1 Introduction

CQUeST-2007-0145

Dynamics of false vacuum bubbles with nonminimal coupling

1   2   3   4   5

Department of Physics, Sogang University, Seoul 121-742, Korea

§Research Institute for Basic Science, Sogang University, Seoul 121-742, Korea

Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea

Department of Physics, and BK21 Division of Advanced Research and Education in Physics, Hanyang University, Seoul 133-791, Korea

Abstract

We study the dynamics of false vacuum bubbles. A nonminimally coupled scalar field gives rise to the effect of negative tension. The mass of a false vacuum bubble from outside observer’s point of view can be positive, zero, or negative. The interior false vacuum has de Sitter geometry while the exterior true vacuum background can have geometry depending on the vacuum energy. We show that there exist expanding false vacuum bubbles without the initial singularity in the past.

PACS numbers: 11.15.Kc, 98.80.Cq, 98.80.Es, 98.80.-k

1 Introduction

Can a false vacuum bubble expand within the true vacuum background? Or, is an expanding false vacuum bubble always inside the horizon of a black hole from outside observer’s point of view? If there was a dynamical spacetime foam structure in the very early universe [1], the detail structure and evolution will depend on the cosmological constant. In the context of a bubble nucleation and dynamics, this phenomenon may be described as follows. A true vacuum bubble can always be nucleated somewhere within the false vacuum background as well as a false vacuum bubble nucleated within the true vacuum bubble background. Some bubbles may expand while some bubbles collapse. Some of them may be connected by wormholes. Then the whole spacetime may have the complicated vacuum or spacetime structure due to the above processes. Earlier works [2, 3, 4, 5] show that the unbound solution representing expanding false vacuum bubble does not exist because the solution for the junction equation can not cover all ranges of . To obtain an expanding false vacuum bubble, the mass of the bubble should be over some critical value. To the observer in the exterior spacetime the expanding false vacuum bubble will be inside the black hole horizon. Only to the observer inside bubble will it appear to expand from a very small size to infinity. However, these bubbles start from an initial singularity. Moreover, there is a puzzle that the entropy of the expanding false vacuum region is greater than the entropy of the black hole surrounding it [5, 6, 7]. Are these descriptions always true? With nonminimal coupling we will show that the unbound solutions representing expanding false vacuum bubbles can exist. On the other hand the idea of the string theory landscape has a vast number of metastable vacua [8]. One of the intriguing features of this landscape is to understand de Sitter universe or tunneling processes in the landscape [9]. Our motivation is an attempt to solve some of these questions within the framework of classical theory of gravity.

The dynamics of the boundary wall of a spherically shaped false vacuum bubble surrounded by true vacuum regions was originally studied in Refs. [10] at the final stage of the true vacuum bubble nucleation in old inflation [11], and was studied systematically in Refs. [2, 3] as an attempt to create a universe in the laboratory by quantum tunneling. They considered the case of interior de Sitter spacetime and exterior Schwarzschild spacetime divided by thin-wall (or domain wall). In Ref. [12] the dynamics of matter distribution that may contaminate a false vacuum bubble was considered. The case of interior de Sitter and exterior Schwarzschild de Sitter spacetime was studied in Ref. [4], where they examined the instability of false vacuum bubbles. The case of interior de Sitter spacetime and exterior Schwarzschild anti-de Sitter spacetime in relation to AdS/CFT correspondence was considered in Ref. [5]. The case of charged false vacuum bubble with interior de Sitter spacetime and exterior Reissner-Nordstrm anti-de Sitter spacetime with arbitrary dimension was considered in Ref. [13]. The possibility of creation of a universe out of a monopole in the laboratory was investigated in Ref. [14], where they have considered the classical and quantum thin-wall dynamics of a magnetic monopole. They have examined the stability of a spherically symmetric self-gravitating magnetic monopole in the thin-wall approximation modeling the interior false vacuum as de Sitter spacetime and the exterior as the Reissner-Nordstrm spacetime as in Ref. [15]. Recently the classification scheme for possible evolution of a vacuum wall in the Schwarzschild-de Sitter geometry was constructed [16]. In addition, there have been studies on the attempts to create a universe in the laboratory [17].

