Threshold of primordial black hole formation

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 = wρc 2 for w ≥ 0. The formula gives δ UH Hc = sin, where δ UH Hc andδ 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 δ UH Hc = w and δ 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 δ UH Hc = sin 2 ( √ 3π/6) ≃ 0.6203 andδ c = (2/3) sin 2 ( √ 3π/6) ≃ 0.4135. We also discuss the maximum amplitude and the cosmological implications of the present result.

Universal threshold for primordial black hole formation

“…The upper inner plot of Fig. 3 shows that δ N c reaches values that are slightly smaller than the 0.41 limit quoted in [10] and later in [3], but they do not drop below the 0.4 limit of our Eq. (7).…”

“…The threshold for the average compaction function (C c = 2/5 = 0.4) is very close to the so-called Harada-Yoo-Kohri (HYK) limit which was analytically found to be ∼ 0.41. The value of the second significant digit is related to assumptions about the Jeans length of the perturbation [10]. As already mentioned, in Fig.…”

“…The abundance of these statistical fluctuation, and in turn of the generated PBHs, is extremely sensitive to the threshold required to form a PBH [6,8,9,12]. To date, analytical estimates of this threshold (see for example [10,13]) are insufficiently accurate, so numerical analyses have been employed (for the latest results see [3,5]).…”

“…where C hom c is the threshold for a homogeneous ball. As a first approximation, one then may consider the functional form [10]C…”

Quantum diffusion during inflation and primordial black holes

“…This can be solved as 19) where W 0 is the principal branch of the Lambert function, and its argument comes with a minus sign in RMI 3 and with a plus sign in RMI 4 . The approximation | ln(φ well /φ 0 )| 1 is satisfied when the argument of the Lambert function in Eq.…”