Ellipsoidal halo finders and implications for models of triaxial halo formation

We describe an algorithm for identifying ellipsoidal haloes in numerical simulations, and quantify how the resulting estimates of halo mass and shape differ with respect to spherical halo finders. Haloes become more prolate when fit with ellipsoids, the difference being most pronounced for the more aspherical objects. Although the ellipsoidal mass is systematically larger, this is less than 10 per cent for most of the haloes. However, even this small difference in mass corresponds to a significant difference in shape. We quantify these effects also on the initial mass and deformation tensors, on which most models of triaxial collapse are based. By studying the properties of protohaloes in the initial conditions, we find that models in which protohaloes are identified in Lagrangian space by three positive eigenvalues of the deformation tensor are tenable only at the masses well above M*. The overdensity δ within almost any protohalo is larger than the critical value associated with spherical collapse (increasing as mass decreases); this is in good qualitative agreement with models which identify haloes requiring that collapse has occurred along all three principal axes, each axis having turned around from the universal expansion at a different time. The distributions of initial values are in agreement with the simplest predictions associated with ellipsoidal collapse, assuming initially spherical protohaloes, collapsed around random positions which were sufficiently overdense. However, most protohaloes are not spherical and departures from sphericity increase as protohalo mass decreases. The mass and deformation tensors are well aligned, in agreement with the fundamental assumption of ellipsoidal collapse, and with models which identify haloes with peaks in the initial density fluctuation field. But the direction of maximum initial compression coincides with the direction of what is initially the longest axis, contrary to what the peaks model predicts. By the final time, it is the shortest axis of the final object which tends to be aligned with the direction of initial maximal compression: the alignment changes during the evolution.