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Evaluating Lyman-$\alpha$ Constraints for General Dark-Matter Velocity Distributions: Multiple Scales and Cautionary Tales

The Lyman-$\alpha$ absorption spectrum associated with photons traversing theintergalactic medium allows us to probe the linear matter power spectrum downto relatively small distance scales. Finding ways of accurately evaluatingLyman-$\alpha$ constraints across large classes of candidate models ofdark-matter physics is thus of paramount importance. While such constraintshave been evaluated for dark-matter models with relatively simple dark-mattervelocity distributions, more complex models -- particularly those withdark-matter velocity distributions stretching across multiple scales -- arereceiving increasing attention. In this paper, we undertake a study of theLyman-$\alpha$ constraints associated with general dark-matter velocitydistributions. Although these constraints are difficult to evaluate inprinciple, in practice there exist two ways of recasting them into forms whichare easier to evaluate and which therefore allow a more rapid determination ofwhether a given dark-matter model is ruled in or out. We utilize both of theserecasts in order to determine the Lyman-$\alpha$ bounds on different classes ofdark-matter velocity distributions. We also develop a general method by whichthe results of these different recasts can be compared. For relatively simpledark-matter velocity distributions, we find that these two classes of recaststend to align and give similar results. However, the situation is far morecomplex for distributions involving multiple velocity scales: while these tworecasts continue to yield similar results within certain regions of parameterspace, they nevertheless yield dramatically different results within preciselythose regions of parameter space which are likely to be phenomenologicallyrelevant. This, then, serves as a cautionary tale regarding the use of suchrecasts for complex dark-matter velocity distributions.

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