Determining the outcome of cosmic bubble collisions in full general relativity

Cosmic bubble collisions provide an important possible observational window on the dynamics of eternal inflation. In eternal inflation, our observable universe is contained in one of many bubbles formed from an inflating metastable vacuum. The collision between bubbles can leave a detectable imprint on the cosmic microwave background radiation. Although phenomenological models of the observational signature have been proposed, to make the theory fully predictive one must determine the bubble collision spacetime, and thus the cosmological observables, from a scalar field theory giving rise to eternal inflation. Because of the intrinsically nonlinear nature of the bubbles and their collision, this requires a numerical treatment incorporating General Relativity. In this paper, we present results from numerical simulations of bubble collisions in full General Relativity. These simulations allow us to accurately determine the outcome of bubble collisions, and examine their effect on the cosmology inside a bubble universe. We confirm the validity of a number of approximations used in previous analytic work, and identify qualitatively new features of bubble collision spacetimes. Both vacuum bubbles and bubbles containing a realistic inflationary cosmology are studied. We identify the constraints on the scalar field potential that must be satisfied in order to obtain collisions that are consistent with our observed cosmology, yet leave detectable signatures.

Figure 1

The generic bubble collision spacetime in the thin-wall approximation is split into four regions. Region I is the false vacuum, region II is the interior of the collision bubble, region III is the future domain of influence of the collision (which might be composed of further subregions), and region IV is the interior of the observation bubble. In the Israel junction condition formalism, the observation and collision bubbles are separated from the false vacuum by infinitesimally thin walls (a and d) characterized only by their tensions. In addition, region III is enclosed (depending on the context) by either domain walls or shells of scalar radiation (b and c), each characterized by their tension and energy density.

Figure 2

The numerically generated instantons for a potential of the form Eq. (d6) with $a=0.1$ for $ϵ=0.0075$. The dimensionless variable $x$ is related to the field $\phi$, $r$ to the metric function $\rho$, and $s$ to $t$ as defined in Eq. (d2). The prediction for the critical radius [based on Eq. (d20) in the thin-wall approximation] is shown as the blue-dashed line.

Figure 3

The generic form of the potentials used in our simulations consists of four components. $T1$ and $T2$ describe the potential barriers between the three vacua of the potential. $C1$ and $C2$ describe regions of the potential leading to the vacua and can drive a period of inflation.

Figure 4

Potentials giving rise to vacuum bubbles. For the potentials on the top row, the false vacuum is located at $\phi =0$, and there are two possible CDL solutions, connecting the false vacuum to positive or negative $\phi$. In both cases, the lowest energy vacuum is the one at positive $\phi$. The potentials on the bottom row have a false vacuum at negative $\phi$. There is only one CDL solution, connecting the false vacuum to the vacuum at $\phi =0$. The lowest energy vacuum for the potential V3 is at positive $\phi$ while the lowest energy vacuum for the potential V4 is at $\phi =0$.

Figure 5

Potentials giving rise to bubbles with an interior cosmology. The potentials on the top row are of the large-field type, both have the false vacuum at $\phi =0$, and both allow for two possible CDL solutions connecting the false vacuum to regions of positive or negative $\phi$. The potential L1 has cosmological evolution inside both bubbles, with a much steeper slope at negative $\phi$ than positive $\phi$. We show both the full potential, and a plot zoomed in around the position of the false vacuum. The potential L2 omits the cosmological evolution at negative $\phi$, but has the same potential at positive $\phi$ as L1. We show only the portion of the potential L2 in the vicinity of the false vacuum. The potentials on the bottom row are of the small-field type, and again, have the false vacuum at $\phi =0$, with two possible CDL solutions. The potential S1 is defined using Eq. (54) while S2 is defined using Eq. (53); inflation occurs in the vicinity of an inflection point for S1 while it occurs in the vicinity of a hilltop for S2.

Figure 6

A contour plot of the field $\phi (x,z)$ for the collision of the two different types of bubbles allowed by the potential V2 shown in Fig. 4. The color bar indicates the value of the field at each point in spacetime. In this case, the postcollision domain wall is repulsive, and accelerates away from the interior of both bubbles.

Figure 7

Demonstrations of the convergence of our code for the simulation shown in Fig. 6. On the left, we show the absolute value of the difference between solutions with discretizations obtained for $p$ and $p+1$, illustrating how the difference decreases with increasing resolution as expected for a convergent implementation. On the right, we plot the estimated convergence rate as calculated from Eq. (55): as the resolution increases, the value obtained approaches 3, which is consistent with the expected value for our implementation.

