Mar 24, 2015 - a,bâ, Bum-Hoon Leed,eâ , Wonwoo Leedâ¡, Jongmann Yanga,b,cÂ§ ... gLeung Center for Cosmology and Particle Astrophysics, ...... One ...

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Basic Science Research Institute, Ewha Womans University, Seoul 120-750, Republic of Korea

b

Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Republic of Korea c d

Department of Physics, Ewha Womans University, Seoul 120-750, Republic of Korea

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

arXiv:1410.4648v2 [gr-qc] 24 Mar 2015

f

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

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan g

Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan

Cosmic bubbles are nucleated through the quantum tunneling process. After nucleation they would expand and undergo collisions with each other. In this paper, we focus in particular on collisions of two equal-sized bubbles and compute gravitational waves emitted from the collisions. First, we study the mechanism of the collisions by means of a real scalar field and its quartic potential. Then, using this model, we compute gravitational waves from the collisions in a straightforward manner. In the quadrupole approximation, time-domain gravitational waveforms are directly obtained by integrating the energy-momentum tensors over the volume of the wave sources, where the energy-momentum tensors are expressed in terms of the scalar field, the local geometry and the potential. We present gravitational waveforms emitted during (i) the initial-to-intermediate stage of strong collisions and (ii) the final stage of weak collisions: the former is obtained numerically, in full General Relativity and the latter analytically, in the flat spacetime approximation. We gain qualitative insights into the time-domain gravitational waveforms from bubble collisions: during (i), the waveforms show the non-linearity of the collisions, characterized by a modulating frequency and cusp-like bumps, whereas during (ii), the waveforms exhibit the linearity of the collisions, featured by smooth monochromatic oscillations.

∗ † ‡ § ¶

[email protected] [email protected] [email protected] [email protected] [email protected]

2 Contents

I. Introduction

2

II. Gravity-scalar field dynamics for colliding bubbles

4

A. Dynamics of bubble collisions

4

B. Solving the scalar field equation

5

III. Gravitational waves from bubble collisions

5

A. Computation of gravitational waves in the quadrupole approximation

8

B. A simplified method to compute gravitational waves in the quadrupole approximation

12

IV. Conclusions

15

Acknowledgments

16

References

17

I.

INTRODUCTION

A detection of signatures of primordial gravitational waves (GWs) in the cosmic microwave background was claimed by the BICEP2 experiment in 2014 [1]. But this was shown to be likely caused by interstellar dust soon thereafter [2], and the search for true signatures of primordial GWs is still ongoing. The detection of the GW signatures, if confirmed, would gain the greatest importance, among other things, from its link to cosmic “inflation”: primordial GWs are seen as the smoking gun for the “Big Bang” expansion. According to the inflation theory, the early universe experienced an extreme burst of expansion, which lasted a tiny fraction of a second, but smoothed out irregularities–inhomogeneities, anisotropies and the curvature of space, and made the universe appear homogeneous and isotropic [3]. It has been suggested that inflationary models of the early universe most likely lead to a “multiverse” [4]. One such model is “eternal inflation” [5]: it proposes that many bubbles of spacetime individually nucleate and grow inside an ever-expanding background multiverse. The nucleation and growth of such bubbles can be modeled by a Coleman-de Luccia (CDL) instanton, a type of quantum transition between two classically disconnected vacua at different energies; the higher energy (false vacuum), the lower energy (true vacuum) [6]. A scalar field initially in the false vacuum state may tunnel quantum mechanically to the true vacuum state. This nucleates bubbles of the true vacuum (new phase) inside of the false vacuum (old phase) background; through a first-order phase transition. These bubbles then expand and collide with each other. The mechanism of bubble collisions can be effectively modeled by the CDL instanton: as bubbles continue to collide repeatedly, the scalar field transitions back and forth repeatedly between the false vacuum and the true vacuum, eventually settling down in the true vacuum as the collision process is gradually terminated.

