violation of the equivalence principle in scalar-tensor ... · violation of the equivalence...
TRANSCRIPT
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.
33