  Recent numerical studies on glassy systems provide evidences for a population
of non-Goldstone modes (NGMs) in the low-frequency spectrum of the vibrational
density of states $D(\omega)$. Similarly to Goldstone modes (GMs), i. e.,
phonons in solids, NGMs are soft low-energy excitations. However, differently
from GMs, NGMs are localized excitations. Here we first show that the parental
temperature $T^*$ modifies the GM/NGM ratio in $D(\omega)$. In particular, the
phonon attenuation is reflected in a parental temperature dependency of the
exponent $s(T^*)$ in the low-frequency power law $D(\omega) \sim
\omega^{s(T^*)}$, with $2 \leq s(T^*) \leq 4 $. Secondly, by comparing $s(T^*)$
with $s(p)$, i. e., the same quantity obtained by pinning \mttp{a} $p$ particle
fraction, we suggest that $s(T^*)$ reflects the presence of dynamical
heterogeneous regions of size $\xi^3 \propto p$. Finally, we provide an
estimate of $\xi$ as a function of $T^*$, finding a mild power law divergence,
$\xi \sim (T^* - T_d)^{-\alpha/3}$, with $T_d$ the dynamical crossover
temperature and $\alpha$ falling in the range $\alpha \in [0.8,1.0]$.
