  The correlation between the fast cage dynamics and structural relaxation is
investigated in a model polymer system. It is shown that the cage vibration
amplitude, as expressed by the Debye-Waller factor, and the relaxation time
$\tau_\alpha$ collapse on a single universal curve with a simple analytic form
when the temperature, the density, the chain length and the monomer-monomer
interaction potential are changed. For the physical states with the same
$\tau_\alpha$ coincidence of the mean-square displacement, the intermediate
scattering function and the non-Gaussian parameter is observed in a wide time
window spanning from the ballistic regime to the onset of the Rouse dynamics
driven by the chain connectivity. The role of the non-Gaussian effects is
discussed.
