Bubble Collision in Curved Spacetime

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Nuclear Physics B (Proc. Suppl.) 246–247 (2014) 196–202 www.elsevier.com/locate/npbps

Bubble Collision in Curved Spacetime Dong-il Hwanga , Bum-Hoon Leea,b , Wonwoo Leea , Dong-han Yeoma b Department

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

Abstract We study vacuum bubble collisions in curved spacetime, in which vacuum bubbles were nucleated in the initial metastable vacuum state by quantum tunneling. The bubbles materialize randomly at different times and then start to grow. It is known that the percolation by true vacuum bubbles is not possible due to the exponential expansion of the space among the bubbles. In this paper, we consider two bubbles of the same size with a preferred axis and assume that two bubbles form very near each other to collide. The two bubbles have the same field value. When the bubbles collide, the collided region oscillates back-and-forth and then the collided region eventually decays and disappears. We discuss radiation and gravitational wave resulting from the collision of two bubbles. Keywords: Vacuum Bubble, Collision, Curved Spacetime

1. Introduction The expanding bubble as our expanding universe was first introduced in Ref. [1], in which the bubbles were defined regions between which no causal influence can ever pass, i.e. they are causally disconnected. One in a bubble can not see any other bubbles unless they collide. The very first picture of an inflationary multiverse scenario [2] was proposed to get our universe without the cosmological singularity problem [3] using an interesting feature of self-reproducting or regenerating exponential expansion of the universe. However, the inflationary spacetimes have a fatal flaw, the spacetime cannot be made complete in the past direction [4], even though the universe is eternal into the future. Thus if we want to understand the origin of our universe we should make a mechanism for the universe with completion in the past direction and with a configuration of low entropy or the universe described as quantum tunneling from nothing [5, 6]. One can also consider a Email addresses: [email protected] (Dong-il Hwang), [email protected] (Bum-Hoon Lee), [email protected] (Wonwoo Lee), [email protected] (Dong-han Yeom)

0920-5632/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.nuclphysbps.2013.10.086

self-creating universe coming from the spacetime with a closed time loop [7]. The eternal inflationary scenario [8, 9, 10, 11] and the string theory landscape scenario [12, 13] were also combined with the multiverse scenario. The eternal inflationary scenario is related to the expanding sea of false vacuum with a positive cosmological constant. The region with the false vacuum state is continuously expanding, even the inflation is ended in one region. This means that the universe can be eternal into the future. The landscape has a huge number of stable or metastable vacua [14] originated from different choices of Calabi-Yau manifolds and generalized magnetic fluxes, in which each local minimum may correspond to the vacuum of a possible stable or metastable universe with different laws of low energy physics. All these above independent scenarios seem to succeed to our ambition of understanding the origin of our universe described in Ref. [1]. The nucleation process of a vacuum bubble has been studied for a long time. The process was first investigated in Ref. [15], developed in both flat [16] and curved spacetime [17, 18]. Other type of transition describing the scalar field jumping simultaneously onto the top of

