Brane Gravity and Brane Gravity and Cosmological ConstantCosmological Constant

Tetsuya ShiromizuTetsuya Shiromizu Tokyo Institute of TechnologyTokyo Institute of Technology

白水 White Water

Plan

1. Warped extra dimension - Randall-Sundrum braneworld -

2. Effective theory

3. D braneworld

4. Summary

1. Warped Extra Dimension

- Randall-Sundrum Braneworld -

Warped extra dimension

/ye

y

flat brane

/ye

constant alcosmologic negativebulk

Randall&Sundrum(1999)

Extra dimension

5-dimensions

The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000

gEGTRgR brane 482

1

4dim Einstein equation

Correction terms

22

525 6

1

2

1 bulkbrane

Cosmological constant on the brane

Bulk cosmological constant Brane tension

For flat brane, we assume Randall-Sundrum tuning 06

1 225 bulk

Two cosmological constants

Bulk cosmological constant

Cosmological constant on brane

Do not confused !Do not confused !

The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000

gEGTRgR brane 82

1

baba

brane

nnCE

TqTTqTTTT

G

)5(

2

22

222

24

1

8

1

12

1

4

16

18

6

1

2

1

Cosmology - FRW model -)( fluidPerfect uuqPuuT

))(2(12

1 uuqPuu

0)()(

0)( 0

PDuuguDuP

uDPDuTD

DuuqPD ))((6

1

0DIf homogeneous and Isotropic

0 ED

spatial derivative

fluid 4-velocity

Cosmology II

0 ,0

EDE

For the homogeneous and isotropic universe, it becomes the energy-momentum of “radiation”.

(Equation of state) (Equation of motion)

)(400 taE

(Dark radiation)

Equation is closed!Mass parameter of Schwarzshild-antideSitter Black Hole

Braneworld adS/CFT (Witten,Gubser,Garriga&Sasaki,Shiromizu&Ida,Tanaka,…)

Scale factor (~radius of universe )

Cosmology III

42

44

2

2

3633

8

aa

KG

a

a N

222

22

)()( Kdr

rf

drdTrfds

Friedmann equation on the brane:

Bulk spacetimes: 5-dim. Schwarzshild-anti deSitter spacetime

22

2

)(r

rKrf

(unit 3-dim. Sphere, plane or hyerboloid(K=1,0, or -1)

Langlois et al, Garriga & Sasaki, Ida, Kraus, Mukohyama et al, …

Problems

How to solve?

?E

✓Linear perturbation( Garriga & Tanaka, Sasaki, Shiromizu & Maeda,…)

✓Cosmological Perturbation( Kodama,Ishibashi&Seto, Mukohyama, Koyama&Soda, Langlois,…)

✓Numerical analysis (Shiromizu&Shibata, Kudoh,Tanaka&Nakamura,…)

✓Gradient expansion( Wiseman, Kanno & Soda, Shiromizu & Koyama,…)

✓Close limit of two branes (Shiromizu,Koyama,Takahashi, de Rham&Webseter,…)

✓Lower dimension (Emparan,Horowitz&Myers)

2. Effective theory

Model Randall-Sundrum I type

)()(2

2

1)(

)(4)(

)(4)5()5(52 branebrane LgxdLgxdRgxdS

)0( y

dxdxxygdyedxdxgds xybaab ),(2),(22

)( 0yy

y

- brane+ brane

Bulk action + Brane action -Brane action

Induced metric & extrinsic curvature

dcdb

caab nqqK

1 , baabbaabab nngnngq

ananan~

paralell transport

Induced metric

hypersurface

Extrinsic curvature

ay

a en )(

qeK

dxdxxyqdyeds

y

xy

2

1

),(2),(22

(1+4)-Decomposition

)(2

|)( )(2

TKK yy

04

3~

12~~

4

3

6 ,

4)(

~~

4

1

)(~~~

2)4(2

22222

traceless)4(

KDKD

RKKK

DDKKKKe

DDDDRKKKe

y

y

(Bulk)

(On Branes)

We solve the evolution equations along the extra dimension y

KKK

4

1~

Junction conditionsJunction conditions

2-1. Low energy effective theory

Wiseman 2002, Kanno & Soda 2002, Shiromizu & Koyama 2003, de Rham 2004

Small parameter

1~||

||2

2

)4(

brane

bulk

LK

R

),(),(),(

0),0( , ),()(),(),()1()0(

)1()1(2

xyKxyKxyK

xgxygxhxyaxyg

We are interested in long wave scale. Then small parameter is the square of ratio of the bulk curvature scale to the brane intrinsic curvature scale:

Sketch

RKy ~

branebrane RK |~|

The junction condition at the branebrane

brane TK 25~|

braneT 25~

Gradient expansion or close limit

braneTRhR 1252

1

0th order

04

3

12~~

4

3

4~~

4

1

0~~

)0()0(

2)0()0(2)0(

2)0()0(2)0()0(

)0()0()0(

KDKD

KKK

KKKKe

KKKe

y

y

2|)(

2)0()0(

yyKK

0th order

),(

)(),(

~

~~)0(

(0))0()0()0(

xyg

xCxyKKKKe y

0),(~

0)( 0x),(~

condition Junction (0)(0) xyKxCyK

ggeK y)0()0()0( 1

2

1

y

xyd

dyexydxhexyg0

),(2

)0( ),( ),(),(

“Integration constant”

0 ,12

4

3 )0(2

2)0( KDK

constraints

Junction conditions 2

6

11

0th order

y xyxyd

dyexydxhexhag

K

0

),(),(

22)0(

2)0(

),( , )()(

6

11with

1

Summary

1st order

TKK

KDKD

gRKK

eDeKKKe

eDDegRKKKe

yy

hh

y

y

)(2

)1(

)1()()1()(

)0()4()1()0(

2)1()1()0()1(

traceless)1()0()4()1()0()1(

2|)(

04

3~

)(2

3

)(2

1

)()(~~~

1st order

2)()()()()(2

2)()()(2

2)()()()4(2

)4(

)(2

111

)(212

)(1

)(

DddDdDdDDa

eDD

DddDdDdDdDDhRa

gR

hhhhhy

hhhhhh

traceless

)()()()(2)4(2)1(4 1)(

~)(

2

~dDdDdDDahRaKa hhhh

yyy

4

traceless

)()()()(2)4(2)1( )(1

)(~

2),(

~

axdDdDdDDahRaxyK hhhh

2)(2)(

2)4(

2)1( )(

11)(

6),( DddD

ahR

axyK hh

・ The traceless part of evolutional equation becomes

solution

Integration constant・ From the Hamiltonian constraint, we obtain

1st order

42)()()(2)()()(2)4(2

)1()1()1()1(

)(2

11)(

2

4

3~

aDddDdDdDdDDahGa

KKKK

hhhhhh

(2) )(2

)(2

(1) )(22

400

20

)4(20)(

)(2

)4()(2

adDDahGagT

hGT

40)2()1( a

)(2

112

)()()()1(

)()1(2

)()(2

0)(

0)(

0)(

02)(

0)()(2

0

)(20

)(2

)4(20

0)()(2

0)4(2

0)(2

0)(

2

DddDdDdDdDDa

hTahThGa

dDDahGahTahT

hhhhhh

hh

Effective gravitational equation

Junction conditions imply

Comment on RSII model

)(2

)( )(2

)4( xThG

undetermined

In principle, it is determined by boundary condition at “y=∞”

It may corresponds to the dark radiation, but we have to confirm that.

)()( 22

)4(

TOEThG

2-2. Close limit

Shiromizu, Koyama and Takahashi, 2003 (primitive version) de Rham and Webster 2005 (elegant version)

Assumption and motivation

Brane collisions is fundamental like particle collisions

Brane collision may give us a new picture of big-bang

1d

The brane distance is much smaller than the bulk curvature scale

What we have to do actually

)(2

12)( 2

2)4(

KKKKKKeeDDeyG y

can be written in terms of energy-momentum tensor on branes (Junction condition)

We must evaluate this

de Rham & Webster’s way

)0(!

1)1(

00

n

nyKn

yK

)/(1ˆ 2 dOKOK ny

ny

2,

,,

, )(:ˆ dSSddSddSO

)0(ˆ

)ˆsinh()0()ˆcosh(

)0()!12(

1)0(

)!2(

1

)0(!

1)1(

0 0

122

0

KO

OKO

Kn

Kn

Kn

K

y

n n

ny

ny

n

ny

)0()ˆtanh(

ˆ)1(

)ˆsinh(

ˆ)0(

K

O

OK

O

OKy

Effective equation

2)()()()()()(

)()()()()(

4)(

)(2

)(2

)()()(

72

1

12

1

8

1

3

1ˆcosech3

1ˆcothˆ

6

2

2

2

||tan

2

||tanh

1

2

||tanh||

11

TTTTT

TTOTTOOA

TTA

d

dddd

d

dd

ddDD

dG

Tensor perturbation

dd

dTh ijij

coth ,~ 22eff

2eff

2

Equation for tensor perturbation is same with 4-dim. one except for gravitational constant

3. D Braneworld

Shiromizu, Koyama, Onda & Torii, 2003 Shiromizu, Koyama & Torii, 2003 Onda, Shiromizu, Koyama & Hayakawa, 2004 Shiromizu, Himemoto, Takahashi, 2004 Iwashita, Shiromizu, Takahashi, Fujii,2005

