We consider dark matter self-interactions mediated by ultralight scalars. We show that effectively massless mediators lead to an enhancement of the matter power spectrum, while heavier mediators lead to a suppression, together with a feature around their Jeans scale. We derive the strongest present constraints by combining Planck and BOSS data. The recent DESI measurements of baryon acoustic oscillations exhibit a mild $2\sigma$ preference for long-range self-interactions, as strong as 4 per mille of the gravitational coupling. Full-shape analyses of forthcoming DESI and Euclid data will confirm or disprove such a hint.

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• Cosmology
• Dark matter
• Large scale structure of the Universe