23/04/091

ブレイン宇宙における重力波の伝播

石原 秀樹大阪市立大学

共同研究者 田中 泉

2

1. Introduction

Brane world is a sub-manifold embedded in a higher dimensional bulk spacetime

Matter fields live only on the brane Gravitational field lives both on the brane

and in the bulk

Gravitational wave is the most hopeful probe to verify the brane universe

Brane-world is a great idea which introduces new aspects on the universe model.

3

Closed strings and Open strings

4

Intrinsic and Extrinsic Geometry

Eratostenes （ B.C.276‐195 ）Riemann

5

Randall-Sundram model

5-dimensional Anti-de Sitter bulk 4-dimensional self-gravitating brane Z2 –symmetry

Phys.Rev.Lett.83,4690,(1999)

Minkowski brane 4-dimensional gravity on the brane

There are lots of works on brane cosmology

6

Basic Equations

5-dimensional Einstein equation

Junction condition

7

Short Cut

8

We study

causal structure of the brane universe,

and

propagation of gravitational waves.

H.Ishihara, Phys. Rev. Lett. 86, 063513 (2001);

H.Ishihara, Phys. Rev. D66 (2002) 023513.

9

2.Geodesics in the Brane Universe

induced metricgeodesic on the branegeodesic in the bulk

Gauss-Weingarten equation

10

Stress tensor

Extrinsic curvature

Vacuum branes are flat in null direction!

Vacuum Brane

11

Friedmann Brane

Stress tensor

Branes are concave in null direction

Extrinsic curvature

Short cuts can exist. Causality in the viewpoint of the brane is apparently violated.

If

12

3. Causal Structure of Friedman Brane

We draw the Penrose’s conformal diagram with respect to the null geodesics in the bulk.

13

Minkowski Brane

14

-10

1

x

-1

0

1

y

0

2

4

t

-1

0

1

y-1

01

x

-1

0

1

y

0

1

2

3

4

t

-1

0

1

y

-10

1x

-1

0

1

y

0

1

2

3

t

-1

0

1

y

Closed Flat Open

Friedmann-brane

15

Short Cuts exist in Friedmann brane !

16

4.Propagation of Gravitational Waves

Perturbations of the 5-dimensional Einstein equation in the bulk

Neumann boundary condition on the brane

17

Gravitational wave perturbations of the Einstein static brane universe.

We consider

Why Einstein Static Brane Universe?

Friedmann-Robertson-Walker brane Concave in null direction

Expanding brane is a moving boundary

18

5-dim. Anti-de Sitter bulk

Einstein Static Brane

Brane is located on = b=const.

Easy to treat perturbations because the boundary is static

19

Einstein Static Brane is concave

Typical dimensions

Stress tensor

Extrinsic curvature

Bulk is negatively curved very much

20

Perturbations

We consider a massless scalar field living in the bulk which mimics the gravitational wave.

avoid gage fixing etc.

21

Wave Equation

Setting we obtain

Neumann boundary condition and regularity

We solve the eigenvalue problem by shooting method.

Klein-Gordon equation

where

22

Mode Functions

n : number of nodes in the extra-dimensionk : wavenumber along the brane: frequency

23

Eigenfunctions

Eigenfunctions for n=0,1,2 and k=1000

We consider cases:n=0,1,2

Amplitude of n=0 mode dominates on the brane.

Expansion of the brane would not modify the waves.

24

Dispersion relation

Frequency on the brane and wavenumber on the brane

phase velocity on the brane and group velocity on the brane

25

Phase Velocity

The phase velocity for n=0,1,2 modes are plotted.

b =5

26

Group Velocity

The group velocity for n=0,1,2 modes are plotted.

b =5

27

Apparent causality vaiolation occurs in the brane universe if null geodesics in the bulk are taken into account.

The phase velocity on the brane for each mode n=0,1,2 exceeds the light velocity.

approaches to the light velocity in large as

The group velocity on the brane for each mode

n=0,1,2 is also larger than the light velocity.

Conclusions

28

ImplicationsPreliminary result :

GW

Change in wave patterns