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arXiv:hep-th/0307073v2 15 Jul 2003
UTAP-454
Exactly solvable model for cosmological perturbations in dilatonic
brane worlds
Kazuya Koyama and Keitaro Takahashi
Department of Physics, University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
Abstract
We construct a model where cosmological perturbations are analytically solved based on dilatonic
brane worlds. A bulk scalar field has an exponential potential in the bulk and an exponential
coupling to the brane tension. The bulk scalar field yields a power-law inflation on the brane.
The exact background metric can be found including the back-reaction of the scalar field. Then
exact solutions for cosmological perturbations which properly satisfy the junction conditions on the
brane are derived. These solutions provide us an interesting model to understand the connection
between the behavior of cosmological perturbations on the brane and the geometry of the bulk.
Using these solutions, the behavior of an anisotropic stress induced on the inflationary brane by
bulk gravitational fields is investigated.
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I. INTRODUCTION
The possibility that our universe is a hypersurface (brane) embedded in a higher-
dimensional spacetime (bulk) has recently attracted much attention [1]. Particularly, a
model proposed by Randall and Sundrum has attractive features for gravity and cosmology
[2]-[4]. In their model, a single positive tension brane is embedded in 5-dimensional Anti-de
Sitter (AdS) spacetime. Ordinary matter fields are assumed to be confined to the brane and
only the gravity can propagate in the bulk. A quite interesting feature of their model is that
a 4-dimensional gravity is recovered in the low energy limit even though the size of the bulk
is infinite. We need no longer a compactification of the extra dimension. Then cosmological
consequences of their model have been intensively investigated. In string models, the grav-
ity enjoys the company of scalar field such as dilaton. Then the extension of Randall and
Sundrum model to dilatonic brane worlds also attracts much interest [5]-[13]. The dilatonic
brane world provides us a new models for inflationary brane world which is often called bulk
inflaton model [14]-[23]. The inflation on the brane is caused by the inflaton in the bulk and
the bulk spacetime itself is not inflating. It has been shown that this model indeed mimics
the 4-dimensional inflation model at low energies where the Hubble horizon on the brane
H−1is sufficiently longer than the curvature radius l in AdS bulk .
One of the most important quantities which should be clarified in brane world models is
the cosmological perturbation because it provides us a possibility to test the brane world
model observationally [24]-[29]. Unfortunately, it is quite difficult to find the solutions for
cosmological perturbations because we should consistently take into account the perturba-
tions in the bulk. Recently, Minamitsuji, Himemoto and Sasaki investigated a behavior of
cosmological perturbations in a model with bulk scalar field in AdS spacetime [22]. They
used a covariant curvature formalism and showed that 4-dimensional results are recovered
at low energies. Thus, to predict signatures specific to the brane world model, we should
investigate the higher energy effects. However, it is quite difficult to perform the calculations
at high energies because we should treat completely a 5-dimensional problem. Technically,
it is necessary to solve complicated coupled partial differential equations to find the behavior
of perturbations.
Hence it is eagerly desired to construct a model where we can solve the equations for
perturbations analytically. Recently, we have developed such a model based on dilatonic
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brane worlds [20]. The action for this model is given by
S =
?
d5x√−g5
?1
2κ2R −1
2∂µφ∂µφ − Λ(φ)
?
−
?
d4x√−g4λ(φ), (1)
where κ2is five-dimensional gravitational constant. The potential of the scalar field in the
bulk and on the brane are taken to be exponential:
κ2Λ(φ) =
?∆
√2λ0e−√2bκφ.
8+ δ
?
λ2
0e−2√2bκφ, (2)
κ2λ(φ) = (3)
Here λ0is the energy scale of the potential, b is the dilaton coupling and we defined
∆ = 4b2−8
3. (4)
We assume the Z2symmetry across the brane. The bulk scalar field φ acts as an inflaton. For
δ ?= 0, the brane undergoes a power-low inflation. The metric for 5-dimensional spacetime is
written by separable functions of time and extra-coordinate y. This is technically essential
because the perturbations can have separable solutions which enables us to derive solutions
analytically. It is not sufficient to derive general solutions for perturbations in the bulk. We
should find particular solutions which satisfy the correct boundary conditions at the brane.
