Abundance of Primordial Black Holes in Peak Theory for an Arbitrary
  Power Spectrum
  We modify the procedure to estimate PBH abundance proposed in
arXiv:1805.03946 so that it can be applied to a broad power spectrum such as
the scale-invariant flat power spectrum. In the new procedure, we focus on
peaks of the Laplacian of the curvature perturbation $\triangle \zeta$ and use
the values of $\triangle \zeta$ and $\triangle \triangle \zeta $ at each peak
to specify the profile of $\zeta$ as a function of the radial coordinate while
the values of $\zeta$ and $\triangle \zeta$ are used in arXiv:1805.03946. The
new procedure decouples the larger-scale environmental effect from the estimate
of PBH abundance. Because the redundant variance due to the environmental
effect is eliminated, we obtain a narrower shape of the mass spectrum compared
to the previous procedure in arXiv:1805.03946. Furthermore, the new procedure
allows us to estimate PBH abundance for the scale-invariant flat power spectrum
by introducing a window function. Although the final result depends on the
choice of the window function, we show that the $k$-space tophat window
minimizes the extra reduction of the mass spectrum due to the window function.
That is, the $k$-space tophat window has the minimum required property in the
theoretical PBH estimation. Our procedure makes it possible to calculate the
PBH mass spectrum for an arbitrary power spectrum by using a plausible PBH
formation criterion with the nonlinear relation taken into account.
