An alternative method to generate pre-initial conditions for
  cosmological $N$-body simulations
  Currently, grid and glass methods are the two most popular choices to
generate uniform particle distributions (i.e., pre-initial conditions) for
cosmological $N$-body simulations. In this article, we introduce an alternative
method called the capacity constrained Voronoi tessellation (CCVT), which
originates from computer graphics. As a geometrical equilibrium state, a CCVT
particle configuration satisfies two constraints: (i) the volume of the Voronoi
cell associated with each particle is equal; (ii) every particle is in the
center-of-mass position of its Voronoi cell. We show that the CCVT
configuration is uniform and isotropic, follows perfectly the minimal power
spectrum, $P(k)\propto k^4$, and is quite stable under gravitational
interactions. It is natural to incorporate periodic boundary conditions during
CCVT making, therefore, we can obtain a larger CCVT by tiling with a small
periodic CCVT. When applying the CCVT pre-initial condition to cosmological
$N$-body simulations, we show that it plays as good as grid and glass schemes.
The CCVT method will be helpful in studying the numerical convergence of
pre-initial conditions in cosmological simulations. It can also be used to set
up pre-initial conditions in smoothed-particle hydrodynamics simulations.
