6D Brane Cosmological Solutions

1

6D Brane Cosmological 6D Brane Cosmological SolutionsSolutions

Masato MinamitsujiMasato Minamitsuji(ASC, LMU, Munich) (ASC, LMU, Munich)

T. Kobayashi & M. Minamitsuji, JCAP0707.016 (2007) [arXiv:0705.3500]T. Kobayashi & M. Minamitsuji, JCAP0707.016 (2007) [arXiv:0705.3500]

M. Minamitsuji, CQG 075019(2008) [arXiv:0801.3080M. Minamitsuji, CQG 075019(2008) [arXiv:0801.3080]]

CENTRA, Lisbon, June 2008CENTRA, Lisbon, June 2008

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ContentsContents ~ Introduction

~ 6D braneworld

~ 6D brane cosmological solutions

~ Tensor perturbations

~ Stability

3

BraneworldIntroductionIntroduction

Matter (SM particles) are confined on the brane

while Gravity can propagate into the bulk

One of the most popular and mostly studied higher-dimensional cosmological scenarios in the last decade

bulkBrane (SM)

Motivated from string / M-theory

(Gravity)

Gauge hierarchy problem, Inflation, Dark energy , …

4

Randall-Sundrum (II) model (RS 1999)

5D braneworld

54 GG

2

44

2

3

8

OG

H)( 22

2

22 ji

ij dxdxdtdzz

ds

3-brane Standard Cosmology

224

1

Vanishing cosmological constant cannot be obtained unless one fine-tunes the value of the brane tension.

04

z

25

6

Localization of gravity by strong warping

5

The property of a codimension 2 brane is quite different from that of the The property of a codimension 2 brane is quite different from that of the codimension 1 brane .codimension 1 brane .

6D braneworld6D braneworld

Codimension 2 brane

~Conical singularity

y

x

Codimension 1

Codimension 2

46M

The tension of the brane is absorbed into the bulk deficit angle and does not curve the brane geometry

Self-tuning of cosmological constant ?

6

Models with the compact bulk

The compact bulk is supported by the magnetic flux Self-tuning of the cosmological constant ? ),( 6 BHH

01 BB

10

however, because of the flux conservation

Caroll & Guica (03), Navarro (03), Aghababaie, et.al (03)

Vinet & Cline (04), Garriga & Poratti (03)

211

200

0

1

2

2

RB

RB

0),( 60 BH

0),( 61 BH

After the sudden phase transition on the brane , it seems to be plausible that the brane keep the initial flat geometry.

We assume that for a given 0B

Rugby-ball shaped bulk

F

7

Nevertheless, as a toy braneworld model with two essential features

Stabilization of extra dimensionsIn comactifying extra dimensions, d.o.f.s associated with the shape and size appear in the 4D effective theory.

Flux stabilized extra dimensionsHigher codimensions

Flux stabilization of extra dimensions would be useful6D model (2D bulk) gives the simplest example

F

C.f. in 5D

d

d is not fixed originally

quantum corrections,…

additional mechanism

8

Northern pole (+-brane)

Southern pole (--brane)

generalization

Static warped solutionsMukohyama et.al (05)Aghababaie, et. al (03), Gibbons, Gueven and Pope

(04)

We derive the cosmological version of these solutions

9

Codimension-2

Codimension-1

4-brane Cap region

Branes in higher co-dimensional bulkCodim-2

Codim >2

Brane tension develops the deficit angle but one cannot put ordinary matter on the brane

One cannot put any kind of matter on the brane

= black holes or curvature singularities

need of regularizations of the brane

2

44

2

1

2

1 8 TgTGR

I

I TGR )(

42 8~

4D GR Scalar mode associated with the compact dimensionLarge distances scales22 RL Recovery of 4D

GR1~ L

Peloso, Sorbo & Tasinato (06), Kobayashi & Minamitsuji (07)

10

1pure Einstein-Maxwell model

0gauged supergravity

10

(D+2)-dimensional Einstein-Maxwell theory

First, we consider seed solutions in higher dimensions

4D

Our purpose is to find brane cosmological solutions in the following 6D Einstein-Maxwell-dilaton theory

Instead of solving coupled Einstein-Maxwell-dilaton system, we start from

ab

abFFeeRgxdS 4

12)(

2

1 266

MND

MNDDDD FFRGXdS )2()2(22)2(

4

12

2

1

6D brane cosmological 6D brane cosmological solutionssolutions

11



12

Dimensional reduction

For a seed (D+2)-dim solution, we consider the dimensional reduction:

Compactified

dim )4( D

0 )( )2()2( mM

Dabab

D FxFF

with some field identifications

D

DDD 4 ,

2

)4(

ab

abFFeeRgxdS 4

12

2

1 266

nmmn

xbaab

xDD dydyedxdxxgeds 22)4(2

2 )(

D6

The effective 6D theory is the same as the one we are interested in

13

(D+2)-dimensional seed solutions

)1()2(2

)2(22

2

QFD

2221

2

1

)1(2

2 )()(2

2

2

2

2

dhh

ddXdXds

Upper bound

Magnetic charge 21

2

1

3

1

5

2

2)1(2

3

1

2

1

1

5

3

2

2

2

2

2

2

Q

2

2

2

2

1

3

1

5

2

22

max

1

1

)5(2

)3)(1(:

,1)(h

D-dimensional Einstein space

has two positive root at

][)( RD

We compactify (D-4) dimensions in

10

14



Warpedgeneralization

1

15

From the (D+2)-dimensional de Sitter brane solutions

)1()2(2 22

2

QF

ln

1

2))(ln(

2,

2 bt

2/1)( a

D-dimensional de Sitter spacetime

Power-law inflationary solutions since

10

))(

)((

2)( 22

22221

2

2 2

dh

h

dbdxdxadds ji

ij

6D cosmological solutions

)(b

16

From the Kasner-de Sitter solutionsnm

mnBji

ijA dydyedxdxedtdXdX

222

)()2(2

1sinh 0

)4(3 ttD

De BDA

)4(3

2

0 )(22

1tanh

D

D

BA ttD

De

Late time cosmology Power-law solutions are always the late-time attractors

2/1)( a )(b

generalizations of solutions found in KK cosmology

qb pa )(

)56(3

162)2(32

22

p 2

22

56

16

q

)41.00 ,22.00 ,063.00.33 ,48.033.0( qqpp

The early time cosmology

Maeda & Nishino (85)

17

0)()1( 2 2

2

2

2

t

e

k

dt

dD

dt

dmmtH

021

2

1

422

mmmd

dh

d

d

)2(2221

)1(22

2

Ddydyedxdxhedt

dXdXgg

nmmn

Htjiijij

Ht

NMMNMN

KK decomposition m

TTijmmij eth )()()(

TT polarization tensor

Tensor perturbations in 6-dim dS solutions

Tensor perturbationsTensor perturbations

= Tensor perturbations in (D+2)-dim dS solutions

18

156.0

)1(4

32

22

c

743.0

4D observers on the brane measure the KK masses 22)4(2 )( m

HtDm eM

The critical mass

Light KK modes may decay slowlyFirst few KK modes Dashed line= critical KK

masses

)1(2)3( 22

~ amc2222 )1(2)3(~ camc

19

For the increasing brane expansion rate, the first KK mass tends to be lighter than the critical one.

)54( 96.0 D)5.4( 33.0 D

Red= The first KK massDashed = The critical KK mass

But one must be careful for the stability of the solutions

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The 6D brane cosmological solutions are stable against the tensor perturbations.

The 6D brane cosmological solutions are derived via the dimensional reduction from the higher-dimensional de Sitter brane solutions

For the larger value of the brane expansion rate, the first KK mass of tensor perturbations becomes lighter than the critical one, below which the mode does not decay during inflation

Summary 1Summary 1

21

Minkowski branes

de Sitter branes

stable

unstable for relatively higher expansion rates

Yoshiguchi, et. al (06), Sendouda, et.al (07)

Lee & Papazoglou (06), Burgess, et.al (06)

Kinoishita, Sendouda & Mukohyama (07)

StabilityStability

Stability of 6-dim dS solutions

= Stability of (D+2)-dim dS solutions

22

Scalar perturbations1

n

nini wx )()( ,

22222

2

12

21

22

2

22

2

cos),(1

3),(21),(),(21

)3()1(2

1),(21

wdxwxwdwxwxw

dxdxxwdsD

wwa ,122122

2)1(

tan

1

1

42

)3()1(2

)2)1((2

KK decomposition

23

The lowest mass eigenvalue is given by

An instability against the scalar perturbations appears in the de Sitter brane solutions with relatively higher expansion rates.

22

222

0 )1(

)3(1)1(

A tachyonic mode appears for the expansion rates

2

1 2max

2

22

3

)1(

inst

inst

24

Dynamical v.s. “thermodynamical” instabilities

Kinoshita, et. al showed the equivalence of

dynamical and “thermodynamical” instabilities

in the 6D warped dS brane solutions with flux compactified bulk

Dynamically unstable solutions

= Thermodynamically unstable solutions

The arguments can be extended to the cases of higher dimensional dS brane solutions.

