From 100 kpc to 10 Gpc: Dark Matter self-interactions before and after DESI

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.