It is in general difficult to find such particular solutions for an expanding brane. But, in
this model, we can successfully find such solutions analytically. The first aim of this paper
is to provide exact and analytic solutions for cosmological perturbations in this model.
Using the solutions for perturbations, we can address the primordial fluctuations gener-
ated during inflation. In a previous paper, we calculated a spectrum of curvature pertur-
bation Rcby quantizing a canonical variable for the second order action [20]. We found
that ,even at high energies, the effects of Kalzua-Klein (KK) modes are negligible at long
wave-length even though the amplitude of the fluctuation is amplified. However, in the
brane worlds, the curvature perturbation alone does not determine the cosmic microwave
background (CMB) anisotropies. The anisotropic stress induced by bulk gravitational fields
affects the CMB anisotropies. Indeed, it is this anisotropic stress which gives distinct fea-
tures in CMB anisotropies in brane worlds from 4-dimensional models [30] [31]. Thus we
should specify the initial condition for the anisotropic stress generated during inflation as
well as the curvature perturbations. The second aim of this paper is to investigate the
behavior of anisotropic stress in this model.
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The structure of the paper is as follows. In section II, the background spacetime is
described. In section III, the solutions for cosmological perturbations are derived. Section
IV is devoted to the investigation of the anisotropic stress on the brane which is generated
during inflation. In section V, we summarize the results. In appendix A, 5-dimensional
Einstein equations for scalar perturbations are shown. In appendix B, the procedure to
derive the solutions for scalar perturbations is presented.
II. BACKGROUND
We first review a background solution [20]. For δ = 0, the static brane solution was
found [8]. The existence of the static brane requires tuning between bulk potential and
brane tension known as Randall-Sunrum tuning. It has been shown that for ∆ ≤ −2, we
can avoid the presence of the naked singularity in the bulk and also ensure the trapping
of the gravity. The reality of the dilaton coupling requires −8/3 ≤ ∆. For ∆ = 8/3, we
recover Randall-Sundrum solution. The value of δ, which is not necessarily small, represents
a deviation from the Randall-Sundrum tuning. This deviation yields an inflation on the
brane.
The solution for background spacetime is found as
ds2= e2W(y)?
φ(t,y) = φ(t) + Ξ(y).
e2√2bκφ(t)dy2− dt2+ e2α(t)δijdxidxj?
,
(5)
The evolution equation for background metric α and φ are given by
˙ α2+
√2bκ˙φ ˙ α =
√2bκ˙φ)˙φ = −4√2bκ−1λ2
1
6κ2˙φ2−1
3λ2
0
∆ + 4
∆
δ
∆e−2√2bκφ.
δe−2√2bκφ,
¨φ + (3 ˙ α +
0
(6)
The solution for α(t) and φ(t) are obtained as
eα(t)= (H0t)
√2bκφ(t)= H0t = (−Hη)
2
3∆+8= (−Hη)
2
3(∆+2), (7)
e
3∆+8
3(∆+2). (8)
where
H0= −3∆ + 8
3(∆ + 2)H,H = −(∆ + 2)
?
−δ
∆λ0, (9)
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and η is a conformal time defined by
η =
3∆ + 8
3(∆ + 2)H
−
0
2
3∆+8
t
3(∆+2)
(3∆+8). (10)
We should notice that power-law inflation occurs on the brane for −8/3 < ∆ < −2. Thus
in the rest of the paper we shall assume −8/3 < ∆ < −2. The solutions for W(y) and Ξ(y)
can be written as
eW(y)= H(y)
2
3(∆+2),eκΞ(y)= H(y)
2√2b
(∆+2), (11)
where
H(y) =
?
−1 −∆
8δsinhHy. (12)
Here we assumed∆
8+δ < 0. At the location of the brane y = y0the solutions should satisfy
junction conditions;
√2
6λ0,
Then the location of the brane is determined by
∂yW(y)|y=y0= −eW(y0)−√2bκΞ(y0)
∂yΞ(y)|y=y0= −eW(y0)−√2bκΞ(y0)bκ−1λ0. (13)
sinhHy0=
?
