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% 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 $\delta$ 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 have occured 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. [Abridged]
