Effective window function for Lagrangian halos

摘要

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 aGaussian; its shape is well described by rounding the edges of the top hat by convolution with a Gaussianthat has a scale length about 5 times smaller. This effective window W_(eff) is particularly simple in Fourierspace, 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 kR_(Lag) ∼ 10, whereR_(lag) is the Lagrangian size of the protohalo. Moreover, with this W_(eff) , all the spectral moments of thepower spectrum are finite, allowing a straightforward estimate of the excursion set peak mass function. Thisestimate 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 onlyaccurate to about 20%, and we discuss why: a top hat in “infall time” towards the protohalo center need notcorrespond to a top hat in the initial spatial distribution, so models in which infall rather than smoothedoverdensity is the relevant variable may be more accurate.