The Ultramarine Simulation: properties of dark matter haloes before redshift 5.5

We introduce the Ultramarine simulation, an extremely large $N$-body simulation of the structure formation and evolution to redshift 5.5 at which cosmic reionization was just completed. The simulation evolves 2.1 trillion particles within a $512~h^{-1}$Mpc cube and has an unprecedented mass and force resolution for large volume simulations of this kind, $5.6\times 10^6 h^{-1}$M$_\odot$ and $1.2~h^{-1}$kpc, respectively. We present some basic statistical results of the simulation, including the halo mass function, halo bias parameter as well as halo mass-concentration relation at high redshifts, and compare them with some existing representative models. We find excellent agreement with some models on the high redshift halo mass functions, but neither the halo bias factor nor halo mass-concentration relation. All halo bias models for comparison over-predicate high redshift halo bias by large factors, an accurate fit to our simulation is given. High redshift dark matter haloes still can be reasonably described with NFW model, the halo mass-concentration relations are monotonic, with more massive haloes having lower concentration, in disfavor of the upturn feature reported by some studies. The mass concentration relation has little evolution between $z=5.5$ to $z=10$, in contrast to strong evolution predicted by most existing models. In addition, concentration parameters of high redshift dark matter haloes are much lower than most model predictions.