VACUUM BUBBLES AND GRAVITY EFFECTS

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Vol. 12 (2012) 330–339 c

World Scientiﬁc Publishing Company

DOI:10.1142/S2010194512006538

VACUUM BUBBLES AND GRAVITY EFFECTS

BUM-HOON LEE

Department of Physics and BK21 Division, and Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea

[email protected] WONWOO LEE

Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea [email protected]

The vacuum bubbles in the curved spacetime are analyzed. We classify the true and false vacuum bubbles with the cosmological constants from the minimum of the potential being

either positive, zero, or negative. We introduce new type of solutions withZ₂symmetry

when the potential has degenerate vacua, i.e., instanton solutions. This new type of solutions are possible only if gravity is taken into account.

Keywords: True vacuum bubbles; False vacuum bubbles; Instanton solutions.

1. Introduction

The landscape paradigm of string theory has many stable as well as unstable vacua1,2,3. Thus, the study of the nucleation of vacuum bubbles in curved spacetime has been one of the key problems to understand the vacuum tunneling in the string theory landscape. In this work, we study the tunneling process in the simple double well-type potential as a toy model.

The theory describing the nucleation of vacuum bubbles or decay of a back-ground vacuum is the subject studied for a long time. The nucleation of a true vacuum bubble withO(4) symmetry within the false vacuum sea at zero temper-ature in flat space was first investigated in Ref.4, developed in flat spacetime in Ref.5, and in curved spacetime in Ref.6. The result for the small true vacuum bub-ble within the large false vacuum background was enlarged by Parke7 to the case with arbitrary cosmological constants. The transition from a positive false vacuum to a negative true vacuum, in the process of the false vacuum energy decreased to zero, was studied in Refs.8,9. The authors found that there is no non-compact solution describing the decay of the false vacuum with zero energy8 and analyzed the solution space of Coleman-De Luccia bounce solutions9.

The mechanism for the nucleation of a false vacuum bubble within the true vac-uum background has also been studied. The nucleation process of the false vacvac-uum bubble in dS space was originally obtained in Ref.10. The nucleation of a small false

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vacuum bubble was explored using a mechanical analogy in the Einstein gravity with a nonminimally coupled scalar field11, in Einstein-Gauss-Bonnet theory12, and in Brans-Dicke type theory13. The nucleation rate as well as classification depending on the size of the false vacuum bubble in de Sitter(dS) space were obtained in Ref.14 in the Einstein theory of gravity. The classification of true and false vacuum bub-bles in the dS background space depends on the value of the cosmological constant. Large bubbles have large value of the cosmological constant, while small bubbles have small value of the cosmological constant. These processes may provide an alternative paradigm in the string theory landscape1,2,3 or eternal inflation15,16,17. Oscillating bounce solutions as another type were also studied18.

The natural question is on the instanton solution between the degenerate vacua and its relation to the bubble. The clue of these solutions could have been seen in Refs.8,9,14,18. There exists the instanton solution withO(4) symmetry between the degenerate vacua in dS space. This can be understood in particle analogy picture due to the change of the damping term into accelerating term. The numerical solution of the scalar field Φ was obtained in Ref.18. The analytic computation and meanings of this solution were further studied in Ref.14. Can we also obtain the instanton solution between the degenerate vacua in both flat and anti-de Sitter(AdS) space? We have studied boundary conditions needed for these solutions not only in dS but also in both flat and AdS space. These boundary conditions respect the Z₂

symmetry, diﬀerent from those studied in bounce solutions4,5,6,7. We have shown that there exist new type of solutions giving rise to the ﬁnite geometry with the

Z₂ symmetry19. These solutions can not be obtained as the limiting cases of the previously known vacuum bubble solutions.

In ﬂat spacetime the bounce solution has exactly one negative mode20. When gravity is taken into account, it is more involved problem21,22,23. On the other hand, our solutions describe quantum mechanical mixing between the degenerate vacua and we expect that the solutions do not have any negative mode.

Quantum mechanically, the tunneling process in a symmetric double-well poten-tial is described by the instanton solution. This tunneling shifts the ground state energy of the classical vacuum due to the presence of an additional potential well lifting the classical degeneracy. The symmetric ground state wave function describes that there is a higher probability of finding it somewhere between the two classical vacua. In the semiclassical approximation, the action is dominated by the instanton configuration. The instanton solution can be interpreted as a particle motion in the inverted potential starting from one vacuum state at the minus infinite Euclidean time and reaching the other one at the plus infinite time. We may consider the multi-instanton solutions as describing the tunneling back and forth between the two vacua. For field theoretical solutions with O(4) symmetry, the equation of motion has an additional term, which can be interpreted as a friction term in the inverted potential. The role of the friction term can be changed into the accelerating term in the presence of gravity when the local maximum value of the potential is positive19.

