  Key aspects of glasses are controlled by the presence of excitations in which
a group of particles can rearrange. Surprisingly, recent observations indicate
that their density is dramatically reduced and their size decreases as the
temperature of the supercooled liquid is lowered. Some theories predict these
excitations to cause a gap in the spectrum of quasi-localised modes of the
Hessian that grows upon cooling, while others predict a pseudo-gap
${D_L(\omega)} \sim \omega^\alpha$. To unify these views and observations, we
generate glassy configurations of controlled gap magnitude $\omega_c$ at
temperature ${T=0}$, using so-called `breathing' particles, and study how such
gapped states respond to thermal fluctuations. We find that \textit{(i)}~the
gap always fills up at finite $T$ with ${D_L(\omega) \approx A_4(T) \,
\omega^4}$ and ${A_4 \sim \exp(-E_a / T)}$ at low $T$, \textit{(ii)}~$E_a$
rapidly grows with $\omega_c$, in reasonable agreement with a simple scaling
prediction ${E_a\sim \omega_c^4}$ and \textit{(iii)}~at larger $\omega_c$
excitations involve fewer particles, as we rationalise, and eventually become
string-like. We propose an interpretation of mean-field theories of the glass
transition, in which the modes beyond the gap act as an excitation reservoir,
from which a pseudo-gap distribution is populated with its magnitude rapidly
decreasing at lower $T$. We discuss how this picture unifies the rarefaction as
well as the decreasing size of excitations upon cooling, together with a
string-like relaxation occurring near the glass transition.
