  We calculate the nonequilibrium local density of states on a vibrational
quantum dot coupled to two electrodes at T=0 using a numerically exact
diagrammatic Monte Carlo method. Our focus is on the interplay between the
electron-phonon interaction strength and the bias voltage. We find that the
spectral density exhibits a significant voltage dependence if the voltage
window includes one or more phonon sidebands. A comparison with
well-established approximate approaches indicates that this effect could be
attributed to the nonequilibrium distribution of the phonons. Moreover, we
discuss the long transient dynamics caused by the electron-phonon coupling.
