Odds of observing the multiverse

Eternal inflation predicts that our observable universe lies within a bubble (or pocket universe) embedded in a volume of inflating space. The interior of the bubble undergoes inflation and standard cosmology, while the bubble walls expand outward and collide with other neighboring bubbles. The collisions provide either an opportunity to make a direct observation of the multiverse or, if they produce unacceptable anisotropy, a threat to inflationary theory. The probability of an observer in our bubble detecting the effects of collisions has an absolute upper bound set by the odds of being in the part of our bubble that lies in the forward light cone of a collision; in the case of collisions with bubbles of identical vacua, this bound is given by the bubble nucleation rate times ($H_{\mathrm{O}}/H_{\mathrm{I}}{)}^2$, where $H_{\mathrm{O}}$ is the Hubble scale outside the bubbles and $H_{\mathrm{I}}$ is the scale of the second round of inflation that occurs inside our bubble. Similar results were obtained by Freigovel et al. using a different method for the case of collisions with bubbles of much larger cosmological constant; here, it is shown to hold in the case of collisions with identical bubbles as well.

Figure 1

An idealized potential $V(\varphi )$ used as the model for eternal inflation in this paper. Over some volume of space, a scalar field $\varphi$ is initially trapped by a barrier in a false vacuum (left), causing the volume to inflate. At an exponentially small rate, bubbles are formed inside which the scalar field jumps to a value just to the other side of the barrier. The field then slow-rolls down the potential, causing a round of inflation on the interior. Eventually, the field approaches a true minimum, oscillates around the minimum, and heats up the bubble interior. In the meantime, the bubble wall expands outward into the eternally inflating space surrounding it and colliding with other bubbles.

Figure 2

A space-time diagram that represents the geometry of how an observer inside a bubble views a collision. (a) The light cone drawn with solid lines represents a bubble whose walls are expanding outward at the speed of light. Outside the bubble, the inflaton field is trapped in the false phase; inside, the field has settled into the true phase. An observer in the interior looks back along his light cone, which ultimately intersects the bubble wall and defines an ellipse. (b) The collision of two bubbles defines a hyperbolic scar on the bubble wall. (c) If the ellipse and hyperbola intersect, they define the part of the observer’s sky that is affected by the collision. (d) Tracing the endpoints back, one can determine the angular scale of the collision.

Figure 3

A single collision divides our bubble into three regions, colored white, yellow, and red. White observers are outside the light cone of the collision and, therefore, see nothing. Yellow observers (light gray, in print) see a collision that occupies less than half their sky, and red observers (dark gray, in print) see a collision that occupies more than half. Because the two bubbles are overlapping, the assignment of the colors yellow and red is not unique; an observer in the other bubble reverses them. An appropriate interpretation is that the red volume has been excised by the collision. The two bubbles are stitched together along the seam, and only white and yellow remain.

Figure 4

A fixed-time slice of our Universe in a compact (Poincaré) representation. Collisions with an infinite number of other bubbles create a fractal web of yellow (light gray) and red (dark gray) regions that surrounds the exterior of our bubble. Because of the compact representation, most of the volume lies around the rim of the disk where the structure is fractal, as can be seen in the inset. To compute the ratio of white to yellow, the plan of attack is to first perform an angular average to determine the fraction of the surface of a “test sphere” of radius $\xi$ that is each color; the radial average is carried out subsequently.

Figure 5

Lorentz invariance is broken by the initial conditions slice, which defines a rest frame for the bubble. Only one observer, represented by the square, is in the center of the bubble in this frame. Other observers, represented by the dot, are far from the center and are more likely to see a collision. A boost that translates such an observer to the center of the bubble also affects the initial conditions slice, which resolves any possible paradox.

Figure 6

A Penrose diagram representing one bubble embedded in an exterior de Sitter space. An observer at position ($\tau ,\xi$) inside the bubble looks back and sees a collision from a bubble that nucleated at ($T,\eta$) in the exterior space. Several time slices inside the bubble are shown in various colors. Each time slice inside the bubble can be identified by a parameter $T_{\mathrm{co}}$, which represents the coordinate $T$ at which the backwards light cone of the central observer intersects the bubble wall, illustrated here for the slice marked “Present.” The “Inflation Ends” slice is drawn sagging slightly below horizontal, because in this figure $H_{\mathrm{I}}$ is assumed to be roughly $H_{\mathrm{O}}$. If $H_{\mathrm{I}}$ were much smaller, this slice would instead bulge way up into the hat; and, likewise, the smaller $\Lambda$ is in comparison to $H_{\mathrm{O}}$, the pointier the hat.

Figure 7

Consider a test sphere or radius $\xi$ drawn about the center of the bubble. As time passes, the yellow and red regions from a single collision encroach into the bubble and increasingly overlap, or paint, the surface of our test sphere. A bubble that collides with ours will paint the surface of our test sphere of radius $\xi$ either yellow (light gray) or red (dark gray). This defines the quantity $\Omega (\xi ,u,v)$.

Figure 8

As its radius $\xi$ is increased, the test sphere increasingly overlaps with the yellow or red region, and is thus “painted” either yellow or red. The number of steradians of solid angle $\Omega$ that have been painted is plotted against the sphere radius $\xi$ (in Hubble units). In this case, the collision bubble is taken to have nucleated at $(u,v)=(.25,1.25)$ and $T_{\mathrm{co}}=\pi /4$.

Figure 9

An infinite number of collisions are considered and weighted by their likelihood. The total number of steradians of solid angle that are painted each color, ${\Omega }_{\mathrm{tot}}$, is plotted as a function of $\xi$ (in Hubble units) for various values of $T_{\mathrm{co}}$. Overlaps are explicitly being ignored, which is why both functions diverge at large $\xi$ instead of asymptoting $4\pi$. The essential feature is that the plots quickly become linear with the same slope. The only attribute that $T_{\mathrm{co}}$ appreciably affects is the distance between the two curves.

Figure 10

Despite the fact that the rim of the Poincaré disk is predominantly red, a few points of measure zero are not, which can occur when a finger of white stretches all the way out to the bubble wall. Because of the diverging volume element at the rim of the disk, an infinite volume of white and yellow is actually contained in the tip and so it is overwhelmingly likely that Earth lies in one.