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the potential barrier was investigated by Hawking and Moss [19] and later in Ref. [20]. As a special case of the true vacuum bubble, a vacuum bubble with a finitesized background after nucleation was studied in Ref. [21]. The decay of false monopoles with a gauge group was also studied using the thin-wall approximation [22]. The bubble or brane resulting from flux tunneling was studied in a six-dimensional Einstein-Maxwell theory [23]. The mechanism for nucleation of a false vacuum bubble in a true vacuum background has also been studied within various contexts. Nucleation of a large false vacuum bubble in dS space was obtained in Ref. [24] and nucleation with a global monopole in Ref. [25]. The mechanism for nucleation of a small false vacuum bubble was obtained in the Einstein gravity with a nonminimally coupled scalar field [26] with the correction of an error term [27], with Gauss-Bonnet term in Ref. [28], and using Brans-Dicke type theory [29]. The classification of vacuum bubbles including false vacuum bubbles in the dS background in the Einstein gravity was obtained in Ref. [30], in which the transition rate and the size of the instanton solution were evaluated in the space, as the limiting case of large true vacuum bubble or large false vacuum bubble. The collision of two vacuum bubbles in flat spacetime was discussed using analytic methods or numerical methods in Ref. [31]. One of the interesting points of colliding bubbles is that the colliding bubble may induce a vacuum transition [32] or a production of particles [33] using the colliding energy. Second, including gravitation, it is highly non-trivial to study the dynamics. We may use the thin-wall approximation [34][35]. In the study of Freivogel, Horowitz, and Shenker [36] (for more advanced review, see [37]), they discussed two colliding true vacuum bubbles in the de Sitter background: one is flat and the other is anti de Sitter. The analysis in itself is very concise and important, but this may not be able to describe the dynamics of fields on the colliding walls. For example, this cannot describe the vacuum transition behaviors. Hence, we may need further numerical studies. There were some numerical studies of bubble collisions with gravitation. Very recently, Johnson, Peiris, and Lehner [38] succeeded in studying bubble collisions with gravitation beyond the thin-wall approximation. They assumed hyperbolic symmetry from the Birkhoff-like theorem of colliding bubbles and assumed initial data from the Coleman-DeLuccia type solutions. They could solve Einstein and field equations numerically and observe and report on symmetric/asymmetric bubble collisions and vacuum transitions. Soon after,

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the authors could do the numerical calculations of bubble collisions with gravity [39], by using double-null formalism [40], where we already applied this method to study general bubble dynamics [41] including semiclassical effects [42]. When the bubbles collide, the collided region oscillates back-and-forth and then the collided region eventually decays and disappears. We expect that radiations and gravitational waves are produced from the collision of two bubbles. The amount of gravitational radiation resulting from the collision of two true vacuum bubbles was numerically computed in Ref. [43] and the envelope approximation to calculate the gravitational radiation was introduced in Ref. [44]. The outline of this paper is as follows: In Sec. 2 we review the nucleation process of a vacuum bubble and the evolution after its materialization in flat Minkowski spacetime. In Sec. 3 we review the collision of two bubbles in the absence of gravity. In Sec. 4 we study bubble collision in the presence of gravity. We summarize and discuss our results in Sec. 5. 2. The Nucleation of a Vacuum Bubble and Evolution in Flat Minkowski Spacetime In this work, we study the system with a scalar field governed by an asymmetric double-well potential. Thus the potential has two non-degenerate minima with lower minima at ΦT , a true vacuum state, and higher minima at ΦF , a false vacuum state. We assume that a system is initially in the false vacuum state Φ = ΦF , i.e. the system has the energy density of the homogeneous and static scalar field everywhere in space. However, the field can inhomogeneously tunnel via the barrier separating the two vacua. That means the initial state is an unstable state, i.e. one whose energy has an imaginary part. The decay rate of the background vacuum state per unit volume and unit time has the form Γ/V  Ae−B . The prefactor A is evaluated from the Gaussian integral over fluctuations around the background classical solution and the leading semiclassical exponent B = S cs − S bg is the difference between the Euclidean action corresponding to the classical solution S cs and the background action S bg . We take Euclidean O(4) symmetry(η2 = τ2 + r2 ) for both Φ and the spacetime metric gμν . The solution with the minimum Euclidean action is assumed to have the highest symmetry [45]. For the flat space the metric with O(4) symmetry has the form ds2 = dη2 + η2 [dχ2 + sin2 χ(dθ2 + sin2 θdφ2 )], (1)

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3. Bubble Collision in the Absence of Gravity

t

If we consider two bubbles of the same size with a preferred axis (say x axis) then the geometry is reduced to O(2, 1) symmetry by the existence of the axis. The situation is therefore invariant under an O(2, 1) group consisting of Lorentz boosts in the y and z directions and spatial rotations in the yz plane. We consider the simple metric having the form

