Gravity in brane world

Takahiro Tanaka (Kyoto univ.)

2008/March @AIU2008

2

Higher dimensional models of particle physics Superstring theory (10dim) , M-theory (11dim)

Our universe is 4-dimensional. → 　 Compactification is necessary!

Compactifiation

yHigher

Dimensionalbulk

flatxm

identify

If the size of the extra-dimensions is very small, we will not notice its presence.

Basic idea:

3

Kaluza-Kleincompactification

Homogeneous in the direction of

extra-dimensions.

BranworldOnly gravity propagates in Higher dimensional spacetime.

Alternative compactification scheme (braneworld)

Red ： gravity flux line

F∝1/rD-2(short ） 　　 1/r2 　　

(long ）

Blue ： EM flux line

F∝1/r2 Since experimental constraints on gravity force are week, relatively large extradimensions are allowed. ～0.1mm　

Gravity naturally propagates in higher dimensional spacetime.

Standard model fields are localized on the brane. 　　　 -otherwise, contradiction with observation.

4

Constraint on the deviation from 1/r2 low

Short rage force /exp1 1212

21 rr

mGmU

Capner et al hep-ph/0611184

5

Large extra dimensions (ADD model)

Effective 4-dim Newton’s constant 　

~1/16p GN

4-dim our universe ⊗2-dim torus

RgydxdG

S n

n

4

4161p

??

Arkani-Hamed, Dimopoulos and Dvali (199 ８ )

444

416Rgxd

GV

n

n p

Assume homogeneity in the direction of extra-dimensions. Volume of

extra-dimensions

nnnpl dMM 2

42

size of extra-dimensions

n=2 , M6 : electroweak scale

1TeV= 103GeVSize of the extra-dimensions

d=1mm ≈ (10-13GeV)-1

(1019GeV)2 ≈ (103GeV)4 (10-13GeV)-2

Hierarchy problem

Example)

6

5

25

43

6 :

Gp

Warped extra dimension

mm dxdxedyds y /222 5-dim anti-de Sitter

Randall Sundrum I model (1999)

y

AdSBulk

5

y

??

y

mx

negative-tensionbrane

positive-tensionbrane

d

Z2-symmetry

Another approach to the hierarchy problem

AdS curvature length5-dim negative cosmological constant

brane tension

7

d

y

mm dxdxgedyds y 4/222

y y

Roughly speaking, 4-dim effective theory can be obtained by substituting Kintaro-candy configuration.

Kintarocandy

54

35 2

2RgxddyMS

代入

444/235 RgxdedyMS

y

y

y

/2/235

2 yy eeM

On the negative tension brane: 3

5/22 1 MeM d

pl

(1019GeV)2 (TeV)2

Hierarchy is explained with d~40{ .

On the positive tension brane: 3

5/22 1 MeM d

pl

Hierarchy is not explained, butwe have finite Mpl even for d →∞.

⇒ new compactification （ RSII ）

This part determines the effective gravitational coupling

8

What was the impact of RSII model on cosmology?Compactification leads to a 4-dim effective massless scalar field corresponding to the size of the extra-dimension. fifth-force (harmful )

Stabilization mechanism to kill this extra scalar d.o.f.. 　　　　　⇒　 The scalar d.o.f. becomes massive.

　Harmless, but the effect of extra-dimensions does not appear at all at the length scale larger than (mass scale)-1 ～ (size of compactification). （￣▽￣）。ｏ０○

Gravity in RSII braneworld

TgTg mmm

211□

½ for 4-dim general relativity. Deviation from ½ is caused by extra scalar degree of

freedom.current bound <10-5

mrer

1Yukawa potential

9

In contrast, compactification is effectively realized due to the warped geometry in RS-II model, although the extra-dimension extends infinitely.

As we do not need stabilization of the volume, gravity at large distance is non-trivial!

mx

Brane

??

