  We investigate the effect of electron-phonon interaction on the phononic
properties in the one-dimensional half-filled Holstein model of spinless
fermions. By means of determinantal Quantum Monte Carlo simulation we show that
the behavior of the phonon dynamics gives a clear signal of the transition to a
charge-ordered phase, and the phase diagram obtained in this way is in
excellent agreement with previous DMRG results. By analyzing the phonon
propagator we extract the renormalized phonon frequency, and study how it first
softens as the transition is approached and then subsequently hardens in the
charge-ordered phase. We then show how anharmonic features develop in the
phonon propagator, and how the interaction induces a sizable dispersion of the
dressed phonon in the non-adiabatic regime.
