曲がる余剰次元と宇宙項曲がる余剰次元と宇宙項- braneworld cosmology -- braneworld cosmology -

8 ?G GT

YITP, Kyoto UYITP, Kyoto UMisao SasakiMisao Sasaki

Plan of this talkPlan of this talk

1.1. IntroductionIntroduction

--- relativist/cosmologist’s view of --- relativist/cosmologist’s view of braneworldbraneworld

--- Randall-Sundrum’s (single) braneworld--- Randall-Sundrum’s (single) braneworld

2. Brane cosmology from the bulk point of 2. Brane cosmology from the bulk point of viewview

3.3. Alternative modelsAlternative models

--- DGP model, Bulk inflaton model, …--- DGP model, Bulk inflaton model, …

4. Einstein Gauss-Bonnet braneworld4. Einstein Gauss-Bonnet braneworld

5. Summary5. Summary

1. Introduction1. Introduction Braneworld Braneworld domain wall with matter fields domain wall with matter fields 3-brane = 3-brane = 4D time like (singular) hypersurface4D time like (singular) hypersurface in in 5D spacetime (bulk).5D spacetime (bulk).

Brane tension (Brane tension (vacuum energy vacuum energy 00 → → self-gravitatingself-gravitating domain walldomain wall

Causality on the brane Causality on the brane causality in the bulkcausality in the bulk Ishihara ‘01Ishihara ‘01

A new picture of the Universe!A new picture of the Universe!

Two different views on gravity in the bulkTwo different views on gravity in the bulk

emission of closed loops spacetime fluctuations

string (~particle) picture geometrical (~wave) picture

This talk is based on a viewThis talk is based on a viewfrom this standpointfrom this standpoint

Need to UNIFY the two pictures!Need to UNIFY the two pictures!

In General Relativity or any theory withIn General Relativity or any theory with

11//rrnn--22 force law in force law in n-n-dim spacetime, dim spacetime,

embedding a embedding a self-gravitatingself-gravitating brane with brane with co-dim > 2co-dim > 2

will generically form a will generically form a black holeblack hole (brane)!(brane)!

ie, object with ie, object with TT~ ~ pp((xx)) where where pp>2>2 in in nn-dim-dim

Example: Example: nn=4=4

• domain wall (domain wall (pp=1) =1) → OK→ OK• string (string (pp=2) → marginally OK=2) → marginally OK• monopole (monopole (pp=3) → =3) → BH!BH!

thin (singular) limitthin (singular) limit

One fact worth keeping in mind ….One fact worth keeping in mind ….

need to consider a thick brane for co-dim>2need to consider a thick brane for co-dim>2(future issue)(future issue)

2

5

6;y

yb e -

Z2-symmetry

······ 5D Einstein theory with 5D Einstein theory with 55<0<0

5 4

5 452

5

12

2 MS d x g R d x q

Randall-Sundrum’s (single-) Randall-Sundrum’s (single-) braneworld braneworld RS ‘99RS ‘99

22 2 25 5for 6yds dy e dx dx

• Warp factor Warp factor bb((yy))ee-|-|yy|/|/ decreases exponentially decreases exponentially as |as |yy||→→..

• Extra dim isExtra dim is effectively effectively compactcompact with with size ~size ~

• gravity is confinedgravity is confined withinwithin | |yy||

5D Anti-de Sitter 5D Anti-de Sitter (AdS(AdS55))

44 on the brane on the brane vacuum energy vacuum energy

On scales > On scales > , gravity reduces to 4D Einstein. , gravity reduces to 4D Einstein.

