The formation and evolution of galaxies is known to be sensitive to tidal processes leading to intrinsic correlations between their shapes and orientations. Such correlations can be measured to high significance today, suggesting that cosmological information can be extracted from them. Among the most pressing questions in particle physics and cosmology is the nature of dark matter. If dark matter is self-interacting, it can leave an imprint on galaxy shapes. In this work, we investigate whether self-interactions can produce a long-lasting imprint on intrinsic galaxy shape correlations. We investigate this observable at low redshift ($z<0.4$) using a state-of-the-art suite of cosmological hydro-dynamical simulations where the dark matter model is varied. We find that dark matter self-interactions induce a mass dependent suppression in the intrinsic alignment signal by up to 50% out to tens of mega-parsecs, showing that self-interactions can impact structure outside the very core of clusters. We find evidence that self-interactions have a scale-dependent impact on the intrinsic alignment signal that is sufficiently different from signatures introduced by differing baryonic physics prescriptions, suggesting that it is detectable with up-coming all-sky surveys.
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