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Encoding beyond cosmological horizons in de Sitter JT gravity

Levine, Adam, Shaghoulian, Edgar·2023·Journal of High Energy Physics
DOI 10.1007/jhep02(2023)179
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Abstract

Black hole event horizons and cosmological event horizons share many properties, making it natural to ask whether our recent advances in understanding black holes generalize to cosmology. To this end, we discuss a paradox that occurs if observers can access what lies beyond their cosmological horizon in the same way that they can access what lies beyond a black hole horizon. In particular, distinct observers with distinct horizons may encode the same portion of spacetime, violating the no-cloning theorem of quantum mechanics. This paradox is due precisely to the observer-dependence of the cosmological horizon — the sharpest difference from a black hole horizon — although we will argue that the gravity path integral avoids the paradox in controlled examples.

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Figures

Figure 1 Solution to JT gravity with n = 3 . The magnitude of the Casimir energy due to matter decreases as n increases, leading to crunching regions which are larger than inflating regions. The Penrose diagram is periodically identified, making the spatial topology that of a circle.
Figure 1 Solution to JT gravity with n = 3 . The magnitude of the Casimir energy due to matter decreases as n increases, leading to crunching regions which are larger than inflating regions. The Penrose diagram is periodically identified, making the spatial topology that of a circle.CC-BY · From the PDF
Figure 2 The lines in this diagram illustrate constant ˜ η and ˜ χ surfaces.
Figure 2 The lines in this diagram illustrate constant ˜ η and ˜ χ surfaces.CC-BY · From the PDF
Figure 3 Island region I for entropy of region R in an n = 3 universe. The entropy on the complement of R ∪ I factorizes into the sum of the entropies on the two intervals; the endpoints of one of the two intervals are given by { σ I , ϕ I } and { σ R , ϕ R } as appears in (3.1).
Figure 3 Island region I for entropy of region R in an n = 3 universe. The entropy on the complement of R ∪ I factorizes into the sum of the entropies on the two intervals; the endpoints of one of the two intervals are given by { σ I , ϕ I } and { σ R , ϕ R } as appears in (3.1).CC-BY · From the PDF
Figure 4 The 'purity' saddle which gives S = 0 for the entropy of one of the two hats in an n = 2 universe.
Figure 4 The 'purity' saddle which gives S = 0 for the entropy of one of the two hats in an n = 2 universe.CC-BY · From the PDF
Figure 5 We imagine taking global de Sitter and freezing two regions near I + pictured in red. We integrate over the geometry away from these intervals. Naively, there is a puzzle since the entanglement wedge for one of the intervals appears to encompass the whole universe, since the spatial cross sections are compact. The naive entanglement wedge for R 2 is pictured in blue.
Figure 5 We imagine taking global de Sitter and freezing two regions near I + pictured in red. We integrate over the geometry away from these intervals. Naively, there is a puzzle since the entanglement wedge for one of the intervals appears to encompass the whole universe, since the spatial cross sections are compact. The naive entanglement wedge for R 2 is pictured in blue.CC-BY · From the PDF
Figure 6 The true saddle for the problem of two regions near I + is actually two disconnected universes, with each region in its own copy of the original spacetime.
Figure 6 The true saddle for the problem of two regions near I + is actually two disconnected universes, with each region in its own copy of the original spacetime.CC-BY · From the PDF
Figure 7 The connected, two-hat saddle which we are interested in studying. Here we have illustrated the Lorentzian to Euclidean continuation of this saddle from the σ = 0 line. This is a hemisphere of curvature R = 2 but with a conical excess at σ = i ∞ , the south pole of the hemisphere. The conical singularity has an opening angle of 4 πn . For n hats, it would have an opening angle of 2 πn .
Figure 7 The connected, two-hat saddle which we are interested in studying. Here we have illustrated the Lorentzian to Euclidean continuation of this saddle from the σ = 0 line. This is a hemisphere of curvature R = 2 but with a conical excess at σ = i ∞ , the south pole of the hemisphere. The conical singularity has an opening angle of 4 πn . For n hats, it would have an opening angle of 2 πn .CC-BY · From the PDF
