Oscillating Fubini instantons in curved space

Transcription

1 Oscillating Fubini instantons in curved space Daeho Ro Department of Physics, Sogang University, , Seoul November 29, 214 Daeho Ro (Sogang University) Inje University November 29, / 2

2 Motivation We are interested in the tunneling process corresponding to the decay of vacuum state with quartic and tachyonic potential. S. Fubini studied about the tunneling process in the absence of gravity with the same potential and the solution is called Fubini instanton. We are interested in the Fubini instanton in curved space. We want to compare the behavior of the existence of solutions in the absence and presence of gravity with the potentials. There are various types of solutions depending on the values of parameters κ, Φ, and U, which correspond to the gravitational constant, initial value of scalar field, and the cosmological constant, respectively. These parameters make a phase diagram and it make several types of solution or not. This presentation is based on arxiv: and arxiv: Daeho Ro (Sogang University) Inje University November 29, / 2

3 Fubini instanton Let us consider the following Euclidean action in the absence of gravity as follows: [ S E = det ηµνd 4 1 x 2 µ Φ µφ + U (Φ)], with the potential M U (Φ) = λ 4 Φ4 + U. In order to find the equation of motion, we assume the O(4) symmetry ds 2 = dη 2 + η 2 [ dχ 2 + sin 2 χ ( dθ 2 + sin 2 θdφ 2)]. We obtain the equation of motion as follows: Φ + 3 η Φ = du dφ, where the boundary conditions are, dφ = and Φ dη η= =. η= Daeho Ro (Sogang University) Inje University November 29, / 2

4 Fubini instanton For the given potential, there is analytic solution which is called Fubini instanton and it has the following form 8 a Φ(η) = λ η 2 + a. 2 Here, η is Euclidean time parameter and a is an arbitrary length scale that characterizes the size of the instanton. (a) 3. (b) a=1 a=3 a= = = = Figure: (a) Analytic solutions for a = 1, a = 3 and a = 5 and (b) numerical solutions for some Φ correspond to a = 1, a = 3 and a = 5. Daeho Ro (Sogang University) Inje University November 29, / 2

5 Fubini instanton in curved space Let us consider the following Euclidean action in the presence of gravity as follows: [ S E = det gµνd 4 x R 2κ + 1 ] 2 µ Φ µφ + U (Φ), M where κ = 8πG and R denotes the scalar curvature of the spacetime M. Cosmological constant is defined by Λ = κu. In order to find the equations of motion, we assume the O(4) symmetry for the geometry and the scalar field: ds 2 = dη 2 + ρ(η) 2 [ dχ 2 + sin 2 χ ( dθ 2 + sin 2 θdφ 2)]. We can obtain the equations of motions and Hamiltonian constraint as follows: Φ + 3 ρ ρ Φ = du dφ, ρ = κ 3 ρ( Φ 2 + U ), ( ρ 2 1 κρ Φ 2 U) =. Daeho Ro (Sogang University) Inje University November 29, / 2

6 Fubini instanton in curved space The Euclidean action with the equations of motion and Ricci scalar are: [ ( ) ] S E = 2π 2 ρ 3 dη 3 1 κ ρ ρ2 2 ρ ρ ρ 2 Φ 2 + U = 2π 2 ρ 3 ( U )dη. In order to solve the equations of motion, we need suitable boundary conditions. We choose the boundary conditions for Λ as follows: dφ =, Φ dη η= =, ρ =, and dρ = 1. η= dη η= η= For Λ >, there is a finite value η max which satisfies ρ(η max) =. Thus we use the boundary conditions for Λ > as follows: dφ dφ dη =, dη =, ρ =, and ρ =. η= η=η max η= η=η max Daeho Ro (Sogang University) Inje University November 29, / 2

7 Details of numerical method We use Runge-Kutta method to solve the equations of motion numerically and for that, we need to make the equations dimensionless. So we change the variables for the given potential: λu U, λφ 2 Φ 2, and κ/λ κ. Equations of motion become dimensionless with same form. Then, we can adopt the result directly to the numerical calculation. Moreover, the equations of motion for Φ and ρ has a singularity at η =. In order to avoid this singularity, we set η initial = ɛ with sufficiently small value and then the initial values are changed by Taylor series expansion as follows: Φ(ɛ) = Φ ɛ2 8 Φ3 +, ρ(ɛ) = ɛ +, Φ(ɛ) = ɛ 4 Φ3 +, ρ(ɛ) = 1 +. Daeho Ro (Sogang University) Inje University November 29, / 2

8 Details of numerical method We have to cut the Euclidean time for the numerical calculation because the evolution parameter is infinite in AdS space. In order to find the initial value of scalar field Φ which satisfy the boundary condition in the equations of motion, we use the shooting method. (a) = = (b).5 =-3.9 = = =-4.5 = Figure: Shooting method examples in ds space Daeho Ro (Sogang University) Inje University November 29, / 2

9 Tunneling probability To get the tunneling probability, we need to calculate Euclidean action [ ] S E = ge d 4 x R E 2κ Φ 2 + U = 2π 2 ρ 3 dη[ U ]. M In the semiclassical approximation, the decay probability is represented as Ae B where the exponent B is the difference between Euclidean action of a bounce solution and background action, B = SE bs S bg E. From the technical reason in AdS space, we change the expression of the exponent B by using constraint equation B = 2π 2 ρ 2 dρ 1 ρ 2 + κ 3 U ( 1 2 Φ 2 U ρ + κ. 2 3 ( U) ) U 1 Daeho Ro (Sogang University) Inje University November 29, / 2

