Bubble Nucleation in Brans Dicke-Type Theory

Transcription

1 Bubble Nucleation in Brans Dicke-Type Theory Young Jae Lee KAIST International School on Numerical Relativity and Gravitational Waves 28 July 2010 APCTP, Pohang Bum-Hoon Lee, Wonwoo Lee and Dong-han Yeom, gr-qc/ Hongsu Kim, Bum-Hoon Lee, Wonwoo Lee, YJ and Dong-han Yeom, in preperation

2 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

3 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

4 Backgrounds In 1977 S. Coleman suggested that the true vacuum bubble can be nucleated within the de Sitter background in the absence of gravity. L = 1 2 µφ µ φ U(φ)

5 Backgrounds In 1977 S. Coleman suggested that the true vacuum bubble can be nucleated within the de Sitter background in the absence of gravity. L = 1 2 µφ µ φ U(φ) By calculating Euclidean action, we can find the bubble nucleation rate. [ ( ) S E = 2π 2 dρρ 3 1 dφ 2 + U(φ)] 0 2 dρ

6 Backgrounds In 1977 S. Coleman suggested that the true vacuum bubble can be nucleated within the de Sitter background in the absence of gravity. L = 1 2 µφ µ φ U(φ) By calculating Euclidean action, we can find the bubble nucleation rate. [ ( ) S E = 2π 2 dρρ 3 1 dφ 2 + U(φ)] 0 2 dρ Γ/V = Ae S E/ [1 + O( )] In fact, the coefficient A is from the first quantum correction. True Vacuum False Vacuum

7 Backgrounds In 1977 S. Coleman suggested that the true vacuum bubble can be nucleated within the de Sitter background in the absence of gravity. L = 1 2 µφ µ φ U(φ) By calculating Euclidean action, we can find the bubble nucleation rate. [ ( ) S E = 2π 2 dρρ 3 1 dφ 2 + U(φ)] 0 2 dρ Γ/V = Ae S E/ [1 + O( )] In fact, the coefficient A is from the first quantum correction. True Vacuum False Vacuum

8 Backgrounds In 1977 S. Coleman suggested that the true vacuum bubble can be nucleated within the de Sitter background in the absence of gravity. L = 1 2 µφ µ φ U(φ) By calculating Euclidean action, we can find the bubble nucleation rate. [ ( ) S E = 2π 2 dρρ 3 1 dφ 2 + U(φ)] 0 2 dρ Γ/V = Ae S E/ [1 + O( )] In fact, the coefficient A is from the first quantum correction. Quantum Tunneling True Vacuum False Vacuum

9 Backgrounds In 1980 S. Coleman and F. De Luccia calculated the true vacuum nucleation rate surrounded by the false vacuum in the presence of gravity. where κ = 8πG. L = 1 2 µφ µ φ U(φ) R 2κ

10 Backgrounds In 1980 S. Coleman and F. De Luccia calculated the true vacuum nucleation rate surrounded by the false vacuum in the presence of gravity. L = 1 2 µφ µ φ U(φ) R 2κ where κ = 8πG. S E = 4π 2 [ dη ρ 3 U(φ) 3ρ ] κ Γ/V = Ae S E/ [1 + O( )]

11 Backgrounds In 1980 S. Coleman and F. De Luccia calculated the true vacuum nucleation rate surrounded by the false vacuum in the presence of gravity. L = 1 2 µφ µ φ U(φ) R 2κ where κ = 8πG. S E = 4π 2 [ dη ρ 3 U(φ) 3ρ ] κ Γ/V = Ae S E/ [1 + O( )] Recently, W. Lee, B. Lee, C. H. Lee, C. Park studied false vacuum bubble nucleation via non-minimal coupling with Einstein gravity. L = R 2κ 1 2 ( φ)2 1 2 ξφ2 R U(φ)

12 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

13 Brans Dicke-Type Gravity The Brans Dicke action with self-interacting potential is S = 1 d 4 x [ g ΦR ω ] 16π Φ Φ;α Φ ;α V (Φ) where c = 8πG 0 = 1.

14 Brans Dicke-Type Gravity The Brans Dicke action with self-interacting potential is S = 1 d 4 x [ g ΦR ω ] 16π Φ Φ;α Φ ;α V (Φ) where c = 8πG 0 = 1. Also, the equation of motion becomes Φ = 1 2ω + 3 [ Φ V ] 2V (Φ) Φ

15 Brans Dicke-Type Gravity The Brans Dicke action with self-interacting potential is S = 1 d 4 x [ g ΦR ω ] 16π Φ Φ;α Φ ;α V (Φ) where c = 8πG 0 = 1. Also, the equation of motion becomes Φ = 1 2ω + 3 [ Φ V ] 2V (Φ) Φ The field equation contains a parameter, ω, called the Brans Dicke coupling constant.