0y

y

10

Static spherical symmetric case

Not exactly Schwarzschild ⇒ ℓ << 0.1mm

Metric perturbations induced on the brane

ijij rrGMh

2

2

312

2

2

00 3212rr

GMh

• For static and spherically sym. configurations second order perturbation is well behaved. 　　

Correction to 4D GR=O (ℓ 2/R2star)

» Giannakis & Ren (’00) exterior» Kudoh, T.T. (’01) interior» Wiseman (’01) numerical

No Schwarzshild-like BH solution?

Gravity on the brane looks like 4D GR approximately, BUT

(Randall Sundrum (‘99) Garriga & T.T. (’99))

11

z

mm

dxdxgdzz

ds Sch 22

22

xgm

Kintarocandy

solution

Black string solution

Metric induced on the braneis exactly Schwarzschild solution.

However, this solution is singular. Cmr Cmr ∝ z 4

Moreover, this solution is unstable. Gregory Laflamme instability

( Chamblin, Hawking, Reall (’00) )

“Black string longer than its radius is unstable.”

12

AdS/CFT correspondence WCFT[q]=SEH+ SGH S1 S2 S3 ( Maldacena (’98) )

( Hawking, Hertog, Reall (’00) )

z0→ 0 limit is well defined with the counter terms

SRS= 2(SEH+ SGH) 2S1 Smatter

= 2S2 Smatter 2(WCFT+ S3)

Boundary metric Counter terms

2

5525

122

1

RgxdSEH

KqxdSGH4

25

1

qxdS 425

13

RqxdS 4425

2 4

"" 243 RS

Brane position

z0 ⇔ cutoff scale parameter

brane tension

4D Einstein-Hilbert action

Evidences for AdS/CFT correspondence

Linear perturbation around flat background (Duff & Liu (’00))

Friedmann cosmology ( Shiromizu & Ida (’01) )

Localized Black hole solution in 3+1 dimensions ( Emparan, Horowitz, Myers

(’00) )

Tensor perturbation around Friedmann ( Tanaka )

13

4D Einstein gravity+CFT quantum

correction equivalent

Classical 5D dynamics in RS II model

GeneralizedAdS/CFT

correspondence

14

4D Einstein+CFT with the lowest order

quantum correction

Classical black hole evaporation conjecture

5D BH on brane4D BH with CFT

equivalent

equivalent

Classical 5D dynamics in RS II model2

4

2

number of field of CFT

Hawking radiation in 4D Einstein+CFT

pictureequivalent

Classical evaporation

of 5D BH

AdS/CFT correspondence

(T.T. (’02), Emparan et al (’02))

Time scale of BH evaporation

32

32

1species

ofNumber

MGMGMM

NN

year100mm1.0 23

SolarMM

15

Metric induced on the brane looks like Schwarzschild solution,

but

2221

22 2121 mm drdrr

dtr

ds

Black Hole solution in 3+1 braneworld

This static solution is not a counter example of the conjecture. Casimir energy of CFT fields on with is given by

“At the lowest order there is no black hole. Hence, absence of Hawking radiation is consistent.”

( Emparan, Horowitz, Myers (’00) )

p

mp 2

234

3/1

21

1

8 33rG

T CFT

pm

m The above metric is a

solution with this effective energy momentum tensor.

3G 22222 drdrdtds 3/123/4 mp

Emparan et al (’02)

Exact solution exists in 3+1-dim.

16

Numerical construction of brane BH

Static and spherical symmetric configuration

T, R and C are functions of z and r.

Kudoh, Nakamura & Tanaka (‘03) Kudoh (’04)

Comparison of 4D areas with 4D and 5D Schwarzschild sols.

p 4A

0 1 2 3 4 5 6Log @L kD

1

1.2

1.4

1.6

1.8

2

2.2

k!!!!!

!!!!!!

A 4p

100*10005D Sch.

4D Sch.

log

4D Sch.

5D Sch.

is surface gravityIt becomes more and more difficult to construct brane BH solutions numerically for larger BHs.

