Tetrahedral collapse: a rotational toy model of simultaneous dark-matter
  halo, filament and wall formation
  We discuss an idealized model of halo formation, in which a collapsing halo
node is tetrahedral, with a filament extruding from each of its four faces, and
with a wall connecting each pair of filaments. In the model, filaments
generally spin when they form, and the halo spins if and only if there is some
rotation in filaments. This is the simplest-possible fully three-dimensional
halo collapse in the 'origami approximation,' in which voids are irrotational,
and the dark-matter sheet out of which dark-matter structures form is allowed
to fold in position-velocity phase space, but not stretch (i.e., it cannot vary
in density along a stream). Up to an overall scaling, the four filament
directions, and only three other quantities, such as filament spins, suffice to
determine all of the collapse's properties: the shape, mass, and spin of the
halo; the densities per unit length and spins of all filaments; and masses per
unit area of the walls. If the filaments are arranged regular-tetrahedrally,
filament properties obey simple laws, reminiscent of angular-momentum
conservation. The model may be most useful in understanding spin correlations
between neighboring galaxies joined by filaments; these correlations would give
intrinsic alignments between galaxies, essential to understand for accurate
cosmological weak-lensing measurements.