16 Brans Dicke-Type Gravity The Brans Dicke action with self-interacting potential is S = 1 d 4 x [ g ΦR ω ] 16π Φ Φ;α Φ ;α V (Φ) where c = 8πG 0 = 1. Also, the equation of motion becomes Φ = 1 2ω + 3 [ Φ V ] 2V (Φ) Φ The field equation contains a parameter, ω, called the Brans Dicke coupling constant. From the solar scale observational tests, ω

17 Brans Dicke-Type Gravity The Brans Dicke action with self-interacting potential is S = 1 d 4 x [ g ΦR ω ] 16π Φ Φ;α Φ ;α V (Φ) where c = 8πG 0 = 1. Also, the equation of motion becomes Φ = 1 2ω + 3 [ Φ V ] 2V (Φ) Φ The field equation contains a parameter, ω, called the Brans Dicke coupling constant. From the solar scale observational tests, ω However, small ω can be allowed in some fundamental theories such as dilaton gravity and brane world models.

18 Euclidean O(4) Bubble Assume that the metric has O(4) symmetry.

19 Euclidean O(4) Bubble Assume that the metric has O(4) symmetry. ds 2 = dη 2 + ρ 2 (η) [ dχ 2 + sin 2 χ(dθ 2 + sin 2 θdφ 2 ) ]

20 Euclidean O(4) Bubble Assume that the metric has O(4) symmetry. ds 2 = dη 2 + ρ 2 (η) [ dχ 2 + sin 2 χ(dθ 2 + sin 2 θdφ 2 ) ] So, the Euclidean action becomes S E = π ( ρ 3 6 dη 8 ρ 2 Φ(ρ ρ + ρ2 1) + ω Φ ) 2 Φ + V

21 Euclidean O(4) Bubble Assume that the metric has O(4) symmetry. ds 2 = dη 2 + ρ 2 (η) [ dχ 2 + sin 2 χ(dθ 2 + sin 2 θdφ 2 ) ] So, the Euclidean action becomes S E = π ( ρ 3 6 dη 8 ρ 2 Φ(ρ ρ + ρ2 1) + ω Φ ) 2 Φ + V Also, the equation of motion is From the Einstein equation, Φ + 3 ρ ρ Φ = ρ = ρ Φ ( 2Φ ± 1 + 2ω 3 [ 1 Φ V ] 2V (Φ) 2w + 3 Φ ) ( ρ Φ ) 2 ρ2 V (Φ) 2Φ 6Φ + 1 1/2

22 Potential The equation of motion for Brans Dicke field is Φ + 3 ρ ρ Φ = 1 2ω + 3 [ Φ V 2V (Φ) Φ ]

23 Potential The equation of motion for Brans Dicke field is Φ + 3 ρ ρ Φ = 1 2ω + 3 [ Φ V 2V (Φ) Φ ] = 1 2ω + 3 F (Φ)

24 Potential The equation of motion for Brans Dicke field is Φ + 3 ρ ρ Φ = 1 2ω + 3 The Lorentzian potential is given by [ Φ V 2V (Φ) Φ V (Φ) = Φ 2 Φ Φ T d ] = Φ F ( Φ) Φ 3 1 2ω + 3 F (Φ)

25 Potential The equation of motion for Brans Dicke field is Φ + 3 ρ ρ Φ = 1 2ω + 3 The Lorentzian potential is given by [ Φ V 2V (Φ) Φ V (Φ) = Φ 2 Φ Φ T d ] = Φ F ( Φ) Φ 3 1 2ω + 3 F (Φ) Also, we can introduce the effective potential as stated in the following. U(Φ) = Φ Φ T d ΦF ( Φ)

26 Potential The equation of motion for Brans Dicke field is Φ + 3 ρ ρ Φ = 1 2ω + 3 The Lorentzian potential is given by [ Φ V 2V (Φ) Φ V (Φ) = Φ 2 Φ Φ T d ] = Φ F ( Φ) Φ 3 1 2ω + 3 F (Φ) Also, we can introduce the effective potential as stated in the following. Φ U(Φ) = d ΦF ( Φ) Φ T As a toy model, we can introduce a simple potential. F (Φ) = A(Φ Φ T )(Φ Φ F )(Φ Φ M + δ)

27 Bubble Nucleations in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

28 Scheme The equation of motions can be expressed as three first-order ODEs. Φ = Ψ ρ = ρψ 2Φ ± Ψ = [ ( 1 + 2ω 3 1 2ω + 3 F (Φ) 3 ρ ρ Ψ ) ( ) ] ρψ 2 1/2 ρ2 V (Φ) 2Φ 6Φ + 1 Solving the equation of motions for η, Φ and ρ, we used the 4th-order Runge Kutta method.

