Acoustic phonon instabilities and central peaks at the Brillouin zone boundary in the organic-inorganic perovskite CH$_3$NH$_3$PbCl$_3$

Lead halide hybrid perovskites consist of an inorganic framework hosting a molecular cation located in the interstitial space. These compounds have been extensively studied as they have been identified as promising materials for photovoltaic applications. In particular, the interaction between the molecular cation and the inorganic framework has been suggested to influence the electronic properties. CH$_3$NH$_3$PbCl$_3$ undergoes several structural transitions associated with both distortion of the octahedra and orientational ordering of the CH$_3$NH$_3$ cation. We have measured the low-frequency lattice dynamics using neutron spectroscopy. We report an anomaly in the acoustic phonon linewidth towards the high-symmetry point $\textbf{Q}_X=(2,\frac{1}{2}, 0)$ when approaching the transitions, and a hardening of the entire phonon branch with decreasing temperature. Measurements at the Brillouin zone edges $\textbf{Q}_X=(2,\frac{1}{2}, 0)$, $\textbf{Q}_M=(\frac{3}{2},\frac{1}{2}, 0)$, and $\textbf{Q}_R=(\frac{3}{2},\frac{3}{2}, \frac{5}{2})$ show central peaks appearing at the lower temperature transition. We discuss the presence of both central peaks and acoustic phonon instabilities as evidence of a strong coupling between the inorganic framework and the molecular cation.