Violation of The Equivalence Principle

in Scalar-Tensor Theories of Gravity

Andrias Fajarudin

Dissertation submitted for

2009-2010 Diploma Course in High Energy Physics

The Abdus Salam

International Centre of Theoretical Physics

Strada Costiera 11, Miramare

34014 Trieste, Italy

Supervisor : Prof. Paolo Creminelli August, 2010

Violation of The Equivalence Principle in Scalar-Tensor Theories of Gravity

by Andrias Fajarudin

Diploma in High Energy Physics (2009-2010)

Supervisor : Prof.Paolo Creminelli

The Abdus Salam International Centre for Theoretical Physics

Strada Costiera 11, Miramare

34014 Trieste, Italy

To be defended on

August 18, 2010

Abstract

In this thesis I study the violation of the equivalence principle induced by two

screening mechanisms in scalar-tensor theories of gravity : the chameleon mech-

anism and the Vainshtein effect. In the chameleon mechanism, the scalar field

acquires a mass which depends on the environment density such that it will be

screened in a high density environments. I discuss the violation of the equivalence

principle both in Einstein and in Jordan frames. In Einstein frame, unscreened

objects will move with bigger acceleration compared to screened objects. In Jor-

dan frame, only unscreened objects feel the chameleon field whereas screened

objects don’t. Consequently, only unscreened objects move on geodesics. This

leads to order unity violation of the equivalence principle. In the Vainshtein

mechanism, the screening of the scalar field comes from derivative interaction

that become large in the vicinity of massive objects. Perturbations of scalar in

such regions acquire a large kinetic term and therefore decouple from matter.

Thus, the scalar screens itself and become invisible to experiments. Vainshtein

mechanism doesn’t lead to order unity violation of the equivalence principle. In

this screening mechanism, equivalence principle violation occurs at a much re-

duced level.

i

Contents

Abstract i

Contents iii

1 Introduction 1

2 The Problem of Motion in General Relativity 3

2.1 Geodesic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Energy-Momentum Conservation Method . . . . . . . . . . . . . . 4

3 The Problem of Motion in Scalar-Tensor Theories 9

3.1 Derivation in Einstein Frame . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Scalar Field without Self-Interaction Potential . . . . . . . 10

3.1.2 Scalar Field with Self-Interaction Potential . . . . . . . . . 13

3.2 Derivation in Jordan Frame . . . . . . . . . . . . . . . . . . . . . 19

3.3 Screening Mechanism in the DGP model by Vainshtein effect . . . 20

4 Conclusion 25

Acknowledgement 27

A Conformal Transformation 28

Bibliography 33

iii

Chapter 1

Introduction

Einstein’s theory of General Relativity has proven spectacularly successful over

90 years of experimental tests. These tests range from millimeter scale tests in

the laboratory to solar system tests and consistency with gravity wave emission

by binary pulsars.

Recently, several attempts to modify General Relativity on cosmological scales

have been made in order to explain something strange that is happening at very

large scales : the acceleration of the universe. Generally, there are two classes

of modified gravity theories. The first class is formed by theories which add

curvature invariants to the Einstein-Hilbert action, such as f(R) theories and

the second class are the theories which give the graviton a mass, such as Dvali-

Gabadadze-Porrati (DGP) braneworld model. All of these modified gravity theo-

ries introduce a light scalar which is only active on cosmological scales. On small

scales, the scalar field must be screened to avoid inconsistencies with solar system

and terrestrial experimental results. These two classes have two different screen-

ing mechanisms. The first class, screens the scalar by the chameleon mechanism.

In this mechanism the scalar acquires a mass that depends on the local density.

The second class, such as DGP, screens the scalar field by the Vainshtein effect

that suppress the scalar on small scales as consequence of derivative interactions.

In this thesis, I study the motion of extended objects in the presence of these two

different screening mechanisms. We will see that the two screening mechanisms

lead to different level of equivalence principle violation. Chameleon screening

could produce O(1) fluctuations in the scalar charge to mass ratio, even for non-

1

relativistic objects. In the theories where the Vainshtein mechanism operates,

there will be no O(1) charge renormalization. Equivalence principle violation in

these theories is of order 1/c2 depending on how relativistic the object’s internal

structure is.

2

Chapter 2

The Problem of Motion in

General Relativity

In this chapter we will discuss the motion of an extended object using energy-

momentum tensor conservation method.

2.1 Geodesic Motion

Usually, to describe the motion of a test particle in the vicinity of a gravitating

body, we use the geodesic assumption. The assumption is that particle do not

significantly affect the field and that it follows geodesics in the field of the gravi-

tating body.

Consider a test particle in the weak gravitational field of a gravitating body.

In the weak field limit we can linearize the theory such that : gµν = ηµν + hµν ,

where hµν ≪ 1. For a non-relativistic particle we have∣

∣

∣

dxi

dt

∣

∣

∣≪ c and

∣

∣

∣

dxi

dτ

∣

∣

∣≪

∣

∣

∣

dx0

dτ

∣

∣

∣

such that dτ ≈ dt. Then, the geodesic equation for this particle becomes :

d2xi

dt2+ Γi

00c2 +O(v) = 0, (2.1)

where the Christoffel symbol is given by Γi00 =

12gik(2∂0gk0−∂kg00)+ 1

2gi0(2∂0g00−

∂0g00). If we assume that the source is moving slowly,∣

∣

∂h..∂x0

∣

∣ ≪∣

∣

∂h..∂xk

∣

∣, we can

neglect the time derivative of the metric. Then, the Christoffel symbol becomes

Γi00 = −1

2∂ih00. The equation of motion for the test particle is given by

ai = −∂iΦ, (2.2)

3

where Φ = −12c2h00. In the Newtonian case, Φ = −GM

rwhich is the Newtonian

potential at the distance r from the source. Then, we have

h00 = −2Φ

c2. (2.3)

Eq.(2.2) describes that the motion of the test particle doesn’t depend on it mass.

Therefore, different test particles with different masses will be accelerated by the

same acceleration in the same environment.

However, treating all objects as a test particle may not be true in general. For

example, consider an extreme case which is the motion of a black hole. In this

case, we can’t treat the black hole like a test particle, because it has several

peculiar features such as singularity and event horizon which make the geodesic

assumption may not be valid anymore.

2.2 Energy-Momentum Conservation Method

In this section we will derive the equation of motion for an extended object using

the energy-momentum tensor conservation method. The idea is to calculate the

momentum flux through a surface (such as a sphere of radius r) enclosing the ex-

tended object. On the surface of the sphere we can assume that the gravitational

field is sufficiently weak such that we can linearize the metric gµν . The radius r

is chosen such that we can ignore the tidal effect from the background.

