Page 1
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.
1
Page 2
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
2
Page 3
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.
3
Page 4
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)
4
Page 5
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)
5
Page 6
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)
6
Page 7
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)
7
Page 8
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)
8
Page 9
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
9
Page 10
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)
10
Page 11
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)
11
Page 12
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
12
Page 13
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
Page 16
×
1
2(−Hη)−
?
Hiν(−pη) −
1
∆ + 2
?
1
iν −∆+3
∆
2(∆+2)Bµ
∆+2
?
Hiν−2(−pη)
?
,
Φ(˜ m)(y0,t) =
1
∆+2(sinhHy0)
−1
2+iν(coshHy0)
×
1
3Hiν(−pη) +
iν −
3∆+7
3(∆+2)
iν −∆+3
∆
2(∆+2)Bµ
∆+2
Hiν−2(−pη)
−1
?
,
δφ(˜ m)(y0,t) = (−Hη)−
1
∆+2(sinhHy0)
√2(∆ + 2)
4b
2+iν(coshHy0)
×
?
−
?
iν +
2
3(∆ + 2)
Hiν(−pη) +
√2b
∆ + 2
?
1
iν −∆+3
∆+2
?
Hiν−2(−pη)
?
.
(69)
These solutions are first main results of this paper. They provide us the connection between
the behavior of the perturbations on the brane and the perturbations in the bulk.
The second order action for scalar perturbations is written in terms of the canonical
variable ωc;
δS(S)=
1
2κ2
?
dydtd3xe3W(y)e
√2bκφ(t)e3α(t)(e−2√2bκφ(t)ω
′2
c− ˙ ω2
c− e−2α(t)p2ω2
c). (70)
This can be verified using the result for the metric Eq.(15) because ωcdoes not change by
the coordinate transformation. This action is the same as the second order action for tensor
perturbations. Then, the massive modes with 0 < m < −H/(∆ + 2) are not normalizable.
Thus there is also mass gap for the scalar perturbations.
IV. PRIMORDIAL FLUCTUATIONS IN THE BULK INFLATON MODEL
In the previous section, we obtained the classical solutions for cosmological perturbations.
These perturbations properly satisfy the boundary conditions at the brane. However, the
boundary conditions on the brane alone do not fix the solutions completely. There remains
a freedom to choose the ”weight” N(˜ m,p) in the summation of KK modes Eq.(61). These
coefficients are fixed once one more boundary condition in the bulk is specified. Because the
brane is inflating, it is natural to specify the boundary conditions for the perturbations by
quantum theory in the same way as the usual 4-dimensional inflationary model. We have
already derived the second order 5-dimensional action for perturbations, the quantization
can be done within the full 5-dimensional theory.
16
Page 17
In the previous paper, we have already carried out the quantization of scalar and tensor
perturbations. It was shown that the KK modes are well suppressed at large scales even
if the energy scale of the inflation is sufficiently higher than the scale of the bulk. More
precisely, the bulk curvature scale and the Hubble constant on the brane are determined by
the bulk potential and the deviation from the RS tuning, respectively. Thus their ratio,
r =
?????
δ
∆/8 + δ
?????, (71)
characterize the behavior of perturbations. If r is large, we expect the five dimensional
nature of the perturbations become important. However, we showed that due to the mass
gap in the KK spectrum, the massive KK modes are hardly excited. Thus, at large scales,
the behavior of tensor perturbation and curvature perturbation defined by
Rc=˙ α
˙φωc, (72)
are essentially four-dimensional except for the amplitude of the perturbations.
Hence, we might expect that this model cannot be distinguish from the usual four-
dimensional inflationary model. However, in the brane world, the curvature perturbation
Rcalone does not determine CMB anisotropies. The anisotropic stress δπEinduced by bulk
gravitational fields also affects the CMB anisotropies. The anisotropic stress is measured by
the difference between Φ and Ψ;
(Ψ − Φ)|y=y0≡ κ2e2αδπE. (73)
where κ4= κλ0. Then we should also determine the initial condition for δπE during the
inflation. From the five-dimensional Einstein equation we find that δπEis related to N;
N|y=y0= κ2e2αδπE. (74)
This implies that it is not sufficient to determine the behavior of the canonical variable ωc
but we need the solutions for all ωi. As already observed, the boundary conditions for ωi
except for ωcare complicated and this reflects the fact that the brane is ”moving”. Thus
we expect that the anisotropic stress can have a distinguishable feature which the curvature
perturbation Rcdoes not possess. Because we can derive the solutions for N, it is possible
to investigate the behavior of δπE.
