Primordial black holes and uncertainties in the choice of the window function

Abstract

Primordial black holes (PBHs) can be produced by the perturbations that exit the horizon during inflationary phase. While inflation models predict the power spectrum of the perturbations in Fourier space, the PBH abundance depends on the probability distribution function (PDF) of density perturbations in real space. In order to estimate the PBH abundance in a given inflation model, we must relate the power spectrum in Fourier space to the PDF in real space by coarse-graining the perturbations with a window function. However, there are uncertainties on what window function should be used, which could change the relation between the PBH abundance and the power spectrum. This is particularly important in considering PBHs with mass 30M that account for the LIGO events because the required power spectrum is severely constrained by the observations. In this paper, we investigate how large influence the uncertainties on the choice of a window function have over the power spectrum required for LIGO PBHs. As a result, it is found that the uncertainties significantly affect the prediction for the stochastic gravitational waves (GWs) induced by the second order effect of the perturbations. In particular, the pulsar timing array constraints on the produced GWs could disappear for the real-space top-hat window function.

Primordial black hole tower: Dark matter, earth-mass, and LIGO black holes

“…As remarked in Ref [54],. the coarse-grained volume slightly depends on the choice of the window function as V (R) = 4πR 3 /3 for the real-space top-hat while V (R) = (2π) 3/2 R 3 for the Guassian window.…”

The primordial black hole formation criterion re-examined: Parametrisation, timing and the choice of window function

“…The sinc function profile given in equation (13) will be used, which has a critical amplitude A ≈ 1.2 in order to collapse [18]. Table I Reference [28] found that, for a fixed A s , ν c varies by around an order of magnitude when different window functions are used -and this result is confirmed when a constant δ c is used.…”

Gravitational waves induced by scalar perturbations as probes of the small-scale primordial spectrum

“…Taking into account the evolutions of Ω GW after the matter-radiation equality and the change in relativistic degrees of freedom, we can derive the relation between the density parameter at η 0 (today) and that at η c as [28] Ω GW (η 0 , k) = 0.83 g c 10.75…”