Towards observable signatures of other bubble universes

We evaluate the possibility of observable effects arising from collisions between vacuum bubbles in a universe undergoing false-vacuum eternal inflation. Contrary to conventional wisdom, we find that under certain assumptions most positions inside a bubble should have access to a large number of collision events. We calculate the expected number and angular size distribution of such collisions on an observer’s “sky,” finding that for typical observers the distribution is anisotropic and includes many bubbles, each of which will affect the majority of the observer’s sky. After a qualitative discussion of the physics involved in collisions between arbitrary bubbles, we evaluate the implications of our results, and outline possible detectable effects. In an optimistic sense, then, the present paper constitutes a first step in an assessment of the possible effects of other bubble universes on the cosmic microwave background and other observables.

Figure 1

On the left is the embedding of two dS spaces of different vacuum energy in 5D Minkowski space (three dimensions suppressed). The construction obtained by matching these two hyperboloids along a plane of constant $X_4$, as shown on the right, corresponds to the one-bubble spacetime shown in Fig. 2 in the limit where the bubble interior is pure dS. The light shaded (green) region represents the false-vacuum exterior spacetime, while the dark shaded (blue) region represents the interior spacetime.

Figure 2

The conformal diagram for a bubble universe. We imagine an observer at some position $({\xi }_o,{\tau }_o,{\theta }_o)$ inside of the observation bubble, which is assumed to nucleate at $t=0$ and expand at the speed of light. The foliation of the bubble interior into constant density, negative curvature, hyperbolic slices is indicated by the solid lines. These spacelike slices denote epochs of cosmological evolution in the open FRW cosmology inside of the bubble. The past light cone of the observer is indicated by the dashed lines. There is a postulated no-bubble surface at some time to the past of the nucleation of the observation bubble. Also shown is a ($\theta ={\theta }_n$, $\varphi ={\varphi }_n$ slice of a) colliding bubble that nucleated at some position $(t_n,r_n,{\theta }_n,{\varphi }_n)$, and intersects the bubble wall within the past light cone of the observer.

Figure 3

A time lapse picture of the null rays reaching an observer from the boundary of the region affected by a collision event in the Poincaré disk representation. The boundaries are located at angles ${\theta }_{1,2}$ from the center of the disk, and at angles ${\alpha }_{1,2}$ from the location of the observer. The total angular scale of the collision event as recorded by the observer, which affects the region of the disk indicated by the double lines, is given by $\psi$.

Figure 4

The effects of the boost. The top cell shows the boosted initial value surface (at $t_0\to -\infty$ in the unboosted frame) for small (left) and large (right) ${\xi }_o$ for a variety of angles (with the bottom curve (red) corresponding to $\theta =0$, the top (yellow) corresponding to $\theta =\pi$, and other lines corresponding to intermediate angles at intervals of $\pi /4$). The bottom cell shows the effects of the boost on points in the exterior spacetime on a slice of constant ($\varphi$, $\theta =0$). Note that even for this very modest boost (${\xi }_o=2$), most of the points are condensed into the wedge created by the past light cone of the nucleation event and the boosted initial value surface.

Figure 5

A spatial slice in the global foliation of the background de Sitter space, and its stereographic projection. The observation bubble is shaded light (yellow) and the colliding bubble is shaded dark (blue). The angle $\psi$ is indicated in the plane of projection.

Figure 6

The three cases of bubble intersection in the plane of projection. The top left cell displays the case where the bubble interior does not encompass the south pole of the projected two-sphere, the top right cell displays the case where the bubble wall intersects the south pole, and the lower cell displays the case where the bubble interior includes the south pole.

Figure 7

The foliation of the exterior de Sitter space into surfaces of constant $\psi$ for junctions with $H_T\sim H_F$ (left) and $H_T\ll H_F$ (right). Dark regions correspond to small $\psi$ and light regions correspond to large $\psi$. Superimposed on this picture is the boosted initial value surface for various ${\theta }_n$ in the limit of large-${\xi }_o$.

Figure 8

The distribution function Eq. (26) for an observer at ${\xi }_o=0$ with (from the blue curve on the bottom to the red curve on top) $T_{\mathrm{co}}=\frac{\pi }{4}$, $\frac{3\pi }{8}$, $\frac{\pi }{2}$ (corresponding to a varying $H_T$), factoring out the overall scale $\lambda H_F^{-4}$. (This factor will in general be astronomically small, but we choose this convention to more clearly display the functional behavior of the distribution function.) This function is independent of ${\theta }_n$ for this observer. As $T_{\mathrm{co}}\to \pi /2$ ($H_T/H_F\to 0$), a divergent peak around $\psi =0$ develops.

Figure 9

The distribution function Eq. (26) for an observer at ${\xi }_o=25$, with $T_{\mathrm{co}}=\frac{\pi }{4}$, for $\theta =\frac{\pi }{10}$, $\frac{\pi }{15}$, $\frac{\pi }{20}$, factoring out the overall scale $\lambda H_F^{-4}$. As ${\theta }_n\to 0$, the position of the peak shifts to larger $\psi$, and increases in amplitude, displaying the predicted anisotropic peak about large angular scales.

Figure 10

The distribution function Eq. (26) with ${\theta }_n=0$ and ${\tau }_{\mathrm{co}}=\frac{3\pi }{8}$ for ${\xi }_o=(1.5,2,100)$, factoring out the overall scale $\lambda H_F^{-4}$. As ${\xi }_o$ gets large, the peak near $\psi \sim 2\pi$ grows, while the peak near $\psi \sim 0$ remains of constant amplitude.

Figure 11

The distribution function Eq. (26) with ${\theta }_n=0$ and ${\xi }_o=2$ for $T_{\mathrm{co}}=(\frac{\pi }{4},\frac{3\pi }{8},\frac{7\pi }{16})$, factoring out the overall scale $\lambda H_F^{-4}$. As $T_{\mathrm{co}}$ grows, the bimodality of the distribution becomes more and more pronounced. Both the peak about $\psi \simeq 0$ and $\psi \simeq 2\pi$ grow, with the growth of the $\psi \simeq 0$ peak eventually overtaking the growth of the $\psi \simeq 2\pi$ peak. The position of the peaks shift as well, with one peak approaching $\psi =0$ and the other $\psi =2\pi$ as $T_{\mathrm{co}}\to \frac{\pi }{2}$.

Figure 12

A log-log plot (calculated numerically) of the total angular area on the sky taken up by late-time collisions with $\psi \simeq 0$.

Figure 13

A general set of situations which might involve collisions between two bubbles in an eternally inflating spacetime. Each cell represents a region near future null infinity (horizontal solid line) of an eternally inflating background dS. True-vacuum bubbles form at very small radius and expand, while false-vacuum bubbles form larger than the exterior horizon size, and contract. Collisions are denoted by filled circles, with the uncertainty of the post-collision spacetime indicated by a question mark.