On scales < On scales < , 5D Einstein gravity is recovered:, 5D Einstein gravity is recovered:

Only gravity propagates in the bulkOnly gravity propagates in the bulk

5 45Newton 2

for a point mass on the brane.

r

GG M M Mr r

Gr

M

22 5

4 48 G

Randall & Sundrum ’99 Randall & Sundrum ’99 Garriga & Tanaka ’99Garriga & Tanaka ’99

22

5

4 5 0 for RS 1

2 1flat br

2ane

cosmology of the early universe is modified significantlycosmology of the early universe is modified significantly

Experimental bound: Experimental bound: <0.1 mm (‘large’ extra-dimension) <0.1 mm (‘large’ extra-dimension)

(( <0.1 mm <0.1 mm t t <10<10-13-13 sec) sec)

2. Brane cosmology from the bulk 2. Brane cosmology from the bulk point of viewpoint of view

T

R

n

"bulk"

Binetruy et al. (’99), Krauss (’99), Ida (’99), ...Binetruy et al. (’99), Krauss (’99), Ida (’99), ...

5D AdS-Schwarzschild as a bulk5D AdS-Schwarzschild as a bulk

22 2 2 2 ;K

dRds A R dT R d

A R 2

5

2

2;

1 2 BH m s, a

0 s

RA R K

K ,R

G M

Induced metric on the braneInduced metric on the brane

2brane

2 2 2

22 2

2 2K

K

RA R T d

A R

R d

ds

d R d

Junction conditionJunction condition

2 25

25

5 1;

6 6

6RS flat brane's valueRSRS

A RT

R

2 22 4

2 2 4

224

31

12

K R KH

R R RR

2

2 54 48 G

• Cosmological evolution deviates from 4D atCosmological evolution deviates from 4D at HH 1 1

dark radiation / Weyl fluid ~ conformal fielddark radiation / Weyl fluid ~ conformal field

• //RR4 4 term term due to BH in AdS bulkdue to BH in AdS bulk(5D Weyl tensor (5D Weyl tensor ‘‘EE’’))

~~ ‘radiation’ ‘radiation’ on the brane on the braneShiromizu, Maeda & MS ‘99Shiromizu, Maeda & MS ‘99

AdS/CFTAdS/CFT

RS braneworld RS braneworld 5D Einstein gravity 5D Einstein gravity• Simplest non-trivial realization of braneworlds.Simplest non-trivial realization of braneworlds.• There are many other possibilities.There are many other possibilities.

3. Alternative models3. Alternative models

• RS two-brane modelRS two-brane model → → existence of existence of ‘radion’ ‘radion’ = distance between 2 branes= distance between 2 branes

mm22~ -4~ -4HH22

Gen & MS (’01)Gen & MS (’01)Frolov & Kofman (’03) Frolov & Kofman (’03)

radion is tachyonicradion is tachyonicin cosmology!in cosmology!

may have interestingmay have interestingimplicationsimplications

e.g., Kanno, Soda & Wands (’05)e.g., Kanno, Soda & Wands (’05)

5 4

5 45 42 25 4

1 1

2 2MS d x g R d x q R

5D gravity on large scales: 5D gravity on large scales: rr > > rr00

4D gravity on small scales: 4D gravity on small scales: r r < < rr00

25

0 24

r

IR modification of gravity: our universe will NOTIR modification of gravity: our universe will NOTbe described by 4D Friedmann if be described by 4D Friedmann if rr00~~HH00

-1-1

Dvali, Gabadadze & Porrati (’00)• DGP braneworldDGP braneworld

induced gravity on the braneinduced gravity on the brane

‘‘Dark energy’Dark energy’ due to 5D gravity, leading to due to 5D gravity, leading to HH=2/=2/rr00

2242

2 20 0 0

1 1 43

Hr r r

• scalar fieldsscalar fields (moduli, dilaton, …) (moduli, dilaton, …) ······ bulk inflaton bulk inflaton modelmodel Kobayashi & Soda, Himemoto & MS, Maeda Wands (’00),…

2 22 25 5

5,eff4,eff

1

2 12 16

Brane can inflate without inflaton on the brane,Brane can inflate without inflaton on the brane,while the bulk is collapsing!while the bulk is collapsing!

