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.
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