ΦT ΦF

ΦF η

η

ds2 = −dt2 + dx2 + dy2 + dz2 .

r

(6)

One can introduce new coordinates ξ, ψ, φ [31] by t = ξ cosh ψ, y = ξ sinh ψ sin φ, z = ξ sinh ψ cos φ,(7)

Figure 1: The figure represents the growth of a vacuum bubble after its materialization.

and the field Φ depends only on the distance η from a certain point in Euclidean space. When the difference between the two vacuum state, , is sufficiently small in comparison with all other parameters of the model, the so-called thin-wall approximation becomes valid. If  is small, the starting point should be very close to ΦT and the particle stays there for a long time with a very small velocity and acceleration so that η grows large with Φ staying near ΦT . As η becomes large, the friction force becomes negligible and Φ quickly goes to ΦF and stays at that point from thereafter. One can calculate the action for the bounce solution to the first order in . The Lorentzian solutions are obtained by applying the analytic continuation π χ → iχ + , 2

(2)

to the Euclidean solution. Then the metric turn out to be ds2 = dη2 +η2 [−dχ2 +cosh2 χ(dθ2 +sin2 θdφ2 )].(3) This metric represents the spherical Rindler-type [46]. The spacetime including a single bubble has O(3, 1) symmetry in Lorentzian. Figure 1 represents the growth of a vacuum bubble after its materialization. The location η¯ means the initial bubble radius. The coordinate transformation r = η cosh χ and t = η sinh χ,

(4)

changes the metric given in Eq. (3) to ds2 = −dt2 + dr2 + r2 dθ2 + r2 sin2 θdφ2 ,

(5)

which obviously reveals that it is indeed the Minkowski metric.

so that ξ2 = t2 − y2 − z2 . In these coordinates the flat space metric takes the form ds2 = −dξ2 + dx2 + ξ2 dH 2 ,

(8)

where dH 2 = dψ2 + sinh2 ψ dφ2 . This spacetime is called hyperbolically symmetric because the orbits of their symmetry group have several properties in common with two-sheeted hyperboloids. The solution representing the colliding bubbles will be independent of the coordinates ψ and φ. By the reflection symmetry, the right-hand wall of the left-hand bubble will be accelerated towards the right in a similar way. Figure 2 represents the growth of two vacuum bubbles of the same size with a preferred axis. They will collide at x = 0 when ξ = ξ1 . If the separation of the bubbles, 2b, is large compared to their radius, the walls will be highly relativistic when they collide. One wall moves to the right and the other to the left away from x = 0, ξ = ξ1 and come back to x = 0 due to the pressure difference. There is a series of oscillations with shorter and shorter periods. The collided region eventually decays and disappears. Thus the kinetic energy of the bubble walls or the collided region with the higher vacuum energy is dissipated and the true vacuum region is reheated to almost the temperature it had before the phase transition [31]. 4. Bubble Collision in the Presence of Gravity We consider the action    √ R 1 α − ∇ Φ∇α Φ − U(Φ) −gd4 x S = 2κ 2 M √ K − Ko , (9) −hd3 x + κ ∂M where κ ≡ 8πG, g = det gμν , h is the induced boundary metric, K and Ko are the traces of the extrinsic curvatures of ∂M for the metric gμν and ημν , respectively. The

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moving and right-moving null directions ξ,u > 0 and ξ,v > 0. Thus, we can show  4ξ,u ξ,v 2m κUξ2 . (16) + = 1+ ξ 3 α2

ξ ΦT

ΦT

ΦF

ΦF

ΦF

b

-b

x

Figure 2: The figure represents the growth of two vacuum bubbles of the same size with a preferred axis.

second term on the right-hand side of the above equation is the boundary term [47, 48]. It is necessary to have a well-posed variational problem including the EinsteinHilbert term. Einstein equations and scalar field equations are as follows: Gμν

