Cosmic Microwave Radiation Anisotropies in brane worlds K. Koyama astro-ph/030310 K. Koyama PRD 66 084003(2002) Kazuya Koyama Tokyo University.
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Cosmic Microwave Radiation Anisotropies in brane worlds
K. Koyama astro-ph/030310
K. Koyama PRD 66 084003(2002)
Kazuya Koyama Tokyo University
Brane World picture
Standard model lives on a brane
Gravity can propagate into bulk
Observational signatures
Extra-dimension can be “observed” only through gravitational interactions
Can we observe extra-dimension using CMB experiments ?
1. Introduction
gravity Standard model
bulkbrane
Gravity on the brane in AdS5
carries the information in the bulk
4D Equations for on the brane
Randall Sundrum model
ETG
2)4(
)( 2TO nnCE )5(:
)0()(,0 2
TDTOEDE
E
E
n6
2
52
2
5
8
12
G
T
symmetry2Z
(Randall, Sundrum)
(Shiromizu, Maeda, Sasaki)
Cosmology
Background universe
),,,(diag2
pppE
0,0
EDE
,043
1
H
p
ijkijkjk
ikk
YYpYva
YvaYE
1
2
0,0
EDE
Perturbations
23
4
4
3
1
1
1
akvHv
vkaH
p
4D equations can determine the behavior of
E)0(
0k
At large scales, Weyl anisotropic stress cannot be determined only by 4D equations
2 2 2 2 2 5/ 2 7 / 2* 4 4
1
4
1 22
5 3
r
r
T
T
k a k a a da
Red shift
photon
(Langlois, Maartens, Sasaki, Wands)
)0(
CMB anisotropies (SW effect)
We cannot predict CMB anisotropies unless the
behavior of Weyl anisotropic stress is known
In this talk, we calculate Weyl anisotropic stress in two branes model at lo
w energies Impact of dark radiation perturbation on CMB ani
sotropies
Anisotropy on the brane
anisotropy in the bulk
anisotropy on the brane
Bulk and brane is coupled
2.2. Large scale anisotropy oLarge scale anisotropy on the branen the brane
)(tR
Black Hole
Brane
Moving brane in BH geometry
Homogeneity and isotropy
= AdS Schwartzshild bulk
( 0)
We know the metric for Ads-SchwartzsildWe know the metric for Ads-Schwartzsild metric near the brane:metric near the brane:
Consider the perturbation of dark radiationConsider the perturbation of dark radiation
22 2 2 / 4 / 24
22 / 2 4 /4
1 ( 1)4
( ) 1 ( 1)12
y l y l
y l y l i jij
lds dy e e dt
le a t e dx dx
AdS spacetime + perturbations (TT gauge)AdS spacetime + perturbations (TT gauge)
2 2 2 / 2
2 / 2
ˆ1 2 ( , )
ˆ( ) 1 2 ( , ) (1 2 ( , ) )
y l
y l i jij ij
ds dy e y t Y dt
e a t y t Y E y t Y dx dx
anisotropic shear
2 24 / 4 / 44 4ˆˆ ( 1), ( 1),
24 8y l y ll l
e e Ca
In TT gauge, anisotropic perturbationIn TT gauge, anisotropic perturbation can exist (5D graviton)can exist (5D graviton)
In TT gauge, the brane location is perturbedIn TT gauge, the brane location is perturbed
( , )E y t
2 /4'' ' 3 0y l a
E E e E El a
( , )E y t
Junction condition relates matter perturbation Junction condition relates matter perturbation on the brane to and perturbations on the brane to and perturbations
Adiabatic condition on matter perturbationsAdiabatic condition on matter perturbations equation for equation for
24 p
2 24 0
2 24 0 0
ˆ6 6 ' | ,
ˆ ˆ2 2 (2 3 ) ' | 2 ' |
y
y y
H H
P H H H
2sP c
2 2 2 2 2 24
1 1(2 3 ) (3 2 3 )
2 3s s sc H H H c H c
24
Solution for : integration constant Solution for : integration constant
Metric perturbations on the brane Metric perturbations on the brane
Curvature perturbationCurvature perturbation
2 4
* ( )
H Ca
H H p
*
2 2
2 2
,
( 2 )
k a HE
k a E HE
tot
H H
H H H H
Curvature perturbation is determined Curvature perturbation is determined independently of independently of
Curvature perturbation = brane dynamicsCurvature perturbation = brane dynamics ( FRW equation = brane dynamics)( FRW equation = brane dynamics)
Solution for curvature perturbation can be derived exactlSolution for curvature perturbation can be derived exactly at large scales (including , at high energies)y at large scales (including , at high energies)
Anisotropic stressAnisotropic stress
anisotropic stress on the braneanisotropic stress on the brane coupled to anisotropic perturbation coupled to anisotropic perturbation
2 1 2 2 2 22 ( 3 )k a k a E HE