17
Page 18
A. Behavior of the canonical variable ωc
We first review the quantization of canonical variable ωc. The second order action for ωc
is nothing but the action for a 5-dimensional massless scalar field. Then the quantization is
easily carried out. The canonical variable ωccan be expressed as
κ−1ωc(t,x,z) =
?
d˜ md3p
?
ap ˜ mθ˜ m(y)χ˜ m(t)eipx+ (h.c.)
?
. (75)
Here ap ˜ mis the annihilation operator and satisfies the following commutation relation,
?
ap ˜ m,a†
p′˜ m′
?
= δ(p − p′)δ(˜ m − ˜ m′). (76)
The modes functions are given by
θ0(y) =
1
√2
?
?
−1 −∆
8δ
?−
?−
1
2(∆+2)??∞
1
2(∆+2)(|ξ|2+ |ζ|2)−1
y0
(sinhHy′)
2
∆+2dy′
?−1
2, (77)
θ˜ m(y) =
H
2
?
−1 −∆
8δ
2(sinhHy)
∆
2(∆+2)Bµ
−1
2+iν(coshHy), (78)
and
χ0(η) =
√π
2H−1
√π
2H−1
2(−Hη)−
1
∆+2H(1)
−
1
∆+2(−pη), (79)
χ˜ m(η) =
2(−Hη)−
1
∆+2e−νπ
2H(1)
iν(−pη), (80)
where
µ = −
∆
2(∆ + 2),
Γ(iν)
Γ(∆+1
ν =
?
˜ m −
1
(∆ + 2)2, (81)
ξ =
∆+2+ iν), (82)
ζ =
Γ(−iν)
Γ(∆+1
∆+2− iν)−
Pµ+1
−1
Qµ+1
−1
2+iν(coshHy0)
2+iν(coshHy0)πeµπiΓ(
1
∆+2+ iν)
Γ(1 + iν)
. (83)
Now the spectrum of the KK modes N(˜ m,p) is determined as
√2π
4
√πb
∆ + 2
N(0,p) = κ
H−1
2
?
−1 −∆
1
iν − 1
8δ
?−
−1 −∆
1
2(∆+2)??∞
y0
(sinhHy′)
2
∆+2dy′
?−1
2,
N(˜ m,p) = −κ
? ? ?
8δ
?−
1
2(∆+2)(|ξ|2+ |ζ|2)−1
2. (84)
The ratio of massive modes and massless mode increases with r. But the ratio becomes
constant for large r. In the previous paper, it was shown that this is caused by the existence
18
Page 19
of mass gap. And also, the amplitude of massive modes are dumped after the horizon
crossing. Thus the spectrum of the massive modes is blue tilted, so the contribution from
massive modes becomes negligible at large scales. We should note that the integration over
˜ m logarithmically diverges. Thus we need some regularization scheme.
B. Behavior of anisotropic stress
Now we turn to the anisotropic stress
κ2
4e2αδπE= (Ψ − Φ)|y=y0. (85)
First let us consider the 0-mode. The 0-mode solution satisfies
κ2
4e2αδπE = N(0) =
√2bδφ(0)
=
?
d3pN(0,p)
3b2
∆ + 3(−Hη)−
4√2b
3(∆ + 4)
?
∆ + 4
4(∆ + 3)(−Hη)−
1
∆+2H(1)
−
1
∆+2(−pη)
−
1
∆+2H(1)
−2∆+5
∆+2(−pη)
?
eipx. (86)
As mentioned in section II, the effective theory for background spacetime is given by the
BD theory. In the BD theory the correspondent equation is given by
Ψ − Φ =δϕBD
ϕBD
=
√2bδφ. (87)
As expected, the 0-mode solution can be described by the BD theory including anisotropic
stress. At late times −pη → 0, N(0) behaves as N(0) = const.
The massive modes also contribute to the anisotropic stress;
κ2
4e2αδπE =
1
2
?
d3p
?∞
−
1
∆+2
d˜ mN(˜ m,p)(sinhHy0)
iν −∆+3
∆
2(∆+2)Bµ
−1
2+iν(coshHy0)(−Hη)−
1
∆+2
×
1
3H(1)
iν(−pη) −
iν −
3∆+5
3(∆+2)
∆+2
H(1)
iν−2(−pη)
eipx. (88)
At the horizon crossing −pη = 1, the ratio of the amplitude of massive mode to 0-mode
modes has similar feature with the curvature perturbation. The ratio increases with r, but
it becomes constant for large r due to the mass gap. Note that the term proportional to
H(1)
iν−2(−pη) does not give an additional divergence in the integration over ˜ m.
