- Email: [email protected]
Birth of the brane world
22 June 2000
Physics Letters B 483 Ž2000. 432–442
Birth of the brane world Kazuya Koyama a , Jiro Soda b a
Graduate School of Human and EnÕironment Studies, Kyoto UniÕersity, Kyoto 606-8501, Japan b Department of Fundamental Sciences, FIHS, Kyoto UniÕersity, Kyoto, 606-8501, Japan Received 4 February 2000; received in revised form 27 April 2000; accepted 27 April 2000 Editor: M. Cveticˇ
Abstract Birth of the brane world is studied using the Hamiltonian approach. It is shown that an inflating brane world can be created from nothing. The wave function of the universe obtained from the Wheeler de-Witt equation and the time-dependent Schrodinger equation for quantized scalar fields on the brane are the same as in the conventional 4-dimensional ¨ quantum cosmology if the bulk is exactly the Anti-de Sitter spacetime. The effect of the massive objects in the bulk is also discussed. This analysis tells us the presence of the extra dimension imprints a nontrivial effect on the quantum cosmology of the brane world. This fact is important for the analysis of the quantum fluctuations in the inflationary scenario of the brane world. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Much attention has been paid to the possibility we are living inside a 3-brane in higher dimensional spacetime w1,2x. The idea of a brane world was renewed by the Horava-Witten theory which relates the strongly coupled E8 = E8 string heterotic theory to the eleven dimensional M-theory compactified on an S1rZ2 orbifold with a set of E8 gauge field at each 10-dimensional fixed plane w3x. In this theory, the pure supergravity lives in the bulk which appears 5-dimensional Anti de-Sitter ŽAdS. spacetime, while the standard model particles are confined to the 3-brane w4x. Phenomenologically, this picture opens up a route towards resolving the mass hierarchy between fundamental scales of particle physics and gravity w5,6x. Recently, a simple model based on this picture was constructed by Randall and Sundrum w7x. In their setting, a 4-dimensional domain wall sits at a 5-dimensional AdS spacetime. It has been shown the Einstein gravity is recovered on the wall with positive tension in the low energy limits w8x. This model gives a new setting for the early universe cosmology, which has been traditionally studied in the framework of the 4-dimensional action. The cosmological evolution of the brane world has been investigated by many authors w9x. It has been shown the Friedman equation is recovered in the low energy limits. In the early universe, the inflationary solution plays an important role. The solution for the inflating brane world has been obtained w10x.
E-mail addresses: [email protected] ŽK. Koyama., [email protected] ŽJ. Soda.. 0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 4 5 - 1
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
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In the traditional 4-dimensional theory, the application of laws of quantum mechanics to the universe shed light on issues such as the initial conditions of the universe. The possibility arises that the inflating universe can be created from nothing by quantum tunneling w11x. Furthermore, the Wheeler de-Witt ŽWDW. equation which describes the evolution of the wave function of the universe leads to the time-dependent Schrodinger equation ¨ for quantized matter in the de Sitter spacetime w12x. This gives the explanation of the origin of the structure of the universe w13x. Then it seems natural to ask how to implement the quantum cosmological idea in the context of the brane world w14x. Can the inflating brane world be created from nothing? What is the wave function of the universe? How to derive the Schrodinger equation for the quantized matter on the brane? ¨ In this letter, we derive the effective action for the brane world from the 5-dimensional action and reply these questions. From the 5-dimensional viewpoint, the creation of the brane is the quantum tunneling of the domain wall. The most promising way to deal with this process is the Hamiltonian approach w15,16x. We derive the Hamiltonian form of the effective action for the brane world and derive the WDW equation. We also discuss the effect of the massive objects in the bulk. This analysis gives the insight to the effect of the deviation of the bulk from the AdS spacetime on the quantum cosmology of the brane world, which is important for the analysis of the quantum fluctuations in the inflationary scenario of the brane world.
2. The effective action In this section we derive the effective action for the brane world. Our starting point is the 5-dimensional action for the bulk gravity plus 4-dimensional domain wall; S s SG q SW s
1
5
Hd x'y g Ž R 16p G
5
m q 2 L. y
2p 2
4
Hwalld A,
Ž 1.
where R 5 is the 5-dimensional Ricci scalar, L is the cosmological constant Ž L ) 0 for the Anti-de Sitter spacetime., G is the Newton constant in the 5-dimensional spacetime, mr2p 2 is the energy per unit volume of the domain wall and Hwall d 4A is the volume of the wall. We assume the spacetime is spherically symmetric. The general spherically symmetric metric can be written as 2
ds 2 s y Ž N t Ž r ,t . dt . q L Ž r ,t .
2
2 Ž dr q N r Ž r ,t . dt . q R Ž r ,t . 2 d V 32 .
Ž 2.
Let us denote the trajectory of the wall in this spacetime as r 0 s r 0 Ž t .. Here r 0 is the coordinate radius of the wall which describes the the degree of the freedom of the wall. Then the action for the wall becomes
(
2
SW s ym dtR 30 N0t 2 y L20 Ž r˙0 q N0r . .
H
Ž 3.
Here the quantities on the wall are denoted like as R 0 s RŽ r s r 0 ,t .. Defining the conjugate momenta
pR s pL s ps
R2 N
t
R2 t
N
ž
2L y R
X
R˙ y L˙ q Ž N r L . q
R
RX N r ,
/
Ž N r RX y R˙ . ,
m R 30 L20 Ž r˙0 q N0r .
(N
2L
t2 2 0 y L0
Ž r˙0 q N0r .
2
,
Ž 4.
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the Hamiltonian form of the action is given by S s dt p r˙0 q dr p L L˙ q p R R˙ y N tHt y N rHr
½
H
H ž
/5,
Ž 5.
where 4G Ht s
3p
ž
p L2 L R
3
p Lp R
y
R
q d Ž r0 y r .
ž
p2 L2
2
/
3p y 4G
ž
yR
2
RX
ž / L
X
RX
2
ž ž //
q LR 1 y
L q
L
3
LR 3
/
1r2
qm
2
R 60
/
,
Hr s yLp LX q RXp R y d Ž r 0 y r . p.
Ž 6.
For a while, we will work in units 4Gr3p s 1. Let the radial coordinate takes the range r 1 F r F r 2 . The spacetime is divided into two regions by the domain wall; V1 :
r1 F r F r 0 ,
V2 :
r0 F r F r2 ,
Ž 7.
We impose the Z2 symmetry across the wall, then r 2 s 2 r 0 y r 1. The spacetime is obtained by deleting the spacetime region from the wall to the boundary and gluing two copies of the remaining spacetime along the 4-sphere at r s r 0 . We assume RŽ r . to be continuous and p R and p L to be free from d functions at the wall. The integration of the constraints implies the following junction conditions at the wall; ^p L s y
p L
,
^ RX s y
E R2
,
Ž 8.
where E s Ž p 2 q m2 L20 R 60 .1r2 is the energy of the wall. We impose the coordinate gauge condition and the slicing condition L s 1,
Rp R s 2p L .
Ž 9.
From these conditions, we obtain the shift function and lapse function as N r s 0,
N tsy
R 2 R˙
pL
.
Ž 10 .
