Cosmological Cosmological Perturbations in Perturbations in the brane worldsthe brane worlds

Kazuya KoyamaKazuya Koyama

Tokyo UniversityTokyo University

JSPS PD fellowJSPS PD fellow

Randall Sundrum model Randall Sundrum model (Randall and Sudrum. ’99)

Simplest model for brane worldSimplest model for brane world

　　 Can we find the brane world signatures in cosCan we find the brane world signatures in cos

mological observations such as CMB, GW ?mological observations such as CMB, GW ?

AdS

Brane worldBrane world

2 2,

0-mode

+ KK modes

c ijR h

2 24

anisotopy

a

CMB

T??

T cR

bulk

??ijh

Bulk inflaton modelBulk inflaton model Exact solutions for perturbationsExact solutions for perturbations

K.K. and K. Takahashi Phys. Rev. D 67 104011(2003)K.K. and K. Takahashi Phys. Rev. D 67 104011(2003) K.K. and K. Takahashi Phys. Rev. D in press (hep-th/03K.K. and K. Takahashi Phys. Rev. D in press (hep-th/03

07073)07073) Tensor perturbationsTensor perturbations Numerical calculationsNumerical calculations

H. Hiramatsu, K.K. and A. Taruya, H. Hiramatsu, K.K. and A. Taruya, Phys. Lett. B in press ( hep-th/0308072) Phys. Lett. B in press ( hep-th/0308072)

CMB anisotropies CMB anisotropies Low energy approximationLow energy approximation

T.Shiromizu and K.K. Phys. Rev. D 67 084022 (2003)T.Shiromizu and K.K. Phys. Rev. D 67 084022 (2003) K.K Phys. Rev. Lett. in press (astro-ph/0303108) K.K Phys. Rev. Lett. in press (astro-ph/0303108)

0-mode0-mode

1.Cosmological 1.Cosmological Gravitational WavesGravitational Waves

Kaluza-Klein(KK) modeKaluza-Klein(KK) mode

？？

2 2H

42 10

(1/ 1 )critf Hz

mm

H

Two ways to see the Two ways to see the bulkbulk

Gaussian normal Gaussian normal coordinatecoordinate

Brane is located at fixed Brane is located at fixed

value of the coordinatevalue of the coordinate

Bulk metric is non-Bulk metric is non-separable with respect to separable with respect to t and yt and y

Gaussian-Normal coordinate

y= const.

Poincare coordinate

Static coordinateStatic coordinate

Bulk metric is Bulk metric is separable with respect separable with respect to t and y to t and y

Brane is moving Brane is moving

1-1 Gaussian Normal 1-1 Gaussian Normal coordinatecoordinate

MetricMetric

Friedmann equationFriedmann equation

Parameter Parameter

2 2 2 2 2( , ) ( , ) i jijds dy n y t dt a y t dx dx

0

( )( , ) ( ) cosh 1 sinh ,

( )( , ) cosh 1 (2 3 ) sinh

ta y t a t y y

tn y t y w y

22 5 ( )

( ) 1 , 3 (1 )3 2

tH t H w

Hiramatsu, Koyama, Taruya, Phys.Lett. B (hep-th/0308072)

* /H ( horizon crossing)

(Binetruy et.al)

Wave equationWave equation

Initial conditionInitial condition near brane/low energy metric is separablenear brane/low energy metric is separable 0-mode + KK-modes0-mode + KK-modes　　

No known brane inflation model predicts No known brane inflation model predicts significant significant

KK modes excitation during inflationKK modes excitation during inflation KK-modes are decreasing at super-horizon scales KK-modes are decreasing at super-horizon scales

We adopt 0-mode initial condition We adopt 0-mode initial condition at at

super-horizon scales ( h=const. ) super-horizon scales ( h=const. )

2 2 22 2

2 2 2

' '3 3 0,

h a n h n h a n hp h n

t a n t a y a n y

(Easther et. al., Battye et. al.)

