Testing approximations for non-linear gravitational clustering

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Abstract
We investigate the accuracy of various analytic approximations for following the evolution of cosmological density fluctuations into the non-linear regime. We compare the results of the application of linear theory, the lognormal approximation and the Zeldovich approximation to a set of initial data with corresponding results obtained by a full three-dimensional N-body computation from the same initial conditions. A cross-correlation technique is used to quantify the agreement between approximated and N-body ‘final’ distributions for different initial power spectra, different normalization epochs and different degrees of smoothing of the final results. We also study the distribution function of mass density. We find that the Zeldovich approximation is consistently the best approximation scheme. It is extremely accurate for power spectra characterized by n ≤ – 1; when the approximation is ‘enhanced’ by truncating highly non-linear Fourier modes the approximation is excellent even for n = + 1. It also matches the distribution function best of all our models, except at the high-density tail where it fails to predict enough high-density regions. The performance of linear theory is less spectrum-dependent but this approximation is less accurate than Zeldovich for all cases because of the failure to treat dynamics. In fact, the truncated Zeldovich approximation is always more accurate than linear theory, even in the linear regime. The lognormal approximation provides generally a very poor fit to the spatial pattern, producing far too ‘clumpy’ a distribution; on the other hand, the distribution of cell occupation densities is fitted reasonably well by a lognormal distribution, though not with the same variance as suggested by Coles & Jones.

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