As for the false vacuum bubble formation, Lee and Weinberg [18] have shown that gravitational effects make it possible for bubbles of a higher-energy false vacuum to nucleate if the vacuum energies are greater than zero. The oscillating bounce solutions, another type of Euclidean solutions, have been studied in Refs. [6, 19, 20]. On the other hand, Kim et al[21] have shown that there exists another decay channel which is described by the false vacuum region of the global monopole formed at the center of a bubble in the high temperature limit. The Hawking-Moss transition [22], as another way of vacuum decay, describes the scalar field jumping simultaneously at the top of the potential barrier. Recently this process has been interpreted in terms of a thermal transition [19, 23]. It has been shown that the false vacuum bubble can be nucleated within the true vacuum background due to a nonminimally coupled scalar field or other similar coupling terms [24]. The quantum nucleation of the vacuum bubble was also studied [25]. In Ref. [26], they found an analytic expression for the tunnelling amplitude and studied the tunnelling between arbitrary (anti-)de Sitter spacetimes in arbitrary spacetime dimensions. The interesting models for bubble collisions in the very early universe are also discussed [27].

In this paper we use the metric junction conditions, which were developed in Ref. [28], to analyze the dynamics of a false vacuum bubble in the theory with nonminimal coupling. There are two types of boundary related to this formalism. One is a “boundary surface” which is a surface that has the stress-energy tensor [29]. The process of star collapsing is a well-known example involving a boundary surface [30]. The other is a “surface layer” which is a thin layer of matter where [31]. In this case is related to the discontinuity of the extrinsic curvature of the surface. In this framework we classify 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 [32]. In the context of the brane cosmology, after Randall and Sundrum’s interesting proposal [33], the junction conditions have become one of methods describing the inflationary cosmology on the brane [34].

Our approach to obtain junction conditions is based on the method of variational principle by Chamblin and Reall [35]. C. Barcelo and M. Visser have obtained the generalized junction conditions using a different approach [36].

The plan of this paper is as follows. In Sec. 2 we present the formalism for the junction conditions in the Einstein theory of gravity with a nonminimally coupled scalar field. In Sec. 3 we study the dynamics of false vacuum bubbles using the junction conditions. In Sec. 4 the bubble wall trajectories in the exterior bulk spacetime are analyzed according to the mass of the false vacuum bubble. Finally, we summarize and discuss our results in Sec. 5.

2 The junction conditions with a nonminimally coupled scalar field

In this section we consider a thin-wall as a surface layer partitioning bulk spacetime into two distinct four-dimensional manifolds and with boundaries and , respectively. To obtain the single glued manifold we demand that the boundaries are identified as follows:

(1)

where the thin-wall boundary is a timelike hypersurface with unit normal .

Let us consider the action

(2)

where is a Nambu-Goto type action on the wall given by , , and . The second term on the right-hand side of the above equation is the boundary term [37] with a nonminimally coupled scalar field. is the scalar field potential, denotes the Ricci curvature of spacetime in , is the trace of the extrinsic curvature of , the term describes the nonminimal coupling of the field to the Ricci curvature and is a dimensionless coupling constant.

We now vary the action to obtain Israel junction conditions. The case of minimal coupling has been considered in Ref. [35].

Varying the nonminimal coupling term in the bulk we get

(3)

and varying the scalar field action we get

(4)

The variation of the boundary term with a nonminimal coupling gives

(5)

and the variation of the wall action gives

(6)

The bulk Einstein equations are

(7)

where is the Ricci tensor and is the matter energy momentum tensor,

(8)

The corresponding scalar field equation on the bulk is written by

(9)

with a boundary condition at the thin-wall

(10)

Here we adopt the notations and sign conventions of Misner, Thorne, and Wheeler [38].

The modified Lanczos equation due to a nonminimally coupled scalar field is given by

(11)

where

(12)

The sign arises because we have chosen the convention that points towards the region of increasing . The is the location of the hypersurface. The signs (+) and (-) represent exterior and interior spacetime, respectively.