−1 −∆
8δ
?−1/2
. (14)
It is quite usefull to note that the above 5-dimensional solution can be obtained by a
coordinate transformation from the metric
ds2= e2Q(z)(dz2− dτ2+ δijdxidxj),eκφ(z)= e3√2bQ(z), (15)
where
eQ(z)= (sinhHy0)−
2
3(∆+2)(Hz)
2
3(∆+2), (16)
by
z = −η sinh(Hy),
τ = −ηcosh(Hy). (17)
Because the metric Eq. (15) is simple, it is convenient to solve the perturbations in the bulk
not directly in Eq. (5) but in Eq. (15) for scalar perturbations.
The background equations on the brane Eqs.(6) can be described by the 4-dimensional
Brans-Dicke theory with the action
S4,eff=
1
2κ2
4
?
d4x√−g4
?
ϕBD
(4)R −ωBD
ϕBD(∂ϕBD)2
?
−
?
d4x√−g4Veff(ϕBD), (18)
where
ϕBD= e
√2bκφ,ωBD=
1
2b2,κ2
4Veff(ϕBD) = −λ2
0δ∆ + 4
∆
1
ϕBD. (19)
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III.COSMOLOGICAL PERTURBATIONS
Now let us consider cosmological perturbations in this background spacetime. Taking
appropriate gauge fixing conditions, the perturbed metric and scalar field is give by
ds2= e2W(y)?
e2√2bκφ(t)dy2− dt2+ e2α(t)(δij+ hij)dxidxj?
, (20)
for tensor perturbations,
ds2= e2W(y)?
for vector perturbations and
e2√2bκφ(t)dy2− dt2+ e2α(t)?
2Tidydxi+ 2Sidtdxi+ δijdxidxj??
, (21)
ds2= e2W(y)?
φ = φ(t) + Ξ(y) + κ−1δφ,
e2√2bκφ(t)(1 + 2N)dy2+ 2Adtdy − (1 + 2Φ)dt2+ e2α(t)(1 − 2Ψ)δijdxidxj)
?
,
(22)
for scalar perturbations, where perturbations are decomposed according to the tensorial type
of perturbations with respect to 3-space metric δij. Here Siand Tiare transverse vector
(∇iSi= 0 and ∇iTi= 0) and hij(y,t,x) is a transverse and traceless tensor(hi
0) where ∇iis the derivative operator on 3-space metric δij. Because each type of variables
obeys an independent closed set of equations in the 5-dimensional Einstein equations, we
i= 0,∇ihij=
derive the solutions for tensor, vector and scalar perturbations separately.
A. Tensor perturbations
The evolution equation for tensor perturbations is simple. The equation for tensor per-
turbation hij(y,t,x) = h(y,t)eipxeijis given by
e2√2bκφ?¨h + (3 ˙ α +
√2bκ˙φ)˙h + e−2αp2h
?
= h′′+ 3W′h′, (23)
where eijis a polarization tensor and dot denotes the derivative with respect to t and prime
denotes the derivative with respect to y. The junction condition for h(y,t) is imposed as
∂yh|y=y0= 0. (24)
We can use the separation of variables to solve this equation. The solution for hijis given
by
hij=
?
dmd3p h(m,p) fm(y)gm(η)eipxeij,(25)
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where
f0(y) = 1,
g0(η) = (−Hη)−
1
∆+2
?
H(1)
−
1
∆+2(−pη) + c(p,0)H(2)
−
1
∆+2(−pη)
?
, (26)
fm(y) = (sinhHy)
∆
2(∆+2)
iν(−pη) + c(p,m)H(2)
Pµ
H(1)
−1
2+iν(coshHy) −
Pµ+1
−1
Qµ+1
−1
iν(−pη)
2+iν(coshHy0)
2+iν(coshHy0)Qµ
?
and H(2)
−1
2+iν(coshHy)
,
gm(η) = (−Hη)−
1
∆+2
?
, (27)
where Pα
βand Qα
βare associated Legendre functions, H(1)
αα
are Hunkel functions
and
µ = −
∆
2(∆ + 2),ν =
?
?
?
?m2
H2−
1
(∆ + 2)2. (28)
The coefficients h(p,m) and c(p,m) are so far arbitrary and these are determined if one
specifies the initial conditions and boundary conditions in the bulk. The canonical variables
for the second-order perturbed action for tensor perturbation is given by
ϕ =
1
2κh.
