Stellar streams and dark substructure: the diffusion regime

The cold dark matter picture predicts an abundance of substructure within the Galactic halo. However, most substructures host no stars and can only be detected indirectly. Stellar streams present a promising probe of this dark substructure. These streams arise from tidally stripped star clusters or dwarf galaxies, and their low dynamical temperature and negligible self-gravity give them a sharp memory of gravitational perturbations caused by passing dark substructures. For this reason, perturbed stellar streams have been the subject of substantial study. While previous studies have been largely numerical, we show here that in the diffusion regime -- where stream stars are subjected to many small velocity kicks -- stream perturbations can be understood on a fully analytic level. In particular, we derive how the (three-dimensional) power spectrum of the substructure density field determines the power spectrum of the (one-dimensional) density of a stellar stream. Our analytic description supplies a clear picture of the behaviour of stream perturbations in response to a perturbing environment, which may include contributions from both dark and luminous substructure. In particular, stream perturbations grow in amplitude initially, settle into a steady state, and ultimately decay. By directly relating stellar stream perturbations to the surrounding matter distribution, this analytic framework represents a versatile new tool for probing the nature of dark matter through astrophysical observations.