested in the so-called class I theories, because theories in other subclasses exhibit some

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

40

pathologies in a cosmological setup [78, 79]. (The class I DHOST theories are conformally/disformally related to
the Horndeski theory [22, 23].) As shown in Sec. 3.3, the class I degeneracy conditions are summarized as
β2 = −6β12 ,a1 + a2 = 0,β3 = −2β1 [2(1 + αH ) + β1 (1 + αT )] ,(5.66)

where
M 2 αT = −4Xa1 ,M 2 = 2(G4 + 2Xa1 ),M 2 β1 = 2X(G4X − a2 + Xa3 ),M 2 αH = −4X(G4X + a1 ),M 2 β2 = 4X[a1 + a2 − 2X(a3 + a4 ) + 4X 2 a5 ],M 2 β3 = −8X(G4X + a1 − Xa4 ).

(5.67)

Here we write the derivative of a function f (X) with respect to X as fX . We thus have 3 constraints among
6 functions (G4 and ai ), leaving 3 free functions in addition to G2 and G3 . These α-parameters are related tolinear perturbations (we will see in next section).
Note that the propagation speed of GWs is given by c2GW = 1+αT . The gravitational wave event GW170817 [52]and its optical counterpart GRB 170817A [53] have placed a tight bound c2GW ≃ 1. We therefore have αT ≃ 0,provided that this constraint is valid at low energies where dark energy/modified gravity models are used [70].
Imposing αT = 0 amounts to taking a1 = a2 = 0, but for the moment we do not require this.

5.2.2 Background equations in shift-symmetric DHOST theories
In the rest of the chapter we focus on the shift-symmetric subclass of DHOST theories, in which the Lagrangianis invariant under a constant shift of the scalar field, namely ϕ → ϕ+ const. This means that the free functions
contained in the Lagrangian are dependent only on the scalar field kinetic term X.
As a matter component we only consider pressureless dust and assume that it is minimally coupled to gravity.
For a homogeneous and isotropic background, ds2 = −dt2 + a2 (t)δij dxi dxj , ϕ = ϕ(t), with the matter energy
density ρm , the gravitational field equations read

−M 2

(

3M 2 H 2 = ρm + ρϕ ,)2Ḣ + 3H 2 = pϕ ,where H = ȧ/a (a dot denotes diﬀerentiation with respect to t), and
(
)
1
2 2
2
ρϕ := ϕ̇J − G2 − M H 6β1 y − β2 y ,
2
[
(
) ]
β2
d
2 2
pϕ := G2 + 2M H (αB + 3β1 )y − β1 +y 2 + 2M 2 β1 (yH) ,4
dt(5.68)(5.69)(5.70)(5.71)

with J being the shift current defined shortly. Here we defined y := ϕ̈/(H ϕ̇) anddM 2,
M 2 H dt()
ϕ̇XG3XαH
β̇1β2
αB := −+− (3 − αM ) β1 ++ β1 +
y.
M 2H
y
H
2

αM :=

1

(5.72)(5.73)The scalar field equation can be written using the shift current as
J˙ + 3HJ = 0,
where

[
2

ϕ̇J = 2XG2X + M H

2(5.74)

(
)
(
) ]
3αM
β̇1
1
β̇2
− 6αB + 6 αM β1 ++ 6β1 y −αM β2 +yy
H
2H+ 6M 2 β1 Ḣ − M 2 β2d(yH) .
dt(5.75)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

41

Equation (5.74) implies that in the expanding Universe J = const/a3 → 0 and hence attractor solutions arecharacterized by J = 0.

... ....
The background equations (5.68), (5.69), and (5.74) contain the higher derivatives ϕ , ϕ , and Ḧ. However,the degeneracy conditions (5.66) allow us to reduce the system to the second-order one. It is not so obvious todemonstrate this explicitly, but one can follow Refs. [84, 126] to see that it is indeed possible to do so.

5.2.3 Density perturbations
Let us study matter density fluctuations in the Newtonian gauge. The perturbed metric in the Newtonian gauge
is given by
ds2 = − [1 + 2Φ(t, x)] dt2 + a2 (t) [1 − 2Ψ(t, x)] δij dxi dxj .

(5.76)

We write the perturbation of the scalar field as
ϕ(t, x) = ϕ(t) + π(t, x).

(5.77)

It is convenient to introduce a dimensionless variable Q := Hπ/ϕ̇, and we will use this instead of π. The density
perturbation is defined byρm (t, x) = ρm (t)[1 + δ(t, x)].

(5.78)

We study the quasi-static evolution of the perturbations inside the sound horizon scale*2 . The quasi-static
approximation indicates that ϵ̇ ∼ Hϵ ≪ ∂i ϵ, where ϵ is any of perturbation variables. This does not mean todrop all the time derivatives and the Hubble parameter, because one may expect that ∇2 Φ/a2 ∼ H 2 δ ∼ H δ̇ ∼ δ̈and hence the time derivatives acting on δ cannot be ignored in general. Expanding the action to second orderin perturbations under the quasi-static approximation, we obtain the following eﬀective action:∫
Seﬀ = d4 x Leﬀ ,
with

where

{
M 2a
Leﬀ =
(c1 Φ + c2 Ψ + c3 Q)∂ 2 Q + 4(1 + αH )Ψ∂ 2 Φ − 2(1 + αT )Ψ∂ 2 Ψ
2
}
[
]
Φ̇
Q̈
Ψ̇
2
2
− β3 Φ∂ Φ + 4αH − 2(2β1 + β3 ) + (4β1 + β3 ) 2 ∂ Q − a3 ρm Φδ,HH
H(5.79)(5.80)[]β3
β̇3c1 := −4 αB − αH + (1 + αM ) +,(5.81)
22H[]α̇Hc2 := 4 αH (1 + αM ) + αM − αT +(5.82),H{()
Ḣ
α̇B − α̇H3ΩmḢc3 := −2
1 + αM + 2 (αB − αH ) +++ 2 + αT − αM
H
H
2
H
[
](
)
(
)2
}
β3
Ḣ
β̇3
Ḣ
Ḣ β̇1Ḣβ3α̇M β3β̈3+ −2 2 β1 + (1 + αM ) +1 + αM − 2 − 2 2+++, (5.83)
H
4
2H
H
H H
H2
2
H 4
4H 2

and
Ωm :=

ρm
.
3M 2 H 2(5.84)*2 The validity of the quasi-static approximation has been discussed in Refs. [127, 128, 133]. See also Refs. [130, 131, 132].

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

42

We have three terms whose coeﬃcients are written solely in terms of β1 and β3 . (The latter can be expressed interms of αH , αT , and β1 using the degeneracy condition given by Eq. (5.66).) These are the new terms in DHOST
theories. The other terms are present in the Horndeski and GLPV theories, but as c1 and c3 are dependent on
β1 and β3 one can see implicitly the contributions of these parameters characterizing DHOST theories.
The field equations are derived by varying the eﬀective action with respect to Φ, Ψ, and Q. Going to Fourier
space, they are given by
β3
2β1 + β3 Q̇
a2
Φ + b1 Q ++ρ δ = 0,2
2
H
2M 2 k 2 m
Q̇(1 + αT )Ψ − (1 + αH )Φ + b2 Q + αH = 0,
H
Ψ̇
Φ̇
Q̇
Q̈
c2 Ψ + c1 Φ + b3 Q + 4αH − 2(2β1 + β3 ) + b4 + 2(4β1 + β3 ) 2 = 0,H
H
H
H
(1 + αH )Ψ −(5.85)(5.86)(5.87)

where k denotes a comoving wavenumber in Fourier space and Φ, Ψ, and Q are now understood as the Fouriercomponents. Here, the coeﬃcients bi (i = 1, 2, 3, 4) are defined as()
c1
1
1 d 2β1 + β3
+ (1 + αM )(2β1 + β3 ) +
,
4
2
2 dt
H
c2
d ( αH )b2 := − + (1 + αM )αH +
,
4 [(
dt
) H
]
Ḣ
α̇M
b3 := 2c3 +1 + αM − 2 (1 + αM ) +(4β1 + β3 )
H
H
(
)
(
)
d 4β1 + β3
d2 2β1 + β3
+ 2,
+ 2(1 + αM )
dt
H
dt
H2
[(
)
(
)]
Ḣ
d 4β1 + β3
b4 := 2 1 + αM − 2 (4β1 + β3 ) +
.
H
dt
H

b1 :=

(5.88)(5.89)(5.90)(5.91)

Since matter is assumed to be minimally coupled to gravity, the fluid equations are the same as the standard
ones, and hence under the quasi-static approximation the matter density fluctuations δ(t, x) and the velocity
field ui (t, x) obey1δ̇ + ∂i [(1 + δ)ui ] = 0,a11u̇i + Hui + uj ∂j ui = − ∂ i Φ.
aa(5.92)(5.93)

At linear order, these equations are combined to giveδ̈ + 2H δ̇ +k2
Φ = 0,
a2(5.94)

where we moved to Fourier space. The eﬀects of modified gravity come into play through the gravitational
potential Φ which is determined by solving Eqs. (5.85)–(5.87).
Let us then solve Eqs. (5.85)–(5.87) to express Φ, Ψ, and Q in terms of δ and its time derivatives. We will
follow the same procedure as that used in [30]. This procedure is feasible thanks to the degeneracy of the theory.
Solving Eqs. (5.85) and (5.86) for Φ and Ψ and substituting these solutions into Eq. (5.87), one finds that Q̈ andQ̇ terms are canceled due to the degeneracy, and hence Q can be expressed in the formδ̇k2
− 2 2 Q = κQ δ + νQ ,a H
H