We can split the Einstein equation to be G(1)µν + G(2)

µν = 8πGTm

µν , where

G(1)µν is the Einstein tensor Gµ

ν at first order of metric perturbations and G(2)µν

contains all the higher terms. We can rewrite the Einstein equation as

G(1)µν = 8πGtµ

ν (2.4)

tµν is the pseudo energy-momentum tensor which is related to the energy mo-

mentum tensor of matter Tmµν by

tµν = Tm

µν − 1

8πGG(2)

µν . (2.5)

We know that Gµν satisfies ∇νGµ

ν = 0. In the first order of metric perturbation

this identity become ∂νG(1)

µν = 0 which implies that tµ

ν is conserved in the flat-

space sense, i.e. ∂νtµν = 0.

4

The linear momentum of the extended object in the ith direction is given by

Pi =

∫

d3x ti0 (2.6)

where we integrate over the whole volume of the sphere. Here, we make the

assumption that ti0 is dominated by the extended object it self with negligible

contribution from the background.

The gravitational force felt by the extended object is

Pi =

∫

d3x∂0ti0 = −

∫

d3∂jtij = −

∮

dSjtij (2.7)

where dSj = dAxj, dA is a surface area element and x is the unit outward normal.

Then the gravitational force is given by the integrated momentum flux through

the surface.

Now, if we take Tmij to be very small at the surface of the sphere, we can neglect

its contribution to tij. Therefore the only one that contribute to ti

j is G(2)ij. If

the surface is located far enough from the extended object, we can assume the

metric perturbations are small such that we can compute G(2)ij up to second

order. The metric in the Newtonian gauge is given by :

ds2 = −(1 + 2Φ)dt2 + (1− 2Ψ)δijdxidxj (2.8)

and by using Gµν = gνβGµβ and gνβ = ηνβ − hνβ, we will get :

G(2)ij = −2Φ(δij∇2Φ− ∂i∂jΦ)− 2Ψ(δij∇2Ψ− ∂i∂jΨ)

+∂iΦ∂jΦ− δij∂kΦ∂kΦ + 3∂iΨ∂jΨ− 2δij∂kΨ∂

kΨ

−∂iΦ∂jΨ− ∂iΨ∂jΦ + 2ΨG(1)ij, (2.9)

where we have ignored time derivatives because we are assuming non-relativistic

motion. The first order Einstein tensor G(1)ij (ignoring time derivatives) is given

by

G(1)ij = δij∇2(Φ−Ψ) + ∂i∂j(Ψ− Φ) (2.10)

Now we split Φ and Ψ around the surface of the sphere to be Φ0,Ψ0 which describe

the large scale fields due to the background and Φ1,Ψ1 which describe the fields

due to the extended object itself. Then we have :

Φ = Φ0 + Φ1(r) (2.11)

5

Ψ = Ψ0 +Ψ1(r), (2.12)

where Φ1 and Ψ1 are the solution of the Einstein equation with the extended

object as the only source, whereas Φ0 and Ψ0 are linear gradient fields that can

always be added to solutions. Notice that we can decompose the solution of Φ and

Ψ as the sum of the object field and background field because we have linearized

the theory. Here, Φ0 and Ψ0 are generated by other sources in the environment

and they vary gently inside of the spherical surface that encloses the object (their

second gradient can be ignored) :

Φ0(~x) ≃ Φ0(0) + ∂iΦ0xi,

Ψ0(~x) ≃ Ψ0(0) + ∂iΨ0xi, (2.13)

where Φ0(0) and Ψ0(0) are the environment fields at the center of the sphere. We

are also assuming that ∂iΦ0 and ∂iΨ0 hardly vary inside of the sphere. On the

other hand Φ1 and Ψ1 are the fields generated by the extended object and they

have large variation within the sphere. For the sake of simplicity, we choose r to

be sufficiently large so that the monopole dominates, therefore we have

Φ1 = Ψ1 ≃−GMr

. (2.14)

Another requirement to choose r is that we have to make sure that r is smaller

than the scale of variation of the background fields. We also assume that the

density of the extended object is much bigger than its immediate environment

such that the total mass inside the enclosing sphere is dominated by the mass of

the extended object.

Now subtitute Φ and Ψ to G(2)i

j and calculate the gravitational force defined on

Eq.(2.7) by using these following assumptions :

• at the surface of the sphere, ∇2Φ1 = ∇2Ψ1 = 0,

• a term such as∮

dSj∂iΦ1∂jΨ1 vanishes because both Φ1 and Ψ1 are spher-

ically symmetric,

• a term such as∮

dSj∂iΦ0∂jΨ0 also vanishes because by assumption, ∂iΦ0

and ∂jΨ0 are both constant on the scale of the sphere,

6

then we will get :

Pi =1

8πG

∮

dSj(2Φ0∂i∂jΦ1 + 2Ψ0∂i∂jΨ1 + ∂iΦ0∂jΦ1 + ∂iΦ1∂jΦ0 − 2δij∂kΦ0∂kΦ1

+3∂iΨ0∂jΨ1 + 3∂iΨ1∂jΨ0 − 4δij∂kΨ0∂kΨ1 − ∂iΦ0∂jΨ1 − ∂iΦ1∂jΨ0

−∂iΨ0∂jΦ1 − ∂iΨ1∂jΦ0 + 2Ψ0∂i∂jΨ1 − 2Ψ0∂i∂jΦ1).

Performing all the integrations in the equation above, yields :

Pi =1

8πG

∮

dSjG(2)

ij =

r2

8G∂iΦ0

[

−16

3

∂Ψ1

∂r− 8

3

∂Φ1

∂r

]

. (2.15)

Only terms proportional to ∂iΦ0 remain, there are no ∂iΨ0 terms. This is not

surprising because by ignoring the time derivatives it means the extended object

is moving non-relativistically with small velocity compared to the speed of light

and so its motion should only be sensitive to the time-time part of the background

metric.

Now subtituting Φ1 and Ψ1 into P we get :

P = −M∂iΦ0, (2.16)

which is the expected GR prediction in the Newtonian limit.

The mass of the object is defined by

M = −∫

d3xt00. (2.17)

Its time derivative is given by M =∫

d3x∂it0i. It can be converted to the surface

integral M =∮

dSit0i. By assuming the energy flux through the surface is small

then we can approximate M as constant. The center mass coordinate of the

object is defined by

X i ≡ −∫

d3xxit00

M. (2.18)

By assuming M is constant then the time derivative of X is given by

X i =

∫

d3xxi∂jt0j

M=

∫

d3x[∂j(xit0

j)− t0i]

M= −

∫

d3xt0i

M, (2.19)

which is precisely Pi/M as defined in Eq.(2.6). Take derivative once again from

both sides we get

MX = −M∂iΦ0. (2.20)

7

Therefore we have shown that the extended object move on the geodesic of

the background fields. We can cancel its inertial mass and gravitational mass

from both sides. It means the motion of an extended object on an external

gravitational field is independent of its mass. Consequently, if there are more

than one extended objects with different masses in the same environment, they

will move in the same way.