3 From the viewpoints of physical cosmology, bubble collisions and GWs emitted from the collisions are interesting in the following contexts: (1) Our primordial inflation would be completed by a second-order (not by a first-order) phase transition. However, there is a possibility that some weaker inflation could occur after the primordial inflation; for example, “thermal inflation” [7]. It is quite probable that the thermal inflation is completed by a first-order phase transition, and therefore bubble collisions could take place through a CDL instanton. Then there would be some signatures of bubble collisions, which would presumably be carried via GWs [8]. (2) Suppose that we live in a single large true vacuum bubble and that the boundary of our bubble would collide with another bubble that is located outside our observable universe [9]. Then there would exist some signatures of bubble collisions and these could be carried via GWs. For scenario (1), the mechanism of bubble collision - GW emission should be modeled stochastically. However, for scenario (2), the mechanism can be well approximated by a two-bubble collision model. There were numerous studies about bubble collisions and GWs emitted from the collisions. Among others, Hawking et al. [10] and Wu [11] studied the mechanism of the collision of two bubbles using the thin-wall approximation. Johnson et al. [12] and Hwang et al. [13] investigated the collision of two bubbles in full General Relativity via numerical computations. Kosowsky et al. [14] computed the GW spectrum resulting from two-bubble collisions in first-order phase transitions in flat spacetime using numerical simulations. Caprini et al. [15] developed a model for the bubble velocity power spectrum to calculate analytically the GW spectrum generated by two-bubble collisions in first-order phase transitions in flat spacetime. In this paper, we focus on collisions of two equal-sized bubbles and compute GWs emitted from the collisions in time domain. Largely, our analysis proceeds in two steps through Sections II and III. In Section II, we study the mechanism of bubble collisions by means of a real scalar field and a quartic potential of this field, building the simplest possible model for a CDL instanton. Einstein equations and a scalar field equation are derived for this system and are solved simultaneously for the full General Relativistic treatment of the collision dynamics. Hwang et al. [13] is closely reviewed for this purpose. In Section III, using the scalar field model from Section II, we compute GWs from the bubble collisions in a straightforward manner. In the quadrupole approximation, time-domain gravitational waveforms are directly obtained by integrating the energy-momentum tensors over the volume of the wave sources, where the energy-momentum tensors are expressed in terms of the scalar field, the local geometry and the potential; therefore, containing all necessary information about the bubble collisions. Part of computational results from Ref. [13] is recycled here to build the energy-momentum tensors. In parallel with the scalar field solutions in Section II, which have been obtained with various false vacuum field values [13], we present gravitational waveforms emitted during (i) the initial-to-intermediate stage of strong collisions and (ii) the final stage of weak collisions: the former is obtained numerically, in full General Relativity and the latter analytically, in the flat spacetime approximation. The thin-wall and quadrupole approximations are assumed to simplify our analysis and the next-to-leading order corrections beyond these approximations are disregarded. However, the approximations serve our purpose well: we aim to provide qualitative illustrations of the time-domain gravitational waveforms from the bubble collisions, which will be useful for constructing the templates for observation in the future. We adopt the unit convention, c = G = 1 for all our computations of

4 GWs.

II.

GRAVITY-SCALAR FIELD DYNAMICS FOR COLLIDING BUBBLES

The mechanism of two equal-sized colliding bubbles can be effectively modeled by means of a CDL instanton [6]. Basically, one can build a model for this, which consists of gravitation, a real scalar field and a potential of the field. In this Section we introduce one such model from Hwang et al. [13], which is built with a quartic potential, the simplest possible one for the CDL instanton.

A.

Dynamics of bubble collisions

A system of Einstein gravity coupled with a scalar field that is governed by a potential can be described by the following action: Z S=

√

1 1 µ d x −g R − ∇µ φ∇ φ − V (φ) , 16π 2 4

(1)

where R denotes the Ricci scalar, φ the scalar field and V (φ) the potential of the scalar field [13]. From this system the Einstein equations are derived: 1 Rµν − Rgµν = 8πTµν , 2

(2)

where the energy-momentum tensors on the right-hand side are written as 1 Tµν = φ;µ φ;ν − φ;ρ φ;σ g ρσ gµν − V (φ) gµν . 2

(3)

Also, the scalar field equation for the system reads ∇2 φ =

dV . dφ

(4)

The Einstein equations (2) and the scalar field equation (4) constitute a scalar field model that effectively describes the mechanism of two colliding bubbles in curved spacetime [13]. Given a potential V (φ), the scalar field solution φ and the geometry solution gµν should be obtained by solving Eqs. (2) and (4) simultaneously 1 . To this end, we prescribe an ansatz for the geometry gµν with the hyperbolic symmetry, using the double-null coordinates: ds2 = −αh2 (u, v) dudv + rh2 (u, v) dH 2 ,

(5)

where dH 2 = dχ2 + sinh2 χdθ2 with 0 ≤ χ < ∞, 0 ≤ θ < 2π [13], and αh (u, v) and rh (u, v) are to be determined by solving Eqs. (2) and (4) simultaneously in the coordinates (u, v, χ, θ). In the flat spacetime limit, the double-null coordinates are defined as u ≡ τ − x and v ≡ τ + x with τ 2 ≡ t2 − y 2 − z 2 , t = τ cosh χ,

1

This inevitably results in the effects of radiation reaction being included in the solutions, φ and gµν [13].

5 y = τ sinh χ sin θ, z = τ sinh χ cos θ: in our analysis, the x-axis of Cartesian coordinates is chosen to coincide with a line adjoining the centers of the two bubbles, and the y-axis and the z-axis lie in a plane perpendicular to the x-axis.

B.

Solving the scalar field equation

To build the simplest model of the CDL instanton for two identical colliding bubbles, we consider the potential in Subsection II A to be V (φ) =

p4 4 p3 3 p2 2 φ + φ + φ , 4! 3! 2!

