Discretising the velocity distribution for directional dark matter
  experiments
  Dark matter (DM) direct detection experiments which are
directionally-sensitive may be the only method of probing the full velocity
distribution function (VDF) of the Galactic DM halo. We present an angular
basis for the DM VDF which can be used to parametrise the distribution in order
to mitigate astrophysical uncertainties in future directional experiments and
extract information about the DM halo. This basis consists of discretising the
VDF in a series of angular bins, with the VDF being only a function of the DM
speed $v$ within each bin. In contrast to other methods, such as spherical
harmonic expansions, the use of this basis allows us to guarantee that the
resulting VDF is everywhere positive and therefore physical. We present a
recipe for calculating the event rates corresponding to the discrete VDF for an
arbitrary number of angular bins $N$ and investigate the discretisation error
which is introduced in this way. For smooth, Standard Halo Model-like
distribution functions, only $N=3$ angular bins are required to achieve an
accuracy of around $10-30\%$ in the number of events in each bin. Shortly after
confirmation of the DM origin of the signal with around 50 events, this
accuracy should be sufficient to allow the discretised velocity distribution to
be employed reliably. For more extreme VDFs (such as streams), the
discretisation error is typically much larger, but can be improved with
increasing $N$. This method paves the way towards an astrophysics-independent
analysis framework for the directional detection of dark matter.