The important feature of this derivation is that it holds independently from the

internal structure of the extended object, such that, Eq.(2.20) remains valid for

the motion of a black hole.

8

Chapter 3

The Problem of Motion in

Scalar-Tensor Theories

In this chapter we will derive the equation of motion of an extended object in

scalar-tensor theories. In these theories ϕ is the mediator of a fifth force and

equivalence principle violation will occur when there are fluctuations in the ratio

of the scalar charge and the mass between objects. If this happens, different

objects will fall at different rates in the same environment.

3.1 Derivation in Einstein Frame

Basically, the equivalence principle violation is easiest to see in the Einstein frame.

This is simply because in Einstein frame, the conformal metric gµν satisfy the

conventional form of Einstein equation and there is a direct coupling of the scalar

ϕ with matter. Notice that in Einstein frame, at the fundamental level, all

particles are coupled in the same way to the scalar, so that there will be no

violation of the equivalence principle for elementary particles. The Einstein frame

action is defined :

S =M2pl

∫

d4x√

−g[12R− 1

2∇µϕ∇µϕ−V (ϕ)]+

∫

d4xLm(ψm,Ω−2(ϕ) ˜gµν) (3.1)

where,˜denotes quantities in Einstein frame, R is Ricci scalar in Einstein frame,

∇µϕ = ∇µϕ = ∂µϕ and ϕ is the scalar that contributes to a fifth force. By

redefining field we can always write the kinetic term of ϕ as Eq.(3.1). In this

notation the scalar field ϕ is dimensionless and V (ϕ) has mass dimension two.

9

The symbol ψm denotes some matter field. The relation between the Einstein

frame metric and the Jordan frame metric is given by

gµν = Ω−2(ϕ)gµν . (3.2)

Since we will perform perturbative computations, Ω2(ϕ) must be close to unity

for the metric perturbations to be small in both frames. Then, at linear order of

ϕ, Ω2(ϕ) can be approximated as

Ω2(ϕ) ≃ 1− 2αϕ (3.3)

where α is a constant and |αϕ| ≪ 1. We also approximate ∂lnΩ2

∂ϕ≈ −2α. Now

we will discuss the motion of an extended object in two different cases, the first

case is when the scalar doesn’t have a self-interaction potential and the second is

when there is a self-interaction potential.

3.1.1 Scalar Field without Self-Interaction Potential

The Einstein equation is :

Gµν = 8πG[Tm

µν + Tϕ

µν ] (3.4)

where,

Tϕµν =

1

8πG[1

2∇µϕ∇µϕ− δµ

ν 1

2∇αϕ∇αϕ], (3.5)

with V (ϕ) = 0. The scalar field equation is :

ϕ = 4πG∂lnΩ2

∂ϕTm

µµ (3.6)

The next step is to perform an object-background split. In the Newtonian gauge

we have :

ds2 = −(1 + 2Φ)dt2 + (1− 2Ψ)δijdxidxj, (3.7)

as before, at the surface of the sphere with radius r from the object, we decompose

Φ, Ψ and ϕ as :

Φ = Φ0 + Φ1(r)

Ψ = Ψ0 + Ψ1(r)

ϕ = ϕ0 + ϕ1(r), (3.8)

10

where Φ0,Ψ0 and ϕ0 are the fields generated by other sources in the environment.

We also assume that their second gradient can be ignored, such that

Φ0(~x) ≈ Φ0(0) + ∂iΦ0(0)xi,

Ψ0(~x) ≈ Ψ0(0) + ∂iΨ0(0)xi,

ϕ0(~x) ≈ ϕ∗ + ∂iϕ0(0)xi, (3.9)

where 0 denotes the origin centered at the object, and ϕ∗ is the background scalar

field value there. We make the same assumption as before that these background

fields vary on a scale much larger than radius of the sphere enclose the object.

To find the solution of the object fields, we have to examine Einstein equation

linearized in metric perturbation :

∇2Ψ = 4πGρ− 1

4∂0ϕ∂

0ϕ+1

4∂iϕ∂

iϕ,

∂0∂iΨ = −1

2∂0ϕ∂iϕ,

(∂i∂j −1

3δij∇2)(Ψ− Φ) = ∂iϕ∂jϕ− 1

3δi

j∂kϕ∂kϕ,

∂20Ψ +1

3∇2(Φ− Ψ) =

1

6(−3

2∂0ϕ∂0ϕ− 1

2∂kϕ∂kϕ) (3.10)

where we have assumed the matter is nonrelativistic and therefore only charac-

terized by its energy density ρ. Here, ϕ is in first order of G, such that we can

ignore all second order terms on the right hand side. Then we get :

∇2Ψ = 4πGρ ∇2(Φ− Ψ) = 0. (3.11)

Assuming that at sufficiently large r the monopole dominates, we have :

Φ1 = Ψ1 = −GMr, (3.12)

where M is the mass of the extended object.

The solution for the scalar field can be derived from Eq.(3.6) which can be

rewritten as

∇2ϕ = α8πGρ, (3.13)

where we have ignored time derivatives and corrections due to metric perturba-

tions, and we have used the approximation ∂lnΩ2

∂ϕ≈ −2α.

11

Therefore, the exterior profile for scalar field ϕ1 sourced by the object

ϕ1 = −2αGM

r(3.14)

with the assumption that at sufficiently large r, the monopole dominates. In this

case, the scalar field is massless inside the object and therefore the exterior scalar

profile is sourced by the whole mass of the object. In other words, there is no

screening.

Now let’s call scalar charge Q

Q = αM, (3.15)

then we can rewrite the scalar profile as ϕ1 = −2QGr

. As before, to find the

equation of motion of an extended object we compute the gravitational force

which given by

Pi = −∮

dSj tij (3.16)

where the integration is over the surface of a sphere enclosing the extended object.

The pseudo-energy-momentum tensor is defined by

tµν = Tm

µν + Tϕ

µν − 1

8πGG(2)

µν . (3.17)

We assume again that at the surface of the sphere Tmi

j is small such that we can

neglect it, and we have already computed the G(2)i

j. The contribution for Tϕij

will give us :

−∮

dSjTϕij = − 1

2Gr2∂iϕ0

∂ϕ1

∂r. (3.18)

Subtituting all the contributions we have

Pi =r2

2G

[

∂iΦ0(−4

3

∂Ψ1

∂r− 2

3

∂Φ1

∂r− ∂iϕ0

∂ϕ1

∂r)

]

. (3.19)

Subtitute all the solutions for Ψ1, Φ1 and ϕ1 to equation above we will get

MX = −M∂iΦ0 − αM∂iϕ0 = −M[

∂iΦ0 + α∂iϕ0

]

, (3.20)

where α is a constant defined by QM. We conclude that if there is no scalar

potential, the ratio of Q/M for all objects will be the same and therefore there

12

is no equivalence principle violation.

