Effective window function for Lagrangian halos

The window function for protohalos in Lagrangian space is often assumed to be a top hat in real space. We measure this profile directly and find that it is more extended than a top hat but less extended than a Gaussian; its shape is well described by rounding the edges of the top hat by convolution with a Gaussian that has a scale length about 5 times smaller. This effective window W_eff is particularly simple in Fourier space, and has an analytic form in real space. Together with the excursion set bias parameters, W_eff describes the scale-dependence of the Lagrangian halo-matter cross correlation up to k R_Lag∼10 , where R_Lag is the Lagrangian size of the protohalo. Moreover, with this W_eff, all the spectral moments of the power spectrum are finite, allowing a straightforward estimate of the excursion set peak mass function. This estimate requires a prescription of the critical overdensity enclosed within a protohalo if it is to collapse, which we calibrate from simulations. We find that the resulting estimate of halo abundances is only accurate to about 20%, and we discuss why: a top hat in "infall time" towards the protohalo center need not correspond to a top hat in the initial spatial distribution, so models in which infall rather than smoothed overdensity is the relevant variable may be more accurate.