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 neighbouring galaxies joined by filaments; these correlations would give intrinsic alignments between galaxies, essential to understand for accurate cosmological weak-lensing measurements.