However, if the scalar field doesn’t have the self-interaction potential V (ϕ),

it will be massless and active at any scale. To avoid inconsistencies with the

experimental results at small scale, the value of constant α must be very small,

α ≪ 1. In this case, the scalar field would be weakly coupled to the matter,

and therefore the fifth force mediated by ϕ will be very small compared to the

Newton force.

3.1.2 Scalar Field with Self-Interaction Potential

In the presence of a scalar potential, the energy-momentum tensor for a scalar

field in the Einstein frame becomes

Tϕµν =

1

8πG[1

2∇µϕ∇µϕ− δµ

ν(1

2∇αϕ∇αϕ+ V )], (3.21)

and the scalar field equation is

ϕ =∂V

∂ϕ+ 4πG

∂lnΩ2

∂ϕTm

µµ. (3.22)

By splitting the object background fields and examining the Einstein equation

linearized in metric perturbation we will get

∇2Ψ = 4πGρ− 1

4∂0ϕ∂

0ϕ+1

4∂iϕ∂

iϕ+1

2V,

∂20Ψ +1

3∇2(Φ− Ψ) =

1

6(−3

2∂0ϕ∂0ϕ− 1

2∂kϕ∂kϕ− 3V ), (3.23)

where the other equations are the same as Eq.(3.10). Here, we ignore again all

second order scalar field terms on the right hand side. We also neglect V with

the assumption that Gρ ≫ V inside the extended object, whereas outside the

object, V or any other sources of energy-momentum tensor are negligible.

Assuming that the metric is sourced only by matter rather than the scalar

field and that at sufficiently large r the monopole dominates, we get the same

solution as Eq.(3.12). Ignoring the time derivatives and corrections due to metric

perturbations we get the scalar field equation d2ϕdr2

+ 2rdϕdr

= ∂V∂ϕ

+ α8πGρ which

can be rewritten asd2ϕ

dr2+

2

r

dϕ

dr=∂Veff∂ϕ

, (3.24)

13

where Veff = V (ϕ) + 8απGρϕ.

The chameleon mechanism could operate for the runaway potential V (ϕ) [1],

which is V (ϕ) ∝ ϕ−n, see Fig. 3.1. It means that ϕ can be trapped at small value

inside the extended object. Inside the object where the object density ρ is large,

Figure 3.1: An effective potential for chameleon mechanism. The effective poten-tial for ϕ is the sum of the potential V (ϕ) which is in the runaway form and thescalar-matter coupling (α8πG)ρϕ

ϕ has a large mass. We will see that there is a screening in the exterior scalar

profile. Which means the chameleon force outside the object is sourced only by

the thin shell at the object’s boundary. Because of this screening, we expect that

the exterior scalar profile, at sufficiently large r, is dominated by a monopole of

the form

ϕ1(r) = −ǫα2GMr

. (3.25)

Screening by the chameleon mechanism

Now we want to derive an approximate solution for ϕ sourced by a static,

spherically symmetric object with homogenous density ρ = Mc4

3πr3c

. We assume

the object is isolated, which means that the effect of the environment can be ne-

glected. Furthermore, this object is immersed in a background with homogenous

density ρ∗.

14

We denote ϕc and ϕ∗ as the field values which minimize Veff for r < rc and

r > rc, such that

dV (ϕ)

dϕ|ϕ=ϕc

+ 8απGρc = 0

dV (ϕ)

dϕ|ϕ=ϕ∗

+ 8απGρ∗ = 0. (3.26)

We also denote mc and m∗ as the massess of the chameleon field inside and out-

side the object. In this case we assume that m∗ ≈ 0. This is because when ρ

become small, the value of ϕ that minimize the effective potential become larger

and the mass of scalar field decreases, as shown on Fig. 3.2. Since Eq.(3.24) is a

Figure 3.2: Chameleon effective potential for large and small ρ.

second order differential equation, it requires two boundary conditions which aredϕdr

= 0 at r = 0 and ϕ→ ϕ∗ as r → ∞.

To derive the exterior solution of Eq.(3.24), it is easy to use a classical me-

chanic analogy. Consider r as a time coordinate and ϕ as the position of the

particle. In this analogy, the particle moves along the inverted potential −Veff .The second term on the left-hand side of Eq.(3.24), proportional to 1

ris consid-

ered as a damping term, whereas the termdVeff

dϕis considered as the driving term.

Here, −Veff depends on time since it contain ρ which depends on r. The effective

potential is discontinous at r = rc since the object density is equal to ρc for r < rc

15

and it jumps suddenly to ρ∗ when r > rc. However, ϕ and dϕdr

are still continuous

at r = rc.

Initially, the particle is at rest at r = 0, dϕdr

= 0 with some initial value ϕi. The

particle remains at rest for small r since at small r the damping term dominates.

When r becomes sufficiently large the driving term,dVeff

dϕ, starts to be effective,

the damping term becomes small and the particle starts to roll down the inverted

potential. The particle keeps rolling down until at ’time’ r = rc when the object

density suddenly change. In this time the particle is climbing up the effective

potential, since at r = rc the effective potential changes shape as shown in Fig.

3.3. The initial value ϕi is chosen such that at r → ∞ ϕ reaches ϕ∗. The screen-

Figure 3.3: The inverted potential −Veff for an object with homogenous densityρc is discontinous at r = rc when the density suddenly changes to ρ∗. The particleinitially at ϕi. It is rolling down until r = rc. At r = rc the particle starts toclimb up the effective potential

.

ing by the chameleon mechanism occurs when ϕi ≈ ϕc. In this case the particle

is at rest at ϕi ≈ ϕc. The particle remains at rest until ’time’ r becomes large,

such that the damping term becomes small enough compared to the driving term.

Consider the particle begins to roll at ’time’ r = rroll. Therefore, we have

ϕ(r) ≈ ϕc 0 < r < rroll. (3.27)

16

When the particle starts to leave ϕc at time r = rroll, the term 8πGαρϕ begins

to dominate such that we can neglect V (ϕ) term, see Fig.3.1. Therefore, on

rroll < r < rc regime we can approximate Eq.(3.24) as

d2ϕ

dr2+

2

r

dϕ

dr≈ 8πGαρ. (3.28)

The solution for equation above with boundary condition ϕ = ϕc anddϕdr

= 0 at

r = rroll is

ϕ(r) =8πGα

3ρ

(

r2

2+r3rollr

)

− 4απGρr2roll + ϕc. (3.29)

Notice that we can separate the solution into two regions 0 < r < rroll and

rroll < r < rc only when rc − rroll ≪ rc, otherwise there will be no rroll.

