# Gravitational waves from cosmic bubble collisions

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## Abstract

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.

## Keywords

Scalar Field Gravitational Wave Flat Spacetime False Vacuum True Vacuum## 1 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, 4, 5].

It has been suggested that inflationary models of the early universe most likely lead to a “multiverse” [6, 7]. One such model is “eternal inflation” [8]: 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) [9, 10, 11]. 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.

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” [12]. 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 [13]. (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 [14]. 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. [15] and Wu [16] studied the mechanism of the collision of two bubbles using the thin-wall approximation. Johnson et al. [17] and Hwang et al. [18] investigated the collision of two bubbles in full General Relativity via numerical computations. Kosowsky et al. [19] computed the GW spectrum resulting from two-bubble collisions in first-order phase transitions in flat spacetime using numerical simulations. Caprini et al. [20] 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 Sects. 2 and 3. In Sect. 2, 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. The 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. [18] is closely reviewed for this purpose. In Sect. 3, using the scalar field model from Sect. 2, 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 the computational results from Ref. [18] are recycled here to build the energy-momentum tensors. In parallel with the scalar field solutions in Sect. 2, which have been obtained with various false vacuum field values [18], we present gravitational waveforms emitted during

(i) the initial-to-intermediate stage of strong collisions, and

(ii) the final stage of weak collisions.

*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 GWs.

## 2 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 [8, 9, 10]. 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. [17], which is built with a quartic potential, the simplest possible one for the CDL instanton.

### 2.1 Dynamics of bubble collisions

^{1}To this end, we prescribe an ansatz for the geometry \(g_{\mu \nu }\) with the hyperbolic symmetry, using the

*double-null*coordinates:

### 2.2 Solving the scalar field equation

## 3 Gravitational waves from bubble collisions

In Sect. 2 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) [18]. Now, we consider that the a present 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, \(t_\mathrm{{R }}=t_\mathrm{{P}}-r/c\ll t_\mathrm{{P}}\), where \(t_\mathrm{{R}}\) denotes the retarded time, \(t_\mathrm{{P}}\) the 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 Fig. 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 Fig. 2 (strong collisions in the initial-to-intermediate stage) and the collisions to be discussed in Sect. 3.2 later (weak collisions in the final stage) should be considered to have taken place within this intersection.^{2}

### 3.1 Computation of gravitational waves in the quadrupole approximation

*qualitative*insights into patterns of GWs from the colliding two-bubble system in time domain, and the next-to-leading order corrections in Eq. (12) are disregarded in our analysis.

^{3}Following Ref. [15], we estimate a bubble wall thickness \(\eta \), assuming that the walls will be highly relativistic when they collide, having the Lorentz factor \(\gamma \):

**Result 1**: The numerical computations of (43) are presented in Fig. 4; with various false vacuum field values, \(S_\mathrm{{f} }=\sqrt{4\pi }\phi _\mathrm{{f}}=\) (1) \(0.1\), (2) \(0.2\), (3) \(0.3\), (4) \(0.4\), in accordance with the scalar field solutions as presented by Fig. 2. Due to Eqs. (29) and (43), the amplitude of our GWs \(_\mathrm{{Q}}h^\mathrm{{TT}}( t) \) scales as \(\phi _\mathrm{{ f}}^{4}\) if the other conditions, \(\epsilon ^{4}\) and \(b\) are kept the same. Thus, with \(S_\mathrm{{f}}=\) (1) \(0.1\), (2) \(0.2\), (3) \(0.3\), (4) \(0.4\), the amplitude scales as (1) \(1\), (2) \(2^{4}\), (3) \(3^{4}\), (4) \(4^{4}\). The frequency of the waves is modulating due to the non-linearity of the collision dynamics in the all four cases of \(S_\mathrm{{f}}\), (1)–(4). However, the modulating frequency increases overall as \(S_\mathrm{{f}}\) increases, which is analogous to the tendency exhibited by \(S( u,v) =\sqrt{4\pi }\phi ( u,v) \) as shown in Fig. 2. In Fig. 4, we present \(_\mathrm{{Q}}h^\mathrm{{TT} }( t) r/( 2\pi \eta ^{2}) \) instead of \(_\mathrm{{Q} }h^\mathrm{{TT}}( t) \), 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 \(T_{ab}\) for Eq. (43) have been obtained via Eq. (3) after solving Eqs. (2) and (4) simultaneously [18]. Therefore, our \(T_{ab}\) contain the full physical information about the bubble collisions in terms of the scalar field \(S=\sqrt{4\pi }\phi \), the geometry \(g_{ab}\) and the potential \(V( S)\); with the radiation reaction effects included in \(S\) and \(g_{ab}\).^{4}

