The Effect of Corner Modes in the Initial Conditions of Cosmological
  Simulations
  In view of future high-precision large-scale structure surveys, it is
important to quantify the percent and subpercent level effects in cosmological
$N$-body simulations from which theoretical predictions are drawn. One such
effect involves deciding whether to zero all modes above the one-dimensional
Nyquist frequency, the so-called "corner" modes, in the initial conditions. We
investigate this effect by comparing power spectra, density distribution
functions, halo mass functions, and halo profiles in simulations with and
without these modes. For a simulation with a mass resolution of $m_p \sim
10^{11}\,h^{-1}\,M_{\odot}$, we find that at $z>6$, the difference in the
matter power spectrum is large at wavenumbers above $\sim 80$\% of
$k_{\rm{Ny}}$, reducing to below 2\% at all scales by $z\sim 3$. Including
corner modes results in a better match between low- and high-resolution
simulations at wavenumbers around the Nyquist frequency of the low-resolution
simulation, but the effect of the corner modes is smaller than the effect of
particle discreteness. The differences in mass functions are 3\% for the
smallest halos at $z=6$ for the $m_p \sim 10^{11}\,h^{-1}\,M_{\odot}$
simulation, but we find no significant difference in the stacked profiles of
well-resolved halos at $z \leq 6$. Thus removing power at
$|\mathbf{k}|>k_{\rm{Ny}}$ in the initial conditions of cosmological
simulations has a small effect on small scales and high redshifts, typically
below a few percent.