At r = rc, ρ suddenly changes from ρc to ρ∗. For r > rc, particle is climbing the

effective potential as shown in Fig.3.3, and hence, Eq.(3.24) can be approximated

asd2ϕ

dr2+

2

r

dϕ

dr≈ 0. (3.30)

The boundary condition is ϕ → ϕ∗ as r → ∞. Therefore the exterior solution is

approximately

ϕ(r) ≈ −kr+ ϕ∗, (3.31)

where k is a constant. We can find the constant k and rroll by matching the value

of ϕ and dϕdr

at r = rc from Eq.(3.28) and Eq.(3.31). By using the approximation

rc − rroll ≪ rc, we find that

rroll =(ϕ∗ − ϕc)r

2c

6GαMc

, (3.32)

k = (ϕ∗ − ϕc)rc, (3.33)

therefore we find

ϕ(r) ≈ −ϕ∗ − ϕc

rrc + ϕ∗ r > rc. (3.34)

Compare with Eq.(3.25), we get

ǫ =ϕ∗ − ϕc

2GαMc

rc ≈ϕ∗

2GαMc

rc (3.35)

17

We also have 3∆rcrc

≈ ϕ∗

2GαMcrc which is the fraction by volume of the object that

sources the exterior scalar field. Therefore the screening occurs for 3∆rcrc

< 1 or

ǫ ≈[

ϕ∗

2α

]

[

GMrc

] < 1. (3.36)

The condition on Eq.(3.36) is known as the thin-shell condition.

However, there is another condition known as the thick-shell condition. This

occurs when 8πGαρ ≫ ∂V∂ϕ

. Here, the particle is released far away from ϕc.

Therefore, the particle starts to roll down immediately after it is released at

r = 0. The solution of the ϕ can be obtained by taking rroll → 0 on Eq.(3.29)

and changing ϕc by ϕi. In this case ǫ = 1 and ϕ∗

2GαMcrc ≥ 1.

To compute the equation of motion for the object we use the same tricks as before.

In this case the scalar charge Q is defined

Q = ǫαM. (3.37)

In the computation of the gravitational force we will have contribution from the

potential term 1G

∮

dSjV ≈ 1G

4πr3

3∂iϕ0

∂ϕi

∂r|ϕ∗

which can be neglected by assum-

ming that the scalar field ϕ1 is dominated by the object rather than its immediate

environment. Therefore, the gravitational force is given by the Eq.(3.19). Plug-

ging in all the fields solution we get

MX = −M∂iΦ0 − ǫαM∂iϕ0 = −M[

∂iΦ0 +Q

M∂iϕ0

]

. (3.38)

Now if we consider both Φ0 and ϕ0 are sourced in the same way such that

∇2Φ0 = 4πGρ∗, ∇2ϕ0 = α8πGρ∗ (3.39)

where ρ∗ is the environment density, then we will get ϕ0 = 2αΦ0. Subtituting

this to the Eq.(3.38) we get

MX = −M∂iΦ0[1 + 2ǫα2]. (3.40)

Now if one has scalar-tensor theory, where 2α2 ≈ 1, then we have MX =

−M∂iΦ0[1 + ǫ]. We conclude that if the scalar field has scalar potential V (ϕ),

there will be O(1) equivalence principle violation which means that the different

objects, screened object with ǫ ≈ 0 and unscreened object with ǫ ≈ 1 would move

differently.

18

3.2 Derivation in Jordan Frame

We can get the Jordan frame action by performing the conformal transformation

defined on Eq.(3.2)

S =M2pl

∫

d4x√−g[1

2Ω−2R− 1

2h(ϕ)∇µϕ∇µϕ− Ω4(ϕ)V ] +

∫

d4xLm(ψm, gµν)

(3.41)

where, h(ϕ) ≡ Ω2[1− 32(∂lnΩ

2

∂ϕ)]. The relation between matter energy-momentum

tensor in the two frames is defined by

Tmµν = Ω2(ϕ)Tm

µν , (3.42)

see Apendix for the detail. We can see from the Jordan frame action that matter

couples only to gµν without direct coupling with to the scalar, therefore it is

supposed tom move on the geodesics of gµν . However, the fact that different

objects move differently in the presence of chameleon field in Einstein frame

should also be true in Jordan frame. Thus, the issue that all objects are move on

the geodesics in Jordan frame is no longer true.

The Jordan frame metric in Newtonian gauge is defined

ds2 = −(1 + 2Φ)dt2 + (1− 2Ψ)δijdxidxj. (3.43)

Using Eq.(3.2) and Eq.(3.6) the relation between Jordan and Einstein frame met-

ric perturbations is given by

Φ = Φ + αϕ, Ψ = Ψ− αϕ. (3.44)

The Einstein equation is not in standard form, but it is given by

Gµν = 8πGΩ−2[Tm

µν + Tϕ

µν ] + Ω−2[∇µ∇µΩ2 − δµ

νΩ2], (3.45)

where

Tϕµν =

1

8πG

[

h∇µϕ∇νϕ− δνµϕ−(

1

2h∇αϕ∇αϕ

)]

, (3.46)

and the value of h depends on the theory. As before, we can split Einstein tensor

and define the pseudo-energy-momentum tensor

G(1)µν = 8πGtµ

ν (3.47)

19

where tµν is defined

tµν = Ω−2(Tm

µν + Tϕ

µν)− 1

8πGG(2)

µν Ω

−2

8πG(∇µ∇νΩ2). (3.48)

As we see, the pseudo-energy-momentum tensor contains scalar field which can

give direct influence to the motion of the object. Therefore, it is not obvious that

the integral of momentum flux should imply geodesic motion in Jordan frame.

Now, we can easily compute the equation of motion for the object by transforming

Eq.(3.40) using Φ0 = Φ0 − αϕ0, we get

MX i = −M∂iΦ0 + (1− ǫ)αM∂iϕ0, (3.49)

where Φ0 and ϕ0 are the background metric perturbation and scalar field. The

screened object has ǫ < 1, whereas the unscreened object has ǫ = 1. In the Jordan

frame the unscreened object would move in geodesic just like an infinitesimal test

particle, since the second term on the right hand side of Eq.(3.49) vanish, on

the other hand, the screened object, would not move on geodesic. Therefore, in

the Jordan frame the unscreened objects don’t feel the scalar force whereas in

the Einstein frame both screened or unscreened objects feel the scalar field with

different proportion.

