Peak and gravity constraints in Gaussian primordial density fields: An application of the Hoffman-Ribak method

Abstract

We develop an algorithm for setting up tailor-made initial conditions for cosmological N-body simulations intended to focus on particular aspects of the structure formation process. The procedure for generating initial Gaussian random density and velocity fields has the ability to specify the presence and characteristics of one or more peaks and dips at arbitrary locations, in combination with the capability to sculpt the matter distribution such that the gravity and tidal fields at the site of the peaks, as well as at arbitrary field locations, have a required strength and orientation. This constrained field approach allows a systematic exploration, at a minimal computational effort and a maximum resolution of the influence of physical quantities relevant to specific issues in the structure formation process. Such specialized studies could for example address the issue of the evolution of the progenitors of galaxies and clusters, or the role of tidal fields in shaping large-scale structure. The described formalism is a specific application of the direct and accurate prescription of Hoffman & Ribak for generating constrained Gaussian random fields. The procedure for imposing peak, gravity and tidal field constraints is an illustration of the general class of constraints that are convolutions with the linear density field. We provide a comprehensive mathematical presentation of the formalism to set up fields with such general convolution-type constraints, in which each of the M constraints has a different character, and concerns a different scale, arbitrary (i.e. non-grid) positions, or different filters. 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) and tidal field constraints are based on a generalization of the formalism developed by Bardeen et al. Also, we provide an analytical expression for the likelihood of the imposed peak constraints in Gaussian random fields. It is shown that the tidal field has a strong tendency to align itself along the principal axes of the mass tensor. The presentation of the procedure includes a detailed assessment of the computational cost of the procedure, valid for any general convolution-type constraints, as well as a graphical illustration of the formalism. This includes an illustration of constraintfield correlation functions and how they add up to the mean fields, and illustrations of the variance characteristics of field realizations. In particular, we concentrate on the consequences for the structure of the mass distribution implied by imposing gravity and tidal field constraints.