1

See the next slide

25

Area of de Sitter horizon

Magnetic flux

Deficit angles (=brane tensions)

2

)( ,

2

)1( ,,

hh

21

2

1

3

1

5

2

2)1(2

3

1

2

1

1

5

3

2

2

2

2

2

2

Q

2

0

2

)1(2

2

)1(2

1

01

1

)2( 2 D

D

D

D

Q

D

DFdd

))1((2

1)2(

2

AHDQ DD

D

)1(2

2

)1(2)2(

11

)2(

2D

DD

D

DHA

Thermodynamical relations

26

,1)(h

D-dimensional de Sitter

has two positive root at

2221

2

1

)1(2

2 )()(2

2

2

2

2

dhh

ddXdXds

Upper bound

2

2

2

2

1

3

1

5

2

22

max

1

1

)5(2

)3)(1(:

][)( RD

10

27

“Thermodynamics”

Intensive variables

The (+)-brane point of view

~ ,~

,~

AA

2

2

2 D

D

DDQH

QH

~2

)1(

1)

~( Qdd

HDAd

D

dd2

1)(

)(1

pdVdET

dS

Somewhat similar to the BH therodynamics

28

The boundary between unstable and stable solutions is given by the curve, which is determined by the breakdown of one-to-one map from plane to conserved quantities .

“Thermodynamical stability” conditions

HD

H

D

HD

DDD

D

D

H

D

H

QHQ

H

QH

HH

QH

HH

1

2

~

),(

)~

,(1~

)(

,),(

)~

,(1

),( H

0~

,0~

DD QHH

0),(

)~

,(

DH

)~

,(

Some Identities

29

1) 6D limit : Special limits

The curve is exactly boundary between dynamically stable and unstable modes

0

2) unwarped limit

The same thing happens in the higher dimensional geometry.

12

2

max 3

)1(2

critinstcrit

D

D 4

Kinoshita, Sendouda & Mukohyama (07)

30

Cosmological evolutions

Cosmological evolutions from (D+2)-dimensional unstable de Sitter brane solutions

Evolution of the radion mode

dc

cdV

c

c

a

aDc eff )(

)1(

)cos()( 22222222 wddwtcdxdxtadtds jiij

)log(4

)1(

))1(2

3(16

12

)3(

)( 22

12

2

122

cccVeff

The potential has one local maximum and one local minimum

1

31

effective potential

Flux conservation relates the initial vacuum to final one.

Two possibilities: toward a stable solution with a smaller radiusdecompactification

1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

32

effective potential



1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

33

effective potential



1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

34

effective potential



1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

35

effective potential



1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

36

effective potential



1

2

122

22

22

222

12

1

)1(4)3(

1

)3(

)1(4

37

a new dS brane solution

an AdS brane solution

AdSinst 1 02

22

22

)3(

)1(4

AdS

max1 AdS 02 The corresponding 6D solution is the collapsing Universe.

The corresponding 6D solution is the stable accelerating, power-law cosmological solutions.

2

22

3

)1(

inst

Inflation Dark Energy Universe ?

38

6D brane cosmological solutions in a class of the Einstein-Maxwell-dilaton theories are obtained via dimensional reduction from the known solutions in higher-dimensional Einstein-Maxwell theory.

Higher-dimensional dS brane solutions (and hence the equivalent 6D solutions) are unstable against scalar perturbations for higher expansion rates. This also has an analogy with the ordinary thermodynamics. The evolution from the unstable to the stable cosmological solutions might be seen as the cosmic evolution from the inflation to the current DE Universe.

SummarySummary

39

Equivalent 6D point of view

4D effective theory for the final stable vacuum

The cosmological evolution may be seen as the evolution from the initial inflation to the current dark energy dominated Universe.

characterizes the effective scalar potential

)1(22)4(

4 22

2ˆ2

1ˆ eqqxdS Reff

AdSinst 102

40

Stability Minkowski branes

de Sitter branes

Einstein-MaxwellSupergravity

Einstein-Maxwell

stable

marginally stable (with one flat direction)

dS brane solutions are unstable for relatively higher expansion rates !

Quantum correctionsGhilencea, et.al (05), Elizalde, Minamitsuji & Naylor (07)

Yoshiguchi, et. al (06), Sendouda, et.al (07)

Lee & Papazoglou (06), Burgess, et.al (06)

Kinoishita, Sendouda & Mukohyama (07)

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6D Brane Cosmological Solutions