Figure 8 If we insert a conical singularity at a position which references only one of the hats, instead of both simultaneously, the dominant saddle will be two disconnected universes illustrated here. If we define the conical singularity with respect to the Hat L , then Hat L will be in a universe with angle 4 π and another, fluctuating asymptotically inflating region. Hat R will be off in its own 2 π universe. We have schematically illustrated the procedure of tying the conical singularity to a hat by the green dashed line. To prevent the saddle illustrated here from dominating the path integral, we need to tie the position of the conical singularity to both hats simultaneously. We discuss some ways of doing this in the main text.
Figure 8 If we insert a conical singularity at a position which references only one of the hats, instead of both simultaneously, the dominant saddle will be two disconnected universes illustrated here. If we define the conical singularity with respect to the Hat L , then Hat L will be in a universe with angle 4 π and another, fluctuating asymptotically inflating region. Hat R will be off in its own 2 π universe. We have schematically illustrated the procedure of tying the conical singularity to a hat by the green dashed line. To prevent the saddle illustrated here from dominating the path integral, we need to tie the position of the conical singularity to both hats simultaneously. We discuss some ways of doing this in the main text.CC-BY · From the PDF
Figure 9 One could make the two-hat connected saddle dominate the path integral by freezing some portion of the geometry in the interior. A natural choice is to freeze all regions with some dilaton value φ ≥ 0 . Here we show the saddle where we have frozen everything filled in with gray. Note that when we include c/ ˜ φ r corrections to this solution then the frozen region actually includes a neighborhood of the conical singularity.
Figure 9 One could make the two-hat connected saddle dominate the path integral by freezing some portion of the geometry in the interior. A natural choice is to freeze all regions with some dilaton value φ ≥ 0 . Here we show the saddle where we have frozen everything filled in with gray. Note that when we include c/ ˜ φ r corrections to this solution then the frozen region actually includes a neighborhood of the conical singularity.CC-BY · From the PDF
Figure 10 This figure illustrates what happens if the frozen portion of the geometry is only connected to one hat, in this case Hat L . The green-dashed lines schematically represent the frozen portion of the geometry. In this case, Hat L from replica 1 can be swapped with Hat L from replica 2 and so the dominant saddle will just be two copies of the saddle which dominates the norm, 〈 HH f | HH f 〉 . In other words, Tr [ ρ 2 ] = 〈 HH f | HH f 〉 2 where ρ is the unnormalized density matrix. The double-headed arrow indicates that the left hats from each replica have been swapped.
Figure 10 This figure illustrates what happens if the frozen portion of the geometry is only connected to one hat, in this case Hat L . The green-dashed lines schematically represent the frozen portion of the geometry. In this case, Hat L from replica 1 can be swapped with Hat L from replica 2 and so the dominant saddle will just be two copies of the saddle which dominates the norm, 〈 HH f | HH f 〉 . In other words, Tr [ ρ 2 ] = 〈 HH f | HH f 〉 2 where ρ is the unnormalized density matrix. The double-headed arrow indicates that the left hats from each replica have been swapped.CC-BY · From the PDF
Figure 11 We illustrate the boundary conditions associated to computing the purity, Tr [ ρ 2 L ] , for the state on Hat L in the Hartle-Hawking state with modified boundary conditions, where the conical singularity (marked by 'X') is inserted in the frozen portion of the geometry. The prime denotes bra vs. ket and the subscript denotes replica number. Since this dressing ties hats of the same replica number to each other, we see that the quantum extremal surface (or its Renyi-2 analog), which gave an answer of S 2 = 0 for each hat, is excluded. The only possibility is the analog of the 'Hawking' saddle, which gives a non-zero answer for S 2 .
Figure 11 We illustrate the boundary conditions associated to computing the purity, Tr [ ρ 2 L ] , for the state on Hat L in the Hartle-Hawking state with modified boundary conditions, where the conical singularity (marked by 'X') is inserted in the frozen portion of the geometry. The prime denotes bra vs. ket and the subscript denotes replica number. Since this dressing ties hats of the same replica number to each other, we see that the quantum extremal surface (or its Renyi-2 analog), which gave an answer of S 2 = 0 for each hat, is excluded. The only possibility is the analog of the 'Hawking' saddle, which gives a non-zero answer for S 2 .CC-BY · From the PDF
