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, σ ≪ 1. For top-hat smoothing and scale-free Gaussian initial fluctuations, the result depends only on the linear variance, σ_linear, and its logarithmic derivative with respect to the filtering scale -(n_linear + 3) = d log σ²_linear/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 σ_eff ≡ σ, where σ 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 σ ≪ 1, and it decreases with increasing σ. A nearly flat plateau is reached when σ ≫ 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 σ2 ≃ 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.