  We investigate the presence and importance of dark matter discs in a sample
of 24 simulated Milky Way galaxies in the APOSTLE project, part of the EAGLE
programme of hydrodynamic simulations in Lambda-CDM cosmology. It has been
suggested that a dark disc in the Milky Way may boost the dark matter density
and modify the velocity modulus relative to a smooth halo at the position of
the Sun, with ramifications for direct detection experiments. From a kinematic
decomposition of the dark matter and a real space analysis of all 24 halos, we
find that only one of the simulated Milky Way analogues has a detectable dark
disc component. This unique event was caused by a merger at late time with an
LMC-mass satellite at very low grazing angle. Considering that even this rare
scenario only enhances the dark matter density at the solar radius by 35% and
affects the high energy tail of the dark matter velocity distribution by less
than 1%, we conclude that the presence of a dark disc in the Milky Way is
unlikely, and is very unlikely to have a significant effect on direct detection
experiments.
