  We inspect the initial and the long time evolution of excitations a Fermi
liquids by analyzing the time behavior of the electron spectral function.
Focusing on the short-time limit we study the electron-boson model for the
homogenous electron gas and apply the first order (in boson propagator)
cumulant expansion of the electron Green's function. In addition to a quadratic
decay in time upon triggering the excitation, we identify non-analytic terms in
the time expansion similar to those found in the Fermi edge singularity
phenomenon. We also demonstrate that the exponential decay in time in the
long-time limit is inconsistent with the GW approximation for the self-energy.
The background for this is the Paley-Wiener theorem of complex analysis. To
reconcile with the Fermi liquid behavior an inclusion of higher order diagrams
(in the screened Coulomb interaction) is required.