The induced metric on the wall is written as ds 2 < wall s ydt 2 q R 02 d V 32 ,
Ž 11 .
where t is the propertime of the wall. The radius at the wall R 0 can be identified with the scale factor of the 3-brane world. We shall reduce the action to have only the degree of the freedom of the brane, that is, R 0 . For this purpose, we shall solve the constraints in the bulk. Using the slicing and gauge fixing conditions Ž9., we can solve the momentum constraints and Hamiltonian constraints; Ht s 0 and Hr s 0. We obtain 2
p R s 2 R Ž r ,t . j j Ž t . , p L s R Ž r ,t . j j Ž t . ,
RX s Ž y1 .
)
s
2
1 q ji Ž t . q
L R Ž r ,t . 6
2
2M y R Ž r ,t .
2
,
Ž 12 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
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where j j Ž t . is the constant of integration with respect to the radial coordinate r in region Vj , and s s 0 for j s 1 and s s 1 for j s 2. M is another constant of integration which is related to the mass of the 5-dimensional AdS-Schwartzschild black hole. Making use of these solutions, we shall rewrite the gravitational part of the action in the bulk; SG s dtdrp R R˙ s dtdr 2 R˙ Ž R j 1 u Ž r 0 y r . q R j 2 u Ž r y r 0 . . .
H
Ž 13 .
H
Here we introduced the step function u Ž x . s 1 for x ) 0 and u Ž x . s 0 for x - 0. Carrying out the integration by parts, this action Ž13. can be rewritten as SG s dt r˙0 R 20 Ž j 2 y j 1 . q dr Ž yj˙1 R 2u Ž r 0 y r . y j˙2 R 2u Ž r y r 0 . . .
H
½
5
H
Ž 14 .
Next consider the contributions from the constraint; SC s dtdr Ž yN tHt y N rHr . .
Ž 15 .
H
In the bulk, since the constraints are satisfied, there is no contribution to SC . On the wall, however, since the solutions of the constraint are not held, RXX and p LX in Ht and Hr give the contribution to Sc ; r 0q e
t
2
XX
Hr ye dr ŽyN Ž R R
. y N r Ž yp LX . . s yN t R 2 ^ RX q N r ^p L .
Ž 16 .
0
Then we obtain SC s dt yN t Ž E q R 2 ^ RX . q N r Ž p q ^p L . 4 .
Ž 17 .
H
The total action Ž5. becomes S s dt dr yj˙1 R 2u 1 y j˙2 R 2u 2 q dt Ž yN t Ž E q R 20 ^ RX . q Ž r˙0 q N r . p˜ . ,
H H ž
/ H
Ž 18 .
where p˜ s p q ^p L ,
)
Ž 19 .
^ RX s y 1 q j 22 q
L R 20
2M y
6
E s Ž m2 R 60 q R 40 Ž j 2 y j 1 .
R 02
2 1r2
.
y
L R 20
)
1 q j 12 q
2M y
6
R 02
,
Ž 20 .
.
Ž 21 .
We shall simplify this action. First, we can drop the term proportional to p˜ from the junction condition p˜ s 0. Next we will simplify the Hamiltonian constraint at the wall. We redefine the lapse function; X 2 ˜t N t ' Ry4 0 N Ž E y R0 ^ R . .
Ž 22 .
Then X 2 4 N t Ž E q R 02 ^ RX . s N˜ t Ry4 0 Ž E y R0 Ž ^ R .
2
.
½
s N˜ t m2 R 20 y 2 1 q j 1 j 2 q
)
q 1 q j 12 q
L R 20 6
2M y
R 02
L R 20 6
)
2M y
R 20
1 q j 22 q
L R 20 6
2M y
R 02
5
.
Ž 23 .
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436
We will take the rest frame of the wall i.e. p s 0. The junction condition implies
j1sj2 'j Ž t. .
Ž 24 .
The constraint can be written by the variable j as Htsm ˜ 2 R 20 q
8M
y 4 Ž1 q j 2 . ,
R 02
m ˜ s m'1 y l ,
Ž 25 .
where we have introduced the parameter l by
ls
2L 3 m2
,
Ž 26 .
which reduces 1 if we take the Randall-Sundrum limit w6x. Finally we shall consider the kinetic term. Using the variable j , the kinetic term becomes r
R0 0 Ž Kinetic term. s HdtH dr Ž y2 j˙R 2 . s yHdt j˙H dR r1
2 R2 RX
R1
.
Ž 27 .
Now we obtain the action describing the dynamics of the brane;
½
S s dt j x˙ y N
H
ž
t
2
j q1y
m ˜2 4
2
R0 Ž x ,j . y
2M R0 Ž x ,j .
2
/5
,
Ž 28 .
where R 0 Ž x , j . is determined by
xs
R0
HR
dR
1
2 R2 RX
s
R0
HR
dR
1
2 R2
(1 q j
2
q L R 2r6 y 2 MrR 2
.
Ž 29 .
3. The birth of the brane world In this section we assume the bulk is exactly AdS spacetime. Then, the effective action Ž28. reduces to the simple form which leads to the conventional Wheeler de Witt ŽWDW. equation. Then we show the creation of the inflating brane world from nothing is possible and derive the wave function of the brane world. 3.1. EffectiÕe action for M s 0 In the effective action Ž28., the canonical variables are Ž x , j .. For M s 0 we can take r 1 s RŽ r 1 . s 0. The integration in the expression x can be done easily, then we obtain the effective action
½
ž
S s dt j x˙ y N t j 2 q 1 y
H
m3 1 y l 8 f Ž l.
x
/5
,
f Ž l. s
1 y
l
ž
1yl
l3r2
/
log
1 q 'l
'1 y l
,
where l is defined in Ž26. and f Ž l. becomes 1 as taking the Randall-Sundrum limit l canonical transformation
xs
2
m
f Ž l . R 20 ,
m js
P
4 f Ž l. R 0
,
Ž 30 .
™ 1. Carrying out the Ž 31 .
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437
we can write the reduced action by R 0 and P; S s dt PR˙ 0 y N tH t 4 ,
H tsP2q
H
G4 s
2G 2m 3p f Ž l .
2
,
H s
2Gm ˜
2
ž /
s
3p
9p 2
ž / 4G42
2G4 3p
R 20 Ž 1 y H 2 R 20 . ,
m Ž 1 y l. f Ž l. .
Ž 32 .
Here we restore the 5-dimensional Newton constant G. This is the Hamiltonian form of the effective action for the brane world. The constant G4 can be identified with the Newton constant in the brane world. To see this fact, we solve the r dependence of the RŽ r,t .. The solution of R in the bulk is given by
° R Ž r ,t . s~
cosh u 1 Ž t .
¢coshu Ž t . 2
6
( ( ( ( L 6
L
sinh
sinh
L 0 - r - r0 ,
r,
6
Ž 33 .
L
Ž 2 r0 y r . ,
6
r0 F r - 2 r0 ,
Here, the integration constants are denoted as u 1 and u 2 . The location of the wall r 0 is determined by the junction condition. We will consider on the rest frame of the wall p s 0. The junction conditions Ž8. become RX Ž r s r 0 " e . s . 12 m R 0 .
p LŽ r s r 0 q e . s p LŽ r s r 0 y e . ,
Ž 34.
From these conditions Ž33., we obtain
L
( (
u 1 s u 2 s u , sinh
6
r0 s
l 1yl
.
Ž 35 .
Thus, the solution is given by RŽ r,t . s aŽ t . W Ž r ., where a Ž t . s Hy1 cosh Ht ,
(
W Ž r . s cosh
t s Hy1u Ž t . ,
L wy
6
1
'l
(
sinh
L 6
< w <,
w s r y r0 .
Ž 36 .
