  Fossil amber offers the unique opportunity of investigating an amorphous
material which has been exploring its energy landscape for more than 110 Myears
of natural aging. By applying different x-ray scattering methods to amber
before and after annealing the sample to erase its thermal history, we identify
a link between the potential energy landscape and the structural and
vibrational properties of glasses. We find that hyperaging induces a depletion
of the vibrational density of states in the THz region, also ruling the sound
dispersion and attenuation properties of the corresponding acoustic waves.
Critically, this is accompanied by a densification with structural implications
different in nature from that caused by hydrostatic compression. Our results,
rationalized within the framework of fluctuating elasticity theory, reveal how
upon approaching the bottom of the potential energy landscape (9% decrease in
the fictive temperature $T_f$) the elastic matrix becomes increasingly less
disordered (6%) and longer-range correlated (22%).
