Primordial black holes and uncertainties in the choice of the window
  function
  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 $30 M_\odot$ 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.