Now if we make the same assumption as before, that both Φ0 = Φ0−αϕ0 and

ϕ0 are sourced in the same way as Eq.(3.39),therefore we get relation, ϕ0 =2αΦ0

1+2α2 .

We can simplify the equation of motion to

MX i = −M(

1 + 2ǫα2

1 + 2α2

)

∂iΦ0. (3.50)

Therefore, in Jordan frame the equivalence principle violation comes because

the unscreened objects move in geodesic whereas the screened objects don’t. The

unscreened object would move with acceleration which is 1 + 2α2 greater than a

screened object.

3.3 Screening Mechanism in the DGP model by

Vainshtein effect

Now we want to study the screening mechanism in the Dvali-Gabadadze-Porrati

model. In the DGP model our world is the 4D boundary of an infinite 5D space-

time. One can integrate out the bulk degree of freedom and find an ’effective’

20

action for the 4D fields. It was found that beside the ordinary graviton, there is

an extra scalar degree of freedom π that plays a crucial role. This is essentially a

brane bending mode contributing to the extrinsic curvature of the boundary like

Kµν ∝ ∂µ∂νπ. However, we can restrict only to the π sector, because there exist

a limit [2], in which all degrees of freedom decouple, and all further interactions

vanish.

The scalar sector of DGP (in Einstein frame) has the lagrangian [3]

L = −3M2P l(∂π)

2 − 2M2

P l

m2(∂π)2π + πTm

µµ, (3.51)

where m is the DGP critical mass scale which is set to the current inverse Hubble

scale.

Now we want to derive an explicit solution for the field generated by a point-

like source. Consider a static point-like source of mass M , located at the origin

such that T = −Mδ3(~x). We want to find a static spherically symmetric solution

π(r), where r is the radial coordinates. We can compute the field equation for

lagrangian on Eq.(3.48)

6π − 2

m2(∂π)2 +

4

m2∂µ(∂

µππ) +Tm

µµ

M2P l

= 0. (3.52)

Now if we see the lagrangian on Eq.(3.51), it is obvious that its source-free part

has a shift symmetry,

π → π + a, (3.53)

therefore we can write the equation of motion of π as the divergence of the

associated Nother’s current

∂µJµ = −Tm

µµ. (3.54)

In our case, the solution is time independent, therefore Eq.(3.52) can be rewritten

as

~∇ ·[

6~∇π − 2

m2~∇(~∇π)2 + 4

m2~∇π~∇2π

]

=M

M2P l

δ3(~x). (3.55)

Now, we define ~E = ~∇π(r) = E(r)r, and we know that in spherical coordinate

~∇ · ~E = dEdr

+ 2rE. Therefore, we can simplify Eq.(3.55) as

~∇ ·[

6Er +8

m2

E2

rr

]

=M

M2P l

δ3(~x). (3.56)

21

We can integrate equation above over a sphere with radius r centered at the origin

such that we get an algebraic equation for E(r)

4πr2[

6E +8

m2

E2

r

]

=M

M2P l

=⇒ 8

m2

E2

r+ 6E − M

4πr2M2P l

= 0, (3.57)

which has the solution

E± =m2

8r

[

±√

9r4 +2

πR3

V r − 3r2

]

, (3.58)

where we call RV as Vainshtein radius

RV =

[

M

M2P lm

2

]1

3

. (3.59)

In the regime r ≪ RV , the two solutions of E can be approximated as

E ≈ ± m2R3/2V

4√2πr1/2

. (3.60)

The solution for π can be obtained by integrating E along r. Therefore we have

π± = ±m2R

3/2V

√r

4√2

. (3.61)

We can check that in this regime the correction to the Newton force is small

Fπ

FNewton

≈ E/MP l

M/M2P lr

2≈

[

r

RV

]3/2

. (3.62)

Therefore at the small r (r ≪ RV ), the force mediated by π is suppressed.

At large distance, r ≫ RV , we will get two solutions of π. The first solution is

π+ ∝ 1rand the second solution is π− ∝ r2. By assuming that at the infinity, i.e in

the absence of localized source we have a trivial solution, we can only consider the

first solution (π+) and neglect the second one. It means, at large r the non-linear

term on Eq.(3.56) is negligible. Then Eq.(3.56) becomes

~∇ · (6Er) = M

M2P l

δ3(~x) (3.63)

by using Gauss’s theorem we can integrate equation above to be

6M2P l

∮

S

Er · d~a ≈M =⇒ E(r) ≈ MG

3r2, (3.64)

22

where S is any surface of sphere centered at the origin with radius much larger

than RV , then by integrating E along r we will get the π profile in the linear

regime

π(r) ≈ −GM3r

, (3.65)

where in this case the scalar charge is M . Unlike in the chameleon case, the

scalar charge for all objects is the same, that means there is no O(1) violation of

the equivalence principle. The only possibility to have an equivalence principle

violation is from how relativistic the source’s internal structure.

Now if we want to compute the force acting on the object using the method

that we use on the previous section, we should be able to decompose π as Eq.(3.8).

Then, to approximate the scalar field sourced by environment to be a linear pure

gradient field we have to draw the surface of the sphere very close to the object.

But, we have some problems here. As we know, if we are very close to the object

this means that we approach the non-linear regime. It is obvious that in the

non-linear regime we can’t add the solution of π with a pure-gradient field to

get another solution. However, it is still possible for us to add another constant-

gradient field to the non linear solution. This is because the source-free part of

the lagrangian on Eq.(3.51) is also invariant under constant shift in the derivative

of π,

∂µπ → ∂µπ + cµ, (3.66)

known as ’Galilean invariance’. Now if the object is smaller than the variation

scale of π0 which is the scalar field from the other source, then we can approximate

π0 as a constant gradient field. Therefore, the full π field is

π = π0 + π1. (3.67)

Now we can draw a sphere with surface S very close to the object and then

calculate the force acting on it.

We still have another problem. If we have an irregular object with sizable

multipoles moment, we don’t even know the non-linear solution for the object

in isolation. To solve this problem, we can use a mathematical trick. Consider

we have two different situations. In the first case we have an object enclosed

by a sphere S on the very small distance from the object. In this case we can

23

Figure 3.4: A mathematical trick to compute the force acting on an irregularobject with sizable multipole moments.

approximate π0, which is the field from the other source as a linear pure-gradient

field. Hence, the full solution of π on S is the sum of π1 and π0, as shown on

Fig. 3.4. (left). Now consider the second case which has a different situation as

shown on Fig. 3.4 (right). In this case the object is enclosed by the same sphere

S, but now the constant-gradient π0 is linear everywhere. indeed the total force

acting on the object in the second case is the same as the total force acting on the

object in the first case. Then we can change the radius of S such that r > RV .

