  Various laboratory-based experiments are underway attempting to detect dark
matter directly. The event rates and detailed signals expected in these
experiments depend on the dark matter phase space distribution on
sub-milliparsec scales. These scales are many orders of magnitude smaller than
those that can be resolved by conventional N-body simulations, so one cannot
hope to use such tools to investigate the effect of mergers in the history of
the Milky Way on the detailed phase-space structure probed by the current
experiments. In this paper we present an alternative approach to investigating
the results of such mergers, by studying a simplified model for a merger of a
sub-halo with a larger parent halo. With an appropriate choice of parent halo
potential, the evolution of material from the sub-halo can be expressed
analytically in action-angle variables, so it is possible to obtain its entire
orbit history very rapidly without numerical integration. Furthermore by
evolving backwards in time, we can obtain arbitrarily-high spatial resolution
for the current velocity distribution at a fixed point. Although this model
cannot provide a detailed quantitative comparison with the Milky Way, its
properties are sufficiently generic that it offers qualitative insight into the
expected structure arising from a merger at a resolution that cannot be
approached with full numerical simulations. Preliminary results indicate that
the velocity-space distribution of dark matter particles remains characterized
by discrete and well-defined peaks over an extended period of time, both for
single and multi-merging systems, in contrast to the simple smooth velocity
distributions sometimes assumed in predicting laboratory experiment detection
rates.
