An analytic description of substructure-induced gravitational perturbations in stellar systems

Perturbations to stellar systems can reflect the gravitational influence of dark matter substructures. On scales much smaller than the size of a stellar system, we point out analytic connections between the stellar and dark matter distributions. In particular, the density and velocity power spectra of the stars are proportional to the density power spectrum of the perturbing dark matter, scaled by $k^{-4}$. This relationship allows easy evaluation of the suitability of a stellar system for detecting dark substructure. As examples, we show that the Galactic stellar halo is expected to be sensitive to cold dark matter substructure at wavenumbers $k\lesssim 0.3$ kpc${}^{-1}$, and the Galactic disk might be sensitive to substructure at wavenumbers $k\sim 4$ kpc${}^{-1}$. The perturbations considered in this work are short-lived, being rapidly erased by the stellar velocity dispersion, so it may be possible to attribute a detection to dark matter substructure without ambiguity.