(29)
Then the second-order perturbed five-dimensional action for the tensor perturbation is given
by
δS(T)=1
2
?
dydtd3xe3W(y)e
√2bκφ(t)e3α(t)(e−2√2bκφ(t)ϕ
′2− ˙ ϕ2− e−2α(t)p2ϕ2). (30)
It should be noted that the modes with 0 < m < −H/(∆ + 2) is not normalizable. Thus
the normalizable modes have a mass gap and the continuous massive modes start from
m = −H/(∆ + 2);
m ≥ −
H
∆ + 2.(31)
B. Vector perturbations
The calculations of the vector perturbations in this model are similar with those given
by Ref.[32]. In order to solve the Einstein equation for vector perturbations, it is convenient
to define a variable
Vi= S′
i−˙Ti= V ei, (32)
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where eiis the polarization vector. Then the Einstein equations of (0,i) and (i,y) compo-
nents are given by
e2α−2√2bκφ(V′+ 3W′V ) = p2S,
√2bκ˙φ)V ) = p2T,
e2α(˙V + (5 ˙ α −
(33)
where we expand the variables by eieipx. The (i,i) component is given by
√2bκ˙φ)S = e−2√2bκφ(T′+ 3W′T′).
˙S + (3 ˙ α +
(34)
We can easily find a master variable from which all perturbations are constructed;
S = e−3W−3α−√2bκφΩ′,
T = e−3W−3α+√2bκφ˙Ω,
V = e−3W−5α+√2bκφp2Ω. (35)
If Ω satisfies the evolution equation,
e2√2bκφ?¨Ω − (3 ˙ α −√2bκ˙φ)˙Ω + e−2αp2?
the 5-dimensional Einstein equation is automatically satisfied. The junction condition for
= Ω′′− 3W′Ω′, (36)
perturbations are imposed as
V |y=y0= T|y=y0= 0. (37)
Thus the master variable should satisfy
Ω|y=y0= 0. (38)
The solution that satisfies the junction condition is obtained as
Ω(t,x,y) =
?
dmd3p Ω(m,p) um(y)vm(t)eipx
(39)
where
u0(y) =
?y
y0e3W(y′)dy′,
v0(η) = (−Hη)
1
∆+2
?
H(1)
−
1
∆+2(−pη) + c(p,0)H(2)
−
1
∆+2(−pη)
?
, (40)
um(y) = (sinhHy)
∆+4
2(∆+2)
Pµ
iν(−pη) + c(p,m)H(2)
−1
2+iν(coshHy) −
Pµ
Qµ
−1
2+iν(coshHy0)
2+iν(coshHy0)Qµ
?
−1
−1
2+iν(coshHy)
,
vm(η) = (−Hη)
1
∆+2
?
H(1)
iν(−pη), (41)
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where
µ = −
∆ + 4
2(∆ + 2),ν =
?
?
?
?m2
H2−
1
(∆ + 2)2. (42)
The second order perturbed action is also written by Ω as
δS(V )=1
2p2
?
dydtd3xe−3W(y)e
√2bκφ(t)e−3α(t)(e−2√2bκφ(t)Ω
′2−˙Ω2− e−2α(t)p2Ω2). (43)
From this action, we can determine the normalization of the perturbations. It should be
noted that the 0-mode solution is not normalizable for vector perturbations.
C. Scalar perturbation
The scalar perturbations are more complicated than tensor and vector perturbations
due to the existence of the scalar field in the bulk. Unlike a maximally symmetric bulk
spacetime, we cannot find a master variable for scalar perturbations and this causes the
difficulty to solve the perturbations. Fortunately, in our background spacetime, there is a
simple coordinate system in the bulk, that is Eq. (15). Thus we can use the metric Eq. (15)
to find general solutions for perturbations in the bulk. In Ref [9], it was shown that there are
variables which make the equations for N,A,Φ,Ψ, and δφ in the bulk to be diagonalized.
Then by performing a coordinate transformation, it is easy to find general solutions for
perturbations in our background spacetime. However, we should proceed further to derive
solutions for perturbations. On the brane, the perturbations should satisfy the boundary
conditions. In terms of the metric perturbations, the boundary conditions are given by
Ψ′|y=y0= −W′(N −
Φ′|y=y0= W′(N −√2bδφ)|y=y0,
δφ′|y=y0= 3√2bW′(N −√2bδφ)|y=y0,
A|y=y0= 0.
