Gravitational waves from cosmic bubble collisions

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 \textit{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.


I. INTRODUCTION
The discovery of signatures of primordial gravitational waves (GW) in the cosmic microwave background has recently been announced by the BICEP2 experiment [1]. If confirmed, the discovery 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 [2].
It has been suggested that inflationary models of the early universe most likely lead to a "multiverse" [3].
One such model is "eternal inflation" [4]: 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 the 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) [5].
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.
There were numerous studies about bubble collisions and GWs emitted from the collisions. Among others, Hawking et al. [6] and Wu [7] studied the mechanism of the collision of two bubbles, using the thin-wall approximation. Johnson et al. [8] and Hwanget al. [9] investigated the collision of two bubbles in full General Relativity via numerical computations. Kosowsky et al. [10] computed the GW spectrum resulting from twobubble collisions in first-order phase transitions in flat spacetime using numerical simulations. Caprini et al. [11] 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 the 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. [9] 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. [9] 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 [9], 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 gain qualitative insights into the time-domain gravitational waveforms from the bubble collisions. We adopt the unit convention, c = G = 1 for all our computations of GWs.

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: where R denotes the Ricci scalar, φ the scalar field and V (φ) the potential of the scalar field [9]. From this system the Einstein equations are derived: where the energy-momentum tensors on the right-hand side are written as Also, the scalar field equation for the system reads 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 [9]. Given a potential V (φ), the scalar field solution φ and the geometry solution g µν should be obtained by solving Eqs. (2) and (4) simultaneously 1 .
To this end, we prescribe an ansatz for the geometry g µν with the hyperbolic symmetry, using the double-null coordinates: where dH 2 = dχ 2 + sinh 2 χdθ 2 with 0 ≤ χ < ∞, 0 ≤ θ < 2π [9], and α h (u, v) and r h (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 doublenull coordinates are defined as u ≡ τ − x and v ≡ τ + x with τ 2 ≡ t 2 − y 2 − z 2 , t = τ cosh χ, 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 where p 2 , p 3 and p 4 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. [9], we may rescale the scalar field, S ≡ √ 4πφ for computational convenience, and can specify p 2 , p 3 and p 4 in terms of the false vacuum field S f = √ 4πφ f , the vacuum energy of the false vacuum V f and a free parameter β. The potential (6) can then be rewritten as With this potential the scalar field equation (4), which is now rescaled, reads  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 (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, , [9], [10]. With the choice of the constants, Eq. (7), the potential takes the forms as given by Figure 1

III. GRAVITATIONAL WAVES FROM BUBBLE COLLISIONS
In the transverse trace-free gauge, GWs, as derived from the perturbed Einstein equations, can be expressed where h ij ≡ h ij − 1 2 δ ij h k k and the unit vector n denotes the propagation direction of the waves, and the projection tensor for gravitational radiation, with Eq. (9) may be expressed in expansion: where r = |x| and t R = t − r/c denotes the retarded time.