(6)

where p2 , p3 and p4 are constants which can be appropriately chosen to tune the shape of the potential. Bubble collisions are represented by the scalar field moving along this potential: the field initially in the false vacuum state (at higher local minimum of potential) tunnels quantum mechanically to the true vacuum state (at lower local minimum of potential), repeating the transitions back and forth between the two states, eventually settling down in the true vacuum state. √ Following Ref. [13], we may rescale the scalar field, S ≡ 4πφ for computational convenience, and can specify √ p2 , p3 and p4 in terms of the false vacuum field Sf = 4πφf , the vacuum energy of the false vacuum Vf and a free parameter β. The potential in Eq. (6) can then be rewritten as V (S) =

3Vf 4 2 (β + 3) Vf 3 3 (β + 1) Vf 2 S − S + S . βSf4 βSf3 βSf2

(7)

With this potential the scalar field equation (4), which is now rescaled, reads ∇2 S =

12Vf 3 6 (β + 3) Vf 2 6 (β + 1) Vf S − S + S. βSf4 βSf3 βSf2

(8)

This is a non-linear wave equation whose analytical solution is not generally known: we normally approach this type of problem with numerical methods. Now, we solve the scalar field equation (8) simultaneously with the Einstein equations (2), using the ansatz given by Eq. (5), in the coordinates (u, v, χ, θ). However, it turns out that our scalar field solution is independent √ of the coordinates χ and θ and is expressed in the coordinates (u, v) only; namely, S (u, v) = 4πφ (u, v) [10, 13, 14]. With the choice of the constants, β = 0.1, Vf = 10−4 and Sf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4 in Eq. (7), the potential takes the forms as given by Figure 1 [13]. With this potential, our numerical solution S (u, v) is obtained as presented by Figure 2 [13]. In each case of Sf , (1) - (4), the bubble wall has a different value of tension due to a different value of Sf as shown by Figure 1. In the top left of Figure 2 the bubble has the lowest tension while in the bottom right it has the highest tension among the four cases of Sf . This results in the wall crossing regions in the top left being relatively wider than those in the bottom right.

III.

GRAVITATIONAL WAVES FROM BUBBLE COLLISIONS

In Section II we have built a system of two equal-sized colliding bubbles in curved spacetime by means of a CDL instanton model, considering the potential given by Eq. (7) [13]. Now, we consider that the a present

6

2(β+3)Vf 3 3Vf 4 f S + 3(β+1)V S 2 ; expressed in terms of the false FIG. 1: (color online). The quartic potential V (S) = βS 4S − βSf3 βSf2 f √ vacuum field Sf = 4πφf , the vacuum energy of the false vacuum Vf and a free parameter β. The patterns of the √ potential are shown for β = 0.1, Vf = 10−4 and Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4. (Credit: Hwang et

al. [13])

√ FIG. 2: (color online). The numerical solutions S (u, v) = 4πφ (u, v) obtained with the potential from Figure 1; with √ various false vacuum field values, Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4. (Credit: Hwang et al. [13])

observer lives in the true vacuum region of one of the two bubbles and that signatures of bubble collisions which took place in the distant past are being carried to the present observer via GWs. Here we assume that the distance between the center of collision region and the observer can be arbitrarily large (within the size of our universe), and thus that the time for a collision event, which is the retarded time to the present observer, can be quite far in the distant past; namely, tR = tP − r/c tP , where tR denotes the retarded time, tP the

7

u T

v F

T

FIG. 3: (color online). A causal relationship between two colliding bubbles and an observer: Gravitational waves from bubble collisions can be observed by a distant observer (red spot) who is in the causal future of a collision event (black spot). The dotted vertical line inside the timelike zone (green-colored region) of a null-cone (green dashed lines) represents the time-axis, along which bubble collisions took place in the distant past; within the intersection of the timelike zone of the null-cone and the ‘diamond’ zone (region enclosed by black dashed lines). u and v represent the double-null coordinates as defined in Section II. T and F denote a true vacuum and a false vacuum, respectively.

present time and r the distance. Therefore, our bubble collision may be regarded as a localized event, as long as the observer is reasonably far away from the collision region. In Figure 3 a causal relationship between two colliding bubbles and an observer is depicted using a null-cone. Here collision events that took place in the distant past are placed within the intersection of the timelike zone (green-colored region) of a null-cone (green dashed lines) and the ‘diamond’ zone (region enclosed by black dashed lines): all the collision events as our GW sources, namely the collisions in Figure 2 (strong collisions in the initial-to-intermediate stage) and the collisions to be discussed in Subsection III B later (weak collisions in the final stage) should be considered to have taken place within this intersection 2 . In the above scenario, our GWs from bubble collisions can be computed in a straightforward manner: by integrating the energy-momentum tensors combined with a Green’s function over the volume of the wave sources, where the energy-momentum tensors are expressed in terms of the scalar field, the local geometry and the potential by means of Eqs. (3), (5) and (8); therefore, containing all necessary information about the bubble collisions. A mathematical description of this computation is given as follows. In the transverse trace-free (TT) gauge, GWs, as derived from the perturbed Einstein equations in linearized gravity, can be expressed as Z 4G 1 |x − x0 | 0 TT 3 0 kl h (t, x) = Λ (n) d x T t − , x , hTT (t, x) = ij,kl ij ij c4 |x − x0 | c