Φ = κT μν

(10)

Φ T μν

= ∇μ Φ∇ν Φ 1 − gμν [ gρσ ∇ρ Φ∇σ Φ − U(Φ)] 2 dU Φ = dΦ

dξ2



+

 2m κUξ2 1+ + dx2 + ξ2 dH 2 , (13) ξ 3

2m ξ

+

κUξ2 3



where m is a continuous parameter permitted to be either positive or negative rather than the AMD mass. This spacetime has a curvature singularity at ξ = 0 when m  0. These are timelike for m < 0 and spacelike for m > 0. The spacetime is not globally asymptotically flat. We define coordinate transformation: dξ = ξ,u du + ξ,v dv, dx =

 α2 dv du − + , 4 ξ,u ξ,v

and use conventions [39] α,v α,u , d≡ , h ≡ α α W ≡ S ,u , Z ≡ S ,v .

(14) (15)

and obtain the double-null metric ds2 = −α2 (u, v)dudv+ ξ2 (u, v)dH 2 . Note that we should choose both left-

f ≡ ξ,u ,

g ≡ ξ,v , (19)

Then Energy-momentum tensor components are φ T uu

=

φ T uv

=

φ T vv

=

φ T χχ

=

(12)

−

1+

Note that usual black hole type solutions in Minkowski vacuum happen for m < 0 limit, and hence, in this work, we are interested in the case m < 0. For convenience, we define √ 4πΦ ≡ S (18)

(11)

The form of Eq. (8) representing the hyperbolic symmetry in the presence of gravity takes the form ds2 =

Therefore, we can identify that the mass function in the double-null coordinate by  4ξ,u ξ,v κUξ2 ξ + . (17) m(u, v) = − 1 − 2 3 α2

1 2 W , 4π α2 U(S ), 2 1 2 Z , 4π ξ2 WZ − ξ2 U(S ). 2πα2

(20) (21) (22) (23)

Therefore, simulation equations can be written as follows: κ φ , (24) f,u = 2 f h − ξT uu 2 κ φ , (25) g,v = 2gd − ξT vv 2 α2 fg κ φ f,v = g,u = + − (26) + ξT uv , 4ξ r 2 f,v κα2 φ h,v = d,u = − 2 T χχ (27) − . ξ 4ξ In addition, we include the scalar field equation: Z,u = W,v = −

f Z gW − − πα2 U  (S ). ξ ξ

(28)

We need initial conditions for all functions (α, h, d, ξ, f, g, S , W, Z) on the initial u = ui and v = vi surfaces, where we set ui = vi = 0. We have gauge freedom to choose the initial r function. Although all constant u and v lines are null, there remains

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freedom to choose the distances between these null lines. Here, we choose ξ(0, 0) = ξ0 , f (u, 0) = ξu0 , and g(0, v) = ξv0 , where ξu0 > 0 and ξv0 > 0 such that the function ξ increases for both of in-going and out-going observers. It is convenient to choose ξu0 = 1/2 and ξv0 = 1/2 and we can choose that the mass function on ui = vi = 0 arbitrarily. Hence, to specify a pure de Sitter background, for given ξ(0, 0) = ξ0 , S (0, 0) = S f , and if the field is at the local minimum, then ⎞−1/2 ⎛ κU(S f )ξ02 2m0 ⎟⎟⎟ ⎜⎜ ⎟⎠ + (29) α(0, 0) = ⎜⎜⎝1 + 3 ξ0 with a free parameter m0 . In this paper, we choose ξ0 = −m0 = 100 for convenience. Now it is possible to assign all of initial conditions along initial u = ui and v = vi surfaces. Initial v = vi surface: We choose ⎧ ⎪ S f u < ushell , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ |S t − S f | G(u) + S t S (u, 0) = ⎪ (30) ⎪ ⎪ ushell ≤ u < ushell + Δu, ⎪ ⎪ ⎪ ⎩ S t ushell + Δu ≤ u, where G(u) is a pasting function which goes from 1 to 0 by a smooth way. We choose G(u) by   π(u − ushell ) . (31) G(u) = 1 − sin2 2Δu Then, we know W(u, 0) = S ,u (u, 0). h(u, 0) is given from Equation (24), since f,u = 0 along the in-going null surface. Then, using h(u, 0), we obtain α(u, 0). We need more information to determine d, g, and Z on the v = 0 surface. We obtain d from Equation (27), g from Equation (26), and Z from Equation (28). Initial u = ui surface: We choose ⎧ ⎪ S f v < vshell , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ |S f − S t | G(v) − S t S (0, v) = ⎪ (32) ⎪ ⎪ vshell ≤ v < vshell + Δv, ⎪ ⎪ ⎪ ⎩ S t vshell + Δv ≤ v, where G(v) = 1 − sin2