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The outline of this paper is as follows: In section 2, the framework of instanton solutions in scalar field theory in one dimension is briefly reviewed. In section 3, we review the nucleation process of a vacuum bubble in scalar field theory. In section 4, we consider the phase transition of a self-gravitating scalar field in the Einstein gravity. In section 5, we consider instaton solutions mediating tunneling between the degenerate vacua in curved space. The results are discussed in section 6. The main work of this article is based on Refs.14,19

2. Instanton solutions in scalar field theory

In this section, we brieﬂy review the framework of an instanton solution in a one-dimensional scalar ﬁeld theory24. The instanton solution in 1 dimension is equivalent to a static soliton in 1 + 1 dimensions. Thus, the method for the kink solution in two dimensions can be employed for the instanton solution in one dimension. The transition amplitude in the Euclidean section is written by

<Φ_f|e−HT|Φ_i>=N

DΦe−SE[Φ]_, ₍₁₎

whereH is the Hamiltonian,T is a positive number, and the Euclidean action has the form S_E[Φ] = _{T /}₂ −T /2 dτ 1 2 ˙ Φ2+V(Φ) . (2)

We consider a symmetric double-well potential with degenerate minima as shown in Fig.1

V(Φ) = λ 8(Φ

2₋_a2₎2_, ₍₃₎

where λ anda are positive parameters. The corresponding Euclidean equation of motion is given by

d2Φ

dτ2 =− d(−V)

dΦ . (4)

The Euclidean equations correspond to a particle moving in an inverted potential. The particle arrives at the tops of the hills at times plus inﬁnity and minus inﬁnity. One obtains the solutions

Φ_s(τ) =±atanh

√

λa(τ−τ_c)

2 , (5)

satisfying the following boundary condition lim

|τ|→∞Φ(τ) =±a. (6)

The sign + indicates instanton,−anti-instanton, andτ_c is an integration constant.

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Fig. 1. The left ﬁgure represents the potential energy and the right the inverted potential energy.

The solutions play a dominant role in evaluating the path integral. The Euclidean action can be easily calculated

S_E[Φ_s] = 2

√ λa3

3 , (7)

where we used the fact that the Euclidean energy of the instanton is equal to zero in this system. This action is identical with the energy of the static solution of the 1+1 dimensional soliton theory. In addition, the action has the same value as the action obtained in connection with the WKB calculation of the splitting in the energies of the two lowest levels for the double well potential. One obtains the energies of the two lowest levels

E_±= √ λa 2 ±Ke −SE_, ₍₈₎ whereK is real.

On the other hand, the energy of an unstable state sitting in the bottom of the well for a bounce solution has the form

E_o= √ λa 2 − i 2|K|e −SE_. ₍₉₎

3. Bounce solutions in scalar field theory: Nucleation of vacuum bubbles

Now we go to the case of a bounce solution in scalar ﬁeld theory5. The bounce solution is related to the vacuum decay process, i.e., the nucleation of vacuum bubbles. The Euclidean action has the form

S_E= M √ g_Ed4x 1 2∇ α_Φ_∇ αΦ +U(Φ) , (10)

whereg_E≡detη_µνE andU(Φ) is the potential with two non-degenerate minima, one corresponding to the metastable vacuum or the false vacuum and the other to the

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Fig. 2. The left ﬁgure represents the potential energy density with two non-degenerate minima,

−a≡Φt anda≡Φ_f, and the right the inverted one.

true vacuum, separated by a potential barrier as shown in Fig.2:

U(Φ) = λ 8(Φ

2₋_a2₎2₊

2a(Φ +a) +Uo, (11)

whereλ,, andaare positive parameters.

In the semiclassical approximation, the background vacuum decay rate or the bubble nucleation rate per unit time per unit volume is given by

Γ/V Ae−B, (12)

where the leading semiclassical exponent B coincides with the difference between Euclidean action corresponding to bubble solution and that of the background, and the sub-leading pre-factorA was studied in Refs.20,25. We are interested in finding the coefficientB.

