An observation of the anisotropy of dark matter interactions in a direction-sensitive detector would provide decisive evidence for the discovery of galactic dark matter. Directional information would also provide a crucial input to understanding its distribution in the local Universe. Most of the existing directional dark matter detectors utilize particle tracking methods in a low-pressure gas time projection chamber. These low pressure detectors require excessively large volumes in order to be competitive in the search for physics beyond the current limit. In order to avoid these volume limitations, we consider a novel proposal, which exploits a columnar recombination effect in a high-pressure gas time projection chamber. The ratio of scintillation to ionization signals observed in the detector carries the angular information of the particle interactions. In this paper, we investigate the sensitivity of a future directional detector focused on the proposed high-pressure Xenon gas time projection chamber. We study the prospect of detecting an anisotropy in the dark matter velocity distribution. We find that tens of events are needed to exclude an isotropic distribution of dark matter interactions at 95% confidence level in the most optimistic case with head-to-tail information. However, one needs at least 10-20 times more events without head-to-tail information for light dark matter below 50 GeV. For an intermediate mass range, we find it challenging to observe an anisotropy of the dark matter distribution. Our results also show that the directional information significantly improves precision measurements of dark matter mass and the elastic scattering cross section for a heavy dark matter.
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