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.