  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 preference for long-range
self-interactions, as strong as 4 per mille of the gravitational coupling.
Forthcoming data from DESI itself and Euclid will confirm or disprove such a
hint.