However, the subsequent evolution of the anisotropic stress is quite different from the
curvature perturbation. After the horizon crossing, the 0-mode remains constant. On the
19
Page 20
other hand, the massive modes increase as (−pη)−(2∆+5)/(∆+2)∝ e−3(2∆+5)α/2due to the term
proportional to Hiν−2(−pη). Thus if ∆ < −5
and it seems to leave significant consequences in the inflationary brane world. Indeed, the
2, the massive modes will dominate over 0-mode
massive modes on the brane Eqs.(69) grow for (−pη) → 0. Then one might worry that this
indicates the gravitational instability of the spacetime. However, the physical amplitude of
the anisotropic stress is given by
δπE∝ e−2αN|y=y0∝ e−(6∆+19)α/2, (89)
which always decreases with time for −2 > ∆ > −8/3. The same situation occurs in the
analysis of the radion in Randall-Sundrum de Sitter two branes. Let us consider two de Sitter
branes in AdS5spacetime. By imposing a fine tuning on the tensions of two branes, the
distance between two branes, the radion, becomes constant. However, if one considers the
perturbation of the radion, the radion has negative mass squared. In ref [33], the effect of the
quantum radion was investigated. They found that the metric perturbations in Longitudinal
gauge grow due to the instability of the radion. However, the physical amplitude of the
anisotropic stress itself decays. The resolution is that the Longitudinal gauge is not really
a good gauge. It is possible to find a gauge where all perturbations do not grow. Thus the
growth of the metric perturbations does not directly imply the instability of the spacetime.
In our case, the same arguments should be applied. In general, the curvature perturbation
Rcis the measure of the linear perturbation amplitude. In our background the curvature
perturbation does not show the instability. Thus the growth of the metric perturbation is
the artifact of the bad choice of the gauge.
Let us explicitly show that we can find a suitable gauge where all perturbations are not
growing. Let us consider a 4-dimensional gauge transformation
η → η − ǫη,xi→ xi− ǫ,i, (90)
By choosing ∂ηǫ = ǫη, the metric and the scalar field are transformed as
ds2= e2α?
˜δφ = δφ − κ(∂ηφ)ǫη,
−(1 + 2˜Φ)dη2+
?
(1 − 2˜Ψ)δij+ E,ij
?
dxidxj?
,
(91)
where
˜Φ = Φ − ∂ηǫη− (∂ηα)ǫη,
20
Page 21
˜Ψ = Ψ + (∂ηα)ǫη,
E = −ǫ. (92)
It is a non-trivial problem to find ǫηthat simultaneously eliminates the growing part of the
Φ(m), Ψ(m) and δφ(m). In this case, it is possible to find such a solution. By taking
ǫη(˜ m) = −1
2H
?
1
iν −∆+3
iν−2(−pη) + H(1)
∆+2
?
(sinhHy0)
∆
2(∆+2)Bµ
−1
2+ıν(coshHy0)(−Hη)
∆+1
∆+2
×
?
H(1)
iν(−pη)
?
, (93)
the resultant metric and the scalar field become
˜Ψ(˜ m) =1
3
?
iν − 1
iν −∆+3
iν − 1
iν −∆+3
√2
4b(∆ + 2)
∆+2
?
(sinhHy0)
∆
2(∆+2)Bµ
−1
2+iν(coshHy0)(−Hη)−
1
∆+2H(1)
iν(−pη),
˜Φ(˜ m) =
2
3
?
∆+2
?
(sinhHy0)
∆
2(∆+2)Bµ
−1
2+iν(coshHy0)(−Hη)−
1
∆+2H(1)
iν(−pη),
˜δφ(˜ m) = −
?
iν −
1
3(∆ + 2)
??
iν − 1
iν −∆+3
∆+2
?
×(sinhHy0)
∆
2(∆+2)Bµ
−1
2+iν(coshHy0)(−Hη)−
1
∆+2H(1)
iν(−pη),
∂ηE(˜ m) = −ǫη
(94)
At late times −pη → 0, E behaves as
E ∝ (−η)−
1
∆+2∝ e−3α/2→ 0. (95)
Thus we can find a gauge where all perturbations remain small. The anisotropic shear E
induced by massive modes decreases in an inflationary background ∆ < −2. Because the
amplitude of E at the horizon crossing is suppressed due to the mass gap, the effect of the
massive modes is always negligible.
