Title: On the temperature dependence of the density of states of liquids at low energies

Abstract: We report neutron-scattering measurements of the density of states (DOS) of water and liquid Fomblin in a wide range of temperatures. In the liquid phase, we confirm the presence of a universal low-energy linear scaling of the experimental DOS as a function of the frequency, $g(\omega)= a(T) \omega$, which persists at all temperatures. The low-frequency scaling of the DOS exhibits a sharp jump at the melting point of water, below which the standard Debye's law, $g(\omega) \propto \omega^2$, is recovered. On the contrary, in Fomblin, we observe a continuous transition between the two exponents reflecting its glassy dynamics, which is confirmed by structure measurements. More importantly, in both systems, we find that the slope $a(T)$ grows with temperature following an exponential Arrhenius-like form, $a(T) \propto \exp(-\langle E \rangle /T)$. We confirm this experimental trend using molecular dynamics simulations and show that the prediction of instantaneous normal mode (INM) theory for $a(T)$ is in qualitative agreement with the experimental data.