  Time-resolved optical lineshapes are calculated using a second-order
inhomogeneous cumulant expansion. The calculation shows that in the
inhomogeneous limit the optical spectra are determined solely by two-time
correlation functions. Therefore, measurements of the Stokes-shift correlation
function and the inhomogeneous linewidth cannot provide information about the
heterogeneity lifetime for systems exhibiting dynamic heterogeneities. The
theoretical results are illustrated using a stochastic model for the optical
transition frequencies. The model rests on the assumption that the transition
frequencies are coupled to the environmental relaxation of the system. The
latter is chosen according to a free-energy landscape model for dynamically
heterogeneous dynamics. The model calculations show that the available
experimental data are fully compatible with a heterogeneity lifetime on the
order of the primary relaxation time.