Figure 12 The impure saddle for region R with n = 2 .
Figure 12 The impure saddle for region R with n = 2 .CC-BY · From the PDF
Figure 13 The impure saddle for region R with n = 3 .
Figure 13 The impure saddle for region R with n = 3 .CC-BY · From the PDF
Figure 14 On the left we have reproduced figure 9, corresponding to the Euclidean preparation of a 4 π universe with frozen regions denoted in gray. On the right we have the Euclidean preparation of the thermofield double in AdS coupled to frozen flat-space 'wing' regions. The solid black line in the middle is the time-reflection symmetric point where the Euclidean manifold is pasted to the Lorentzian manifold. These two situations are analogous.
Figure 14 On the left we have reproduced figure 9, corresponding to the Euclidean preparation of a 4 π universe with frozen regions denoted in gray. On the right we have the Euclidean preparation of the thermofield double in AdS coupled to frozen flat-space 'wing' regions. The solid black line in the middle is the time-reflection symmetric point where the Euclidean manifold is pasted to the Lorentzian manifold. These two situations are analogous.CC-BY · From the PDF
Figure 15 On the right we have a pure-state black hole with an end-of-the-world brane behind the horizon. This means that the single wing system (plus boundary) encodes the entire bulk, and has fine-grained entropy S = 0 . This corresponds to unfreezing one of the two frozen regions in the TFD in figure 14. It is analogous to a single frozen hat region in dS, shown on the left. In this case the theory in the hat has fine-grained entropy S = 0 and encodes the entire universe on its own.
Figure 15 On the right we have a pure-state black hole with an end-of-the-world brane behind the horizon. This means that the single wing system (plus boundary) encodes the entire bulk, and has fine-grained entropy S = 0 . This corresponds to unfreezing one of the two frozen regions in the TFD in figure 14. It is analogous to a single frozen hat region in dS, shown on the left. In this case the theory in the hat has fine-grained entropy S = 0 and encodes the entire universe on its own.CC-BY · From the PDF
Figure 16 In the figure on the left, we imagine attaching two semi-infinite cylinders to holes at the north and south poles of the sphere. Here we have exaggerated the size of the holes relative to the size of the sphere. This prepares the CFT matter state in the vacuum on the circle, as was considered in section 2. On the right, we imagine making the CFT state thermal. This will raise the energy in the spacetime and make the wormholes longer upon analytic continuation. The red dots denote points of maximal dilaton and the black dots denote points of minimal dilaton along the σ = 0 time slice.
Figure 16 In the figure on the left, we imagine attaching two semi-infinite cylinders to holes at the north and south poles of the sphere. Here we have exaggerated the size of the holes relative to the size of the sphere. This prepares the CFT matter state in the vacuum on the circle, as was considered in section 2. On the right, we imagine making the CFT state thermal. This will raise the energy in the spacetime and make the wormholes longer upon analytic continuation. The red dots denote points of maximal dilaton and the black dots denote points of minimal dilaton along the σ = 0 time slice.CC-BY · From the PDF
Figure 17 Here we illustrate the Lorentzian interpretations of the Euclidean saddles shown in figure 16. The idea is that for finite τ 0 these saddles describe a situation where a closed dS 2 universe with multiple inflating and crunching regions is encoded in a CFT 2 on a spatial circle. The dashed lines signify the thermal entanglement between the closed universe and the CFT.
Figure 17 Here we illustrate the Lorentzian interpretations of the Euclidean saddles shown in figure 16. The idea is that for finite τ 0 these saddles describe a situation where a closed dS 2 universe with multiple inflating and crunching regions is encoded in a CFT 2 on a spatial circle. The dashed lines signify the thermal entanglement between the closed universe and the CFT.CC-BY · From the PDF

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Citations

Cites

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Metadata

Authors
Levine, A., & Shaghoulian, E.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2023-02-17
Volume
2023
Issue
2
ISSN
1029-8479
DOI
10.1007/jhep02(2023)179
Cited by
22
References
8
Language
en
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