(9)

where hij ≡ hij − 12 δij hkk and the unit vector n denotes the propagation direction of the waves, and the

2

In principle, the diamond zone can be extended to cover the longer evolution of bubble collision. This will result in the larger intersection area with the timelike zone of the null-cone. However, no matter how large the intersection area is, it should still be regarded as a well-localized region for our GW sources: we assume that a present observer is reasonably far away from the sources in our computations of GWs. This naturally renders our results convergent. While we consider only the retarded field for our ‘time-domain’ GWs, Kosowsky et al. [14] take both the retarded and advanced fields for their ‘frequency-domain’ GWs. On account of this, their computation domain is unbounded, but they obtain convergent results using a method of ‘phenomenological cutoff’.

8 projection tensor for gravitational radiation, 1 Λij,kl (n) ≡ Pik Pjl − Pij Pkl 2

(10)

Pij = δij − ni nj .

(11)

with

We find that the computation would be technically quite difficult with the integral as it is in Eq. (9): the way the source point x0 (integration variable) is combined with the field point x in the integrand would make our calculation quite intractable. However, expressing the integral in expansion, we obtain a more computationally favorable form: hTT ij (t, x) =

Z 4G d3 x0 T kl (tR , x0 ) Λ (n) ij,kl c4 r Z 1 m d + n d3 x0 T kl (tR , x0 ) x0m c dtR Z 1 m p d2 3 0 kl 0 0 0 d x T (tR , x ) xm xp + · · · , + 2n n 2 2c dtR tR =t−r/c

(12)

where r = |x| and tR = t − r/c denotes the retarded time. In particular, the computation resulting from the first term alone in the square bracket in Eq. (12) is called the “quadrupole approximation”. The next terms will provide corrections to this computation.

A.

Computation of gravitational waves in the quadrupole approximation

The complete information about the motion of the colliding two-bubble system is encoded in the scalar field √ solution S = 4πφ, as given by Figure 2, and thus is carried by the energy-momentum tensors through Eq. (3): to be precise, the energy-momentum tensors are comprised of the scalar field S and the geometry gµν , which are obtained by solving Eqs. (2) and (4) simultaneously [13]. As described by Eqs. (9) and (12), GWs from the system are computed with the energy-momentum tensors being the sources. It is believed that the two bubbles will be in highly relativistic motion when they collide [10]. In view of this, corrections due to the next-to-leading order terms in Eq. (12) should not be disregarded if one aims to compute GWs from the system accurately. However, although not perfectly accurate, the leading order term alone in Eq. (12) provides the “quadrupole approximation” of GWs: TT Q hij

(t, x) =

4 Λij,kl (n) Ikl (tR ) , r

where we have adopted the unit convention c = G = 1, and Z Ikl (tR ) ≡ d3 x0 T kl (tR , x0 ) .

(13)

(14)

Throughout this paper our computation is carried out only from this piece. Our main purpose is to provide qualitative insights into patterns of GWs from the colliding two-bubble system in time domain, and the nextto-leading order corrections in Eq. (12) are disregarded in our analysis.

9 Following Ref. [14], we can reduce the amount of computation in a great deal. As described in Subsection II A, we choose the x-axis to coincide with the line adjoining the centers of the two bubbles. With the axial symmetry about the x-axis, the off-diagonal components are zero and we can put Ikl in the form, Ikl = Dδkl + 4δkx δlx .

(15)

Here, the first term turns out to be D=

1 (Iyy + Izz ) , 2

(16)

which does not contribute to gravitational radiation due to Eqs. (10) and (11). The second term is given by 4 = Ixx −

1 (Iyy + Izz ) . 2

Therefore, Ikl is practically equivalent to 4δkx δlx : 1 Ikl ∼ δkx δlx Ixx − (Iyy + Izz ) . 2 Then by Eqs. (14) and (18) we may express Z 1 Ikl (tR ) = δkx δlx d3 x0 T xx (tR , x0 ) − [T yy (tR , x0 ) + T zz (tR , x0 )] . 2

(17)

(18)

(19)

Now, recall from Subsection II A that in the flat spacetime we define the hyperbolic coordinates τ , χ, θ by tR = τ cosh χ ,

(20)

y = τ sinh χ sin θ ,

(21)

z = τ sinh χ cos θ ,

(22)

τ 2 = t2R − ρ2 ,

(23)

so that

where 0 ≤ χ < ∞, 0 ≤ θ < 2π and ρ ≡

p

y 2 + z 2 . In these coordinates the flat spacetime metric takes the

form, ds2 = −dτ 2 + dx2 + τ 2 dχ2 + sinh2 χdθ2 .

(24)

In this geometry, however, the scalar field solution is independent of the coordinates χ and θ and is expressed in the coordinates (τ, x) only; namely, φ (τ, x) [10, 14]. Taking this into account, we should rewrite the volume element for the integral (19) as d3 x = dxρdρdθ = −dxτ dτ dθ ,

(25)

which is defined at the instant tR by means of Eq. (23). From Eq. (23) we see that τ has an upper bound tR with ρ = 0. This represents the exterior surface of the bubble walls, i.e. τ = tR .

