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We consider a self-interacting dark matter model in which the massive darkphoton mediating the self-interaction decays to light dark fermions to avoidover-closing the universe. We find that if the model is constrained to explainthe dark matter halos inferred for spiral galaxies and galaxy clusterssimultaneously, there is a strong indication that dark matter is producedasymmetrically in the early universe. It also implies the presence of darkradiation, late kinetic decoupling for dark matter, and a suppressed linearpower spectrum due to dark acoustic damping. The Lyman-$\alpha$ forest powerspectrum measurements put a strong upper limit on the damping scale and themodel has little room to reduce the abundances of satellite galaxies. Futureobservations in the matter power spectrum and the CMB, in tandem with theimpact of self-interactions in galactic halos, makes it possible to measure thegauge coupling and masses of the dark sector particles even when signals inconventional dark matter searches are absent.

Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB