Threshold of primordial black hole formation
  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.
