Simulating the complexity of the dark matter sheet I: numerical algorithms

At early times, dark matter has a thermal velocity dispersion of unknown amplitude which, for warm dark matter (WDM) models, can influence the formation of non-linear structure on observable scales. We propose a new scheme to simulate cosmologies with a small-scale suppression of perturbations that combines two previous methods in a way that avoids the numerical artefacts which have so far prevented either from producing fully reliable results. At low densities and throughout most of the cosmological volume, we represent the dark matter phase sheet directly using high-accuracy interpolation, thereby avoiding the artificial fragmentation which afflicts particle-based methods in this regime. Such phase-sheet methods are, however, unable to follow the rapidly increasing complexity of the denser regions of dark matter haloes, so for these we switch to an N-body scheme which uses the geodesic deviation equation to track phase-sheet properties local to each particle. In addition, we present a novel high-resolution force calculation scheme based on an oct-tree of cubic force resolution elements which is well suited to approximate the force field of our combined sheet+particle distribution. Our hybrid simulation scheme enables the first reliable simulations of the internal structure of low-mass haloes in a WDM cosmology.