The bounce with the minimum Euclidean action is assumed to have the highest symmetry26 and Coleman considered the solution with O(4) symmetry. Then Φ depends only onη(=τ2+|x|2), and the Euclidean ﬁeld equation becomes

d2Φ dη2 + 3 η dΦ dη =− d(−U) dΦ . (13)

Equation (13) is formally equivalent to a one-particle equation of motion with η

playing the role of time in the corresponding potential well−U(Φ) in the presence of the friction term. The boundary conditions for the solution are

lim η→∞Φ(η) = Φf and dΦ dη η=0 = 0, (14)

where the second condition is required for smoothness of the field near η = 0. Our first mission is to find the numerical solution of Eq. (13) with the bound-ary conditions (14) and second to obtain the coefficient B by using the thin-wall approximation.

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Multiplying Eq. (13) by d_dηΦ and rearranging the terms, one obtains d dη 1 2 dΦ dη 2 −U =−3 η dΦ dη 2 . (15)

The quantity in the square brackets here can be interpreted as the total energy of the particle with the potential energy−U, the term on the right hand side as the dissipation rate of the total energy. The nucleation of a true vacuum bubble within the false vacuum background corresponds to the particle starting at some point near Φ_tatη = 0 with zero velocity and reaching Φ_f atη=∞. WhenU(Φ_f)−U(Φ_t) is suﬃciently small in comparison with all other parameters of the model, the so-called thin-wall approximation becomes valid. Ifis small, the starting point should be very close to Φ_t and the particle stays there for a long time with a very small velocity and acceleration so thatηgrows large with Φ staying near Φ_t. Asηbecomes large, the friction force becomes negligible and Φ quickly goes to Φ_f and stays at that point from thereafter.

We take the case where/λa4is small and approximate the quantity to the ﬁrst order of this parameter. In the thin-wall approximation, it is possible to neglect the second term 3

ηddηΦ and the action

B=S_Eb −S_Ef.v. (16)

can be divided into three parts:B=B_out+B_wall+B_in. The action for the critical size bubble is obtained as

B= 27π2S_o4/23. (17)

For small, the exponentB is large so that the nucleation process of the vacuum bubble is strongly suppressed.

4. Nucleation of vacuum bubbles in the Einstein gravity

We consider the phase transition of a self-gravitating scalar ﬁeld in the Einstein gravity6. Let us consider the following action

S= M √ −gd4x R 2κ− 1 2∇ α_Φ_∇ αΦ−U(Φ) + ∂M √ −hd3xK−Ko κ , (18)

where κ ≡ 8πG, g ≡detg_µν, K and K_o are traces of the extrinsic curvatures of

∂M in the metric g_µν and η_µν, respectively. The second term on the right-hand side is the boundary term27,28. Here we adopt the notations and sign conventions in Ref.29.

We take Euclidean O(4) symmetry for both Φ and the spacetime metric g_µν. Then, the Euclidean ﬁeld equations for Φ andρhave the form

Φ+3ρ ρ Φ ₌dU dΦ and ρ ₌₋κ 3ρ(Φ 2₊_U₎_, ₍₁₉₎

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where the role of the friction term can be changed into the accelerating term when the local maximum value of the potential is positive19. The Hamiltonian constraint is given by ρ2−1−κρ 2 3 1 2Φ 2₋_U_{= 0}_, ₍₂₀₎

where the prime denotes the derivative with respect toη. The boundary conditions for above equations are given by

lim η→ηmax Φ(η)Φ_{f /t}, dΦ dη η=0 = 0, ρ|_η₌₀= 0, and ˙ρ|_η₌₀= ˙ρ_o, (21)

where Φ_f is for a true vacuum bubble nucleation and Φ_tis for a false one. Φ(η_max) is exponentially approaching to but not reaching Φ_{f /t}if the initial space is de Sitter. In the recent work14, we have studied the possible types of vacuum bubbles and calculated the radius and the nucleation rate. We have obtained some numerical solutions as well as analytic computation using the thin-wall approximation. We consider the onlyU_f >0 so that the exterior geometry of the bubble will remain to be de Sitter (dS) space. The true vacuum bubble can be classified according to the interior geometry and the size of the bubble. The interior geometry of a true vacuum bubble can be different depending on U_t = 0(flat), U_t < 0(AdS), andU_t>0(dS). There are nine types of true vacuum bubbles, three false vacuum bubbles and Hawking-Moss transition30 in which the thin-wall approximation is not considered. We confirmed that Parkes formula7 is applicable to all types of true vacuum bubbles even thoughB_in is different. In addition, we have obtained the single formula that applies also to all types of false vacuum bubbles. There are some conditions for classifying the bubbles. The conditions areU_f−U_t>3κS_o2/4 for small bubbles,U_f−U_t= 3κS_o2/4 for half bubbles, andU_f−U_t<3κS_o2/4 for large bubbles. The coefficientB and the radius of the former is continuously connected to those of the latter.