Small BH case ( –1 < ℓ ) is beyond the range of validity of the AdS/CFT correspondence.

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We need to solve only the Hamiltonian constraint to obtain a time-symmetric initial data:

easier!

Time-symmetric initial data for brane BH

Tanahashi & Tanaka (to appear in JHEP)

1) Even an initial data might be difficult to construct for large AH area.

1) It was possible to construct an initial data with large AH area. 2) We failed to obtain an initial data with MBH<MBS for the same AH area,

Initial data is not unique, but

2) If there is a stable static BH, we expect MBH<MBS for the same horizon area. Results:

Next step is its time evolution!

which is consistent with “classical BH evaporation conjecture”.

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Dvali-Gabadadze-Porrati modelAction:

(Phys. Lett. B485, 208 (2000))

For r rc , 4-dim induced gravity term dominates?Extension is infinite, but 4-dim GR seems to be recovered for r rc .

mx

Brane

??y

MinkowskiBulk

mattLRMgxdRgxdMS 424

44535

crMM 2/24

35

Critical length scale

yconstant

induced gravity term

19

Flat Friedmann equation

In early universe , cosmic expansion is normal.

Late time behavior for e = +1

Cosmology in DGP model

HrHM cer 2

243

1

1 crH

1 crH

1e

self-accelerationin the limit r → 0

figure taken from Chamousis et.al. hep-th/0604086

1e

1e

1e

(Deffayet (2006))

20

Self-acceleration has a Ghost

2Lnormal

2Lghost

Spontaneous pair production of ghost and normal particles

unstable vacuum.Once a channel opens, Lorentz invariance leads to divergence.

Maybe we need non-Lorentz invariant cutoff.

21

Ghost in self-accelerating branch in DGP model

The mass of the lowest KK graviton in self-accelerating branch

m2 = 2H2 for r = 0, 0 < m2 < 2H2 for r > 0.

A massive graviton in de Sitter space with 0 < m2 < 2H2 contains a spin-2 ghost mode in general.

(Higuchi Nucl. Phys. B282, 397 (1987))

Marginal, but there is a ghost mode in DGP. (e.g. Gorbunov, Koyama and Sibiryakov, Phys. Rev. D73, 044016 (2006))

Spin2 ghost: ,0; mh 0

hhelicity decomposition (0, 1, 2)

(scalar, vector, tensor) in cosmological perturbation 1 2 2 This mode becomes a ghost

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Can we erase the ghost simply by putting the second regulator brane in the bulk?

The idea is: if the distance between two branes becomes closer, the KK mass will increase.

m2 > 2H2

The ghost will disappear.

Can we erase the ghost?

Bulk is Rindler wedge of Minkowski space

t

r

y=y y=y

24

222deSitterDdsydyds

23

In fact, there is no ghost in spin-2 sector once the second brane (or negative energy density) is introduced. t

ry=y y=y

y

H 1

2m

22H

self-acceleration H1/rc

r 0: H1/rc

1 crH

far limi

t

close limitHowever, at the point where the spin 2 ghost

disappears, spin 0 (brane bending mode) ghost appears instead. (K. Izumi, K. Koyama & T.T, 2007)

(single brane: Charmousis et al. 2006)

r 0: H1/rc

spin-2 ghost exists

Stubborn ghost

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Spontaneous pair production of ghost and normal particles

Vacuum becomes unstable. particle production rate diverges due to UV contribution,

but it is a bit strange that UV behavior is affected by the value of cosmological constant.

In H→0 limit there is no ghost. If there is a non-covariant cutoff scale, the pair production rate becomes finite. Then, the model might be saved.

Do we really need to be afraid of

spin 2 ghost in de Sitter space?

Here we discuss general massive gravity theory in de Sitter background.

25

Strong coupling scale If a mode has a small quadratic kinetic term 2232 LS

→ 0 limit strong coupling.

