Islands in Closed and Open Universes

TL;DR

The paper investigates how spatial curvature affects entanglement islands in cosmology using a thermofield-double purified FRW setup and the quantum extremal surface framework. It shows that arbitrarily small positive curvature makes the entire spacetime an island, while proper-subset islands require a negative cosmological constant with sufficiently large magnitude and constraints on the turnaround dynamics; in open universes, islands arise for Λ<0 provided the curvature radius is large enough and curvature remains subdominant (quantified by γ<1/2). The analysis hinges on four curvature- and geometry-sensitive island conditions applied on time-symmetric slices and then extended to full spacetime solutions, yielding a rich dichotomy between open/closed and Λ signs. These results imply that the entanglement structure of cosmological spacetimes can host islands under broad curvature conditions, with potential relevance for the holographic interpretation of cosmology and multiverse scenarios, even when the observed universe is approximately flat.

Abstract

We show that spatial curvature has a significant effect on the existence of entanglement islands in cosmology. We consider a homogeneous, isotropic universe with thermal radiation purified by a reference spacetime. Arbitrarily small positive curvature guarantees that the entire universe is an island. Proper subsets of the time-symmetric slice of a closed or open universe can be islands, but only if the cosmological constant is negative and sufficiently large in magnitude.

Figures (7)

• Figure 1: Penrose diagram of a closed recollapsing universe. The entire universe can always be an island. For a proper subset $I$ to be an island, it must lie near the turnaround slice, with boundary within a certain angular range. This range is nonvanishing only if the cosmological constant is negative and sufficiently large.
• Figure 2: Penrose diagram of an open universe. A sufficiently large region $I$ at the turnaround time is an island, if the cosmological constant is negative and large enough for curvature not to dominate below the critical radius $\chi_3$.
• Figure 3: Regions satisfying the four island conditions are shown for a closed universe with $\Lambda=0$. The radiation temperature is $\beta^{-1}$ at the turnaround time $t=\eta=0$. Chosen for display is $t_C=170 t_P$. Top and bottom cutoffs are chosen so as to eliminate artifacts of the Planck regime near the big bang and big crunch. The lack of four-way overlap shows that no region satisfies all four conditions, so there cannot be any islands.
• Figure 4: Island conditions in a closed universe with $\Lambda<0$, $t_C \gg t_\Lambda$. Chosen for display is $t_C=25000t_P$, $t_\Lambda=400t_P$. All conditions overlap in a region centered on the equator with temporal width of order $\beta$ around the turnaround time. We verify explicitly that any region at $t=0$ whose only boundary lies in this overlap is an island.
• Figure 5: Island conditions in a closed universe with $\Lambda>0$, $t_C \gg t_\Lambda$. Chosen for display is $t_C=1000t_P$, $t_\Lambda=20t_P$. Such a universe expands eternally and thus has no time-symmetric slice. There is no region in which all four conditions are satisfied, meaning there cannot be islands.