Probability distribution functions of the cosine of the angle between the direction toward the DM offset and the orientation of the stellar bar, defined as the major axis of the stellar density ellipsoid inward of 1 kpc. The three panels show the distributions split by cosmic time: early times, prior to the appearance of the DM offset (top panel), an intermediate time period over which the DM offset grows in magnitude (middle), and late times, during which the DM offset remains more or less constant at ~340 pc (bottom). The early distribution is consistent with uniform in |cos  θ|, whereas once the DM offset becomes pronounced the distributions prefer large values of |cos  θ|, indicating alignment between the offset peak and the stellar bar.