### 3.2 A simplified method to compute gravitational waves in the quadrupole approximation

**Result 2**: Toward the end of the bubble collisions, \(\tau \gg 1\), the scalar field oscillates around the true vacuum state, i.e. \(\vert \phi \vert \ll 1\), being nearly

*monochromatic*. Then we can approximate Eq. (8) as

^{5}With the help of Refs. [22, 23], we obtain a solution for Eq. (50):

## 4 Conclusions

- (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.

One of the notable differences between the waveforms emitted during (i) and during (ii) is the sign, as can be seen from Figs. 4 and 5. This is due to the difference between Eqs. (43) and (49): the integral in (49) is always positive while its counterpart in (43) is not necessarily. This has to do with the composition of the integrands in the two expressions. The integrand in (43) consists of the energy-momentum tensors \(T_{ab}\) which have been obtained via Eq. (3) after solving Eqs. (2) and (4) simultaneously [18]: thus \(T_{ab}\) contain the full physical information of bubble collisions in terms of the scalar field \(\phi \), the geometry \(g_{ab}\) and the potential \(V(\phi )\); with the radiation reaction effects included in \(\phi \) and \(g_{ab}\). However, as explained in the beginning of Sect. 3.2, the integrand in (49) comes only from the first term, with the second and third terms being disregarded in (3) as the gravitational effects on the bubbles are assumed to be neglected, following Ref. [19]. This, combined with the thin-wall approximation, results in the integrand in (49) being positive, which leads to the integral being also positive. But this is not the case for the integral in (43) due to the minus signs appearing in (3) and in the integrand in (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. A huge amount of computation will be involved in this task, and we leave it for follow-up studies.

## Footnotes

- 1.
This inevitably results in the effects of radiation reaction being included in the solutions, \(\phi \) and \(g_{\mu \nu }\) [18].

- 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. [19] take both the retarded and the 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’.

- 3.
By Eqs. (23) and (25) the volume piece can also be viewed as \( \mathcal {V}=\Delta x\rho \Delta \rho \Delta \theta =\frac{\pi }{4}\eta ^{2}\Delta x[ 1+\mathcal {O}( \eta ^{2}/t_\mathrm{{R}}^{2}) ] \). 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) \(\eta \) surrounding the \(x\)-axis. This is in agreement with the statement from Ref. [15]: “The kinetic energy of the bubble walls will be concentrated in a small region around the \(x\)-axis of a wall thickness \(\eta \)”.

- 4.
The way our GWs are calculated here resembles a “semi-relativistic treatment”, due to Ruffini and Sasaki [21], in the following senses: (a) the field (\(h_{ab}\)) radiates as if it were in flat spacetime, (b) the source (\(T_{ab}\)) contains the General Relativistic information about its local spacetime. In Eq. (9) we see that our GWs \(h_{ab}\) result from distant sources \(T_{ab}\), 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.

- 5.
A similar analysis is found in Ref. [17], in which the scalar field equation is solved in hyperbolic “de Sitter” spacetime in the limit, \(\tau \gg H^{-1}\), where \(H\) is the Hubble parameter. The solution shows fluctuations of decreasing amplitude and increasing period (or decreasing frequency) in \( \tau \). However, in our analysis, the equation is solved in hyperbolic “flat” spacetime and our solution given by (51) has fluctuations of decreasing amplitude and fixed period (or single frequency; monochromatic) in \(\tau \).

## Notes

### 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).

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