STATISTICS OF PRIMORDIAL DENSITY PERTURBATIONS FROM DISCRETE SEED MASSES

We examine the statistics of density perturbations for general distributions of seed masses with arbitrary matter accretion. Formal expressions for the power spectrum, the N-point correlation functions, and the density distribution function are derived. These results are applied to the case of uncorrelated seed masses, and power spectra are derived for accretion of both hot and cold dark matter plus baryons. We compute the reduced moments (cumulants) of the density distribution and use them to obtain a series expansion for the density distribution function. We obtain analytic results for the density distribution function in the case of a distribution of seed masses with a spherical tophat accretion pattern. More generally, our formalism allows us to give a complete characterization of the statistical properties of any random field generated from a discrete linear superposition of kernels. In particular, our results can be applied to density fields derived by smoothing a discrete set of points with a window function.