Figure 8

The numerical simulation of a single observation bubble formed from the potential V1. On the left, we show a contour plot of $\phi (x,z)$. As this is a vacuum bubble, the field is constant on either side of the bubble wall. The yellow-dashed line is the wall trajectory predicted by the thin-wall approximation, illustrating the excellent agreement between the numerical and junction-condition solutions. In the middle, we show a contour plot of the numerical solution for $\alpha (x,z)$ (black-solid lines) as well as the junction-condition solution (red-dashed lines). The agreement is excellent, except in the vicinity of the bubble walls at large $z$. On the right, we show a contour plot of the numerical solution for $a(x,z)$ as well as the junction-condition solution (red-dashed lines). The disagreement between these two solutions is due to the fact that in the junction-condition formalism, the $x$ coordinate is discontinuous across the wall, while in the simulation it is treated as a continuous coordinate. The solutions agree in the bubble center and the bubble exterior, which is expected. In the numerical solution, $a$ must continuously interpolate between the solution in the bubble center and the bubble exterior.

Figure 9

Numerical simulation of a single observation bubble formed from the potential L1. On the left, we show a contour plot of $\phi (x,z)$. The trajectory for the wall in the junction-condition solution (yellow-dashed line) can be found under the assumption that the bubble interior has a constant vacuum energy; the agreement with the numerical solution is excellent. The field is in the false vacuum outside the bubble, and evolves on the bubble interior down the potential towards increasingly positive values of $\phi$. The numerical solution has the correct qualitative behavior except in the vicinity of the bubble walls. As the resolution is increased, we expect that the contours in the bubble interior to go up to $z\to \infty$. In the right panel, we plot the field in the center of the simulated bubble $\phi (x=0,z)$ as a function of the elapsed proper time (black-solid line). The slow-roll solution obtained by integrating the field and Friedmann equations is overplotted (dashed-yellow line); the agreement is excellent.

Figure 10

A contour plot of the field for a collision between identical bubbles formed in the potential V1. The solid lines and color bar indicate the value of the field. In the immediate aftermath of the collision, a pocket of the false vacuum is formed. This pocket expands, recollapses, and then reforms. After a few cycles of this, a coherent, oscillating field configuration known as an oscillon is formed.

Figure 11

A contour plot of the metric functions $a$ (left) and $\alpha$ (right) for the collision shown in Fig. 10.

Figure 12

The field on constant $z\simeq 3$ slices for simulated collisions between identical bubbles formed in the potential V1 for three initial separations $H_F{x}_c=0.25$ (black-solid line), $H_F{x}_c=0.50$ (red-dashed line), and $H_F{x}_c=1.00$ (blue dot-dashed line). The solutions have been translated so that the collisions each occur at $H_{F}x=0$. On subsequent time slices, the profile retains its shape, but oscillates in amplitude. Such structures are known as oscillons.

Figure 13

Cutting through the center of the intermediate oscillon profile shown in Fig. 12, we plot the field as a function of $z$. Overplotted (red-dashed lines) is the analytic solution Eq. (58). The qualitative agreement is quite good, although the analytic solution does not get the drift of the frequency of oscillation exactly correct. As the initial bubble separation is increased, the packet amplitude becomes larger.

Figure 14

The metric function $\alpha$ evaluated at $x=0.375$ (left) and $x=0.280$ (right) for the collision between two identical bubbles formed in the potential V1 and initially separated by $x_c=0.75$. The plot on the left cuts through the region that underwent a transition back to the false vacuum. We overplot the expectation for $\alpha$ from the thin-wall approximation (blue-dashed line) as well as the metric in empty hyperbolic de Sitter outside (solid-red line) and inside (dot-dashed purple line) the bubble. The agreement between the thin-wall solution to the immediate future of the collision (at $z\simeq 0.4$) and the numerics is excellent. The plot on the right cuts through the region to the future of the outgoing scalar radiation, but outside the pocket of false vacuum. We find the best fit for the mass parameter, assuming that $\alpha =A_I^{-1/2}$ [see Eq. (2)] in this region, obtaining $M_r\simeq 3.0\times {10}^{-3}$. This function is overplotted on the simulation as the blue-dashed line, and in addition we show the metric function for empty HdS inside (dot-dashed purple line) and outside (solid-red line) the bubble.

Figure 15

The field on two constant-time slices (left and right) just before and just after the collision of two identical bubbles formed in the potential V3. The field linearly superposes in the vicinity of the collision. This value of the field is in the vicinity of a third vacuum, inducing a classical transition.

