Framework for the string theory landscape

It seems likely that string theory has a landscape of vacua that includes very many metastable de Sitter spaces. However, as emphasized by Banks, Dine, and Gorbatov, no current framework exists for examining these metastable vacua in string theory. In this paper we attempt to correct this situation by introducing an eternally inflating background in which the entire collection of accelerating cosmologies is present as intermediate states. The background is a classical solution which consists of a bubble of zero cosmological constant inside de Sitter space, separated by a domain wall. At early and late times the flat space region becomes infinitely big, so an S-matrix can be defined. Quantum mechanically, the system can tunnel to an intermediate state which is pure de Sitter space. We present evidence that a string theory S-matrix makes sense in this background, and that it contains metastable de Sitter space as an intermediate state.

Figure 1

On the left, the Penrose diagram for de Sitter space. On the right, the conformal diagram for $1+1$ dimensional de Sitter space.

Figure 2

On the left, a bubble of smaller cosmological constant forming inside de Sitter space. On the right, a bubble of flat space forming inside de Sitter space.

Figure 3

A potential which has a metastable minimum with positive energy and a stable minimum at infinity with zero energy.

Figure 4

Wick rotation of the Euclidean geometry yields a Lorentzian geometry with this Penrose diagram. The geometry is not geodesically complete and can be analytically continued to yield the full Lorentzian solution.

Figure 5

On the left, the Penrose diagram for the full Lorentzian geometry. On the right, the value of the scalar field is shown. At the top right of the diagram, the scalar field rolls to its true minimum giving $\Lambda =0$; at the top left the field is in the false vacuum and $\Lambda >0$.

Figure 6

The thin-wall limit of the geometry consists of flat space separated by a domain wall from de Sitter space. The true Penrose diagram is half of the figure. The diamond-shaped region is the causal patch of an observer at the center.

Figure 7

Both the Euclidean and Lorentzian geometries can be embedded in higher dimensional flat space. On the left, the Euclidean version is the surface of a sphere intersected by a plane. On the right, the Lorentzian version is a hyperboloid intersected by a plane. In both cases, the domain wall is at the boundary between the curved part and the flat part. The cosmological constant jumps across the brane.

Figure 8

The causal patch of our observer can be foliated by time slices as shown. The true Penrose diagram is half of the figure.

Figure 9

There are similarities between the Penrose diagram for the bounce background (left) and the Schwarzschild black hole (right). Constant time surfaces are shown. In both cases, we believe that unitary evolution exists between part of the $t=-\infty$ surface and part of the $t=+\infty$ surface.

Figure 10

Ordinary scattering behind the black hole horizon can be translated into a very complicated process occurring outside the horizon.

Figure 11

There is no bound on the entropy contained in the upper “hat,” as this Bousso-Penrose diagram shows.

Figure 12

The effective potential for the motion of the domain wall has a barrier. The classical evolution corresponds to coming in from infinity, bouncing off the barrier, and going back out. Tunneling in to the center means that the brane reaches 0 radius: it disappears, leaving pure de Sitter space.

Figure 13

Adding a mass inside the bubble causes the causal patch of the observer at the center to “expand” as shown in the figure. The new causal patch is shown in bold.

Figure 14

A “solution” in which the brane tunnels into the minimum at zero size for a time can be illustrated by this conformal diagram. The evolution is classical except at the points where the brane disappears and is nucleated. It is not really nucleated at zero size unless the tension is zero.

Figure 15

Escher’s famous figure illustrates properties of the hyperbolic plane, which is the spatial geometry seen by the FRW observer.

Figure 16

The S-matrix of a quantum mechanical particle moving in this potential exhibits a pole at the energy of the bound state.

Figure 17

In the limit of zero brane tension, the domain wall always moves at the speed of light. Conformal diagrams for the classical solution (left) and the tunneling solution (right) are shown.

Figure 18

At left, a slice through the embedding shows the classical solution in the limit of zero brane tension. The hyperboloid is sliced through by a plane (thick vertical line). At right, a tunneling solution. The hyperboloid is sliced by two semi-infinite planes (thick rays). In both cases, time slices in the rest frame of the bubble are shown. They are horizontal lines on the left; on the right they are boosted so they are not horizontal anymore. One can see that the area between the top and bottom horizontal lines on the left is the same as the area between the top and bottom boosted horizontal lines on the right, so the action is equal for the two configurations aside from the tunneling factor.