Now we are on the linear regime of π and therefore in this regime, the object’s

multipoles decay faster than the monopole, and the computation is precisely the

same as linear scalar field.

24

Chapter 4

Conclusion

We have discussed two screening mechanisms in scalar-tensor theories. The first

mechanism is the chameleon mechanism. This mechanism screen the scalar field

by giving a mass that depends on the local density. In the high density region,

the scalar will be massive and therefore it blends with the environment and

become essentially invisible to search for equivalence principle violation and fifth

force. In the small density region, chameleon mechanism leads toO(1) equivalence

principle violation.

In Einstein frame, unscreened objects move with an acceleration that is larger

than screened objects. Screened objects have ǫ < 1, while screened objects have

ǫ = 1. ǫ is the screening parameter which is controlled by

ǫ ≃ ϕ∗

2α(GM/rc), (4.1)

where M is the mass of the object, rc is the size of the object and ϕ∗ is the

external scalar field.

In Jordan frame it is no longer true that all objects move in geodesic of metric

gµν . Only unscreened objects move in geodesic. Screened objects which have

ǫ < 1 will feel the the effect of the fifth force and therefore they don’t move in

geodesic of gµν . In the absence of self-interaction potential, the scalar is massless,

therefore it can be active in anyscales. To avoid inconsistencies with small scale

experimental results, the value of α on Eq.(3.14) must be very small, α ≪ 1.

In the Vainsthein effect, compact source creates scalar profile that scales like

1/r at large distances. In approaching the source, the non-linear term in the EOM

of the scalar becomes important and then changes the dynamical of the scalar

25

to√r. This suppress the fifth force mediated by the scalar field for all objects

that are inside a halo with radius RV = (M/M2P lm

2)1/3, where M is the mass

of the object and m is the graviton mass (about Hubble scale today). Unlike

the chameleon mechanism, Vainsthein effect doesn’t lead to O(1) equivalence

principle violation. The equivalence principle violation could occurs at order

O(1/c2), depending on how relativistic the source’s internal structure is.

26

Acknowledgement

I would like to thank my supervisor, Prof. Paolo creminelli for the support,

invaluable assistance and kindness. For all other lectures : Prof. Randjabar

Daemi, Prof. Narain, Prof. Edi Gava, Prof. Smirnov, Prof. Goran, Prof.Bobby

thanks for teaching me everything. It is special pleasure for me to study physics

from all of you. I thank Hani for all the helps and useful discussions, all of my

friends in High Energy : Alejandro, Ammar, Hameda, Homero, Mutib, monireh

and Talal for sharing everything to me during the hard time in ICTP. Special

thanks to Elis Anitasari for supporting me in every condition, I owe you many

things.

Thanks to The Abdus Salam International Centre for Theoretical Physics for

giving me invaluable opportunity to join the Diploma Programme in High Energy

Physics 2009/2010.

Trieste, August 2010

Andrias Fajarudin

27

Appendix A

Conformal Transformation

Usualy, conformal transformation is used to bring a theory, such as scalar-tensor

theory, in to a form that looks like conventional general relativity. We define

conformal metric :

˜gµν = ω2(x)gµν → gµν = ω−2 ˜gµν . (A.1)

Now we want to know how quantities in the original metric gµν are related to

those in the conformal metric gµν .

Christoffel Symbol

Suppose after performing conformal transformation the Christoffel symbol is Γρµν .

Then the difference between Γρµν and Γρ

µν is a tensor, say Cρµν , defined by

Cρµν = Γρ

µν − Γρµν . (A.2)

The Christoffel symbol is defined by

Γρµν =

1

2gρσ(∂µgνσ + ∂νgσµ − ∂σgµν). (A.3)

Subtitute the conformal metric Γρµν defined on Eq.(A.1), we get

Γρµν =

1

2ω−2gρσ(2ω∂µωgνσ + ω2∂µgνσ + 2ω∂νωgσµ +

ω2∂νgσµ − 2ω∂σωgµν − ω2∂σgµν). (A.4)

Now we can calculate tensor Cρµν ,

Cρµν =

1

2ω−2gρσ(2ω∂µωgνσ + ω2∂µgνσ + 2ω∂νωgσµ + ω2∂νgσµ −

2ω∂σωgµν − ω2∂σgµν)−1

2gρσ(∂µgνσ + ∂νgσµ − ∂σgµν)

= ω−1(δρν∇µω + δρµ∇νω − gµνgρσ∇σω) (A.5)

28

Riemann Tensor

Riemann tensor in the conformal frame is defined by

Rρσµν = ∂µΓ

ρνσ + Γρ

µλΓλνσ − ∂νΓ

ρµσ − Γρ

νλΓλµσ = Rρ

σµν + ∂µCρνσ + Cρ

µλCλνσ

+ΓλνσC

ρµλ + Γρ

µλCλνσ − ∂νC

ρµσ − Cρ

νλCλµσ − Γλ

µσCρνλ − Γρ

νλCλµσ,(A.6)

and we know that

∇µCρνσ = ∂µC

ρνσ + Γρ

µλCλνσ − Γλ

µνCρλσ − Γλ

µσCρνλ,

∇νCρµσ = ∂νC

ρµσ + Γρ

νλCλµσ − Γλ

νµCρλσ − Γλ

νσCρµλ,

then we can rewrite Eq.(A.6) as

Rρσµν = Rρ

σµν +∇µCρνσ − ∂νC

ρµσ + Cρ

µλCλνσ − Cρ

νλCλµσ. (A.7)

Now subtitute Eq.(A.5) to Rρσµν , after some cancellations we get

Rρσµν = Rρ

σµν + ω−2(gνσgραδβµ − gµσg

ραδβν + 2δρµδαν δ

βσ + δρσδ

αµδ

βν

−δρµgνσgλβδαλ − gµνgραδβσ + gραgνσδ

βµ − 2δρνδ

αµδ

βσ − δρσδ

αν δ

βµ

+δρνgµσgλβδαλ + gµνg

ραδβσ − gραgµσδβν )∇αω∇βω − ω−1(δρµδ

αν δ

βσ

−δρνδαµδβσ + gνσgραδβµ − gµσg

ραδβν )∇αω∇βω, (A.8)

where we use the fact that (gµσgρβδαν −gµσgραδβν )∇αω∇βω = 0 and also the similar

term with µ → ν. To get the Ricci tensor we have to contract ρ and µ indices,

we get

Rσν = Rσν+ω−2(4δασδ

βν −gνσgαβ)∇αω∇βω−ω−1(2δασδ

βν +gνσg

αβ)∇α∇βω. (A.9)