(26)

10 However, the interior surface is found from Eq. (23) to be η2 τ = tR − +O 8tR

η4 t3R

,

(27)

given a wall thickness η ∼ 2ρ tR . Then from Eqs. (19), (25), (26) and (27) we can compute Ikl (tR ) effectively out of a volume piece V: 1 d3 x0 Txx − (Tyy + Tzz ) 2 V 4 Z tR Z xo η 1 0 0 0 = 2πδkx δlx dτ τ dx Txx − (Tyy + Tzz ) + O 3 2 t 2 tR −η /(8tR ) −xo R 4 Z xo π η 1 = δkx δlx η 2 dx0 Txx − (Tyy + Tzz ) +O 3 , 4 2 tR −xo τ =tR Z

Ikl (tR ) = δkx δlx

(28)

where the volume piece V is defined from a thin cylindrical shell in motion, having the thickness η 2 / (8tR ), extending along the x-axis, by means of Eqs. (26) and (27): from tR − η 2 / (8tR ) + O η 4 /t3R ≤ τ ≤ tR we find V = ∆x |τ ∆τ | ∆θ = π4 η 2 ∆x 1 + O η 2 /t2R , and the limit of the integral xo = ∆x/2 should be chosen to be sufficiently large such that collision effects be fully covered in numerical integration 3 . Following Ref. [10], we estimate a bubble wall thickness η, assuming that the walls will be highly relativistic when they collide, having the Lorentz factor γ: η∼

3 3 φf / ξ 2 γ ∼ φ2f / b4 , 2 2

(29)

where φf denotes the scalar field value at the false vacuum and ξ 4 the effective height of the potential barrier between the two minima, and the Lorentz factor γ = b4 / ξ 2 φf with 2b representing the separation of the bubbles and 4 the potential difference between the two minima (which is equivalent to Vf in our analysis in Subsection II B) [10]. However, as described in Subsection II B, our scalar field S =

√

4πφ is obtained by solving Eqs. (2) and (8)

simultaneously, using the ansatz given by Eq. (5), in the coordinates (u, v, χ, θ). Then by Eq. (3) the energymomentum tensors should be expressed in the same coordinates. Now, due to the definitions of u and v in the flat spacetime limit, and by Eqs. (20), (21) and (22) we have u = τ −x= v = τ +x= χ = tanh−1

3

q

t2R − (y 2 + z 2 ) − x ,

(30)

t2R − (y 2 + z 2 ) + x , s ! y2 + z2 . t2R

(31)

q

(32)

By Eqs. (23) and (25) the volume piece can also be viewed as V = ∆xρ∆ρ∆θ = π4 η 2 ∆x 1 + O η 2 /t2R . Then it may be stated that the volume integral in Eq. (28) will be equivalently evaluated out of this volume piece, whose shape is a long thin cylinder with the diameter (thickness) η surrounding the x-axis. This is in agreement with the statement from Ref. [10]: “The kinetic energy of the bubble walls will be concentrated in a small region around the x-axis of a wall thickness η.”

11 Using these relations, we find 2 2 2 ∂u ∂v ∂v ∂χ ∂u + 2Tuv + Tvv + Tχχ Txx = Tuu ∂x ∂x ∂x ∂x ∂x = Tuu − 2Tuv + Tvv , 2 2 2 ∂u ∂u ∂v ∂v ∂χ Tyy = Tuu + 2Tuv + Tvv + Tχχ ∂y ∂y ∂y ∂y ∂y 2 t2R y 2 y Tχχ , (T + 2T + T ) + = 2 uu uv vv 2 tR − (y 2 + z 2 ) [t2R − (y 2 + z 2 )] (y 2 + z 2 ) 2 2 2 ∂u ∂u ∂v ∂χ ∂v Tzz = Tuu + 2Tuv + Tχχ + Tvv ∂z ∂z ∂z ∂z ∂z 2 z t2R z 2 Tχχ . = 2 (T + 2T + T ) + uu uv vv 2 tR − (y 2 + z 2 ) [t2R − (y 2 + z 2 )] (y 2 + z 2 ) Substituting Eqs. (33), (34) and (35) into Eq. (28), we obtain 4 Z xo η π 1 2 0 Ikl (tR ) = δkx δlx η +O 3 , dx Tuu − 2Tuv + Tvv − 2 Tχχ 4 2t t −xo R R tR

(33)

(34)

(35)

(36)

where the subscript tR outside the square bracket means that the double-null coordinates (u, v) are defined at τ = tR ; namely, u = tR − x and v = tR + x. In the actual computation of Eq. (36), we integrate Tuu (u, v), √ Tuv (u, v), Tvv (u, v) and Tχχ (u, v), which are constructed out of the scalar field solution S (u, v) = 4πφ (u, v), the geometry solution guv (u, v), gχχ (u, v), gθθ (u, v) and the potential V (S) via Eq. (3). Then we need to change the variable of integration, from x to u or v. Using the relations u = tR − x and v = tR + x, we can convert dx = −du or dx = dv. Then we may rewrite Z xo Z dx0 Tab (u, v) = −xo

tR +xo

(37)

tR +xo

Z du Tab (u, 2tR − u) =

dv Tab (2tR − v, v)

tR −xo Z xo

tR −xo Z xo

du Tab (tR + u, tR − u) = 2

= 2

dv Tab (tR − v, tR + v) ,

0

(38)

