Extended Perturbation Theory for the Local Density Distribution Function

Perturbation theory makes it possible to calculate the probability distribution function (PDF) of the large scale density eld in the small variance limit, 1. For top hat smoothing and scale-free Gaussian initial uctuations, the result depends only on the linear variance, linear, and its logarithmic derivative with respect to the ltering scale (nlinear + 3) = d log 2 linear =d log ` (Bernardeau 1994a). In this paper, we measure the PDF and its low-order moments in scale-free simulations evolved well into the nonlinear regime and compare the results with the above predictions, assuming that the spectral index and the variance are adjustable parameters, ne and e , where is the true, nonlinear variance. With these additional degrees of freedom, results from perturbation theory provide a good t of the PDFs, even in the highly nonlinear regime. The value of ne is of course equal to nlinear when 1, and it decreases with increasing . A nearly at plateau is reached when 1. In this regime, the di erence between ne and nlinear increases when nlinear decreases. For initial power-spectra with nlinear = 2; 1; 0;+1, we nd ne ' 9; 3; 1; 0:5 when 2 ' 100. It is worth noting that (3+ne ) is di erent from the logarithmic derivative of the nonlinear variance with respect to the ltering scale. Consequently, it is not straightforward to determine the nonlinearly evolved PDF from arbitrary (scale-dependent) initial conditions, such as Cold Dark Matter, although we propose a simple method that makes this feasible. Thus, estimates of the variance (using, for example, the prescription proposed by Hamilton et al. 1991) and of ne as functions of scale for a given power spectrum makes it possible to calculate the local density PDF at any time from the initial conditions.