After Eq. (10) is substituted in Eq. (11) the junction conditions become

(13)

Actually, the second term on the right-hand side of Eq. (13) vanishes because and vanish in the exterier and in the interior spacetime of the wall, respectively.

In order to find the gravitational field and the motion of a wall we must first find two sets, both inside and outside of the wall, of solutions of the bulk Einstein equation and scalar field equation. So if the bulk solutions are given we only need to match the junction conditions to determine the motion of the wall. In the next section, we will only consider the junction equations because bulk solutions are easily given; the bulk solution of is already known in Ref. [24], while the case for corresponds to Schwarzschild solution because of Birkhoff’s theorem [39]. For the case of , the mass of a false vacuum becomes effectively negative, which is possible due to nonminimal coupling.

3 Dynamics of a false vacuum bubble

For applications of the modified junction equations on the false vacuum bubble, the bulk spacetime geometry for the inside and outside of the wall have a spherically symmetric spacetime

(14)

where

(15)

and is the mass or the total energy of a false vacuum bubble. The constant is related to a cosmological constant; for de Sitter spacetime , for Minkowski spacetime, and for anti-de Sitter spacetime. Since we consider a false vacuum bubble, .

We take the energy-momentum tensor as the form

(16)

where and is the positive surface energy density, or surface tension, of the wall without nonminimal coupling. For bubble walls is a constant having the same value at all events on the timelike surface [2, 40]. Note that the stress-energy tensor of the surface can be defined as the integral over the thickness, , of the surface in the limit as goes to zero

(17)

Then, because the internal structure of the wall is neglected in thin-wall limit and .

To keep the analysis as simple as possible, we take the position of a false vacuum in the potential as zero, that is (see Ref. [24]). In this case Eq. (13) becomes

(18)

By spherical symmetry, the extrinsic curvature has only two components, and . The junction equation is related to and the covariant acceleration in the normal direction is related to .

We introduce the Gaussian normal coordinate system near the wall

(19)

where and . It must agree with the coordinate of the interior and exterior coordinate systems. The angle variables can be taken to be invariant in all regions. In this coordinate system the induced metric on the wall is given by

(20)

where is the proper time measured by an observer at rest with respect to the wall and is the proper radius of . The following relation is satisfied

(21)

In these treatments, the condition becomes

(22)

or more generally

(23)

where . Hereafter denotes . We are using the dot notation to refer to derivatives with respect to . are if the outward normal to the wall is pointing towards increasing and if towards decreasing [13, 41]. There are parameter regions where both and are positive in all ranges of . This situation is similar to the case of the evolution of a true vacuum bubble. In earlier works, a sign change of was needed to cover all ranges of for the solution. This is because the positive signs for covered only partial ranges of ; thus the interesting unbound solutions were excluded. To obtain an expanding false vacuum bubble, the mass of the bubble should be greater than some critical value. These bubbles start from an initial singularity. The second term on the right-hand side of Eq. (23) can be interpreted as the negative tension of the wall due to a nonminimal coupling term.

After squaring twice, the Eq. (22) can be written in the form

(24)

where the effective potential is

(25)

with

(26)

Eq. (24) formally coincides with the equation describing one-dimensional motion of a unit-mass particle moving in the corresponding potential with zero total energy. The properties of the trajectory of the wall can be read off directly from the shape of . In the next section we will discuss the details of trajectories of the wall.

4 The bubble wall trajectories

In this section we consider the bubble wall trajectories according to the mass of a false vacuum bubble in the exterior bulk spacetime. The modified junction equations with nonminimal coupling determine the trajectories. From the shape of we can obtain the behavior of solutions without solving the equations exactly. We consider only bubble solutions without black holes. In other words, we consider the size of a false vacuum bubble larger than the black hole horizon.

