Linear response theory and damped modes of stellar clusters

Because all stars contribute to its gravitational potential, stellar clusters amplify perturbations collectively. In the limit of small fluctuations, this is described through linear response theory, via the so-called response matrix. While the evaluation of this matrix is somewhat straightforward for unstable modes (i.e. with a positive growth rate), it requires a careful analytic continuation for damped modes (i.e. with a negative growth rate). We present a generic method to perform such a calculation in spherically symmetric stellar clusters. When applied to an isotropic isochrone cluster, we recover the presence of a low-frequency weakly damped $\ell = 1$ mode. We finally use a set of direct $N$-body simulations to test explicitly this prediction through the statistics of the correlated random walk undergone by a cluster's density centre.