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 g µν , which are obtained by solving Eqs. (2) and (4) simultaneously [9]. As described by Eq. (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 [6]. 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.
approximation" of GWs: where we have adopted the unit convention c = G = 1, and 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.
Following Ref. [10], 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 I kl in the form, Here, the first term turns out to be which does not contribute to gravitational radiation due to Eqs. (10) and (11). The second term is given by Therefore, I kl is practically equivalent to δ kx δ lx : Then by Eqs. (14) and (18) we may express Now, recall from Subsection II A that in the flat spacetime we define the hyperbolic coordinates τ , χ, θ by so that where 0 ≤ χ < ∞, 0 ≤ θ < 2π and ρ ≡ y 2 + z 2 . In these coordinates the flat spacetime metric takes the form, 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) [6], [10]. Taking this into account, we should rewrite the volume element for the integral (19) by means of Eq. (23): According to Ref. [6], the kinetic energy of the bubble walls will be concentrated in a small region around the x-axis of a wall thickness η. Then the volume integral in Eq. (19) will be effectively computed out of a volume piece ∆V ∼ π 4 η 2 ∆x, whose shape is a long thin cylinder surrounding the x-axis. We estimate the wall thickness η, assuming that the walls will be highly relativistic when they collide and will have the Lorentz factor γ: where φ f 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 = b 4 / ξ 2 φ f with 2b representing the separation of the bubbles and 4 the potential difference between the two minima (which is equivalent to V f in our analysis in Subsection II B) [6]. In Figure 3 we show the plots of the kinetic energy of the bubble walls against ρ and t = τ 2 − ρ 2 (for ρ = 0 and ρ = η/2), created by following Ref. [6].
From Eq. (23) we find given a wall thickness η ∼ 2ρ t R . Then from Eqs. (19), (25) and (27) we can compute I kl (t R ) out of a volume piece ∆V : where the volume piece is ∆V = ∆xρ∆ρ∆θ = ∆x |τ ∆τ | ∆θ = π 4 η 2 ∆x 1 + O η 2 t 2 R , and the limit of the integral x o should be chosen to be sufficiently large such that collision effects be fully covered in numerical integration.
However, as described in Subsection II B, our scalar field S = √ 4πφ is obtained by solving Eqs. (2) and (8) simultaneously, using the ansatz (5), in the coordinates (u, v, χ, θ). Then by Eq.  where K is concentrated; in a small region around the x-axis of a wall thickness 2ρ ∼ η [6]. <Right> The kinetic energy of the bubble walls K plotted against t = τ 2 − ρ 2 . The two curves here represent K (t) for ρ = 0 (solid line) and ρ = η/2 (dotted line), respectively. For 0 < ρ < η/2, K (t) should be placed somewhere between these two curves.
Using these relations, we find T yy = T uu ∂u ∂y 2 + 2T uv ∂u ∂y ∂v ∂y + T vv ∂v ∂y Substituting Eqs. (32), (33) and (34) into Eq. (28), we obtain where the subscript t R outside the square bracket means that the double-null coordinates (u, v) are defined at In the actual computation of Eq. (35), we integrate T uu (u, v), T uv (u, v), T vv (u, v) and T χχ (u, v), which are constructed out of the scalar field solution S (u, v) = √ 4πφ (u, v), the geometry solution g uv (u, v), g χχ (u, v), g θθ (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 = t R − x and v = t R + x, we can Then we may rewrite where T ab represents any of T uu , T uv , T vv and T χχ , and the expressions in the second line have been obtained Then by Eqs. (35) and (37) I kl (t R ) can be expressed as If the wall thickness η can be taken sufficiently small in Eq. (38), then by Eq. (13) we can compute the bubble-collision-induced GWs in the quadrupole approximation as Now, without loss of generality we may choose n =(n x , n y , n z ) = (cos ϑ, sin ϑ, 0) , where ϑ denotes the angle of propagation taken from the x-axis. From this it follows that due to Eqs. (10) and (11). Substituting this into Eq. (39), we finally express where t R = t − r = t − |x|, and the wall thickness η can be specified by means of (26); namely, in terms of the quantities for the bubble collision profiles, such as the false vacuum field φ f (equivalent to S f / √ 4π), the potential difference between the two minima 4 (equivalent to V f ) and half the separation of the bubbles b [6]. One should note here that our actual numerical data of the energy-momentum tensors T ab for Eq. (42) have been obtained via Eq. (3) after solving Eqs. (2) and (4) simultaneously [9]. Therefore, our T ab contain the full physical information about the bubble collisions in terms of the scalar field S = √ 4πφ, the geometry g ab and the potential V (S); with the radiation reaction effects included in S and g ab .
Ref. [10] presents a simplified method to compute the GWs Q h TT ij (t, x) of Eq. (13) by neglecting the gravitational effects on the bubbles: namely, g µν in Eq. (3) is replaced by η µν , assuming that the bubbles are in flat spacetime. Then Eq. (14) can be simplified as where the energy-momentum tensors from Eq. (3) have been reduced; T ij → ∂ i φ∂ j φ because the terms proportional to δ ij in T ij makes no contribution to gravitational radiation (13) due to the property of Eq. (10); namely, Λ ij,kl δ ij = 0 [10].

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 proportional relationship, i.e. ω t ∼ 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 φ f affects the amplitude of the waveforms. During (i), the amplitude scales as η 2 ∼ φ 4 f , whereas during (ii), the amplitude scales as η 2 ω −1 t ∼ φ 5 f , 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  2) and (4) simultaneously [9]: thus T ab contain the full physical information of bubble collisions in terms of the scalar field φ, the geometry g ab and the potential V (φ); with the radiation reaction effects included in φ and g ab . However, as explained in the beginning of Subsection III B, the integrand in (48) 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. [10].
This, combined with the thin-wall approximation, results in the integrand in (48) being positive, which leads to the integral being also positive. But this is not the case for the integral in (42) due to the minus signs appearing in (3) and in the integrand in (42).
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.