(5.95)

where the explicit expressions for the time-dependent coeﬃcients κQ and νQ are presented in Appendix A. Finally,
substituting this back into Eqs. (5.85) and (5.86), the gravitational potentials Φ and Ψ can be expressed in terms

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

43of δ, δ̇, and δ̈ ask2δ̇δ̈− 2 2 Φ = κΦ δ + νΦ + µΦ 2 ,
a H
H
H
δ̇
δ̈
k2
− 2 2 Ψ = κΨ δ + νΨ + µΨ 2 .
a H
HH(5.96)(5.97)The explicit expressions for the time-dependent coeﬃcients µi , νi , and κi (i = Φ, Ψ) are also shown in Appendix A.
Within the Horndeski theory we have µi = νi = 0 and in the GLPV theory we still have µΨ = 0. That is, µΨ
first appears in DHOST theories beyond GLPV. Equation (5.96) allows us to eliminate Φ from Eq. (5.94) and
we obtain the closed-form equation for δ as3
δ̈ + (2 + ς)H δ̇ − Ωm ΞΦ H 2 δ = 0,2

(5.98)

where the additional friction ς and the eﬀective gravitational coupling (multiplied by 8πM 2 ) ΞΦ are written interms of µΦ , νΦ , and κΦ as
2µΦ − νΦ,1 − µΦ
2
κΦΞΦ =.
3Ωm 1 − µΦς=(5.99)(5.100)

These two functions characterize modification of gravity. The evolution equation (5.98) has essentially the same
form as that in DHOST theories with c2GW = 1 [84] and in the GLPV theory [107, 133]. Whether or not c2GW = 1
does not play an important role in determining the qualitative form of Eq. (5.98). In the case of the Horndeski
theory (αH = β1 = 0), the additional friction term vanishes, ς = 0, and the result of Ref. [134] is recovered.
Equation (5.98) tells us that, even in DHOST theories under the quasi-static approximation, the evolution of
the matter density fluctuations is independent of the wavenumber, so that as usual (see Eq. (5.57)) we can write
the growing solution to Eq. (5.98) asδ(t, k) = D+ (t)δL (k),(5.101)

where δL (k) represents the initial density field. The eﬀect of the modified evolution of the density perturbations
is thus imprinted in the growth factor, D+ (t). Introducing the linear growth rate f := ddlnlnDa+ , the evolution
equation can be written as()
df
d ln H
3+ 2+ς +
f + f 2 − Ωm ΞΦ = 0.
d ln ad ln a2(5.102)Given the expansion history and the dynamics of the scalar field, one can obtain the evolution of the linear growthrate by solving the above equation.

5.3 Modeling DHOST cosmology in the matter dominated era
We consider possible cosmological constraints on DHOST theories from observables during the matter dominatedera and in the early stage of the dark energy dominated era. To do so, we assume that during these stages y, G2 ,αi (i =H, M, B, T), and β1 can be expressed as a series expansion form in terms of ε := 1 − Ωm (≪ 1) as
y = y0 + O(ε),( )G2 = g2 M H ε + O ε2 ,( )αi = ci ε + O ε2 ,( )β1 = βε + O ε2 ,2

2

(5.103)(5.104)(5.105)(5.106)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

44

where y0 , g2 , ci , and β are constants. Note that the expansion of αi and β1 starts at O(ε), as modifications ofgravity are supposed not to be significant at early times. As seen below, the background equations are consistent
with Eqs. (5.103)–(5.106). The expansion coeﬃcients (y0 , g2 , ci , β) are not all independent parameters. We will
express some of them in terms of the other coeﬃcients and the parameters of an underlying model.
Substituting Eqs. (5.103)–(5.106) to Eqs. (5.70) and (5.71), one finds, for the attractor solutions (J = 0), that( )ρϕ = − (g2 + 6βy0 ) M 2 H 2 ε + O ε2 ,(5.107)[]()()
pϕ = g2 + 2 cB + 3β y0 − 2βy02 M 2 H 2 ε + 2M 2 βy0 Ḣε + O ε2 .
(5.108)
Noting that 3M 2 H 2 − ρ̄m = 3M 2 H 2 ε = ρϕ , we have
g2 = −3 (1 + 2βy0 ) .

(5.109)The eﬀective dark energy equation of state parameter, wϕ := pϕ /ρϕ , can be expanded as
wϕ = w(0) + O(ε).
From Eqs. (5.107)–(5.109) and Ḣ/H 2 = −3/2 + O(ε) we obtain()2
3
w(0) = −1 +cB y0 − βy0 − βy02 .
32(5.110)(5.111)

Using the above expression for w(0) , one has the following useful formulas valid up to O(ε):
(
)
)
Ḣ
3(
ε̇
(0)
(0)
2
=
−
=
c
−
3w
(5.112)
1
+
w
ε
+
O(ε
),
ε + O(ε2 ).
M
H22H
To proceed further, let us assume that G2 ∝ X p , where p is a constant model parameter. This assumption
leads to the relation XG2X = pG2 . Using this assumption and Eq. (5.112), we find
[
]
(
)
(
)
0 = ϕ̇J = 2pg2 + 3 y0−1 cM − 2cB − 9β + 6β cM − 3w(0) + 6βy0 M 2 H 2 ε + O(ε2 ).

(5.113)Equations (5.111) and (5.113) givey0
(cH + 2p) ,
3
(
)
cH
3
cB = −p −+ β y0 +.
22

w(0) = −1 −(5.114)(5.115)Thus, w(0) and cB are expressed in terms of the model parameter p and the other coeﬃcients.
In the following we consider tracker solutions characterized by the condition
H ϕ̇2q = const ,(5.116)

where q is a constant. Such tracker solutions have been studied in the context of the Horndeski theory [71, 72]
and its extensions [84, 117, 126]. For instance, the cosmological solution discussed in [84, 126] corresponds to thecase with p = 2 and q = 1. In this thesis we regard q as another model parameter. For the solutions satisfying
Eq. (5.116), it is easy to see that
y0 =

3
.
4q

(5.117)

In what follows we will use q instead of y0
So far we have not imposed c2GW = 1 (⇔ αT = 0), as αT does not appear explicitly in the background equations.
Upon imposing αT = 0, it follows from the definitions that M 2 = 2G4 and M 2 αH = −4XG4X , which implies
another relation between the parameters:
αM = −yαH⇒cM = −y0 cH .

(5.118)Thus, under the assumption of c2GW = 1, we have four independent parameters, (p, q, cH , β), in terms of which
g2 , cM , cB , as well as w(0) , can be expressed.

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

45

5.4 Constraining DHOST cosmology
5.4.1 Growth index
Let us derive the solution to (5.102) in a series expansion form in terms of ε. We start with expanding ς and ΞΦ
in terms of ε. Since αi = O(ε) and β1 = O(ε), we have ς → 0 and ΞΦ → 1 for ε → 0, so that, to O(ε), ς and ΞΦ
can be written asς = ς (1) ε + O(ε2 ),(1)ΞΦ = 1 + ΞΦ ε + O(ε2 ),(5.119)(1)

where ς (1) and ΞΦ can be written in terms of the parameters introduced in the previous section. See Appendix A
for their explicit expressions. Then, Eq. (5.102) reduces to
[
(
) ]
[
(
) ]
(
) df
1
3
3
(1)
cM − 3w(0) ε ++ ς (1) − w(0) ε f + f 2 − 1 − 1 − ΞΦ ε + O(ε2 ) = 0,
dε
2
2
2

(5.120)where we used Eq. (5.112). The solution to this equation is given by
[ (
]
)
(1)
3 1 − w(0) + 2ς (1) − 3ΞΦf =1−
ε + O(ε2 ).
5 − 6w(0) + 2cM(5.121)

From the solution (5.121) we immediately obtain(1)

γ=3(1 − w(0) ) + 2ς (1) − 3ΞΦ+ O(ε).
5 − 6w(0) + 2cM(5.122)(1)

It is easy to see that the standard result γ = 6/11 is recovered for w(0) = −1, cM = ς (1) = ΞΦ = 0. Substituting(1)

the explicit expressions for ΞΦ and ς (1) [Eqs. (A.13) and (A.14)] into Eq. (5.122), one can evaluate an approximate
form of the growth index γ during the matter dominated era and the early stage of the dark energy dominatedera satisfying ε ≪ 1:
[(
)
]
(
)] 2
3 1 − w(0) − cT
2[
(0)
γ=
−
c
−
c
+
c
−
β
c
−
3w
B
M
T
M
Σ
5 − 6w(0) + 2cM
{
[
) (
)
(
)]
cH + β (6 1 + w(0) + cM − cT 1 − 2 cM − 3w(0)
+
Σ
}
[
(
)][
(
)]
(
)(
)
(0)
(0)
(0)
+ cB − β cM − 3w
5 − 2 cM − 3w
+ 5 cH + β cM − 3w
+ O(ε) ,(5.123)

where
Σ=

}
{
[
(
) (
)
(
)][
)]
1(
5 − 6w(0) + 2cM 3 1 + w(0) + 2 (cM − cT ) + 1 − 2 cM − 3w(0) cB − cH − β cM − 3w(0) + 1.
3(5.124)

The first two terms in Eq. (5.123) are the generalization of the previous results derived in the case of the
Horndeski theory [162] and the third term appears when at least either of cH and β is nonvanishing, namely
when one considers theories beyond Horndeski. Equation (5.123) is general in the sense that we have not yet
imposed αT = 0. Now, imposing αT = 0 (⇒ cT = 0), as discussed around Eq. (5.118), γ can be written in termsof (p, q, cH , β) as
γ=

{[
]
3
(p + 4q) (−3 + 6p + 10q) − 8pq 22(−3 + 6p + 10q)(3p + 11q)}]
3q(1 + 2q)(−3 + 6p + 16q)(cH + β)21[+ (−3 + 6p + 10q) + 8q (3p + 2q) cH +.
22pq + 3qcH + (3p + 5q)β(5.125)

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

46

1.0

cH

0.5

0.0

-0.5 (p, q)(1, 1 / 2)
(1, 3 / 2)
-1.0
(3, 1 / 2)
-1.0

-0.5

0.0
β

0.5

1.0

Fig. 5.1 The allowed parameter region in the β-cH plane obtained from the gravitational growth index in
the shift-symmetric quadratic DHOST cosmology after GW170817. The parameters are given by (p, q) =(1, 1/2) (red), (1, 3) (green), and (3, 1/2) (blue). The first set of the parameters corresponds to the solution
discussed in [84, 126].

