OPTIMIZING THE ZELDOVICH APPROXIMATION

We have recently learned that the Zel'dovich 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 by Coles, Melott & Shandarin (hereafter CMS) the accuracy of several analytic approximations to gravitational clustering was studied in the mildly non-linear regime. We found that what was called the 'truncated Zel'dovich approximation' (TZA) was better than any other (except, in one case, the ordinary Zel'dovich approximation) over a wide range from linear to mildy non-linear (sigma approximately 3) regimes. TZA was specified by setting Fourier amplitudes equal to zero for all wavenumbers greater than k(nl), where k(nl) marks the transition to the non-linear regime. Here we study the cross-correlation of generalized TZA with a group of N-body simulations for three shapes of window function: sharp k-truncation (as in CMS), a top-hat in coordinate space, and a Gaussian. We also study the variation in the cross-correlation as a function of initial truncation scale within each type. We find that k-truncation, which was so much better than other things tried in CMS, is the worst of these three window shapes. We find that a Gaussian window exp (-k2/2 k(G)2) applied to the initial Fourier amplitudes is the best choice. It produces a greatly improved cross-correlation in those cases that most needed improvement, e.g. those with more small-scale power in the initial conditions. The optimum choice of kG for the Gaussian window is (somewhat spectrum-dependent) 1 to 1.5 times k(nl), where k(nl) is defined by equation (3). Although all three windows produce similar power spectra and density distribution functions after application of the Zerdovich approximation, the agreement of the phases of the Fourier components with the N-body simulation is better for the Gaussian window. We therefore ascribe the success of the best-choice Gaussian window to its superior treatment of phases in the nonlinear regime. We also report on the accuracy of particle positions and velocities produced by TZA.