4.1 The case of

Figure 1: The effective potential for various in the case of . The time evolution of the wall can be interpreted as the motion of a particle coming from infinity, reflecting at the barrier, and then going back to infinity. The three curves are (i) dotted curve: ; (ii) dashed curve: ; (iii) solid curve: in (a), (b) and (c). Figure (d) indicates the potential with different background at . There exist only ”unbound” solutions.
Figure 2: The effective potential for various in the case of . The three curves are (i) dotted curve: ; (ii) dashed curve: ; (iii) solid curve: in (a), (b) and (c). There exist only ”unbound” solutions.

This case is related to the results in Ref. [24]. If a false vacuum bubble can be nucleated within the true vacuum background without changing the exterior spacetime, the surface density becomes negative and/or junction conditions itself need to be modified. Our results show how the surface tension as well as junction conditions are modified by nonminimal coupling. From now on we scale the dimension of a vacuum energy density to , that of to , that of to , that of to , and that of to , to make all terms in Eq. (24) dimensionless. For the sake of simplicity we take , , and . In this case the effective potential function is at and there exist only ”unbound” solutions. We consider three particular cases: (case 1) the interior as well as the exterior spacetime is de Sitter; (case 2) the interior false vacuum as de Sitter and the exterior as flat Minkowski spacetime; (case 3) the interior false vacuum as de Sitter and the exterior as anti-de Sitter spacetime. The shapes of the effective potential as a function of are shown in Fig. 1. These figures indicate only unbound trajectories are possible. That is, the bound and monotonic trajectories do not appear as classical solutions.

We see that the allowed minimum size of a false vacuum bubble is diminished as is decreased. So if the radius of a nucleated false vacuum bubble is greater than the allowed minimum size then the false vacuum bubble can expand within the true vacuum background. These expanding bubbles have no initial singularity, as we can see from the Fig. 1. So it is possible to create a universe by an expanding false vacuum bubble nucleated by a proper mechanism.

4.2 The case of

Figure 3: The effective potential for various in the case of . The three curves are (i) dotted curve: ; (ii) dashed curve: ; (iii) solid curve: in (a), (b) and (c). There exist ”unbound” solutions as well as ”bound” solutions.

In this case the mass of the false vacuum bubble is taken as a constant parameter. The portion of the potential is inherently restricted since the allowed region of from Eq. (25) is given by

(27)

In the case , we consider three particular cases: (case 1) the interior spacetime as de Sitter and the exterior as Schwarzschild de Sitter; (case 2) the interior as de Sitter and the exterior as Schwarzschild spacetime; (case 3) the interior as de Sitter and the exterior as Schwarzschild anti-de Sitter spacetime. For these cases massive bubbles can be formed in the early universe. These cases have been studied by many authors in the pure Einstein theory of gravity [2, 3, 4, 5]. However, there are different features between their models and ours. One is related to the sign of . Unlike previous works unbound solutions are allowed in our model since there are parameter regions where both and are positive in all ranges of . Note that there is the restricted region of as in Eq. (27). The other is related to the junction equation Eq. (24). Our approach is somewhat different from their works. We consider the case of positive as well as zero mass. It seems not appropriate that the case of zero mass is applied in their formalism. We can consider the junction equation regardless of the mass. The shapes of the effective potential as a function of are shown in Fig. 2. In these cases, the false vacuum bubble can also expand within the true vacuum background.

4.3 The case of

The case of negative mass bubble is allowed in this framework. In this case we assume the geometry of outside spacetime with spherical symmetry is similarly written by Eq. (14). These objects are considered in different contexts in Refs. [42].

In the case of , we consider three particular cases: (case 1) the interior spacetime as de Sitter and the exterior as Schwarzschild, with negative mass, de Sitter; (case 2) the interior as de Sitter and the exterior as Schwarzschild spacetime; (case 3) the interior as de Sitter and the exterior as Schwarzschild anti-de Sitter spacetime. Although the case of negative mass is physically unclear, since it does not satisfy the positive energy condition, it still gives rise to the solutions of the Einstein equations. So we proceed to analyze the case of . The shapes of the effective potential as a function of are shown in Fig. 3. There also exist expanding false vacuum bubbles.

Figure 4: The magnitude of the negative tension of the wall, , due to a nonminimal coupling term in the spacetime with different mass sign. Here denotes the allowed minimum size of a false vacuum bubble.