Boundary conditionBoundary condition

There is a There is a coordinate singularitycoordinate singularity in the in the bulkbulk

y

t

( )cy y t

( )cy y t

( )cy y t

00y y

h

( )

0c ty y

n h

Physical brane

Regulator brane

Results ( )Results ( )

Time evolution on the brane

KK-mode

0-mode

Solution on t = const. surface

* 10.2p

aH

H

Amplitude of GW decreases Amplitude of GW decreases due to KK modes excitationdue to KK modes excitation

suppression at ?suppression at ?damping

critf f

Damping factor 42 10critf Hz

Coordinate singularity in GN coordinateCoordinate singularity in GN coordinate high energy region is difficult to treathigh energy region is difficult to treat

Poincare coordinate is well behaved Poincare coordinate is well behaved

The general solutions for GW are easily derivedThe general solutions for GW are easily derived

2

2 2 21( )i j

ijds dz d dx dxz

1-2 Poincare coordinate 1-2 Poincare coordinate (work in progres(work in progress)s)

22( , ) ( ) ( )( ( ) )i i ipxh z dmh m z Z mz e c m e e

(1) (2)2 2 2( ) ( ) ( ) ( )Z mz H mz b m H mz

2 2 2( )m p

Brane is movingBrane is moving

z

Motion of the brane

1, ( )

( )z T t

a t

2 22 5 ( )

( ) 13 2

a tH t

a

211 ( )

( )T H

a t

cosmic timethigh energy

low energy

2

( ) ( )2( , ) ( ) ( )

( ) ( )i T t i T tm

h t dm h m Z e c m ea t a t

Junction conditionJunction condition

Particular solutionsParticular solutionsWe should determine unknown coefficients from We should determine unknown coefficients from

initialinitial

conditions and boundary conditions conditions and boundary conditions

( )a t z

1/0H h

t

(1) (2)2 2 2( ) ( ) ( ) ( )Z mz H mz b m H mz

Recovery of “0-mode” solutionRecovery of “0-mode” solution Due to the moving of the brane, “0-mode” on Due to the moving of the brane, “0-mode” on

FRW brane does NOT correspond to m=0 FRW brane does NOT correspond to m=0

Junction condition at late times Junction condition at late times

1/

4 42

2

( )22

2, log

( ) i T t

Ht

m i H m m mdm O

a a a a a

mh m Z e

a a

( ( ) )T t

Non-local terms

(“CFT” part in the context of AdS/CFT correspondence) 2 2

2

12 (1/ , ) 0p h t

a H

Numerical solution forNumerical solution for

Naïve boundary and initial conditions Naïve boundary and initial conditions “ “no incoming radiation” at Cauchy horizonno incoming radiation” at Cauchy horizon

Initial condition Initial condition

( )h m

2

(1) (1) ( )2

2

(2) (2) ( )2

( , ) ( )( ) ( )

( )( ) ( )

i T t

i T t

mh t dm h m H e

a t a t

mh m H e

a t a t

(1) ( ) (2) ( )( ) ( ) , ( , )im z im zdm h m e h m e m z

(1/ , ) 0, ( / 0)ih t p aH

Numerical results for low energyNumerical results for low energy

The resultant solution The resultant solution

20 40 60 80 100

-0.2

-0.1

0.1

0.2

0.3 Re[ ( )]h m

(1/ , )h t

/ 0.001H

( )i

m

a t 1

10 15 20 30 500.00001

0.0001

0.001

0.01

0.1

1

4D

4

1( ) ip

Dh t e

0( ) ipmh t e

Initial/boundary conditionsInitial/boundary conditions

De Sitter brane

(GN coordinate)

Initial condition

(Battye et. al.)

We define a vacuum state during inflation We define a vacuum state during inflation

““Mode mixing”Mode mixing”

Initial condition Initial condition

2 2 2 24sinhds dy Hy ds

( , ) ( ) ( ) ( ) ipxph y d N f y g e

22( ) ( ) ( , ) ( ) i

pf y g dm V m z Z mz e

sinh

cosh

Hy

z Hy

22( , ) ( ) ( ) ii ipx

ih z dmh m z Z mz e e ( ) ( ) ( , )h m d N V m

Quantum theory

(Gorbunov et. al, T. Kobayashi et. al.)

z

Prediction of GW at high Prediction of GW at high frequenciesfrequencies

(in the near future)(in the near future)

2. CMB anisotropies2. CMB anisotropies

KK modesKK modes At decoupling time,At decoupling time,

KK modes are unlikely to be excited KK modes are unlikely to be excited

Dark radiationDark radiation In homogeneous and isotropic universe, the bulk In homogeneous and isotropic universe, the bulk

BH can BH can

affect the dynamics of the brane at late timesaffect the dynamics of the brane at late times

Effects of dark radiation Effects of dark radiation perturbation on CMB ?perturbation on CMB ?

