The false vacuum bubble : - formation and evolution - in collaboration with Chul H. Lee(Hanyang), Wonwoo Lee, Siyong Nam, and Chanyong Park (CQUeST) Based on PRD74, (2006), PRD75, (2007), arXiv: [hep-th] Bum-Hoon Lee ( 李 範 勳 ) Center for Quantum SpaceTime (CQUeST) Sogang University ( 西江大學校 ), Seoul, Korea Workshop on String theory and Cosmology KITPC, Beijing, Nov 5-9, 2007

The plan of this talk  Motivations, Basics – bubble nucleation in the flat spacetime 2. Bubble nucleation in the Einstein theory of gravity 3. False vacuum bubble nucleation due to a nonminimally coupled scalar field 1) Numerical calculation 2) Thin-wall approximation 4. The dynamics of a false vacuum bubble : the junction equations 5. Summary and discussions

1. Motivations (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)

Basics : Bubble formation -tunneling in flat spacetime - Vacuum-to-vacuum phase transition rate B : Euclidean Action (semiclassical approx.) S. Coleman, PRD 15, 2929 (1977) 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 dimension (unit mass) - Lagrangian Quantum Tunneling :(Euclidean time) The particle penetrates the potential barrier and materializes at the escape point,, with zero kinetic energy, Classical Propagation :(back to Minkowski time) Time evolution after tunneling where = classical Euclidean action Tunneling probability per unit volume

The minimization problem is equivalent to solving the Euler-Lagrange equations for the Euclidean Lagrangian b oundary conditions (*)Euclidean action analytic continuation : prime denotes differentiation with respect to Euclidean time

(2) Tunneling in multidimension Lagrangian To find the WKB approximation to the tunneling rate, one considers all possible paths that start at a point on one side of the barrier and end at a point on the other side, with the requirement that Then the tunneling exponent is given by where The leading approximation to the tunneling rate is obtained from the path and endpoints that minimize B.

This minimization problem for the tunneling is equivalent to finding a solution of the Euler-Lagrange equations for the Euclidean Lagrangian where the prime denotes differentiation with respect to Euclidean time. b oundary conditions Time evolution after the tunneling is classical with the ordinary Minkowski time.

The coefficient B is the total Euclidean action for the bounce, (Note : ) Thus, to find the coefficient B, we need only find the bounce. from where

(3) Tunneling in field theory We consider a theory with single scalar field where False vacuumTrue vacuum

Equation for the bounce from is with boundary conditions Tunneling rate : Note : Quantum fluctuation make true vacuum bubble appear someplace. Far from the bubble, the false vacuum persists. for finite B

We consider O(4)-symmetry : Rotationally invariant Euclidean metric The Euclidean field equations : boundary conditions Tunneling probability factor

“ Particle” Analogy : “Particle” moving in the potential –U, with the damping force inversely proportional to time with 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

Evolution of the bubble The false vacuum makes a quantum tunneling into a true vacuum bubble at time t = 0, described by Afterwards, it evolves according to the classical equation of motion The solution (by analytic continuation)

2. Bubble nucleation in the Einstein gravity Action Einstein equations S. Coleman and F. De Luccia, PRD21, 3305 (1980)

Potential

We consider O(4)-symmetry : Rotationally invariant Euclidean metric The Euclidean field equations, boundary conditions Bubble nucleation rate

(i) From de Sitter to flat spacetime the radius of a true vacuum bubble where the nucleation rate of a true vacuum bubble (ii) From flat to Anti-de Sitter spacetime the radius of a true vacuum bubble the nucleation rate of a true vacuum bubble

(iii) the case of arbitrary vacuum energy S. Parke, PLB121, 313 (1983) Evolution of the bubble  via analytic continuation back to Lorentzian time Ex) de Sitter -> de Sitter : A. Brown & E. Weinberg, PRD 2007

Case 1.

Case 2.

Case 3.

Case 4.

Case 5.

False-to-trueTrue-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?

3. The Einstein theory of gravity with a nonminimally coupled scalar field Vacuum-to-vacuum phase transition rate Action Einstein equations

curvature scalar Potential Rotationally invariant Euclidean metric : O(4)-symmetry The Euclidean field equations boundary conditions

Our main idea (during the phase transition)

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-trueTrue-to-false De Sitter – de Sitter OO Flat – de Sitter OO Anti-de Sitter – de Sitter OO Anti-de Sitter – flat OO Anti-de Sitter – Anti- de Sitter OO

2.Thin-wall approximation B is the difference In this approximation Outside the wall

In the wall where inside the wall

(a) false vacuum nucleation if

The coefficient B (b) true vacuum nucleation

The coefficient B if (by S. Parke) where

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

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

After squaring twice, the equation turns out to be where the effective potential is with

(1) M = 0 DS – DS

DS – FLAT

DS – ADS

(2) M > 0 DS – SDS

DS – S

DS – SADS

5. 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?