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 field in the small variance limit. For top hat smoothing and scale-free Gaussian initial fluctuations, the result depends only on the linear variance, sigmaₗinear, and its logarithmic derivative with respect to the filtering scale - (nₗinear+3) =dlog sigmaₗinear²/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, nₑff and sigmaₑff=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 nₑff is of course equal to nₗinear when sigma 1, and it decreases with increasing sigma. A nearly flat plateau is reached when sigma 1. In this regime, the difference between nₑff and nₗinear increases when nₗinear decreases. For initial power-spectra with nₗinear=-2, -1, 0, +1, we find nₑff ~ -9, -3, -1, -0. 5 when sigma² ~ 100.