Probing the shape of the Milky Way dark matter halo with hypervelocity
  stars: a new method
  We propose a new method to determine the shape of the gravitational potential
of the dark matter (DM) halo of the Milky Way (MW) with the galactocentric
tangential velocities of a sample of hypervelocity stars (HVSs). We compute the
trajectories of different samples of HVSs in a MW where the baryon distribution
is axisymmetric and the DM potential either is spherical or is spheroidal or
triaxial with radial-dependent axis ratios. We determine the shape of the DM
potential with the distribution of the latitudinal velocity $|v_{\vartheta}|$
in axisymmetric Galactic potentials, or with the distribution of
$|v_{\vartheta}|$ and of a function $\bar v_{\varphi}$ of the azimuthal
velocity in non-axisymmetric Galactic potentials. We recover the correct shape
of the DM potential by comparing the distribution of $|v_{\vartheta}|$ and
$\bar v_{\varphi}$ against the corresponding distributions of mock samples of
HVSs that traveled in DM halos of different shapes. We use the largest possible
sample of $\sim 800$ HVSs of $4~M_\odot$ ejected with the Hills mechanism at a
rate $\sim 10^{-4}$ yr$^{-1}$, currently outgoing, and located at more than 10
kpc from the Galactic center. In our ideal case of galactocentric velocities
with null uncertainties and no observational limitations, our method recovers
the correct shape of the DM potential with a success rate $S\gtrsim 89\%$ in
axisymmetric Galactic potentials, and $S > 96\%$ in the explored
non-axisymmetric cases. The unsuccessful cases yield axis ratios of the DM
potential that are off by $\pm 0.1$. The success rate decreases with decreasing
sample size: for example, for a spherical DM halo, $S$ drops from $\sim 98\%$
to $\sim 38\%$ when the sample size decreases from $\sim 800$ to $\sim 40$
HVSs. A robust determination of the shape of the DM potential thus requires the
measure of the galactocentric velocity of a few hundred genuine HVSs.
