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)