  We investigate the thermal correlator of the trace of the energy-momentum
tensor in the SU(3) Yang-Mills theory. Our goal is to constrain the spectral
function in that channel, whose low-frequency part determines the bulk
viscosity. We focus on the thermal modification of the spectral function,
$\rho(\omega,T)-\rho(\omega,0)$. Using the operator-product expansion we give
the high-frequency behavior of this difference in terms of thermodynamic
potentials. We take into account the presence of an exact delta function
located at the origin, which had been missed in previous analyses. We then
combine the bulk sum rule and a Monte-Carlo evaluation of the Euclidean
correlator to determine the intervals of frequency where the spectral density
is enhanced or depleted by thermal effects. We find evidence that the thermal
spectral density is non-zero for frequencies below the scalar glueball mass $m$
and is significantly depleted for $m\lesssim\omega\lesssim 3m$.
