The false vacuum bubble :

- formation and evolution -

in collaboration withBum-Hoon Lee, Chul H. Lee, Siyong Nam, and Chanyong Park

Based on PRD74, 123520 (2006), PRD75, 103506 (2007), arXiv:0710.4599 [hep-th]

Wonwoo LeeSogang University

HaengDang Symposium 2007- Recent Studies in Astro-Particle Physics

- Nov. 30.(Fri.) 2007

The plan of this talk

1. Motivations, 2. False vacuum bubble nucleation due to a

nonminimally coupled scalar field 1) Numerical calculation 2) Thin-wall approximation

3. The dynamics of a false vacuum bubble : the junction equations

4. Summary and discussions

1. Motivations -I

What is the origin of our universe?

1) Is our universe created from nothing?

2) Is our universe created from something?

3) Can the universe create itself?

1. Motivations -II(1) What did the spacetime look like in the very early universe? - Wheeler’s spacetime foam structure - The cosmological constant as a dynamical variable Can we obtain the mechanism for the nucleation of a false vacuum bubble? Can a false vacuum bubble expand within the true vacuum background?

(2) The idea of the string theory landscape has a vast number of metastable vacua. Which mechanism worked to select our universe in this landscape?

Can we be in the vacuum with positive cosmological constant ?(through an alternative way to KKLT, for example)

2. The Einstein theory of gravity with a nonminimally

coupled scalar fieldVacuum-to-vacuum phase transition rate

Action

Einstein equations

/ exp[ / ]V A B

4 21 1[ ( )]2 2 2

RS gd x R U

1

2R g R T

2 22

1 1[ ( ) ( )]

1 2T g U g

boundaryS

curvature scalar

Potential

Rotationally invariant Euclidean metric : O(4)-symmetry

The Euclidean field equations

boundary conditions

2 2( ) ( 2 ) ( 2 )8 2 oU b b U

b

2 2 2 2 2 2 2 2( )[ sin ( sin )]ds d d d d

2

2

1

]3)(4[

U

R

3 ''' ' E

dUR

d

22 2

2

1' 1 ( ' )

3(1 ) 2U

0|,)( 0(max)

lim

d

dT

Our main idea

(during the phase transition)

3 ''ER

1) Numerical calculation

(Case 1) from de Sitter to de Sitter

(Case 2) from flat to de Sitter

(Case 3) from anti-de Sitter to de Sitter

(Case 4) from anti-de Sitter to flat

(Case 5) from anti-de Sitter to anti-de Sitter

False-to-true True-to-false

De Sitter – de Sitter O O

Flat – de Sitter O O

Anti-de Sitter – de Sitter

O O

Anti-de Sitter – flat O O

Anti-de Sitter – Anti-de Sitter O O

2)Thin-wall approximation

B is the difference

In this approximation

Outside the wall

b TE EB S S

in wall outB B B B

( ) ( ) 0out E T E TB S S

In the wall

where

inside the wall

dUUS

F

TTo )]()([2

412 4(1 2ln )

b bC

22 2 3/ 2

22

2

( )(1 ) {[1 ] 1}

3(1 )12[ ( )]

( )

FF

Fin F T

F

U

BU

),(22

32

C

SB owall

(a) false vacuum bubble nucleation

if

6816

3

)2(4

4)82( 242

22 CS

bbU

UbS

bH ooo

o

23

8

3

8

3

2

8

1

29

222222ooooo SUbUbUS

E

3

8256

3

264

2364 CSb

bCS

bD oo

22 H H ED

E

o

4

2

2

1

22

2221

oo

op

/3 oo S )](/3[21 TF UU 2

2 [3 / ( )]F TU U

The coefficient B

(b) true vacuum bubble nucleation

1

)41(31

)41(1

31

1122/3

22

2222/32

2

2

b

U

U

bU

UB T

T

F

F

6816

3

)2(4

4)82( 1242

22

1

CSbb

UUb

SbH oo

oo

3

8256

3

264 1

23614

1

CSbb

CSbD oo

23

8

3

8

3

2

8

1

29

222222ooooo SUbUbUS

E

)

4ln21(

1241

b

bC

E

EDHH 12112

1

The coefficient B

if (by S. Parke)

where

o

4

2

2

1

22

2221

oo

op

13

11

)41(311

)41(122/322/3

22

222

2

2F

F

T

T

U

Ub

U

U

bB

2/14

2

2

1

2

1

2

4

2

2/14

2

2

1

2

1

22211

2

2221

212

ooo

oooo

p

B

B

342 2/27 oo SB

Two types related to this formalism

(1) Boundary surface

(2) Surface layer In this case it is related to the discontinuity of the extrinsic curvature of the surface. We consider thin-wall partitions bulk spacetime into two distinct manifolds

and with boundaries and , respectively. To obtain the single glued manifold

we demand that the boundaries are identified as follows:

0ijS

0ijS

3. The dynamics of a false vacuum bubble : the junction

equations

M

M

MMM

We consider the action

where

In this framework, junction condition becomes

or

where , a effective negative tension of the wall

There are parameter regions including that both and are positive in all ranges of

4 21 1[ ( )]2 2 2

RS gd x R U

twSK

xdh )1( 23

),(3

UxdhStw,8 G gg det

rHrHr2

1)1( 222

)(2

1 _22

rHrHr

Hr

r2

2_ 2

,1)( 2rAinH r

GMrAoutH

21)( 2

r

After squaring twice, the equation turns out to be

where the effective potential is

with

P

PQTTrVeff 2)(

2

0)(2

1 2

rVr eff

222

2

22422222

222222222

222222222

42222222

222222222

22222222

])1(1[

,4

)1(4

)1]()1(1[

])1(1[4)1](4

1)1[(

}2

1]

4

1)1][()1(1{[2

}]4

1)1{[(

,}2

1]

4

1)1][()1(1{[

)1(])1(1[2

])1(1[

P

r

MG

r

GM

GMrAA

rAA

rAAAQ

rAA

r

GMT

(1) M = 0

DS – DS

DS – FLAT

DS – ADS

(2) M > 0

DS – SDS

DS – S

DS – SADS

ER

4. Summary and Discussions• The false vacuum bubble can be nucleated within the true vacuum background with a nonminimally coupled scalar field

• expect the phenomenon be possible in many other theories of gravity with similar terms. • A false vacuum bubble with minimal coupling, without singularity in their past, can expand within the true vacuum background with nonminimal coupling.• An expanding false vacuum bubble is not inside the horizon of a black hole from outside observer’s point of view. • Can it be a model for the accelerating expanding universe?

Thank you for

your attention!