Using this solution the lapse function can be written as N t s u˙ Hy1 W Ž r .. Then the classical solution of the spacetime is obtained ds 2 s dw 2 q W 2 Ž w . g i j dx i dx j ,
2
g i j dx i dx j s ydt 2 q a Ž t . d V 32 .
Ž 37 .
The factor W Ž w . is often called the warp factor. The effective action for 4-dimensional gravity is given by w6x Se f f ;
1
Hd G
4
™
ž
x 2
r0
H0
dw Ž W Ž w . .
2
/
1
'g Rg s G Hd 4 x'g Rg .
Ž 38 .
4
Here Rg is the Ricci scalar made out of g i j . Hence G4 can be identified with the Newton constant in the brane world. In the l 1 limits, the brane world becomes Minkowski spacetime. In this case, G4 can be written as G4 s GŽ Lr6.1r2 w6x.
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
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3.2. The birth of the brane world Based on the effective action Ž32., we discuss the creation of the brane world and it’s wave function. There is no classical solutions for l G 1. Then we assume l - 1. The phase space Ž R 0 , P . is described as follows: Hy1 - R 0 ,
classically allowed region,
0 - R 0 - Hy1 ,
classically forbidden region.
Ž 39 .
Choosing the lapse function appropriately, the classical equation of motion is obtained as follows R˙ 0
2
ž /
1 sy
R 02
R0
qH 2.
Ž 40 .
The universe can contract from the infinite size and bounce at a minimum radius Hy1 and then re-expand classically. We shall quantize this system as a one-dimensional particle system. We take the wave function of the brane world as C Ž R 0 .. Using the representation P
™ yi dRd ,
Ž 41 .
0
we write the constraint as a Wheler de-Witt ŽWDW. equation HtC Ž R 0 . s 0;
ž
y 12 R 0
d dR 0
d
Ry1 0
dR 0
/
q V Ž R 0 . C Ž R 0 . s 0,
V Ž R0 . s
1 2
9p 2
ž / 4G42
R 20 Ž 1 y H 2 R 20 . ,
Ž 42 .
where the choice of the factor ordering is made, which is not important at the semiclassical level. Consider the boundary conditions of this Eq. Ž42.. In the classical forbidden region, the tunneling from R 0 s 0 to R 0 s Hy1 is possible. From the 5-dimensional point of view, this is a tunneling of the domain wall. The boundary condition that selects the wave function representing the tunneling requires that C Ž R 0 . should be only an outgoing wave at R 0 `. In the classical forbidden region the wave function is given by the linear combination of the growing and the decaying solutions. The wave function corresponds to this boundary condition is given by
™
C Ž R 0 . A Ai z Ž R 0 . Ai z Ž Hy1 . q i Bi z Ž R 0 . Bi z Ž Hy1 . , 2 . 3r2 Ž
where Ai and Bi is the Airy functions and z Ž R . s Ž3pr4G4 H 1yH decaying solution dominates. The semiclassical wave function is given by
ž
C Ž R 0 . ; exp y
R0
H0
/
(
dR 2V Ž R . .
Ž 43 . 2
R 02 .
w11x. Under-barrier region, the
Ž 44 .
Then the tunneling probability is obtained as P;
C Ž Hy1 . C Ž 0.
2
ž
; exp y
p G4 H 2
/
.
Ž 45 .
From the four dimensional point of view, a tunneling from R 0 s 0 to R 0 s Hy1 represents the creation of the brane world from nothing. If we identify G4 as the Newton constant in our brane world, this tunneling probability agrees with the one traditionally obtained in the 4-dimensional quantum cosmology w14x. The wave function of the brane world coincides with the Vilenkin’s tunneling wave function. We shall include the effect of the matter confined to the wall. It is usefull to consider a scalar field as the matter confined to the wall. For simplicity, we shall consider a scalar field '2 pf with potential 2p 2 UŽ f .. We
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
439
assume there is no back reaction to the geometry of the bulk and the junction conditions from this scalar filed, that is UŽ f .rm < 1. The 4-dimensional gravity couples to the matter on the wall with the coupling G4 . The WDW equation including the scalar field on the wall is given by
ž
y 12 R 0
Hf s
d
Ry1 0
dR 0 d3 x
H 2p
2
ž
d
q V Ž R0 . y
dR 0
R 20
E
1 y
2 R 20
Ef
q
2
3p 2G4
/
Hf Ž f , R 0 . C Ž R 0 . s 0,
2 Ž E i f . q R 04 U Ž f . .
/
2
Ž 46 .
We put the WKB solution of the WDW equation of the form
C Ž R 0 , f . s C Ž R 0 . eyi SŽ R 0 .c Ž R 0 , f . ,
SŽ R0 . s
R0
H0
(
dR y 2V Ž R . .
Ž 47 .
Then we find c Ž R 0 , f . satisfies the time-dependent Schrodinger equation ¨ Ec dt i s Hf c , s R 0 , R 0 Ž t . s Hy1 cosh Ht . Eh dh
Ž 48 .
This is a Schrodinger equation for a quantized scalar field in the de Sitter spacetime. ¨ 4. Effect of the black hole in the bulk 4.1. EffectiÕe action for M / 0 In this section we discuss the effect of the spherically symmetric objects in the bulk. The bulk deviates from the AdS spacetime to the AdS-Schwartzschild spacetimes. For M / 0, there is an event horizon at r s r h at which R h Ž rh . s
3
L
4ML
ž (
1q
y1 q
3
/
.
Ž 49 .
We will take some r 1 at which RŽ r 1 . ) R h . We rewrite the effective action by the canonical variable R 0 . We must use slightly different canonical transformation from the previous one. This is related to the fact RŽ r,t . cannot be separable into the warp factor and the scale factor as is done for M s 0. We use the similar method developed by Kolitch and Eardley w16x. By taking the integration by part, the kinetic term can be written as R
0 Ž kinetic term. s yHdt j˙H dR
2 R2 RX
° 2R cosh u ¢ 3 (cosh u q L R r6 y 2 MrR
H
q
R0
dR
R0
3
s yHdt u˙~
2
2 R3
ž /ž 3
2
LR
2M q
6
R3
/Ž
2
cosh u cosh2u q L R 2r6 y 2 MrR 2 .
¶• ß.
3r2
Ž 50 .
Here we changed the variables as j s sinh u . It is convenient to introduce the quantity T ; T s ysinh u
H
R0
dR
ž
Ž 2 R 3r3 . Ž L Rr6 q 2 MrR 3 . (cosh2u q L R 2r6 y 2 MrR 2 Ž1 q L R 2r6 y 2 MrR 2 .
/
.
Ž 51 .
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We can show the total derivative of T with respect to t is given by 2 R 30
T˙s Ž kinetic term . q
F Ž R 0 ,u . ,
3
Ž 52 .
where cosh u
F Ž R 0 , u . s u˙
(cosh u q L R r6 y 2 MrR 2
y R˙ 0
2 0
2 0
Ž L R 0r6 q 2 MrR 30 . sinh u (cosh2u q L R 20r6 y 2 MrR 20 Ž1 q L R 20r6 y 2 MrR 20 . .
Ž 53 .
We introduce the variable c by sinh u
sinh c s
.
(
1 q L R 20r6 y 2 MrR 20
Ž 54 .
The total derivative of the variable is written as
c˙ s F Ž R 0 , u . ,
Ž 55 .
Then defining z s 2 R 30r3, we can write the effective action up to the total derivative
½
S s dt cz˙y N t m2 Ž 32 z
H
ž
.