In summary, in the anisotropic stress, the contribution of the massive modes will dom-
inate over 0-mode and this causes the growth of metric perturbations in the Longitudinal
gauge on the brane. However, we should carefully choose the gauge in evaluating the effect
of these massive modes on metric perturbations. It is possible to find a ”good” gauge where
all perturbations remain small. In this gauge, the contribution of massive modes are not
strong enough to cause the instability on the brane and massive modes do not leave signif-
icant consequences. So we conclude that, in our bulk inflaton model, the long-wavelength
21
Page 22
perturbations are quite similar with the perturbations in the BD theory described by the
0-mode solution even for the high energy inflation.
V.CONCLUSION
In this paper, we derived the exact and analytic solutions for cosmological perturbations
in dilatonic brane worlds. We used a background spacetime where the brane undergoes a
power-law expansion due to the bulk scalar field. The effective theory on the brane is given
by a Brans-Dicke theory. The interesting feature of this model is that we can derive the
exact background solutions including the back-reaction of the bulk scalar field. Moreover the
spacetime metric is separable with respect to the brane coordinate and the bulk coordinate.
Then it is possible to solve the cosmological perturbations analytically.
Scalar perturbations are quite complicated because of the existence of the bulk scalar
field. We can find variables which make the equations in the bulk to be diagonalized. But
the boundary conditions for these variables are not diagonalized and also they effectively
contain the time derivatives of the variables. The exception is the canonical variable ωcof
the action which is related to the curvature perturbation Rcon the brane. The evolution
equation for ωcand the junction condition on the brane are decoupled from other variables.
However, if one wants to derive the solutions for all metric perturbations, we should solve the
complicated boundary conditions. Because this complexity is caused by the expansion of the
brane, the derivation of the solutions for this problem is a central part of the calculations
of cosmological perturbations in the brane world. This difficulty has prevented us from
understanding the behavior of scalar perturbations on the brane. In this background, it
is possible to derive the solution analytically. Then we found the solutions for all metric
perturbations which properly satisfy the junction conditions on the brane.
As an application, we have investigated the behavior of the anisotropic stress on the
brane induced by the bulk perturbations. We used the quantum theory for the 5-dimensional
perturbations to determine the amplitude of the perturbations. As was shown in our previous
paper, the massive KK modes do not significantly contribute to the curvature perturbations
even in the high energy inflation where the Hubble scale on the brane is larger than the
curvature scale in the bulk. In the anisotropic stress, the contribution of the massive modes
is suppressed at the horizon crossing. Remarkably, however, the subsequent evolution of
22
Page 23
the anisotropic stress is quite different from the curvature perturbations where massive
modes rapidly decays. In the anisotropic stress, the contribution of massive modes seems to
dominate over 0-mode after the horizon crossing for ∆ < −5/2. The difference comes from
the junction conditions. The junction conditions effectively include the time derivative of
the variables except for the junction condition for curvature perturbation. This causes the
growth of the massive modes. However, it also causes the growth of metric perturbations
in the Longitudinal gauge. Thus a careful choice of the gauge was required to discuss the
effects of these massive modes. We found a suitable gauge where all perturbations remain
small. In this gauge, there is an anisotropic shear which describes the contribution of the
massive modes. It was shown that the anisotropic shear decays in our spacetime. Thus, the
contribution of massive modes are not strong enough to cause the instability on the brane
and massive modes do not leave significant consequences. We concluded that, in our bulk
inflaton model, the perturbations are quite similar with the perturbations in the BD theory
described by the 0-mode solution at large scales even for the high energy inflation.
Our analysis indicates that the behavior of anisotropic stress is quite complicated com-
pared with the curvature perturbation. The behavior of the curvature perturbation can be
determined by partially solving the perturbations. But in order to determine the anisotropic
stress, we needed to derive full solutions for perturbations. This is indeed the generic feature
of the brane world cosmological perturbations. For example, in the Randall-Sundrum model,
the behavior of the curvature perturbation is determined only by the conservation of the
energy-momentum tensor on the brane at large scales. But the anisotropic stress can be de-
termined only if the gravitational field in the bulk is completely specified. So far the analysis
of this anisotropic stress is very limited due to the difficulty of solving the full 5-dimensional
perturbations. Our solutions would provide an interesting toy model for the investigation
about the relation between the behavior the anisotropic stress on the brane and the bulk
gravitational field. In this paper, we determine the boundary condition for perturbations in
the bulk by quantum theory. But it is also possible to consider other boundary conditions.
For example, it might be interesting to consider the boundary condition which allows the
existence of dark radiation on the brane. Of course, in the inflationary background, the
dark radiation does not play a role, but it is certainly important to fully understand the
relation between the geometry of the bulk and the behavior of anisotropic stress, because it
plays an essential role in the calculation of the CMB anisotropies in the brane worlds. For
23
Page 24
this purpose, it may be useful to re-derive our results using a covariant curvature formalism
because the geometrical meanings is manifest in this formulation. We will report these issues
in the near future [34].
