Effective Window Function for Lagrangian Halos
  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. Moreover, with this $W_{\rm eff}$, 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.
