Constraining Peaks in Gaussian Primordial Density Fields: an Application of the Hoffman-Ribak Method

Abstract

We develop an algorithm for setting up initial Gaussian random density and velocity fields containing one or more peaks or dips, in an arbitrary cosmological scenario. The intention is to generate appropriate initial conditions for cosmological N-body simulations that focus on the evolution of the progenitors of the present-day galaxies and clusters. The procedure is an application of the direct and accurate prescription of Hoffman & Ribak (1991) for generating constrained random fields. For each peak a total of 21 physical characteristics can be specified, including its scale, position, density Hessian, velocity, and velocity gradient. The velocity (or, equivalently, gravity) field constrants are based on a generalization of the formalism developed by Bardeen et al. (1986). The resulting density field is sculpted such that it induces the desired amount of net gravitational and tidal forces. We provide a detailed mathematical presentation of the formalism. Afterwards we provide analytical estimates of the likelihood of the imposed constraints. Amongst others, it is shown that the tidal field has a strong tendency to align itself along the principal axes of the mass tensor. The method is illustrated by means of some concrete examples. In addition to the illustration of constraint-field correlation functions and how they add up to the mean fields, followed by illustrations of the variance characteristics of field realizations, we concentrate in particular on the consequences of imposing gravitational field constraints (or, equivalent in the linear regime for growing mode fluctuations, peculiar velocity field constraints).