Bounded islands in dS$_{2}$ multiverse model

Abstract: The cosmological event horizons are observer-dependent, which might bring a paradox. As an example, in dS$_{2}$ multiverse model there are entanglement islands in crunching regions encoding the information of regions near future infinity of inflating or Minkowski bubbles, however, for two observers in different bubbles, since their island regions overlap, both observers will be able to get access to the information encoded in the overlapping region, indicating a violation of no-cloning theorem. In this paper, we present a different resolution to this paradox. Based on the Petz $\mathrm{R\acute{e}nyi}$ mutual information, we show that besides the quantum extremal surfaces there might be another boundary for the island in corresponding spacetime so that the island regions are bounded by ``division points" rather than extending to the rest of the entire spacetime. We also discuss the implications of our result.