√2bδφ)|y=y0,
(44)
A problem is that if we rewrite these conditions in terms of variables that make the bulk
equations to be diagonalized, the boundary conditions become complicated. Indeed, we will
find that the boundary conditions are not diagonalized and also they effectively contain
time derivatives of the variables. Thus unlike vector and tensor perturbations, imposing
the boundary conditions is not so easy. This is in contrast to the case for the static brane
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which sits at constant value of the z in the coordinate Eq.(15). For the static brane, the
boundary conditions for variables which make the equations in the bulk to be diagonalized
also make the boundary conditions to be diagonalized and also they do not contain the time
derivative of the variables. Thus, this complexity of the boundary conditions reflects the
fact that our brane is moving in the coordinate Eq.(15). This movement of the brane causes
the cosmological expansion on the brane. Hence, it is an essential part of the calculations
to find particular solutions which satisfy the boundary conditions at the brane. In fact, it
has been recognized that this is the central part of the calculations of scalar cosmological
perturbations. In general, the bulk perturbations are not written by separable functions
with respect to a brane coordinate and a bulk coordinate. Thus there is almost no hope
to find particular solutions analytically and it has prevented us from understanding the
behavior of scalar perturbations on the cosmological brane. In our background spacetime,
the bulk perturbations are obtained analytically by separable functions and it enables us to
derive the solutions which properly satisfy the boundary conditions on the brane.
Now, we will describe our procedure to derive solutions. We first solve the perturbation
in a static coordinate;
ds2= e2Q(z)((1 + 2Γ)dz2− (1 + 2φ)dτ2+ 2Gdzdτ + (1 − 2ψ)δijdxidxj),
φ = φ(t) + κ−1δφ. (45)
It is possible to find variables which make the equations diagonal [9] (see Appendix A-2)
ωc = δφ + 3√2bψ,
ωψ = Γ − 2ψ,
ωN = Γ −√2bδφ,
ωA = G.(46)
The evolution equations in the bulk are obtained from 5-dimensional Einstein equation as
25ωc = 0,
25ωψ = 0,
25ωN =
2(∆ + 4)
∆ + 2
2
∆ + 2
1
z2ωN,
25ωA =
1
z2ωA, (47)
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where
25=
∂2
∂z2+
2
∆ + 2
1
z
∂
∂z−
?∂2
∂τ2+ p2
?
. (48)
By performing a coordinate transformation, the evolution equation in our background space-
time can be derived
25ωc = 0,
25ωψ = 0,
25ωN =
2(∆ + 4)
∆ + 2
2
∆ + 2
H2
(−Hη)2sinh2HyωN,
H2
(−Hη)2sinh2HyωA,
25ωA =
(49)
where
25=
?
1
−Hη
?2?∂2
∂y2+
2
∆ + 2H cothHy∂
∂y
?
−
?∂2
∂η2+∆ + 4
∆ + 2
1
η
∂
∂η+ p2
?
. (50)
The solutions for ωiare given by
ωc =
?
×(−Hη)−
?
×(−Hη)−
?
×(−Hη)−
?
×(−Hη)−
dmcNc(mc,p)(sinhHy)
1
∆+2Hiν(mc)(−pη),
dmψNψ(mψ,p)(sinhHy)
1
∆+2Hiν(mψ)(−pη),
dmANA(mA,p)(sinhHy)
1
∆+2Hiν(mA)(−pη),
dmNNN(mN,p)(sinhHy)
1
∆+2Hiν(mN)(−pη),
∆
2(∆+2)
?
Pµ
−1
2+iν(mc)(coshHy) + CcQµ
−1
2+iν(mc)(cosh(Hy))
?
ωψ =
∆
2(∆+2)
?
Pµ
−1
2+iν(mψ)(coshHy) + CψQµ
−1
2+iν(mψ)(cosh(Hy))
?
ωA =
∆
2(∆+2)
?
Pµ+1
−1
2+iν(mA)(coshHy) + CAQµ+1
−1
2+iν(mA)(cosh(Hy))
?