5.4.2 Observational constraints
In this section, we investigate constraints on DHOST theories based on current observational limits on thegravitational growth index γ. For instance, clustering measurements from the BOSS DR12 give the limit as
γ = 0.52 ± 0.10 in Ref. [136] (based on the analysis in Fourier space) and γ = 0.609 ± 0.079 in Ref. [137] (based on
the analysis in configuration space). The constraints from BOSS DR14 are given as γ = 0.55 ± 0.19 in Ref. [138]and γ = 0.580 ± 0.082 in Ref. [139] (by adding tomographic analysis). Since the typical value of the deviationfrom the central value of γ in the current observations as shown above can be roughly estimated as ≲ O(0.1), let
us employ γ = 6/11 ± 0.1 as a conservative constraint. For a given set of the model parameters (p, q), this can betranslated into constraints on (β, cH ) using Eq. (5.125). The parameter regions in the β-cH plane allowed by theconstraint γ = 6/11 ± 0.1 are plotted in Fig. 5.1 for (p, q) = (1, 1/2) (red), (1, 3/2) (green), and (3, 1/2) (blue).
One finds from Fig. 5.1 that a constant-γ curve for fixed p and q is a hyperbola in the β-cH plane for (p, q) andγ that we are considering. This means that we have degeneracy between cH and β in the observations of thegrowth index. In contrast, in the GLPV theory we have β = 0, and hence we can obtain for instance the following
constraints on cH : −0.4 ≤ cH ≤ 0.4 for (p, q) = (1 , 1/2), −0.4 ≤ cH ≤ 0.5 for (1 , 3/2), and −1.1 ≤ cH ≤ 0.7 for(3 ,1/2). Deriving the constraints for other values of (p, q) is straightforward. It should be emphasized that theconstraints we have obtained in Fig. 5.1 are those at high redshifts satisfying Ωm ≃ 1.
To compare our results with previously known constraints, it is necessary to make further assumptions thatconnect the series expansion of αH and β1 to their present values. Specifically, we assume that αH = cH (1 − Ωm ),β1 = β (1 − Ωm ), and the leading order expression of γ [Eq. (5.125)] are valid all the way up to the present time.
Hereafter we focus on the specific parameter values (p, q) = (1, 1/2), which corresponds to the model discussed
in [84, 126], and demonstrate the allowed parameter region. Though details of constraints will be diﬀerent for
diﬀerent choices of (p, q), we expect that the order of the bounds is approximately the same.
Existing constraints on DHOST theories mainly come from the Newtonian stellar structure modified due tothe partial breaking of the Vainshtein mechanism, which is characterized by a single parameter Υ1 := −2(αH +Chapter 5 Constraining DHOST theories with linear growth of density fluctuations

47

1.5

γ-

1.0

6/

11

<0
.1

cH

0.5

γ-

0.0

Υ1 < 1.6

-0.5
-1.0

6/

11

<0

.1

Υ1 > -2 / 3
-4

-3

-2

-1
β

0

1

2

Fig. 5.2 The allowed parameter region obtained from the gravitational growth index in the shift-symmetricquadratic DHOST cosmology after GW170817 is shown by the red area in the β-cH plane. The parameters
are given by p = 1, q = 1/2. For comparison, the existing constraints −2/3 ≤ Υ1 ≤ 1.6 [85, 86] are shownby the gray area.

β1 )2 /(αH + 2β1 ) (the definition here is for theories with c2GW = 1) [80, 82, 83]. The lower bound on Υ1 has beenobtained from the requirement that gravity is attractive at the stellar center: Υ1 > −2/3 [85]. The upper boundis given by comparing the minimum mass of stars with the hydrogen burning with the minimum mass of observed
red dwarfs: Υ1 < 1.6 [86].
There are several attempts for improving the above bounds [87, 88, 89], including the one concerning the
speed of sound in the atmosphere of the Earth [90]. Aside from the constraints from the Newtonian stellar
structure, another constraint has been proposed, which comes from precise observations of the Hulse-Taylor
pulsar. This can severely constrain the eﬀective parameters through the coupling of GWs to matter [83, 121]:−7.5 × 10−3 ≤ αH + 3β1 ≤ 2.5 × 10−3 . However, when deriving this result, several assumptions have been madeand the resultant constraint would depend on the details of how the screening mechanism operates in a binary
system. In this thesis, we try to constrain the eﬀective parameters without taking into account these potentiallymore stringent bounds, and use the most conservative constraint: −2/3 < Υ1 < 1.6.
We plot in Fig. 5.2 the allowed parameter region in the β-cH plane obtained from the constraints on the growthindex (red) and stellar structure (black). As shown in Fig. 5.2, combining our results and the conservative
constraints discussed above can break the degeneracy between cH and β without using the Halse-Taylor pulsar
bound. The overlap region between these gives the constraints on both parameters: −1.0 ≤ cH ≤ 1.7 and−4.7 ≤ β ≤ 1.8.
Note that recently it was pointed out in Ref. [92] that the absence of gravitational wave decay into scalar modes
requires αH + 2β1 = 0. As seen from the fact that the denominator of Υ1 vanishes when this is satisfied, this isa special case which has not been explored so far. It would be interesting to investigate the behavior of gravity
in this limiting case in detail, but it is beyond the scope of this thesis, and we do not consider this constraint.

5.5 Summary
In this chapter, we have considered a possibility to constrain degenerate higher-order scalar-tensor (DHOST)
theories by using the information about the linear growth of matter density fluctuations. In DHOST theories,

Chapter 5 Constraining DHOST theories with linear growth of density fluctuations48

the evolution equation for the linear matter density fluctuations is modified in such a way that the eﬀective
gravitational coupling is changed by the factor ΞΦ and the friction term has an additional contribution ςH, bothof which can be expressed in terms of the eﬀective parameters αi and βi used in the literature.
We have constructed cosmological models in DHOST theories as a series expansion in terms of 1 − Ωm . In
doing so, we have assumed for simplicity that cosmological solutions under consideration are attractors in shiftsymmetric theories and subject to the tracker ansatz. The resultant cosmology is characterized by two model
parameters (p, q) and four independent eﬀective parameters in general (i.e., six parameters in total), and upon
imposing c2GW = 1 the number of independent parameters reduces to four in total. Our construction thus
provides a concise description of DHOST cosmology during the matter dominated era and the early stage of thedark energy dominated era.
We have then explicitly expressed the gravitational growth index γ in terms of (p, q) and the eﬀective parameters. We have found that the constant-γ curve in the β-cH plane generically is a hyperbola for c2GW = 1 and fixed(p, q). One can thus obtain constraints on a certain combination of the eﬀective parameters at high redshifts byusing the observations of the growth index alone.
Under the additional assumption that our leading order results in 1 − Ωm expansion can be extrapolated all
the way to the present time, we have compared the constraints from the growth index with the previously known
bounds. Combining our results and the constraints from modifications of the gravitational law inside stellar
objects, we have shown that the parameter degeneracy between αH /(1 − Ωm ) and β1 /(1 − Ωm ) could be broken
without using the Hulse-Taylor pulsar constraint, though our results slightly depend on the model parameters.
Future-planned observations for large-scale structure would exclude the currently allowed region of the parameter
space and serve as tests of the viability of DHOST theories.

