Fully relativistic treatment of decaying cold dark matter in $N$-body
  simulations
  We present $N$-body simulations in which either all, or a fraction of, the
cold dark matter decays non-relativistically to a relativistic, non-interacting
dark radiation component. All effects from radiation and general relativity are
self-consistently included at the level of linear perturbation theory, and our
simulation results therefore match those from linear Einstein-Boltzmann codes
such as CLASS in the appropriate large-scale limit. We also find that standard,
Newtonian $N$-body simulations adequately describe the non-linear evolution at
smaller scales ($k \gtrsim 0.1 \, h/{\mathrm{Mpc}}$) in this type of model,
provided that the mass of the decaying component is modified correctly, and
that the background evolution is correctly treated. That is, for studies of
small scales, effects from general relativity and radiation can be safely
neglected.
