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, sigma << 1. 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) = d log sigma(linear)(2)/d log l. In this paper, we measure the PDF and its low-order moments in scale-free simulations evolved well into the non-linear 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 a is the true, non-linear variance. With these additional degrees of freedom, results from perturbation theory provide a good fit of the PDFs, even in the highly non-linear 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) similar or equal to -9, -3, -1, -0.5 when sigma(2) similar or equal to 100. It is worth noting that -(3 + n(eff)) is different from the logarithmic derivative of the non-linear variance with respect to the filtering scale. Consequently, it is not straightforward to determine the non-linearly 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.) and of n(eff) as functions of scale for a given power spectrum make it possible to calculate the local density PDF at any time from the initial conditions.