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Here is the simple simple answer. If dark matter did interact via the electromagnetic force, it would EMIT some light since light is the tell-tale evidence of electromagnetic interaction. This is known to not be the case. Also, and perhaps more importantly, if dark matter did interact electromagnetically it would ABSORB light from distant stars. This is known to not be the case since we don't see unexpected absorption lines in the spectra from stars in our galaxy or other galaxies.

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