5. Instaton solutions mediating tunneling between the degenerate vacua in curved space

Now we go back to the case of instaton solutions with O(4) symmetry in curved space19.

We have to impose the boundary conditions to solve Eqs. (19) and (20). In this work, we have to consider 4 conditions for two equations of 2nd order. In the case of Euclidean dS background with compact geometry, the following choice of boundary conditions is also possible: Φ= 0 andρ= 0 atη= 0 andη_max= 018. This boundary condition hasZ₂symmetry under the exchange of two points corresponding toη= 0 andη_max= 0. In general, for a potential with degenerate vacua, we can choose the

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Fig. 3. Each ﬁgure represents the solution ofρin dS, ﬂat, and AdS space.

boundary condition withZ₂ symmetry as follows

dΦ dη η=0 = 0, dΦ dη η=max

= 0, ρ|_η₌₀= 0, andρ|_η₌_η_max= ˜ρ(η_max) = 0, (22) where η_max will have a ﬁnite value in the cases we will consider. We adopt this boundary condition for our instanton solutions. We also choose ˜ρ(η_max) = 0 for both ﬂat and AdS space to imposeZ₂ symmetry. This allow new solutions withZ₂

symmetry even both flat and AdS space. Due toZ₂ symmetry about the wall, the inside geometry will be identical to the outside one. As a result, the whole geometry is finite. This is very different from the known solutions with infinite geometry5,6,7. We explore in more detail the possibility for the existence of the instanton solu-tion. To understand solutions qualitatively, we rearrange the terms in Eq. (19) after multiplying by d_dηΦ ηmax 0 d 1 2Φ 2₋_U₌₋ ηmax 0 dη3ρ ρ Φ 2_. ₍₂₃₎

The quantity in the square brackets here can be interpreted as the total energy of the particle with the potential energy−U. The term on the right hand side can be considered as the dissipation rate of the total energy as long asρ >0. However, the role of the term can be changed from damping to acceleration ifρ changes the sign during the transition. For the tunneling between the degenerate vacua the region forρ >0 and ρ <0 will be equally divided due to the Z₂ symmetry, hence the term on the right hand side will be vanished. Thus, the total energy at both ends,

η= 0 andη=η_max, is conserved as we can see from Eq. (23). The condition for the change of the sign depends on the maximum value of the potential. To allow the change during the transition, the local maximum value of the potentialU(0) must be positive. This can be seen from Eq. (20). That is to say,U(0) = _κρ32+1₂Φ|2_(Φ=0)>0.

In other words, the sign of the second term of the ﬁrst equation in Eq. (19) can be changed during the transition for−λa4/8< U_o.

The numerical solutions for the equations ofρare illustrated in Fig.3. We take the dimensionless variables ˆU_o = 0.01 and ˆκ 0.0474 in dS space, ˆU_o = 0 and ˆ

κ0.1805 in ﬂat space, and ˆU_o=−0.01 and ˆκ0.2392 in AdS space.

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6. Summary and Discussions

In this work, we have studied bounce and instanton solutions withO(4) symmetry in curved space. we have obtained numerical solutions as well as performed the analytic computations using the thin-wall approximation. We studied the possible types of vacuum bubbles in dS background. There are nine types of true vacuum bubbles and three false vacuum bubbles. The instanton solutions withO(4) symmetry are possible only if gravity is taken into account. We explored the condition for the existence of the instaton solution quantitatively.

We propose the solutions with exactZ₂ symmetry can represent the nucleation of the braneworld-like object if the mechanism is applied in higher-dimensional theory. The braneworld having the ﬁnite size with the exactZ₂ symmetry can be nucleated not only in dS but also in ﬂat and AdS bulk spacetime, and then expand, seen from observer’s point of view on the wall, without eating up bulk(inside and outside) spacetime.

Acknowledgments

This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021.

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