APPENDIX A: 5-DIMENSIONAL EINSTEIN EQUATION
1. 5-dimensional Einstein equation
For convenience, we present the 5-dimensional Einstein equation for the scalar perturba-
tions of the metric
ds2= −e2β(t,y)(1+2Φ)dt2+2e2β(t,y)Adtdy+e2γ(t,y)(1+2N)dy2+e2α(t,y)(1−2Ψ)dxidxi. (A1)
(t,t) component
− 3e−2γΨ′′+ e−2α∇2N − 2e−2α∇2Ψ
− 3e−2β˙ α˙N + 3e−2β(2 ˙ α + ˙ γ)˙Ψ − 3e−2γα′N′− 3e−2γ(4α′− γ′)Ψ′+ 3e−2γ˙ αA′
− 6e−2γ?
= κ2(e−2β(Φ˙φ2−˙φ˙δφ) − Λ′δφ + e−2γ(Nφ′2− φ′δφ′)).
α′′+ 2(α′)2− α′γ′?
N + 6e−2β˙ α( ˙ α + ˙ γ)Φ + 3e−2γ[ ˙ α′+ ˙ α(3α′+ β′− γ′)]A
(A2)
(t,i) component
−
− e−2β( ˙ α − ˙ γ)N − e−2β(2 ˙ α + ˙ γ)Φ −1
= −κ2e−2β˙φδφ.
1
2e−2γA′+ e−2β˙N − 2e−2β˙Ψ
2e−2γ(α′+ 3β′− γ′)A
(A3)
(t,y) component
− 3e−2β˙Ψ′− 3e−2βα′˙N − 3e−2β(α′− β′)˙Ψ − 3e−2β˙ αΦ′+1
− 6e−2β( ˙ α′+ ˙ αα′− ˙ αβ′− α′˙ γ)Φ − 3e−2γ?
= κ2e−2β?
(i,i) component
2e−2α∇2A
α′′+ (α′)2− α′(β′+ γ′)
?
A
Aφ′2+ 2˙φφ′Φ −˙φδφ′− φ′δ˙φ
?
.(A4)
− e−2β¨ N + 2e−2β¨Ψ − 2e−2γΨ′′+ e−2γΦ′′+ e−2α∇2N − e−2α∇2Ψ + e−2α∇2Φ + e−2γ˙A′
24
Page 25
− e−2β(2 ˙ α −˙β + 2˙ γ)˙N + 2e−2β(3 ˙ α −˙β + ˙ γ)˙Ψ + e−2β(2 ˙ α + ˙ γ)˙Φ + e−2γ(2α′+ 2β′− γ′)˙A
− e−2γ(2α′+ β′)N′− 2e−2γ(3α′+ β′− γ′)Ψ′+ e−2γ(2α′+ 2β′− γ′)Φ′+ e−2γ(2 ˙ α +˙β)A′
− 2e−2γ?
+ 2e−2β?
+ e−2γ?
= κ2?
(i ?= j) component
N − Ψ + Φ = 0.
2α′′+ β′′+ 3(α′)2+ 2α′β′− 2α′γ′+ (β′)2− β′γ′?
2¨ α + ¨ γ + 3 ˙ α2− 2 ˙ α˙β + 2 ˙ α˙ γ −˙β ˙ γ + ˙ γ2?
4 ˙ α′+ 2˙β′+ 2 ˙ α(3α′+ β′+ γ′) +˙β(2α′+ 2β′− γ′) − ˙ γ(2α′+ β′)
−e−2β(Φ˙φ2−˙φ˙δφ) − Λ′δφ − e−2β˙φφ′A + e−2γ(Nφ
N
Φ
?
A
′2− φ′δφ′)
?
. (A5)
(A6)
(i,y) component
2 Ψ′− Φ′−1
= κ2φ′δφ.
2
˙A + (2α′+ β′)N + (α′− β′)Φ −1
2( ˙ α +˙β + ˙ γ)A
(A7)
(y,y) component
3 e−2β¨Ψ − 2e−2α∇2Ψ + e−2α∇2Φ + 3e−2β(4 ˙ α −˙β)˙Ψ + 3e−2β˙ α˙Φ + 3e−2γα′˙A
− 3e−2γ(2α′+ β′)Ψ′+ 3e−2γα′Φ′− 6e−2γα′(α′+ β′)N
+ 6e−2β(¨ α + 2 ˙ α2− ˙ α˙β)Φ + 3e−2γ?