0

where Tab represents any of Tuu , Tuv , Tvv and Tχχ , and the expressions in the second line have been obtained via translations, u → u − tR and v → v − tR . Then by Eqs. (36) and (38) Ikl (tR ) can be expressed as Z xo Z xo π du Tuu (tR ± u, tR ∓ u) − 2 du Tuv (tR ± u, tR ∓ u) Ikl (tR ) = δkx δlx η 2 2 0 0 4 Z xo Z xo 1 η + du Tvv (tR ± u, tR ∓ u) − 2 du Tχχ (tR ± u, tR ∓ u) + O 3 . 2t t 0 R 0 R

(39)

If the wall thickness η can be taken sufficiently small in Eq. (39), then by Eq. (13) we can compute the bubble-collision-induced GWs in the quadrupole approximation as TT Q hij

(t, x) ≈

2π 2 η Λij,kl (n) δkx δlx r Z xo

xo

Z

du Tuu (tR ± u, tR ∓ u) − 2

× Z + 0

0 xo

du Tuv (tR ± u, tR ∓ u) 0

1 du Tvv (tR ± u, tR ∓ u) − 2 2tR

Z 0

xo

du Tχχ (tR ± u, tR ∓ u) .

(40)

12 Now, without loss of generality we may choose n =(nx , ny , nz ) = (cos ϑ, sin ϑ, 0) ,

(41)

where ϑ denotes the angle of propagation taken from the x-axis. From this it follows that 1 1 Λij,kl (n) δkx δlx = δix δjx − 2δix nj cos ϑ + ni nj 1 + cos2 ϑ − δij sin2 ϑ , 2 2 due to Eqs. (10) and (11). Substituting this into Eq. (40), we finally express 1 2π 2 1 2 TT 2 η δix δjx − 2δix nj cos ϑ + ni nj 1 + cos ϑ − δij sin ϑ Q hij (t, x) ≈ r 2 2 Z xo Z xo du Tuu (tR ± u, tR ∓ u) − 2 du Tuv (tR ± u, tR ∓ u) × 0 0 Z xo Z xo 1 du Tvv (tR ± u, tR ∓ u) − 2 + du Tχχ (tR ± u, tR ∓ u) , 2tR 0 0

(42)

(43)

where tR = t − r = t − |x|, and the wall thickness η can be specified by means of Eq. (29); namely, in terms √ of the quantities for the bubble collision profiles, such as the false vacuum field φf (equivalent to Sf / 4π), the potential difference between the two minima 4 (equivalent to Vf ) and half the separation of the bubbles b [10]. RESULT 1: The numerical computations of Eq. (43) are presented in Figure 4; with various false vacuum √ field values, Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4, in accordance with the scalar field solutions as presented by Figure 2. Due to Eqs. (29) and (43), the amplitude of our GWs

TT Qh

(t) scales as φ4f if the

other conditions, 4 and b are kept the same. Thus, with Sf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4, the amplitude scales as (1) 1, (2) 24 , (3) 34 , (4) 44 . The frequency of the waves is modulating due to the non-linearity of the collision dynamics in the all four cases of Sf , (1) - (4). However, the modulating frequency increases overall as √ Sf increases, which is analogous to the tendency exhibited by S (u, v) = 4πφ (u, v) as shown in Figure 2. In Figure 4, we present Q hTT (t) r/ 2πη 2 instead of Q hTT (t), and thus all the waveforms are plotted in the same scale. One should note here that our actual numerical data of the energy-momentum tensors Tab for Eq. (43) have been obtained via Eq. (3) after solving Eqs. (2) and (4) simultaneously [13]. Therefore, our Tab contain √ the full physical information about the bubble collisions in terms of the scalar field S = 4πφ, the geometry gab and the potential V (S); with the radiation reaction effects included in S and gab 4 .

B.

A simplified method to compute gravitational waves in the quadrupole approximation

Ref. [14] presents a simplified method to compute the GWs

TT Q hij

(t, x) of Eq. (13) by neglecting the gravi-

tational effects on the bubbles: namely, gµν in Eq. (3) is replaced by ηµν , assuming that the bubbles are in flat

4

The way our GWs are calculated here resembles a “semi-relativistic treatment”, originated by Ruffini and Sasaki [16], in the following senses: (a) the field (hab ) radiates as if it were in flat spacetime, (b) the source (Tab ) contains the General Relativistic information about its local spacetime. In Eq. (9) we see that our GWs hab result from distant sources Tab , which are composed of the scalar field and the local geometry given via Eqs. (3), (5) and (8), thus containing the full General Relativistic information, including the radiation reaction effects.

