  The dispersion relation $\epsilon(k)$ of the elementary excitations of
superfluid $^4$He has been measured at very low temperatures, from saturated
vapor pressure up to solidification, using a high flux time-of-flight neutron
scattering spectrometer equipped with a high spatial resolution detector
(10$^5$ 'pixels'). A complete determination of $\epsilon(k)$ is achieved, from
very low wave-vectors up to the end of Pitaeskii's plateau. The results compare
favorably in the whole the wave-vector range with the predictions of the
dynamic many-body theory (DMBT). At low wave-vectors, bridging the gap between
ultrasonic data and former neutron measurements, the evolution with the
pressure from anomalous to normal dispersion, as well as the peculiar
wave-vector dependence of the phase and group velocities, are accurately
characterized. The thermodynamic properties have been calculated analytically,
developing Landau's model, using the measured dispersion curve. A good
agreement is found below 0.85 K between direct heat capacity measurements and
the calculated specific heat, if thermodynamically consistent power series
expansions are used. The thermodynamic properties have also been calculated
numerically; in this case, the results are applicable with excellent accuracy
up to 1.3 K, a temperature above which the dispersion relation itself becomes
temperature dependent.
