The false vacuum bubble, the true vacuum bubble, and the instanton solution

in curved space

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APCTP 2010 YongPyong : Astro-Particle and Conformal Topical PhysicsFeb. 23-27 , 2010

Bum-Hoon Lee and Wonwoo Lee (CQUeST & Sogang University)

Based on

arXiv: 0910.1653 by Bum-Hoon Lee, WL(CQUeST & Sogang), Chul H. Lee, and Changheon Oh(Hanyang),

CQG 26:225002 (2009) by Bum-Hoon Lee, WL(CQUeST & Sogang)

The plan of this talk1. Motivations

2. Our works related to the motivations

3. Bubble nucleation in the Einstein theory of gravity

4. Instanton solution mediating tunneling between the degenerate vacua

5. Time evolution

6. Summary and discussions 2/23

1. Motivations(1) What did the spacetime look like in the very early universe?

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(2) The idea of the string theory landscape has a vast number of metastable vacua.

(4) Can we obtain the tunneling solution with O(4) symmetry betweenthe degenerate vacua in the double well potential in de Sitter, flat oranti-de Sitter space?

(3) Multiverse scenario

• We would like to understand the mechanism how the complicated spacetime structure could be created and tunneling process occur.

• The tunneling process becomes a remarkable event in these framework.

• The calculation of bubble nucleation rates becomes one of the key problems.

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2. Our works related to the motivations

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In 4 dimension

(1) We have obtained the mechanism for the nucleation of a false vacuum bubble within the true vacuum background in the Einstein theory of gravity with a nonminimally coupled scalar field.

W. Lee, B.-H. Lee, C. H. Lee, and C. Park, PRD 74, 123520 (2006).(2) We have obtained an expanding false vacuum bubble, without

the initial singularity in the past, with an effective negative tension due to the nonminimal coupling within the true vacuum background.

B.-H. Lee, C. H. Lee, W. Lee, S. Nam, and C. Park, PRD 77, 063502 (2008) .

(3) We classified the possible types of vacuum bubbles and calculated the radius and the nucleation rate. We present some numerical solutions as well as analytic computation using the thin-wall approximation.

B.-H. Lee and W. Lee, CQG 26, 225002 (2009).(4) We study the tunneling transition between the degenerate

vacua in flat and anti-de Sitter space. We obtain O(4)-symmetric bubble solution in these background. The nontrivial solution corresponding to the tunneling is possible only if gravity is taken into account. We propose a new formation mechanism of the dynamical geometry which has the finite size with the exact Z_2 symmetry.

B.-H. Lee, C. H. Lee, W. Lee, and C. Oh, arXiv: 0910.1653.

In 5 dimension

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(1) We classified the cosmological behaviors from the viewpoint of an observer on the domain wall and find a solution with multiple accelerations in five-dimension in the Einstein theory of gravity.

B.-H. Lee, W. Lee, S. Nam, and C. Park, PRD 75, 103506 (2007).

(2) We studied the dynamics of a domain wall universe embedded into the charged black hole spacetime of the Einstein-Born-Infeld (EBI) theory. There are four kinds of possible spacetime structures, i.e., those with no horizon, the extremal one, those with two horizons (as the RN black hole), and those with a single horizon (as the Schwarzshild black hole). We derive the effective cosmological equations on the wall.

B.-H. Lee , W. Lee and M. Minamitsuji, PLB 679, 160 (2009).

3. Bubble nucleation in the Einstein gravity

The bubble nucleation rate or the decay rate of background vacuum

B : Euclidean Action (semiclassical approx.) S. Coleman, PRD 15, 2929 (1977) S. Coleman and F. De Luccia, PRD21, 3305 (1980) S. Parke, PLB121, 313 (1983)

A : determinant factor from the quantum correction C. G. Callan and S. Coleman, PRD 16, 1762 (1977)

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Action

We take O(4) symmetry for both scalar field and metric, excepting

its dominant contribution

then and

Thus the Euclidean action becomes

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The Euclidean field equations

The boundary conditions for the solutions (de Sitter background) The asymptotic value of scalar field

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potential The typical vacuum bubble profile

with the wall

False vacuum

True vacuu

m

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Flat-ds

dS

flat

AdS

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For the purpose of analytic computation, we do integration by parts where the surface term drops out and then substitute the field equation in the action

, In the thin-wall approximation scheme, the action can be divided into three parts. Outside the wall : On the wall :

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3.1 True vacuum bubblesWe now compute the contribution from inside the wall

This can be seen by the relation

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Numerical solutions for several types of true vacuum bubbles.

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Large background (Parke): ( )

Half background : ( , )

Small background : ( , )

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3.2 False vacuum bubbles(i) Reflected diagram of (3-1)

(ii) Reflected diagram of (3-2)

(iii) Reflected diagram of (3-3)

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It is possible that the tunneling occurs via

the potential with degenerate vacua in de

Sitter space. The numerical solution of

This tunneling is possible due to the

changing role of the second termin Eucildean equation from

damping toaccelerating during the transition.

The radius of the bubble and nucleation rate

4. Instanton solution mediating tunneling between the degenerate vacua

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dS - dS

Two observer’s point of view

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flat -flat

The radius of the solution and transition rate

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AdS-AdS

The radius of the solution and transition rate

where

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as the initial value of goes to increase, the exponent becomes

5. Time evolution in LorentzianAnalytic continuation

After the analytic continuation

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analytic continuation

If the whole spacetime has the exact symmetry

by the junction condition

effective potential

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6. Summary and DiscussionsWe classified the possible types of vacuum bubbles and calculated the

radius and the nucleation rate. We present some numerical solutions as well as analytic computation using the thin-wall approximation. We considered the pair creation of black holes in the background of bubble solutions. We obtained static bubble wall solutions of junction equation with black hole pair.

[B.-H. Lee and W. Lee, CQG 26, 225002 (2009)]

We study the tunneling process between the degenerate vacua in curved space. We show that there exist O(4)-symmetric solutions in not only de Sitter but also both flat and anti-de Sitter space. The nontrivial solution corresponding to the tunneling is possible only if gravity is taken into account. The numerical solutions as well as the analytic computations are presented. From the analysis of the instanton solutions, we propose a new formation mechanism of the dynamical geometry which has the finite size with the exact $Z_2$ symmetry.

[B.-H. Lee, C. H. Lee, W. Lee, and C. Oh, arXiv:0910.1653]

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