Momentum space sampling of neutrinos in $N$-body simulations
  Including massive neutrinos in $N$-body simulations is a challenging task due
to the large thermal velocities of the neutrinos. In particle based codes this
leads to problems of shot-noise due to insufficient sampling of the neutrino
momentum distribution function. In this paper we investigate the benefits and
drawbacks of a scheme first suggested in a paper by Banerjee et al. in which
the initial neutrino distribution is symmetrised in momentum space. We confirm
that this method reduce shot-noise significantly, but we also find that it
generates some spurious power in the neutrino power spectrum at intermediate
and small scales. We speculate that this happens because many neutrinos in the
simulation sample the same underlying dark matter structures while moving
through the simulation. By carefully tuning the number of directions in
momentum space of the initial neutrino distribution we show that some
improvements can be made over the case where initial neutrino directions are
purely random. At redshifts $z\gtrsim 3$ the method works very well, but at
smaller redshifts significant improvements are not possible due to the spurious
power generation.