( , )E y t
( , )E y t
Evolution equation forEvolution equation for
Junction condition at the brane Junction condition at the brane
Gradient expansionGradient expansion
2 /4'' ' 3 0y lE E e E HEl
2 2'(0, )E t k la
2 2( '/ ) ( ) 1E E Hl 2
2 / 4 /0 0 0
1( , ) ( ) ( 3 )( 1) ( 1)
4(
4)y l y ll
E y t FE t E E e etH
integration cons( ) nt: taF t
( , )E y t
Junction condition on the braneJunction condition on the brane
One more boundary condition is needed toOne more boundary condition is needed to specify specify
Anisotropic stress is nothing but Anisotropic stress is nothing but
Behavior of anisotropic stress completely depeBehavior of anisotropic stress completely depends on the boundary condition in the bulknds on the boundary condition in the bulk
2 2'(0, )E t k la 2 2
0 03 2 ( )2E HE k a F t
( )F t
( )F t2 1 2 2 2 2
2 2
2 ( 3
( )
)
2
k a k a
F
E HE
k a t
The simplest case where we can determineThe simplest case where we can determine another brane in the bulkanother brane in the bulk
Junction conditionsJunction conditions
( )t
( )F t
( )c t*d
2 2
2 2*
'(0, ) ,
'( , ) c
E t k la
E d t k la
( )F t
Solution for , Solution for ,
Anisotropic stressAnisotropic stress
( )t ( )c t
*
* * *
2* *
3(1 ) 1,
3 1 9 1
3(1 ) 1,
3 1 9 1
r
r
drc c
wC C
w w
we C
w w
*
2 1 2 2* *2
*
2 3(1 )2 ( )
1 3 1
1
9 1
cd
r
wk a
e w
Cw
*2d
0
(for e 1)
Two branes modelTwo branes model Shadow matter, radion affect the anisotropy of the Shadow matter, radion affect the anisotropy of the
brane at large scalesbrane at large scales
CMB anisotropies provide CMB anisotropies provide useful constraints on the modeluseful constraints on the model
Realistic model should be consideredRealistic model should be considered
case is interesting case is interesting Anisotropic stress associated with dark Anisotropic stress associated with dark
radiation does not depend onradiation does not depend on
* const.d
2k
aH
Toy model for anisotropic stress
(Rhodes, van der Bruck, Brax, Davis)
* const.d
3.Effect of dark radiation on CMB anisotropies
Dark radiation perturbation induces the isocurvature perturbation
Weyl anisotropic stress gives distinguishable features to CMB anisotropies
We use the result for to investigate
the effect of dark radiation on CMB anisotropies
2 2 2 2 2 5/ 2 7 / 2* 4 4
1 22
5 3
Tk a k a a da
T
* const.d
Covariant formalism
)(6
)(0
,)(2
11
)(2)1(
2)(
/22
20
10
2
2000
02
0/2
2
/20
0
0
0
0
TTeDddDE
DddDdD
dDdDDTTee
yE
d
yy
dd
:')(0
0
),'(0
y xyedyxd
T
Radion
T
0
0d
1/),(2 00 xyde
1
Parameter
amplitude of dark radiation perturbation
isocurvature perturbation, anisotropic stress
Due to Weyl anisotropic stress, the resultant CMB spectrum is different from simple mixtures of adiabatic and isocurvature perturbations
Likelihood analysis Malkov Chain Monte Calro approach
*cdark=0.25 /r
0.16 dark 0.06 95%c (K. Koyama and K Ichiki in preparation)
SummarySummary
Two branes modelTwo branes model Shadow matter, radion affect large scale CMB Shadow matter, radion affect large scale CMB
anisotropiesanisotropies
Extensions to general cases Extensions to general cases Construct realistic modelsConstruct realistic models stabilization mechanismstabilization mechanism or time dependent fundamental constant or time dependent fundamental constant
0 00, ( )d d t (K. Koyama, R. Maartens, work in progress)
One brane modelOne brane model We should carefully investigate the We should carefully investigate the boundary condition in the bulkboundary condition in the bulk
Anisotropic stress boundary conditionAnisotropic stress boundary condition Validity of our modeling ?Validity of our modeling ?
– Anisotropic AdS-Schwartzshild spacetimeAnisotropic AdS-Schwartzshild spacetime– Toy model where the above relation can beToy model where the above relation can be analytically examined in a whole spacetimeanalytically examined in a whole spacetime (KK and K. Takahashi hep-th/0307073)(KK and K. Takahashi hep-th/0307073)
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