Optimizing the Zel'dovich Approximation

We have recently learned that the Zeldovich approximation can be successfully used for a far wider range of gravitational instability scenarios than formerly proposed; we study here how to extend this range. In previous work we studied the accuracy of several analytic approximations to gravitational clustering in the mildly nonlinear regime. We found that the ``truncated Zel'dovich approximation" (TZA) was better than any other (except in one case the ordinary Zeldovich approximation) over a wide range from linear to mildly nonlinear ($σ\sim 3$) regimes. TZA sets Fourier amplitudes equal to zero for {\it all} wavenumbers greater than $k_{n\ell}$, where $k_{n\ell}$ marks the transition to the nonlinear regime. Here, we study crosscorrelation of generalized TZA with a group of $n$--body simulations for three shapes of window function: sharp $k$--truncation (as in CMS), tophat in coordinate space, or a Gaussian. We also study the crosscorrelation as a function of initial scale within each window type. We find $k$--truncation, which was so much better than other things tried in CMS, is the {it worst} of these three window shapes. We find that a Gaussian window $e^{-k^2/2k_G^2}$ applied to the intial Fourier amplitudes is the best choice. It produces a greatly improved crosscorrelation all cases we studied. The optimum choice of $k_G$ for the Gaussian window is (spectrum-- dependent) 1--1.5 times $k_{n\ell}$, with $k_{n\ell}$ defined by (3). Although all three windows produce similar power spectra and density distribution functions after application of the Zeldovich approximation, phase agreement with the $n$--body simulation is better for the Gaussian window. We ascribe Gaussian window success to its superior treatment of phase evolution.

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