= κ2?
˙ α′+ α′(3 ˙ α +˙β − ˙ γ)
?
A
e−2β(˙φ˙δφ − Φ˙φ2) − Λ′δφ − e−2γ(φ′2N − φ′δφ′)
?
.(A8)
2.Equations for ωi
In order to derive the evolution equations for ωiin the bulk, it is easy to use the coordinate
Eq.(15). By combining the Einstein equations and the equation of motion for the scalar field,
we first get the evolution equations for metric perturbations ψ,φ,Γ,G and δφ;
∂2
zψ + 3(∂zQ)∂zψ − p2ψ − ∂2
∂2
τψ = −2(∂2
zQ)Γ − 2κ(∂zQ)(∂zφ)δφ +2
∂2
3κ−1dΛ
dφe2Qδφ,
zΓ + 3(∂zQ)∂zΓ − p2Γ − ∂2
τΓ = −
?
zQ + 3(∂zQ)2− κ2(∂zφ)2?
?
τδφ = 2κ(∂2
Γ
−κ(∂zQ)(∂zφ) − ∂2
zφ)Γ + 2κ2(∂zφ)2δφ + κ−2d2Λ
zφ
?
δφ −1
3κ−1dΛ
dφe2Qδφ,
∂2
zδφ + 3(∂zQ)∂zδφ − p2δφ − ∂2
∂2
dφ2e2Qδφ,
zG + 3(∂zQ)∂zG − p2G − ∂2
τG = −3(∂2
zQ)G (A9)
25
Page 26
These equations can be diagonalized using ωidefined in Eqs. (46) [9]. Using
∂zQ =
2
3(∆ + 2)
1
z,
κ∂zφ = 3√2b∂zQ,
dΛ
dφe2Q= −2√2bκ
∆
(∆ + 2)2
1
z2, (A10)
we get the evolution equations for ωi.
The constraint equations are also easy to be derived using the metric Eq.(15). The (τ,i)
component of Einstein equation is given by
−1
2(∂zG + 3(∂zQ)G) − 2∂τψ + ∂τΓ = 0. (A11)
Rewriting these equations by ωi, we get
−1
2(∂zωA+ 3(∂zQ)ωA) + ∂τωψ= 0. (A12)
Then performing the coordinate transformation, we get Eq.(60). The remaining two con-
straint equations can be derived in the same way or can be obtained directly in our spacetime.
APPENDIX B: DERIVATION OF SOLUTIONS FOR SCALAR PERTURBA-
TIONS
1. 0-mode
In order to derive the solution the formula for the derivative of Hunkel functions is usefull;
d
dη
d
dη
?
(−Hη)−
(−Hη)−
1
∆+2H−
1
∆+2(−pη)
?
?
= −p(−Hη)−
1
∆+2H−∆+3
∆+2(−pη)
?
1
∆+2H−2∆+5
∆+2(−pη)= p(−Hη)−
2(∆ + 3)
∆ + 2
1
∆+2H−∆+3
∆+2(−pη)
+
H(−Hη)−∆+3
∆+2H−2∆+5
∆+2(−pη).
(B1)
Let us find the solution for ωiwith
ωc= (−Hη)−
1
∆+2H−
1
∆+2(−pη). (B2)
First we use the constraint equation Eq.(60). Because this equation contains the first deriva-
tive with respect to time, the solution for ωinecessarily includes H−2∆+5
only the choice that can eliminate H−∆+3
∆+2(−pη) because it is
∆+2(−pη) which arises when we take the time derivative
26
Page 27
of H−
the fact that ωishould satisfy the evolution equation in the bulk, the y-dependence of the
1
∆+2(−pη). Then we assume that ωicontains H−
1
∆+2(−pη) and H−2∆+5
∆+2(−pη). Using
variables is automatically determined. Then substituting the ansatz for the variables into
the junction conditions, we can determine all coefficients except for an over-all normaliza-
tion. It is a non-trivial check that these solutions indeed satisfy three constraint equations.
It is easy to verify that these solutions indeed satisfy the constraint equations.
2. Massive modes
For massive modes, the following formula is useful;
d
dη
?
(−Hη)−
1
∆+2Hiν(−pη)
?
= −p(−Hη)−
?
= p(−Hη)−
?2∆ + 5
1
∆+2Hiν−1(−pη)
?
1
∆+2Hiν−1(−pη)
?
+ H
1
∆ + 2+ iν(−Hη)−∆+3
∆+2Hiν(−pη).
d
dη
?
(−Hη)−
1
∆+2H−2+iν(−pη)
?
