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=wρc² for w≥0. The formula gives δHcUH=sin²[π√w/(1+3w)] and δc=[3(1+w)/(5+3w)]sin²[π√w/(1+3w)], where δHcUH and δc 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 δHcUH=w and δc=3w(1+w)/0ex0ex(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 δHcUH=sin²(√3π/6)0.6203 and δc=(2/3)sin²(√3π/6)0.4135. We also discuss the maximum amplitude and the cosmological implications of the present result.