2r3
L
ž
y4 1q
Ž 32 z .
6
2r3
y 2 M Ž 32 z
y2r3
.
/
cosh2c
/5
.
Ž 56 .
Now, performing the canonical transformation
cs
P 2 R 20
z s 23 R 30 ,
,
Ž 57 .
we rewrite the constraint as Ps"
3p
ž /
)
R 20 arcsinh
2G
yR 20 q H 2 R 04 q Ž 8Gr3p . M R 02 q Ž Lr6 . R 04 y Ž 8Gr3p . M
.
Ž 58 .
There is an ambiguity in the choice of the sign. The choice of the sign determines the boundary condition of the wave function. We will choose the sign so that the wave function represents the tunneling. After doing this procedure, we obtain
° ¢
S s Hdt~PR˙ y N 0
t
ž
Py
3p
ž / 2G
R 02
)
arcsinh
yR 20 q H 2 R 40 q Ž 8Gr3p . M R 20 q
Ž Lr6.
R 40 y
Ž 8Gr3p . M
¶• ß,
/
Ž 59 .
where we restore the 5-dimensional Newton constant G. 4.2. Effect of bulk black hole The turning point where P s 0 is given by R "s
(
1 " 1 y Ž 32Gr3p . H 2 M 2H2
.
Ž 60 .
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441
For 1 - Ž32Gr3p . H 2 M there is no turning point. Then we set 0 - Ž32Gr3p . H 2 M - 1. The phase space is described as follows: R 1 - R 0 - Ry ,
classically allowed region,
Ry- R 0 - Rq ,
classically forbidden region,
Rq- R 0 ,
classically allowed region.
Ž 61 .
Classical equation of motion is obtained by R˙ 0
2
ž /
1 sy
R 02
R0
qH 2q
8G M 3p R 04
.
Ž 62 .
The additional term proportional to M behaves as the radiation. Taking the momentum as a differential operators acting on the wave function of the brane world C Ž R 0 . P
™ yi E RE .
Ž 63 .
0
we obtain the WDW equation
E E R0
C Ž R0 . s y
žž
3p 2G
/
)
R 20 arcsin
R 20 y H 2 R 40 y Ž 8Gr3p . M R 20 q Ž Lr6 . R 40 y Ž 8Gr3p . M
/
C Ž R 0 . ' yS Ž R 0 . C Ž R 0 . .
Ž 64 . The tunneling rate at the semiclassical order can be evaluated by
ž
P ; exp y2
Rq
HR
/
dR S Ž R . .
y
Ž 65 .
™
We can verify this agrees with the result obtained in the previous section for the limit M 0. For M / 0, the inflating brane world is created by the tunneling from the closed FRW universe driven by the massive objects in the bulk. For a small radius of the brane world R 0 - Ž8GMr3p H 2 .1r4 , the wave function is different from the one obtained in 4-dimensional quantum cosmology. This implies the presence of the extra dimension gives the nontrivial effect on the quantum cosmology of the 4-dimensional universe if the bulk deviates from the AdS spacetime. Technically, this comes from the fact the r dependence in RŽ r,t . cannot be separable. For large R 0 , the effect of the massive objects in the bulk on the evolution of the brane world becomes negligible, then the wave function of the universe becomes the familiar one obtained in the previous section for M s 0.
5. Summary and discussion In this letter, we derived the effective action for the brane world from the 5-dimensional action for the bulk gravity plus 4-dimensional domain wall. Based on this action, we derived the WDW equation for the brane world. If the bulk is AdS spacetime, the brane world can be created from nothing. The WDW equation and the time-dependent Schrodinger equation for the quantized matter on the inflating brane are the same as those in the ¨ conventional 4-dimensional quantum cosmology. It is easy to extend our analysis to the many brane case. In that case, the ‘‘time’’ is defined in each brane world as is suggested by the derivation of the Schrodinger equation. ¨ We discussed the effect of the spherically symmetric massive objects in the bulk. For a small radius of the brane world, the bulk deviates from the AdS spacetime largely. In this case we found the presence of the extra
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dimension affects the quantum cosmology of the brane world, then the wave function is different from the one obtained in 4-dimensional quantum cosmology. The most important success of the inflationary scenario is that this gives the origin of the structure of the universe. The analysis of the evolution of the quantum fluctuations is needed to explore this issue in the context of the brane world. We must include the effect of the back reaction to the geometry of the bulk in the Schrodinger equation for the quantized matter on the wall. It has been shown the deviation of the bulk spacetime ¨ from the exact AdS spacetime is essential to describe the inhomogeneous brane world w8x. The analysis for the bulk with black holes tells us if the bulk deviates from the AdS spacetime, the presence of the extra dimension imprints a non-trivial effects on the quantum cosmology of the brane world. A detailed investigation of the evolution of the quantum fluctuations will be left for the future work w17x.
Acknowledgements The work of J.S. was supported by Monbusho Grant-in-Aid No.10740118 and the work of K.K. was supported by JSPS Research Fellowships for Young Scientist No.4687. K.K. also thanks Kayoko Maeda for useful discussions.
References w1x V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 Ž1983. 136. w2x K. Akama, Pregeometry in: K. Kikkawa, N. Nakanishi, H. Nariai ŽEds.., Lecture Notes in Physics, 176, Gauge Theory and Gravitation, Proc., Nara, Springer, Berlin, 1982, 267-271, hep-thr0001113. w3x P. Horava, E. Witten, Nucl. Phys. B 460 Ž1996. 506; Nucl. Phys. B 475 Ž1996. 94. w4x A. Lukas, B.A. Ovrut, K.S. Stelle, D. Waldram, Phys. Rev. D 59 Ž1999. 086001. w5x N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 Ž1998. 163; Phys. Rev. D 59 Ž1999. 086004; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 Ž1998. 257. w6x L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 3370. w7x L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 4690. w8x T. Shiromizu, K. Maeda, M. Sasaki, gr-qcr9910076. w9x P. Binetruy, C. Deffayet, U. Ellwanger, D. Langlois, hep-thr9910219; E.E. Flanagan, S.-H.H. Tye, I. Wasserman, hep-phr9910498; ´ P. Kraus, hep-thr9910149; S. Mukohyama hep-phr9911165; D. Ida, gr-qcr9912002; see also C. Csaki, M. Graesser, L. Randall, J. Terning, hep-phr9911406 and references there in. w10x N. Kaloper, Phys. Rev. D 60 Ž1999. 123506; T. Nihei, Phys. Lett. B 465 Ž1999. 81; H.B. Kim, H.D. Kim, hep-thr9906542. w11x A. Vilenkin, Phys. Lett. B 117 Ž1982. 25; Phys. Rev. D 30 Ž1984. 509; Phys. Rev. D 33 Ž1986. 3560. w12x V.A. Rubakov, Phys. Lett. B 148 Ž1984. 280. w13x J.J. Halliwell, S.W. Hawking, Phys. Rev. D 31 Ž1985. 1777. w14x J. Garriga, M. Sasaki, hep-THr9912118. w15x W. Fischler, D. Morgan, J. Polchinski, Phys. Rev. D 42 Ž1990. 4042. w16x S.J. Kolitch, D.M. Eardley, Phys. Rev. D 56 Ž1997. 4651; Phys. Rev. D 56 Ž1997. 4663. w17x K. Koyama, J. Soda, in preparation.