+ H
∆ + 2− iν(−Hη)−∆+3
∆+2H−2+iν(−pη).
(B3)
and
d
dy
?
−cothHy∆ + 4
(sinhHy)
∆
2(∆+2)Bµ+2
β
(coshHy)
?
= H(sinhHy)
∆
2(∆+2)
×
d
dy
d
dy
?
∆ + 2Bµ+2
β
(coshHy) +
?
β −
∆ + 4
2(∆ + 2)
??
β +
3∆ + 8
2(∆ + 2)
?
Bµ+1
β
(coshHy)
?
,
?
?
(sinhHy)
∆
2(∆+2)Bµ
β(coshHy)
?
= H(sinhHy)
∆
2(∆+2)Bµ+1
β
(coshHy),
(sinhHy)
∆
2(∆+2)Bµ+1
β
(coshHy)
?
= H(sinhHy)
∆
2(∆+2)
?
cothHyBµ+1
β
(coshHy) + Bµ+2
β
(coshHy)
?
,
= H(sinhHy)
??
∆ + 2
∆
2(∆+2)
×−
2
?
cothHyBµ+1
β
(coshHy) +
?
β +
∆
2(∆ + 2)
??
β +
∆ + 4
2(∆ + 2)
?
Bµ
β(coshHy)
?
.
(B4)
As for the 0-mode, the ansatz for the solutions are determined so that they can satisfy the
constraint equation Eq.(60). We asssume
ωc = (−Hη)−
1
∆+2(sinhHy)
∆
2(∆+2)CcBµ
−1
2+iνHiν(−pη)
27
Page 28
ωψ = (−Hη)−
×
ωA = (−Hη)−
×
ωN = (−Hη)−
×
1
∆+2(sinhHy)
∆
2(∆+2)
?
CψBµ
−1
2+iν(coshHy)Hiν(−pη) + DψBµ
1
∆+2(sinhHy)
−5
2+iν(coshHy)Hiν−2
?
∆
2(∆+2)
?
CABµ+1
−1
2+iν(coshHy)Hiν(−pη) + DABµ+1
1
∆+2(sinhHy)
−5
2+iν(coshHy)Hiν−2
?
∆
2(∆+2)
?
CNBµ+2
−1
2+iν(coshHy)Hiν(−pη) + DNBµ+2
−5
2+iν(coshHy)Hiν−2
?
(B5)
where Bα
βis given by Eq.(64). First let us consider the constraint equation Eq.(60). Because
this equation only contains ωψ and ωA, it is easy to determine the coefficients using the
formula for derivatives. We get
Cψ= −1
2
?
iν −
1
∆ + 2
?
CA,Dψ=1
2
?
iν −2∆ + 3
∆ + 2
?
DA,DA= −
?iν −∆+1
iν −∆+3
∆+2
∆+2
?
CA.
(B6)
Next we use the junction conditions. From Eq.(58), DNis determined as
DN= −1
2
?
1
iν −∆+3
∆+2
?
CA. (B7)
From Eq.(57), CNis given by
CN=∆ + 2
2
CA. (B8)
We should note that at this time, the problem becomes non-trivial because the equation
should be satisfied by the coefficients which have already determined. The point is that, on
the brane, Bµ+1
−1
2+iν(coshHy0) = 0. Using this fact, we can show that
sinhHy0coshHy0Bµ+2
−5
2+iν(coshHy0) =
1
2(iν − 1)
−
?
iν −∆ + 1
∆ + 2
?
? ?
iν −2∆ + 5
∆ + 2
?
Bµ+1
−5
2+iν(coshHy0)
?
iν −∆ + 1
∆ + 2
cosh2Hy0Bµ+1
−5
2+iν(coshHy0).(B9)
Using this relation, it is shown that the junction condition Eq.(58) can be satisfied. Finally,
we use Eq.(57). We get
Cc= −
√2
4b(∆ + 2)
?
iν −
1
∆ + 2
?
(iν − 1). (B10)
Here we again used Eq.(B9) in order to show that the junction condition is satisfied. Now all
coefficients are determined except for an over-all normalization. It remains a task to verify
28
Page 29
that these solutions satisfy the remaining two constraint equations. In order to show that,
we use the formula for associate Legendre function and Hankel function
Bµ+2
β
(z) + 2(µ + 1)z(z2− 1)−1
2Bµ+1
β
(z) = (β + µ)(β + µ + 1)Bµ
β(z),
Hβ−1(z) + Hβ+1(z) = 2βz−1Hβ(z). (B11)
Then after long calculations, it is possible to show that the above solution indeed satisfy the
constraint equations.
