Island and Page curve for one-sided asymptotically flat black hole

Great breakthrough in solving black hole information paradox took place when semiclassical island rule for entanglement entropy of Hawking radiation was proposed in recent years. Up to now, most papers which discussed island rule of asymptotic flat black hole with $D \ge 4$ focus on eternal black hole. In this paper, we take one more step further by discussing island of "in" vacuum state which describes one-sided asymptotically flat black hole formed by gravitational collapse in $D \ge 4$. We find that island $I$ emerges at late time and saves entropy bound. And boundary of island $\partial I$ depends on the position of cutoff surface. When cutoff surface is far from horizon, $\partial I$ is inside and near horizon. When cutoff surface is set to be near horizon, $\partial I$ is outside and near horizon. This is different from the case of eternal black hole in which $\partial I$ is always outside horizon no matter cutoff surface is far from or near horizon. We will see that different states will manifestly affect $S_{\text{ent}}$ in island formula when cutoff surface is far from horizon and thus have different result for Page time.
Comment: v1:22 pages, 2 figures,v2: typos corrected, an appendix on s-wave approximation is added, references added, published version