Physics Letters B 483 Ž2000. 432–442
Birth of the brane world Kazuya Koyama a , Jiro Soda b a
Graduate School of Human and EnÕironment Studies, Kyoto UniÕersity, Kyoto 606-8501, Japan b Department of Fundamental Sciences, FIHS, Kyoto UniÕersity, Kyoto, 606-8501, Japan Received 4 February 2000; received in revised form 27 April 2000; accepted 27 April 2000 Editor: M. Cveticˇ
Abstract Birth of the brane world is studied using the Hamiltonian approach. It is shown that an inflating brane world can be created from nothing. The wave function of the universe obtained from the Wheeler de-Witt equation and the time-dependent Schrodinger equation for quantized scalar fields on the brane are the same as in the conventional 4-dimensional ¨ quantum cosmology if the bulk is exactly the Anti-de Sitter spacetime. The effect of the massive objects in the bulk is also discussed. This analysis tells us the presence of the extra dimension imprints a nontrivial effect on the quantum cosmology of the brane world. This fact is important for the analysis of the quantum fluctuations in the inflationary scenario of the brane world. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Much attention has been paid to the possibility we are living inside a 3-brane in higher dimensional spacetime w1,2x. The idea of a brane world was renewed by the Horava-Witten theory which relates the strongly coupled E8 = E8 string heterotic theory to the eleven dimensional M-theory compactified on an S1rZ2 orbifold with a set of E8 gauge field at each 10-dimensional fixed plane w3x. In this theory, the pure supergravity lives in the bulk which appears 5-dimensional Anti de-Sitter ŽAdS. spacetime, while the standard model particles are confined to the 3-brane w4x. Phenomenologically, this picture opens up a route towards resolving the mass hierarchy between fundamental scales of particle physics and gravity w5,6x. Recently, a simple model based on this picture was constructed by Randall and Sundrum w7x. In their setting, a 4-dimensional domain wall sits at a 5-dimensional AdS spacetime. It has been shown the Einstein gravity is recovered on the wall with positive tension in the low energy limits w8x. This model gives a new setting for the early universe cosmology, which has been traditionally studied in the framework of the 4-dimensional action. The cosmological evolution of the brane world has been investigated by many authors w9x. It has been shown the Friedman equation is recovered in the low energy limits. In the early universe, the inflationary solution plays an important role. The solution for the inflating brane world has been obtained w10x.
E-mail addresses: [email protected] ŽK. Koyama., [email protected] ŽJ. Soda.. 0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 5 4 5 - 1
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
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In the traditional 4-dimensional theory, the application of laws of quantum mechanics to the universe shed light on issues such as the initial conditions of the universe. The possibility arises that the inflating universe can be created from nothing by quantum tunneling w11x. Furthermore, the Wheeler de-Witt ŽWDW. equation which describes the evolution of the wave function of the universe leads to the time-dependent Schrodinger equation ¨ for quantized matter in the de Sitter spacetime w12x. This gives the explanation of the origin of the structure of the universe w13x. Then it seems natural to ask how to implement the quantum cosmological idea in the context of the brane world w14x. Can the inflating brane world be created from nothing? What is the wave function of the universe? How to derive the Schrodinger equation for the quantized matter on the brane? ¨ In this letter, we derive the effective action for the brane world from the 5-dimensional action and reply these questions. From the 5-dimensional viewpoint, the creation of the brane is the quantum tunneling of the domain wall. The most promising way to deal with this process is the Hamiltonian approach w15,16x. We derive the Hamiltonian form of the effective action for the brane world and derive the WDW equation. We also discuss the effect of the massive objects in the bulk. This analysis gives the insight to the effect of the deviation of the bulk from the AdS spacetime on the quantum cosmology of the brane world, which is important for the analysis of the quantum fluctuations in the inflationary scenario of the brane world.
2. The effective action In this section we derive the effective action for the brane world. Our starting point is the 5-dimensional action for the bulk gravity plus 4-dimensional domain wall; S s SG q SW s
1
5
Hd x'y g Ž R 16p G
5
m q 2 L. y
2p 2
4
Hwalld A,
Ž 1.
where R 5 is the 5-dimensional Ricci scalar, L is the cosmological constant Ž L ) 0 for the Anti-de Sitter spacetime., G is the Newton constant in the 5-dimensional spacetime, mr2p 2 is the energy per unit volume of the domain wall and Hwall d 4A is the volume of the wall. We assume the spacetime is spherically symmetric. The general spherically symmetric metric can be written as 2
ds 2 s y Ž N t Ž r ,t . dt . q L Ž r ,t .
2
2 Ž dr q N r Ž r ,t . dt . q R Ž r ,t . 2 d V 32 .
Ž 2.
Let us denote the trajectory of the wall in this spacetime as r 0 s r 0 Ž t .. Here r 0 is the coordinate radius of the wall which describes the the degree of the freedom of the wall. Then the action for the wall becomes
(
2
SW s ym dtR 30 N0t 2 y L20 Ž r˙0 q N0r . .
H
Ž 3.
Here the quantities on the wall are denoted like as R 0 s RŽ r s r 0 ,t .. Defining the conjugate momenta
pR s pL s ps
R2 N
t
R2 t
N
ž
2L y R
X
R˙ y L˙ q Ž N r L . q
R
RX N r ,
/
Ž N r RX y R˙ . ,
m R 30 L20 Ž r˙0 q N0r .
(N
2L
t2 2 0 y L0
Ž r˙0 q N0r .
2
,
Ž 4.
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
434
the Hamiltonian form of the action is given by S s dt p r˙0 q dr p L L˙ q p R R˙ y N tHt y N rHr
½
H
H ž
/5,
Ž 5.
where 4G Ht s
3p
ž
p L2 L R
3
p Lp R
y
R
q d Ž r0 y r .
ž
p2 L2
2
/
3p y 4G
ž
yR
2
RX
ž / L
X
RX
2
ž ž //
q LR 1 y
L q
L
3
LR 3
/
1r2
qm
2
R 60
/
,
Hr s yLp LX q RXp R y d Ž r 0 y r . p.
Ž 6.
For a while, we will work in units 4Gr3p s 1. Let the radial coordinate takes the range r 1 F r F r 2 . The spacetime is divided into two regions by the domain wall; V1 :
r1 F r F r 0 ,
V2 :
r0 F r F r2 ,
Ž 7.
We impose the Z2 symmetry across the wall, then r 2 s 2 r 0 y r 1. The spacetime is obtained by deleting the spacetime region from the wall to the boundary and gluing two copies of the remaining spacetime along the 4-sphere at r s r 0 . We assume RŽ r . to be continuous and p R and p L to be free from d functions at the wall. The integration of the constraints implies the following junction conditions at the wall; ^p L s y
p L
,
^ RX s y
E R2
,
Ž 8.
where E s Ž p 2 q m2 L20 R 60 .1r2 is the energy of the wall. We impose the coordinate gauge condition and the slicing condition L s 1,
Rp R s 2p L .
Ž 9.
From these conditions, we obtain the shift function and lapse function as N r s 0,
N tsy
R 2 R˙
pL
.
Ž 10 .
The induced metric on the wall is written as ds 2 < wall s ydt 2 q R 02 d V 32 ,
Ž 11 .
where t is the propertime of the wall. The radius at the wall R 0 can be identified with the scale factor of the 3-brane world. We shall reduce the action to have only the degree of the freedom of the brane, that is, R 0 . For this purpose, we shall solve the constraints in the bulk. Using the slicing and gauge fixing conditions Ž9., we can solve the momentum constraints and Hamiltonian constraints; Ht s 0 and Hr s 0. We obtain 2
p R s 2 R Ž r ,t . j j Ž t . , p L s R Ž r ,t . j j Ž t . ,
RX s Ž y1 .
