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 2019/4/28

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

Closed strings and Open strings

Intrinsic and Extrinsic Geometry Riemann Eratostenes （B.C.276‐195）

Randall-Sundram model Phys.Rev.Lett.83,4690,(1999) 5-dimensional Anti-de Sitter bulk 4-dimensional self-gravitating brane Z2 –symmetry Minkowski brane 4-dimensional gravity on the brane There are lots of works on brane cosmology

5-dimensional Einstein equation Basic Equations 5-dimensional Einstein equation Junction condition

Short Cut

causal structure of the brane universe, We study causal structure of the brane universe, H.Ishihara, Phys. Rev. Lett. 86, 063513 (2001); H.Ishihara, Phys. Rev. D66 (2002) 023513. and propagation of gravitational waves.

2.Geodesics in the Brane Universe induced metric geodesic on the brane geodesic in the bulk Gauss-Weingarten equation

Vacuum Brane Stress tensor Extrinsic curvature Vacuum branes are flat in null direction!

Friedmann Brane Stress tensor Extrinsic curvature Branes are concave in null direction Short cuts can exist. Causality in the viewpoint of the brane is apparently violated. If

3. Causal Structure of Friedman Brane We draw the Penrose’s conformal diagram with respect to the null geodesics in the bulk.

Minkowski Brane

Friedmann-brane Closed Flat Open

Short Cuts exist in Friedmann brane !

4.Propagation of Gravitational Waves Perturbations of the 5-dimensional Einstein equation in the bulk Neumann boundary condition on the brane

We consider Gravitational wave perturbations of the Einstein static brane universe. Why Einstein Static Brane Universe? Friedmann-Robertson-Walker brane Concave in null direction Expanding brane is a moving boundary

Einstein Static Brane 5-dim. Anti-de Sitter bulk Brane is located on c = cb=const. Easy to treat perturbations because the boundary is static

Einstein Static Brane is concave Stress tensor Extrinsic curvature Typical dimensions Bulk is negatively curved very much

Perturbations We consider a massless scalar field ,F, living in the bulk which mimics the gravitational wave. avoid gage fixing etc.

Wave Equation Klein-Gordon equation Setting we obtain where Neumann boundary condition and regularity We solve the eigenvalue problem by shooting method.

Mode Functions n : number of nodes in the extra-dimension k : wavenumber along the brane w : frequency

Eigenfunctions We consider cases: n=0,1,2 Eigenfunctions for n=0,1,2 and k=1000 Amplitude of n=0 mode dominates on the brane. Expansion of the brane would not modify the waves.

Dispersion relation Frequency on the brane and wavenumber on the brane phase velocity on the brane and group velocity on the brane

Phase Velocity cb =5 The phase velocity for n=0,1,2 modes are plotted.

Group Velocity cb =5 The group velocity for n=0,1,2 modes are plotted.

Conclusions 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.

Implications Preliminary result : GW Change in wave patterns