10 Solutions We numerically solve the coupled equations of the scalar field Φ and scale factor ρ which comes from gravity. As we said before, there are three parameters κ, U and Φ. Each parameters correspond to the gravitational constant, cosmological constant, and initial value of scalar field, respectively. In AdS space, any set of parameter values always give the solutions with finite number of oscillation. So, we are focused to classify the number of solutions. Especially, we are interested in the oscillating solutions which correspond to the boundary of oscillating numbers. In ds space, specific set of parameter values give the solutions with different number of oscillation. Thus, we are focused to find the solutions. Interestingly, there are two types of solutions which are symmetric and asymmetric. For each solutions, we get the action difference B. It might be infinite or finite. Daeho Ro (Sogang University) Inje University November 29, / 2

11 Solutions in AdS space (a) s 1 b 1 s 2 b 2 s 3 (b) s 1 b 1 s 2 b 2 s (c) ' s 1 b 1 s 2 b 2 s 3 (d) s 1 b 1 s 2 b 2 s Figure: (a) Numerical solutions for Φ, (b) for ρ, (c) phase diagram of Φ vs. Φ, and (d) Euclidean energy E ξ in AdS space. We take κ =.3 and U =.3. Daeho Ro (Sogang University) Inje University November 29, / 2

12 Solutions in AdS space Late time behavior of Φ is linear in log-log scale for all solutions. This means that the solutions are approaching to zero when the time goes to infinity. From the analysis of action difference B, we notice that the marginal solutions only have the finite action difference and the others looks have infinity. (a) =-3., =-3.69, =-3.698, = , = , = , = , = , = , = , = (b) B =-3., =-3.69, =-3.698, = , = , = , = , = , = , = , = Figure: Log-Log graph of (a) Φ and (b) B versus ρ for AdS solutions. Daeho Ro (Sogang University) Inje University November 29, / 2

13 Solutions in ds space (a) 6 3 (b) z, = z 1, = z 2, = z 3, = z, = z 1, = z 2, = z 3, = (c) ' z, = z 1, = z 2, = z 3, = (d) -1-2 z, = z 1, = z 2, = z 3, = Figure: (a) Numerical solutions for Φ, (b) for ρ, (c) phase diagram of Φ vs. Φ, and (d) Euclidean energy E ξ of Z 2 symmetric cases in ds space. Daeho Ro (Sogang University) Inje University November 29, / 2

14 Solutions in ds space (a) 4 (b) a 1L a 1R a 1L a 1R (c) 4 (d) 2-1 ' a 1L a 1R -3 a 1L a 1R Figure: (a) Numerical solutions for Φ, (b) for ρ, (c) phase diagram of Φ vs. Φ, and (d) Euclidean energy E ξ of Z 2 asymmetric cases in ds space. Daeho Ro (Sogang University) Inje University November 29, / 2

15 Solutions in ds space (a) 8 4 (b) a 3L a 3R a 3L a 3R (c) 1 (d) 5-1 ' a 3L a 3R -3-4 a 3L a 3R Figure: (a) Numerical solutions for Φ, (b) for ρ, (c) phase diagram of Φ vs. Φ, and (d) Euclidean energy E ξ of Z 2 asymmetric cases in ds space. Daeho Ro (Sogang University) Inje University November 29, / 2

16 Details of numerical method We are tested the behavior of solutions with different parameter values. By using this result, we draw a map which visualize the information about solutions. Figure: Matrix plots in (a) AdS space with white, gray and black colors which are correspond to the different number of oscillation such as 1, 2, and 3, respectively, and in (b) ds space with white and black colors which are correspond to the direction of divergence such as negative and positive, respectively Through the matrix plot, we modify our code to find the phase diagram easily. Daeho Ro (Sogang University) Inje University November 29, / 2

17 Phase diagram in AdS space (a) (b) U -1. U b 1 b 2 (c) (d) b 1 b U -1. U b 1 b b 1 b Figure: Parametric phase diagram in AdS space with (a) κ =.5, (b) κ =.1, (c) κ =.3, and (d) κ =.5, respectively Daeho Ro (Sogang University) Inje University November 29, / 2

18 Phase diagram in ds space (a) 1..5 U a 1 z 1 z 2 a 3 z 3 z 4 (b) U a 1 z 1 z 2 z 3 z (c) a 1 z 1 z 2 z 3 z 4 (d) a 1 z 1 z 2 z 3 z 4 U 1. U Figure: Parametric phase diagrams in ds space with (a) κ =.5, (b) κ =.1, (c) κ =.3, and (d) κ =.5, respectively Daeho Ro (Sogang University) Inje University November 29, / 2

19 Summary In the absence of gravity, a quartic and tachyonic potential makes infinitely many solutions which is called Fubini instanton. However, inclusion of the gravity changes the situation abruptly. Depending on the sign of Λ, solution space changed or get reduced. In AdS space, there always exist the oscillating solutions but the oscillating numbers are fixed by the choice of parameters. This means that the solution space is changed. Interestingly, the action difference B is finite when the solution is a marginal solution. The other solutions looks have infinite action difference. In ds space, there are Z 2 symmetric and asymmetric solutions with specific set of parameter values. This means that the solution space is reduced. In this case, the action difference is finite for all solutions. Daeho Ro (Sogang University) Inje University November 29, / 2

20 The end Thanks! Daeho Ro (Sogang University) Inje University November 29, / 2