D brane cosmology

Brane world is motivated by D braneD brane

deSitter/Inflation model in warped flux compactification

Kachru,Kallosh,Linde,Trivedi, 2003

Kachru, Kallosh, Linde, Maldacena, McAlister, Trivedi (KKLMMT), 2003

Gravity on D brane

D brane

55 SadS compactification

Tension T = Charge Q

Bulk: IIB supergravity compactified on S^5 (~ 5-dim. theory)

Z_2 symmetry

H_3=dB_2, F_3=dC_2, G_5=dD_4

B_2

D_4Brane: Born Infeld action + Chern Simons

From Ten　 to Five

334210

25

23

21

23

210

210

10210

10

4

1

|~

|4

1|

~|

2

1||

2

1||

2

1)(4

2

1

FHC

FFFHRegxdS

GFDCC

dedxdxxgds

HCFFHCFFdCFdBH pp

~~ ,~ ,~

)(

~ ,

~ , ,

540

25

22

325530331223

IIB Supergravity

Theory

5on ty supergravi IIB type:Bulk S

termSimons-Chernaction Infeld-Born :Brane

25

23

24

52

322)5(4

52)5(5

2|

~||

~|)(

2

1||

2

15)(

4

5)(4

2

1GFeHRegxdS

222244

brane 2

1)det( BBBCDQBhexdTS

form-5 form-3 S of size dilaton 5

324532323 ~~

,~~

,~ HCdDGHdCFdBH

Way to look at gravity on brane gK y~

) tensorstressbulk (~ RKy

branebranebrane RK |) tensorstressbulk (|~|

The junction condition at the brane )(~| 25

branebrane TThK

braneT 25~

Replaced by stress tensor on the brane using the junction condition for bulk form fields

braneT 25~

cancel

Gradient expansion or close limit

5-dim Einstein equation

Anti-de Sitter curvature radius

Gravity on brane

The gravitational theory at low energy

0~2

1RgR

Z_2 symmetry

H_3=dB_2, F_3=dC_2, G_5=dD_4

B_2

D_4B_2 on the brane is not the source of the brane gravity

Tension T = Charge Q

Breaks “supersymmetry”

T≠Q

Introduction of anti-D brane

T ≠ Q

braneTTgQTRgR 112 )(2

1

rebraneTgQTRgR ,1212 )(2

1

branerebrane TTQTT 1, )(:

Tension T ≠ Charge QTension T ≠ Charge Q

Z_2 symmetry

H_3=dB_2, F_3=dC_2, G_5=dD_4

B_2

D_4

No cancellationNo cancellation

Cosmological constant appears at the Cosmological constant appears at the same time as B_2 is a source of the same time as B_2 is a source of the gravitygravity

Shiromizu, Koyama, Torii, 2003, Iwashita, Shiromizu, Takahashi, Fujii, 2005

4. Summary

Summary

Braneworld effective theoryBraneworld effective theory

Gradient expansionGradient expansion

Close limitClose limit

Braneworld based on D-braneBraneworld based on D-brane

The gauge fields on The gauge fields on BPS D braneBPS D brane is is not sourcenot source of gravity on brane. of gravity on brane.

The gauge fields on the The gauge fields on the deSitter (anti-) D branedeSitter (anti-) D brane is is sourcesource of gravity on brane. of gravity on brane.

Remaining Issues

Higher co-dimension, Warped flux compactifHigher co-dimension, Warped flux compactification( KKLT, KKLMMT model)ication( KKLT, KKLMMT model)

other fieldsother fields

stabilisationstabilisation

Basics of higher dimensional spacetimesBasics of higher dimensional spacetimes

Mukohyama et al, hep-th/0506050

Fin

Thank you very much

Anti-D brane

Randall-Sundrum type modelRandall-Sundrum type model

T_+=Q_+ T_-=Q_-=-T_+=

D brane with positive tension

D brane with negative tension

Inflating anti-D brane in warped extra dimension

T_+=Q_+ T_-=Q_-≠-T_+

D brane with positive tension

D brane with negative tension

anti-D brane T_0=-Q_0

Koyama & Koyama, hep-th/0505256

cosmological constant is cosmological constant is induced on anti-D brane induced on anti-D brane

Note: D branes is not deSitter brane

If the brane distances are stabilised, D branes would be inflating

Gravity on anti-D brane?Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005

Koyama & KoyamaKoyama & Koyama:

“anti-D brane is inflating”

Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005:

“non-BPS brane is inflating at the same time as gauge field on the brane is a source of the gravity on the brane”

The gauge field on the anti-D brane The gauge field on the anti-D brane would be a source of the gravity on the would be a source of the gravity on the anti-D brane. anti-D brane.

