  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 Pitaevskii's 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.
