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Dark and thermal reservoir contributions to polariton sound velocity

Exciton-polaritons in an optical microcavity can form a macroscopically coherent state despite being an inherently driven-dissipative system. In comparison with equilibrium bosonic fluids, polaritonic condensates possess multiple peculiarities that make them behave differently from well-known textbook examples. One such peculiarity is the presence of dark excitons which are created by the pump together with optically-active particles. They can considerably affect the spectrum of elementary excitations of the condensate and hence change its superfluid properties. Here, we theoretically analyze the influence of the bright and dark ``reservoir'' populations on the sound velocity $c_s$ of incoherently-driven polaritons. Both pulsed and continuous-wave pumping schemes characterized by essentially different condensate-to-reservoir ratio are considered. We show that the dark exciton contribution leads to considerable lowering of $c_s$ and to its deviation from the square-root-like behavior on the system's chemical potential (measurable condensate blueshift). Importantly, our model allows to unambiguously define the density of dark excitons in the system by experimentally tracking $c_s$ against the condensate blueshift and fitting the dependence at a given temperature.