Therefore the Ricci scalar in the conformal frame is given by

R = ω−2R− 6ω−3gαβ∇α∇βω. (A.10)

Covariant Derivative of scalar ϕ

The first derivative of scalar in the conformal metric and the original metric are

the same since both of them are equal to partial derivative

∇µϕ = ∇µϕ = ∂µϕ. (A.11)

29

The second derivative, however contain the Christoffel symbol. Using Eq.(A.3)

and Eq.(A.4) we get

∇µ∇νϕ = ∇µ∇νϕ− ω−1(δαµδβν + δαν δ

βµ − gαβgµν)∇αω∇βϕ. (A.12)

Now to express all quantities in the original metric in terms of the conformal

metric we simply changing gµν → gµν and ∇ → ∇, and use the facts that

• gνσgρα = gνσg

ρα

• ∇αω = ∇αω

• ∇α∇βω = ∇α∇βω − ω−1(2∇αω∇βω − gγθgαβ∇γω∇θω).

The Riemann tensor becomes

Rρσµν = Rρ

σµν + ω−2(−gνσgραδβµ − gµσgραδβν − δρσδ

αµδ

βν + gµν g

ραδβσ + gραgνσδβµ

+δρσδαν δ

βµ − gµν g

ραδβσ + gραgµσδβν − gνσg

αβδρµ + gµσgαβδρν)∇αω∇βω

−ω−1(δρµδαν δ

βσ − δρνδ

αµδ

βσ + gνσg

ραδβµ − gµσgραδβν )∇αω∇βω,

contracting the Ricci tensor with gσν , we get Ricci scalar in terms of conformal

quantities

R = ω2R− 12gαβ∇αω∇βω + 6ωgαβ∇α∇βω. (A.13)

The covariant derivative of scalar ϕ is given by

∇µ∇νϕ = ∇µ∇νϕ+ (δαµδβν + δβµδ

αν − gµν g

αβ)ω−1∇αω∇βϕ. (A.14)

Jordan frame and Einstein frame in scalar-tensor theories

Now we want to study the action of scalar-tensor theories in Jordan frame and

Einstein frame. In the Jordan frame, the action is a sum of gravitational piece,

a pure scalar piece and a matter piece

S = SfR + Sλ + SM (A.15)

The gravitational piece is defined as

SfR =

∫

d4xf(λ)R. (A.16)

30

To convert this action to the Einstein frame action we have to use conformal

transformation defined on Eq.(A.1). Using the conformal factor ω2 = 16πGf(λ)

and Eq.(A.13) the gravitational piece of the action in Einstein frame becomes

SfR =

∫

d4x√

−g(16πGf)−2f [ω2R− 12gαβ∇αω∇βω + 6ωgαβ∇α∇βω]

=

∫

d4x√

−g(16πG)−1[R− 12f−1gαβ∇αf1/2∇βf

1/2

+6gαβf−1/2∇α∇βf1/2]. (A.17)

Now integrate by part and discard the surface term from the last term of Eq.(A.17)

we get :

SfR =

∫

d4x√

−g(16πG)−1[R− 3

2f−2gαβ

(

df

dλ

)2

∇αλ∇βλ], (A.18)

which known as Einstein frame action.

The pure-scalar action in Jordan frame is defined

Sλ =

∫

d4x√−g

[

−1

2h(λ)gµν∂µλ∂νλ− U(λ)

]

. (A.19)

In the Einstein frame it becomes

Sλ =

∫

d4x√

−g(16πGf)−2

[

−1

2h(λ)16πGf gµν∂µλ∂νλ− U(λ)

]

=

∫

d4x√

−g[

−1

2h(λ)(16πG)−1f−1gµν∇µλ∇νλ− U(λ)

(16πG)2f 2

]

. (A.20)

Therefore we have

SfR + Sλ =

∫

d4x√

−g[(16πG)−1R− 1

2gαβ∇αλ∇βλ

1

16πGf 2(3f ′2 + fh)

− U(λ)

(16πG)2f 2].(A.21)

Now call k(λ) = 116πGf2

(fh+ 3f ′2) and define a new scalar via

ϕ =

∫

dλk1/2, (A.22)

we can simplify Eq.(A.21) as

SfR + Sλ =

∫

d4x√

−g[

R

16πG− 1

2gαβ∇αϕ∇βϕ− V (ϕ)

]

, (A.23)

31

where V (ϕ) = U(λ(ϕ))

(16πG)2f2(λ(ϕ)).

The action for matter in Jordan frame is defined as

SM =

∫

d4x√−gLM(gµν , ψi), (A.24)

where matter is coupled to the metric gµν . While, in the Einstein frame, the

matter is not only coupled to the gµν but also with scalar function

SM =

∫

d4x√

−gLM(16πGf(λ)gµν , ψi). (A.25)

The stress-energy momentum tensor in the Einstein frame is defined by

T µν = −21

−√g

δSM

δgµν=

−2√−g(16πGf)

(16πGf)2δSM

δgµν= (16πG)−1Tµν . (A.26)

The coupling of matter to ϕ in Einstein frame is coming from varying the

matter action w.r.t ϕ

δSM

δϕ=∂gαβ

∂ϕ

δSM

δgαβ= − 1

2f

df

dϕ

√

−gTM , (A.27)

where we have used Eq.(A.26).

Taking variaton w.r.t to gµν we will get the Einstein equation in the standard

form

Gµν = 8πG(T (M)µν + T (ϕ)

µν ), (A.28)

where

T (ϕ)µν =

1

8πG

(

∇µϕ∇νϕ− gµν [1

2gρσ∇ρϕ∇σϕ+ V (ϕ)]

)

. (A.29)

To get the equation of motion for scalar field, we have to vary the total action

w.r.t ϕ, yields

ϕ− dV

dϕ=

1

2f

df

dϕTM . (A.30)

32

Bibliography

[1] Justin Khoury and Amanda Weltman, Chameleon Cosmology. Phys.Rev.D

69, 044026 (2004).

[2] Alberto Nicolis and Riccardo Rattazzi, Classical and Quantum Consistency

of the DGP Model. arXiv : hep-th/0404159v1 21 Apr 2004.

[3] Lam Hui, Alberto Nicolis and Christopher W.Stubbs, Equivalence Principle

Implications of Modified Gravity Models. Phys.Rev D 80, 104002 (2009).

[4] Bhuvnesh Jain and Justin Khoury, Cosmological Test of Gravity. arXiv :

1004.3294v1 [astro-ph.CO] 19 Apr 2010.

[5] Sean Carrol, Spacetime and Geometry, An Introduction to General Relativity.

Addison Wesley, 2004.

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