  We use a recently developed high-resolution coherent probe spectroscopy
method to investigate the dispersion of collective excitations of a polaritonic
quantum fluid. We measure the dispersion relation with high energy and
wavenumber resolution, which allows us to determine the speed of sound in the
fluid and to evidence the contribution of an excitonic reservoir. We report on
the generation of collective excitations at negative energies, on the ghost
branch of the dispersion curve. Precursors of dynamical instabilities are also
identified. Our methods open the way to the precise study of quantum
hydrodynamics of quantum fluids of light.
