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Recent cosmological tensions have rekindled the search for models beyond$\Lambda$CDM that cause a suppression of the matter power spectrum. Due to thesmall scales accessible to Lyman-$\alpha$ data they are an excellent additionaltool to probe such models. In this work we extend a recently-developed approachfor using Lyman-$\alpha$ data to constrain the power spectrum suppressioncaused by almost any mixture of cold and non-standard dark matter. We highlightthe steps involved in the development of a corresponding likelihood that willbe publicly released upon publication of this work. We study three examples ofmodels suppressing the power spectrum, namely feebly interacting dark matter,dark matter interacting with baryons, and mixed cold+warm dark matter. Thelatter two can be well constrained from Lyman-$\alpha$ data, and we derivenovel conclusions on the cosmologically allowed parameter spaces, includingfinding a mild preference for non-zero interactions between dark matter andbaryons. The consistency of the constraints obtained on these models highlightthe robustness and flexibility of the likelihood developed here.

One likelihood to bind them all: Lyman-$\alpha$ constraints on non-standard dark matter