29 Scheme The equation of motions can be expressed as three first-order ODEs. Φ = Ψ ρ = ρψ 2Φ ± Ψ = [ ( 1 + 2ω 3 1 2ω + 3 F (Φ) 3 ρ ρ Ψ ) ( ) ] ρψ 2 1/2 ρ2 V (Φ) 2Φ 6Φ + 1 Solving the equation of motions for η, Φ and ρ, we used the 4th-order Runge Kutta method. Also, the initial condition comes from the Euclidean dynamics. In the case of true vacuum bubble nucleation, Φ 0 = Φ T, ρ 0 = 0, Ψ 0 = 0 In the case of false vacuum bubble nucleation, Φ 0 = Φ F, ρ 0 = 0, Ψ 0 = 0

30 True Vacuum Bubble Nucleation: 2ω + 3 > 0 True Vacuum False Vacuum Figure: On the Lorentzian Potential

31 True Vacuum Bubble Nucleation: 2ω + 3 > 0 True Vacuum False Vacuum Figure: On the Lorentzian Potential

32 True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential

33 True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Figure: On the Euclidean Potential

34 True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Figure: On the Euclidean Potential

35 True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Classical Dynamics Figure: On the Euclidean Potential

36 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum True Vacuum Figure: On the Lorentzian Potential

37 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum True Vacuum Figure: On the Lorentzian Potential

38 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential

39 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Figure: On the Euclidean Potential

40 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Figure: On the Euclidean Potential

41 False Vacuum Bubble Nucleation: 2ω + 3 < 0 False Vacuum Quantum Tunneling True Vacuum Figure: On the Lorentzian Potential Classical Dynamics Figure: On the Euclidean Potential

42 Result Figure: True Vacuum Bubble Φ(η), ρ(η) and ρ Φ/ρ from de Sitter Background: ω = 10 Figure: False Vacuum Bubble Φ(η), ρ(η) and ρ Φ/ρ from Flat Background: ω = 2

43 Bubble Nucleations in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

44 Thin Wall Approximation: False Vacuum Bubble Assume that the transition region is sufficiently thin. Φ ρ ρ 1

45 Thin Wall Approximation: False Vacuum Bubble Assume that the transition region is sufficiently thin. Φ ρ ρ 1 The approximation above and the Einstein s equation gives us ρ ρ2 ( ) 6Φ 2 ω Φ 2 ΦV

46 Thin Wall Approximation: False Vacuum Bubble Assume that the transition region is sufficiently thin. Φ ρ ρ 1 The approximation above and the Einstein s equation gives us Therefore, ρ ρ2 ( ) 6Φ 2 ω Φ 2 ΦV dρ dη 1 + ρ2 ( ) 6Φ 2 ω Φ 2 ΦV

47 Thin Wall Approximation: False Vacuum Bubble Assume that the transition region is sufficiently thin. Φ ρ ρ 1 The approximation above and the Einstein s equation gives us Therefore, ρ ρ2 ( ) 6Φ 2 ω Φ 2 ΦV dρ dη 1 + ρ2 ( ) 6Φ 2 ω Φ 2 ΦV Also, the Euclidean equation of motion becomes [ ] dφ dη 2 1/2 2ω + 3 (U(Φ) U(Φ i))

48 Thin Wall Approximation: False Vacuum Bubble We set where ρ is the position of the shell. { 1 ρ(η) > ρ Φ(η) = Φ F ρ(η) < ρ

49 Thin Wall Approximation: False Vacuum Bubble We set { 1 ρ(η) > ρ Φ(η) = Φ F ρ(η) < ρ where ρ is the position of the shell. Also, { 0 Φ = 1 V (Φ) = Λ Φ = Φ F