Gravity on anti-D brane

)partraidon (22

1 25

0

25 braneTgTRgR

Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005

Set-up

)(g

MNg

)(2B

)(2C

)(4D

23 dBH

23 dCF

45 dDG

)(2F

tenson=charge )( )(

)(2B

)(2C

)(4D

)(2F

tenson=charge )( )(

)(g

dxdxxygdydxdxgds NMMN ),(22

0~)(xy 0~)( yxy

brane-D brane-D

Toy Model

2||

2

1||

2

1||

2

1)(

2

1

2

1 25

23

23

2)5()5(52

GFHRgxdS

)(2

)(2

)(2

)(2

)(4)(

)(4)()brane(

)(2

)(2

)(2

)(2

)(4)(

)(4)()brane(

2

1)det(

2

1)det(

fffCDfqxdS

fffCDfhxdS

)(2/1)(

)()( ||

FBf

06

52 2

)(4 Background spacetime can be pure anti-deSitter+ flat branes

assumption

Equations

0~

2

1

~~

0~

2

1~~

0~

2

1

~

4

1~~

3

4

22

)5()5(2)4(

2)5()5(2)4(

yyyy

yyy

yy

yyy

yy

yyy

y

MMy

FHKD

GKG

GHFKF

GFKXX

KKTTRK

KTTRKe

MNKLAB

NMKLABMNKL

NMKLMNKL

NMKLMNNMMN

yyy

gGGFgFFHgHHgT

FHX

2 2 2)5(2 ~~

96

1|

~|

~~

4

1||

4

1)(

2

1

2

1

~

0~

0~

0

~~

4

3

2

1

)5(2

)5(22)4(

y

y

y

y

yy

GD

FD

XD

TKDKD

TKKKR

0~ FH

For simplicity, we assume

(Evolutional equations)

(Constraint equations)

Junction conditions

)()()(

2

)(2

)()(

2

)()(

2

)()(

2

)()(

8),(

),(~

2),(

~

),(

3

1

21),(

ffx

xG

fxF

fxH

TgxyK

y

y

y

y

branes-on D

)(

)(

)()()(

4

1 ffffT

Long wave approximation

12

L

scale curvature intrinsic brane :

scale curvaturebulk :

L

)1()0(

)1(2 )(),(),(

KKK

gxhxyaxyg

0th order

)(),( ,1 2

2)0()0(

xhexyagKy

)(2

)(2

6

1

6

11

)(24 )(

~ yaGy

)()(:

1st order

hhfeaxyF

feaxyH

hhGFa

H

hhGHa

F

y

y

yyyy

yyyy

)(62)2/1(

)(62)2/1(

)2/1(

4

)2/1(

)2/1(

4

)2/1(

2),(

),(

0~~

2

1

0~

2

1~

)(140

)()(60

)( TaTfaf

/0

0yea

)2/1()2/1(

,~

yy HF

brane-Don condition junction

brane-Don condition junction

1st order )1(K

)(~

)1(2

11

)(2

)(~

2

1

2

2)(

~

2

1

2

)(2

)(~

2),(

~

)(~~~

)4(20

)()()()(40

)()()()(

40

)(

160

2)4(2

0

)()()()()(

2

)(

2)4(

)()()()()(

2

4)(

162

2)1(

)(

1624)4(2)1()0()1(

hRaDDDDaDDDD

xaTahRaDDDDT

ThRDDDDT

xaTahRaxyK

TahRaKKK

tracelesstraceless

traceless

traceless

y

0)(1

)(1

)(3

2)(

3

2

)(6

)(1

)(3

2

)(6

)(1

)(3

2

)(6

),(

2220

22)()(

2)()(

2

)4(20

22)()(

2

)4(22)()(

2

)4(2

)1(

DDaDD

hRa

DD

hRDD

hRa

xyK

Traceless part

Trace part

Junction conditions

Junction conditions

Effective Equation at 1st order

2)(

)(

)()()(2

)(

)()(20

)()()(40)()(

2

)()4(2

0

2

112

)()()1(

DdgdDdDdDgdDDa

gagGa

0)(3

2)(

3

2)(

1 20)()(

2)()(

22)()(

220

adDdDa

Effective Equation at 2nd order

)()(2)(

)()()(

)()(20

140

1

)(20

120

140)()(

12

2)(

)(

)()()(2

)(

)()(20

)()()(20

40)()(

2

)()4(2

0

)1(12

1)1(

14

1

8

3

4

33

28

81

7

3)(

2

112

))(1(1)()()1(

TDTDDTDDaa

Taaa

DdgdDdDdDgdDDa

gaagGa

hhffa

aadDdDa

)()(2

0)()(

2

2)()(

20

20)()(

2)()(

22)()(

220

)1)((2

))(1(9

2)(

3

2)(

3

2)(

1