Wide binaries in ultra-faint galaxies: a window onto dark matter on the
  smallest scales
  We carry out controlled $N$-body simulations that follow the dynamical
evolution of binary stars in the dark matter (DM) haloes of ultra-faint dwarf
spheroidals (dSphs). We find that wide binaries with semi-major axes $a\gtrsim
a_t$ tend to be quickly disrupted by the tidal field of the halo. In smooth
potentials the truncation scale, $a_t$, is mainly governed by (i) the mass
enclosed within the dwarf half-light radius ($R_h$) and (ii) the slope of the
DM halo profile at $R\approx R_h$, and is largely independent of the initial
eccentricity distribution of the binary systems and the anisotropy of the
stellar orbits about the galactic potential. For the reported velocity
dispersion and half-light radius of Segue I, the closest ultra-faint, our
models predict $a_t$ values that are a factor 2--3 smaller in cuspy haloes than
in any of the cored models considered here. Using mock observations of Segue I
we show that measuring the projected two-point correlation function of stellar
pairs with sub-arcsecond resolution may provide a useful tool to constrain the
amount and distribution of DM in the smallest and most DM-dominated galaxies.
