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

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