  In a recent publication we established an analogy between the free energy of
a hard sphere system and the energy of an elastic network [1]. This result
enables one to study the free energy landscape of hard spheres, in particular
to define normal modes. In this Letter we use these tools to analyze the
activated transitions between meta-bassins, both in the aging regime deep in
the glass phase and near the glass transition. We observe numerically that
structural relaxation occurs mostly along a very small number of
nearly-unstable extended modes. This number decays for denser packing and is
significantly lowered as the system undergoes the glass transition. This
observation supports that structural relaxation and marginal modes share common
properties. In particular theoretical results [2, 3] show that these modes
extend at least on some length scale $l^*\sim (\phi_c-\phi)^{-1/2}$ where
$\phi_c$ corresponds to the maximum packing fraction, i.e. the jamming
transition. This prediction is consistent with very recent numerical
observations of sheared systems near the jamming threshold [4], where a similar
exponent is found, and with the commonly observed growth of the rearranging
regions with compression near the glass transition.