49

Chapter 6

Matter bispectrum beyond Horndeski theories
In this chapter, we study the matter bispectrum of large scale structure as a probe of these modified gravity
theories, focusing in particular on the eﬀect of the terms that newly appear in the so-called “beyond Horndeski”
theories. This topic is based on S. Hirano, T. Kobayashi, H. Tashiro and S. Yokoyama, “Matter bispectrum
beyond Horndeski theories,” Phys. Rev. D 97 (2018) no.10, 103517 [arXiv:1801.07885 [astro-ph.CO]] [30].
Since the Horndeski theory [18, 19, 20] shares the same structure of nonlinear derivative interaction as theGalileon theory [50], the Vainshtein screening mechanism can naturally be implemented [64, 65, 66]. It is expectedthat this derivative nonlinearity is imprinted in the one-loop dark-matter power spectrum and the bispectrum.
This point has been investigated within the Horndeski theory in Refs. [141, 142, 143]. Higher derivative operators
arise in DHOST theories beyond Horndeski, and one of the interesting eﬀects due to them is the partial breakingof Vainshtein screening inside matter [80]. These new interactions will also participate in the one-loop matter
power spectrum and the bispectrum, which could be a probe of modified gravity theories beyond Horndeski. See
Refs.[83, 85, 86, 87, 88, 109, 110, 111, 112, 113] for other probes of DHOST theories.
The purpose of this chapter is to investigate the impact of the new operators of the Gleyzes-Langlois-PiazzaVernizzi (GLPV) theory [106, 107] on the matter bispectrum. As the GLPV theory (without the so-called F5term) is the simplest extension of the Horndeski theory in the context of degenerate theories, this work is a firststep to study how new nonlinear interactions beyond Horndeski aﬀect non-Gaussianity of large scale structure.
This chapter is organized as follows. In the next section, we derive our basic equations for the matter density
perturbations δ in the GLPV theory. We then give a second-order solution for δ in Sec. 6.2. In Sec. IV, thematter bispectrum in the GLPV theory is evaluated and its particular feature is emphasized. In Sec. V, we givea short comment on the implication of the recent GWs constraints for the theory. We draw our conclusions in
Sec. VI.

6.1 Basic Equations
6.1.1 The GLPV theory
The action of the GLPV theory is given by [106, 107]∫√
S = d4 x −g (L + Lm ) ,(6.1)

Chapter 6 Matter bispectrum beyond Horndeski theories

50

where*1
L = G2 (ϕ, X) − G3 (ϕ, X)□ϕ[]
+ G4 (ϕ, X)R + G4X (□ϕ)2 − ϕ2µν
1
+ G5 (ϕ, X)Gµν ϕµν − G5X [(□ϕ)3 − 3(□ϕ)ϕ2µν + 2ϕ3µν ]
6
′
′
′
1
− F4 (ϕ, X)ϵµνρσ ϵµ′ ν ′ ρ′ σ ϕµ ϕµ ϕνν ϕρρ
2
′
′
′
′
1
− F5 (ϕ, X)ϵµνρσ ϵµ′ ν ′ ρ′ σ′ ϕµ ϕµ ϕνν ϕρρ ϕσσ ,
3

(6.2)

and Lm is the Lagrangian of the matter components. Here we use the notation ϕµ := ∇µ ϕ, ϕµν := ∇µ ∇ν ϕ,
GX := ∂G/∂X, and ϵµνρσ is the totally antisymmetric Levi-Civita tensor. The above Lagrangian has six arbitrary
functions, Gi (i = 2, 3, 4, 5) and Fj (j = 4, 5), of ϕ and X := (−1/2)ϕµ ϕµ . The GLPV theory is an extension of
the Horndeski theory, and Eq. (6.2) reduces to the Horndeski Lagrangian in the case of F4 = F5 = 0.
Among wide classes of theories described by the GLPV action, we focus on those with G5 = F5 = 0 in thepresent thesis. This is a reasonable restriction because the G5 term not only hinders the recovery of the Newtonian
behavior of the gravitational potentials on small scales in a cosmological background [64], but also causes someinstabilities inside the Vainshtein radius [66]. Since the F5 term has the structure similar to the G5 term, thesame pathologies are expected, though this has not been confirmed explicitly so far. In the absence of G5 and F5 ,the GLPV theory is degenerate without further conditions [22], so that there are at most 3 propagating degrees
of freedom in any background spacetime. This nature is desirable in view of Ostrogradsky instabilities.
One of the interesting consequences of the F4 term is the partial breaking of the Vainshtein screening mechanism
inside matter sources [80, 83, 85, 86, 87, 88, 109, 110, 111, 112, 113], where derivative nonlinearities are significant.
It turns out that the partial breaking of the Vainshtein mechanism generically occurs in degenerate higher-orderscalar-tensor theories [81, 82, 83, 84]. In the present thesis, we study the impact of the nonlinearities of the F4term on the matter bispectrum. Some studies in this direction have already been undertaken in the context ofthe Horndeski theory in Refs. [141, 142, 143], and this work is an extension of [141].

6.1.2 Eﬀective action under the quasi-static approximation
We consider cosmological perturbations in a homogeneous and isotropic cosmological background. The field
equations governing the background evolution are found in Ref. [80]. As we are not interested in the evolutionof the universe in a particular modified gravity model, here we simply assume that the field equations admit asolution that is very close to the usual ΛCDM model. This is in principle possible because we have the four freefunctions in the theory that can be tuned if necessary.
The perturbed metric in the Newtonian gauge is given by
ds2 = −(1 + 2Φ)dt2 + a2 (t)(1 − 2Ψ)dx2 ,(6.3)

and the perturbed scalar field is written as
ϕ(t, x) = ϕ̄(t) + π(t, x),(6.4)

where a barred variable denotes the background quantity. It is convenient to introduce the dimensionless scalar
˙ We only consider nonrelativistic matter and write its energy density as
field perturbation as Q(t, x) := Hπ/ϕ̄.
ρm (t, x) = ρ̄m (t)[1 + δ(t, x)],

(6.5)

*1 Concerning the factors in front of F4 and F5 , we follow the convention of Ref. [80] which is diﬀerent from the one used inRef. [106, 107].

Chapter 6 Matter bispectrum beyond Horndeski theories

51

where δ is a density contrast.
We expand the action (6.1) in terms of the perturbations. Since we are interested in the evolution of the density
perturbations inside the (sound) horizon, we employ the quasi-static approximation, ∇i ϵ ≫ ϵ̇ ∼ Hϵ, where ∇iis the spatial derivative, a dot stands for the time derivative, and ϵ is any of Φ, Ψ, or π. As we have shownin Sec. 5.4, this does not mean to drop all the time derivatives and the Hubble parameter and hence the time
derivatives acting on δ cannot be ignored in general. In the case of the GLPV theory, we will also have terms
like ∇2 Ψ̇ in the perturbation equations, which must be retained as well.
The crucial point in the perturbative expansion is that, in the Horndeski and GLPV theories, the second
derivatives of perturbations can be large on small scales even though the first and zeroth derivatives are small, sothat the terms nonlinear in the second derivatives cannot be neglected. This is the very reason why the Vainshtein
screening mechanism (partially) works. This is also the key nonlinearity for the matter bispectrum.
Noting that the matter Lagrangian can be written as Lm = −Φρ̄m δ, we have the following eﬀective action
governing the perturbation evolution in the quasi-static regime [80]:∫[]Seﬀ = dtd3 x a3 L(2) + L(NL) ,

(6.6)

where
∇2 Ψ
∇2 Φ
L(2) = −M 2 (1 + αT )Ψ 2 + 2M 2 (1 + αH )Ψ 2
a
a
(
[
)
Ḣ
3Ωm
Ḣ
+
+ 1 + αM + 2 (αB − αH )
− M2
H2
2
H
]
α̇B − α̇H
∇2 Q+
+ (αT − αM ) Q 2
H
a
2
∇ Q
− 2M 2 (αB − αH )Φ 2
a
[
]
α̇H
∇2 Q
2
+ 2M αH (1 + αM ) + αM − αT +Ψ 2Ha− ρ̄m Φδ + 2M 2 αHΨ̇ ∇2 Q
,
H a2

(6.7)

and
L(NL) =[]M2
α̇G + α̇H L3
α
−
3(α
−
α
)
+
4α
−
α
(2
+
α
+
α
)
−
G
H
T
B
M
G
H
2H 2
H
a4
M2
Q(2)
M2
Q(2)
2M 2
∇i Ψ∇j Q∇i ∇j Q
(αG − αH )Φ 4 +αT Ψ 4 −
αH
2
2
2
2H
a
2H
a
H
a4
2
M
L4
+
(αG − αH + αT ) 6 ,
4
2H
a

+

(6.8)

with
1
L3 = − (∇Q)2 ∇2 Q,
2
1
L4 = − (∇Q)2 Q(2) ,2
Q(2) = (∇2 Q)2 − (∇i ∇j Q)2 .

(6.9)(6.10)(6.11)

Chapter 6 Matter bispectrum beyond Horndeski theories

52

The time-dependent parameters in the coeﬃcients are defined by
M 2 = 2(G4 − 2XG4X − 2X 2 F4 ),d ln M 2,
dt
HM 2 αB = −ϕ̇(XG3X − G4ϕ − 2XG4ϕX ) − 4HX
αM = H −1

× (G4X + 2XG4XX + 4XF4 + 2X 2 F4X ),M 2 αT = 4X(G4X + XF4 ),2

2(6.12)(6.13)(6.14)(6.15)(6.16)M αH = 4X F4 ,
and
Ωm :=

ρ̄m
,
3M 2 H 2(6.17)

which were introduced and used in Refs. [107, 144, 145, 146]. (we follow the convention of Ref. [107].) We havedefined another useful parameter as
M 2 αG = 4X(G4X + 2XG4XX + 4XF4 + 2X 2 F4X ),(6.18)

which first appears in the cubic order action.
The physical meanings of those parameters are as follows: M is the eﬀective Planck mass, αM is its evolution
rate, αB is the braiding parameter that characterizes the kinetic mixing of the scalar field and the metric, andαT parameterizes the deviation of the speed of GWs from that of light. The αH parameter signals novel eﬀects
compared to the Horndeski theory. The last term in Eq. (6.7) and the last term in the second line in Eq. (6.8),
which generate third-order derivatives in the equations of motion, are proportional solely to this parameter andhence appear for the first time in the GLPV theory. Note that Ωm cannot always be interpreted as the familiar
density parameter, because the Friedmann equation is modified and we do not necessarily have the equation of
the form 3M 2 H 2 = ρ̄m + the energy density of the scalar field. This is related to the fact that the distinction
between the geometry (the “left hand side” of the gravitational field equations) and the energy-momentum tensor
is ambiguous in the presence of nonminimal coupling.
If all the α parameters vanish and M = MPl (the Planck mass), the nonlinear part of the Lagrangian, L(NL) ,vanishes and the quadratic Lagrangian L(2) reduces to the standard expression in GR. In view of this, we assume
that
αM , αB , αT , αH , αG ≪ 1,

(6.19)

in the early stage of the matter-dominant universe, so that standard cosmology is recovered. In the late-time
universe, however, the eﬀect of modification of gravity emerges, which is assumed to be responsible for theaccelerated expansion. In this stage we assume O(1) modification from GR, i.e.,αM , αB , αT , αH , αG = O(1).

