  The unconventional thermal properties of jammed amorphous solids are directly
related to their density of vibrational states. While the vibrational spectrum
of jammed soft sphere solids has been fully described, the vibrational spectrum
of hard spheres, a model for colloidal glasses, is still unknown due to the
difficulty of treating the non-analytic interaction potential. We bypass this
difficulty using the recently described effective interaction potential for the
free energy of thermal hard spheres. By minimizing this effective free energy
we mimic a quench and produce typical configurations of low temperature
colloidal glasses. We measure the resulting vibrational spectrum and
characterize its evolution towards the jamming point where configurations of
hard and soft spheres are trivially unified. For densities approaching jamming
from below, we observe low frequency modes which agree with those found in
numerical simulations of jammed soft spheres. Our measurements of the
vibrational structure demonstrate that the jamming universality extends away
from jamming: hard sphere thermal systems below jamming exhibit the same
vibrational spectra as thermal and athermal soft sphere systems above the
transition.
