Peaks and dips in Gaussian random fields: a new algorithm for the shear eigenvalues, and the excursion set theory

We present a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, called the `peak/dip excursion-set-based' algorithm, at positions which correspond to peaks or dips of the correlated density field. The computational procedure is based on a new formula which extends Doroshkevich's unconditional distribution for the eigenvalues of the linear tidal field, to account for the fact that halos and voids may correspond to maxima or minima of the density field. The ability to differentiate between random positions and special points in space around which halos or voids may form (peaks/dips), encoded in the new formula and reflected in the algorithm, naturally leads to a straightforward implementation of an excursion set model for peaks and dips in Gaussian random fields - one of the key advantages of this sampling procedure. In addition, it offers novel insights into the statistical description of the cosmic web. As a first physical application, we show how the standard distributions of shear ellipticity and prolateness in triaxial models of structure formation are modified by the constraint. In particular, we provide a new expression for the conditional distribution of shape parameters given the density peak constraint, which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halo shapes, in relation to their initial shape distribution. We also test and confirm our theoretical predictions for the individual distributions of eigenvalues subjected to the extremum constraint, along with other directly related conditional probabilities. Finally, we indicate how the proposed sampling procedure naturally integrates into the standard excursion set model, potentially solving some of its well-known problems, and into the ellipsoidal collapse framework. (abridged)