(6.20)

This is equivalent to assuming that−1 −2
2ϕ̇ ∼ MPl H0 , G2 ∼ MPl
H02 , G3X ∼ MPl
H0 ,
−2 −4
2
G4 ∼ MPl, F4 ∼ MPl
H0 , · · ·in the late-time universe, where the Hubble parameter is roughly given by its present value, H0 .

(6.21)

Chapter 6 Matter bispectrum beyond Horndeski theories

53

6.1.3 Field equations in Fourier space
Now we move to the field equations that can be derived by varying the eﬀective action (6.6) with respect to Ψ, Φ,and Q. They are given, in Fourier space,*3 by[]
Q̇(t, p)B1
− p FT Ψ(t, p) − GT Φ(t, p) − A3 Q(t, p) + M αH
= 2 2 Γ[t, p; Q, Q]
H
2a H
∫
2
M αH 1
d3 k1 d3 k2 δ (3) (k1 + k2 − p)k12 k22 β(k1 , k2 )Q(t, k1 )Q(t, k2 ), (6.22)+ 2 2
a H (2π)3a2
B2− p2 [GT Ψ(t, p) + A2 Q(t, p)] − ρ̄m δ(t, p) = − 2 2 Γ[t, p; Q, Q],(6.23)
2
2a H
]
[
B0
B1
Ψ̇(t, p)= − 2 2 Γ[t, p; Q, Q] + 2 2 Γ[t, p; Q, Ψ]− p2 A0 Q(t, p) − A1 Ψ(t, p) − A2 Φ(t, p) − M 2 αH
H
a H
a H
2

2

B2
Γ[t, p; Q, Φ]
a2 H 2∫M 2 αH 1− 2 2
d3 k1 d3 k2 δ (3) (k1 + k2 − p)k12 k22 α(k1 , k2 )Q(t, k1 )Ψ(t, k2 )a H (2π)3
∫
C0
1
+ 4 4
d3 k1 d3 k2 d3 k3 δ (3) (k1 + k2 + k3 − p)a H (2π)6+× [−k12 k22 k32 + 3k12 (k2 · k3 )2 − 2(k1 · k2 )(k2 · k3 )(k3 · k1 )]Q(t, k1 )Q(t, k2 )Q(t, k3 ),(6.24)
where for Y, Z = Ψ, Φ, Q we defined1
Γ[t, p; Y, Z] =(2π)3∫d3 k1 d3 k2 δ (3) (k1 + k2 − p)k12 k22 γ(k1 · k2 )Y (t, k1 )Z(t, k2 ),(6.25)

and we introduced(k1 · k2 ),k22
(k1 · k2 )|k1 + k2 |2β(k1 , k2 ) =,2k12 k22(k1 · k2 )2γ(k1 , k2 ) = 1 −.
k12 k22α(k1 , k2 ) = 1 +(6.26)(6.27)(6.28)

The coeﬃcients FT , GT , A1 , A2 , · · · all have the dimension of (mass)2 and are written in terms of M 2 and theα parameters as presented explicitly in Appendix B. One finds that there are four terms proportional to αH in
Eqs. (6.22)–(6.24) (the fourth term in the left hand side of Eq. (6.22), the second term in the right hand sideof Eq. (6.22), the fourth term in the left hand side of Eq. (6.24), and the fourth term in the right hand side ofEq. (6.24)). Those are the new terms beyond Horndeski. The other coeﬃcients contain αH , but they are not
new in the sense that even in the case of αH = 0 those coeﬃcients do not vanish and just reduce to the known
expressions in the Horndeski theory [141].

6.1.4 Fluid equations
Since it is assumed that matter is minimally coupled to gravity, the fluid equations are the same as the usual ones.
Under the quasi-static approximation, the conservation and Euler equations for nonrelativistic matter expressed*3 Our convention for the Fourier transform isf (t, x) =

1
(2π)3

∫
d3 p f (t, p)eip·x .

Chapter 6 Matter bispectrum beyond Horndeski theories

54

in terms of the density contrast δ and the velocity field ui are given by1δ̇ + ∇i [(1 + δ)ui ] = 0,a11u̇i + Hui + uj ∇j ui = − ∇i Φ.
aa(6.29)(6.30)

Modification of gravity comes into play in the evolution of matter density perturbations through the gravitational
potential Φ in Eq. (6.30), which is determined by Eqs. (6.22), (6.23), and (6.24). Going to Fourier space,
Eqs. (6.29) and (6.30) are written as∫δ̇(t, p)1
+ θ(t, p) = −d3 k1 d3 k2 δ (3) (k1 + k2 − p)α(k1 , k2 )θ(t, k1 )δ(t, k2 ),
H
(2π)3
(
)
θ̇(t, p)Ḣ
p2
+ 2 + 2 θ(t, p) − 2 2 Φ(t, p)
H
H
a H
∫
1
=−
d3 k1 d3 k2 δ (3) (k1 + k2 − p)β(k1 , k2 )θ(t, k1 )θ(t, k2 ),(2π)3(6.31)(6.32)

where we introduced a scalar function defined as θ = ∇i ui /aH.

6.2 Matter density perturbations in GLPV theory
Based on the set of the equations obtained in the previous section, here, we derive the bispectrum of the matter
density perturbations, δ, and highlight the impact of the new operators in the GLPV theory as the simplest
extension of the Horndeski theory . In order to investigate the matter bispectrum at the tree level, we needto consider the perturbations up to second order under the assumption that the perturbations initially obey
Gaussian statistics. Before deriving the matter bispectrum from the second-order perturbations, let us begin
with giving a linear evolution equation for the matter density perturbations.

6.2.1 Linear perturbations
Since we are considering the minimally-coupled matter there is not any modification in the continuity and Euler
equations even in modified theories of gravity such as the GLPV theory. Thus, the linear evolution equation forthe matter density perturbations in Fourier space is given by the standard one as Eq. (5.94). The modificationof gravity is encoded in Φ that is determined from the modified Poisson equation.
As we have shown in Sec. 5.4, we truncate Eqs. (6.22), (6.23), and (6.24) at the linear order and solve themfor Φ, Ψ, and Q, we obtain the modified Poisson equation. Even under the quasi-static approximation, thoseequations contain Ψ̇ and Q̇. Thanks to the degeneracy of the system, it can be straightforward to express Φ (andthe other two variables) in terms of δ and its derivatives. The final result one thus arrives at is:δ̇p2− 2 2 Q = κQ δ + νQ ,
a H
H
p2
δ̇
− 2 2 Ψ = κΨ δ + νΨ ,
a H
H
p2
δ̇
δ̈
− 2 2 Φ = κΦ δ + νΦ + µΦ 2 ,a H
HH(6.33)(6.34)(6.35)The explicit forms of the coeﬃcients are given in Appendix B. The diﬀerence from that in the result of DHOST
theories is the absence of µΨ (see Eq. (5.97)). Equation (6.35) allows us to eliminate Φ from Eq. (5.94), leaving
a closed-form, second-order evolution equation for δ is given by the same equation, Eq. (5.98). Therefore, we canalso obtain the same growing solution to the equation as Eq. (5.101).

Chapter 6 Matter bispectrum beyond Horndeski theories

55

Using Eq. (5.94), one can eliminate δ̈ from Eq. (6.35). Then, replacing δ̇ with f Hδ, we can rewrite Eqs. (6.33)–(6.35) asp2− 2 2 Q = (κQ + f νQ ) δ =: KQ δ,
a H
p2− 2 2 Ψ = (κΨ + f νΨ ) δ =: KΨ δ,
a H
(
)
p2
3
− 2 2Φ =Ωm ΞΦ − ςf δ =: KΦ δ,
a H
2(6.36)(6.37)(6.38)

where we recall that the definitions of ΞΦ , ς, and f are given in Eqs. (5.99), (5.100), and (5.63) respectively.
These equations are convenient for the second-order analysis in the next subsection.