Here we discuss again the term which can be interpreted as the negative tension of the wall. This effect is different from earlier works [2, 3, 4, 5]. In Fig. 4, we see that the magnitude of the negative tension of the wall, , due to a nonminimal coupling term is a constant only in the case of . In other cases, the magnitudes approach the value in the case of as is increased. On the other hand the magnitude is increased as is decreased in the case of and decreased as is decreased in the case of . Unlike other cases, there exist bound solutions in narrow range of in the case of although we are not interested in this. In Fig. 4 we take only unbound solutions. denotes the allowed minimum size of a false vacuum bubble in this framework.

5 Summary and Discussions

In this paper we have shown that there can be an expanding false vacuum bubble within the true vacuum background. We have presented the formalism for the junction conditions with nonminimal coupling in section 2.

In section 3, we have studied the dynamics of a false vacuum bubble using the modified junction conditions. The nonminimal coupling term can be interpreted as the negative tension of the wall in the junction conditions. In this treatment the mass of a false vacuum bubble from outside observer’s point of view can be positive, zero, or negative. The solutions in our model does not have black holes. In other words, the size of a false vacuum bubble is larger than the black hole horizon. The mass of a false vacuum bubble has been treated as a parameter in this work.

In section 4, the bubble wall trajectories in the exterior bulk spacetime are analyzed according to the mass of false vacuum bubbles. We have obtained the behavior of solutions in various cases. In this framework there are parameter regions where both and are positive in all ranges of . This situation is similar to the case of the evolution of a true vacuum bubble. In the case of , we have considered three particular cases: (case 1) the interior as well as the exterior spacetime is de Sitter; (case 2) the interior false vacuum as de Sitter and the exterior as flat Minkowski spacetime; (case 3) the interior false vacuum as de Sitter and the exterior as anti-de Sitter spacetime. In these cases only unbound trajectories are possible. In the case , we have considered three particular cases: (case 1) the interior spacetime as de Sitter and the exterior as Schwarzschild de Sitter; (case 2) the interior as de Sitter and the exterior as Schwarzschild spacetime; (case 3) the interior as de Sitter and the exterior as Schwarzschild anti-de Sitter spacetime. In these cases also only unbound trajectories are possible. The portion of the potential is inherently restricted since the allowed region of is given by Eq. (27). For the case of , there also exist expanding false vacuum bubbles. These objects are considered in different contexts in Ref. [42].

In earlier works on the expanding false vacuum bubbles [2, 3, 4, 5], they have the initial singularity in the past. In Ref. [24], it was shown that a false vacuum bubble can be nucleated within the true vacuum background due to a nonminimally coupled scalar field. In order to keep the outside geometry invariant, after a false vacuum bubble is nucleated, the junction conditions need to be modified. In our model the false vacuum bubbles with minimal coupling can expand within the true vacuum background with nonminimal coupling. It will be interesting if this solution can be related to tunnelling from nothing to de Sitter space [43] or related to a kind of eternal inflation [44]. Our model is within a framework of classical theory of gravity. It will be interesting if this framework can be embedded in the superstring theory.

Acknowledgements

We would like to thank Yun Soo Myung, Yongsung Yoon, Hongsu Kim, Sunggeun Lee, John J. Oh, Myungseok Yoon and Changheon Oh for valuable discussions and kind comments. We would like to thank S. Ansoldi and S. Sasaki for their constructive comments and M. Sasaki for his hospitality at the YITP-KIAS joint workshop on String phenomenology and Cosmology, September, 2007. We would like to thank E. J. Weinberg for helpful discussions. This work was supported by the Science Research Center Program of the Korea Science and Engineering Foundation through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number R11 - 2005- 021 and by the Sogang University Research Grant in 2007. WL was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2007-355-C00014).

Footnotes

  1. email:bhl@sogang.ac.kr
  2. email:chulhoon@hanyang.ac.kr
  3. email:warrior@sogang.ac.kr
  4. email:stringphy@gmail.com
  5. email:cyong21@sogang.ac.kr

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