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Perturbations to stellar systems can reflect the gravitational influence ofdark matter substructures. Whereas perturbations to cold stellar systems arethe most commonly studied, the sources of perturbations to dynamically hotsystems are less ambiguous because such systems cannot support persistentinhomogeneity on small scales. We point out a simple algebraic relationshipbetween the two-point statistics of a hot stellar system and those of theperturbing matter. The density and velocity power spectra of the stars areproportional to the density power spectrum of the perturbers, scaled by$k^{-4}$. This relationship allows easy evaluation of the suitability of astellar system for detecting dark substructure. As examples, we show that theGalactic stellar halo is expected to be sensitive to cold dark mattersubstructure at wave numbers $k\lesssim 0.4$ kpc$^{-1}$, and the Galactic diskmight be sensitive to substructure at wave numbers $k\sim 4$ kpc$^{-1}$. Thesesystems could provide direct measurements of the nonlinear matter powerspectrum at these wave numbers.

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