6.2.2 Second-order perturbations
To investigate the bispectrum of δ at the tree level, we need to solve the perturbation equations up to second
order. Let us now move to the second-order analysis of the matter density perturbations based on the equations
derived in the previous section. Substituting the first-order solutions (6.36)–(6.38) to the right hand sides ofEqs. (6.22)–(6.24), we obtain, up to second order in δ,
)
(
2 2
Q̇
B1 2
2 a H
2
2
FT Ψ − GT Φ − A3 Q + M αH
= −D+ 2K Wγ (p) ,(6.39)M αH KQ Wβ (p) +
H
p
2 Q
2 2
a2
2 a H B2GT Ψ + A2 Q + 2 ρm δ = D+K 2 Wγ (p),(6.40)
2p
p2 2 Q
2 2 [
(
)
]
Ψ̇
2 a H
2
A0 Q − A1 Ψ − A2 Φ − M 2 αH= D+
M 2 αH KQ KΨ Wα (p) + B0 KQ− B1 KΨ KQ − B2 KΦ KQ Wγ (p) ,2H
p(6.41)
2

where Wα (p) := I[p; αs (k1 , k2 )], Wβ (p) := I[p; β(k1 , k2 )], and Wγ (p) := I[p; γ(k1 , k2 )], with∫
1
I[p; Y (k1 , k2 )] :=d3 k1 d3 k2 δ (3) (k1 + k2 − p) Y (k1 , k2 )δL (k1 )δL (k2 ).
(2π)3(6.42)

Here we introduced a symmetrized version of α(k1 , k2 ) as
αs (k1 , k2 ) = 1 +(k1 · k2 )(k12 + k22 ).
2k12 k22(6.43)

Note that we have the following relation: Wβ (p) = Wα (p) − Wγ (p). The functions Wα , Wβ , and Wγ are
dependent on the initial density field δL (k), but not on modification of gravity.
From the nonlinear fluid equations (6.31) and (6.32) with the analysis of the linear perturbations in 6.2.1, we
can obtain the following equation up to the second order in δL :
δ̈ + 2H δ̇ +

(
)
p2
2 2
Φ
=
H
D
S̃
W
−
S̃
W
,
α
α
γ
γ
+
a2

(6.44)

where3S̃α = 2f 2 + Ωm ΞΦ − ς f,2S̃γ = f 2 .

(6.45)(6.46)The second-order nonlinearity due to the modification of gravity, which appears in the right hand sides ofEqs. (6.39)–(6.41), is introduced through the gravitational potential Φ as follows. Repeating the same procedureChapter 6 Matter bispectrum beyond Horndeski theories

56

as in the linear analysis, we can write Q, Φ, and Ψ in terms of δ, its first and second derivatives, and thesecond-order terms in the right hand sides of Eqs. (6.39)–(6.41) asp2
δ̇2− 2 2 Q = κQ δ + νQ + D+
(τQα Wα − τQγ Wγ ) ,
a H
H
p2
δ̇
2
− 2 2 Ψ = κΨ δ + νΨ + D+(τΨα Wα − τΨγ Wγ ) ,
a H
H
p2
δ̇
δ̈
2
− 2 2 Φ = κΦ δ + νΦ + µΦ 2 + D+(τΦα Wα − τΦγ Wγ ) ,a H
HH(6.47)(6.48)(6.49)

where)
M 2 αH (2
A2 GT KQ
+ GT2 KQ KΨ ,
Z
{[
(
(
))
]
B
B
2
Ḣ
1
1
2
2
B0 GT2 +A2 G T +T + 3M 2 αH GT 1 +τQγ =− M 2 αH A2 GT KQ
Z
2
2
3 H2
}
2
2 ·
B2 KQ
)
M 2 αH GT (D+22−B1 GT KΨ KQ − B2 GT KΦ KQ +
,
2H
2
D+

τQα = −)
M 2 αH ( 2 2A2 KQ + A2 GT KQ KΨ ,
Z
2
B2 KQ
− 2A2 τQγ
τΨγ =
2GT
{[
(
(
))
]
1
B1 2 B2
2 Ḣ
2
2
2
2
=−
B0 A2 GT +A +−S + 3M αH A2 1 +− M αH A2 KQ
Z
2 2
2
3 H2
}
2 ·
2
)
B2 K QM 2 αH A2 (D+−B1 A2 GT KΨ KQ − B2 A2 GT KΦ KQ +
,
2H
2
D+
)]
}
{
[
(
1
Ḣ
τ̇Qα
2
2
2
2
τΦα =
τQα + M αH
− M αH KQ ,FT τΨα − A3 − 2M αH 1 + f + 2
GT
H
H
)]
{
[
(
}
(
)
1
τ̇
B
Ḣ
Qγ
1
2
τΦγ =
τQγ + M 2 αH−
− M 2 αH KQ
FT τΨγ − A3 − 2M 2 αH 1 + f + 2
.
GT
H
H
2

τΨα =

(6.50)(6.51)(6.52)(6.53)(6.54)(6.55)We then eliminate Φ from Eq. (6.44) and obtain the evolution equation for δ capturing the eﬀect of the secondorder nonlinearity of the scalar field:
32
H 2 (Sα Wα − Sγ Wγ ) .
δ̈ + (2 + ς)H δ̇ − Ωm ΞΦ H 2 δ = D+2

(6.56)In the right hand side we defined Sα and Sγ by(1 − µΦ ) Sα (t) := S̃α + τΦα ,(6.57)(1 − µΦ ) Sγ (t) := S̃γ + τΦγ ,(6.58)The second-order nonlinearity due to modification of gravity appears in all of these τ coeﬃcients, but it shouldbe emphasized that τΦα = 0 for αH = 0 (i.e., in the Horndeski theory ), while τΦγ ̸= 0 in general if gravity is
modified anyway (see Table 6.1). In other words, τΦα is a new term beyond Horndeski. The solution to Eq. (6.56)
up to second order in δL can be written as2
δ(t, p) = D+ (t)δL (p) + D+
(t)

1
(2π)3

∫
d3 k1 d3 k2 δ (3) (k1 + k2 − p)F2 (t, k1 , k2 )δL (k1 )δL (k2 ),(6.59)

with the second-order kernel defined as
2F2 (t, k1 , k2 ) := κ(t)αs (k1 , k2 ) − λ(t)γ(k1 , k2 ),7(6.60)

Chapter 6 Matter bispectrum beyond Horndeski theories

57

where κ(t) and λ(t) are the solutions of the following second-order diﬀerential equations,
)
(
3
2
2
κ̈ + [4f + (2 + ς)]H κ̇ + H 2f + Ωm ΞΦ κ = H 2 Sα ,
2
)
(
3
7
2
2
λ̈ + [4f + (2 + ς)]H λ̇ + H 2f + Ωm ΞΦ λ = H 2 Sγ ,22(6.61)(6.62)

supplemented with the condition that κ, λ → 1 in the early time (it is easy to check that κ = λ = 1 indeed solves
Eqs. (6.61) and (6.62) if all the α parameters are negligibly small and Ωm = 1). In the Horndeski limit (αH = 0),
these expressions reproduce the result of Ref. [141]. Especially, since ς = 0 and τΦα = 0 in the Horndeski theory()(see Table 6.1), the right hand side of Eq. (6.61) reduces to H 2 2f 2 + 3Ωm ΞΦ /2 , so that κ(t) = 1 at any time.
In this case the second-order kernel (6.60) therefore depends only on λ(t) [141]. Thus, we find that a new featurein the GLPV theory beyond Horndeski is the κ term that is diﬀerent from 1 and is time-dependent in general.
This is the main result of this chapter.
ΛCDM

Horndeski

beyond

ς

0

0

✓

ΞΦ

1

✓

✓

µΦ

0

0

✓

τΦα

0

0

✓

τΦγ

0

✓

✓

Table 6.1 Summary of the parameters in the second-order evolution equation for δ, (6.56) with Eqs. (6.57) and (6.58).

6.3 Matter bispectrum

Fig. 6.1 (color online) The reduced bispectrum as a function of θ12 , with fixed k1 and k2 . We adopt theisosceles triangular configuration with k1 = k2 = 0.01h/Mpc in the left panel (a), and the distorted triangle
with k1 = 5k2 = 0.05h/Mpc in the right panel (b). In both panels, we take a diﬀerent value for κ(t) to be
1.0 (gray solid line), 0.9 (blue dashed line), and 1.1 (orange dashed line), while λ(t) is fixed to be 1.

Finally, let us investigate the matter bispectrum as an observable for probing such quasi-nonlinear evolution
based on the above analysis for the matter density perturbations up to second order. The power spectrum andChapter 6 Matter bispectrum beyond Horndeski theories

58

Fig. 6.2 (color online) The reduced bispectrum as a function of θ12 , with fixed k1 and k2 . We adopt theisosceles triangular configuration with k1 = k2 = 0.01h/Mpc in the left panel (a), and the distorted triangle
with k1 = 5k2 = 0.05h/Mpc in the right panel (b). In both panels, we take a diﬀerent value for λ(t) to be
1.0 (gray solid line), 0.9 (blue dashed line), and 1.1 (orange dashed line), while κ(t) is fixed to be 1.

the bispectrum of the matter density perturbations are respectively defined by⟨δ(t, k1 )δ(t, k2 )⟩ =: (2π)3 δ (3) (k1 + k2 )P (t, k1 ),(6.63)⟨δ(t, k1 )δ(t, k2 )δ(t, k3 )⟩ =: (2π)3 δ (3) (k1 + k2 + k3 )× B(t, k1 , k2 , k3 ).

(6.64)

Here, for simplicity we assume that the initial density field δL obeys Gaussian statistics, and, by making use of
the expression (6.59), the matter bispectrum at the tree-level can be evaluated as−4D+(t)B(t, k1 , k2 , k3 )= 2[F2 (t, k1 , k2 )P11 (k1 )P11 (k2 ) + 2 cyclic terms],(6.65)

where P11 represents the power spectrum of the initial density field defined by⟨δL (k1 )δL (k2 )⟩ =: (2π)3 δ (3) (k1 + k2 )P11 (k1 ).