potential U1 in Ref. [39] and vary the field value of the false vacuum S f = 0.1, 0.2, 0.3, 0.4. We used the second order Runge-Kutta method. Figure 3 shows the results when ξ0 = −m0 = 100 and S f = 0.1, 0.2, 0.3, 0.4 cases. Here, we follow the conventions of [39]. Figure 4 shows Φ/Φmax along the diagonal line of Figure 3, where we choose Φmax so that Φ/Φmax = 1 for the initial point. x-axis is number of printed steps (proportional to the time parameter). Thick curve is for S f = 0.1, dashed curve is for S f = 0.2, dotted curve is for S f = 0.3, and dashed-dot curve is for S f = 0.4. Figure 5 shows T uu component for S f = 0.1 (thick curve) and S f = 0.2 (dashed curve) along a diagonal line. After the bubble collides, there appear periodically damping peaks. This can give some intuitions for the effects to the gravitational radiation via bubble collisions. 5. Summary and Discussions



π(v − vshell ) . 2Δv

Figure 3: Field dynamics during the bubble collision. Yellow region is the false vacuum region and skyblue region is the true vacuum region.

(33)

We obtain d(0, v) from Equation (25), since g,v (0, v) = 0. By integrating d along v, we have α(0, v). We need more information for h, f, and W on the u = 0 surface. We obtain h from Equation (27), f from Equation (26), and W from Equation (28). This finishes the assignments of the initial conditions. For convenience, we fix Δu = Δv = 20 and ushell = vshell = 30 and S t = 0. We follow the convention of the

In this work, we have studied vacuum bubble collisions in curved spacetime by numerical calculations. We used the double-null formalism to implement numerical simulations. We used the hyperbolic symmetry to describe the collisions. When the bubbles collide, the collided region oscillates back-and-forth and then the collided region eventually decays and disappears. Thus we can expect that the collided region with the higher vacuum energy decays into radiations and gravitational waves and the true vacuum region is filled with those. In

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Figure 4: The field value along the diagonal line. Figure 5: The energy-momentum tensor along the diagonal line is damping and oscillating.

this work, we observed more general initial conditions than Coleman-DeLuccia type bubbles. This was easily done, because we used the double-null formalism. One interesting observation is that as tension increases walls slowly moves and they may not be collided. The smooth transitions of metric functions (especially ξ) are realized. This cannot be done by the thin-wall approximation. We obtained future boundary formation from smooth initial data without assuming thin-wall and classified various causal structures. There can be further applications including gravitational wave production and there will be many other models and issues for bubble collisions. We leave these for future work.

Acknowledgements

We would like to thank Dong-Hoon Kim for useful comments. We also would like to thank Pauchy WY. Hwang, Sang Pyo Kim, Yu-Hsiang Lin, and Lisa Lin for their hospitality at the 9th International Symposium on Cosmology and Particle Astrophysics in Taiwan, 13-17 Nov 2012. 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. WL was upported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2012R1A1A2043908).

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