Effective Window Function for Lagrangian Halos

Abstract

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 $W_{\mathrm{eff}}$ is particularly simple in Fourier space, and has an analytic form in real space. Together with the excursion set bias parameters, $W_{\mathrm{eff}}$ describes the scale-dependence of the Lagrangian halo-matter cross correlation up to $k{R}_{\mathrm{Lag}}\sim 10$, where $R_{\mathrm{Lag}}$ is the Lagrangian size of the protohalo. Moreover, with this $W_{\mathrm{eff}}$, 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 tophat in `infall time' towards the protohalo center need not correspond to a tophat in the initial spatial distribution, so models in which infall rather than smoothed overdensity is the relevant variable may be more accurate.

15 pages, 14 figures, matched the published version, discussion and presentation significantly improved, conclusions unchanged

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