Small Schwarzschild de Sitter black holes, the future boundary and islands

We continue the study of 4-dimensional Schwarzschild de Sitter black holes in the regime where the black hole mass is small compared with the de Sitter scale, following arXiv:2207.10724 [hep-th]. The de Sitter temperature is very low compared with that of the black hole. We consider the future boundary as the location where the black hole Hawking radiation is collected. Using 2-dimensional tools, we find unbounded growth of the entanglement entropy of radiation as the radiation region approaches the entire future boundary. Self-consistently including appropriate late time islands emerging just inside the black hole horizon leads to a reasonable Page curve. We also discuss other potential island solutions which show inconsistencies.