  Using the O(N) model as an example, we investigate the self-interaction
effects of inflaton on the dynamics of the primordial perturbations. When
taking interactions into account, it is essential to employ a self-consistent
off-equilibrium formalism to study the evolution of the inflationary background
field and its fluctuations with the back-reaction effects. Within the Hartree
factorization scheme, we show that the O(N) model has at least two observable
remains left behind the off-equilibrium processes: the running spectral index
of primordial density perturbations and the correlations between perturbation
modes in phase space. We find that the running of the spectral index is fully
determined by the rate of the energy transfer from the inflationary background
field to its fluctuations via particle creation processes as well as the
dynamics of the background field itself. Furthermore, the amplitude of the
field fluctuations turns out to be scale-dependent due to the off-equilibrium
evolution. As a consequence, the scale-dependence of fluctuations yields a
correlation between the phase space modes of energy density perturbations,
while the one-point function of the fluctuations in each Hartree mode is still
Gaussian. More importantly, the mode-mode correlation of the primordial
perturbations depends upon the dynamics of the self-interaction {\it as well
as} the initial conditions of the inflation. Hence, we propose that the running
spectral index and the correlation between phase-space modes would be two
observable fossils to probe the epoch of inflation, even beyond.