ωN =
∆
2(∆+2)
?
Pµ+2
−1
2+iν(mN)(coshHy) + CNQµ+2
−1
2+iν(mN)(cosh(Hy))
?
(51)
where
µ = −
∆
2(∆ + 2),ν(m) =
?
?
?
?m2
H2−
1
(∆ + 2)2.(52)
The perturbations Ψ,δφ,N,A and Φ are related to ψ,δφ,Γ,G and φ by a coordinate trans-
formation;
Ψ = ψ,δφ = δφ,Φ = Ψ − N,
N = Γcosh2Hy − φsinh2Hy + GsinhHy coshHy,
A = −(−Hη)
3∆+8
3(∆+2)
?
2(Γ − φ)coshHysinhHy + G(sinh2Hy + cosh2Hy)
?
. (53)
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From Eq.(46), ψ,δφ,Γ,G and φ are written in terms of ωias
ψ = −
2
3(∆ + 4)(ωψ− ωN−
2√2b
∆ + 4
4
3(∆ + 4)(ωN+
φ = ψ − Γ.
√2bωc),
δφ =
?
ωψ− ωN+
√2
3bωc
?
,
Γ =
√2bωc+ 3b2ωψ),
(54)
Then the general solutions for the metric perturbations in the bulk are obtained as
Ψ = −
2
3(∆ + 4)(ωψ− ωN−√2bωc),
2√2b
∆ + 4(ωψ− ωN+
2
3(∆ + 4)(2 + 3sinh2Hy)ωN+
2√2b
3(∆ + 4)(2 + 3sinh2Hy)ωc+ sinhHycoshHyωA,
?
+ (1 + 2sinh2Hy)ωA
.
δφ =
√2
3bωc),
N =
?
4b2
∆ + 4+2(∆ + 3) ∆ + 4
sinh2Hy
?
ωψ
+
A = −(−Hη)
3∆+8
3(∆+2)
2sinhHy coshHy
?
2
∆ + 4ωN+2(∆ + 3)
∆ + 4
ωψ+2√2b
∆ + 4ωc
?
?
(55)
We should impose boundary conditions on the brane. In terms of ωi, the junction condi-
tions become
ω′
ωA = −2coshHy0sinhHy0
1 + 2sinh2Hy0
∆ + 4
∆ + 2H cothHy0
?
∆ + 4
c= 0, (56)
?
2
∆ + 4ωN+2(∆ + 3)
ωN+1
2tanhHy0ωA
?
∆ + 4
ωψ+2√2b
∆ + 4ωc
?
, (57)
ω′
ψ− ω′
N=
??
, (58)
2
∆ + 4sinh2Hy0ω′
N+ 1 +2(∆ + 3)sinh2Hy0
ω′
ψ+ sinhHy0coshHy0ω′
A= 0,(59)
where above equations should be evaluated on the brane y = y0. The variables should
also satisfy the ”constraint equations” which do not include the second derivatives of the
variables with respect to t and y in 5-dimensional Einstein equations, that is, (t,i), (y,i) and
(y,t) components of Einstein equaitons. Among them, the equation obtained by combining
(0,i) and (y,i) components of the 5-dimensional Einstein equation (see Appendix A-2 for
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derivation)
(1 + sinh2Hy)ω′
A+
2
∆ + 2H cothHyωA+ 2sinhHy coshHyω′
= −(−Hη)
ψ
?
coshHy sinhHy∂ηωA+ 2(1 + sinh2Hy)∂ηωψ
?
, (60)
will be useful to find solutions. Because ωAis a variable which is associated with Z2odd
variable A, ω′
Ain the junction conditions would be eliminated using constraint equations.
Indeed, by projecting Eq.(60) on the brane, we find that ω′
A(y0) can be rewritten in terms
of ωA,ω′
ψand ωψ. An important point is that this equation contains the time derivative of
the variables. Thus effectively, the boundary conditions contain the time derivative of the
metric perturbations. It should be noted the junction condition for ωcis decoupled from
other variables. This variable ωcis the canonical variable for the second order action and it
is directly related to the curvature perturbations on the brane. Thus we do not need to know
the full solutions for perturbations in deriving the solutions for curvature perturbation. On
the other hand, in order to derive the anisotropic stress induced by bulk perturbations, we
should know the solutions for all variables, as we will observe later. This indicates that the
anisotropic stress on the brane is complicatedly coupled to bulk perturbations compared
with the curvature perturbation.
