The Lyman-$alpha$ forest is a valuable probe of dark matter models featuring a scale-dependent suppression of the power spectrum as compared to $Lambda$CDM. In this work, we present a new estimator of the Lyman-$alpha$ flux power spectrum that does not rely on hydrodynamical simulations. Our framework is characterized by nuisance parameters that encapsulate the complex physics of the intergalactic medium and sensitivity to highly non-linear small-scale modes. After validating the approach based on high-resolution hydrodynamical simulations for $Lambda$CDM, we derive conservative constraints on interacting dark matter models from BOSS Lyman-$alpha$ data on large scales, k<0.02(km/s)^(-1), with the relevant nuisance parameters left free in the model fit. The estimator yields lower bounds on the mass of cannibal dark matter, where freeze-out occurs through 3-to-2 annihilation, in the MeV range. Furthermore, we find that models of dark matter interacting with dark radiation, which have been argued to address the $H_0$ and $sigma_8$ tensions, are compatible with BOSS Lyman-$alpha$ data.
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