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Perturbation theory makes it possible to calculate the probability distribution function (PDF) of the large scale density field in the small variance limit. For top hat smoothing and scale-free Gaussian initial fluctuations, the result depends only on the linear variance, sigma_linear, and its logarithmic derivative with respect to the filtering scale -(n_linear+3)=dlog sigma_linear^2/dlog L (Bernardeau 1994). 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, n_eff and sigma_eff=sigma, where sigma is the true, nonlinear variance. With these additional degrees of freedom, results from perturbation theory provide a good fit of the PDFs, even in the highly nonlinear regime. The value of n_eff is of course equal to n_linear when sigma 1, and it decreases with increasing sigma. A nearly flat plateau is reached when sigma 1. In this regime, the difference between n_eff and n_linear increases when n_linear decreases. For initial power-spectra with n_linear=-2,-1,0,+1, we find n_eff ~ -9,-3,-1,-0.5 when sigma^2 ~ 100

Extended perturbation theory for the local density distribution function