50 Thin Wall Approximation: False Vacuum Bubble We set { 1 ρ(η) > ρ Φ(η) = Φ F ρ(η) < ρ where ρ is the position of the shell. Also, { 0 Φ = 1 V (Φ) = Λ Φ = Φ F Outside the bubble, Inside the bubble, dρ dη 1 + ρ2 ( 6Φ 2 ω Φ ) 2 ΦV = 1 dρ dη 1 + ρ2 ( 6Φ 2 ω Φ ) 2 ΦV = 1 Λ ρ 2 6Φ F The solutions of these equations are { η ρ(η) > ρ ρ = 6ΦF sin η ρ(η) < ρ Λ

51 Thin Wall Approximation: False Vacuum Bubble The Euclidean action becomes ( S E = 2π 2 ρ 3 dη 1 ΦR + ω Φ ) 2 16π Φ + V = π ( ρ 3 6 dη 8 ρ 2 Φ(ρ ρ + ρ2 1) + ω Φ ) 2 Φ + V = π ( dη 6 8 Φ ρρ 2 6Φρ ρ 2 6Φρ + ωρ Φ ) 3 2 Φ + ρ3 V + S B

52 Thin Wall Approximation: False Vacuum Bubble The Euclidean action becomes ( S E = 2π 2 ρ 3 dη 1 ΦR + ω Φ ) 2 16π Φ + V = π ( ρ 3 6 dη 8 ρ 2 Φ(ρ ρ + ρ2 1) + ω Φ ) 2 Φ + V = π ( dη 6 8 Φ ρρ 2 6Φρ ρ 2 6Φρ + ωρ Φ ) 3 2 Φ + ρ3 V + S B Finally, we obtain S E = π dη ( ρ 3 V 6ρΦ ) 4

53 Thin Wall Approximation: False Vacuum Bubble The Euclidean action becomes ( S E = 2π 2 ρ 3 dη 1 ΦR + ω Φ ) 2 16π Φ + V = π ( ρ 3 6 dη 8 ρ 2 Φ(ρ ρ + ρ2 1) + ω Φ ) 2 Φ + V = π ( dη 6 8 Φ ρρ 2 6Φρ ρ 2 6Φρ + ωρ Φ ) 3 2 Φ + ρ3 V + S B Finally, we obtain S E = π 4 dη ( ρ 3 V 6ρΦ ) This result is consistent with the Coleman and De Luccia s. ( S E = 4π 2 dη ρ 3 V 3ρ ) 8πG

54 Therefore, we can find the coefficient B by the simple calculation. B outside = 0

55 Therefore, we can find the coefficient B by the simple calculation. B outside = 0 B shell = π 4 = π 4 dη ( ρ 3 V (Φ) 6 ρφ + 6 ρ ) 2ω π 2 ρ 3 σ(ω, ρ) 1 Φ F dφ ( ρ 3 V (Φ) 6 ρφ + 6 ρ ) U(Φ) U(ΦF )

56 Therefore, we can find the coefficient B by the simple calculation. B outside = 0 B shell = π 4 = π 4 dη ( ρ 3 V (Φ) 6 ρφ + 6 ρ ) 2ω Φ F dφ ( ρ 3 V (Φ) 6 ρφ + 6 ρ ) U(Φ) U(ΦF ) 2π 2 ρ 3 σ(ω, ρ) B inside = π dη ( ρ 3 Λ 6ρΦ F + 6ρ ) 4 [ ( = 3π ( 2Φ F 1 1 Λ ) ) ] 3/2 ρ 2 ρ2 2 Λ 6Φ F 2

57 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

58 Conclusion i. w > 3 ; Flat true vacuum bubble nucleation within de Sitter background 2 ii. w < 3 ; de Sitter false vacuum bubble nucleation within Minkovski background 2 iii. Thin wall approximation holds iv. Negative energy at the shell of the false vacuum bubble

59 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List

60 To-do List i. O(4) bubble nucleation with the various parameters, ω, A, and δ ii. Violation of weak equivalence principle with the condition w < 3 2 iii. The bubble nucleation at finite temperature iv. Other modified gravitational theories

61 Bubble Nucleation in Brans Dicke-Type Theory Backgrounds Methods Brans Dicke-Type Gravity Euclidean O(4) Bubble Potential Numerical Solution Scheme True Vacuum Bubble Nucleation: 2ω + 3 > 0 False Vacuum Bubble Nucleation: 2ω + 3 < 0 Result Thin Wall Approximation: False Vacuum Bubble Conclusion To-do List