Extracting dark-matter velocities from halo masses: A reconstruction conjecture

Increasing attention has recently focused on nontraditional dark-matter production mechanisms which result in primordial dark-matter velocity distributions with highly nonthermal shapes. In this paper, we undertake an assessment of how the detailed shape of a general dark-matter velocity distribution impacts structure formation in the nonlinear regime. In particular, we investigate the impact on the halo-mass and subhalo-mass functions, as well as on astrophysical observables such as satellite and cluster-number counts. We find that many of the standard expectations no longer hold in situations in which this velocity distribution takes a highly nontrivial, even multimodal shape. For example, we find that the nominal free-streaming scale alone becomes insufficient to characterize the effect of free-streaming on structure formation. In addition, we propose a simple one-line conjecture which can be used to "reconstruct" the primordial dark-matter velocity distribution directly from the shape of the halo-mass function. Although our conjecture is completely heuristic, we show that it successfully reproduces the salient features of the underlying dark-matter velocity distribution even for nontrivial distributions which are highly nonthermal and/or multimodal, such as might occur for nonminimal dark sectors. Moreover, since our approach relies only on the halo-mass function, our conjecture provides a method of probing dark-matter properties even for scenarios in which the dark and visible sectors interact only gravitationally.