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In this paper we present an efficient method for constraining both thermaland non-thermal Dark Matter (DM) scenarios with the Lyman-$\alpha$ forest,based on a simple and flexible parametrisation capable to reproduce the smallscale clustering signal of a large set of non-cold DM (nCDM) models. We extractnew limits on the fundamental DM properties, through an extensive analysis ofthe high resolution, high redshift data obtained by the MIKE/HIRESspectrographs. By using a large suite of hydrodynamical simulations, wedetermine constraints on both astrophysical, cosmological, and nCDM parametersby performing a full Monte Carlo Markov Chain (MCMC) analysis. We obtain amarginalised upper limit on the largest possible scale at which a powersuppression induced by nearly any nCDM scenario can occur, i.e.$\alpha<0.03~{\rm{Mpc}}/h$ (2$\sigma$ C.L.). We explicitly describe how to testseveral of the most viable nCDM scenarios without the need to run any specificnumerical simulations, due to the novel parametrisation proposed, and due to anew scheme that interpolates between the cosmological models explored. Theshape of the linear matter power spectrum for standard thermal warm DM modelsappear to be in mild tension ($\sim 2\sigma$ C.L.) with the data, compared tonon-thermal scenarios. We show that a DM fluid composed by both a warm(thermal) and a cold component is also in tension with the Lyman-$\alpha$forest, at least for large $\alpha$ values. This is the first study that allowsto probe the linear small scale shape of the DM power spectrum for a large setof nCDM models.

Novel constraints on non-cold (non-thermal) Dark Matter from Lyman-$\alpha$ forest data