(6.66)

As we have mentioned before, we assume that standard cosmology is recovered in the early stage of the matterdominant universe. Thus, here, we calculate P11 (k) adopting the best fit cosmological parameters taken from
Planck data [147].
As usual, in order to investigate the shape of the bispectrum in Fourier space, let us introduce a reduced
bispectrum which is defined by
B(t, k1 , k2 , k3 )4 (t)[P (k )P (k ) + 2 cyclic terms] .
D+
11 1
11 2

(6.67)

2[F2 (t, k1 , k2 )P11 (k1 )P11 (k2 ) + 2 cyclic terms].
[P11 (k1 )P11 (k2 ) + 2 cyclic terms](6.68)

Q123 (t, k1 , k2 , k3 ) :=From Eq. (6.65), we have
Q123 (t, k1 , k2 , k3 ) =Thus, the reduced bispectrum does not depend on the linear growth function D+ and the eﬀect of modificationof gravity is encoded in the second-order kernel, F2 (t, k1 , k2 ). As we have discussed, the characteristic feature of
the GLPV theory beyond Horndeski with αH ̸= 0 is that κ in the second-order kernel is diﬀerent from 1 and istime-dependent in general.

Chapter 6 Matter bispectrum beyond Horndeski theories

59

To demonstrate how this new feature beyond Horndeski distorts the shape of the bispectrum, we plot in
Figs. 6.1(a) and 6.1(b) the reduced bispectrum as a function of θ12 which is the angle between k1 and k2 , with k1
and k2 fixed. In these figures, we take diﬀerent values of κ as κ = 1.0 (gray solid line), 0.9 (blue dashed line), and
1.1 (orange dashed line) while we fix λ = 1.0. As one can see, except for the squeezed configurations (θ12 → π in
Fig. 6.1(a)), the reduced bispectrum becomes larger for κ > 1 and smaller for κ < 1.
As a comparison, we show in Figs. 6.2(a) and 6.2(b) the reduced bispectrum for diﬀerent values of λ. In thesefigures, κ is fixed to be 1 which corresponds to the case with αH = 0. Compared with Figs. 6.1(a) and 6.1(b), one
finds that the deviation of λ from unity would give a large eﬀect on the reduced bispectrum only for θ12 ≃ 2π/3
in the left panel and θ12 ≃ π/2 in the right panel. Thus, the eﬀect of the GLPV theory on the matter bispectrum
is significant for θ12 → 0. In other words, the matter bispectrum with θ12 = 0 is considered to be a powerfulprobe of the GLPV theory beyond Horndeski.

6.4 GWs constatints
The gravitational wave event GW170817 [52] and its optical counterpart GRB 170817A [53] placed a tightconstraint on the propagation speed of GWs, |cGW − 1| < O(10−15 ). The consequences of this constraint on the
Galileon theory, the Horndeski theory, and its extensions have been discussed in Refs. [73, 81, 82, 83, 84, 98, 99,
100, 101, 102, 148, 149, 150, 151, 152, 153, 154].*2 In terms of the functions in the action, the constraint reads|αT | < O(10−15 )⇒G4X + XF4 ≃ 0.

(6.69)

This must hold at least in the late-time universe. Upon imposing αT = 0, we have αH = αG , while αM , αB , andαH itself are still allowed to be O(0.1) − O(1) [99]. A further constraint can be obtained from the Hulse-Taylor
pulsar under the additional assumption that the scalar radiation does not take part in the energy loss, whichleads to |αH | < O(10−3 ) [83]. This implies that O(10−3 )–O(1) deviation of κ from its Horndeski value (κ = 1) is
still possible, depending on the assumption one makes. This fact also is true in DHOST theories [164]. As shownin Sec. 3.5, generic quadratic DHOST theories have the catastrophic decay channel of GWs into dark energy
field. In GLPV theories, the decay rate in this channel is proportional to αH [164]. Thus, GLPV theories needs
to be generalized to DHOST theories in order to evade this catastrophic decay.

6.5 Discussion and Summary
In this chapter, we have studied the matter bispectrum of large scale structure as a probe of the so-called“beyond Horndeski” theory or the GLPV theory of modified gravity. We focused on the nonlinearity generated
from derivative interactions of the metric perturbations and the scalar degree of freedom and derived a secondorder solution of the matter density perturbations δ(t, k). We have shown that a new, time-dependent coeﬃcientκ appears in the second-order kernel in the GLPV theory. Since we have κ := 1 in GR and even in the Horndeski
theory [141], this is certainly a characteristic feature of the theory beyond Horndeski. Based on this secondorder solution, we have evaluated the matter bispectrum and found that the eﬀect of nonstandard values of κ
can be seen in the bispectrum at the folded configurations (k1 + k2 = k3 ). We thus conclude that a deformed
matter bispectrum at the folded configurations can be a unique probe of “beyond Horndeski” operators. Notethat there exist several scenarios where the primordial curvature perturbations would acquire the folded-type
non-Gaussianity during inflation (see, e.g., Ref. [157]) and such a type of primordial non-Gaussianity could also
deform the matter bispectrum at the folded configurations. However, if we can precisely measure not only the*2 See Refs. [95, 96, 97, 121, 155, 156] for earlier works before this event on the prospects of measuring cT

Chapter 6 Matter bispectrum beyond Horndeski theories

60

dependence of θ12 but also the scale dependence of the matter bispectrum, it would help us discriminate thesignature of beyond Horndeski from such a folded-type primordial non-Gaussianity.
It is not suﬃcient to detect the shape dependence of matter bispectrum in current observations (for example,see [158]). In Refs. [159, 160, 161], combining matter power spectrum and bispectrum, these authors havediscussed the improvements of cosmological parameter fitting in galaxy surveys. In Ref. [162], they estimate theconstraint for the growth index γ and λ within Horndeski theories from the future galaxy surveys, SKA andEuclid surveys. Using the same analysis, we might estimate the constraint for λ and κ for DHOST theories
from future galaxy surveys. Alternatively, the possibility to detect the signals of λ and κ is discussed from CMB
lensing [163].
In light of the recent GWs constraints, there is a growing interest in the so-called DHOST theories which
are more general than the one considered in this chapter but evade the stringent constraints. In Ref. [164], theauthors study matter bispectrum in DHOST theories. That feature is the same as that in GLPV theories. We
have caught the typical feature of matter bispectrum beyond Horndeski theories.

61

Chapter 7

Conclusions
In this thesis, we have studied the properties of Degenerate Higher-Order Scalar-Tensor (DHOST) theories on
small and large scales toward its tests as alternatives to dark energy.
In Ch. 2, we overviewed the late-time acceleration based on General Relativity (GR). We introduced modified
gravity as an interesting one of the possibilities to explain the late-time acceleration which is consistent with ouruniverse.
In Ch. 3, we overviewed DHOST theories and introduced its viable classes evading gravitational wave constraints.
In Ch. 4, we have studied the screening mechanism in a particular subclass of DHOST theories in which the
speed of Gravitational Waves (GWs) is equal to the speed of light and gravitons do not decay into scalar fluctuations. By inspecting a spherically symmetric gravitational field, we have found that the screening mechanism
operates in a very diﬀerent way from that in generic DHOST theories [80, 81, 82, 83]. First, the fine-tuning is required so that solar-system tests are evaded in the vacuum exterior region. This is in contrast to generic DHOST
theories, in which the implementation of the Vainshtein screening mechanism outside the matter distribution israther automatic. Second, the way of the Vainshtein breaking inside extended objects is also diﬀerent from thatin generic DHOST theories. We have shown that in the interior region the metric potentials obey the standard
inverse power law, but the two do not coincide. Moreover, the eﬀective gravitational constant diﬀers from its
exterior value. However, the current most stringent bound comes from the fact that the eﬀective gravitational
coupling for GWs is diﬀerent from the Newtonian constant [83, 121], rather than from the above interesting
phenomenology. We conclude that the allowed parameter space is small for DHOST theories as alternatives todark energy evading gravitational wave constraints.
In Ch. 5, we have considered a possibility to constrain DHOST theories by using the information about thelinear growth of matter density fluctuations. In DHOST theories, the evolution equation for the linear matter
density fluctuations is modified in such a way that the eﬀective gravitational coupling is changed and the friction
term has an additional contribution. We have constructed cosmological models in DHOST theories as a series
expansion in terms of 1 − Ωm . In doing so, we have assumed for simplicity that cosmological solutions underconsideration are attractors in shift-symmetric theories and subject to the tracker ansatz. Our construction
provides a concise description of DHOST cosmology during the matter-dominated era and the early stage ofthe dark energy dominated era. We have then explicitly expressed the gravitational growth index γ in terms of
the eﬀective parameters in DHOST theories. One can thus obtain constraints on a certain combination of theeﬀective parameters at high redshifts by using the observations of the growth index alone. Under the additional
assumption that our leading-order results in 1−Ωm expansion can be extrapolated all the way to the present time,
we have compared the constraints from the growth index with the previously known bounds. Combining our
results and the constraints from modifications of the gravitational law inside stellar objects, we have shown thatthe parameter degeneracy between αH /(1 − Ωm ) and β1 /(1 − Ωm ) could be broken without using the Hulse-Taylor

Chapter 7 Conclusions

62

pulsar constraint, though our results slightly depend on the model parameters. Future-planned observations forlarge scale structure would exclude the currently allowed region of the parameter space and serve as tests of theviability of DHOST theories.
In Ch. 6, we have studied the matter bispectrum of large scale structure as a probe of the GLPV theory ofmodified gravity. The GLPV theory is the simplest extension of the Horndeski theory in DHOST theories. We
focused on the nonlinearity generated from derivative interactions of the metric perturbations and the scalardegree of freedom and derived a second-order solution of the matter density perturbations. We have shown thata new time-dependent coeﬃcient κ appears in the second-order kernel in the GLPV theory. Since we have κ := 1
in GR and even in the Horndeski theory [141], this is certainly a characteristic feature of the theory beyond
Horndeski. Based on this second-order solution, we have evaluated the matter bispectrum and found that theeﬀect of nonstandard values of κ can be seen in the bispectrum at the folded configurations. We thus conclude
that a deformed matter bispectrum at the folded configurations can be a unique probe of “beyond Horndeski”
operators. In light of the recent GWs constraints, there is a growing interest in DHOST theories which are moregeneral than the one considered in this chapter but evade the stringent constraints. In Ref. [164], the authors
study matter bispectrum in DHOST theories. That feature is the same as that in GLPV theories. We havecaught the typical feature of matter bispectrum beyond Horndeski theories.
As future directions, we had better investigate the one-loop matter power spectrum in DHOST theories.
Because the scale dependence in the amplitude of matter bispectrum appears, we expect that the higher-ordercorrections to the power spectrum are larger than that of Horndeski theories of GR. In future observations oflarge scale structure, the detectability of higher-order correlation functions will be improved. In this sense, theanalysis of the one-loop power spectrum in DHOST theories would be motivated.
In this thesis, we focus only on the density fluctuations deeply inside the horizon. Future surveys, such as
the SKA project, could detect the correlations near the cosmological horizon. At the horizon scale, (general)
relativistic corrections appear, for example, the time derivative of gravitational potentials. Then, the galaxy
number count can be aﬀected. In scalar-tensor theories, one assumes that quasi-static approximation is valid
on the evolution of the scalar field, and the time derivatives of the scalar field are negligible deeply inside thehorizon. At the horizon scale, the time derivatives of the scalar field cannot be negligible, so the eﬀect of the scalar
field could appear on observables such as the galaxy number count. We would like to formulate the (general)
relativistic eﬀect of the scalar field in Horndeski theories or DHOST theories based on Ref. [165]. Also, we would
analyze the scale dependence of bias parameters between dark matter and baryon fluctuations due to non-linear
scalar self-interactions based on Ref. [166].

63

Acknowledgement
First of all, I would like to express the appreciation to Prof. Tsutomu Kobayashi for fruitful discussions, proper
comments and collaborations. I have obtained many things as a researcher. I am grateful to my collaborators
Katsuki Aoki, Shingo Akama, Sakine Nishi, Hiroyuki Tashiro, Junsei Tokuda, Daisuke Yamauchi, and Shuichiro
Yokoyama. The examiners, Tomohiro Harada, Shingo Kameda, and Tsutomu Kobayashi give me a lot of usefulcomments and suggestions. I would like to thank all the colleagues in the department of physics in Rikkyo university, in particular Yasuyuki Hatsuda, Takashi Hiramatsu, Takahisa Igata, Kumiko Inagawa, Masashi Kimura, Tomotaka Kitamura, Takafumi Kokubu, Yu Nakayama, and Hidekazu Tanaka. Shinji Mukohyama, Atsushi Taruya,
and Kazuhiro Yamamoto give me great hospitality in my visit of their university . There are many thanks tomy friends and ex-colleagues Yuji Akita, Kazuyuki Akitsu, Shun Arai, Toru Eguchi, Tomohiro Fujita, Souichiro
Hashiba, Keisuke Inomata, Asuka Itoh, Rampei Kimura, Akira Matsumura, Taisaku Mori, Tomohiro Nakamura,
Shintaro Nakamura, Atsushi Naruko, Kota Ogasawara, Hiromu Ogawa, Seiga Sato, Kazufumi Takahashi, Takuma
Tsukamoto, Koji Yajima, Masashi Yamazaki, Daiske Yoshida, and Daisuke Yoshida for conversations, discussion,and drinking. I sincerely thank my family, Susumu, Masako, and Katsuhiko. S.H. was supported by JSPS Young
Scientists Research fellowship.

64

Appendix A

Explicit expressions for some coeﬃcients in
Chapter 5
Let us write down explicitly the coeﬃcients in Eqs. (5.95)–(5.97). The coeﬃcients in Eq. (5.95) are given by
3Ωm
NΦ ,
2Z {
(
)]
(
)
3Ωm [
κQ =
c1 + 2 2β1 + β3 FΦ + c2 − 4αH FΨ
8Z
[
(
)
(
)]}
) d FΦ
M2 (d FΨ
−2 2β1 + β3
−
4α
,
H
H
dt M 2
dt M 2
νQ =

(A.1)

(A.2)

where we have defined the some dimensionless parameters as
(
) 1(
)(
)
SNΦ = αH 1 + αH − 1 + αT 2β1 + β3 ,2SFΦ = 1 + αT , SFΨ = 1 + αH ,)()2 1 (
S = 1 + αH − 1 + αT β3 ,2The denominator Z can be written as{[]}1
2(2β1 + β3 )4αHZ=EΦ c1 + EΨ c2 − b3 +
ĖΦ −
ĖΨ
4
H
H
where c1 , c2 , and b3 were defined in Eqs (5.81), (5.82), and (5.90). We have also defined()()
SEΦ = b1 1 + αT − b2 1 + αH ,() 1SEΨ = b1 1 + αH − b2 β3 .
2

(A.3)
(A.4)
(A.5)

(A.6)

(A.7)
(A.8)

The coeﬃcients in Eqs. (5.96) and (5.97) areµa = Na νQ ,[
νa = −Ea νQ + Na κQ +])
1 d ( 2aHν,Q
a2 H 2 dt
3
Na d ( 2 2 )κa = Ωm Fa − Ea κQ + 2 3
a H κQ ,2
a H dt(A.9)(A.10)(A.11)

for a = Ψ , Φ, where()
1SNΨ = − 1 + αH β1 − β3 .
2(A.12)

Appendix A Explicit expressions for some coeﬃcients in Chapter 5

65

The coeﬃcients in Eq. (5.119) are given by)32((cH + β) cM − 3w(0) ,
Z
(
)]2
2[
(1)
ΞΦ = cT +cB − cM + cT − β cM − 3w(0)
Z {
) (
)[
(
)]
cH + β (6 1 + w(0) + cM − cT 1 − 2 cM − 3w(0)
−
Z
}
[
)
(
)(
)][
(
)]
(
(0)
(0)
(0)
,
5 − 2 cM − 3w
+ 3 cH + β cM − 3w
+ cB − β cM − 3w

ς (1) =

(A.13)(A.14)

where
[
(
)
(
)][
(
)]
Z = 3 1 + w(0) + 2cM + 1 − 2 cM − 3w(0) cB − cH − β cM − 3w(0) + 1 .

(A.15)

We can then finally obtain the explicit expression γ in the main text by sustituting Eqs. (A.13) and (A.14) into
Eq. (5.122).

66

Appendix B

The coeﬃcients of Eqs. (6.22)–(6.24) andEqs. (6.33)–(6.35) in Chapter 6
The coeﬃcients of Eqs. (6.22)–(6.24) are given in terms of the α parameters by
FT = M 2 (1 + αT ), GT = M 2 (1 + αH ),
(
)
[
]
Ḣ
3Ωm
α̇B − α̇H
2 ḢA0 = M
+
1
+
α
+
+
(α
−
α
)
+
(α
−
α
)
+
M
T
M ,
B
H
H2
2
H2
H
[
]
α̇H
2
A1 = M αH (1 + αM ) + αM − αT +,H
A2 = −M 2 (αB − αH ),()
ḢA3 = M
αM − αT + 2 αH ,H[]2
α̇G + α̇HM
αG − 3(αH − αT ) + 4αB − αM (2 + αG + αH ) −,B0 = −
4
H
M2
M2
B1 =
αT , B2 =(αG − αH ),
2
2
M2
C0 =
(αG − αH + αT ).
4
2

(B.1)

The coeﬃcients of Eqs. (6.33)–(6.35) are given by3 2
M 2 αH GT
M Ωm
,
2
Z
3
T
κQ = M 2 Ω m ,2Z
3
M 2 αH A2,
ν Ψ = − M 2 Ωm
2Z
3SκΨ = M 2 Ωm ,2Z
M 2 αH
µΦ =
νQ ,
GT
{
1
νΦ =
FT νΨ − A3 νQ
GT
[
]}
)·
1 (
+ M 2 αH κQ + 2 2 a2 HνQ
,
a H
}
{
)·
1
M 2 αH (κΦ =.
FT κΨ − A3 κQ + 2 3 a2 H 2 κQ
GT
a H
νQ =

(B.2)
(B.3)
(B.4)
(B.5)
(B.6)

(B.7)
(B.8)

Appendix B The coeﬃcients of Eqs. (6.22)–(6.24) and Eqs. (6.33)–(6.35) in Chapter 6

67

with
)
ĠT
,
T := A2 FT + A1 GT − M αH GT +
H
(
)
Ȧ
2
S := A0 GT + A2 A3 + M 2 αH A2 +
,
H
(

2

M
Z := A0 GT2 + A2 (A1 + A3 )GT + A22 FT +(B.9)

(B.10)
2

αH 2
GT
H

(

A2
GT

)·
.

(B.11)

68

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