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.