23/ 10H

Cosmology with dark radiationCosmology with dark radiation

Creation of the dark radiationCreation of the dark radiationEmission to the bulk creates dark radiation Emission to the bulk creates dark radiation bulk field (reheating in bulk inflaton model)bulk field (reheating in bulk inflaton model) graviton emission in high energy eragraviton emission in high energy era

Cosmological observationsCosmological observations BBN constraints asBBN constraints as Dark radiation induces isocurvature perturbationDark radiation induces isocurvature perturbation

Results of WMAP on CMB anisotropies strongly reResults of WMAP on CMB anisotropies strongly restrictstrict

the existence of isocurvature modesthe existence of isocurvature modes

(Himemoto, Tanaka)

(Langlois, Sorbo)

/ 1.23 / 0.11 (Ichiki et. al.)

CMB anisotropies (SW effect)CMB anisotropies (SW effect)

for adiabatic perturbationfor adiabatic perturbation Longitudinal metric perturbationsLongitudinal metric perturbations

Curvature perturbationCurvature perturbation can be calculated can be calculated without solvingwithout solving

bulk perturbations but bulk perturbations but anisotropic stressanisotropic stress cannot becannot be

predicted unless bulk perturbations are knownpredicted unless bulk perturbations are known

m

T

T

2 1 2 2

,c

HR

H H

k a

Red shift

photonconst.m

(Langlois, Maartens, Sasaki, Wands)

Large scale perturbationsLarge scale perturbations - view from the bulk -- view from the bulk -

Isotopic and homogeneous

bulk= AdS-Schwarzshild

Anisotropy on the brane

anisotropy in the bulk

anisotropy on the brane

Bulk and brane is coupled

Gaussian-Normal coordinate for Ads-SchwarzsGaussian-Normal coordinate for Ads-Schwarzshildhild

Consider the perturbation of dark radiationConsider the perturbation of dark radiation

2 2 2 2 2( , ) ( , ) i jijds dy n y t dt a y t dx dx

C C AdS spacetime + perturbations AdS spacetime + perturbations

2 2 2 2 2

2

ˆ1 2 ( , ) 2 ( , )

ˆ1 2 ( , ((1 2 ), ))

ii

i jij ij

ds dy n y t Y dt a B y t Y dx d

E y t

t

a y t Y Y dx dx

2 42

2 2 00 2 2

( , ) ( ) cosh(2 ) cosh(2 ) 1 1 sinh(2 )2

H CaHa y t a t y y y

2

2 450

( )( ) 1 ( )

3 2

tH t Ca t

Solutions for trace partSolutions for trace part

Equations for Equations for

1

2 2 40

2 22

1 ( / ) sinh(2 )ˆ ( , ) ,

2cosh( ) 1 ( / ) sinh( )

ˆ ˆ ˆ( , ) ( , ) ( , )

H y Cay t

y H y

ay t y t y t

a

2( ), ( )B O p E O p

2

2 2 2

' 'ˆˆ 2 " 5 ' 0,

' '" 3 ' 3

ˆ ˆ( ) 3

a a np B B

a a n

n a n aE E n E E

n a n a

n aa p n p B B

n a

In this gauge, the brane location is In this gauge, the brane location is perturbedperturbed

Perform infinitesimal coordinate Perform infinitesimal coordinate transformation and impose junction transformation and impose junction conditionsconditions

Matter perturbations

Junction condition relates matter Junction condition relates matter perturbation on the brane to and perturbation on the brane to and bulk perturbations bulk perturbations

Adiabatic condition on matter Adiabatic condition on matter perturbationsperturbations

equation for equation for

2 24 0

2 24 0 0

ˆ6 6 ' | ,

ˆ ˆ2 2 (2 3 ) ' | 2 ' |

y

y y

H H

P H H H

2 44 0Ca

2 44 0

1

3Ca

2sP c

2 2 2 2

2 2 44 0

(2 3 ) (3 2 3 )

1 1

2 3

s s

s

c H H H c H

c Ca

( 1)H

Solution for : integration Solution for : integration constant constant

Metric perturbations on the brane Metric perturbations on the brane

Curvature perturbationCurvature perturbation

2 4

* *( )( ) m

H Ca

H H p

*

2 20 0

2 20 0

, ( ( ) (0, ))

