Surviving the crash: Assessing the aftermath of cosmic bubble collisions

This paper is the third in a series investigating the possibility that if we reside in an inflationary “bubble universe,” we might observe the effects of collisions with other such bubbles. Here, we study the interior structure of a bubble collision spacetime, focusing on the issue of where observers can reside. Numerical simulations indicate that if the interbubble domain wall accelerates away, infinite spacelike surfaces of homogeneity develop to the future of the collision; this strongly suggests that observers can have collisions to their past, and previous results then imply that this is very likely. However, for observers at nearly all locations, the restoration of homogeneity relegates any observable effects to a vanishingly small region on the sky. We find that bubble collisions may also play an important role in defining measures in inflation: a potentially infinite relative volume factor arises between two bubble types depending on the sign of the acceleration of the domain wall between them; this may in turn correlate with observables such as the scale or type of inflation.

Figure 1

The relation between the scalar potential and the bubble spacetime. The open patch with metric (4) fills the forward light cone emanating from the nucleation center, where $\tau$ labels successive spacelike hyperboloids and $\xi$ is the coordinate distance from the origin on each hyperboloid. Constant-$\xi$ trajectories can also be viewed as geodesics passing through the nucleation center with various boosts, as indicated by the congruence at the bottom of the picture. Region 1, which comprises the bubble wall, is determined by the analytic continuation of the instanton, whose end point (black dot on the potential) analytically continues to the light cone (dashed line between regions 1 and 2). Region 2 is the period of curvature domination, which then gives way to inflation in region 3 as the universe expands. Inflation ends in region 4, and the field oscillates and reheats the universe. In region 5, the field settles into its minimum and standard big-bang FLRW cosmology begins.

Figure 2

An accelerated worldline in the background de Sitter spacetime (solid red curve) starts at rest at a nucleation point $P_{\alpha }$ and meets an observer at some $({\tau }_{\mathrm{obs}},{\xi }_{\mathrm{obs}})$ in an open universe that also nucleates at $X_0=0$. The trajectory lies on the intersection of the hyperboloid and a hyperplane that is a distance $\alpha$ from the $X_0$ axis and parallel to it. The angle ${\chi }_{\alpha }$ between the plane and the $X_4$ axis is fixed by the observer position and $\alpha$.

Figure 3

A $2+1\mathrm{D}$ slice of the collision between two bubbles in Minkowski space (plotting several constant-field surfaces from the fiducial simulation below) showing the relation between the Cartesian coordinates $(t,x,y,w)$ (with $w=0$ suppressed) and the hyperbolic coordinates $(z,x,\chi ,\phi =0)$. The depicted lines of constant $z$ are hyperbolae in the Cartesian coordinates. Given the SO(2,1) symmetry, the simulation region represents the region above the planes $y=\pm t$, or can alternatively be taken as the slice ($y=w=0$), i.e. the Cartesian $t-x$ plane.

Figure 4

The rescaled large-field potential used in numerically evolving $\varphi$. In order to neglect gravitational effects (so that the bubble radius is much smaller than a false-vacuum Hubble volume), the inflationary region of the potential is very broad compared to the barrier region in the vicinity of the false vacuum at $\varphi =0$. On top, we zoom in on the potential in the barrier region. Important field values are labeled with a C or O superscript for the collision or observation bubble, a min or max superscript for local minima or maxima, and a tun superscript for values of the (posttunneling) instanton end point. A selection of these values are also labeled in the plots of $\varphi (x,z)$ and $\varphi (r,t)$. This potential is qualitatively similar to the potential in Fig. 1.

Figure 5

The rescaled small-field potential used in numerically evolving $\varphi$. Important field values are labeled with a C or O superscript for the collision or observation bubble, a min or max superscript for local minima or maxima, and a tun superscript for values of the (posttunneling) instanton end point.

Figure 6

Evolution of $\varphi (\tau )$ for a direct integration of the field equation for homogeneous slices of constant $\tau$ with (dotted line) or without (dashed line) gravity. The solid line is the simulation field at $\xi =0$; the corner at $\tau \sim 50$ is the impact of the colliding bubble.

Figure 7

An SO(3,1) invariant bubble solution. The spacelike surfaces of homogeneity foliate the interior of the light cone emanating from the nucleation center.

Figure 8

An expanding bubble obtained by perturbing the initial data for a static O(3)-invariant solution. It can be seen that at late times, there are spacelike hyperbolic surfaces of homogeneity that fill the expanding bubble wall. The surfaces of constant field (black lines) are fit to hyperbolas (white dashed lines), with origins indicated by the diamonds.

Figure 9

The collision of two identical bubbles. Each point in the simulation plane corresponds to a 2-hyperbola of radius $z$. Numerical fitting of surfaces that look hyperboloidal shows that in general, they actually are hyperboloids to excellent accuracy.

Figure 10

Collision of two bubbles. Each point corresponds to a 2-hyperbola of radius $z$. The observation bubble is centered on $x=0$; it interpolates between the false minimum ${\varphi }_{\mathrm{F},\min }$ and ${\varphi }_{\mathrm{O},\mathrm{tun}}$ across the right barrier (see Fig. 4). The collision bubble is centered on $x=50$; it interpolates between the false minimum and ${\varphi }_{\mathrm{C},\mathrm{tun}}$ across the left barrier. The invariant distance from the origin, $\Delta s^2=H^{-2}$ which corresponds to the false-vacuum Hubble radius, is depicted; we expect that gravitational effects will be negligible within this region.

Figure 11

The same collision shown in Fig. 10, but boosted by $\gamma =8$ in the $-x$ direction and showing only the ($x-z$) plane (see text). The boundaries of the region correspond to the (highly boosted) edges of the simulation in the original frame, and the region shown corresponds to a region very near the domain wall in Fig. 10. As in Fig. 8, the surfaces of constant field (black lines) are fit to hyperbolas (white dashed lines), with origins indicated by the diamonds. The (small) scatter in these origins is caused by the lower resolution available in this highly boosted frame.

Figure 12

A collisions between bubbles in an accidental inflation scenario. Prior to the collision, the field is essentially homogeneous inside the observation bubble; but in the region hit by the bubble, the inflection point no longer coincides with zero kinetic energy in the field, and inflation is lost.

Figure 13

A toy model for accidental open inflation with $\{v_0=1.36$, ${\lambda }_1=0.65$, ${\lambda }_2=-1.2$, ${\lambda }_4=1$, ${ϵ}_1=0.788$, ${ϵ}_2=9.778$, ${ϵ}_3=-2\pi$, ${ϵ}_4=1$, $g=0.12\}$. The field tunnels in the $\psi$ direction, beginning an epoch of open inflation near the inflection point in the $\varphi$ direction.