Window function dependence of the novel mass function of primordial black holes

The mass function of primordial black holes (PBHs) has been discussed by many authors so far and, recently, the novel formulation of the PBH mass function has been proposed to avoid uncertainty of the mass in the conventional mass function. However, the novel formulation still has the ambiguity originating from the choice of the window function. In this paper, we discuss the window function dependence of the novel mass function, focusing on the exponential factor in the mass function. In particular, we consider the top-hat power spectrum of the curvature perturbation as a fiducial example. As a result, we find that while the window function dependence of the exponential factor in the novel mass function is the same as the one in the conventional mass function around the top-hat scale, the dependence is different on other scales, which leads to the narrower mass function in the novel formulation for some window functions.