)
s
2
1 q ji Ž t . q
L R Ž r ,t . 6
2
2M y R Ž r ,t .
2
,
Ž 12 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
435
where j j Ž t . is the constant of integration with respect to the radial coordinate r in region Vj , and s s 0 for j s 1 and s s 1 for j s 2. M is another constant of integration which is related to the mass of the 5-dimensional AdS-Schwartzschild black hole. Making use of these solutions, we shall rewrite the gravitational part of the action in the bulk; SG s dtdrp R R˙ s dtdr 2 R˙ Ž R j 1 u Ž r 0 y r . q R j 2 u Ž r y r 0 . . .
H
Ž 13 .
H
Here we introduced the step function u Ž x . s 1 for x ) 0 and u Ž x . s 0 for x - 0. Carrying out the integration by parts, this action Ž13. can be rewritten as SG s dt r˙0 R 20 Ž j 2 y j 1 . q dr Ž yj˙1 R 2u Ž r 0 y r . y j˙2 R 2u Ž r y r 0 . . .
H
½
5
H
Ž 14 .
Next consider the contributions from the constraint; SC s dtdr Ž yN tHt y N rHr . .
Ž 15 .
H
In the bulk, since the constraints are satisfied, there is no contribution to SC . On the wall, however, since the solutions of the constraint are not held, RXX and p LX in Ht and Hr give the contribution to Sc ; r 0q e
t
2
XX
Hr ye dr ŽyN Ž R R
. y N r Ž yp LX . . s yN t R 2 ^ RX q N r ^p L .
Ž 16 .
0
Then we obtain SC s dt yN t Ž E q R 2 ^ RX . q N r Ž p q ^p L . 4 .
Ž 17 .
H
The total action Ž5. becomes S s dt dr yj˙1 R 2u 1 y j˙2 R 2u 2 q dt Ž yN t Ž E q R 20 ^ RX . q Ž r˙0 q N r . p˜ . ,
H H ž
/ H
Ž 18 .
where p˜ s p q ^p L ,
)
Ž 19 .
^ RX s y 1 q j 22 q
L R 20
2M y
6
E s Ž m2 R 60 q R 40 Ž j 2 y j 1 .
R 02
2 1r2
.
y
L R 20
)
1 q j 12 q
2M y
6
R 02
,
Ž 20 .
.
Ž 21 .
We shall simplify this action. First, we can drop the term proportional to p˜ from the junction condition p˜ s 0. Next we will simplify the Hamiltonian constraint at the wall. We redefine the lapse function; X 2 ˜t N t ' Ry4 0 N Ž E y R0 ^ R . .
Ž 22 .
Then X 2 4 N t Ž E q R 02 ^ RX . s N˜ t Ry4 0 Ž E y R0 Ž ^ R .
2
.
½
s N˜ t m2 R 20 y 2 1 q j 1 j 2 q
)
q 1 q j 12 q
L R 20 6
2M y
R 02
L R 20 6
)
2M y
R 20
1 q j 22 q
L R 20 6
2M y
R 02
5
.
Ž 23 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
436
We will take the rest frame of the wall i.e. p s 0. The junction condition implies
j1sj2 'j Ž t. .
Ž 24 .
The constraint can be written by the variable j as Htsm ˜ 2 R 20 q
8M
y 4 Ž1 q j 2 . ,
R 02
m ˜ s m'1 y l ,
Ž 25 .
where we have introduced the parameter l by
ls
2L 3 m2
,
Ž 26 .
which reduces 1 if we take the Randall-Sundrum limit w6x. Finally we shall consider the kinetic term. Using the variable j , the kinetic term becomes r
R0 0 Ž Kinetic term. s HdtH dr Ž y2 j˙R 2 . s yHdt j˙H dR r1
2 R2 RX
R1
.
Ž 27 .
Now we obtain the action describing the dynamics of the brane;
½
S s dt j x˙ y N
H
ž
t
2
j q1y
m ˜2 4
2
R0 Ž x ,j . y
2M R0 Ž x ,j .
2
/5
,
Ž 28 .
where R 0 Ž x , j . is determined by
xs
R0
HR
dR
1
2 R2 RX
s
R0
HR
dR
1
2 R2
(1 q j
2
q L R 2r6 y 2 MrR 2
.
Ž 29 .
3. The birth of the brane world In this section we assume the bulk is exactly AdS spacetime. Then, the effective action Ž28. reduces to the simple form which leads to the conventional Wheeler de Witt ŽWDW. equation. Then we show the creation of the inflating brane world from nothing is possible and derive the wave function of the brane world. 3.1. EffectiÕe action for M s 0 In the effective action Ž28., the canonical variables are Ž x , j .. For M s 0 we can take r 1 s RŽ r 1 . s 0. The integration in the expression x can be done easily, then we obtain the effective action
½
ž
S s dt j x˙ y N t j 2 q 1 y
H
m3 1 y l 8 f Ž l.
x
/5
,
f Ž l. s
1 y
l
ž
1yl
l3r2
/
log
1 q 'l
'1 y l
,
where l is defined in Ž26. and f Ž l. becomes 1 as taking the Randall-Sundrum limit l canonical transformation
xs
2
m
f Ž l . R 20 ,
m js
P
4 f Ž l. R 0
,
Ž 30 .
™ 1. Carrying out the Ž 31 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
437
we can write the reduced action by R 0 and P; S s dt PR˙ 0 y N tH t 4 ,
H tsP2q
H
G4 s
2G 2m 3p f Ž l .
2
,
H s
2Gm ˜
2
ž /
s
3p
9p 2
ž / 4G42
2G4 3p
R 20 Ž 1 y H 2 R 20 . ,
m Ž 1 y l. f Ž l. .
Ž 32 .
Here we restore the 5-dimensional Newton constant G. This is the Hamiltonian form of the effective action for the brane world. The constant G4 can be identified with the Newton constant in the brane world. To see this fact, we solve the r dependence of the RŽ r,t .. The solution of R in the bulk is given by
° R Ž r ,t . s~
cosh u 1 Ž t .
¢coshu Ž t . 2
6
( ( ( ( L 6
L
sinh
sinh
L 0 - r - r0 ,
r,
6
Ž 33 .
L
Ž 2 r0 y r . ,
6
r0 F r - 2 r0 ,
Here, the integration constants are denoted as u 1 and u 2 . The location of the wall r 0 is determined by the junction condition. We will consider on the rest frame of the wall p s 0. The junction conditions Ž8. become RX Ž r s r 0 " e . s . 12 m R 0 .
p LŽ r s r 0 q e . s p LŽ r s r 0 y e . ,
Ž 34.
From these conditions Ž33., we obtain
L
( (
u 1 s u 2 s u , sinh
6
r0 s
l 1yl
.
Ž 35 .
Thus, the solution is given by RŽ r,t . s aŽ t . W Ž r ., where a Ž t . s Hy1 cosh Ht ,
(
W Ž r . s cosh
t s Hy1u Ž t . ,
L wy
6
1
'l
(
sinh
L 6
< w <,
w s r y r0 .
Ž 36 .