In order to derive the solutions for metric perturbations on the brane, we used Eq. (B11)
and the following equations;
?
?
iν −2∆ + 3
∆ + 2
? ?
? ?
iν −∆ + 1
∆ + 2
?
Bµ
−5
2+iν(coshHy0)
iν −∆ + 3
= iν −
1
∆ + 2
2(iν − 1)sinh2Hy0+
?
∆ + 2
??
Bµ
−1
2+iν(coshHy0), (B12)
Bµ+2
−5
2+iν(coshHy0) =
?
iν −
1
∆ + 2
? ??
iν +
1
∆ + 2
?
+ 2(iν − 1)sinh2Hy0
?
Bµ
−1
2+iν(coshHy0),
(B13)
which can be derived using Bµ+1
−1
2+iν(coshHy0) = 0.
[1] P. Hoˇ rava and E. Witten, Nucl. Phys. B 460 (1996) 506; 475 (1996) 94.
[2] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370.
[3] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690.
[4] K. i. Maeda and M. Sasaki, (eds.), Brane world:New perspective in comsology, Prog.Theor.
Phys. Supplement, 148 (2002).
[5] A. Lukas, B. A. Ovrut and D. Waldram, Phys. Rev. D 60 (1999) 086001.
[6] K. i. Maeda and D. Wands, Phys. Rev. D 62 (2000) 124009.
[7] S. Nojiri, O. Obregon, S. D. Odintsov, Phys. Rev. D 62 (2000) 104003; S. Nojiri, O. Obregon,
S. D. Odintsov, V. I. Tkach, Phys. Rev. D 64 (2001) 043505.
[8] M. Cvetiˇ c, H. L¨ u and C. N. Pope, Phys. Rev. D 63 (2001) 086004.
[9] V. Bozza, M. Gasperini, G. Veneziano, Nucl. Phys. B 619 (2001) 191-210.
[10] O. Seto and H. Kodama, Phys. Rev. D 63 (2001) 123506.
29
Page 30
[11] D. Langlois and M Rodr´iguez-Mart´inez, Phys. Rev. D64 (2001) 123507.
[12] H. Ochiai and K. Sato, Phys. Lett. B 503 (2001) 404.
[13] S. Kobayashi and K. Koyama, JHEP 12 (2002) 056.
[14] S. Kobayashi, K. Koyama and J. Soda, Phys. Lett. B 501 (2001) 157.
[15] Y. Himemoto and M. Sasaki, Phys. Rev. D 63 (2001) 044015.
[16] J. Yokoyama and Y. Himemoto, Phys. Rev. D 64 (2001) 083511.
[17] N. Sago, Y. Himemoto and M. Sasaki, Phys. Rev. D 65 (2002) 024014.
[18] Y. Himemoto, T. Tanaka and M. Sasaki, Phys. Rev. D 65 (2002) 104020.
[19] Y. Himemoto and T. Tanaka, Phys.Rev. D67 (2003) 084014; T. Tanaka and Y. Himemoto,
Phys.Rev. D67 (2003) 104007.
[20] K. Koyama and K. Takahashi, Phys. Rev. D67 (2003) 103503.
[21] D. Langlois and M. Sasaki, gr-qc/0302069
[22] M. Minamitsuji, Y. Himemoto and M.Sasaki, gr-qc/0303108.
[23] S. Kanno and J. Soda, hep-th/0303203
[24] S. Mukohyama, Phys. Rev. D 62 (2000) 084015.
[25] H. Kodama, A. Ishibashi and O. Seto, Phys. Rev. D 62 (2000) 064022 .
[26] R. Maartens, Phys. Rev. D 62 (2000) 084023.
[27] D. Langlois, Phys. Rev. D 62 (2000) 126012.
[28] C. van de Bruck, M. Dorca, R. H. Brandenberger and A. Lukas, Phys. Rev. D 62 (2000)
123515.
[29] K. Koyama and J. Soda, Phys. Rev. D 62 (2000) 123502; K. Koyama and J. Soda, Phys. Rev.
D 65 (2002) 023514; K. Koyama, Phys. Rev. D 66 (2002) 084003; and references therein.
[30] D Langlois, R Maartens, M Sasaki, D Wands, Phys. Rev. D 63, (2001) 084009.
[31] K. Koyama, astro-ph/0303108.
[32] H. A. Bridgman, K. A. Malik, D. Wands Phys. Rev. D 63 (2001) 084012.
[33] G. Uchida and M. Sasaki, Prog. Theor. Phys. 108 (2002) 471-494.
[34] H. Yoshiguchi, K. Koyama and Y. Himemoto, in preparation.
30
Download full-text