Holography for de Sitter bubble geometries

Abstract

We generalize the de Sitter static patch holographic proposal and the bilayer holographic entanglement entropy prescription to de Sitter geometries containing a bubble of smaller positive or vanishing cosmological constant. When the causal patch of an observer at the center of the bubble overlaps with the ``parent'' de Sitter region, we propose that the full spacetime can be holographically encoded on two holographic screens. The leading geometrical entanglement entropy between the two screens, which can be constant or time-dependent in some cases, never exceeds the Gibbons-Hawking entropy associated with the ``parent'' de Sitter space. When the causal patch of the observer at the center of the bubble is causally disconnected from the ``parent'' de Sitter region, the holographic proposal no longer applies, and more than two holographic screens are required to encode the whole spacetime.

Figures (16)

• Figure 1: Penrose diagram for the de Sitter spacetime. Spacelike slices have the topology of a sphere and the worldlines of two antipodal observers follow the left and right edges of the diagram. Holographic degrees of freedom lie on the two cosmological horizons depicted by the diagonal lines. The black dots depict the intersections of the cosmological horizons with a bulk Cauchy slice $\rm{\Sigma}$, on which the holographic screens ${\cal S}_{\rm L}$ and ${\cal S}_{\rm R}$ are located.
• Figure 2: The three de Sitter bubble geometries considered in this paper. The blue (red) region is the interior (exterior) of the bubble, which is a part of dS$_{n+1}$ with a smaller (larger) cosmological constant. Each region is bounded by a thin domain wall trajectory (brown lines). The gray-shaded region is not a part of the spacetime. The worldlines of the two antipodal observers correspond to the left and right vertical edges of the diagrams. Each of them has a causal patch bounded by future and past cosmological horizons depicted by the thick dashed lines.
• Figure 3: We cut two different solutions of the Euclidean Einstein's equations with different cosmological constants, $\Lambda_L < \Lambda_R$ ($R_L>R_R$), along a cycle (brown lines) of the same size. In the thin-wall approximation, we glue the parts of the two solutions (shaded regions) together along this cycle. Note that for a specific choice of $R_B$, there are different choices of spherical caps - smaller or larger than a hemisphere - to glue together. Each geometry can be labeled by two signs $\varepsilon_L=\pm 1$, $\varepsilon_R=\pm 1$, as defined in Eq. \ref{['eq:lorenzian_dom_walls']}.
• Figure 4: The Penrose diagrams for the $(\varepsilon_L,\varepsilon_R)=(+1,+1)$ bubble geometry. The blue (red) region is the interior (exterior) of the bubble, which is a part of dS$_{n+1}$ with a smaller (larger) cosmological constant. Appropriate boundary conditions enforce the identification of the points on the left and right domain wall trajectories bounding the two de Sitter regions. The left and right vertical edges correspond to the worldlines of two antipodal observers. Each of them has a causal patch bounded by future and past cosmological horizons depicted by the thick dashed lines. The apparent horizons associated with the left observer at the center of the bubble are depicted by the thin dotted lines. Bousso wedges are drawn in each region of the diagrams.
• Figure 5: The Penrose diagrams for the $(\varepsilon_L,\varepsilon_R)=(+1,-1)$ bubble geometry. The blue (red) region is the interior (exterior) of the bubble, which is a part of dS$_{n+1}$ with a smaller (larger) cosmological constant. Appropriate boundary conditions enforce the identification of the points on the left and right domain wall trajectories bounding the two de Sitter regions. The causal patches of the pode and antipode observers are bounded by future and past cosmological horizons, depicted by the thick dashed lines. Bousso wedges are drawn in each region of the diagrams.