( 2 )

k a HE E t E t

k a E HE

c

H HR

H H H H

(Koyama. ’02)

Curvature perturbation is determined only by Curvature perturbation is determined only by

Curvature perturbation = brane dynamicsCurvature perturbation = brane dynamics ( FRW equation = brane daynamics)( FRW equation = brane daynamics)

Solution for curvature perturbation can be derived exactly Solution for curvature perturbation can be derived exactly at large scales (including , at high energies)at large scales (including , at high energies)

Anisotropic stressAnisotropic stress

anisotropic stress on the braneanisotropic stress on the brane coupled to anisotropic shear in the bulk coupled to anisotropic shear in the bulk

2 1 2 2 2 20 02 ( 3 )k a k a E HE

C

Equations for anisotropic shearEquations for anisotropic shear

Junction conditionJunction condition

The problem is to solve the wave The problem is to solve the wave equations for equations for

with source and junction conditions with source and junction conditions

2

2 2 2

' 'ˆˆ 2 " 5 ' 0,

' '" 3 ' 3

ˆ ˆ( ) 3

a a np B B

a a n

n a n aE E n E E

n a n a

n aa p n p B B

n a

2 20'(0, )E t p a

( , )E y t

Low energy/gradient expansion Low energy/gradient expansion AssumingAssuming

Solution Solution

Junction condition Junction condition

Anisotropic stressAnisotropic stress

2 40/ ' , / , / 1E E Ca H

2 /'' 4 ' 3 0y lE E e E HE

2 40 0 02

1 1( , ) ( ) ( 3 )( 1) ( 1)

4)

4(y yE y t E t E HE F te e

( )F t : Integration constant

2 20 03 2 ( )2E HE k a F t 2 2

0'(0, )E t p a

2 1 2 2 2 20 0 ( )2p a F tp a

Junction condition completely determine Junction condition completely determine

This is equivalent to use the effective theoryThis is equivalent to use the effective theory

4 ( , )d y xE e e

0

0

22 24

0 02

20 0 0

2( )

( 1) 2

1( )

2

dcdE T e T D D d D d

e

D d D d Dd

(Shiromizu, Koyama)

( )t ( )c t0d

2 2

2 20

'(0, ) ,

'( , ) c

E t p a

E d t p a

( )F t

(4) 24G T E

( )F t

Two branes model at low energiesTwo branes model at low energies

Prediction of CMB anisotopiesPrediction of CMB anisotopies in two branes modelin two branes model

K.K Phys. Rev. Lett. in press (astro-ph/030310K.K Phys. Rev. Lett. in press (astro-ph/0303108)8)

0( 0, const.)C d Detailed analysis of CMB anisotropiesDetailed analysis of CMB anisotropies in two branes model (work in progress)in two branes model (work in progress)

GeneralizationsGeneralizations

Comparison with observationsComparison with observations

Stabilization mechanismStabilization mechanism

00, const.C d

In Gaussian Normal coordinate it is again difficult to address thIn Gaussian Normal coordinate it is again difficult to address the boundary condition due to the coordinate singularity e boundary condition due to the coordinate singularity

Formulation in Poincare coordinateFormulation in Poincare coordinate

Anisotropic stress depends on boundary/initial condition on bulAnisotropic stress depends on boundary/initial condition on bulk gravitational fieldk gravitational field

What is the “natural” boundary/initial condition What is the “natural” boundary/initial condition

withwith dark radiation ? ( de Sitter vacuum for GW)dark radiation ? ( de Sitter vacuum for GW)

One brane modelOne brane model

(Koyama, Soda 00)

( )F t(Cf. )

We should understand the relation between the We should understand the relation between the choice of the boundary condition in the bulk andchoice of the boundary condition in the bulk and the behavior of anisotropic stress on the branethe behavior of anisotropic stress on the brane

Anisotropic stress boundary conditionAnisotropic stress boundary condition

NumericallyNumerically Toy model where this relation can beToy model where this relation can be analytically examined in a whole bulk spacetimeanalytically examined in a whole bulk spacetime

(Koyama, Takahashi, hep-th/0307073)

2 444 0

2 444 0

ˆ ( 1),8

3ˆ ( 1)8

y

y

Cae

Cae

2 40 0 02

1 1( , ) ( ) ( 3 )( 1) ( 1)

4)

4(y yE y t E t E HE F te e

2 1 2 2 2 20 0 ( )2p a F tp a