Extracting Dark-Matter Velocities from Halo Masses: A Reconstruction
  Conjecture
  The distribution of primordial dark-matter velocities can significantly
influence the growth of cosmological structure. In principle, one can therefore
exploit the halo-mass distribution in order to learn about the dark sector. In
practice, however, this task is both theoretically and computationally
intractable. In this paper, 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
non-trivial distributions which are highly non-thermal and/or multi-modal, such
as might occur for non-minimal dark sectors. Our conjecture therefore provides
an operational tool for probing the dark sector which does not rely on the
existence of non-gravitational couplings between dark and visible states.
