Dark matter doesn't interact with electromagnetic radiation, but it, at least, participates in gravitational interactions as known from the discovery of dark matter. But does dark matter exist in a spiral shape around our galaxy?
Answer
In current cosmological models, the Milky Way resides in a 'halo' of dark matter. Halo is a technical term - in this case, it means a spherically symmetric collection of dark matter. Since dark matter is not self-interacting and does not interact with other matter, it doesn't experience any sort of collisions or friction, and therefore never flattens out into a disk the way normal (baryonic) matter does. So, dark matter does not trace out a disk and does not follow spiral arms.
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