Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues rho(lambda) has a sharp maximum near (but not at) lambda = 0 and decreases monotonically with |lambda| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for rho(lambda), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of rho(lambda) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T = 0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.