We describe the procedure to derive the solutions which satisfy the boundary conditions
and constrained equations in Appendix B. In this section we only show the results. The
solutions are written as the summation of KK modes with mass m;
ωi=
?
d3pd˜ mN(˜ m,p)ωi(˜ m)(y,t)eipx, (61)
where ˜ m = m/H. For 0-mode with m = 0, we get
ωc(0) = (−Hη)−
1
∆+2H−
1
∆+2(−pη),
ωψ(0) = −
√2b
∆ + 3
?
?
(−Hη)−
1
∆+2H−
1
∆+2(−pη)
?
+ 1 +2(∆ + 3)
∆ + 4
sinh2Hy(−Hη)−
1
∆+2H−2∆+5
∆+2(−pη)
?
,
ωN(0) = −2√2b
∆ + 4sinh2Hy(−Hη)−
4√2b
∆ + 4sinhHycoshHy(−Hη)−
1
∆+2H−2∆+5
∆+2(−pη),
ωA(0) =
1
∆+2H−2∆+5
∆+2(−pη).(62)
13
Page 14
For massive modes with m ≥ −H/(∆ + 2)
√2(∆ + 2)
4b
ωc(˜ m) = −
(iν − 1)(sinhHy)
∆
2(∆+2)Bµ
−1
2+iν(coshHy)(−Hη)−
1
∆+2Hiν(−pη),
ωψ(˜ m) = (sinhHy)
∆
2(∆+2)(−Hη)−
? ?iν −∆+1
iν −∆+3
∆
2(∆+2)(−Hη)−
1
∆+2
?
?
−1
2Bµ
−1
2+iν(coshHy)Hiν(−pη)
−
1
2
?iν −2∆+3
iν −
∆+2
1
∆+2
∆+2
∆+2
Bµ
−5
2+iν(coshHy)Hiν−2(−pη)
?
,
ωN(˜ m) = (sinhHy)
1
∆+2
?∆ + 2
2
?
1
iν −
1
∆+2
?
Bµ+2
−1
2+iν(coshHy)Hiν(−pη)
−
1
2
?
1
iν −
1
∆+2
??
1
iν −∆+3
∆+2
?
Bµ+2
−5
2+iν(coshHy)Hiν−2(−pη)
?
,
ωA(˜ m) = (sinhHy)
∆
2(∆+2)(−Hη)−
1
∆+2
??
1
iν −
1
∆+2
?
Bµ+1
−1
2+iν(coshHy)Hiν(−pη)
−
?
1
iν −
1
∆+2
??iν −∆+1
iν −∆+3
∆+2
∆+2
?
Bµ+1
−5
2+iν(coshHy)Hiν−2(−pη)
?
, (63)
where
Bα
β(coshHy) = Pα
β(coshHy) −
Pµ+1
−1
Qµ+1
−1
2+iν(coshHy0)
2+iν(coshHy0)Qα
β(coshHy),(64)
and Hαis the arbitrary combination of Hunkel functions H(1)
α
and H(2)
α. Then the solutions
for metric perturbations are derived as
Ψ(y,t,x) =
?
?
?
?
d3pd˜ mN(˜ m,p)Φ(˜ m)(y,t)eipx,
d3pd˜ mN(˜ m,p)δφ(˜ m)(y,t)eipx,
d3pd˜ mN(˜ m,p)N(˜ m)(y,t)eipx,
d3pd˜ mN(˜ m,p)A(˜ m)(y,t)eipx,
δφ(y,t,x) =
N(y,t,x) =
A(y,t,x) =
Φ(y,t,x) = Ψ(y,t,x) − N(y,t,x). (65)
where
Ψ(0) =
2√2b
3(∆ + 4)
4
3(∆ + 4)
√2bδφ(0),
?∆ + 4
?
∆ + 3(−Hη)−
∆ + 4
4(∆ + 3)(−Hη)−
1
∆+2H−
1
∆+2(−pη) +
1
∆ + 3(−Hη)−
3b2
∆ + 3(−Hη)−
1
∆+2H−2∆+5
∆+2(−pη)
?
