Stochastic binary tree method computes compaction function in inflation to distinguish type I/II PBH fluctuations, finding broader mass distributions and type-II dominance in quantum regimes of a toy model.
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Threshold of primordial black hole formation
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abstract
Based on a physical argument, we derive a new analytic formula for the amplitude of density perturbation at the threshold of primordial black hole formation in the universe dominated by a perfect fluid with the equation of state $p=w\rho c^{2}$ for $w\ge 0$. The formula gives $\delta^{\rm UH}_{H c}=\sin^{2}[\pi \sqrt{w}/(1+3w)]$ and $\tilde{\delta}_{c}=[3(1+w)/(5+3w)]\sin^{2}[\pi\sqrt{w}/(1+3w)]$, where $\delta^{\rm UH}_{H c}$ and $\tilde{\delta}_{c}$ are the amplitude of the density perturbation at the horizon crossing time in the uniform Hubble slice and the amplitude measure used in numerical simulations, respectively, while the conventional one gives $\delta^{\rm UH}_{H c}=w$ and $\tilde{\delta}_{c}=3w(1+w)/(5+3w)$. Our formula shows a much better agreement with the result of recent numerical simulations both qualitatively and quantitatively than the conventional formula. For a radiation fluid, our formula gives $\delta^{\rm UH}_{H c}=\sin^{2}(\sqrt{3}\pi/6)\simeq 0.6203$ and $\tilde{\delta}_{c}=(2/3)\sin^{2}(\sqrt{3}\pi/6)\simeq 0.4135$. We also discuss the maximum amplitude and the cosmological implications of the present result.
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