3

664

3

2

243 1

pLpp

pL

xx

x x

When spin-2 ghost marginally appears (m2= 2H2), all the scales are necessarily in the strong coupling regime!

strong coupling scale = L/1/2

may introduce a natural cutoff scale??

vertex propagator loop integral

26

We may justify 3-momentum cutoff?Spin2 ghost:

helicity decomposition (0, 1, 2) (scalar, vector, tensor) in cosmological perturbation

1 2 2 ghost

Action for spin 2, helicity-0 mode

dsHms

kHmmMS kflatk

kghost

2

2

4

22224 22 □

*

22 ss

ksh

i

m

Action for helicity-0 mode, depending on 3-momentum k, is not covariant, but spin 2 graviton in total is covariant classical mechanically.

≡background covariance ≡de Sitter invariance

(K. Izumi & T.T, 2007)

27

However, quantum mechanical state will lose covariance.

222 ˆ2

ˆ21ˆ xpH

†aeaex titi ˆˆ21ˆ

†aeaeixp titi ˆˆ2

ˆˆ

222

2xxL

pi

xae ti ˆ1ˆˆ2

Quantization of a ghost)

00ˆ a 0,10

txx x

2

0 2exp, xtx

21ˆˆˆ aaH †

21nEn

⇒ -ve (ghost case)

⇒ +ve (normal case)

00ˆ †aTo make the ground state wave function normalizable,

2

0 2exp, xtx

0ˆ nan †&

21nEn

0ˆ nan negative energy statesQuantization which avoids negative norm distinguishes ghost from normal case.

28

Self-acceleration branch of DGP model has a ghost.

Ghost is composed of helicity 0-mode in spin-2 sector.

Quantization of this ghost breaks Lorentz invariance.

The strong coupling energy scale is low.

It’s not completely clear if violent particle production occurs because the relevant modes are in the strong coupling regime.

Why did the ghost appear?

2929

Correction to gravity in DGP normal branch

Perturbation equation in weakly non-linear regime:

5,

5,

24

5 ˆˆˆ3rm

mrrm TMrc

□

524

ˆ1212 mmmm □

crTTMDh

5̂

Substituting the linearised version of the above equation into ★,

TTMDh mmm

312 2

4

□□ 1crD

Once non-linear term becomes important, one can neglect in Eq.★. Then, 4D GR is reproduced.

5̂

★

Is 4D GR a good approximation even for strongly gravitating system ？　How about BH solutions ？

Since this coefficient is extremely small, non-linear terms becomes important even for weakly gravitating system.

:brane bending d.o.f.

relatively large strong coupling scale

3030

Gravity in higher co-dimension brane world

In general gravitational potential becomes singular at the brane. )dimentions-co of(#2 r

very shortly

Some regularization is necessary.co-dimension 1 brane + KK compactification.

~similar to 5-dim cases

Gauss-Bonnet term in the bulk (6-dim).

~does not seem to work as is initially proposed.

Nested brane world with induced gravity terms.

Ghost appears but it is claimed that the ghost can be erased by putting sufficiently large 4-dim tension.Not as stubborn as the ghost in self-accelerating branch?

mattLMNPLMNP

LMLM LgxdRRRRRRgxdMS 44253

5 42

mattLRMgxdRgxdMRgxdMS 424

4455535

646

(Bostock, Gregory, Navarro, Santiago(2003))

(de Rham, Dvali, Hofmann, Khoury, Pujolas, Redi, Tolley, arXiv:0711.2072) 

31

SummaryGravity is quite non-trivial in several brane world models. RS-II model

Extension is infinite, but effectively 4-dim gravity is realized. 1/r3 potential: correction is not exponentially suppressed. Stationary black hole solution may not exist.

(classical BH evaporation conjecture)Induced gravity (DGP model)

Self-acceleration branch has a ghost (helicity 0 mode of massive

graviton), which might be less harmful than the usual

ghost. Normal branch is also abnormal.

Non-linear terms are important for the recovery of 4-dim GR.Superluminal motion around the gravitating body. Black hole solution is not found.

Higher derivative, Higher co-dimensions, etc.