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 >= 0. The formula gives delta(UH)(Hc) = sin(2)[pi root w/(1 + 3w)] and (delta) over tilde (c) = [3(1 + w)/(5 + 3w)]sin(2)[pi root w/(1 + 3w)], where delta(UH)(Hc) and (delta) over tilde (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(UH)(Hc) = w and (delta) over tilde (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(UH)(Hc) = sin(2)(root 3 pi/6) similar or equal to 0.6203 and (delta) over tilde (c) = (2/3)sin(2)(root 3 pi/6) similar or equal to 0.4135. We also discuss the maximum amplitude and the cosmological implications of the present result.