Wide binaries in ultra-faint dwarf galaxies as a probe of extended dark objects

Wide stellar binaries are sensitive dynamical probes of dark matter substructure. We generalize existing point-mass disruption constraints to spherical extended dark objects and apply the resulting framework to the population of wide-binary candidates identified in the ultra-faint dwarf galaxy Bootes I. We derive 95% confidence limits on the dark matter fraction as a function of the perturber mass, the radius enclosing 90% of the mass, and the density profile, considering Navarro-Frenk-White, r^-3/2, and r^-9/4 models. Sufficiently compact objects converge to the point-mass limit, whereas finite size suppresses binary disruption when the perturber becomes comparable to the relevant encounter and binary scales. The constraints remain sensitive to dark matter fractions well below unity over a broad region of the mass-radius plane and extend to objects substantially larger than those accessible to conventional microlensing searches. As an application, we map the r^-3/2 results onto ultracompact minihalos and derive an illustrative constraint on the primordial curvature power spectrum. Ultra-faint-dwarf wide binaries therefore offer a purely gravitational probe of the abundance of extended dark matter objects and of the small-scale primordial curvature power spectrum.