Low energy (Low energy (HH 11) effective theory on the brane) effective theory on the brane = 4D Einstein-scalar system= 4D Einstein-scalar system

HH22==4,eff4,eff/3/3 ··· Hubble parameter on the brane ··· Hubble parameter on the brane

1/1/22=-=-5,eff5,eff/6 ··· Effective AdS curvature in the bulk/6 ··· Effective AdS curvature in the bulk

Minamitsuji, Himemoto & MS, Kanno & Soda (’03)

5,eff5,eff determined by determined by 5D dynamics of 5D dynamics of

• 5D theory may not be pure Einstein:5D theory may not be pure Einstein:

Signatures of 5D bulk appear only ifSignatures of 5D bulk appear only if HH 1 1

• bulk degrees of freedom = Kaluza-Klein (KK) modesbulk degrees of freedom = Kaluza-Klein (KK) modes

0 p py,x x dp f y x

~ massive fields on the brane~ massive fields on the brane

KK modes are non-negligible when KK modes are non-negligible when HH 11

‘‘zero’ modezero’ mode KK modesKK modes

• important difference from KK cosmology:important difference from KK cosmology:

KK effects on the brane KK effects on the brane KK modes in KK cosmologyKK modes in KK cosmology

22 29

4p Hm

(classical) KK modes act as(classical) KK modes act asnegative energy dustnegative energy dust on the brane! on the brane!

Minamitsuji, MS Minamitsuji, MS & Langlois ‘05& Langlois ‘05

• 5D action with higher derivatives5D action with higher derivatives 　　Gauss-Bonnet term Gauss-Bonnet term ··· ghost-free; inspired by string theory··· ghost-free; inspired by string theory

Einstein Gauss-Bonnet (EGB) gravity:Einstein Gauss-Bonnet (EGB) gravity:

5

552

5

12

2 M GBS d x g R L

Deruelle & Delezel ‘00, Charmousis & Dufaux ‘02Davis ’02, Gravanis & Willison ’02, ……..

Lovelock ’71

2 4 ab abcdGB ab abcdL R R R R R

• GB term is topological in 4D. EGB is non-trivial only for GB term is topological in 4D. EGB is non-trivial only for DD55..

• Natural extension of Einstein theory in the sense thatNatural extension of Einstein theory in the sense that no derivatives higher than secondno derivatives higher than second appear in the field eqns. appear in the field eqns.

• A brane can be consistently embedded in the EGB bulkA brane can be consistently embedded in the EGB bulk despite the presence of despite the presence of RR22 terms. terms.

e.g., Mavromatos & Papantonopoulos ‘05

4. Einstein Gauss-Bonnet (EGB) 4. Einstein Gauss-Bonnet (EGB) braneworldbraneworld

5 4

5 45 brane2

5

12

2 GM BS d x g R d xL qL

5 52 2

4 41 1 1 11 1 or 1 1

4 3 4 3

- +

• EGB also admits AdS bulk, but with curvature radius given byEGB also admits AdS bulk, but with curvature radius given by

convenient parameter (We assume convenient parameter (We assume >0 & >0 & 55<0)<0)2

4

0 1 for , 1 for - +

‘‘+’ branch is known to be unstable +’ branch is known to be unstable (but may have interesting cosmological implications)(but may have interesting cosmological implications)

1/1/2 2 → → 1/1/0022 ≡≡--55/6/6

‘‘-’ branch reduces to AdS in Einstein in the limit -’ branch reduces to AdS in Einstein in the limit →0:→0:

• (Z(Z22-symmetric)-symmetric) de Sitter brane in AdS de Sitter brane in AdS bulkbulk

2 2 2 2 2 23

2 coshdtds dy b y H Ht d

sinhb y H y /

t =const. hypersurface in 5D(two identical H4 cut and pasted at radius y=y0 , b(y0)=1)

de Sitter with radius H-1

2 22

-1/22 2

5 0

1 1 ;3 2

1 2 6

1RS RSH H

tension:tension:

• Flat (Minkowski) brane if Flat (Minkowski) brane if is tuned: is tuned:

1 2

for 1 13 2

/

RS

• Gravity on the brane reduces to 4D Einstein onGravity on the brane reduces to 4D Einstein on

scales > scales > , as in the case of RS braneworld., as in the case of RS braneworld.