13

√ FIG. 4: The numerical plots of Q hTT (t) r/ 2πη 2 with various false vacuum field values, Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4. As η 2 ∼ φ4f , the amplitude of

Qh

TT

(t) should scale as (1) 1, (2) 24 , (3) 34 , (4) 44 .

spacetime. Then Eq. (14) can be simplified as Z Z Ikl (tR ) = d3 x0 T kl (tR , x0 ) = d3 x0 ∂k φ (tR , x0 ) ∂l φ (tR , x0 ) ,

(44)

where the energy-momentum tensors from Eq. (3) have been reduced; Tij → ∂i φ∂j φ because the terms proportional to δij in Tij makes no contribution to gravitational radiation in Eq. (13) due to the property of Eq. (10); namely, Λij,kl δij = 0 [14]. By Eqs. (18) and (44) we have "

Z

Ikl

∂φ ∂x0

2

1 − 2

∂φ ∂y 0

2

1 − = δkx δlx d3 x0 2 " # 2 2 Z ∂φ 1 ∂φ = δkx δlx d3 x0 − . ∂x0 2 ∂ρ0

∂φ ∂z 0

2 #

(45)

Using Eq. (23), we can modify

∂φ ∂ρ

2

=

∂φ ∂τ

2

∂τ ∂ρ

2 =

t2R − τ 2 τ2

∂φ ∂τ

2 .

(46)

Then in the same manner as described above by Eq. (28), the integral Ikl (tR ) is computed out of the volume

14 π 2 4 η ∆x

1 + O η 2 /t2R : " 2 2 # Z 1 ∂φ ∂φ 3 0 − Ikl (tR ) = δkx δlx d x ∂x0 2 ∂ρ0 V " 4 2 2 # Z xo Z tR 2 02 t − τ η ∂φ ∂φ 0 0 0 R − +O 3 dx = 2πδkx δlx dτ τ 0 02 0 ∂x 2τ ∂τ t 2 −xo tR −η /(8tR ) R 2 4 Z xo π ∂φ η = δkx δlx η 2 dx0 +O 3 . 0 2 ∂x τ =tR tR 0

piece V = ∆x |τ ∆τ | ∆θ =

(47)

If the wall thickness η can be taken sufficiently small in Eq. (47), by Eqs. (13) and (47) we can compute the bubble-collision-induced GWs in the quadrupole approximation as 2 Z xo 2π ∂φ TT 2 0 . Λij,kl (n) δkx δlx η dx Q hij (t, x) ≈ r ∂x0 τ =tR 0 Substituting Eq. (42) into Eq. (48), we finally express 1 2π 2 1 2 TT 2 η δix δjx − 2δix nj cos ϑ + ni nj 1 + cos ϑ − δij sin ϑ Q hij (t, x) ≈ r 2 2 2 Z xo ∂φ × dx0 , ∂x0 τ =tR 0

(48)

(49)

where tR = t − r = t − |x|, and η is specified by Eq. (29). RESULT 2: Toward the end of the bubble collisions, τ 1, the scalar field oscillates around the true vacuum state, i.e. |φ| 1, being nearly monochromatic. Then we can approximate Eq. (8) as φ (τ, x) ≈

6 (β + 1) Vf φ (τ, x) , βSf2

(50)

where we have replaced the curved spacetime Laplacian ∇2 by the flat spacetime d’Alembertian ≡ −∂ 2 /∂τ 2 − (2/τ ) ∂/∂τ + ∂ 2 /∂x2 , neglecting the gravitational effects on the bubbles to simplify the problem 5 . With the help of Ref. [17], we obtain a solution for Eq. (50): √ J1 ωt τ 2 − x2 √ φ (τ, x) = φo , (51) ωt τ 2 − x2 p where Jn denotes the Bessel function of the first kind, ωt ≡ 6 (β + 1) Vf / (βSf2 ) represents the ‘terminal’ frequency of the bubble collisions, and φo is the amplitude which is determined by the initial conditions of the field. Substituting Eq. (51) into Eq. (49), the GWs emitted from the bubble collisions in the final stage can be computed. Figure 5 shows the GWs Q hTT (t) r/ 2πη 2 computed with various false vacuum field values, √ Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4. Corresponding to the field values are the frequencies, ωt ' (1) 0.8124, (2) 0.4062, (3) 0.2708, (4) 0.2031 ∼ (1) 1, (2) 1/2, (3) 1/3, (4) 1/4, as can be seen from Figure 5. Also, the amplitude of Q hTT (t) r/ 2πη 2 scales as (1) 1, (2) 2, (3) 3, (4) 4 due to the factor ωt−1 ∼ Sf , as can be seen from Figure 5. Then the amplitude of Q hTT (t) should scale as η 2 ωt−1 ∼ Sf5 ∼ (1) 1, (2) 25 , (3) 35 , (4) 45 .