Using this solution the lapse function can be written as N t s u˙ Hy1 W Ž r .. Then the classical solution of the spacetime is obtained ds 2 s dw 2 q W 2 Ž w . g i j dx i dx j ,
2
g i j dx i dx j s ydt 2 q a Ž t . d V 32 .
Ž 37 .
The factor W Ž w . is often called the warp factor. The effective action for 4-dimensional gravity is given by w6x Se f f ;
1
Hd G
4
™
ž
x 2
r0
H0
dw Ž W Ž w . .
2
/
1
'g Rg s G Hd 4 x'g Rg .
Ž 38 .
4
Here Rg is the Ricci scalar made out of g i j . Hence G4 can be identified with the Newton constant in the brane world. In the l 1 limits, the brane world becomes Minkowski spacetime. In this case, G4 can be written as G4 s GŽ Lr6.1r2 w6x.
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
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3.2. The birth of the brane world Based on the effective action Ž32., we discuss the creation of the brane world and it’s wave function. There is no classical solutions for l G 1. Then we assume l - 1. The phase space Ž R 0 , P . is described as follows: Hy1 - R 0 ,
classically allowed region,
0 - R 0 - Hy1 ,
classically forbidden region.
Ž 39 .
Choosing the lapse function appropriately, the classical equation of motion is obtained as follows R˙ 0
2
ž /
1 sy
R 02
R0
qH 2.
Ž 40 .
The universe can contract from the infinite size and bounce at a minimum radius Hy1 and then re-expand classically. We shall quantize this system as a one-dimensional particle system. We take the wave function of the brane world as C Ž R 0 .. Using the representation P
™ yi dRd ,
Ž 41 .
0
we write the constraint as a Wheler de-Witt ŽWDW. equation HtC Ž R 0 . s 0;
ž
y 12 R 0
d dR 0
d
Ry1 0
dR 0
/
q V Ž R 0 . C Ž R 0 . s 0,
V Ž R0 . s
1 2
9p 2
ž / 4G42
R 20 Ž 1 y H 2 R 20 . ,
Ž 42 .
where the choice of the factor ordering is made, which is not important at the semiclassical level. Consider the boundary conditions of this Eq. Ž42.. In the classical forbidden region, the tunneling from R 0 s 0 to R 0 s Hy1 is possible. From the 5-dimensional point of view, this is a tunneling of the domain wall. The boundary condition that selects the wave function representing the tunneling requires that C Ž R 0 . should be only an outgoing wave at R 0 `. In the classical forbidden region the wave function is given by the linear combination of the growing and the decaying solutions. The wave function corresponds to this boundary condition is given by
™
C Ž R 0 . A Ai z Ž R 0 . Ai z Ž Hy1 . q i Bi z Ž R 0 . Bi z Ž Hy1 . , 2 . 3r2 Ž
where Ai and Bi is the Airy functions and z Ž R . s Ž3pr4G4 H 1yH decaying solution dominates. The semiclassical wave function is given by
ž
C Ž R 0 . ; exp y
R0
H0
/
(
dR 2V Ž R . .
Ž 43 . 2
R 02 .
w11x. Under-barrier region, the
Ž 44 .
Then the tunneling probability is obtained as P;
C Ž Hy1 . C Ž 0.
2
ž
; exp y
p G4 H 2
/
.
Ž 45 .
From the four dimensional point of view, a tunneling from R 0 s 0 to R 0 s Hy1 represents the creation of the brane world from nothing. If we identify G4 as the Newton constant in our brane world, this tunneling probability agrees with the one traditionally obtained in the 4-dimensional quantum cosmology w14x. The wave function of the brane world coincides with the Vilenkin’s tunneling wave function. We shall include the effect of the matter confined to the wall. It is usefull to consider a scalar field as the matter confined to the wall. For simplicity, we shall consider a scalar field '2 pf with potential 2p 2 UŽ f .. We
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
439
assume there is no back reaction to the geometry of the bulk and the junction conditions from this scalar filed, that is UŽ f .rm < 1. The 4-dimensional gravity couples to the matter on the wall with the coupling G4 . The WDW equation including the scalar field on the wall is given by
ž
y 12 R 0
Hf s
d
Ry1 0
dR 0 d3 x
H 2p
2
ž
d
q V Ž R0 . y
dR 0
R 20
E
1 y
2 R 20
Ef
q
2
3p 2G4
/
Hf Ž f , R 0 . C Ž R 0 . s 0,
2 Ž E i f . q R 04 U Ž f . .
/
2
Ž 46 .
We put the WKB solution of the WDW equation of the form
C Ž R 0 , f . s C Ž R 0 . eyi SŽ R 0 .c Ž R 0 , f . ,
SŽ R0 . s
R0
H0
(
dR y 2V Ž R . .
Ž 47 .
Then we find c Ž R 0 , f . satisfies the time-dependent Schrodinger equation ¨ Ec dt i s Hf c , s R 0 , R 0 Ž t . s Hy1 cosh Ht . Eh dh
Ž 48 .
This is a Schrodinger equation for a quantized scalar field in the de Sitter spacetime. ¨ 4. Effect of the black hole in the bulk 4.1. EffectiÕe action for M / 0 In this section we discuss the effect of the spherically symmetric objects in the bulk. The bulk deviates from the AdS spacetime to the AdS-Schwartzschild spacetimes. For M / 0, there is an event horizon at r s r h at which R h Ž rh . s
3
L
4ML
ž (
1q
y1 q
3
/
.
Ž 49 .
We will take some r 1 at which RŽ r 1 . ) R h . We rewrite the effective action by the canonical variable R 0 . We must use slightly different canonical transformation from the previous one. This is related to the fact RŽ r,t . cannot be separable into the warp factor and the scale factor as is done for M s 0. We use the similar method developed by Kolitch and Eardley w16x. By taking the integration by part, the kinetic term can be written as R
0 Ž kinetic term. s yHdt j˙H dR
2 R2 RX
° 2R cosh u ¢ 3 (cosh u q L R r6 y 2 MrR
H
q
R0
dR
R0
3
s yHdt u˙~
2
2 R3
ž /ž 3
2
LR
2M q
6
R3
/Ž
2
cosh u cosh2u q L R 2r6 y 2 MrR 2 .
¶• ß.
3r2
Ž 50 .
Here we changed the variables as j s sinh u . It is convenient to introduce the quantity T ; T s ysinh u
H
R0
dR
ž
Ž 2 R 3r3 . Ž L Rr6 q 2 MrR 3 . (cosh2u q L R 2r6 y 2 MrR 2 Ž1 q L R 2r6 y 2 MrR 2 .
/
.
Ž 51 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
440
We can show the total derivative of T with respect to t is given by 2 R 30
T˙s Ž kinetic term . q
F Ž R 0 ,u . ,
3
Ž 52 .
where cosh u
F Ž R 0 , u . s u˙
(cosh u q L R r6 y 2 MrR 2
y R˙ 0
2 0
2 0
Ž L R 0r6 q 2 MrR 30 . sinh u (cosh2u q L R 20r6 y 2 MrR 20 Ž1 q L R 20r6 y 2 MrR 20 . .
Ž 53 .
We introduce the variable c by sinh u
sinh c s
.
(
1 q L R 20r6 y 2 MrR 20
Ž 54 .
The total derivative of the variable is written as
c˙ s F Ž R 0 , u . ,
Ž 55 .
Then defining z s 2 R 30r3, we can write the effective action up to the total derivative
½
S s dt cz˙y N t m2 Ž 32 z
H
ž
.
2r3
L
ž
y4 1q
Ž 32 z .