However, ……..However, ……..

22 2 yds dy e dx dx

(because the extra-dimension is essentially compact)(because the extra-dimension is essentially compact)

Gravity never becomes 5D even at short distances!Gravity never becomes 5D even at short distances!

effNewton ;

G x M rx

r

0

0 3.

0 9.

0 6.

2

4

Deruelle & MS ‘03

Experimental bound on Experimental bound on is relaxed substantially: is relaxed substantially:

Davis ’04

scalar-tensor scalar-tensor (~Brans-Dicke)(~Brans-Dicke) theory at theory at rr < <

1km < 1km < < 100km is also possible if < 100km is also possible if ~ 1~ 1~ ~

Thus, 5D EGB leads to intrigueing possibilities thatThus, 5D EGB leads to intrigueing possibilities that

• Size (curvature radius) of the extra-dim may beSize (curvature radius) of the extra-dim may be macroscopic.macroscopic.

~10 km is an interesting scale for SN/NS/GW physics~10 km is an interesting scale for SN/NS/GW physics

• Scalar-tensor gravity on the brane at short distances,Scalar-tensor gravity on the brane at short distances, while Einstein (tensor) gravity at long distances.while Einstein (tensor) gravity at long distances.

Emergence of an (effectively massive) 4D scalar fieldEmergence of an (effectively massive) 4D scalar field Impact on the early universe cosmology?Impact on the early universe cosmology?

~10 km ~10 km t t ~10~10-4 -4 secsec

Linearized gravity on de Sitter braneLinearized gravity on de Sitter brane

2

2 2 254 4 ; 2 1

6 1H H

T

□

Integro-differential equationsIntegro-differential equations

25 2

0 0;1 p

y p,dpGh y Ty

22 5

2 2

242

2 2 0

2

32 1

31

1 1

1

1

2y

G h TH hH

D D HH

h yH

□

spin 2 part of matter spin 2 part of matter stress tensorstress tensor

1

pure Einstein?

fluctuation offluctuation ofbrane positionbrane position

5D (KK) effect5D (KK) effect

‘‘KK propagator’KK propagator’

negligible at negligible at rr → → BD gravityBD gravity

Minamitsuji & MS ‘04Minamitsuji & MS ‘04

55. Summary. Summary

KK modes summed up to give 4D-like behaviorExtra dim can be as large as ~ 10km~ 10km

difference can appear at short or long distances

Infinitely large extra dimension“dark energy” without dark energy

Braneworld givesBraneworld gives a new picture of the universe a new picture of the universe

Different models can lead toDifferent models can lead to a variety of predictions a variety of predictions

In DGP braneworld, 4D at In DGP braneworld, 4D at r r rr00 , , 5D at 5D at r r rr00

In EGB braneworld,In EGB braneworld, BD at BD at r r , , Einstein at Einstein at r r

Effect of KK modes (5D gravity)Effect of KK modes (5D gravity)

in the braneworlds is highly non-trivial:in the braneworlds is highly non-trivial:

brane geometry

effective 4D geometry in conventionalKK cosmology

KK modes lead toKK modes lead to

• cosmological constant in DGPcosmological constant in DGP

• massive scalar in EGBmassive scalar in EGB

• negative energy dust in RSnegative energy dust in RS

• … …. .

observational predictions!any hint of deviation from Einstein?

After all, instabilities are needed in cosmology!

string-inspired models, cosmological perturbations,‘thick’ braneworld, higher co-dimensions,

black holes, quantum effects, ….and

There are surely much more to be doneThere are surely much more to be donein braneworld cosmology.in braneworld cosmology.

Models with instabilities should not be rejected at once.Models with instabilities should not be rejected at once.