Gravity effects to the Vacuum Bubbles Based on PRD74, (2006), PRD75, (2007), PRD77, (2008), arXiv: [hep-th] & works in preparation Bum-Hoon Lee Center for Quantum SpaceTime (CQUeST) Sogang University, Seoul, Korea Gamow Memorial International Conference (GMIC’105) Astrophysics & Cosmology after Gamow : Recent Progress & New Horizons ( 17 – 23 August 2009, Odessa, Ukraine)

Our works related to this talk 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, Phys. Rev. D 74, (2006), hep-th/ ) 2. 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, Phys. Rev. D 75, (2007), hep-th/ ) 3. 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, Phys. Rev. D 77, (2008), arXiv: ) 4. 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, arXiv: ) 5. 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. To get the nontrivial solution corresponding to the tunneling is possible if gravity is taken into account. The numerical solutions are presented. (B.-H. Lee, C. H. Lee, W. Lee, and C. Oh, in preparation)

Contents 1. Motivations, Basics – bubble nucleation in the flat spacetime 2. Bubble nucleation in the Einstein gravity 3. True & False vacuum bubble nucleation 3.1 True vacuum bubbles 3.2 False vacuum bubbles 3.3 Tunneling between the degenerate vacua 4. Summary and discussions

1. Motivations (1) The string theory landscape has a vast number of metastable vacua. How to select our universe with positive cosmological constants? (2) In the early universe with or without cosmological constants, Can we obtain the mechanism for the nucleation of a false vacuum bubble? Can a false vacuum bubble expand within the true vacuum background?  Can we be in the vacuum with positive cosmological constant ? (an alternative way to KKLT(Kachru,Kallosh,Linde,Trivedi, PRD 2003), for example)  Revisit the gravity effect in cosmological phase transitions.

Basics : Bubble formation Vacuum-to-vacuum phase transition rate 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)

(1) Tunneling in Quantum Mechanics - particle in one dim. with unit mass - Lagrangian Quantum Tunneling :(Euclidean time) The particle penetrates the potential barrier and materializes at the escape point,, with zero kinetic energy, where =classical Euclidean action Eq. of motion : boundary conditions The bounce solution is unstable (exists a mode with negative eigenvalue) Tunneling probability per unit volume Time evolution after tunneling : Classical Propagation :(back to Minkowski time)

(2) Tunneling in multidimension Lagrangian The leading approx. to the tunneling rate is obtained from the path and endpoints that minimize the tunneling exponent B. Boundary conditions for the bounce Time evolution after the tunneling is classical with the ordinary Minkowski time.

(3) Tunneling in field theory (in flat spacetime ) Theory with single scalar field where False vacuumTrue vacuum Equation for the bounce from with boundary conditions Tunneling rate : (Finite size true vacuum bubble)

O(4)-symmetry : Rotationally invariant Euclidean metric Tunneling probability factor The Euclidean field equations boundary conditions “Particle” Analogy : “Particle” moving in the potential –U, with the damping force inversely proportional to time At time 0, the particle is released at rest (The initial position should be chosen such that) at time infinity, the particle will come to rest at. The motion of a particle located at position phi at time eta

Thin-wall approximation B is the difference In this approximation Outside the wall potential (Epsilon : small parameter) True vacuum False vacuum -U R Large 4dim. spherical bubble with radius R and thin wall

In the wall where inside the wall the radius of a true vacuum bubble the nucleation rate of a true vacuum bubble

2. Bubble nucleation in the Einstein gravity Action Einstein equations S. Coleman and F. De Luccia, PRD21, 3305 (1980) O(4)-symmetric Ansatz : Rotationally invariant Euclidean metric The Euclidean field equations boundary conditions (Scalar eq. of motion & Einstein eq.), Bubble nucleation rate

3.1 True vacuum bubbles Large background (Parke): Half background : Small background : For the de Sitter background space the size of the background space (or bubble) will be called “large'' (or “small'' ) if its size is larger than half of the de Sitter space itself. (with the de Sitter (dS) exterior geometry)

3.2 False vacuum bubbles (i) Reflected diagram of (3-1) (ii) Reflected diagram of (3-2) (iii) Reflected diagram of (3-3) (iv) Hawking-Moss transition

False-to- true True-to- false De Sitter – De Sitter OO De Sitter – Flat O? De Sitter – Anti-de Sitter O? Flat – Anti-de Sitter O? Anti-de Sitter – Anti-de Sitter O? In the Einstein theory of gravity with a nonminimally coupled scalar field, there can exist the false vacua :

It is possible that the tunneling occurs via the potential with degenerate vacua in de Sitter space. The numerical solution of this tunneling was only obtained by Hackworth and Weinberg. The analytic computation and interpretation : (B.-H. Lee and W. Lee, arXiv: , ‘The vacuum bubble and black hole pair creation’) This tunneling is possible due to the changing role of the second term in Eucildean equation from damping to accelerating during the phase transition Tunneling between the degenerate vacua

dS - dS 10 Two observer’s point of view

flat -flat 11 AdS-AdS

4. Summary and Discussions We reviewed the formulation of the bubble. Classified the types of vacuum bubbles for & calculated the radius and the nucleation rate. Nine types of true vacuum bubbles & conditions three false vacuum bubbles, & Hawking-Moss transition. False vacua exist e.g., in non-minimally coupled theory. The tunneling of degenerate vacua in dS, flat, & AdS. Obtained the transition rate and the radius of a bubble. Can it be a model for the accelerating expanding universe?