,
δφ(0) =
1
∆+2H−
1
∆+2(−pη) −
1
∆+2H−2∆+5
∆+2(−pη)
?
,
N(0) =
A(0) = 0, (66)
and
Ψ(˜ m)(y,t) = −
2
3(∆ + 4)(−Hη)−
1
∆+2(sinhHy)
∆
2(∆+2)
14
Page 15
×
?∆ + 2
2
?
−
1
iν −
??
1
∆+2
Bµ+2
−1
2+iν(coshHy) +
?
iν −∆ + 3
∆ + 2
?
Bµ
−1
2+iν(coshHy)
?
Hiν(−pη)
+
?
1
iν −∆+3
∆+2
1
iν −
1
∆+2
??1
2Bµ+2
−5
2+iν(coshHy) −1
2
?
iν −2∆ + 3
∆ + 2
??
iν −∆ + 1
∆ + 2
?
× Bµ
−5
2+iν(coshHy)
2√2b
∆ + 4(−Hη)−
?
iν −
?
iν −∆+3
× Bµ
?
Hiν−2(−pη)
?
,
δφ(˜ m)(y,t) =
1
∆+2(sinhHy)
∆
2(∆+2)
?∆ + 2
1
3b2
2
?
×−
1
1
∆+2
Bµ+2
−1
??
2+iν(coshHy) −
iν −
∆
4(∆ + 2)
?
Bµ
−1
2+iν(coshHy)
?
Hiν(−pη)
+
1
∆+2
1
iν −
1
∆+2
??1
2Bµ+2
−5
2+iν(coshHy) −1
2
?
iν −2∆ + 3
∆ + 2
??
iν −∆ + 1
∆ + 2
?
−5
2+iν(coshHy)
?
Hiν−2(−pη)
?
,
N(˜ m)(y,t) = (−Hη)−
1
∆+2(sinhHy)
∆
2(∆+2)
?2(∆ + 2)
3(∆ + 4)
?
×
?
1
iν −
1
∆+2
Bµ+2
−1
??
2+iν(coshHy) −
iν −
∆
4(∆ + 2)
?
Bµ
−1
2+iν(coshHy)
?
Hiν(−pη)
+
?
1
iν −∆+3
?
Bµ+2
−5
∆+2
1
iν −
? ?
1
∆+2
iν −2∆ + 3
∆ + 2
?
??
−
2
∆ + 4
?1
3Bµ+2
−5
2+iν(coshHy)
+ b2
iν −∆ + 1
∆ + 2
?
Bµ
−5
? ?
2+iν(coshHy)
iν −2∆ + 3
?
?
+∆ + 2
∆ + 4(iν − 1)
×
× Hiν−2(−pη)],
?
2+iν(coshHy) −
iν −∆ + 1
∆ + 2∆ + 2
Bµ
−5
2+iν(coshHy)
?
sinh2Hy
?
A(˜ m)(y,t) = −(−Hη)
3∆+8
3(∆+2)
?
1
iν −
?
1
∆+2
??
−Bµ+1
−1
2+iν(coshHy)Hiν(−pη)
+ 2sinhHycoshHy
1
iν −∆+3
? ?
1
iν −∆+3
∆+2
iν −∆ + 1
∆ + 2
??
??
−
1
∆ + 4Bµ+2
−5
2+iν(coshHy)
−
∆ + 3
∆ + 4
?
iν −2∆ + 3
∆ + 2
?
Bµ
−5
2+iν(coshHy)
?
Hiν−2(−pη)
+ (1 − 2cosh2Hy)
?
∆+2
iν −∆ + 1
∆ + 2
?
Bµ+1
−5
2+iν(coshHy)Hiν−2(−pη)
?
. (67)
The solutions for massive modes evaluated on the brane become somewhat simple because
the function Bα
βsatisfies
Bµ+1
−1
2+iν(coshHy0) = 0. (68)
Using the equations presented in Appendix B, we get
Ψ(˜ m)(y0,t) =
1
3(−Hη)−
1
∆+2(sinhHy0)
∆
2(∆+2)Bµ
−1
2+iν(coshHy0)
15
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