Effective Window Function for Lagrangian Halos

Understand

The window function for protohalos in Lagrangian space is often assumed to be a tophat in real space.

• We measure this profile directly and find that it is more extended than a tophat but less extended than a Gaussian; its shape is well-described by rounding the edges of the tophat by convolution with a Gaussian that has a scale length about 5 times smaller.
• This effective window $W_{\rm eff}$ is particularly simple in Fourier space, and has an analytic form in real space.
• Together with the excursion set bias parameters, $W_{\rm eff}$ describes the scale-dependence of the Lagrangian halo-matter cross correlation up to $kR_{\rm Lag} \sim 10 $, where $R_{\rm Lag}$ is the Lagrangian size of the protohalo.