5

A similar analysis is found in Ref. [12], in which the scalar field equation is solved in hyperbolic “de Sitter” spacetime in the limit, τ H −1 , where H is the Hubble parameter. The solution shows fluctuations of decreasing amplitude and increasing period (or decreasing frequency) in τ . However, in our analysis, the equation is solved in hyperbolic “flat” spacetime and our solution given by Eq. (51) has fluctuations of decreasing amplitude and fixed period (or single frequency; monochromatic) in τ .

15

(1)

(2)

(3)

(4)

(t) r/ 2πη 2 computed via the expression in Eq. (49), with φ given by Eq. (51). Corresponding √ to the false vacuum field values, Sf = 4πφf = (1) 0.1, (2) 0.2, (3) 0.3, (4) 0.4, are the frequencies, ωt ' (1) 0.8124, (2) 0.4062, (3) 0.2708, (4) 0.2031 ∼ (1) 1, (2) 1/2, (3) 1/3, (4) 1/4. And the amplitude of Q hTT (t) r/ 2πη 2 scales as (1)

FIG. 5: Plots of

Qh

TT

1, (2) 2, (3) 3, (4) 4 due to the factor ωt−1 ∼ Sf . Then the amplitude of

Qh

TT

(t) should scale as η 2 ωt−1 ∼ Sf5 ∼ (1) 1,

(2) 25 , (3) 35 , (4) 45 .

IV.

CONCLUSIONS

We have computed GWs emitted from collisions of two equal-sized bubbles in time domain. The waveforms have been obtained for (i) the initial-to-intermediate stage of strong collisions and (ii) the final stage of weak collisions, in full General Relativity and in the flat spacetime approximation, using numerical and analytical methods, respectively. During (i), the waveforms show the non-linearity of the collisions, characterized by a modulating frequency and cusp-like bumps, whereas during (ii), the waveforms exhibit the linearity of the collisions, featured by a constant frequency and smooth oscillations, as can be checked from Figures 4 and 5, respectively. Also, depending on the false vacuum field value φf , the waveforms have different scales of frequency. During (i), the modulating frequency increases overall as the false vacuum field value φf increases, whereas during (ii), the frequency ωt decreases as the false vacuum field value φf increases in an inversely proportional relationship, i.e. ωt ∼ φ−1 f . It is interesting to note that the relationship between the false vacuum field value and the frequency during (i) changes almost inversely during (ii). In addition, the false vacuum field value φf affects the amplitude of the waveforms. During (i), the amplitude scales as η 2 ∼ φ4f , whereas during (ii), the amplitude scales as η 2 ωt−1 ∼ φ5f , where η is a bubble wall thickness. One of the notable differences between the waveforms emitted during (i) and during (ii) is the sign, as can be seen from Figures 4 and 5. This is due to the difference between Eqs. (43) and (49): the integral in Eq. (49)

16 is always positive while its counterpart in Eq. (43) is not necessarily. This has to do with the composition of the integrands in the two expressions. The integrand in Eq. (43) consists of the energy-momentum tensors Tab which have been obtained via Eq. (3) after solving Eqs. (2) and (4) simultaneously [13]: thus Tab contain the full physical information of bubble collisions in terms of the scalar field φ, the geometry gab and the potential V (φ); with the radiation reaction effects included in φ and gab . However, as explained in the beginning of Subsection III B, the integrand in Eq. (49) comes only from the first term, with the second and third terms being disregarded in Eq. (3) as the gravitational effects on the bubbles are assumed to be neglected, following Ref. [14]. This, combined with the thin-wall approximation, results in the integrand in Eq. (49) being positive, which leads to the integral being also positive. But this is not the case for the integral in Eq. (43) due to the minus signs appearing in Eq. (3) and in the integrand in Eq. (43). Throughout the paper, we used the thin-wall and quadrupole approximations to simplify our computations. These approximations served our purpose well in that we were able to gain some qualitative insights into the time-domain gravitational waveforms emitted from bubble collisions. However, to obtain more physically reasonable waveforms, taking into account a generic thickness and relativistic motion of bubble wall, it will be inevitable to include in our computations the next-to-leading order corrections beyond each approximation. Huge amount of computation will be involved in this task, and we leave it for follow-up studies.

Acknowledgments

The authors would like to thank Dong-il Hwang for his valuable comments and assistance during an early stage of this work. The authors also would like to thank Hongsu Kim, Sang Pyo Kim, Hyung Won Lee, Gungwon Kang and Inyong Cho for fruitful discussions and helpful comments. BHL, WL and DY appreciate Pauchy W. Y. Hwang and Sang Pyo Kim for their hospitality at the 9th International Symposium on Cosmology and Particle Astrophysics in Taiwan, 13-17 November, 2012. DHK and WL appreciate APCTP for its hospitality during completion of this work. DHK and JY were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2008901 and 2013R1A1A2A10004883). BHL was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2014R1A2A1A01002306). WL was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A2043908). DY was supported by the JSPS Grant-in-Aid for Scientific Research (A) (No. 21244033) and also supported by Leung Center for Cosmology and Particle Astrophysics (LeCosPA) of National Taiwan University (103R4000).

17

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