The dynamic structure factor of superfluid $^4$He has been investigated at very low temperatures by inelastic neutron scattering. The measurements combine different incoming energies resulting in an unprecedentedly large dynamic range with excellent energy resolution, covering wave vectors $Q$ up to 5 AA$^{-1}$ and energies $omega$ up to 15 meV. A detailed description of the dynamics of superfluid $^4$He is obtained from saturated vapor pressure up to solidification. The single-excitation spectrum is substantially modified at high pressures, as the maxon energy exceeds the roton-roton decay threshold. A highly structured multi-excitation spectrum is observed at low energies, where clear thresholds and branches have been identified. Strong phonon emission branches are observed when the phonon or roton group velocities exceed the sound velocity. The spectrum is found to display strong multi-excitations whenever the single-excitations face disintegration following Pitaevskiis type a or b criteria. At intermediate energies, an interesting pattern in the dynamic structure factor is observed in the vicinity of the recoil energy. All these features, which evolve significantly with pressure, are in very good agreement with the Dynamic Many-body calculations, even at the highest densities, where the correlations are strongest.
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