Figure 16

Contour plots of the field for the collision of two identical vacuum bubbles formed in the potential V3 (left) and V4 (right). In the right panel, the boundaries of the simulation lie at the center of each bubble. In both cases, the collision causes a classical transition to a region containing another vacuum. On the left, the new vacuum has lower energy than either bubble interior, and therefore the region that underwent the classical transition expands. On the right, the new vacuum has higher energy than either bubble interior, but because the domain walls surrounding the new vacuum are repulsive, the region that underwent the classical transition survives.

Figure 17

On the left, we show a contour plot of $\phi (x,z)$ for the collision between the two different bubbles allowed by the potential V1. On the right, the potential barrier in the region $T1$ has been compressed by a factor of 2 relative to the potential V1. In the free-passage approximation, the structure of the potential determines the position of the field in the immediate aftermath of the collision. For the collision on the left, the potential barriers are nearly symmetric, and the sum of the wall profiles is close to $\phi =0$. This leads to the production of an initially expanding pocket of false vacuum inside the postcollision domain wall. The collision on the right was generated from a potential where the width of $T1$ is roughly half that of $T2$. The sum of the wall profiles is at positive $\phi$ near the vacuum of the observation bubble. Therefore, the postcollision domain wall is formed moving away from the observation bubble interior, and no internal modes of the postcollision domain wall are excited. In both cases at late times the postcollision domain wall accelerates from the observation bubble interior, as predicted by the junction condition formalism in Eq. (4).

Figure 18

A contour plot of the field $\phi$ for the collision of two identical bubbles in the large-field potential L1. The field in the future of the collision is moved down the inflationary potential, as predicted by the free-passage approximation.

Figure 19

The field on two constant $z$ slices just before and just after the collision in the simulation shown in Fig. 18. The red-dashed line is the sum of the distance in field space between the false vacuum and the position the endpoint of the CDL instanton. In agreement with the free-passage approximation, the field profiles in the immediate future of the collision simply superpose.

Figure 20

A contour plot of $\phi$ for the collision of two different bubbles formed from the potential L2. In the immediate future of the collision event, the inflaton inside the observation bubble is perturbed, and a pocket of the false vacuum is formed. This pocket of false vacuum is a breathing mode of the postcollision domain wall, which once formed, accelerates out of the observation bubble. The breathing mode further perturbs the inflaton inside the observation bubble. The amplitude of perturbations to the inflaton is small compared to the total field excursion during inflation in this model.

Figure 21

On the left, we show a contour plot of $\phi$ for the collision of bubbles in a potential with a barrier $T1$ twice as large as that of the potential L2. Initially, the postcollision domain wall moves into the observation bubble, but then turns around. There is very little disturbance to the inflaton inside the observation bubble. On the right, we show a contour plot of $\phi$ for the collision of bubbles in a potential with a barrier $T1$ half as large as that of the potential L2. The postcollision domain wall forms accelerating out of the observation bubble, and no breathing modes are excited. These two examples are consistent with the predictions of the free-passage approximation. In both panels, the boundaries of the simulation lie at the center of each bubble.

Figure 22

A contour plot of $\phi$ for the collision of bubbles in the potential L1. Because the slope $C1$ away from the potential barrier is much steeper inside the collision bubble than inside the observation bubble, the postcollision domain wall moves into the observation bubble during its inflationary phase.

Figure 23

Contour plots of $\phi$ for the collision of bubbles in the potential S1. On the top left, we simulate the collision between two identical bubbles. In the future light cone of the collision, inflation is completely disrupted (as can be seen by comparing the color bar to the field values on the potential in Fig. 5). On the top right, we simulate the collision of two different bubbles in the case where the width of the barrier $T1$ is adjusted to be a quarter of the width of the barrier $T2$. Again, inflation is completely disrupted by the collision. On the bottom, we simulate the collision of two different bubbles in the case where the width of the barrier $T1$ is adjusted to be 4 times the width of the barrier $T2$. In this case, the inflaton inside the observation bubble is hardly perturbed, and inflation continues to the future of the collision. In all cases, the boundaries of the simulation lie at the center of each bubble.

Figure 24

$v$ (black line) and $v_0$ (red-dashed line) with $a=0.1$ and $C=2$.

Figure 25

The tunnelling exponent as a function of $ϵ$ computed for a few examples numerically (dots) and from Eq. (d21) using the upper and lower estimates of the tension (red-dashed lines).

Figure 26

The numerically computed instanton action (dots) plotted against the background subtraction (red-dashed line). Both are growing like ${ϵ}^{-4}$ as $ϵ\to 0$.