Non‐Gaussian tails of cosmological density distribution function from dark halo approach

We present a simple model based on the dark halo approach which provides a useful way to understand key points that determine the shape of the non‐Gaussian tails of the dark matter one‐point probability distribution function (PDF). In particular, using scale‐free models with a power‐law profile of dark haloes, we derive a simple analytic expression for the one‐point PDF. It is found that the shape of the PDF changes at a characteristic value of δ * , which is defined by the smoothed density of a halo with characteristic mass M * at the epoch. In cold dark matter models with top‐hat smoothing filters, the characteristic smoothed density at the present time typically takes the value δ * ≫ 1 for a small smoothing scale R th ∼ 1 h −1 Mpc and conversely δ * ≪ 1 for a large smoothing scale R th > 10 h −1 Mpc . In the range δ/δ * < 1 , the shape of the PDF is almost solely determined by the outer slope of haloes and scales as a power law. The resultant non‐Gaussian tails of the PDF then resemble log‐normal PDFs in that range and show good agreement with N ‐body simulations, which can be ascribed to the universality of the outer slope of the halo profile. In contrast, the tails of the one‐point PDF in the range δ/δ * > 1 basically follow the steep exponential tails of the halo mass function, which exhibit a strong sensitivity on both the outer slope of the halo profile and the initial power spectrum. Based on these results, a discussion on the PDF of galaxy distribution and application to weak lensing statistics are also presented.