6
2r3
y 2 M Ž 32 z
y2r3
.
/
cosh2c
/5
.
Ž 56 .
Now, performing the canonical transformation
cs
P 2 R 20
z s 23 R 30 ,
,
Ž 57 .
we rewrite the constraint as Ps"
3p
ž /
)
R 20 arcsinh
2G
yR 20 q H 2 R 04 q Ž 8Gr3p . M R 02 q Ž Lr6 . R 04 y Ž 8Gr3p . M
.
Ž 58 .
There is an ambiguity in the choice of the sign. The choice of the sign determines the boundary condition of the wave function. We will choose the sign so that the wave function represents the tunneling. After doing this procedure, we obtain
° ¢
S s Hdt~PR˙ y N 0
t
ž
Py
3p
ž / 2G
R 02
)
arcsinh
yR 20 q H 2 R 40 q Ž 8Gr3p . M R 20 q
Ž Lr6.
R 40 y
Ž 8Gr3p . M
¶• ß,
/
Ž 59 .
where we restore the 5-dimensional Newton constant G. 4.2. Effect of bulk black hole The turning point where P s 0 is given by R "s
(
1 " 1 y Ž 32Gr3p . H 2 M 2H2
.
Ž 60 .
K. Koyama, J. Soda r Physics Letters B 483 (2000) 432–442
441
For 1 - Ž32Gr3p . H 2 M there is no turning point. Then we set 0 - Ž32Gr3p . H 2 M - 1. The phase space is described as follows: R 1 - R 0 - Ry ,
classically allowed region,
Ry- R 0 - Rq ,
classically forbidden region,
Rq- R 0 ,
classically allowed region.
Ž 61 .
Classical equation of motion is obtained by R˙ 0
2
ž /
1 sy
R 02
R0
qH 2q
8G M 3p R 04
.
Ž 62 .
The additional term proportional to M behaves as the radiation. Taking the momentum as a differential operators acting on the wave function of the brane world C Ž R 0 . P
™ yi E RE .
Ž 63 .
0
we obtain the WDW equation
E E R0
C Ž R0 . s y
žž
3p 2G
/
)
R 20 arcsin
R 20 y H 2 R 40 y Ž 8Gr3p . M R 20 q Ž Lr6 . R 40 y Ž 8Gr3p . M
/
C Ž R 0 . ' yS Ž R 0 . C Ž R 0 . .
Ž 64 . The tunneling rate at the semiclassical order can be evaluated by
ž
P ; exp y2
Rq
HR
/
dR S Ž R . .
y
Ž 65 .
™
We can verify this agrees with the result obtained in the previous section for the limit M 0. For M / 0, the inflating brane world is created by the tunneling from the closed FRW universe driven by the massive objects in the bulk. For a small radius of the brane world R 0 - Ž8GMr3p H 2 .1r4 , the wave function is different from the one obtained in 4-dimensional quantum cosmology. This implies the presence of the extra dimension gives the nontrivial effect on the quantum cosmology of the 4-dimensional universe if the bulk deviates from the AdS spacetime. Technically, this comes from the fact the r dependence in RŽ r,t . cannot be separable. For large R 0 , the effect of the massive objects in the bulk on the evolution of the brane world becomes negligible, then the wave function of the universe becomes the familiar one obtained in the previous section for M s 0.
5. Summary and discussion In this letter, we derived the effective action for the brane world from the 5-dimensional action for the bulk gravity plus 4-dimensional domain wall. Based on this action, we derived the WDW equation for the brane world. If the bulk is AdS spacetime, the brane world can be created from nothing. The WDW equation and the time-dependent Schrodinger equation for the quantized matter on the inflating brane are the same as those in the ¨ conventional 4-dimensional quantum cosmology. It is easy to extend our analysis to the many brane case. In that case, the ‘‘time’’ is defined in each brane world as is suggested by the derivation of the Schrodinger equation. ¨ We discussed the effect of the spherically symmetric massive objects in the bulk. For a small radius of the brane world, the bulk deviates from the AdS spacetime largely. In this case we found the presence of the extra
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dimension affects the quantum cosmology of the brane world, then the wave function is different from the one obtained in 4-dimensional quantum cosmology. The most important success of the inflationary scenario is that this gives the origin of the structure of the universe. The analysis of the evolution of the quantum fluctuations is needed to explore this issue in the context of the brane world. We must include the effect of the back reaction to the geometry of the bulk in the Schrodinger equation for the quantized matter on the wall. It has been shown the deviation of the bulk spacetime ¨ from the exact AdS spacetime is essential to describe the inhomogeneous brane world w8x. The analysis for the bulk with black holes tells us if the bulk deviates from the AdS spacetime, the presence of the extra dimension imprints a non-trivial effects on the quantum cosmology of the brane world. A detailed investigation of the evolution of the quantum fluctuations will be left for the future work w17x.
Acknowledgements The work of J.S. was supported by Monbusho Grant-in-Aid No.10740118 and the work of K.K. was supported by JSPS Research Fellowships for Young Scientist No.4687. K.K. also thanks Kayoko Maeda for useful discussions.
References w1x V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 Ž1983. 136. w2x K. Akama, Pregeometry in: K. Kikkawa, N. Nakanishi, H. Nariai ŽEds.., Lecture Notes in Physics, 176, Gauge Theory and Gravitation, Proc., Nara, Springer, Berlin, 1982, 267-271, hep-thr0001113. w3x P. Horava, E. Witten, Nucl. Phys. B 460 Ž1996. 506; Nucl. Phys. B 475 Ž1996. 94. w4x A. Lukas, B.A. Ovrut, K.S. Stelle, D. Waldram, Phys. Rev. D 59 Ž1999. 086001. w5x N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 Ž1998. 163; Phys. Rev. D 59 Ž1999. 086004; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 Ž1998. 257. w6x L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 3370. w7x L. Randall, R. Sundrum, Phys. Rev. Lett. 83 Ž1999. 4690. w8x T. Shiromizu, K. Maeda, M. Sasaki, gr-qcr9910076. w9x P. Binetruy, C. Deffayet, U. Ellwanger, D. Langlois, hep-thr9910219; E.E. Flanagan, S.-H.H. Tye, I. Wasserman, hep-phr9910498; ´ P. Kraus, hep-thr9910149; S. Mukohyama hep-phr9911165; D. Ida, gr-qcr9912002; see also C. Csaki, M. Graesser, L. Randall, J. Terning, hep-phr9911406 and references there in. w10x N. Kaloper, Phys. Rev. D 60 Ž1999. 123506; T. Nihei, Phys. Lett. B 465 Ž1999. 81; H.B. Kim, H.D. Kim, hep-thr9906542. w11x A. Vilenkin, Phys. Lett. B 117 Ž1982. 25; Phys. Rev. D 30 Ž1984. 509; Phys. Rev. D 33 Ž1986. 3560. w12x V.A. Rubakov, Phys. Lett. B 148 Ž1984. 280. w13x J.J. Halliwell, S.W. Hawking, Phys. Rev. D 31 Ž1985. 1777. w14x J. Garriga, M. Sasaki, hep-THr9912118. w15x W. Fischler, D. Morgan, J. Polchinski, Phys. Rev. D 42 Ž1990. 4042. w16x S.J. Kolitch, D.M. Eardley, Phys. Rev. D 56 Ž1997. 4651; Phys. Rev. D 56 Ž1997. 4663. w17x K. Koyama, J. Soda, in preparation.