Origami constraints on the initial-conditions arrangement of dark-matter caustics and streams

In a cold dark matter universe, cosmological structure formation proceeds in rough analogy to origami folding. Dark matter occupies a three-dimensional ‘sheet’ of free-fall observers, non-intersecting in six-dimensional velocity–position phase space. At early times, the sheet was flat like an origami sheet, i.e. velocities were essentially zero, but as time passes, the sheet folds up to form cosmic structure. This paper further illustrates this analogy, and clarifies a Lagrangian definition of caustics and streams: caustics are two-dimensional surfaces in this initial sheet along which it folds, tessellating Lagrangian space into a set of three-dimensional regions, i.e. streams. The main scientific result of this paper is that streams may be coloured by only two colours, with no two neighbouring streams (i.e. streams on either side of a caustic surface) coloured the same. The two colours correspond to positive and negative parities of local Lagrangian volumes. This is a severe restriction on the connectivity and therefore arrangement of streams in Lagrangian space, since arbitrarily many colours can be necessary to colour a general arrangement of three-dimensional regions. This stream two-colourability has consequences from graph theory, which we explain. Then, using N-body simulations, we test how these caustics correspond in Lagrangian space to the boundaries of haloes, filaments and walls. We also test how well outer caustics correspond to a Zel'dovich-approximation prediction.