Grey-body Factors, Irreversibility and Multiple Island Saddles

We consider the effect of grey-body factors on the entanglement island prescription for computing the entropy of an arbitrary subset of the Hawking radiation of an evaporating black hole. When there is a non-trivial grey-body factor, the modes reflected back into the black hole affect the position of the quantum extremal surfaces at a subleading level with respect to the scrambling time. The grey-body factor allows us to analyse the role of irreversibility in the evaporation. In particular, we show that irreversibility allows multiple saddles to dominate the entropy, rather than just two as expected on the basis of Page's theorem. We show that these multiple saddles can be derived from a generalization of Page's theorem that involves a nested temporal sequence of unitary averages. We then consider how irreversibility affects the monogamy entanglement problem.