Extending the halo mass resolution of $N$-body simulations
  We present a scheme to extend the halo mass resolution of N-body simulations
of the hierarchical clustering of dark matter. The method uses the density
field of the simulation to predict the number of sub-resolution dark matter
haloes expected in different regions. The technique requires as input the
abundance of haloes of a given mass and their average clustering, as expressed
through the linear and higher order bias factors. These quantities can be
computed analytically or, more accurately, derived from a higher resolution
simulation as done here. Our method can recover the abundance and clustering in
real- and redshift-space of haloes with mass below $\sim 7.5 \times
10^{13}h^{-1}M_{\odot}$ at $z=0$ to better than 10%. We demonstrate the
technique by applying it to an ensemble of 50 low resolution, large-volume
$N$-body simulations to compute the correlation function and covariance matrix
of luminous red galaxies (LRGs). The limited resolution of the original
simulations results in them resolving just two thirds of the LRG population. We
extend the resolution of the simulations by a factor of 30 in halo mass in
order to recover all LRGs. With existing simulations it is possible to generate
a halo catalogue equivalent to that which would be obtained from a $N$-body
simulation using more than 20 trillion particles; a direct simulation of this
size is likely to remain unachievable for many years. Using our method it is
now feasible to build the large numbers of high-resolution large volume mock
galaxy catalogues required to compute the covariance matrices necessary to
analyse upcoming galaxy surveys designed to probe dark energy.
